%------------------------------------------------------------------------------ % File : DT2H2X---1.9.5 % Problem : ITP236^2 : TPTP v9.2.1. Bugfixed v9.2.1. % Transfm : none % Format : tptp:raw % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300 % Computer : n025.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 : Sat Feb 28 09:08:36 AM UTC 2026 % Result : Theorem 18.50s 6.83s % Output : Refutation 18.50s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.12 % Problem : ITP236^2 : TPTP v9.2.1. Bugfixed v9.2.1. % 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300 % 0.15/0.34 % Computer : n025.cluster.edu % 0.15/0.34 % Model : x86_64 x86_64 % 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.15/0.34 % Memory : 8042.1875MB % 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.15/0.34 % CPULimit : 300 % 0.15/0.34 % WCLimit : 300 % 0.15/0.34 % DateTime : Thu Feb 26 15:54:27 EST 2026 % 0.15/0.34 % CPUTime : % 0.15/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300 % 0.30/0.47 ---- Original DTF file --- % 0.30/0.47 thf(spec,logic,$$dhol). % 0.30/0.47 %------------------------------------------------------------------------------ % 0.30/0.47 % File : ITP236^2 : TPTP v9.2.1. Bugfixed v9.2.1. % 0.30/0.47 % Domain : Interactive Theorem Proving % 0.30/0.47 % Problem : Sledgehammer problem VEBT_MinMax 00214_010433 % 0.30/0.47 % Version : [Des22] axioms. % 0.30/0.47 % English : % 0.30/0.47 % 0.30/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal % 0.30/0.47 % : [Des22] Desharnais (2022), Email to Geoff Sutcliffe % 0.30/0.47 % Source : [Des22] % 0.30/0.47 % Names : 0068_VEBT_MinMax_00214_010433 [Des22] % 0.30/0.47 % 0.30/0.47 % Status : Theorem % 0.30/0.47 % Rating : ? v9.2.1 % 0.30/0.47 % Syntax : Number of formulae : 9644 (3028 unt; 639 typ; 0 def) % 0.30/0.47 % Number of atoms : 28865 (9780 equ; 5 cnn) % 0.30/0.47 % Maximal formula atoms : 71 ( 3 avg) % 0.30/0.47 % Number of connectives : 174781 (2571 ~; 357 |;2440 &;155889 @) % 0.30/0.47 % ( 0 <=>;13524 =>; 0 <=; 0 <~>) % 0.30/0.47 % Maximal formula depth : 40 ( 8 avg) % 0.30/0.47 % Number of types : 12 ( 11 usr) % 0.30/0.47 % Number of type conns : 4247 (4247 >; 0 *; 0 +; 0 <<) % 0.30/0.47 % Number of symbols : 632 ( 628 usr; 23 con; 0-9 aty) % 0.30/0.47 % Number of variables : 31643 (2756 ^;27301 !; 988 ?;31643 :) % 0.30/0.47 % ( 598 !>; 0 ?*; 0 @-; 0 @+) % 0.30/0.47 % SPC : TH1_THM_EQU_NAR % 0.30/0.47 % 0.30/0.47 % Comments : This file was generated by Isabelle (most likely Sledgehammer) % 0.30/0.47 % from the van Emde Boas Trees session in the Archive of Formal % 0.30/0.47 % proofs - % 0.30/0.47 % www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees % 0.30/0.47 % 2022-02-17 22:12:25.402 % 0.30/0.47 % Bugfixes : Added ()s around negated sides of equations. % 0.30/0.47 %------------------------------------------------------------------------------ % 0.30/0.47 % Could-be-implicit typings (18) % 0.30/0.47 thf(ty_t_VEBT__Definitions_OVEBT,type, % 0.30/0.47 vEBT_VEBT: $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_Code__Numeral_Ointeger,type, % 0.30/0.47 code_integer: $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_Code__Evaluation_Oterm,type, % 0.30/0.47 code_term: $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_Product__Type_Ounit,type, % 0.30/0.47 product_unit: $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_Product__Type_Oprod,type, % 0.30/0.47 product_prod: $tType > $tType > $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_Extended__Nat_Oenat,type, % 0.30/0.47 extended_enat: $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_Complex_Ocomplex,type, % 0.30/0.47 complex: $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_Sum__Type_Osum,type, % 0.30/0.47 sum_sum: $tType > $tType > $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_Option_Ooption,type, % 0.30/0.47 option: $tType > $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_Filter_Ofilter,type, % 0.30/0.47 filter: $tType > $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_Real_Oreal,type, % 0.30/0.47 real: $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_List_Olist,type, % 0.30/0.47 list: $tType > $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_Set_Oset,type, % 0.30/0.47 set: $tType > $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_Rat_Orat,type, % 0.30/0.47 rat: $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_Num_Onum,type, % 0.30/0.47 num: $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_Nat_Onat,type, % 0.30/0.47 nat: $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_Int_Oint,type, % 0.30/0.47 int: $tType ). % 0.30/0.47 % 0.30/0.47 thf(ty_t_itself,type, % 0.30/0.47 itself: $tType > $tType ). % 0.30/0.47 % 0.30/0.47 % Explicit typings (621) % 0.30/0.47 thf(sy_cl_HOL_Otype,type, % 0.30/0.47 type: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Nat_Osize,type, % 0.30/0.47 size: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Odvd,type, % 0.30/0.47 dvd: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Oone,type, % 0.30/0.47 one: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Oidom,type, % 0.30/0.47 idom: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Oring,type, % 0.30/0.47 ring: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Oplus,type, % 0.30/0.47 plus: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Ozero,type, % 0.30/0.47 zero: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Num_Onumeral,type, % 0.30/0.47 numeral: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Power_Opower,type, % 0.30/0.47 power: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Fields_Ofield,type, % 0.30/0.47 field: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_GCD_Oring__gcd,type, % 0.30/0.47 ring_gcd: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Oabs__if,type, % 0.30/0.47 abs_if: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Oring__1,type, % 0.30/0.47 ring_1: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Ouminus,type, % 0.30/0.47 uminus: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Orderings_Obot,type, % 0.30/0.47 bot: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Orderings_Oord,type, % 0.30/0.47 ord: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Orderings_Otop,type, % 0.30/0.47 top: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Osemidom,type, % 0.30/0.47 semidom: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Fields_Oinverse,type, % 0.30/0.47 inverse: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Osemiring,type, % 0.30/0.47 semiring: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Nat_Oring__char__0,type, % 0.30/0.47 ring_char_0: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Num_Oneg__numeral,type, % 0.30/0.47 neg_numeral: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Orderings_Oorder,type, % 0.30/0.47 order: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Ocomm__ring,type, % 0.30/0.47 comm_ring: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Omult__zero,type, % 0.30/0.47 mult_zero: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_GCD_Osemiring__Gcd,type, % 0.30/0.47 semiring_Gcd: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_GCD_Osemiring__gcd,type, % 0.30/0.47 semiring_gcd: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Ogroup__add,type, % 0.30/0.47 group_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Lattices_Olattice,type, % 0.30/0.47 lattice: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Orderings_Ono__bot,type, % 0.30/0.47 no_bot: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Orderings_Ono__top,type, % 0.30/0.47 no_top: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Osemiring__0,type, % 0.30/0.47 semiring_0: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Osemiring__1,type, % 0.30/0.47 semiring_1: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Finite__Set_Ofinite,type, % 0.30/0.47 finite_finite: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Omonoid__add,type, % 0.30/0.47 monoid_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Ocomm__ring__1,type, % 0.30/0.47 comm_ring_1: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Oidom__divide,type, % 0.30/0.47 idom_divide: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Oidom__modulo,type, % 0.30/0.47 idom_modulo: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Transcendental_Oln,type, % 0.30/0.47 ln: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Omonoid__mult,type, % 0.30/0.47 monoid_mult: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Orderings_Olinorder,type, % 0.30/0.47 linorder: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Orderings_Opreorder,type, % 0.30/0.47 preorder: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Parity_Oring__parity,type, % 0.30/0.47 ring_parity: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Oidom__abs__sgn,type, % 0.30/0.47 idom_abs_sgn: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Oordered__ring,type, % 0.30/0.47 ordered_ring: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Ozero__neq__one,type, % 0.30/0.47 zero_neq_one: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Enum_Ofinite__lattice,type, % 0.30/0.47 finite_lattice: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Fields_Ofield__char__0,type, % 0.30/0.47 field_char_0: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Oab__group__add,type, % 0.30/0.47 ab_group_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Nat_Osemiring__char__0,type, % 0.30/0.47 semiring_char_0: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Orderings_Oorder__bot,type, % 0.30/0.47 order_bot: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Orderings_Oorder__top,type, % 0.30/0.47 order_top: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Orderings_Owellorder,type, % 0.30/0.47 wellorder: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Ocomm__semiring,type, % 0.30/0.47 comm_semiring: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Ozero__less__one,type, % 0.30/0.47 zero_less_one: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Fields_Odivision__ring,type, % 0.30/0.47 division_ring: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Fields_Ofield__abs__sgn,type, % 0.30/0.47 field_abs_sgn: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Osemigroup__add,type, % 0.30/0.47 semigroup_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Num_Osemiring__numeral,type, % 0.30/0.47 semiring_numeral: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Osemidom__divide,type, % 0.30/0.47 semidom_divide: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Osemidom__modulo,type, % 0.30/0.47 semidom_modulo: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Osemigroup__mult,type, % 0.30/0.47 semigroup_mult: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Orderings_Odense__order,type, % 0.30/0.47 dense_order: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Ocomm__semiring__0,type, % 0.30/0.47 comm_semiring_0: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Ocomm__semiring__1,type, % 0.30/0.47 comm_semiring_1: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Olinordered__idom,type, % 0.30/0.47 linordered_idom: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Olinordered__ring,type, % 0.30/0.47 linordered_ring: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Osemiring__modulo,type, % 0.30/0.47 semiring_modulo: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Ocomm__monoid__add,type, % 0.30/0.47 comm_monoid_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Parity_Osemiring__parity,type, % 0.30/0.47 semiring_parity: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Oordered__ring__abs,type, % 0.30/0.47 ordered_ring_abs: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Oordered__semiring,type, % 0.30/0.47 ordered_semiring: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Fields_Olinordered__field,type, % 0.30/0.47 linordered_field: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Oab__semigroup__add,type, % 0.30/0.47 ab_semigroup_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Ocomm__monoid__diff,type, % 0.30/0.47 comm_monoid_diff: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Ocomm__monoid__mult,type, % 0.30/0.47 comm_monoid_mult: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Oalgebraic__semidom,type, % 0.30/0.47 algebraic_semidom: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Osemiring__1__cancel,type, % 0.30/0.47 semiring_1_cancel: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Oab__semigroup__mult,type, % 0.30/0.47 ab_semigroup_mult: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Lattices_Obounded__lattice,type, % 0.30/0.47 bounded_lattice: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Lattices_Odistrib__lattice,type, % 0.30/0.47 distrib_lattice: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Lattices_Osemilattice__inf,type, % 0.30/0.47 semilattice_inf: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Lattices_Osemilattice__sup,type, % 0.30/0.47 semilattice_sup: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Orderings_Odense__linorder,type, % 0.30/0.47 dense_linorder: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Olinordered__semidom,type, % 0.30/0.47 linordered_semidom: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Oordered__semiring__0,type, % 0.30/0.47 ordered_semiring_0: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Real__Vector__Spaces_Obanach,type, % 0.30/0.47 real_Vector_banach: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Olinordered__semiring,type, % 0.30/0.47 linordered_semiring: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Complete__Partial__Order_Occpo,type, % 0.30/0.47 comple9053668089753744459l_ccpo: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Enum_Ofinite__distrib__lattice,type, % 0.30/0.47 finite8700451911770168679attice: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Ocancel__semigroup__add,type, % 0.30/0.47 cancel_semigroup_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Oordered__ab__group__add,type, % 0.30/0.47 ordered_ab_group_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Olinordered__semiring__1,type, % 0.30/0.47 linord6961819062388156250ring_1: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Oordered__comm__semiring,type, % 0.30/0.47 ordere2520102378445227354miring: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Topological__Spaces_Ot1__space,type, % 0.30/0.47 topological_t1_space: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Topological__Spaces_Ot2__space,type, % 0.30/0.47 topological_t2_space: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Topological__Spaces_Ot3__space,type, % 0.30/0.47 topological_t3_space: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Topological__Spaces_Ot4__space,type, % 0.30/0.47 topological_t4_space: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Bit__Operations_Osemiring__bits,type, % 0.30/0.47 bit_semiring_bits: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Lattices_Obounded__lattice__bot,type, % 0.30/0.47 bounded_lattice_bot: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Limits_Otopological__group__add,type, % 0.30/0.47 topolo1633459387980952147up_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type, % 0.30/0.47 comm_s4317794764714335236cancel: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Olinordered__ring__strict,type, % 0.30/0.47 linord4710134922213307826strict: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type, % 0.30/0.47 cancel1802427076303600483id_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Limits_Otopological__monoid__add,type, % 0.30/0.47 topolo6943815403480290642id_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Real__Vector__Spaces_Oreal__field,type, % 0.30/0.47 real_V7773925162809079976_field: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Oring__1__no__zero__divisors,type, % 0.30/0.47 ring_15535105094025558882visors: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type, % 0.30/0.47 cancel2418104881723323429up_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Olinordered__ab__group__add,type, % 0.30/0.47 linord5086331880401160121up_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Oordered__comm__monoid__add,type, % 0.30/0.47 ordere6911136660526730532id_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type, % 0.30/0.47 real_V4867850818363320053vector: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type, % 0.30/0.47 archim2362893244070406136eiling: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_GCD_Osemiring__gcd__mult__normalize,type, % 0.30/0.47 semiri6843258321239162965malize: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Oordered__ab__group__add__abs,type, % 0.30/0.47 ordere166539214618696060dd_abs: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type, % 0.30/0.47 ordere6658533253407199908up_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Limits_Otopological__ab__group__add,type, % 0.30/0.47 topolo1287966508704411220up_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Real__Vector__Spaces_Ometric__space,type, % 0.30/0.47 real_V7819770556892013058_space: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type, % 0.30/0.47 semiri3467727345109120633visors: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type, % 0.30/0.47 boolea8198339166811842893lgebra: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Olinordered__semiring__strict,type, % 0.30/0.47 linord8928482502909563296strict: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Topological__Spaces_Operfect__space,type, % 0.30/0.47 topolo8386298272705272623_space: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Topological__Spaces_Ouniform__space,type, % 0.30/0.47 topolo7287701948861334536_space: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Limits_Otopological__semigroup__mult,type, % 0.30/0.47 topolo4211221413907600880p_mult: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Real__Vector__Spaces_Ocomplete__space,type, % 0.30/0.47 real_V8037385150606011577_space: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Real__Vector__Spaces_Oreal__algebra__1,type, % 0.30/0.47 real_V2191834092415804123ebra_1: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type, % 0.30/0.47 linord181362715937106298miring: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type, % 0.30/0.47 semiri2026040879449505780visors: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Topological__Spaces_Oorder__topology,type, % 0.30/0.47 topolo2564578578187576103pology: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Bit__Operations_Oring__bit__operations,type, % 0.30/0.47 bit_ri3973907225187159222ations: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type, % 0.30/0.47 comple6319245703460814977attice: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type, % 0.30/0.47 linord4140545234300271783up_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Limits_Otopological__comm__monoid__add,type, % 0.30/0.47 topolo5987344860129210374id_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Orderings_Ounbounded__dense__linorder,type, % 0.30/0.47 unboun7993243217541854897norder: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Real__Vector__Spaces_Ouniformity__dist,type, % 0.30/0.47 real_V768167426530841204y_dist: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Olinordered__semiring__1__strict,type, % 0.30/0.47 linord715952674999750819strict: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type, % 0.30/0.47 archim462609752435547400_field: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type, % 0.30/0.47 comple5582772986160207858norder: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Real__Vector__Spaces_Oreal__div__algebra,type, % 0.30/0.47 real_V5047593784448816457lgebra: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type, % 0.30/0.47 bounde4346867609351753570nf_top: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type, % 0.30/0.47 bounde4967611905675639751up_bot: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__field,type, % 0.30/0.47 real_V3459762299906320749_field: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Topological__Spaces_Odiscrete__topology,type, % 0.30/0.47 topolo8865339358273720382pology: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Topological__Spaces_Olinorder__topology,type, % 0.30/0.47 topolo1944317154257567458pology: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Topological__Spaces_Otopological__space,type, % 0.30/0.47 topolo4958980785337419405_space: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type, % 0.30/0.47 euclid3725896446679973847miring: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type, % 0.30/0.47 canoni5634975068530333245id_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type, % 0.30/0.47 ordere8940638589300402666id_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type, % 0.30/0.47 strict7427464778891057005id_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type, % 0.30/0.47 real_V822414075346904944vector: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type, % 0.30/0.47 linord2810124833399127020strict: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type, % 0.30/0.47 bit_se359711467146920520ations: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type, % 0.30/0.47 ordere2412721322843649153imp_le: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type, % 0.30/0.47 ordere580206878836729694up_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type, % 0.30/0.47 ordere1170586879665033532d_diff: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type, % 0.30/0.47 strict9044650504122735259up_add: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Real__Vector__Spaces_Oordered__real__vector,type, % 0.30/0.47 real_V5355595471888546746vector: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra,type, % 0.30/0.47 real_V4412858255891104859lgebra: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type, % 0.30/0.47 semiri6575147826004484403cancel: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type, % 0.30/0.47 euclid8851590272496341667cancel: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra__1,type, % 0.30/0.47 real_V2822296259951069270ebra_1: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type, % 0.30/0.47 unique1627219031080169319umeral: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type, % 0.30/0.47 comple592849572758109894attice: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__div__algebra,type, % 0.30/0.47 real_V8999393235501362500lgebra: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type, % 0.30/0.47 topolo3112930676232923870pology: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type, % 0.30/0.47 euclid4440199948858584721cancel: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type, % 0.30/0.47 euclid3128863361964157862miring: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Topological__Spaces_Olinear__continuum__topology,type, % 0.30/0.47 topolo8458572112393995274pology: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type, % 0.30/0.47 ordere1937475149494474687imp_le: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type, % 0.30/0.47 condit5016429287641298734tinuum: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type, % 0.30/0.47 euclid5411537665997757685th_nat: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Countable__Complete__Lattices_Ocountable__complete__lattice,type, % 0.30/0.47 counta3822494911875563373attice: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type, % 0.30/0.47 semiri1453513574482234551roduct: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type, % 0.30/0.47 bit_un5681908812861735899ations: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type, % 0.30/0.47 condit1219197933456340205attice: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type, % 0.30/0.47 condit6923001295902523014norder: % 0.30/0.47 !>[A: $tType] : $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Archimedean__Field_Oceiling,type, % 0.30/0.47 archimedean_ceiling: % 0.30/0.47 !>[A: $tType] : ( A > int ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type, % 0.30/0.47 archim6421214686448440834_floor: % 0.30/0.47 !>[A: $tType] : ( A > int ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Archimedean__Field_Ofrac,type, % 0.30/0.47 archimedean_frac: % 0.30/0.47 !>[A: $tType] : ( A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Archimedean__Field_Oround,type, % 0.30/0.47 archimedean_round: % 0.30/0.47 !>[A: $tType] : ( A > int ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal,type, % 0.30/0.47 bNF_Ca7293521722713021262ofinal: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type, % 0.30/0.47 bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type, % 0.30/0.47 bNF_Ca8459412986667044542atLess: set @ ( product_prod @ nat @ nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type, % 0.30/0.47 bNF_Ca3754400796208372196lChain: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_BNF__Def_OGr,type, % 0.30/0.47 bNF_Gr: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type, % 0.30/0.47 bNF_Greatest_image2: % 0.30/0.47 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_BNF__Wellorder__Constructions_OFunc,type, % 0.30/0.47 bNF_Wellorder_Func: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( A > B ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_BNF__Wellorder__Constructions_OFunc__map,type, % 0.30/0.47 bNF_We4925052301507509544nc_map: % 0.30/0.47 !>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set @ B ) > ( C > A ) > ( B > D ) > ( D > C ) > B > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_BNF__Wellorder__Constructions_Obsqr,type, % 0.30/0.47 bNF_Wellorder_bsqr: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type, % 0.30/0.47 bNF_We1388413361240627857o_max2: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type, % 0.30/0.47 basic_BNF_size_prod: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Binomial_Obinomial,type, % 0.30/0.47 binomial: nat > nat > nat ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Binomial_Ogbinomial,type, % 0.30/0.47 gbinomial: % 0.30/0.47 !>[A: $tType] : ( A > nat > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Oand__int__rel,type, % 0.30/0.47 bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Oand__not__num,type, % 0.30/0.47 bit_and_not_num: num > num > ( option @ num ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Oconcat__bit,type, % 0.30/0.47 bit_concat_bit: nat > int > int > int ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type, % 0.30/0.47 bit_ri4277139882892585799ns_not: % 0.30/0.47 !>[A: $tType] : ( A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type, % 0.30/0.47 bit_ri4674362597316999326ke_bit: % 0.30/0.47 !>[A: $tType] : ( nat > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type, % 0.30/0.47 bit_se5824344872417868541ns_and: % 0.30/0.47 !>[A: $tType] : ( A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type, % 0.30/0.47 bit_se4197421643247451524op_bit: % 0.30/0.47 !>[A: $tType] : ( nat > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type, % 0.30/0.47 bit_se8732182000553998342ip_bit: % 0.30/0.47 !>[A: $tType] : ( nat > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type, % 0.30/0.47 bit_se2239418461657761734s_mask: % 0.30/0.47 !>[A: $tType] : ( nat > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type, % 0.30/0.47 bit_se1065995026697491101ons_or: % 0.30/0.47 !>[A: $tType] : ( A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type, % 0.30/0.47 bit_se4730199178511100633sh_bit: % 0.30/0.47 !>[A: $tType] : ( nat > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type, % 0.30/0.47 bit_se5668285175392031749et_bit: % 0.30/0.47 !>[A: $tType] : ( nat > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type, % 0.30/0.47 bit_se2584673776208193580ke_bit: % 0.30/0.47 !>[A: $tType] : ( nat > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type, % 0.30/0.47 bit_se2638667681897837118et_bit: % 0.30/0.47 !>[A: $tType] : ( nat > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type, % 0.30/0.47 bit_se5824344971392196577ns_xor: % 0.30/0.47 !>[A: $tType] : ( A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type, % 0.30/0.47 bit_se5641148757651400278ts_bit: % 0.30/0.47 !>[A: $tType] : ( A > nat > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Otake__bit__num,type, % 0.30/0.47 bit_take_bit_num: nat > num > ( option @ num ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type, % 0.30/0.47 bit_un7362597486090784418nd_num: num > num > ( option @ num ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type, % 0.30/0.47 bit_un2480387367778600638or_num: num > num > ( option @ num ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type, % 0.30/0.47 boolea2506097494486148201lgebra: % 0.30/0.47 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type, % 0.30/0.47 boolea3799213064322606851m_diff: % 0.30/0.47 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Code__Numeral_Obit__cut__integer,type, % 0.30/0.47 code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Code__Numeral_Odivmod__abs,type, % 0.30/0.47 code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Code__Numeral_Odivmod__integer,type, % 0.30/0.47 code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type, % 0.30/0.47 code_int_of_integer: code_integer > int ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type, % 0.30/0.47 code_integer_of_int: int > code_integer ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Code__Numeral_Onat__of__integer,type, % 0.30/0.47 code_nat_of_integer: code_integer > nat ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Complete__Lattices_OInf__class_OInf,type, % 0.30/0.47 complete_Inf_Inf: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Complete__Lattices_OSup__class_OSup,type, % 0.30/0.47 complete_Sup_Sup: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Complete__Partial__Order_Ochain,type, % 0.30/0.47 comple1602240252501008431_chain: % 0.30/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Complex_OArg,type, % 0.30/0.47 arg: complex > real ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Complex_Ocis,type, % 0.30/0.47 cis: real > complex ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Complex_Ocomplex_OComplex,type, % 0.30/0.47 complex2: real > real > complex ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type, % 0.30/0.47 condit941137186595557371_above: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type, % 0.30/0.47 condit1013018076250108175_below: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Countable__Set_Oto__nat__on,type, % 0.30/0.47 countable_to_nat_on: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > A > nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Deriv_Odifferentiable,type, % 0.30/0.47 differentiable: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Deriv_Ohas__derivative,type, % 0.30/0.47 has_derivative: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( filter @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Deriv_Ohas__field__derivative,type, % 0.30/0.47 has_field_derivative: % 0.30/0.47 !>[A: $tType] : ( ( A > A ) > A > ( filter @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Divides_Oadjust__div,type, % 0.30/0.47 adjust_div: ( product_prod @ int @ int ) > int ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Divides_Odivmod__nat,type, % 0.30/0.47 divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Divides_Oeucl__rel__int,type, % 0.30/0.47 eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type, % 0.30/0.47 unique5940410009612947441es_aux: % 0.30/0.47 !>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type, % 0.30/0.47 unique8689654367752047608divmod: % 0.30/0.47 !>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type, % 0.30/0.47 unique1321980374590559556d_step: % 0.30/0.47 !>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type, % 0.30/0.47 comm_s3205402744901411588hammer: % 0.30/0.47 !>[A: $tType] : ( A > nat > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type, % 0.30/0.47 semiring_char_0_fact: % 0.30/0.47 !>[A: $tType] : ( nat > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fields_Oinverse__class_Oinverse,type, % 0.30/0.47 inverse_inverse: % 0.30/0.47 !>[A: $tType] : ( A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Filter_Oat__bot,type, % 0.30/0.47 at_bot: % 0.30/0.47 !>[A: $tType] : ( filter @ A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Filter_Oat__top,type, % 0.30/0.47 at_top: % 0.30/0.47 !>[A: $tType] : ( filter @ A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Filter_Oeventually,type, % 0.30/0.47 eventually: % 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Filter_Ofiltercomap,type, % 0.30/0.47 filtercomap: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Filter_Ofilterlim,type, % 0.30/0.47 filterlim: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Filter_Ofiltermap,type, % 0.30/0.47 filtermap: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Filter_Ofinite__subsets__at__top,type, % 0.30/0.47 finite5375528669736107172at_top: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( filter @ ( set @ A ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Filter_Omap__filter__on,type, % 0.30/0.47 map_filter_on: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Filter_Oprincipal,type, % 0.30/0.47 principal: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Filter_Oprod__filter,type, % 0.30/0.47 prod_filter: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( filter @ A ) > ( filter @ B ) > ( filter @ ( product_prod @ A @ B ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Finite__Set_OFpow,type, % 0.30/0.47 finite_Fpow: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Finite__Set_Ocard,type, % 0.30/0.47 finite_card: % 0.30/0.47 !>[B: $tType] : ( ( set @ B ) > nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Finite__Set_Ocomp__fun__commute,type, % 0.30/0.47 finite6289374366891150609ommute: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Finite__Set_Ocomp__fun__commute__on,type, % 0.30/0.47 finite4664212375090638736ute_on: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Finite__Set_Ocomp__fun__idem__on,type, % 0.30/0.47 finite673082921795544331dem_on: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Finite__Set_Ofinite,type, % 0.30/0.47 finite_finite2: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Finite__Set_Ofold,type, % 0.30/0.47 finite_fold: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Finite__Set_Ofolding__idem__on,type, % 0.30/0.47 finite1890593828518410140dem_on: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Finite__Set_Ofolding__on,type, % 0.30/0.47 finite_folding_on: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Finite__Set_Ofolding__on_OF,type, % 0.30/0.47 finite_folding_F: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun_Obij__betw,type, % 0.30/0.47 bij_betw: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun_Ocomp,type, % 0.30/0.47 comp: % 0.30/0.47 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun_Ofun__upd,type, % 0.30/0.47 fun_upd: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun_Oid,type, % 0.30/0.47 id: % 0.30/0.47 !>[A: $tType] : ( A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun_Oinj__on,type, % 0.30/0.47 inj_on: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun_Omap__fun,type, % 0.30/0.47 map_fun: % 0.30/0.47 !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun_Ostrict__mono__on,type, % 0.30/0.47 strict_mono_on: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun_Othe__inv__into,type, % 0.30/0.47 the_inv_into: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun__Def_Ois__measure,type, % 0.30/0.47 fun_is_measure: % 0.30/0.47 !>[A: $tType] : ( ( A > nat ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun__Def_Omax__strict,type, % 0.30/0.47 fun_max_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun__Def_Omax__weak,type, % 0.30/0.47 fun_max_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun__Def_Omin__strict,type, % 0.30/0.47 fun_min_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun__Def_Omin__weak,type, % 0.30/0.47 fun_min_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun__Def_Opair__leq,type, % 0.30/0.47 fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun__Def_Opair__less,type, % 0.30/0.47 fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Fun__Def_Oreduction__pair,type, % 0.30/0.47 fun_reduction_pair: % 0.30/0.47 !>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_GCD_OGcd__class_OGcd,type, % 0.30/0.47 gcd_Gcd: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_GCD_Obezw,type, % 0.30/0.47 bezw: nat > nat > ( product_prod @ int @ int ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_GCD_Obezw__rel,type, % 0.30/0.47 bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_GCD_Ogcd__class_Ogcd,type, % 0.30/0.47 gcd_gcd: % 0.30/0.47 !>[A: $tType] : ( A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_GCD_Ogcd__nat__rel,type, % 0.30/0.47 gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_GCD_Osemiring__1__class_Osemiring__char,type, % 0.30/0.47 semiri4206861660011772517g_char: % 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Groups_Oabs__class_Oabs,type, % 0.30/0.47 abs_abs: % 0.30/0.47 !>[A: $tType] : ( A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Groups_Ominus__class_Ominus,type, % 0.30/0.47 minus_minus: % 0.30/0.47 !>[A: $tType] : ( A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Groups_Oone__class_Oone,type, % 0.30/0.47 one_one: % 0.30/0.47 !>[A: $tType] : A ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Groups_Oplus__class_Oplus,type, % 0.30/0.47 plus_plus: % 0.30/0.47 !>[A: $tType] : ( A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Groups_Osgn__class_Osgn,type, % 0.30/0.47 sgn_sgn: % 0.30/0.47 !>[A: $tType] : ( A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Groups_Otimes__class_Otimes,type, % 0.30/0.47 times_times: % 0.30/0.47 !>[A: $tType] : ( A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Groups_Ouminus__class_Ouminus,type, % 0.30/0.47 uminus_uminus: % 0.30/0.47 !>[A: $tType] : ( A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type, % 0.30/0.47 zero_zero: % 0.30/0.47 !>[A: $tType] : A ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type, % 0.30/0.47 groups7311177749621191930dd_sum: % 0.30/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type, % 0.30/0.47 groups1027152243600224163dd_sum: % 0.30/0.47 !>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type, % 0.30/0.47 groups7121269368397514597t_prod: % 0.30/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type, % 0.30/0.47 groups1962203154675924110t_prod: % 0.30/0.47 !>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type, % 0.30/0.47 groups4207007520872428315er_sum: % 0.30/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type, % 0.30/0.47 groups8242544230860333062m_list: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_HOL_ONO__MATCH,type, % 0.30/0.47 nO_MATCH: % 0.30/0.47 !>[A: $tType,B: $tType] : ( A > B > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_HOL_OThe,type, % 0.30/0.47 the: % 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_HOL_Oundefined,type, % 0.30/0.47 undefined: % 0.30/0.47 !>[A: $tType] : A ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_If,type, % 0.30/0.47 if: % 0.30/0.47 !>[A: $tType] : ( $o > A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Int_OAbs__Integ,type, % 0.30/0.47 abs_Integ: ( product_prod @ nat @ nat ) > int ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Int_ORep__Integ,type, % 0.30/0.47 rep_Integ: int > ( product_prod @ nat @ nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Int_Oint__ge__less__than,type, % 0.30/0.47 int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Int_Oint__ge__less__than2,type, % 0.30/0.47 int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Int_Onat,type, % 0.30/0.47 nat2: int > nat ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Int_Opower__int,type, % 0.30/0.47 power_int: % 0.30/0.47 !>[A: $tType] : ( A > int > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Int_Oring__1__class_OInts,type, % 0.30/0.47 ring_1_Ints: % 0.30/0.47 !>[A: $tType] : ( set @ A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Int_Oring__1__class_Oof__int,type, % 0.30/0.47 ring_1_of_int: % 0.30/0.47 !>[A: $tType] : ( int > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Lattices_Oinf__class_Oinf,type, % 0.30/0.47 inf_inf: % 0.30/0.47 !>[A: $tType] : ( A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Lattices_Osemilattice__neutr__order,type, % 0.30/0.47 semila1105856199041335345_order: % 0.30/0.47 !>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Lattices_Osup__class_Osup,type, % 0.30/0.47 sup_sup: % 0.30/0.47 !>[A: $tType] : ( A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Lattices__Big_Olinorder_OMax,type, % 0.30/0.47 lattices_Max: % 0.30/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Lattices__Big_Olinorder_OMin,type, % 0.30/0.47 lattices_Min: % 0.30/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Lattices__Big_Olinorder__class_OMax,type, % 0.30/0.47 lattic643756798349783984er_Max: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Lattices__Big_Olinorder__class_OMin,type, % 0.30/0.47 lattic643756798350308766er_Min: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type, % 0.30/0.47 lattic7623131987881927897min_on: % 0.30/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type, % 0.30/0.47 lattic7752659483105999362nf_fin: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type, % 0.30/0.47 lattic5882676163264333800up_fin: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Limits_OBfun,type, % 0.30/0.47 bfun: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Limits_Oat__infinity,type, % 0.30/0.47 at_infinity: % 0.30/0.47 !>[A: $tType] : ( filter @ A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oappend,type, % 0.30/0.47 append: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oarg__min__list,type, % 0.30/0.47 arg_min_list: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Obind,type, % 0.30/0.47 bind: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oconcat,type, % 0.30/0.47 concat: % 0.30/0.47 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Ocount__list,type, % 0.30/0.47 count_list: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > A > nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Odistinct,type, % 0.30/0.47 distinct: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Odrop,type, % 0.30/0.47 drop: % 0.30/0.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_OdropWhile,type, % 0.30/0.47 dropWhile: % 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oenumerate,type, % 0.30/0.47 enumerate: % 0.30/0.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oextract,type, % 0.30/0.47 extract: % 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Ofilter,type, % 0.30/0.47 filter2: % 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Ofind,type, % 0.30/0.47 find: % 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Ofolding__insort__key,type, % 0.30/0.47 folding_insort_key: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Ofoldl,type, % 0.30/0.47 foldl: % 0.30/0.47 !>[B: $tType,A: $tType] : ( ( B > A > B ) > B > ( list @ A ) > B ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Ofoldr,type, % 0.30/0.47 foldr: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Ogen__length,type, % 0.30/0.47 gen_length: % 0.30/0.47 !>[A: $tType] : ( nat > ( list @ A ) > nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olenlex,type, % 0.30/0.47 lenlex: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olex,type, % 0.30/0.47 lex: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olexn,type, % 0.30/0.47 lexn: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olexord,type, % 0.30/0.47 lexord: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olexordp,type, % 0.30/0.47 lexordp: % 0.30/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olinorder_Oinsort__key,type, % 0.30/0.47 insort_key: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olinorder_Osorted__key__list__of__set,type, % 0.30/0.47 sorted8670434370408473282of_set: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( set @ B ) > ( list @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olinorder__class_Oinsort__key,type, % 0.30/0.47 linorder_insort_key: % 0.30/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olinorder__class_Osorted__key__list__of__set,type, % 0.30/0.47 linord144544945434240204of_set: % 0.30/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > ( list @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type, % 0.30/0.47 linord4507533701916653071of_set: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olist_OCons,type, % 0.30/0.47 cons: % 0.30/0.47 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olist_ONil,type, % 0.30/0.47 nil: % 0.30/0.47 !>[A: $tType] : ( list @ A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olist_Ocase__list,type, % 0.30/0.47 case_list: % 0.30/0.47 !>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olist_Ohd,type, % 0.30/0.47 hd: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olist_Omap,type, % 0.30/0.47 map: % 0.30/0.47 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olist_Oset,type, % 0.30/0.47 set2: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olist_Osize__list,type, % 0.30/0.47 size_list: % 0.30/0.47 !>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olist_Otl,type, % 0.30/0.47 tl: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olist__update,type, % 0.30/0.47 list_update: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olistrel,type, % 0.30/0.47 listrel: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olistrel1,type, % 0.30/0.47 listrel1: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olistrel1p,type, % 0.30/0.47 listrel1p: % 0.30/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olistrelp,type, % 0.30/0.47 listrelp: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Olistset,type, % 0.30/0.47 listset: % 0.30/0.47 !>[A: $tType] : ( ( list @ ( set @ A ) ) > ( set @ ( list @ A ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Omap__filter,type, % 0.30/0.47 map_filter: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Omap__project,type, % 0.30/0.47 map_project: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > ( set @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Omin__list,type, % 0.30/0.47 min_list: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Omin__list__rel,type, % 0.30/0.47 min_list_rel: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_On__lists,type, % 0.30/0.47 n_lists: % 0.30/0.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Onth,type, % 0.30/0.47 nth: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > nat > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Onths,type, % 0.30/0.47 nths: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Opartition,type, % 0.30/0.47 partition: % 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oproduct,type, % 0.30/0.47 product: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oproduct__lists,type, % 0.30/0.47 product_lists: % 0.30/0.47 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oremdups,type, % 0.30/0.47 remdups: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oremdups__adj,type, % 0.30/0.47 remdups_adj: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oremdups__adj__rel,type, % 0.30/0.47 remdups_adj_rel: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oremove1,type, % 0.30/0.47 remove1: % 0.30/0.47 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_OremoveAll,type, % 0.30/0.47 removeAll: % 0.30/0.47 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oreplicate,type, % 0.30/0.47 replicate: % 0.30/0.47 !>[A: $tType] : ( nat > A > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Orev,type, % 0.30/0.47 rev: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Orotate,type, % 0.30/0.47 rotate: % 0.30/0.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Orotate1,type, % 0.30/0.47 rotate1: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oset__Cons,type, % 0.30/0.47 set_Cons: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) > ( set @ ( list @ A ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oshuffles,type, % 0.30/0.47 shuffles: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Osorted__wrt,type, % 0.30/0.47 sorted_wrt: % 0.30/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Osubseqs,type, % 0.30/0.47 subseqs: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Otake,type, % 0.30/0.47 take: % 0.30/0.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_OtakeWhile,type, % 0.30/0.47 takeWhile: % 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Otranspose,type, % 0.30/0.47 transpose: % 0.30/0.47 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Otranspose__rel,type, % 0.30/0.47 transpose_rel: % 0.30/0.47 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Ounion,type, % 0.30/0.47 union: % 0.30/0.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oupt,type, % 0.30/0.47 upt: nat > nat > ( list @ nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oupto,type, % 0.30/0.47 upto: int > int > ( list @ int ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oupto__aux,type, % 0.30/0.47 upto_aux: int > int > ( list @ int ) > ( list @ int ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Oupto__rel,type, % 0.30/0.47 upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_List_Ozip,type, % 0.30/0.47 zip: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Map_Odom,type, % 0.30/0.47 dom: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Map_Ograph,type, % 0.30/0.47 graph: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Map_Omap__of,type, % 0.30/0.47 map_of: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Map_Omap__upds,type, % 0.30/0.47 map_upds: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Map_Oran,type, % 0.30/0.47 ran: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Map_Orestrict__map,type, % 0.30/0.47 restrict_map: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat_OSuc,type, % 0.30/0.47 suc: nat > nat ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat_Ocompow,type, % 0.30/0.47 compow: % 0.30/0.47 !>[A: $tType] : ( nat > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat_Ofunpow,type, % 0.30/0.47 funpow: % 0.30/0.47 !>[A: $tType] : ( nat > ( A > A ) > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat_Onat_Ocase__nat,type, % 0.30/0.47 case_nat: % 0.30/0.47 !>[A: $tType] : ( A > ( nat > A ) > nat > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat_Onat_Opred,type, % 0.30/0.47 pred: nat > nat ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat_Oold_Onat_Orec__nat,type, % 0.30/0.47 rec_nat: % 0.30/0.47 !>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat_Oold_Onat_Orec__set__nat,type, % 0.30/0.47 rec_set_nat: % 0.30/0.47 !>[T: $tType] : ( T > ( nat > T > T ) > nat > T > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat_Osemiring__1__class_ONats,type, % 0.30/0.47 semiring_1_Nats: % 0.30/0.47 !>[A: $tType] : ( set @ A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type, % 0.30/0.47 semiring_1_of_nat: % 0.30/0.47 !>[A: $tType] : ( nat > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type, % 0.30/0.47 semiri8178284476397505188at_aux: % 0.30/0.47 !>[A: $tType] : ( ( A > A ) > nat > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize,type, % 0.30/0.47 size_size: % 0.30/0.47 !>[A: $tType] : ( A > nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat__Bijection_Olist__encode,type, % 0.30/0.47 nat_list_encode: ( list @ nat ) > nat ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat__Bijection_Olist__encode__rel,type, % 0.30/0.47 nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type, % 0.30/0.47 nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type, % 0.30/0.47 nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat__Bijection_Oprod__encode,type, % 0.30/0.47 nat_prod_encode: ( product_prod @ nat @ nat ) > nat ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat__Bijection_Oset__decode,type, % 0.30/0.47 nat_set_decode: nat > ( set @ nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat__Bijection_Oset__encode,type, % 0.30/0.47 nat_set_encode: ( set @ nat ) > nat ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Nat__Bijection_Otriangle,type, % 0.30/0.47 nat_triangle: nat > nat ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_NthRoot_Oroot,type, % 0.30/0.47 root: nat > real > real ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_NthRoot_Osqrt,type, % 0.30/0.47 sqrt: real > real ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Num_OBitM,type, % 0.30/0.47 bitM: num > num ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Num_Oinc,type, % 0.30/0.47 inc: num > num ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type, % 0.30/0.47 neg_numeral_dbl_dec: % 0.30/0.47 !>[A: $tType] : ( A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Num_Oneg__numeral__class_Osub,type, % 0.30/0.47 neg_numeral_sub: % 0.30/0.47 !>[A: $tType] : ( num > num > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Num_Onum_OBit0,type, % 0.30/0.47 bit0: num > num ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Num_Onum_OBit1,type, % 0.30/0.47 bit1: num > num ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Num_Onum_OOne,type, % 0.30/0.47 one2: num ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Num_Onum_Ocase__num,type, % 0.30/0.47 case_num: % 0.30/0.47 !>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Num_Onum_Osize__num,type, % 0.30/0.47 size_num: num > nat ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Num_Onum__of__nat,type, % 0.30/0.47 num_of_nat: nat > num ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Num_Onumeral__class_Onumeral,type, % 0.30/0.47 numeral_numeral: % 0.30/0.47 !>[A: $tType] : ( num > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Num_Opow,type, % 0.30/0.47 pow: num > num > num ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Num_Opred__numeral,type, % 0.30/0.47 pred_numeral: num > nat ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Option_Ooption_ONone,type, % 0.30/0.47 none: % 0.30/0.47 !>[A: $tType] : ( option @ A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Option_Ooption_OSome,type, % 0.30/0.47 some: % 0.30/0.47 !>[A: $tType] : ( A > ( option @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Option_Ooption_Ocase__option,type, % 0.30/0.47 case_option: % 0.30/0.47 !>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Option_Ooption_Omap__option,type, % 0.30/0.47 map_option: % 0.30/0.47 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( option @ A ) > ( option @ Aa ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Option_Ooption_Orec__option,type, % 0.30/0.47 rec_option: % 0.30/0.47 !>[C: $tType,A: $tType] : ( C > ( A > C ) > ( option @ A ) > C ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Option_Ooption_Osize__option,type, % 0.30/0.47 size_option: % 0.30/0.47 !>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Option_Ooption_Othe,type, % 0.30/0.47 the2: % 0.30/0.47 !>[A: $tType] : ( ( option @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Option_Othese,type, % 0.30/0.47 these: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Order__Continuity_Ocountable__complete__lattice__class_Occlfp,type, % 0.30/0.47 order_532582986084564980_cclfp: % 0.30/0.47 !>[A: $tType] : ( ( A > A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Order__Relation_OAbove,type, % 0.30/0.47 order_Above: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Order__Relation_OUnder,type, % 0.30/0.47 order_Under: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Order__Relation_OUnderS,type, % 0.30/0.47 order_UnderS: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Order__Relation_OaboveS,type, % 0.30/0.47 order_aboveS: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Order__Relation_Olinear__order__on,type, % 0.30/0.47 order_679001287576687338der_on: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Order__Relation_Opartial__order__on,type, % 0.30/0.47 order_7125193373082350890der_on: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Order__Relation_Ounder,type, % 0.30/0.47 order_under: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Order__Relation_OunderS,type, % 0.30/0.47 order_underS: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Orderings_Obot__class_Obot,type, % 0.30/0.47 bot_bot: % 0.30/0.47 !>[A: $tType] : A ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Orderings_Oord_OLeast,type, % 0.30/0.47 least: % 0.30/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Orderings_Oord_Omax,type, % 0.30/0.47 max: % 0.30/0.47 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Orderings_Oord_Omin,type, % 0.30/0.47 min: % 0.30/0.47 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Orderings_Oord__class_Oless,type, % 0.30/0.47 ord_less: % 0.30/0.47 !>[A: $tType] : ( A > A > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type, % 0.30/0.47 ord_less_eq: % 0.30/0.47 !>[A: $tType] : ( A > A > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Orderings_Oord__class_Omax,type, % 0.30/0.47 ord_max: % 0.30/0.47 !>[A: $tType] : ( A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Orderings_Oord__class_Omin,type, % 0.30/0.47 ord_min: % 0.30/0.47 !>[A: $tType] : ( A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Orderings_Oorder__class_OGreatest,type, % 0.30/0.47 order_Greatest: % 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Orderings_Oorder__class_Oantimono,type, % 0.30/0.47 order_antimono: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Orderings_Oorder__class_Omono,type, % 0.30/0.47 order_mono: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type, % 0.30/0.47 order_strict_mono: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Orderings_Otop__class_Otop,type, % 0.30/0.47 top_top: % 0.30/0.47 !>[A: $tType] : A ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Partial__Function_Oflat__lub,type, % 0.30/0.47 partial_flat_lub: % 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Power_Opower__class_Opower,type, % 0.30/0.47 power_power: % 0.30/0.47 !>[A: $tType] : ( A > nat > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Product__Type_OPair,type, % 0.30/0.47 product_Pair: % 0.30/0.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Product__Type_OSigma,type, % 0.30/0.47 product_Sigma: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Product__Type_Oapfst,type, % 0.30/0.47 product_apfst: % 0.30/0.47 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Product__Type_Oapsnd,type, % 0.30/0.47 product_apsnd: % 0.30/0.47 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type, % 0.30/0.47 product_rec_prod: % 0.30/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type, % 0.30/0.47 product_rec_set_prod: % 0.30/0.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Product__Type_Oprod_Ocase__prod,type, % 0.30/0.47 product_case_prod: % 0.30/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Product__Type_Oprod_Ofst,type, % 0.30/0.47 product_fst: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Product__Type_Oprod_Osnd,type, % 0.30/0.47 product_snd: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Product__Type_Oproduct,type, % 0.30/0.47 product_product: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Rat_OFract,type, % 0.30/0.47 fract: int > int > rat ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Rat_OFrct,type, % 0.30/0.47 frct: ( product_prod @ int @ int ) > rat ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Rat_ORep__Rat,type, % 0.30/0.47 rep_Rat: rat > ( product_prod @ int @ int ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Rat_Ofield__char__0__class_ORats,type, % 0.30/0.47 field_char_0_Rats: % 0.30/0.47 !>[A: $tType] : ( set @ A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Rat_Onormalize,type, % 0.30/0.47 normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Rat_Oof__int,type, % 0.30/0.47 of_int: int > rat ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Rat_Opositive,type, % 0.30/0.47 positive: rat > $o ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Rat_Oquotient__of,type, % 0.30/0.47 quotient_of: rat > ( product_prod @ int @ int ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Real__Vector__Spaces_Obounded__linear,type, % 0.30/0.47 real_V3181309239436604168linear: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type, % 0.30/0.47 real_V557655796197034286t_dist: % 0.30/0.47 !>[A: $tType] : ( A > A > real ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type, % 0.30/0.47 real_V7770717601297561774m_norm: % 0.30/0.47 !>[A: $tType] : ( A > real ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Real__Vector__Spaces_Oof__real,type, % 0.30/0.47 real_Vector_of_real: % 0.30/0.47 !>[A: $tType] : ( real > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type, % 0.30/0.47 real_V8093663219630862766scaleR: % 0.30/0.47 !>[A: $tType] : ( real > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Relation_OField,type, % 0.30/0.47 field2: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Relation_OId,type, % 0.30/0.47 id2: % 0.30/0.47 !>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Relation_OId__on,type, % 0.30/0.47 id_on: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Relation_Oirrefl,type, % 0.30/0.47 irrefl: % 0.30/0.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Relation_Orefl__on,type, % 0.30/0.47 refl_on: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Relation_Orelcomp,type, % 0.30/0.47 relcomp: % 0.30/0.47 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Relation_Ototal__on,type, % 0.30/0.47 total_on: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Rings_Odivide__class_Odivide,type, % 0.30/0.47 divide_divide: % 0.30/0.47 !>[A: $tType] : ( A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Rings_Odvd__class_Odvd,type, % 0.30/0.47 dvd_dvd: % 0.30/0.47 !>[A: $tType] : ( A > A > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Rings_Omodulo__class_Omodulo,type, % 0.30/0.47 modulo_modulo: % 0.30/0.47 !>[A: $tType] : ( A > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type, % 0.30/0.47 zero_neq_one_of_bool: % 0.30/0.47 !>[A: $tType] : ( $o > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Series_Osuminf,type, % 0.30/0.47 suminf: % 0.30/0.47 !>[A: $tType] : ( ( nat > A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Series_Osummable,type, % 0.30/0.47 summable: % 0.30/0.47 !>[A: $tType] : ( ( nat > A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Series_Osums,type, % 0.30/0.47 sums: % 0.30/0.47 !>[A: $tType] : ( ( nat > A ) > A > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set_OBall,type, % 0.30/0.47 ball: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set_OBex,type, % 0.30/0.47 bex: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set_OCollect,type, % 0.30/0.47 collect: % 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set_OPow,type, % 0.30/0.47 pow2: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set_Ofilter,type, % 0.30/0.47 filter3: % 0.30/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set_Oimage,type, % 0.30/0.47 image: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set_Oinsert,type, % 0.30/0.47 insert: % 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set_Ois__singleton,type, % 0.30/0.47 is_singleton: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set_Oremove,type, % 0.30/0.47 remove: % 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set_Othe__elem,type, % 0.30/0.47 the_elem: % 0.30/0.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set_Ovimage,type, % 0.30/0.47 vimage: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type, % 0.30/0.47 set_fo6178422350223883121st_nat: % 0.30/0.47 !>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set__Interval_Oord__class_OatLeast,type, % 0.30/0.47 set_ord_atLeast: % 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type, % 0.30/0.47 set_or1337092689740270186AtMost: % 0.30/0.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type, % 0.30/0.47 set_or7035219750837199246ssThan: % 0.30/0.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set__Interval_Oord__class_OatMost,type, % 0.30/0.47 set_ord_atMost: % 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type, % 0.30/0.47 set_ord_greaterThan: % 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type, % 0.30/0.47 set_or3652927894154168847AtMost: % 0.30/0.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type, % 0.30/0.47 set_or5935395276787703475ssThan: % 0.30/0.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Set__Interval_Oord__class_OlessThan,type, % 0.30/0.47 set_ord_lessThan: % 0.30/0.47 !>[A: $tType] : ( A > ( set @ A ) ) ). % 0.30/0.47 % 0.30/0.47 thf(sy_c_Topological__Spaces_Ocontinuous,type, % 0.30/0.47 topolo3448309680560233919inuous: % 0.30/0.47 !>[A: $tType,B: $tType] : ( ( filter @ A ) > ( A > B ) > $o ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Topological__Spaces_Ocontinuous__on,type, % 0.30/0.48 topolo81223032696312382ous_on: % 0.30/0.48 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $o ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Topological__Spaces_Omonoseq,type, % 0.30/0.48 topological_monoseq: % 0.30/0.48 !>[A: $tType] : ( ( nat > A ) > $o ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Topological__Spaces_Oopen__class_Oopen,type, % 0.30/0.48 topolo1002775350975398744n_open: % 0.30/0.48 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Topological__Spaces_Ot2__space__class_OLim,type, % 0.30/0.48 topolo3827282254853284352ce_Lim: % 0.30/0.48 !>[F: $tType,A: $tType] : ( ( filter @ F ) > ( F > A ) > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type, % 0.30/0.48 topolo174197925503356063within: % 0.30/0.48 !>[A: $tType] : ( A > ( set @ A ) > ( filter @ A ) ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Topological__Spaces_Otopological__space__class_Oclosed,type, % 0.30/0.48 topolo7761053866217962861closed: % 0.30/0.48 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Topological__Spaces_Otopological__space__class_Ocompact,type, % 0.30/0.48 topolo2193935891317330818ompact: % 0.30/0.48 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type, % 0.30/0.48 topolo7230453075368039082e_nhds: % 0.30/0.48 !>[A: $tType] : ( A > ( filter @ A ) ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type, % 0.30/0.48 topolo3814608138187158403Cauchy: % 0.30/0.48 !>[A: $tType] : ( ( nat > A ) > $o ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type, % 0.30/0.48 topolo6773858410816713723filter: % 0.30/0.48 !>[A: $tType] : ( ( filter @ A ) > $o ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocomplete,type, % 0.30/0.48 topolo2479028161051973599mplete: % 0.30/0.48 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type, % 0.30/0.48 topolo6688025880775521714ounded: % 0.30/0.48 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity,type, % 0.30/0.48 topolo7806501430040627800ormity: % 0.30/0.48 !>[A: $tType] : ( filter @ ( product_prod @ A @ A ) ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Topological__Spaces_Ouniformly__continuous__on,type, % 0.30/0.48 topolo6026614971017936543ous_on: % 0.30/0.48 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $o ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Oarccos,type, % 0.30/0.48 arccos: real > real ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Oarcosh,type, % 0.30/0.48 arcosh: % 0.30/0.48 !>[A: $tType] : ( A > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Oarcsin,type, % 0.30/0.48 arcsin: real > real ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Oarctan,type, % 0.30/0.48 arctan: real > real ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Oarsinh,type, % 0.30/0.48 arsinh: % 0.30/0.48 !>[A: $tType] : ( A > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Oartanh,type, % 0.30/0.48 artanh: % 0.30/0.48 !>[A: $tType] : ( A > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Ocos,type, % 0.30/0.48 cos: % 0.30/0.48 !>[A: $tType] : ( A > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Ocos__coeff,type, % 0.30/0.48 cos_coeff: nat > real ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Ocosh,type, % 0.30/0.48 cosh: % 0.30/0.48 !>[A: $tType] : ( A > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Ocot,type, % 0.30/0.48 cot: % 0.30/0.48 !>[A: $tType] : ( A > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Odiffs,type, % 0.30/0.48 diffs: % 0.30/0.48 !>[A: $tType] : ( ( nat > A ) > nat > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Oexp,type, % 0.30/0.48 exp: % 0.30/0.48 !>[A: $tType] : ( A > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Oln__class_Oln,type, % 0.30/0.48 ln_ln: % 0.30/0.48 !>[A: $tType] : ( A > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Olog,type, % 0.30/0.48 log: real > real > real ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Opi,type, % 0.30/0.48 pi: real ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Opowr,type, % 0.30/0.48 powr: % 0.30/0.48 !>[A: $tType] : ( A > A > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Osin,type, % 0.30/0.48 sin: % 0.30/0.48 !>[A: $tType] : ( A > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Osin__coeff,type, % 0.30/0.48 sin_coeff: nat > real ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Osinh,type, % 0.30/0.48 sinh: % 0.30/0.48 !>[A: $tType] : ( A > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Otan,type, % 0.30/0.48 tan: % 0.30/0.48 !>[A: $tType] : ( A > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transcendental_Otanh,type, % 0.30/0.48 tanh: % 0.30/0.48 !>[A: $tType] : ( A > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transitive__Closure_Ontrancl,type, % 0.30/0.48 transitive_ntrancl: % 0.30/0.48 !>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transitive__Closure_Ortrancl,type, % 0.30/0.48 transitive_rtrancl: % 0.30/0.48 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Transitive__Closure_Otrancl,type, % 0.30/0.48 transitive_trancl: % 0.30/0.48 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type, % 0.30/0.48 vEBT_Leaf: $o > $o > vEBT_VEBT ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT_ONode,type, % 0.30/0.48 vEBT_Node: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > vEBT_VEBT ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type, % 0.30/0.48 vEBT_size_VEBT: vEBT_VEBT > nat ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type, % 0.30/0.48 vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type, % 0.30/0.48 vEBT_VEBT_high: nat > nat > nat ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type, % 0.30/0.48 vEBT_V5917875025757280293ildren: nat > ( list @ vEBT_VEBT ) > nat > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type, % 0.30/0.48 vEBT_VEBT_low: nat > nat > nat ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type, % 0.30/0.48 vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type, % 0.30/0.48 vEBT_V4351362008482014158ma_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type, % 0.30/0.48 vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type, % 0.30/0.48 vEBT_V5765760719290551771er_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type, % 0.30/0.48 vEBT_VEBT_valid: vEBT_VEBT > nat > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type, % 0.30/0.48 vEBT_VEBT_valid_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type, % 0.30/0.48 vEBT_invar_vebt: vEBT_VEBT > nat > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_Oset__vebt,type, % 0.30/0.48 vEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_Ovebt__buildup,type, % 0.30/0.48 vEBT_vebt_buildup: nat > vEBT_VEBT ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type, % 0.30/0.48 vEBT_v4011308405150292612up_rel: nat > nat > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type, % 0.30/0.48 vEBT_VEBT_bit_concat: nat > nat > nat > nat ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type, % 0.30/0.48 vEBT_VEBT_minNull: vEBT_VEBT > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type, % 0.30/0.48 vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type, % 0.30/0.48 vEBT_VEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Member_Ovebt__member,type, % 0.30/0.48 vEBT_vebt_member: vEBT_VEBT > nat > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__Member_Ovebt__member__rel,type, % 0.30/0.48 vEBT_vebt_member_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type, % 0.30/0.48 vEBT_VEBT_add: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type, % 0.30/0.48 vEBT_VEBT_greater: ( option @ nat ) > ( option @ nat ) > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type, % 0.30/0.48 vEBT_VEBT_less: ( option @ nat ) > ( option @ nat ) > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type, % 0.30/0.48 vEBT_VEBT_lesseq: ( option @ nat ) > ( option @ nat ) > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type, % 0.30/0.48 vEBT_VEBT_max_in_set: ( set @ nat ) > nat > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type, % 0.30/0.48 vEBT_VEBT_min_in_set: ( set @ nat ) > nat > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type, % 0.30/0.48 vEBT_VEBT_mul: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift,type, % 0.30/0.48 vEBT_V2048590022279873568_shift: % 0.30/0.48 !>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type, % 0.30/0.48 vEBT_VEBT_power: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__MinMax_Ovebt__maxt,type, % 0.30/0.48 vEBT_vebt_maxt: vEBT_VEBT > ( option @ nat ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type, % 0.30/0.48 vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__MinMax_Ovebt__mint,type, % 0.30/0.48 vEBT_vebt_mint: vEBT_VEBT > ( option @ nat ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type, % 0.30/0.48 vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Wellfounded_Oaccp,type, % 0.30/0.48 accp: % 0.30/0.48 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Wellfounded_Ofinite__psubset,type, % 0.30/0.48 finite_psubset: % 0.30/0.48 !>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Wellfounded_Olex__prod,type, % 0.30/0.48 lex_prod: % 0.30/0.48 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Wellfounded_Omax__ext,type, % 0.30/0.48 max_ext: % 0.30/0.48 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Wellfounded_Omax__extp,type, % 0.30/0.48 max_extp: % 0.30/0.48 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ A ) > $o ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Wellfounded_Omeasure,type, % 0.30/0.48 measure: % 0.30/0.48 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Wellfounded_Omin__ext,type, % 0.30/0.48 min_ext: % 0.30/0.48 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Wellfounded_Omlex__prod,type, % 0.30/0.48 mlex_prod: % 0.30/0.48 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Wellfounded_Opred__nat,type, % 0.30/0.48 pred_nat: set @ ( product_prod @ nat @ nat ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Wellfounded_Owf,type, % 0.30/0.48 wf: % 0.30/0.48 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Wfrec_Osame__fst,type, % 0.30/0.48 same_fst: % 0.30/0.48 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_Zorn_Ochains,type, % 0.30/0.48 chains: % 0.30/0.48 !>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ ( set @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_fChoice,type, % 0.30/0.48 fChoice: % 0.30/0.48 !>[A: $tType] : ( ( A > $o ) > A ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_c_member,type, % 0.30/0.48 member: % 0.30/0.48 !>[A: $tType] : ( A > ( set @ A ) > $o ) ). % 0.30/0.48 % 0.30/0.48 thf(sy_v_deg,type, % 0.30/0.48 deg: nat ). % 0.30/0.48 % 0.30/0.48 thf(sy_v_m____,type, % 0.30/0.48 m: nat ). % 0.30/0.48 % 0.30/0.48 thf(sy_v_ma,type, % 0.30/0.48 ma: nat ). % 0.30/0.48 % 0.30/0.48 thf(sy_v_maxs____,type, % 0.30/0.48 maxs: nat ). % 0.30/0.48 % 0.30/0.48 thf(sy_v_mi,type, % 0.30/0.48 mi: nat ). % 0.30/0.48 % 0.30/0.48 thf(sy_v_n____,type, % 0.30/0.48 n: nat ). % 0.30/0.48 % 0.30/0.48 thf(sy_v_summary,type, % 0.30/0.48 summary: vEBT_VEBT ). % 0.30/0.48 % 0.30/0.48 thf(sy_v_thesis____,type, % 0.30/0.48 thesis: $o ). % 0.30/0.48 % 0.30/0.48 thf(sy_v_treeList,type, % 0.30/0.48 treeList: list @ vEBT_VEBT ). % 0.30/0.48 % 0.30/0.48 % Relevant facts (8186) % 0.30/0.48 thf(fact_0__092_060open_062_092_060exists_062x_O_Aboth__member__options_A_ItreeList_A_B_Amaxs_J_Ax_092_060close_062,axiom, % 0.30/0.48 ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ maxs ) @ X_1 ) ). % 0.30/0.48 % 0.30/0.48 % \<open>\<exists>x. both_member_options (treeList ! maxs) x\<close> % 0.30/0.48 thf(fact_1_not__min__Null__member,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT] : % 0.30/0.48 ( ~ ( vEBT_VEBT_minNull @ T2 ) % 0.30/0.48 => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_1 ) ) ). % 0.30/0.48 % 0.30/0.48 % not_min_Null_member % 0.30/0.48 thf(fact_2__092_060open_062Some_Amaxs_A_061_Avebt__maxt_Asummary_092_060close_062,axiom, % 0.30/0.48 ( ( some @ nat @ maxs ) % 0.30/0.48 = ( vEBT_vebt_maxt @ summary ) ) ). % 0.30/0.48 % 0.30/0.48 % \<open>Some maxs = vebt_maxt summary\<close> % 0.30/0.48 thf(fact_3_both__member__options__def,axiom, % 0.30/0.48 ( vEBT_V8194947554948674370ptions % 0.30/0.48 = ( ^ [T3: vEBT_VEBT,X: nat] : % 0.30/0.48 ( ( vEBT_V5719532721284313246member @ T3 @ X ) % 0.30/0.48 | ( vEBT_VEBT_membermima @ T3 @ X ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % both_member_options_def % 0.30/0.48 thf(fact_4__C5_C_I7_J,axiom, % 0.30/0.48 ( ( mi = ma ) % 0.30/0.48 => ! [X2: vEBT_VEBT] : % 0.30/0.48 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ treeList ) ) % 0.30/0.48 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % "5"(7) % 0.30/0.48 thf(fact_5__C5_C_I1_J,axiom, % 0.30/0.48 ! [X2: vEBT_VEBT] : % 0.30/0.48 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ treeList ) ) % 0.30/0.48 => ( vEBT_invar_vebt @ X2 @ n ) ) ). % 0.30/0.48 % 0.30/0.48 % "5"(1) % 0.30/0.48 thf(fact_6_in__children__def,axiom, % 0.30/0.48 ( vEBT_V5917875025757280293ildren % 0.30/0.48 = ( ^ [N: nat,TreeList: list @ vEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ N ) ) @ ( vEBT_VEBT_low @ X @ N ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % in_children_def % 0.30/0.48 thf(fact_7_both__member__options__equiv__member,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,N2: nat,X3: nat] : % 0.30/0.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.48 => ( ( vEBT_V8194947554948674370ptions @ T2 @ X3 ) % 0.30/0.48 = ( vEBT_vebt_member @ T2 @ X3 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % both_member_options_equiv_member % 0.30/0.48 thf(fact_8_valid__member__both__member__options,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,N2: nat,X3: nat] : % 0.30/0.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.48 => ( ( vEBT_V8194947554948674370ptions @ T2 @ X3 ) % 0.30/0.48 => ( vEBT_vebt_member @ T2 @ X3 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % valid_member_both_member_options % 0.30/0.48 thf(fact_9_intind,axiom, % 0.30/0.48 ! [A: $tType,I: nat,N2: nat,P: A > $o,X3: A] : % 0.30/0.48 ( ( ord_less @ nat @ I @ N2 ) % 0.30/0.48 => ( ( P @ X3 ) % 0.30/0.48 => ( P @ ( nth @ A @ ( replicate @ A @ N2 @ X3 ) @ I ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % intind % 0.30/0.48 thf(fact_10_assms_I2_J,axiom, % 0.30/0.48 mi != ma ). % 0.30/0.48 % 0.30/0.48 % assms(2) % 0.30/0.48 thf(fact_11_list__update__id,axiom, % 0.30/0.48 ! [A: $tType,Xs: list @ A,I: nat] : % 0.30/0.48 ( ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ I ) ) % 0.30/0.48 = Xs ) ). % 0.30/0.48 % 0.30/0.48 % list_update_id % 0.30/0.48 thf(fact_12_nth__list__update__neq,axiom, % 0.30/0.48 ! [A: $tType,I: nat,J: nat,Xs: list @ A,X3: A] : % 0.30/0.48 ( ( I != J ) % 0.30/0.48 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ J ) % 0.30/0.48 = ( nth @ A @ Xs @ J ) ) ) ). % 0.30/0.48 % 0.30/0.48 % nth_list_update_neq % 0.30/0.48 thf(fact_13_min__Null__member,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,X3: nat] : % 0.30/0.48 ( ( vEBT_VEBT_minNull @ T2 ) % 0.30/0.48 => ~ ( vEBT_vebt_member @ T2 @ X3 ) ) ). % 0.30/0.48 % 0.30/0.48 % min_Null_member % 0.30/0.48 thf(fact_14__C5_C_I2_J,axiom, % 0.30/0.48 vEBT_invar_vebt @ summary @ m ). % 0.30/0.48 % 0.30/0.48 % "5"(2) % 0.30/0.48 thf(fact_15_bit__split__inv,axiom, % 0.30/0.48 ! [X3: nat,D2: nat] : % 0.30/0.48 ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X3 @ D2 ) @ ( vEBT_VEBT_low @ X3 @ D2 ) @ D2 ) % 0.30/0.48 = X3 ) ). % 0.30/0.48 % 0.30/0.48 % bit_split_inv % 0.30/0.48 thf(fact_16__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxs_O_ASome_Amaxs_A_061_Avebt__maxt_Asummary_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom, % 0.30/0.48 ~ ! [Maxs: nat] : % 0.30/0.48 ( ( some @ nat @ Maxs ) % 0.30/0.48 != ( vEBT_vebt_maxt @ summary ) ) ). % 0.30/0.48 % 0.30/0.48 % \<open>\<And>thesis. (\<And>maxs. Some maxs = vebt_maxt summary \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close> % 0.30/0.48 thf(fact_17_member__valid__both__member__options,axiom, % 0.30/0.48 ! [Tree: vEBT_VEBT,N2: nat,X3: nat] : % 0.30/0.48 ( ( vEBT_invar_vebt @ Tree @ N2 ) % 0.30/0.48 => ( ( vEBT_vebt_member @ Tree @ X3 ) % 0.30/0.48 => ( ( vEBT_V5719532721284313246member @ Tree @ X3 ) % 0.30/0.48 | ( vEBT_VEBT_membermima @ Tree @ X3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % member_valid_both_member_options % 0.30/0.48 thf(fact_18_maxt__member,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,N2: nat,Maxi: nat] : % 0.30/0.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.48 => ( ( ( vEBT_vebt_maxt @ T2 ) % 0.30/0.48 = ( some @ nat @ Maxi ) ) % 0.30/0.48 => ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ). % 0.30/0.48 % 0.30/0.48 % maxt_member % 0.30/0.48 thf(fact_19_list__update__overwrite,axiom, % 0.30/0.48 ! [A: $tType,Xs: list @ A,I: nat,X3: A,Y: A] : % 0.30/0.48 ( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ I @ Y ) % 0.30/0.48 = ( list_update @ A @ Xs @ I @ Y ) ) ). % 0.30/0.48 % 0.30/0.48 % list_update_overwrite % 0.30/0.48 thf(fact_20_member__correct,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,N2: nat,X3: nat] : % 0.30/0.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.48 => ( ( vEBT_vebt_member @ T2 @ X3 ) % 0.30/0.48 = ( member @ nat @ X3 @ ( vEBT_set_vebt @ T2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % member_correct % 0.30/0.48 thf(fact_21__092_060open_062both__member__options_Asummary_A_Ihigh_Ama_An_J_092_060close_062,axiom, % 0.30/0.48 vEBT_V8194947554948674370ptions @ summary @ ( vEBT_VEBT_high @ ma @ n ) ). % 0.30/0.48 % 0.30/0.48 % \<open>both_member_options summary (high ma n)\<close> % 0.30/0.48 thf(fact_22_nth__replicate,axiom, % 0.30/0.48 ! [A: $tType,I: nat,N2: nat,X3: A] : % 0.30/0.48 ( ( ord_less @ nat @ I @ N2 ) % 0.30/0.48 => ( ( nth @ A @ ( replicate @ A @ N2 @ X3 ) @ I ) % 0.30/0.48 = X3 ) ) ). % 0.30/0.48 % 0.30/0.48 % nth_replicate % 0.30/0.48 thf(fact_23__C5_C_I8_J,axiom, % 0.30/0.48 ord_less_eq @ nat @ mi @ ma ). % 0.30/0.48 % 0.30/0.48 % "5"(8) % 0.30/0.48 thf(fact_24_greater__shift,axiom, % 0.30/0.48 ( ( ord_less @ nat ) % 0.30/0.48 = ( ^ [Y2: nat,X: nat] : ( vEBT_VEBT_greater @ ( some @ nat @ X ) @ ( some @ nat @ Y2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % greater_shift % 0.30/0.48 thf(fact_25_less__shift,axiom, % 0.30/0.48 ( ( ord_less @ nat ) % 0.30/0.48 = ( ^ [X: nat,Y2: nat] : ( vEBT_VEBT_less @ ( some @ nat @ X ) @ ( some @ nat @ Y2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_shift % 0.30/0.48 thf(fact_26__092_060open_062high_Ama_An_A_060_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062,axiom, % 0.30/0.48 ord_less @ nat @ ( vEBT_VEBT_high @ ma @ n ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ summary ) ) ). % 0.30/0.48 % 0.30/0.48 % \<open>high ma n < the (vebt_maxt summary)\<close> % 0.30/0.48 thf(fact_27_mint__member,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,N2: nat,Maxi: nat] : % 0.30/0.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.48 => ( ( ( vEBT_vebt_mint @ T2 ) % 0.30/0.48 = ( some @ nat @ Maxi ) ) % 0.30/0.48 => ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ). % 0.30/0.48 % 0.30/0.48 % mint_member % 0.30/0.48 thf(fact_28_list__update__swap,axiom, % 0.30/0.48 ! [A: $tType,I: nat,I2: nat,Xs: list @ A,X3: A,X4: A] : % 0.30/0.48 ( ( I != I2 ) % 0.30/0.48 => ( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ I2 @ X4 ) % 0.30/0.48 = ( list_update @ A @ ( list_update @ A @ Xs @ I2 @ X4 ) @ I @ X3 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % list_update_swap % 0.30/0.48 thf(fact_29_maxt__corr__help,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,N2: nat,Maxi: nat,X3: nat] : % 0.30/0.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.48 => ( ( ( vEBT_vebt_maxt @ T2 ) % 0.30/0.48 = ( some @ nat @ Maxi ) ) % 0.30/0.48 => ( ( vEBT_vebt_member @ T2 @ X3 ) % 0.30/0.48 => ( ord_less_eq @ nat @ X3 @ Maxi ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % maxt_corr_help % 0.30/0.48 thf(fact_30_maxt__corr,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,N2: nat,X3: nat] : % 0.30/0.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.48 => ( ( ( vEBT_vebt_maxt @ T2 ) % 0.30/0.48 = ( some @ nat @ X3 ) ) % 0.30/0.48 => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % maxt_corr % 0.30/0.48 thf(fact_31_maxt__sound,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,N2: nat,X3: nat] : % 0.30/0.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.48 => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 ) % 0.30/0.48 => ( ( vEBT_vebt_maxt @ T2 ) % 0.30/0.48 = ( some @ nat @ X3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % maxt_sound % 0.30/0.48 thf(fact_32_buildup__nothing__in__leaf,axiom, % 0.30/0.48 ! [N2: nat,X3: nat] : % 0.30/0.48 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X3 ) ). % 0.30/0.48 % 0.30/0.48 % buildup_nothing_in_leaf % 0.30/0.48 thf(fact_33_buildup__nothing__in__min__max,axiom, % 0.30/0.48 ! [N2: nat,X3: nat] : % 0.30/0.48 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X3 ) ). % 0.30/0.48 % 0.30/0.48 % buildup_nothing_in_min_max % 0.30/0.48 thf(fact_34_option_Oinject,axiom, % 0.30/0.48 ! [A: $tType,X22: A,Y22: A] : % 0.30/0.48 ( ( ( some @ A @ X22 ) % 0.30/0.48 = ( some @ A @ Y22 ) ) % 0.30/0.48 = ( X22 = Y22 ) ) ). % 0.30/0.48 % 0.30/0.48 % option.inject % 0.30/0.48 thf(fact_35_valid__eq,axiom, % 0.30/0.48 vEBT_VEBT_valid = vEBT_invar_vebt ). % 0.30/0.48 % 0.30/0.48 % valid_eq % 0.30/0.48 thf(fact_36_valid__eq1,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,D2: nat] : % 0.30/0.48 ( ( vEBT_invar_vebt @ T2 @ D2 ) % 0.30/0.48 => ( vEBT_VEBT_valid @ T2 @ D2 ) ) ). % 0.30/0.48 % 0.30/0.48 % valid_eq1 % 0.30/0.48 thf(fact_37_valid__eq2,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,D2: nat] : % 0.30/0.48 ( ( vEBT_VEBT_valid @ T2 @ D2 ) % 0.30/0.48 => ( vEBT_invar_vebt @ T2 @ D2 ) ) ). % 0.30/0.48 % 0.30/0.48 % valid_eq2 % 0.30/0.48 thf(fact_38_inthall,axiom, % 0.30/0.48 ! [A: $tType,Xs: list @ A,P: A > $o,N2: nat] : % 0.30/0.48 ( ! [X5: A] : % 0.30/0.48 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.30/0.48 => ( P @ X5 ) ) % 0.30/0.48 => ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.48 => ( P @ ( nth @ A @ Xs @ N2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % inthall % 0.30/0.48 thf(fact_39_deg__not__0,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,N2: nat] : % 0.30/0.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.48 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % deg_not_0 % 0.30/0.48 thf(fact_40__092_060open_062high_Ama_An_A_092_060le_062_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062,axiom, % 0.30/0.48 ord_less_eq @ nat @ ( vEBT_VEBT_high @ ma @ n ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ summary ) ) ). % 0.30/0.48 % 0.30/0.48 % \<open>high ma n \<le> the (vebt_maxt summary)\<close> % 0.30/0.48 thf(fact_41_valid__tree__deg__neq__0,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT] : % 0.30/0.48 ~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ). % 0.30/0.48 % 0.30/0.48 % valid_tree_deg_neq_0 % 0.30/0.48 thf(fact_42_valid__0__not,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT] : % 0.30/0.48 ~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ). % 0.30/0.48 % 0.30/0.48 % valid_0_not % 0.30/0.48 thf(fact_43_min__in__set__def,axiom, % 0.30/0.48 ( vEBT_VEBT_min_in_set % 0.30/0.48 = ( ^ [Xs2: set @ nat,X: nat] : % 0.30/0.48 ( ( member @ nat @ X @ Xs2 ) % 0.30/0.48 & ! [Y2: nat] : % 0.30/0.48 ( ( member @ nat @ Y2 @ Xs2 ) % 0.30/0.48 => ( ord_less_eq @ nat @ X @ Y2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % min_in_set_def % 0.30/0.48 thf(fact_44_max__in__set__def,axiom, % 0.30/0.48 ( vEBT_VEBT_max_in_set % 0.30/0.48 = ( ^ [Xs2: set @ nat,X: nat] : % 0.30/0.48 ( ( member @ nat @ X @ Xs2 ) % 0.30/0.48 & ! [Y2: nat] : % 0.30/0.48 ( ( member @ nat @ Y2 @ Xs2 ) % 0.30/0.48 => ( ord_less_eq @ nat @ Y2 @ X ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % max_in_set_def % 0.30/0.48 thf(fact_45_mem__Collect__eq,axiom, % 0.30/0.48 ! [A: $tType,A2: A,P: A > $o] : % 0.30/0.48 ( ( member @ A @ A2 @ ( collect @ A @ P ) ) % 0.30/0.48 = ( P @ A2 ) ) ). % 0.30/0.48 % 0.30/0.48 % mem_Collect_eq % 0.30/0.48 thf(fact_46_Collect__mem__eq,axiom, % 0.30/0.48 ! [A: $tType,A3: set @ A] : % 0.30/0.48 ( ( collect @ A % 0.30/0.48 @ ^ [X: A] : ( member @ A @ X @ A3 ) ) % 0.30/0.48 = A3 ) ). % 0.30/0.48 % 0.30/0.48 % Collect_mem_eq % 0.30/0.48 thf(fact_47_Collect__cong,axiom, % 0.30/0.48 ! [A: $tType,P: A > $o,Q: A > $o] : % 0.30/0.48 ( ! [X5: A] : % 0.30/0.48 ( ( P @ X5 ) % 0.30/0.48 = ( Q @ X5 ) ) % 0.30/0.48 => ( ( collect @ A @ P ) % 0.30/0.48 = ( collect @ A @ Q ) ) ) ). % 0.30/0.48 % 0.30/0.48 % Collect_cong % 0.30/0.48 thf(fact_48_ext,axiom, % 0.30/0.48 ! [B: $tType,A: $tType,F2: A > B,G: A > B] : % 0.30/0.48 ( ! [X5: A] : % 0.30/0.48 ( ( F2 @ X5 ) % 0.30/0.48 = ( G @ X5 ) ) % 0.30/0.48 => ( F2 = G ) ) ). % 0.30/0.48 % 0.30/0.48 % ext % 0.30/0.48 thf(fact_49_buildup__gives__valid,axiom, % 0.30/0.48 ! [N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.48 => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N2 ) @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % buildup_gives_valid % 0.30/0.48 thf(fact_50_set__vebt__set__vebt_H__valid,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,N2: nat] : % 0.30/0.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.48 => ( ( vEBT_set_vebt @ T2 ) % 0.30/0.48 = ( vEBT_VEBT_set_vebt @ T2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % set_vebt_set_vebt'_valid % 0.30/0.48 thf(fact_51_mint__corr__help,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,N2: nat,Mini: nat,X3: nat] : % 0.30/0.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.48 => ( ( ( vEBT_vebt_mint @ T2 ) % 0.30/0.48 = ( some @ nat @ Mini ) ) % 0.30/0.48 => ( ( vEBT_vebt_member @ T2 @ X3 ) % 0.30/0.48 => ( ord_less_eq @ nat @ Mini @ X3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % mint_corr_help % 0.30/0.48 thf(fact_52_replicate__eq__replicate,axiom, % 0.30/0.48 ! [A: $tType,M: nat,X3: A,N2: nat,Y: A] : % 0.30/0.48 ( ( ( replicate @ A @ M @ X3 ) % 0.30/0.48 = ( replicate @ A @ N2 @ Y ) ) % 0.30/0.48 = ( ( M = N2 ) % 0.30/0.48 & ( ( M % 0.30/0.48 != ( zero_zero @ nat ) ) % 0.30/0.48 => ( X3 = Y ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % replicate_eq_replicate % 0.30/0.48 thf(fact_53_length__replicate,axiom, % 0.30/0.48 ! [A: $tType,N2: nat,X3: A] : % 0.30/0.48 ( ( size_size @ ( list @ A ) @ ( replicate @ A @ N2 @ X3 ) ) % 0.30/0.48 = N2 ) ). % 0.30/0.48 % 0.30/0.48 % length_replicate % 0.30/0.48 thf(fact_54_length__list__update,axiom, % 0.30/0.48 ! [A: $tType,Xs: list @ A,I: nat,X3: A] : % 0.30/0.48 ( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I @ X3 ) ) % 0.30/0.48 = ( size_size @ ( list @ A ) @ Xs ) ) ). % 0.30/0.48 % 0.30/0.48 % length_list_update % 0.30/0.48 thf(fact_55_mint__sound,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,N2: nat,X3: nat] : % 0.30/0.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.48 => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 ) % 0.30/0.48 => ( ( vEBT_vebt_mint @ T2 ) % 0.30/0.48 = ( some @ nat @ X3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % mint_sound % 0.30/0.48 thf(fact_56_mint__corr,axiom, % 0.30/0.48 ! [T2: vEBT_VEBT,N2: nat,X3: nat] : % 0.30/0.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.48 => ( ( ( vEBT_vebt_mint @ T2 ) % 0.30/0.48 = ( some @ nat @ X3 ) ) % 0.30/0.48 => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % mint_corr % 0.30/0.48 thf(fact_57_Ball__set__replicate,axiom, % 0.30/0.48 ! [A: $tType,N2: nat,A2: A,P: A > $o] : % 0.30/0.48 ( ( ! [X: A] : % 0.30/0.48 ( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N2 @ A2 ) ) ) % 0.30/0.48 => ( P @ X ) ) ) % 0.30/0.48 = ( ( P @ A2 ) % 0.30/0.48 | ( N2 % 0.30/0.48 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % Ball_set_replicate % 0.30/0.48 thf(fact_58_Bex__set__replicate,axiom, % 0.30/0.48 ! [A: $tType,N2: nat,A2: A,P: A > $o] : % 0.30/0.48 ( ( ? [X: A] : % 0.30/0.48 ( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N2 @ A2 ) ) ) % 0.30/0.48 & ( P @ X ) ) ) % 0.30/0.48 = ( ( P @ A2 ) % 0.30/0.48 & ( N2 % 0.30/0.48 != ( zero_zero @ nat ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % Bex_set_replicate % 0.30/0.48 thf(fact_59_in__set__replicate,axiom, % 0.30/0.48 ! [A: $tType,X3: A,N2: nat,Y: A] : % 0.30/0.48 ( ( member @ A @ X3 @ ( set2 @ A @ ( replicate @ A @ N2 @ Y ) ) ) % 0.30/0.48 = ( ( X3 = Y ) % 0.30/0.48 & ( N2 % 0.30/0.48 != ( zero_zero @ nat ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % in_set_replicate % 0.30/0.48 thf(fact_60_list__update__beyond,axiom, % 0.30/0.48 ! [A: $tType,Xs: list @ A,I: nat,X3: A] : % 0.30/0.48 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I ) % 0.30/0.48 => ( ( list_update @ A @ Xs @ I @ X3 ) % 0.30/0.48 = Xs ) ) ). % 0.30/0.48 % 0.30/0.48 % list_update_beyond % 0.30/0.48 thf(fact_61__C5_C_I4_J,axiom, % 0.30/0.48 ( m % 0.30/0.48 = ( suc @ n ) ) ). % 0.30/0.48 % 0.30/0.48 % "5"(4) % 0.30/0.48 thf(fact_62_nth__list__update__eq,axiom, % 0.30/0.48 ! [A: $tType,I: nat,Xs: list @ A,X3: A] : % 0.30/0.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.48 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ I ) % 0.30/0.48 = X3 ) ) ). % 0.30/0.48 % 0.30/0.48 % nth_list_update_eq % 0.30/0.48 thf(fact_63_lesseq__shift,axiom, % 0.30/0.48 ( ( ord_less_eq @ nat ) % 0.30/0.48 = ( ^ [X: nat,Y2: nat] : ( vEBT_VEBT_lesseq @ ( some @ nat @ X ) @ ( some @ nat @ Y2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % lesseq_shift % 0.30/0.48 thf(fact_64_set__swap,axiom, % 0.30/0.48 ! [A: $tType,I: nat,Xs: list @ A,J: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.48 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.48 => ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) ) % 0.30/0.48 = ( set2 @ A @ Xs ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % set_swap % 0.30/0.48 thf(fact_65_Ex__list__of__length,axiom, % 0.30/0.48 ! [A: $tType,N2: nat] : % 0.30/0.48 ? [Xs3: list @ A] : % 0.30/0.48 ( ( size_size @ ( list @ A ) @ Xs3 ) % 0.30/0.48 = N2 ) ). % 0.30/0.48 % 0.30/0.48 % Ex_list_of_length % 0.30/0.48 thf(fact_66_neq__if__length__neq,axiom, % 0.30/0.48 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.30/0.48 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.30/0.48 != ( size_size @ ( list @ A ) @ Ys ) ) % 0.30/0.48 => ( Xs != Ys ) ) ). % 0.30/0.48 % 0.30/0.48 % neq_if_length_neq % 0.30/0.48 thf(fact_67_option_Osel,axiom, % 0.30/0.48 ! [A: $tType,X22: A] : % 0.30/0.48 ( ( the2 @ A @ ( some @ A @ X22 ) ) % 0.30/0.48 = X22 ) ). % 0.30/0.48 % 0.30/0.48 % option.sel % 0.30/0.48 thf(fact_68_length__pos__if__in__set,axiom, % 0.30/0.48 ! [A: $tType,X3: A,Xs: list @ A] : % 0.30/0.48 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.30/0.48 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ). % 0.30/0.48 % 0.30/0.48 % length_pos_if_in_set % 0.30/0.48 thf(fact_69_length__induct,axiom, % 0.30/0.48 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] : % 0.30/0.48 ( ! [Xs3: list @ A] : % 0.30/0.48 ( ! [Ys2: list @ A] : % 0.30/0.48 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) % 0.30/0.48 => ( P @ Ys2 ) ) % 0.30/0.48 => ( P @ Xs3 ) ) % 0.30/0.48 => ( P @ Xs ) ) ). % 0.30/0.48 % 0.30/0.48 % length_induct % 0.30/0.48 thf(fact_70_list__eq__iff__nth__eq,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ^ [Y3: list @ A,Z: list @ A] : ( Y3 = Z ) ) % 0.30/0.48 = ( ^ [Xs2: list @ A,Ys3: list @ A] : % 0.30/0.48 ( ( ( size_size @ ( list @ A ) @ Xs2 ) % 0.30/0.48 = ( size_size @ ( list @ A ) @ Ys3 ) ) % 0.30/0.48 & ! [I3: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) % 0.30/0.48 => ( ( nth @ A @ Xs2 @ I3 ) % 0.30/0.48 = ( nth @ A @ Ys3 @ I3 ) ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % list_eq_iff_nth_eq % 0.30/0.48 thf(fact_71_Skolem__list__nth,axiom, % 0.30/0.48 ! [A: $tType,K: nat,P: nat > A > $o] : % 0.30/0.48 ( ( ! [I3: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I3 @ K ) % 0.30/0.48 => ? [X6: A] : ( P @ I3 @ X6 ) ) ) % 0.30/0.48 = ( ? [Xs2: list @ A] : % 0.30/0.48 ( ( ( size_size @ ( list @ A ) @ Xs2 ) % 0.30/0.48 = K ) % 0.30/0.48 & ! [I3: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I3 @ K ) % 0.30/0.48 => ( P @ I3 @ ( nth @ A @ Xs2 @ I3 ) ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % Skolem_list_nth % 0.30/0.48 thf(fact_72_nth__equalityI,axiom, % 0.30/0.48 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.30/0.48 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.30/0.48 = ( size_size @ ( list @ A ) @ Ys ) ) % 0.30/0.48 => ( ! [I4: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.48 => ( ( nth @ A @ Xs @ I4 ) % 0.30/0.48 = ( nth @ A @ Ys @ I4 ) ) ) % 0.30/0.48 => ( Xs = Ys ) ) ) ). % 0.30/0.48 % 0.30/0.48 % nth_equalityI % 0.30/0.48 thf(fact_73_replicate__length__same,axiom, % 0.30/0.48 ! [A: $tType,Xs: list @ A,X3: A] : % 0.30/0.48 ( ! [X5: A] : % 0.30/0.48 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.30/0.48 => ( X5 = X3 ) ) % 0.30/0.48 => ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X3 ) % 0.30/0.48 = Xs ) ) ). % 0.30/0.48 % 0.30/0.48 % replicate_length_same % 0.30/0.48 thf(fact_74_replicate__eqI,axiom, % 0.30/0.48 ! [A: $tType,Xs: list @ A,N2: nat,X3: A] : % 0.30/0.48 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.30/0.48 = N2 ) % 0.30/0.48 => ( ! [Y4: A] : % 0.30/0.48 ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) ) % 0.30/0.48 => ( Y4 = X3 ) ) % 0.30/0.48 => ( Xs % 0.30/0.48 = ( replicate @ A @ N2 @ X3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % replicate_eqI % 0.30/0.48 thf(fact_75_all__set__conv__all__nth,axiom, % 0.30/0.48 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.30/0.48 ( ( ! [X: A] : % 0.30/0.48 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.30/0.48 => ( P @ X ) ) ) % 0.30/0.48 = ( ! [I3: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.48 => ( P @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % all_set_conv_all_nth % 0.30/0.48 thf(fact_76_all__nth__imp__all__set,axiom, % 0.30/0.48 ! [A: $tType,Xs: list @ A,P: A > $o,X3: A] : % 0.30/0.48 ( ! [I4: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.48 => ( P @ ( nth @ A @ Xs @ I4 ) ) ) % 0.30/0.48 => ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.30/0.48 => ( P @ X3 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % all_nth_imp_all_set % 0.30/0.48 thf(fact_77_in__set__conv__nth,axiom, % 0.30/0.48 ! [A: $tType,X3: A,Xs: list @ A] : % 0.30/0.48 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.30/0.48 = ( ? [I3: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.48 & ( ( nth @ A @ Xs @ I3 ) % 0.30/0.48 = X3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % in_set_conv_nth % 0.30/0.48 thf(fact_78_list__ball__nth,axiom, % 0.30/0.48 ! [A: $tType,N2: nat,Xs: list @ A,P: A > $o] : % 0.30/0.48 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.48 => ( ! [X5: A] : % 0.30/0.48 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.30/0.48 => ( P @ X5 ) ) % 0.30/0.48 => ( P @ ( nth @ A @ Xs @ N2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % list_ball_nth % 0.30/0.48 thf(fact_79_nth__mem,axiom, % 0.30/0.48 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.30/0.48 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.48 => ( member @ A @ ( nth @ A @ Xs @ N2 ) @ ( set2 @ A @ Xs ) ) ) ). % 0.30/0.48 % 0.30/0.48 % nth_mem % 0.30/0.48 thf(fact_80_set__update__memI,axiom, % 0.30/0.48 ! [A: $tType,N2: nat,Xs: list @ A,X3: A] : % 0.30/0.48 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.48 => ( member @ A @ X3 @ ( set2 @ A @ ( list_update @ A @ Xs @ N2 @ X3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % set_update_memI % 0.30/0.48 thf(fact_81_list__update__same__conv,axiom, % 0.30/0.48 ! [A: $tType,I: nat,Xs: list @ A,X3: A] : % 0.30/0.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.48 => ( ( ( list_update @ A @ Xs @ I @ X3 ) % 0.30/0.48 = Xs ) % 0.30/0.48 = ( ( nth @ A @ Xs @ I ) % 0.30/0.48 = X3 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % list_update_same_conv % 0.30/0.48 thf(fact_82_nth__list__update,axiom, % 0.30/0.48 ! [A: $tType,I: nat,Xs: list @ A,J: nat,X3: A] : % 0.30/0.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.48 => ( ( ( I = J ) % 0.30/0.48 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ J ) % 0.30/0.48 = X3 ) ) % 0.30/0.48 & ( ( I != J ) % 0.30/0.48 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ J ) % 0.30/0.48 = ( nth @ A @ Xs @ J ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % nth_list_update % 0.30/0.48 thf(fact_83_le0,axiom, % 0.30/0.48 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ). % 0.30/0.48 % 0.30/0.48 % le0 % 0.30/0.48 thf(fact_84_bot__nat__0_Oextremum,axiom, % 0.30/0.48 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ). % 0.30/0.48 % 0.30/0.48 % bot_nat_0.extremum % 0.30/0.48 thf(fact_85_less__nat__zero__code,axiom, % 0.30/0.48 ! [N2: nat] : % 0.30/0.48 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ). % 0.30/0.48 % 0.30/0.48 % less_nat_zero_code % 0.30/0.48 thf(fact_86_neq0__conv,axiom, % 0.30/0.48 ! [N2: nat] : % 0.30/0.48 ( ( N2 % 0.30/0.48 != ( zero_zero @ nat ) ) % 0.30/0.48 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % neq0_conv % 0.30/0.48 thf(fact_87_bot__nat__0_Onot__eq__extremum,axiom, % 0.30/0.48 ! [A2: nat] : % 0.30/0.48 ( ( A2 % 0.30/0.48 != ( zero_zero @ nat ) ) % 0.30/0.48 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ). % 0.30/0.48 % 0.30/0.48 % bot_nat_0.not_eq_extremum % 0.30/0.48 thf(fact_88_not__gr__zero,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( canoni5634975068530333245id_add @ A ) % 0.30/0.48 => ! [N2: A] : % 0.30/0.48 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) % 0.30/0.48 = ( N2 % 0.30/0.48 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % not_gr_zero % 0.30/0.48 thf(fact_89_le__zero__eq,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( canoni5634975068530333245id_add @ A ) % 0.30/0.48 => ! [N2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) ) % 0.30/0.48 = ( N2 % 0.30/0.48 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % le_zero_eq % 0.30/0.48 thf(fact_90__C5_C_I5_J,axiom, % 0.30/0.48 ( deg % 0.30/0.48 = ( plus_plus @ nat @ n @ m ) ) ). % 0.30/0.48 % 0.30/0.48 % "5"(5) % 0.30/0.48 thf(fact_91_ex__least__nat__le,axiom, % 0.30/0.48 ! [P: nat > $o,N2: nat] : % 0.30/0.48 ( ( P @ N2 ) % 0.30/0.48 => ( ~ ( P @ ( zero_zero @ nat ) ) % 0.30/0.48 => ? [K2: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ K2 @ N2 ) % 0.30/0.48 & ! [I5: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I5 @ K2 ) % 0.30/0.48 => ~ ( P @ I5 ) ) % 0.30/0.48 & ( P @ K2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % ex_least_nat_le % 0.30/0.48 thf(fact_92_buildup__gives__empty,axiom, % 0.30/0.48 ! [N2: nat] : % 0.30/0.48 ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) ) % 0.30/0.48 = ( bot_bot @ ( set @ nat ) ) ) ). % 0.30/0.48 % 0.30/0.48 % buildup_gives_empty % 0.30/0.48 thf(fact_93_dual__order_Orefl,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( preorder @ A ) % 0.30/0.48 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ). % 0.30/0.48 % 0.30/0.48 % dual_order.refl % 0.30/0.48 thf(fact_94_even__odd__cases,axiom, % 0.30/0.48 ! [X3: nat] : % 0.30/0.48 ( ! [N3: nat] : % 0.30/0.48 ( X3 % 0.30/0.48 != ( plus_plus @ nat @ N3 @ N3 ) ) % 0.30/0.48 => ~ ! [N3: nat] : % 0.30/0.48 ( X3 % 0.30/0.48 != ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % even_odd_cases % 0.30/0.48 thf(fact_95_add__shift,axiom, % 0.30/0.48 ! [X3: nat,Y: nat,Z2: nat] : % 0.30/0.48 ( ( ( plus_plus @ nat @ X3 @ Y ) % 0.30/0.48 = Z2 ) % 0.30/0.48 = ( ( vEBT_VEBT_add @ ( some @ nat @ X3 ) @ ( some @ nat @ Y ) ) % 0.30/0.48 = ( some @ nat @ Z2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_shift % 0.30/0.48 thf(fact_96_order__refl,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( preorder @ A ) % 0.30/0.48 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ). % 0.30/0.48 % 0.30/0.48 % order_refl % 0.30/0.48 thf(fact_97_add__left__cancel,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( cancel_semigroup_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( ( plus_plus @ A @ A2 @ B2 ) % 0.30/0.48 = ( plus_plus @ A @ A2 @ C2 ) ) % 0.30/0.48 = ( B2 = C2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_left_cancel % 0.30/0.48 thf(fact_98_add__right__cancel,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( cancel_semigroup_add @ A ) % 0.30/0.48 => ! [B2: A,A2: A,C2: A] : % 0.30/0.48 ( ( ( plus_plus @ A @ B2 @ A2 ) % 0.30/0.48 = ( plus_plus @ A @ C2 @ A2 ) ) % 0.30/0.48 = ( B2 = C2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_right_cancel % 0.30/0.48 thf(fact_99_bot__apply,axiom, % 0.30/0.48 ! [C: $tType,D: $tType] : % 0.30/0.48 ( ( bot @ C ) % 0.30/0.48 => ( ( bot_bot @ ( D > C ) ) % 0.30/0.48 = ( ^ [X: D] : ( bot_bot @ C ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % bot_apply % 0.30/0.48 thf(fact_100_nat_Oinject,axiom, % 0.30/0.48 ! [X22: nat,Y22: nat] : % 0.30/0.48 ( ( ( suc @ X22 ) % 0.30/0.48 = ( suc @ Y22 ) ) % 0.30/0.48 = ( X22 = Y22 ) ) ). % 0.30/0.48 % 0.30/0.48 % nat.inject % 0.30/0.48 thf(fact_101_old_Onat_Oinject,axiom, % 0.30/0.48 ! [Nat: nat,Nat2: nat] : % 0.30/0.48 ( ( ( suc @ Nat ) % 0.30/0.48 = ( suc @ Nat2 ) ) % 0.30/0.48 = ( Nat = Nat2 ) ) ). % 0.30/0.48 % 0.30/0.48 % old.nat.inject % 0.30/0.48 thf(fact_102_add__le__cancel__left,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere2412721322843649153imp_le @ A ) % 0.30/0.48 => ! [C2: A,A2: A,B2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) % 0.30/0.48 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_le_cancel_left % 0.30/0.48 thf(fact_103_add__le__cancel__right,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere2412721322843649153imp_le @ A ) % 0.30/0.48 => ! [A2: A,C2: A,B2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) % 0.30/0.48 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_le_cancel_right % 0.30/0.48 thf(fact_104_add_Oright__neutral,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( monoid_add @ A ) % 0.30/0.48 => ! [A2: A] : % 0.30/0.48 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.48 = A2 ) ) ). % 0.30/0.48 % 0.30/0.48 % add.right_neutral % 0.30/0.48 thf(fact_105_double__zero__sym,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linord5086331880401160121up_add @ A ) % 0.30/0.48 => ! [A2: A] : % 0.30/0.48 ( ( ( zero_zero @ A ) % 0.30/0.48 = ( plus_plus @ A @ A2 @ A2 ) ) % 0.30/0.48 = ( A2 % 0.30/0.48 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % double_zero_sym % 0.30/0.48 thf(fact_106_add__cancel__left__left,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( cancel1802427076303600483id_add @ A ) % 0.30/0.48 => ! [B2: A,A2: A] : % 0.30/0.48 ( ( ( plus_plus @ A @ B2 @ A2 ) % 0.30/0.48 = A2 ) % 0.30/0.48 = ( B2 % 0.30/0.48 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_cancel_left_left % 0.30/0.48 thf(fact_107_add__cancel__left__right,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( cancel1802427076303600483id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ( plus_plus @ A @ A2 @ B2 ) % 0.30/0.48 = A2 ) % 0.30/0.48 = ( B2 % 0.30/0.48 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_cancel_left_right % 0.30/0.48 thf(fact_108_add__cancel__right__left,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( cancel1802427076303600483id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( A2 % 0.30/0.48 = ( plus_plus @ A @ B2 @ A2 ) ) % 0.30/0.48 = ( B2 % 0.30/0.48 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_cancel_right_left % 0.30/0.48 thf(fact_109_add__cancel__right__right,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( cancel1802427076303600483id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( A2 % 0.30/0.48 = ( plus_plus @ A @ A2 @ B2 ) ) % 0.30/0.48 = ( B2 % 0.30/0.48 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_cancel_right_right % 0.30/0.48 thf(fact_110_add__eq__0__iff__both__eq__0,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( canoni5634975068530333245id_add @ A ) % 0.30/0.48 => ! [X3: A,Y: A] : % 0.30/0.48 ( ( ( plus_plus @ A @ X3 @ Y ) % 0.30/0.48 = ( zero_zero @ A ) ) % 0.30/0.48 = ( ( X3 % 0.30/0.48 = ( zero_zero @ A ) ) % 0.30/0.48 & ( Y % 0.30/0.48 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_eq_0_iff_both_eq_0 % 0.30/0.48 thf(fact_111_zero__eq__add__iff__both__eq__0,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( canoni5634975068530333245id_add @ A ) % 0.30/0.48 => ! [X3: A,Y: A] : % 0.30/0.48 ( ( ( zero_zero @ A ) % 0.30/0.48 = ( plus_plus @ A @ X3 @ Y ) ) % 0.30/0.48 = ( ( X3 % 0.30/0.48 = ( zero_zero @ A ) ) % 0.30/0.48 & ( Y % 0.30/0.48 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % zero_eq_add_iff_both_eq_0 % 0.30/0.48 thf(fact_112_add__0,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( monoid_add @ A ) % 0.30/0.48 => ! [A2: A] : % 0.30/0.48 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.48 = A2 ) ) ). % 0.30/0.48 % 0.30/0.48 % add_0 % 0.30/0.48 thf(fact_113_add__less__cancel__left,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere2412721322843649153imp_le @ A ) % 0.30/0.48 => ! [C2: A,A2: A,B2: A] : % 0.30/0.48 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) % 0.30/0.48 = ( ord_less @ A @ A2 @ B2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_less_cancel_left % 0.30/0.48 thf(fact_114_add__less__cancel__right,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere2412721322843649153imp_le @ A ) % 0.30/0.48 => ! [A2: A,C2: A,B2: A] : % 0.30/0.48 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) % 0.30/0.48 = ( ord_less @ A @ A2 @ B2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_less_cancel_right % 0.30/0.48 thf(fact_115_lessI,axiom, % 0.30/0.48 ! [N2: nat] : ( ord_less @ nat @ N2 @ ( suc @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % lessI % 0.30/0.48 thf(fact_116_Suc__mono,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.48 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % Suc_mono % 0.30/0.48 thf(fact_117_Suc__less__eq,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) % 0.30/0.48 = ( ord_less @ nat @ M @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % Suc_less_eq % 0.30/0.48 thf(fact_118_Suc__le__mono,axiom, % 0.30/0.48 ! [N2: nat,M: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( suc @ M ) ) % 0.30/0.48 = ( ord_less_eq @ nat @ N2 @ M ) ) ). % 0.30/0.48 % 0.30/0.48 % Suc_le_mono % 0.30/0.48 thf(fact_119_add__Suc__right,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( plus_plus @ nat @ M @ ( suc @ N2 ) ) % 0.30/0.48 = ( suc @ ( plus_plus @ nat @ M @ N2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_Suc_right % 0.30/0.48 thf(fact_120_add__is__0,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ( plus_plus @ nat @ M @ N2 ) % 0.30/0.48 = ( zero_zero @ nat ) ) % 0.30/0.48 = ( ( M % 0.30/0.48 = ( zero_zero @ nat ) ) % 0.30/0.48 & ( N2 % 0.30/0.48 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_is_0 % 0.30/0.48 thf(fact_121_Nat_Oadd__0__right,axiom, % 0.30/0.48 ! [M: nat] : % 0.30/0.48 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) ) % 0.30/0.48 = M ) ). % 0.30/0.48 % 0.30/0.48 % Nat.add_0_right % 0.30/0.48 thf(fact_122_nat__add__left__cancel__less,axiom, % 0.30/0.48 ! [K: nat,M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) ) % 0.30/0.48 = ( ord_less @ nat @ M @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % nat_add_left_cancel_less % 0.30/0.48 thf(fact_123_nat__add__left__cancel__le,axiom, % 0.30/0.48 ! [K: nat,M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) ) % 0.30/0.48 = ( ord_less_eq @ nat @ M @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % nat_add_left_cancel_le % 0.30/0.48 thf(fact_124_add__le__same__cancel1,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere1937475149494474687imp_le @ A ) % 0.30/0.48 => ! [B2: A,A2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 ) % 0.30/0.48 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_le_same_cancel1 % 0.30/0.48 thf(fact_125_add__le__same__cancel2,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere1937475149494474687imp_le @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 ) % 0.30/0.48 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_le_same_cancel2 % 0.30/0.48 thf(fact_126_le__add__same__cancel1,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere1937475149494474687imp_le @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.30/0.48 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % le_add_same_cancel1 % 0.30/0.48 thf(fact_127_le__add__same__cancel2,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere1937475149494474687imp_le @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) ) % 0.30/0.48 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % le_add_same_cancel2 % 0.30/0.48 thf(fact_128_double__add__le__zero__iff__single__add__le__zero,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linord5086331880401160121up_add @ A ) % 0.30/0.48 => ! [A2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) ) % 0.30/0.48 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % double_add_le_zero_iff_single_add_le_zero % 0.30/0.48 thf(fact_129_zero__le__double__add__iff__zero__le__single__add,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linord5086331880401160121up_add @ A ) % 0.30/0.48 => ! [A2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) ) % 0.30/0.48 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % zero_le_double_add_iff_zero_le_single_add % 0.30/0.48 thf(fact_130_zero__less__double__add__iff__zero__less__single__add,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linord5086331880401160121up_add @ A ) % 0.30/0.48 => ! [A2: A] : % 0.30/0.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) ) % 0.30/0.48 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % zero_less_double_add_iff_zero_less_single_add % 0.30/0.48 thf(fact_131_double__add__less__zero__iff__single__add__less__zero,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linord5086331880401160121up_add @ A ) % 0.30/0.48 => ! [A2: A] : % 0.30/0.48 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) ) % 0.30/0.48 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % double_add_less_zero_iff_single_add_less_zero % 0.30/0.48 thf(fact_132_less__add__same__cancel2,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere1937475149494474687imp_le @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) ) % 0.30/0.48 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_add_same_cancel2 % 0.30/0.48 thf(fact_133_less__add__same__cancel1,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere1937475149494474687imp_le @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.30/0.48 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_add_same_cancel1 % 0.30/0.48 thf(fact_134_add__less__same__cancel2,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere1937475149494474687imp_le @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 ) % 0.30/0.48 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_less_same_cancel2 % 0.30/0.48 thf(fact_135_add__less__same__cancel1,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere1937475149494474687imp_le @ A ) % 0.30/0.48 => ! [B2: A,A2: A] : % 0.30/0.48 ( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 ) % 0.30/0.48 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_less_same_cancel1 % 0.30/0.48 thf(fact_136_zero__less__Suc,axiom, % 0.30/0.48 ! [N2: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % zero_less_Suc % 0.30/0.48 thf(fact_137_less__Suc0,axiom, % 0.30/0.48 ! [N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.48 = ( N2 % 0.30/0.48 = ( zero_zero @ nat ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_Suc0 % 0.30/0.48 thf(fact_138_add__gr__0,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N2 ) ) % 0.30/0.48 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.30/0.48 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_gr_0 % 0.30/0.48 thf(fact_139_one__is__add,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ( suc @ ( zero_zero @ nat ) ) % 0.30/0.48 = ( plus_plus @ nat @ M @ N2 ) ) % 0.30/0.48 = ( ( ( M % 0.30/0.48 = ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.48 & ( N2 % 0.30/0.48 = ( zero_zero @ nat ) ) ) % 0.30/0.48 | ( ( M % 0.30/0.48 = ( zero_zero @ nat ) ) % 0.30/0.48 & ( N2 % 0.30/0.48 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % one_is_add % 0.30/0.48 thf(fact_140_add__is__1,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ( plus_plus @ nat @ M @ N2 ) % 0.30/0.48 = ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.48 = ( ( ( M % 0.30/0.48 = ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.48 & ( N2 % 0.30/0.48 = ( zero_zero @ nat ) ) ) % 0.30/0.48 | ( ( M % 0.30/0.48 = ( zero_zero @ nat ) ) % 0.30/0.48 & ( N2 % 0.30/0.48 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_is_1 % 0.30/0.48 thf(fact_141_ab__semigroup__add__class_Oadd__ac_I1_J,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ab_semigroup_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.48 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % ab_semigroup_add_class.add_ac(1) % 0.30/0.48 thf(fact_142_add__mono__thms__linordered__semiring_I4_J,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6658533253407199908up_add @ A ) % 0.30/0.48 => ! [I: A,J: A,K: A,L: A] : % 0.30/0.48 ( ( ( I = J ) % 0.30/0.48 & ( K = L ) ) % 0.30/0.48 => ( ( plus_plus @ A @ I @ K ) % 0.30/0.48 = ( plus_plus @ A @ J @ L ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_mono_thms_linordered_semiring(4) % 0.30/0.48 thf(fact_143_nat__arith_Osuc1,axiom, % 0.30/0.48 ! [A3: nat,K: nat,A2: nat] : % 0.30/0.48 ( ( A3 % 0.30/0.48 = ( plus_plus @ nat @ K @ A2 ) ) % 0.30/0.48 => ( ( suc @ A3 ) % 0.30/0.48 = ( plus_plus @ nat @ K @ ( suc @ A2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % nat_arith.suc1 % 0.30/0.48 thf(fact_144_add__Suc,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( plus_plus @ nat @ ( suc @ M ) @ N2 ) % 0.30/0.48 = ( suc @ ( plus_plus @ nat @ M @ N2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_Suc % 0.30/0.48 thf(fact_145_group__cancel_Oadd1,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( comm_monoid_add @ A ) % 0.30/0.48 => ! [A3: A,K: A,A2: A,B2: A] : % 0.30/0.48 ( ( A3 % 0.30/0.48 = ( plus_plus @ A @ K @ A2 ) ) % 0.30/0.48 => ( ( plus_plus @ A @ A3 @ B2 ) % 0.30/0.48 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % group_cancel.add1 % 0.30/0.48 thf(fact_146_group__cancel_Oadd2,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( comm_monoid_add @ A ) % 0.30/0.48 => ! [B3: A,K: A,B2: A,A2: A] : % 0.30/0.48 ( ( B3 % 0.30/0.48 = ( plus_plus @ A @ K @ B2 ) ) % 0.30/0.48 => ( ( plus_plus @ A @ A2 @ B3 ) % 0.30/0.48 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % group_cancel.add2 % 0.30/0.48 thf(fact_147_add_Oassoc,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( semigroup_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.48 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add.assoc % 0.30/0.48 thf(fact_148_add_Oleft__cancel,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( group_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( ( plus_plus @ A @ A2 @ B2 ) % 0.30/0.48 = ( plus_plus @ A @ A2 @ C2 ) ) % 0.30/0.48 = ( B2 = C2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add.left_cancel % 0.30/0.48 thf(fact_149_add_Oright__cancel,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( group_add @ A ) % 0.30/0.48 => ! [B2: A,A2: A,C2: A] : % 0.30/0.48 ( ( ( plus_plus @ A @ B2 @ A2 ) % 0.30/0.48 = ( plus_plus @ A @ C2 @ A2 ) ) % 0.30/0.48 = ( B2 = C2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add.right_cancel % 0.30/0.48 thf(fact_150_add_Ocommute,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ab_semigroup_add @ A ) % 0.30/0.48 => ( ( plus_plus @ A ) % 0.30/0.48 = ( ^ [A4: A,B4: A] : ( plus_plus @ A @ B4 @ A4 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add.commute % 0.30/0.48 thf(fact_151_add_Oleft__commute,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ab_semigroup_add @ A ) % 0.30/0.48 => ! [B2: A,A2: A,C2: A] : % 0.30/0.48 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) % 0.30/0.48 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add.left_commute % 0.30/0.48 thf(fact_152_add__left__imp__eq,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( cancel_semigroup_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( ( plus_plus @ A @ A2 @ B2 ) % 0.30/0.48 = ( plus_plus @ A @ A2 @ C2 ) ) % 0.30/0.48 => ( B2 = C2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_left_imp_eq % 0.30/0.48 thf(fact_153_add__right__imp__eq,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( cancel_semigroup_add @ A ) % 0.30/0.48 => ! [B2: A,A2: A,C2: A] : % 0.30/0.48 ( ( ( plus_plus @ A @ B2 @ A2 ) % 0.30/0.48 = ( plus_plus @ A @ C2 @ A2 ) ) % 0.30/0.48 => ( B2 = C2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_right_imp_eq % 0.30/0.48 thf(fact_154_Suc__inject,axiom, % 0.30/0.48 ! [X3: nat,Y: nat] : % 0.30/0.48 ( ( ( suc @ X3 ) % 0.30/0.48 = ( suc @ Y ) ) % 0.30/0.48 => ( X3 = Y ) ) ). % 0.30/0.48 % 0.30/0.48 % Suc_inject % 0.30/0.48 thf(fact_155_n__not__Suc__n,axiom, % 0.30/0.48 ! [N2: nat] : % 0.30/0.48 ( N2 % 0.30/0.48 != ( suc @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % n_not_Suc_n % 0.30/0.48 thf(fact_156_add__Suc__shift,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( plus_plus @ nat @ ( suc @ M ) @ N2 ) % 0.30/0.48 = ( plus_plus @ nat @ M @ ( suc @ N2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_Suc_shift % 0.30/0.48 thf(fact_157_bot__fun__def,axiom, % 0.30/0.48 ! [B: $tType,A: $tType] : % 0.30/0.48 ( ( bot @ B ) % 0.30/0.48 => ( ( bot_bot @ ( A > B ) ) % 0.30/0.48 = ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % bot_fun_def % 0.30/0.48 thf(fact_158_less__imp__Suc__add,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.48 => ? [K2: nat] : % 0.30/0.48 ( N2 % 0.30/0.48 = ( suc @ ( plus_plus @ nat @ M @ K2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_imp_Suc_add % 0.30/0.48 thf(fact_159_less__iff__Suc__add,axiom, % 0.30/0.48 ( ( ord_less @ nat ) % 0.30/0.48 = ( ^ [M2: nat,N: nat] : % 0.30/0.48 ? [K3: nat] : % 0.30/0.48 ( N % 0.30/0.48 = ( suc @ ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_iff_Suc_add % 0.30/0.48 thf(fact_160_less__add__Suc2,axiom, % 0.30/0.48 ! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_add_Suc2 % 0.30/0.48 thf(fact_161_less__add__Suc1,axiom, % 0.30/0.48 ! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_add_Suc1 % 0.30/0.48 thf(fact_162_less__natE,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.48 => ~ ! [Q2: nat] : % 0.30/0.48 ( N2 % 0.30/0.48 != ( suc @ ( plus_plus @ nat @ M @ Q2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_natE % 0.30/0.48 thf(fact_163_bot_Oextremum,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order_bot @ A ) % 0.30/0.48 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ). % 0.30/0.48 % 0.30/0.48 % bot.extremum % 0.30/0.48 thf(fact_164_bot_Oextremum__unique,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order_bot @ A ) % 0.30/0.48 => ! [A2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) ) % 0.30/0.48 = ( A2 % 0.30/0.48 = ( bot_bot @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % bot.extremum_unique % 0.30/0.48 thf(fact_165_bot_Oextremum__uniqueI,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order_bot @ A ) % 0.30/0.48 => ! [A2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) ) % 0.30/0.48 => ( A2 % 0.30/0.48 = ( bot_bot @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % bot.extremum_uniqueI % 0.30/0.48 thf(fact_166_bot_Oextremum__strict,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order_bot @ A ) % 0.30/0.48 => ! [A2: A] : % 0.30/0.48 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ). % 0.30/0.48 % 0.30/0.48 % bot.extremum_strict % 0.30/0.48 thf(fact_167_bot_Onot__eq__extremum,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order_bot @ A ) % 0.30/0.48 => ! [A2: A] : % 0.30/0.48 ( ( A2 % 0.30/0.48 != ( bot_bot @ A ) ) % 0.30/0.48 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % bot.not_eq_extremum % 0.30/0.48 thf(fact_168_add__mono__thms__linordered__semiring_I3_J,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6658533253407199908up_add @ A ) % 0.30/0.48 => ! [I: A,J: A,K: A,L: A] : % 0.30/0.48 ( ( ( ord_less_eq @ A @ I @ J ) % 0.30/0.48 & ( K = L ) ) % 0.30/0.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_mono_thms_linordered_semiring(3) % 0.30/0.48 thf(fact_169_add__mono__thms__linordered__semiring_I2_J,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6658533253407199908up_add @ A ) % 0.30/0.48 => ! [I: A,J: A,K: A,L: A] : % 0.30/0.48 ( ( ( I = J ) % 0.30/0.48 & ( ord_less_eq @ A @ K @ L ) ) % 0.30/0.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_mono_thms_linordered_semiring(2) % 0.30/0.48 thf(fact_170_add__mono__thms__linordered__semiring_I1_J,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6658533253407199908up_add @ A ) % 0.30/0.48 => ! [I: A,J: A,K: A,L: A] : % 0.30/0.48 ( ( ( ord_less_eq @ A @ I @ J ) % 0.30/0.48 & ( ord_less_eq @ A @ K @ L ) ) % 0.30/0.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_mono_thms_linordered_semiring(1) % 0.30/0.48 thf(fact_171_add__mono,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6658533253407199908up_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.48 => ( ( ord_less_eq @ A @ C2 @ D2 ) % 0.30/0.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_mono % 0.30/0.48 thf(fact_172_add__left__mono,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6658533253407199908up_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_left_mono % 0.30/0.48 thf(fact_173_less__eqE,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( canoni5634975068530333245id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.48 => ~ ! [C3: A] : % 0.30/0.48 ( B2 % 0.30/0.48 != ( plus_plus @ A @ A2 @ C3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_eqE % 0.30/0.48 thf(fact_174_add__right__mono,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6658533253407199908up_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_right_mono % 0.30/0.48 thf(fact_175_le__iff__add,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( canoni5634975068530333245id_add @ A ) % 0.30/0.48 => ( ( ord_less_eq @ A ) % 0.30/0.48 = ( ^ [A4: A,B4: A] : % 0.30/0.48 ? [C4: A] : % 0.30/0.48 ( B4 % 0.30/0.48 = ( plus_plus @ A @ A4 @ C4 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % le_iff_add % 0.30/0.48 thf(fact_176_add__le__imp__le__left,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere2412721322843649153imp_le @ A ) % 0.30/0.48 => ! [C2: A,A2: A,B2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) % 0.30/0.48 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_le_imp_le_left % 0.30/0.48 thf(fact_177_add__le__imp__le__right,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere2412721322843649153imp_le @ A ) % 0.30/0.48 => ! [A2: A,C2: A,B2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) % 0.30/0.48 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_le_imp_le_right % 0.30/0.48 thf(fact_178_comm__monoid__add__class_Oadd__0,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( comm_monoid_add @ A ) % 0.30/0.48 => ! [A2: A] : % 0.30/0.48 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.48 = A2 ) ) ). % 0.30/0.48 % 0.30/0.48 % comm_monoid_add_class.add_0 % 0.30/0.48 thf(fact_179_add_Ocomm__neutral,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( comm_monoid_add @ A ) % 0.30/0.48 => ! [A2: A] : % 0.30/0.48 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.48 = A2 ) ) ). % 0.30/0.48 % 0.30/0.48 % add.comm_neutral % 0.30/0.48 thf(fact_180_add_Ogroup__left__neutral,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( group_add @ A ) % 0.30/0.48 => ! [A2: A] : % 0.30/0.48 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.48 = A2 ) ) ). % 0.30/0.48 % 0.30/0.48 % add.group_left_neutral % 0.30/0.48 thf(fact_181_add__mono__thms__linordered__field_I5_J,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere580206878836729694up_add @ A ) % 0.30/0.48 => ! [I: A,J: A,K: A,L: A] : % 0.30/0.48 ( ( ( ord_less @ A @ I @ J ) % 0.30/0.48 & ( ord_less @ A @ K @ L ) ) % 0.30/0.48 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_mono_thms_linordered_field(5) % 0.30/0.48 thf(fact_182_add__mono__thms__linordered__field_I2_J,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere580206878836729694up_add @ A ) % 0.30/0.48 => ! [I: A,J: A,K: A,L: A] : % 0.30/0.48 ( ( ( I = J ) % 0.30/0.48 & ( ord_less @ A @ K @ L ) ) % 0.30/0.48 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_mono_thms_linordered_field(2) % 0.30/0.48 thf(fact_183_add__mono__thms__linordered__field_I1_J,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere580206878836729694up_add @ A ) % 0.30/0.48 => ! [I: A,J: A,K: A,L: A] : % 0.30/0.48 ( ( ( ord_less @ A @ I @ J ) % 0.30/0.48 & ( K = L ) ) % 0.30/0.48 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_mono_thms_linordered_field(1) % 0.30/0.48 thf(fact_184_add__strict__mono,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( strict9044650504122735259up_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.48 => ( ( ord_less @ A @ C2 @ D2 ) % 0.30/0.48 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_strict_mono % 0.30/0.48 thf(fact_185_add__strict__left__mono,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere580206878836729694up_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.48 => ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_strict_left_mono % 0.30/0.48 thf(fact_186_add__strict__right__mono,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere580206878836729694up_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.48 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_strict_right_mono % 0.30/0.48 thf(fact_187_add__less__imp__less__left,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere2412721322843649153imp_le @ A ) % 0.30/0.48 => ! [C2: A,A2: A,B2: A] : % 0.30/0.48 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) % 0.30/0.48 => ( ord_less @ A @ A2 @ B2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_less_imp_less_left % 0.30/0.48 thf(fact_188_add__less__imp__less__right,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere2412721322843649153imp_le @ A ) % 0.30/0.48 => ! [A2: A,C2: A,B2: A] : % 0.30/0.48 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) % 0.30/0.48 => ( ord_less @ A @ A2 @ B2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_less_imp_less_right % 0.30/0.48 thf(fact_189_nat_Odistinct_I1_J,axiom, % 0.30/0.48 ! [X22: nat] : % 0.30/0.48 ( ( zero_zero @ nat ) % 0.30/0.48 != ( suc @ X22 ) ) ). % 0.30/0.48 % 0.30/0.48 % nat.distinct(1) % 0.30/0.48 thf(fact_190_old_Onat_Odistinct_I2_J,axiom, % 0.30/0.48 ! [Nat3: nat] : % 0.30/0.48 ( ( suc @ Nat3 ) % 0.30/0.48 != ( zero_zero @ nat ) ) ). % 0.30/0.48 % 0.30/0.48 % old.nat.distinct(2) % 0.30/0.48 thf(fact_191_old_Onat_Odistinct_I1_J,axiom, % 0.30/0.48 ! [Nat2: nat] : % 0.30/0.48 ( ( zero_zero @ nat ) % 0.30/0.48 != ( suc @ Nat2 ) ) ). % 0.30/0.48 % 0.30/0.48 % old.nat.distinct(1) % 0.30/0.48 thf(fact_192_nat_OdiscI,axiom, % 0.30/0.48 ! [Nat: nat,X22: nat] : % 0.30/0.48 ( ( Nat % 0.30/0.48 = ( suc @ X22 ) ) % 0.30/0.48 => ( Nat % 0.30/0.48 != ( zero_zero @ nat ) ) ) ). % 0.30/0.48 % 0.30/0.48 % nat.discI % 0.30/0.48 thf(fact_193_old_Onat_Oexhaust,axiom, % 0.30/0.48 ! [Y: nat] : % 0.30/0.48 ( ( Y % 0.30/0.48 != ( zero_zero @ nat ) ) % 0.30/0.48 => ~ ! [Nat4: nat] : % 0.30/0.48 ( Y % 0.30/0.48 != ( suc @ Nat4 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % old.nat.exhaust % 0.30/0.48 thf(fact_194_nat__induct,axiom, % 0.30/0.48 ! [P: nat > $o,N2: nat] : % 0.30/0.48 ( ( P @ ( zero_zero @ nat ) ) % 0.30/0.48 => ( ! [N3: nat] : % 0.30/0.48 ( ( P @ N3 ) % 0.30/0.48 => ( P @ ( suc @ N3 ) ) ) % 0.30/0.48 => ( P @ N2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % nat_induct % 0.30/0.48 thf(fact_195_diff__induct,axiom, % 0.30/0.48 ! [P: nat > nat > $o,M: nat,N2: nat] : % 0.30/0.48 ( ! [X5: nat] : ( P @ X5 @ ( zero_zero @ nat ) ) % 0.30/0.48 => ( ! [Y4: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y4 ) ) % 0.30/0.48 => ( ! [X5: nat,Y4: nat] : % 0.30/0.48 ( ( P @ X5 @ Y4 ) % 0.30/0.48 => ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) ) % 0.30/0.48 => ( P @ M @ N2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % diff_induct % 0.30/0.48 thf(fact_196_zero__induct,axiom, % 0.30/0.48 ! [P: nat > $o,K: nat] : % 0.30/0.48 ( ( P @ K ) % 0.30/0.48 => ( ! [N3: nat] : % 0.30/0.48 ( ( P @ ( suc @ N3 ) ) % 0.30/0.48 => ( P @ N3 ) ) % 0.30/0.48 => ( P @ ( zero_zero @ nat ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % zero_induct % 0.30/0.48 thf(fact_197_Suc__neq__Zero,axiom, % 0.30/0.48 ! [M: nat] : % 0.30/0.48 ( ( suc @ M ) % 0.30/0.48 != ( zero_zero @ nat ) ) ). % 0.30/0.48 % 0.30/0.48 % Suc_neq_Zero % 0.30/0.48 thf(fact_198_Zero__neq__Suc,axiom, % 0.30/0.48 ! [M: nat] : % 0.30/0.48 ( ( zero_zero @ nat ) % 0.30/0.48 != ( suc @ M ) ) ). % 0.30/0.48 % 0.30/0.48 % Zero_neq_Suc % 0.30/0.48 thf(fact_199_Zero__not__Suc,axiom, % 0.30/0.48 ! [M: nat] : % 0.30/0.48 ( ( zero_zero @ nat ) % 0.30/0.48 != ( suc @ M ) ) ). % 0.30/0.48 % 0.30/0.48 % Zero_not_Suc % 0.30/0.48 thf(fact_200_not0__implies__Suc,axiom, % 0.30/0.48 ! [N2: nat] : % 0.30/0.48 ( ( N2 % 0.30/0.48 != ( zero_zero @ nat ) ) % 0.30/0.48 => ? [M3: nat] : % 0.30/0.48 ( N2 % 0.30/0.48 = ( suc @ M3 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % not0_implies_Suc % 0.30/0.48 thf(fact_201_plus__nat_Oadd__0,axiom, % 0.30/0.48 ! [N2: nat] : % 0.30/0.48 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.48 = N2 ) ). % 0.30/0.48 % 0.30/0.48 % plus_nat.add_0 % 0.30/0.48 thf(fact_202_add__eq__self__zero,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ( plus_plus @ nat @ M @ N2 ) % 0.30/0.48 = M ) % 0.30/0.48 => ( N2 % 0.30/0.48 = ( zero_zero @ nat ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_eq_self_zero % 0.30/0.48 thf(fact_203_Nat_OlessE,axiom, % 0.30/0.48 ! [I: nat,K: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I @ K ) % 0.30/0.48 => ( ( K % 0.30/0.48 != ( suc @ I ) ) % 0.30/0.48 => ~ ! [J2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I @ J2 ) % 0.30/0.48 => ( K % 0.30/0.48 != ( suc @ J2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % Nat.lessE % 0.30/0.48 thf(fact_204_Suc__lessD,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ ( suc @ M ) @ N2 ) % 0.30/0.48 => ( ord_less @ nat @ M @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % Suc_lessD % 0.30/0.48 thf(fact_205_Suc__lessE,axiom, % 0.30/0.48 ! [I: nat,K: nat] : % 0.30/0.48 ( ( ord_less @ nat @ ( suc @ I ) @ K ) % 0.30/0.48 => ~ ! [J2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I @ J2 ) % 0.30/0.48 => ( K % 0.30/0.48 != ( suc @ J2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % Suc_lessE % 0.30/0.48 thf(fact_206_Suc__lessI,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.48 => ( ( ( suc @ M ) % 0.30/0.48 != N2 ) % 0.30/0.48 => ( ord_less @ nat @ ( suc @ M ) @ N2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % Suc_lessI % 0.30/0.48 thf(fact_207_less__SucE,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ M @ ( suc @ N2 ) ) % 0.30/0.48 => ( ~ ( ord_less @ nat @ M @ N2 ) % 0.30/0.48 => ( M = N2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_SucE % 0.30/0.48 thf(fact_208_less__SucI,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.48 => ( ord_less @ nat @ M @ ( suc @ N2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_SucI % 0.30/0.48 thf(fact_209_Ex__less__Suc,axiom, % 0.30/0.48 ! [N2: nat,P: nat > $o] : % 0.30/0.48 ( ( ? [I3: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I3 @ ( suc @ N2 ) ) % 0.30/0.48 & ( P @ I3 ) ) ) % 0.30/0.48 = ( ( P @ N2 ) % 0.30/0.48 | ? [I3: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I3 @ N2 ) % 0.30/0.48 & ( P @ I3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % Ex_less_Suc % 0.30/0.48 thf(fact_210_less__Suc__eq,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ M @ ( suc @ N2 ) ) % 0.30/0.48 = ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.48 | ( M = N2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_Suc_eq % 0.30/0.48 thf(fact_211_not__less__eq,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ~ ( ord_less @ nat @ M @ N2 ) ) % 0.30/0.48 = ( ord_less @ nat @ N2 @ ( suc @ M ) ) ) ). % 0.30/0.48 % 0.30/0.48 % not_less_eq % 0.30/0.48 thf(fact_212_All__less__Suc,axiom, % 0.30/0.48 ! [N2: nat,P: nat > $o] : % 0.30/0.48 ( ( ! [I3: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I3 @ ( suc @ N2 ) ) % 0.30/0.48 => ( P @ I3 ) ) ) % 0.30/0.48 = ( ( P @ N2 ) % 0.30/0.48 & ! [I3: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I3 @ N2 ) % 0.30/0.48 => ( P @ I3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % All_less_Suc % 0.30/0.48 thf(fact_213_Suc__less__eq2,axiom, % 0.30/0.48 ! [N2: nat,M: nat] : % 0.30/0.48 ( ( ord_less @ nat @ ( suc @ N2 ) @ M ) % 0.30/0.48 = ( ? [M4: nat] : % 0.30/0.48 ( ( M % 0.30/0.48 = ( suc @ M4 ) ) % 0.30/0.48 & ( ord_less @ nat @ N2 @ M4 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % Suc_less_eq2 % 0.30/0.48 thf(fact_214_less__antisym,axiom, % 0.30/0.48 ! [N2: nat,M: nat] : % 0.30/0.48 ( ~ ( ord_less @ nat @ N2 @ M ) % 0.30/0.48 => ( ( ord_less @ nat @ N2 @ ( suc @ M ) ) % 0.30/0.48 => ( M = N2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_antisym % 0.30/0.48 thf(fact_215_Suc__less__SucD,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) % 0.30/0.48 => ( ord_less @ nat @ M @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % Suc_less_SucD % 0.30/0.48 thf(fact_216_less__trans__Suc,axiom, % 0.30/0.48 ! [I: nat,J: nat,K: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I @ J ) % 0.30/0.48 => ( ( ord_less @ nat @ J @ K ) % 0.30/0.48 => ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_trans_Suc % 0.30/0.48 thf(fact_217_less__Suc__induct,axiom, % 0.30/0.48 ! [I: nat,J: nat,P: nat > nat > $o] : % 0.30/0.48 ( ( ord_less @ nat @ I @ J ) % 0.30/0.48 => ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) ) % 0.30/0.48 => ( ! [I4: nat,J2: nat,K2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I4 @ J2 ) % 0.30/0.48 => ( ( ord_less @ nat @ J2 @ K2 ) % 0.30/0.48 => ( ( P @ I4 @ J2 ) % 0.30/0.48 => ( ( P @ J2 @ K2 ) % 0.30/0.48 => ( P @ I4 @ K2 ) ) ) ) ) % 0.30/0.48 => ( P @ I @ J ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_Suc_induct % 0.30/0.48 thf(fact_218_strict__inc__induct,axiom, % 0.30/0.48 ! [I: nat,J: nat,P: nat > $o] : % 0.30/0.48 ( ( ord_less @ nat @ I @ J ) % 0.30/0.48 => ( ! [I4: nat] : % 0.30/0.48 ( ( J % 0.30/0.48 = ( suc @ I4 ) ) % 0.30/0.48 => ( P @ I4 ) ) % 0.30/0.48 => ( ! [I4: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I4 @ J ) % 0.30/0.48 => ( ( P @ ( suc @ I4 ) ) % 0.30/0.48 => ( P @ I4 ) ) ) % 0.30/0.48 => ( P @ I ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % strict_inc_induct % 0.30/0.48 thf(fact_219_not__less__less__Suc__eq,axiom, % 0.30/0.48 ! [N2: nat,M: nat] : % 0.30/0.48 ( ~ ( ord_less @ nat @ N2 @ M ) % 0.30/0.48 => ( ( ord_less @ nat @ N2 @ ( suc @ M ) ) % 0.30/0.48 = ( N2 = M ) ) ) ). % 0.30/0.48 % 0.30/0.48 % not_less_less_Suc_eq % 0.30/0.48 thf(fact_220_add__lessD1,axiom, % 0.30/0.48 ! [I: nat,J: nat,K: nat] : % 0.30/0.48 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) % 0.30/0.48 => ( ord_less @ nat @ I @ K ) ) ). % 0.30/0.48 % 0.30/0.48 % add_lessD1 % 0.30/0.48 thf(fact_221_add__less__mono,axiom, % 0.30/0.48 ! [I: nat,J: nat,K: nat,L: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I @ J ) % 0.30/0.48 => ( ( ord_less @ nat @ K @ L ) % 0.30/0.48 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_less_mono % 0.30/0.48 thf(fact_222_not__add__less1,axiom, % 0.30/0.48 ! [I: nat,J: nat] : % 0.30/0.48 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ). % 0.30/0.48 % 0.30/0.48 % not_add_less1 % 0.30/0.48 thf(fact_223_not__add__less2,axiom, % 0.30/0.48 ! [J: nat,I: nat] : % 0.30/0.48 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ). % 0.30/0.48 % 0.30/0.48 % not_add_less2 % 0.30/0.48 thf(fact_224_add__less__mono1,axiom, % 0.30/0.48 ! [I: nat,J: nat,K: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I @ J ) % 0.30/0.48 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_less_mono1 % 0.30/0.48 thf(fact_225_trans__less__add1,axiom, % 0.30/0.48 ! [I: nat,J: nat,M: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I @ J ) % 0.30/0.48 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ). % 0.30/0.48 % 0.30/0.48 % trans_less_add1 % 0.30/0.48 thf(fact_226_trans__less__add2,axiom, % 0.30/0.48 ! [I: nat,J: nat,M: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I @ J ) % 0.30/0.48 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ). % 0.30/0.48 % 0.30/0.48 % trans_less_add2 % 0.30/0.48 thf(fact_227_less__add__eq__less,axiom, % 0.30/0.48 ! [K: nat,L: nat,M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ K @ L ) % 0.30/0.48 => ( ( ( plus_plus @ nat @ M @ L ) % 0.30/0.48 = ( plus_plus @ nat @ K @ N2 ) ) % 0.30/0.48 => ( ord_less @ nat @ M @ N2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_add_eq_less % 0.30/0.48 thf(fact_228_Suc__leD,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 ) % 0.30/0.48 => ( ord_less_eq @ nat @ M @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % Suc_leD % 0.30/0.48 thf(fact_229_le__SucE,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) % 0.30/0.48 => ( ~ ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.48 => ( M % 0.30/0.48 = ( suc @ N2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % le_SucE % 0.30/0.48 thf(fact_230_le__SucI,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.48 => ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % le_SucI % 0.30/0.48 thf(fact_231_Suc__le__D,axiom, % 0.30/0.48 ! [N2: nat,M5: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ ( suc @ N2 ) @ M5 ) % 0.30/0.48 => ? [M3: nat] : % 0.30/0.48 ( M5 % 0.30/0.48 = ( suc @ M3 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % Suc_le_D % 0.30/0.48 thf(fact_232_le__Suc__eq,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) % 0.30/0.48 = ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.48 | ( M % 0.30/0.48 = ( suc @ N2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % le_Suc_eq % 0.30/0.48 thf(fact_233_Suc__n__not__le__n,axiom, % 0.30/0.48 ! [N2: nat] : % 0.30/0.48 ~ ( ord_less_eq @ nat @ ( suc @ N2 ) @ N2 ) ). % 0.30/0.48 % 0.30/0.48 % Suc_n_not_le_n % 0.30/0.48 thf(fact_234_not__less__eq__eq,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ~ ( ord_less_eq @ nat @ M @ N2 ) ) % 0.30/0.48 = ( ord_less_eq @ nat @ ( suc @ N2 ) @ M ) ) ). % 0.30/0.48 % 0.30/0.48 % not_less_eq_eq % 0.30/0.48 thf(fact_235_full__nat__induct,axiom, % 0.30/0.48 ! [P: nat > $o,N2: nat] : % 0.30/0.48 ( ! [N3: nat] : % 0.30/0.48 ( ! [M6: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ ( suc @ M6 ) @ N3 ) % 0.30/0.48 => ( P @ M6 ) ) % 0.30/0.48 => ( P @ N3 ) ) % 0.30/0.48 => ( P @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % full_nat_induct % 0.30/0.48 thf(fact_236_nat__induct__at__least,axiom, % 0.30/0.48 ! [M: nat,N2: nat,P: nat > $o] : % 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.48 => ( ( P @ M ) % 0.30/0.48 => ( ! [N3: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N3 ) % 0.30/0.48 => ( ( P @ N3 ) % 0.30/0.48 => ( P @ ( suc @ N3 ) ) ) ) % 0.30/0.48 => ( P @ N2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % nat_induct_at_least % 0.30/0.48 thf(fact_237_transitive__stepwise__le,axiom, % 0.30/0.48 ! [M: nat,N2: nat,R: nat > nat > $o] : % 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.48 => ( ! [X5: nat] : ( R @ X5 @ X5 ) % 0.30/0.48 => ( ! [X5: nat,Y4: nat,Z3: nat] : % 0.30/0.48 ( ( R @ X5 @ Y4 ) % 0.30/0.48 => ( ( R @ Y4 @ Z3 ) % 0.30/0.48 => ( R @ X5 @ Z3 ) ) ) % 0.30/0.48 => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) ) % 0.30/0.48 => ( R @ M @ N2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % transitive_stepwise_le % 0.30/0.48 thf(fact_238_add__leE,axiom, % 0.30/0.48 ! [M: nat,K: nat,N2: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N2 ) % 0.30/0.48 => ~ ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.48 => ~ ( ord_less_eq @ nat @ K @ N2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_leE % 0.30/0.48 thf(fact_239_le__add1,axiom, % 0.30/0.48 ! [N2: nat,M: nat] : ( ord_less_eq @ nat @ N2 @ ( plus_plus @ nat @ N2 @ M ) ) ). % 0.30/0.48 % 0.30/0.48 % le_add1 % 0.30/0.48 thf(fact_240_le__add2,axiom, % 0.30/0.48 ! [N2: nat,M: nat] : ( ord_less_eq @ nat @ N2 @ ( plus_plus @ nat @ M @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % le_add2 % 0.30/0.48 thf(fact_241_add__leD1,axiom, % 0.30/0.48 ! [M: nat,K: nat,N2: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N2 ) % 0.30/0.48 => ( ord_less_eq @ nat @ M @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % add_leD1 % 0.30/0.48 thf(fact_242_add__leD2,axiom, % 0.30/0.48 ! [M: nat,K: nat,N2: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N2 ) % 0.30/0.48 => ( ord_less_eq @ nat @ K @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % add_leD2 % 0.30/0.48 thf(fact_243_le__Suc__ex,axiom, % 0.30/0.48 ! [K: nat,L: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ K @ L ) % 0.30/0.48 => ? [N3: nat] : % 0.30/0.48 ( L % 0.30/0.48 = ( plus_plus @ nat @ K @ N3 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % le_Suc_ex % 0.30/0.48 thf(fact_244_add__le__mono,axiom, % 0.30/0.48 ! [I: nat,J: nat,K: nat,L: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.48 => ( ( ord_less_eq @ nat @ K @ L ) % 0.30/0.48 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_le_mono % 0.30/0.48 thf(fact_245_add__le__mono1,axiom, % 0.30/0.48 ! [I: nat,J: nat,K: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.48 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_le_mono1 % 0.30/0.48 thf(fact_246_trans__le__add1,axiom, % 0.30/0.48 ! [I: nat,J: nat,M: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.48 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ). % 0.30/0.48 % 0.30/0.48 % trans_le_add1 % 0.30/0.48 thf(fact_247_trans__le__add2,axiom, % 0.30/0.48 ! [I: nat,J: nat,M: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.48 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ). % 0.30/0.48 % 0.30/0.48 % trans_le_add2 % 0.30/0.48 thf(fact_248_nat__le__iff__add,axiom, % 0.30/0.48 ( ( ord_less_eq @ nat ) % 0.30/0.48 = ( ^ [M2: nat,N: nat] : % 0.30/0.48 ? [K3: nat] : % 0.30/0.48 ( N % 0.30/0.48 = ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % nat_le_iff_add % 0.30/0.48 thf(fact_249_subset__code_I1_J,axiom, % 0.30/0.48 ! [A: $tType,Xs: list @ A,B3: set @ A] : % 0.30/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B3 ) % 0.30/0.48 = ( ! [X: A] : % 0.30/0.48 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.30/0.48 => ( member @ A @ X @ B3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % subset_code(1) % 0.30/0.48 thf(fact_250_add__decreasing,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.48 => ! [A2: A,C2: A,B2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.48 => ( ( ord_less_eq @ A @ C2 @ B2 ) % 0.30/0.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_decreasing % 0.30/0.48 thf(fact_251_add__increasing,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ C2 ) % 0.30/0.48 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_increasing % 0.30/0.48 thf(fact_252_add__decreasing2,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.48 => ! [C2: A,A2: A,B2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.48 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_decreasing2 % 0.30/0.48 thf(fact_253_add__increasing2,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.48 => ! [C2: A,B2: A,A2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.30/0.48 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_increasing2 % 0.30/0.48 thf(fact_254_add__nonneg__nonneg,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_nonneg_nonneg % 0.30/0.48 thf(fact_255_add__nonpos__nonpos,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) % 0.30/0.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_nonpos_nonpos % 0.30/0.48 thf(fact_256_add__nonneg__eq__0__iff,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.48 => ! [X3: A,Y: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.48 => ( ( ( plus_plus @ A @ X3 @ Y ) % 0.30/0.48 = ( zero_zero @ A ) ) % 0.30/0.48 = ( ( X3 % 0.30/0.48 = ( zero_zero @ A ) ) % 0.30/0.48 & ( Y % 0.30/0.48 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_nonneg_eq_0_iff % 0.30/0.48 thf(fact_257_add__nonpos__eq__0__iff,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.48 => ! [X3: A,Y: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 0.30/0.48 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) ) % 0.30/0.48 => ( ( ( plus_plus @ A @ X3 @ Y ) % 0.30/0.48 = ( zero_zero @ A ) ) % 0.30/0.48 = ( ( X3 % 0.30/0.48 = ( zero_zero @ A ) ) % 0.30/0.48 & ( Y % 0.30/0.48 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_nonpos_eq_0_iff % 0.30/0.48 thf(fact_258_add__less__le__mono,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere580206878836729694up_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.48 => ( ( ord_less_eq @ A @ C2 @ D2 ) % 0.30/0.48 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_less_le_mono % 0.30/0.48 thf(fact_259_add__le__less__mono,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere580206878836729694up_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.48 => ( ( ord_less @ A @ C2 @ D2 ) % 0.30/0.48 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_le_less_mono % 0.30/0.48 thf(fact_260_add__mono__thms__linordered__field_I3_J,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere580206878836729694up_add @ A ) % 0.30/0.48 => ! [I: A,J: A,K: A,L: A] : % 0.30/0.48 ( ( ( ord_less @ A @ I @ J ) % 0.30/0.48 & ( ord_less_eq @ A @ K @ L ) ) % 0.30/0.48 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_mono_thms_linordered_field(3) % 0.30/0.48 thf(fact_261_add__mono__thms__linordered__field_I4_J,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere580206878836729694up_add @ A ) % 0.30/0.48 => ! [I: A,J: A,K: A,L: A] : % 0.30/0.48 ( ( ( ord_less_eq @ A @ I @ J ) % 0.30/0.48 & ( ord_less @ A @ K @ L ) ) % 0.30/0.48 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_mono_thms_linordered_field(4) % 0.30/0.48 thf(fact_262_pos__add__strict,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( strict7427464778891057005id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.48 => ( ( ord_less @ A @ B2 @ C2 ) % 0.30/0.48 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % pos_add_strict % 0.30/0.48 thf(fact_263_canonically__ordered__monoid__add__class_OlessE,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( canoni5634975068530333245id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.48 => ~ ! [C3: A] : % 0.30/0.48 ( ( B2 % 0.30/0.48 = ( plus_plus @ A @ A2 @ C3 ) ) % 0.30/0.48 => ( C3 % 0.30/0.48 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % canonically_ordered_monoid_add_class.lessE % 0.30/0.48 thf(fact_264_add__pos__pos,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.48 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.48 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_pos_pos % 0.30/0.48 thf(fact_265_add__neg__neg,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.48 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 0.30/0.48 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_neg_neg % 0.30/0.48 thf(fact_266_lift__Suc__mono__less,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order @ A ) % 0.30/0.48 => ! [F2: nat > A,N2: nat,N4: nat] : % 0.30/0.48 ( ! [N3: nat] : ( ord_less @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) ) % 0.30/0.48 => ( ( ord_less @ nat @ N2 @ N4 ) % 0.30/0.48 => ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ N4 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % lift_Suc_mono_less % 0.30/0.48 thf(fact_267_lift__Suc__mono__less__iff,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order @ A ) % 0.30/0.48 => ! [F2: nat > A,N2: nat,M: nat] : % 0.30/0.48 ( ! [N3: nat] : ( ord_less @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) ) % 0.30/0.48 => ( ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ M ) ) % 0.30/0.48 = ( ord_less @ nat @ N2 @ M ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % lift_Suc_mono_less_iff % 0.30/0.48 thf(fact_268_lift__Suc__mono__le,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order @ A ) % 0.30/0.48 => ! [F2: nat > A,N2: nat,N4: nat] : % 0.30/0.48 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) ) % 0.30/0.48 => ( ( ord_less_eq @ nat @ N2 @ N4 ) % 0.30/0.48 => ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( F2 @ N4 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % lift_Suc_mono_le % 0.30/0.48 thf(fact_269_lift__Suc__antimono__le,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order @ A ) % 0.30/0.48 => ! [F2: nat > A,N2: nat,N4: nat] : % 0.30/0.48 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) ) % 0.30/0.48 => ( ( ord_less_eq @ nat @ N2 @ N4 ) % 0.30/0.48 => ( ord_less_eq @ A @ ( F2 @ N4 ) @ ( F2 @ N2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % lift_Suc_antimono_le % 0.30/0.48 thf(fact_270_less__Suc__eq__0__disj,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ M @ ( suc @ N2 ) ) % 0.30/0.48 = ( ( M % 0.30/0.48 = ( zero_zero @ nat ) ) % 0.30/0.48 | ? [J3: nat] : % 0.30/0.48 ( ( M % 0.30/0.48 = ( suc @ J3 ) ) % 0.30/0.48 & ( ord_less @ nat @ J3 @ N2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_Suc_eq_0_disj % 0.30/0.48 thf(fact_271_gr0__implies__Suc,axiom, % 0.30/0.48 ! [N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.48 => ? [M3: nat] : % 0.30/0.48 ( N2 % 0.30/0.48 = ( suc @ M3 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % gr0_implies_Suc % 0.30/0.48 thf(fact_272_All__less__Suc2,axiom, % 0.30/0.48 ! [N2: nat,P: nat > $o] : % 0.30/0.48 ( ( ! [I3: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I3 @ ( suc @ N2 ) ) % 0.30/0.48 => ( P @ I3 ) ) ) % 0.30/0.48 = ( ( P @ ( zero_zero @ nat ) ) % 0.30/0.48 & ! [I3: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I3 @ N2 ) % 0.30/0.48 => ( P @ ( suc @ I3 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % All_less_Suc2 % 0.30/0.48 thf(fact_273_gr0__conv__Suc,axiom, % 0.30/0.48 ! [N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.48 = ( ? [M2: nat] : % 0.30/0.48 ( N2 % 0.30/0.48 = ( suc @ M2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % gr0_conv_Suc % 0.30/0.48 thf(fact_274_Ex__less__Suc2,axiom, % 0.30/0.48 ! [N2: nat,P: nat > $o] : % 0.30/0.48 ( ( ? [I3: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I3 @ ( suc @ N2 ) ) % 0.30/0.48 & ( P @ I3 ) ) ) % 0.30/0.48 = ( ( P @ ( zero_zero @ nat ) ) % 0.30/0.48 | ? [I3: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I3 @ N2 ) % 0.30/0.48 & ( P @ ( suc @ I3 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % Ex_less_Suc2 % 0.30/0.48 thf(fact_275_less__imp__add__positive,axiom, % 0.30/0.48 ! [I: nat,J: nat] : % 0.30/0.48 ( ( ord_less @ nat @ I @ J ) % 0.30/0.48 => ? [K2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 ) % 0.30/0.48 & ( ( plus_plus @ nat @ I @ K2 ) % 0.30/0.48 = J ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_imp_add_positive % 0.30/0.48 thf(fact_276_le__imp__less__Suc,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.48 => ( ord_less @ nat @ M @ ( suc @ N2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % le_imp_less_Suc % 0.30/0.48 thf(fact_277_less__eq__Suc__le,axiom, % 0.30/0.48 ( ( ord_less @ nat ) % 0.30/0.48 = ( ^ [N: nat] : ( ord_less_eq @ nat @ ( suc @ N ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_eq_Suc_le % 0.30/0.48 thf(fact_278_less__Suc__eq__le,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ M @ ( suc @ N2 ) ) % 0.30/0.48 = ( ord_less_eq @ nat @ M @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % less_Suc_eq_le % 0.30/0.48 thf(fact_279_le__less__Suc__eq,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.48 => ( ( ord_less @ nat @ N2 @ ( suc @ M ) ) % 0.30/0.48 = ( N2 = M ) ) ) ). % 0.30/0.48 % 0.30/0.48 % le_less_Suc_eq % 0.30/0.48 thf(fact_280_Suc__le__lessD,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 ) % 0.30/0.48 => ( ord_less @ nat @ M @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % Suc_le_lessD % 0.30/0.48 thf(fact_281_inc__induct,axiom, % 0.30/0.48 ! [I: nat,J: nat,P: nat > $o] : % 0.30/0.48 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.48 => ( ( P @ J ) % 0.30/0.48 => ( ! [N3: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ I @ N3 ) % 0.30/0.48 => ( ( ord_less @ nat @ N3 @ J ) % 0.30/0.48 => ( ( P @ ( suc @ N3 ) ) % 0.30/0.48 => ( P @ N3 ) ) ) ) % 0.30/0.48 => ( P @ I ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % inc_induct % 0.30/0.48 thf(fact_282_dec__induct,axiom, % 0.30/0.48 ! [I: nat,J: nat,P: nat > $o] : % 0.30/0.48 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.48 => ( ( P @ I ) % 0.30/0.48 => ( ! [N3: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ I @ N3 ) % 0.30/0.48 => ( ( ord_less @ nat @ N3 @ J ) % 0.30/0.48 => ( ( P @ N3 ) % 0.30/0.48 => ( P @ ( suc @ N3 ) ) ) ) ) % 0.30/0.48 => ( P @ J ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % dec_induct % 0.30/0.48 thf(fact_283_Suc__le__eq,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 ) % 0.30/0.48 = ( ord_less @ nat @ M @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % Suc_le_eq % 0.30/0.48 thf(fact_284_Suc__leI,axiom, % 0.30/0.48 ! [M: nat,N2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.48 => ( ord_less_eq @ nat @ ( suc @ M ) @ N2 ) ) ). % 0.30/0.48 % 0.30/0.48 % Suc_leI % 0.30/0.48 thf(fact_285_mono__nat__linear__lb,axiom, % 0.30/0.48 ! [F2: nat > nat,M: nat,K: nat] : % 0.30/0.48 ( ! [M3: nat,N3: nat] : % 0.30/0.48 ( ( ord_less @ nat @ M3 @ N3 ) % 0.30/0.48 => ( ord_less @ nat @ ( F2 @ M3 ) @ ( F2 @ N3 ) ) ) % 0.30/0.48 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus @ nat @ M @ K ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % mono_nat_linear_lb % 0.30/0.48 thf(fact_286_add__strict__increasing2,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere8940638589300402666id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.48 => ( ( ord_less @ A @ B2 @ C2 ) % 0.30/0.48 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_strict_increasing2 % 0.30/0.48 thf(fact_287_add__strict__increasing,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere8940638589300402666id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ C2 ) % 0.30/0.48 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_strict_increasing % 0.30/0.48 thf(fact_288_add__pos__nonneg,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.48 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_pos_nonneg % 0.30/0.48 thf(fact_289_add__nonpos__neg,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.48 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 0.30/0.48 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_nonpos_neg % 0.30/0.48 thf(fact_290_add__nonneg__pos,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.48 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.48 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_nonneg_pos % 0.30/0.48 thf(fact_291_add__neg__nonpos,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) % 0.30/0.48 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % add_neg_nonpos % 0.30/0.48 thf(fact_292_ex__least__nat__less,axiom, % 0.30/0.48 ! [P: nat > $o,N2: nat] : % 0.30/0.48 ( ( P @ N2 ) % 0.30/0.48 => ( ~ ( P @ ( zero_zero @ nat ) ) % 0.30/0.48 => ? [K2: nat] : % 0.30/0.48 ( ( ord_less @ nat @ K2 @ N2 ) % 0.30/0.48 & ! [I5: nat] : % 0.30/0.48 ( ( ord_less_eq @ nat @ I5 @ K2 ) % 0.30/0.48 => ~ ( P @ I5 ) ) % 0.30/0.48 & ( P @ ( suc @ K2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % ex_least_nat_less % 0.30/0.48 thf(fact_293_set__update__subsetI,axiom, % 0.30/0.48 ! [A: $tType,Xs: list @ A,A3: set @ A,X3: A,I: nat] : % 0.30/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 ) % 0.30/0.48 => ( ( member @ A @ X3 @ A3 ) % 0.30/0.48 => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X3 ) ) @ A3 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % set_update_subsetI % 0.30/0.48 thf(fact_294_nle__le,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linorder @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) % 0.30/0.48 = ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.30/0.48 & ( B2 != A2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % nle_le % 0.30/0.48 thf(fact_295_le__cases3,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linorder @ A ) % 0.30/0.48 => ! [X3: A,Y: A,Z2: A] : % 0.30/0.48 ( ( ( ord_less_eq @ A @ X3 @ Y ) % 0.30/0.48 => ~ ( ord_less_eq @ A @ Y @ Z2 ) ) % 0.30/0.48 => ( ( ( ord_less_eq @ A @ Y @ X3 ) % 0.30/0.48 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) ) % 0.30/0.48 => ( ( ( ord_less_eq @ A @ X3 @ Z2 ) % 0.30/0.48 => ~ ( ord_less_eq @ A @ Z2 @ Y ) ) % 0.30/0.48 => ( ( ( ord_less_eq @ A @ Z2 @ Y ) % 0.30/0.48 => ~ ( ord_less_eq @ A @ Y @ X3 ) ) % 0.30/0.48 => ( ( ( ord_less_eq @ A @ Y @ Z2 ) % 0.30/0.48 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) ) % 0.30/0.48 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 ) % 0.30/0.48 => ~ ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % le_cases3 % 0.30/0.48 thf(fact_296_order__class_Oorder__eq__iff,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order @ A ) % 0.30/0.48 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) ) % 0.30/0.48 = ( ^ [X: A,Y2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y2 ) % 0.30/0.48 & ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % order_class.order_eq_iff % 0.30/0.48 thf(fact_297_ord__eq__le__trans,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ord @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( A2 = B2 ) % 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ C2 ) % 0.30/0.48 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % ord_eq_le_trans % 0.30/0.48 thf(fact_298_ord__le__eq__trans,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ord @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.48 => ( ( B2 = C2 ) % 0.30/0.48 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % ord_le_eq_trans % 0.30/0.48 thf(fact_299_order__antisym,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order @ A ) % 0.30/0.48 => ! [X3: A,Y: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.30/0.48 => ( ( ord_less_eq @ A @ Y @ X3 ) % 0.30/0.48 => ( X3 = Y ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % order_antisym % 0.30/0.48 thf(fact_300_order_Otrans,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( preorder @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ C2 ) % 0.30/0.48 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % order.trans % 0.30/0.48 thf(fact_301_order__trans,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( preorder @ A ) % 0.30/0.48 => ! [X3: A,Y: A,Z2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.30/0.48 => ( ( ord_less_eq @ A @ Y @ Z2 ) % 0.30/0.48 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % order_trans % 0.30/0.48 thf(fact_302_linorder__wlog,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linorder @ A ) % 0.30/0.48 => ! [P: A > A > $o,A2: A,B2: A] : % 0.30/0.48 ( ! [A5: A,B5: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A5 @ B5 ) % 0.30/0.48 => ( P @ A5 @ B5 ) ) % 0.30/0.48 => ( ! [A5: A,B5: A] : % 0.30/0.48 ( ( P @ B5 @ A5 ) % 0.30/0.48 => ( P @ A5 @ B5 ) ) % 0.30/0.48 => ( P @ A2 @ B2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % linorder_wlog % 0.30/0.48 thf(fact_303_dual__order_Oeq__iff,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order @ A ) % 0.30/0.48 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) ) % 0.30/0.48 = ( ^ [A4: A,B4: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ B4 @ A4 ) % 0.30/0.48 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % dual_order.eq_iff % 0.30/0.48 thf(fact_304_dual__order_Oantisym,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order @ A ) % 0.30/0.48 => ! [B2: A,A2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.30/0.48 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.48 => ( A2 = B2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % dual_order.antisym % 0.30/0.48 thf(fact_305_dual__order_Otrans,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( preorder @ A ) % 0.30/0.48 => ! [B2: A,A2: A,C2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.30/0.48 => ( ( ord_less_eq @ A @ C2 @ B2 ) % 0.30/0.48 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % dual_order.trans % 0.30/0.48 thf(fact_306_antisym,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.30/0.48 => ( A2 = B2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % antisym % 0.30/0.48 thf(fact_307_le__funD,axiom, % 0.30/0.48 ! [B: $tType,A: $tType] : % 0.30/0.48 ( ( ord @ B ) % 0.30/0.48 => ! [F2: A > B,G: A > B,X3: A] : % 0.30/0.48 ( ( ord_less_eq @ ( A > B ) @ F2 @ G ) % 0.30/0.48 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % le_funD % 0.30/0.48 thf(fact_308_le__funE,axiom, % 0.30/0.48 ! [B: $tType,A: $tType] : % 0.30/0.48 ( ( ord @ B ) % 0.30/0.48 => ! [F2: A > B,G: A > B,X3: A] : % 0.30/0.48 ( ( ord_less_eq @ ( A > B ) @ F2 @ G ) % 0.30/0.48 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % le_funE % 0.30/0.48 thf(fact_309_le__funI,axiom, % 0.30/0.48 ! [B: $tType,A: $tType] : % 0.30/0.48 ( ( ord @ B ) % 0.30/0.48 => ! [F2: A > B,G: A > B] : % 0.30/0.48 ( ! [X5: A] : ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G @ X5 ) ) % 0.30/0.48 => ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ). % 0.30/0.48 % 0.30/0.48 % le_funI % 0.30/0.48 thf(fact_310_le__fun__def,axiom, % 0.30/0.48 ! [B: $tType,A: $tType] : % 0.30/0.48 ( ( ord @ B ) % 0.30/0.48 => ( ( ord_less_eq @ ( A > B ) ) % 0.30/0.48 = ( ^ [F3: A > B,G2: A > B] : % 0.30/0.48 ! [X: A] : ( ord_less_eq @ B @ ( F3 @ X ) @ ( G2 @ X ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % le_fun_def % 0.30/0.48 thf(fact_311_Orderings_Oorder__eq__iff,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order @ A ) % 0.30/0.48 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) ) % 0.30/0.48 = ( ^ [A4: A,B4: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ A4 @ B4 ) % 0.30/0.48 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % Orderings.order_eq_iff % 0.30/0.48 thf(fact_312_order__subst1,axiom, % 0.30/0.48 ! [A: $tType,B: $tType] : % 0.30/0.48 ( ( ( order @ B ) % 0.30/0.48 & ( order @ A ) ) % 0.30/0.48 => ! [A2: A,F2: B > A,B2: B,C2: B] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) ) % 0.30/0.48 => ( ( ord_less_eq @ B @ B2 @ C2 ) % 0.30/0.48 => ( ! [X5: B,Y4: B] : % 0.30/0.48 ( ( ord_less_eq @ B @ X5 @ Y4 ) % 0.30/0.48 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.30/0.48 => ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % order_subst1 % 0.30/0.48 thf(fact_313_order__subst2,axiom, % 0.30/0.48 ! [A: $tType,C: $tType] : % 0.30/0.48 ( ( ( order @ C ) % 0.30/0.48 & ( order @ A ) ) % 0.30/0.48 => ! [A2: A,B2: A,F2: A > C,C2: C] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.48 => ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 ) % 0.30/0.48 => ( ! [X5: A,Y4: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ X5 @ Y4 ) % 0.30/0.48 => ( ord_less_eq @ C @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.30/0.48 => ( ord_less_eq @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % order_subst2 % 0.30/0.48 thf(fact_314_order__eq__refl,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( preorder @ A ) % 0.30/0.48 => ! [X3: A,Y: A] : % 0.30/0.48 ( ( X3 = Y ) % 0.30/0.48 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ). % 0.30/0.48 % 0.30/0.48 % order_eq_refl % 0.30/0.48 thf(fact_315_linorder__linear,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linorder @ A ) % 0.30/0.48 => ! [X3: A,Y: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.30/0.48 | ( ord_less_eq @ A @ Y @ X3 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % linorder_linear % 0.30/0.48 thf(fact_316_ord__eq__le__subst,axiom, % 0.30/0.48 ! [A: $tType,B: $tType] : % 0.30/0.48 ( ( ( ord @ B ) % 0.30/0.48 & ( ord @ A ) ) % 0.30/0.48 => ! [A2: A,F2: B > A,B2: B,C2: B] : % 0.30/0.48 ( ( A2 % 0.30/0.48 = ( F2 @ B2 ) ) % 0.30/0.48 => ( ( ord_less_eq @ B @ B2 @ C2 ) % 0.30/0.48 => ( ! [X5: B,Y4: B] : % 0.30/0.48 ( ( ord_less_eq @ B @ X5 @ Y4 ) % 0.30/0.48 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.30/0.48 => ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % ord_eq_le_subst % 0.30/0.48 thf(fact_317_ord__le__eq__subst,axiom, % 0.30/0.48 ! [A: $tType,B: $tType] : % 0.30/0.48 ( ( ( ord @ B ) % 0.30/0.48 & ( ord @ A ) ) % 0.30/0.48 => ! [A2: A,B2: A,F2: A > B,C2: B] : % 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.48 => ( ( ( F2 @ B2 ) % 0.30/0.48 = C2 ) % 0.30/0.48 => ( ! [X5: A,Y4: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ X5 @ Y4 ) % 0.30/0.48 => ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.30/0.48 => ( ord_less_eq @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % ord_le_eq_subst % 0.30/0.48 thf(fact_318_linorder__le__cases,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linorder @ A ) % 0.30/0.48 => ! [X3: A,Y: A] : % 0.30/0.48 ( ~ ( ord_less_eq @ A @ X3 @ Y ) % 0.30/0.48 => ( ord_less_eq @ A @ Y @ X3 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % linorder_le_cases % 0.30/0.48 thf(fact_319_order__antisym__conv,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order @ A ) % 0.30/0.48 => ! [Y: A,X3: A] : % 0.30/0.48 ( ( ord_less_eq @ A @ Y @ X3 ) % 0.30/0.48 => ( ( ord_less_eq @ A @ X3 @ Y ) % 0.30/0.48 = ( X3 = Y ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % order_antisym_conv % 0.30/0.48 thf(fact_320_zero__reorient,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( zero @ A ) % 0.30/0.48 => ! [X3: A] : % 0.30/0.48 ( ( ( zero_zero @ A ) % 0.30/0.48 = X3 ) % 0.30/0.48 = ( X3 % 0.30/0.48 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % zero_reorient % 0.30/0.48 thf(fact_321_lt__ex,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( no_bot @ A ) % 0.30/0.48 => ! [X3: A] : % 0.30/0.48 ? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ). % 0.30/0.48 % 0.30/0.48 % lt_ex % 0.30/0.48 thf(fact_322_gt__ex,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( no_top @ A ) % 0.30/0.48 => ! [X3: A] : % 0.30/0.48 ? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ). % 0.30/0.48 % 0.30/0.48 % gt_ex % 0.30/0.48 thf(fact_323_dense,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( dense_order @ A ) % 0.30/0.48 => ! [X3: A,Y: A] : % 0.30/0.48 ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.48 => ? [Z3: A] : % 0.30/0.48 ( ( ord_less @ A @ X3 @ Z3 ) % 0.30/0.48 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % dense % 0.30/0.48 thf(fact_324_less__imp__neq,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order @ A ) % 0.30/0.48 => ! [X3: A,Y: A] : % 0.30/0.48 ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.48 => ( X3 != Y ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_imp_neq % 0.30/0.48 thf(fact_325_order_Oasym,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( preorder @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.48 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % order.asym % 0.30/0.48 thf(fact_326_ord__eq__less__trans,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ord @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( A2 = B2 ) % 0.30/0.48 => ( ( ord_less @ A @ B2 @ C2 ) % 0.30/0.48 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % ord_eq_less_trans % 0.30/0.48 thf(fact_327_ord__less__eq__trans,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( ord @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.48 => ( ( B2 = C2 ) % 0.30/0.48 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % ord_less_eq_trans % 0.30/0.48 thf(fact_328_less__induct,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( wellorder @ A ) % 0.30/0.48 => ! [P: A > $o,A2: A] : % 0.30/0.48 ( ! [X5: A] : % 0.30/0.48 ( ! [Y5: A] : % 0.30/0.48 ( ( ord_less @ A @ Y5 @ X5 ) % 0.30/0.48 => ( P @ Y5 ) ) % 0.30/0.48 => ( P @ X5 ) ) % 0.30/0.48 => ( P @ A2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % less_induct % 0.30/0.48 thf(fact_329_antisym__conv3,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linorder @ A ) % 0.30/0.48 => ! [Y: A,X3: A] : % 0.30/0.48 ( ~ ( ord_less @ A @ Y @ X3 ) % 0.30/0.48 => ( ( ~ ( ord_less @ A @ X3 @ Y ) ) % 0.30/0.48 = ( X3 = Y ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % antisym_conv3 % 0.30/0.48 thf(fact_330_linorder__cases,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linorder @ A ) % 0.30/0.48 => ! [X3: A,Y: A] : % 0.30/0.48 ( ~ ( ord_less @ A @ X3 @ Y ) % 0.30/0.48 => ( ( X3 != Y ) % 0.30/0.48 => ( ord_less @ A @ Y @ X3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % linorder_cases % 0.30/0.48 thf(fact_331_dual__order_Oasym,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( preorder @ A ) % 0.30/0.48 => ! [B2: A,A2: A] : % 0.30/0.48 ( ( ord_less @ A @ B2 @ A2 ) % 0.30/0.48 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % dual_order.asym % 0.30/0.48 thf(fact_332_dual__order_Oirrefl,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( preorder @ A ) % 0.30/0.48 => ! [A2: A] : % 0.30/0.48 ~ ( ord_less @ A @ A2 @ A2 ) ) ). % 0.30/0.48 % 0.30/0.48 % dual_order.irrefl % 0.30/0.48 thf(fact_333_exists__least__iff,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( wellorder @ A ) % 0.30/0.48 => ( ( ^ [P2: A > $o] : % 0.30/0.48 ? [X7: A] : ( P2 @ X7 ) ) % 0.30/0.48 = ( ^ [P3: A > $o] : % 0.30/0.48 ? [N: A] : % 0.30/0.48 ( ( P3 @ N ) % 0.30/0.48 & ! [M2: A] : % 0.30/0.48 ( ( ord_less @ A @ M2 @ N ) % 0.30/0.48 => ~ ( P3 @ M2 ) ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % exists_least_iff % 0.30/0.48 thf(fact_334_linorder__less__wlog,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linorder @ A ) % 0.30/0.48 => ! [P: A > A > $o,A2: A,B2: A] : % 0.30/0.48 ( ! [A5: A,B5: A] : % 0.30/0.48 ( ( ord_less @ A @ A5 @ B5 ) % 0.30/0.48 => ( P @ A5 @ B5 ) ) % 0.30/0.48 => ( ! [A5: A] : ( P @ A5 @ A5 ) % 0.30/0.48 => ( ! [A5: A,B5: A] : % 0.30/0.48 ( ( P @ B5 @ A5 ) % 0.30/0.48 => ( P @ A5 @ B5 ) ) % 0.30/0.48 => ( P @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % linorder_less_wlog % 0.30/0.48 thf(fact_335_order_Ostrict__trans,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( preorder @ A ) % 0.30/0.48 => ! [A2: A,B2: A,C2: A] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.48 => ( ( ord_less @ A @ B2 @ C2 ) % 0.30/0.48 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % order.strict_trans % 0.30/0.48 thf(fact_336_not__less__iff__gr__or__eq,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linorder @ A ) % 0.30/0.48 => ! [X3: A,Y: A] : % 0.30/0.48 ( ( ~ ( ord_less @ A @ X3 @ Y ) ) % 0.30/0.48 = ( ( ord_less @ A @ Y @ X3 ) % 0.30/0.48 | ( X3 = Y ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % not_less_iff_gr_or_eq % 0.30/0.48 thf(fact_337_dual__order_Ostrict__trans,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( preorder @ A ) % 0.30/0.48 => ! [B2: A,A2: A,C2: A] : % 0.30/0.48 ( ( ord_less @ A @ B2 @ A2 ) % 0.30/0.48 => ( ( ord_less @ A @ C2 @ B2 ) % 0.30/0.48 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % dual_order.strict_trans % 0.30/0.48 thf(fact_338_order_Ostrict__implies__not__eq,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.48 => ( A2 != B2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % order.strict_implies_not_eq % 0.30/0.48 thf(fact_339_dual__order_Ostrict__implies__not__eq,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( order @ A ) % 0.30/0.48 => ! [B2: A,A2: A] : % 0.30/0.48 ( ( ord_less @ A @ B2 @ A2 ) % 0.30/0.48 => ( A2 != B2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % dual_order.strict_implies_not_eq % 0.30/0.48 thf(fact_340_measure__induct,axiom, % 0.30/0.48 ! [B: $tType,A: $tType] : % 0.30/0.48 ( ( wellorder @ B ) % 0.30/0.48 => ! [F2: A > B,P: A > $o,A2: A] : % 0.30/0.48 ( ! [X5: A] : % 0.30/0.48 ( ! [Y5: A] : % 0.30/0.48 ( ( ord_less @ B @ ( F2 @ Y5 ) @ ( F2 @ X5 ) ) % 0.30/0.48 => ( P @ Y5 ) ) % 0.30/0.48 => ( P @ X5 ) ) % 0.30/0.48 => ( P @ A2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % measure_induct % 0.30/0.48 thf(fact_341_measure__induct__rule,axiom, % 0.30/0.48 ! [B: $tType,A: $tType] : % 0.30/0.48 ( ( wellorder @ B ) % 0.30/0.48 => ! [F2: A > B,P: A > $o,A2: A] : % 0.30/0.48 ( ! [X5: A] : % 0.30/0.48 ( ! [Y5: A] : % 0.30/0.48 ( ( ord_less @ B @ ( F2 @ Y5 ) @ ( F2 @ X5 ) ) % 0.30/0.48 => ( P @ Y5 ) ) % 0.30/0.48 => ( P @ X5 ) ) % 0.30/0.48 => ( P @ A2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % measure_induct_rule % 0.30/0.48 thf(fact_342_linorder__neqE,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linorder @ A ) % 0.30/0.48 => ! [X3: A,Y: A] : % 0.30/0.48 ( ( X3 != Y ) % 0.30/0.48 => ( ~ ( ord_less @ A @ X3 @ Y ) % 0.30/0.48 => ( ord_less @ A @ Y @ X3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % linorder_neqE % 0.30/0.48 thf(fact_343_order__less__asym,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( preorder @ A ) % 0.30/0.48 => ! [X3: A,Y: A] : % 0.30/0.48 ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.48 => ~ ( ord_less @ A @ Y @ X3 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % order_less_asym % 0.30/0.48 thf(fact_344_linorder__neq__iff,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( linorder @ A ) % 0.30/0.48 => ! [X3: A,Y: A] : % 0.30/0.48 ( ( X3 != Y ) % 0.30/0.48 = ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.48 | ( ord_less @ A @ Y @ X3 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % linorder_neq_iff % 0.30/0.48 thf(fact_345_order__less__asym_H,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( preorder @ A ) % 0.30/0.48 => ! [A2: A,B2: A] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.48 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ). % 0.30/0.48 % 0.30/0.48 % order_less_asym' % 0.30/0.48 thf(fact_346_order__less__trans,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( preorder @ A ) % 0.30/0.48 => ! [X3: A,Y: A,Z2: A] : % 0.30/0.48 ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.48 => ( ( ord_less @ A @ Y @ Z2 ) % 0.30/0.48 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % order_less_trans % 0.30/0.48 thf(fact_347_ord__eq__less__subst,axiom, % 0.30/0.48 ! [A: $tType,B: $tType] : % 0.30/0.48 ( ( ( ord @ B ) % 0.30/0.48 & ( ord @ A ) ) % 0.30/0.48 => ! [A2: A,F2: B > A,B2: B,C2: B] : % 0.30/0.48 ( ( A2 % 0.30/0.48 = ( F2 @ B2 ) ) % 0.30/0.48 => ( ( ord_less @ B @ B2 @ C2 ) % 0.30/0.48 => ( ! [X5: B,Y4: B] : % 0.30/0.48 ( ( ord_less @ B @ X5 @ Y4 ) % 0.30/0.48 => ( ord_less @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.30/0.48 => ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % ord_eq_less_subst % 0.30/0.48 thf(fact_348_ord__less__eq__subst,axiom, % 0.30/0.48 ! [A: $tType,B: $tType] : % 0.30/0.48 ( ( ( ord @ B ) % 0.30/0.48 & ( ord @ A ) ) % 0.30/0.48 => ! [A2: A,B2: A,F2: A > B,C2: B] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.48 => ( ( ( F2 @ B2 ) % 0.30/0.48 = C2 ) % 0.30/0.48 => ( ! [X5: A,Y4: A] : % 0.30/0.48 ( ( ord_less @ A @ X5 @ Y4 ) % 0.30/0.48 => ( ord_less @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.30/0.48 => ( ord_less @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % ord_less_eq_subst % 0.30/0.48 thf(fact_349_order__less__irrefl,axiom, % 0.30/0.48 ! [A: $tType] : % 0.30/0.48 ( ( preorder @ A ) % 0.30/0.48 => ! [X3: A] : % 0.30/0.48 ~ ( ord_less @ A @ X3 @ X3 ) ) ). % 0.30/0.48 % 0.30/0.48 % order_less_irrefl % 0.30/0.48 thf(fact_350_order__less__subst1,axiom, % 0.30/0.48 ! [A: $tType,B: $tType] : % 0.30/0.48 ( ( ( order @ B ) % 0.30/0.48 & ( order @ A ) ) % 0.30/0.48 => ! [A2: A,F2: B > A,B2: B,C2: B] : % 0.30/0.48 ( ( ord_less @ A @ A2 @ ( F2 @ B2 ) ) % 0.30/0.48 => ( ( ord_less @ B @ B2 @ C2 ) % 0.30/0.48 => ( ! [X5: B,Y4: B] : % 0.30/0.48 ( ( ord_less @ B @ X5 @ Y4 ) % 0.30/0.48 => ( ord_less @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.30/0.48 => ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ). % 0.30/0.48 % 0.30/0.48 % order_less_subst1 % 0.30/0.48 thf(fact_351_order__less__subst2,axiom, % 0.30/0.48 ! [A: $tType,C: $tType] : % 0.30/0.48 ( ( ( order @ C ) % 0.30/0.48 & ( order @ A ) ) % 0.30/0.49 => ! [A2: A,B2: A,F2: A > C,C2: C] : % 0.30/0.49 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.49 => ( ( ord_less @ C @ ( F2 @ B2 ) @ C2 ) % 0.30/0.49 => ( ! [X5: A,Y4: A] : % 0.30/0.49 ( ( ord_less @ A @ X5 @ Y4 ) % 0.30/0.49 => ( ord_less @ C @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.30/0.49 => ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_less_subst2 % 0.30/0.49 thf(fact_352_order__less__not__sym,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( preorder @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.49 => ~ ( ord_less @ A @ Y @ X3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_less_not_sym % 0.30/0.49 thf(fact_353_order__less__imp__triv,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( preorder @ A ) % 0.30/0.49 => ! [X3: A,Y: A,P: $o] : % 0.30/0.49 ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.49 => ( ( ord_less @ A @ Y @ X3 ) % 0.30/0.49 => P ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_less_imp_triv % 0.30/0.49 thf(fact_354_linorder__less__linear,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.49 | ( X3 = Y ) % 0.30/0.49 | ( ord_less @ A @ Y @ X3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % linorder_less_linear % 0.30/0.49 thf(fact_355_order__less__imp__not__eq,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.49 => ( X3 != Y ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_less_imp_not_eq % 0.30/0.49 thf(fact_356_order__less__imp__not__eq2,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.49 => ( Y != X3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_less_imp_not_eq2 % 0.30/0.49 thf(fact_357_order__less__imp__not__less,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( preorder @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.49 => ~ ( ord_less @ A @ Y @ X3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_less_imp_not_less % 0.30/0.49 thf(fact_358_nat__neq__iff,axiom, % 0.30/0.49 ! [M: nat,N2: nat] : % 0.30/0.49 ( ( M != N2 ) % 0.30/0.49 = ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.49 | ( ord_less @ nat @ N2 @ M ) ) ) ). % 0.30/0.49 % 0.30/0.49 % nat_neq_iff % 0.30/0.49 thf(fact_359_less__not__refl,axiom, % 0.30/0.49 ! [N2: nat] : % 0.30/0.49 ~ ( ord_less @ nat @ N2 @ N2 ) ). % 0.30/0.49 % 0.30/0.49 % less_not_refl % 0.30/0.49 thf(fact_360_less__not__refl2,axiom, % 0.30/0.49 ! [N2: nat,M: nat] : % 0.30/0.49 ( ( ord_less @ nat @ N2 @ M ) % 0.30/0.49 => ( M != N2 ) ) ). % 0.30/0.49 % 0.30/0.49 % less_not_refl2 % 0.30/0.49 thf(fact_361_less__not__refl3,axiom, % 0.30/0.49 ! [S: nat,T2: nat] : % 0.30/0.49 ( ( ord_less @ nat @ S @ T2 ) % 0.30/0.49 => ( S != T2 ) ) ). % 0.30/0.49 % 0.30/0.49 % less_not_refl3 % 0.30/0.49 thf(fact_362_less__irrefl__nat,axiom, % 0.30/0.49 ! [N2: nat] : % 0.30/0.49 ~ ( ord_less @ nat @ N2 @ N2 ) ). % 0.30/0.49 % 0.30/0.49 % less_irrefl_nat % 0.30/0.49 thf(fact_363_nat__less__induct,axiom, % 0.30/0.49 ! [P: nat > $o,N2: nat] : % 0.30/0.49 ( ! [N3: nat] : % 0.30/0.49 ( ! [M6: nat] : % 0.30/0.49 ( ( ord_less @ nat @ M6 @ N3 ) % 0.30/0.49 => ( P @ M6 ) ) % 0.30/0.49 => ( P @ N3 ) ) % 0.30/0.49 => ( P @ N2 ) ) ). % 0.30/0.49 % 0.30/0.49 % nat_less_induct % 0.30/0.49 thf(fact_364_infinite__descent,axiom, % 0.30/0.49 ! [P: nat > $o,N2: nat] : % 0.30/0.49 ( ! [N3: nat] : % 0.30/0.49 ( ~ ( P @ N3 ) % 0.30/0.49 => ? [M6: nat] : % 0.30/0.49 ( ( ord_less @ nat @ M6 @ N3 ) % 0.30/0.49 & ~ ( P @ M6 ) ) ) % 0.30/0.49 => ( P @ N2 ) ) ). % 0.30/0.49 % 0.30/0.49 % infinite_descent % 0.30/0.49 thf(fact_365_linorder__neqE__nat,axiom, % 0.30/0.49 ! [X3: nat,Y: nat] : % 0.30/0.49 ( ( X3 != Y ) % 0.30/0.49 => ( ~ ( ord_less @ nat @ X3 @ Y ) % 0.30/0.49 => ( ord_less @ nat @ Y @ X3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % linorder_neqE_nat % 0.30/0.49 thf(fact_366_infinite__descent__measure,axiom, % 0.30/0.49 ! [A: $tType,P: A > $o,V: A > nat,X3: A] : % 0.30/0.49 ( ! [X5: A] : % 0.30/0.49 ( ~ ( P @ X5 ) % 0.30/0.49 => ? [Y5: A] : % 0.30/0.49 ( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X5 ) ) % 0.30/0.49 & ~ ( P @ Y5 ) ) ) % 0.30/0.49 => ( P @ X3 ) ) ). % 0.30/0.49 % 0.30/0.49 % infinite_descent_measure % 0.30/0.49 thf(fact_367_le__refl,axiom, % 0.30/0.49 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ). % 0.30/0.49 % 0.30/0.49 % le_refl % 0.30/0.49 thf(fact_368_le__trans,axiom, % 0.30/0.49 ! [I: nat,J: nat,K: nat] : % 0.30/0.49 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.49 => ( ( ord_less_eq @ nat @ J @ K ) % 0.30/0.49 => ( ord_less_eq @ nat @ I @ K ) ) ) ). % 0.30/0.49 % 0.30/0.49 % le_trans % 0.30/0.49 thf(fact_369_eq__imp__le,axiom, % 0.30/0.49 ! [M: nat,N2: nat] : % 0.30/0.49 ( ( M = N2 ) % 0.30/0.49 => ( ord_less_eq @ nat @ M @ N2 ) ) ). % 0.30/0.49 % 0.30/0.49 % eq_imp_le % 0.30/0.49 thf(fact_370_le__antisym,axiom, % 0.30/0.49 ! [M: nat,N2: nat] : % 0.30/0.49 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.49 => ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.49 => ( M = N2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % le_antisym % 0.30/0.49 thf(fact_371_nat__le__linear,axiom, % 0.30/0.49 ! [M: nat,N2: nat] : % 0.30/0.49 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.49 | ( ord_less_eq @ nat @ N2 @ M ) ) ). % 0.30/0.49 % 0.30/0.49 % nat_le_linear % 0.30/0.49 thf(fact_372_Nat_Oex__has__greatest__nat,axiom, % 0.30/0.49 ! [P: nat > $o,K: nat,B2: nat] : % 0.30/0.49 ( ( P @ K ) % 0.30/0.49 => ( ! [Y4: nat] : % 0.30/0.49 ( ( P @ Y4 ) % 0.30/0.49 => ( ord_less_eq @ nat @ Y4 @ B2 ) ) % 0.30/0.49 => ? [X5: nat] : % 0.30/0.49 ( ( P @ X5 ) % 0.30/0.49 & ! [Y5: nat] : % 0.30/0.49 ( ( P @ Y5 ) % 0.30/0.49 => ( ord_less_eq @ nat @ Y5 @ X5 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % Nat.ex_has_greatest_nat % 0.30/0.49 thf(fact_373_size__neq__size__imp__neq,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( size @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ( ( size_size @ A @ X3 ) % 0.30/0.49 != ( size_size @ A @ Y ) ) % 0.30/0.49 => ( X3 != Y ) ) ) ). % 0.30/0.49 % 0.30/0.49 % size_neq_size_imp_neq % 0.30/0.49 thf(fact_374_zero__le,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( canoni5634975068530333245id_add @ A ) % 0.30/0.49 => ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ). % 0.30/0.49 % 0.30/0.49 % zero_le % 0.30/0.49 thf(fact_375_leD,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [Y: A,X3: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ Y @ X3 ) % 0.30/0.49 => ~ ( ord_less @ A @ X3 @ Y ) ) ) ). % 0.30/0.49 % 0.30/0.49 % leD % 0.30/0.49 thf(fact_376_leI,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ~ ( ord_less @ A @ X3 @ Y ) % 0.30/0.49 => ( ord_less_eq @ A @ Y @ X3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % leI % 0.30/0.49 thf(fact_377_nless__le,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [A2: A,B2: A] : % 0.30/0.49 ( ( ~ ( ord_less @ A @ A2 @ B2 ) ) % 0.30/0.49 = ( ~ ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.49 | ( A2 = B2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % nless_le % 0.30/0.49 thf(fact_378_antisym__conv1,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ~ ( ord_less @ A @ X3 @ Y ) % 0.30/0.49 => ( ( ord_less_eq @ A @ X3 @ Y ) % 0.30/0.49 = ( X3 = Y ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % antisym_conv1 % 0.30/0.49 thf(fact_379_antisym__conv2,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.30/0.49 => ( ( ~ ( ord_less @ A @ X3 @ Y ) ) % 0.30/0.49 = ( X3 = Y ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % antisym_conv2 % 0.30/0.49 thf(fact_380_dense__ge,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( dense_linorder @ A ) % 0.30/0.49 => ! [Z2: A,Y: A] : % 0.30/0.49 ( ! [X5: A] : % 0.30/0.49 ( ( ord_less @ A @ Z2 @ X5 ) % 0.30/0.49 => ( ord_less_eq @ A @ Y @ X5 ) ) % 0.30/0.49 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % dense_ge % 0.30/0.49 thf(fact_381_dense__le,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( dense_linorder @ A ) % 0.30/0.49 => ! [Y: A,Z2: A] : % 0.30/0.49 ( ! [X5: A] : % 0.30/0.49 ( ( ord_less @ A @ X5 @ Y ) % 0.30/0.49 => ( ord_less_eq @ A @ X5 @ Z2 ) ) % 0.30/0.49 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % dense_le % 0.30/0.49 thf(fact_382_less__le__not__le,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( preorder @ A ) % 0.30/0.49 => ( ( ord_less @ A ) % 0.30/0.49 = ( ^ [X: A,Y2: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ X @ Y2 ) % 0.30/0.49 & ~ ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % less_le_not_le % 0.30/0.49 thf(fact_383_not__le__imp__less,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [Y: A,X3: A] : % 0.30/0.49 ( ~ ( ord_less_eq @ A @ Y @ X3 ) % 0.30/0.49 => ( ord_less @ A @ X3 @ Y ) ) ) ). % 0.30/0.49 % 0.30/0.49 % not_le_imp_less % 0.30/0.49 thf(fact_384_order_Oorder__iff__strict,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ( ( ord_less_eq @ A ) % 0.30/0.49 = ( ^ [A4: A,B4: A] : % 0.30/0.49 ( ( ord_less @ A @ A4 @ B4 ) % 0.30/0.49 | ( A4 = B4 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order.order_iff_strict % 0.30/0.49 thf(fact_385_order_Ostrict__iff__order,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ( ( ord_less @ A ) % 0.30/0.49 = ( ^ [A4: A,B4: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ A4 @ B4 ) % 0.30/0.49 & ( A4 != B4 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order.strict_iff_order % 0.30/0.49 thf(fact_386_order_Ostrict__trans1,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( preorder @ A ) % 0.30/0.49 => ! [A2: A,B2: A,C2: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.49 => ( ( ord_less @ A @ B2 @ C2 ) % 0.30/0.49 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order.strict_trans1 % 0.30/0.49 thf(fact_387_order_Ostrict__trans2,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( preorder @ A ) % 0.30/0.49 => ! [A2: A,B2: A,C2: A] : % 0.30/0.49 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.49 => ( ( ord_less_eq @ A @ B2 @ C2 ) % 0.30/0.49 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order.strict_trans2 % 0.30/0.49 thf(fact_388_order_Ostrict__iff__not,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( preorder @ A ) % 0.30/0.49 => ( ( ord_less @ A ) % 0.30/0.49 = ( ^ [A4: A,B4: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ A4 @ B4 ) % 0.30/0.49 & ~ ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order.strict_iff_not % 0.30/0.49 thf(fact_389_dense__ge__bounded,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( dense_linorder @ A ) % 0.30/0.49 => ! [Z2: A,X3: A,Y: A] : % 0.30/0.49 ( ( ord_less @ A @ Z2 @ X3 ) % 0.30/0.49 => ( ! [W: A] : % 0.30/0.49 ( ( ord_less @ A @ Z2 @ W ) % 0.30/0.49 => ( ( ord_less @ A @ W @ X3 ) % 0.30/0.49 => ( ord_less_eq @ A @ Y @ W ) ) ) % 0.30/0.49 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % dense_ge_bounded % 0.30/0.49 thf(fact_390_dense__le__bounded,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( dense_linorder @ A ) % 0.30/0.49 => ! [X3: A,Y: A,Z2: A] : % 0.30/0.49 ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.49 => ( ! [W: A] : % 0.30/0.49 ( ( ord_less @ A @ X3 @ W ) % 0.30/0.49 => ( ( ord_less @ A @ W @ Y ) % 0.30/0.49 => ( ord_less_eq @ A @ W @ Z2 ) ) ) % 0.30/0.49 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % dense_le_bounded % 0.30/0.49 thf(fact_391_dual__order_Oorder__iff__strict,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ( ( ord_less_eq @ A ) % 0.30/0.49 = ( ^ [B4: A,A4: A] : % 0.30/0.49 ( ( ord_less @ A @ B4 @ A4 ) % 0.30/0.49 | ( A4 = B4 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % dual_order.order_iff_strict % 0.30/0.49 thf(fact_392_dual__order_Ostrict__iff__order,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ( ( ord_less @ A ) % 0.30/0.49 = ( ^ [B4: A,A4: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ B4 @ A4 ) % 0.30/0.49 & ( A4 != B4 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % dual_order.strict_iff_order % 0.30/0.49 thf(fact_393_dual__order_Ostrict__trans1,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( preorder @ A ) % 0.30/0.49 => ! [B2: A,A2: A,C2: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.30/0.49 => ( ( ord_less @ A @ C2 @ B2 ) % 0.30/0.49 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % dual_order.strict_trans1 % 0.30/0.49 thf(fact_394_dual__order_Ostrict__trans2,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( preorder @ A ) % 0.30/0.49 => ! [B2: A,A2: A,C2: A] : % 0.30/0.49 ( ( ord_less @ A @ B2 @ A2 ) % 0.30/0.49 => ( ( ord_less_eq @ A @ C2 @ B2 ) % 0.30/0.49 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % dual_order.strict_trans2 % 0.30/0.49 thf(fact_395_dual__order_Ostrict__iff__not,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( preorder @ A ) % 0.30/0.49 => ( ( ord_less @ A ) % 0.30/0.49 = ( ^ [B4: A,A4: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ B4 @ A4 ) % 0.30/0.49 & ~ ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % dual_order.strict_iff_not % 0.30/0.49 thf(fact_396_order_Ostrict__implies__order,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( preorder @ A ) % 0.30/0.49 => ! [A2: A,B2: A] : % 0.30/0.49 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.49 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order.strict_implies_order % 0.30/0.49 thf(fact_397_dual__order_Ostrict__implies__order,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( preorder @ A ) % 0.30/0.49 => ! [B2: A,A2: A] : % 0.30/0.49 ( ( ord_less @ A @ B2 @ A2 ) % 0.30/0.49 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % dual_order.strict_implies_order % 0.30/0.49 thf(fact_398_order__le__less,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ( ( ord_less_eq @ A ) % 0.30/0.49 = ( ^ [X: A,Y2: A] : % 0.30/0.49 ( ( ord_less @ A @ X @ Y2 ) % 0.30/0.49 | ( X = Y2 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_le_less % 0.30/0.49 thf(fact_399_order__less__le,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ( ( ord_less @ A ) % 0.30/0.49 = ( ^ [X: A,Y2: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ X @ Y2 ) % 0.30/0.49 & ( X != Y2 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_less_le % 0.30/0.49 thf(fact_400_linorder__not__le,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ( ~ ( ord_less_eq @ A @ X3 @ Y ) ) % 0.30/0.49 = ( ord_less @ A @ Y @ X3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % linorder_not_le % 0.30/0.49 thf(fact_401_linorder__not__less,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ( ~ ( ord_less @ A @ X3 @ Y ) ) % 0.30/0.49 = ( ord_less_eq @ A @ Y @ X3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % linorder_not_less % 0.30/0.49 thf(fact_402_order__less__imp__le,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( preorder @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.49 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_less_imp_le % 0.30/0.49 thf(fact_403_order__le__neq__trans,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [A2: A,B2: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.49 => ( ( A2 != B2 ) % 0.30/0.49 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_le_neq_trans % 0.30/0.49 thf(fact_404_order__neq__le__trans,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [A2: A,B2: A] : % 0.30/0.49 ( ( A2 != B2 ) % 0.30/0.49 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.49 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_neq_le_trans % 0.30/0.49 thf(fact_405_order__le__less__trans,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( preorder @ A ) % 0.30/0.49 => ! [X3: A,Y: A,Z2: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.30/0.49 => ( ( ord_less @ A @ Y @ Z2 ) % 0.30/0.49 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_le_less_trans % 0.30/0.49 thf(fact_406_order__less__le__trans,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( preorder @ A ) % 0.30/0.49 => ! [X3: A,Y: A,Z2: A] : % 0.30/0.49 ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.49 => ( ( ord_less_eq @ A @ Y @ Z2 ) % 0.30/0.49 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_less_le_trans % 0.30/0.49 thf(fact_407_order__le__less__subst1,axiom, % 0.30/0.49 ! [A: $tType,B: $tType] : % 0.30/0.49 ( ( ( order @ B ) % 0.30/0.49 & ( order @ A ) ) % 0.30/0.49 => ! [A2: A,F2: B > A,B2: B,C2: B] : % 0.30/0.49 ( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) ) % 0.30/0.49 => ( ( ord_less @ B @ B2 @ C2 ) % 0.30/0.49 => ( ! [X5: B,Y4: B] : % 0.30/0.49 ( ( ord_less @ B @ X5 @ Y4 ) % 0.30/0.49 => ( ord_less @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.30/0.49 => ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_le_less_subst1 % 0.30/0.49 thf(fact_408_order__le__less__subst2,axiom, % 0.30/0.49 ! [A: $tType,C: $tType] : % 0.30/0.49 ( ( ( order @ C ) % 0.30/0.49 & ( order @ A ) ) % 0.30/0.49 => ! [A2: A,B2: A,F2: A > C,C2: C] : % 0.30/0.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.49 => ( ( ord_less @ C @ ( F2 @ B2 ) @ C2 ) % 0.30/0.49 => ( ! [X5: A,Y4: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ X5 @ Y4 ) % 0.30/0.49 => ( ord_less_eq @ C @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.30/0.49 => ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_le_less_subst2 % 0.30/0.49 thf(fact_409_order__less__le__subst1,axiom, % 0.30/0.49 ! [A: $tType,B: $tType] : % 0.30/0.49 ( ( ( order @ B ) % 0.30/0.49 & ( order @ A ) ) % 0.30/0.49 => ! [A2: A,F2: B > A,B2: B,C2: B] : % 0.30/0.49 ( ( ord_less @ A @ A2 @ ( F2 @ B2 ) ) % 0.30/0.49 => ( ( ord_less_eq @ B @ B2 @ C2 ) % 0.30/0.49 => ( ! [X5: B,Y4: B] : % 0.30/0.49 ( ( ord_less_eq @ B @ X5 @ Y4 ) % 0.30/0.49 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.30/0.49 => ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_less_le_subst1 % 0.30/0.49 thf(fact_410_order__less__le__subst2,axiom, % 0.30/0.49 ! [A: $tType,C: $tType] : % 0.30/0.49 ( ( ( order @ C ) % 0.30/0.49 & ( order @ A ) ) % 0.30/0.49 => ! [A2: A,B2: A,F2: A > C,C2: C] : % 0.30/0.49 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.49 => ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 ) % 0.30/0.49 => ( ! [X5: A,Y4: A] : % 0.30/0.49 ( ( ord_less @ A @ X5 @ Y4 ) % 0.30/0.49 => ( ord_less @ C @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.30/0.49 => ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_less_le_subst2 % 0.30/0.49 thf(fact_411_linorder__le__less__linear,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.30/0.49 | ( ord_less @ A @ Y @ X3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % linorder_le_less_linear % 0.30/0.49 thf(fact_412_order__le__imp__less__or__eq,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.30/0.49 => ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.49 | ( X3 = Y ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % order_le_imp_less_or_eq % 0.30/0.49 thf(fact_413_gr__zeroI,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( canoni5634975068530333245id_add @ A ) % 0.30/0.49 => ! [N2: A] : % 0.30/0.49 ( ( N2 % 0.30/0.49 != ( zero_zero @ A ) ) % 0.30/0.49 => ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % gr_zeroI % 0.30/0.49 thf(fact_414_not__less__zero,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( canoni5634975068530333245id_add @ A ) % 0.30/0.49 => ! [N2: A] : % 0.30/0.49 ~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ). % 0.30/0.49 % 0.30/0.49 % not_less_zero % 0.30/0.49 thf(fact_415_gr__implies__not__zero,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( canoni5634975068530333245id_add @ A ) % 0.30/0.49 => ! [M: A,N2: A] : % 0.30/0.49 ( ( ord_less @ A @ M @ N2 ) % 0.30/0.49 => ( N2 % 0.30/0.49 != ( zero_zero @ A ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % gr_implies_not_zero % 0.30/0.49 thf(fact_416_zero__less__iff__neq__zero,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( canoni5634975068530333245id_add @ A ) % 0.30/0.49 => ! [N2: A] : % 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) % 0.30/0.49 = ( N2 % 0.30/0.49 != ( zero_zero @ A ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % zero_less_iff_neq_zero % 0.30/0.49 thf(fact_417_bot__nat__0_Oextremum__strict,axiom, % 0.30/0.49 ! [A2: nat] : % 0.30/0.49 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ). % 0.30/0.49 % 0.30/0.49 % bot_nat_0.extremum_strict % 0.30/0.49 thf(fact_418_gr0I,axiom, % 0.30/0.49 ! [N2: nat] : % 0.30/0.49 ( ( N2 % 0.30/0.49 != ( zero_zero @ nat ) ) % 0.30/0.49 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ). % 0.30/0.49 % 0.30/0.49 % gr0I % 0.30/0.49 thf(fact_419_not__gr0,axiom, % 0.30/0.49 ! [N2: nat] : % 0.30/0.49 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.30/0.49 = ( N2 % 0.30/0.49 = ( zero_zero @ nat ) ) ) ). % 0.30/0.49 % 0.30/0.49 % not_gr0 % 0.30/0.49 thf(fact_420_not__less0,axiom, % 0.30/0.49 ! [N2: nat] : % 0.30/0.49 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ). % 0.30/0.49 % 0.30/0.49 % not_less0 % 0.30/0.49 thf(fact_421_less__zeroE,axiom, % 0.30/0.49 ! [N2: nat] : % 0.30/0.49 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ). % 0.30/0.49 % 0.30/0.49 % less_zeroE % 0.30/0.49 thf(fact_422_gr__implies__not0,axiom, % 0.30/0.49 ! [M: nat,N2: nat] : % 0.30/0.49 ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.49 => ( N2 % 0.30/0.49 != ( zero_zero @ nat ) ) ) ). % 0.30/0.49 % 0.30/0.49 % gr_implies_not0 % 0.30/0.49 thf(fact_423_infinite__descent0,axiom, % 0.30/0.49 ! [P: nat > $o,N2: nat] : % 0.30/0.49 ( ( P @ ( zero_zero @ nat ) ) % 0.30/0.49 => ( ! [N3: nat] : % 0.30/0.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 0.30/0.49 => ( ~ ( P @ N3 ) % 0.30/0.49 => ? [M6: nat] : % 0.30/0.49 ( ( ord_less @ nat @ M6 @ N3 ) % 0.30/0.49 & ~ ( P @ M6 ) ) ) ) % 0.30/0.49 => ( P @ N2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % infinite_descent0 % 0.30/0.49 thf(fact_424_infinite__descent0__measure,axiom, % 0.30/0.49 ! [A: $tType,V: A > nat,P: A > $o,X3: A] : % 0.30/0.49 ( ! [X5: A] : % 0.30/0.49 ( ( ( V @ X5 ) % 0.30/0.49 = ( zero_zero @ nat ) ) % 0.30/0.49 => ( P @ X5 ) ) % 0.30/0.49 => ( ! [X5: A] : % 0.30/0.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X5 ) ) % 0.30/0.49 => ( ~ ( P @ X5 ) % 0.30/0.49 => ? [Y5: A] : % 0.30/0.49 ( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X5 ) ) % 0.30/0.49 & ~ ( P @ Y5 ) ) ) ) % 0.30/0.49 => ( P @ X3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % infinite_descent0_measure % 0.30/0.49 thf(fact_425_less__eq__nat_Osimps_I1_J,axiom, % 0.30/0.49 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ). % 0.30/0.49 % 0.30/0.49 % less_eq_nat.simps(1) % 0.30/0.49 thf(fact_426_bot__nat__0_Oextremum__unique,axiom, % 0.30/0.49 ! [A2: nat] : % 0.30/0.49 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) ) % 0.30/0.49 = ( A2 % 0.30/0.49 = ( zero_zero @ nat ) ) ) ). % 0.30/0.49 % 0.30/0.49 % bot_nat_0.extremum_unique % 0.30/0.49 thf(fact_427_bot__nat__0_Oextremum__uniqueI,axiom, % 0.30/0.49 ! [A2: nat] : % 0.30/0.49 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) ) % 0.30/0.49 => ( A2 % 0.30/0.49 = ( zero_zero @ nat ) ) ) ). % 0.30/0.49 % 0.30/0.49 % bot_nat_0.extremum_uniqueI % 0.30/0.49 thf(fact_428_le__0__eq,axiom, % 0.30/0.49 ! [N2: nat] : % 0.30/0.49 ( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) ) % 0.30/0.49 = ( N2 % 0.30/0.49 = ( zero_zero @ nat ) ) ) ). % 0.30/0.49 % 0.30/0.49 % le_0_eq % 0.30/0.49 thf(fact_429_nat__less__le,axiom, % 0.30/0.49 ( ( ord_less @ nat ) % 0.30/0.49 = ( ^ [M2: nat,N: nat] : % 0.30/0.49 ( ( ord_less_eq @ nat @ M2 @ N ) % 0.30/0.49 & ( M2 != N ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % nat_less_le % 0.30/0.49 thf(fact_430_less__imp__le__nat,axiom, % 0.30/0.49 ! [M: nat,N2: nat] : % 0.30/0.49 ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.49 => ( ord_less_eq @ nat @ M @ N2 ) ) ). % 0.30/0.49 % 0.30/0.49 % less_imp_le_nat % 0.30/0.49 thf(fact_431_le__eq__less__or__eq,axiom, % 0.30/0.49 ( ( ord_less_eq @ nat ) % 0.30/0.49 = ( ^ [M2: nat,N: nat] : % 0.30/0.49 ( ( ord_less @ nat @ M2 @ N ) % 0.30/0.49 | ( M2 = N ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % le_eq_less_or_eq % 0.30/0.49 thf(fact_432_less__or__eq__imp__le,axiom, % 0.30/0.49 ! [M: nat,N2: nat] : % 0.30/0.49 ( ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.49 | ( M = N2 ) ) % 0.30/0.49 => ( ord_less_eq @ nat @ M @ N2 ) ) ). % 0.30/0.49 % 0.30/0.49 % less_or_eq_imp_le % 0.30/0.49 thf(fact_433_le__neq__implies__less,axiom, % 0.30/0.49 ! [M: nat,N2: nat] : % 0.30/0.49 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.49 => ( ( M != N2 ) % 0.30/0.49 => ( ord_less @ nat @ M @ N2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % le_neq_implies_less % 0.30/0.49 thf(fact_434_less__mono__imp__le__mono,axiom, % 0.30/0.49 ! [F2: nat > nat,I: nat,J: nat] : % 0.30/0.49 ( ! [I4: nat,J2: nat] : % 0.30/0.49 ( ( ord_less @ nat @ I4 @ J2 ) % 0.30/0.49 => ( ord_less @ nat @ ( F2 @ I4 ) @ ( F2 @ J2 ) ) ) % 0.30/0.49 => ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.49 => ( ord_less_eq @ nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % less_mono_imp_le_mono % 0.30/0.49 thf(fact_435_double__eq__0__iff,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linord5086331880401160121up_add @ A ) % 0.30/0.49 => ! [A2: A] : % 0.30/0.49 ( ( ( plus_plus @ A @ A2 @ A2 ) % 0.30/0.49 = ( zero_zero @ A ) ) % 0.30/0.49 = ( A2 % 0.30/0.49 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % double_eq_0_iff % 0.30/0.49 thf(fact_436_assms_I1_J,axiom, % 0.30/0.49 vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ). % 0.30/0.49 % 0.30/0.49 % assms(1) % 0.30/0.49 thf(fact_437_mint__corr__help__empty,axiom, % 0.30/0.49 ! [T2: vEBT_VEBT,N2: nat] : % 0.30/0.49 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.49 => ( ( ( vEBT_vebt_mint @ T2 ) % 0.30/0.49 = ( none @ nat ) ) % 0.30/0.49 => ( ( vEBT_VEBT_set_vebt @ T2 ) % 0.30/0.49 = ( bot_bot @ ( set @ nat ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % mint_corr_help_empty % 0.30/0.49 thf(fact_438_maxt__corr__help__empty,axiom, % 0.30/0.49 ! [T2: vEBT_VEBT,N2: nat] : % 0.30/0.49 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.49 => ( ( ( vEBT_vebt_maxt @ T2 ) % 0.30/0.49 = ( none @ nat ) ) % 0.30/0.49 => ( ( vEBT_VEBT_set_vebt @ T2 ) % 0.30/0.49 = ( bot_bot @ ( set @ nat ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % maxt_corr_help_empty % 0.30/0.49 thf(fact_439_field__le__epsilon,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linordered_field @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ! [E: A] : % 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ E ) % 0.30/0.49 => ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ Y @ E ) ) ) % 0.30/0.49 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ). % 0.30/0.49 % 0.30/0.49 % field_le_epsilon % 0.30/0.49 thf(fact_440_ex__has__greatest__nat__lemma,axiom, % 0.30/0.49 ! [A: $tType,P: A > $o,K: A,F2: A > nat,N2: nat] : % 0.30/0.49 ( ( P @ K ) % 0.30/0.49 => ( ! [X5: A] : % 0.30/0.49 ( ( P @ X5 ) % 0.30/0.49 => ? [Y5: A] : % 0.30/0.49 ( ( P @ Y5 ) % 0.30/0.49 & ~ ( ord_less_eq @ nat @ ( F2 @ Y5 ) @ ( F2 @ X5 ) ) ) ) % 0.30/0.49 => ? [Y4: A] : % 0.30/0.49 ( ( P @ Y4 ) % 0.30/0.49 & ~ ( ord_less @ nat @ ( F2 @ Y4 ) @ ( plus_plus @ nat @ ( F2 @ K ) @ N2 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % ex_has_greatest_nat_lemma % 0.30/0.49 thf(fact_441_option_Osize__gen_I2_J,axiom, % 0.30/0.49 ! [A: $tType,X3: A > nat,X22: A] : % 0.30/0.49 ( ( size_option @ A @ X3 @ ( some @ A @ X22 ) ) % 0.30/0.49 = ( plus_plus @ nat @ ( X3 @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % option.size_gen(2) % 0.30/0.49 thf(fact_442_add__less__zeroD,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linordered_idom @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ( ord_less @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) % 0.30/0.49 => ( ( ord_less @ A @ X3 @ ( zero_zero @ A ) ) % 0.30/0.49 | ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % add_less_zeroD % 0.30/0.49 thf(fact_443_empty__subsetI,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ). % 0.30/0.49 % 0.30/0.49 % empty_subsetI % 0.30/0.49 thf(fact_444_subset__empty,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) % 0.30/0.49 = ( A3 % 0.30/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % subset_empty % 0.30/0.49 thf(fact_445_empty__iff,axiom, % 0.30/0.49 ! [A: $tType,C2: A] : % 0.30/0.49 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ). % 0.30/0.49 % 0.30/0.49 % empty_iff % 0.30/0.49 thf(fact_446_all__not__in__conv,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A] : % 0.30/0.49 ( ( ! [X: A] : % 0.30/0.49 ~ ( member @ A @ X @ A3 ) ) % 0.30/0.49 = ( A3 % 0.30/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % all_not_in_conv % 0.30/0.49 thf(fact_447_deg__deg__n,axiom, % 0.30/0.49 ! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat] : % 0.30/0.49 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 ) % 0.30/0.49 => ( Deg = N2 ) ) ). % 0.30/0.49 % 0.30/0.49 % deg_deg_n % 0.30/0.49 thf(fact_448_deg__SUcn__Node,axiom, % 0.30/0.49 ! [Tree: vEBT_VEBT,N2: nat] : % 0.30/0.49 ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) ) % 0.30/0.49 => ? [Info2: option @ ( product_prod @ nat @ nat ),TreeList3: list @ vEBT_VEBT,S2: vEBT_VEBT] : % 0.30/0.49 ( Tree % 0.30/0.49 = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N2 ) ) @ TreeList3 @ S2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % deg_SUcn_Node % 0.30/0.49 thf(fact_449_empty__Collect__eq,axiom, % 0.30/0.49 ! [A: $tType,P: A > $o] : % 0.30/0.49 ( ( ( bot_bot @ ( set @ A ) ) % 0.30/0.49 = ( collect @ A @ P ) ) % 0.30/0.49 = ( ! [X: A] : % 0.30/0.49 ~ ( P @ X ) ) ) ). % 0.30/0.49 % 0.30/0.49 % empty_Collect_eq % 0.30/0.49 thf(fact_450_Collect__empty__eq,axiom, % 0.30/0.49 ! [A: $tType,P: A > $o] : % 0.30/0.49 ( ( ( collect @ A @ P ) % 0.30/0.49 = ( bot_bot @ ( set @ A ) ) ) % 0.30/0.49 = ( ! [X: A] : % 0.30/0.49 ~ ( P @ X ) ) ) ). % 0.30/0.49 % 0.30/0.49 % Collect_empty_eq % 0.30/0.49 thf(fact_451_psubsetI,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.49 => ( ( A3 != B3 ) % 0.30/0.49 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % psubsetI % 0.30/0.49 thf(fact_452_subset__antisym,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 0.30/0.49 => ( A3 = B3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % subset_antisym % 0.30/0.49 thf(fact_453_subsetI,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.49 ( ! [X5: A] : % 0.30/0.49 ( ( member @ A @ X5 @ A3 ) % 0.30/0.49 => ( member @ A @ X5 @ B3 ) ) % 0.30/0.49 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ). % 0.30/0.49 % 0.30/0.49 % subsetI % 0.30/0.49 thf(fact_454_mi__eq__ma__no__ch,axiom, % 0.30/0.49 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] : % 0.30/0.49 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) % 0.30/0.49 => ( ( Mi = Ma ) % 0.30/0.49 => ( ! [X2: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 0.30/0.49 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) % 0.30/0.49 & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % mi_eq_ma_no_ch % 0.30/0.49 thf(fact_455_not__Some__eq,axiom, % 0.30/0.49 ! [A: $tType,X3: option @ A] : % 0.30/0.49 ( ( ! [Y2: A] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( some @ A @ Y2 ) ) ) % 0.30/0.49 = ( X3 % 0.30/0.49 = ( none @ A ) ) ) ). % 0.30/0.49 % 0.30/0.49 % not_Some_eq % 0.30/0.49 thf(fact_456_not__None__eq,axiom, % 0.30/0.49 ! [A: $tType,X3: option @ A] : % 0.30/0.49 ( ( X3 % 0.30/0.49 != ( none @ A ) ) % 0.30/0.49 = ( ? [Y2: A] : % 0.30/0.49 ( X3 % 0.30/0.49 = ( some @ A @ Y2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % not_None_eq % 0.30/0.49 thf(fact_457_option_Ocollapse,axiom, % 0.30/0.49 ! [A: $tType,Option: option @ A] : % 0.30/0.49 ( ( Option % 0.30/0.49 != ( none @ A ) ) % 0.30/0.49 => ( ( some @ A @ ( the2 @ A @ Option ) ) % 0.30/0.49 = Option ) ) ). % 0.30/0.49 % 0.30/0.49 % option.collapse % 0.30/0.49 thf(fact_458_add__def,axiom, % 0.30/0.49 ( vEBT_VEBT_add % 0.30/0.49 = ( vEBT_V2048590022279873568_shift @ nat @ ( plus_plus @ nat ) ) ) ). % 0.30/0.49 % 0.30/0.49 % add_def % 0.30/0.49 thf(fact_459_psubsetE,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.49 ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.49 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.49 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % psubsetE % 0.30/0.49 thf(fact_460_psubset__eq,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( ord_less @ ( set @ A ) ) % 0.30/0.49 = ( ^ [A6: set @ A,B6: set @ A] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) % 0.30/0.49 & ( A6 != B6 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % psubset_eq % 0.30/0.49 thf(fact_461_less__fun__def,axiom, % 0.30/0.49 ! [B: $tType,A: $tType] : % 0.30/0.49 ( ( ord @ B ) % 0.30/0.49 => ( ( ord_less @ ( A > B ) ) % 0.30/0.49 = ( ^ [F3: A > B,G2: A > B] : % 0.30/0.49 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 ) % 0.30/0.49 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % less_fun_def % 0.30/0.49 thf(fact_462_psubset__imp__subset,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.49 ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.49 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ). % 0.30/0.49 % 0.30/0.49 % psubset_imp_subset % 0.30/0.49 thf(fact_463_psubset__subset__trans,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.30/0.49 ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 ) % 0.30/0.49 => ( ord_less @ ( set @ A ) @ A3 @ C5 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % psubset_subset_trans % 0.30/0.49 thf(fact_464_subset__not__subset__eq,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( ord_less @ ( set @ A ) ) % 0.30/0.49 = ( ^ [A6: set @ A,B6: set @ A] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) % 0.30/0.49 & ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % subset_not_subset_eq % 0.30/0.49 thf(fact_465_subset__psubset__trans,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.49 => ( ( ord_less @ ( set @ A ) @ B3 @ C5 ) % 0.30/0.49 => ( ord_less @ ( set @ A ) @ A3 @ C5 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % subset_psubset_trans % 0.30/0.49 thf(fact_466_subset__iff__psubset__eq,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) ) % 0.30/0.49 = ( ^ [A6: set @ A,B6: set @ A] : % 0.30/0.49 ( ( ord_less @ ( set @ A ) @ A6 @ B6 ) % 0.30/0.49 | ( A6 = B6 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % subset_iff_psubset_eq % 0.30/0.49 thf(fact_467_Collect__mono__iff,axiom, % 0.30/0.49 ! [A: $tType,P: A > $o,Q: A > $o] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) % 0.30/0.49 = ( ! [X: A] : % 0.30/0.49 ( ( P @ X ) % 0.30/0.49 => ( Q @ X ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % Collect_mono_iff % 0.30/0.49 thf(fact_468_set__eq__subset,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( ^ [Y3: set @ A,Z: set @ A] : ( Y3 = Z ) ) % 0.30/0.49 = ( ^ [A6: set @ A,B6: set @ A] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) % 0.30/0.49 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % set_eq_subset % 0.30/0.49 thf(fact_469_subset__trans,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 ) % 0.30/0.49 => ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % subset_trans % 0.30/0.49 thf(fact_470_Collect__mono,axiom, % 0.30/0.49 ! [A: $tType,P: A > $o,Q: A > $o] : % 0.30/0.49 ( ! [X5: A] : % 0.30/0.49 ( ( P @ X5 ) % 0.30/0.49 => ( Q @ X5 ) ) % 0.30/0.49 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ). % 0.30/0.49 % 0.30/0.49 % Collect_mono % 0.30/0.49 thf(fact_471_subset__refl,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ). % 0.30/0.49 % 0.30/0.49 % subset_refl % 0.30/0.49 thf(fact_472_subset__iff,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) ) % 0.30/0.49 = ( ^ [A6: set @ A,B6: set @ A] : % 0.30/0.49 ! [T3: A] : % 0.30/0.49 ( ( member @ A @ T3 @ A6 ) % 0.30/0.49 => ( member @ A @ T3 @ B6 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % subset_iff % 0.30/0.49 thf(fact_473_equalityD2,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.49 ( ( A3 = B3 ) % 0.30/0.49 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ). % 0.30/0.49 % 0.30/0.49 % equalityD2 % 0.30/0.49 thf(fact_474_equalityD1,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.49 ( ( A3 = B3 ) % 0.30/0.49 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ). % 0.30/0.49 % 0.30/0.49 % equalityD1 % 0.30/0.49 thf(fact_475_subset__eq,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) ) % 0.30/0.49 = ( ^ [A6: set @ A,B6: set @ A] : % 0.30/0.49 ! [X: A] : % 0.30/0.49 ( ( member @ A @ X @ A6 ) % 0.30/0.49 => ( member @ A @ X @ B6 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % subset_eq % 0.30/0.49 thf(fact_476_equalityE,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.49 ( ( A3 = B3 ) % 0.30/0.49 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.49 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % equalityE % 0.30/0.49 thf(fact_477_subsetD,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.49 => ( ( member @ A @ C2 @ A3 ) % 0.30/0.49 => ( member @ A @ C2 @ B3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % subsetD % 0.30/0.49 thf(fact_478_in__mono,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A,X3: A] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.49 => ( ( member @ A @ X3 @ A3 ) % 0.30/0.49 => ( member @ A @ X3 @ B3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % in_mono % 0.30/0.49 thf(fact_479_bot__set__def,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( bot_bot @ ( set @ A ) ) % 0.30/0.49 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % bot_set_def % 0.30/0.49 thf(fact_480_bot__nat__def,axiom, % 0.30/0.49 ( ( bot_bot @ nat ) % 0.30/0.49 = ( zero_zero @ nat ) ) ). % 0.30/0.49 % 0.30/0.49 % bot_nat_def % 0.30/0.49 thf(fact_481_combine__options__cases,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,X3: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] : % 0.30/0.49 ( ( ( X3 % 0.30/0.49 = ( none @ A ) ) % 0.30/0.49 => ( P @ X3 @ Y ) ) % 0.30/0.49 => ( ( ( Y % 0.30/0.49 = ( none @ B ) ) % 0.30/0.49 => ( P @ X3 @ Y ) ) % 0.30/0.49 => ( ! [A5: A,B5: B] : % 0.30/0.49 ( ( X3 % 0.30/0.49 = ( some @ A @ A5 ) ) % 0.30/0.49 => ( ( Y % 0.30/0.49 = ( some @ B @ B5 ) ) % 0.30/0.49 => ( P @ X3 @ Y ) ) ) % 0.30/0.49 => ( P @ X3 @ Y ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % combine_options_cases % 0.30/0.49 thf(fact_482_split__option__all,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( ^ [P2: ( option @ A ) > $o] : % 0.30/0.49 ! [X7: option @ A] : ( P2 @ X7 ) ) % 0.30/0.49 = ( ^ [P3: ( option @ A ) > $o] : % 0.30/0.49 ( ( P3 @ ( none @ A ) ) % 0.30/0.49 & ! [X: A] : ( P3 @ ( some @ A @ X ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % split_option_all % 0.30/0.49 thf(fact_483_split__option__ex,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( ^ [P2: ( option @ A ) > $o] : % 0.30/0.49 ? [X7: option @ A] : ( P2 @ X7 ) ) % 0.30/0.49 = ( ^ [P3: ( option @ A ) > $o] : % 0.30/0.49 ( ( P3 @ ( none @ A ) ) % 0.30/0.49 | ? [X: A] : ( P3 @ ( some @ A @ X ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % split_option_ex % 0.30/0.49 thf(fact_484_option_Oexhaust,axiom, % 0.30/0.49 ! [A: $tType,Y: option @ A] : % 0.30/0.49 ( ( Y % 0.30/0.49 != ( none @ A ) ) % 0.30/0.49 => ~ ! [X23: A] : % 0.30/0.49 ( Y % 0.30/0.49 != ( some @ A @ X23 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % option.exhaust % 0.30/0.49 thf(fact_485_option_OdiscI,axiom, % 0.30/0.49 ! [A: $tType,Option: option @ A,X22: A] : % 0.30/0.49 ( ( Option % 0.30/0.49 = ( some @ A @ X22 ) ) % 0.30/0.49 => ( Option % 0.30/0.49 != ( none @ A ) ) ) ). % 0.30/0.49 % 0.30/0.49 % option.discI % 0.30/0.49 thf(fact_486_option_Odistinct_I1_J,axiom, % 0.30/0.49 ! [A: $tType,X22: A] : % 0.30/0.49 ( ( none @ A ) % 0.30/0.49 != ( some @ A @ X22 ) ) ). % 0.30/0.49 % 0.30/0.49 % option.distinct(1) % 0.30/0.49 thf(fact_487_vebt__maxt_Osimps_I3_J,axiom, % 0.30/0.49 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] : % 0.30/0.49 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) ) % 0.30/0.49 = ( some @ nat @ Ma ) ) ). % 0.30/0.49 % 0.30/0.49 % vebt_maxt.simps(3) % 0.30/0.49 thf(fact_488_vebt__mint_Osimps_I3_J,axiom, % 0.30/0.49 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] : % 0.30/0.49 ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) ) % 0.30/0.49 = ( some @ nat @ Mi ) ) ). % 0.30/0.49 % 0.30/0.49 % vebt_mint.simps(3) % 0.30/0.49 thf(fact_489_option_Oexpand,axiom, % 0.30/0.49 ! [A: $tType,Option: option @ A,Option2: option @ A] : % 0.30/0.49 ( ( ( Option % 0.30/0.49 = ( none @ A ) ) % 0.30/0.49 = ( Option2 % 0.30/0.49 = ( none @ A ) ) ) % 0.30/0.49 => ( ( ( Option % 0.30/0.49 != ( none @ A ) ) % 0.30/0.49 => ( ( Option2 % 0.30/0.49 != ( none @ A ) ) % 0.30/0.49 => ( ( the2 @ A @ Option ) % 0.30/0.49 = ( the2 @ A @ Option2 ) ) ) ) % 0.30/0.49 => ( Option = Option2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % option.expand % 0.30/0.49 thf(fact_490_option_Osize_I3_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( size_size @ ( option @ A ) @ ( none @ A ) ) % 0.30/0.49 = ( suc @ ( zero_zero @ nat ) ) ) ). % 0.30/0.49 % 0.30/0.49 % option.size(3) % 0.30/0.49 thf(fact_491_option_Osize__gen_I1_J,axiom, % 0.30/0.49 ! [A: $tType,X3: A > nat] : % 0.30/0.49 ( ( size_option @ A @ X3 @ ( none @ A ) ) % 0.30/0.49 = ( suc @ ( zero_zero @ nat ) ) ) ). % 0.30/0.49 % 0.30/0.49 % option.size_gen(1) % 0.30/0.49 thf(fact_492_option_Oexhaust__sel,axiom, % 0.30/0.49 ! [A: $tType,Option: option @ A] : % 0.30/0.49 ( ( Option % 0.30/0.49 != ( none @ A ) ) % 0.30/0.49 => ( Option % 0.30/0.49 = ( some @ A @ ( the2 @ A @ Option ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % option.exhaust_sel % 0.30/0.49 thf(fact_493_option_Osize__neq,axiom, % 0.30/0.49 ! [A: $tType,X3: option @ A] : % 0.30/0.49 ( ( size_size @ ( option @ A ) @ X3 ) % 0.30/0.49 != ( zero_zero @ nat ) ) ). % 0.30/0.49 % 0.30/0.49 % option.size_neq % 0.30/0.49 thf(fact_494_linorder__neqE__linordered__idom,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linordered_idom @ A ) % 0.30/0.49 => ! [X3: A,Y: A] : % 0.30/0.49 ( ( X3 != Y ) % 0.30/0.49 => ( ~ ( ord_less @ A @ X3 @ Y ) % 0.30/0.49 => ( ord_less @ A @ Y @ X3 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % linorder_neqE_linordered_idom % 0.30/0.49 thf(fact_495_linordered__field__no__ub,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linordered_field @ A ) % 0.30/0.49 => ! [X2: A] : % 0.30/0.49 ? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ). % 0.30/0.49 % 0.30/0.49 % linordered_field_no_ub % 0.30/0.49 thf(fact_496_linordered__field__no__lb,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linordered_field @ A ) % 0.30/0.49 => ! [X2: A] : % 0.30/0.49 ? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ). % 0.30/0.49 % 0.30/0.49 % linordered_field_no_lb % 0.30/0.49 thf(fact_497_not__psubset__empty,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A] : % 0.30/0.49 ~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ). % 0.30/0.49 % 0.30/0.49 % not_psubset_empty % 0.30/0.49 thf(fact_498_ex__in__conv,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A] : % 0.30/0.49 ( ( ? [X: A] : ( member @ A @ X @ A3 ) ) % 0.30/0.49 = ( A3 % 0.30/0.49 != ( bot_bot @ ( set @ A ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % ex_in_conv % 0.30/0.49 thf(fact_499_equals0I,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A] : % 0.30/0.49 ( ! [Y4: A] : % 0.30/0.49 ~ ( member @ A @ Y4 @ A3 ) % 0.30/0.49 => ( A3 % 0.30/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % equals0I % 0.30/0.49 thf(fact_500_equals0D,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,A2: A] : % 0.30/0.49 ( ( A3 % 0.30/0.49 = ( bot_bot @ ( set @ A ) ) ) % 0.30/0.49 => ~ ( member @ A @ A2 @ A3 ) ) ). % 0.30/0.49 % 0.30/0.49 % equals0D % 0.30/0.49 thf(fact_501_emptyE,axiom, % 0.30/0.49 ! [A: $tType,A2: A] : % 0.30/0.49 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ). % 0.30/0.49 % 0.30/0.49 % emptyE % 0.30/0.49 thf(fact_502_ex__has__least__nat,axiom, % 0.30/0.49 ! [A: $tType,P: A > $o,K: A,M: A > nat] : % 0.30/0.49 ( ( P @ K ) % 0.30/0.49 => ? [X5: A] : % 0.30/0.49 ( ( P @ X5 ) % 0.30/0.49 & ! [Y5: A] : % 0.30/0.49 ( ( P @ Y5 ) % 0.30/0.49 => ( ord_less_eq @ nat @ ( M @ X5 ) @ ( M @ Y5 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % ex_has_least_nat % 0.30/0.49 thf(fact_503_Lattices__Big_Oex__has__greatest__nat,axiom, % 0.30/0.49 ! [A: $tType,P: A > $o,K: A,F2: A > nat,B2: nat] : % 0.30/0.49 ( ( P @ K ) % 0.30/0.49 => ( ! [Y4: A] : % 0.30/0.49 ( ( P @ Y4 ) % 0.30/0.49 => ( ord_less @ nat @ ( F2 @ Y4 ) @ B2 ) ) % 0.30/0.49 => ? [X5: A] : % 0.30/0.49 ( ( P @ X5 ) % 0.30/0.49 & ! [Y5: A] : % 0.30/0.49 ( ( P @ Y5 ) % 0.30/0.49 => ( ord_less_eq @ nat @ ( F2 @ Y5 ) @ ( F2 @ X5 ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % Lattices_Big.ex_has_greatest_nat % 0.30/0.49 thf(fact_504_option_Osize_I4_J,axiom, % 0.30/0.49 ! [A: $tType,X22: A] : % 0.30/0.49 ( ( size_size @ ( option @ A ) @ ( some @ A @ X22 ) ) % 0.30/0.49 = ( suc @ ( zero_zero @ nat ) ) ) ). % 0.30/0.49 % 0.30/0.49 % option.size(4) % 0.30/0.49 thf(fact_505_VEBT__internal_Omembermima_Osimps_I3_J,axiom, % 0.30/0.49 ! [Mi: nat,Ma: nat,Va: list @ vEBT_VEBT,Vb: vEBT_VEBT,X3: nat] : % 0.30/0.49 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ Va @ Vb ) @ X3 ) % 0.30/0.49 = ( ( X3 = Mi ) % 0.30/0.49 | ( X3 = Ma ) ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.membermima.simps(3) % 0.30/0.49 thf(fact_506_vebt__member_Osimps_I4_J,axiom, % 0.30/0.49 ! [V2: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] : % 0.30/0.49 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X3 ) ). % 0.30/0.49 % 0.30/0.49 % vebt_member.simps(4) % 0.30/0.49 thf(fact_507_vebt__member_Osimps_I3_J,axiom, % 0.30/0.49 ! [V2: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X3: nat] : % 0.30/0.49 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X3 ) ). % 0.30/0.49 % 0.30/0.49 % vebt_member.simps(3) % 0.30/0.49 thf(fact_508_VEBT__internal_OminNull_Osimps_I5_J,axiom, % 0.30/0.49 ! [Uz: product_prod @ nat @ nat,Va: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] : % 0.30/0.49 ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va @ Vb @ Vc ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.minNull.simps(5) % 0.30/0.49 thf(fact_509_VEBT__internal_Onaive__member_Osimps_I2_J,axiom, % 0.30/0.49 ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : % 0.30/0.49 ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Ux ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.naive_member.simps(2) % 0.30/0.49 thf(fact_510_VEBT_Oinject_I1_J,axiom, % 0.30/0.49 ! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,Y11: option @ ( product_prod @ nat @ nat ),Y12: nat,Y13: list @ vEBT_VEBT,Y14: vEBT_VEBT] : % 0.30/0.49 ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) % 0.30/0.49 = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) ) % 0.30/0.49 = ( ( X11 = Y11 ) % 0.30/0.49 & ( X12 = Y12 ) % 0.30/0.49 & ( X13 = Y13 ) % 0.30/0.49 & ( X14 = Y14 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT.inject(1) % 0.30/0.49 thf(fact_511_old_Oprod_Oinject,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] : % 0.30/0.49 ( ( ( product_Pair @ A @ B @ A2 @ B2 ) % 0.30/0.49 = ( product_Pair @ A @ B @ A7 @ B7 ) ) % 0.30/0.49 = ( ( A2 = A7 ) % 0.30/0.49 & ( B2 = B7 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % old.prod.inject % 0.30/0.49 thf(fact_512_prod_Oinject,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] : % 0.30/0.49 ( ( ( product_Pair @ A @ B @ X1 @ X22 ) % 0.30/0.49 = ( product_Pair @ A @ B @ Y1 @ Y22 ) ) % 0.30/0.49 = ( ( X1 = Y1 ) % 0.30/0.49 & ( X22 = Y22 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % prod.inject % 0.30/0.49 thf(fact_513_nth__enumerate__eq,axiom, % 0.30/0.49 ! [A: $tType,M: nat,Xs: list @ A,N2: nat] : % 0.30/0.49 ( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.49 => ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) @ M ) % 0.30/0.49 = ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N2 @ M ) @ ( nth @ A @ Xs @ M ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % nth_enumerate_eq % 0.30/0.49 thf(fact_514_set__vebt__finite,axiom, % 0.30/0.49 ! [T2: vEBT_VEBT,N2: nat] : % 0.30/0.49 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.49 => ( finite_finite2 @ nat @ ( vEBT_VEBT_set_vebt @ T2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % set_vebt_finite % 0.30/0.49 thf(fact_515_Euclid__induct,axiom, % 0.30/0.49 ! [P: nat > nat > $o,A2: nat,B2: nat] : % 0.30/0.49 ( ! [A5: nat,B5: nat] : % 0.30/0.49 ( ( P @ A5 @ B5 ) % 0.30/0.49 = ( P @ B5 @ A5 ) ) % 0.30/0.49 => ( ! [A5: nat] : ( P @ A5 @ ( zero_zero @ nat ) ) % 0.30/0.49 => ( ! [A5: nat,B5: nat] : % 0.30/0.49 ( ( P @ A5 @ B5 ) % 0.30/0.49 => ( P @ A5 @ ( plus_plus @ nat @ A5 @ B5 ) ) ) % 0.30/0.49 => ( P @ A2 @ B2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % Euclid_induct % 0.30/0.49 thf(fact_516_List_Ofinite__set,axiom, % 0.30/0.49 ! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ). % 0.30/0.49 % 0.30/0.49 % List.finite_set % 0.30/0.49 thf(fact_517_length__enumerate,axiom, % 0.30/0.49 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.30/0.49 ( ( size_size @ ( list @ ( product_prod @ nat @ A ) ) @ ( enumerate @ A @ N2 @ Xs ) ) % 0.30/0.49 = ( size_size @ ( list @ A ) @ Xs ) ) ). % 0.30/0.49 % 0.30/0.49 % length_enumerate % 0.30/0.49 thf(fact_518_psubset__trans,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.30/0.49 ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.49 => ( ( ord_less @ ( set @ A ) @ B3 @ C5 ) % 0.30/0.49 => ( ord_less @ ( set @ A ) @ A3 @ C5 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % psubset_trans % 0.30/0.49 thf(fact_519_psubsetD,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] : % 0.30/0.49 ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.49 => ( ( member @ A @ C2 @ A3 ) % 0.30/0.49 => ( member @ A @ C2 @ B3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % psubsetD % 0.30/0.49 thf(fact_520_VEBT__internal_Ooption__shift_Ocases,axiom, % 0.30/0.49 ! [A: $tType,X3: product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] : % 0.30/0.49 ( ! [Uu2: A > A > A,Uv2: option @ A] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) ) % 0.30/0.49 => ( ! [Uw2: A > A > A,V3: A] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) ) % 0.30/0.49 => ~ ! [F4: A > A > A,A5: A,B5: A] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F4 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ A5 ) @ ( some @ A @ B5 ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.option_shift.cases % 0.30/0.49 thf(fact_521_VEBT__internal_Ooption__comp__shift_Ocases,axiom, % 0.30/0.49 ! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] : % 0.30/0.49 ( ! [Uu2: A > A > $o,Uv2: option @ A] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) ) % 0.30/0.49 => ( ! [Uw2: A > A > $o,V3: A] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) ) % 0.30/0.49 => ~ ! [F4: A > A > $o,X5: A,Y4: A] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F4 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X5 ) @ ( some @ A @ Y4 ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.option_comp_shift.cases % 0.30/0.49 thf(fact_522_finite__list,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A] : % 0.30/0.49 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.49 => ? [Xs3: list @ A] : % 0.30/0.49 ( ( set2 @ A @ Xs3 ) % 0.30/0.49 = A3 ) ) ). % 0.30/0.49 % 0.30/0.49 % finite_list % 0.30/0.49 thf(fact_523_vebt__member_Osimps_I2_J,axiom, % 0.30/0.49 ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X3: nat] : % 0.30/0.49 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X3 ) ). % 0.30/0.49 % 0.30/0.49 % vebt_member.simps(2) % 0.30/0.49 thf(fact_524_VEBT__internal_OminNull_Osimps_I4_J,axiom, % 0.30/0.49 ! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.minNull.simps(4) % 0.30/0.49 thf(fact_525_ex__min__if__finite,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [S3: set @ A] : % 0.30/0.49 ( ( finite_finite2 @ A @ S3 ) % 0.30/0.49 => ( ( S3 % 0.30/0.49 != ( bot_bot @ ( set @ A ) ) ) % 0.30/0.49 => ? [X5: A] : % 0.30/0.49 ( ( member @ A @ X5 @ S3 ) % 0.30/0.49 & ~ ? [Xa: A] : % 0.30/0.49 ( ( member @ A @ Xa @ S3 ) % 0.30/0.49 & ( ord_less @ A @ Xa @ X5 ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % ex_min_if_finite % 0.30/0.49 thf(fact_526_infinite__growing,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [X8: set @ A] : % 0.30/0.49 ( ( X8 % 0.30/0.49 != ( bot_bot @ ( set @ A ) ) ) % 0.30/0.49 => ( ! [X5: A] : % 0.30/0.49 ( ( member @ A @ X5 @ X8 ) % 0.30/0.49 => ? [Xa: A] : % 0.30/0.49 ( ( member @ A @ Xa @ X8 ) % 0.30/0.49 & ( ord_less @ A @ X5 @ Xa ) ) ) % 0.30/0.49 => ~ ( finite_finite2 @ A @ X8 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % infinite_growing % 0.30/0.49 thf(fact_527_VEBT__internal_Omembermima_Osimps_I2_J,axiom, % 0.30/0.49 ! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : % 0.30/0.49 ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Uz ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.membermima.simps(2) % 0.30/0.49 thf(fact_528_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom, % 0.30/0.49 ! [A: $tType,F2: A > A > A,A2: A,B2: A] : % 0.30/0.49 ( ( vEBT_V2048590022279873568_shift @ A @ F2 @ ( some @ A @ A2 ) @ ( some @ A @ B2 ) ) % 0.30/0.49 = ( some @ A @ ( F2 @ A2 @ B2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.option_shift.simps(3) % 0.30/0.49 thf(fact_529_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom, % 0.30/0.49 ! [A: $tType,Uu: A > A > A,Uv: option @ A] : % 0.30/0.49 ( ( vEBT_V2048590022279873568_shift @ A @ Uu @ ( none @ A ) @ Uv ) % 0.30/0.49 = ( none @ A ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.option_shift.simps(1) % 0.30/0.49 thf(fact_530_old_Oprod_Oexhaust,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] : % 0.30/0.49 ~ ! [A5: A,B5: B] : % 0.30/0.49 ( Y % 0.30/0.49 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ). % 0.30/0.49 % 0.30/0.49 % old.prod.exhaust % 0.30/0.49 thf(fact_531_surj__pair,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,P4: product_prod @ A @ B] : % 0.30/0.49 ? [X5: A,Y4: B] : % 0.30/0.49 ( P4 % 0.30/0.49 = ( product_Pair @ A @ B @ X5 @ Y4 ) ) ). % 0.30/0.49 % 0.30/0.49 % surj_pair % 0.30/0.49 thf(fact_532_prod__cases,axiom, % 0.30/0.49 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] : % 0.30/0.49 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) ) % 0.30/0.49 => ( P @ P4 ) ) ). % 0.30/0.49 % 0.30/0.49 % prod_cases % 0.30/0.49 thf(fact_533_Pair__inject,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] : % 0.30/0.49 ( ( ( product_Pair @ A @ B @ A2 @ B2 ) % 0.30/0.49 = ( product_Pair @ A @ B @ A7 @ B7 ) ) % 0.30/0.49 => ~ ( ( A2 = A7 ) % 0.30/0.49 => ( B2 != B7 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % Pair_inject % 0.30/0.49 thf(fact_534_prod__cases3,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] : % 0.30/0.49 ~ ! [A5: A,B5: B,C3: C] : % 0.30/0.49 ( Y % 0.30/0.49 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % prod_cases3 % 0.30/0.49 thf(fact_535_prod__cases4,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] : % 0.30/0.49 ~ ! [A5: A,B5: B,C3: C,D3: D] : % 0.30/0.49 ( Y % 0.30/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D3 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % prod_cases4 % 0.30/0.49 thf(fact_536_prod__cases5,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] : % 0.30/0.49 ~ ! [A5: A,B5: B,C3: C,D3: D,E: E2] : % 0.30/0.49 ( Y % 0.30/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D3 @ E ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % prod_cases5 % 0.30/0.49 thf(fact_537_prod__cases6,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) )] : % 0.30/0.49 ~ ! [A5: A,B5: B,C3: C,D3: D,E: E2,F4: F] : % 0.30/0.49 ( Y % 0.30/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F ) @ D3 @ ( product_Pair @ E2 @ F @ E @ F4 ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % prod_cases6 % 0.30/0.49 thf(fact_538_prod__cases7,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) )] : % 0.30/0.49 ~ ! [A5: A,B5: B,C3: C,D3: D,E: E2,F4: F,G4: G3] : % 0.30/0.49 ( Y % 0.30/0.49 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F @ G3 ) @ E @ ( product_Pair @ F @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % prod_cases7 % 0.30/0.49 thf(fact_539_prod__induct3,axiom, % 0.30/0.49 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] : % 0.30/0.49 ( ! [A5: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) % 0.30/0.49 => ( P @ X3 ) ) ). % 0.30/0.49 % 0.30/0.49 % prod_induct3 % 0.30/0.49 thf(fact_540_prod__induct4,axiom, % 0.30/0.49 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] : % 0.30/0.49 ( ! [A5: A,B5: B,C3: C,D3: 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 @ C3 @ D3 ) ) ) ) % 0.30/0.49 => ( P @ X3 ) ) ). % 0.30/0.49 % 0.30/0.49 % prod_induct4 % 0.30/0.49 thf(fact_541_prod__induct5,axiom, % 0.30/0.49 ! [E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] : % 0.30/0.49 ( ! [A5: A,B5: B,C3: C,D3: D,E: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D3 @ E ) ) ) ) ) % 0.30/0.49 => ( P @ X3 ) ) ). % 0.30/0.49 % 0.30/0.49 % prod_induct5 % 0.30/0.49 thf(fact_542_prod__induct6,axiom, % 0.30/0.49 ! [F: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) )] : % 0.30/0.49 ( ! [A5: A,B5: B,C3: C,D3: D,E: E2,F4: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F ) @ D3 @ ( product_Pair @ E2 @ F @ E @ F4 ) ) ) ) ) ) % 0.30/0.49 => ( P @ X3 ) ) ). % 0.30/0.49 % 0.30/0.49 % prod_induct6 % 0.30/0.49 thf(fact_543_prod__induct7,axiom, % 0.30/0.49 ! [G3: $tType,F: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) )] : % 0.30/0.49 ( ! [A5: A,B5: B,C3: C,D3: D,E: E2,F4: F,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F @ G3 ) @ E @ ( product_Pair @ F @ G3 @ F4 @ G4 ) ) ) ) ) ) ) % 0.30/0.49 => ( P @ X3 ) ) ). % 0.30/0.49 % 0.30/0.49 % prod_induct7 % 0.30/0.49 thf(fact_544_vebt__maxt_Osimps_I2_J,axiom, % 0.30/0.49 ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] : % 0.30/0.49 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) % 0.30/0.49 = ( none @ nat ) ) ). % 0.30/0.49 % 0.30/0.49 % vebt_maxt.simps(2) % 0.30/0.49 thf(fact_545_vebt__mint_Osimps_I2_J,axiom, % 0.30/0.49 ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] : % 0.30/0.49 ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) % 0.30/0.49 = ( none @ nat ) ) ). % 0.30/0.49 % 0.30/0.49 % vebt_mint.simps(2) % 0.30/0.49 thf(fact_546_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom, % 0.30/0.49 ! [A: $tType,Uw: A > A > A,V2: A] : % 0.30/0.49 ( ( vEBT_V2048590022279873568_shift @ A @ Uw @ ( some @ A @ V2 ) @ ( none @ A ) ) % 0.30/0.49 = ( none @ A ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.option_shift.simps(2) % 0.30/0.49 thf(fact_547_VEBT__internal_Ooption__shift_Oelims,axiom, % 0.30/0.49 ! [A: $tType,X3: A > A > A,Xa2: option @ A,Xb: option @ A,Y: option @ A] : % 0.30/0.49 ( ( ( vEBT_V2048590022279873568_shift @ A @ X3 @ Xa2 @ Xb ) % 0.30/0.49 = Y ) % 0.30/0.49 => ( ( ( Xa2 % 0.30/0.49 = ( none @ A ) ) % 0.30/0.49 => ( Y % 0.30/0.49 != ( none @ A ) ) ) % 0.30/0.49 => ( ( ? [V3: A] : % 0.30/0.49 ( Xa2 % 0.30/0.49 = ( some @ A @ V3 ) ) % 0.30/0.49 => ( ( Xb % 0.30/0.49 = ( none @ A ) ) % 0.30/0.49 => ( Y % 0.30/0.49 != ( none @ A ) ) ) ) % 0.30/0.49 => ~ ! [A5: A] : % 0.30/0.49 ( ( Xa2 % 0.30/0.49 = ( some @ A @ A5 ) ) % 0.30/0.49 => ! [B5: A] : % 0.30/0.49 ( ( Xb % 0.30/0.49 = ( some @ A @ B5 ) ) % 0.30/0.49 => ( Y % 0.30/0.49 != ( some @ A @ ( X3 @ A5 @ B5 ) ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.option_shift.elims % 0.30/0.49 thf(fact_548_vebt__buildup_Ocases,axiom, % 0.30/0.49 ! [X3: nat] : % 0.30/0.49 ( ( X3 % 0.30/0.49 != ( zero_zero @ nat ) ) % 0.30/0.49 => ( ( X3 % 0.30/0.49 != ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.49 => ~ ! [Va2: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( suc @ ( suc @ Va2 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % vebt_buildup.cases % 0.30/0.49 thf(fact_549_old_Oprod_Orec,axiom, % 0.30/0.49 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] : % 0.30/0.49 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) % 0.30/0.49 = ( F1 @ A2 @ B2 ) ) ). % 0.30/0.49 % 0.30/0.49 % old.prod.rec % 0.30/0.49 thf(fact_550_finite__has__minimal,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [A3: set @ A] : % 0.30/0.49 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.49 => ( ( A3 % 0.30/0.49 != ( bot_bot @ ( set @ A ) ) ) % 0.30/0.49 => ? [X5: A] : % 0.30/0.49 ( ( member @ A @ X5 @ A3 ) % 0.30/0.49 & ! [Xa: A] : % 0.30/0.49 ( ( member @ A @ Xa @ A3 ) % 0.30/0.49 => ( ( ord_less_eq @ A @ Xa @ X5 ) % 0.30/0.49 => ( X5 = Xa ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % finite_has_minimal % 0.30/0.49 thf(fact_551_finite__has__maximal,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [A3: set @ A] : % 0.30/0.49 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.49 => ( ( A3 % 0.30/0.49 != ( bot_bot @ ( set @ A ) ) ) % 0.30/0.49 => ? [X5: A] : % 0.30/0.49 ( ( member @ A @ X5 @ A3 ) % 0.30/0.49 & ! [Xa: A] : % 0.30/0.49 ( ( member @ A @ Xa @ A3 ) % 0.30/0.49 => ( ( ord_less_eq @ A @ X5 @ Xa ) % 0.30/0.49 => ( X5 = Xa ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % finite_has_maximal % 0.30/0.49 thf(fact_552_finite__code,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( finite_finite @ A ) % 0.30/0.49 => ( ( finite_finite2 @ A ) % 0.30/0.49 = ( ^ [A6: set @ A] : $true ) ) ) ). % 0.30/0.49 % 0.30/0.49 % finite_code % 0.30/0.49 thf(fact_553_bot__empty__eq,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( bot_bot @ ( A > $o ) ) % 0.30/0.49 = ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % bot_empty_eq % 0.30/0.49 thf(fact_554_Collect__empty__eq__bot,axiom, % 0.30/0.49 ! [A: $tType,P: A > $o] : % 0.30/0.49 ( ( ( collect @ A @ P ) % 0.30/0.49 = ( bot_bot @ ( set @ A ) ) ) % 0.30/0.49 = ( P % 0.30/0.49 = ( bot_bot @ ( A > $o ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % Collect_empty_eq_bot % 0.30/0.49 thf(fact_555_infinite__nat__iff__unbounded__le,axiom, % 0.30/0.49 ! [S3: set @ nat] : % 0.30/0.49 ( ( ~ ( finite_finite2 @ nat @ S3 ) ) % 0.30/0.49 = ( ! [M2: nat] : % 0.30/0.49 ? [N: nat] : % 0.30/0.49 ( ( ord_less_eq @ nat @ M2 @ N ) % 0.30/0.49 & ( member @ nat @ N @ S3 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % infinite_nat_iff_unbounded_le % 0.30/0.49 thf(fact_556_finite__nat__set__iff__bounded__le,axiom, % 0.30/0.49 ( ( finite_finite2 @ nat ) % 0.30/0.49 = ( ^ [N5: set @ nat] : % 0.30/0.49 ? [M2: nat] : % 0.30/0.49 ! [X: nat] : % 0.30/0.49 ( ( member @ nat @ X @ N5 ) % 0.30/0.49 => ( ord_less_eq @ nat @ X @ M2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % finite_nat_set_iff_bounded_le % 0.30/0.49 thf(fact_557_unbounded__k__infinite,axiom, % 0.30/0.49 ! [K: nat,S3: set @ nat] : % 0.30/0.49 ( ! [M3: nat] : % 0.30/0.49 ( ( ord_less @ nat @ K @ M3 ) % 0.30/0.49 => ? [N6: nat] : % 0.30/0.49 ( ( ord_less @ nat @ M3 @ N6 ) % 0.30/0.49 & ( member @ nat @ N6 @ S3 ) ) ) % 0.30/0.49 => ~ ( finite_finite2 @ nat @ S3 ) ) ). % 0.30/0.49 % 0.30/0.49 % unbounded_k_infinite % 0.30/0.49 thf(fact_558_bounded__nat__set__is__finite,axiom, % 0.30/0.49 ! [N7: set @ nat,N2: nat] : % 0.30/0.49 ( ! [X5: nat] : % 0.30/0.49 ( ( member @ nat @ X5 @ N7 ) % 0.30/0.49 => ( ord_less @ nat @ X5 @ N2 ) ) % 0.30/0.49 => ( finite_finite2 @ nat @ N7 ) ) ). % 0.30/0.49 % 0.30/0.49 % bounded_nat_set_is_finite % 0.30/0.49 thf(fact_559_finite__maxlen,axiom, % 0.30/0.49 ! [A: $tType,M7: set @ ( list @ A )] : % 0.30/0.49 ( ( finite_finite2 @ ( list @ A ) @ M7 ) % 0.30/0.49 => ? [N3: nat] : % 0.30/0.49 ! [X2: list @ A] : % 0.30/0.49 ( ( member @ ( list @ A ) @ X2 @ M7 ) % 0.30/0.49 => ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X2 ) @ N3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % finite_maxlen % 0.30/0.49 thf(fact_560_subrelI,axiom, % 0.30/0.49 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] : % 0.30/0.49 ( ! [X5: A,Y4: B] : % 0.30/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ R2 ) % 0.30/0.49 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ S ) ) % 0.30/0.49 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ). % 0.30/0.49 % 0.30/0.49 % subrelI % 0.30/0.49 thf(fact_561_finite__set__choice,axiom, % 0.30/0.49 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] : % 0.30/0.49 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.49 => ( ! [X5: A] : % 0.30/0.49 ( ( member @ A @ X5 @ A3 ) % 0.30/0.49 => ? [X_12: B] : ( P @ X5 @ X_12 ) ) % 0.30/0.49 => ? [F4: A > B] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( member @ A @ X2 @ A3 ) % 0.30/0.49 => ( P @ X2 @ ( F4 @ X2 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % finite_set_choice % 0.30/0.49 thf(fact_562_finite,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( finite_finite @ A ) % 0.30/0.49 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ). % 0.30/0.49 % 0.30/0.49 % finite % 0.30/0.49 thf(fact_563_bounded__Max__nat,axiom, % 0.30/0.49 ! [P: nat > $o,X3: nat,M7: nat] : % 0.30/0.49 ( ( P @ X3 ) % 0.30/0.49 => ( ! [X5: nat] : % 0.30/0.49 ( ( P @ X5 ) % 0.30/0.49 => ( ord_less_eq @ nat @ X5 @ M7 ) ) % 0.30/0.49 => ~ ! [M3: nat] : % 0.30/0.49 ( ( P @ M3 ) % 0.30/0.49 => ~ ! [X2: nat] : % 0.30/0.49 ( ( P @ X2 ) % 0.30/0.49 => ( ord_less_eq @ nat @ X2 @ M3 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % bounded_Max_nat % 0.30/0.49 thf(fact_564_fold__atLeastAtMost__nat_Ocases,axiom, % 0.30/0.49 ! [A: $tType,X3: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] : % 0.30/0.49 ~ ! [F4: nat > A > A,A5: nat,B5: nat,Acc: A] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F4 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A5 @ ( product_Pair @ nat @ A @ B5 @ Acc ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % fold_atLeastAtMost_nat.cases % 0.30/0.49 thf(fact_565_finite__has__minimal2,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [A3: set @ A,A2: A] : % 0.30/0.49 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.49 => ( ( member @ A @ A2 @ A3 ) % 0.30/0.49 => ? [X5: A] : % 0.30/0.49 ( ( member @ A @ X5 @ A3 ) % 0.30/0.49 & ( ord_less_eq @ A @ X5 @ A2 ) % 0.30/0.49 & ! [Xa: A] : % 0.30/0.49 ( ( member @ A @ Xa @ A3 ) % 0.30/0.49 => ( ( ord_less_eq @ A @ Xa @ X5 ) % 0.30/0.49 => ( X5 = Xa ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % finite_has_minimal2 % 0.30/0.49 thf(fact_566_finite__has__maximal2,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [A3: set @ A,A2: A] : % 0.30/0.49 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.49 => ( ( member @ A @ A2 @ A3 ) % 0.30/0.49 => ? [X5: A] : % 0.30/0.49 ( ( member @ A @ X5 @ A3 ) % 0.30/0.49 & ( ord_less_eq @ A @ A2 @ X5 ) % 0.30/0.49 & ! [Xa: A] : % 0.30/0.49 ( ( member @ A @ Xa @ A3 ) % 0.30/0.49 => ( ( ord_less_eq @ A @ X5 @ Xa ) % 0.30/0.49 => ( X5 = Xa ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % finite_has_maximal2 % 0.30/0.49 thf(fact_567_finite__transitivity__chain,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,R: A > A > $o] : % 0.30/0.49 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.49 => ( ! [X5: A] : % 0.30/0.49 ~ ( R @ X5 @ X5 ) % 0.30/0.49 => ( ! [X5: A,Y4: A,Z3: A] : % 0.30/0.49 ( ( R @ X5 @ Y4 ) % 0.30/0.49 => ( ( R @ Y4 @ Z3 ) % 0.30/0.49 => ( R @ X5 @ Z3 ) ) ) % 0.30/0.49 => ( ! [X5: A] : % 0.30/0.49 ( ( member @ A @ X5 @ A3 ) % 0.30/0.49 => ? [Y5: A] : % 0.30/0.49 ( ( member @ A @ Y5 @ A3 ) % 0.30/0.49 & ( R @ X5 @ Y5 ) ) ) % 0.30/0.49 => ( A3 % 0.30/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % finite_transitivity_chain % 0.30/0.49 thf(fact_568_infinite__imp__nonempty,axiom, % 0.30/0.49 ! [A: $tType,S3: set @ A] : % 0.30/0.49 ( ~ ( finite_finite2 @ A @ S3 ) % 0.30/0.49 => ( S3 % 0.30/0.49 != ( bot_bot @ ( set @ A ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % infinite_imp_nonempty % 0.30/0.49 thf(fact_569_finite_OemptyI,axiom, % 0.30/0.49 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ). % 0.30/0.49 % 0.30/0.49 % finite.emptyI % 0.30/0.49 thf(fact_570_rev__finite__subset,axiom, % 0.30/0.49 ! [A: $tType,B3: set @ A,A3: set @ A] : % 0.30/0.49 ( ( finite_finite2 @ A @ B3 ) % 0.30/0.49 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.49 => ( finite_finite2 @ A @ A3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % rev_finite_subset % 0.30/0.49 thf(fact_571_infinite__super,axiom, % 0.30/0.49 ! [A: $tType,S3: set @ A,T4: set @ A] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ S3 @ T4 ) % 0.30/0.49 => ( ~ ( finite_finite2 @ A @ S3 ) % 0.30/0.49 => ~ ( finite_finite2 @ A @ T4 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % infinite_super % 0.30/0.49 thf(fact_572_finite__subset,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.49 => ( ( finite_finite2 @ A @ B3 ) % 0.30/0.49 => ( finite_finite2 @ A @ A3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % finite_subset % 0.30/0.49 thf(fact_573_finite__psubset__induct,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] : % 0.30/0.49 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.49 => ( ! [A8: set @ A] : % 0.30/0.49 ( ( finite_finite2 @ A @ A8 ) % 0.30/0.49 => ( ! [B8: set @ A] : % 0.30/0.49 ( ( ord_less @ ( set @ A ) @ B8 @ A8 ) % 0.30/0.49 => ( P @ B8 ) ) % 0.30/0.49 => ( P @ A8 ) ) ) % 0.30/0.49 => ( P @ A3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % finite_psubset_induct % 0.30/0.49 thf(fact_574_finite__nat__set__iff__bounded,axiom, % 0.30/0.49 ( ( finite_finite2 @ nat ) % 0.30/0.49 = ( ^ [N5: set @ nat] : % 0.30/0.49 ? [M2: nat] : % 0.30/0.49 ! [X: nat] : % 0.30/0.49 ( ( member @ nat @ X @ N5 ) % 0.30/0.49 => ( ord_less @ nat @ X @ M2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % finite_nat_set_iff_bounded % 0.30/0.49 thf(fact_575_infinite__nat__iff__unbounded,axiom, % 0.30/0.49 ! [S3: set @ nat] : % 0.30/0.49 ( ( ~ ( finite_finite2 @ nat @ S3 ) ) % 0.30/0.49 = ( ! [M2: nat] : % 0.30/0.49 ? [N: nat] : % 0.30/0.49 ( ( ord_less @ nat @ M2 @ N ) % 0.30/0.49 & ( member @ nat @ N @ S3 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % infinite_nat_iff_unbounded % 0.30/0.49 thf(fact_576_arg__min__if__finite_I2_J,axiom, % 0.30/0.49 ! [B: $tType,A: $tType] : % 0.30/0.49 ( ( order @ B ) % 0.30/0.49 => ! [S3: set @ A,F2: A > B] : % 0.30/0.49 ( ( finite_finite2 @ A @ S3 ) % 0.30/0.49 => ( ( S3 % 0.30/0.49 != ( bot_bot @ ( set @ A ) ) ) % 0.30/0.49 => ~ ? [X2: A] : % 0.30/0.49 ( ( member @ A @ X2 @ S3 ) % 0.30/0.49 & ( ord_less @ B @ ( F2 @ X2 ) @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S3 ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % arg_min_if_finite(2) % 0.30/0.49 thf(fact_577_arg__min__least,axiom, % 0.30/0.49 ! [B: $tType,A: $tType] : % 0.30/0.49 ( ( linorder @ B ) % 0.30/0.49 => ! [S3: set @ A,Y: A,F2: A > B] : % 0.30/0.49 ( ( finite_finite2 @ A @ S3 ) % 0.30/0.49 => ( ( S3 % 0.30/0.49 != ( bot_bot @ ( set @ A ) ) ) % 0.30/0.49 => ( ( member @ A @ Y @ S3 ) % 0.30/0.49 => ( ord_less_eq @ B @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S3 ) ) @ ( F2 @ Y ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % arg_min_least % 0.30/0.49 thf(fact_578_nat__descend__induct,axiom, % 0.30/0.49 ! [N2: nat,P: nat > $o,M: nat] : % 0.30/0.49 ( ! [K2: nat] : % 0.30/0.49 ( ( ord_less @ nat @ N2 @ K2 ) % 0.30/0.49 => ( P @ K2 ) ) % 0.30/0.49 => ( ! [K2: nat] : % 0.30/0.49 ( ( ord_less_eq @ nat @ K2 @ N2 ) % 0.30/0.49 => ( ! [I5: nat] : % 0.30/0.49 ( ( ord_less @ nat @ K2 @ I5 ) % 0.30/0.49 => ( P @ I5 ) ) % 0.30/0.49 => ( P @ K2 ) ) ) % 0.30/0.49 => ( P @ M ) ) ) ). % 0.30/0.49 % 0.30/0.49 % nat_descend_induct % 0.30/0.49 thf(fact_579_subset__emptyI,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ A] : % 0.30/0.49 ( ! [X5: A] : % 0.30/0.49 ~ ( member @ A @ X5 @ A3 ) % 0.30/0.49 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % subset_emptyI % 0.30/0.49 thf(fact_580_exists__least__lemma,axiom, % 0.30/0.49 ! [P: nat > $o] : % 0.30/0.49 ( ~ ( P @ ( zero_zero @ nat ) ) % 0.30/0.49 => ( ? [X_12: nat] : ( P @ X_12 ) % 0.30/0.49 => ? [N3: nat] : % 0.30/0.49 ( ~ ( P @ N3 ) % 0.30/0.49 & ( P @ ( suc @ N3 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % exists_least_lemma % 0.30/0.49 thf(fact_581_dependent__nat__choice,axiom, % 0.30/0.49 ! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] : % 0.30/0.49 ( ? [X_12: A] : ( P @ ( zero_zero @ nat ) @ X_12 ) % 0.30/0.49 => ( ! [X5: A,N3: nat] : % 0.30/0.49 ( ( P @ N3 @ X5 ) % 0.30/0.49 => ? [Y5: A] : % 0.30/0.49 ( ( P @ ( suc @ N3 ) @ Y5 ) % 0.30/0.49 & ( Q @ N3 @ X5 @ Y5 ) ) ) % 0.30/0.49 => ? [F4: nat > A] : % 0.30/0.49 ! [N6: nat] : % 0.30/0.49 ( ( P @ N6 @ ( F4 @ N6 ) ) % 0.30/0.49 & ( Q @ N6 @ ( F4 @ N6 ) @ ( F4 @ ( suc @ N6 ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % dependent_nat_choice % 0.30/0.49 thf(fact_582_list__decode_Ocases,axiom, % 0.30/0.49 ! [X3: nat] : % 0.30/0.49 ( ( X3 % 0.30/0.49 != ( zero_zero @ nat ) ) % 0.30/0.49 => ~ ! [N3: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( suc @ N3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % list_decode.cases % 0.30/0.49 thf(fact_583_verit__sum__simplify,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( cancel1802427076303600483id_add @ A ) % 0.30/0.49 => ! [A2: A] : % 0.30/0.49 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.49 = A2 ) ) ). % 0.30/0.49 % 0.30/0.49 % verit_sum_simplify % 0.30/0.49 thf(fact_584_add__0__iff,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( semiri1453513574482234551roduct @ A ) % 0.30/0.49 => ! [B2: A,A2: A] : % 0.30/0.49 ( ( B2 % 0.30/0.49 = ( plus_plus @ A @ B2 @ A2 ) ) % 0.30/0.49 = ( A2 % 0.30/0.49 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % add_0_iff % 0.30/0.49 thf(fact_585_field__lbound__gt__zero,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linordered_field @ A ) % 0.30/0.49 => ! [D1: A,D22: A] : % 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 ) % 0.30/0.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 ) % 0.30/0.49 => ? [E: A] : % 0.30/0.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ E ) % 0.30/0.49 & ( ord_less @ A @ E @ D1 ) % 0.30/0.49 & ( ord_less @ A @ E @ D22 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % field_lbound_gt_zero % 0.30/0.49 thf(fact_586_less__by__empty,axiom, % 0.30/0.49 ! [A: $tType,A3: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] : % 0.30/0.49 ( ( A3 % 0.30/0.49 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) % 0.30/0.49 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B3 ) ) ). % 0.30/0.49 % 0.30/0.49 % less_by_empty % 0.30/0.49 thf(fact_587_verit__comp__simplify1_I2_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ). % 0.30/0.49 % 0.30/0.49 % verit_comp_simplify1(2) % 0.30/0.49 thf(fact_588_verit__la__disequality,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [A2: A,B2: A] : % 0.30/0.49 ( ( A2 = B2 ) % 0.30/0.49 | ~ ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.49 | ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % verit_la_disequality % 0.30/0.49 thf(fact_589_verit__comp__simplify1_I1_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( order @ A ) % 0.30/0.49 => ! [A2: A] : % 0.30/0.49 ~ ( ord_less @ A @ A2 @ A2 ) ) ). % 0.30/0.49 % 0.30/0.49 % verit_comp_simplify1(1) % 0.30/0.49 thf(fact_590_ssubst__Pair__rhs,axiom, % 0.30/0.49 ! [B: $tType,A: $tType,R2: A,S: B,R: set @ ( product_prod @ A @ B ),S4: B] : % 0.30/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S ) @ R ) % 0.30/0.49 => ( ( S4 = S ) % 0.30/0.49 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S4 ) @ R ) ) ) ). % 0.30/0.49 % 0.30/0.49 % ssubst_Pair_rhs % 0.30/0.49 thf(fact_591_prod__decode__aux_Ocases,axiom, % 0.30/0.49 ! [X3: product_prod @ nat @ nat] : % 0.30/0.49 ~ ! [K2: nat,M3: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ nat @ nat @ K2 @ M3 ) ) ). % 0.30/0.49 % 0.30/0.49 % prod_decode_aux.cases % 0.30/0.49 thf(fact_592_arg__min__if__finite_I1_J,axiom, % 0.30/0.49 ! [B: $tType,A: $tType] : % 0.30/0.49 ( ( order @ B ) % 0.30/0.49 => ! [S3: set @ A,F2: A > B] : % 0.30/0.49 ( ( finite_finite2 @ A @ S3 ) % 0.30/0.49 => ( ( S3 % 0.30/0.49 != ( bot_bot @ ( set @ A ) ) ) % 0.30/0.49 => ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S3 ) @ S3 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % arg_min_if_finite(1) % 0.30/0.49 thf(fact_593_verit__comp__simplify1_I3_J,axiom, % 0.30/0.49 ! [B: $tType] : % 0.30/0.49 ( ( linorder @ B ) % 0.30/0.49 => ! [B7: B,A7: B] : % 0.30/0.49 ( ( ~ ( ord_less_eq @ B @ B7 @ A7 ) ) % 0.30/0.49 = ( ord_less @ B @ A7 @ B7 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % verit_comp_simplify1(3) % 0.30/0.49 thf(fact_594_triangle__Suc,axiom, % 0.30/0.49 ! [N2: nat] : % 0.30/0.49 ( ( nat_triangle @ ( suc @ N2 ) ) % 0.30/0.49 = ( plus_plus @ nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % triangle_Suc % 0.30/0.49 thf(fact_595_less__numeral__extra_I3_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linord181362715937106298miring @ A ) % 0.30/0.49 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.49 % 0.30/0.49 % less_numeral_extra(3) % 0.30/0.49 thf(fact_596_minf_I8_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [T2: A] : % 0.30/0.49 ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ X2 @ Z3 ) % 0.30/0.49 => ~ ( ord_less_eq @ A @ T2 @ X2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % minf(8) % 0.30/0.49 thf(fact_597_minf_I6_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [T2: A] : % 0.30/0.49 ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ X2 @ Z3 ) % 0.30/0.49 => ( ord_less_eq @ A @ X2 @ T2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % minf(6) % 0.30/0.49 thf(fact_598_pinf_I8_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [T2: A] : % 0.30/0.49 ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ Z3 @ X2 ) % 0.30/0.49 => ( ord_less_eq @ A @ T2 @ X2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % pinf(8) % 0.30/0.49 thf(fact_599_pinf_I6_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [T2: A] : % 0.30/0.49 ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ Z3 @ X2 ) % 0.30/0.49 => ~ ( ord_less_eq @ A @ X2 @ T2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % pinf(6) % 0.30/0.49 thf(fact_600_complete__interval,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( condit6923001295902523014norder @ A ) % 0.30/0.49 => ! [A2: A,B2: A,P: A > $o] : % 0.30/0.49 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.49 => ( ( P @ A2 ) % 0.30/0.49 => ( ~ ( P @ B2 ) % 0.30/0.49 => ? [C3: A] : % 0.30/0.49 ( ( ord_less_eq @ A @ A2 @ C3 ) % 0.30/0.49 & ( ord_less_eq @ A @ C3 @ B2 ) % 0.30/0.49 & ! [X2: A] : % 0.30/0.49 ( ( ( ord_less_eq @ A @ A2 @ X2 ) % 0.30/0.49 & ( ord_less @ A @ X2 @ C3 ) ) % 0.30/0.49 => ( P @ X2 ) ) % 0.30/0.49 & ! [D4: A] : % 0.30/0.49 ( ! [X5: A] : % 0.30/0.49 ( ( ( ord_less_eq @ A @ A2 @ X5 ) % 0.30/0.49 & ( ord_less @ A @ X5 @ D4 ) ) % 0.30/0.49 => ( P @ X5 ) ) % 0.30/0.49 => ( ord_less_eq @ A @ D4 @ C3 ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % complete_interval % 0.30/0.49 thf(fact_601_le__numeral__extra_I3_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linord181362715937106298miring @ A ) % 0.30/0.49 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.49 % 0.30/0.49 % le_numeral_extra(3) % 0.30/0.49 thf(fact_602_find__Some__iff,axiom, % 0.30/0.49 ! [A: $tType,P: A > $o,Xs: list @ A,X3: A] : % 0.30/0.49 ( ( ( find @ A @ P @ Xs ) % 0.30/0.49 = ( some @ A @ X3 ) ) % 0.30/0.49 = ( ? [I3: nat] : % 0.30/0.49 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.49 & ( P @ ( nth @ A @ Xs @ I3 ) ) % 0.30/0.49 & ( X3 % 0.30/0.49 = ( nth @ A @ Xs @ I3 ) ) % 0.30/0.49 & ! [J3: nat] : % 0.30/0.49 ( ( ord_less @ nat @ J3 @ I3 ) % 0.30/0.49 => ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % find_Some_iff % 0.30/0.49 thf(fact_603_find__Some__iff2,axiom, % 0.30/0.49 ! [A: $tType,X3: A,P: A > $o,Xs: list @ A] : % 0.30/0.49 ( ( ( some @ A @ X3 ) % 0.30/0.49 = ( find @ A @ P @ Xs ) ) % 0.30/0.49 = ( ? [I3: nat] : % 0.30/0.49 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.49 & ( P @ ( nth @ A @ Xs @ I3 ) ) % 0.30/0.49 & ( X3 % 0.30/0.49 = ( nth @ A @ Xs @ I3 ) ) % 0.30/0.49 & ! [J3: nat] : % 0.30/0.49 ( ( ord_less @ nat @ J3 @ I3 ) % 0.30/0.49 => ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % find_Some_iff2 % 0.30/0.49 thf(fact_604_triangle__0,axiom, % 0.30/0.49 ( ( nat_triangle @ ( zero_zero @ nat ) ) % 0.30/0.49 = ( zero_zero @ nat ) ) ). % 0.30/0.49 % 0.30/0.49 % triangle_0 % 0.30/0.49 thf(fact_605_find__cong,axiom, % 0.30/0.49 ! [A: $tType,Xs: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] : % 0.30/0.49 ( ( Xs = Ys ) % 0.30/0.49 => ( ! [X5: A] : % 0.30/0.49 ( ( member @ A @ X5 @ ( set2 @ A @ Ys ) ) % 0.30/0.49 => ( ( P @ X5 ) % 0.30/0.49 = ( Q @ X5 ) ) ) % 0.30/0.49 => ( ( find @ A @ P @ Xs ) % 0.30/0.49 = ( find @ A @ Q @ Ys ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % find_cong % 0.30/0.49 thf(fact_606_ex__gt__or__lt,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( condit5016429287641298734tinuum @ A ) % 0.30/0.49 => ! [A2: A] : % 0.30/0.49 ? [B5: A] : % 0.30/0.49 ( ( ord_less @ A @ A2 @ B5 ) % 0.30/0.49 | ( ord_less @ A @ B5 @ A2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % ex_gt_or_lt % 0.30/0.49 thf(fact_607_pinf_I1_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] : % 0.30/0.49 ( ? [Z4: A] : % 0.30/0.49 ! [X5: A] : % 0.30/0.49 ( ( ord_less @ A @ Z4 @ X5 ) % 0.30/0.49 => ( ( P @ X5 ) % 0.30/0.49 = ( P5 @ X5 ) ) ) % 0.30/0.49 => ( ? [Z4: A] : % 0.30/0.49 ! [X5: A] : % 0.30/0.49 ( ( ord_less @ A @ Z4 @ X5 ) % 0.30/0.49 => ( ( Q @ X5 ) % 0.30/0.49 = ( Q3 @ X5 ) ) ) % 0.30/0.49 => ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ Z3 @ X2 ) % 0.30/0.49 => ( ( ( P @ X2 ) % 0.30/0.49 & ( Q @ X2 ) ) % 0.30/0.49 = ( ( P5 @ X2 ) % 0.30/0.49 & ( Q3 @ X2 ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % pinf(1) % 0.30/0.49 thf(fact_608_pinf_I2_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] : % 0.30/0.49 ( ? [Z4: A] : % 0.30/0.49 ! [X5: A] : % 0.30/0.49 ( ( ord_less @ A @ Z4 @ X5 ) % 0.30/0.49 => ( ( P @ X5 ) % 0.30/0.49 = ( P5 @ X5 ) ) ) % 0.30/0.49 => ( ? [Z4: A] : % 0.30/0.49 ! [X5: A] : % 0.30/0.49 ( ( ord_less @ A @ Z4 @ X5 ) % 0.30/0.49 => ( ( Q @ X5 ) % 0.30/0.49 = ( Q3 @ X5 ) ) ) % 0.30/0.49 => ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ Z3 @ X2 ) % 0.30/0.49 => ( ( ( P @ X2 ) % 0.30/0.49 | ( Q @ X2 ) ) % 0.30/0.49 = ( ( P5 @ X2 ) % 0.30/0.49 | ( Q3 @ X2 ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % pinf(2) % 0.30/0.49 thf(fact_609_pinf_I3_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [T2: A] : % 0.30/0.49 ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ Z3 @ X2 ) % 0.30/0.49 => ( X2 != T2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % pinf(3) % 0.30/0.49 thf(fact_610_pinf_I4_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [T2: A] : % 0.30/0.49 ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ Z3 @ X2 ) % 0.30/0.49 => ( X2 != T2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % pinf(4) % 0.30/0.49 thf(fact_611_pinf_I5_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [T2: A] : % 0.30/0.49 ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ Z3 @ X2 ) % 0.30/0.49 => ~ ( ord_less @ A @ X2 @ T2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % pinf(5) % 0.30/0.49 thf(fact_612_pinf_I7_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [T2: A] : % 0.30/0.49 ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ Z3 @ X2 ) % 0.30/0.49 => ( ord_less @ A @ T2 @ X2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % pinf(7) % 0.30/0.49 thf(fact_613_pinf_I11_J,axiom, % 0.30/0.49 ! [C: $tType,D: $tType] : % 0.30/0.49 ( ( ord @ C ) % 0.30/0.49 => ! [F5: D] : % 0.30/0.49 ? [Z3: C] : % 0.30/0.49 ! [X2: C] : % 0.30/0.49 ( ( ord_less @ C @ Z3 @ X2 ) % 0.30/0.49 => ( F5 = F5 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % pinf(11) % 0.30/0.49 thf(fact_614_minf_I1_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] : % 0.30/0.49 ( ? [Z4: A] : % 0.30/0.49 ! [X5: A] : % 0.30/0.49 ( ( ord_less @ A @ X5 @ Z4 ) % 0.30/0.49 => ( ( P @ X5 ) % 0.30/0.49 = ( P5 @ X5 ) ) ) % 0.30/0.49 => ( ? [Z4: A] : % 0.30/0.49 ! [X5: A] : % 0.30/0.49 ( ( ord_less @ A @ X5 @ Z4 ) % 0.30/0.49 => ( ( Q @ X5 ) % 0.30/0.49 = ( Q3 @ X5 ) ) ) % 0.30/0.49 => ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ X2 @ Z3 ) % 0.30/0.49 => ( ( ( P @ X2 ) % 0.30/0.49 & ( Q @ X2 ) ) % 0.30/0.49 = ( ( P5 @ X2 ) % 0.30/0.49 & ( Q3 @ X2 ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % minf(1) % 0.30/0.49 thf(fact_615_minf_I2_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] : % 0.30/0.49 ( ? [Z4: A] : % 0.30/0.49 ! [X5: A] : % 0.30/0.49 ( ( ord_less @ A @ X5 @ Z4 ) % 0.30/0.49 => ( ( P @ X5 ) % 0.30/0.49 = ( P5 @ X5 ) ) ) % 0.30/0.49 => ( ? [Z4: A] : % 0.30/0.49 ! [X5: A] : % 0.30/0.49 ( ( ord_less @ A @ X5 @ Z4 ) % 0.30/0.49 => ( ( Q @ X5 ) % 0.30/0.49 = ( Q3 @ X5 ) ) ) % 0.30/0.49 => ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ X2 @ Z3 ) % 0.30/0.49 => ( ( ( P @ X2 ) % 0.30/0.49 | ( Q @ X2 ) ) % 0.30/0.49 = ( ( P5 @ X2 ) % 0.30/0.49 | ( Q3 @ X2 ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % minf(2) % 0.30/0.49 thf(fact_616_minf_I3_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [T2: A] : % 0.30/0.49 ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ X2 @ Z3 ) % 0.30/0.49 => ( X2 != T2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % minf(3) % 0.30/0.49 thf(fact_617_minf_I4_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [T2: A] : % 0.30/0.49 ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ X2 @ Z3 ) % 0.30/0.49 => ( X2 != T2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % minf(4) % 0.30/0.49 thf(fact_618_minf_I5_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [T2: A] : % 0.30/0.49 ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ X2 @ Z3 ) % 0.30/0.49 => ( ord_less @ A @ X2 @ T2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % minf(5) % 0.30/0.49 thf(fact_619_minf_I7_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linorder @ A ) % 0.30/0.49 => ! [T2: A] : % 0.30/0.49 ? [Z3: A] : % 0.30/0.49 ! [X2: A] : % 0.30/0.49 ( ( ord_less @ A @ X2 @ Z3 ) % 0.30/0.49 => ~ ( ord_less @ A @ T2 @ X2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % minf(7) % 0.30/0.49 thf(fact_620_minf_I11_J,axiom, % 0.30/0.49 ! [C: $tType,D: $tType] : % 0.30/0.49 ( ( ord @ C ) % 0.30/0.49 => ! [F5: D] : % 0.30/0.49 ? [Z3: C] : % 0.30/0.49 ! [X2: C] : % 0.30/0.49 ( ( ord_less @ C @ X2 @ Z3 ) % 0.30/0.49 => ( F5 = F5 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % minf(11) % 0.30/0.49 thf(fact_621_is__num__normalize_I1_J,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( neg_numeral @ A ) % 0.30/0.49 => ! [A2: A,B2: A,C2: A] : % 0.30/0.49 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.49 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % is_num_normalize(1) % 0.30/0.49 thf(fact_622_find__None__iff2,axiom, % 0.30/0.49 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.30/0.49 ( ( ( none @ A ) % 0.30/0.49 = ( find @ A @ P @ Xs ) ) % 0.30/0.49 = ( ~ ? [X: A] : % 0.30/0.49 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.30/0.49 & ( P @ X ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % find_None_iff2 % 0.30/0.49 thf(fact_623_find__None__iff,axiom, % 0.30/0.49 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.30/0.49 ( ( ( find @ A @ P @ Xs ) % 0.30/0.49 = ( none @ A ) ) % 0.30/0.49 = ( ~ ? [X: A] : % 0.30/0.49 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.30/0.49 & ( P @ X ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % find_None_iff % 0.30/0.49 thf(fact_624_nth__zip,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys: list @ B] : % 0.30/0.49 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.49 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) ) % 0.30/0.49 => ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I ) % 0.30/0.49 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys @ I ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % nth_zip % 0.30/0.49 thf(fact_625__C5_C_I10_J,axiom, % 0.30/0.49 ( ( mi != ma ) % 0.30/0.49 => ! [I5: nat] : % 0.30/0.49 ( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) ) % 0.30/0.49 => ( ( ( ( vEBT_VEBT_high @ ma @ n ) % 0.30/0.49 = I5 ) % 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I5 ) @ ( vEBT_VEBT_low @ ma @ n ) ) ) % 0.30/0.49 & ! [X2: nat] : % 0.30/0.49 ( ( ( ( vEBT_VEBT_high @ X2 @ n ) % 0.30/0.49 = I5 ) % 0.30/0.49 & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I5 ) @ ( vEBT_VEBT_low @ X2 @ n ) ) ) % 0.30/0.49 => ( ( ord_less @ nat @ mi @ X2 ) % 0.30/0.49 & ( ord_less_eq @ nat @ X2 @ ma ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % "5"(10) % 0.30/0.49 thf(fact_626_vebt__member_Ocases,axiom, % 0.30/0.49 ! [X3: product_prod @ vEBT_VEBT @ nat] : % 0.30/0.49 ( ! [A5: $o,B5: $o,X5: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X5 ) ) % 0.30/0.49 => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X5 ) ) % 0.30/0.49 => ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ X5 ) ) % 0.30/0.49 => ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ X5 ) ) % 0.30/0.49 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % vebt_member.cases % 0.30/0.49 thf(fact_627_VEBT__internal_Omembermima_Ocases,axiom, % 0.30/0.49 ! [X3: product_prod @ vEBT_VEBT @ nat] : % 0.30/0.49 ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) ) % 0.30/0.49 => ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Uz2 ) ) % 0.30/0.49 => ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ X5 ) ) % 0.30/0.49 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ X5 ) ) % 0.30/0.49 => ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT,X5: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ X5 ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.membermima.cases % 0.30/0.49 thf(fact_628_count__notin,axiom, % 0.30/0.49 ! [A: $tType,X3: A,Xs: list @ A] : % 0.30/0.49 ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.30/0.49 => ( ( count_list @ A @ Xs @ X3 ) % 0.30/0.49 = ( zero_zero @ nat ) ) ) ). % 0.30/0.49 % 0.30/0.49 % count_notin % 0.30/0.49 thf(fact_629_listrel1__iff__update,axiom, % 0.30/0.49 ! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.30/0.49 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) % 0.30/0.49 = ( ? [Y2: A,N: nat] : % 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N ) @ Y2 ) @ R2 ) % 0.30/0.49 & ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.30/0.49 & ( Ys % 0.30/0.49 = ( list_update @ A @ Xs @ N @ Y2 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % listrel1_iff_update % 0.30/0.49 thf(fact_630_pair__lessI2,axiom, % 0.30/0.49 ! [A2: nat,B2: nat,S: nat,T2: nat] : % 0.30/0.49 ( ( ord_less_eq @ nat @ A2 @ B2 ) % 0.30/0.49 => ( ( ord_less @ nat @ S @ T2 ) % 0.30/0.49 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ) ). % 0.30/0.49 % 0.30/0.49 % pair_lessI2 % 0.30/0.49 thf(fact_631_pair__less__iff1,axiom, % 0.30/0.49 ! [X3: nat,Y: nat,Z2: nat] : % 0.30/0.49 ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X3 @ Y ) @ ( product_Pair @ nat @ nat @ X3 @ Z2 ) ) @ fun_pair_less ) % 0.30/0.49 = ( ord_less @ nat @ Y @ Z2 ) ) ). % 0.30/0.49 % 0.30/0.49 % pair_less_iff1 % 0.30/0.49 thf(fact_632_gen__length__def,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( gen_length @ A ) % 0.30/0.49 = ( ^ [N: nat,Xs2: list @ A] : ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % gen_length_def % 0.30/0.49 thf(fact_633_set__encode__empty,axiom, % 0.30/0.49 ( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) ) % 0.30/0.49 = ( zero_zero @ nat ) ) ). % 0.30/0.49 % 0.30/0.49 % set_encode_empty % 0.30/0.49 thf(fact_634_Leaf__0__not,axiom, % 0.30/0.49 ! [A2: $o,B2: $o] : % 0.30/0.49 ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) ) ). % 0.30/0.49 % 0.30/0.49 % Leaf_0_not % 0.30/0.49 thf(fact_635_power__shift,axiom, % 0.30/0.49 ! [X3: nat,Y: nat,Z2: nat] : % 0.30/0.49 ( ( ( power_power @ nat @ X3 @ Y ) % 0.30/0.49 = Z2 ) % 0.30/0.49 = ( ( vEBT_VEBT_power @ ( some @ nat @ X3 ) @ ( some @ nat @ Y ) ) % 0.30/0.49 = ( some @ nat @ Z2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % power_shift % 0.30/0.49 thf(fact_636__C5_C_I3_J,axiom, % 0.30/0.49 ( ( size_size @ ( list @ vEBT_VEBT ) @ treeList ) % 0.30/0.49 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) ) ). % 0.30/0.49 % 0.30/0.49 % "5"(3) % 0.30/0.49 thf(fact_637__C5_C_I9_J,axiom, % 0.30/0.49 ord_less @ nat @ ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ) ). % 0.30/0.49 % 0.30/0.49 % "5"(9) % 0.30/0.49 thf(fact_638_VEBT_Oinject_I2_J,axiom, % 0.30/0.49 ! [X21: $o,X222: $o,Y21: $o,Y222: $o] : % 0.30/0.49 ( ( ( vEBT_Leaf @ X21 @ X222 ) % 0.30/0.49 = ( vEBT_Leaf @ Y21 @ Y222 ) ) % 0.30/0.49 = ( ( X21 = Y21 ) % 0.30/0.49 & ( X222 = Y222 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT.inject(2) % 0.30/0.49 thf(fact_639_local_Opower__def,axiom, % 0.30/0.49 ( vEBT_VEBT_power % 0.30/0.49 = ( vEBT_V2048590022279873568_shift @ nat @ ( power_power @ nat ) ) ) ). % 0.30/0.49 % 0.30/0.49 % local.power_def % 0.30/0.49 thf(fact_640_high__bound__aux,axiom, % 0.30/0.49 ! [Ma: nat,N2: nat,M: nat] : % 0.30/0.49 ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) ) % 0.30/0.49 => ( ord_less @ nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ). % 0.30/0.49 % 0.30/0.49 % high_bound_aux % 0.30/0.49 thf(fact_641_member__bound,axiom, % 0.30/0.49 ! [Tree: vEBT_VEBT,X3: nat,N2: nat] : % 0.30/0.49 ( ( vEBT_vebt_member @ Tree @ X3 ) % 0.30/0.49 => ( ( vEBT_invar_vebt @ Tree @ N2 ) % 0.30/0.49 => ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % member_bound % 0.30/0.49 thf(fact_642__C5_C_I6_J,axiom, % 0.30/0.49 ! [I5: nat] : % 0.30/0.49 ( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) ) % 0.30/0.49 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I5 ) @ X6 ) ) % 0.30/0.49 = ( vEBT_V8194947554948674370ptions @ summary @ I5 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % "5"(6) % 0.30/0.49 thf(fact_643_numeral__le__iff,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linord181362715937106298miring @ A ) % 0.30/0.49 => ! [M: num,N2: num] : % 0.30/0.49 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) % 0.30/0.49 = ( ord_less_eq @ num @ M @ N2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % numeral_le_iff % 0.30/0.49 thf(fact_644_numeral__less__iff,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linord181362715937106298miring @ A ) % 0.30/0.49 => ! [M: num,N2: num] : % 0.30/0.49 ( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) % 0.30/0.49 = ( ord_less @ num @ M @ N2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % numeral_less_iff % 0.30/0.49 thf(fact_645_numeral__plus__numeral,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( numeral @ A ) % 0.30/0.49 => ! [M: num,N2: num] : % 0.30/0.49 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) % 0.30/0.49 = ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % numeral_plus_numeral % 0.30/0.49 thf(fact_646_add__numeral__left,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( numeral @ A ) % 0.30/0.49 => ! [V2: num,W2: num,Z2: A] : % 0.30/0.49 ( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Z2 ) ) % 0.30/0.49 = ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W2 ) ) @ Z2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % add_numeral_left % 0.30/0.49 thf(fact_647_mi__ma__2__deg,axiom, % 0.30/0.49 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat] : % 0.30/0.49 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 ) % 0.30/0.49 => ( ( ord_less_eq @ nat @ Mi @ Ma ) % 0.30/0.49 & ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % mi_ma_2_deg % 0.30/0.49 thf(fact_648_Suc__numeral,axiom, % 0.30/0.49 ! [N2: num] : % 0.30/0.49 ( ( suc @ ( numeral_numeral @ nat @ N2 ) ) % 0.30/0.49 = ( numeral_numeral @ nat @ ( plus_plus @ num @ N2 @ one2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % Suc_numeral % 0.30/0.49 thf(fact_649_add__2__eq__Suc_H,axiom, % 0.30/0.49 ! [N2: nat] : % 0.30/0.49 ( ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.49 = ( suc @ ( suc @ N2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % add_2_eq_Suc' % 0.30/0.49 thf(fact_650_add__2__eq__Suc,axiom, % 0.30/0.49 ! [N2: nat] : % 0.30/0.49 ( ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.49 = ( suc @ ( suc @ N2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % add_2_eq_Suc % 0.30/0.49 thf(fact_651_numeral__Bit0,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( numeral @ A ) % 0.30/0.49 => ! [N2: num] : % 0.30/0.49 ( ( numeral_numeral @ A @ ( bit0 @ N2 ) ) % 0.30/0.49 = ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % numeral_Bit0 % 0.30/0.49 thf(fact_652_numeral__2__eq__2,axiom, % 0.30/0.49 ( ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) % 0.30/0.49 = ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % numeral_2_eq_2 % 0.30/0.49 thf(fact_653_less__2__cases,axiom, % 0.30/0.49 ! [N2: nat] : % 0.30/0.49 ( ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.49 => ( ( N2 % 0.30/0.49 = ( zero_zero @ nat ) ) % 0.30/0.49 | ( N2 % 0.30/0.49 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % less_2_cases % 0.30/0.49 thf(fact_654_less__2__cases__iff,axiom, % 0.30/0.49 ! [N2: nat] : % 0.30/0.49 ( ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.49 = ( ( N2 % 0.30/0.49 = ( zero_zero @ nat ) ) % 0.30/0.49 | ( N2 % 0.30/0.49 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % less_2_cases_iff % 0.30/0.49 thf(fact_655_VEBT__internal_Oexp__split__high__low_I1_J,axiom, % 0.30/0.49 ! [X3: nat,N2: nat,M: nat] : % 0.30/0.49 ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) ) % 0.30/0.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.30/0.49 => ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.exp_split_high_low(1) % 0.30/0.49 thf(fact_656_VEBT__internal_Oexp__split__high__low_I2_J,axiom, % 0.30/0.49 ! [X3: nat,N2: nat,M: nat] : % 0.30/0.49 ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) ) % 0.30/0.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.30/0.49 => ( ord_less @ nat @ ( vEBT_VEBT_low @ X3 @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.exp_split_high_low(2) % 0.30/0.49 thf(fact_657_numeral__1__eq__Suc__0,axiom, % 0.30/0.49 ( ( numeral_numeral @ nat @ one2 ) % 0.30/0.49 = ( suc @ ( zero_zero @ nat ) ) ) ). % 0.30/0.49 % 0.30/0.49 % numeral_1_eq_Suc_0 % 0.30/0.49 thf(fact_658_Suc__nat__number__of__add,axiom, % 0.30/0.49 ! [V2: num,N2: nat] : % 0.30/0.49 ( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N2 ) ) % 0.30/0.49 = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V2 @ one2 ) ) @ N2 ) ) ). % 0.30/0.49 % 0.30/0.49 % Suc_nat_number_of_add % 0.30/0.49 thf(fact_659_VEBT_Osize_I4_J,axiom, % 0.30/0.49 ! [X21: $o,X222: $o] : % 0.30/0.49 ( ( size_size @ vEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) ) % 0.30/0.49 = ( zero_zero @ nat ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT.size(4) % 0.30/0.49 thf(fact_660_VEBT__internal_Ovalid_H_Ocases,axiom, % 0.30/0.49 ! [X3: product_prod @ vEBT_VEBT @ nat] : % 0.30/0.49 ( ! [Uu2: $o,Uv2: $o,D3: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D3 ) ) % 0.30/0.49 => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.valid'.cases % 0.30/0.49 thf(fact_661_VEBT_Odistinct_I1_J,axiom, % 0.30/0.49 ! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] : % 0.30/0.49 ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) % 0.30/0.49 != ( vEBT_Leaf @ X21 @ X222 ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT.distinct(1) % 0.30/0.49 thf(fact_662_VEBT_Oexhaust,axiom, % 0.30/0.49 ! [Y: vEBT_VEBT] : % 0.30/0.49 ( ! [X112: option @ ( product_prod @ nat @ nat ),X122: nat,X132: list @ vEBT_VEBT,X142: vEBT_VEBT] : % 0.30/0.49 ( Y % 0.30/0.49 != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) ) % 0.30/0.49 => ~ ! [X212: $o,X223: $o] : % 0.30/0.49 ( Y % 0.30/0.49 != ( vEBT_Leaf @ X212 @ X223 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT.exhaust % 0.30/0.49 thf(fact_663_zero__neq__numeral,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( semiring_char_0 @ A ) % 0.30/0.49 => ! [N2: num] : % 0.30/0.49 ( ( zero_zero @ A ) % 0.30/0.49 != ( numeral_numeral @ A @ N2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % zero_neq_numeral % 0.30/0.49 thf(fact_664_VEBT__internal_Omembermima_Osimps_I1_J,axiom, % 0.30/0.49 ! [Uu: $o,Uv: $o,Uw: nat] : % 0.30/0.49 ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.membermima.simps(1) % 0.30/0.49 thf(fact_665_num_Osize_I4_J,axiom, % 0.30/0.49 ( ( size_size @ num @ one2 ) % 0.30/0.49 = ( zero_zero @ nat ) ) ). % 0.30/0.49 % 0.30/0.49 % num.size(4) % 0.30/0.49 thf(fact_666_listrel1__eq__len,axiom, % 0.30/0.49 ! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.30/0.49 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) % 0.30/0.49 => ( ( size_size @ ( list @ A ) @ Xs ) % 0.30/0.49 = ( size_size @ ( list @ A ) @ Ys ) ) ) ). % 0.30/0.49 % 0.30/0.49 % listrel1_eq_len % 0.30/0.49 thf(fact_667_listrel1__mono,axiom, % 0.30/0.49 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] : % 0.30/0.49 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) % 0.30/0.49 => ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R2 ) @ ( listrel1 @ A @ S ) ) ) ). % 0.30/0.49 % 0.30/0.49 % listrel1_mono % 0.30/0.49 thf(fact_668_VEBT__internal_OminNull_Osimps_I3_J,axiom, % 0.30/0.49 ! [Uu: $o] : % 0.30/0.49 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.minNull.simps(3) % 0.30/0.49 thf(fact_669_VEBT__internal_OminNull_Osimps_I2_J,axiom, % 0.30/0.49 ! [Uv: $o] : % 0.30/0.49 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.minNull.simps(2) % 0.30/0.49 thf(fact_670_VEBT__internal_OminNull_Osimps_I1_J,axiom, % 0.30/0.49 vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.minNull.simps(1) % 0.30/0.49 thf(fact_671_invar__vebt_Ointros_I2_J,axiom, % 0.30/0.49 ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] : % 0.30/0.49 ( ! [X5: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 0.30/0.49 => ( vEBT_invar_vebt @ X5 @ N2 ) ) % 0.30/0.49 => ( ( vEBT_invar_vebt @ Summary @ M ) % 0.30/0.49 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) % 0.30/0.49 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 0.30/0.49 => ( ( M = N2 ) % 0.30/0.49 => ( ( Deg % 0.30/0.49 = ( plus_plus @ nat @ N2 @ M ) ) % 0.30/0.49 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) % 0.30/0.49 => ( ! [X5: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 0.30/0.49 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) % 0.30/0.49 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % invar_vebt.intros(2) % 0.30/0.49 thf(fact_672_invar__vebt_Ointros_I3_J,axiom, % 0.30/0.49 ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] : % 0.30/0.49 ( ! [X5: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 0.30/0.49 => ( vEBT_invar_vebt @ X5 @ N2 ) ) % 0.30/0.49 => ( ( vEBT_invar_vebt @ Summary @ M ) % 0.30/0.49 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) % 0.30/0.49 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 0.30/0.49 => ( ( M % 0.30/0.49 = ( suc @ N2 ) ) % 0.30/0.49 => ( ( Deg % 0.30/0.49 = ( plus_plus @ nat @ N2 @ M ) ) % 0.30/0.49 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) % 0.30/0.49 => ( ! [X5: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 0.30/0.49 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) % 0.30/0.49 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % invar_vebt.intros(3) % 0.30/0.49 thf(fact_673_zip__same,axiom, % 0.30/0.49 ! [A: $tType,A2: A,B2: A,Xs: list @ A] : % 0.30/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Xs ) ) ) % 0.30/0.49 = ( ( member @ A @ A2 @ ( set2 @ A @ Xs ) ) % 0.30/0.49 & ( A2 = B2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % zip_same % 0.30/0.49 thf(fact_674_in__set__zipE,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,X3: A,Y: B,Xs: list @ A,Ys: list @ B] : % 0.30/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) % 0.30/0.49 => ~ ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.30/0.49 => ~ ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % in_set_zipE % 0.30/0.49 thf(fact_675_set__zip__leftD,axiom, % 0.30/0.49 ! [B: $tType,A: $tType,X3: A,Y: B,Xs: list @ A,Ys: list @ B] : % 0.30/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) % 0.30/0.49 => ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ). % 0.30/0.49 % 0.30/0.49 % set_zip_leftD % 0.30/0.49 thf(fact_676_set__zip__rightD,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,X3: A,Y: B,Xs: list @ A,Ys: list @ B] : % 0.30/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) % 0.30/0.49 => ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ). % 0.30/0.49 % 0.30/0.49 % set_zip_rightD % 0.30/0.49 thf(fact_677_zip__update,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,Xs: list @ A,I: nat,X3: A,Ys: list @ B,Y: B] : % 0.30/0.49 ( ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ X3 ) @ ( list_update @ B @ Ys @ I @ Y ) ) % 0.30/0.49 = ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I @ ( product_Pair @ A @ B @ X3 @ Y ) ) ) ). % 0.30/0.49 % 0.30/0.49 % zip_update % 0.30/0.49 thf(fact_678_zero__le__numeral,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linord181362715937106298miring @ A ) % 0.30/0.49 => ! [N2: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % zero_le_numeral % 0.30/0.49 thf(fact_679_not__numeral__le__zero,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linord181362715937106298miring @ A ) % 0.30/0.49 => ! [N2: num] : % 0.30/0.49 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.49 % 0.30/0.49 % not_numeral_le_zero % 0.30/0.49 thf(fact_680_not__numeral__less__zero,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linord181362715937106298miring @ A ) % 0.30/0.49 => ! [N2: num] : % 0.30/0.49 ~ ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.49 % 0.30/0.49 % not_numeral_less_zero % 0.30/0.49 thf(fact_681_zero__less__numeral,axiom, % 0.30/0.49 ! [A: $tType] : % 0.30/0.49 ( ( linord181362715937106298miring @ A ) % 0.30/0.49 => ! [N2: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ). % 0.30/0.49 % 0.30/0.49 % zero_less_numeral % 0.30/0.49 thf(fact_682_set__encode__eq,axiom, % 0.30/0.49 ! [A3: set @ nat,B3: set @ nat] : % 0.30/0.49 ( ( finite_finite2 @ nat @ A3 ) % 0.30/0.49 => ( ( finite_finite2 @ nat @ B3 ) % 0.30/0.49 => ( ( ( nat_set_encode @ A3 ) % 0.30/0.49 = ( nat_set_encode @ B3 ) ) % 0.30/0.49 = ( A3 = B3 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % set_encode_eq % 0.30/0.49 thf(fact_683_vebt__buildup_Osimps_I1_J,axiom, % 0.30/0.49 ( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) ) % 0.30/0.49 = ( vEBT_Leaf @ $false @ $false ) ) ). % 0.30/0.49 % 0.30/0.49 % vebt_buildup.simps(1) % 0.30/0.49 thf(fact_684_in__set__impl__in__set__zip2,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Y: B] : % 0.30/0.49 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.30/0.49 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.30/0.49 => ( ( member @ B @ Y @ ( set2 @ B @ Ys ) ) % 0.30/0.49 => ~ ! [X5: A] : % 0.30/0.49 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % in_set_impl_in_set_zip2 % 0.30/0.49 thf(fact_685_in__set__impl__in__set__zip1,axiom, % 0.30/0.49 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X3: A] : % 0.30/0.49 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.30/0.49 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.30/0.49 => ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.30/0.49 => ~ ! [Y4: B] : % 0.30/0.49 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % in_set_impl_in_set_zip1 % 0.30/0.49 thf(fact_686_VEBT__internal_Onaive__member_Ocases,axiom, % 0.30/0.49 ! [X3: product_prod @ vEBT_VEBT @ nat] : % 0.30/0.49 ( ! [A5: $o,B5: $o,X5: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X5 ) ) % 0.30/0.49 => ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Ux2 ) ) % 0.30/0.49 => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT,S2: vEBT_VEBT,X5: nat] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) @ X5 ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.naive_member.cases % 0.30/0.49 thf(fact_687_invar__vebt_Ointros_I1_J,axiom, % 0.30/0.49 ! [A2: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ). % 0.30/0.49 % 0.30/0.49 % invar_vebt.intros(1) % 0.30/0.49 thf(fact_688_vebt__buildup_Osimps_I2_J,axiom, % 0.30/0.49 ( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.49 = ( vEBT_Leaf @ $false @ $false ) ) ). % 0.30/0.49 % 0.30/0.49 % vebt_buildup.simps(2) % 0.30/0.49 thf(fact_689_invar__vebt_Osimps,axiom, % 0.30/0.49 ( vEBT_invar_vebt % 0.30/0.49 = ( ^ [A1: vEBT_VEBT,A22: nat] : % 0.30/0.49 ( ( ? [A4: $o,B4: $o] : % 0.30/0.49 ( A1 % 0.30/0.49 = ( vEBT_Leaf @ A4 @ B4 ) ) % 0.30/0.49 & ( A22 % 0.30/0.49 = ( suc @ ( zero_zero @ nat ) ) ) ) % 0.30/0.49 | ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT] : % 0.30/0.49 ( ( A1 % 0.30/0.49 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A22 @ TreeList @ Summary3 ) ) % 0.30/0.49 & ! [X: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 0.30/0.49 => ( vEBT_invar_vebt @ X @ N ) ) % 0.30/0.49 & ( vEBT_invar_vebt @ Summary3 @ N ) % 0.30/0.49 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) % 0.30/0.49 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) % 0.30/0.49 & ( A22 % 0.30/0.49 = ( plus_plus @ nat @ N @ N ) ) % 0.30/0.49 & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 ) % 0.30/0.49 & ! [X: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 0.30/0.49 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.30/0.49 | ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT] : % 0.30/0.49 ( ( A1 % 0.30/0.49 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A22 @ TreeList @ Summary3 ) ) % 0.30/0.49 & ! [X: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 0.30/0.49 => ( vEBT_invar_vebt @ X @ N ) ) % 0.30/0.49 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) ) % 0.30/0.49 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) % 0.30/0.49 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) % 0.30/0.49 & ( A22 % 0.30/0.49 = ( plus_plus @ nat @ N @ ( suc @ N ) ) ) % 0.30/0.49 & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 ) % 0.30/0.49 & ! [X: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 0.30/0.49 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.30/0.49 | ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] : % 0.30/0.49 ( ( A1 % 0.30/0.49 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) ) % 0.30/0.49 & ! [X: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 0.30/0.49 => ( vEBT_invar_vebt @ X @ N ) ) % 0.30/0.49 & ( vEBT_invar_vebt @ Summary3 @ N ) % 0.30/0.49 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) % 0.30/0.49 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) % 0.30/0.49 & ( A22 % 0.30/0.49 = ( plus_plus @ nat @ N @ N ) ) % 0.30/0.49 & ! [I3: nat] : % 0.30/0.49 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) % 0.30/0.49 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X6 ) ) % 0.30/0.49 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) ) % 0.30/0.49 & ( ( Mi3 = Ma3 ) % 0.30/0.49 => ! [X: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 0.30/0.49 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.30/0.49 & ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 0.30/0.49 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A22 ) ) % 0.30/0.49 & ( ( Mi3 != Ma3 ) % 0.30/0.49 => ! [I3: nat] : % 0.30/0.49 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) % 0.30/0.49 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N ) % 0.30/0.49 = I3 ) % 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) ) % 0.30/0.49 & ! [X: nat] : % 0.30/0.49 ( ( ( ( vEBT_VEBT_high @ X @ N ) % 0.30/0.49 = I3 ) % 0.30/0.49 & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X @ N ) ) ) % 0.30/0.49 => ( ( ord_less @ nat @ Mi3 @ X ) % 0.30/0.49 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) % 0.30/0.49 | ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] : % 0.30/0.49 ( ( A1 % 0.30/0.49 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) ) % 0.30/0.49 & ! [X: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 0.30/0.49 => ( vEBT_invar_vebt @ X @ N ) ) % 0.30/0.49 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) ) % 0.30/0.49 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) % 0.30/0.49 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) % 0.30/0.49 & ( A22 % 0.30/0.49 = ( plus_plus @ nat @ N @ ( suc @ N ) ) ) % 0.30/0.49 & ! [I3: nat] : % 0.30/0.49 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) % 0.30/0.49 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X6 ) ) % 0.30/0.49 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) ) % 0.30/0.49 & ( ( Mi3 = Ma3 ) % 0.30/0.49 => ! [X: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 0.30/0.49 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.30/0.49 & ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 0.30/0.49 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A22 ) ) % 0.30/0.49 & ( ( Mi3 != Ma3 ) % 0.30/0.49 => ! [I3: nat] : % 0.30/0.49 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) % 0.30/0.49 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N ) % 0.30/0.49 = I3 ) % 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) ) % 0.30/0.49 & ! [X: nat] : % 0.30/0.49 ( ( ( ( vEBT_VEBT_high @ X @ N ) % 0.30/0.49 = I3 ) % 0.30/0.49 & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X @ N ) ) ) % 0.30/0.49 => ( ( ord_less @ nat @ Mi3 @ X ) % 0.30/0.49 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % invar_vebt.simps % 0.30/0.49 thf(fact_690_invar__vebt_Ocases,axiom, % 0.30/0.49 ! [A12: vEBT_VEBT,A23: nat] : % 0.30/0.49 ( ( vEBT_invar_vebt @ A12 @ A23 ) % 0.30/0.49 => ( ( ? [A5: $o,B5: $o] : % 0.30/0.49 ( A12 % 0.30/0.49 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.49 => ( A23 % 0.30/0.49 != ( suc @ ( zero_zero @ nat ) ) ) ) % 0.30/0.49 => ( ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] : % 0.30/0.49 ( ( A12 % 0.30/0.49 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) ) % 0.30/0.49 => ( ( A23 = Deg2 ) % 0.30/0.49 => ( ! [X2: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.30/0.49 => ( vEBT_invar_vebt @ X2 @ N3 ) ) % 0.30/0.49 => ( ( vEBT_invar_vebt @ Summary2 @ M3 ) % 0.30/0.49 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 0.30/0.49 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 0.30/0.49 => ( ( M3 = N3 ) % 0.30/0.49 => ( ( Deg2 % 0.30/0.49 = ( plus_plus @ nat @ N3 @ M3 ) ) % 0.30/0.49 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 ) % 0.30/0.49 => ~ ! [X2: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.30/0.49 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) ) % 0.30/0.49 => ( ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] : % 0.30/0.49 ( ( A12 % 0.30/0.49 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) ) % 0.30/0.49 => ( ( A23 = Deg2 ) % 0.30/0.49 => ( ! [X2: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.30/0.49 => ( vEBT_invar_vebt @ X2 @ N3 ) ) % 0.30/0.49 => ( ( vEBT_invar_vebt @ Summary2 @ M3 ) % 0.30/0.49 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 0.30/0.49 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 0.30/0.49 => ( ( M3 % 0.30/0.49 = ( suc @ N3 ) ) % 0.30/0.49 => ( ( Deg2 % 0.30/0.49 = ( plus_plus @ nat @ N3 @ M3 ) ) % 0.30/0.49 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 ) % 0.30/0.49 => ~ ! [X2: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.30/0.49 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) ) % 0.30/0.49 => ( ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] : % 0.30/0.49 ( ( A12 % 0.30/0.49 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) ) % 0.30/0.49 => ( ( A23 = Deg2 ) % 0.30/0.49 => ( ! [X2: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.30/0.49 => ( vEBT_invar_vebt @ X2 @ N3 ) ) % 0.30/0.49 => ( ( vEBT_invar_vebt @ Summary2 @ M3 ) % 0.30/0.49 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 0.30/0.49 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 0.30/0.49 => ( ( M3 = N3 ) % 0.30/0.49 => ( ( Deg2 % 0.30/0.49 = ( plus_plus @ nat @ N3 @ M3 ) ) % 0.30/0.49 => ( ! [I5: nat] : % 0.30/0.49 ( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 0.30/0.49 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ X6 ) ) % 0.30/0.49 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) ) % 0.30/0.49 => ( ( ( Mi2 = Ma2 ) % 0.30/0.49 => ! [X2: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.30/0.49 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) % 0.30/0.49 => ( ( ord_less_eq @ nat @ Mi2 @ Ma2 ) % 0.30/0.49 => ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 0.30/0.49 => ~ ( ( Mi2 != Ma2 ) % 0.30/0.49 => ! [I5: nat] : % 0.30/0.49 ( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 0.30/0.49 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 ) % 0.30/0.49 = I5 ) % 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) ) % 0.30/0.49 & ! [X2: nat] : % 0.30/0.49 ( ( ( ( vEBT_VEBT_high @ X2 @ N3 ) % 0.30/0.49 = I5 ) % 0.30/0.49 & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) ) % 0.30/0.49 => ( ( ord_less @ nat @ Mi2 @ X2 ) % 0.30/0.49 & ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) % 0.30/0.49 => ~ ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] : % 0.30/0.49 ( ( A12 % 0.30/0.49 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) ) % 0.30/0.49 => ( ( A23 = Deg2 ) % 0.30/0.49 => ( ! [X2: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.30/0.49 => ( vEBT_invar_vebt @ X2 @ N3 ) ) % 0.30/0.49 => ( ( vEBT_invar_vebt @ Summary2 @ M3 ) % 0.30/0.49 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 0.30/0.49 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 0.30/0.49 => ( ( M3 % 0.30/0.49 = ( suc @ N3 ) ) % 0.30/0.49 => ( ( Deg2 % 0.30/0.49 = ( plus_plus @ nat @ N3 @ M3 ) ) % 0.30/0.49 => ( ! [I5: nat] : % 0.30/0.49 ( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 0.30/0.49 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ X6 ) ) % 0.30/0.49 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) ) % 0.30/0.49 => ( ( ( Mi2 = Ma2 ) % 0.30/0.49 => ! [X2: vEBT_VEBT] : % 0.30/0.49 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.30/0.49 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) % 0.30/0.49 => ( ( ord_less_eq @ nat @ Mi2 @ Ma2 ) % 0.30/0.49 => ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 0.30/0.49 => ~ ( ( Mi2 != Ma2 ) % 0.30/0.49 => ! [I5: nat] : % 0.30/0.49 ( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 0.30/0.49 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 ) % 0.30/0.49 = I5 ) % 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) ) % 0.30/0.49 & ! [X2: nat] : % 0.30/0.49 ( ( ( ( vEBT_VEBT_high @ X2 @ N3 ) % 0.30/0.49 = I5 ) % 0.30/0.49 & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) ) % 0.30/0.49 => ( ( ord_less @ nat @ Mi2 @ X2 ) % 0.30/0.49 & ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % invar_vebt.cases % 0.30/0.49 thf(fact_691_VEBT__internal_OminNull_Ocases,axiom, % 0.30/0.49 ! [X3: vEBT_VEBT] : % 0.30/0.49 ( ( X3 % 0.30/0.49 != ( vEBT_Leaf @ $false @ $false ) ) % 0.30/0.49 => ( ! [Uv2: $o] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( vEBT_Leaf @ $true @ Uv2 ) ) % 0.30/0.49 => ( ! [Uu2: $o] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( vEBT_Leaf @ Uu2 @ $true ) ) % 0.30/0.49 => ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) % 0.30/0.49 => ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 0.30/0.49 ( X3 % 0.30/0.49 != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ). % 0.30/0.49 % 0.30/0.49 % VEBT_internal.minNull.cases % 0.30/0.49 thf(fact_692_num_Osize_I5_J,axiom, % 0.30/0.49 ! [X22: num] : % 0.30/0.49 ( ( size_size @ num @ ( bit0 @ X22 ) ) % 0.30/0.49 = ( plus_plus @ nat @ ( size_size @ num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % num.size(5) % 0.30/0.50 thf(fact_693_invar__vebt_Ointros_I4_J,axiom, % 0.30/0.50 ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] : % 0.30/0.50 ( ! [X5: vEBT_VEBT] : % 0.30/0.50 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 0.30/0.50 => ( vEBT_invar_vebt @ X5 @ N2 ) ) % 0.30/0.50 => ( ( vEBT_invar_vebt @ Summary @ M ) % 0.30/0.50 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) % 0.30/0.50 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 0.30/0.50 => ( ( M = N2 ) % 0.30/0.50 => ( ( Deg % 0.30/0.50 = ( plus_plus @ nat @ N2 @ M ) ) % 0.30/0.50 => ( ! [I4: nat] : % 0.30/0.50 ( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 0.30/0.50 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) ) % 0.30/0.50 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) ) % 0.30/0.50 => ( ( ( Mi = Ma ) % 0.30/0.50 => ! [X5: vEBT_VEBT] : % 0.30/0.50 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 0.30/0.50 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) % 0.30/0.50 => ( ( ord_less_eq @ nat @ Mi @ Ma ) % 0.30/0.50 => ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) % 0.30/0.50 => ( ( ( Mi != Ma ) % 0.30/0.50 => ! [I4: nat] : % 0.30/0.50 ( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 0.30/0.50 => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 ) % 0.30/0.50 = I4 ) % 0.30/0.50 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) ) % 0.30/0.50 & ! [X5: nat] : % 0.30/0.50 ( ( ( ( vEBT_VEBT_high @ X5 @ N2 ) % 0.30/0.50 = I4 ) % 0.30/0.50 & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) ) % 0.30/0.50 => ( ( ord_less @ nat @ Mi @ X5 ) % 0.30/0.50 & ( ord_less_eq @ nat @ X5 @ Ma ) ) ) ) ) ) % 0.30/0.50 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % invar_vebt.intros(4) % 0.30/0.50 thf(fact_694_VEBT__internal_OminNull_Oelims_I3_J,axiom, % 0.30/0.50 ! [X3: vEBT_VEBT] : % 0.30/0.50 ( ~ ( vEBT_VEBT_minNull @ X3 ) % 0.30/0.50 => ( ! [Uv2: $o] : % 0.30/0.50 ( X3 % 0.30/0.50 != ( vEBT_Leaf @ $true @ Uv2 ) ) % 0.30/0.50 => ( ! [Uu2: $o] : % 0.30/0.50 ( X3 % 0.30/0.50 != ( vEBT_Leaf @ Uu2 @ $true ) ) % 0.30/0.50 => ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 0.30/0.50 ( X3 % 0.30/0.50 != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % VEBT_internal.minNull.elims(3) % 0.30/0.50 thf(fact_695_VEBT__internal_OminNull_Oelims_I2_J,axiom, % 0.30/0.50 ! [X3: vEBT_VEBT] : % 0.30/0.50 ( ( vEBT_VEBT_minNull @ X3 ) % 0.30/0.50 => ( ( X3 % 0.30/0.50 != ( vEBT_Leaf @ $false @ $false ) ) % 0.30/0.50 => ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 0.30/0.50 ( X3 % 0.30/0.50 != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % VEBT_internal.minNull.elims(2) % 0.30/0.50 thf(fact_696_invar__vebt_Ointros_I5_J,axiom, % 0.30/0.50 ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] : % 0.30/0.50 ( ! [X5: vEBT_VEBT] : % 0.30/0.50 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 0.30/0.50 => ( vEBT_invar_vebt @ X5 @ N2 ) ) % 0.30/0.50 => ( ( vEBT_invar_vebt @ Summary @ M ) % 0.30/0.50 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) % 0.30/0.50 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 0.30/0.50 => ( ( M % 0.30/0.50 = ( suc @ N2 ) ) % 0.30/0.50 => ( ( Deg % 0.30/0.50 = ( plus_plus @ nat @ N2 @ M ) ) % 0.30/0.50 => ( ! [I4: nat] : % 0.30/0.50 ( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 0.30/0.50 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) ) % 0.30/0.50 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) ) % 0.30/0.50 => ( ( ( Mi = Ma ) % 0.30/0.50 => ! [X5: vEBT_VEBT] : % 0.30/0.50 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 0.30/0.50 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) % 0.30/0.50 => ( ( ord_less_eq @ nat @ Mi @ Ma ) % 0.30/0.50 => ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) % 0.30/0.50 => ( ( ( Mi != Ma ) % 0.30/0.50 => ! [I4: nat] : % 0.30/0.50 ( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 0.30/0.50 => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 ) % 0.30/0.50 = I4 ) % 0.30/0.50 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) ) % 0.30/0.50 & ! [X5: nat] : % 0.30/0.50 ( ( ( ( vEBT_VEBT_high @ X5 @ N2 ) % 0.30/0.50 = I4 ) % 0.30/0.50 & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) ) % 0.30/0.50 => ( ( ord_less @ nat @ Mi @ X5 ) % 0.30/0.50 & ( ord_less_eq @ nat @ X5 @ Ma ) ) ) ) ) ) % 0.30/0.50 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % invar_vebt.intros(5) % 0.30/0.50 thf(fact_697_set__encode__inf,axiom, % 0.30/0.50 ! [A3: set @ nat] : % 0.30/0.50 ( ~ ( finite_finite2 @ nat @ A3 ) % 0.30/0.50 => ( ( nat_set_encode @ A3 ) % 0.30/0.50 = ( zero_zero @ nat ) ) ) ). % 0.30/0.50 % 0.30/0.50 % set_encode_inf % 0.30/0.50 thf(fact_698_pair__lessI1,axiom, % 0.30/0.50 ! [A2: nat,B2: nat,S: nat,T2: nat] : % 0.30/0.50 ( ( ord_less @ nat @ A2 @ B2 ) % 0.30/0.50 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ). % 0.30/0.50 % 0.30/0.50 % pair_lessI1 % 0.30/0.50 thf(fact_699_count__le__length,axiom, % 0.30/0.50 ! [A: $tType,Xs: list @ A,X3: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X3 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ). % 0.30/0.50 % 0.30/0.50 % count_le_length % 0.30/0.50 thf(fact_700_vebt__mint_Ocases,axiom, % 0.30/0.50 ! [X3: vEBT_VEBT] : % 0.30/0.50 ( ! [A5: $o,B5: $o] : % 0.30/0.50 ( X3 % 0.30/0.50 != ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.50 => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 0.30/0.50 ( X3 % 0.30/0.50 != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 0.30/0.50 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 0.30/0.50 ( X3 % 0.30/0.50 != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % vebt_mint.cases % 0.30/0.50 thf(fact_701_VEBT__internal_OminNull_Oelims_I1_J,axiom, % 0.30/0.50 ! [X3: vEBT_VEBT,Y: $o] : % 0.30/0.50 ( ( ( vEBT_VEBT_minNull @ X3 ) % 0.30/0.50 = Y ) % 0.30/0.50 => ( ( ( X3 % 0.30/0.50 = ( vEBT_Leaf @ $false @ $false ) ) % 0.30/0.50 => ~ Y ) % 0.30/0.50 => ( ( ? [Uv2: $o] : % 0.30/0.50 ( X3 % 0.30/0.50 = ( vEBT_Leaf @ $true @ Uv2 ) ) % 0.30/0.50 => Y ) % 0.30/0.50 => ( ( ? [Uu2: $o] : % 0.30/0.50 ( X3 % 0.30/0.50 = ( vEBT_Leaf @ Uu2 @ $true ) ) % 0.30/0.50 => Y ) % 0.30/0.50 => ( ( ? [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 0.30/0.50 ( X3 % 0.30/0.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) % 0.30/0.50 => ~ Y ) % 0.30/0.50 => ~ ( ? [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 0.30/0.50 ( X3 % 0.30/0.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) % 0.30/0.50 => Y ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % VEBT_internal.minNull.elims(1) % 0.30/0.50 thf(fact_702_length__code,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( size_size @ ( list @ A ) ) % 0.30/0.50 = ( gen_length @ A @ ( zero_zero @ nat ) ) ) ). % 0.30/0.50 % 0.30/0.50 % length_code % 0.30/0.50 thf(fact_703_sum__power2__eq__zero__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_idom @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 = ( ( X3 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 & ( Y % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % sum_power2_eq_zero_iff % 0.30/0.50 thf(fact_704_zero__less__power2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_idom @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.30/0.50 = ( A2 % 0.30/0.50 != ( zero_zero @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_less_power2 % 0.30/0.50 thf(fact_705_power2__less__eq__zero__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_idom @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) % 0.30/0.50 = ( A2 % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power2_less_eq_zero_iff % 0.30/0.50 thf(fact_706_power2__eq__iff__nonneg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.50 => ( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.50 = ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.30/0.50 = ( X3 = Y ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power2_eq_iff_nonneg % 0.30/0.50 thf(fact_707_zero__eq__power2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiri2026040879449505780visors @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 = ( A2 % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_eq_power2 % 0.30/0.50 thf(fact_708_power__mono__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,B2: A,N2: nat] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.50 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) % 0.30/0.50 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_mono_iff % 0.30/0.50 thf(fact_709_power__eq__0__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiri2026040879449505780visors @ A ) % 0.30/0.50 => ! [A2: A,N2: nat] : % 0.30/0.50 ( ( ( power_power @ A @ A2 @ N2 ) % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 = ( ( A2 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_eq_0_iff % 0.30/0.50 thf(fact_710_member__inv,axiom, % 0.30/0.50 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : % 0.30/0.50 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 ) % 0.30/0.50 => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) % 0.30/0.50 & ( ( X3 = Mi ) % 0.30/0.50 | ( X3 = Ma ) % 0.30/0.50 | ( ( ord_less @ nat @ X3 @ Ma ) % 0.30/0.50 & ( ord_less @ nat @ Mi @ X3 ) % 0.30/0.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) % 0.30/0.50 & ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % member_inv % 0.30/0.50 thf(fact_711_both__member__options__ding,axiom, % 0.30/0.50 ! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X3: nat] : % 0.30/0.50 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 ) % 0.30/0.50 => ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) % 0.30/0.50 => ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.30/0.50 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % both_member_options_ding % 0.30/0.50 thf(fact_712_nat__zero__less__power__iff,axiom, % 0.30/0.50 ! [X3: nat,N2: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X3 @ N2 ) ) % 0.30/0.50 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 ) % 0.30/0.50 | ( N2 % 0.30/0.50 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nat_zero_less_power_iff % 0.30/0.50 thf(fact_713_pow__sum,axiom, % 0.30/0.50 ! [A2: nat,B2: nat] : % 0.30/0.50 ( ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) ) % 0.30/0.50 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) ) ). % 0.30/0.50 % 0.30/0.50 % pow_sum % 0.30/0.50 thf(fact_714_high__def,axiom, % 0.30/0.50 ( vEBT_VEBT_high % 0.30/0.50 = ( ^ [X: nat,N: nat] : ( divide_divide @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % high_def % 0.30/0.50 thf(fact_715_division__ring__divide__zero,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( division_ring @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 = ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % division_ring_divide_zero % 0.30/0.50 thf(fact_716_divide__cancel__right,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [A2: A,C2: A,B2: A] : % 0.30/0.50 ( ( ( divide_divide @ A @ A2 @ C2 ) % 0.30/0.50 = ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.50 = ( ( C2 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 | ( A2 = B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_cancel_right % 0.30/0.50 thf(fact_717_divide__cancel__left,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [C2: A,A2: A,B2: A] : % 0.30/0.50 ( ( ( divide_divide @ A @ C2 @ A2 ) % 0.30/0.50 = ( divide_divide @ A @ C2 @ B2 ) ) % 0.30/0.50 = ( ( C2 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 | ( A2 = B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_cancel_left % 0.30/0.50 thf(fact_718_div__by__0,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semidom_divide @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 = ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_by_0 % 0.30/0.50 thf(fact_719_divide__eq__0__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ( divide_divide @ A @ A2 @ B2 ) % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 = ( ( A2 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 | ( B2 % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_eq_0_iff % 0.30/0.50 thf(fact_720_div__0,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semidom_divide @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 = ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_0 % 0.30/0.50 thf(fact_721_power__0__Suc,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiring_1 @ A ) % 0.30/0.50 => ! [N2: nat] : % 0.30/0.50 ( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N2 ) ) % 0.30/0.50 = ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_0_Suc % 0.30/0.50 thf(fact_722_power__zero__numeral,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiring_1 @ A ) % 0.30/0.50 => ! [K: num] : % 0.30/0.50 ( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ K ) ) % 0.30/0.50 = ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_zero_numeral % 0.30/0.50 thf(fact_723_power__Suc0__right,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( monoid_mult @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.50 = A2 ) ) ). % 0.30/0.50 % 0.30/0.50 % power_Suc0_right % 0.30/0.50 thf(fact_724_power__Suc__0,axiom, % 0.30/0.50 ! [N2: nat] : % 0.30/0.50 ( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 0.30/0.50 = ( suc @ ( zero_zero @ nat ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_Suc_0 % 0.30/0.50 thf(fact_725_nat__power__eq__Suc__0__iff,axiom, % 0.30/0.50 ! [X3: nat,M: nat] : % 0.30/0.50 ( ( ( power_power @ nat @ X3 @ M ) % 0.30/0.50 = ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.50 = ( ( M % 0.30/0.50 = ( zero_zero @ nat ) ) % 0.30/0.50 | ( X3 % 0.30/0.50 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nat_power_eq_Suc_0_iff % 0.30/0.50 thf(fact_726_add__divide__distrib,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( division_ring @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.50 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % add_divide_distrib % 0.30/0.50 thf(fact_727_divide__right__mono__neg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_right_mono_neg % 0.30/0.50 thf(fact_728_divide__nonpos__nonpos,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_nonpos_nonpos % 0.30/0.50 thf(fact_729_divide__nonpos__nonneg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.50 => ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_nonpos_nonneg % 0.30/0.50 thf(fact_730_divide__nonneg__nonpos,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_nonneg_nonpos % 0.30/0.50 thf(fact_731_divide__nonneg__nonneg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_nonneg_nonneg % 0.30/0.50 thf(fact_732_zero__le__divide__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) ) % 0.30/0.50 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) % 0.30/0.50 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_le_divide_iff % 0.30/0.50 thf(fact_733_divide__right__mono,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.50 => ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_right_mono % 0.30/0.50 thf(fact_734_divide__le__0__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) % 0.30/0.50 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) % 0.30/0.50 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_le_0_iff % 0.30/0.50 thf(fact_735_divide__neg__neg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less @ A @ X3 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_neg_neg % 0.30/0.50 thf(fact_736_divide__neg__pos,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less @ A @ X3 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.50 => ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_neg_pos % 0.30/0.50 thf(fact_737_divide__pos__neg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.50 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_pos_neg % 0.30/0.50 thf(fact_738_divide__pos__pos,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.50 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_pos_pos % 0.30/0.50 thf(fact_739_divide__less__0__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) % 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) % 0.30/0.50 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_less_0_iff % 0.30/0.50 thf(fact_740_divide__less__cancel,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,C2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.50 => ( ord_less @ A @ A2 @ B2 ) ) % 0.30/0.50 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less @ A @ B2 @ A2 ) ) % 0.30/0.50 & ( C2 % 0.30/0.50 != ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_less_cancel % 0.30/0.50 thf(fact_741_zero__less__divide__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) ) % 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) % 0.30/0.50 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_less_divide_iff % 0.30/0.50 thf(fact_742_divide__strict__right__mono,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.50 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_strict_right_mono % 0.30/0.50 thf(fact_743_divide__strict__right__mono__neg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [B2: A,A2: A,C2: A] : % 0.30/0.50 ( ( ord_less @ A @ B2 @ A2 ) % 0.30/0.50 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_strict_right_mono_neg % 0.30/0.50 thf(fact_744_frac__le,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [Y: A,X3: A,W2: A,Z2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.50 => ( ( ord_less_eq @ A @ X3 @ Y ) % 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ W2 @ Z2 ) % 0.30/0.50 => ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % frac_le % 0.30/0.50 thf(fact_745_frac__less,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A,Y: A,W2: A,Z2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.50 => ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ W2 @ Z2 ) % 0.30/0.50 => ( ord_less @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % frac_less % 0.30/0.50 thf(fact_746_frac__less2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A,Y: A,W2: A,Z2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ X3 @ Y ) % 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 ) % 0.30/0.50 => ( ( ord_less @ A @ W2 @ Z2 ) % 0.30/0.50 => ( ord_less @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % frac_less2 % 0.30/0.50 thf(fact_747_divide__le__cancel,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,C2: A,B2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.50 => ( ord_less_eq @ A @ A2 @ B2 ) ) % 0.30/0.50 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_le_cancel % 0.30/0.50 thf(fact_748_divide__nonneg__neg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.50 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_nonneg_neg % 0.30/0.50 thf(fact_749_divide__nonneg__pos,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_nonneg_pos % 0.30/0.50 thf(fact_750_divide__nonpos__neg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_nonpos_neg % 0.30/0.50 thf(fact_751_divide__nonpos__pos,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.50 => ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_nonpos_pos % 0.30/0.50 thf(fact_752_field__sum__of__halves,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A] : % 0.30/0.50 ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) % 0.30/0.50 = X3 ) ) ). % 0.30/0.50 % 0.30/0.50 % field_sum_of_halves % 0.30/0.50 thf(fact_753_half__gt__zero,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % half_gt_zero % 0.30/0.50 thf(fact_754_half__gt__zero__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) % 0.30/0.50 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % half_gt_zero_iff % 0.30/0.50 thf(fact_755_field__less__half__sum,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less @ A @ X3 @ Y ) % 0.30/0.50 => ( ord_less @ A @ X3 @ ( divide_divide @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % field_less_half_sum % 0.30/0.50 thf(fact_756_power__not__zero,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiri2026040879449505780visors @ A ) % 0.30/0.50 => ! [A2: A,N2: nat] : % 0.30/0.50 ( ( A2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( power_power @ A @ A2 @ N2 ) % 0.30/0.50 != ( zero_zero @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_not_zero % 0.30/0.50 thf(fact_757_zero__le__power,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,N2: nat] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_le_power % 0.30/0.50 thf(fact_758_power__mono,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,B2: A,N2: nat] : % 0.30/0.50 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_mono % 0.30/0.50 thf(fact_759_zero__less__power,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,N2: nat] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_less_power % 0.30/0.50 thf(fact_760_nat__power__less__imp__less,axiom, % 0.30/0.50 ! [I: nat,M: nat,N2: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I ) % 0.30/0.50 => ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N2 ) ) % 0.30/0.50 => ( ord_less @ nat @ M @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nat_power_less_imp_less % 0.30/0.50 thf(fact_761_power__less__imp__less__base,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,N2: nat,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.50 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_less_imp_less_base % 0.30/0.50 thf(fact_762_power__le__imp__le__base,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,N2: nat,B2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N2 ) ) @ ( power_power @ A @ B2 @ ( suc @ N2 ) ) ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.50 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_le_imp_le_base % 0.30/0.50 thf(fact_763_power__inject__base,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,N2: nat,B2: A] : % 0.30/0.50 ( ( ( power_power @ A @ A2 @ ( suc @ N2 ) ) % 0.30/0.50 = ( power_power @ A @ B2 @ ( suc @ N2 ) ) ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.50 => ( A2 = B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_inject_base % 0.30/0.50 thf(fact_764_zero__power,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiring_1 @ A ) % 0.30/0.50 => ! [N2: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.50 => ( ( power_power @ A @ ( zero_zero @ A ) @ N2 ) % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_power % 0.30/0.50 thf(fact_765_power__gt__expt,axiom, % 0.30/0.50 ! [N2: nat,K: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 0.30/0.50 => ( ord_less @ nat @ K @ ( power_power @ nat @ N2 @ K ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_gt_expt % 0.30/0.50 thf(fact_766_nat__one__le__power,axiom, % 0.30/0.50 ! [I: nat,N2: nat] : % 0.30/0.50 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I ) % 0.30/0.50 => ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nat_one_le_power % 0.30/0.50 thf(fact_767_power__eq__iff__eq__base,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [N2: nat,A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.50 => ( ( ( power_power @ A @ A2 @ N2 ) % 0.30/0.50 = ( power_power @ A @ B2 @ N2 ) ) % 0.30/0.50 = ( A2 = B2 ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_eq_iff_eq_base % 0.30/0.50 thf(fact_768_power__eq__imp__eq__base,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,N2: nat,B2: A] : % 0.30/0.50 ( ( ( power_power @ A @ A2 @ N2 ) % 0.30/0.50 = ( power_power @ A @ B2 @ N2 ) ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.50 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.50 => ( A2 = B2 ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_eq_imp_eq_base % 0.30/0.50 thf(fact_769_zero__power2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiring_1 @ A ) % 0.30/0.50 => ( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.50 = ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_power2 % 0.30/0.50 thf(fact_770_less__exp,axiom, % 0.30/0.50 ! [N2: nat] : ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 0.30/0.50 % 0.30/0.50 % less_exp % 0.30/0.50 thf(fact_771_power2__nat__le__imp__le,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : % 0.30/0.50 ( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N2 ) % 0.30/0.50 => ( ord_less_eq @ nat @ M @ N2 ) ) ). % 0.30/0.50 % 0.30/0.50 % power2_nat_le_imp_le % 0.30/0.50 thf(fact_772_power2__nat__le__eq__le,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : % 0.30/0.50 ( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.30/0.50 = ( ord_less_eq @ nat @ M @ N2 ) ) ). % 0.30/0.50 % 0.30/0.50 % power2_nat_le_eq_le % 0.30/0.50 thf(fact_773_self__le__ge2__pow,axiom, % 0.30/0.50 ! [K: nat,M: nat] : % 0.30/0.50 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) % 0.30/0.50 => ( ord_less_eq @ nat @ M @ ( power_power @ nat @ K @ M ) ) ) ). % 0.30/0.50 % 0.30/0.50 % self_le_ge2_pow % 0.30/0.50 thf(fact_774_power2__le__imp__le,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.50 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power2_le_imp_le % 0.30/0.50 thf(fact_775_power2__eq__imp__eq,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.50 = ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.50 => ( X3 = Y ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power2_eq_imp_eq % 0.30/0.50 thf(fact_776_zero__le__power2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_idom @ A ) % 0.30/0.50 => ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_le_power2 % 0.30/0.50 thf(fact_777_power__strict__mono,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,B2: A,N2: nat] : % 0.30/0.50 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.50 => ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_strict_mono % 0.30/0.50 thf(fact_778_power2__less__0,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_idom @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ~ ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power2_less_0 % 0.30/0.50 thf(fact_779_power2__less__imp__less,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.50 => ( ord_less @ A @ X3 @ Y ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power2_less_imp_less % 0.30/0.50 thf(fact_780_sum__power2__le__zero__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_idom @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) % 0.30/0.50 = ( ( X3 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 & ( Y % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % sum_power2_le_zero_iff % 0.30/0.50 thf(fact_781_sum__power2__ge__zero,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_idom @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % sum_power2_ge_zero % 0.30/0.50 thf(fact_782_not__sum__power2__lt__zero,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_idom @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % not_sum_power2_lt_zero % 0.30/0.50 thf(fact_783_sum__power2__gt__zero__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_idom @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.30/0.50 = ( ( X3 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 | ( Y % 0.30/0.50 != ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % sum_power2_gt_zero_iff % 0.30/0.50 thf(fact_784_both__member__options__from__chilf__to__complete__tree,axiom, % 0.30/0.50 ! [X3: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] : % 0.30/0.50 ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) % 0.30/0.50 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg ) % 0.30/0.50 => ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.30/0.50 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % both_member_options_from_chilf_to_complete_tree % 0.30/0.50 thf(fact_785_add__self__div__2,axiom, % 0.30/0.50 ! [M: nat] : % 0.30/0.50 ( ( divide_divide @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.50 = M ) ). % 0.30/0.50 % 0.30/0.50 % add_self_div_2 % 0.30/0.50 thf(fact_786_div2__Suc__Suc,axiom, % 0.30/0.50 ! [M: nat] : % 0.30/0.50 ( ( divide_divide @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.50 = ( suc @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div2_Suc_Suc % 0.30/0.50 thf(fact_787_both__member__options__from__complete__tree__to__child,axiom, % 0.30/0.50 ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : % 0.30/0.50 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg ) % 0.30/0.50 => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 ) % 0.30/0.50 => ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.30/0.50 | ( X3 = Mi ) % 0.30/0.50 | ( X3 = Ma ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % both_member_options_from_complete_tree_to_child % 0.30/0.50 thf(fact_788_semiring__norm_I76_J,axiom, % 0.30/0.50 ! [N2: num] : ( ord_less @ num @ one2 @ ( bit0 @ N2 ) ) ). % 0.30/0.50 % 0.30/0.50 % semiring_norm(76) % 0.30/0.50 thf(fact_789_div__less,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : % 0.30/0.50 ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.50 => ( ( divide_divide @ nat @ M @ N2 ) % 0.30/0.50 = ( zero_zero @ nat ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_less % 0.30/0.50 thf(fact_790_div__by__Suc__0,axiom, % 0.30/0.50 ! [M: nat] : % 0.30/0.50 ( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.50 = M ) ). % 0.30/0.50 % 0.30/0.50 % div_by_Suc_0 % 0.30/0.50 thf(fact_791_set__n__deg__not__0,axiom, % 0.30/0.50 ! [TreeList2: list @ vEBT_VEBT,N2: nat,M: nat] : % 0.30/0.50 ( ! [X5: vEBT_VEBT] : % 0.30/0.50 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 0.30/0.50 => ( vEBT_invar_vebt @ X5 @ N2 ) ) % 0.30/0.50 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) % 0.30/0.50 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 0.30/0.50 => ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % set_n_deg_not_0 % 0.30/0.50 thf(fact_792_Suc__n__div__2__gt__zero,axiom, % 0.30/0.50 ! [N2: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.50 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % Suc_n_div_2_gt_zero % 0.30/0.50 thf(fact_793_div__2__gt__zero,axiom, % 0.30/0.50 ! [N2: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 0.30/0.50 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_2_gt_zero % 0.30/0.50 thf(fact_794_deg1Leaf,axiom, % 0.30/0.50 ! [T2: vEBT_VEBT] : % 0.30/0.50 ( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) ) % 0.30/0.50 = ( ? [A4: $o,B4: $o] : % 0.30/0.50 ( T2 % 0.30/0.50 = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % deg1Leaf % 0.30/0.50 thf(fact_795_deg__1__Leaf,axiom, % 0.30/0.50 ! [T2: vEBT_VEBT] : % 0.30/0.50 ( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) ) % 0.30/0.50 => ? [A5: $o,B5: $o] : % 0.30/0.50 ( T2 % 0.30/0.50 = ( vEBT_Leaf @ A5 @ B5 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % deg_1_Leaf % 0.30/0.50 thf(fact_796_deg__1__Leafy,axiom, % 0.30/0.50 ! [T2: vEBT_VEBT,N2: nat] : % 0.30/0.50 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.50 => ( ( N2 % 0.30/0.50 = ( one_one @ nat ) ) % 0.30/0.50 => ? [A5: $o,B5: $o] : % 0.30/0.50 ( T2 % 0.30/0.50 = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % deg_1_Leafy % 0.30/0.50 thf(fact_797_div__by__1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semidom_divide @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( divide_divide @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.50 = A2 ) ) ). % 0.30/0.50 % 0.30/0.50 % div_by_1 % 0.30/0.50 thf(fact_798_semiring__norm_I78_J,axiom, % 0.30/0.50 ! [M: num,N2: num] : % 0.30/0.50 ( ( ord_less @ num @ ( bit0 @ M ) @ ( bit0 @ N2 ) ) % 0.30/0.50 = ( ord_less @ num @ M @ N2 ) ) ). % 0.30/0.50 % 0.30/0.50 % semiring_norm(78) % 0.30/0.50 thf(fact_799_semiring__norm_I75_J,axiom, % 0.30/0.50 ! [M: num] : % 0.30/0.50 ~ ( ord_less @ num @ M @ one2 ) ). % 0.30/0.50 % 0.30/0.50 % semiring_norm(75) % 0.30/0.50 thf(fact_800_divide__eq__1__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ( divide_divide @ A @ A2 @ B2 ) % 0.30/0.50 = ( one_one @ A ) ) % 0.30/0.50 = ( ( B2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 & ( A2 = B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_eq_1_iff % 0.30/0.50 thf(fact_801_div__self,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semidom_divide @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( A2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ A2 @ A2 ) % 0.30/0.50 = ( one_one @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_self % 0.30/0.50 thf(fact_802_one__eq__divide__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ( one_one @ A ) % 0.30/0.50 = ( divide_divide @ A @ A2 @ B2 ) ) % 0.30/0.50 = ( ( B2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 & ( A2 = B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % one_eq_divide_iff % 0.30/0.50 thf(fact_803_divide__self,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( division_ring @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( A2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ A2 @ A2 ) % 0.30/0.50 = ( one_one @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_self % 0.30/0.50 thf(fact_804_divide__self__if,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( division_ring @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( ( A2 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ A2 @ A2 ) % 0.30/0.50 = ( zero_zero @ A ) ) ) % 0.30/0.50 & ( ( A2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ A2 @ A2 ) % 0.30/0.50 = ( one_one @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_self_if % 0.30/0.50 thf(fact_805_divide__eq__eq__1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [B2: A,A2: A] : % 0.30/0.50 ( ( ( divide_divide @ A @ B2 @ A2 ) % 0.30/0.50 = ( one_one @ A ) ) % 0.30/0.50 = ( ( A2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 & ( A2 = B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_eq_eq_1 % 0.30/0.50 thf(fact_806_eq__divide__eq__1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [B2: A,A2: A] : % 0.30/0.50 ( ( ( one_one @ A ) % 0.30/0.50 = ( divide_divide @ A @ B2 @ A2 ) ) % 0.30/0.50 = ( ( A2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 & ( A2 = B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % eq_divide_eq_1 % 0.30/0.50 thf(fact_807_one__divide__eq__0__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 = ( A2 % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % one_divide_eq_0_iff % 0.30/0.50 thf(fact_808_zero__eq__1__divide__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( ( zero_zero @ A ) % 0.30/0.50 = ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) % 0.30/0.50 = ( A2 % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_eq_1_divide_iff % 0.30/0.50 thf(fact_809_power__inject__exp,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,M: nat,N2: nat] : % 0.30/0.50 ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.50 => ( ( ( power_power @ A @ A2 @ M ) % 0.30/0.50 = ( power_power @ A @ A2 @ N2 ) ) % 0.30/0.50 = ( M = N2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_inject_exp % 0.30/0.50 thf(fact_810_less__one,axiom, % 0.30/0.50 ! [N2: nat] : % 0.30/0.50 ( ( ord_less @ nat @ N2 @ ( one_one @ nat ) ) % 0.30/0.50 = ( N2 % 0.30/0.50 = ( zero_zero @ nat ) ) ) ). % 0.30/0.50 % 0.30/0.50 % less_one % 0.30/0.50 thf(fact_811_divide__le__0__1__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) ) % 0.30/0.50 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_le_0_1_iff % 0.30/0.50 thf(fact_812_zero__le__divide__1__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) % 0.30/0.50 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_le_divide_1_iff % 0.30/0.50 thf(fact_813_zero__less__divide__1__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) % 0.30/0.50 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_less_divide_1_iff % 0.30/0.50 thf(fact_814_less__divide__eq__1__pos,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) ) % 0.30/0.50 = ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % less_divide_eq_1_pos % 0.30/0.50 thf(fact_815_less__divide__eq__1__neg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) ) % 0.30/0.50 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % less_divide_eq_1_neg % 0.30/0.50 thf(fact_816_divide__less__eq__1__pos,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) ) % 0.30/0.50 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_less_eq_1_pos % 0.30/0.50 thf(fact_817_divide__less__eq__1__neg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) ) % 0.30/0.50 = ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_less_eq_1_neg % 0.30/0.50 thf(fact_818_divide__less__0__1__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) ) % 0.30/0.50 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_less_0_1_iff % 0.30/0.50 thf(fact_819_power__strict__increasing__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [B2: A,X3: nat,Y: nat] : % 0.30/0.50 ( ( ord_less @ A @ ( one_one @ A ) @ B2 ) % 0.30/0.50 => ( ( ord_less @ A @ ( power_power @ A @ B2 @ X3 ) @ ( power_power @ A @ B2 @ Y ) ) % 0.30/0.50 = ( ord_less @ nat @ X3 @ Y ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_strict_increasing_iff % 0.30/0.50 thf(fact_820_le__divide__eq__1__pos,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) ) % 0.30/0.50 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % le_divide_eq_1_pos % 0.30/0.50 thf(fact_821_le__divide__eq__1__neg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) ) % 0.30/0.50 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % le_divide_eq_1_neg % 0.30/0.50 thf(fact_822_divide__le__eq__1__pos,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) ) % 0.30/0.50 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_le_eq_1_pos % 0.30/0.50 thf(fact_823_divide__le__eq__1__neg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) ) % 0.30/0.50 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_le_eq_1_neg % 0.30/0.50 thf(fact_824_one__add__one,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( numeral @ A ) % 0.30/0.50 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) % 0.30/0.50 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % one_add_one % 0.30/0.50 thf(fact_825_power__strict__decreasing__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [B2: A,M: nat,N2: nat] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.50 => ( ( ord_less @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.50 => ( ( ord_less @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N2 ) ) % 0.30/0.50 = ( ord_less @ nat @ N2 @ M ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_strict_decreasing_iff % 0.30/0.50 thf(fact_826_power__increasing__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [B2: A,X3: nat,Y: nat] : % 0.30/0.50 ( ( ord_less @ A @ ( one_one @ A ) @ B2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X3 ) @ ( power_power @ A @ B2 @ Y ) ) % 0.30/0.50 = ( ord_less_eq @ nat @ X3 @ Y ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_increasing_iff % 0.30/0.50 thf(fact_827_Suc__1,axiom, % 0.30/0.50 ( ( suc @ ( one_one @ nat ) ) % 0.30/0.50 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % Suc_1 % 0.30/0.50 thf(fact_828_one__plus__numeral,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( numeral @ A ) % 0.30/0.50 => ! [N2: num] : % 0.30/0.50 ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) ) % 0.30/0.50 = ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % one_plus_numeral % 0.30/0.50 thf(fact_829_numeral__plus__one,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( numeral @ A ) % 0.30/0.50 => ! [N2: num] : % 0.30/0.50 ( ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) ) % 0.30/0.50 = ( numeral_numeral @ A @ ( plus_plus @ num @ N2 @ one2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % numeral_plus_one % 0.30/0.50 thf(fact_830_numeral__le__one__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord181362715937106298miring @ A ) % 0.30/0.50 => ! [N2: num] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) ) % 0.30/0.50 = ( ord_less_eq @ num @ N2 @ one2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % numeral_le_one_iff % 0.30/0.50 thf(fact_831_one__less__numeral__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord181362715937106298miring @ A ) % 0.30/0.50 => ! [N2: num] : % 0.30/0.50 ( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) ) % 0.30/0.50 = ( ord_less @ num @ one2 @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % one_less_numeral_iff % 0.30/0.50 thf(fact_832_one__div__two__eq__zero,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( euclid5411537665997757685th_nat @ A ) % 0.30/0.50 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.50 = ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % one_div_two_eq_zero % 0.30/0.50 thf(fact_833_power__decreasing__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [B2: A,M: nat,N2: nat] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.50 => ( ( ord_less @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N2 ) ) % 0.30/0.50 = ( ord_less_eq @ nat @ N2 @ M ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_decreasing_iff % 0.30/0.50 thf(fact_834_one__reorient,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( one @ A ) % 0.30/0.50 => ! [X3: A] : % 0.30/0.50 ( ( ( one_one @ A ) % 0.30/0.50 = X3 ) % 0.30/0.50 = ( X3 % 0.30/0.50 = ( one_one @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % one_reorient % 0.30/0.50 thf(fact_835_le__numeral__extra_I4_J,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord181362715937106298miring @ A ) % 0.30/0.50 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % le_numeral_extra(4) % 0.30/0.50 thf(fact_836_zero__neq__one,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( zero_neq_one @ A ) % 0.30/0.50 => ( ( zero_zero @ A ) % 0.30/0.50 != ( one_one @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_neq_one % 0.30/0.50 thf(fact_837_less__numeral__extra_I4_J,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord181362715937106298miring @ A ) % 0.30/0.50 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % less_numeral_extra(4) % 0.30/0.50 thf(fact_838_div__add__self2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.50 => ! [B2: A,A2: A] : % 0.30/0.50 ( ( B2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 ) % 0.30/0.50 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_add_self2 % 0.30/0.50 thf(fact_839_div__add__self1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.50 => ! [B2: A,A2: A] : % 0.30/0.50 ( ( B2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 ) % 0.30/0.50 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_add_self1 % 0.30/0.50 thf(fact_840_zero__less__one__class_Ozero__le__one,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( zero_less_one @ A ) % 0.30/0.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_less_one_class.zero_le_one % 0.30/0.50 thf(fact_841_linordered__nonzero__semiring__class_Ozero__le__one,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord181362715937106298miring @ A ) % 0.30/0.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % linordered_nonzero_semiring_class.zero_le_one % 0.30/0.50 thf(fact_842_not__one__le__zero,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord181362715937106298miring @ A ) % 0.30/0.50 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % not_one_le_zero % 0.30/0.50 thf(fact_843_not__one__less__zero,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord181362715937106298miring @ A ) % 0.30/0.50 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % not_one_less_zero % 0.30/0.50 thf(fact_844_zero__less__one,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( zero_less_one @ A ) % 0.30/0.50 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_less_one % 0.30/0.50 thf(fact_845_less__numeral__extra_I1_J,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord181362715937106298miring @ A ) % 0.30/0.50 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % less_numeral_extra(1) % 0.30/0.50 thf(fact_846_one__le__numeral,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord181362715937106298miring @ A ) % 0.30/0.50 => ! [N2: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % one_le_numeral % 0.30/0.50 thf(fact_847_not__numeral__less__one,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord181362715937106298miring @ A ) % 0.30/0.50 => ! [N2: num] : % 0.30/0.50 ~ ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % not_numeral_less_one % 0.30/0.50 thf(fact_848_add__mono1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord181362715937106298miring @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.50 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % add_mono1 % 0.30/0.50 thf(fact_849_less__add__one,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % less_add_one % 0.30/0.50 thf(fact_850_right__inverse__eq,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( division_ring @ A ) % 0.30/0.50 => ! [B2: A,A2: A] : % 0.30/0.50 ( ( B2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ( divide_divide @ A @ A2 @ B2 ) % 0.30/0.50 = ( one_one @ A ) ) % 0.30/0.50 = ( A2 = B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % right_inverse_eq % 0.30/0.50 thf(fact_851_one__plus__numeral__commute,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( numeral @ A ) % 0.30/0.50 => ! [X3: num] : % 0.30/0.50 ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X3 ) ) % 0.30/0.50 = ( plus_plus @ A @ ( numeral_numeral @ A @ X3 ) @ ( one_one @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % one_plus_numeral_commute % 0.30/0.50 thf(fact_852_one__le__power,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,N2: nat] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.50 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % one_le_power % 0.30/0.50 thf(fact_853_power__0,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( power @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( power_power @ A @ A2 @ ( zero_zero @ nat ) ) % 0.30/0.50 = ( one_one @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_0 % 0.30/0.50 thf(fact_854_div__eq__dividend__iff,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.30/0.50 => ( ( ( divide_divide @ nat @ M @ N2 ) % 0.30/0.50 = M ) % 0.30/0.50 = ( N2 % 0.30/0.50 = ( one_one @ nat ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_eq_dividend_iff % 0.30/0.50 thf(fact_855_div__less__dividend,axiom, % 0.30/0.50 ! [N2: nat,M: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( one_one @ nat ) @ N2 ) % 0.30/0.50 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.30/0.50 => ( ord_less @ nat @ ( divide_divide @ nat @ M @ N2 ) @ M ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_less_dividend % 0.30/0.50 thf(fact_856_One__nat__def,axiom, % 0.30/0.50 ( ( one_one @ nat ) % 0.30/0.50 = ( suc @ ( zero_zero @ nat ) ) ) ). % 0.30/0.50 % 0.30/0.50 % One_nat_def % 0.30/0.50 thf(fact_857_Suc__eq__plus1,axiom, % 0.30/0.50 ( suc % 0.30/0.50 = ( ^ [N: nat] : ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % Suc_eq_plus1 % 0.30/0.50 thf(fact_858_plus__1__eq__Suc,axiom, % 0.30/0.50 ( ( plus_plus @ nat @ ( one_one @ nat ) ) % 0.30/0.50 = suc ) ). % 0.30/0.50 % 0.30/0.50 % plus_1_eq_Suc % 0.30/0.50 thf(fact_859_Suc__eq__plus1__left,axiom, % 0.30/0.50 ( suc % 0.30/0.50 = ( plus_plus @ nat @ ( one_one @ nat ) ) ) ). % 0.30/0.50 % 0.30/0.50 % Suc_eq_plus1_left % 0.30/0.50 thf(fact_860_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom, % 0.30/0.50 ! [Uu: $o,Uv: $o,D2: nat] : % 0.30/0.50 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D2 ) % 0.30/0.50 = ( D2 % 0.30/0.50 = ( one_one @ nat ) ) ) ). % 0.30/0.50 % 0.30/0.50 % VEBT_internal.valid'.simps(1) % 0.30/0.50 thf(fact_861_zero__less__two,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord181362715937106298miring @ A ) % 0.30/0.50 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_less_two % 0.30/0.50 thf(fact_862_power__le__one,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,N2: nat] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.50 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_le_one % 0.30/0.50 thf(fact_863_less__divide__eq__1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [B2: A,A2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) ) % 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 & ( ord_less @ A @ A2 @ B2 ) ) % 0.30/0.50 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 & ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % less_divide_eq_1 % 0.30/0.50 thf(fact_864_divide__less__eq__1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [B2: A,A2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) ) % 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 & ( ord_less @ A @ B2 @ A2 ) ) % 0.30/0.50 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 & ( ord_less @ A @ A2 @ B2 ) ) % 0.30/0.50 | ( A2 % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_less_eq_1 % 0.30/0.50 thf(fact_865_less__half__sum,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.50 => ( ord_less @ A @ A2 @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % less_half_sum % 0.30/0.50 thf(fact_866_gt__half__sum,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.50 => ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % gt_half_sum % 0.30/0.50 thf(fact_867_power__0__left,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiring_1 @ A ) % 0.30/0.50 => ! [N2: nat] : % 0.30/0.50 ( ( ( N2 % 0.30/0.50 = ( zero_zero @ nat ) ) % 0.30/0.50 => ( ( power_power @ A @ ( zero_zero @ A ) @ N2 ) % 0.30/0.50 = ( one_one @ A ) ) ) % 0.30/0.50 & ( ( N2 % 0.30/0.50 != ( zero_zero @ nat ) ) % 0.30/0.50 => ( ( power_power @ A @ ( zero_zero @ A ) @ N2 ) % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_0_left % 0.30/0.50 thf(fact_868_power__gt1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,N2: nat] : % 0.30/0.50 ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.50 => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( suc @ N2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_gt1 % 0.30/0.50 thf(fact_869_power__less__imp__less__exp,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,M: nat,N2: nat] : % 0.30/0.50 ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) % 0.30/0.50 => ( ord_less @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_less_imp_less_exp % 0.30/0.50 thf(fact_870_power__strict__increasing,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [N2: nat,N7: nat,A2: A] : % 0.30/0.50 ( ( ord_less @ nat @ N2 @ N7 ) % 0.30/0.50 => ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.50 => ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ A2 @ N7 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_strict_increasing % 0.30/0.50 thf(fact_871_power__increasing,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [N2: nat,N7: nat,A2: A] : % 0.30/0.50 ( ( ord_less_eq @ nat @ N2 @ N7 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.50 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ A2 @ N7 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_increasing % 0.30/0.50 thf(fact_872_nat__induct__non__zero,axiom, % 0.30/0.50 ! [N2: nat,P: nat > $o] : % 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.50 => ( ( P @ ( one_one @ nat ) ) % 0.30/0.50 => ( ! [N3: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 0.30/0.50 => ( ( P @ N3 ) % 0.30/0.50 => ( P @ ( suc @ N3 ) ) ) ) % 0.30/0.50 => ( P @ N2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nat_induct_non_zero % 0.30/0.50 thf(fact_873_vebt__member_Osimps_I1_J,axiom, % 0.30/0.50 ! [A2: $o,B2: $o,X3: nat] : % 0.30/0.50 ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A2 @ B2 ) @ X3 ) % 0.30/0.50 = ( ( ( X3 % 0.30/0.50 = ( zero_zero @ nat ) ) % 0.30/0.50 => A2 ) % 0.30/0.50 & ( ( X3 % 0.30/0.50 != ( zero_zero @ nat ) ) % 0.30/0.50 => ( ( ( X3 % 0.30/0.50 = ( one_one @ nat ) ) % 0.30/0.50 => B2 ) % 0.30/0.50 & ( X3 % 0.30/0.50 = ( one_one @ nat ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % vebt_member.simps(1) % 0.30/0.50 thf(fact_874_VEBT__internal_Onaive__member_Osimps_I1_J,axiom, % 0.30/0.50 ! [A2: $o,B2: $o,X3: nat] : % 0.30/0.50 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A2 @ B2 ) @ X3 ) % 0.30/0.50 = ( ( ( X3 % 0.30/0.50 = ( zero_zero @ nat ) ) % 0.30/0.50 => A2 ) % 0.30/0.50 & ( ( X3 % 0.30/0.50 != ( zero_zero @ nat ) ) % 0.30/0.50 => ( ( ( X3 % 0.30/0.50 = ( one_one @ nat ) ) % 0.30/0.50 => B2 ) % 0.30/0.50 & ( X3 % 0.30/0.50 = ( one_one @ nat ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % VEBT_internal.naive_member.simps(1) % 0.30/0.50 thf(fact_875_le__divide__eq__1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [B2: A,A2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) ) % 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 & ( ord_less_eq @ A @ A2 @ B2 ) ) % 0.30/0.50 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 & ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % le_divide_eq_1 % 0.30/0.50 thf(fact_876_divide__le__eq__1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [B2: A,A2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) ) % 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 & ( ord_less_eq @ A @ B2 @ A2 ) ) % 0.30/0.50 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 & ( ord_less_eq @ A @ A2 @ B2 ) ) % 0.30/0.50 | ( A2 % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_le_eq_1 % 0.30/0.50 thf(fact_877_power__Suc__le__self,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,N2: nat] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.50 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_Suc_le_self % 0.30/0.50 thf(fact_878_power__Suc__less__one,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,N2: nat] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.50 => ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ N2 ) ) @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_Suc_less_one % 0.30/0.50 thf(fact_879_power__strict__decreasing,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [N2: nat,N7: nat,A2: A] : % 0.30/0.50 ( ( ord_less @ nat @ N2 @ N7 ) % 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.50 => ( ord_less @ A @ ( power_power @ A @ A2 @ N7 ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_strict_decreasing % 0.30/0.50 thf(fact_880_power__decreasing,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [N2: nat,N7: nat,A2: A] : % 0.30/0.50 ( ( ord_less_eq @ nat @ N2 @ N7 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.50 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N7 ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_decreasing % 0.30/0.50 thf(fact_881_power__le__imp__le__exp,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,M: nat,N2: nat] : % 0.30/0.50 ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) % 0.30/0.50 => ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % power_le_imp_le_exp % 0.30/0.50 thf(fact_882_self__le__power,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,N2: nat] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.50 => ( ord_less_eq @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % self_le_power % 0.30/0.50 thf(fact_883_one__less__power,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_semidom @ A ) % 0.30/0.50 => ! [A2: A,N2: nat] : % 0.30/0.50 ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.50 => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % one_less_power % 0.30/0.50 thf(fact_884_nat__1__add__1,axiom, % 0.30/0.50 ( ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) % 0.30/0.50 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nat_1_add_1 % 0.30/0.50 thf(fact_885_vebt__maxt_Osimps_I1_J,axiom, % 0.30/0.50 ! [B2: $o,A2: $o] : % 0.30/0.50 ( ( B2 % 0.30/0.50 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) ) % 0.30/0.50 = ( some @ nat @ ( one_one @ nat ) ) ) ) % 0.30/0.50 & ( ~ B2 % 0.30/0.50 => ( ( A2 % 0.30/0.50 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) ) % 0.30/0.50 = ( some @ nat @ ( zero_zero @ nat ) ) ) ) % 0.30/0.50 & ( ~ A2 % 0.30/0.50 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) ) % 0.30/0.50 = ( none @ nat ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % vebt_maxt.simps(1) % 0.30/0.50 thf(fact_886_vebt__mint_Osimps_I1_J,axiom, % 0.30/0.50 ! [A2: $o,B2: $o] : % 0.30/0.50 ( ( A2 % 0.30/0.50 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) ) % 0.30/0.50 = ( some @ nat @ ( zero_zero @ nat ) ) ) ) % 0.30/0.50 & ( ~ A2 % 0.30/0.50 => ( ( B2 % 0.30/0.50 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) ) % 0.30/0.50 = ( some @ nat @ ( one_one @ nat ) ) ) ) % 0.30/0.50 & ( ~ B2 % 0.30/0.50 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) ) % 0.30/0.50 = ( none @ nat ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % vebt_mint.simps(1) % 0.30/0.50 thf(fact_887_div__le__mono,axiom, % 0.30/0.50 ! [M: nat,N2: nat,K: nat] : % 0.30/0.50 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.50 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N2 @ K ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_le_mono % 0.30/0.50 thf(fact_888_div__le__dividend,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N2 ) @ M ) ). % 0.30/0.50 % 0.30/0.50 % div_le_dividend % 0.30/0.50 thf(fact_889_ex__power__ivl2,axiom, % 0.30/0.50 ! [B2: nat,K: nat] : % 0.30/0.50 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) % 0.30/0.50 => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) % 0.30/0.50 => ? [N3: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K ) % 0.30/0.50 & ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % ex_power_ivl2 % 0.30/0.50 thf(fact_890_ex__power__ivl1,axiom, % 0.30/0.50 ! [B2: nat,K: nat] : % 0.30/0.50 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) % 0.30/0.50 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K ) % 0.30/0.50 => ? [N3: nat] : % 0.30/0.50 ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K ) % 0.30/0.50 & ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % ex_power_ivl1 % 0.30/0.50 thf(fact_891_Euclidean__Division_Odiv__eq__0__iff,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : % 0.30/0.50 ( ( ( divide_divide @ nat @ M @ N2 ) % 0.30/0.50 = ( zero_zero @ nat ) ) % 0.30/0.50 = ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.50 | ( N2 % 0.30/0.50 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % Euclidean_Division.div_eq_0_iff % 0.30/0.50 thf(fact_892_Suc__div__le__mono,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N2 ) @ ( divide_divide @ nat @ ( suc @ M ) @ N2 ) ) ). % 0.30/0.50 % 0.30/0.50 % Suc_div_le_mono % 0.30/0.50 thf(fact_893_vebt__maxt_Oelims,axiom, % 0.30/0.50 ! [X3: vEBT_VEBT,Y: option @ nat] : % 0.30/0.50 ( ( ( vEBT_vebt_maxt @ X3 ) % 0.30/0.50 = Y ) % 0.30/0.50 => ( ! [A5: $o,B5: $o] : % 0.30/0.50 ( ( X3 % 0.30/0.50 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.50 => ~ ( ( B5 % 0.30/0.50 => ( Y % 0.30/0.50 = ( some @ nat @ ( one_one @ nat ) ) ) ) % 0.30/0.50 & ( ~ B5 % 0.30/0.50 => ( ( A5 % 0.30/0.50 => ( Y % 0.30/0.50 = ( some @ nat @ ( zero_zero @ nat ) ) ) ) % 0.30/0.50 & ( ~ A5 % 0.30/0.50 => ( Y % 0.30/0.50 = ( none @ nat ) ) ) ) ) ) ) % 0.30/0.50 => ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 0.30/0.50 ( X3 % 0.30/0.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 0.30/0.50 => ( Y % 0.30/0.50 != ( none @ nat ) ) ) % 0.30/0.50 => ~ ! [Mi2: nat,Ma2: nat] : % 0.30/0.50 ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 0.30/0.50 ( X3 % 0.30/0.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) % 0.30/0.50 => ( Y % 0.30/0.50 != ( some @ nat @ Ma2 ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % vebt_maxt.elims % 0.30/0.50 thf(fact_894_vebt__mint_Oelims,axiom, % 0.30/0.50 ! [X3: vEBT_VEBT,Y: option @ nat] : % 0.30/0.50 ( ( ( vEBT_vebt_mint @ X3 ) % 0.30/0.50 = Y ) % 0.30/0.50 => ( ! [A5: $o,B5: $o] : % 0.30/0.50 ( ( X3 % 0.30/0.50 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.50 => ~ ( ( A5 % 0.30/0.50 => ( Y % 0.30/0.50 = ( some @ nat @ ( zero_zero @ nat ) ) ) ) % 0.30/0.50 & ( ~ A5 % 0.30/0.50 => ( ( B5 % 0.30/0.50 => ( Y % 0.30/0.50 = ( some @ nat @ ( one_one @ nat ) ) ) ) % 0.30/0.50 & ( ~ B5 % 0.30/0.50 => ( Y % 0.30/0.50 = ( none @ nat ) ) ) ) ) ) ) % 0.30/0.50 => ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 0.30/0.50 ( X3 % 0.30/0.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 0.30/0.50 => ( Y % 0.30/0.50 != ( none @ nat ) ) ) % 0.30/0.50 => ~ ! [Mi2: nat] : % 0.30/0.50 ( ? [Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 0.30/0.50 ( X3 % 0.30/0.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) % 0.30/0.50 => ( Y % 0.30/0.50 != ( some @ nat @ Mi2 ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % vebt_mint.elims % 0.30/0.50 thf(fact_895_div__greater__zero__iff,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N2 ) ) % 0.30/0.50 = ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.50 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_greater_zero_iff % 0.30/0.50 thf(fact_896_div__le__mono2,axiom, % 0.30/0.50 ! [M: nat,N2: nat,K: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.30/0.50 => ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.50 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N2 ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_le_mono2 % 0.30/0.50 thf(fact_897_bits__1__div__2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( bit_semiring_bits @ A ) % 0.30/0.50 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.50 = ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % bits_1_div_2 % 0.30/0.50 thf(fact_898_nat__induct2,axiom, % 0.30/0.50 ! [P: nat > $o,N2: nat] : % 0.30/0.50 ( ( P @ ( zero_zero @ nat ) ) % 0.30/0.50 => ( ( P @ ( one_one @ nat ) ) % 0.30/0.50 => ( ! [N3: nat] : % 0.30/0.50 ( ( P @ N3 ) % 0.30/0.50 => ( P @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.30/0.50 => ( P @ N2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nat_induct2 % 0.30/0.50 thf(fact_899_div__exp__eq,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( bit_semiring_bits @ A ) % 0.30/0.50 => ! [A2: A,M: nat,N2: nat] : % 0.30/0.50 ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.50 = ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_exp_eq % 0.30/0.50 thf(fact_900_exp__add__not__zero__imp__right,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( bit_semiring_bits @ A ) % 0.30/0.50 => ! [M: nat,N2: nat] : % 0.30/0.50 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) ) % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.50 != ( zero_zero @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % exp_add_not_zero_imp_right % 0.30/0.50 thf(fact_901_exp__add__not__zero__imp__left,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( bit_semiring_bits @ A ) % 0.30/0.50 => ! [M: nat,N2: nat] : % 0.30/0.50 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) ) % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) % 0.30/0.50 != ( zero_zero @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % exp_add_not_zero_imp_left % 0.30/0.50 thf(fact_902_bit__concat__def,axiom, % 0.30/0.50 ( vEBT_VEBT_bit_concat % 0.30/0.50 = ( ^ [H: nat,L2: nat,D5: nat] : ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ D5 ) ) @ L2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % bit_concat_def % 0.30/0.50 thf(fact_903_low__inv,axiom, % 0.30/0.50 ! [X3: nat,N2: nat,Y: nat] : % 0.30/0.50 ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.50 => ( ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ X3 ) @ N2 ) % 0.30/0.50 = X3 ) ) ). % 0.30/0.50 % 0.30/0.50 % low_inv % 0.30/0.50 thf(fact_904_high__inv,axiom, % 0.30/0.50 ! [X3: nat,N2: nat,Y: nat] : % 0.30/0.50 ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.50 => ( ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ X3 ) @ N2 ) % 0.30/0.50 = Y ) ) ). % 0.30/0.50 % 0.30/0.50 % high_inv % 0.30/0.50 thf(fact_905_enat__ord__number_I1_J,axiom, % 0.30/0.50 ! [M: num,N2: num] : % 0.30/0.50 ( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N2 ) ) % 0.30/0.50 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % enat_ord_number(1) % 0.30/0.50 thf(fact_906_enat__ord__number_I2_J,axiom, % 0.30/0.50 ! [M: num,N2: num] : % 0.30/0.50 ( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N2 ) ) % 0.30/0.50 = ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % enat_ord_number(2) % 0.30/0.50 thf(fact_907_mul__shift,axiom, % 0.30/0.50 ! [X3: nat,Y: nat,Z2: nat] : % 0.30/0.50 ( ( ( times_times @ nat @ X3 @ Y ) % 0.30/0.50 = Z2 ) % 0.30/0.50 = ( ( vEBT_VEBT_mul @ ( some @ nat @ X3 ) @ ( some @ nat @ Y ) ) % 0.30/0.50 = ( some @ nat @ Z2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mul_shift % 0.30/0.50 thf(fact_908_mul__def,axiom, % 0.30/0.50 ( vEBT_VEBT_mul % 0.30/0.50 = ( vEBT_V2048590022279873568_shift @ nat @ ( times_times @ nat ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mul_def % 0.30/0.50 thf(fact_909_i0__less,axiom, % 0.30/0.50 ! [N2: extended_enat] : % 0.30/0.50 ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 ) % 0.30/0.50 = ( N2 % 0.30/0.50 != ( zero_zero @ extended_enat ) ) ) ). % 0.30/0.50 % 0.30/0.50 % i0_less % 0.30/0.50 thf(fact_910_mult__cancel__right,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiri6575147826004484403cancel @ A ) % 0.30/0.50 => ! [A2: A,C2: A,B2: A] : % 0.30/0.50 ( ( ( times_times @ A @ A2 @ C2 ) % 0.30/0.50 = ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.50 = ( ( C2 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 | ( A2 = B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_cancel_right % 0.30/0.50 thf(fact_911_mult__cancel__left,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiri6575147826004484403cancel @ A ) % 0.30/0.50 => ! [C2: A,A2: A,B2: A] : % 0.30/0.50 ( ( ( times_times @ A @ C2 @ A2 ) % 0.30/0.50 = ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.50 = ( ( C2 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 | ( A2 = B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_cancel_left % 0.30/0.50 thf(fact_912_mult__eq__0__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiri3467727345109120633visors @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ( times_times @ A @ A2 @ B2 ) % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 = ( ( A2 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 | ( B2 % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_eq_0_iff % 0.30/0.50 thf(fact_913_mult__zero__right,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( mult_zero @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( times_times @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 = ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_zero_right % 0.30/0.50 thf(fact_914_mult__zero__left,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( mult_zero @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( times_times @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 = ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_zero_left % 0.30/0.50 thf(fact_915_bits__div__by__0,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( bit_semiring_bits @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 = ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % bits_div_by_0 % 0.30/0.50 thf(fact_916_bits__div__0,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( bit_semiring_bits @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 = ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % bits_div_0 % 0.30/0.50 thf(fact_917_mult_Oright__neutral,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( monoid_mult @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( times_times @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.50 = A2 ) ) ). % 0.30/0.50 % 0.30/0.50 % mult.right_neutral % 0.30/0.50 thf(fact_918_mult__1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( monoid_mult @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( times_times @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.50 = A2 ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_1 % 0.30/0.50 thf(fact_919_half__negative__int__iff,axiom, % 0.30/0.50 ! [K: int] : % 0.30/0.50 ( ( ord_less @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ int ) ) % 0.30/0.50 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ). % 0.30/0.50 % 0.30/0.50 % half_negative_int_iff % 0.30/0.50 thf(fact_920_times__divide__eq__left,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [B2: A,C2: A,A2: A] : % 0.30/0.50 ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 0.30/0.50 = ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % times_divide_eq_left % 0.30/0.50 thf(fact_921_divide__divide__eq__left,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.50 = ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_divide_eq_left % 0.30/0.50 thf(fact_922_divide__divide__eq__right,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.50 = ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_divide_eq_right % 0.30/0.50 thf(fact_923_times__divide__eq__right,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( division_ring @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.50 = ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % times_divide_eq_right % 0.30/0.50 thf(fact_924_mult__cancel2,axiom, % 0.30/0.50 ! [M: nat,K: nat,N2: nat] : % 0.30/0.50 ( ( ( times_times @ nat @ M @ K ) % 0.30/0.50 = ( times_times @ nat @ N2 @ K ) ) % 0.30/0.50 = ( ( M = N2 ) % 0.30/0.50 | ( K % 0.30/0.50 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_cancel2 % 0.30/0.50 thf(fact_925_mult__cancel1,axiom, % 0.30/0.50 ! [K: nat,M: nat,N2: nat] : % 0.30/0.50 ( ( ( times_times @ nat @ K @ M ) % 0.30/0.50 = ( times_times @ nat @ K @ N2 ) ) % 0.30/0.50 = ( ( M = N2 ) % 0.30/0.50 | ( K % 0.30/0.50 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_cancel1 % 0.30/0.50 thf(fact_926_mult__0__right,axiom, % 0.30/0.50 ! [M: nat] : % 0.30/0.50 ( ( times_times @ nat @ M @ ( zero_zero @ nat ) ) % 0.30/0.50 = ( zero_zero @ nat ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_0_right % 0.30/0.50 thf(fact_927_mult__is__0,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : % 0.30/0.50 ( ( ( times_times @ nat @ M @ N2 ) % 0.30/0.50 = ( zero_zero @ nat ) ) % 0.30/0.50 = ( ( M % 0.30/0.50 = ( zero_zero @ nat ) ) % 0.30/0.50 | ( N2 % 0.30/0.50 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_is_0 % 0.30/0.50 thf(fact_928_nat__1__eq__mult__iff,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : % 0.30/0.50 ( ( ( one_one @ nat ) % 0.30/0.50 = ( times_times @ nat @ M @ N2 ) ) % 0.30/0.50 = ( ( M % 0.30/0.50 = ( one_one @ nat ) ) % 0.30/0.50 & ( N2 % 0.30/0.50 = ( one_one @ nat ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nat_1_eq_mult_iff % 0.30/0.50 thf(fact_929_nat__mult__eq__1__iff,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : % 0.30/0.50 ( ( ( times_times @ nat @ M @ N2 ) % 0.30/0.50 = ( one_one @ nat ) ) % 0.30/0.50 = ( ( M % 0.30/0.50 = ( one_one @ nat ) ) % 0.30/0.50 & ( N2 % 0.30/0.50 = ( one_one @ nat ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nat_mult_eq_1_iff % 0.30/0.50 thf(fact_930_mult__cancel__right2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ring_15535105094025558882visors @ A ) % 0.30/0.50 => ! [A2: A,C2: A] : % 0.30/0.50 ( ( ( times_times @ A @ A2 @ C2 ) % 0.30/0.50 = C2 ) % 0.30/0.50 = ( ( C2 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 | ( A2 % 0.30/0.50 = ( one_one @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_cancel_right2 % 0.30/0.50 thf(fact_931_mult__cancel__right1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ring_15535105094025558882visors @ A ) % 0.30/0.50 => ! [C2: A,B2: A] : % 0.30/0.50 ( ( C2 % 0.30/0.50 = ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.50 = ( ( C2 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 | ( B2 % 0.30/0.50 = ( one_one @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_cancel_right1 % 0.30/0.50 thf(fact_932_mult__cancel__left2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ring_15535105094025558882visors @ A ) % 0.30/0.50 => ! [C2: A,A2: A] : % 0.30/0.50 ( ( ( times_times @ A @ C2 @ A2 ) % 0.30/0.50 = C2 ) % 0.30/0.50 = ( ( C2 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 | ( A2 % 0.30/0.50 = ( one_one @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_cancel_left2 % 0.30/0.50 thf(fact_933_mult__cancel__left1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ring_15535105094025558882visors @ A ) % 0.30/0.50 => ! [C2: A,B2: A] : % 0.30/0.50 ( ( C2 % 0.30/0.50 = ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.50 = ( ( C2 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 | ( B2 % 0.30/0.50 = ( one_one @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_cancel_left1 % 0.30/0.50 thf(fact_934_sum__squares__eq__zero__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord4710134922213307826strict @ A ) % 0.30/0.50 => ! [X3: A,Y: A] : % 0.30/0.50 ( ( ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 = ( ( X3 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 & ( Y % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % sum_squares_eq_zero_iff % 0.30/0.50 thf(fact_935_nonzero__mult__divide__mult__cancel__right2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [C2: A,A2: A,B2: A] : % 0.30/0.50 ( ( C2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.50 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nonzero_mult_divide_mult_cancel_right2 % 0.30/0.50 thf(fact_936_nonzero__mult__div__cancel__right,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semidom_divide @ A ) % 0.30/0.50 => ! [B2: A,A2: A] : % 0.30/0.50 ( ( B2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 ) % 0.30/0.50 = A2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nonzero_mult_div_cancel_right % 0.30/0.50 thf(fact_937_nonzero__mult__divide__mult__cancel__right,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [C2: A,A2: A,B2: A] : % 0.30/0.50 ( ( C2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.50 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nonzero_mult_divide_mult_cancel_right % 0.30/0.50 thf(fact_938_nonzero__mult__divide__mult__cancel__left2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [C2: A,A2: A,B2: A] : % 0.30/0.50 ( ( C2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.50 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nonzero_mult_divide_mult_cancel_left2 % 0.30/0.50 thf(fact_939_nonzero__mult__div__cancel__left,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semidom_divide @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( A2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ A2 ) % 0.30/0.50 = B2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nonzero_mult_div_cancel_left % 0.30/0.50 thf(fact_940_nonzero__mult__divide__mult__cancel__left,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [C2: A,A2: A,B2: A] : % 0.30/0.50 ( ( C2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.50 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nonzero_mult_divide_mult_cancel_left % 0.30/0.50 thf(fact_941_mult__divide__mult__cancel__left__if,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [C2: A,A2: A,B2: A] : % 0.30/0.50 ( ( ( C2 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.50 = ( zero_zero @ A ) ) ) % 0.30/0.50 & ( ( C2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.50 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_divide_mult_cancel_left_if % 0.30/0.50 thf(fact_942_div__mult__mult1__if,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.50 => ! [C2: A,A2: A,B2: A] : % 0.30/0.50 ( ( ( C2 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.50 = ( zero_zero @ A ) ) ) % 0.30/0.50 & ( ( C2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.50 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_mult_mult1_if % 0.30/0.50 thf(fact_943_div__mult__mult2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.50 => ! [C2: A,A2: A,B2: A] : % 0.30/0.50 ( ( C2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.50 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_mult_mult2 % 0.30/0.50 thf(fact_944_div__mult__mult1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.50 => ! [C2: A,A2: A,B2: A] : % 0.30/0.50 ( ( C2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.50 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_mult_mult1 % 0.30/0.50 thf(fact_945_distrib__right__numeral,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ( numeral @ A ) % 0.30/0.50 & ( semiring @ A ) ) % 0.30/0.50 => ! [A2: A,B2: A,V2: num] : % 0.30/0.50 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( numeral_numeral @ A @ V2 ) ) % 0.30/0.50 = ( plus_plus @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % distrib_right_numeral % 0.30/0.50 thf(fact_946_distrib__left__numeral,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ( numeral @ A ) % 0.30/0.50 & ( semiring @ A ) ) % 0.30/0.50 => ! [V2: num,B2: A,C2: A] : % 0.30/0.50 ( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) % 0.30/0.50 = ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % distrib_left_numeral % 0.30/0.50 thf(fact_947_mult__eq__1__iff,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : % 0.30/0.50 ( ( ( times_times @ nat @ M @ N2 ) % 0.30/0.50 = ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.50 = ( ( M % 0.30/0.50 = ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.50 & ( N2 % 0.30/0.50 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_eq_1_iff % 0.30/0.50 thf(fact_948_one__eq__mult__iff,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : % 0.30/0.50 ( ( ( suc @ ( zero_zero @ nat ) ) % 0.30/0.50 = ( times_times @ nat @ M @ N2 ) ) % 0.30/0.50 = ( ( M % 0.30/0.50 = ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.50 & ( N2 % 0.30/0.50 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % one_eq_mult_iff % 0.30/0.50 thf(fact_949_mult__less__cancel2,axiom, % 0.30/0.50 ! [M: nat,K: nat,N2: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) % 0.30/0.50 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.30/0.50 & ( ord_less @ nat @ M @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_less_cancel2 % 0.30/0.50 thf(fact_950_nat__0__less__mult__iff,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N2 ) ) % 0.30/0.50 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.30/0.50 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nat_0_less_mult_iff % 0.30/0.50 thf(fact_951_nat__mult__less__cancel__disj,axiom, % 0.30/0.50 ! [K: nat,M: nat,N2: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 0.30/0.50 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.30/0.50 & ( ord_less @ nat @ M @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nat_mult_less_cancel_disj % 0.30/0.50 thf(fact_952_mult__Suc__right,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : % 0.30/0.50 ( ( times_times @ nat @ M @ ( suc @ N2 ) ) % 0.30/0.50 = ( plus_plus @ nat @ M @ ( times_times @ nat @ M @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_Suc_right % 0.30/0.50 thf(fact_953_nat__mult__div__cancel__disj,axiom, % 0.30/0.50 ! [K: nat,M: nat,N2: nat] : % 0.30/0.50 ( ( ( K % 0.30/0.50 = ( zero_zero @ nat ) ) % 0.30/0.50 => ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 0.30/0.50 = ( zero_zero @ nat ) ) ) % 0.30/0.50 & ( ( K % 0.30/0.50 != ( zero_zero @ nat ) ) % 0.30/0.50 => ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 0.30/0.50 = ( divide_divide @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nat_mult_div_cancel_disj % 0.30/0.50 thf(fact_954_divide__le__eq__numeral1_I1_J,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [B2: A,W2: num,A2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A2 ) % 0.30/0.50 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_le_eq_numeral1(1) % 0.30/0.50 thf(fact_955_le__divide__eq__numeral1_I1_J,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A,W2: num] : % 0.30/0.50 ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) ) % 0.30/0.50 = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % le_divide_eq_numeral1(1) % 0.30/0.50 thf(fact_956_divide__eq__eq__numeral1_I1_J,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( division_ring @ A ) % 0.30/0.50 => ! [B2: A,W2: num,A2: A] : % 0.30/0.50 ( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) % 0.30/0.50 = A2 ) % 0.30/0.50 = ( ( ( ( numeral_numeral @ A @ W2 ) % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( B2 % 0.30/0.50 = ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) ) % 0.30/0.50 & ( ( ( numeral_numeral @ A @ W2 ) % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 => ( A2 % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_eq_eq_numeral1(1) % 0.30/0.50 thf(fact_957_eq__divide__eq__numeral1_I1_J,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( division_ring @ A ) % 0.30/0.50 => ! [A2: A,B2: A,W2: num] : % 0.30/0.50 ( ( A2 % 0.30/0.50 = ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) ) % 0.30/0.50 = ( ( ( ( numeral_numeral @ A @ W2 ) % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) % 0.30/0.50 = B2 ) ) % 0.30/0.50 & ( ( ( numeral_numeral @ A @ W2 ) % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 => ( A2 % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % eq_divide_eq_numeral1(1) % 0.30/0.50 thf(fact_958_less__divide__eq__numeral1_I1_J,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [A2: A,B2: A,W2: num] : % 0.30/0.50 ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) ) % 0.30/0.50 = ( ord_less @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % less_divide_eq_numeral1(1) % 0.30/0.50 thf(fact_959_divide__less__eq__numeral1_I1_J,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_field @ A ) % 0.30/0.50 => ! [B2: A,W2: num,A2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A2 ) % 0.30/0.50 = ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_less_eq_numeral1(1) % 0.30/0.50 thf(fact_960_nonzero__divide__mult__cancel__left,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( A2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) ) % 0.30/0.50 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nonzero_divide_mult_cancel_left % 0.30/0.50 thf(fact_961_nonzero__divide__mult__cancel__right,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [B2: A,A2: A] : % 0.30/0.50 ( ( B2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A2 @ B2 ) ) % 0.30/0.50 = ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nonzero_divide_mult_cancel_right % 0.30/0.50 thf(fact_962_div__mult__self1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.50 => ! [B2: A,A2: A,C2: A] : % 0.30/0.50 ( ( B2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 ) % 0.30/0.50 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_mult_self1 % 0.30/0.50 thf(fact_963_div__mult__self2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.50 => ! [B2: A,A2: A,C2: A] : % 0.30/0.50 ( ( B2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 ) % 0.30/0.50 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_mult_self2 % 0.30/0.50 thf(fact_964_div__mult__self3,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.50 => ! [B2: A,C2: A,A2: A] : % 0.30/0.50 ( ( B2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 ) % 0.30/0.50 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_mult_self3 % 0.30/0.50 thf(fact_965_div__mult__self4,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.50 => ! [B2: A,C2: A,A2: A] : % 0.30/0.50 ( ( B2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 ) % 0.30/0.50 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % div_mult_self4 % 0.30/0.50 thf(fact_966_one__le__mult__iff,axiom, % 0.30/0.50 ! [M: nat,N2: nat] : % 0.30/0.50 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N2 ) ) % 0.30/0.50 = ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M ) % 0.30/0.50 & ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % one_le_mult_iff % 0.30/0.50 thf(fact_967_mult__le__cancel2,axiom, % 0.30/0.50 ! [M: nat,K: nat,N2: nat] : % 0.30/0.50 ( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) % 0.30/0.50 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.30/0.50 => ( ord_less_eq @ nat @ M @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_le_cancel2 % 0.30/0.50 thf(fact_968_nat__mult__le__cancel__disj,axiom, % 0.30/0.50 ! [K: nat,M: nat,N2: nat] : % 0.30/0.50 ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 0.30/0.50 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.30/0.50 => ( ord_less_eq @ nat @ M @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nat_mult_le_cancel_disj % 0.30/0.50 thf(fact_969_div__mult__self__is__m,axiom, % 0.30/0.50 ! [N2: nat,M: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.50 => ( ( divide_divide @ nat @ ( times_times @ nat @ M @ N2 ) @ N2 ) % 0.30/0.50 = M ) ) ). % 0.30/0.50 % 0.30/0.50 % div_mult_self_is_m % 0.30/0.50 thf(fact_970_div__mult__self1__is__m,axiom, % 0.30/0.50 ! [N2: nat,M: nat] : % 0.30/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.50 => ( ( divide_divide @ nat @ ( times_times @ nat @ N2 @ M ) @ N2 ) % 0.30/0.50 = M ) ) ). % 0.30/0.50 % 0.30/0.50 % div_mult_self1_is_m % 0.30/0.50 thf(fact_971_ab__semigroup__mult__class_Omult__ac_I1_J,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ab_semigroup_mult @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.50 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % ab_semigroup_mult_class.mult_ac(1) % 0.30/0.50 thf(fact_972_mult_Oassoc,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semigroup_mult @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.50 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult.assoc % 0.30/0.50 thf(fact_973_mult_Ocommute,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ab_semigroup_mult @ A ) % 0.30/0.50 => ( ( times_times @ A ) % 0.30/0.50 = ( ^ [A4: A,B4: A] : ( times_times @ A @ B4 @ A4 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult.commute % 0.30/0.50 thf(fact_974_mult_Oleft__commute,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ab_semigroup_mult @ A ) % 0.30/0.50 => ! [B2: A,A2: A,C2: A] : % 0.30/0.50 ( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) % 0.30/0.50 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult.left_commute % 0.30/0.50 thf(fact_975_not__iless0,axiom, % 0.30/0.50 ! [N2: extended_enat] : % 0.30/0.50 ~ ( ord_less @ extended_enat @ N2 @ ( zero_zero @ extended_enat ) ) ). % 0.30/0.50 % 0.30/0.50 % not_iless0 % 0.30/0.50 thf(fact_976_enat__less__induct,axiom, % 0.30/0.50 ! [P: extended_enat > $o,N2: extended_enat] : % 0.30/0.50 ( ! [N3: extended_enat] : % 0.30/0.50 ( ! [M6: extended_enat] : % 0.30/0.50 ( ( ord_less @ extended_enat @ M6 @ N3 ) % 0.30/0.50 => ( P @ M6 ) ) % 0.30/0.50 => ( P @ N3 ) ) % 0.30/0.50 => ( P @ N2 ) ) ). % 0.30/0.50 % 0.30/0.50 % enat_less_induct % 0.30/0.50 thf(fact_977_mult__right__cancel,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiri6575147826004484403cancel @ A ) % 0.30/0.50 => ! [C2: A,A2: A,B2: A] : % 0.30/0.50 ( ( C2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ( times_times @ A @ A2 @ C2 ) % 0.30/0.50 = ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.50 = ( A2 = B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_right_cancel % 0.30/0.50 thf(fact_978_mult__left__cancel,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiri6575147826004484403cancel @ A ) % 0.30/0.50 => ! [C2: A,A2: A,B2: A] : % 0.30/0.50 ( ( C2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ( times_times @ A @ C2 @ A2 ) % 0.30/0.50 = ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.50 = ( A2 = B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_left_cancel % 0.30/0.50 thf(fact_979_no__zero__divisors,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiri3467727345109120633visors @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( A2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( B2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( times_times @ A @ A2 @ B2 ) % 0.30/0.50 != ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % no_zero_divisors % 0.30/0.50 thf(fact_980_divisors__zero,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiri3467727345109120633visors @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ( times_times @ A @ A2 @ B2 ) % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 => ( ( A2 % 0.30/0.50 = ( zero_zero @ A ) ) % 0.30/0.50 | ( B2 % 0.30/0.50 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divisors_zero % 0.30/0.50 thf(fact_981_mult__not__zero,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( mult_zero @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ( times_times @ A @ A2 @ B2 ) % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 => ( ( A2 % 0.30/0.50 != ( zero_zero @ A ) ) % 0.30/0.50 & ( B2 % 0.30/0.50 != ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_not_zero % 0.30/0.50 thf(fact_982_comm__monoid__mult__class_Omult__1,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( comm_monoid_mult @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( times_times @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.50 = A2 ) ) ). % 0.30/0.50 % 0.30/0.50 % comm_monoid_mult_class.mult_1 % 0.30/0.50 thf(fact_983_mult_Ocomm__neutral,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( comm_monoid_mult @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ( ( times_times @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.50 = A2 ) ) ). % 0.30/0.50 % 0.30/0.50 % mult.comm_neutral % 0.30/0.50 thf(fact_984_combine__common__factor,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiring @ A ) % 0.30/0.50 => ! [A2: A,E3: A,B2: A,C2: A] : % 0.30/0.50 ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ C2 ) ) % 0.30/0.50 = ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ E3 ) @ C2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % combine_common_factor % 0.30/0.50 thf(fact_985_distrib__right,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiring @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.50 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % distrib_right % 0.30/0.50 thf(fact_986_distrib__left,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiring @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) % 0.30/0.50 = ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % distrib_left % 0.30/0.50 thf(fact_987_comm__semiring__class_Odistrib,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( comm_semiring @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.50 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % comm_semiring_class.distrib % 0.30/0.50 thf(fact_988_ring__class_Oring__distribs_I1_J,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ring @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) % 0.30/0.50 = ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % ring_class.ring_distribs(1) % 0.30/0.50 thf(fact_989_ring__class_Oring__distribs_I2_J,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ring @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.50 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % ring_class.ring_distribs(2) % 0.30/0.50 thf(fact_990_crossproduct__noteq,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiri1453513574482234551roduct @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.50 ( ( ( A2 != B2 ) % 0.30/0.50 & ( C2 != D2 ) ) % 0.30/0.50 = ( ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) % 0.30/0.50 != ( plus_plus @ A @ ( times_times @ A @ A2 @ D2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % crossproduct_noteq % 0.30/0.50 thf(fact_991_crossproduct__eq,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( semiri1453513574482234551roduct @ A ) % 0.30/0.50 => ! [W2: A,Y: A,X3: A,Z2: A] : % 0.30/0.50 ( ( ( plus_plus @ A @ ( times_times @ A @ W2 @ Y ) @ ( times_times @ A @ X3 @ Z2 ) ) % 0.30/0.50 = ( plus_plus @ A @ ( times_times @ A @ W2 @ Z2 ) @ ( times_times @ A @ X3 @ Y ) ) ) % 0.30/0.50 = ( ( W2 = X3 ) % 0.30/0.50 | ( Y = Z2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % crossproduct_eq % 0.30/0.50 thf(fact_992_times__divide__times__eq,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [X3: A,Y: A,Z2: A,W2: A] : % 0.30/0.50 ( ( times_times @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ Z2 @ W2 ) ) % 0.30/0.50 = ( divide_divide @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ Y @ W2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % times_divide_times_eq % 0.30/0.50 thf(fact_993_divide__divide__times__eq,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( field @ A ) % 0.30/0.50 => ! [X3: A,Y: A,Z2: A,W2: A] : % 0.30/0.50 ( ( divide_divide @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ Z2 @ W2 ) ) % 0.30/0.50 = ( divide_divide @ A @ ( times_times @ A @ X3 @ W2 ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_divide_times_eq % 0.30/0.50 thf(fact_994_divide__divide__eq__left_H,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( division_ring @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.50 = ( divide_divide @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % divide_divide_eq_left' % 0.30/0.50 thf(fact_995_Suc__mult__cancel1,axiom, % 0.30/0.50 ! [K: nat,M: nat,N2: nat] : % 0.30/0.50 ( ( ( times_times @ nat @ ( suc @ K ) @ M ) % 0.30/0.50 = ( times_times @ nat @ ( suc @ K ) @ N2 ) ) % 0.30/0.50 = ( M = N2 ) ) ). % 0.30/0.50 % 0.30/0.50 % Suc_mult_cancel1 % 0.30/0.50 thf(fact_996_mult__0,axiom, % 0.30/0.50 ! [N2: nat] : % 0.30/0.50 ( ( times_times @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.50 = ( zero_zero @ nat ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_0 % 0.30/0.50 thf(fact_997_nat__mult__eq__cancel__disj,axiom, % 0.30/0.50 ! [K: nat,M: nat,N2: nat] : % 0.30/0.50 ( ( ( times_times @ nat @ K @ M ) % 0.30/0.50 = ( times_times @ nat @ K @ N2 ) ) % 0.30/0.50 = ( ( K % 0.30/0.50 = ( zero_zero @ nat ) ) % 0.30/0.50 | ( M = N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % nat_mult_eq_cancel_disj % 0.30/0.50 thf(fact_998_mult__le__mono2,axiom, % 0.30/0.50 ! [I: nat,J: nat,K: nat] : % 0.30/0.50 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.50 => ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_le_mono2 % 0.30/0.50 thf(fact_999_mult__le__mono1,axiom, % 0.30/0.50 ! [I: nat,J: nat,K: nat] : % 0.30/0.50 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.50 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_le_mono1 % 0.30/0.50 thf(fact_1000_mult__le__mono,axiom, % 0.30/0.50 ! [I: nat,J: nat,K: nat,L: nat] : % 0.30/0.50 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.50 => ( ( ord_less_eq @ nat @ K @ L ) % 0.30/0.50 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_le_mono % 0.30/0.50 thf(fact_1001_le__square,axiom, % 0.30/0.50 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ). % 0.30/0.50 % 0.30/0.50 % le_square % 0.30/0.50 thf(fact_1002_le__cube,axiom, % 0.30/0.50 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ). % 0.30/0.50 % 0.30/0.50 % le_cube % 0.30/0.50 thf(fact_1003_add__mult__distrib2,axiom, % 0.30/0.50 ! [K: nat,M: nat,N2: nat] : % 0.30/0.50 ( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N2 ) ) % 0.30/0.50 = ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % add_mult_distrib2 % 0.30/0.50 thf(fact_1004_add__mult__distrib,axiom, % 0.30/0.50 ! [M: nat,N2: nat,K: nat] : % 0.30/0.50 ( ( times_times @ nat @ ( plus_plus @ nat @ M @ N2 ) @ K ) % 0.30/0.50 = ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ). % 0.30/0.50 % 0.30/0.50 % add_mult_distrib % 0.30/0.50 thf(fact_1005_left__add__mult__distrib,axiom, % 0.30/0.50 ! [I: nat,U: nat,J: nat,K: nat] : % 0.30/0.50 ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) ) % 0.30/0.50 = ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ). % 0.30/0.50 % 0.30/0.50 % left_add_mult_distrib % 0.30/0.50 thf(fact_1006_nat__mult__1,axiom, % 0.30/0.50 ! [N2: nat] : % 0.30/0.50 ( ( times_times @ nat @ ( one_one @ nat ) @ N2 ) % 0.30/0.50 = N2 ) ). % 0.30/0.50 % 0.30/0.50 % nat_mult_1 % 0.30/0.50 thf(fact_1007_nat__mult__1__right,axiom, % 0.30/0.50 ! [N2: nat] : % 0.30/0.50 ( ( times_times @ nat @ N2 @ ( one_one @ nat ) ) % 0.30/0.50 = N2 ) ). % 0.30/0.50 % 0.30/0.50 % nat_mult_1_right % 0.30/0.50 thf(fact_1008_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ordere2520102378445227354miring @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.50 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % ordered_comm_semiring_class.comm_mult_left_mono % 0.30/0.50 thf(fact_1009_zero__le__mult__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord4710134922213307826strict @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 0.30/0.50 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) % 0.30/0.50 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_le_mult_iff % 0.30/0.50 thf(fact_1010_mult__nonneg__nonpos2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ordered_semiring_0 @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_nonneg_nonpos2 % 0.30/0.50 thf(fact_1011_mult__nonpos__nonneg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ordered_semiring_0 @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.50 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_nonpos_nonneg % 0.30/0.50 thf(fact_1012_mult__nonneg__nonpos,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ordered_semiring_0 @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_nonneg_nonpos % 0.30/0.50 thf(fact_1013_mult__nonneg__nonneg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ordered_semiring_0 @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_nonneg_nonneg % 0.30/0.50 thf(fact_1014_split__mult__neg__le,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ordered_semiring_0 @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) % 0.30/0.50 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) % 0.30/0.50 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % split_mult_neg_le % 0.30/0.50 thf(fact_1015_mult__le__0__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord4710134922213307826strict @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) % 0.30/0.50 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) % 0.30/0.50 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_le_0_iff % 0.30/0.50 thf(fact_1016_mult__right__mono,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ordered_semiring @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.50 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_right_mono % 0.30/0.50 thf(fact_1017_mult__right__mono__neg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ordered_ring @ A ) % 0.30/0.50 => ! [B2: A,A2: A,C2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_right_mono_neg % 0.30/0.50 thf(fact_1018_mult__left__mono,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ordered_semiring @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.50 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_left_mono % 0.30/0.50 thf(fact_1019_mult__nonpos__nonpos,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ordered_ring @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_nonpos_nonpos % 0.30/0.50 thf(fact_1020_mult__left__mono__neg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ordered_ring @ A ) % 0.30/0.50 => ! [B2: A,A2: A,C2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_left_mono_neg % 0.30/0.50 thf(fact_1021_split__mult__pos__le,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ordered_ring @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) % 0.30/0.50 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) % 0.30/0.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % split_mult_pos_le % 0.30/0.50 thf(fact_1022_zero__le__square,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_ring @ A ) % 0.30/0.50 => ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ A2 ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_le_square % 0.30/0.50 thf(fact_1023_mult__mono_H,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ordered_semiring @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ C2 @ D2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.50 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_mono' % 0.30/0.50 thf(fact_1024_mult__mono,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( ordered_semiring @ A ) % 0.30/0.50 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.50 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ C2 @ D2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.50 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_mono % 0.30/0.50 thf(fact_1025_mult__neg__neg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord4710134922213307826strict @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_neg_neg % 0.30/0.50 thf(fact_1026_not__square__less__zero,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linordered_ring @ A ) % 0.30/0.50 => ! [A2: A] : % 0.30/0.50 ~ ( ord_less @ A @ ( times_times @ A @ A2 @ A2 ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.50 % 0.30/0.50 % not_square_less_zero % 0.30/0.50 thf(fact_1027_mult__less__0__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord4710134922213307826strict @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) % 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) % 0.30/0.50 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_less_0_iff % 0.30/0.50 thf(fact_1028_mult__neg__pos,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord8928482502909563296strict @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.50 => ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_neg_pos % 0.30/0.50 thf(fact_1029_mult__pos__neg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord8928482502909563296strict @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_pos_neg % 0.30/0.50 thf(fact_1030_mult__pos__pos,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord8928482502909563296strict @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.50 => ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_pos_pos % 0.30/0.50 thf(fact_1031_mult__pos__neg2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord8928482502909563296strict @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ord_less @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_pos_neg2 % 0.30/0.50 thf(fact_1032_zero__less__mult__iff,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord4710134922213307826strict @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 0.30/0.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) % 0.30/0.50 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.50 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_less_mult_iff % 0.30/0.50 thf(fact_1033_zero__less__mult__pos,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord8928482502909563296strict @ A ) % 0.30/0.50 => ! [A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_less_mult_pos % 0.30/0.50 thf(fact_1034_zero__less__mult__pos2,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord8928482502909563296strict @ A ) % 0.30/0.50 => ! [B2: A,A2: A] : % 0.30/0.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A2 ) ) % 0.30/0.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.50 => ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % zero_less_mult_pos2 % 0.30/0.50 thf(fact_1035_mult__less__cancel__left__neg,axiom, % 0.30/0.50 ! [A: $tType] : % 0.30/0.50 ( ( linord4710134922213307826strict @ A ) % 0.30/0.50 => ! [C2: A,A2: A,B2: A] : % 0.30/0.50 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.50 => ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.50 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ). % 0.30/0.50 % 0.30/0.50 % mult_less_cancel_left_neg % 0.30/0.51 thf(fact_1036_mult__less__cancel__left__pos,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord4710134922213307826strict @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.51 = ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_less_cancel_left_pos % 0.30/0.51 thf(fact_1037_mult__strict__left__mono__neg,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord4710134922213307826strict @ A ) % 0.30/0.51 => ! [B2: A,A2: A,C2: A] : % 0.30/0.51 ( ( ord_less @ A @ B2 @ A2 ) % 0.30/0.51 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_strict_left_mono_neg % 0.30/0.51 thf(fact_1038_mult__strict__left__mono,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord8928482502909563296strict @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_strict_left_mono % 0.30/0.51 thf(fact_1039_mult__less__cancel__left__disj,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord4710134922213307826strict @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 & ( ord_less @ A @ A2 @ B2 ) ) % 0.30/0.51 | ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 & ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_less_cancel_left_disj % 0.30/0.51 thf(fact_1040_mult__strict__right__mono__neg,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord4710134922213307826strict @ A ) % 0.30/0.51 => ! [B2: A,A2: A,C2: A] : % 0.30/0.51 ( ( ord_less @ A @ B2 @ A2 ) % 0.30/0.51 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_strict_right_mono_neg % 0.30/0.51 thf(fact_1041_mult__strict__right__mono,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord8928482502909563296strict @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_strict_right_mono % 0.30/0.51 thf(fact_1042_mult__less__cancel__right__disj,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord4710134922213307826strict @ A ) % 0.30/0.51 => ! [A2: A,C2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 & ( ord_less @ A @ A2 @ B2 ) ) % 0.30/0.51 | ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 & ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_less_cancel_right_disj % 0.30/0.51 thf(fact_1043_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord2810124833399127020strict @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono % 0.30/0.51 thf(fact_1044_add__scale__eq__noteq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semiri1453513574482234551roduct @ A ) % 0.30/0.51 => ! [R2: A,A2: A,B2: A,C2: A,D2: A] : % 0.30/0.51 ( ( R2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( ( A2 = B2 ) % 0.30/0.51 & ( C2 != D2 ) ) % 0.30/0.51 => ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R2 @ C2 ) ) % 0.30/0.51 != ( plus_plus @ A @ B2 @ ( times_times @ A @ R2 @ D2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % add_scale_eq_noteq % 0.30/0.51 thf(fact_1045_less__1__mult,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_semidom @ A ) % 0.30/0.51 => ! [M: A,N2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( one_one @ A ) @ M ) % 0.30/0.51 => ( ( ord_less @ A @ ( one_one @ A ) @ N2 ) % 0.30/0.51 => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % less_1_mult % 0.30/0.51 thf(fact_1046_frac__eq__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( field @ A ) % 0.30/0.51 => ! [Y: A,Z2: A,X3: A,W2: A] : % 0.30/0.51 ( ( Y % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( Z2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( ( divide_divide @ A @ X3 @ Y ) % 0.30/0.51 = ( divide_divide @ A @ W2 @ Z2 ) ) % 0.30/0.51 = ( ( times_times @ A @ X3 @ Z2 ) % 0.30/0.51 = ( times_times @ A @ W2 @ Y ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % frac_eq_eq % 0.30/0.51 thf(fact_1047_divide__eq__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( division_ring @ A ) % 0.30/0.51 => ! [B2: A,C2: A,A2: A] : % 0.30/0.51 ( ( ( divide_divide @ A @ B2 @ C2 ) % 0.30/0.51 = A2 ) % 0.30/0.51 = ( ( ( C2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( B2 % 0.30/0.51 = ( times_times @ A @ A2 @ C2 ) ) ) % 0.30/0.51 & ( ( C2 % 0.30/0.51 = ( zero_zero @ A ) ) % 0.30/0.51 => ( A2 % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % divide_eq_eq % 0.30/0.51 thf(fact_1048_eq__divide__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( division_ring @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( A2 % 0.30/0.51 = ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ( ( C2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( times_times @ A @ A2 @ C2 ) % 0.30/0.51 = B2 ) ) % 0.30/0.51 & ( ( C2 % 0.30/0.51 = ( zero_zero @ A ) ) % 0.30/0.51 => ( A2 % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % eq_divide_eq % 0.30/0.51 thf(fact_1049_divide__eq__imp,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( division_ring @ A ) % 0.30/0.51 => ! [C2: A,B2: A,A2: A] : % 0.30/0.51 ( ( C2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( B2 % 0.30/0.51 = ( times_times @ A @ A2 @ C2 ) ) % 0.30/0.51 => ( ( divide_divide @ A @ B2 @ C2 ) % 0.30/0.51 = A2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % divide_eq_imp % 0.30/0.51 thf(fact_1050_eq__divide__imp,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( division_ring @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( C2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( ( times_times @ A @ A2 @ C2 ) % 0.30/0.51 = B2 ) % 0.30/0.51 => ( A2 % 0.30/0.51 = ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % eq_divide_imp % 0.30/0.51 thf(fact_1051_nonzero__divide__eq__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( division_ring @ A ) % 0.30/0.51 => ! [C2: A,B2: A,A2: A] : % 0.30/0.51 ( ( C2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( ( divide_divide @ A @ B2 @ C2 ) % 0.30/0.51 = A2 ) % 0.30/0.51 = ( B2 % 0.30/0.51 = ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % nonzero_divide_eq_eq % 0.30/0.51 thf(fact_1052_nonzero__eq__divide__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( division_ring @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( C2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( A2 % 0.30/0.51 = ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ( times_times @ A @ A2 @ C2 ) % 0.30/0.51 = B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % nonzero_eq_divide_eq % 0.30/0.51 thf(fact_1053_power__Suc,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( power @ A ) % 0.30/0.51 => ! [A2: A,N2: nat] : % 0.30/0.51 ( ( power_power @ A @ A2 @ ( suc @ N2 ) ) % 0.30/0.51 = ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % power_Suc % 0.30/0.51 thf(fact_1054_power__Suc2,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( monoid_mult @ A ) % 0.30/0.51 => ! [A2: A,N2: nat] : % 0.30/0.51 ( ( power_power @ A @ A2 @ ( suc @ N2 ) ) % 0.30/0.51 = ( times_times @ A @ ( power_power @ A @ A2 @ N2 ) @ A2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % power_Suc2 % 0.30/0.51 thf(fact_1055_Suc__mult__less__cancel1,axiom, % 0.30/0.51 ! [K: nat,M: nat,N2: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N2 ) ) % 0.30/0.51 = ( ord_less @ nat @ M @ N2 ) ) ). % 0.30/0.51 % 0.30/0.51 % Suc_mult_less_cancel1 % 0.30/0.51 thf(fact_1056_power__add,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( monoid_mult @ A ) % 0.30/0.51 => ! [A2: A,M: nat,N2: nat] : % 0.30/0.51 ( ( power_power @ A @ A2 @ ( plus_plus @ nat @ M @ N2 ) ) % 0.30/0.51 = ( times_times @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % power_add % 0.30/0.51 thf(fact_1057_mult__less__mono1,axiom, % 0.30/0.51 ! [I: nat,J: nat,K: nat] : % 0.30/0.51 ( ( ord_less @ nat @ I @ J ) % 0.30/0.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.30/0.51 => ( ord_less @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_less_mono1 % 0.30/0.51 thf(fact_1058_mult__less__mono2,axiom, % 0.30/0.51 ! [I: nat,J: nat,K: nat] : % 0.30/0.51 ( ( ord_less @ nat @ I @ J ) % 0.30/0.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.30/0.51 => ( ord_less @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_less_mono2 % 0.30/0.51 thf(fact_1059_nat__mult__eq__cancel1,axiom, % 0.30/0.51 ! [K: nat,M: nat,N2: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.30/0.51 => ( ( ( times_times @ nat @ K @ M ) % 0.30/0.51 = ( times_times @ nat @ K @ N2 ) ) % 0.30/0.51 = ( M = N2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % nat_mult_eq_cancel1 % 0.30/0.51 thf(fact_1060_nat__mult__less__cancel1,axiom, % 0.30/0.51 ! [K: nat,M: nat,N2: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.30/0.51 => ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 0.30/0.51 = ( ord_less @ nat @ M @ N2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % nat_mult_less_cancel1 % 0.30/0.51 thf(fact_1061_Suc__mult__le__cancel1,axiom, % 0.30/0.51 ! [K: nat,M: nat,N2: nat] : % 0.30/0.51 ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N2 ) ) % 0.30/0.51 = ( ord_less_eq @ nat @ M @ N2 ) ) ). % 0.30/0.51 % 0.30/0.51 % Suc_mult_le_cancel1 % 0.30/0.51 thf(fact_1062_mult__Suc,axiom, % 0.30/0.51 ! [M: nat,N2: nat] : % 0.30/0.51 ( ( times_times @ nat @ ( suc @ M ) @ N2 ) % 0.30/0.51 = ( plus_plus @ nat @ N2 @ ( times_times @ nat @ M @ N2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_Suc % 0.30/0.51 thf(fact_1063_mult__eq__self__implies__10,axiom, % 0.30/0.51 ! [M: nat,N2: nat] : % 0.30/0.51 ( ( M % 0.30/0.51 = ( times_times @ nat @ M @ N2 ) ) % 0.30/0.51 => ( ( N2 % 0.30/0.51 = ( one_one @ nat ) ) % 0.30/0.51 | ( M % 0.30/0.51 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_eq_self_implies_10 % 0.30/0.51 thf(fact_1064_less__mult__imp__div__less,axiom, % 0.30/0.51 ! [M: nat,I: nat,N2: nat] : % 0.30/0.51 ( ( ord_less @ nat @ M @ ( times_times @ nat @ I @ N2 ) ) % 0.30/0.51 => ( ord_less @ nat @ ( divide_divide @ nat @ M @ N2 ) @ I ) ) ). % 0.30/0.51 % 0.30/0.51 % less_mult_imp_div_less % 0.30/0.51 thf(fact_1065_times__div__less__eq__dividend,axiom, % 0.30/0.51 ! [N2: nat,M: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ ( divide_divide @ nat @ M @ N2 ) ) @ M ) ). % 0.30/0.51 % 0.30/0.51 % times_div_less_eq_dividend % 0.30/0.51 thf(fact_1066_div__times__less__eq__dividend,axiom, % 0.30/0.51 ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N2 ) @ N2 ) @ M ) ). % 0.30/0.51 % 0.30/0.51 % div_times_less_eq_dividend % 0.30/0.51 thf(fact_1067_power__odd__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( monoid_mult @ A ) % 0.30/0.51 => ! [A2: A,N2: nat] : % 0.30/0.51 ( ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 0.30/0.51 = ( times_times @ A @ A2 @ ( power_power @ A @ ( power_power @ A @ A2 @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % power_odd_eq % 0.30/0.51 thf(fact_1068_Suc__double__not__eq__double,axiom, % 0.30/0.51 ! [M: nat,N2: nat] : % 0.30/0.51 ( ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 0.30/0.51 != ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 0.30/0.51 % 0.30/0.51 % Suc_double_not_eq_double % 0.30/0.51 thf(fact_1069_double__not__eq__Suc__double,axiom, % 0.30/0.51 ! [M: nat,N2: nat] : % 0.30/0.51 ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) % 0.30/0.51 != ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % double_not_eq_Suc_double % 0.30/0.51 thf(fact_1070_mult__le__cancel__left,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord4710134922213307826strict @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less_eq @ A @ A2 @ B2 ) ) % 0.30/0.51 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_le_cancel_left % 0.30/0.51 thf(fact_1071_mult__le__cancel__right,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord4710134922213307826strict @ A ) % 0.30/0.51 => ! [A2: A,C2: A,B2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less_eq @ A @ A2 @ B2 ) ) % 0.30/0.51 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_le_cancel_right % 0.30/0.51 thf(fact_1072_mult__left__less__imp__less,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_semiring @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_left_less_imp_less % 0.30/0.51 thf(fact_1073_mult__strict__mono,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord8928482502909563296strict @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.51 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( ord_less @ A @ C2 @ D2 ) % 0.30/0.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_strict_mono % 0.30/0.51 thf(fact_1074_mult__less__cancel__left,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord4710134922213307826strict @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.51 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ A2 @ B2 ) ) % 0.30/0.51 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_less_cancel_left % 0.30/0.51 thf(fact_1075_mult__right__less__imp__less,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_semiring @ A ) % 0.30/0.51 => ! [A2: A,C2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_right_less_imp_less % 0.30/0.51 thf(fact_1076_mult__strict__mono_H,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord8928482502909563296strict @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.51 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( ord_less @ A @ C2 @ D2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_strict_mono' % 0.30/0.51 thf(fact_1077_mult__less__cancel__right,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord4710134922213307826strict @ A ) % 0.30/0.51 => ! [A2: A,C2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ A2 @ B2 ) ) % 0.30/0.51 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_less_cancel_right % 0.30/0.51 thf(fact_1078_mult__le__cancel__left__neg,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord4710134922213307826strict @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.51 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_le_cancel_left_neg % 0.30/0.51 thf(fact_1079_mult__le__cancel__left__pos,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord4710134922213307826strict @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.51 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_le_cancel_left_pos % 0.30/0.51 thf(fact_1080_mult__left__le__imp__le,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord8928482502909563296strict @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_left_le_imp_le % 0.30/0.51 thf(fact_1081_mult__right__le__imp__le,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord8928482502909563296strict @ A ) % 0.30/0.51 => ! [A2: A,C2: A,B2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_right_le_imp_le % 0.30/0.51 thf(fact_1082_mult__le__less__imp__less,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord8928482502909563296strict @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( ord_less @ A @ C2 @ D2 ) % 0.30/0.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_le_less_imp_less % 0.30/0.51 thf(fact_1083_mult__less__le__imp__less,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord8928482502909563296strict @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.51 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ C2 @ D2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_less_le_imp_less % 0.30/0.51 thf(fact_1084_mult__left__le,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord181362715937106298miring @ A ) % 0.30/0.51 => ! [C2: A,A2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.51 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ A2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_left_le % 0.30/0.51 thf(fact_1085_mult__le__one,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord181362715937106298miring @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_le_one % 0.30/0.51 thf(fact_1086_mult__right__le__one__le,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [X3: A,Y: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.51 => ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( times_times @ A @ X3 @ Y ) @ X3 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_right_le_one_le % 0.30/0.51 thf(fact_1087_mult__left__le__one__le,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [X3: A,Y: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.51 => ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( times_times @ A @ Y @ X3 ) @ X3 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_left_le_one_le % 0.30/0.51 thf(fact_1088_sum__squares__le__zero__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord4710134922213307826strict @ A ) % 0.30/0.51 => ! [X3: A,Y: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) % 0.30/0.51 = ( ( X3 % 0.30/0.51 = ( zero_zero @ A ) ) % 0.30/0.51 & ( Y % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % sum_squares_le_zero_iff % 0.30/0.51 thf(fact_1089_sum__squares__ge__zero,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_ring @ A ) % 0.30/0.51 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % sum_squares_ge_zero % 0.30/0.51 thf(fact_1090_sum__squares__gt__zero__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord4710134922213307826strict @ A ) % 0.30/0.51 => ! [X3: A,Y: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) ) % 0.30/0.51 = ( ( X3 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 | ( Y % 0.30/0.51 != ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % sum_squares_gt_zero_iff % 0.30/0.51 thf(fact_1091_not__sum__squares__lt__zero,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_ring @ A ) % 0.30/0.51 => ! [X3: A,Y: A] : % 0.30/0.51 ~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.51 % 0.30/0.51 % not_sum_squares_lt_zero % 0.30/0.51 thf(fact_1092_divide__less__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [B2: A,C2: A,A2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % divide_less_eq % 0.30/0.51 thf(fact_1093_less__divide__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % less_divide_eq % 0.30/0.51 thf(fact_1094_neg__divide__less__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [C2: A,B2: A,A2: A] : % 0.30/0.51 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 0.30/0.51 = ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % neg_divide_less_eq % 0.30/0.51 thf(fact_1095_neg__less__divide__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % neg_less_divide_eq % 0.30/0.51 thf(fact_1096_pos__divide__less__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [C2: A,B2: A,A2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 0.30/0.51 = ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % pos_divide_less_eq % 0.30/0.51 thf(fact_1097_pos__less__divide__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % pos_less_divide_eq % 0.30/0.51 thf(fact_1098_mult__imp__div__pos__less,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [Y: A,X3: A,Z2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.51 => ( ( ord_less @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) ) % 0.30/0.51 => ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ Z2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_imp_div_pos_less % 0.30/0.51 thf(fact_1099_mult__imp__less__div__pos,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [Y: A,Z2: A,X3: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.51 => ( ( ord_less @ A @ ( times_times @ A @ Z2 @ Y ) @ X3 ) % 0.30/0.51 => ( ord_less @ A @ Z2 @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_imp_less_div_pos % 0.30/0.51 thf(fact_1100_divide__strict__left__mono,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [B2: A,A2: A,C2: A] : % 0.30/0.51 ( ( ord_less @ A @ B2 @ A2 ) % 0.30/0.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 0.30/0.51 => ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % divide_strict_left_mono % 0.30/0.51 thf(fact_1101_divide__strict__left__mono__neg,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 0.30/0.51 => ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % divide_strict_left_mono_neg % 0.30/0.51 thf(fact_1102_divide__eq__eq__numeral_I1_J,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( division_ring @ A ) % 0.30/0.51 => ! [B2: A,C2: A,W2: num] : % 0.30/0.51 ( ( ( divide_divide @ A @ B2 @ C2 ) % 0.30/0.51 = ( numeral_numeral @ A @ W2 ) ) % 0.30/0.51 = ( ( ( C2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( B2 % 0.30/0.51 = ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) ) % 0.30/0.51 & ( ( C2 % 0.30/0.51 = ( zero_zero @ A ) ) % 0.30/0.51 => ( ( numeral_numeral @ A @ W2 ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % divide_eq_eq_numeral(1) % 0.30/0.51 thf(fact_1103_eq__divide__eq__numeral_I1_J,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( division_ring @ A ) % 0.30/0.51 => ! [W2: num,B2: A,C2: A] : % 0.30/0.51 ( ( ( numeral_numeral @ A @ W2 ) % 0.30/0.51 = ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ( ( C2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) % 0.30/0.51 = B2 ) ) % 0.30/0.51 & ( ( C2 % 0.30/0.51 = ( zero_zero @ A ) ) % 0.30/0.51 => ( ( numeral_numeral @ A @ W2 ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % eq_divide_eq_numeral(1) % 0.30/0.51 thf(fact_1104_add__divide__eq__if__simps_I2_J,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( division_ring @ A ) % 0.30/0.51 => ! [Z2: A,A2: A,B2: A] : % 0.30/0.51 ( ( ( Z2 % 0.30/0.51 = ( zero_zero @ A ) ) % 0.30/0.51 => ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 ) % 0.30/0.51 = B2 ) ) % 0.30/0.51 & ( ( Z2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 ) % 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % add_divide_eq_if_simps(2) % 0.30/0.51 thf(fact_1105_add__divide__eq__if__simps_I1_J,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( division_ring @ A ) % 0.30/0.51 => ! [Z2: A,A2: A,B2: A] : % 0.30/0.51 ( ( ( Z2 % 0.30/0.51 = ( zero_zero @ A ) ) % 0.30/0.51 => ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) ) % 0.30/0.51 = A2 ) ) % 0.30/0.51 & ( ( Z2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) ) % 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % add_divide_eq_if_simps(1) % 0.30/0.51 thf(fact_1106_add__frac__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( field @ A ) % 0.30/0.51 => ! [Y: A,Z2: A,X3: A,W2: A] : % 0.30/0.51 ( ( Y % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( Z2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) ) % 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % add_frac_eq % 0.30/0.51 thf(fact_1107_add__frac__num,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( field @ A ) % 0.30/0.51 => ! [Y: A,X3: A,Z2: A] : % 0.30/0.51 ( ( Y % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Y ) @ Z2 ) % 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % add_frac_num % 0.30/0.51 thf(fact_1108_add__num__frac,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( field @ A ) % 0.30/0.51 => ! [Y: A,Z2: A,X3: A] : % 0.30/0.51 ( ( Y % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ X3 @ Y ) ) % 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % add_num_frac % 0.30/0.51 thf(fact_1109_add__divide__eq__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( division_ring @ A ) % 0.30/0.51 => ! [Z2: A,X3: A,Y: A] : % 0.30/0.51 ( ( Z2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( plus_plus @ A @ X3 @ ( divide_divide @ A @ Y @ Z2 ) ) % 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % add_divide_eq_iff % 0.30/0.51 thf(fact_1110_divide__add__eq__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( division_ring @ A ) % 0.30/0.51 => ! [Z2: A,X3: A,Y: A] : % 0.30/0.51 ( ( Z2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ Y ) % 0.30/0.51 = ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % divide_add_eq_iff % 0.30/0.51 thf(fact_1111_power__gt1__lemma,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_semidom @ A ) % 0.30/0.51 => ! [A2: A,N2: nat] : % 0.30/0.51 ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.51 => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % power_gt1_lemma % 0.30/0.51 thf(fact_1112_power__less__power__Suc,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_semidom @ A ) % 0.30/0.51 => ! [A2: A,N2: nat] : % 0.30/0.51 ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.51 => ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % power_less_power_Suc % 0.30/0.51 thf(fact_1113_one__less__mult,axiom, % 0.30/0.51 ! [N2: nat,M: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 0.30/0.51 => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M ) % 0.30/0.51 => ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % one_less_mult % 0.30/0.51 thf(fact_1114_n__less__m__mult__n,axiom, % 0.30/0.51 ! [N2: nat,M: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.51 => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M ) % 0.30/0.51 => ( ord_less @ nat @ N2 @ ( times_times @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % n_less_m_mult_n % 0.30/0.51 thf(fact_1115_n__less__n__mult__m,axiom, % 0.30/0.51 ! [N2: nat,M: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.51 => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M ) % 0.30/0.51 => ( ord_less @ nat @ N2 @ ( times_times @ nat @ N2 @ M ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % n_less_n_mult_m % 0.30/0.51 thf(fact_1116_nat__mult__le__cancel1,axiom, % 0.30/0.51 ! [K: nat,M: nat,N2: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.30/0.51 => ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 0.30/0.51 = ( ord_less_eq @ nat @ M @ N2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % nat_mult_le_cancel1 % 0.30/0.51 thf(fact_1117_nat__mult__div__cancel1,axiom, % 0.30/0.51 ! [K: nat,M: nat,N2: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.30/0.51 => ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 0.30/0.51 = ( divide_divide @ nat @ M @ N2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % nat_mult_div_cancel1 % 0.30/0.51 thf(fact_1118_div__less__iff__less__mult,axiom, % 0.30/0.51 ! [Q4: nat,M: nat,N2: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 ) % 0.30/0.51 => ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q4 ) @ N2 ) % 0.30/0.51 = ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ Q4 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_less_iff_less_mult % 0.30/0.51 thf(fact_1119_mult__le__cancel__left1,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [C2: A,B2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) ) % 0.30/0.51 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_le_cancel_left1 % 0.30/0.51 thf(fact_1120_mult__le__cancel__left2,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [C2: A,A2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) ) % 0.30/0.51 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_le_cancel_left2 % 0.30/0.51 thf(fact_1121_mult__le__cancel__right1,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [C2: A,B2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) ) % 0.30/0.51 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_le_cancel_right1 % 0.30/0.51 thf(fact_1122_mult__le__cancel__right2,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [A2: A,C2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) ) % 0.30/0.51 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_le_cancel_right2 % 0.30/0.51 thf(fact_1123_mult__less__cancel__left1,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [C2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.51 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ ( one_one @ A ) @ B2 ) ) % 0.30/0.51 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_less_cancel_left1 % 0.30/0.51 thf(fact_1124_mult__less__cancel__left2,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [C2: A,A2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 ) % 0.30/0.51 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ A2 @ ( one_one @ A ) ) ) % 0.30/0.51 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_less_cancel_left2 % 0.30/0.51 thf(fact_1125_mult__less__cancel__right1,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [C2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ ( one_one @ A ) @ B2 ) ) % 0.30/0.51 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_less_cancel_right1 % 0.30/0.51 thf(fact_1126_mult__less__cancel__right2,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [A2: A,C2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 ) % 0.30/0.51 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ A2 @ ( one_one @ A ) ) ) % 0.30/0.51 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_less_cancel_right2 % 0.30/0.51 thf(fact_1127_field__le__mult__one__interval,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [X3: A,Y: A] : % 0.30/0.51 ( ! [Z3: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 ) % 0.30/0.51 => ( ( ord_less @ A @ Z3 @ ( one_one @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X3 ) @ Y ) ) ) % 0.30/0.51 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ). % 0.30/0.51 % 0.30/0.51 % field_le_mult_one_interval % 0.30/0.51 thf(fact_1128_divide__le__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [B2: A,C2: A,A2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % divide_le_eq % 0.30/0.51 thf(fact_1129_le__divide__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % le_divide_eq % 0.30/0.51 thf(fact_1130_divide__left__mono,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [B2: A,A2: A,C2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % divide_left_mono % 0.30/0.51 thf(fact_1131_neg__divide__le__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [C2: A,B2: A,A2: A] : % 0.30/0.51 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 0.30/0.51 = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % neg_divide_le_eq % 0.30/0.51 thf(fact_1132_neg__le__divide__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % neg_le_divide_eq % 0.30/0.51 thf(fact_1133_pos__divide__le__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [C2: A,B2: A,A2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 0.30/0.51 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % pos_divide_le_eq % 0.30/0.51 thf(fact_1134_pos__le__divide__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % pos_le_divide_eq % 0.30/0.51 thf(fact_1135_mult__imp__div__pos__le,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [Y: A,X3: A,Z2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.51 => ( ( ord_less_eq @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ Z2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_imp_div_pos_le % 0.30/0.51 thf(fact_1136_mult__imp__le__div__pos,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [Y: A,Z2: A,X3: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ Y ) @ X3 ) % 0.30/0.51 => ( ord_less_eq @ A @ Z2 @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_imp_le_div_pos % 0.30/0.51 thf(fact_1137_divide__left__mono__neg,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % divide_left_mono_neg % 0.30/0.51 thf(fact_1138_convex__bound__le,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord6961819062388156250ring_1 @ A ) % 0.30/0.51 => ! [X3: A,A2: A,Y: A,U: A,V2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ X3 @ A2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ Y @ A2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 ) % 0.30/0.51 => ( ( ( plus_plus @ A @ U @ V2 ) % 0.30/0.51 = ( one_one @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X3 ) @ ( times_times @ A @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % convex_bound_le % 0.30/0.51 thf(fact_1139_divide__less__eq__numeral_I1_J,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [B2: A,C2: A,W2: num] : % 0.30/0.51 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % divide_less_eq_numeral(1) % 0.30/0.51 thf(fact_1140_less__divide__eq__numeral_I1_J,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [W2: num,B2: A,C2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % less_divide_eq_numeral(1) % 0.30/0.51 thf(fact_1141_power__Suc__less,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_semidom @ A ) % 0.30/0.51 => ! [A2: A,N2: nat] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.51 => ( ( ord_less @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ord_less @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % power_Suc_less % 0.30/0.51 thf(fact_1142_mult__2,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semiring_numeral @ A ) % 0.30/0.51 => ! [Z2: A] : % 0.30/0.51 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) % 0.30/0.51 = ( plus_plus @ A @ Z2 @ Z2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_2 % 0.30/0.51 thf(fact_1143_mult__2__right,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semiring_numeral @ A ) % 0.30/0.51 => ! [Z2: A] : % 0.30/0.51 ( ( times_times @ A @ Z2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.51 = ( plus_plus @ A @ Z2 @ Z2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_2_right % 0.30/0.51 thf(fact_1144_left__add__twice,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semiring_numeral @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.30/0.51 = ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % left_add_twice % 0.30/0.51 thf(fact_1145_div__nat__eqI,axiom, % 0.30/0.51 ! [N2: nat,Q4: nat,M: nat] : % 0.30/0.51 ( ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ Q4 ) @ M ) % 0.30/0.51 => ( ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ ( suc @ Q4 ) ) ) % 0.30/0.51 => ( ( divide_divide @ nat @ M @ N2 ) % 0.30/0.51 = Q4 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_nat_eqI % 0.30/0.51 thf(fact_1146_less__eq__div__iff__mult__less__eq,axiom, % 0.30/0.51 ! [Q4: nat,M: nat,N2: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 ) % 0.30/0.51 => ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N2 @ Q4 ) ) % 0.30/0.51 = ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q4 ) @ N2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % less_eq_div_iff_mult_less_eq % 0.30/0.51 thf(fact_1147_split__div,axiom, % 0.30/0.51 ! [P: nat > $o,M: nat,N2: nat] : % 0.30/0.51 ( ( P @ ( divide_divide @ nat @ M @ N2 ) ) % 0.30/0.51 = ( ( ( N2 % 0.30/0.51 = ( zero_zero @ nat ) ) % 0.30/0.51 => ( P @ ( zero_zero @ nat ) ) ) % 0.30/0.51 & ( ( N2 % 0.30/0.51 != ( zero_zero @ nat ) ) % 0.30/0.51 => ! [I3: nat,J3: nat] : % 0.30/0.51 ( ( ord_less @ nat @ J3 @ N2 ) % 0.30/0.51 => ( ( M % 0.30/0.51 = ( plus_plus @ nat @ ( times_times @ nat @ N2 @ I3 ) @ J3 ) ) % 0.30/0.51 => ( P @ I3 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % split_div % 0.30/0.51 thf(fact_1148_dividend__less__div__times,axiom, % 0.30/0.51 ! [N2: nat,M: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.51 => ( ord_less @ nat @ M @ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N2 ) @ N2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dividend_less_div_times % 0.30/0.51 thf(fact_1149_dividend__less__times__div,axiom, % 0.30/0.51 ! [N2: nat,M: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.51 => ( ord_less @ nat @ M @ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ N2 @ ( divide_divide @ nat @ M @ N2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dividend_less_times_div % 0.30/0.51 thf(fact_1150_nat__bit__induct,axiom, % 0.30/0.51 ! [P: nat > $o,N2: nat] : % 0.30/0.51 ( ( P @ ( zero_zero @ nat ) ) % 0.30/0.51 => ( ! [N3: nat] : % 0.30/0.51 ( ( P @ N3 ) % 0.30/0.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 0.30/0.51 => ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) % 0.30/0.51 => ( ! [N3: nat] : % 0.30/0.51 ( ( P @ N3 ) % 0.30/0.51 => ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) % 0.30/0.51 => ( P @ N2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % nat_bit_induct % 0.30/0.51 thf(fact_1151_convex__bound__lt,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linord715952674999750819strict @ A ) % 0.30/0.51 => ! [X3: A,A2: A,Y: A,U: A,V2: A] : % 0.30/0.51 ( ( ord_less @ A @ X3 @ A2 ) % 0.30/0.51 => ( ( ord_less @ A @ Y @ A2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 ) % 0.30/0.51 => ( ( ( plus_plus @ A @ U @ V2 ) % 0.30/0.51 = ( one_one @ A ) ) % 0.30/0.51 => ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X3 ) @ ( times_times @ A @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % convex_bound_lt % 0.30/0.51 thf(fact_1152_divide__le__eq__numeral_I1_J,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [B2: A,C2: A,W2: num] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % divide_le_eq_numeral(1) % 0.30/0.51 thf(fact_1153_le__divide__eq__numeral_I1_J,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [W2: num,B2: A,C2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) ) % 0.30/0.51 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % le_divide_eq_numeral(1) % 0.30/0.51 thf(fact_1154_split__div_H,axiom, % 0.30/0.51 ! [P: nat > $o,M: nat,N2: nat] : % 0.30/0.51 ( ( P @ ( divide_divide @ nat @ M @ N2 ) ) % 0.30/0.51 = ( ( ( N2 % 0.30/0.51 = ( zero_zero @ nat ) ) % 0.30/0.51 & ( P @ ( zero_zero @ nat ) ) ) % 0.30/0.51 | ? [Q5: nat] : % 0.30/0.51 ( ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ Q5 ) @ M ) % 0.30/0.51 & ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ ( suc @ Q5 ) ) ) % 0.30/0.51 & ( P @ Q5 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % split_div' % 0.30/0.51 thf(fact_1155_power2__sum,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_semiring_1 @ A ) % 0.30/0.51 => ! [X3: A,Y: A] : % 0.30/0.51 ( ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.51 = ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) @ Y ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % power2_sum % 0.30/0.51 thf(fact_1156_zero__le__even__power_H,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [A2: A,N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % zero_le_even_power' % 0.30/0.51 thf(fact_1157_triangle__def,axiom, % 0.30/0.51 ( nat_triangle % 0.30/0.51 = ( ^ [N: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % triangle_def % 0.30/0.51 thf(fact_1158_odd__0__le__power__imp__0__le,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [A2: A,N2: nat] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) % 0.30/0.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % odd_0_le_power_imp_0_le % 0.30/0.51 thf(fact_1159_odd__power__less__zero,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [A2: A,N2: nat] : % 0.30/0.51 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.51 => ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % odd_power_less_zero % 0.30/0.51 thf(fact_1160_arith__geo__mean,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [U: A,X3: A,Y: A] : % 0.30/0.51 ( ( ( power_power @ A @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.51 = ( times_times @ A @ X3 @ Y ) ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.51 => ( ord_less_eq @ A @ U @ ( divide_divide @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % arith_geo_mean % 0.30/0.51 thf(fact_1161_sum__squares__bound,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_field @ A ) % 0.30/0.51 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) @ Y ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % sum_squares_bound % 0.30/0.51 thf(fact_1162_set__bit__0,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A2 ) % 0.30/0.51 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % set_bit_0 % 0.30/0.51 thf(fact_1163_pos2,axiom, % 0.30/0.51 ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ). % 0.30/0.51 % 0.30/0.51 % pos2 % 0.30/0.51 thf(fact_1164_unset__bit__0,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A2 ) % 0.30/0.51 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unset_bit_0 % 0.30/0.51 thf(fact_1165_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.51 => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unique_euclidean_semiring_numeral_class.div_mult2_eq % 0.30/0.51 thf(fact_1166_discrete,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.51 => ( ( ord_less @ A ) % 0.30/0.51 = ( ^ [A4: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % discrete % 0.30/0.51 thf(fact_1167_unique__euclidean__semiring__numeral__class_Odiv__less,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.51 => ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( divide_divide @ A @ A2 @ B2 ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unique_euclidean_semiring_numeral_class.div_less % 0.30/0.51 thf(fact_1168_div__positive,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.51 => ! [B2: A,A2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.30/0.51 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_positive % 0.30/0.51 thf(fact_1169_zle__add1__eq__le,axiom, % 0.30/0.51 ! [W2: int,Z2: int] : % 0.30/0.51 ( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) ) % 0.30/0.51 = ( ord_less_eq @ int @ W2 @ Z2 ) ) ). % 0.30/0.51 % 0.30/0.51 % zle_add1_eq_le % 0.30/0.51 thf(fact_1170_set__bit__negative__int__iff,axiom, % 0.30/0.51 ! [N2: nat,K: int] : % 0.30/0.51 ( ( ord_less @ int @ ( bit_se5668285175392031749et_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) % 0.30/0.51 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ). % 0.30/0.51 % 0.30/0.51 % set_bit_negative_int_iff % 0.30/0.51 thf(fact_1171_unset__bit__negative__int__iff,axiom, % 0.30/0.51 ! [N2: nat,K: int] : % 0.30/0.51 ( ( ord_less @ int @ ( bit_se2638667681897837118et_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) % 0.30/0.51 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unset_bit_negative_int_iff % 0.30/0.51 thf(fact_1172_div__pos__pos__trivial,axiom, % 0.30/0.51 ! [K: int,L: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.51 => ( ( ord_less @ int @ K @ L ) % 0.30/0.51 => ( ( divide_divide @ int @ K @ L ) % 0.30/0.51 = ( zero_zero @ int ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_pos_pos_trivial % 0.30/0.51 thf(fact_1173_div__neg__neg__trivial,axiom, % 0.30/0.51 ! [K: int,L: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) ) % 0.30/0.51 => ( ( ord_less @ int @ L @ K ) % 0.30/0.51 => ( ( divide_divide @ int @ K @ L ) % 0.30/0.51 = ( zero_zero @ int ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_neg_neg_trivial % 0.30/0.51 thf(fact_1174_nonneg1__imp__zdiv__pos__iff,axiom, % 0.30/0.51 ! [A2: int,B2: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) % 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) ) % 0.30/0.51 = ( ( ord_less_eq @ int @ B2 @ A2 ) % 0.30/0.51 & ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % nonneg1_imp_zdiv_pos_iff % 0.30/0.51 thf(fact_1175_pos__imp__zdiv__nonneg__iff,axiom, % 0.30/0.51 ! [B2: int,A2: int] : % 0.30/0.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) ) % 0.30/0.51 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % pos_imp_zdiv_nonneg_iff % 0.30/0.51 thf(fact_1176_neg__imp__zdiv__nonneg__iff,axiom, % 0.30/0.51 ! [B2: int,A2: int] : % 0.30/0.51 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) ) % 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) ) % 0.30/0.51 = ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % neg_imp_zdiv_nonneg_iff % 0.30/0.51 thf(fact_1177_int__one__le__iff__zero__less,axiom, % 0.30/0.51 ! [Z2: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 ) % 0.30/0.51 = ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ). % 0.30/0.51 % 0.30/0.51 % int_one_le_iff_zero_less % 0.30/0.51 thf(fact_1178_pos__imp__zdiv__pos__iff,axiom, % 0.30/0.51 ! [K: int,I: int] : % 0.30/0.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I @ K ) ) % 0.30/0.51 = ( ord_less_eq @ int @ K @ I ) ) ) ). % 0.30/0.51 % 0.30/0.51 % pos_imp_zdiv_pos_iff % 0.30/0.51 thf(fact_1179_div__nonpos__pos__le0,axiom, % 0.30/0.51 ! [A2: int,B2: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) % 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.30/0.51 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_nonpos_pos_le0 % 0.30/0.51 thf(fact_1180_div__nonneg__neg__le0,axiom, % 0.30/0.51 ! [A2: int,B2: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) % 0.30/0.51 => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) ) % 0.30/0.51 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_nonneg_neg_le0 % 0.30/0.51 thf(fact_1181_div__positive__int,axiom, % 0.30/0.51 ! [L: int,K: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ L @ K ) % 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ L ) % 0.30/0.51 => ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_positive_int % 0.30/0.51 thf(fact_1182_div__int__pos__iff,axiom, % 0.30/0.51 ! [K: int,L: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) ) % 0.30/0.51 = ( ( K % 0.30/0.51 = ( zero_zero @ int ) ) % 0.30/0.51 | ( L % 0.30/0.51 = ( zero_zero @ int ) ) % 0.30/0.51 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.51 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) % 0.30/0.51 | ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 0.30/0.51 & ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_int_pos_iff % 0.30/0.51 thf(fact_1183_zless__imp__add1__zle,axiom, % 0.30/0.51 ! [W2: int,Z2: int] : % 0.30/0.51 ( ( ord_less @ int @ W2 @ Z2 ) % 0.30/0.51 => ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z2 ) ) ). % 0.30/0.51 % 0.30/0.51 % zless_imp_add1_zle % 0.30/0.51 thf(fact_1184_zdiv__mono2__neg,axiom, % 0.30/0.51 ! [A2: int,B7: int,B2: int] : % 0.30/0.51 ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) % 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 ) % 0.30/0.51 => ( ( ord_less_eq @ int @ B7 @ B2 ) % 0.30/0.51 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B7 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % zdiv_mono2_neg % 0.30/0.51 thf(fact_1185_zdiv__mono1__neg,axiom, % 0.30/0.51 ! [A2: int,A7: int,B2: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ A2 @ A7 ) % 0.30/0.51 => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) ) % 0.30/0.51 => ( ord_less_eq @ int @ ( divide_divide @ int @ A7 @ B2 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % zdiv_mono1_neg % 0.30/0.51 thf(fact_1186_int__div__pos__eq,axiom, % 0.30/0.51 ! [A2: int,B2: int,Q4: int,R2: int] : % 0.30/0.51 ( ( A2 % 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) ) % 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 ) % 0.30/0.51 => ( ( ord_less @ int @ R2 @ B2 ) % 0.30/0.51 => ( ( divide_divide @ int @ A2 @ B2 ) % 0.30/0.51 = Q4 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % int_div_pos_eq % 0.30/0.51 thf(fact_1187_int__div__neg__eq,axiom, % 0.30/0.51 ! [A2: int,B2: int,Q4: int,R2: int] : % 0.30/0.51 ( ( A2 % 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) ) % 0.30/0.51 => ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) ) % 0.30/0.51 => ( ( ord_less @ int @ B2 @ R2 ) % 0.30/0.51 => ( ( divide_divide @ int @ A2 @ B2 ) % 0.30/0.51 = Q4 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % int_div_neg_eq % 0.30/0.51 thf(fact_1188_zdiv__eq__0__iff,axiom, % 0.30/0.51 ! [I: int,K: int] : % 0.30/0.51 ( ( ( divide_divide @ int @ I @ K ) % 0.30/0.51 = ( zero_zero @ int ) ) % 0.30/0.51 = ( ( K % 0.30/0.51 = ( zero_zero @ int ) ) % 0.30/0.51 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I ) % 0.30/0.51 & ( ord_less @ int @ I @ K ) ) % 0.30/0.51 | ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) % 0.30/0.51 & ( ord_less @ int @ K @ I ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % zdiv_eq_0_iff % 0.30/0.51 thf(fact_1189_zdiv__mono2,axiom, % 0.30/0.51 ! [A2: int,B7: int,B2: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) % 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 ) % 0.30/0.51 => ( ( ord_less_eq @ int @ B7 @ B2 ) % 0.30/0.51 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B7 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % zdiv_mono2 % 0.30/0.51 thf(fact_1190_zdiv__mono1,axiom, % 0.30/0.51 ! [A2: int,A7: int,B2: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ A2 @ A7 ) % 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.30/0.51 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A7 @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % zdiv_mono1 % 0.30/0.51 thf(fact_1191_split__zdiv,axiom, % 0.30/0.51 ! [P: int > $o,N2: int,K: int] : % 0.30/0.51 ( ( P @ ( divide_divide @ int @ N2 @ K ) ) % 0.30/0.51 = ( ( ( K % 0.30/0.51 = ( zero_zero @ int ) ) % 0.30/0.51 => ( P @ ( zero_zero @ int ) ) ) % 0.30/0.51 & ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.51 => ! [I3: int,J3: int] : % 0.30/0.51 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 ) % 0.30/0.51 & ( ord_less @ int @ J3 @ K ) % 0.30/0.51 & ( N2 % 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) ) % 0.30/0.51 => ( P @ I3 ) ) ) % 0.30/0.51 & ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 0.30/0.51 => ! [I3: int,J3: int] : % 0.30/0.51 ( ( ( ord_less @ int @ K @ J3 ) % 0.30/0.51 & ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) ) % 0.30/0.51 & ( N2 % 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) ) % 0.30/0.51 => ( P @ I3 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % split_zdiv % 0.30/0.51 thf(fact_1192_le__imp__0__less,axiom, % 0.30/0.51 ! [Z2: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 0.30/0.51 => ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % le_imp_0_less % 0.30/0.51 thf(fact_1193_add1__zle__eq,axiom, % 0.30/0.51 ! [W2: int,Z2: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z2 ) % 0.30/0.51 = ( ord_less @ int @ W2 @ Z2 ) ) ). % 0.30/0.51 % 0.30/0.51 % add1_zle_eq % 0.30/0.51 thf(fact_1194_odd__less__0__iff,axiom, % 0.30/0.51 ! [Z2: int] : % 0.30/0.51 ( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 ) @ ( zero_zero @ int ) ) % 0.30/0.51 = ( ord_less @ int @ Z2 @ ( zero_zero @ int ) ) ) ). % 0.30/0.51 % 0.30/0.51 % odd_less_0_iff % 0.30/0.51 thf(fact_1195_pos__zmult__eq__1__iff,axiom, % 0.30/0.51 ! [M: int,N2: int] : % 0.30/0.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ M ) % 0.30/0.51 => ( ( ( times_times @ int @ M @ N2 ) % 0.30/0.51 = ( one_one @ int ) ) % 0.30/0.51 = ( ( M % 0.30/0.51 = ( one_one @ int ) ) % 0.30/0.51 & ( N2 % 0.30/0.51 = ( one_one @ int ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % pos_zmult_eq_1_iff % 0.30/0.51 thf(fact_1196_zless__add1__eq,axiom, % 0.30/0.51 ! [W2: int,Z2: int] : % 0.30/0.51 ( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) ) % 0.30/0.51 = ( ( ord_less @ int @ W2 @ Z2 ) % 0.30/0.51 | ( W2 = Z2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % zless_add1_eq % 0.30/0.51 thf(fact_1197_int__gr__induct,axiom, % 0.30/0.51 ! [K: int,I: int,P: int > $o] : % 0.30/0.51 ( ( ord_less @ int @ K @ I ) % 0.30/0.51 => ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) % 0.30/0.51 => ( ! [I4: int] : % 0.30/0.51 ( ( ord_less @ int @ K @ I4 ) % 0.30/0.51 => ( ( P @ I4 ) % 0.30/0.51 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) ) % 0.30/0.51 => ( P @ I ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % int_gr_induct % 0.30/0.51 thf(fact_1198_pos__imp__zdiv__neg__iff,axiom, % 0.30/0.51 ! [B2: int,A2: int] : % 0.30/0.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.30/0.51 => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) % 0.30/0.51 = ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % pos_imp_zdiv_neg_iff % 0.30/0.51 thf(fact_1199_neg__imp__zdiv__neg__iff,axiom, % 0.30/0.51 ! [B2: int,A2: int] : % 0.30/0.51 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) ) % 0.30/0.51 => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) % 0.30/0.51 = ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % neg_imp_zdiv_neg_iff % 0.30/0.51 thf(fact_1200_int__div__less__self,axiom, % 0.30/0.51 ! [X3: int,K: int] : % 0.30/0.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ X3 ) % 0.30/0.51 => ( ( ord_less @ int @ ( one_one @ int ) @ K ) % 0.30/0.51 => ( ord_less @ int @ ( divide_divide @ int @ X3 @ K ) @ X3 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % int_div_less_self % 0.30/0.51 thf(fact_1201_div__neg__pos__less0,axiom, % 0.30/0.51 ! [A2: int,B2: int] : % 0.30/0.51 ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) % 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.30/0.51 => ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_neg_pos_less0 % 0.30/0.51 thf(fact_1202_odd__nonzero,axiom, % 0.30/0.51 ! [Z2: int] : % 0.30/0.51 ( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 ) % 0.30/0.51 != ( zero_zero @ int ) ) ). % 0.30/0.51 % 0.30/0.51 % odd_nonzero % 0.30/0.51 thf(fact_1203_times__int__code_I2_J,axiom, % 0.30/0.51 ! [L: int] : % 0.30/0.51 ( ( times_times @ int @ ( zero_zero @ int ) @ L ) % 0.30/0.51 = ( zero_zero @ int ) ) ). % 0.30/0.51 % 0.30/0.51 % times_int_code(2) % 0.30/0.51 thf(fact_1204_times__int__code_I1_J,axiom, % 0.30/0.51 ! [K: int] : % 0.30/0.51 ( ( times_times @ int @ K @ ( zero_zero @ int ) ) % 0.30/0.51 = ( zero_zero @ int ) ) ). % 0.30/0.51 % 0.30/0.51 % times_int_code(1) % 0.30/0.51 thf(fact_1205_bot__enat__def,axiom, % 0.30/0.51 ( ( bot_bot @ extended_enat ) % 0.30/0.51 = ( zero_zero @ extended_enat ) ) ). % 0.30/0.51 % 0.30/0.51 % bot_enat_def % 0.30/0.51 thf(fact_1206_incr__mult__lemma,axiom, % 0.30/0.51 ! [D2: int,P: int > $o,K: int] : % 0.30/0.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 ) % 0.30/0.51 => ( ! [X5: int] : % 0.30/0.51 ( ( P @ X5 ) % 0.30/0.51 => ( P @ ( plus_plus @ int @ X5 @ D2 ) ) ) % 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.51 => ! [X2: int] : % 0.30/0.51 ( ( P @ X2 ) % 0.30/0.51 => ( P @ ( plus_plus @ int @ X2 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % incr_mult_lemma % 0.30/0.51 thf(fact_1207_zmult__zless__mono2,axiom, % 0.30/0.51 ! [I: int,J: int,K: int] : % 0.30/0.51 ( ( ord_less @ int @ I @ J ) % 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.51 => ( ord_less @ int @ ( times_times @ int @ K @ I ) @ ( times_times @ int @ K @ J ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % zmult_zless_mono2 % 0.30/0.51 thf(fact_1208_less__int__code_I1_J,axiom, % 0.30/0.51 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ). % 0.30/0.51 % 0.30/0.51 % less_int_code(1) % 0.30/0.51 thf(fact_1209_q__pos__lemma,axiom, % 0.30/0.51 ! [B7: int,Q6: int,R3: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B7 @ Q6 ) @ R3 ) ) % 0.30/0.51 => ( ( ord_less @ int @ R3 @ B7 ) % 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 ) % 0.30/0.51 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q6 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % q_pos_lemma % 0.30/0.51 thf(fact_1210_zdiv__mono2__lemma,axiom, % 0.30/0.51 ! [B2: int,Q4: int,R2: int,B7: int,Q6: int,R3: int] : % 0.30/0.51 ( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) % 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ B7 @ Q6 ) @ R3 ) ) % 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B7 @ Q6 ) @ R3 ) ) % 0.30/0.51 => ( ( ord_less @ int @ R3 @ B7 ) % 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 ) % 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 ) % 0.30/0.51 => ( ( ord_less_eq @ int @ B7 @ B2 ) % 0.30/0.51 => ( ord_less_eq @ int @ Q4 @ Q6 ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % zdiv_mono2_lemma % 0.30/0.51 thf(fact_1211_zdiv__mono2__neg__lemma,axiom, % 0.30/0.51 ! [B2: int,Q4: int,R2: int,B7: int,Q6: int,R3: int] : % 0.30/0.51 ( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) % 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ B7 @ Q6 ) @ R3 ) ) % 0.30/0.51 => ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B7 @ Q6 ) @ R3 ) @ ( zero_zero @ int ) ) % 0.30/0.51 => ( ( ord_less @ int @ R2 @ B2 ) % 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 ) % 0.30/0.51 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 ) % 0.30/0.51 => ( ( ord_less_eq @ int @ B7 @ B2 ) % 0.30/0.51 => ( ord_less_eq @ int @ Q6 @ Q4 ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % zdiv_mono2_neg_lemma % 0.30/0.51 thf(fact_1212_unique__quotient__lemma,axiom, % 0.30/0.51 ! [B2: int,Q6: int,R3: int,Q4: int,R2: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R3 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) ) % 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 ) % 0.30/0.51 => ( ( ord_less @ int @ R3 @ B2 ) % 0.30/0.51 => ( ( ord_less @ int @ R2 @ B2 ) % 0.30/0.51 => ( ord_less_eq @ int @ Q6 @ Q4 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unique_quotient_lemma % 0.30/0.51 thf(fact_1213_unique__quotient__lemma__neg,axiom, % 0.30/0.51 ! [B2: int,Q6: int,R3: int,Q4: int,R2: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R3 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) ) % 0.30/0.51 => ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) ) % 0.30/0.51 => ( ( ord_less @ int @ B2 @ R2 ) % 0.30/0.51 => ( ( ord_less @ int @ B2 @ R3 ) % 0.30/0.51 => ( ord_less_eq @ int @ Q4 @ Q6 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unique_quotient_lemma_neg % 0.30/0.51 thf(fact_1214_enat__0__less__mult__iff,axiom, % 0.30/0.51 ! [M: extended_enat,N2: extended_enat] : % 0.30/0.51 ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ ( times_times @ extended_enat @ M @ N2 ) ) % 0.30/0.51 = ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ M ) % 0.30/0.51 & ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % enat_0_less_mult_iff % 0.30/0.51 thf(fact_1215_realpow__pos__nth2,axiom, % 0.30/0.51 ! [A2: real,N2: nat] : % 0.30/0.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.30/0.51 => ? [R4: real] : % 0.30/0.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 ) % 0.30/0.51 & ( ( power_power @ real @ R4 @ ( suc @ N2 ) ) % 0.30/0.51 = A2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % realpow_pos_nth2 % 0.30/0.51 thf(fact_1216_realpow__pos__nth__unique,axiom, % 0.30/0.51 ! [N2: nat,A2: real] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.30/0.51 => ? [X5: real] : % 0.30/0.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X5 ) % 0.30/0.51 & ( ( power_power @ real @ X5 @ N2 ) % 0.30/0.51 = A2 ) % 0.30/0.51 & ! [Y5: real] : % 0.30/0.51 ( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) % 0.30/0.51 & ( ( power_power @ real @ Y5 @ N2 ) % 0.30/0.51 = A2 ) ) % 0.30/0.51 => ( Y5 = X5 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % realpow_pos_nth_unique % 0.30/0.51 thf(fact_1217_realpow__pos__nth,axiom, % 0.30/0.51 ! [N2: nat,A2: real] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.30/0.51 => ? [R4: real] : % 0.30/0.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 ) % 0.30/0.51 & ( ( power_power @ real @ R4 @ N2 ) % 0.30/0.51 = A2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % realpow_pos_nth % 0.30/0.51 thf(fact_1218_mult__le__cancel__iff2,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [Z2: A,X3: A,Y: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ X3 ) @ ( times_times @ A @ Z2 @ Y ) ) % 0.30/0.51 = ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_le_cancel_iff2 % 0.30/0.51 thf(fact_1219_mult__le__cancel__iff1,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [Z2: A,X3: A,Y: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 ) % 0.30/0.51 => ( ( ord_less_eq @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ Y @ Z2 ) ) % 0.30/0.51 = ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_le_cancel_iff1 % 0.30/0.51 thf(fact_1220_divides__aux__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.51 => ! [Q4: A,R2: A] : % 0.30/0.51 ( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q4 @ R2 ) ) % 0.30/0.51 = ( R2 % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % divides_aux_eq % 0.30/0.51 thf(fact_1221_low__def,axiom, % 0.30/0.51 ( vEBT_VEBT_low % 0.30/0.51 = ( ^ [X: nat,N: nat] : ( modulo_modulo @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % low_def % 0.30/0.51 thf(fact_1222_even__succ__div__exp,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( bit_semiring_bits @ A ) % 0.30/0.51 => ! [A2: A,N2: nat] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.51 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.51 = ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % even_succ_div_exp % 0.30/0.51 thf(fact_1223_set__decode__Suc,axiom, % 0.30/0.51 ! [N2: nat,X3: nat] : % 0.30/0.51 ( ( member @ nat @ ( suc @ N2 ) @ ( nat_set_decode @ X3 ) ) % 0.30/0.51 = ( member @ nat @ N2 @ ( nat_set_decode @ ( divide_divide @ nat @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % set_decode_Suc % 0.30/0.51 thf(fact_1224_length__product,axiom, % 0.30/0.51 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 0.30/0.51 ( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs @ Ys ) ) % 0.30/0.51 = ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ). % 0.30/0.51 % 0.30/0.51 % length_product % 0.30/0.51 thf(fact_1225_mult__less__iff1,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [Z2: A,X3: A,Y: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 ) % 0.30/0.51 => ( ( ord_less @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ Y @ Z2 ) ) % 0.30/0.51 = ( ord_less @ A @ X3 @ Y ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_less_iff1 % 0.30/0.51 thf(fact_1226_num_Osize__gen_I2_J,axiom, % 0.30/0.51 ! [X22: num] : % 0.30/0.51 ( ( size_num @ ( bit0 @ X22 ) ) % 0.30/0.51 = ( plus_plus @ nat @ ( size_num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % num.size_gen(2) % 0.30/0.51 thf(fact_1227_nat__dvd__1__iff__1,axiom, % 0.30/0.51 ! [M: nat] : % 0.30/0.51 ( ( dvd_dvd @ nat @ M @ ( one_one @ nat ) ) % 0.30/0.51 = ( M % 0.30/0.51 = ( one_one @ nat ) ) ) ). % 0.30/0.51 % 0.30/0.51 % nat_dvd_1_iff_1 % 0.30/0.51 thf(fact_1228_dvd__0__left__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_semiring_1 @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.51 = ( A2 % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_0_left_iff % 0.30/0.51 thf(fact_1229_dvd__0__right,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_semiring_1 @ A ) % 0.30/0.51 => ! [A2: A] : ( dvd_dvd @ A @ A2 @ ( zero_zero @ A ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_0_right % 0.30/0.51 thf(fact_1230_dvd__add__triv__right__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_s4317794764714335236cancel @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) ) % 0.30/0.51 = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_add_triv_right_iff % 0.30/0.51 thf(fact_1231_dvd__add__triv__left__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_s4317794764714335236cancel @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.30/0.51 = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_add_triv_left_iff % 0.30/0.51 thf(fact_1232_bits__mod__0,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( bit_semiring_bits @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ). % 0.30/0.51 % 0.30/0.51 % bits_mod_0 % 0.30/0.51 thf(fact_1233_mod__self,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semidom_modulo @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ A2 @ A2 ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_self % 0.30/0.51 thf(fact_1234_mod__by__0,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semidom_modulo @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.51 = A2 ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_by_0 % 0.30/0.51 thf(fact_1235_mod__0,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semidom_modulo @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_0 % 0.30/0.51 thf(fact_1236_dvd__1__left,axiom, % 0.30/0.51 ! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ). % 0.30/0.51 % 0.30/0.51 % dvd_1_left % 0.30/0.51 thf(fact_1237_dvd__1__iff__1,axiom, % 0.30/0.51 ! [M: nat] : % 0.30/0.51 ( ( dvd_dvd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.51 = ( M % 0.30/0.51 = ( suc @ ( zero_zero @ nat ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_1_iff_1 % 0.30/0.51 thf(fact_1238_div__dvd__div,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ A2 @ C2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( divide_divide @ A @ C2 @ A2 ) ) % 0.30/0.51 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_dvd_div % 0.30/0.51 thf(fact_1239_not__real__square__gt__zero,axiom, % 0.30/0.51 ! [X3: real] : % 0.30/0.51 ( ( ~ ( ord_less @ real @ ( zero_zero @ real ) @ ( times_times @ real @ X3 @ X3 ) ) ) % 0.30/0.51 = ( X3 % 0.30/0.51 = ( zero_zero @ real ) ) ) ). % 0.30/0.51 % 0.30/0.51 % not_real_square_gt_zero % 0.30/0.51 thf(fact_1240_mod__add__self2,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 ) % 0.30/0.51 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_add_self2 % 0.30/0.51 thf(fact_1241_mod__add__self1,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.51 => ! [B2: A,A2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 ) % 0.30/0.51 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_add_self1 % 0.30/0.51 thf(fact_1242_nat__mult__dvd__cancel__disj,axiom, % 0.30/0.51 ! [K: nat,M: nat,N2: nat] : % 0.30/0.51 ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 0.30/0.51 = ( ( K % 0.30/0.51 = ( zero_zero @ nat ) ) % 0.30/0.51 | ( dvd_dvd @ nat @ M @ N2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % nat_mult_dvd_cancel_disj % 0.30/0.51 thf(fact_1243_mod__less,axiom, % 0.30/0.51 ! [M: nat,N2: nat] : % 0.30/0.51 ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.51 => ( ( modulo_modulo @ nat @ M @ N2 ) % 0.30/0.51 = M ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_less % 0.30/0.51 thf(fact_1244_dvd__mult__cancel__left,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( idom @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.51 = ( ( C2 % 0.30/0.51 = ( zero_zero @ A ) ) % 0.30/0.51 | ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_mult_cancel_left % 0.30/0.51 thf(fact_1245_dvd__mult__cancel__right,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( idom @ A ) % 0.30/0.51 => ! [A2: A,C2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( ( C2 % 0.30/0.51 = ( zero_zero @ A ) ) % 0.30/0.51 | ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_mult_cancel_right % 0.30/0.51 thf(fact_1246_dvd__times__left__cancel__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( A2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) % 0.30/0.51 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_times_left_cancel_iff % 0.30/0.51 thf(fact_1247_dvd__times__right__cancel__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( A2 % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 => ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) ) % 0.30/0.51 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_times_right_cancel_iff % 0.30/0.51 thf(fact_1248_unit__prod,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.51 => ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unit_prod % 0.30/0.51 thf(fact_1249_dvd__add__times__triv__right__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_s4317794764714335236cancel @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) ) ) % 0.30/0.51 = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_add_times_triv_right_iff % 0.30/0.51 thf(fact_1250_dvd__add__times__triv__left__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_s4317794764714335236cancel @ A ) % 0.30/0.51 => ! [A2: A,C2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A2 ) @ B2 ) ) % 0.30/0.51 = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_add_times_triv_left_iff % 0.30/0.51 thf(fact_1251_mod__mult__self1__is__0,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.51 => ! [B2: A,A2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A2 ) @ B2 ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_mult_self1_is_0 % 0.30/0.51 thf(fact_1252_mod__mult__self2__is__0,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_mult_self2_is_0 % 0.30/0.51 thf(fact_1253_bits__mod__by__1,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( bit_semiring_bits @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ). % 0.30/0.51 % 0.30/0.51 % bits_mod_by_1 % 0.30/0.51 thf(fact_1254_mod__by__1,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semidom_modulo @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_by_1 % 0.30/0.51 thf(fact_1255_dvd__div__mult__self,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A2 ) @ A2 ) % 0.30/0.51 = B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_div_mult_self % 0.30/0.51 thf(fact_1256_dvd__mult__div__cancel,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ A2 ) ) % 0.30/0.51 = B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_mult_div_cancel % 0.30/0.51 thf(fact_1257_unit__div,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.51 => ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unit_div % 0.30/0.51 thf(fact_1258_unit__div__1__unit,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.51 => ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( one_one @ A ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unit_div_1_unit % 0.30/0.51 thf(fact_1259_unit__div__1__div__1,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) % 0.30/0.51 = A2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unit_div_1_div_1 % 0.30/0.51 thf(fact_1260_div__add,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ C2 @ A2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ C2 @ B2 ) % 0.30/0.51 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.51 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_add % 0.30/0.51 thf(fact_1261_bits__mod__div__trivial,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( bit_semiring_bits @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ). % 0.30/0.51 % 0.30/0.51 % bits_mod_div_trivial % 0.30/0.51 thf(fact_1262_mod__div__trivial,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_div_trivial % 0.30/0.51 thf(fact_1263_mod__mult__self4,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.51 => ! [B2: A,C2: A,A2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 ) % 0.30/0.51 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_mult_self4 % 0.30/0.51 thf(fact_1264_mod__mult__self3,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.51 => ! [C2: A,B2: A,A2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 ) % 0.30/0.51 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_mult_self3 % 0.30/0.51 thf(fact_1265_mod__mult__self2,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 ) % 0.30/0.51 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_mult_self2 % 0.30/0.51 thf(fact_1266_mod__mult__self1,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.51 => ! [A2: A,C2: A,B2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 ) % 0.30/0.51 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_mult_self1 % 0.30/0.51 thf(fact_1267_dvd__imp__mod__0,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semidom_modulo @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( modulo_modulo @ A @ B2 @ A2 ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_imp_mod_0 % 0.30/0.51 thf(fact_1268_mod__by__Suc__0,axiom, % 0.30/0.51 ! [M: nat] : % 0.30/0.51 ( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.51 = ( zero_zero @ nat ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_by_Suc_0 % 0.30/0.51 thf(fact_1269_set__decode__zero,axiom, % 0.30/0.51 ( ( nat_set_decode @ ( zero_zero @ nat ) ) % 0.30/0.51 = ( bot_bot @ ( set @ nat ) ) ) ). % 0.30/0.51 % 0.30/0.51 % set_decode_zero % 0.30/0.51 thf(fact_1270_set__encode__inverse,axiom, % 0.30/0.51 ! [A3: set @ nat] : % 0.30/0.51 ( ( finite_finite2 @ nat @ A3 ) % 0.30/0.51 => ( ( nat_set_decode @ ( nat_set_encode @ A3 ) ) % 0.30/0.51 = A3 ) ) ). % 0.30/0.51 % 0.30/0.51 % set_encode_inverse % 0.30/0.51 thf(fact_1271_unit__div__mult__self,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A2 ) @ A2 ) % 0.30/0.51 = B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unit_div_mult_self % 0.30/0.51 thf(fact_1272_unit__mult__div__div,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) % 0.30/0.51 = ( divide_divide @ A @ B2 @ A2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unit_mult_div_div % 0.30/0.51 thf(fact_1273_even__Suc,axiom, % 0.30/0.51 ! [N2: nat] : % 0.30/0.51 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) % 0.30/0.51 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % even_Suc % 0.30/0.51 thf(fact_1274_even__Suc__Suc__iff,axiom, % 0.30/0.51 ! [N2: nat] : % 0.30/0.51 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ N2 ) ) ) % 0.30/0.51 = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 0.30/0.51 % 0.30/0.51 % even_Suc_Suc_iff % 0.30/0.51 thf(fact_1275_pow__divides__pow__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semiring_gcd @ A ) % 0.30/0.51 => ! [N2: nat,A2: A,B2: A] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) % 0.30/0.51 = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % pow_divides_pow_iff % 0.30/0.51 thf(fact_1276_Suc__mod__mult__self4,axiom, % 0.30/0.51 ! [N2: nat,K: nat,M: nat] : % 0.30/0.51 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ K ) @ M ) ) @ N2 ) % 0.30/0.51 = ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ). % 0.30/0.51 % 0.30/0.51 % Suc_mod_mult_self4 % 0.30/0.51 thf(fact_1277_Suc__mod__mult__self3,axiom, % 0.30/0.51 ! [K: nat,N2: nat,M: nat] : % 0.30/0.51 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N2 ) @ M ) ) @ N2 ) % 0.30/0.51 = ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ). % 0.30/0.51 % 0.30/0.51 % Suc_mod_mult_self3 % 0.30/0.51 thf(fact_1278_Suc__mod__mult__self2,axiom, % 0.30/0.51 ! [M: nat,N2: nat,K: nat] : % 0.30/0.51 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N2 @ K ) ) ) @ N2 ) % 0.30/0.51 = ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ). % 0.30/0.51 % 0.30/0.51 % Suc_mod_mult_self2 % 0.30/0.51 thf(fact_1279_Suc__mod__mult__self1,axiom, % 0.30/0.51 ! [M: nat,K: nat,N2: nat] : % 0.30/0.51 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ K @ N2 ) ) ) @ N2 ) % 0.30/0.51 = ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ). % 0.30/0.51 % 0.30/0.51 % Suc_mod_mult_self1 % 0.30/0.51 thf(fact_1280_even__add,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semiring_parity @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.30/0.51 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.51 = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % even_add % 0.30/0.51 thf(fact_1281_odd__add,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semiring_parity @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) % 0.30/0.51 = ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) % 0.30/0.51 != ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % odd_add % 0.30/0.51 thf(fact_1282_odd__Suc__div__two,axiom, % 0.30/0.51 ! [N2: nat] : % 0.30/0.51 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.51 => ( ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.51 = ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % odd_Suc_div_two % 0.30/0.51 thf(fact_1283_even__Suc__div__two,axiom, % 0.30/0.51 ! [N2: nat] : % 0.30/0.51 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.51 => ( ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.51 = ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % even_Suc_div_two % 0.30/0.51 thf(fact_1284_mod2__Suc__Suc,axiom, % 0.30/0.51 ! [M: nat] : % 0.30/0.51 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.51 = ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod2_Suc_Suc % 0.30/0.51 thf(fact_1285_Suc__times__numeral__mod__eq,axiom, % 0.30/0.51 ! [K: num,N2: nat] : % 0.30/0.51 ( ( ( numeral_numeral @ nat @ K ) % 0.30/0.51 != ( one_one @ nat ) ) % 0.30/0.51 => ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N2 ) ) @ ( numeral_numeral @ nat @ K ) ) % 0.30/0.51 = ( one_one @ nat ) ) ) ). % 0.30/0.51 % 0.30/0.51 % Suc_times_numeral_mod_eq % 0.30/0.51 thf(fact_1286_set__decode__0,axiom, % 0.30/0.51 ! [X3: nat] : % 0.30/0.51 ( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X3 ) ) % 0.30/0.51 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X3 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % set_decode_0 % 0.30/0.51 thf(fact_1287_zero__le__power__eq__numeral,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [A2: A,W2: num] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) ) % 0.30/0.51 = ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) ) % 0.30/0.51 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) ) % 0.30/0.51 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % zero_le_power_eq_numeral % 0.30/0.51 thf(fact_1288_power__less__zero__eq__numeral,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [A2: A,W2: num] : % 0.30/0.51 ( ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) ) % 0.30/0.51 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) ) % 0.30/0.51 & ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % power_less_zero_eq_numeral % 0.30/0.51 thf(fact_1289_power__less__zero__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [A2: A,N2: nat] : % 0.30/0.51 ( ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( zero_zero @ A ) ) % 0.30/0.51 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.51 & ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % power_less_zero_eq % 0.30/0.51 thf(fact_1290_even__plus__one__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semiring_parity @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) % 0.30/0.51 = ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % even_plus_one_iff % 0.30/0.51 thf(fact_1291_not__mod__2__eq__1__eq__0,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semiring_parity @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.51 != ( one_one @ A ) ) % 0.30/0.51 = ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % not_mod_2_eq_1_eq_0 % 0.30/0.51 thf(fact_1292_not__mod__2__eq__0__eq__1,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semiring_parity @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.51 != ( zero_zero @ A ) ) % 0.30/0.51 = ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.51 = ( one_one @ A ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % not_mod_2_eq_0_eq_1 % 0.30/0.51 thf(fact_1293_not__mod2__eq__Suc__0__eq__0,axiom, % 0.30/0.51 ! [N2: nat] : % 0.30/0.51 ( ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.51 != ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.51 = ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.51 = ( zero_zero @ nat ) ) ) ). % 0.30/0.51 % 0.30/0.51 % not_mod2_eq_Suc_0_eq_0 % 0.30/0.51 thf(fact_1294_add__self__mod__2,axiom, % 0.30/0.51 ! [M: nat] : % 0.30/0.51 ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.51 = ( zero_zero @ nat ) ) ). % 0.30/0.51 % 0.30/0.51 % add_self_mod_2 % 0.30/0.51 thf(fact_1295_zero__less__power__eq__numeral,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [A2: A,W2: num] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) ) % 0.30/0.51 = ( ( ( numeral_numeral @ nat @ W2 ) % 0.30/0.51 = ( zero_zero @ nat ) ) % 0.30/0.51 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) ) % 0.30/0.51 & ( A2 % 0.30/0.51 != ( zero_zero @ A ) ) ) % 0.30/0.51 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) ) % 0.30/0.51 & ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % zero_less_power_eq_numeral % 0.30/0.51 thf(fact_1296_even__succ__div__two,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid5411537665997757685th_nat @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.51 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.51 = ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % even_succ_div_two % 0.30/0.51 thf(fact_1297_odd__succ__div__two,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid5411537665997757685th_nat @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.51 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.51 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % odd_succ_div_two % 0.30/0.51 thf(fact_1298_even__succ__div__2,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( bit_semiring_bits @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.51 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.51 = ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % even_succ_div_2 % 0.30/0.51 thf(fact_1299_even__power,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semiring_parity @ A ) % 0.30/0.51 => ! [A2: A,N2: nat] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A2 @ N2 ) ) % 0.30/0.51 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.51 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % even_power % 0.30/0.51 thf(fact_1300_mod2__gr__0,axiom, % 0.30/0.51 ! [M: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.30/0.51 = ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.51 = ( one_one @ nat ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod2_gr_0 % 0.30/0.51 thf(fact_1301_odd__two__times__div__two__succ,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid5411537665997757685th_nat @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.51 => ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) ) % 0.30/0.51 = A2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % odd_two_times_div_two_succ % 0.30/0.51 thf(fact_1302_power__le__zero__eq__numeral,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( linordered_idom @ A ) % 0.30/0.51 => ! [A2: A,W2: num] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) ) % 0.30/0.51 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W2 ) ) % 0.30/0.51 & ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) ) % 0.30/0.51 & ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) % 0.30/0.51 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) ) % 0.30/0.51 & ( A2 % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % power_le_zero_eq_numeral % 0.30/0.51 thf(fact_1303_even__succ__mod__exp,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( bit_semiring_bits @ A ) % 0.30/0.51 => ! [A2: A,N2: nat] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.51 => ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.51 = ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % even_succ_mod_exp % 0.30/0.51 thf(fact_1304_real__arch__pow,axiom, % 0.30/0.51 ! [X3: real,Y: real] : % 0.30/0.51 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 0.30/0.51 => ? [N3: nat] : ( ord_less @ real @ Y @ ( power_power @ real @ X3 @ N3 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % real_arch_pow % 0.30/0.51 thf(fact_1305_real__arch__pow__inv,axiom, % 0.30/0.51 ! [Y: real,X3: real] : % 0.30/0.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 0.30/0.51 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 0.30/0.51 => ? [N3: nat] : ( ord_less @ real @ ( power_power @ real @ X3 @ N3 ) @ Y ) ) ) ). % 0.30/0.51 % 0.30/0.51 % real_arch_pow_inv % 0.30/0.51 thf(fact_1306_int__distrib_I2_J,axiom, % 0.30/0.51 ! [W2: int,Z1: int,Z22: int] : % 0.30/0.51 ( ( times_times @ int @ W2 @ ( plus_plus @ int @ Z1 @ Z22 ) ) % 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % int_distrib(2) % 0.30/0.51 thf(fact_1307_int__distrib_I1_J,axiom, % 0.30/0.51 ! [Z1: int,Z22: int,W2: int] : % 0.30/0.51 ( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W2 ) % 0.30/0.51 = ( plus_plus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % int_distrib(1) % 0.30/0.51 thf(fact_1308_zdvd__period,axiom, % 0.30/0.51 ! [A2: int,D2: int,X3: int,T2: int,C2: int] : % 0.30/0.51 ( ( dvd_dvd @ int @ A2 @ D2 ) % 0.30/0.51 => ( ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ X3 @ T2 ) ) % 0.30/0.51 = ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ ( plus_plus @ int @ X3 @ ( times_times @ int @ C2 @ D2 ) ) @ T2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % zdvd_period % 0.30/0.51 thf(fact_1309_zdvd__reduce,axiom, % 0.30/0.51 ! [K: int,N2: int,M: int] : % 0.30/0.51 ( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N2 @ ( times_times @ int @ K @ M ) ) ) % 0.30/0.51 = ( dvd_dvd @ int @ K @ N2 ) ) ). % 0.30/0.51 % 0.30/0.51 % zdvd_reduce % 0.30/0.51 thf(fact_1310_less__eq__real__def,axiom, % 0.30/0.51 ( ( ord_less_eq @ real ) % 0.30/0.51 = ( ^ [X: real,Y2: real] : % 0.30/0.51 ( ( ord_less @ real @ X @ Y2 ) % 0.30/0.51 | ( X = Y2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % less_eq_real_def % 0.30/0.51 thf(fact_1311_mod__eq__0__iff__dvd,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semidom_modulo @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( ( modulo_modulo @ A @ A2 @ B2 ) % 0.30/0.51 = ( zero_zero @ A ) ) % 0.30/0.51 = ( dvd_dvd @ A @ B2 @ A2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_eq_0_iff_dvd % 0.30/0.51 thf(fact_1312_dvd__eq__mod__eq__0,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semidom_modulo @ A ) % 0.30/0.51 => ( ( dvd_dvd @ A ) % 0.30/0.51 = ( ^ [A4: A,B4: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ B4 @ A4 ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_eq_mod_eq_0 % 0.30/0.51 thf(fact_1313_mod__0__imp__dvd,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semiring_modulo @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( ( modulo_modulo @ A @ A2 @ B2 ) % 0.30/0.51 = ( zero_zero @ A ) ) % 0.30/0.51 => ( dvd_dvd @ A @ B2 @ A2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_0_imp_dvd % 0.30/0.51 thf(fact_1314_dvd__refl,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_monoid_mult @ A ) % 0.30/0.51 => ! [A2: A] : ( dvd_dvd @ A @ A2 @ A2 ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_refl % 0.30/0.51 thf(fact_1315_dvd__trans,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_monoid_mult @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ B2 @ C2 ) % 0.30/0.51 => ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_trans % 0.30/0.51 thf(fact_1316_dvd__mod__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semidom_modulo @ A ) % 0.30/0.51 => ! [C2: A,B2: A,A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ C2 @ B2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) % 0.30/0.51 = ( dvd_dvd @ A @ C2 @ A2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_mod_iff % 0.30/0.51 thf(fact_1317_dvd__mod__imp__dvd,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semidom_modulo @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) % 0.30/0.51 => ( ( dvd_dvd @ A @ C2 @ B2 ) % 0.30/0.51 => ( dvd_dvd @ A @ C2 @ A2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_mod_imp_dvd % 0.30/0.51 thf(fact_1318_int__ge__induct,axiom, % 0.30/0.51 ! [K: int,I: int,P: int > $o] : % 0.30/0.51 ( ( ord_less_eq @ int @ K @ I ) % 0.30/0.51 => ( ( P @ K ) % 0.30/0.51 => ( ! [I4: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ K @ I4 ) % 0.30/0.51 => ( ( P @ I4 ) % 0.30/0.51 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) ) % 0.30/0.51 => ( P @ I ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % int_ge_induct % 0.30/0.51 thf(fact_1319_unit__imp__mod__eq__0,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid3725896446679973847miring @ A ) % 0.30/0.51 => ! [B2: A,A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ( modulo_modulo @ A @ A2 @ B2 ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unit_imp_mod_eq_0 % 0.30/0.51 thf(fact_1320_mod__greater__zero__iff__not__dvd,axiom, % 0.30/0.51 ! [M: nat,N2: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ N2 ) ) % 0.30/0.51 = ( ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_greater_zero_iff_not_dvd % 0.30/0.51 thf(fact_1321_mod__add__right__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) % 0.30/0.51 = ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_add_right_eq % 0.30/0.51 thf(fact_1322_mod__add__left__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.51 => ! [A2: A,C2: A,B2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 ) % 0.30/0.51 = ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_add_left_eq % 0.30/0.51 thf(fact_1323_mod__add__cong,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.51 => ! [A2: A,C2: A,A7: A,B2: A,B7: A] : % 0.30/0.51 ( ( ( modulo_modulo @ A @ A2 @ C2 ) % 0.30/0.51 = ( modulo_modulo @ A @ A7 @ C2 ) ) % 0.30/0.51 => ( ( ( modulo_modulo @ A @ B2 @ C2 ) % 0.30/0.51 = ( modulo_modulo @ A @ B7 @ C2 ) ) % 0.30/0.51 => ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.51 = ( modulo_modulo @ A @ ( plus_plus @ A @ A7 @ B7 ) @ C2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_add_cong % 0.30/0.51 thf(fact_1324_mod__add__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.51 => ! [A2: A,C2: A,B2: A] : % 0.30/0.51 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) % 0.30/0.51 = ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_add_eq % 0.30/0.51 thf(fact_1325_dvd__0__left,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_semiring_1 @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.51 => ( A2 % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_0_left % 0.30/0.51 thf(fact_1326_dvd__field__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( field @ A ) % 0.30/0.51 => ( ( dvd_dvd @ A ) % 0.30/0.51 = ( ^ [A4: A,B4: A] : % 0.30/0.51 ( ( A4 % 0.30/0.51 = ( zero_zero @ A ) ) % 0.30/0.51 => ( B4 % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_field_iff % 0.30/0.51 thf(fact_1327_mod__Suc__Suc__eq,axiom, % 0.30/0.51 ! [M: nat,N2: nat] : % 0.30/0.51 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) ) @ N2 ) % 0.30/0.51 = ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ N2 ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_Suc_Suc_eq % 0.30/0.51 thf(fact_1328_mod__Suc__eq,axiom, % 0.30/0.51 ! [M: nat,N2: nat] : % 0.30/0.51 ( ( modulo_modulo @ nat @ ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) @ N2 ) % 0.30/0.51 = ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_Suc_eq % 0.30/0.51 thf(fact_1329_dvd__productE,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semiring_gcd @ A ) % 0.30/0.51 => ! [P4: A,A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ P4 @ ( times_times @ A @ A2 @ B2 ) ) % 0.30/0.51 => ~ ! [X5: A,Y4: A] : % 0.30/0.51 ( ( P4 % 0.30/0.51 = ( times_times @ A @ X5 @ Y4 ) ) % 0.30/0.51 => ( ( dvd_dvd @ A @ X5 @ A2 ) % 0.30/0.51 => ~ ( dvd_dvd @ A @ Y4 @ B2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_productE % 0.30/0.51 thf(fact_1330_division__decomp,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semiring_gcd @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.51 => ? [B9: A,C6: A] : % 0.30/0.51 ( ( A2 % 0.30/0.51 = ( times_times @ A @ B9 @ C6 ) ) % 0.30/0.51 & ( dvd_dvd @ A @ B9 @ B2 ) % 0.30/0.51 & ( dvd_dvd @ A @ C6 @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % division_decomp % 0.30/0.51 thf(fact_1331_dvdE,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( dvd @ A ) % 0.30/0.51 => ! [B2: A,A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ B2 @ A2 ) % 0.30/0.51 => ~ ! [K2: A] : % 0.30/0.51 ( A2 % 0.30/0.51 != ( times_times @ A @ B2 @ K2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvdE % 0.30/0.51 thf(fact_1332_dvdI,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( dvd @ A ) % 0.30/0.51 => ! [A2: A,B2: A,K: A] : % 0.30/0.51 ( ( A2 % 0.30/0.51 = ( times_times @ A @ B2 @ K ) ) % 0.30/0.51 => ( dvd_dvd @ A @ B2 @ A2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvdI % 0.30/0.51 thf(fact_1333_dvd__def,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( dvd @ A ) % 0.30/0.51 => ( ( dvd_dvd @ A ) % 0.30/0.51 = ( ^ [B4: A,A4: A] : % 0.30/0.51 ? [K3: A] : % 0.30/0.51 ( A4 % 0.30/0.51 = ( times_times @ A @ B4 @ K3 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_def % 0.30/0.51 thf(fact_1334_dvd__mult,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_monoid_mult @ A ) % 0.30/0.51 => ! [A2: A,C2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ C2 ) % 0.30/0.51 => ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_mult % 0.30/0.51 thf(fact_1335_dvd__mult2,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_monoid_mult @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ B2 ) % 0.30/0.51 => ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_mult2 % 0.30/0.51 thf(fact_1336_dvd__mult__left,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_monoid_mult @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.51 => ( dvd_dvd @ A @ A2 @ C2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_mult_left % 0.30/0.51 thf(fact_1337_dvd__triv__left,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_monoid_mult @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_triv_left % 0.30/0.51 thf(fact_1338_mult__dvd__mono,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_monoid_mult @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ C2 @ D2 ) % 0.30/0.51 => ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_dvd_mono % 0.30/0.51 thf(fact_1339_dvd__mult__right,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_monoid_mult @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.51 => ( dvd_dvd @ A @ B2 @ C2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_mult_right % 0.30/0.51 thf(fact_1340_dvd__triv__right,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_monoid_mult @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_triv_right % 0.30/0.51 thf(fact_1341_one__dvd,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_monoid_mult @ A ) % 0.30/0.51 => ! [A2: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A2 ) ) ). % 0.30/0.51 % 0.30/0.51 % one_dvd % 0.30/0.51 thf(fact_1342_unit__imp__dvd,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [B2: A,A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.51 => ( dvd_dvd @ A @ B2 @ A2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unit_imp_dvd % 0.30/0.51 thf(fact_1343_dvd__unit__imp__unit,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.51 => ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_unit_imp_unit % 0.30/0.51 thf(fact_1344_dvd__add__right__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_s4317794764714335236cancel @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_add_right_iff % 0.30/0.51 thf(fact_1345_dvd__add__left__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_s4317794764714335236cancel @ A ) % 0.30/0.51 => ! [A2: A,C2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ C2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_add_left_iff % 0.30/0.51 thf(fact_1346_dvd__add,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( comm_semiring_1 @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ B2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ A2 @ C2 ) % 0.30/0.51 => ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_add % 0.30/0.51 thf(fact_1347_dvd__div__eq__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semidom_divide @ A ) % 0.30/0.51 => ! [C2: A,A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ C2 @ A2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ C2 @ B2 ) % 0.30/0.51 => ( ( ( divide_divide @ A @ A2 @ C2 ) % 0.30/0.51 = ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( A2 = B2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_div_eq_iff % 0.30/0.51 thf(fact_1348_dvd__div__eq__cancel,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semidom_divide @ A ) % 0.30/0.51 => ! [A2: A,C2: A,B2: A] : % 0.30/0.51 ( ( ( divide_divide @ A @ A2 @ C2 ) % 0.30/0.51 = ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.51 => ( ( dvd_dvd @ A @ C2 @ A2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ C2 @ B2 ) % 0.30/0.51 => ( A2 = B2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_div_eq_cancel % 0.30/0.51 thf(fact_1349_div__div__div__same,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [D2: A,B2: A,A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ D2 @ B2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ B2 @ A2 ) % 0.30/0.51 => ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) ) % 0.30/0.51 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_div_div_same % 0.30/0.51 thf(fact_1350_mod__less__eq__dividend,axiom, % 0.30/0.51 ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N2 ) @ M ) ). % 0.30/0.51 % 0.30/0.51 % mod_less_eq_dividend % 0.30/0.51 thf(fact_1351_gcd__nat_Oextremum__uniqueI,axiom, % 0.30/0.51 ! [A2: nat] : % 0.30/0.51 ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 ) % 0.30/0.51 => ( A2 % 0.30/0.51 = ( zero_zero @ nat ) ) ) ). % 0.30/0.51 % 0.30/0.51 % gcd_nat.extremum_uniqueI % 0.30/0.51 thf(fact_1352_gcd__nat_Onot__eq__extremum,axiom, % 0.30/0.51 ! [A2: nat] : % 0.30/0.51 ( ( A2 % 0.30/0.51 != ( zero_zero @ nat ) ) % 0.30/0.51 = ( ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) ) % 0.30/0.51 & ( A2 % 0.30/0.51 != ( zero_zero @ nat ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % gcd_nat.not_eq_extremum % 0.30/0.51 thf(fact_1353_gcd__nat_Oextremum__unique,axiom, % 0.30/0.51 ! [A2: nat] : % 0.30/0.51 ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 ) % 0.30/0.51 = ( A2 % 0.30/0.51 = ( zero_zero @ nat ) ) ) ). % 0.30/0.51 % 0.30/0.51 % gcd_nat.extremum_unique % 0.30/0.51 thf(fact_1354_gcd__nat_Oextremum__strict,axiom, % 0.30/0.51 ! [A2: nat] : % 0.30/0.51 ~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 ) % 0.30/0.51 & ( ( zero_zero @ nat ) % 0.30/0.51 != A2 ) ) ). % 0.30/0.51 % 0.30/0.51 % gcd_nat.extremum_strict % 0.30/0.51 thf(fact_1355_gcd__nat_Oextremum,axiom, % 0.30/0.51 ! [A2: nat] : ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) ) ). % 0.30/0.51 % 0.30/0.51 % gcd_nat.extremum % 0.30/0.51 thf(fact_1356_zdvd__antisym__nonneg,axiom, % 0.30/0.51 ! [M: int,N2: int] : % 0.30/0.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M ) % 0.30/0.51 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) % 0.30/0.51 => ( ( dvd_dvd @ int @ M @ N2 ) % 0.30/0.51 => ( ( dvd_dvd @ int @ N2 @ M ) % 0.30/0.51 => ( M = N2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % zdvd_antisym_nonneg % 0.30/0.51 thf(fact_1357_zdvd__mult__cancel,axiom, % 0.30/0.51 ! [K: int,M: int,N2: int] : % 0.30/0.51 ( ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N2 ) ) % 0.30/0.51 => ( ( K % 0.30/0.51 != ( zero_zero @ int ) ) % 0.30/0.51 => ( dvd_dvd @ int @ M @ N2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % zdvd_mult_cancel % 0.30/0.51 thf(fact_1358_even__iff__mod__2__eq__zero,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semiring_parity @ A ) % 0.30/0.51 => ! [A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.51 = ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % even_iff_mod_2_eq_zero % 0.30/0.51 thf(fact_1359_subset__decode__imp__le,axiom, % 0.30/0.51 ! [M: nat,N2: nat] : % 0.30/0.51 ( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) ) % 0.30/0.51 => ( ord_less_eq @ nat @ M @ N2 ) ) ). % 0.30/0.51 % 0.30/0.51 % subset_decode_imp_le % 0.30/0.51 thf(fact_1360_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.51 => ( ord_less_eq @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ A2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend % 0.30/0.51 thf(fact_1361_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.51 => ! [B2: A,A2: A] : % 0.30/0.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.51 => ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unique_euclidean_semiring_numeral_class.pos_mod_bound % 0.30/0.51 thf(fact_1362_mod__eq__self__iff__div__eq__0,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid3725896446679973847miring @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( ( modulo_modulo @ A @ A2 @ B2 ) % 0.30/0.51 = A2 ) % 0.30/0.51 = ( ( divide_divide @ A @ A2 @ B2 ) % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_eq_self_iff_div_eq_0 % 0.30/0.51 thf(fact_1363_mod__eqE,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid8851590272496341667cancel @ A ) % 0.30/0.51 => ! [A2: A,C2: A,B2: A] : % 0.30/0.51 ( ( ( modulo_modulo @ A @ A2 @ C2 ) % 0.30/0.51 = ( modulo_modulo @ A @ B2 @ C2 ) ) % 0.30/0.51 => ~ ! [D3: A] : % 0.30/0.51 ( B2 % 0.30/0.51 != ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ D3 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_eqE % 0.30/0.51 thf(fact_1364_div__add1__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( euclid3128863361964157862miring @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.51 = ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_add1_eq % 0.30/0.51 thf(fact_1365_not__is__unit__0,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ~ ( dvd_dvd @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ). % 0.30/0.51 % 0.30/0.51 % not_is_unit_0 % 0.30/0.51 thf(fact_1366_minf_I10_J,axiom, % 0.30/0.51 ! [B: $tType] : % 0.30/0.51 ( ( ( plus @ B ) % 0.30/0.51 & ( linorder @ B ) % 0.30/0.51 & ( dvd @ B ) ) % 0.30/0.51 => ! [D2: B,S: B] : % 0.30/0.51 ? [Z3: B] : % 0.30/0.51 ! [X2: B] : % 0.30/0.51 ( ( ord_less @ B @ X2 @ Z3 ) % 0.30/0.51 => ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) % 0.30/0.51 = ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % minf(10) % 0.30/0.51 thf(fact_1367_minf_I9_J,axiom, % 0.30/0.51 ! [B: $tType] : % 0.30/0.51 ( ( ( plus @ B ) % 0.30/0.51 & ( linorder @ B ) % 0.30/0.51 & ( dvd @ B ) ) % 0.30/0.51 => ! [D2: B,S: B] : % 0.30/0.51 ? [Z3: B] : % 0.30/0.51 ! [X2: B] : % 0.30/0.51 ( ( ord_less @ B @ X2 @ Z3 ) % 0.30/0.51 => ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) % 0.30/0.51 = ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % minf(9) % 0.30/0.51 thf(fact_1368_pinf_I10_J,axiom, % 0.30/0.51 ! [B: $tType] : % 0.30/0.51 ( ( ( plus @ B ) % 0.30/0.51 & ( linorder @ B ) % 0.30/0.51 & ( dvd @ B ) ) % 0.30/0.51 => ! [D2: B,S: B] : % 0.30/0.51 ? [Z3: B] : % 0.30/0.51 ! [X2: B] : % 0.30/0.51 ( ( ord_less @ B @ Z3 @ X2 ) % 0.30/0.51 => ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) % 0.30/0.51 = ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % pinf(10) % 0.30/0.51 thf(fact_1369_pinf_I9_J,axiom, % 0.30/0.51 ! [B: $tType] : % 0.30/0.51 ( ( ( plus @ B ) % 0.30/0.51 & ( linorder @ B ) % 0.30/0.51 & ( dvd @ B ) ) % 0.30/0.51 => ! [D2: B,S: B] : % 0.30/0.51 ? [Z3: B] : % 0.30/0.51 ! [X2: B] : % 0.30/0.51 ( ( ord_less @ B @ Z3 @ X2 ) % 0.30/0.51 => ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) % 0.30/0.51 = ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % pinf(9) % 0.30/0.51 thf(fact_1370_dvd__div__eq__0__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( semidom_divide @ A ) % 0.30/0.51 => ! [B2: A,A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ B2 @ A2 ) % 0.30/0.51 => ( ( ( divide_divide @ A @ A2 @ B2 ) % 0.30/0.51 = ( zero_zero @ A ) ) % 0.30/0.51 = ( A2 % 0.30/0.51 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_div_eq_0_iff % 0.30/0.51 thf(fact_1371_unit__mult__right__cancel,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ( ( times_times @ A @ B2 @ A2 ) % 0.30/0.51 = ( times_times @ A @ C2 @ A2 ) ) % 0.30/0.51 = ( B2 = C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unit_mult_right_cancel % 0.30/0.51 thf(fact_1372_unit__mult__left__cancel,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ( ( times_times @ A @ A2 @ B2 ) % 0.30/0.51 = ( times_times @ A @ A2 @ C2 ) ) % 0.30/0.51 = ( B2 = C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % unit_mult_left_cancel % 0.30/0.51 thf(fact_1373_mult__unit__dvd__iff_H,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A,B2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.51 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_unit_dvd_iff' % 0.30/0.51 thf(fact_1374_dvd__mult__unit__iff_H,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [B2: A,A2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_mult_unit_iff' % 0.30/0.51 thf(fact_1375_mult__unit__dvd__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [B2: A,A2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.51 = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mult_unit_dvd_iff % 0.30/0.51 thf(fact_1376_dvd__mult__unit__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [B2: A,A2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) % 0.30/0.51 = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_mult_unit_iff % 0.30/0.51 thf(fact_1377_is__unit__mult__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) % 0.30/0.51 = ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.51 & ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % is_unit_mult_iff % 0.30/0.51 thf(fact_1378_mod__Suc,axiom, % 0.30/0.51 ! [M: nat,N2: nat] : % 0.30/0.51 ( ( ( ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) % 0.30/0.51 = N2 ) % 0.30/0.51 => ( ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) % 0.30/0.51 = ( zero_zero @ nat ) ) ) % 0.30/0.51 & ( ( ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) % 0.30/0.51 != N2 ) % 0.30/0.51 => ( ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) % 0.30/0.51 = ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_Suc % 0.30/0.51 thf(fact_1379_finite__set__decode,axiom, % 0.30/0.51 ! [N2: nat] : ( finite_finite2 @ nat @ ( nat_set_decode @ N2 ) ) ). % 0.30/0.51 % 0.30/0.51 % finite_set_decode % 0.30/0.51 thf(fact_1380_mod__induct,axiom, % 0.30/0.51 ! [P: nat > $o,N2: nat,P4: nat,M: nat] : % 0.30/0.51 ( ( P @ N2 ) % 0.30/0.51 => ( ( ord_less @ nat @ N2 @ P4 ) % 0.30/0.51 => ( ( ord_less @ nat @ M @ P4 ) % 0.30/0.51 => ( ! [N3: nat] : % 0.30/0.51 ( ( ord_less @ nat @ N3 @ P4 ) % 0.30/0.51 => ( ( P @ N3 ) % 0.30/0.51 => ( P @ ( modulo_modulo @ nat @ ( suc @ N3 ) @ P4 ) ) ) ) % 0.30/0.51 => ( P @ M ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_induct % 0.30/0.51 thf(fact_1381_dvd__div__mult,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [C2: A,B2: A,A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ C2 @ B2 ) % 0.30/0.51 => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 0.30/0.51 = ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_div_mult % 0.30/0.51 thf(fact_1382_div__mult__swap,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [C2: A,B2: A,A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ C2 @ B2 ) % 0.30/0.51 => ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_mult_swap % 0.30/0.51 thf(fact_1383_div__div__eq__right,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [C2: A,B2: A,A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ C2 @ B2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ B2 @ A2 ) % 0.30/0.51 => ( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_div_eq_right % 0.30/0.51 thf(fact_1384_dvd__div__mult2__eq,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [B2: A,C2: A,A2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) % 0.30/0.51 => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.51 = ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_div_mult2_eq % 0.30/0.51 thf(fact_1385_dvd__mult__imp__div,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [A2: A,C2: A,B2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) % 0.30/0.51 => ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_mult_imp_div % 0.30/0.51 thf(fact_1386_div__mult__div__if__dvd,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [B2: A,A2: A,D2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ B2 @ A2 ) % 0.30/0.51 => ( ( dvd_dvd @ A @ D2 @ C2 ) % 0.30/0.51 => ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( divide_divide @ A @ C2 @ D2 ) ) % 0.30/0.51 = ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % div_mult_div_if_dvd % 0.30/0.51 thf(fact_1387_gcd__nat__induct,axiom, % 0.30/0.51 ! [P: nat > nat > $o,M: nat,N2: nat] : % 0.30/0.51 ( ! [M3: nat] : ( P @ M3 @ ( zero_zero @ nat ) ) % 0.30/0.51 => ( ! [M3: nat,N3: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 0.30/0.51 => ( ( P @ N3 @ ( modulo_modulo @ nat @ M3 @ N3 ) ) % 0.30/0.51 => ( P @ M3 @ N3 ) ) ) % 0.30/0.51 => ( P @ M @ N2 ) ) ) ). % 0.30/0.51 % 0.30/0.51 % gcd_nat_induct % 0.30/0.51 thf(fact_1388_mod__less__divisor,axiom, % 0.30/0.51 ! [N2: nat,M: nat] : % 0.30/0.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.51 => ( ord_less @ nat @ ( modulo_modulo @ nat @ M @ N2 ) @ N2 ) ) ). % 0.30/0.51 % 0.30/0.51 % mod_less_divisor % 0.30/0.51 thf(fact_1389_dvd__div__unit__iff,axiom, % 0.30/0.51 ! [A: $tType] : % 0.30/0.51 ( ( algebraic_semidom @ A ) % 0.30/0.51 => ! [B2: A,A2: A,C2: A] : % 0.30/0.51 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.51 => ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ C2 @ B2 ) ) % 0.30/0.51 = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.51 % 0.30/0.51 % dvd_div_unit_iff % 0.30/0.51 thf(fact_1390_div__unit__dvd__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [B2: A,A2: A,C2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.52 => ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.52 = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % div_unit_dvd_iff % 0.30/0.52 thf(fact_1391_unit__div__cancel,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.52 => ( ( ( divide_divide @ A @ B2 @ A2 ) % 0.30/0.52 = ( divide_divide @ A @ C2 @ A2 ) ) % 0.30/0.52 = ( B2 = C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % unit_div_cancel % 0.30/0.52 thf(fact_1392_div__plus__div__distrib__dvd__right,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.52 => ! [C2: A,B2: A,A2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ C2 @ B2 ) % 0.30/0.52 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.52 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % div_plus_div_distrib_dvd_right % 0.30/0.52 thf(fact_1393_div__plus__div__distrib__dvd__left,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.52 => ! [C2: A,A2: A,B2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ C2 @ A2 ) % 0.30/0.52 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.52 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % div_plus_div_distrib_dvd_left % 0.30/0.52 thf(fact_1394_mod__Suc__le__divisor,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ ( suc @ N2 ) ) @ N2 ) ). % 0.30/0.52 % 0.30/0.52 % mod_Suc_le_divisor % 0.30/0.52 thf(fact_1395_mod__eq__0D,axiom, % 0.30/0.52 ! [M: nat,D2: nat] : % 0.30/0.52 ( ( ( modulo_modulo @ nat @ M @ D2 ) % 0.30/0.52 = ( zero_zero @ nat ) ) % 0.30/0.52 => ? [Q2: nat] : % 0.30/0.52 ( M % 0.30/0.52 = ( times_times @ nat @ D2 @ Q2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_eq_0D % 0.30/0.52 thf(fact_1396_le__imp__power__dvd,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( comm_semiring_1 @ A ) % 0.30/0.52 => ! [M: nat,N2: nat,A2: A] : % 0.30/0.52 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.52 => ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % le_imp_power_dvd % 0.30/0.52 thf(fact_1397_power__le__dvd,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( comm_semiring_1 @ A ) % 0.30/0.52 => ! [A2: A,N2: nat,B2: A,M: nat] : % 0.30/0.52 ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N2 ) @ B2 ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.52 => ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % power_le_dvd % 0.30/0.52 thf(fact_1398_dvd__power__le,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( comm_semiring_1 @ A ) % 0.30/0.52 => ! [X3: A,Y: A,N2: nat,M: nat] : % 0.30/0.52 ( ( dvd_dvd @ A @ X3 @ Y ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.52 => ( dvd_dvd @ A @ ( power_power @ A @ X3 @ N2 ) @ ( power_power @ A @ Y @ M ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_power_le % 0.30/0.52 thf(fact_1399_nat__mod__eq__iff,axiom, % 0.30/0.52 ! [X3: nat,N2: nat,Y: nat] : % 0.30/0.52 ( ( ( modulo_modulo @ nat @ X3 @ N2 ) % 0.30/0.52 = ( modulo_modulo @ nat @ Y @ N2 ) ) % 0.30/0.52 = ( ? [Q1: nat,Q22: nat] : % 0.30/0.52 ( ( plus_plus @ nat @ X3 @ ( times_times @ nat @ N2 @ Q1 ) ) % 0.30/0.52 = ( plus_plus @ nat @ Y @ ( times_times @ nat @ N2 @ Q22 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % nat_mod_eq_iff % 0.30/0.52 thf(fact_1400_nat__dvd__not__less,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.30/0.52 => ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.52 => ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ). % 0.30/0.52 % 0.30/0.52 % nat_dvd_not_less % 0.30/0.52 thf(fact_1401_dvd__pos__nat,axiom, % 0.30/0.52 ! [N2: nat,M: nat] : % 0.30/0.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ M @ N2 ) % 0.30/0.52 => ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_pos_nat % 0.30/0.52 thf(fact_1402_zdvd__imp__le,axiom, % 0.30/0.52 ! [Z2: int,N2: int] : % 0.30/0.52 ( ( dvd_dvd @ int @ Z2 @ N2 ) % 0.30/0.52 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 ) % 0.30/0.52 => ( ord_less_eq @ int @ Z2 @ N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % zdvd_imp_le % 0.30/0.52 thf(fact_1403_zdvd__not__zless,axiom, % 0.30/0.52 ! [M: int,N2: int] : % 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ M ) % 0.30/0.52 => ( ( ord_less @ int @ M @ N2 ) % 0.30/0.52 => ~ ( dvd_dvd @ int @ N2 @ M ) ) ) ). % 0.30/0.52 % 0.30/0.52 % zdvd_not_zless % 0.30/0.52 thf(fact_1404_bezout__add__nat,axiom, % 0.30/0.52 ! [A2: nat,B2: nat] : % 0.30/0.52 ? [D3: nat,X5: nat,Y4: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ D3 @ A2 ) % 0.30/0.52 & ( dvd_dvd @ nat @ D3 @ B2 ) % 0.30/0.52 & ( ( ( times_times @ nat @ A2 @ X5 ) % 0.30/0.52 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D3 ) ) % 0.30/0.52 | ( ( times_times @ nat @ B2 @ X5 ) % 0.30/0.52 = ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y4 ) @ D3 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % bezout_add_nat % 0.30/0.52 thf(fact_1405_bezout__lemma__nat,axiom, % 0.30/0.52 ! [D2: nat,A2: nat,B2: nat,X3: nat,Y: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ D2 @ A2 ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ D2 @ B2 ) % 0.30/0.52 => ( ( ( ( times_times @ nat @ A2 @ X3 ) % 0.30/0.52 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y ) @ D2 ) ) % 0.30/0.52 | ( ( times_times @ nat @ B2 @ X3 ) % 0.30/0.52 = ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y ) @ D2 ) ) ) % 0.30/0.52 => ? [X5: nat,Y4: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ D2 @ A2 ) % 0.30/0.52 & ( dvd_dvd @ nat @ D2 @ ( plus_plus @ nat @ A2 @ B2 ) ) % 0.30/0.52 & ( ( ( times_times @ nat @ A2 @ X5 ) % 0.30/0.52 = ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ Y4 ) @ D2 ) ) % 0.30/0.52 | ( ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ X5 ) % 0.30/0.52 = ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y4 ) @ D2 ) ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % bezout_lemma_nat % 0.30/0.52 thf(fact_1406_parity__cases,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_parity @ A ) % 0.30/0.52 => ! [A2: A] : % 0.30/0.52 ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.52 => ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.52 != ( zero_zero @ A ) ) ) % 0.30/0.52 => ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.52 => ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.52 != ( one_one @ A ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % parity_cases % 0.30/0.52 thf(fact_1407_mod2__eq__if,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_parity @ A ) % 0.30/0.52 => ! [A2: A] : % 0.30/0.52 ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.52 => ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.52 = ( zero_zero @ A ) ) ) % 0.30/0.52 & ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.52 => ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.52 = ( one_one @ A ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mod2_eq_if % 0.30/0.52 thf(fact_1408_unique__euclidean__semiring__numeral__class_Omod__less,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.52 => ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( modulo_modulo @ A @ A2 @ B2 ) % 0.30/0.52 = A2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % unique_euclidean_semiring_numeral_class.mod_less % 0.30/0.52 thf(fact_1409_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.52 => ! [B2: A,A2: A] : % 0.30/0.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % unique_euclidean_semiring_numeral_class.pos_mod_sign % 0.30/0.52 thf(fact_1410_cong__exp__iff__simps_I2_J,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.52 => ! [N2: num,Q4: num] : % 0.30/0.52 ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) % 0.30/0.52 = ( zero_zero @ A ) ) % 0.30/0.52 = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ Q4 ) ) % 0.30/0.52 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % cong_exp_iff_simps(2) % 0.30/0.52 thf(fact_1411_cong__exp__iff__simps_I1_J,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.52 => ! [N2: num] : % 0.30/0.52 ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ one2 ) ) % 0.30/0.52 = ( zero_zero @ A ) ) ) ). % 0.30/0.52 % 0.30/0.52 % cong_exp_iff_simps(1) % 0.30/0.52 thf(fact_1412_mult__div__mod__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_modulo @ A ) % 0.30/0.52 => ! [B2: A,A2: A] : % 0.30/0.52 ( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) % 0.30/0.52 = A2 ) ) ). % 0.30/0.52 % 0.30/0.52 % mult_div_mod_eq % 0.30/0.52 thf(fact_1413_mod__mult__div__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_modulo @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) % 0.30/0.52 = A2 ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_mult_div_eq % 0.30/0.52 thf(fact_1414_mod__div__mult__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_modulo @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) ) % 0.30/0.52 = A2 ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_div_mult_eq % 0.30/0.52 thf(fact_1415_div__mult__mod__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_modulo @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) % 0.30/0.52 = A2 ) ) ). % 0.30/0.52 % 0.30/0.52 % div_mult_mod_eq % 0.30/0.52 thf(fact_1416_mod__div__decomp,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_modulo @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( A2 % 0.30/0.52 = ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_div_decomp % 0.30/0.52 thf(fact_1417_cancel__div__mod__rules_I1_J,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semidom_modulo @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 ) % 0.30/0.52 = ( plus_plus @ A @ A2 @ C2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % cancel_div_mod_rules(1) % 0.30/0.52 thf(fact_1418_cancel__div__mod__rules_I2_J,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semidom_modulo @ A ) % 0.30/0.52 => ! [B2: A,A2: A,C2: A] : % 0.30/0.52 ( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 ) % 0.30/0.52 = ( plus_plus @ A @ A2 @ C2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % cancel_div_mod_rules(2) % 0.30/0.52 thf(fact_1419_div__mult1__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( euclid3128863361964157862miring @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.52 = ( plus_plus @ A @ ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % div_mult1_eq % 0.30/0.52 thf(fact_1420_unit__dvdE,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.52 => ~ ( ( A2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ! [C3: A] : % 0.30/0.52 ( B2 % 0.30/0.52 != ( times_times @ A @ A2 @ C3 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % unit_dvdE % 0.30/0.52 thf(fact_1421_unity__coeff__ex,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ( dvd @ A ) % 0.30/0.52 & ( semiring_0 @ A ) ) % 0.30/0.52 => ! [P: A > $o,L: A] : % 0.30/0.52 ( ( ? [X: A] : ( P @ ( times_times @ A @ L @ X ) ) ) % 0.30/0.52 = ( ? [X: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X @ ( zero_zero @ A ) ) ) % 0.30/0.52 & ( P @ X ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % unity_coeff_ex % 0.30/0.52 thf(fact_1422_dvd__div__eq__mult,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( A2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( dvd_dvd @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( ( divide_divide @ A @ B2 @ A2 ) % 0.30/0.52 = C2 ) % 0.30/0.52 = ( B2 % 0.30/0.52 = ( times_times @ A @ C2 @ A2 ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_div_eq_mult % 0.30/0.52 thf(fact_1423_div__dvd__iff__mult,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [B2: A,A2: A,C2: A] : % 0.30/0.52 ( ( B2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( dvd_dvd @ A @ B2 @ A2 ) % 0.30/0.52 => ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.52 = ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % div_dvd_iff_mult % 0.30/0.52 thf(fact_1424_dvd__div__iff__mult,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [C2: A,B2: A,A2: A] : % 0.30/0.52 ( ( C2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( dvd_dvd @ A @ C2 @ B2 ) % 0.30/0.52 => ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.52 = ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_div_iff_mult % 0.30/0.52 thf(fact_1425_dvd__div__div__eq__mult,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [A2: A,C2: A,B2: A,D2: A] : % 0.30/0.52 ( ( A2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( C2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( dvd_dvd @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( dvd_dvd @ A @ C2 @ D2 ) % 0.30/0.52 => ( ( ( divide_divide @ A @ B2 @ A2 ) % 0.30/0.52 = ( divide_divide @ A @ D2 @ C2 ) ) % 0.30/0.52 = ( ( times_times @ A @ B2 @ C2 ) % 0.30/0.52 = ( times_times @ A @ A2 @ D2 ) ) ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_div_div_eq_mult % 0.30/0.52 thf(fact_1426_unit__div__eq__0__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [B2: A,A2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.52 => ( ( ( divide_divide @ A @ A2 @ B2 ) % 0.30/0.52 = ( zero_zero @ A ) ) % 0.30/0.52 = ( A2 % 0.30/0.52 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % unit_div_eq_0_iff % 0.30/0.52 thf(fact_1427_is__unit__div__mult2__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [B2: A,C2: A,A2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.52 => ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) % 0.30/0.52 => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.52 = ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % is_unit_div_mult2_eq % 0.30/0.52 thf(fact_1428_unit__div__mult__swap,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [C2: A,A2: A,B2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) % 0.30/0.52 => ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.52 = ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % unit_div_mult_swap % 0.30/0.52 thf(fact_1429_unit__div__commute,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [B2: A,A2: A,C2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.52 => ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.52 = ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % unit_div_commute % 0.30/0.52 thf(fact_1430_div__mult__unit2,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [C2: A,B2: A,A2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) % 0.30/0.52 => ( ( dvd_dvd @ A @ B2 @ A2 ) % 0.30/0.52 => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.52 = ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % div_mult_unit2 % 0.30/0.52 thf(fact_1431_unit__eq__div2,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [B2: A,A2: A,C2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.52 => ( ( A2 % 0.30/0.52 = ( divide_divide @ A @ C2 @ B2 ) ) % 0.30/0.52 = ( ( times_times @ A @ A2 @ B2 ) % 0.30/0.52 = C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % unit_eq_div2 % 0.30/0.52 thf(fact_1432_unit__eq__div1,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [B2: A,A2: A,C2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.52 => ( ( ( divide_divide @ A @ A2 @ B2 ) % 0.30/0.52 = C2 ) % 0.30/0.52 = ( A2 % 0.30/0.52 = ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % unit_eq_div1 % 0.30/0.52 thf(fact_1433_mod__le__divisor,axiom, % 0.30/0.52 ! [N2: nat,M: nat] : % 0.30/0.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.52 => ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N2 ) @ N2 ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_le_divisor % 0.30/0.52 thf(fact_1434_is__unit__power__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [A2: A,N2: nat] : % 0.30/0.52 ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N2 ) @ ( one_one @ A ) ) % 0.30/0.52 = ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.52 | ( N2 % 0.30/0.52 = ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % is_unit_power_iff % 0.30/0.52 thf(fact_1435_div__less__mono,axiom, % 0.30/0.52 ! [A3: nat,B3: nat,N2: nat] : % 0.30/0.52 ( ( ord_less @ nat @ A3 @ B3 ) % 0.30/0.52 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.52 => ( ( ( modulo_modulo @ nat @ A3 @ N2 ) % 0.30/0.52 = ( zero_zero @ nat ) ) % 0.30/0.52 => ( ( ( modulo_modulo @ nat @ B3 @ N2 ) % 0.30/0.52 = ( zero_zero @ nat ) ) % 0.30/0.52 => ( ord_less @ nat @ ( divide_divide @ nat @ A3 @ N2 ) @ ( divide_divide @ nat @ B3 @ N2 ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % div_less_mono % 0.30/0.52 thf(fact_1436_mod__eq__nat1E,axiom, % 0.30/0.52 ! [M: nat,Q4: nat,N2: nat] : % 0.30/0.52 ( ( ( modulo_modulo @ nat @ M @ Q4 ) % 0.30/0.52 = ( modulo_modulo @ nat @ N2 @ Q4 ) ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.52 => ~ ! [S2: nat] : % 0.30/0.52 ( M % 0.30/0.52 != ( plus_plus @ nat @ N2 @ ( times_times @ nat @ Q4 @ S2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_eq_nat1E % 0.30/0.52 thf(fact_1437_mod__eq__nat2E,axiom, % 0.30/0.52 ! [M: nat,Q4: nat,N2: nat] : % 0.30/0.52 ( ( ( modulo_modulo @ nat @ M @ Q4 ) % 0.30/0.52 = ( modulo_modulo @ nat @ N2 @ Q4 ) ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.52 => ~ ! [S2: nat] : % 0.30/0.52 ( N2 % 0.30/0.52 != ( plus_plus @ nat @ M @ ( times_times @ nat @ Q4 @ S2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_eq_nat2E % 0.30/0.52 thf(fact_1438_nat__mod__eq__lemma,axiom, % 0.30/0.52 ! [X3: nat,N2: nat,Y: nat] : % 0.30/0.52 ( ( ( modulo_modulo @ nat @ X3 @ N2 ) % 0.30/0.52 = ( modulo_modulo @ nat @ Y @ N2 ) ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ Y @ X3 ) % 0.30/0.52 => ? [Q2: nat] : % 0.30/0.52 ( X3 % 0.30/0.52 = ( plus_plus @ nat @ Y @ ( times_times @ nat @ N2 @ Q2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % nat_mod_eq_lemma % 0.30/0.52 thf(fact_1439_dvd__imp__le,axiom, % 0.30/0.52 ! [K: nat,N2: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ K @ N2 ) % 0.30/0.52 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.52 => ( ord_less_eq @ nat @ K @ N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_imp_le % 0.30/0.52 thf(fact_1440_div__mod__decomp,axiom, % 0.30/0.52 ! [A3: nat,N2: nat] : % 0.30/0.52 ( A3 % 0.30/0.52 = ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A3 @ N2 ) @ N2 ) @ ( modulo_modulo @ nat @ A3 @ N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % div_mod_decomp % 0.30/0.52 thf(fact_1441_mod__mult2__eq,axiom, % 0.30/0.52 ! [M: nat,N2: nat,Q4: nat] : % 0.30/0.52 ( ( modulo_modulo @ nat @ M @ ( times_times @ nat @ N2 @ Q4 ) ) % 0.30/0.52 = ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M @ N2 ) @ Q4 ) ) @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_mult2_eq % 0.30/0.52 thf(fact_1442_dvd__mult__cancel,axiom, % 0.30/0.52 ! [K: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 0.30/0.52 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.30/0.52 => ( dvd_dvd @ nat @ M @ N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_mult_cancel % 0.30/0.52 thf(fact_1443_nat__mult__dvd__cancel1,axiom, % 0.30/0.52 ! [K: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 0.30/0.52 = ( dvd_dvd @ nat @ M @ N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % nat_mult_dvd_cancel1 % 0.30/0.52 thf(fact_1444_bezout__add__strong__nat,axiom, % 0.30/0.52 ! [A2: nat,B2: nat] : % 0.30/0.52 ( ( A2 % 0.30/0.52 != ( zero_zero @ nat ) ) % 0.30/0.52 => ? [D3: nat,X5: nat,Y4: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ D3 @ A2 ) % 0.30/0.52 & ( dvd_dvd @ nat @ D3 @ B2 ) % 0.30/0.52 & ( ( times_times @ nat @ A2 @ X5 ) % 0.30/0.52 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D3 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % bezout_add_strong_nat % 0.30/0.52 thf(fact_1445_even__zero,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_parity @ A ) % 0.30/0.52 => ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.52 % 0.30/0.52 % even_zero % 0.30/0.52 thf(fact_1446_is__unitE,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [A2: A,C2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.52 => ~ ( ( A2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ! [B5: A] : % 0.30/0.52 ( ( B5 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( dvd_dvd @ A @ B5 @ ( one_one @ A ) ) % 0.30/0.52 => ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.52 = B5 ) % 0.30/0.52 => ( ( ( divide_divide @ A @ ( one_one @ A ) @ B5 ) % 0.30/0.52 = A2 ) % 0.30/0.52 => ( ( ( times_times @ A @ A2 @ B5 ) % 0.30/0.52 = ( one_one @ A ) ) % 0.30/0.52 => ( ( divide_divide @ A @ C2 @ A2 ) % 0.30/0.52 != ( times_times @ A @ C2 @ B5 ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % is_unitE % 0.30/0.52 thf(fact_1447_is__unit__div__mult__cancel__left,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( A2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.52 => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) ) % 0.30/0.52 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % is_unit_div_mult_cancel_left % 0.30/0.52 thf(fact_1448_is__unit__div__mult__cancel__right,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( A2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 0.30/0.52 => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) ) % 0.30/0.52 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % is_unit_div_mult_cancel_right % 0.30/0.52 thf(fact_1449_odd__even__add,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_parity @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.52 => ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) % 0.30/0.52 => ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % odd_even_add % 0.30/0.52 thf(fact_1450_dvd__power__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( algebraic_semidom @ A ) % 0.30/0.52 => ! [X3: A,M: nat,N2: nat] : % 0.30/0.52 ( ( X3 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( dvd_dvd @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ X3 @ N2 ) ) % 0.30/0.52 = ( ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) % 0.30/0.52 | ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_power_iff % 0.30/0.52 thf(fact_1451_dvd__power,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( comm_semiring_1 @ A ) % 0.30/0.52 => ! [N2: nat,X3: A] : % 0.30/0.52 ( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.52 | ( X3 % 0.30/0.52 = ( one_one @ A ) ) ) % 0.30/0.52 => ( dvd_dvd @ A @ X3 @ ( power_power @ A @ X3 @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_power % 0.30/0.52 thf(fact_1452_split__mod,axiom, % 0.30/0.52 ! [P: nat > $o,M: nat,N2: nat] : % 0.30/0.52 ( ( P @ ( modulo_modulo @ nat @ M @ N2 ) ) % 0.30/0.52 = ( ( ( N2 % 0.30/0.52 = ( zero_zero @ nat ) ) % 0.30/0.52 => ( P @ M ) ) % 0.30/0.52 & ( ( N2 % 0.30/0.52 != ( zero_zero @ nat ) ) % 0.30/0.52 => ! [I3: nat,J3: nat] : % 0.30/0.52 ( ( ord_less @ nat @ J3 @ N2 ) % 0.30/0.52 => ( ( M % 0.30/0.52 = ( plus_plus @ nat @ ( times_times @ nat @ N2 @ I3 ) @ J3 ) ) % 0.30/0.52 => ( P @ J3 ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % split_mod % 0.30/0.52 thf(fact_1453_dvd__mult__cancel1,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ M @ N2 ) @ M ) % 0.30/0.52 = ( N2 % 0.30/0.52 = ( one_one @ nat ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_mult_cancel1 % 0.30/0.52 thf(fact_1454_dvd__mult__cancel2,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ N2 @ M ) @ M ) % 0.30/0.52 = ( N2 % 0.30/0.52 = ( one_one @ nat ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_mult_cancel2 % 0.30/0.52 thf(fact_1455_power__dvd__imp__le,axiom, % 0.30/0.52 ! [I: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N2 ) ) % 0.30/0.52 => ( ( ord_less @ nat @ ( one_one @ nat ) @ I ) % 0.30/0.52 => ( ord_less_eq @ nat @ M @ N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % power_dvd_imp_le % 0.30/0.52 thf(fact_1456_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.52 => ! [C2: A,A2: A,B2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.52 => ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) % 0.30/0.52 = ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % unique_euclidean_semiring_numeral_class.mod_mult2_eq % 0.30/0.52 thf(fact_1457_product__nth,axiom, % 0.30/0.52 ! [A: $tType,B: $tType,N2: nat,Xs: list @ A,Ys: list @ B] : % 0.30/0.52 ( ( ord_less @ nat @ N2 @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) % 0.30/0.52 => ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) @ N2 ) % 0.30/0.52 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % product_nth % 0.30/0.52 thf(fact_1458_power__mono__odd,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_idom @ A ) % 0.30/0.52 => ! [N2: nat,A2: A,B2: A] : % 0.30/0.52 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.52 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.52 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % power_mono_odd % 0.30/0.52 thf(fact_1459_Suc__times__mod__eq,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M ) % 0.30/0.52 => ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M @ N2 ) ) @ M ) % 0.30/0.52 = ( one_one @ nat ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Suc_times_mod_eq % 0.30/0.52 thf(fact_1460_odd__pos,axiom, % 0.30/0.52 ! [N2: nat] : % 0.30/0.52 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.52 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ). % 0.30/0.52 % 0.30/0.52 % odd_pos % 0.30/0.52 thf(fact_1461_dvd__power__iff__le,axiom, % 0.30/0.52 ! [K: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N2 ) ) % 0.30/0.52 = ( ord_less_eq @ nat @ M @ N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_power_iff_le % 0.30/0.52 thf(fact_1462_even__unset__bit__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.52 => ! [M: nat,A2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) ) % 0.30/0.52 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.52 | ( M % 0.30/0.52 = ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % even_unset_bit_iff % 0.30/0.52 thf(fact_1463_even__set__bit__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.52 => ! [M: nat,A2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) ) % 0.30/0.52 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.52 & ( M % 0.30/0.52 != ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % even_set_bit_iff % 0.30/0.52 thf(fact_1464_num_Osize__gen_I1_J,axiom, % 0.30/0.52 ( ( size_num @ one2 ) % 0.30/0.52 = ( zero_zero @ nat ) ) ). % 0.30/0.52 % 0.30/0.52 % num.size_gen(1) % 0.30/0.52 thf(fact_1465_divmod__digit__0_I2_J,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.52 => ! [B2: A,A2: A] : % 0.30/0.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.52 => ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 ) % 0.30/0.52 => ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) % 0.30/0.52 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % divmod_digit_0(2) % 0.30/0.52 thf(fact_1466_bits__stable__imp__add__self,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( bit_semiring_bits @ A ) % 0.30/0.52 => ! [A2: A] : % 0.30/0.52 ( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.52 = A2 ) % 0.30/0.52 => ( ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) % 0.30/0.52 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % bits_stable_imp_add_self % 0.30/0.52 thf(fact_1467_div__exp__mod__exp__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( bit_semiring_bits @ A ) % 0.30/0.52 => ! [A2: A,N2: nat,M: nat] : % 0.30/0.52 ( ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) % 0.30/0.52 = ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % div_exp_mod_exp_eq % 0.30/0.52 thf(fact_1468_oddE,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_parity @ A ) % 0.30/0.52 => ! [A2: A] : % 0.30/0.52 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.52 => ~ ! [B5: A] : % 0.30/0.52 ( A2 % 0.30/0.52 != ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % oddE % 0.30/0.52 thf(fact_1469_zero__le__even__power,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_idom @ A ) % 0.30/0.52 => ! [N2: nat,A2: A] : % 0.30/0.52 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % zero_le_even_power % 0.30/0.52 thf(fact_1470_zero__le__odd__power,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_idom @ A ) % 0.30/0.52 => ! [N2: nat,A2: A] : % 0.30/0.52 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) % 0.30/0.52 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % zero_le_odd_power % 0.30/0.52 thf(fact_1471_zero__le__power__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_idom @ A ) % 0.30/0.52 => ! [A2: A,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) % 0.30/0.52 = ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.52 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.52 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % zero_le_power_eq % 0.30/0.52 thf(fact_1472_verit__le__mono__div,axiom, % 0.30/0.52 ! [A3: nat,B3: nat,N2: nat] : % 0.30/0.52 ( ( ord_less @ nat @ A3 @ B3 ) % 0.30/0.52 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.52 => ( ord_less_eq @ nat % 0.30/0.52 @ ( plus_plus @ nat @ ( divide_divide @ nat @ A3 @ N2 ) % 0.30/0.52 @ ( if @ nat % 0.30/0.52 @ ( ( modulo_modulo @ nat @ B3 @ N2 ) % 0.30/0.52 = ( zero_zero @ nat ) ) % 0.30/0.52 @ ( one_one @ nat ) % 0.30/0.52 @ ( zero_zero @ nat ) ) ) % 0.30/0.52 @ ( divide_divide @ nat @ B3 @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % verit_le_mono_div % 0.30/0.52 thf(fact_1473_even__set__encode__iff,axiom, % 0.30/0.52 ! [A3: set @ nat] : % 0.30/0.52 ( ( finite_finite2 @ nat @ A3 ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A3 ) ) % 0.30/0.52 = ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % even_set_encode_iff % 0.30/0.52 thf(fact_1474_divmod__digit__0_I1_J,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.52 => ! [B2: A,A2: A] : % 0.30/0.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.52 => ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 ) % 0.30/0.52 => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) % 0.30/0.52 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % divmod_digit_0(1) % 0.30/0.52 thf(fact_1475_zero__less__power__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_idom @ A ) % 0.30/0.52 => ! [A2: A,N2: nat] : % 0.30/0.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) % 0.30/0.52 = ( ( N2 % 0.30/0.52 = ( zero_zero @ nat ) ) % 0.30/0.52 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.52 & ( A2 % 0.30/0.52 != ( zero_zero @ A ) ) ) % 0.30/0.52 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.52 & ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % zero_less_power_eq % 0.30/0.52 thf(fact_1476_mod__double__modulus,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.52 => ! [M: A,X3: A] : % 0.30/0.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ M ) % 0.30/0.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.52 => ( ( ( modulo_modulo @ A @ X3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) % 0.30/0.52 = ( modulo_modulo @ A @ X3 @ M ) ) % 0.30/0.52 | ( ( modulo_modulo @ A @ X3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) % 0.30/0.52 = ( plus_plus @ A @ ( modulo_modulo @ A @ X3 @ M ) @ M ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_double_modulus % 0.30/0.52 thf(fact_1477_unset__bit__Suc,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.52 => ! [N2: nat,A2: A] : % 0.30/0.52 ( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N2 ) @ A2 ) % 0.30/0.52 = ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % unset_bit_Suc % 0.30/0.52 thf(fact_1478_set__bit__Suc,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.52 => ! [N2: nat,A2: A] : % 0.30/0.52 ( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N2 ) @ A2 ) % 0.30/0.52 = ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % set_bit_Suc % 0.30/0.52 thf(fact_1479_power__le__zero__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_idom @ A ) % 0.30/0.52 => ! [A2: A,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( zero_zero @ A ) ) % 0.30/0.52 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.52 & ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.52 & ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) % 0.30/0.52 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.52 & ( A2 % 0.30/0.52 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % power_le_zero_eq % 0.30/0.52 thf(fact_1480_divmod__digit__1_I1_J,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.52 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.52 => ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) % 0.30/0.52 => ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) ) % 0.30/0.52 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % divmod_digit_1(1) % 0.30/0.52 thf(fact_1481_flip__bit__Suc,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.52 => ! [N2: nat,A2: A] : % 0.30/0.52 ( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N2 ) @ A2 ) % 0.30/0.52 = ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % flip_bit_Suc % 0.30/0.52 thf(fact_1482_pos__eucl__rel__int__mult__2,axiom, % 0.30/0.52 ! [B2: int,A2: int,Q4: int,R2: int] : % 0.30/0.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 ) % 0.30/0.52 => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 0.30/0.52 => ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q4 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % pos_eucl_rel_int_mult_2 % 0.30/0.52 thf(fact_1483_signed__take__bit__Suc,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( bit_ri3973907225187159222ations @ A ) % 0.30/0.52 => ! [N2: nat,A2: A] : % 0.30/0.52 ( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N2 ) @ A2 ) % 0.30/0.52 = ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % signed_take_bit_Suc % 0.30/0.52 thf(fact_1484_even__mult__exp__div__exp__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( bit_semiring_bits @ A ) % 0.30/0.52 => ! [A2: A,M: nat,N2: nat] : % 0.30/0.52 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 0.30/0.52 = ( ( ord_less @ nat @ N2 @ M ) % 0.30/0.52 | ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.52 = ( zero_zero @ A ) ) % 0.30/0.52 | ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.52 & ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % even_mult_exp_div_exp_iff % 0.30/0.52 thf(fact_1485_even__flip__bit__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.52 => ! [M: nat,A2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) ) % 0.30/0.52 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.52 != ( M % 0.30/0.52 = ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % even_flip_bit_iff % 0.30/0.52 thf(fact_1486_one__mod__2__pow__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( euclid5411537665997757685th_nat @ A ) % 0.30/0.52 => ! [N2: nat] : % 0.30/0.52 ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.52 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % one_mod_2_pow_eq % 0.30/0.52 thf(fact_1487_set__encode__insert,axiom, % 0.30/0.52 ! [A3: set @ nat,N2: nat] : % 0.30/0.52 ( ( finite_finite2 @ nat @ A3 ) % 0.30/0.52 => ( ~ ( member @ nat @ N2 @ A3 ) % 0.30/0.52 => ( ( nat_set_encode @ ( insert @ nat @ N2 @ A3 ) ) % 0.30/0.52 = ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( nat_set_encode @ A3 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % set_encode_insert % 0.30/0.52 thf(fact_1488_insert__absorb2,axiom, % 0.30/0.52 ! [A: $tType,X3: A,A3: set @ A] : % 0.30/0.52 ( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A3 ) ) % 0.30/0.52 = ( insert @ A @ X3 @ A3 ) ) ). % 0.30/0.52 % 0.30/0.52 % insert_absorb2 % 0.30/0.52 thf(fact_1489_insert__iff,axiom, % 0.30/0.52 ! [A: $tType,A2: A,B2: A,A3: set @ A] : % 0.30/0.52 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) ) % 0.30/0.52 = ( ( A2 = B2 ) % 0.30/0.52 | ( member @ A @ A2 @ A3 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % insert_iff % 0.30/0.52 thf(fact_1490_insertCI,axiom, % 0.30/0.52 ! [A: $tType,A2: A,B3: set @ A,B2: A] : % 0.30/0.52 ( ( ~ ( member @ A @ A2 @ B3 ) % 0.30/0.52 => ( A2 = B2 ) ) % 0.30/0.52 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % insertCI % 0.30/0.52 thf(fact_1491_diff__self,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( group_add @ A ) % 0.30/0.52 => ! [A2: A] : % 0.30/0.52 ( ( minus_minus @ A @ A2 @ A2 ) % 0.30/0.52 = ( zero_zero @ A ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_self % 0.30/0.52 thf(fact_1492_diff__0__right,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( group_add @ A ) % 0.30/0.52 => ! [A2: A] : % 0.30/0.52 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.52 = A2 ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_0_right % 0.30/0.52 thf(fact_1493_zero__diff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( comm_monoid_diff @ A ) % 0.30/0.52 => ! [A2: A] : % 0.30/0.52 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.52 = ( zero_zero @ A ) ) ) ). % 0.30/0.52 % 0.30/0.52 % zero_diff % 0.30/0.52 thf(fact_1494_diff__zero,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( cancel1802427076303600483id_add @ A ) % 0.30/0.52 => ! [A2: A] : % 0.30/0.52 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.52 = A2 ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_zero % 0.30/0.52 thf(fact_1495_cancel__comm__monoid__add__class_Odiff__cancel,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( cancel1802427076303600483id_add @ A ) % 0.30/0.52 => ! [A2: A] : % 0.30/0.52 ( ( minus_minus @ A @ A2 @ A2 ) % 0.30/0.52 = ( zero_zero @ A ) ) ) ). % 0.30/0.52 % 0.30/0.52 % cancel_comm_monoid_add_class.diff_cancel % 0.30/0.52 thf(fact_1496_add__diff__cancel,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 ) % 0.30/0.52 = A2 ) ) ). % 0.30/0.52 % 0.30/0.52 % add_diff_cancel % 0.30/0.52 thf(fact_1497_diff__add__cancel,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 ) % 0.30/0.52 = A2 ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_add_cancel % 0.30/0.52 thf(fact_1498_add__diff__cancel__left,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( cancel2418104881723323429up_add @ A ) % 0.30/0.52 => ! [C2: A,A2: A,B2: A] : % 0.30/0.52 ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) % 0.30/0.52 = ( minus_minus @ A @ A2 @ B2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % add_diff_cancel_left % 0.30/0.52 thf(fact_1499_add__diff__cancel__left_H,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( cancel2418104881723323429up_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 ) % 0.30/0.52 = B2 ) ) ). % 0.30/0.52 % 0.30/0.52 % add_diff_cancel_left' % 0.30/0.52 thf(fact_1500_add__diff__cancel__right,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( cancel2418104881723323429up_add @ A ) % 0.30/0.52 => ! [A2: A,C2: A,B2: A] : % 0.30/0.52 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) % 0.30/0.52 = ( minus_minus @ A @ A2 @ B2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % add_diff_cancel_right % 0.30/0.52 thf(fact_1501_add__diff__cancel__right_H,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( cancel2418104881723323429up_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 ) % 0.30/0.52 = A2 ) ) ). % 0.30/0.52 % 0.30/0.52 % add_diff_cancel_right' % 0.30/0.52 thf(fact_1502_singletonI,axiom, % 0.30/0.52 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % singletonI % 0.30/0.52 thf(fact_1503_finite__insert,axiom, % 0.30/0.52 ! [A: $tType,A2: A,A3: set @ A] : % 0.30/0.52 ( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) ) % 0.30/0.52 = ( finite_finite2 @ A @ A3 ) ) ). % 0.30/0.52 % 0.30/0.52 % finite_insert % 0.30/0.52 thf(fact_1504_insert__subset,axiom, % 0.30/0.52 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] : % 0.30/0.52 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 ) % 0.30/0.52 = ( ( member @ A @ X3 @ B3 ) % 0.30/0.52 & ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % insert_subset % 0.30/0.52 thf(fact_1505_diff__Suc__Suc,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) % 0.30/0.52 = ( minus_minus @ nat @ M @ N2 ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_Suc_Suc % 0.30/0.52 thf(fact_1506_Suc__diff__diff,axiom, % 0.30/0.52 ! [M: nat,N2: nat,K: nat] : % 0.30/0.52 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) ) % 0.30/0.52 = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K ) ) ). % 0.30/0.52 % 0.30/0.52 % Suc_diff_diff % 0.30/0.52 thf(fact_1507_diff__0__eq__0,axiom, % 0.30/0.52 ! [N2: nat] : % 0.30/0.52 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.52 = ( zero_zero @ nat ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_0_eq_0 % 0.30/0.52 thf(fact_1508_diff__self__eq__0,axiom, % 0.30/0.52 ! [M: nat] : % 0.30/0.52 ( ( minus_minus @ nat @ M @ M ) % 0.30/0.52 = ( zero_zero @ nat ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_self_eq_0 % 0.30/0.52 thf(fact_1509_diff__diff__cancel,axiom, % 0.30/0.52 ! [I: nat,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ I @ N2 ) % 0.30/0.52 => ( ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ N2 @ I ) ) % 0.30/0.52 = I ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_diff_cancel % 0.30/0.52 thf(fact_1510_diff__diff__left,axiom, % 0.30/0.52 ! [I: nat,J: nat,K: nat] : % 0.30/0.52 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K ) % 0.30/0.52 = ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_diff_left % 0.30/0.52 thf(fact_1511_flip__bit__negative__int__iff,axiom, % 0.30/0.52 ! [N2: nat,K: int] : % 0.30/0.52 ( ( ord_less @ int @ ( bit_se8732182000553998342ip_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) % 0.30/0.52 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ). % 0.30/0.52 % 0.30/0.52 % flip_bit_negative_int_iff % 0.30/0.52 thf(fact_1512_of__bool__less__eq__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linord181362715937106298miring @ A ) % 0.30/0.52 => ! [P: $o,Q: $o] : % 0.30/0.52 ( ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) % 0.30/0.52 = ( P % 0.30/0.52 => Q ) ) ) ). % 0.30/0.52 % 0.30/0.52 % of_bool_less_eq_iff % 0.30/0.52 thf(fact_1513_of__bool__eq_I1_J,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( zero_neq_one @ A ) % 0.30/0.52 => ( ( zero_neq_one_of_bool @ A @ $false ) % 0.30/0.52 = ( zero_zero @ A ) ) ) ). % 0.30/0.52 % 0.30/0.52 % of_bool_eq(1) % 0.30/0.52 thf(fact_1514_of__bool__eq__0__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( zero_neq_one @ A ) % 0.30/0.52 => ! [P: $o] : % 0.30/0.52 ( ( ( zero_neq_one_of_bool @ A @ P ) % 0.30/0.52 = ( zero_zero @ A ) ) % 0.30/0.52 = ( ~ P ) ) ) ). % 0.30/0.52 % 0.30/0.52 % of_bool_eq_0_iff % 0.30/0.52 thf(fact_1515_of__bool__less__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linord181362715937106298miring @ A ) % 0.30/0.52 => ! [P: $o,Q: $o] : % 0.30/0.52 ( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) % 0.30/0.52 = ( ~ P % 0.30/0.52 & Q ) ) ) ). % 0.30/0.52 % 0.30/0.52 % of_bool_less_iff % 0.30/0.52 thf(fact_1516_of__bool__eq__1__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( zero_neq_one @ A ) % 0.30/0.52 => ! [P: $o] : % 0.30/0.52 ( ( ( zero_neq_one_of_bool @ A @ P ) % 0.30/0.52 = ( one_one @ A ) ) % 0.30/0.52 = P ) ) ). % 0.30/0.52 % 0.30/0.52 % of_bool_eq_1_iff % 0.30/0.52 thf(fact_1517_of__bool__eq_I2_J,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( zero_neq_one @ A ) % 0.30/0.52 => ( ( zero_neq_one_of_bool @ A @ $true ) % 0.30/0.52 = ( one_one @ A ) ) ) ). % 0.30/0.52 % 0.30/0.52 % of_bool_eq(2) % 0.30/0.52 thf(fact_1518_signed__take__bit__of__0,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( bit_ri3973907225187159222ations @ A ) % 0.30/0.52 => ! [N2: nat] : % 0.30/0.52 ( ( bit_ri4674362597316999326ke_bit @ A @ N2 @ ( zero_zero @ A ) ) % 0.30/0.52 = ( zero_zero @ A ) ) ) ). % 0.30/0.52 % 0.30/0.52 % signed_take_bit_of_0 % 0.30/0.52 thf(fact_1519_diff__ge__0__iff__ge,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) ) % 0.30/0.52 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_ge_0_iff_ge % 0.30/0.52 thf(fact_1520_diff__gt__0__iff__gt,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) ) % 0.30/0.52 = ( ord_less @ A @ B2 @ A2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_gt_0_iff_gt % 0.30/0.52 thf(fact_1521_le__add__diff__inverse2,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_semidom @ A ) % 0.30/0.52 => ! [B2: A,A2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.30/0.52 => ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 ) % 0.30/0.52 = A2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % le_add_diff_inverse2 % 0.30/0.52 thf(fact_1522_le__add__diff__inverse,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_semidom @ A ) % 0.30/0.52 => ! [B2: A,A2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.30/0.52 => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) ) % 0.30/0.52 = A2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % le_add_diff_inverse % 0.30/0.52 thf(fact_1523_diff__numeral__special_I9_J,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( neg_numeral @ A ) % 0.30/0.52 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) % 0.30/0.52 = ( zero_zero @ A ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_numeral_special(9) % 0.30/0.52 thf(fact_1524_diff__add__zero,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( comm_monoid_diff @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.30/0.52 = ( zero_zero @ A ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_add_zero % 0.30/0.52 thf(fact_1525_div__diff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( idom_modulo @ A ) % 0.30/0.52 => ! [C2: A,A2: A,B2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ C2 @ A2 ) % 0.30/0.52 => ( ( dvd_dvd @ A @ C2 @ B2 ) % 0.30/0.52 => ( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.52 = ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % div_diff % 0.30/0.52 thf(fact_1526_singleton__insert__inj__eq_H,axiom, % 0.30/0.52 ! [A: $tType,A2: A,A3: set @ A,B2: A] : % 0.30/0.52 ( ( ( insert @ A @ A2 @ A3 ) % 0.30/0.52 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.30/0.52 = ( ( A2 = B2 ) % 0.30/0.52 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % singleton_insert_inj_eq' % 0.30/0.52 thf(fact_1527_singleton__insert__inj__eq,axiom, % 0.30/0.52 ! [A: $tType,B2: A,A2: A,A3: set @ A] : % 0.30/0.52 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) % 0.30/0.52 = ( insert @ A @ A2 @ A3 ) ) % 0.30/0.52 = ( ( A2 = B2 ) % 0.30/0.52 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % singleton_insert_inj_eq % 0.30/0.52 thf(fact_1528_zero__less__of__bool__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_semidom @ A ) % 0.30/0.52 => ! [P: $o] : % 0.30/0.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) % 0.30/0.52 = P ) ) ). % 0.30/0.52 % 0.30/0.52 % zero_less_of_bool_iff % 0.30/0.52 thf(fact_1529_zero__less__diff,axiom, % 0.30/0.52 ! [N2: nat,M: nat] : % 0.30/0.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) % 0.30/0.52 = ( ord_less @ nat @ M @ N2 ) ) ). % 0.30/0.52 % 0.30/0.52 % zero_less_diff % 0.30/0.52 thf(fact_1530_diff__is__0__eq,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ( ( minus_minus @ nat @ M @ N2 ) % 0.30/0.52 = ( zero_zero @ nat ) ) % 0.30/0.52 = ( ord_less_eq @ nat @ M @ N2 ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_is_0_eq % 0.30/0.52 thf(fact_1531_diff__is__0__eq_H,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.52 => ( ( minus_minus @ nat @ M @ N2 ) % 0.30/0.52 = ( zero_zero @ nat ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_is_0_eq' % 0.30/0.52 thf(fact_1532_of__bool__less__one__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_semidom @ A ) % 0.30/0.52 => ! [P: $o] : % 0.30/0.52 ( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) ) % 0.30/0.52 = ( ~ P ) ) ) ). % 0.30/0.52 % 0.30/0.52 % of_bool_less_one_iff % 0.30/0.52 thf(fact_1533_of__bool__not__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ring_1 @ A ) % 0.30/0.52 => ! [P: $o] : % 0.30/0.52 ( ( zero_neq_one_of_bool @ A @ ~ P ) % 0.30/0.52 = ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % of_bool_not_iff % 0.30/0.52 thf(fact_1534_Nat_Odiff__diff__right,axiom, % 0.30/0.52 ! [K: nat,J: nat,I: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ K @ J ) % 0.30/0.52 => ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) % 0.30/0.52 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Nat.diff_diff_right % 0.30/0.52 thf(fact_1535_Nat_Oadd__diff__assoc2,axiom, % 0.30/0.52 ! [K: nat,J: nat,I: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ K @ J ) % 0.30/0.52 => ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) % 0.30/0.52 = ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Nat.add_diff_assoc2 % 0.30/0.52 thf(fact_1536_Nat_Oadd__diff__assoc,axiom, % 0.30/0.52 ! [K: nat,J: nat,I: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ K @ J ) % 0.30/0.52 => ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) % 0.30/0.52 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Nat.add_diff_assoc % 0.30/0.52 thf(fact_1537_diff__Suc__1,axiom, % 0.30/0.52 ! [N2: nat] : % 0.30/0.52 ( ( minus_minus @ nat @ ( suc @ N2 ) @ ( one_one @ nat ) ) % 0.30/0.52 = N2 ) ). % 0.30/0.52 % 0.30/0.52 % diff_Suc_1 % 0.30/0.52 thf(fact_1538_Suc__0__mod__eq,axiom, % 0.30/0.52 ! [N2: nat] : % 0.30/0.52 ( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 0.30/0.52 = ( zero_neq_one_of_bool @ nat % 0.30/0.52 @ ( N2 % 0.30/0.52 != ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Suc_0_mod_eq % 0.30/0.52 thf(fact_1539_mod__pos__pos__trivial,axiom, % 0.30/0.52 ! [K: int,L: int] : % 0.30/0.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.52 => ( ( ord_less @ int @ K @ L ) % 0.30/0.52 => ( ( modulo_modulo @ int @ K @ L ) % 0.30/0.52 = K ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_pos_pos_trivial % 0.30/0.52 thf(fact_1540_mod__neg__neg__trivial,axiom, % 0.30/0.52 ! [K: int,L: int] : % 0.30/0.52 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) ) % 0.30/0.52 => ( ( ord_less @ int @ L @ K ) % 0.30/0.52 => ( ( modulo_modulo @ int @ K @ L ) % 0.30/0.52 = K ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_neg_neg_trivial % 0.30/0.52 thf(fact_1541_signed__take__bit__Suc__1,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( bit_ri3973907225187159222ations @ A ) % 0.30/0.52 => ! [N2: nat] : % 0.30/0.52 ( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N2 ) @ ( one_one @ A ) ) % 0.30/0.52 = ( one_one @ A ) ) ) ). % 0.30/0.52 % 0.30/0.52 % signed_take_bit_Suc_1 % 0.30/0.52 thf(fact_1542_Suc__pred,axiom, % 0.30/0.52 ! [N2: nat] : % 0.30/0.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.52 => ( ( suc @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) % 0.30/0.52 = N2 ) ) ). % 0.30/0.52 % 0.30/0.52 % Suc_pred % 0.30/0.52 thf(fact_1543_diff__Suc__diff__eq2,axiom, % 0.30/0.52 ! [K: nat,J: nat,I: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ K @ J ) % 0.30/0.52 => ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J @ K ) ) @ I ) % 0.30/0.52 = ( minus_minus @ nat @ ( suc @ J ) @ ( plus_plus @ nat @ K @ I ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_Suc_diff_eq2 % 0.30/0.52 thf(fact_1544_diff__Suc__diff__eq1,axiom, % 0.30/0.52 ! [K: nat,J: nat,I: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ K @ J ) % 0.30/0.52 => ( ( minus_minus @ nat @ I @ ( suc @ ( minus_minus @ nat @ J @ K ) ) ) % 0.30/0.52 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ ( suc @ J ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_Suc_diff_eq1 % 0.30/0.52 thf(fact_1545_signed__take__bit__Suc__bit0,axiom, % 0.30/0.52 ! [N2: nat,K: num] : % 0.30/0.52 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) % 0.30/0.52 = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % signed_take_bit_Suc_bit0 % 0.30/0.52 thf(fact_1546_Suc__diff__1,axiom, % 0.30/0.52 ! [N2: nat] : % 0.30/0.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.52 => ( ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) % 0.30/0.52 = N2 ) ) ). % 0.30/0.52 % 0.30/0.52 % Suc_diff_1 % 0.30/0.52 thf(fact_1547_set__replicate,axiom, % 0.30/0.52 ! [A: $tType,N2: nat,X3: A] : % 0.30/0.52 ( ( N2 % 0.30/0.52 != ( zero_zero @ nat ) ) % 0.30/0.52 => ( ( set2 @ A @ ( replicate @ A @ N2 @ X3 ) ) % 0.30/0.52 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % set_replicate % 0.30/0.52 thf(fact_1548_even__diff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ring_parity @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A2 @ B2 ) ) % 0.30/0.52 = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % even_diff % 0.30/0.52 thf(fact_1549_of__bool__half__eq__0,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( euclid5411537665997757685th_nat @ A ) % 0.30/0.52 => ! [B2: $o] : % 0.30/0.52 ( ( divide_divide @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.52 = ( zero_zero @ A ) ) ) ). % 0.30/0.52 % 0.30/0.52 % of_bool_half_eq_0 % 0.30/0.52 thf(fact_1550_odd__Suc__minus__one,axiom, % 0.30/0.52 ! [N2: nat] : % 0.30/0.52 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.52 => ( ( suc @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) % 0.30/0.52 = N2 ) ) ). % 0.30/0.52 % 0.30/0.52 % odd_Suc_minus_one % 0.30/0.52 thf(fact_1551_even__diff__nat,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N2 ) ) % 0.30/0.52 = ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.52 | ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % even_diff_nat % 0.30/0.52 thf(fact_1552_semiring__parity__class_Oeven__mask__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_parity @ A ) % 0.30/0.52 => ! [N2: nat] : % 0.30/0.52 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) ) % 0.30/0.52 = ( N2 % 0.30/0.52 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % semiring_parity_class.even_mask_iff % 0.30/0.52 thf(fact_1553_one__div__2__pow__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( euclid5411537665997757685th_nat @ A ) % 0.30/0.52 => ! [N2: nat] : % 0.30/0.52 ( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.52 = ( zero_neq_one_of_bool @ A % 0.30/0.52 @ ( N2 % 0.30/0.52 = ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % one_div_2_pow_eq % 0.30/0.52 thf(fact_1554_bits__1__div__exp,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( bit_semiring_bits @ A ) % 0.30/0.52 => ! [N2: nat] : % 0.30/0.52 ( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.52 = ( zero_neq_one_of_bool @ A % 0.30/0.52 @ ( N2 % 0.30/0.52 = ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % bits_1_div_exp % 0.30/0.52 thf(fact_1555_flip__bit__0,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.52 => ! [A2: A] : % 0.30/0.52 ( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A2 ) % 0.30/0.52 = ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % flip_bit_0 % 0.30/0.52 thf(fact_1556_mod__int__unique,axiom, % 0.30/0.52 ! [K: int,L: int,Q4: int,R2: int] : % 0.30/0.52 ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 0.30/0.52 => ( ( modulo_modulo @ int @ K @ L ) % 0.30/0.52 = R2 ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_int_unique % 0.30/0.52 thf(fact_1557_gcd__nat_Oasym,axiom, % 0.30/0.52 ! [A2: nat,B2: nat] : % 0.30/0.52 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.30/0.52 & ( A2 != B2 ) ) % 0.30/0.52 => ~ ( ( dvd_dvd @ nat @ B2 @ A2 ) % 0.30/0.52 & ( B2 != A2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % gcd_nat.asym % 0.30/0.52 thf(fact_1558_gcd__nat_Orefl,axiom, % 0.30/0.52 ! [A2: nat] : ( dvd_dvd @ nat @ A2 @ A2 ) ). % 0.30/0.52 % 0.30/0.52 % gcd_nat.refl % 0.30/0.52 thf(fact_1559_gcd__nat_Otrans,axiom, % 0.30/0.52 ! [A2: nat,B2: nat,C2: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ B2 @ C2 ) % 0.30/0.52 => ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % gcd_nat.trans % 0.30/0.52 thf(fact_1560_gcd__nat_Oeq__iff,axiom, % 0.30/0.52 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) ) % 0.30/0.52 = ( ^ [A4: nat,B4: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ A4 @ B4 ) % 0.30/0.52 & ( dvd_dvd @ nat @ B4 @ A4 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % gcd_nat.eq_iff % 0.30/0.52 thf(fact_1561_gcd__nat_Oirrefl,axiom, % 0.30/0.52 ! [A2: nat] : % 0.30/0.52 ~ ( ( dvd_dvd @ nat @ A2 @ A2 ) % 0.30/0.52 & ( A2 != A2 ) ) ). % 0.30/0.52 % 0.30/0.52 % gcd_nat.irrefl % 0.30/0.52 thf(fact_1562_gcd__nat_Oantisym,axiom, % 0.30/0.52 ! [A2: nat,B2: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ B2 @ A2 ) % 0.30/0.52 => ( A2 = B2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % gcd_nat.antisym % 0.30/0.52 thf(fact_1563_gcd__nat_Ostrict__trans,axiom, % 0.30/0.52 ! [A2: nat,B2: nat,C2: nat] : % 0.30/0.52 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.30/0.52 & ( A2 != B2 ) ) % 0.30/0.52 => ( ( ( dvd_dvd @ nat @ B2 @ C2 ) % 0.30/0.52 & ( B2 != C2 ) ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ A2 @ C2 ) % 0.30/0.52 & ( A2 != C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % gcd_nat.strict_trans % 0.30/0.52 thf(fact_1564_gcd__nat_Ostrict__trans1,axiom, % 0.30/0.52 ! [A2: nat,B2: nat,C2: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.30/0.52 => ( ( ( dvd_dvd @ nat @ B2 @ C2 ) % 0.30/0.52 & ( B2 != C2 ) ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ A2 @ C2 ) % 0.30/0.52 & ( A2 != C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % gcd_nat.strict_trans1 % 0.30/0.52 thf(fact_1565_gcd__nat_Ostrict__trans2,axiom, % 0.30/0.52 ! [A2: nat,B2: nat,C2: nat] : % 0.30/0.52 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.30/0.52 & ( A2 != B2 ) ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ B2 @ C2 ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ A2 @ C2 ) % 0.30/0.52 & ( A2 != C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % gcd_nat.strict_trans2 % 0.30/0.52 thf(fact_1566_gcd__nat_Ostrict__iff__not,axiom, % 0.30/0.52 ! [A2: nat,B2: nat] : % 0.30/0.52 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.30/0.52 & ( A2 != B2 ) ) % 0.30/0.52 = ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.30/0.52 & ~ ( dvd_dvd @ nat @ B2 @ A2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % gcd_nat.strict_iff_not % 0.30/0.52 thf(fact_1567_gcd__nat_Oorder__iff__strict,axiom, % 0.30/0.52 ( ( dvd_dvd @ nat ) % 0.30/0.52 = ( ^ [A4: nat,B4: nat] : % 0.30/0.52 ( ( ( dvd_dvd @ nat @ A4 @ B4 ) % 0.30/0.52 & ( A4 != B4 ) ) % 0.30/0.52 | ( A4 = B4 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % gcd_nat.order_iff_strict % 0.30/0.52 thf(fact_1568_gcd__nat_Ostrict__iff__order,axiom, % 0.30/0.52 ! [A2: nat,B2: nat] : % 0.30/0.52 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.30/0.52 & ( A2 != B2 ) ) % 0.30/0.52 = ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.30/0.52 & ( A2 != B2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % gcd_nat.strict_iff_order % 0.30/0.52 thf(fact_1569_gcd__nat_Ostrict__implies__order,axiom, % 0.30/0.52 ! [A2: nat,B2: nat] : % 0.30/0.52 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.30/0.52 & ( A2 != B2 ) ) % 0.30/0.52 => ( dvd_dvd @ nat @ A2 @ B2 ) ) ). % 0.30/0.52 % 0.30/0.52 % gcd_nat.strict_implies_order % 0.30/0.52 thf(fact_1570_gcd__nat_Ostrict__implies__not__eq,axiom, % 0.30/0.52 ! [A2: nat,B2: nat] : % 0.30/0.52 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.30/0.52 & ( A2 != B2 ) ) % 0.30/0.52 => ( A2 != B2 ) ) ). % 0.30/0.52 % 0.30/0.52 % gcd_nat.strict_implies_not_eq % 0.30/0.52 thf(fact_1571_gcd__nat_Onot__eq__order__implies__strict,axiom, % 0.30/0.52 ! [A2: nat,B2: nat] : % 0.30/0.52 ( ( A2 != B2 ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.30/0.52 & ( A2 != B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % gcd_nat.not_eq_order_implies_strict % 0.30/0.52 thf(fact_1572_dvd__antisym,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ M @ N2 ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ N2 @ M ) % 0.30/0.52 => ( M = N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_antisym % 0.30/0.52 thf(fact_1573_dvd__diff__nat,axiom, % 0.30/0.52 ! [K: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ K @ M ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ K @ N2 ) % 0.30/0.52 => ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_diff_nat % 0.30/0.52 thf(fact_1574_eucl__rel__int,axiom, % 0.30/0.52 ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K @ L ) @ ( modulo_modulo @ int @ K @ L ) ) ) ). % 0.30/0.52 % 0.30/0.52 % eucl_rel_int % 0.30/0.52 thf(fact_1575_mk__disjoint__insert,axiom, % 0.30/0.52 ! [A: $tType,A2: A,A3: set @ A] : % 0.30/0.52 ( ( member @ A @ A2 @ A3 ) % 0.30/0.52 => ? [B10: set @ A] : % 0.30/0.52 ( ( A3 % 0.30/0.52 = ( insert @ A @ A2 @ B10 ) ) % 0.30/0.52 & ~ ( member @ A @ A2 @ B10 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mk_disjoint_insert % 0.30/0.52 thf(fact_1576_insert__commute,axiom, % 0.30/0.52 ! [A: $tType,X3: A,Y: A,A3: set @ A] : % 0.30/0.52 ( ( insert @ A @ X3 @ ( insert @ A @ Y @ A3 ) ) % 0.30/0.52 = ( insert @ A @ Y @ ( insert @ A @ X3 @ A3 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % insert_commute % 0.30/0.52 thf(fact_1577_insert__eq__iff,axiom, % 0.30/0.52 ! [A: $tType,A2: A,A3: set @ A,B2: A,B3: set @ A] : % 0.30/0.52 ( ~ ( member @ A @ A2 @ A3 ) % 0.30/0.52 => ( ~ ( member @ A @ B2 @ B3 ) % 0.30/0.52 => ( ( ( insert @ A @ A2 @ A3 ) % 0.30/0.52 = ( insert @ A @ B2 @ B3 ) ) % 0.30/0.52 = ( ( ( A2 = B2 ) % 0.30/0.52 => ( A3 = B3 ) ) % 0.30/0.52 & ( ( A2 != B2 ) % 0.30/0.52 => ? [C7: set @ A] : % 0.30/0.52 ( ( A3 % 0.30/0.52 = ( insert @ A @ B2 @ C7 ) ) % 0.30/0.52 & ~ ( member @ A @ B2 @ C7 ) % 0.30/0.52 & ( B3 % 0.30/0.52 = ( insert @ A @ A2 @ C7 ) ) % 0.30/0.52 & ~ ( member @ A @ A2 @ C7 ) ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % insert_eq_iff % 0.30/0.52 thf(fact_1578_insert__absorb,axiom, % 0.30/0.52 ! [A: $tType,A2: A,A3: set @ A] : % 0.30/0.52 ( ( member @ A @ A2 @ A3 ) % 0.30/0.52 => ( ( insert @ A @ A2 @ A3 ) % 0.30/0.52 = A3 ) ) ). % 0.30/0.52 % 0.30/0.52 % insert_absorb % 0.30/0.52 thf(fact_1579_insert__ident,axiom, % 0.30/0.52 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] : % 0.30/0.52 ( ~ ( member @ A @ X3 @ A3 ) % 0.30/0.52 => ( ~ ( member @ A @ X3 @ B3 ) % 0.30/0.52 => ( ( ( insert @ A @ X3 @ A3 ) % 0.30/0.52 = ( insert @ A @ X3 @ B3 ) ) % 0.30/0.52 = ( A3 = B3 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % insert_ident % 0.30/0.52 thf(fact_1580_Set_Oset__insert,axiom, % 0.30/0.52 ! [A: $tType,X3: A,A3: set @ A] : % 0.30/0.52 ( ( member @ A @ X3 @ A3 ) % 0.30/0.52 => ~ ! [B10: set @ A] : % 0.30/0.52 ( ( A3 % 0.30/0.52 = ( insert @ A @ X3 @ B10 ) ) % 0.30/0.52 => ( member @ A @ X3 @ B10 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Set.set_insert % 0.30/0.52 thf(fact_1581_insertI2,axiom, % 0.30/0.52 ! [A: $tType,A2: A,B3: set @ A,B2: A] : % 0.30/0.52 ( ( member @ A @ A2 @ B3 ) % 0.30/0.52 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % insertI2 % 0.30/0.52 thf(fact_1582_insertI1,axiom, % 0.30/0.52 ! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ). % 0.30/0.52 % 0.30/0.52 % insertI1 % 0.30/0.52 thf(fact_1583_insertE,axiom, % 0.30/0.52 ! [A: $tType,A2: A,B2: A,A3: set @ A] : % 0.30/0.52 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) ) % 0.30/0.52 => ( ( A2 != B2 ) % 0.30/0.52 => ( member @ A @ A2 @ A3 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % insertE % 0.30/0.52 thf(fact_1584_of__bool__eq__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( zero_neq_one @ A ) % 0.30/0.52 => ! [P4: $o,Q4: $o] : % 0.30/0.52 ( ( ( zero_neq_one_of_bool @ A @ P4 ) % 0.30/0.52 = ( zero_neq_one_of_bool @ A @ Q4 ) ) % 0.30/0.52 = ( P4 = Q4 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % of_bool_eq_iff % 0.30/0.52 thf(fact_1585_diff__commute,axiom, % 0.30/0.52 ! [I: nat,J: nat,K: nat] : % 0.30/0.52 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K ) % 0.30/0.52 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_commute % 0.30/0.52 thf(fact_1586_diff__right__commute,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( cancel2418104881723323429up_add @ A ) % 0.30/0.52 => ! [A2: A,C2: A,B2: A] : % 0.30/0.52 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) % 0.30/0.52 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_right_commute % 0.30/0.52 thf(fact_1587_diff__eq__diff__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.52 ( ( ( minus_minus @ A @ A2 @ B2 ) % 0.30/0.52 = ( minus_minus @ A @ C2 @ D2 ) ) % 0.30/0.52 => ( ( A2 = B2 ) % 0.30/0.52 = ( C2 = D2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_eq_diff_eq % 0.30/0.52 thf(fact_1588_full__exhaustive__int_H_Ocases,axiom, % 0.30/0.52 ! [X3: product_prod @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] : % 0.30/0.52 ~ ! [F4: ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: int,I4: int] : % 0.30/0.52 ( X3 % 0.30/0.52 != ( product_Pair @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D3 @ I4 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % full_exhaustive_int'.cases % 0.30/0.52 thf(fact_1589_exhaustive__int_H_Ocases,axiom, % 0.30/0.52 ! [X3: product_prod @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] : % 0.30/0.52 ~ ! [F4: int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: int,I4: int] : % 0.30/0.52 ( X3 % 0.30/0.52 != ( product_Pair @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D3 @ I4 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % exhaustive_int'.cases % 0.30/0.52 thf(fact_1590_small__lazy_H_Ocases,axiom, % 0.30/0.52 ! [X3: product_prod @ int @ int] : % 0.30/0.52 ~ ! [D3: int,I4: int] : % 0.30/0.52 ( X3 % 0.30/0.52 != ( product_Pair @ int @ int @ D3 @ I4 ) ) ). % 0.30/0.52 % 0.30/0.52 % small_lazy'.cases % 0.30/0.52 thf(fact_1591_unique__quotient,axiom, % 0.30/0.52 ! [A2: int,B2: int,Q4: int,R2: int,Q6: int,R3: int] : % 0.30/0.52 ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 0.30/0.52 => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q6 @ R3 ) ) % 0.30/0.52 => ( Q4 = Q6 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % unique_quotient % 0.30/0.52 thf(fact_1592_unique__remainder,axiom, % 0.30/0.52 ! [A2: int,B2: int,Q4: int,R2: int,Q6: int,R3: int] : % 0.30/0.52 ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 0.30/0.52 => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q6 @ R3 ) ) % 0.30/0.52 => ( R2 = R3 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % unique_remainder % 0.30/0.52 thf(fact_1593_of__bool__conj,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_1 @ A ) % 0.30/0.52 => ! [P: $o,Q: $o] : % 0.30/0.52 ( ( zero_neq_one_of_bool @ A % 0.30/0.52 @ ( P % 0.30/0.52 & Q ) ) % 0.30/0.52 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % of_bool_conj % 0.30/0.52 thf(fact_1594_diff__mono,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A,D2: A,C2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( ord_less_eq @ A @ D2 @ C2 ) % 0.30/0.52 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_mono % 0.30/0.52 thf(fact_1595_diff__left__mono,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ! [B2: A,A2: A,C2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.30/0.52 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_left_mono % 0.30/0.52 thf(fact_1596_diff__right__mono,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.52 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_right_mono % 0.30/0.52 thf(fact_1597_diff__eq__diff__less__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.52 ( ( ( minus_minus @ A @ A2 @ B2 ) % 0.30/0.52 = ( minus_minus @ A @ C2 @ D2 ) ) % 0.30/0.52 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.52 = ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_eq_diff_less_eq % 0.30/0.52 thf(fact_1598_eq__iff__diff__eq__0,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( group_add @ A ) % 0.30/0.52 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) ) % 0.30/0.52 = ( ^ [A4: A,B4: A] : % 0.30/0.52 ( ( minus_minus @ A @ A4 @ B4 ) % 0.30/0.52 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % eq_iff_diff_eq_0 % 0.30/0.52 thf(fact_1599_diff__strict__mono,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A,D2: A,C2: A] : % 0.30/0.52 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( ord_less @ A @ D2 @ C2 ) % 0.30/0.52 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_strict_mono % 0.30/0.52 thf(fact_1600_diff__eq__diff__less,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.52 ( ( ( minus_minus @ A @ A2 @ B2 ) % 0.30/0.52 = ( minus_minus @ A @ C2 @ D2 ) ) % 0.30/0.52 => ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.52 = ( ord_less @ A @ C2 @ D2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_eq_diff_less % 0.30/0.52 thf(fact_1601_diff__strict__left__mono,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ! [B2: A,A2: A,C2: A] : % 0.30/0.52 ( ( ord_less @ A @ B2 @ A2 ) % 0.30/0.52 => ( ord_less @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_strict_left_mono % 0.30/0.52 thf(fact_1602_diff__strict__right__mono,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.52 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_strict_right_mono % 0.30/0.52 thf(fact_1603_right__diff__distrib_H,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( comm_s4317794764714335236cancel @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) ) % 0.30/0.52 = ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % right_diff_distrib' % 0.30/0.52 thf(fact_1604_left__diff__distrib_H,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( comm_s4317794764714335236cancel @ A ) % 0.30/0.52 => ! [B2: A,C2: A,A2: A] : % 0.30/0.52 ( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A2 ) % 0.30/0.52 = ( minus_minus @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % left_diff_distrib' % 0.30/0.52 thf(fact_1605_right__diff__distrib,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ring @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) ) % 0.30/0.52 = ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % right_diff_distrib % 0.30/0.52 thf(fact_1606_left__diff__distrib,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ring @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.52 = ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % left_diff_distrib % 0.30/0.52 thf(fact_1607_add__diff__add,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ab_group_add @ A ) % 0.30/0.52 => ! [A2: A,C2: A,B2: A,D2: A] : % 0.30/0.52 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) % 0.30/0.52 = ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( minus_minus @ A @ C2 @ D2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % add_diff_add % 0.30/0.52 thf(fact_1608_group__cancel_Osub1,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ab_group_add @ A ) % 0.30/0.52 => ! [A3: A,K: A,A2: A,B2: A] : % 0.30/0.52 ( ( A3 % 0.30/0.52 = ( plus_plus @ A @ K @ A2 ) ) % 0.30/0.52 => ( ( minus_minus @ A @ A3 @ B2 ) % 0.30/0.52 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % group_cancel.sub1 % 0.30/0.52 thf(fact_1609_diff__eq__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( ( minus_minus @ A @ A2 @ B2 ) % 0.30/0.52 = C2 ) % 0.30/0.52 = ( A2 % 0.30/0.52 = ( plus_plus @ A @ C2 @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_eq_eq % 0.30/0.52 thf(fact_1610_eq__diff__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( group_add @ A ) % 0.30/0.52 => ! [A2: A,C2: A,B2: A] : % 0.30/0.52 ( ( A2 % 0.30/0.52 = ( minus_minus @ A @ C2 @ B2 ) ) % 0.30/0.52 = ( ( plus_plus @ A @ A2 @ B2 ) % 0.30/0.52 = C2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % eq_diff_eq % 0.30/0.52 thf(fact_1611_add__diff__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) ) % 0.30/0.52 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % add_diff_eq % 0.30/0.52 thf(fact_1612_diff__diff__eq2,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) ) % 0.30/0.52 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_diff_eq2 % 0.30/0.52 thf(fact_1613_diff__add__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ab_group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.52 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_add_eq % 0.30/0.52 thf(fact_1614_diff__add__eq__diff__diff__swap,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) % 0.30/0.52 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_add_eq_diff_diff_swap % 0.30/0.52 thf(fact_1615_add__implies__diff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( cancel1802427076303600483id_add @ A ) % 0.30/0.52 => ! [C2: A,B2: A,A2: A] : % 0.30/0.52 ( ( ( plus_plus @ A @ C2 @ B2 ) % 0.30/0.52 = A2 ) % 0.30/0.52 => ( C2 % 0.30/0.52 = ( minus_minus @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % add_implies_diff % 0.30/0.52 thf(fact_1616_diff__diff__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( cancel2418104881723323429up_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.52 = ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_diff_eq % 0.30/0.52 thf(fact_1617_diff__divide__distrib,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( division_ring @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.52 = ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_divide_distrib % 0.30/0.52 thf(fact_1618_singletonD,axiom, % 0.30/0.52 ! [A: $tType,B2: A,A2: A] : % 0.30/0.52 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.30/0.52 => ( B2 = A2 ) ) ). % 0.30/0.52 % 0.30/0.52 % singletonD % 0.30/0.52 thf(fact_1619_singleton__iff,axiom, % 0.30/0.52 ! [A: $tType,B2: A,A2: A] : % 0.30/0.52 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.30/0.52 = ( B2 = A2 ) ) ). % 0.30/0.52 % 0.30/0.52 % singleton_iff % 0.30/0.52 thf(fact_1620_doubleton__eq__iff,axiom, % 0.30/0.52 ! [A: $tType,A2: A,B2: A,C2: A,D2: A] : % 0.30/0.52 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.30/0.52 = ( insert @ A @ C2 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.30/0.52 = ( ( ( A2 = C2 ) % 0.30/0.52 & ( B2 = D2 ) ) % 0.30/0.52 | ( ( A2 = D2 ) % 0.30/0.52 & ( B2 = C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % doubleton_eq_iff % 0.30/0.52 thf(fact_1621_insert__not__empty,axiom, % 0.30/0.52 ! [A: $tType,A2: A,A3: set @ A] : % 0.30/0.52 ( ( insert @ A @ A2 @ A3 ) % 0.30/0.52 != ( bot_bot @ ( set @ A ) ) ) ). % 0.30/0.52 % 0.30/0.52 % insert_not_empty % 0.30/0.52 thf(fact_1622_singleton__inject,axiom, % 0.30/0.52 ! [A: $tType,A2: A,B2: A] : % 0.30/0.52 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) % 0.30/0.52 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.30/0.52 => ( A2 = B2 ) ) ). % 0.30/0.52 % 0.30/0.52 % singleton_inject % 0.30/0.52 thf(fact_1623_finite_OinsertI,axiom, % 0.30/0.52 ! [A: $tType,A3: set @ A,A2: A] : % 0.30/0.52 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.52 => ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % finite.insertI % 0.30/0.52 thf(fact_1624_insert__subsetI,axiom, % 0.30/0.52 ! [A: $tType,X3: A,A3: set @ A,X8: set @ A] : % 0.30/0.52 ( ( member @ A @ X3 @ A3 ) % 0.30/0.52 => ( ( ord_less_eq @ ( set @ A ) @ X8 @ A3 ) % 0.30/0.52 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ X8 ) @ A3 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % insert_subsetI % 0.30/0.52 thf(fact_1625_insert__mono,axiom, % 0.30/0.52 ! [A: $tType,C5: set @ A,D6: set @ A,A2: A] : % 0.30/0.52 ( ( ord_less_eq @ ( set @ A ) @ C5 @ D6 ) % 0.30/0.52 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C5 ) @ ( insert @ A @ A2 @ D6 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % insert_mono % 0.30/0.52 thf(fact_1626_subset__insert,axiom, % 0.30/0.52 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] : % 0.30/0.52 ( ~ ( member @ A @ X3 @ A3 ) % 0.30/0.52 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) ) % 0.30/0.52 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % subset_insert % 0.30/0.52 thf(fact_1627_subset__insertI,axiom, % 0.30/0.52 ! [A: $tType,B3: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( insert @ A @ A2 @ B3 ) ) ). % 0.30/0.52 % 0.30/0.52 % subset_insertI % 0.30/0.52 thf(fact_1628_subset__insertI2,axiom, % 0.30/0.52 ! [A: $tType,A3: set @ A,B3: set @ A,B2: A] : % 0.30/0.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.52 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B3 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % subset_insertI2 % 0.30/0.52 thf(fact_1629_dvd__diff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( comm_ring_1 @ A ) % 0.30/0.52 => ! [X3: A,Y: A,Z2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ X3 @ Y ) % 0.30/0.52 => ( ( dvd_dvd @ A @ X3 @ Z2 ) % 0.30/0.52 => ( dvd_dvd @ A @ X3 @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_diff % 0.30/0.52 thf(fact_1630_zero__induct__lemma,axiom, % 0.30/0.52 ! [P: nat > $o,K: nat,I: nat] : % 0.30/0.52 ( ( P @ K ) % 0.30/0.52 => ( ! [N3: nat] : % 0.30/0.52 ( ( P @ ( suc @ N3 ) ) % 0.30/0.52 => ( P @ N3 ) ) % 0.30/0.52 => ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % zero_induct_lemma % 0.30/0.52 thf(fact_1631_minus__nat_Odiff__0,axiom, % 0.30/0.52 ! [M: nat] : % 0.30/0.52 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) ) % 0.30/0.52 = M ) ). % 0.30/0.52 % 0.30/0.52 % minus_nat.diff_0 % 0.30/0.52 thf(fact_1632_diffs0__imp__equal,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ( ( minus_minus @ nat @ M @ N2 ) % 0.30/0.52 = ( zero_zero @ nat ) ) % 0.30/0.52 => ( ( ( minus_minus @ nat @ N2 @ M ) % 0.30/0.52 = ( zero_zero @ nat ) ) % 0.30/0.52 => ( M = N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diffs0_imp_equal % 0.30/0.52 thf(fact_1633_diff__less__mono2,axiom, % 0.30/0.52 ! [M: nat,N2: nat,L: nat] : % 0.30/0.52 ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.52 => ( ( ord_less @ nat @ M @ L ) % 0.30/0.52 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_less_mono2 % 0.30/0.52 thf(fact_1634_less__imp__diff__less,axiom, % 0.30/0.52 ! [J: nat,K: nat,N2: nat] : % 0.30/0.52 ( ( ord_less @ nat @ J @ K ) % 0.30/0.52 => ( ord_less @ nat @ ( minus_minus @ nat @ J @ N2 ) @ K ) ) ). % 0.30/0.52 % 0.30/0.52 % less_imp_diff_less % 0.30/0.52 thf(fact_1635_dvd__minus__self,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ M ) ) % 0.30/0.52 = ( ( ord_less @ nat @ N2 @ M ) % 0.30/0.52 | ( dvd_dvd @ nat @ M @ N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_minus_self % 0.30/0.52 thf(fact_1636_eq__diff__iff,axiom, % 0.30/0.52 ! [K: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ K @ M ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ K @ N2 ) % 0.30/0.52 => ( ( ( minus_minus @ nat @ M @ K ) % 0.30/0.52 = ( minus_minus @ nat @ N2 @ K ) ) % 0.30/0.52 = ( M = N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % eq_diff_iff % 0.30/0.52 thf(fact_1637_le__diff__iff,axiom, % 0.30/0.52 ! [K: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ K @ M ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ K @ N2 ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) ) % 0.30/0.52 = ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % le_diff_iff % 0.30/0.52 thf(fact_1638_Nat_Odiff__diff__eq,axiom, % 0.30/0.52 ! [K: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ K @ M ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ K @ N2 ) % 0.30/0.52 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) ) % 0.30/0.52 = ( minus_minus @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Nat.diff_diff_eq % 0.30/0.52 thf(fact_1639_diff__le__mono,axiom, % 0.30/0.52 ! [M: nat,N2: nat,L: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.52 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N2 @ L ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_le_mono % 0.30/0.52 thf(fact_1640_diff__le__self,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ). % 0.30/0.52 % 0.30/0.52 % diff_le_self % 0.30/0.52 thf(fact_1641_le__diff__iff_H,axiom, % 0.30/0.52 ! [A2: nat,C2: nat,B2: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ A2 @ C2 ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ B2 @ C2 ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A2 ) @ ( minus_minus @ nat @ C2 @ B2 ) ) % 0.30/0.52 = ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % le_diff_iff' % 0.30/0.52 thf(fact_1642_diff__le__mono2,axiom, % 0.30/0.52 ! [M: nat,N2: nat,L: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.52 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_le_mono2 % 0.30/0.52 thf(fact_1643_dvd__diffD,axiom, % 0.30/0.52 ! [K: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ K @ N2 ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.52 => ( dvd_dvd @ nat @ K @ M ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_diffD % 0.30/0.52 thf(fact_1644_dvd__diffD1,axiom, % 0.30/0.52 ! [K: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ K @ M ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.52 => ( dvd_dvd @ nat @ K @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_diffD1 % 0.30/0.52 thf(fact_1645_less__eq__dvd__minus,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ M @ N2 ) % 0.30/0.52 = ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % less_eq_dvd_minus % 0.30/0.52 thf(fact_1646_Nat_Odiff__cancel,axiom, % 0.30/0.52 ! [K: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) ) % 0.30/0.52 = ( minus_minus @ nat @ M @ N2 ) ) ). % 0.30/0.52 % 0.30/0.52 % Nat.diff_cancel % 0.30/0.52 thf(fact_1647_diff__cancel2,axiom, % 0.30/0.52 ! [M: nat,K: nat,N2: nat] : % 0.30/0.52 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) % 0.30/0.52 = ( minus_minus @ nat @ M @ N2 ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_cancel2 % 0.30/0.52 thf(fact_1648_diff__add__inverse,axiom, % 0.30/0.52 ! [N2: nat,M: nat] : % 0.30/0.52 ( ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ N2 ) % 0.30/0.52 = M ) ). % 0.30/0.52 % 0.30/0.52 % diff_add_inverse % 0.30/0.52 thf(fact_1649_diff__add__inverse2,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ N2 ) % 0.30/0.52 = M ) ). % 0.30/0.52 % 0.30/0.52 % diff_add_inverse2 % 0.30/0.52 thf(fact_1650_diff__mult__distrib2,axiom, % 0.30/0.52 ! [K: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) ) % 0.30/0.52 = ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_mult_distrib2 % 0.30/0.52 thf(fact_1651_diff__mult__distrib,axiom, % 0.30/0.52 ! [M: nat,N2: nat,K: nat] : % 0.30/0.52 ( ( times_times @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K ) % 0.30/0.52 = ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_mult_distrib % 0.30/0.52 thf(fact_1652_bezout1__nat,axiom, % 0.30/0.52 ! [A2: nat,B2: nat] : % 0.30/0.52 ? [D3: nat,X5: nat,Y4: nat] : % 0.30/0.52 ( ( dvd_dvd @ nat @ D3 @ A2 ) % 0.30/0.52 & ( dvd_dvd @ nat @ D3 @ B2 ) % 0.30/0.52 & ( ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X5 ) @ ( times_times @ nat @ B2 @ Y4 ) ) % 0.30/0.52 = D3 ) % 0.30/0.52 | ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X5 ) @ ( times_times @ nat @ A2 @ Y4 ) ) % 0.30/0.52 = D3 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % bezout1_nat % 0.30/0.52 thf(fact_1653_eucl__rel__int__by0,axiom, % 0.30/0.52 ! [K: int] : ( eucl_rel_int @ K @ ( zero_zero @ int ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K ) ) ). % 0.30/0.52 % 0.30/0.52 % eucl_rel_int_by0 % 0.30/0.52 thf(fact_1654_div__int__unique,axiom, % 0.30/0.52 ! [K: int,L: int,Q4: int,R2: int] : % 0.30/0.52 ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 0.30/0.52 => ( ( divide_divide @ int @ K @ L ) % 0.30/0.52 = Q4 ) ) ). % 0.30/0.52 % 0.30/0.52 % div_int_unique % 0.30/0.52 thf(fact_1655_zero__less__eq__of__bool,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_semidom @ A ) % 0.30/0.52 => ! [P: $o] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ). % 0.30/0.52 % 0.30/0.52 % zero_less_eq_of_bool % 0.30/0.52 thf(fact_1656_of__bool__less__eq__one,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_semidom @ A ) % 0.30/0.52 => ! [P: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) ) ) ). % 0.30/0.52 % 0.30/0.52 % of_bool_less_eq_one % 0.30/0.52 thf(fact_1657_split__of__bool__asm,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( zero_neq_one @ A ) % 0.30/0.52 => ! [P: A > $o,P4: $o] : % 0.30/0.52 ( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) ) % 0.30/0.52 = ( ~ ( ( P4 % 0.30/0.52 & ~ ( P @ ( one_one @ A ) ) ) % 0.30/0.52 | ( ~ P4 % 0.30/0.52 & ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % split_of_bool_asm % 0.30/0.52 thf(fact_1658_split__of__bool,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( zero_neq_one @ A ) % 0.30/0.52 => ! [P: A > $o,P4: $o] : % 0.30/0.52 ( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) ) % 0.30/0.52 = ( ( P4 % 0.30/0.52 => ( P @ ( one_one @ A ) ) ) % 0.30/0.52 & ( ~ P4 % 0.30/0.52 => ( P @ ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % split_of_bool % 0.30/0.52 thf(fact_1659_of__bool__def,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( zero_neq_one @ A ) % 0.30/0.52 => ( ( zero_neq_one_of_bool @ A ) % 0.30/0.52 = ( ^ [P6: $o] : ( if @ A @ P6 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % of_bool_def % 0.30/0.52 thf(fact_1660_le__iff__diff__le__0,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ( ( ord_less_eq @ A ) % 0.30/0.52 = ( ^ [A4: A,B4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % le_iff_diff_le_0 % 0.30/0.52 thf(fact_1661_less__iff__diff__less__0,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ( ( ord_less @ A ) % 0.30/0.52 = ( ^ [A4: A,B4: A] : ( ord_less @ A @ ( minus_minus @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % less_iff_diff_less_0 % 0.30/0.52 thf(fact_1662_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordere1170586879665033532d_diff @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( ( minus_minus @ A @ B2 @ A2 ) % 0.30/0.52 = C2 ) % 0.30/0.52 = ( B2 % 0.30/0.52 = ( plus_plus @ A @ C2 @ A2 ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add % 0.30/0.52 thf(fact_1663_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordere1170586879665033532d_diff @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ A2 ) ) % 0.30/0.52 = B2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse % 0.30/0.52 thf(fact_1664_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordere1170586879665033532d_diff @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) % 0.30/0.52 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.diff_diff_right % 0.30/0.52 thf(fact_1665_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordere1170586879665033532d_diff @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) % 0.30/0.52 = ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2 % 0.30/0.52 thf(fact_1666_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordere1170586879665033532d_diff @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 ) % 0.30/0.52 = ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2 % 0.30/0.52 thf(fact_1667_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordere1170586879665033532d_diff @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 ) % 0.30/0.52 = ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc % 0.30/0.52 thf(fact_1668_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordere1170586879665033532d_diff @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) % 0.30/0.52 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc % 0.30/0.52 thf(fact_1669_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordere1170586879665033532d_diff @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) % 0.30/0.52 = ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2 % 0.30/0.52 thf(fact_1670_le__add__diff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordere1170586879665033532d_diff @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.52 => ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % le_add_diff % 0.30/0.52 thf(fact_1671_diff__add,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordere1170586879665033532d_diff @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ A2 ) % 0.30/0.52 = B2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_add % 0.30/0.52 thf(fact_1672_le__diff__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ! [A2: A,C2: A,B2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) ) % 0.30/0.52 = ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % le_diff_eq % 0.30/0.52 thf(fact_1673_diff__le__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.52 = ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_le_eq % 0.30/0.52 thf(fact_1674_add__le__add__imp__diff__le,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_semidom @ A ) % 0.30/0.52 => ! [I: A,K: A,N2: A,J: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N2 ) % 0.30/0.52 => ( ( ord_less_eq @ A @ N2 @ ( plus_plus @ A @ J @ K ) ) % 0.30/0.52 => ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N2 ) % 0.30/0.52 => ( ( ord_less_eq @ A @ N2 @ ( plus_plus @ A @ J @ K ) ) % 0.30/0.52 => ( ord_less_eq @ A @ ( minus_minus @ A @ N2 @ K ) @ J ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % add_le_add_imp_diff_le % 0.30/0.52 thf(fact_1675_add__le__imp__le__diff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_semidom @ A ) % 0.30/0.52 => ! [I: A,K: A,N2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N2 ) % 0.30/0.52 => ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N2 @ K ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % add_le_imp_le_diff % 0.30/0.52 thf(fact_1676_less__diff__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ! [A2: A,C2: A,B2: A] : % 0.30/0.52 ( ( ord_less @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) ) % 0.30/0.52 = ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % less_diff_eq % 0.30/0.52 thf(fact_1677_diff__less__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ab_group_add @ A ) % 0.30/0.52 => ! [A2: A,B2: A,C2: A] : % 0.30/0.52 ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 0.30/0.52 = ( ord_less @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_less_eq % 0.30/0.52 thf(fact_1678_linordered__semidom__class_Oadd__diff__inverse,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_semidom @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ~ ( ord_less @ A @ A2 @ B2 ) % 0.30/0.52 => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) ) % 0.30/0.52 = A2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % linordered_semidom_class.add_diff_inverse % 0.30/0.52 thf(fact_1679_mult__diff__mult,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ring @ A ) % 0.30/0.52 => ! [X3: A,Y: A,A2: A,B2: A] : % 0.30/0.52 ( ( minus_minus @ A @ ( times_times @ A @ X3 @ Y ) @ ( times_times @ A @ A2 @ B2 ) ) % 0.30/0.52 = ( plus_plus @ A @ ( times_times @ A @ X3 @ ( minus_minus @ A @ Y @ B2 ) ) @ ( times_times @ A @ ( minus_minus @ A @ X3 @ A2 ) @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mult_diff_mult % 0.30/0.52 thf(fact_1680_square__diff__square__factored,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( comm_ring @ A ) % 0.30/0.52 => ! [X3: A,Y: A] : % 0.30/0.52 ( ( minus_minus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) % 0.30/0.52 = ( times_times @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( minus_minus @ A @ X3 @ Y ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % square_diff_square_factored % 0.30/0.52 thf(fact_1681_eq__add__iff2,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ring @ A ) % 0.30/0.52 => ! [A2: A,E3: A,C2: A,B2: A,D2: A] : % 0.30/0.52 ( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) % 0.30/0.52 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) ) % 0.30/0.52 = ( C2 % 0.30/0.52 = ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % eq_add_iff2 % 0.30/0.52 thf(fact_1682_eq__add__iff1,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ring @ A ) % 0.30/0.52 => ! [A2: A,E3: A,C2: A,B2: A,D2: A] : % 0.30/0.52 ( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) % 0.30/0.52 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) ) % 0.30/0.52 = ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C2 ) % 0.30/0.52 = D2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % eq_add_iff1 % 0.30/0.52 thf(fact_1683_finite_Ocases,axiom, % 0.30/0.52 ! [A: $tType,A2: set @ A] : % 0.30/0.52 ( ( finite_finite2 @ A @ A2 ) % 0.30/0.52 => ( ( A2 % 0.30/0.52 != ( bot_bot @ ( set @ A ) ) ) % 0.30/0.52 => ~ ! [A8: set @ A] : % 0.30/0.52 ( ? [A5: A] : % 0.30/0.52 ( A2 % 0.30/0.52 = ( insert @ A @ A5 @ A8 ) ) % 0.30/0.52 => ~ ( finite_finite2 @ A @ A8 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % finite.cases % 0.30/0.52 thf(fact_1684_finite_Osimps,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( finite_finite2 @ A ) % 0.30/0.52 = ( ^ [A4: set @ A] : % 0.30/0.52 ( ( A4 % 0.30/0.52 = ( bot_bot @ ( set @ A ) ) ) % 0.30/0.52 | ? [A6: set @ A,B4: A] : % 0.30/0.52 ( ( A4 % 0.30/0.52 = ( insert @ A @ B4 @ A6 ) ) % 0.30/0.52 & ( finite_finite2 @ A @ A6 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % finite.simps % 0.30/0.52 thf(fact_1685_finite__induct,axiom, % 0.30/0.52 ! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] : % 0.30/0.52 ( ( finite_finite2 @ A @ F5 ) % 0.30/0.52 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 0.30/0.52 => ( ! [X5: A,F6: set @ A] : % 0.30/0.52 ( ( finite_finite2 @ A @ F6 ) % 0.30/0.52 => ( ~ ( member @ A @ X5 @ F6 ) % 0.30/0.52 => ( ( P @ F6 ) % 0.30/0.52 => ( P @ ( insert @ A @ X5 @ F6 ) ) ) ) ) % 0.30/0.52 => ( P @ F5 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % finite_induct % 0.30/0.52 thf(fact_1686_finite__ne__induct,axiom, % 0.30/0.52 ! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] : % 0.30/0.52 ( ( finite_finite2 @ A @ F5 ) % 0.30/0.52 => ( ( F5 % 0.30/0.52 != ( bot_bot @ ( set @ A ) ) ) % 0.30/0.52 => ( ! [X5: A] : ( P @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.30/0.52 => ( ! [X5: A,F6: set @ A] : % 0.30/0.52 ( ( finite_finite2 @ A @ F6 ) % 0.30/0.52 => ( ( F6 % 0.30/0.52 != ( bot_bot @ ( set @ A ) ) ) % 0.30/0.52 => ( ~ ( member @ A @ X5 @ F6 ) % 0.30/0.52 => ( ( P @ F6 ) % 0.30/0.52 => ( P @ ( insert @ A @ X5 @ F6 ) ) ) ) ) ) % 0.30/0.52 => ( P @ F5 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % finite_ne_induct % 0.30/0.52 thf(fact_1687_infinite__finite__induct,axiom, % 0.30/0.52 ! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] : % 0.30/0.52 ( ! [A8: set @ A] : % 0.30/0.52 ( ~ ( finite_finite2 @ A @ A8 ) % 0.30/0.52 => ( P @ A8 ) ) % 0.30/0.52 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 0.30/0.52 => ( ! [X5: A,F6: set @ A] : % 0.30/0.52 ( ( finite_finite2 @ A @ F6 ) % 0.30/0.52 => ( ~ ( member @ A @ X5 @ F6 ) % 0.30/0.52 => ( ( P @ F6 ) % 0.30/0.52 => ( P @ ( insert @ A @ X5 @ F6 ) ) ) ) ) % 0.30/0.52 => ( P @ A3 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % infinite_finite_induct % 0.30/0.52 thf(fact_1688_Euclidean__Division_Opos__mod__bound,axiom, % 0.30/0.52 ! [L: int,K: int] : % 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ L ) % 0.30/0.52 => ( ord_less @ int @ ( modulo_modulo @ int @ K @ L ) @ L ) ) ). % 0.30/0.52 % 0.30/0.52 % Euclidean_Division.pos_mod_bound % 0.30/0.52 thf(fact_1689_neg__mod__bound,axiom, % 0.30/0.52 ! [L: int,K: int] : % 0.30/0.52 ( ( ord_less @ int @ L @ ( zero_zero @ int ) ) % 0.30/0.52 => ( ord_less @ int @ L @ ( modulo_modulo @ int @ K @ L ) ) ) ). % 0.30/0.52 % 0.30/0.52 % neg_mod_bound % 0.30/0.52 thf(fact_1690_subset__singleton__iff,axiom, % 0.30/0.52 ! [A: $tType,X8: set @ A,A2: A] : % 0.30/0.52 ( ( ord_less_eq @ ( set @ A ) @ X8 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.30/0.52 = ( ( X8 % 0.30/0.52 = ( bot_bot @ ( set @ A ) ) ) % 0.30/0.52 | ( X8 % 0.30/0.52 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % subset_singleton_iff % 0.30/0.52 thf(fact_1691_subset__singletonD,axiom, % 0.30/0.52 ! [A: $tType,A3: set @ A,X3: A] : % 0.30/0.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.30/0.52 => ( ( A3 % 0.30/0.52 = ( bot_bot @ ( set @ A ) ) ) % 0.30/0.52 | ( A3 % 0.30/0.52 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % subset_singletonD % 0.30/0.52 thf(fact_1692_dvd__minus__mod,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semidom_modulo @ A ) % 0.30/0.52 => ! [B2: A,A2: A] : ( dvd_dvd @ A @ B2 @ ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_minus_mod % 0.30/0.52 thf(fact_1693_diff__less__Suc,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N2 ) @ ( suc @ M ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_less_Suc % 0.30/0.52 thf(fact_1694_Suc__diff__Suc,axiom, % 0.30/0.52 ! [N2: nat,M: nat] : % 0.30/0.52 ( ( ord_less @ nat @ N2 @ M ) % 0.30/0.52 => ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N2 ) ) ) % 0.30/0.52 = ( minus_minus @ nat @ M @ N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Suc_diff_Suc % 0.30/0.52 thf(fact_1695_diff__less,axiom, % 0.30/0.52 ! [N2: nat,M: nat] : % 0.30/0.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.52 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.30/0.52 => ( ord_less @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_less % 0.30/0.52 thf(fact_1696_Suc__diff__le,axiom, % 0.30/0.52 ! [N2: nat,M: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.52 => ( ( minus_minus @ nat @ ( suc @ M ) @ N2 ) % 0.30/0.52 = ( suc @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Suc_diff_le % 0.30/0.52 thf(fact_1697_diff__less__mono,axiom, % 0.30/0.52 ! [A2: nat,B2: nat,C2: nat] : % 0.30/0.52 ( ( ord_less @ nat @ A2 @ B2 ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ C2 @ A2 ) % 0.30/0.52 => ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_less_mono % 0.30/0.52 thf(fact_1698_less__diff__iff,axiom, % 0.30/0.52 ! [K: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ K @ M ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ K @ N2 ) % 0.30/0.52 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) ) % 0.30/0.52 = ( ord_less @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % less_diff_iff % 0.30/0.52 thf(fact_1699_diff__add__0,axiom, % 0.30/0.52 ! [N2: nat,M: nat] : % 0.30/0.52 ( ( minus_minus @ nat @ N2 @ ( plus_plus @ nat @ N2 @ M ) ) % 0.30/0.52 = ( zero_zero @ nat ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_add_0 % 0.30/0.52 thf(fact_1700_add__diff__inverse__nat,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ~ ( ord_less @ nat @ M @ N2 ) % 0.30/0.52 => ( ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ M @ N2 ) ) % 0.30/0.52 = M ) ) ). % 0.30/0.52 % 0.30/0.52 % add_diff_inverse_nat % 0.30/0.52 thf(fact_1701_less__diff__conv,axiom, % 0.30/0.52 ! [I: nat,J: nat,K: nat] : % 0.30/0.52 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) % 0.30/0.52 = ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ). % 0.30/0.52 % 0.30/0.52 % less_diff_conv % 0.30/0.52 thf(fact_1702_Nat_Ole__imp__diff__is__add,axiom, % 0.30/0.52 ! [I: nat,J: nat,K: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.52 => ( ( ( minus_minus @ nat @ J @ I ) % 0.30/0.52 = K ) % 0.30/0.52 = ( J % 0.30/0.52 = ( plus_plus @ nat @ K @ I ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Nat.le_imp_diff_is_add % 0.30/0.52 thf(fact_1703_Nat_Odiff__add__assoc2,axiom, % 0.30/0.52 ! [K: nat,J: nat,I: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ K @ J ) % 0.30/0.52 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) % 0.30/0.52 = ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Nat.diff_add_assoc2 % 0.30/0.52 thf(fact_1704_Nat_Odiff__add__assoc,axiom, % 0.30/0.52 ! [K: nat,J: nat,I: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ K @ J ) % 0.30/0.52 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) % 0.30/0.52 = ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Nat.diff_add_assoc % 0.30/0.52 thf(fact_1705_Nat_Ole__diff__conv2,axiom, % 0.30/0.52 ! [K: nat,J: nat,I: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ K @ J ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) % 0.30/0.52 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Nat.le_diff_conv2 % 0.30/0.52 thf(fact_1706_le__diff__conv,axiom, % 0.30/0.52 ! [J: nat,K: nat,I: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) % 0.30/0.52 = ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ). % 0.30/0.52 % 0.30/0.52 % le_diff_conv % 0.30/0.52 thf(fact_1707_diff__Suc__eq__diff__pred,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ( minus_minus @ nat @ M @ ( suc @ N2 ) ) % 0.30/0.52 = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N2 ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_Suc_eq_diff_pred % 0.30/0.52 thf(fact_1708_mod__geq,axiom, % 0.30/0.52 ! [M: nat,N2: nat] : % 0.30/0.52 ( ~ ( ord_less @ nat @ M @ N2 ) % 0.30/0.52 => ( ( modulo_modulo @ nat @ M @ N2 ) % 0.30/0.52 = ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_geq % 0.30/0.52 thf(fact_1709_mod__if,axiom, % 0.30/0.52 ( ( modulo_modulo @ nat ) % 0.30/0.52 = ( ^ [M2: nat,N: nat] : ( if @ nat @ ( ord_less @ nat @ M2 @ N ) @ M2 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_if % 0.30/0.52 thf(fact_1710_le__mod__geq,axiom, % 0.30/0.52 ! [N2: nat,M: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.52 => ( ( modulo_modulo @ nat @ M @ N2 ) % 0.30/0.52 = ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % le_mod_geq % 0.30/0.52 thf(fact_1711_mod__eq__dvd__iff__nat,axiom, % 0.30/0.52 ! [N2: nat,M: nat,Q4: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.52 => ( ( ( modulo_modulo @ nat @ M @ Q4 ) % 0.30/0.52 = ( modulo_modulo @ nat @ N2 @ Q4 ) ) % 0.30/0.52 = ( dvd_dvd @ nat @ Q4 @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_eq_dvd_iff_nat % 0.30/0.52 thf(fact_1712_ordered__ring__class_Ole__add__iff2,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ring @ A ) % 0.30/0.52 => ! [A2: A,E3: A,C2: A,B2: A,D2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) ) % 0.30/0.52 = ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % ordered_ring_class.le_add_iff2 % 0.30/0.52 thf(fact_1713_ordered__ring__class_Ole__add__iff1,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ring @ A ) % 0.30/0.52 => ! [A2: A,E3: A,C2: A,B2: A,D2: A] : % 0.30/0.52 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) ) % 0.30/0.52 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C2 ) @ D2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % ordered_ring_class.le_add_iff1 % 0.30/0.52 thf(fact_1714_less__add__iff2,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ring @ A ) % 0.30/0.52 => ! [A2: A,E3: A,C2: A,B2: A,D2: A] : % 0.30/0.52 ( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) ) % 0.30/0.52 = ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % less_add_iff2 % 0.30/0.52 thf(fact_1715_less__add__iff1,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ordered_ring @ A ) % 0.30/0.52 => ! [A2: A,E3: A,C2: A,B2: A,D2: A] : % 0.30/0.52 ( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) ) % 0.30/0.52 = ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C2 ) @ D2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % less_add_iff1 % 0.30/0.52 thf(fact_1716_divide__diff__eq__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( division_ring @ A ) % 0.30/0.52 => ! [Z2: A,X3: A,Y: A] : % 0.30/0.52 ( ( Z2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( minus_minus @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ Y ) % 0.30/0.52 = ( divide_divide @ A @ ( minus_minus @ A @ X3 @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % divide_diff_eq_iff % 0.30/0.52 thf(fact_1717_diff__divide__eq__iff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( division_ring @ A ) % 0.30/0.52 => ! [Z2: A,X3: A,Y: A] : % 0.30/0.52 ( ( Z2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( minus_minus @ A @ X3 @ ( divide_divide @ A @ Y @ Z2 ) ) % 0.30/0.52 = ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_divide_eq_iff % 0.30/0.52 thf(fact_1718_diff__frac__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( field @ A ) % 0.30/0.52 => ! [Y: A,Z2: A,X3: A,W2: A] : % 0.30/0.52 ( ( Y % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( Z2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( minus_minus @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) ) % 0.30/0.52 = ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_frac_eq % 0.30/0.52 thf(fact_1719_add__divide__eq__if__simps_I4_J,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( division_ring @ A ) % 0.30/0.52 => ! [Z2: A,A2: A,B2: A] : % 0.30/0.52 ( ( ( Z2 % 0.30/0.52 = ( zero_zero @ A ) ) % 0.30/0.52 => ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) ) % 0.30/0.52 = A2 ) ) % 0.30/0.52 & ( ( Z2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) ) % 0.30/0.52 = ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A2 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % add_divide_eq_if_simps(4) % 0.30/0.52 thf(fact_1720_square__diff__one__factored,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ring_1 @ A ) % 0.30/0.52 => ! [X3: A] : % 0.30/0.52 ( ( minus_minus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( one_one @ A ) ) % 0.30/0.52 = ( times_times @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) @ ( minus_minus @ A @ X3 @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % square_diff_one_factored % 0.30/0.52 thf(fact_1721_finite__ranking__induct,axiom, % 0.30/0.52 ! [A: $tType,B: $tType] : % 0.30/0.52 ( ( linorder @ A ) % 0.30/0.52 => ! [S3: set @ B,P: ( set @ B ) > $o,F2: B > A] : % 0.30/0.52 ( ( finite_finite2 @ B @ S3 ) % 0.30/0.52 => ( ( P @ ( bot_bot @ ( set @ B ) ) ) % 0.30/0.52 => ( ! [X5: B,S5: set @ B] : % 0.30/0.52 ( ( finite_finite2 @ B @ S5 ) % 0.30/0.52 => ( ! [Y5: B] : % 0.30/0.52 ( ( member @ B @ Y5 @ S5 ) % 0.30/0.52 => ( ord_less_eq @ A @ ( F2 @ Y5 ) @ ( F2 @ X5 ) ) ) % 0.30/0.52 => ( ( P @ S5 ) % 0.30/0.52 => ( P @ ( insert @ B @ X5 @ S5 ) ) ) ) ) % 0.30/0.52 => ( P @ S3 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % finite_ranking_induct % 0.30/0.52 thf(fact_1722_finite__linorder__min__induct,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linorder @ A ) % 0.30/0.52 => ! [A3: set @ A,P: ( set @ A ) > $o] : % 0.30/0.52 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.52 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 0.30/0.52 => ( ! [B5: A,A8: set @ A] : % 0.30/0.52 ( ( finite_finite2 @ A @ A8 ) % 0.30/0.52 => ( ! [X2: A] : % 0.30/0.52 ( ( member @ A @ X2 @ A8 ) % 0.30/0.52 => ( ord_less @ A @ B5 @ X2 ) ) % 0.30/0.52 => ( ( P @ A8 ) % 0.30/0.52 => ( P @ ( insert @ A @ B5 @ A8 ) ) ) ) ) % 0.30/0.52 => ( P @ A3 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % finite_linorder_min_induct % 0.30/0.52 thf(fact_1723_finite__linorder__max__induct,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linorder @ A ) % 0.30/0.52 => ! [A3: set @ A,P: ( set @ A ) > $o] : % 0.30/0.52 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.52 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 0.30/0.52 => ( ! [B5: A,A8: set @ A] : % 0.30/0.52 ( ( finite_finite2 @ A @ A8 ) % 0.30/0.52 => ( ! [X2: A] : % 0.30/0.52 ( ( member @ A @ X2 @ A8 ) % 0.30/0.52 => ( ord_less @ A @ X2 @ B5 ) ) % 0.30/0.52 => ( ( P @ A8 ) % 0.30/0.52 => ( P @ ( insert @ A @ B5 @ A8 ) ) ) ) ) % 0.30/0.52 => ( P @ A3 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % finite_linorder_max_induct % 0.30/0.52 thf(fact_1724_inf__period_I4_J,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ( comm_ring @ A ) % 0.30/0.52 & ( dvd @ A ) ) % 0.30/0.52 => ! [D2: A,D6: A,T2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ D2 @ D6 ) % 0.30/0.52 => ! [X2: A,K4: A] : % 0.30/0.52 ( ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X2 @ T2 ) ) ) % 0.30/0.52 = ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K4 @ D6 ) ) @ T2 ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % inf_period(4) % 0.30/0.52 thf(fact_1725_inf__period_I3_J,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( ( comm_ring @ A ) % 0.30/0.52 & ( dvd @ A ) ) % 0.30/0.52 => ! [D2: A,D6: A,T2: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ D2 @ D6 ) % 0.30/0.52 => ! [X2: A,K4: A] : % 0.30/0.52 ( ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X2 @ T2 ) ) % 0.30/0.52 = ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K4 @ D6 ) ) @ T2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % inf_period(3) % 0.30/0.52 thf(fact_1726_finite__subset__induct,axiom, % 0.30/0.52 ! [A: $tType,F5: set @ A,A3: set @ A,P: ( set @ A ) > $o] : % 0.30/0.52 ( ( finite_finite2 @ A @ F5 ) % 0.30/0.52 => ( ( ord_less_eq @ ( set @ A ) @ F5 @ A3 ) % 0.30/0.52 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 0.30/0.52 => ( ! [A5: A,F6: set @ A] : % 0.30/0.52 ( ( finite_finite2 @ A @ F6 ) % 0.30/0.52 => ( ( member @ A @ A5 @ A3 ) % 0.30/0.52 => ( ~ ( member @ A @ A5 @ F6 ) % 0.30/0.52 => ( ( P @ F6 ) % 0.30/0.52 => ( P @ ( insert @ A @ A5 @ F6 ) ) ) ) ) ) % 0.30/0.52 => ( P @ F5 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % finite_subset_induct % 0.30/0.52 thf(fact_1727_finite__subset__induct_H,axiom, % 0.30/0.52 ! [A: $tType,F5: set @ A,A3: set @ A,P: ( set @ A ) > $o] : % 0.30/0.52 ( ( finite_finite2 @ A @ F5 ) % 0.30/0.52 => ( ( ord_less_eq @ ( set @ A ) @ F5 @ A3 ) % 0.30/0.52 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 0.30/0.52 => ( ! [A5: A,F6: set @ A] : % 0.30/0.52 ( ( finite_finite2 @ A @ F6 ) % 0.30/0.52 => ( ( member @ A @ A5 @ A3 ) % 0.30/0.52 => ( ( ord_less_eq @ ( set @ A ) @ F6 @ A3 ) % 0.30/0.52 => ( ~ ( member @ A @ A5 @ F6 ) % 0.30/0.52 => ( ( P @ F6 ) % 0.30/0.52 => ( P @ ( insert @ A @ A5 @ F6 ) ) ) ) ) ) ) % 0.30/0.52 => ( P @ F5 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % finite_subset_induct' % 0.30/0.52 thf(fact_1728_minus__mult__div__eq__mod,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_modulo @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) % 0.30/0.52 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % minus_mult_div_eq_mod % 0.30/0.52 thf(fact_1729_minus__mod__eq__mult__div,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_modulo @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) % 0.30/0.52 = ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % minus_mod_eq_mult_div % 0.30/0.52 thf(fact_1730_minus__mod__eq__div__mult,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_modulo @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) % 0.30/0.52 = ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % minus_mod_eq_div_mult % 0.30/0.52 thf(fact_1731_minus__div__mult__eq__mod,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semiring_modulo @ A ) % 0.30/0.52 => ! [A2: A,B2: A] : % 0.30/0.52 ( ( minus_minus @ A @ A2 @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) ) % 0.30/0.52 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % minus_div_mult_eq_mod % 0.30/0.52 thf(fact_1732_eucl__rel__int__dividesI,axiom, % 0.30/0.52 ! [L: int,K: int,Q4: int] : % 0.30/0.52 ( ( L % 0.30/0.52 != ( zero_zero @ int ) ) % 0.30/0.52 => ( ( K % 0.30/0.52 = ( times_times @ int @ Q4 @ L ) ) % 0.30/0.52 => ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % eucl_rel_int_dividesI % 0.30/0.52 thf(fact_1733_Euclidean__Division_Opos__mod__sign,axiom, % 0.30/0.52 ! [L: int,K: int] : % 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ L ) % 0.30/0.52 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Euclidean_Division.pos_mod_sign % 0.30/0.52 thf(fact_1734_neg__mod__sign,axiom, % 0.30/0.52 ! [L: int,K: int] : % 0.30/0.52 ( ( ord_less @ int @ L @ ( zero_zero @ int ) ) % 0.30/0.52 => ( ord_less_eq @ int @ ( modulo_modulo @ int @ K @ L ) @ ( zero_zero @ int ) ) ) ). % 0.30/0.52 % 0.30/0.52 % neg_mod_sign % 0.30/0.52 thf(fact_1735_neg__mod__conj,axiom, % 0.30/0.52 ! [B2: int,A2: int] : % 0.30/0.52 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) ) % 0.30/0.52 => ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) % 0.30/0.52 & ( ord_less @ int @ B2 @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % neg_mod_conj % 0.30/0.52 thf(fact_1736_pos__mod__conj,axiom, % 0.30/0.52 ! [B2: int,A2: int] : % 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.30/0.52 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A2 @ B2 ) ) % 0.30/0.52 & ( ord_less @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % pos_mod_conj % 0.30/0.52 thf(fact_1737_zmod__trivial__iff,axiom, % 0.30/0.52 ! [I: int,K: int] : % 0.30/0.52 ( ( ( modulo_modulo @ int @ I @ K ) % 0.30/0.52 = I ) % 0.30/0.52 = ( ( K % 0.30/0.52 = ( zero_zero @ int ) ) % 0.30/0.52 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I ) % 0.30/0.52 & ( ord_less @ int @ I @ K ) ) % 0.30/0.52 | ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) % 0.30/0.52 & ( ord_less @ int @ K @ I ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % zmod_trivial_iff % 0.30/0.52 thf(fact_1738_mod__int__pos__iff,axiom, % 0.30/0.52 ! [K: int,L: int] : % 0.30/0.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) ) % 0.30/0.52 = ( ( dvd_dvd @ int @ L @ K ) % 0.30/0.52 | ( ( L % 0.30/0.52 = ( zero_zero @ int ) ) % 0.30/0.52 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) % 0.30/0.52 | ( ord_less @ int @ ( zero_zero @ int ) @ L ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_int_pos_iff % 0.30/0.52 thf(fact_1739_diff__Suc__less,axiom, % 0.30/0.52 ! [N2: nat,I: nat] : % 0.30/0.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.52 => ( ord_less @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ). % 0.30/0.52 % 0.30/0.52 % diff_Suc_less % 0.30/0.52 thf(fact_1740_zdiv__mono__strict,axiom, % 0.30/0.52 ! [A3: int,B3: int,N2: int] : % 0.30/0.52 ( ( ord_less @ int @ A3 @ B3 ) % 0.30/0.52 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 ) % 0.30/0.52 => ( ( ( modulo_modulo @ int @ A3 @ N2 ) % 0.30/0.52 = ( zero_zero @ int ) ) % 0.30/0.52 => ( ( ( modulo_modulo @ int @ B3 @ N2 ) % 0.30/0.52 = ( zero_zero @ int ) ) % 0.30/0.52 => ( ord_less @ int @ ( divide_divide @ int @ A3 @ N2 ) @ ( divide_divide @ int @ B3 @ N2 ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % zdiv_mono_strict % 0.30/0.52 thf(fact_1741_nat__diff__split__asm,axiom, % 0.30/0.52 ! [P: nat > $o,A2: nat,B2: nat] : % 0.30/0.52 ( ( P @ ( minus_minus @ nat @ A2 @ B2 ) ) % 0.30/0.52 = ( ~ ( ( ( ord_less @ nat @ A2 @ B2 ) % 0.30/0.52 & ~ ( P @ ( zero_zero @ nat ) ) ) % 0.30/0.52 | ? [D5: nat] : % 0.30/0.52 ( ( A2 % 0.30/0.52 = ( plus_plus @ nat @ B2 @ D5 ) ) % 0.30/0.52 & ~ ( P @ D5 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % nat_diff_split_asm % 0.30/0.52 thf(fact_1742_nat__diff__split,axiom, % 0.30/0.52 ! [P: nat > $o,A2: nat,B2: nat] : % 0.30/0.52 ( ( P @ ( minus_minus @ nat @ A2 @ B2 ) ) % 0.30/0.52 = ( ( ( ord_less @ nat @ A2 @ B2 ) % 0.30/0.52 => ( P @ ( zero_zero @ nat ) ) ) % 0.30/0.52 & ! [D5: nat] : % 0.30/0.52 ( ( A2 % 0.30/0.52 = ( plus_plus @ nat @ B2 @ D5 ) ) % 0.30/0.52 => ( P @ D5 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % nat_diff_split % 0.30/0.52 thf(fact_1743_less__diff__conv2,axiom, % 0.30/0.52 ! [K: nat,J: nat,I: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ K @ J ) % 0.30/0.52 => ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) % 0.30/0.52 = ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % less_diff_conv2 % 0.30/0.52 thf(fact_1744_nat__diff__add__eq2,axiom, % 0.30/0.52 ! [I: nat,J: nat,U: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.52 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 0.30/0.52 = ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % nat_diff_add_eq2 % 0.30/0.52 thf(fact_1745_nat__diff__add__eq1,axiom, % 0.30/0.52 ! [J: nat,I: nat,U: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ J @ I ) % 0.30/0.52 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 0.30/0.52 = ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % nat_diff_add_eq1 % 0.30/0.52 thf(fact_1746_nat__le__add__iff2,axiom, % 0.30/0.52 ! [I: nat,J: nat,U: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 0.30/0.52 = ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % nat_le_add_iff2 % 0.30/0.52 thf(fact_1747_nat__le__add__iff1,axiom, % 0.30/0.52 ! [J: nat,I: nat,U: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ J @ I ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 0.30/0.52 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % nat_le_add_iff1 % 0.30/0.52 thf(fact_1748_nat__eq__add__iff2,axiom, % 0.30/0.52 ! [I: nat,J: nat,U: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.52 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) % 0.30/0.52 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 0.30/0.52 = ( M % 0.30/0.52 = ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % nat_eq_add_iff2 % 0.30/0.52 thf(fact_1749_nat__eq__add__iff1,axiom, % 0.30/0.52 ! [J: nat,I: nat,U: nat,M: nat,N2: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ J @ I ) % 0.30/0.52 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) % 0.30/0.52 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 0.30/0.52 = ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) % 0.30/0.52 = N2 ) ) ) ). % 0.30/0.52 % 0.30/0.52 % nat_eq_add_iff1 % 0.30/0.52 thf(fact_1750_dvd__minus__add,axiom, % 0.30/0.52 ! [Q4: nat,N2: nat,R2: nat,M: nat] : % 0.30/0.52 ( ( ord_less_eq @ nat @ Q4 @ N2 ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ Q4 @ ( times_times @ nat @ R2 @ M ) ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ Q4 ) ) % 0.30/0.52 = ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ ( times_times @ nat @ R2 @ M ) @ Q4 ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % dvd_minus_add % 0.30/0.52 thf(fact_1751_set__update__subset__insert,axiom, % 0.30/0.52 ! [A: $tType,Xs: list @ A,I: nat,X3: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X3 ) ) @ ( insert @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ). % 0.30/0.52 % 0.30/0.52 % set_update_subset_insert % 0.30/0.52 thf(fact_1752_mod__nat__eqI,axiom, % 0.30/0.52 ! [R2: nat,N2: nat,M: nat] : % 0.30/0.52 ( ( ord_less @ nat @ R2 @ N2 ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ R2 @ M ) % 0.30/0.52 => ( ( dvd_dvd @ nat @ N2 @ ( minus_minus @ nat @ M @ R2 ) ) % 0.30/0.52 => ( ( modulo_modulo @ nat @ M @ N2 ) % 0.30/0.52 = R2 ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_nat_eqI % 0.30/0.52 thf(fact_1753_exp__div__exp__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( bit_semiring_bits @ A ) % 0.30/0.52 => ! [M: nat,N2: nat] : % 0.30/0.52 ( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.52 = ( times_times @ A % 0.30/0.52 @ ( zero_neq_one_of_bool @ A % 0.30/0.52 @ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 & ( ord_less_eq @ nat @ N2 @ M ) ) ) % 0.30/0.52 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % exp_div_exp_eq % 0.30/0.52 thf(fact_1754_frac__le__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_field @ A ) % 0.30/0.52 => ! [Y: A,Z2: A,X3: A,W2: A] : % 0.30/0.52 ( ( Y % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( Z2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) ) % 0.30/0.52 = ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % frac_le_eq % 0.30/0.52 thf(fact_1755_frac__less__eq,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( linordered_field @ A ) % 0.30/0.52 => ! [Y: A,Z2: A,X3: A,W2: A] : % 0.30/0.52 ( ( Y % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( Z2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) ) % 0.30/0.52 = ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % frac_less_eq % 0.30/0.52 thf(fact_1756_signed__take__bit__int__less__exp,axiom, % 0.30/0.52 ! [N2: nat,K: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 0.30/0.52 % 0.30/0.52 % signed_take_bit_int_less_exp % 0.30/0.52 thf(fact_1757_power__diff,axiom, % 0.30/0.52 ! [A: $tType] : % 0.30/0.52 ( ( semidom_divide @ A ) % 0.30/0.52 => ! [A2: A,N2: nat,M: nat] : % 0.30/0.52 ( ( A2 % 0.30/0.52 != ( zero_zero @ A ) ) % 0.30/0.52 => ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.52 => ( ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N2 ) ) % 0.30/0.52 = ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % power_diff % 0.30/0.52 thf(fact_1758_mod__pos__neg__trivial,axiom, % 0.30/0.52 ! [K: int,L: int] : % 0.30/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.52 => ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) ) % 0.30/0.52 => ( ( modulo_modulo @ int @ K @ L ) % 0.30/0.52 = ( plus_plus @ int @ K @ L ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % mod_pos_neg_trivial % 0.30/0.52 thf(fact_1759_Suc__diff__eq__diff__pred,axiom, % 0.30/0.52 ! [N2: nat,M: nat] : % 0.30/0.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.52 => ( ( minus_minus @ nat @ ( suc @ M ) @ N2 ) % 0.30/0.52 = ( minus_minus @ nat @ M @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Suc_diff_eq_diff_pred % 0.30/0.52 thf(fact_1760_Suc__pred_H,axiom, % 0.30/0.52 ! [N2: nat] : % 0.30/0.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.52 => ( N2 % 0.30/0.52 = ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % Suc_pred' % 0.30/0.52 thf(fact_1761_div__geq,axiom, % 0.30/0.52 ! [N2: nat,M: nat] : % 0.30/0.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.52 => ( ~ ( ord_less @ nat @ M @ N2 ) % 0.30/0.52 => ( ( divide_divide @ nat @ M @ N2 ) % 0.30/0.52 = ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % div_geq % 0.30/0.52 thf(fact_1762_div__if,axiom, % 0.30/0.52 ( ( divide_divide @ nat ) % 0.30/0.52 = ( ^ [M2: nat,N: nat] : % 0.30/0.52 ( if @ nat % 0.30/0.52 @ ( ( ord_less @ nat @ M2 @ N ) % 0.30/0.52 | ( N % 0.30/0.52 = ( zero_zero @ nat ) ) ) % 0.30/0.52 @ ( zero_zero @ nat ) % 0.30/0.52 @ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % div_if % 0.30/0.52 thf(fact_1763_set__replicate__Suc,axiom, % 0.30/0.52 ! [A: $tType,N2: nat,X3: A] : % 0.30/0.52 ( ( set2 @ A @ ( replicate @ A @ ( suc @ N2 ) @ X3 ) ) % 0.30/0.52 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % set_replicate_Suc % 0.30/0.52 thf(fact_1764_add__eq__if,axiom, % 0.30/0.52 ( ( plus_plus @ nat ) % 0.30/0.52 = ( ^ [M2: nat,N: nat] : % 0.30/0.52 ( if @ nat % 0.30/0.52 @ ( M2 % 0.30/0.52 = ( zero_zero @ nat ) ) % 0.30/0.52 @ N % 0.30/0.52 @ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N ) ) ) ) ) ). % 0.30/0.52 % 0.30/0.52 % add_eq_if % 0.30/0.52 thf(fact_1765_set__replicate__conv__if,axiom, % 0.30/0.52 ! [A: $tType,N2: nat,X3: A] : % 0.30/0.53 ( ( ( N2 % 0.30/0.53 = ( zero_zero @ nat ) ) % 0.30/0.53 => ( ( set2 @ A @ ( replicate @ A @ N2 @ X3 ) ) % 0.30/0.53 = ( bot_bot @ ( set @ A ) ) ) ) % 0.30/0.53 & ( ( N2 % 0.30/0.53 != ( zero_zero @ nat ) ) % 0.30/0.53 => ( ( set2 @ A @ ( replicate @ A @ N2 @ X3 ) ) % 0.30/0.53 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % set_replicate_conv_if % 0.30/0.53 thf(fact_1766_nat__less__add__iff2,axiom, % 0.30/0.53 ! [I: nat,J: nat,U: nat,M: nat,N2: nat] : % 0.30/0.53 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.53 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 0.30/0.53 = ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % nat_less_add_iff2 % 0.30/0.53 thf(fact_1767_nat__less__add__iff1,axiom, % 0.30/0.53 ! [J: nat,I: nat,U: nat,M: nat,N2: nat] : % 0.30/0.53 ( ( ord_less_eq @ nat @ J @ I ) % 0.30/0.53 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 0.30/0.53 = ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % nat_less_add_iff1 % 0.30/0.53 thf(fact_1768_mult__eq__if,axiom, % 0.30/0.53 ( ( times_times @ nat ) % 0.30/0.53 = ( ^ [M2: nat,N: nat] : % 0.30/0.53 ( if @ nat % 0.30/0.53 @ ( M2 % 0.30/0.53 = ( zero_zero @ nat ) ) % 0.30/0.53 @ ( zero_zero @ nat ) % 0.30/0.53 @ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % mult_eq_if % 0.30/0.53 thf(fact_1769_scaling__mono,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [U: A,V2: A,R2: A,S: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ U @ V2 ) % 0.30/0.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 ) % 0.30/0.53 => ( ( ord_less_eq @ A @ R2 @ S ) % 0.30/0.53 => ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R2 @ ( minus_minus @ A @ V2 @ U ) ) @ S ) ) @ V2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % scaling_mono % 0.30/0.53 thf(fact_1770_exp__not__zero__imp__exp__diff__not__zero,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_semiring_bits @ A ) % 0.30/0.53 => ! [N2: nat,M: nat] : % 0.30/0.53 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) % 0.30/0.53 != ( zero_zero @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % exp_not_zero_imp_exp_diff_not_zero % 0.30/0.53 thf(fact_1771_signed__take__bit__int__greater__eq__self__iff,axiom, % 0.30/0.53 ! [K: int,N2: nat] : % 0.30/0.53 ( ( ord_less_eq @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) ) % 0.30/0.53 = ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % signed_take_bit_int_greater_eq_self_iff % 0.30/0.53 thf(fact_1772_signed__take__bit__int__less__self__iff,axiom, % 0.30/0.53 ! [N2: nat,K: int] : % 0.30/0.53 ( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ K ) % 0.30/0.53 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ). % 0.30/0.53 % 0.30/0.53 % signed_take_bit_int_less_self_iff % 0.30/0.53 thf(fact_1773_power__diff__power__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( euclid4440199948858584721cancel @ A ) % 0.30/0.53 => ! [A2: A,N2: nat,M: nat] : % 0.30/0.53 ( ( A2 % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.53 => ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) % 0.30/0.53 = ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N2 ) ) ) ) % 0.30/0.53 & ( ~ ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.53 => ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) % 0.30/0.53 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % power_diff_power_eq % 0.30/0.53 thf(fact_1774_split__zmod,axiom, % 0.30/0.53 ! [P: int > $o,N2: int,K: int] : % 0.30/0.53 ( ( P @ ( modulo_modulo @ int @ N2 @ K ) ) % 0.30/0.53 = ( ( ( K % 0.30/0.53 = ( zero_zero @ int ) ) % 0.30/0.53 => ( P @ N2 ) ) % 0.30/0.53 & ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.53 => ! [I3: int,J3: int] : % 0.30/0.53 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 ) % 0.30/0.53 & ( ord_less @ int @ J3 @ K ) % 0.30/0.53 & ( N2 % 0.30/0.53 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) ) % 0.30/0.53 => ( P @ J3 ) ) ) % 0.30/0.53 & ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 0.30/0.53 => ! [I3: int,J3: int] : % 0.30/0.53 ( ( ( ord_less @ int @ K @ J3 ) % 0.30/0.53 & ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) ) % 0.30/0.53 & ( N2 % 0.30/0.53 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) ) % 0.30/0.53 => ( P @ J3 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % split_zmod % 0.30/0.53 thf(fact_1775_int__mod__neg__eq,axiom, % 0.30/0.53 ! [A2: int,B2: int,Q4: int,R2: int] : % 0.30/0.53 ( ( A2 % 0.30/0.53 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) ) % 0.30/0.53 => ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) ) % 0.30/0.53 => ( ( ord_less @ int @ B2 @ R2 ) % 0.30/0.53 => ( ( modulo_modulo @ int @ A2 @ B2 ) % 0.30/0.53 = R2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % int_mod_neg_eq % 0.30/0.53 thf(fact_1776_int__mod__pos__eq,axiom, % 0.30/0.53 ! [A2: int,B2: int,Q4: int,R2: int] : % 0.30/0.53 ( ( A2 % 0.30/0.53 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) ) % 0.30/0.53 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 ) % 0.30/0.53 => ( ( ord_less @ int @ R2 @ B2 ) % 0.30/0.53 => ( ( modulo_modulo @ int @ A2 @ B2 ) % 0.30/0.53 = R2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % int_mod_pos_eq % 0.30/0.53 thf(fact_1777_power__eq__if,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( power @ A ) % 0.30/0.53 => ( ( power_power @ A ) % 0.30/0.53 = ( ^ [P6: A,M2: nat] : % 0.30/0.53 ( if @ A % 0.30/0.53 @ ( M2 % 0.30/0.53 = ( zero_zero @ nat ) ) % 0.30/0.53 @ ( one_one @ A ) % 0.30/0.53 @ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % power_eq_if % 0.30/0.53 thf(fact_1778_power__minus__mult,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( monoid_mult @ A ) % 0.30/0.53 => ! [N2: nat,A2: A] : % 0.30/0.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.53 => ( ( times_times @ A @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ A2 ) % 0.30/0.53 = ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % power_minus_mult % 0.30/0.53 thf(fact_1779_diff__le__diff__pow,axiom, % 0.30/0.53 ! [K: nat,M: nat,N2: nat] : % 0.30/0.53 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) % 0.30/0.53 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N2 ) @ ( minus_minus @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % diff_le_diff_pow % 0.30/0.53 thf(fact_1780_le__div__geq,axiom, % 0.30/0.53 ! [N2: nat,M: nat] : % 0.30/0.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.53 => ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.53 => ( ( divide_divide @ nat @ M @ N2 ) % 0.30/0.53 = ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % le_div_geq % 0.30/0.53 thf(fact_1781_bits__induct,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_semiring_bits @ A ) % 0.30/0.53 => ! [P: A > $o,A2: A] : % 0.30/0.53 ( ! [A5: A] : % 0.30/0.53 ( ( ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.53 = A5 ) % 0.30/0.53 => ( P @ A5 ) ) % 0.30/0.53 => ( ! [A5: A,B5: $o] : % 0.30/0.53 ( ( P @ A5 ) % 0.30/0.53 => ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.53 = A5 ) % 0.30/0.53 => ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) ) ) ) % 0.30/0.53 => ( P @ A2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % bits_induct % 0.30/0.53 thf(fact_1782_verit__le__mono__div__int,axiom, % 0.30/0.53 ! [A3: int,B3: int,N2: int] : % 0.30/0.53 ( ( ord_less @ int @ A3 @ B3 ) % 0.30/0.53 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 ) % 0.30/0.53 => ( ord_less_eq @ int % 0.30/0.53 @ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ N2 ) % 0.30/0.53 @ ( if @ int % 0.30/0.53 @ ( ( modulo_modulo @ int @ B3 @ N2 ) % 0.30/0.53 = ( zero_zero @ int ) ) % 0.30/0.53 @ ( one_one @ int ) % 0.30/0.53 @ ( zero_zero @ int ) ) ) % 0.30/0.53 @ ( divide_divide @ int @ B3 @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % verit_le_mono_div_int % 0.30/0.53 thf(fact_1783_split__pos__lemma,axiom, % 0.30/0.53 ! [K: int,P: int > int > $o,N2: int] : % 0.30/0.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.53 => ( ( P @ ( divide_divide @ int @ N2 @ K ) @ ( modulo_modulo @ int @ N2 @ K ) ) % 0.30/0.53 = ( ! [I3: int,J3: int] : % 0.30/0.53 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 ) % 0.30/0.53 & ( ord_less @ int @ J3 @ K ) % 0.30/0.53 & ( N2 % 0.30/0.53 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) ) % 0.30/0.53 => ( P @ I3 @ J3 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % split_pos_lemma % 0.30/0.53 thf(fact_1784_split__neg__lemma,axiom, % 0.30/0.53 ! [K: int,P: int > int > $o,N2: int] : % 0.30/0.53 ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 0.30/0.53 => ( ( P @ ( divide_divide @ int @ N2 @ K ) @ ( modulo_modulo @ int @ N2 @ K ) ) % 0.30/0.53 = ( ! [I3: int,J3: int] : % 0.30/0.53 ( ( ( ord_less @ int @ K @ J3 ) % 0.30/0.53 & ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) ) % 0.30/0.53 & ( N2 % 0.30/0.53 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) ) % 0.30/0.53 => ( P @ I3 @ J3 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % split_neg_lemma % 0.30/0.53 thf(fact_1785_int__power__div__base,axiom, % 0.30/0.53 ! [M: nat,K: int] : % 0.30/0.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.30/0.53 => ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.53 => ( ( divide_divide @ int @ ( power_power @ int @ K @ M ) @ K ) % 0.30/0.53 = ( power_power @ int @ K @ ( minus_minus @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % int_power_div_base % 0.30/0.53 thf(fact_1786_set__decode__plus__power__2,axiom, % 0.30/0.53 ! [N2: nat,Z2: nat] : % 0.30/0.53 ( ~ ( member @ nat @ N2 @ ( nat_set_decode @ Z2 ) ) % 0.30/0.53 => ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ Z2 ) ) % 0.30/0.53 = ( insert @ nat @ N2 @ ( nat_set_decode @ Z2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % set_decode_plus_power_2 % 0.30/0.53 thf(fact_1787_exp__mod__exp,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( euclid5411537665997757685th_nat @ A ) % 0.30/0.53 => ! [M: nat,N2: nat] : % 0.30/0.53 ( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.53 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M @ N2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % exp_mod_exp % 0.30/0.53 thf(fact_1788_power2__diff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( comm_ring_1 @ A ) % 0.30/0.53 => ! [X3: A,Y: A] : % 0.30/0.53 ( ( power_power @ A @ ( minus_minus @ A @ X3 @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.53 = ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) @ Y ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % power2_diff % 0.30/0.53 thf(fact_1789_mult__exp__mod__exp__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_semiring_bits @ A ) % 0.30/0.53 => ! [M: nat,N2: nat,A2: A] : % 0.30/0.53 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.53 => ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.53 = ( times_times @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % mult_exp_mod_exp_eq % 0.30/0.53 thf(fact_1790_even__mod__4__div__2,axiom, % 0.30/0.53 ! [N2: nat] : % 0.30/0.53 ( ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) % 0.30/0.53 = ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.53 => ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % even_mod_4_div_2 % 0.30/0.53 thf(fact_1791_eucl__rel__int__iff,axiom, % 0.30/0.53 ! [K: int,L: int,Q4: int,R2: int] : % 0.30/0.53 ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 0.30/0.53 = ( ( K % 0.30/0.53 = ( plus_plus @ int @ ( times_times @ int @ L @ Q4 ) @ R2 ) ) % 0.30/0.53 & ( ( ord_less @ int @ ( zero_zero @ int ) @ L ) % 0.30/0.53 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 ) % 0.30/0.53 & ( ord_less @ int @ R2 @ L ) ) ) % 0.30/0.53 & ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L ) % 0.30/0.53 => ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) ) % 0.30/0.53 => ( ( ord_less @ int @ L @ R2 ) % 0.30/0.53 & ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) ) ) ) % 0.30/0.53 & ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) ) % 0.30/0.53 => ( Q4 % 0.30/0.53 = ( zero_zero @ int ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % eucl_rel_int_iff % 0.30/0.53 thf(fact_1792_divmod__digit__1_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.53 => ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) % 0.30/0.53 => ( ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 ) % 0.30/0.53 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % divmod_digit_1(2) % 0.30/0.53 thf(fact_1793_even__mask__div__iff_H,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( euclid5411537665997757685th_nat @ A ) % 0.30/0.53 => ! [M: nat,N2: nat] : % 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 0.30/0.53 = ( ord_less_eq @ nat @ M @ N2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % even_mask_div_iff' % 0.30/0.53 thf(fact_1794_even__mask__div__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_semiring_bits @ A ) % 0.30/0.53 => ! [M: nat,N2: nat] : % 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 0.30/0.53 = ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.53 = ( zero_zero @ A ) ) % 0.30/0.53 | ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % even_mask_div_iff % 0.30/0.53 thf(fact_1795_divmod__step__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.53 => ! [L: num,R2: A,Q4: A] : % 0.30/0.53 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 ) % 0.30/0.53 => ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R2 ) ) % 0.30/0.53 = ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R2 @ ( numeral_numeral @ A @ L ) ) ) ) ) % 0.30/0.53 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 ) % 0.30/0.53 => ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R2 ) ) % 0.30/0.53 = ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ R2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % divmod_step_eq % 0.30/0.53 thf(fact_1796_inrange,axiom, % 0.30/0.53 ! [T2: vEBT_VEBT,N2: nat] : % 0.30/0.53 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 0.30/0.53 => ( ord_less_eq @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % inrange % 0.30/0.53 thf(fact_1797_Suc__if__eq,axiom, % 0.30/0.53 ! [A: $tType,F2: nat > A,H2: nat > A,G: A,N2: nat] : % 0.30/0.53 ( ! [N3: nat] : % 0.30/0.53 ( ( F2 @ ( suc @ N3 ) ) % 0.30/0.53 = ( H2 @ N3 ) ) % 0.30/0.53 => ( ( ( F2 @ ( zero_zero @ nat ) ) % 0.30/0.53 = G ) % 0.30/0.53 => ( ( ( N2 % 0.30/0.53 = ( zero_zero @ nat ) ) % 0.30/0.53 => ( ( F2 @ N2 ) % 0.30/0.53 = G ) ) % 0.30/0.53 & ( ( N2 % 0.30/0.53 != ( zero_zero @ nat ) ) % 0.30/0.53 => ( ( F2 @ N2 ) % 0.30/0.53 = ( H2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Suc_if_eq % 0.30/0.53 thf(fact_1798_Divides_Oadjust__div__eq,axiom, % 0.30/0.53 ! [Q4: int,R2: int] : % 0.30/0.53 ( ( adjust_div @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 0.30/0.53 = ( plus_plus @ int @ Q4 % 0.30/0.53 @ ( zero_neq_one_of_bool @ int % 0.30/0.53 @ ( R2 % 0.30/0.53 != ( zero_zero @ int ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Divides.adjust_div_eq % 0.30/0.53 thf(fact_1799_signed__take__bit__rec,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A ) % 0.30/0.53 => ( ( bit_ri4674362597316999326ke_bit @ A ) % 0.30/0.53 = ( ^ [N: nat,A4: A] : % 0.30/0.53 ( if @ A % 0.30/0.53 @ ( N % 0.30/0.53 = ( zero_zero @ nat ) ) % 0.30/0.53 @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) % 0.30/0.53 @ ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % signed_take_bit_rec % 0.30/0.53 thf(fact_1800_neg__eucl__rel__int__mult__2,axiom, % 0.30/0.53 ! [B2: int,A2: int,Q4: int,R2: int] : % 0.30/0.53 ( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) ) % 0.30/0.53 => ( ( eucl_rel_int @ ( plus_plus @ int @ A2 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 0.30/0.53 => ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q4 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) @ ( one_one @ int ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_eucl_rel_int_mult_2 % 0.30/0.53 thf(fact_1801_diff__shunt__var,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( boolea8198339166811842893lgebra @ A ) % 0.30/0.53 => ! [X3: A,Y: A] : % 0.30/0.53 ( ( ( minus_minus @ A @ X3 @ Y ) % 0.30/0.53 = ( bot_bot @ A ) ) % 0.30/0.53 = ( ord_less_eq @ A @ X3 @ Y ) ) ) ). % 0.30/0.53 % 0.30/0.53 % diff_shunt_var % 0.30/0.53 thf(fact_1802_take__bit__rec,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A ) % 0.30/0.53 = ( ^ [N: nat,A4: A] : % 0.30/0.53 ( if @ A % 0.30/0.53 @ ( N % 0.30/0.53 = ( zero_zero @ nat ) ) % 0.30/0.53 @ ( zero_zero @ A ) % 0.30/0.53 @ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_rec % 0.30/0.53 thf(fact_1803_add_Oinverse__inverse,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) ) % 0.30/0.53 = A2 ) ) ). % 0.30/0.53 % 0.30/0.53 % add.inverse_inverse % 0.30/0.53 thf(fact_1804_neg__equal__iff__equal,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ( uminus_uminus @ A @ A2 ) % 0.30/0.53 = ( uminus_uminus @ A @ B2 ) ) % 0.30/0.53 = ( A2 = B2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_equal_iff_equal % 0.30/0.53 thf(fact_1805_Diff__cancel,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A] : % 0.30/0.53 ( ( minus_minus @ ( set @ A ) @ A3 @ A3 ) % 0.30/0.53 = ( bot_bot @ ( set @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Diff_cancel % 0.30/0.53 thf(fact_1806_empty__Diff,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A] : % 0.30/0.53 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) % 0.30/0.53 = ( bot_bot @ ( set @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % empty_Diff % 0.30/0.53 thf(fact_1807_Diff__empty,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A] : % 0.30/0.53 ( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) % 0.30/0.53 = A3 ) ). % 0.30/0.53 % 0.30/0.53 % Diff_empty % 0.30/0.53 thf(fact_1808_finite__Diff2,axiom, % 0.30/0.53 ! [A: $tType,B3: set @ A,A3: set @ A] : % 0.30/0.53 ( ( finite_finite2 @ A @ B3 ) % 0.30/0.53 => ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 0.30/0.53 = ( finite_finite2 @ A @ A3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % finite_Diff2 % 0.30/0.53 thf(fact_1809_finite__Diff,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.53 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.53 => ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % finite_Diff % 0.30/0.53 thf(fact_1810_Compl__subset__Compl__iff,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) % 0.30/0.53 = ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ). % 0.30/0.53 % 0.30/0.53 % Compl_subset_Compl_iff % 0.30/0.53 thf(fact_1811_Compl__anti__mono,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.53 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Compl_anti_mono % 0.30/0.53 thf(fact_1812_Diff__insert0,axiom, % 0.30/0.53 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] : % 0.30/0.53 ( ~ ( member @ A @ X3 @ A3 ) % 0.30/0.53 => ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) ) % 0.30/0.53 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Diff_insert0 % 0.30/0.53 thf(fact_1813_insert__Diff1,axiom, % 0.30/0.53 ! [A: $tType,X3: A,B3: set @ A,A3: set @ A] : % 0.30/0.53 ( ( member @ A @ X3 @ B3 ) % 0.30/0.53 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 ) % 0.30/0.53 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % insert_Diff1 % 0.30/0.53 thf(fact_1814_compl__le__compl__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( boolea8198339166811842893lgebra @ A ) % 0.30/0.53 => ! [X3: A,Y: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) ) % 0.30/0.53 = ( ord_less_eq @ A @ Y @ X3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % compl_le_compl_iff % 0.30/0.53 thf(fact_1815_neg__le__iff__le,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ordered_ab_group_add @ A ) % 0.30/0.53 => ! [B2: A,A2: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) % 0.30/0.53 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_le_iff_le % 0.30/0.53 thf(fact_1816_neg__equal__zero,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linord5086331880401160121up_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( ( uminus_uminus @ A @ A2 ) % 0.30/0.53 = A2 ) % 0.30/0.53 = ( A2 % 0.30/0.53 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_equal_zero % 0.30/0.53 thf(fact_1817_equal__neg__zero,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linord5086331880401160121up_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( A2 % 0.30/0.53 = ( uminus_uminus @ A @ A2 ) ) % 0.30/0.53 = ( A2 % 0.30/0.53 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % equal_neg_zero % 0.30/0.53 thf(fact_1818_neg__equal__0__iff__equal,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( ( uminus_uminus @ A @ A2 ) % 0.30/0.53 = ( zero_zero @ A ) ) % 0.30/0.53 = ( A2 % 0.30/0.53 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_equal_0_iff_equal % 0.30/0.53 thf(fact_1819_neg__0__equal__iff__equal,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( ( zero_zero @ A ) % 0.30/0.53 = ( uminus_uminus @ A @ A2 ) ) % 0.30/0.53 = ( ( zero_zero @ A ) % 0.30/0.53 = A2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_0_equal_iff_equal % 0.30/0.53 thf(fact_1820_add_Oinverse__neutral,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % add.inverse_neutral % 0.30/0.53 thf(fact_1821_compl__less__compl__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( boolea8198339166811842893lgebra @ A ) % 0.30/0.53 => ! [X3: A,Y: A] : % 0.30/0.53 ( ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) ) % 0.30/0.53 = ( ord_less @ A @ Y @ X3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % compl_less_compl_iff % 0.30/0.53 thf(fact_1822_neg__less__iff__less,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ordered_ab_group_add @ A ) % 0.30/0.53 => ! [B2: A,A2: A] : % 0.30/0.53 ( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) % 0.30/0.53 = ( ord_less @ A @ A2 @ B2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_less_iff_less % 0.30/0.53 thf(fact_1823_mult__minus__left,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ring @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 0.30/0.53 = ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % mult_minus_left % 0.30/0.53 thf(fact_1824_minus__mult__minus,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ring @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) % 0.30/0.53 = ( times_times @ A @ A2 @ B2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_mult_minus % 0.30/0.53 thf(fact_1825_mult__minus__right,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ring @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) % 0.30/0.53 = ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % mult_minus_right % 0.30/0.53 thf(fact_1826_minus__add__distrib,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ab_group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.30/0.53 = ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_add_distrib % 0.30/0.53 thf(fact_1827_minus__add__cancel,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.30/0.53 = B2 ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_add_cancel % 0.30/0.53 thf(fact_1828_add__minus__cancel,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) % 0.30/0.53 = B2 ) ) ). % 0.30/0.53 % 0.30/0.53 % add_minus_cancel % 0.30/0.53 thf(fact_1829_minus__diff__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) ) % 0.30/0.53 = ( minus_minus @ A @ B2 @ A2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_diff_eq % 0.30/0.53 thf(fact_1830_dvd__minus__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( comm_ring_1 @ A ) % 0.30/0.53 => ! [X3: A,Y: A] : % 0.30/0.53 ( ( dvd_dvd @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) % 0.30/0.53 = ( dvd_dvd @ A @ X3 @ Y ) ) ) ). % 0.30/0.53 % 0.30/0.53 % dvd_minus_iff % 0.30/0.53 thf(fact_1831_minus__dvd__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( comm_ring_1 @ A ) % 0.30/0.53 => ! [X3: A,Y: A] : % 0.30/0.53 ( ( dvd_dvd @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) % 0.30/0.53 = ( dvd_dvd @ A @ X3 @ Y ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_dvd_iff % 0.30/0.53 thf(fact_1832_Diff__eq__empty__iff,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.53 ( ( ( minus_minus @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.53 = ( bot_bot @ ( set @ A ) ) ) % 0.30/0.53 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ). % 0.30/0.53 % 0.30/0.53 % Diff_eq_empty_iff % 0.30/0.53 thf(fact_1833_atLeastAtMost__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ord @ A ) % 0.30/0.53 => ! [I: A,L: A,U: A] : % 0.30/0.53 ( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L @ U ) ) % 0.30/0.53 = ( ( ord_less_eq @ A @ L @ I ) % 0.30/0.53 & ( ord_less_eq @ A @ I @ U ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % atLeastAtMost_iff % 0.30/0.53 thf(fact_1834_Icc__eq__Icc,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( order @ A ) % 0.30/0.53 => ! [L: A,H2: A,L3: A,H3: A] : % 0.30/0.53 ( ( ( set_or1337092689740270186AtMost @ A @ L @ H2 ) % 0.30/0.53 = ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) % 0.30/0.53 = ( ( ( L = L3 ) % 0.30/0.53 & ( H2 = H3 ) ) % 0.30/0.53 | ( ~ ( ord_less_eq @ A @ L @ H2 ) % 0.30/0.53 & ~ ( ord_less_eq @ A @ L3 @ H3 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Icc_eq_Icc % 0.30/0.53 thf(fact_1835_insert__Diff__single,axiom, % 0.30/0.53 ! [A: $tType,A2: A,A3: set @ A] : % 0.30/0.53 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.30/0.53 = ( insert @ A @ A2 @ A3 ) ) ). % 0.30/0.53 % 0.30/0.53 % insert_Diff_single % 0.30/0.53 thf(fact_1836_finite__Diff__insert,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] : % 0.30/0.53 ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) ) % 0.30/0.53 = ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % finite_Diff_insert % 0.30/0.53 thf(fact_1837_take__bit__of__0,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.53 => ! [N2: nat] : % 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( zero_zero @ A ) ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_of_0 % 0.30/0.53 thf(fact_1838_finite__atLeastAtMost,axiom, % 0.30/0.53 ! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ). % 0.30/0.53 % 0.30/0.53 % finite_atLeastAtMost % 0.30/0.53 thf(fact_1839_neg__less__eq__nonneg,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linord5086331880401160121up_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 ) % 0.30/0.53 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_less_eq_nonneg % 0.30/0.53 thf(fact_1840_less__eq__neg__nonpos,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linord5086331880401160121up_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) ) % 0.30/0.53 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % less_eq_neg_nonpos % 0.30/0.53 thf(fact_1841_neg__le__0__iff__le,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ordered_ab_group_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) ) % 0.30/0.53 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_le_0_iff_le % 0.30/0.53 thf(fact_1842_neg__0__le__iff__le,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ordered_ab_group_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) ) % 0.30/0.53 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_0_le_iff_le % 0.30/0.53 thf(fact_1843_less__neg__neg,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linord5086331880401160121up_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) ) % 0.30/0.53 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % less_neg_neg % 0.30/0.53 thf(fact_1844_neg__less__pos,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linord5086331880401160121up_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 ) % 0.30/0.53 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_less_pos % 0.30/0.53 thf(fact_1845_neg__0__less__iff__less,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ordered_ab_group_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) ) % 0.30/0.53 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_0_less_iff_less % 0.30/0.53 thf(fact_1846_neg__less__0__iff__less,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ordered_ab_group_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) ) % 0.30/0.53 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_less_0_iff_less % 0.30/0.53 thf(fact_1847_add_Oright__inverse,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( plus_plus @ A @ A2 @ ( uminus_uminus @ A @ A2 ) ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % add.right_inverse % 0.30/0.53 thf(fact_1848_ab__left__minus,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % ab_left_minus % 0.30/0.53 thf(fact_1849_verit__minus__simplify_I3_J,axiom, % 0.30/0.53 ! [B: $tType] : % 0.30/0.53 ( ( group_add @ B ) % 0.30/0.53 => ! [B2: B] : % 0.30/0.53 ( ( minus_minus @ B @ ( zero_zero @ B ) @ B2 ) % 0.30/0.53 = ( uminus_uminus @ B @ B2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % verit_minus_simplify(3) % 0.30/0.53 thf(fact_1850_diff__0,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.53 = ( uminus_uminus @ A @ A2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % diff_0 % 0.30/0.53 thf(fact_1851_add__neg__numeral__simps_I3_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( neg_numeral @ A ) % 0.30/0.53 => ! [M: num,N2: num] : % 0.30/0.53 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) % 0.30/0.53 = ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % add_neg_numeral_simps(3) % 0.30/0.53 thf(fact_1852_uminus__add__conv__diff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ab_group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 0.30/0.53 = ( minus_minus @ A @ B2 @ A2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % uminus_add_conv_diff % 0.30/0.53 thf(fact_1853_diff__minus__eq__add,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( minus_minus @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) % 0.30/0.53 = ( plus_plus @ A @ A2 @ B2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % diff_minus_eq_add % 0.30/0.53 thf(fact_1854_divide__minus1,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( field @ A ) % 0.30/0.53 => ! [X3: A] : % 0.30/0.53 ( ( divide_divide @ A @ X3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.30/0.53 = ( uminus_uminus @ A @ X3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % divide_minus1 % 0.30/0.53 thf(fact_1855_atLeastatMost__empty__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( preorder @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 0.30/0.53 = ( bot_bot @ ( set @ A ) ) ) % 0.30/0.53 = ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % atLeastatMost_empty_iff % 0.30/0.53 thf(fact_1856_atLeastatMost__empty__iff2,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( preorder @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ( bot_bot @ ( set @ A ) ) % 0.30/0.53 = ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.30/0.53 = ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % atLeastatMost_empty_iff2 % 0.30/0.53 thf(fact_1857_atLeastatMost__subset__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( preorder @ A ) % 0.30/0.53 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) ) % 0.30/0.53 = ( ~ ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.53 | ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.30/0.53 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % atLeastatMost_subset_iff % 0.30/0.53 thf(fact_1858_atLeastatMost__empty,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( order @ A ) % 0.30/0.53 => ! [B2: A,A2: A] : % 0.30/0.53 ( ( ord_less @ A @ B2 @ A2 ) % 0.30/0.53 => ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 0.30/0.53 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % atLeastatMost_empty % 0.30/0.53 thf(fact_1859_infinite__Icc__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( dense_linorder @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) ) % 0.30/0.53 = ( ord_less @ A @ A2 @ B2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % infinite_Icc_iff % 0.30/0.53 thf(fact_1860_subset__Compl__singleton,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,B2: A] : % 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.30/0.53 = ( ~ ( member @ A @ B2 @ A3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % subset_Compl_singleton % 0.30/0.53 thf(fact_1861_take__bit__0,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( zero_zero @ nat ) @ A2 ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_0 % 0.30/0.53 thf(fact_1862_atLeastAtMost__singleton,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( order @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( set_or1337092689740270186AtMost @ A @ A2 @ A2 ) % 0.30/0.53 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % atLeastAtMost_singleton % 0.30/0.53 thf(fact_1863_atLeastAtMost__singleton__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( order @ A ) % 0.30/0.53 => ! [A2: A,B2: A,C2: A] : % 0.30/0.53 ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 0.30/0.53 = ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.30/0.53 = ( ( A2 = B2 ) % 0.30/0.53 & ( B2 = C2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % atLeastAtMost_singleton_iff % 0.30/0.53 thf(fact_1864_take__bit__Suc__1,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.53 => ! [N2: nat] : % 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( one_one @ A ) ) % 0.30/0.53 = ( one_one @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_Suc_1 % 0.30/0.53 thf(fact_1865_add__neg__numeral__special_I8_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( neg_numeral @ A ) % 0.30/0.53 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % add_neg_numeral_special(8) % 0.30/0.53 thf(fact_1866_add__neg__numeral__special_I7_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( neg_numeral @ A ) % 0.30/0.53 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % add_neg_numeral_special(7) % 0.30/0.53 thf(fact_1867_diff__numeral__special_I12_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( neg_numeral @ A ) % 0.30/0.53 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % diff_numeral_special(12) % 0.30/0.53 thf(fact_1868_mod__minus1__right,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( euclid8851590272496341667cancel @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( modulo_modulo @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % mod_minus1_right % 0.30/0.53 thf(fact_1869_take__bit__of__1__eq__0__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.53 => ! [N2: nat] : % 0.30/0.53 ( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( one_one @ A ) ) % 0.30/0.53 = ( zero_zero @ A ) ) % 0.30/0.53 = ( N2 % 0.30/0.53 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_of_1_eq_0_iff % 0.30/0.53 thf(fact_1870_semiring__norm_I168_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( neg_numeral @ A ) % 0.30/0.53 => ! [V2: num,W2: num,Y: A] : % 0.30/0.53 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) ) % 0.30/0.53 = ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W2 ) ) ) @ Y ) ) ) ). % 0.30/0.53 % 0.30/0.53 % semiring_norm(168) % 0.30/0.53 thf(fact_1871_neg__numeral__le__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num,N2: num] : % 0.30/0.53 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) % 0.30/0.53 = ( ord_less_eq @ num @ N2 @ M ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_numeral_le_iff % 0.30/0.53 thf(fact_1872_neg__numeral__less__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num,N2: num] : % 0.30/0.53 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) % 0.30/0.53 = ( ord_less @ num @ N2 @ M ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_numeral_less_iff % 0.30/0.53 thf(fact_1873_zle__diff1__eq,axiom, % 0.30/0.53 ! [W2: int,Z2: int] : % 0.30/0.53 ( ( ord_less_eq @ int @ W2 @ ( minus_minus @ int @ Z2 @ ( one_one @ int ) ) ) % 0.30/0.53 = ( ord_less @ int @ W2 @ Z2 ) ) ). % 0.30/0.53 % 0.30/0.53 % zle_diff1_eq % 0.30/0.53 thf(fact_1874_take__bit__of__Suc__0,axiom, % 0.30/0.53 ! [N2: nat] : % 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.53 = ( zero_neq_one_of_bool @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_of_Suc_0 % 0.30/0.53 thf(fact_1875_not__neg__one__le__neg__numeral__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num] : % 0.30/0.53 ( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) % 0.30/0.53 = ( M != one2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % not_neg_one_le_neg_numeral_iff % 0.30/0.53 thf(fact_1876_divide__le__eq__numeral1_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [B2: A,W2: num,A2: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A2 ) % 0.30/0.53 = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % divide_le_eq_numeral1(2) % 0.30/0.53 thf(fact_1877_le__divide__eq__numeral1_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [A2: A,B2: A,W2: num] : % 0.30/0.53 ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) % 0.30/0.53 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % le_divide_eq_numeral1(2) % 0.30/0.53 thf(fact_1878_divide__eq__eq__numeral1_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [B2: A,W2: num,A2: A] : % 0.30/0.53 ( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) % 0.30/0.53 = A2 ) % 0.30/0.53 = ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( B2 % 0.30/0.53 = ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) % 0.30/0.53 & ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) % 0.30/0.53 = ( zero_zero @ A ) ) % 0.30/0.53 => ( A2 % 0.30/0.53 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % divide_eq_eq_numeral1(2) % 0.30/0.53 thf(fact_1879_eq__divide__eq__numeral1_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [A2: A,B2: A,W2: num] : % 0.30/0.53 ( ( A2 % 0.30/0.53 = ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) % 0.30/0.53 = ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) % 0.30/0.53 = B2 ) ) % 0.30/0.53 & ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) % 0.30/0.53 = ( zero_zero @ A ) ) % 0.30/0.53 => ( A2 % 0.30/0.53 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % eq_divide_eq_numeral1(2) % 0.30/0.53 thf(fact_1880_neg__numeral__less__neg__one__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num] : % 0.30/0.53 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.30/0.53 = ( M != one2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_numeral_less_neg_one_iff % 0.30/0.53 thf(fact_1881_less__divide__eq__numeral1_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [A2: A,B2: A,W2: num] : % 0.30/0.53 ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) % 0.30/0.53 = ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % less_divide_eq_numeral1(2) % 0.30/0.53 thf(fact_1882_divide__less__eq__numeral1_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [B2: A,W2: num,A2: A] : % 0.30/0.53 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A2 ) % 0.30/0.53 = ( ord_less @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % divide_less_eq_numeral1(2) % 0.30/0.53 thf(fact_1883_take__bit__of__1,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.53 => ! [N2: nat] : % 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( one_one @ A ) ) % 0.30/0.53 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_of_1 % 0.30/0.53 thf(fact_1884_add__neg__numeral__special_I9_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( neg_numeral @ A ) % 0.30/0.53 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.30/0.53 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % add_neg_numeral_special(9) % 0.30/0.53 thf(fact_1885_even__take__bit__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.53 => ! [N2: nat,A2: A] : % 0.30/0.53 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) % 0.30/0.53 = ( ( N2 % 0.30/0.53 = ( zero_zero @ nat ) ) % 0.30/0.53 | ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % even_take_bit_eq % 0.30/0.53 thf(fact_1886_signed__take__bit__Suc__minus__bit0,axiom, % 0.30/0.53 ! [N2: nat,K: num] : % 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) ) % 0.30/0.53 = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % signed_take_bit_Suc_minus_bit0 % 0.30/0.53 thf(fact_1887_take__bit__Suc__0,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ ( zero_zero @ nat ) ) @ A2 ) % 0.30/0.53 = ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_Suc_0 % 0.30/0.53 thf(fact_1888_signed__take__bit__0,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A2 ) % 0.30/0.53 = ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % signed_take_bit_0 % 0.30/0.53 thf(fact_1889_take__bit__of__exp,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.53 => ! [M: nat,N2: nat] : % 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.53 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N2 @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_of_exp % 0.30/0.53 thf(fact_1890_take__bit__of__2,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.53 => ! [N2: nat] : % 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.53 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_of_2 % 0.30/0.53 thf(fact_1891_minus__int__code_I2_J,axiom, % 0.30/0.53 ! [L: int] : % 0.30/0.53 ( ( minus_minus @ int @ ( zero_zero @ int ) @ L ) % 0.30/0.53 = ( uminus_uminus @ int @ L ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_int_code(2) % 0.30/0.53 thf(fact_1892_compl__mono,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( boolea8198339166811842893lgebra @ A ) % 0.30/0.53 => ! [X3: A,Y: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.30/0.53 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % compl_mono % 0.30/0.53 thf(fact_1893_compl__le__swap1,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( boolea8198339166811842893lgebra @ A ) % 0.30/0.53 => ! [Y: A,X3: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X3 ) ) % 0.30/0.53 => ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % compl_le_swap1 % 0.30/0.53 thf(fact_1894_compl__le__swap2,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( boolea8198339166811842893lgebra @ A ) % 0.30/0.53 => ! [Y: A,X3: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X3 ) % 0.30/0.53 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) ) ) ). % 0.30/0.53 % 0.30/0.53 % compl_le_swap2 % 0.30/0.53 thf(fact_1895_equation__minus__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( A2 % 0.30/0.53 = ( uminus_uminus @ A @ B2 ) ) % 0.30/0.53 = ( B2 % 0.30/0.53 = ( uminus_uminus @ A @ A2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % equation_minus_iff % 0.30/0.53 thf(fact_1896_minus__equation__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ( uminus_uminus @ A @ A2 ) % 0.30/0.53 = B2 ) % 0.30/0.53 = ( ( uminus_uminus @ A @ B2 ) % 0.30/0.53 = A2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_equation_iff % 0.30/0.53 thf(fact_1897_compl__less__swap1,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( boolea8198339166811842893lgebra @ A ) % 0.30/0.53 => ! [Y: A,X3: A] : % 0.30/0.53 ( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X3 ) ) % 0.30/0.53 => ( ord_less @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % compl_less_swap1 % 0.30/0.53 thf(fact_1898_compl__less__swap2,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( boolea8198339166811842893lgebra @ A ) % 0.30/0.53 => ! [Y: A,X3: A] : % 0.30/0.53 ( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X3 ) % 0.30/0.53 => ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) ) ) ). % 0.30/0.53 % 0.30/0.53 % compl_less_swap2 % 0.30/0.53 thf(fact_1899_Compl__insert,axiom, % 0.30/0.53 ! [A: $tType,X3: A,A3: set @ A] : % 0.30/0.53 ( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) ) % 0.30/0.53 = ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Compl_insert % 0.30/0.53 thf(fact_1900_take__bit__add,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.53 => ! [N2: nat,A2: A,B2: A] : % 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ B2 ) ) ) % 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_add % 0.30/0.53 thf(fact_1901_take__bit__tightened,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.53 => ! [N2: nat,A2: A,B2: A,M: nat] : % 0.30/0.53 ( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) % 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ A @ N2 @ B2 ) ) % 0.30/0.53 => ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) % 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ A @ M @ B2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_tightened % 0.30/0.53 thf(fact_1902_take__bit__nat__less__eq__self,axiom, % 0.30/0.53 ! [N2: nat,M: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) @ M ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_nat_less_eq_self % 0.30/0.53 thf(fact_1903_take__bit__tightened__less__eq__nat,axiom, % 0.30/0.53 ! [M: nat,N2: nat,Q4: nat] : % 0.30/0.53 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.53 => ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M @ Q4 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ Q4 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_tightened_less_eq_nat % 0.30/0.53 thf(fact_1904_le__imp__neg__le,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ordered_ab_group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.53 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % le_imp_neg_le % 0.30/0.53 thf(fact_1905_minus__le__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ordered_ab_group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 0.30/0.53 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_le_iff % 0.30/0.53 thf(fact_1906_le__minus__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ordered_ab_group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) % 0.30/0.53 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % le_minus_iff % 0.30/0.53 thf(fact_1907_minus__less__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ordered_ab_group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 0.30/0.53 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_less_iff % 0.30/0.53 thf(fact_1908_less__minus__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ordered_ab_group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) % 0.30/0.53 = ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % less_minus_iff % 0.30/0.53 thf(fact_1909_verit__negate__coefficient_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ordered_ab_group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.53 => ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % verit_negate_coefficient(2) % 0.30/0.53 thf(fact_1910_square__eq__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( idom @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ( times_times @ A @ A2 @ A2 ) % 0.30/0.53 = ( times_times @ A @ B2 @ B2 ) ) % 0.30/0.53 = ( ( A2 = B2 ) % 0.30/0.53 | ( A2 % 0.30/0.53 = ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % square_eq_iff % 0.30/0.53 thf(fact_1911_minus__mult__commute,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ring @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 0.30/0.53 = ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_mult_commute % 0.30/0.53 thf(fact_1912_is__num__normalize_I8_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( neg_numeral @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.30/0.53 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % is_num_normalize(8) % 0.30/0.53 thf(fact_1913_add_Oinverse__distrib__swap,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.30/0.53 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % add.inverse_distrib_swap % 0.30/0.53 thf(fact_1914_group__cancel_Oneg1,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ab_group_add @ A ) % 0.30/0.53 => ! [A3: A,K: A,A2: A] : % 0.30/0.53 ( ( A3 % 0.30/0.53 = ( plus_plus @ A @ K @ A2 ) ) % 0.30/0.53 => ( ( uminus_uminus @ A @ A3 ) % 0.30/0.53 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % group_cancel.neg1 % 0.30/0.53 thf(fact_1915_minus__diff__commute,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ab_group_add @ A ) % 0.30/0.53 => ! [B2: A,A2: A] : % 0.30/0.53 ( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) % 0.30/0.53 = ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_diff_commute % 0.30/0.53 thf(fact_1916_minus__divide__right,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( field @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 0.30/0.53 = ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_divide_right % 0.30/0.53 thf(fact_1917_minus__divide__divide,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( field @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) % 0.30/0.53 = ( divide_divide @ A @ A2 @ B2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_divide_divide % 0.30/0.53 thf(fact_1918_minus__divide__left,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 0.30/0.53 = ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_divide_left % 0.30/0.53 thf(fact_1919_Diff__infinite__finite,axiom, % 0.30/0.53 ! [A: $tType,T4: set @ A,S3: set @ A] : % 0.30/0.53 ( ( finite_finite2 @ A @ T4 ) % 0.30/0.53 => ( ~ ( finite_finite2 @ A @ S3 ) % 0.30/0.53 => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S3 @ T4 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Diff_infinite_finite % 0.30/0.53 thf(fact_1920_double__diff,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.53 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 ) % 0.30/0.53 => ( ( minus_minus @ ( set @ A ) @ B3 @ ( minus_minus @ ( set @ A ) @ C5 @ A3 ) ) % 0.30/0.53 = A3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % double_diff % 0.30/0.53 thf(fact_1921_Diff__subset,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ A3 ) ). % 0.30/0.53 % 0.30/0.53 % Diff_subset % 0.30/0.53 thf(fact_1922_Diff__mono,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,C5: set @ A,D6: set @ A,B3: set @ A] : % 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) % 0.30/0.53 => ( ( ord_less_eq @ ( set @ A ) @ D6 @ B3 ) % 0.30/0.53 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ C5 @ D6 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Diff_mono % 0.30/0.53 thf(fact_1923_insert__Diff__if,axiom, % 0.30/0.53 ! [A: $tType,X3: A,B3: set @ A,A3: set @ A] : % 0.30/0.53 ( ( ( member @ A @ X3 @ B3 ) % 0.30/0.53 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 ) % 0.30/0.53 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) % 0.30/0.53 & ( ~ ( member @ A @ X3 @ B3 ) % 0.30/0.53 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 ) % 0.30/0.53 = ( insert @ A @ X3 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % insert_Diff_if % 0.30/0.53 thf(fact_1924_minus__int__code_I1_J,axiom, % 0.30/0.53 ! [K: int] : % 0.30/0.53 ( ( minus_minus @ int @ K @ ( zero_zero @ int ) ) % 0.30/0.53 = K ) ). % 0.30/0.53 % 0.30/0.53 % minus_int_code(1) % 0.30/0.53 thf(fact_1925_uminus__int__code_I1_J,axiom, % 0.30/0.53 ( ( uminus_uminus @ int @ ( zero_zero @ int ) ) % 0.30/0.53 = ( zero_zero @ int ) ) ). % 0.30/0.53 % 0.30/0.53 % uminus_int_code(1) % 0.30/0.53 thf(fact_1926_int__distrib_I4_J,axiom, % 0.30/0.53 ! [W2: int,Z1: int,Z22: int] : % 0.30/0.53 ( ( times_times @ int @ W2 @ ( minus_minus @ int @ Z1 @ Z22 ) ) % 0.30/0.53 = ( minus_minus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % int_distrib(4) % 0.30/0.53 thf(fact_1927_int__distrib_I3_J,axiom, % 0.30/0.53 ! [Z1: int,Z22: int,W2: int] : % 0.30/0.53 ( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W2 ) % 0.30/0.53 = ( minus_minus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % int_distrib(3) % 0.30/0.53 thf(fact_1928_psubset__imp__ex__mem,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.53 ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.53 => ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % psubset_imp_ex_mem % 0.30/0.53 thf(fact_1929_zdvd__zdiffD,axiom, % 0.30/0.53 ! [K: int,M: int,N2: int] : % 0.30/0.53 ( ( dvd_dvd @ int @ K @ ( minus_minus @ int @ M @ N2 ) ) % 0.30/0.53 => ( ( dvd_dvd @ int @ K @ N2 ) % 0.30/0.53 => ( dvd_dvd @ int @ K @ M ) ) ) ). % 0.30/0.53 % 0.30/0.53 % zdvd_zdiffD % 0.30/0.53 thf(fact_1930_zmod__minus1,axiom, % 0.30/0.53 ! [B2: int] : % 0.30/0.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.30/0.53 => ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 ) % 0.30/0.53 = ( minus_minus @ int @ B2 @ ( one_one @ int ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % zmod_minus1 % 0.30/0.53 thf(fact_1931_take__bit__minus__small__eq,axiom, % 0.30/0.53 ! [K: int,N2: nat] : % 0.30/0.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.53 => ( ( ord_less_eq @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ K ) ) % 0.30/0.53 = ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_minus_small_eq % 0.30/0.53 thf(fact_1932_infinite__Icc,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( dense_linorder @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.53 => ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % infinite_Icc % 0.30/0.53 thf(fact_1933_atLeastAtMost__singleton_H,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( order @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( A2 = B2 ) % 0.30/0.53 => ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 0.30/0.53 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % atLeastAtMost_singleton' % 0.30/0.53 thf(fact_1934_take__bit__tightened__less__eq__int,axiom, % 0.30/0.53 ! [M: nat,N2: nat,K: int] : % 0.30/0.53 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.53 => ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_tightened_less_eq_int % 0.30/0.53 thf(fact_1935_take__bit__int__greater__self__iff,axiom, % 0.30/0.53 ! [K: int,N2: nat] : % 0.30/0.53 ( ( ord_less @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) % 0.30/0.53 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_int_greater_self_iff % 0.30/0.53 thf(fact_1936_not__take__bit__negative,axiom, % 0.30/0.53 ! [N2: nat,K: int] : % 0.30/0.53 ~ ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) ). % 0.30/0.53 % 0.30/0.53 % not_take_bit_negative % 0.30/0.53 thf(fact_1937_signed__take__bit__eq__iff__take__bit__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A ) % 0.30/0.53 => ! [N2: nat,A2: A,B2: A] : % 0.30/0.53 ( ( ( bit_ri4674362597316999326ke_bit @ A @ N2 @ A2 ) % 0.30/0.53 = ( bit_ri4674362597316999326ke_bit @ A @ N2 @ B2 ) ) % 0.30/0.53 = ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A2 ) % 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ B2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % signed_take_bit_eq_iff_take_bit_eq % 0.30/0.53 thf(fact_1938_signed__take__bit__take__bit,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A ) % 0.30/0.53 => ! [M: nat,N2: nat,A2: A] : % 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) % 0.30/0.53 = ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N2 @ M ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 ) @ ( bit_ri4674362597316999326ke_bit @ A @ M ) @ A2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % signed_take_bit_take_bit % 0.30/0.53 thf(fact_1939_ex__nat__less,axiom, % 0.30/0.53 ! [N2: nat,P: nat > $o] : % 0.30/0.53 ( ( ? [M2: nat] : % 0.30/0.53 ( ( ord_less_eq @ nat @ M2 @ N2 ) % 0.30/0.53 & ( P @ M2 ) ) ) % 0.30/0.53 = ( ? [X: nat] : % 0.30/0.53 ( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.30/0.53 & ( P @ X ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % ex_nat_less % 0.30/0.53 thf(fact_1940_all__nat__less,axiom, % 0.30/0.53 ! [N2: nat,P: nat > $o] : % 0.30/0.53 ( ( ! [M2: nat] : % 0.30/0.53 ( ( ord_less_eq @ nat @ M2 @ N2 ) % 0.30/0.53 => ( P @ M2 ) ) ) % 0.30/0.53 = ( ! [X: nat] : % 0.30/0.53 ( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.30/0.53 => ( P @ X ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % all_nat_less % 0.30/0.53 thf(fact_1941_neg__numeral__le__numeral,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num,N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_numeral_le_numeral % 0.30/0.53 thf(fact_1942_not__numeral__le__neg__numeral,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num,N2: num] : % 0.30/0.53 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % not_numeral_le_neg_numeral % 0.30/0.53 thf(fact_1943_zero__neq__neg__numeral,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ring_char_0 @ A ) % 0.30/0.53 => ! [N2: num] : % 0.30/0.53 ( ( zero_zero @ A ) % 0.30/0.53 != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % zero_neq_neg_numeral % 0.30/0.53 thf(fact_1944_not__numeral__less__neg__numeral,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num,N2: num] : % 0.30/0.53 ~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % not_numeral_less_neg_numeral % 0.30/0.53 thf(fact_1945_neg__numeral__less__numeral,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num,N2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_numeral_less_numeral % 0.30/0.53 thf(fact_1946_le__minus__one__simps_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % le_minus_one_simps(2) % 0.30/0.53 thf(fact_1947_le__minus__one__simps_I4_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % le_minus_one_simps(4) % 0.30/0.53 thf(fact_1948_zero__neq__neg__one,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ring_char_0 @ A ) % 0.30/0.53 => ( ( zero_zero @ A ) % 0.30/0.53 != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % zero_neq_neg_one % 0.30/0.53 thf(fact_1949_add__eq__0__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ( plus_plus @ A @ A2 @ B2 ) % 0.30/0.53 = ( zero_zero @ A ) ) % 0.30/0.53 = ( B2 % 0.30/0.53 = ( uminus_uminus @ A @ A2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % add_eq_0_iff % 0.30/0.53 thf(fact_1950_ab__group__add__class_Oab__left__minus,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ab_group_add @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % ab_group_add_class.ab_left_minus % 0.30/0.53 thf(fact_1951_add_Oinverse__unique,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ( plus_plus @ A @ A2 @ B2 ) % 0.30/0.53 = ( zero_zero @ A ) ) % 0.30/0.53 => ( ( uminus_uminus @ A @ A2 ) % 0.30/0.53 = B2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % add.inverse_unique % 0.30/0.53 thf(fact_1952_eq__neg__iff__add__eq__0,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( A2 % 0.30/0.53 = ( uminus_uminus @ A @ B2 ) ) % 0.30/0.53 = ( ( plus_plus @ A @ A2 @ B2 ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % eq_neg_iff_add_eq_0 % 0.30/0.53 thf(fact_1953_neg__eq__iff__add__eq__0,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ( uminus_uminus @ A @ A2 ) % 0.30/0.53 = B2 ) % 0.30/0.53 = ( ( plus_plus @ A @ A2 @ B2 ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_eq_iff_add_eq_0 % 0.30/0.53 thf(fact_1954_less__minus__one__simps_I4_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % less_minus_one_simps(4) % 0.30/0.53 thf(fact_1955_less__minus__one__simps_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % less_minus_one_simps(2) % 0.30/0.53 thf(fact_1956_nonzero__minus__divide__right,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [B2: A,A2: A] : % 0.30/0.53 ( ( B2 % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 0.30/0.53 = ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % nonzero_minus_divide_right % 0.30/0.53 thf(fact_1957_nonzero__minus__divide__divide,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [B2: A,A2: A] : % 0.30/0.53 ( ( B2 % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) % 0.30/0.53 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % nonzero_minus_divide_divide % 0.30/0.53 thf(fact_1958_square__eq__1__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ring_15535105094025558882visors @ A ) % 0.30/0.53 => ! [X3: A] : % 0.30/0.53 ( ( ( times_times @ A @ X3 @ X3 ) % 0.30/0.53 = ( one_one @ A ) ) % 0.30/0.53 = ( ( X3 % 0.30/0.53 = ( one_one @ A ) ) % 0.30/0.53 | ( X3 % 0.30/0.53 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % square_eq_1_iff % 0.30/0.53 thf(fact_1959_group__cancel_Osub2,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ab_group_add @ A ) % 0.30/0.53 => ! [B3: A,K: A,B2: A,A2: A] : % 0.30/0.53 ( ( B3 % 0.30/0.53 = ( plus_plus @ A @ K @ B2 ) ) % 0.30/0.53 => ( ( minus_minus @ A @ A2 @ B3 ) % 0.30/0.53 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % group_cancel.sub2 % 0.30/0.53 thf(fact_1960_diff__conv__add__uminus,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( group_add @ A ) % 0.30/0.53 => ( ( minus_minus @ A ) % 0.30/0.53 = ( ^ [A4: A,B4: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % diff_conv_add_uminus % 0.30/0.53 thf(fact_1961_ab__group__add__class_Oab__diff__conv__add__uminus,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ab_group_add @ A ) % 0.30/0.53 => ( ( minus_minus @ A ) % 0.30/0.53 = ( ^ [A4: A,B4: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % ab_group_add_class.ab_diff_conv_add_uminus % 0.30/0.53 thf(fact_1962_take__bit__unset__bit__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.53 => ! [N2: nat,M: nat,A2: A] : % 0.30/0.53 ( ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) ) % 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) % 0.30/0.53 & ( ~ ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) ) % 0.30/0.53 = ( bit_se2638667681897837118et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_unset_bit_eq % 0.30/0.53 thf(fact_1963_take__bit__set__bit__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.53 => ! [N2: nat,M: nat,A2: A] : % 0.30/0.53 ( ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) ) % 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) % 0.30/0.53 & ( ~ ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) ) % 0.30/0.53 = ( bit_se5668285175392031749et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_set_bit_eq % 0.30/0.53 thf(fact_1964_take__bit__flip__bit__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.53 => ! [N2: nat,M: nat,A2: A] : % 0.30/0.53 ( ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) ) % 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) % 0.30/0.53 & ( ~ ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) ) % 0.30/0.53 = ( bit_se8732182000553998342ip_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_flip_bit_eq % 0.30/0.53 thf(fact_1965_dvd__neg__div,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( idom_divide @ A ) % 0.30/0.53 => ! [B2: A,A2: A] : % 0.30/0.53 ( ( dvd_dvd @ A @ B2 @ A2 ) % 0.30/0.53 => ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 0.30/0.53 = ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % dvd_neg_div % 0.30/0.53 thf(fact_1966_dvd__div__neg,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( idom_divide @ A ) % 0.30/0.53 => ! [B2: A,A2: A] : % 0.30/0.53 ( ( dvd_dvd @ A @ B2 @ A2 ) % 0.30/0.53 => ( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) % 0.30/0.53 = ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % dvd_div_neg % 0.30/0.53 thf(fact_1967_subset__Compl__self__eq,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A] : % 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) % 0.30/0.53 = ( A3 % 0.30/0.53 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % subset_Compl_self_eq % 0.30/0.53 thf(fact_1968_Diff__insert__absorb,axiom, % 0.30/0.53 ! [A: $tType,X3: A,A3: set @ A] : % 0.30/0.53 ( ~ ( member @ A @ X3 @ A3 ) % 0.30/0.53 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.30/0.53 = A3 ) ) ). % 0.30/0.53 % 0.30/0.53 % Diff_insert_absorb % 0.30/0.53 thf(fact_1969_Diff__insert2,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] : % 0.30/0.53 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) % 0.30/0.53 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ). % 0.30/0.53 % 0.30/0.53 % Diff_insert2 % 0.30/0.53 thf(fact_1970_insert__Diff,axiom, % 0.30/0.53 ! [A: $tType,A2: A,A3: set @ A] : % 0.30/0.53 ( ( member @ A @ A2 @ A3 ) % 0.30/0.53 => ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.30/0.53 = A3 ) ) ). % 0.30/0.53 % 0.30/0.53 % insert_Diff % 0.30/0.53 thf(fact_1971_Diff__insert,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] : % 0.30/0.53 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) % 0.30/0.53 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Diff_insert % 0.30/0.53 thf(fact_1972_subset__Diff__insert,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,B3: set @ A,X3: A,C5: set @ A] : % 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ ( insert @ A @ X3 @ C5 ) ) ) % 0.30/0.53 = ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ C5 ) ) % 0.30/0.53 & ~ ( member @ A @ X3 @ A3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % subset_Diff_insert % 0.30/0.53 thf(fact_1973_real__0__less__add__iff,axiom, % 0.30/0.53 ! [X3: real,Y: real] : % 0.30/0.53 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X3 @ Y ) ) % 0.30/0.53 = ( ord_less @ real @ ( uminus_uminus @ real @ X3 ) @ Y ) ) ). % 0.30/0.53 % 0.30/0.53 % real_0_less_add_iff % 0.30/0.53 thf(fact_1974_real__add__less__0__iff,axiom, % 0.30/0.53 ! [X3: real,Y: real] : % 0.30/0.53 ( ( ord_less @ real @ ( plus_plus @ real @ X3 @ Y ) @ ( zero_zero @ real ) ) % 0.30/0.53 = ( ord_less @ real @ Y @ ( uminus_uminus @ real @ X3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % real_add_less_0_iff % 0.30/0.53 thf(fact_1975_zminus1__lemma,axiom, % 0.30/0.53 ! [A2: int,B2: int,Q4: int,R2: int] : % 0.30/0.53 ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 0.30/0.53 => ( ( B2 % 0.30/0.53 != ( zero_zero @ int ) ) % 0.30/0.53 => ( eucl_rel_int @ ( uminus_uminus @ int @ A2 ) @ B2 % 0.30/0.53 @ ( product_Pair @ int @ int % 0.30/0.53 @ ( if @ int % 0.30/0.53 @ ( R2 % 0.30/0.53 = ( zero_zero @ int ) ) % 0.30/0.53 @ ( uminus_uminus @ int @ Q4 ) % 0.30/0.53 @ ( minus_minus @ int @ ( uminus_uminus @ int @ Q4 ) @ ( one_one @ int ) ) ) % 0.30/0.53 @ ( if @ int % 0.30/0.53 @ ( R2 % 0.30/0.53 = ( zero_zero @ int ) ) % 0.30/0.53 @ ( zero_zero @ int ) % 0.30/0.53 @ ( minus_minus @ int @ B2 @ R2 ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % zminus1_lemma % 0.30/0.53 thf(fact_1976_int__le__induct,axiom, % 0.30/0.53 ! [I: int,K: int,P: int > $o] : % 0.30/0.53 ( ( ord_less_eq @ int @ I @ K ) % 0.30/0.53 => ( ( P @ K ) % 0.30/0.53 => ( ! [I4: int] : % 0.30/0.53 ( ( ord_less_eq @ int @ I4 @ K ) % 0.30/0.53 => ( ( P @ I4 ) % 0.30/0.53 => ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) ) % 0.30/0.53 => ( P @ I ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % int_le_induct % 0.30/0.53 thf(fact_1977_int__less__induct,axiom, % 0.30/0.53 ! [I: int,K: int,P: int > $o] : % 0.30/0.53 ( ( ord_less @ int @ I @ K ) % 0.30/0.53 => ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) ) % 0.30/0.53 => ( ! [I4: int] : % 0.30/0.53 ( ( ord_less @ int @ I4 @ K ) % 0.30/0.53 => ( ( P @ I4 ) % 0.30/0.53 => ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) ) % 0.30/0.53 => ( P @ I ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % int_less_induct % 0.30/0.53 thf(fact_1978_pos__zmult__eq__1__iff__lemma,axiom, % 0.30/0.53 ! [M: int,N2: int] : % 0.30/0.53 ( ( ( times_times @ int @ M @ N2 ) % 0.30/0.53 = ( one_one @ int ) ) % 0.30/0.53 => ( ( M % 0.30/0.53 = ( one_one @ int ) ) % 0.30/0.53 | ( M % 0.30/0.53 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % pos_zmult_eq_1_iff_lemma % 0.30/0.53 thf(fact_1979_zmult__eq__1__iff,axiom, % 0.30/0.53 ! [M: int,N2: int] : % 0.30/0.53 ( ( ( times_times @ int @ M @ N2 ) % 0.30/0.53 = ( one_one @ int ) ) % 0.30/0.53 = ( ( ( M % 0.30/0.53 = ( one_one @ int ) ) % 0.30/0.53 & ( N2 % 0.30/0.53 = ( one_one @ int ) ) ) % 0.30/0.53 | ( ( M % 0.30/0.53 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) % 0.30/0.53 & ( N2 % 0.30/0.53 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % zmult_eq_1_iff % 0.30/0.53 thf(fact_1980_signed__take__bit__eq__take__bit__shift,axiom, % 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int ) % 0.30/0.53 = ( ^ [N: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( plus_plus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % signed_take_bit_eq_take_bit_shift % 0.30/0.53 thf(fact_1981_take__bit__Suc__minus__bit0,axiom, % 0.30/0.53 ! [N2: nat,K: num] : % 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) ) % 0.30/0.53 = ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_Suc_minus_bit0 % 0.30/0.53 thf(fact_1982_take__bit__int__less__eq,axiom, % 0.30/0.53 ! [N2: nat,K: int] : % 0.30/0.53 ( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) % 0.30/0.53 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.53 => ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_int_less_eq % 0.30/0.53 thf(fact_1983_atLeastatMost__psubset__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( preorder @ A ) % 0.30/0.53 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.30/0.53 ( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) ) % 0.30/0.53 = ( ( ~ ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.53 | ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.30/0.53 & ( ord_less_eq @ A @ B2 @ D2 ) % 0.30/0.53 & ( ( ord_less @ A @ C2 @ A2 ) % 0.30/0.53 | ( ord_less @ A @ B2 @ D2 ) ) ) ) % 0.30/0.53 & ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % atLeastatMost_psubset_iff % 0.30/0.53 thf(fact_1984_take__bit__signed__take__bit,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A ) % 0.30/0.53 => ! [M: nat,N2: nat,A2: A] : % 0.30/0.53 ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) % 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N2 @ A2 ) ) % 0.30/0.53 = ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_signed_take_bit % 0.30/0.53 thf(fact_1985_not__zero__le__neg__numeral,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [N2: num] : % 0.30/0.53 ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % not_zero_le_neg_numeral % 0.30/0.53 thf(fact_1986_neg__numeral__le__zero,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_numeral_le_zero % 0.30/0.53 thf(fact_1987_neg__numeral__less__zero,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [N2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_numeral_less_zero % 0.30/0.53 thf(fact_1988_not__zero__less__neg__numeral,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [N2: num] : % 0.30/0.53 ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % not_zero_less_neg_numeral % 0.30/0.53 thf(fact_1989_le__minus__one__simps_I1_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % le_minus_one_simps(1) % 0.30/0.53 thf(fact_1990_le__minus__one__simps_I3_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % le_minus_one_simps(3) % 0.30/0.53 thf(fact_1991_less__minus__one__simps_I3_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % less_minus_one_simps(3) % 0.30/0.53 thf(fact_1992_less__minus__one__simps_I1_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % less_minus_one_simps(1) % 0.30/0.53 thf(fact_1993_neg__numeral__le__one,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_numeral_le_one % 0.30/0.53 thf(fact_1994_neg__one__le__numeral,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_one_le_numeral % 0.30/0.53 thf(fact_1995_neg__numeral__le__neg__one,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_numeral_le_neg_one % 0.30/0.53 thf(fact_1996_not__numeral__le__neg__one,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num] : % 0.30/0.53 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % not_numeral_le_neg_one % 0.30/0.53 thf(fact_1997_not__one__le__neg__numeral,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num] : % 0.30/0.53 ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % not_one_le_neg_numeral % 0.30/0.53 thf(fact_1998_not__neg__one__less__neg__numeral,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num] : % 0.30/0.53 ~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % not_neg_one_less_neg_numeral % 0.30/0.53 thf(fact_1999_not__one__less__neg__numeral,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num] : % 0.30/0.53 ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % not_one_less_neg_numeral % 0.30/0.53 thf(fact_2000_not__numeral__less__neg__one,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num] : % 0.30/0.53 ~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % not_numeral_less_neg_one % 0.30/0.53 thf(fact_2001_neg__one__less__numeral,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_one_less_numeral % 0.30/0.53 thf(fact_2002_neg__numeral__less__one,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_numeral_less_one % 0.30/0.53 thf(fact_2003_eq__minus__divide__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [A2: A,B2: A,C2: A] : % 0.30/0.53 ( ( A2 % 0.30/0.53 = ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) ) % 0.30/0.53 = ( ( ( C2 % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( ( times_times @ A @ A2 @ C2 ) % 0.30/0.53 = ( uminus_uminus @ A @ B2 ) ) ) % 0.30/0.53 & ( ( C2 % 0.30/0.53 = ( zero_zero @ A ) ) % 0.30/0.53 => ( A2 % 0.30/0.53 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % eq_minus_divide_eq % 0.30/0.53 thf(fact_2004_minus__divide__eq__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [B2: A,C2: A,A2: A] : % 0.30/0.53 ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.53 = A2 ) % 0.30/0.53 = ( ( ( C2 % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( ( uminus_uminus @ A @ B2 ) % 0.30/0.53 = ( times_times @ A @ A2 @ C2 ) ) ) % 0.30/0.53 & ( ( C2 % 0.30/0.53 = ( zero_zero @ A ) ) % 0.30/0.53 => ( A2 % 0.30/0.53 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_divide_eq_eq % 0.30/0.53 thf(fact_2005_nonzero__neg__divide__eq__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [B2: A,A2: A,C2: A] : % 0.30/0.53 ( ( B2 % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 0.30/0.53 = C2 ) % 0.30/0.53 = ( ( uminus_uminus @ A @ A2 ) % 0.30/0.53 = ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % nonzero_neg_divide_eq_eq % 0.30/0.53 thf(fact_2006_nonzero__neg__divide__eq__eq2,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [B2: A,C2: A,A2: A] : % 0.30/0.53 ( ( B2 % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( ( C2 % 0.30/0.53 = ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) % 0.30/0.53 = ( ( times_times @ A @ C2 @ B2 ) % 0.30/0.53 = ( uminus_uminus @ A @ A2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % nonzero_neg_divide_eq_eq2 % 0.30/0.53 thf(fact_2007_divide__eq__minus__1__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( field @ A ) % 0.30/0.53 => ! [A2: A,B2: A] : % 0.30/0.53 ( ( ( divide_divide @ A @ A2 @ B2 ) % 0.30/0.53 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.30/0.53 = ( ( B2 % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 & ( A2 % 0.30/0.53 = ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % divide_eq_minus_1_iff % 0.30/0.53 thf(fact_2008_finite__empty__induct,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] : % 0.30/0.53 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.53 => ( ( P @ A3 ) % 0.30/0.53 => ( ! [A5: A,A8: set @ A] : % 0.30/0.53 ( ( finite_finite2 @ A @ A8 ) % 0.30/0.53 => ( ( member @ A @ A5 @ A8 ) % 0.30/0.53 => ( ( P @ A8 ) % 0.30/0.53 => ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) % 0.30/0.53 => ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % finite_empty_induct % 0.30/0.53 thf(fact_2009_infinite__coinduct,axiom, % 0.30/0.53 ! [A: $tType,X8: ( set @ A ) > $o,A3: set @ A] : % 0.30/0.53 ( ( X8 @ A3 ) % 0.30/0.53 => ( ! [A8: set @ A] : % 0.30/0.53 ( ( X8 @ A8 ) % 0.30/0.53 => ? [X2: A] : % 0.30/0.53 ( ( member @ A @ X2 @ A8 ) % 0.30/0.53 & ( ( X8 @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.30/0.53 | ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) % 0.30/0.53 => ~ ( finite_finite2 @ A @ A3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % infinite_coinduct % 0.30/0.53 thf(fact_2010_infinite__remove,axiom, % 0.30/0.53 ! [A: $tType,S3: set @ A,A2: A] : % 0.30/0.53 ( ~ ( finite_finite2 @ A @ S3 ) % 0.30/0.53 => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % infinite_remove % 0.30/0.53 thf(fact_2011_Diff__single__insert,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,X3: A,B3: set @ A] : % 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) % 0.30/0.53 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Diff_single_insert % 0.30/0.53 thf(fact_2012_subset__insert__iff,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,X3: A,B3: set @ A] : % 0.30/0.53 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) ) % 0.30/0.53 = ( ( ( member @ A @ X3 @ A3 ) % 0.30/0.53 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) % 0.30/0.53 & ( ~ ( member @ A @ X3 @ A3 ) % 0.30/0.53 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % subset_insert_iff % 0.30/0.53 thf(fact_2013_take__bit__Suc__minus__1__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A ) % 0.30/0.53 => ! [N2: nat] : % 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.30/0.53 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) @ ( one_one @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_Suc_minus_1_eq % 0.30/0.53 thf(fact_2014_atLeast0__atMost__Suc,axiom, % 0.30/0.53 ! [N2: nat] : % 0.30/0.53 ( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) % 0.30/0.53 = ( insert @ nat @ ( suc @ N2 ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % atLeast0_atMost_Suc % 0.30/0.53 thf(fact_2015_mod__pos__geq,axiom, % 0.30/0.53 ! [L: int,K: int] : % 0.30/0.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ L ) % 0.30/0.53 => ( ( ord_less_eq @ int @ L @ K ) % 0.30/0.53 => ( ( modulo_modulo @ int @ K @ L ) % 0.30/0.53 = ( modulo_modulo @ int @ ( minus_minus @ int @ K @ L ) @ L ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % mod_pos_geq % 0.30/0.53 thf(fact_2016_Icc__eq__insert__lb__nat,axiom, % 0.30/0.53 ! [M: nat,N2: nat] : % 0.30/0.53 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.53 => ( ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) % 0.30/0.53 = ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Icc_eq_insert_lb_nat % 0.30/0.53 thf(fact_2017_atLeastAtMostSuc__conv,axiom, % 0.30/0.53 ! [M: nat,N2: nat] : % 0.30/0.53 ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) % 0.30/0.53 => ( ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) % 0.30/0.53 = ( insert @ nat @ ( suc @ N2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % atLeastAtMostSuc_conv % 0.30/0.53 thf(fact_2018_atLeastAtMost__insertL,axiom, % 0.30/0.53 ! [M: nat,N2: nat] : % 0.30/0.53 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.53 => ( ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) % 0.30/0.53 = ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % atLeastAtMost_insertL % 0.30/0.53 thf(fact_2019_plusinfinity,axiom, % 0.30/0.53 ! [D2: int,P5: int > $o,P: int > $o] : % 0.30/0.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 ) % 0.30/0.53 => ( ! [X5: int,K2: int] : % 0.30/0.53 ( ( P5 @ X5 ) % 0.30/0.53 = ( P5 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D2 ) ) ) ) % 0.30/0.53 => ( ? [Z4: int] : % 0.30/0.53 ! [X5: int] : % 0.30/0.53 ( ( ord_less @ int @ Z4 @ X5 ) % 0.30/0.53 => ( ( P @ X5 ) % 0.30/0.53 = ( P5 @ X5 ) ) ) % 0.30/0.53 => ( ? [X_12: int] : ( P5 @ X_12 ) % 0.30/0.53 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % plusinfinity % 0.30/0.53 thf(fact_2020_minusinfinity,axiom, % 0.30/0.53 ! [D2: int,P1: int > $o,P: int > $o] : % 0.30/0.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 ) % 0.30/0.53 => ( ! [X5: int,K2: int] : % 0.30/0.53 ( ( P1 @ X5 ) % 0.30/0.53 = ( P1 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D2 ) ) ) ) % 0.30/0.53 => ( ? [Z4: int] : % 0.30/0.53 ! [X5: int] : % 0.30/0.53 ( ( ord_less @ int @ X5 @ Z4 ) % 0.30/0.53 => ( ( P @ X5 ) % 0.30/0.53 = ( P1 @ X5 ) ) ) % 0.30/0.53 => ( ? [X_12: int] : ( P1 @ X_12 ) % 0.30/0.53 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minusinfinity % 0.30/0.53 thf(fact_2021_int__induct,axiom, % 0.30/0.53 ! [P: int > $o,K: int,I: int] : % 0.30/0.53 ( ( P @ K ) % 0.30/0.53 => ( ! [I4: int] : % 0.30/0.53 ( ( ord_less_eq @ int @ K @ I4 ) % 0.30/0.53 => ( ( P @ I4 ) % 0.30/0.53 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) ) % 0.30/0.53 => ( ! [I4: int] : % 0.30/0.53 ( ( ord_less_eq @ int @ I4 @ K ) % 0.30/0.53 => ( ( P @ I4 ) % 0.30/0.53 => ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) ) % 0.30/0.53 => ( P @ I ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % int_induct % 0.30/0.53 thf(fact_2022_subset__eq__atLeast0__atMost__finite,axiom, % 0.30/0.53 ! [N7: set @ nat,N2: nat] : % 0.30/0.53 ( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.30/0.53 => ( finite_finite2 @ nat @ N7 ) ) ). % 0.30/0.53 % 0.30/0.53 % subset_eq_atLeast0_atMost_finite % 0.30/0.53 thf(fact_2023_take__bit__int__less__exp,axiom, % 0.30/0.53 ! [N2: nat,K: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_int_less_exp % 0.30/0.53 thf(fact_2024_pos__minus__divide__less__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [C2: A,B2: A,A2: A] : % 0.30/0.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 ) % 0.30/0.53 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % pos_minus_divide_less_eq % 0.30/0.53 thf(fact_2025_pos__less__minus__divide__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [C2: A,A2: A,B2: A] : % 0.30/0.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) ) % 0.30/0.53 = ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % pos_less_minus_divide_eq % 0.30/0.53 thf(fact_2026_neg__minus__divide__less__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [C2: A,B2: A,A2: A] : % 0.30/0.53 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 ) % 0.30/0.53 = ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_minus_divide_less_eq % 0.30/0.53 thf(fact_2027_neg__less__minus__divide__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [C2: A,A2: A,B2: A] : % 0.30/0.53 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) ) % 0.30/0.53 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_less_minus_divide_eq % 0.30/0.53 thf(fact_2028_minus__divide__less__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [B2: A,C2: A,A2: A] : % 0.30/0.53 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 ) % 0.30/0.53 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_divide_less_eq % 0.30/0.53 thf(fact_2029_less__minus__divide__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [A2: A,B2: A,C2: A] : % 0.30/0.53 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) ) % 0.30/0.53 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % less_minus_divide_eq % 0.30/0.53 thf(fact_2030_divide__eq__eq__numeral_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [B2: A,C2: A,W2: num] : % 0.30/0.53 ( ( ( divide_divide @ A @ B2 @ C2 ) % 0.30/0.53 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) % 0.30/0.53 = ( ( ( C2 % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( B2 % 0.30/0.53 = ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) ) % 0.30/0.53 & ( ( C2 % 0.30/0.53 = ( zero_zero @ A ) ) % 0.30/0.53 => ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % divide_eq_eq_numeral(2) % 0.30/0.53 thf(fact_2031_eq__divide__eq__numeral_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [W2: num,B2: A,C2: A] : % 0.30/0.53 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) % 0.30/0.53 = ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.53 = ( ( ( C2 % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) % 0.30/0.53 = B2 ) ) % 0.30/0.53 & ( ( C2 % 0.30/0.53 = ( zero_zero @ A ) ) % 0.30/0.53 => ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % eq_divide_eq_numeral(2) % 0.30/0.53 thf(fact_2032_minus__divide__add__eq__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [Z2: A,X3: A,Y: A] : % 0.30/0.53 ( ( Z2 % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X3 @ Z2 ) ) @ Y ) % 0.30/0.53 = ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X3 ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_divide_add_eq_iff % 0.30/0.53 thf(fact_2033_add__divide__eq__if__simps_I3_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [Z2: A,A2: A,B2: A] : % 0.30/0.53 ( ( ( Z2 % 0.30/0.53 = ( zero_zero @ A ) ) % 0.30/0.53 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 ) % 0.30/0.53 = B2 ) ) % 0.30/0.53 & ( ( Z2 % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 ) % 0.30/0.53 = ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % add_divide_eq_if_simps(3) % 0.30/0.53 thf(fact_2034_add__divide__eq__if__simps_I6_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [Z2: A,A2: A,B2: A] : % 0.30/0.53 ( ( ( Z2 % 0.30/0.53 = ( zero_zero @ A ) ) % 0.30/0.53 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 ) % 0.30/0.53 = ( uminus_uminus @ A @ B2 ) ) ) % 0.30/0.53 & ( ( Z2 % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 ) % 0.30/0.53 = ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % add_divide_eq_if_simps(6) % 0.30/0.53 thf(fact_2035_add__divide__eq__if__simps_I5_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [Z2: A,A2: A,B2: A] : % 0.30/0.53 ( ( ( Z2 % 0.30/0.53 = ( zero_zero @ A ) ) % 0.30/0.53 => ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 ) % 0.30/0.53 = ( uminus_uminus @ A @ B2 ) ) ) % 0.30/0.53 & ( ( Z2 % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 ) % 0.30/0.53 = ( divide_divide @ A @ ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % add_divide_eq_if_simps(5) % 0.30/0.53 thf(fact_2036_minus__divide__diff__eq__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( division_ring @ A ) % 0.30/0.53 => ! [Z2: A,X3: A,Y: A] : % 0.30/0.53 ( ( Z2 % 0.30/0.53 != ( zero_zero @ A ) ) % 0.30/0.53 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X3 @ Z2 ) ) @ Y ) % 0.30/0.53 = ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X3 ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_divide_diff_eq_iff % 0.30/0.53 thf(fact_2037_remove__induct,axiom, % 0.30/0.53 ! [A: $tType,P: ( set @ A ) > $o,B3: set @ A] : % 0.30/0.53 ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 0.30/0.53 => ( ( ~ ( finite_finite2 @ A @ B3 ) % 0.30/0.53 => ( P @ B3 ) ) % 0.30/0.53 => ( ! [A8: set @ A] : % 0.30/0.53 ( ( finite_finite2 @ A @ A8 ) % 0.30/0.53 => ( ( A8 % 0.30/0.53 != ( bot_bot @ ( set @ A ) ) ) % 0.30/0.53 => ( ( ord_less_eq @ ( set @ A ) @ A8 @ B3 ) % 0.30/0.53 => ( ! [X2: A] : % 0.30/0.53 ( ( member @ A @ X2 @ A8 ) % 0.30/0.53 => ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) % 0.30/0.53 => ( P @ A8 ) ) ) ) ) % 0.30/0.53 => ( P @ B3 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % remove_induct % 0.30/0.53 thf(fact_2038_finite__remove__induct,axiom, % 0.30/0.53 ! [A: $tType,B3: set @ A,P: ( set @ A ) > $o] : % 0.30/0.53 ( ( finite_finite2 @ A @ B3 ) % 0.30/0.53 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 0.30/0.53 => ( ! [A8: set @ A] : % 0.30/0.53 ( ( finite_finite2 @ A @ A8 ) % 0.30/0.53 => ( ( A8 % 0.30/0.53 != ( bot_bot @ ( set @ A ) ) ) % 0.30/0.53 => ( ( ord_less_eq @ ( set @ A ) @ A8 @ B3 ) % 0.30/0.53 => ( ! [X2: A] : % 0.30/0.53 ( ( member @ A @ X2 @ A8 ) % 0.30/0.53 => ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) % 0.30/0.53 => ( P @ A8 ) ) ) ) ) % 0.30/0.53 => ( P @ B3 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % finite_remove_induct % 0.30/0.53 thf(fact_2039_signed__take__bit__int__greater__self__iff,axiom, % 0.30/0.53 ! [K: int,N2: nat] : % 0.30/0.53 ( ( ord_less @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) ) % 0.30/0.53 = ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % signed_take_bit_int_greater_self_iff % 0.30/0.53 thf(fact_2040_finite__induct__select,axiom, % 0.30/0.53 ! [A: $tType,S3: set @ A,P: ( set @ A ) > $o] : % 0.30/0.53 ( ( finite_finite2 @ A @ S3 ) % 0.30/0.53 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 0.30/0.53 => ( ! [T5: set @ A] : % 0.30/0.53 ( ( ord_less @ ( set @ A ) @ T5 @ S3 ) % 0.30/0.53 => ( ( P @ T5 ) % 0.30/0.53 => ? [X2: A] : % 0.30/0.53 ( ( member @ A @ X2 @ ( minus_minus @ ( set @ A ) @ S3 @ T5 ) ) % 0.30/0.53 & ( P @ ( insert @ A @ X2 @ T5 ) ) ) ) ) % 0.30/0.53 => ( P @ S3 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % finite_induct_select % 0.30/0.53 thf(fact_2041_psubset__insert__iff,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,X3: A,B3: set @ A] : % 0.30/0.53 ( ( ord_less @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) ) % 0.30/0.53 = ( ( ( member @ A @ X3 @ B3 ) % 0.30/0.53 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) % 0.30/0.53 & ( ~ ( member @ A @ X3 @ B3 ) % 0.30/0.53 => ( ( ( member @ A @ X3 @ A3 ) % 0.30/0.53 => ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) % 0.30/0.53 & ( ~ ( member @ A @ X3 @ A3 ) % 0.30/0.53 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % psubset_insert_iff % 0.30/0.53 thf(fact_2042_decr__mult__lemma,axiom, % 0.30/0.53 ! [D2: int,P: int > $o,K: int] : % 0.30/0.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 ) % 0.30/0.53 => ( ! [X5: int] : % 0.30/0.53 ( ( P @ X5 ) % 0.30/0.53 => ( P @ ( minus_minus @ int @ X5 @ D2 ) ) ) % 0.30/0.53 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.53 => ! [X2: int] : % 0.30/0.53 ( ( P @ X2 ) % 0.30/0.53 => ( P @ ( minus_minus @ int @ X2 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % decr_mult_lemma % 0.30/0.53 thf(fact_2043_verit__less__mono__div__int2,axiom, % 0.30/0.53 ! [A3: int,B3: int,N2: int] : % 0.30/0.53 ( ( ord_less_eq @ int @ A3 @ B3 ) % 0.30/0.53 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N2 ) ) % 0.30/0.53 => ( ord_less_eq @ int @ ( divide_divide @ int @ B3 @ N2 ) @ ( divide_divide @ int @ A3 @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % verit_less_mono_div_int2 % 0.30/0.53 thf(fact_2044_div__eq__minus1,axiom, % 0.30/0.53 ! [B2: int] : % 0.30/0.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.30/0.53 => ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 ) % 0.30/0.53 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % div_eq_minus1 % 0.30/0.53 thf(fact_2045_take__bit__Suc__bit0,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.53 => ! [N2: nat,K: num] : % 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) % 0.30/0.53 = ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_Suc_bit0 % 0.30/0.53 thf(fact_2046_take__bit__nat__eq__self,axiom, % 0.30/0.53 ! [M: nat,N2: nat] : % 0.30/0.53 ( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) % 0.30/0.53 = M ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_nat_eq_self % 0.30/0.53 thf(fact_2047_take__bit__nat__less__exp,axiom, % 0.30/0.53 ! [N2: nat,M: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_nat_less_exp % 0.30/0.53 thf(fact_2048_take__bit__nat__eq__self__iff,axiom, % 0.30/0.53 ! [N2: nat,M: nat] : % 0.30/0.53 ( ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) % 0.30/0.53 = M ) % 0.30/0.53 = ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_nat_eq_self_iff % 0.30/0.53 thf(fact_2049_take__bit__int__less__self__iff,axiom, % 0.30/0.53 ! [N2: nat,K: int] : % 0.30/0.53 ( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ K ) % 0.30/0.53 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_int_less_self_iff % 0.30/0.53 thf(fact_2050_take__bit__int__greater__eq__self__iff,axiom, % 0.30/0.53 ! [K: int,N2: nat] : % 0.30/0.53 ( ( ord_less_eq @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) % 0.30/0.53 = ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_int_greater_eq_self_iff % 0.30/0.53 thf(fact_2051_le__minus__divide__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [A2: A,B2: A,C2: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) ) % 0.30/0.53 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % le_minus_divide_eq % 0.30/0.53 thf(fact_2052_minus__divide__le__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [B2: A,C2: A,A2: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 ) % 0.30/0.53 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % minus_divide_le_eq % 0.30/0.53 thf(fact_2053_neg__le__minus__divide__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [C2: A,A2: A,B2: A] : % 0.30/0.53 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) ) % 0.30/0.53 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_le_minus_divide_eq % 0.30/0.53 thf(fact_2054_neg__minus__divide__le__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [C2: A,B2: A,A2: A] : % 0.30/0.53 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 ) % 0.30/0.53 = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_minus_divide_le_eq % 0.30/0.53 thf(fact_2055_pos__le__minus__divide__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [C2: A,A2: A,B2: A] : % 0.30/0.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) ) % 0.30/0.53 = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % pos_le_minus_divide_eq % 0.30/0.53 thf(fact_2056_pos__minus__divide__le__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [C2: A,B2: A,A2: A] : % 0.30/0.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 ) % 0.30/0.53 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % pos_minus_divide_le_eq % 0.30/0.53 thf(fact_2057_divide__less__eq__numeral_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [B2: A,C2: A,W2: num] : % 0.30/0.53 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) % 0.30/0.53 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % divide_less_eq_numeral(2) % 0.30/0.53 thf(fact_2058_less__divide__eq__numeral_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [W2: num,B2: A,C2: A] : % 0.30/0.53 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.53 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % less_divide_eq_numeral(2) % 0.30/0.53 thf(fact_2059_signed__take__bit__int__less__eq,axiom, % 0.30/0.53 ! [N2: nat,K: int] : % 0.30/0.53 ( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) % 0.30/0.53 => ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % signed_take_bit_int_less_eq % 0.30/0.53 thf(fact_2060_neg__one__power__add__eq__neg__one__power__diff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ring_1 @ A ) % 0.30/0.53 => ! [K: nat,N2: nat] : % 0.30/0.53 ( ( ord_less_eq @ nat @ K @ N2 ) % 0.30/0.53 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N2 @ K ) ) % 0.30/0.53 = ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % neg_one_power_add_eq_neg_one_power_diff % 0.30/0.53 thf(fact_2061_signed__take__bit__int__eq__self,axiom, % 0.30/0.53 ! [N2: nat,K: int] : % 0.30/0.53 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ K ) % 0.30/0.53 => ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.53 => ( ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) % 0.30/0.53 = K ) ) ) ). % 0.30/0.53 % 0.30/0.53 % signed_take_bit_int_eq_self % 0.30/0.53 thf(fact_2062_signed__take__bit__int__eq__self__iff,axiom, % 0.30/0.53 ! [N2: nat,K: int] : % 0.30/0.53 ( ( ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) % 0.30/0.53 = K ) % 0.30/0.53 = ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ K ) % 0.30/0.53 & ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % signed_take_bit_int_eq_self_iff % 0.30/0.53 thf(fact_2063_take__bit__eq__0__iff,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.53 => ! [N2: nat,A2: A] : % 0.30/0.53 ( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) % 0.30/0.53 = ( zero_zero @ A ) ) % 0.30/0.53 = ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ A2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_eq_0_iff % 0.30/0.53 thf(fact_2064_take__bit__nat__less__self__iff,axiom, % 0.30/0.53 ! [N2: nat,M: nat] : % 0.30/0.53 ( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) @ M ) % 0.30/0.53 = ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_nat_less_self_iff % 0.30/0.53 thf(fact_2065_take__bit__int__eq__self,axiom, % 0.30/0.53 ! [K: int,N2: nat] : % 0.30/0.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.53 => ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) % 0.30/0.53 = K ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_int_eq_self % 0.30/0.53 thf(fact_2066_take__bit__int__eq__self__iff,axiom, % 0.30/0.53 ! [N2: nat,K: int] : % 0.30/0.53 ( ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) % 0.30/0.53 = K ) % 0.30/0.53 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.53 & ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_int_eq_self_iff % 0.30/0.53 thf(fact_2067_divide__le__eq__numeral_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [B2: A,C2: A,W2: num] : % 0.30/0.53 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) % 0.30/0.53 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % divide_le_eq_numeral(2) % 0.30/0.53 thf(fact_2068_le__divide__eq__numeral_I2_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_field @ A ) % 0.30/0.53 => ! [W2: num,B2: A,C2: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) ) % 0.30/0.53 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.30/0.53 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) ) % 0.30/0.53 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 0.30/0.53 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % le_divide_eq_numeral(2) % 0.30/0.53 thf(fact_2069_square__le__1,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( linordered_idom @ A ) % 0.30/0.53 => ! [X3: A] : % 0.30/0.53 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X3 ) % 0.30/0.53 => ( ( ord_less_eq @ A @ X3 @ ( one_one @ A ) ) % 0.30/0.53 => ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % square_le_1 % 0.30/0.53 thf(fact_2070_div__pos__geq,axiom, % 0.30/0.53 ! [L: int,K: int] : % 0.30/0.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ L ) % 0.30/0.53 => ( ( ord_less_eq @ int @ L @ K ) % 0.30/0.53 => ( ( divide_divide @ int @ K @ L ) % 0.30/0.53 = ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % div_pos_geq % 0.30/0.53 thf(fact_2071_div__pos__neg__trivial,axiom, % 0.30/0.53 ! [K: int,L: int] : % 0.30/0.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.53 => ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) ) % 0.30/0.53 => ( ( divide_divide @ int @ K @ L ) % 0.30/0.53 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % div_pos_neg_trivial % 0.30/0.53 thf(fact_2072_take__bit__int__greater__eq,axiom, % 0.30/0.53 ! [K: int,N2: nat] : % 0.30/0.53 ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 0.30/0.53 => ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_int_greater_eq % 0.30/0.53 thf(fact_2073_power__minus1__odd,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( ring_1 @ A ) % 0.30/0.53 => ! [N2: nat] : % 0.30/0.53 ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 0.30/0.53 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % power_minus1_odd % 0.30/0.53 thf(fact_2074_take__bit__Suc,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.53 => ! [N2: nat,A2: A] : % 0.30/0.53 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A2 ) % 0.30/0.53 = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % take_bit_Suc % 0.30/0.53 thf(fact_2075_signed__take__bit__int__greater__eq,axiom, % 0.30/0.53 ! [K: int,N2: nat] : % 0.30/0.53 ( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 0.30/0.53 => ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) ) ) ). % 0.30/0.53 % 0.30/0.53 % signed_take_bit_int_greater_eq % 0.30/0.53 thf(fact_2076_stable__imp__take__bit__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.53 => ! [A2: A,N2: nat] : % 0.30/0.53 ( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.53 = A2 ) % 0.30/0.53 => ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) % 0.30/0.53 = ( zero_zero @ A ) ) ) % 0.30/0.53 & ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.53 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) % 0.30/0.53 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % stable_imp_take_bit_eq % 0.30/0.53 thf(fact_2077_signed__take__bit__Suc__minus__bit1,axiom, % 0.30/0.53 ! [N2: nat,K: num] : % 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) ) % 0.30/0.53 = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ). % 0.30/0.53 % 0.30/0.53 % signed_take_bit_Suc_minus_bit1 % 0.30/0.53 thf(fact_2078_concat__bit__Suc,axiom, % 0.30/0.53 ! [N2: nat,K: int,L: int] : % 0.30/0.53 ( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L ) % 0.30/0.53 = ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % concat_bit_Suc % 0.30/0.53 thf(fact_2079_Bolzano,axiom, % 0.30/0.53 ! [A2: real,B2: real,P: real > real > $o] : % 0.30/0.53 ( ( ord_less_eq @ real @ A2 @ B2 ) % 0.30/0.53 => ( ! [A5: real,B5: real,C3: real] : % 0.30/0.53 ( ( P @ A5 @ B5 ) % 0.30/0.53 => ( ( P @ B5 @ C3 ) % 0.30/0.53 => ( ( ord_less_eq @ real @ A5 @ B5 ) % 0.30/0.53 => ( ( ord_less_eq @ real @ B5 @ C3 ) % 0.30/0.53 => ( P @ A5 @ C3 ) ) ) ) ) % 0.30/0.53 => ( ! [X5: real] : % 0.30/0.53 ( ( ord_less_eq @ real @ A2 @ X5 ) % 0.30/0.53 => ( ( ord_less_eq @ real @ X5 @ B2 ) % 0.30/0.53 => ? [D4: real] : % 0.30/0.53 ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 ) % 0.30/0.53 & ! [A5: real,B5: real] : % 0.30/0.53 ( ( ( ord_less_eq @ real @ A5 @ X5 ) % 0.30/0.53 & ( ord_less_eq @ real @ X5 @ B5 ) % 0.30/0.53 & ( ord_less @ real @ ( minus_minus @ real @ B5 @ A5 ) @ D4 ) ) % 0.30/0.53 => ( P @ A5 @ B5 ) ) ) ) ) % 0.30/0.53 => ( P @ A2 @ B2 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Bolzano % 0.30/0.53 thf(fact_2080_signed__take__bit__Suc__bit1,axiom, % 0.30/0.53 ! [N2: nat,K: num] : % 0.30/0.53 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) % 0.30/0.53 = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ). % 0.30/0.53 % 0.30/0.53 % signed_take_bit_Suc_bit1 % 0.30/0.53 thf(fact_2081_and__int_Oelims,axiom, % 0.30/0.53 ! [X3: int,Xa2: int,Y: int] : % 0.30/0.53 ( ( ( bit_se5824344872417868541ns_and @ int @ X3 @ Xa2 ) % 0.30/0.53 = Y ) % 0.30/0.53 => ( ( ( ( member @ int @ X3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 0.30/0.53 & ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 0.30/0.53 => ( Y % 0.30/0.53 = ( uminus_uminus @ int % 0.30/0.53 @ ( zero_neq_one_of_bool @ int % 0.30/0.53 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 ) % 0.30/0.53 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) ) ) ) % 0.30/0.53 & ( ~ ( ( member @ int @ X3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 0.30/0.53 & ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 0.30/0.53 => ( Y % 0.30/0.53 = ( plus_plus @ int % 0.30/0.53 @ ( zero_neq_one_of_bool @ int % 0.30/0.53 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 ) % 0.30/0.53 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) % 0.30/0.53 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % and_int.elims % 0.30/0.53 thf(fact_2082_and__int_Osimps,axiom, % 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ int ) % 0.30/0.53 = ( ^ [K3: int,L2: int] : % 0.30/0.53 ( if @ int % 0.30/0.53 @ ( ( member @ int @ K3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 0.30/0.53 & ( member @ int @ L2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 0.30/0.53 @ ( uminus_uminus @ int % 0.30/0.53 @ ( zero_neq_one_of_bool @ int % 0.30/0.53 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) % 0.30/0.53 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) ) % 0.30/0.53 @ ( plus_plus @ int % 0.30/0.53 @ ( zero_neq_one_of_bool @ int % 0.30/0.53 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) % 0.30/0.53 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) % 0.30/0.53 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % and_int.simps % 0.30/0.53 thf(fact_2083_Diff__idemp,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.53 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ B3 ) % 0.30/0.53 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ). % 0.30/0.53 % 0.30/0.53 % Diff_idemp % 0.30/0.53 thf(fact_2084_Diff__iff,axiom, % 0.30/0.53 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 0.30/0.53 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 0.30/0.53 = ( ( member @ A @ C2 @ A3 ) % 0.30/0.53 & ~ ( member @ A @ C2 @ B3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Diff_iff % 0.30/0.53 thf(fact_2085_DiffI,axiom, % 0.30/0.53 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 0.30/0.53 ( ( member @ A @ C2 @ A3 ) % 0.30/0.53 => ( ~ ( member @ A @ C2 @ B3 ) % 0.30/0.53 => ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % DiffI % 0.30/0.53 thf(fact_2086_ComplI,axiom, % 0.30/0.53 ! [A: $tType,C2: A,A3: set @ A] : % 0.30/0.53 ( ~ ( member @ A @ C2 @ A3 ) % 0.30/0.53 => ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % ComplI % 0.30/0.53 thf(fact_2087_Compl__iff,axiom, % 0.30/0.53 ! [A: $tType,C2: A,A3: set @ A] : % 0.30/0.53 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) % 0.30/0.53 = ( ~ ( member @ A @ C2 @ A3 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Compl_iff % 0.30/0.53 thf(fact_2088_Compl__eq__Compl__iff,axiom, % 0.30/0.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.30/0.53 ( ( ( uminus_uminus @ ( set @ A ) @ A3 ) % 0.30/0.53 = ( uminus_uminus @ ( set @ A ) @ B3 ) ) % 0.30/0.53 = ( A3 = B3 ) ) ). % 0.30/0.53 % 0.30/0.53 % Compl_eq_Compl_iff % 0.30/0.53 thf(fact_2089_and__zero__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % and_zero_eq % 0.30/0.53 thf(fact_2090_zero__and__eq,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.53 => ! [A2: A] : % 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % zero_and_eq % 0.30/0.53 thf(fact_2091_bit_Oconj__zero__left,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A ) % 0.30/0.53 => ! [X3: A] : % 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % bit.conj_zero_left % 0.30/0.53 thf(fact_2092_bit_Oconj__zero__right,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_ri3973907225187159222ations @ A ) % 0.30/0.53 => ! [X3: A] : % 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ X3 @ ( zero_zero @ A ) ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % bit.conj_zero_right % 0.30/0.53 thf(fact_2093_concat__bit__0,axiom, % 0.30/0.53 ! [K: int,L: int] : % 0.30/0.53 ( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L ) % 0.30/0.53 = L ) ). % 0.30/0.53 % 0.30/0.53 % concat_bit_0 % 0.30/0.53 thf(fact_2094_semiring__norm_I80_J,axiom, % 0.30/0.53 ! [M: num,N2: num] : % 0.30/0.53 ( ( ord_less @ num @ ( bit1 @ M ) @ ( bit1 @ N2 ) ) % 0.30/0.53 = ( ord_less @ num @ M @ N2 ) ) ). % 0.30/0.53 % 0.30/0.53 % semiring_norm(80) % 0.30/0.53 thf(fact_2095_and__negative__int__iff,axiom, % 0.30/0.53 ! [K: int,L: int] : % 0.30/0.53 ( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ ( zero_zero @ int ) ) % 0.30/0.53 = ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 0.30/0.53 & ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ). % 0.30/0.53 % 0.30/0.53 % and_negative_int_iff % 0.30/0.53 thf(fact_2096_semiring__norm_I81_J,axiom, % 0.30/0.53 ! [M: num,N2: num] : % 0.30/0.53 ( ( ord_less @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) ) % 0.30/0.53 = ( ord_less @ num @ M @ N2 ) ) ). % 0.30/0.53 % 0.30/0.53 % semiring_norm(81) % 0.30/0.53 thf(fact_2097_semiring__norm_I77_J,axiom, % 0.30/0.53 ! [N2: num] : ( ord_less @ num @ one2 @ ( bit1 @ N2 ) ) ). % 0.30/0.53 % 0.30/0.53 % semiring_norm(77) % 0.30/0.53 thf(fact_2098_concat__bit__negative__iff,axiom, % 0.30/0.53 ! [N2: nat,K: int,L: int] : % 0.30/0.53 ( ( ord_less @ int @ ( bit_concat_bit @ N2 @ K @ L ) @ ( zero_zero @ int ) ) % 0.30/0.53 = ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ). % 0.30/0.53 % 0.30/0.53 % concat_bit_negative_iff % 0.30/0.53 thf(fact_2099_semiring__norm_I79_J,axiom, % 0.30/0.53 ! [M: num,N2: num] : % 0.30/0.53 ( ( ord_less @ num @ ( bit0 @ M ) @ ( bit1 @ N2 ) ) % 0.30/0.53 = ( ord_less_eq @ num @ M @ N2 ) ) ). % 0.30/0.53 % 0.30/0.53 % semiring_norm(79) % 0.30/0.53 thf(fact_2100_semiring__norm_I74_J,axiom, % 0.30/0.53 ! [M: num,N2: num] : % 0.30/0.53 ( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) ) % 0.30/0.53 = ( ord_less @ num @ M @ N2 ) ) ). % 0.30/0.53 % 0.30/0.53 % semiring_norm(74) % 0.30/0.53 thf(fact_2101_and__numerals_I5_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.53 => ! [X3: num] : % 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( one_one @ A ) ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % and_numerals(5) % 0.30/0.53 thf(fact_2102_and__numerals_I1_J,axiom, % 0.30/0.53 ! [A: $tType] : % 0.30/0.53 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.53 => ! [Y: num] : % 0.30/0.53 ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) ) % 0.30/0.53 = ( zero_zero @ A ) ) ) ). % 0.30/0.53 % 0.30/0.53 % and_numerals(1) % 0.30/0.53 thf(fact_2103_div__Suc__eq__div__add3,axiom, % 0.30/0.53 ! [M: nat,N2: nat] : % 0.30/0.53 ( ( divide_divide @ nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) ) % 0.30/0.53 = ( divide_divide @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % div_Suc_eq_div_add3 % 0.30/0.53 thf(fact_2104_Suc__div__eq__add3__div__numeral,axiom, % 0.30/0.53 ! [M: nat,V2: num] : % 0.30/0.53 ( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V2 ) ) % 0.30/0.53 = ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V2 ) ) ) ). % 0.30/0.53 % 0.30/0.53 % Suc_div_eq_add3_div_numeral % 0.30/0.53 thf(fact_2105_mod__Suc__eq__mod__add3,axiom, % 0.30/0.53 ! [M: nat,N2: nat] : % 0.30/0.53 ( ( modulo_modulo @ nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) ) % 0.30/0.53 = ( modulo_modulo @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % mod_Suc_eq_mod_add3 % 0.30/0.54 thf(fact_2106_Suc__mod__eq__add3__mod__numeral,axiom, % 0.30/0.54 ! [M: nat,V2: num] : % 0.30/0.54 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V2 ) ) % 0.30/0.54 = ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % Suc_mod_eq_add3_mod_numeral % 0.30/0.54 thf(fact_2107_and__numerals_I7_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.54 => ! [X3: num,Y: num] : % 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) % 0.30/0.54 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % and_numerals(7) % 0.30/0.54 thf(fact_2108_DiffD2,axiom, % 0.30/0.54 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 0.30/0.54 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 0.30/0.54 => ~ ( member @ A @ C2 @ B3 ) ) ). % 0.30/0.54 % 0.30/0.54 % DiffD2 % 0.30/0.54 thf(fact_2109_DiffD1,axiom, % 0.30/0.54 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 0.30/0.54 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 0.30/0.54 => ( member @ A @ C2 @ A3 ) ) ). % 0.30/0.54 % 0.30/0.54 % DiffD1 % 0.30/0.54 thf(fact_2110_DiffE,axiom, % 0.30/0.54 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 0.30/0.54 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 0.30/0.54 => ~ ( ( member @ A @ C2 @ A3 ) % 0.30/0.54 => ( member @ A @ C2 @ B3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % DiffE % 0.30/0.54 thf(fact_2111_ComplD,axiom, % 0.30/0.54 ! [A: $tType,C2: A,A3: set @ A] : % 0.30/0.54 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) % 0.30/0.54 => ~ ( member @ A @ C2 @ A3 ) ) ). % 0.30/0.54 % 0.30/0.54 % ComplD % 0.30/0.54 thf(fact_2112_double__complement,axiom, % 0.30/0.54 ! [A: $tType,A3: set @ A] : % 0.30/0.54 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) % 0.30/0.54 = A3 ) ). % 0.30/0.54 % 0.30/0.54 % double_complement % 0.30/0.54 thf(fact_2113_concat__bit__assoc,axiom, % 0.30/0.54 ! [N2: nat,K: int,M: nat,L: int,R2: int] : % 0.30/0.54 ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L @ R2 ) ) % 0.30/0.54 = ( bit_concat_bit @ ( plus_plus @ nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R2 ) ) ). % 0.30/0.54 % 0.30/0.54 % concat_bit_assoc % 0.30/0.54 thf(fact_2114_xor__num_Ocases,axiom, % 0.30/0.54 ! [X3: product_prod @ num @ num] : % 0.30/0.54 ( ( X3 % 0.30/0.54 != ( product_Pair @ num @ num @ one2 @ one2 ) ) % 0.30/0.54 => ( ! [N3: num] : % 0.30/0.54 ( X3 % 0.30/0.54 != ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) % 0.30/0.54 => ( ! [N3: num] : % 0.30/0.54 ( X3 % 0.30/0.54 != ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) % 0.30/0.54 => ( ! [M3: num] : % 0.30/0.54 ( X3 % 0.30/0.54 != ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ one2 ) ) % 0.30/0.54 => ( ! [M3: num,N3: num] : % 0.30/0.54 ( X3 % 0.30/0.54 != ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) ) % 0.30/0.54 => ( ! [M3: num,N3: num] : % 0.30/0.54 ( X3 % 0.30/0.54 != ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) ) % 0.30/0.54 => ( ! [M3: num] : % 0.30/0.54 ( X3 % 0.30/0.54 != ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ one2 ) ) % 0.30/0.54 => ( ! [M3: num,N3: num] : % 0.30/0.54 ( X3 % 0.30/0.54 != ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) ) % 0.30/0.54 => ~ ! [M3: num,N3: num] : % 0.30/0.54 ( X3 % 0.30/0.54 != ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % xor_num.cases % 0.30/0.54 thf(fact_2115_and__less__eq,axiom, % 0.30/0.54 ! [L: int,K: int] : % 0.30/0.54 ( ( ord_less @ int @ L @ ( zero_zero @ int ) ) % 0.30/0.54 => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ K ) ) ). % 0.30/0.54 % 0.30/0.54 % and_less_eq % 0.30/0.54 thf(fact_2116_AND__upper1_H_H,axiom, % 0.30/0.54 ! [Y: int,Z2: int,Ya: int] : % 0.30/0.54 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y ) % 0.30/0.54 => ( ( ord_less @ int @ Y @ Z2 ) % 0.30/0.54 => ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya ) @ Z2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % AND_upper1'' % 0.30/0.54 thf(fact_2117_AND__upper2_H_H,axiom, % 0.30/0.54 ! [Y: int,Z2: int,X3: int] : % 0.30/0.54 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y ) % 0.30/0.54 => ( ( ord_less @ int @ Y @ Z2 ) % 0.30/0.54 => ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X3 @ Y ) @ Z2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % AND_upper2'' % 0.30/0.54 thf(fact_2118_numeral__Bit1,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( numeral @ A ) % 0.30/0.54 => ! [N2: num] : % 0.30/0.54 ( ( numeral_numeral @ A @ ( bit1 @ N2 ) ) % 0.30/0.54 = ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( one_one @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % numeral_Bit1 % 0.30/0.54 thf(fact_2119_simp__from__to,axiom, % 0.30/0.54 ( ( set_or1337092689740270186AtMost @ int ) % 0.30/0.54 = ( ^ [I3: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I3 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I3 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % simp_from_to % 0.30/0.54 thf(fact_2120_eval__nat__numeral_I3_J,axiom, % 0.30/0.54 ! [N2: num] : % 0.30/0.54 ( ( numeral_numeral @ nat @ ( bit1 @ N2 ) ) % 0.30/0.54 = ( suc @ ( numeral_numeral @ nat @ ( bit0 @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % eval_nat_numeral(3) % 0.30/0.54 thf(fact_2121_cong__exp__iff__simps_I3_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.54 => ! [N2: num,Q4: num] : % 0.30/0.54 ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) % 0.30/0.54 != ( zero_zero @ A ) ) ) ). % 0.30/0.54 % 0.30/0.54 % cong_exp_iff_simps(3) % 0.30/0.54 thf(fact_2122_numeral__3__eq__3,axiom, % 0.30/0.54 ( ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) % 0.30/0.54 = ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % numeral_3_eq_3 % 0.30/0.54 thf(fact_2123_Suc3__eq__add__3,axiom, % 0.30/0.54 ! [N2: nat] : % 0.30/0.54 ( ( suc @ ( suc @ ( suc @ N2 ) ) ) % 0.30/0.54 = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N2 ) ) ). % 0.30/0.54 % 0.30/0.54 % Suc3_eq_add_3 % 0.30/0.54 thf(fact_2124_num_Osize_I6_J,axiom, % 0.30/0.54 ! [X32: num] : % 0.30/0.54 ( ( size_size @ num @ ( bit1 @ X32 ) ) % 0.30/0.54 = ( plus_plus @ nat @ ( size_size @ num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % num.size(6) % 0.30/0.54 thf(fact_2125_num_Osize__gen_I3_J,axiom, % 0.30/0.54 ! [X32: num] : % 0.30/0.54 ( ( size_num @ ( bit1 @ X32 ) ) % 0.30/0.54 = ( plus_plus @ nat @ ( size_num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % num.size_gen(3) % 0.30/0.54 thf(fact_2126_cong__exp__iff__simps_I11_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.54 => ! [M: num,Q4: num] : % 0.30/0.54 ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) % 0.30/0.54 = ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) ) % 0.30/0.54 = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q4 ) ) % 0.30/0.54 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % cong_exp_iff_simps(11) % 0.30/0.54 thf(fact_2127_cong__exp__iff__simps_I7_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.54 => ! [Q4: num,N2: num] : % 0.30/0.54 ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) % 0.30/0.54 = ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) ) % 0.30/0.54 = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ Q4 ) ) % 0.30/0.54 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % cong_exp_iff_simps(7) % 0.30/0.54 thf(fact_2128_periodic__finite__ex,axiom, % 0.30/0.54 ! [D2: int,P: int > $o] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 ) % 0.30/0.54 => ( ! [X5: int,K2: int] : % 0.30/0.54 ( ( P @ X5 ) % 0.30/0.54 = ( P @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D2 ) ) ) ) % 0.30/0.54 => ( ( ? [X6: int] : ( P @ X6 ) ) % 0.30/0.54 = ( ? [X: int] : % 0.30/0.54 ( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) ) % 0.30/0.54 & ( P @ X ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % periodic_finite_ex % 0.30/0.54 thf(fact_2129_bset_I3_J,axiom, % 0.30/0.54 ! [D6: int,T2: int,B3: set @ int] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 0.30/0.54 => ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B3 ) % 0.30/0.54 => ! [X2: int] : % 0.30/0.54 ( ! [Xa3: int] : % 0.30/0.54 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 => ! [Xb2: int] : % 0.30/0.54 ( ( member @ int @ Xb2 @ B3 ) % 0.30/0.54 => ( X2 % 0.30/0.54 != ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) ) % 0.30/0.54 => ( ( X2 = T2 ) % 0.30/0.54 => ( ( minus_minus @ int @ X2 @ D6 ) % 0.30/0.54 = T2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bset(3) % 0.30/0.54 thf(fact_2130_bset_I4_J,axiom, % 0.30/0.54 ! [D6: int,T2: int,B3: set @ int] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 0.30/0.54 => ( ( member @ int @ T2 @ B3 ) % 0.30/0.54 => ! [X2: int] : % 0.30/0.54 ( ! [Xa3: int] : % 0.30/0.54 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 => ! [Xb2: int] : % 0.30/0.54 ( ( member @ int @ Xb2 @ B3 ) % 0.30/0.54 => ( X2 % 0.30/0.54 != ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) ) % 0.30/0.54 => ( ( X2 != T2 ) % 0.30/0.54 => ( ( minus_minus @ int @ X2 @ D6 ) % 0.30/0.54 != T2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bset(4) % 0.30/0.54 thf(fact_2131_bset_I5_J,axiom, % 0.30/0.54 ! [D6: int,B3: set @ int,T2: int] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 0.30/0.54 => ! [X2: int] : % 0.30/0.54 ( ! [Xa3: int] : % 0.30/0.54 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 => ! [Xb2: int] : % 0.30/0.54 ( ( member @ int @ Xb2 @ B3 ) % 0.30/0.54 => ( X2 % 0.30/0.54 != ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) ) % 0.30/0.54 => ( ( ord_less @ int @ X2 @ T2 ) % 0.30/0.54 => ( ord_less @ int @ ( minus_minus @ int @ X2 @ D6 ) @ T2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bset(5) % 0.30/0.54 thf(fact_2132_bset_I7_J,axiom, % 0.30/0.54 ! [D6: int,T2: int,B3: set @ int] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 0.30/0.54 => ( ( member @ int @ T2 @ B3 ) % 0.30/0.54 => ! [X2: int] : % 0.30/0.54 ( ! [Xa3: int] : % 0.30/0.54 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 => ! [Xb2: int] : % 0.30/0.54 ( ( member @ int @ Xb2 @ B3 ) % 0.30/0.54 => ( X2 % 0.30/0.54 != ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) ) % 0.30/0.54 => ( ( ord_less @ int @ T2 @ X2 ) % 0.30/0.54 => ( ord_less @ int @ T2 @ ( minus_minus @ int @ X2 @ D6 ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bset(7) % 0.30/0.54 thf(fact_2133_aset_I3_J,axiom, % 0.30/0.54 ! [D6: int,T2: int,A3: set @ int] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 0.30/0.54 => ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 ) % 0.30/0.54 => ! [X2: int] : % 0.30/0.54 ( ! [Xa3: int] : % 0.30/0.54 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 => ! [Xb2: int] : % 0.30/0.54 ( ( member @ int @ Xb2 @ A3 ) % 0.30/0.54 => ( X2 % 0.30/0.54 != ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) ) % 0.30/0.54 => ( ( X2 = T2 ) % 0.30/0.54 => ( ( plus_plus @ int @ X2 @ D6 ) % 0.30/0.54 = T2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % aset(3) % 0.30/0.54 thf(fact_2134_aset_I4_J,axiom, % 0.30/0.54 ! [D6: int,T2: int,A3: set @ int] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 0.30/0.54 => ( ( member @ int @ T2 @ A3 ) % 0.30/0.54 => ! [X2: int] : % 0.30/0.54 ( ! [Xa3: int] : % 0.30/0.54 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 => ! [Xb2: int] : % 0.30/0.54 ( ( member @ int @ Xb2 @ A3 ) % 0.30/0.54 => ( X2 % 0.30/0.54 != ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) ) % 0.30/0.54 => ( ( X2 != T2 ) % 0.30/0.54 => ( ( plus_plus @ int @ X2 @ D6 ) % 0.30/0.54 != T2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % aset(4) % 0.30/0.54 thf(fact_2135_aset_I5_J,axiom, % 0.30/0.54 ! [D6: int,T2: int,A3: set @ int] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 0.30/0.54 => ( ( member @ int @ T2 @ A3 ) % 0.30/0.54 => ! [X2: int] : % 0.30/0.54 ( ! [Xa3: int] : % 0.30/0.54 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 => ! [Xb2: int] : % 0.30/0.54 ( ( member @ int @ Xb2 @ A3 ) % 0.30/0.54 => ( X2 % 0.30/0.54 != ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) ) % 0.30/0.54 => ( ( ord_less @ int @ X2 @ T2 ) % 0.30/0.54 => ( ord_less @ int @ ( plus_plus @ int @ X2 @ D6 ) @ T2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % aset(5) % 0.30/0.54 thf(fact_2136_aset_I7_J,axiom, % 0.30/0.54 ! [D6: int,A3: set @ int,T2: int] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 0.30/0.54 => ! [X2: int] : % 0.30/0.54 ( ! [Xa3: int] : % 0.30/0.54 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 => ! [Xb2: int] : % 0.30/0.54 ( ( member @ int @ Xb2 @ A3 ) % 0.30/0.54 => ( X2 % 0.30/0.54 != ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) ) % 0.30/0.54 => ( ( ord_less @ int @ T2 @ X2 ) % 0.30/0.54 => ( ord_less @ int @ T2 @ ( plus_plus @ int @ X2 @ D6 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % aset(7) % 0.30/0.54 thf(fact_2137_Suc__div__eq__add3__div,axiom, % 0.30/0.54 ! [M: nat,N2: nat] : % 0.30/0.54 ( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 ) % 0.30/0.54 = ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N2 ) ) ). % 0.30/0.54 % 0.30/0.54 % Suc_div_eq_add3_div % 0.30/0.54 thf(fact_2138_Suc__mod__eq__add3__mod,axiom, % 0.30/0.54 ! [M: nat,N2: nat] : % 0.30/0.54 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 ) % 0.30/0.54 = ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N2 ) ) ). % 0.30/0.54 % 0.30/0.54 % Suc_mod_eq_add3_mod % 0.30/0.54 thf(fact_2139_aset_I8_J,axiom, % 0.30/0.54 ! [D6: int,A3: set @ int,T2: int] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 0.30/0.54 => ! [X2: int] : % 0.30/0.54 ( ! [Xa3: int] : % 0.30/0.54 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 => ! [Xb2: int] : % 0.30/0.54 ( ( member @ int @ Xb2 @ A3 ) % 0.30/0.54 => ( X2 % 0.30/0.54 != ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) ) % 0.30/0.54 => ( ( ord_less_eq @ int @ T2 @ X2 ) % 0.30/0.54 => ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X2 @ D6 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % aset(8) % 0.30/0.54 thf(fact_2140_aset_I6_J,axiom, % 0.30/0.54 ! [D6: int,T2: int,A3: set @ int] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 0.30/0.54 => ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 ) % 0.30/0.54 => ! [X2: int] : % 0.30/0.54 ( ! [Xa3: int] : % 0.30/0.54 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 => ! [Xb2: int] : % 0.30/0.54 ( ( member @ int @ Xb2 @ A3 ) % 0.30/0.54 => ( X2 % 0.30/0.54 != ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) ) % 0.30/0.54 => ( ( ord_less_eq @ int @ X2 @ T2 ) % 0.30/0.54 => ( ord_less_eq @ int @ ( plus_plus @ int @ X2 @ D6 ) @ T2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % aset(6) % 0.30/0.54 thf(fact_2141_bset_I8_J,axiom, % 0.30/0.54 ! [D6: int,T2: int,B3: set @ int] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 0.30/0.54 => ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B3 ) % 0.30/0.54 => ! [X2: int] : % 0.30/0.54 ( ! [Xa3: int] : % 0.30/0.54 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 => ! [Xb2: int] : % 0.30/0.54 ( ( member @ int @ Xb2 @ B3 ) % 0.30/0.54 => ( X2 % 0.30/0.54 != ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) ) % 0.30/0.54 => ( ( ord_less_eq @ int @ T2 @ X2 ) % 0.30/0.54 => ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X2 @ D6 ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bset(8) % 0.30/0.54 thf(fact_2142_bset_I6_J,axiom, % 0.30/0.54 ! [D6: int,B3: set @ int,T2: int] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 0.30/0.54 => ! [X2: int] : % 0.30/0.54 ( ! [Xa3: int] : % 0.30/0.54 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 => ! [Xb2: int] : % 0.30/0.54 ( ( member @ int @ Xb2 @ B3 ) % 0.30/0.54 => ( X2 % 0.30/0.54 != ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) ) % 0.30/0.54 => ( ( ord_less_eq @ int @ X2 @ T2 ) % 0.30/0.54 => ( ord_less_eq @ int @ ( minus_minus @ int @ X2 @ D6 ) @ T2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bset(6) % 0.30/0.54 thf(fact_2143_cppi,axiom, % 0.30/0.54 ! [D6: int,P: int > $o,P5: int > $o,A3: set @ int] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 0.30/0.54 => ( ? [Z4: int] : % 0.30/0.54 ! [X5: int] : % 0.30/0.54 ( ( ord_less @ int @ Z4 @ X5 ) % 0.30/0.54 => ( ( P @ X5 ) % 0.30/0.54 = ( P5 @ X5 ) ) ) % 0.30/0.54 => ( ! [X5: int] : % 0.30/0.54 ( ! [Xa: int] : % 0.30/0.54 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 => ! [Xb3: int] : % 0.30/0.54 ( ( member @ int @ Xb3 @ A3 ) % 0.30/0.54 => ( X5 % 0.30/0.54 != ( minus_minus @ int @ Xb3 @ Xa ) ) ) ) % 0.30/0.54 => ( ( P @ X5 ) % 0.30/0.54 => ( P @ ( plus_plus @ int @ X5 @ D6 ) ) ) ) % 0.30/0.54 => ( ! [X5: int,K2: int] : % 0.30/0.54 ( ( P5 @ X5 ) % 0.30/0.54 = ( P5 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D6 ) ) ) ) % 0.30/0.54 => ( ( ? [X6: int] : ( P @ X6 ) ) % 0.30/0.54 = ( ? [X: int] : % 0.30/0.54 ( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 & ( P5 @ X ) ) % 0.30/0.54 | ? [X: int] : % 0.30/0.54 ( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 & ? [Y2: int] : % 0.30/0.54 ( ( member @ int @ Y2 @ A3 ) % 0.30/0.54 & ( P @ ( minus_minus @ int @ Y2 @ X ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % cppi % 0.30/0.54 thf(fact_2144_cpmi,axiom, % 0.30/0.54 ! [D6: int,P: int > $o,P5: int > $o,B3: set @ int] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 0.30/0.54 => ( ? [Z4: int] : % 0.30/0.54 ! [X5: int] : % 0.30/0.54 ( ( ord_less @ int @ X5 @ Z4 ) % 0.30/0.54 => ( ( P @ X5 ) % 0.30/0.54 = ( P5 @ X5 ) ) ) % 0.30/0.54 => ( ! [X5: int] : % 0.30/0.54 ( ! [Xa: int] : % 0.30/0.54 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 => ! [Xb3: int] : % 0.30/0.54 ( ( member @ int @ Xb3 @ B3 ) % 0.30/0.54 => ( X5 % 0.30/0.54 != ( plus_plus @ int @ Xb3 @ Xa ) ) ) ) % 0.30/0.54 => ( ( P @ X5 ) % 0.30/0.54 => ( P @ ( minus_minus @ int @ X5 @ D6 ) ) ) ) % 0.30/0.54 => ( ! [X5: int,K2: int] : % 0.30/0.54 ( ( P5 @ X5 ) % 0.30/0.54 = ( P5 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D6 ) ) ) ) % 0.30/0.54 => ( ( ? [X6: int] : ( P @ X6 ) ) % 0.30/0.54 = ( ? [X: int] : % 0.30/0.54 ( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 & ( P5 @ X ) ) % 0.30/0.54 | ? [X: int] : % 0.30/0.54 ( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 0.30/0.54 & ? [Y2: int] : % 0.30/0.54 ( ( member @ int @ Y2 @ B3 ) % 0.30/0.54 & ( P @ ( plus_plus @ int @ Y2 @ X ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % cpmi % 0.30/0.54 thf(fact_2145_take__bit__Suc__bit1,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.54 => ! [N2: nat,K: num] : % 0.30/0.54 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) ) % 0.30/0.54 = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % take_bit_Suc_bit1 % 0.30/0.54 thf(fact_2146_odd__mod__4__div__2,axiom, % 0.30/0.54 ! [N2: nat] : % 0.30/0.54 ( ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) % 0.30/0.54 = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) % 0.30/0.54 => ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % odd_mod_4_div_2 % 0.30/0.54 thf(fact_2147_mod__exhaust__less__4,axiom, % 0.30/0.54 ! [M: nat] : % 0.30/0.54 ( ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) % 0.30/0.54 = ( zero_zero @ nat ) ) % 0.30/0.54 | ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) % 0.30/0.54 = ( one_one @ nat ) ) % 0.30/0.54 | ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) % 0.30/0.54 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.54 | ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) % 0.30/0.54 = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % mod_exhaust_less_4 % 0.30/0.54 thf(fact_2148_divmod__algorithm__code_I7_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.54 => ! [M: num,N2: num] : % 0.30/0.54 ( ( ( ord_less_eq @ num @ M @ N2 ) % 0.30/0.54 => ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N2 ) ) % 0.30/0.54 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M ) ) ) ) ) % 0.30/0.54 & ( ~ ( ord_less_eq @ num @ M @ N2 ) % 0.30/0.54 => ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N2 ) ) % 0.30/0.54 = ( unique1321980374590559556d_step @ A @ ( bit1 @ N2 ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % divmod_algorithm_code(7) % 0.30/0.54 thf(fact_2149_divmod__algorithm__code_I8_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.54 => ! [M: num,N2: num] : % 0.30/0.54 ( ( ( ord_less @ num @ M @ N2 ) % 0.30/0.54 => ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N2 ) ) % 0.30/0.54 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M ) ) ) ) ) % 0.30/0.54 & ( ~ ( ord_less @ num @ M @ N2 ) % 0.30/0.54 => ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N2 ) ) % 0.30/0.54 = ( unique1321980374590559556d_step @ A @ ( bit1 @ N2 ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % divmod_algorithm_code(8) % 0.30/0.54 thf(fact_2150_take__bit__Suc__minus__bit1,axiom, % 0.30/0.54 ! [N2: nat,K: num] : % 0.30/0.54 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) ) % 0.30/0.54 = ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ). % 0.30/0.54 % 0.30/0.54 % take_bit_Suc_minus_bit1 % 0.30/0.54 thf(fact_2151_and__int_Opsimps,axiom, % 0.30/0.54 ! [K: int,L: int] : % 0.30/0.54 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) ) % 0.30/0.54 => ( ( ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 0.30/0.54 & ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 0.30/0.54 => ( ( bit_se5824344872417868541ns_and @ int @ K @ L ) % 0.30/0.54 = ( uminus_uminus @ int % 0.30/0.54 @ ( zero_neq_one_of_bool @ int % 0.30/0.54 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) % 0.30/0.54 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) ) % 0.30/0.54 & ( ~ ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 0.30/0.54 & ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 0.30/0.54 => ( ( bit_se5824344872417868541ns_and @ int @ K @ L ) % 0.30/0.54 = ( plus_plus @ int % 0.30/0.54 @ ( zero_neq_one_of_bool @ int % 0.30/0.54 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) % 0.30/0.54 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) % 0.30/0.54 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % and_int.psimps % 0.30/0.54 thf(fact_2152_and__int_Opelims,axiom, % 0.30/0.54 ! [X3: int,Xa2: int,Y: int] : % 0.30/0.54 ( ( ( bit_se5824344872417868541ns_and @ int @ X3 @ Xa2 ) % 0.30/0.54 = Y ) % 0.30/0.54 => ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X3 @ Xa2 ) ) % 0.30/0.54 => ~ ( ( ( ( ( member @ int @ X3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 0.30/0.54 & ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 0.30/0.54 => ( Y % 0.30/0.54 = ( uminus_uminus @ int % 0.30/0.54 @ ( zero_neq_one_of_bool @ int % 0.30/0.54 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 ) % 0.30/0.54 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) ) ) ) % 0.30/0.54 & ( ~ ( ( member @ int @ X3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 0.30/0.54 & ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 0.30/0.54 => ( Y % 0.30/0.54 = ( plus_plus @ int % 0.30/0.54 @ ( zero_neq_one_of_bool @ int % 0.30/0.54 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 ) % 0.30/0.54 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) % 0.30/0.54 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) % 0.30/0.54 => ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X3 @ Xa2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % and_int.pelims % 0.30/0.54 thf(fact_2153_finite__atLeastAtMost__int,axiom, % 0.30/0.54 ! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_atLeastAtMost_int % 0.30/0.54 thf(fact_2154_and__nat__numerals_I1_J,axiom, % 0.30/0.54 ! [Y: num] : % 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) ) % 0.30/0.54 = ( zero_zero @ nat ) ) ). % 0.30/0.54 % 0.30/0.54 % and_nat_numerals(1) % 0.30/0.54 thf(fact_2155_and__nat__numerals_I3_J,axiom, % 0.30/0.54 ! [X3: num] : % 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.54 = ( zero_zero @ nat ) ) ). % 0.30/0.54 % 0.30/0.54 % and_nat_numerals(3) % 0.30/0.54 thf(fact_2156_and__nat__numerals_I2_J,axiom, % 0.30/0.54 ! [Y: num] : % 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) % 0.30/0.54 = ( one_one @ nat ) ) ). % 0.30/0.54 % 0.30/0.54 % and_nat_numerals(2) % 0.30/0.54 thf(fact_2157_and__nat__numerals_I4_J,axiom, % 0.30/0.54 ! [X3: num] : % 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.54 = ( one_one @ nat ) ) ). % 0.30/0.54 % 0.30/0.54 % and_nat_numerals(4) % 0.30/0.54 thf(fact_2158_divmod__algorithm__code_I2_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.54 => ! [M: num] : % 0.30/0.54 ( ( unique8689654367752047608divmod @ A @ M @ one2 ) % 0.30/0.54 = ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M ) @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % divmod_algorithm_code(2) % 0.30/0.54 thf(fact_2159_add__neg__numeral__special_I6_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( neg_numeral @ A ) % 0.30/0.54 => ! [M: num] : % 0.30/0.54 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % add_neg_numeral_special(6) % 0.30/0.54 thf(fact_2160_add__neg__numeral__special_I5_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( neg_numeral @ A ) % 0.30/0.54 => ! [N2: num] : % 0.30/0.54 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) % 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % add_neg_numeral_special(5) % 0.30/0.54 thf(fact_2161_and__Suc__0__eq,axiom, % 0.30/0.54 ! [N2: nat] : % 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.54 = ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % and_Suc_0_eq % 0.30/0.54 thf(fact_2162_Suc__0__and__eq,axiom, % 0.30/0.54 ! [N2: nat] : % 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 0.30/0.54 = ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % Suc_0_and_eq % 0.30/0.54 thf(fact_2163_divmod__algorithm__code_I3_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.54 => ! [N2: num] : % 0.30/0.54 ( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N2 ) ) % 0.30/0.54 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % divmod_algorithm_code(3) % 0.30/0.54 thf(fact_2164_divmod__algorithm__code_I4_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.54 => ! [N2: num] : % 0.30/0.54 ( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N2 ) ) % 0.30/0.54 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % divmod_algorithm_code(4) % 0.30/0.54 thf(fact_2165_numeral__inc,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( numeral @ A ) % 0.30/0.54 => ! [X3: num] : % 0.30/0.54 ( ( numeral_numeral @ A @ ( inc @ X3 ) ) % 0.30/0.54 = ( plus_plus @ A @ ( numeral_numeral @ A @ X3 ) @ ( one_one @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % numeral_inc % 0.30/0.54 thf(fact_2166_divmod__int__def,axiom, % 0.30/0.54 ( ( unique8689654367752047608divmod @ int ) % 0.30/0.54 = ( ^ [M2: num,N: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % divmod_int_def % 0.30/0.54 thf(fact_2167_divmod__def,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.54 => ( ( unique8689654367752047608divmod @ A ) % 0.30/0.54 = ( ^ [M2: num,N: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % divmod_def % 0.30/0.54 thf(fact_2168_divmod_H__nat__def,axiom, % 0.30/0.54 ( ( unique8689654367752047608divmod @ nat ) % 0.30/0.54 = ( ^ [M2: num,N: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % divmod'_nat_def % 0.30/0.54 thf(fact_2169_and__nat__unfold,axiom, % 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ nat ) % 0.30/0.54 = ( ^ [M2: nat,N: nat] : % 0.30/0.54 ( if @ nat % 0.30/0.54 @ ( ( M2 % 0.30/0.54 = ( zero_zero @ nat ) ) % 0.30/0.54 | ( N % 0.30/0.54 = ( zero_zero @ nat ) ) ) % 0.30/0.54 @ ( zero_zero @ nat ) % 0.30/0.54 @ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % and_nat_unfold % 0.30/0.54 thf(fact_2170_and__nat__rec,axiom, % 0.30/0.54 ( ( bit_se5824344872417868541ns_and @ nat ) % 0.30/0.54 = ( ^ [M2: nat,N: nat] : % 0.30/0.54 ( plus_plus @ nat % 0.30/0.54 @ ( zero_neq_one_of_bool @ nat % 0.30/0.54 @ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) % 0.30/0.54 & ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) % 0.30/0.54 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % and_nat_rec % 0.30/0.54 thf(fact_2171_and__int_Opinduct,axiom, % 0.30/0.54 ! [A0: int,A12: int,P: int > int > $o] : % 0.30/0.54 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) ) % 0.30/0.54 => ( ! [K2: int,L4: int] : % 0.30/0.54 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L4 ) ) % 0.30/0.54 => ( ( ~ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 0.30/0.54 & ( member @ int @ L4 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 0.30/0.54 => ( P @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) % 0.30/0.54 => ( P @ K2 @ L4 ) ) ) % 0.30/0.54 => ( P @ A0 @ A12 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % and_int.pinduct % 0.30/0.54 thf(fact_2172_divmod__divmod__step,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.54 => ( ( unique8689654367752047608divmod @ A ) % 0.30/0.54 = ( ^ [M2: num,N: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M2 @ N ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M2 ) ) @ ( unique1321980374590559556d_step @ A @ N @ ( unique8689654367752047608divmod @ A @ M2 @ ( bit0 @ N ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % divmod_divmod_step % 0.30/0.54 thf(fact_2173_upto_Opinduct,axiom, % 0.30/0.54 ! [A0: int,A12: int,P: int > int > $o] : % 0.30/0.54 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) ) % 0.30/0.54 => ( ! [I4: int,J2: int] : % 0.30/0.54 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I4 @ J2 ) ) % 0.30/0.54 => ( ( ( ord_less_eq @ int @ I4 @ J2 ) % 0.30/0.54 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J2 ) ) % 0.30/0.54 => ( P @ I4 @ J2 ) ) ) % 0.30/0.54 => ( P @ A0 @ A12 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % upto.pinduct % 0.30/0.54 thf(fact_2174_signed__take__bit__eq__take__bit__minus,axiom, % 0.30/0.54 ( ( bit_ri4674362597316999326ke_bit @ int ) % 0.30/0.54 = ( ^ [N: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K3 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % signed_take_bit_eq_take_bit_minus % 0.30/0.54 thf(fact_2175_take__bit__numeral__bit1,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.54 => ! [L: num,K: num] : % 0.30/0.54 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) ) % 0.30/0.54 = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % take_bit_numeral_bit1 % 0.30/0.54 thf(fact_2176_of__int__less__neg__numeral__power__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [A2: int,X3: num,N2: nat] : % 0.30/0.54 ( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) ) % 0.30/0.54 = ( ord_less @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_less_neg_numeral_power_cancel_iff % 0.30/0.54 thf(fact_2177_neg__numeral__power__less__of__int__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [X3: num,N2: nat,A2: int] : % 0.30/0.54 ( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) ) % 0.30/0.54 = ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) @ A2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % neg_numeral_power_less_of_int_cancel_iff % 0.30/0.54 thf(fact_2178_of__int__le__neg__numeral__power__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [A2: int,X3: num,N2: nat] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) ) % 0.30/0.54 = ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_le_neg_numeral_power_cancel_iff % 0.30/0.54 thf(fact_2179_neg__numeral__power__le__of__int__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [X3: num,N2: nat,A2: int] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) ) % 0.30/0.54 = ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) @ A2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % neg_numeral_power_le_of_int_cancel_iff % 0.30/0.54 thf(fact_2180_of__int__eq__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_char_0 @ A ) % 0.30/0.54 => ! [W2: int,Z2: int] : % 0.30/0.54 ( ( ( ring_1_of_int @ A @ W2 ) % 0.30/0.54 = ( ring_1_of_int @ A @ Z2 ) ) % 0.30/0.54 = ( W2 = Z2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_eq_iff % 0.30/0.54 thf(fact_2181_bit__0__eq,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_semiring_bits @ A ) % 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ A @ ( zero_zero @ A ) ) % 0.30/0.54 = ( bot_bot @ ( nat > $o ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_0_eq % 0.30/0.54 thf(fact_2182_of__int__0,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_1 @ A ) % 0.30/0.54 => ( ( ring_1_of_int @ A @ ( zero_zero @ int ) ) % 0.30/0.54 = ( zero_zero @ A ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_0 % 0.30/0.54 thf(fact_2183_of__int__0__eq__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_char_0 @ A ) % 0.30/0.54 => ! [Z2: int] : % 0.30/0.54 ( ( ( zero_zero @ A ) % 0.30/0.54 = ( ring_1_of_int @ A @ Z2 ) ) % 0.30/0.54 = ( Z2 % 0.30/0.54 = ( zero_zero @ int ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_0_eq_iff % 0.30/0.54 thf(fact_2184_of__int__eq__0__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_char_0 @ A ) % 0.30/0.54 => ! [Z2: int] : % 0.30/0.54 ( ( ( ring_1_of_int @ A @ Z2 ) % 0.30/0.54 = ( zero_zero @ A ) ) % 0.30/0.54 = ( Z2 % 0.30/0.54 = ( zero_zero @ int ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_eq_0_iff % 0.30/0.54 thf(fact_2185_of__int__le__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [W2: int,Z2: int] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) % 0.30/0.54 = ( ord_less_eq @ int @ W2 @ Z2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_le_iff % 0.30/0.54 thf(fact_2186_of__int__eq__numeral__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_char_0 @ A ) % 0.30/0.54 => ! [Z2: int,N2: num] : % 0.30/0.54 ( ( ( ring_1_of_int @ A @ Z2 ) % 0.30/0.54 = ( numeral_numeral @ A @ N2 ) ) % 0.30/0.54 = ( Z2 % 0.30/0.54 = ( numeral_numeral @ int @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_eq_numeral_iff % 0.30/0.54 thf(fact_2187_of__int__numeral,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_1 @ A ) % 0.30/0.54 => ! [K: num] : % 0.30/0.54 ( ( ring_1_of_int @ A @ ( numeral_numeral @ int @ K ) ) % 0.30/0.54 = ( numeral_numeral @ A @ K ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_numeral % 0.30/0.54 thf(fact_2188_of__int__less__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [W2: int,Z2: int] : % 0.30/0.54 ( ( ord_less @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) % 0.30/0.54 = ( ord_less @ int @ W2 @ Z2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_less_iff % 0.30/0.54 thf(fact_2189_of__int__eq__1__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_char_0 @ A ) % 0.30/0.54 => ! [Z2: int] : % 0.30/0.54 ( ( ( ring_1_of_int @ A @ Z2 ) % 0.30/0.54 = ( one_one @ A ) ) % 0.30/0.54 = ( Z2 % 0.30/0.54 = ( one_one @ int ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_eq_1_iff % 0.30/0.54 thf(fact_2190_of__int__1,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_1 @ A ) % 0.30/0.54 => ( ( ring_1_of_int @ A @ ( one_one @ int ) ) % 0.30/0.54 = ( one_one @ A ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_1 % 0.30/0.54 thf(fact_2191_of__int__mult,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_1 @ A ) % 0.30/0.54 => ! [W2: int,Z2: int] : % 0.30/0.54 ( ( ring_1_of_int @ A @ ( times_times @ int @ W2 @ Z2 ) ) % 0.30/0.54 = ( times_times @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_mult % 0.30/0.54 thf(fact_2192_of__int__add,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_1 @ A ) % 0.30/0.54 => ! [W2: int,Z2: int] : % 0.30/0.54 ( ( ring_1_of_int @ A @ ( plus_plus @ int @ W2 @ Z2 ) ) % 0.30/0.54 = ( plus_plus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_add % 0.30/0.54 thf(fact_2193_of__int__minus,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_1 @ A ) % 0.30/0.54 => ! [Z2: int] : % 0.30/0.54 ( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z2 ) ) % 0.30/0.54 = ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_minus % 0.30/0.54 thf(fact_2194_of__int__diff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_1 @ A ) % 0.30/0.54 => ! [W2: int,Z2: int] : % 0.30/0.54 ( ( ring_1_of_int @ A @ ( minus_minus @ int @ W2 @ Z2 ) ) % 0.30/0.54 = ( minus_minus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_diff % 0.30/0.54 thf(fact_2195_pred__numeral__simps_I1_J,axiom, % 0.30/0.54 ( ( pred_numeral @ one2 ) % 0.30/0.54 = ( zero_zero @ nat ) ) ). % 0.30/0.54 % 0.30/0.54 % pred_numeral_simps(1) % 0.30/0.54 thf(fact_2196_eq__numeral__Suc,axiom, % 0.30/0.54 ! [K: num,N2: nat] : % 0.30/0.54 ( ( ( numeral_numeral @ nat @ K ) % 0.30/0.54 = ( suc @ N2 ) ) % 0.30/0.54 = ( ( pred_numeral @ K ) % 0.30/0.54 = N2 ) ) ). % 0.30/0.54 % 0.30/0.54 % eq_numeral_Suc % 0.30/0.54 thf(fact_2197_Suc__eq__numeral,axiom, % 0.30/0.54 ! [N2: nat,K: num] : % 0.30/0.54 ( ( ( suc @ N2 ) % 0.30/0.54 = ( numeral_numeral @ nat @ K ) ) % 0.30/0.54 = ( N2 % 0.30/0.54 = ( pred_numeral @ K ) ) ) ). % 0.30/0.54 % 0.30/0.54 % Suc_eq_numeral % 0.30/0.54 thf(fact_2198_of__int__power__eq__of__int__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_char_0 @ A ) % 0.30/0.54 => ! [X3: int,B2: int,W2: nat] : % 0.30/0.54 ( ( ( ring_1_of_int @ A @ X3 ) % 0.30/0.54 = ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) ) % 0.30/0.54 = ( X3 % 0.30/0.54 = ( power_power @ int @ B2 @ W2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_power_eq_of_int_cancel_iff % 0.30/0.54 thf(fact_2199_of__int__eq__of__int__power__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_char_0 @ A ) % 0.30/0.54 => ! [B2: int,W2: nat,X3: int] : % 0.30/0.54 ( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) % 0.30/0.54 = ( ring_1_of_int @ A @ X3 ) ) % 0.30/0.54 = ( ( power_power @ int @ B2 @ W2 ) % 0.30/0.54 = X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_eq_of_int_power_cancel_iff % 0.30/0.54 thf(fact_2200_of__int__power,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_1 @ A ) % 0.30/0.54 => ! [Z2: int,N2: nat] : % 0.30/0.54 ( ( ring_1_of_int @ A @ ( power_power @ int @ Z2 @ N2 ) ) % 0.30/0.54 = ( power_power @ A @ ( ring_1_of_int @ A @ Z2 ) @ N2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_power % 0.30/0.54 thf(fact_2201_of__int__of__bool,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_1 @ A ) % 0.30/0.54 => ! [P: $o] : % 0.30/0.54 ( ( ring_1_of_int @ A @ ( zero_neq_one_of_bool @ int @ P ) ) % 0.30/0.54 = ( zero_neq_one_of_bool @ A @ P ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_of_bool % 0.30/0.54 thf(fact_2202_bit__numeral__Bit0__Suc__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.54 => ! [M: num,N2: nat] : % 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( suc @ N2 ) ) % 0.30/0.54 = ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_numeral_Bit0_Suc_iff % 0.30/0.54 thf(fact_2203_bit__numeral__Bit1__Suc__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.54 => ! [M: num,N2: nat] : % 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( suc @ N2 ) ) % 0.30/0.54 = ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_numeral_Bit1_Suc_iff % 0.30/0.54 thf(fact_2204_less__numeral__Suc,axiom, % 0.30/0.54 ! [K: num,N2: nat] : % 0.30/0.54 ( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) ) % 0.30/0.54 = ( ord_less @ nat @ ( pred_numeral @ K ) @ N2 ) ) ). % 0.30/0.54 % 0.30/0.54 % less_numeral_Suc % 0.30/0.54 thf(fact_2205_less__Suc__numeral,axiom, % 0.30/0.54 ! [N2: nat,K: num] : % 0.30/0.54 ( ( ord_less @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) ) % 0.30/0.54 = ( ord_less @ nat @ N2 @ ( pred_numeral @ K ) ) ) ). % 0.30/0.54 % 0.30/0.54 % less_Suc_numeral % 0.30/0.54 thf(fact_2206_le__numeral__Suc,axiom, % 0.30/0.54 ! [K: num,N2: nat] : % 0.30/0.54 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) ) % 0.30/0.54 = ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N2 ) ) ). % 0.30/0.54 % 0.30/0.54 % le_numeral_Suc % 0.30/0.54 thf(fact_2207_le__Suc__numeral,axiom, % 0.30/0.54 ! [N2: nat,K: num] : % 0.30/0.54 ( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) ) % 0.30/0.54 = ( ord_less_eq @ nat @ N2 @ ( pred_numeral @ K ) ) ) ). % 0.30/0.54 % 0.30/0.54 % le_Suc_numeral % 0.30/0.54 thf(fact_2208_diff__Suc__numeral,axiom, % 0.30/0.54 ! [N2: nat,K: num] : % 0.30/0.54 ( ( minus_minus @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) ) % 0.30/0.54 = ( minus_minus @ nat @ N2 @ ( pred_numeral @ K ) ) ) ). % 0.30/0.54 % 0.30/0.54 % diff_Suc_numeral % 0.30/0.54 thf(fact_2209_diff__numeral__Suc,axiom, % 0.30/0.54 ! [K: num,N2: nat] : % 0.30/0.54 ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) ) % 0.30/0.54 = ( minus_minus @ nat @ ( pred_numeral @ K ) @ N2 ) ) ). % 0.30/0.54 % 0.30/0.54 % diff_numeral_Suc % 0.30/0.54 thf(fact_2210_signed__take__bit__negative__iff,axiom, % 0.30/0.54 ! [N2: nat,K: int] : % 0.30/0.54 ( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) % 0.30/0.54 = ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ). % 0.30/0.54 % 0.30/0.54 % signed_take_bit_negative_iff % 0.30/0.54 thf(fact_2211_of__int__0__le__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [Z2: int] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) % 0.30/0.54 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_0_le_iff % 0.30/0.54 thf(fact_2212_of__int__le__0__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [Z2: int] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( zero_zero @ A ) ) % 0.30/0.54 = ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_le_0_iff % 0.30/0.54 thf(fact_2213_of__int__less__0__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [Z2: int] : % 0.30/0.54 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( zero_zero @ A ) ) % 0.30/0.54 = ( ord_less @ int @ Z2 @ ( zero_zero @ int ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_less_0_iff % 0.30/0.54 thf(fact_2214_of__int__0__less__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [Z2: int] : % 0.30/0.54 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) % 0.30/0.54 = ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_0_less_iff % 0.30/0.54 thf(fact_2215_of__int__le__numeral__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [Z2: int,N2: num] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( numeral_numeral @ A @ N2 ) ) % 0.30/0.54 = ( ord_less_eq @ int @ Z2 @ ( numeral_numeral @ int @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_le_numeral_iff % 0.30/0.54 thf(fact_2216_of__int__numeral__le__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [N2: num,Z2: int] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_of_int @ A @ Z2 ) ) % 0.30/0.54 = ( ord_less_eq @ int @ ( numeral_numeral @ int @ N2 ) @ Z2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_numeral_le_iff % 0.30/0.54 thf(fact_2217_of__int__less__numeral__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [Z2: int,N2: num] : % 0.30/0.54 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( numeral_numeral @ A @ N2 ) ) % 0.30/0.54 = ( ord_less @ int @ Z2 @ ( numeral_numeral @ int @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_less_numeral_iff % 0.30/0.54 thf(fact_2218_of__int__numeral__less__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [N2: num,Z2: int] : % 0.30/0.54 ( ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_of_int @ A @ Z2 ) ) % 0.30/0.54 = ( ord_less @ int @ ( numeral_numeral @ int @ N2 ) @ Z2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_numeral_less_iff % 0.30/0.54 thf(fact_2219_of__int__le__1__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [Z2: int] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) % 0.30/0.54 = ( ord_less_eq @ int @ Z2 @ ( one_one @ int ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_le_1_iff % 0.30/0.54 thf(fact_2220_of__int__1__le__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [Z2: int] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) % 0.30/0.54 = ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_1_le_iff % 0.30/0.54 thf(fact_2221_of__int__less__1__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [Z2: int] : % 0.30/0.54 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) % 0.30/0.54 = ( ord_less @ int @ Z2 @ ( one_one @ int ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_less_1_iff % 0.30/0.54 thf(fact_2222_of__int__1__less__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [Z2: int] : % 0.30/0.54 ( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) % 0.30/0.54 = ( ord_less @ int @ ( one_one @ int ) @ Z2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_1_less_iff % 0.30/0.54 thf(fact_2223_of__int__le__of__int__power__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [B2: int,W2: nat,X3: int] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X3 ) ) % 0.30/0.54 = ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W2 ) @ X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_le_of_int_power_cancel_iff % 0.30/0.54 thf(fact_2224_of__int__power__le__of__int__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [X3: int,B2: int,W2: nat] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) ) % 0.30/0.54 = ( ord_less_eq @ int @ X3 @ ( power_power @ int @ B2 @ W2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_power_le_of_int_cancel_iff % 0.30/0.54 thf(fact_2225_of__int__eq__numeral__power__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_char_0 @ A ) % 0.30/0.54 => ! [Y: int,X3: num,N2: nat] : % 0.30/0.54 ( ( ( ring_1_of_int @ A @ Y ) % 0.30/0.54 = ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) ) % 0.30/0.54 = ( Y % 0.30/0.54 = ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_eq_numeral_power_cancel_iff % 0.30/0.54 thf(fact_2226_numeral__power__eq__of__int__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_char_0 @ A ) % 0.30/0.54 => ! [X3: num,N2: nat,Y: int] : % 0.30/0.54 ( ( ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) % 0.30/0.54 = ( ring_1_of_int @ A @ Y ) ) % 0.30/0.54 = ( ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) % 0.30/0.54 = Y ) ) ) ). % 0.30/0.54 % 0.30/0.54 % numeral_power_eq_of_int_cancel_iff % 0.30/0.54 thf(fact_2227_of__int__less__of__int__power__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [B2: int,W2: nat,X3: int] : % 0.30/0.54 ( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X3 ) ) % 0.30/0.54 = ( ord_less @ int @ ( power_power @ int @ B2 @ W2 ) @ X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_less_of_int_power_cancel_iff % 0.30/0.54 thf(fact_2228_of__int__power__less__of__int__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [X3: int,B2: int,W2: nat] : % 0.30/0.54 ( ( ord_less @ A @ ( ring_1_of_int @ A @ X3 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) ) % 0.30/0.54 = ( ord_less @ int @ X3 @ ( power_power @ int @ B2 @ W2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_power_less_of_int_cancel_iff % 0.30/0.54 thf(fact_2229_bit__minus__numeral__Bit0__Suc__iff,axiom, % 0.30/0.54 ! [W2: num,N2: nat] : % 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) ) @ ( suc @ N2 ) ) % 0.30/0.54 = ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ N2 ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_minus_numeral_Bit0_Suc_iff % 0.30/0.54 thf(fact_2230_bit__minus__numeral__Bit1__Suc__iff,axiom, % 0.30/0.54 ! [W2: num,N2: nat] : % 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) ) @ ( suc @ N2 ) ) % 0.30/0.54 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ N2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_minus_numeral_Bit1_Suc_iff % 0.30/0.54 thf(fact_2231_bit__0,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_semiring_bits @ A ) % 0.30/0.54 => ! [A2: A] : % 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( zero_zero @ nat ) ) % 0.30/0.54 = ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_0 % 0.30/0.54 thf(fact_2232_bit__mod__2__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_semiring_bits @ A ) % 0.30/0.54 => ! [A2: A,N2: nat] : % 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 ) % 0.30/0.54 = ( ( N2 % 0.30/0.54 = ( zero_zero @ nat ) ) % 0.30/0.54 & ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_mod_2_iff % 0.30/0.54 thf(fact_2233_numeral__power__le__of__int__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [X3: num,N2: nat,A2: int] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) ) % 0.30/0.54 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) @ A2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % numeral_power_le_of_int_cancel_iff % 0.30/0.54 thf(fact_2234_of__int__le__numeral__power__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [A2: int,X3: num,N2: nat] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) ) % 0.30/0.54 = ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_le_numeral_power_cancel_iff % 0.30/0.54 thf(fact_2235_numeral__power__less__of__int__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [X3: num,N2: nat,A2: int] : % 0.30/0.54 ( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) ) % 0.30/0.54 = ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) @ A2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % numeral_power_less_of_int_cancel_iff % 0.30/0.54 thf(fact_2236_of__int__less__numeral__power__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [A2: int,X3: num,N2: nat] : % 0.30/0.54 ( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) ) % 0.30/0.54 = ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_less_numeral_power_cancel_iff % 0.30/0.54 thf(fact_2237_of__int__eq__neg__numeral__power__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_char_0 @ A ) % 0.30/0.54 => ! [Y: int,X3: num,N2: nat] : % 0.30/0.54 ( ( ( ring_1_of_int @ A @ Y ) % 0.30/0.54 = ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) ) % 0.30/0.54 = ( Y % 0.30/0.54 = ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_eq_neg_numeral_power_cancel_iff % 0.30/0.54 thf(fact_2238_neg__numeral__power__eq__of__int__cancel__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_char_0 @ A ) % 0.30/0.54 => ! [X3: num,N2: nat,Y: int] : % 0.30/0.54 ( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) % 0.30/0.54 = ( ring_1_of_int @ A @ Y ) ) % 0.30/0.54 = ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) % 0.30/0.54 = Y ) ) ) ). % 0.30/0.54 % 0.30/0.54 % neg_numeral_power_eq_of_int_cancel_iff % 0.30/0.54 thf(fact_2239_bit__disjunctive__add__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_semiring_bits @ A ) % 0.30/0.54 => ! [A2: A,B2: A,N2: nat] : % 0.30/0.54 ( ! [N3: nat] : % 0.30/0.54 ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 ) % 0.30/0.54 | ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) ) % 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N2 ) % 0.30/0.54 = ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) % 0.30/0.54 | ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_disjunctive_add_iff % 0.30/0.54 thf(fact_2240_ex__le__of__int,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim462609752435547400_field @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ? [Z3: int] : ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ex_le_of_int % 0.30/0.54 thf(fact_2241_ex__less__of__int,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim462609752435547400_field @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ? [Z3: int] : ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ Z3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ex_less_of_int % 0.30/0.54 thf(fact_2242_ex__of__int__less,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim462609752435547400_field @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ? [Z3: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X3 ) ) ). % 0.30/0.54 % 0.30/0.54 % ex_of_int_less % 0.30/0.54 thf(fact_2243_mult__of__int__commute,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_1 @ A ) % 0.30/0.54 => ! [X3: int,Y: A] : % 0.30/0.54 ( ( times_times @ A @ ( ring_1_of_int @ A @ X3 ) @ Y ) % 0.30/0.54 = ( times_times @ A @ Y @ ( ring_1_of_int @ A @ X3 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % mult_of_int_commute % 0.30/0.54 thf(fact_2244_not__bit__1__Suc,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.54 => ! [N2: nat] : % 0.30/0.54 ~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % not_bit_1_Suc % 0.30/0.54 thf(fact_2245_bit__1__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_semiring_bits @ A ) % 0.30/0.54 => ! [N2: nat] : % 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N2 ) % 0.30/0.54 = ( N2 % 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_1_iff % 0.30/0.54 thf(fact_2246_bit__take__bit__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.54 => ! [M: nat,A2: A,N2: nat] : % 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) @ N2 ) % 0.30/0.54 = ( ( ord_less @ nat @ N2 @ M ) % 0.30/0.54 & ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_take_bit_iff % 0.30/0.54 thf(fact_2247_bit__of__bool__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_semiring_bits @ A ) % 0.30/0.54 => ! [B2: $o,N2: nat] : % 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ N2 ) % 0.30/0.54 = ( B2 % 0.30/0.54 & ( N2 % 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_of_bool_iff % 0.30/0.54 thf(fact_2248_numeral__eq__Suc,axiom, % 0.30/0.54 ( ( numeral_numeral @ nat ) % 0.30/0.54 = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % numeral_eq_Suc % 0.30/0.54 thf(fact_2249_of__int__nonneg,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [Z2: int] : % 0.30/0.54 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 0.30/0.54 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_nonneg % 0.30/0.54 thf(fact_2250_of__int__pos,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( linordered_idom @ A ) % 0.30/0.54 => ! [Z2: int] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) % 0.30/0.54 => ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_pos % 0.30/0.54 thf(fact_2251_floor__exists,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim462609752435547400_field @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ? [Z3: int] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X3 ) % 0.30/0.54 & ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % floor_exists % 0.30/0.54 thf(fact_2252_floor__exists1,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim462609752435547400_field @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ? [X5: int] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X5 ) @ X3 ) % 0.30/0.54 & ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X5 @ ( one_one @ int ) ) ) ) % 0.30/0.54 & ! [Y5: int] : % 0.30/0.54 ( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y5 ) @ X3 ) % 0.30/0.54 & ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y5 @ ( one_one @ int ) ) ) ) ) % 0.30/0.54 => ( Y5 = X5 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % floor_exists1 % 0.30/0.54 thf(fact_2253_of__int__neg__numeral,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ring_1 @ A ) % 0.30/0.54 => ! [K: num] : % 0.30/0.54 ( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) % 0.30/0.54 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_neg_numeral % 0.30/0.54 thf(fact_2254_bit__imp__take__bit__positive,axiom, % 0.30/0.54 ! [N2: nat,M: nat,K: int] : % 0.30/0.54 ( ( ord_less @ nat @ N2 @ M ) % 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) % 0.30/0.54 => ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_imp_take_bit_positive % 0.30/0.54 thf(fact_2255_int__le__real__less,axiom, % 0.30/0.54 ( ( ord_less_eq @ int ) % 0.30/0.54 = ( ^ [N: int,M2: int] : ( ord_less @ real @ ( ring_1_of_int @ real @ N ) @ ( plus_plus @ real @ ( ring_1_of_int @ real @ M2 ) @ ( one_one @ real ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % int_le_real_less % 0.30/0.54 thf(fact_2256_int__less__real__le,axiom, % 0.30/0.54 ( ( ord_less @ int ) % 0.30/0.54 = ( ^ [N: int,M2: int] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ M2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % int_less_real_le % 0.30/0.54 thf(fact_2257_bit__concat__bit__iff,axiom, % 0.30/0.54 ! [M: nat,K: int,L: int,N2: nat] : % 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K @ L ) @ N2 ) % 0.30/0.54 = ( ( ( ord_less @ nat @ N2 @ M ) % 0.30/0.54 & ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) % 0.30/0.54 | ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.54 & ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_concat_bit_iff % 0.30/0.54 thf(fact_2258_exp__eq__0__imp__not__bit,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_semiring_bits @ A ) % 0.30/0.54 => ! [N2: nat,A2: A] : % 0.30/0.54 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.54 = ( zero_zero @ A ) ) % 0.30/0.54 => ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % exp_eq_0_imp_not_bit % 0.30/0.54 thf(fact_2259_bit__Suc,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_semiring_bits @ A ) % 0.30/0.54 => ! [A2: A,N2: nat] : % 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ N2 ) ) % 0.30/0.54 = ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_Suc % 0.30/0.54 thf(fact_2260_int__bit__bound,axiom, % 0.30/0.54 ! [K: int] : % 0.30/0.54 ~ ! [N3: nat] : % 0.30/0.54 ( ! [M6: nat] : % 0.30/0.54 ( ( ord_less_eq @ nat @ N3 @ M6 ) % 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ M6 ) % 0.30/0.54 = ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) % 0.30/0.54 => ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) % 0.30/0.54 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % int_bit_bound % 0.30/0.54 thf(fact_2261_and__exp__eq__0__iff__not__bit,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.54 => ! [A2: A,N2: nat] : % 0.30/0.54 ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.54 = ( zero_zero @ A ) ) % 0.30/0.54 = ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % and_exp_eq_0_iff_not_bit % 0.30/0.54 thf(fact_2262_even__bit__succ__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_semiring_bits @ A ) % 0.30/0.54 => ! [A2: A,N2: nat] : % 0.30/0.54 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ N2 ) % 0.30/0.54 = ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) % 0.30/0.54 | ( N2 % 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % even_bit_succ_iff % 0.30/0.54 thf(fact_2263_odd__bit__iff__bit__pred,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_semiring_bits @ A ) % 0.30/0.54 => ! [A2: A,N2: nat] : % 0.30/0.54 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) % 0.30/0.54 = ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ N2 ) % 0.30/0.54 | ( N2 % 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % odd_bit_iff_bit_pred % 0.30/0.54 thf(fact_2264_bit__sum__mult__2__cases,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.54 => ! [A2: A,B2: A,N2: nat] : % 0.30/0.54 ( ! [J2: nat] : % 0.30/0.54 ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ J2 ) ) % 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N2 ) % 0.30/0.54 = ( ( ( N2 % 0.30/0.54 = ( zero_zero @ nat ) ) % 0.30/0.54 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) % 0.30/0.54 & ( ( N2 % 0.30/0.54 != ( zero_zero @ nat ) ) % 0.30/0.54 => ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N2 ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_sum_mult_2_cases % 0.30/0.54 thf(fact_2265_bit__rec,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_semiring_bits @ A ) % 0.30/0.54 => ( ( bit_se5641148757651400278ts_bit @ A ) % 0.30/0.54 = ( ^ [A4: A,N: nat] : % 0.30/0.54 ( ( ( N % 0.30/0.54 = ( zero_zero @ nat ) ) % 0.30/0.54 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) % 0.30/0.54 & ( ( N % 0.30/0.54 != ( zero_zero @ nat ) ) % 0.30/0.54 => ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_rec % 0.30/0.54 thf(fact_2266_take__bit__Suc__from__most,axiom, % 0.30/0.54 ! [N2: nat,K: int] : % 0.30/0.54 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ K ) % 0.30/0.54 = ( plus_plus @ int @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ). % 0.30/0.54 % 0.30/0.54 % take_bit_Suc_from_most % 0.30/0.54 thf(fact_2267_round__unique,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A,Y: int] : % 0.30/0.54 ( ( ord_less @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y ) ) % 0.30/0.54 => ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y ) @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) % 0.30/0.54 => ( ( archimedean_round @ A @ X3 ) % 0.30/0.54 = Y ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % round_unique % 0.30/0.54 thf(fact_2268_of__int__round__gt,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A] : ( ord_less @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_round_gt % 0.30/0.54 thf(fact_2269_of__int__round__ge,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_round_ge % 0.30/0.54 thf(fact_2270_of__int__round__le,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_round_le % 0.30/0.54 thf(fact_2271_divmod__BitM__2__eq,axiom, % 0.30/0.54 ! [M: num] : % 0.30/0.54 ( ( unique8689654367752047608divmod @ int @ ( bitM @ M ) @ ( bit0 @ one2 ) ) % 0.30/0.54 = ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ). % 0.30/0.54 % 0.30/0.54 % divmod_BitM_2_eq % 0.30/0.54 thf(fact_2272_mask__numeral,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.54 => ! [N2: num] : % 0.30/0.54 ( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N2 ) ) % 0.30/0.54 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N2 ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % mask_numeral % 0.30/0.54 thf(fact_2273_neg__numeral__le__ceiling,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [V2: num,X3: A] : % 0.30/0.54 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X3 ) ) % 0.30/0.54 = ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % neg_numeral_le_ceiling % 0.30/0.54 thf(fact_2274_mask__nat__positive__iff,axiom, % 0.30/0.54 ! [N2: nat] : % 0.30/0.54 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) % 0.30/0.54 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ). % 0.30/0.54 % 0.30/0.54 % mask_nat_positive_iff % 0.30/0.54 thf(fact_2275_ceiling__zero,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) ) % 0.30/0.54 = ( zero_zero @ int ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_zero % 0.30/0.54 thf(fact_2276_mask__eq__0__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.54 => ! [N2: nat] : % 0.30/0.54 ( ( ( bit_se2239418461657761734s_mask @ A @ N2 ) % 0.30/0.54 = ( zero_zero @ A ) ) % 0.30/0.54 = ( N2 % 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % mask_eq_0_iff % 0.30/0.54 thf(fact_2277_mask__0,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.54 => ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) ) % 0.30/0.54 = ( zero_zero @ A ) ) ) ). % 0.30/0.54 % 0.30/0.54 % mask_0 % 0.30/0.54 thf(fact_2278_round__0,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ( ( archimedean_round @ A @ ( zero_zero @ A ) ) % 0.30/0.54 = ( zero_zero @ int ) ) ) ). % 0.30/0.54 % 0.30/0.54 % round_0 % 0.30/0.54 thf(fact_2279_mask__Suc__0,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.54 => ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.54 = ( one_one @ A ) ) ) ). % 0.30/0.54 % 0.30/0.54 % mask_Suc_0 % 0.30/0.54 thf(fact_2280_ceiling__add__of__int,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A,Z2: int] : % 0.30/0.54 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) ) % 0.30/0.54 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_add_of_int % 0.30/0.54 thf(fact_2281_ceiling__le__zero,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( zero_zero @ int ) ) % 0.30/0.54 = ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_le_zero % 0.30/0.54 thf(fact_2282_zero__less__ceiling,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X3 ) ) % 0.30/0.54 = ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % zero_less_ceiling % 0.30/0.54 thf(fact_2283_ceiling__le__numeral,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A,V2: num] : % 0.30/0.54 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) % 0.30/0.54 = ( ord_less_eq @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_le_numeral % 0.30/0.54 thf(fact_2284_ceiling__less__one,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) ) % 0.30/0.54 = ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_less_one % 0.30/0.54 thf(fact_2285_one__le__ceiling,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X3 ) ) % 0.30/0.54 = ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % one_le_ceiling % 0.30/0.54 thf(fact_2286_numeral__less__ceiling,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [V2: num,X3: A] : % 0.30/0.54 ( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X3 ) ) % 0.30/0.54 = ( ord_less @ A @ ( numeral_numeral @ A @ V2 ) @ X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % numeral_less_ceiling % 0.30/0.54 thf(fact_2287_ceiling__le__one,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) ) % 0.30/0.54 = ( ord_less_eq @ A @ X3 @ ( one_one @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_le_one % 0.30/0.54 thf(fact_2288_one__less__ceiling,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X3 ) ) % 0.30/0.54 = ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % one_less_ceiling % 0.30/0.54 thf(fact_2289_ceiling__add__numeral,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A,V2: num] : % 0.30/0.54 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) ) % 0.30/0.54 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_add_numeral % 0.30/0.54 thf(fact_2290_ceiling__add__one,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) ) % 0.30/0.54 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_add_one % 0.30/0.54 thf(fact_2291_ceiling__less__zero,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( zero_zero @ int ) ) % 0.30/0.54 = ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_less_zero % 0.30/0.54 thf(fact_2292_zero__le__ceiling,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X3 ) ) % 0.30/0.54 = ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % zero_le_ceiling % 0.30/0.54 thf(fact_2293_ceiling__less__numeral,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A,V2: num] : % 0.30/0.54 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) % 0.30/0.54 = ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_less_numeral % 0.30/0.54 thf(fact_2294_numeral__le__ceiling,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [V2: num,X3: A] : % 0.30/0.54 ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X3 ) ) % 0.30/0.54 = ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % numeral_le_ceiling % 0.30/0.54 thf(fact_2295_ceiling__le__neg__numeral,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A,V2: num] : % 0.30/0.54 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) % 0.30/0.54 = ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_le_neg_numeral % 0.30/0.54 thf(fact_2296_neg__numeral__less__ceiling,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [V2: num,X3: A] : % 0.30/0.54 ( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X3 ) ) % 0.30/0.54 = ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % neg_numeral_less_ceiling % 0.30/0.54 thf(fact_2297_ceiling__less__neg__numeral,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A,V2: num] : % 0.30/0.54 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) % 0.30/0.54 = ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_less_neg_numeral % 0.30/0.54 thf(fact_2298_less__eq__mask,axiom, % 0.30/0.54 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) ). % 0.30/0.54 % 0.30/0.54 % less_eq_mask % 0.30/0.54 thf(fact_2299_bit__Suc__0__iff,axiom, % 0.30/0.54 ! [N2: nat] : % 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 0.30/0.54 = ( N2 % 0.30/0.54 = ( zero_zero @ nat ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_Suc_0_iff % 0.30/0.54 thf(fact_2300_not__bit__Suc__0__Suc,axiom, % 0.30/0.54 ! [N2: nat] : % 0.30/0.54 ~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( suc @ N2 ) ) ). % 0.30/0.54 % 0.30/0.54 % not_bit_Suc_0_Suc % 0.30/0.54 thf(fact_2301_ceiling__mono,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [Y: A,X3: A] : % 0.30/0.54 ( ( ord_less_eq @ A @ Y @ X3 ) % 0.30/0.54 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_mono % 0.30/0.54 thf(fact_2302_le__of__int__ceiling,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % le_of_int_ceiling % 0.30/0.54 thf(fact_2303_ceiling__less__cancel,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A,Y: A] : % 0.30/0.54 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( archimedean_ceiling @ A @ Y ) ) % 0.30/0.54 => ( ord_less @ A @ X3 @ Y ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_less_cancel % 0.30/0.54 thf(fact_2304_not__mask__negative__int,axiom, % 0.30/0.54 ! [N2: nat] : % 0.30/0.54 ~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N2 ) @ ( zero_zero @ int ) ) ). % 0.30/0.54 % 0.30/0.54 % not_mask_negative_int % 0.30/0.54 thf(fact_2305_not__bit__Suc__0__numeral,axiom, % 0.30/0.54 ! [N2: num] : % 0.30/0.54 ~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ N2 ) ) ). % 0.30/0.54 % 0.30/0.54 % not_bit_Suc_0_numeral % 0.30/0.54 thf(fact_2306_ceiling__le__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A,Z2: int] : % 0.30/0.54 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z2 ) % 0.30/0.54 = ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_le_iff % 0.30/0.54 thf(fact_2307_ceiling__le,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A,A2: int] : % 0.30/0.54 ( ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ A2 ) ) % 0.30/0.54 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ A2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_le % 0.30/0.54 thf(fact_2308_less__ceiling__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [Z2: int,X3: A] : % 0.30/0.54 ( ( ord_less @ int @ Z2 @ ( archimedean_ceiling @ A @ X3 ) ) % 0.30/0.54 = ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % less_ceiling_iff % 0.30/0.54 thf(fact_2309_ceiling__add__le,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A,Y: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( archimedean_ceiling @ A @ Y ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_add_le % 0.30/0.54 thf(fact_2310_round__mono,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A,Y: A] : % 0.30/0.54 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.30/0.54 => ( ord_less_eq @ int @ ( archimedean_round @ A @ X3 ) @ ( archimedean_round @ A @ Y ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % round_mono % 0.30/0.54 thf(fact_2311_less__mask,axiom, % 0.30/0.54 ! [N2: nat] : % 0.30/0.54 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 0.30/0.54 => ( ord_less @ nat @ N2 @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % less_mask % 0.30/0.54 thf(fact_2312_eval__nat__numeral_I2_J,axiom, % 0.30/0.54 ! [N2: num] : % 0.30/0.54 ( ( numeral_numeral @ nat @ ( bit0 @ N2 ) ) % 0.30/0.54 = ( suc @ ( numeral_numeral @ nat @ ( bitM @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % eval_nat_numeral(2) % 0.30/0.54 thf(fact_2313_of__int__ceiling__le__add__one,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [R2: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R2 ) ) @ ( plus_plus @ A @ R2 @ ( one_one @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_ceiling_le_add_one % 0.30/0.54 thf(fact_2314_of__int__ceiling__diff__one__le,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [R2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R2 ) ) @ ( one_one @ A ) ) @ R2 ) ) ). % 0.30/0.54 % 0.30/0.54 % of_int_ceiling_diff_one_le % 0.30/0.54 thf(fact_2315_ceiling__split,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [P: int > $o,T2: A] : % 0.30/0.54 ( ( P @ ( archimedean_ceiling @ A @ T2 ) ) % 0.30/0.54 = ( ! [I3: int] : % 0.30/0.54 ( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) @ T2 ) % 0.30/0.54 & ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I3 ) ) ) % 0.30/0.54 => ( P @ I3 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_split % 0.30/0.54 thf(fact_2316_ceiling__eq__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A,A2: int] : % 0.30/0.54 ( ( ( archimedean_ceiling @ A @ X3 ) % 0.30/0.54 = A2 ) % 0.30/0.54 = ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) @ X3 ) % 0.30/0.54 & ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ A2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_eq_iff % 0.30/0.54 thf(fact_2317_ceiling__unique,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [Z2: int,X3: A] : % 0.30/0.54 ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X3 ) % 0.30/0.54 => ( ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) % 0.30/0.54 => ( ( archimedean_ceiling @ A @ X3 ) % 0.30/0.54 = Z2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_unique % 0.30/0.54 thf(fact_2318_ceiling__correct,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) @ ( one_one @ A ) ) @ X3 ) % 0.30/0.54 & ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_correct % 0.30/0.54 thf(fact_2319_mult__ceiling__le,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [A2: A,B2: A] : % 0.30/0.54 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.54 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.54 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A2 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A2 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % mult_ceiling_le % 0.30/0.54 thf(fact_2320_ceiling__less__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [X3: A,Z2: int] : % 0.30/0.54 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z2 ) % 0.30/0.54 = ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_less_iff % 0.30/0.54 thf(fact_2321_le__ceiling__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [Z2: int,X3: A] : % 0.30/0.54 ( ( ord_less_eq @ int @ Z2 @ ( archimedean_ceiling @ A @ X3 ) ) % 0.30/0.54 = ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % le_ceiling_iff % 0.30/0.54 thf(fact_2322_ceiling__divide__upper,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [Q4: A,P4: A] : % 0.30/0.54 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 ) % 0.30/0.54 => ( ord_less_eq @ A @ P4 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_divide_upper % 0.30/0.54 thf(fact_2323_Suc__mask__eq__exp,axiom, % 0.30/0.54 ! [N2: nat] : % 0.30/0.54 ( ( suc @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) % 0.30/0.54 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 0.30/0.54 % 0.30/0.54 % Suc_mask_eq_exp % 0.30/0.54 thf(fact_2324_mask__nat__less__exp,axiom, % 0.30/0.54 ! [N2: nat] : ( ord_less @ nat @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 0.30/0.54 % 0.30/0.54 % mask_nat_less_exp % 0.30/0.54 thf(fact_2325_semiring__bit__operations__class_Oeven__mask__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.54 => ! [N2: nat] : % 0.30/0.54 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) % 0.30/0.54 = ( N2 % 0.30/0.54 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % semiring_bit_operations_class.even_mask_iff % 0.30/0.54 thf(fact_2326_ceiling__divide__lower,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [Q4: A,P4: A] : % 0.30/0.54 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 ) % 0.30/0.54 => ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) @ P4 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_divide_lower % 0.30/0.54 thf(fact_2327_ceiling__eq,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.54 => ! [N2: int,X3: A] : % 0.30/0.54 ( ( ord_less @ A @ ( ring_1_of_int @ A @ N2 ) @ X3 ) % 0.30/0.54 => ( ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N2 ) @ ( one_one @ A ) ) ) % 0.30/0.54 => ( ( archimedean_ceiling @ A @ X3 ) % 0.30/0.54 = ( plus_plus @ int @ N2 @ ( one_one @ int ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % ceiling_eq % 0.30/0.54 thf(fact_2328_Suc__0__xor__eq,axiom, % 0.30/0.54 ! [N2: nat] : % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 0.30/0.54 = ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 0.30/0.54 @ ( zero_neq_one_of_bool @ nat % 0.30/0.54 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % Suc_0_xor_eq % 0.30/0.54 thf(fact_2329_xor__Suc__0__eq,axiom, % 0.30/0.54 ! [N2: nat] : % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.54 = ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 0.30/0.54 @ ( zero_neq_one_of_bool @ nat % 0.30/0.54 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % xor_Suc_0_eq % 0.30/0.54 thf(fact_2330_horner__sum__of__bool__2__less,axiom, % 0.30/0.54 ! [Bs: list @ $o] : ( ord_less @ int @ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Bs ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ $o ) @ Bs ) ) ) ). % 0.30/0.54 % 0.30/0.54 % horner_sum_of_bool_2_less % 0.30/0.54 thf(fact_2331_dbl__dec__simps_I2_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( neg_numeral @ A ) % 0.30/0.54 => ( ( neg_numeral_dbl_dec @ A @ ( zero_zero @ A ) ) % 0.30/0.54 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % dbl_dec_simps(2) % 0.30/0.54 thf(fact_2332_the__elem__eq,axiom, % 0.30/0.54 ! [A: $tType,X3: A] : % 0.30/0.54 ( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.30/0.54 = X3 ) ). % 0.30/0.54 % 0.30/0.54 % the_elem_eq % 0.30/0.54 thf(fact_2333_vebt__member_Oelims_I1_J,axiom, % 0.30/0.54 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 0.30/0.54 ( ( ( vEBT_vebt_member @ X3 @ Xa2 ) % 0.30/0.54 = Y ) % 0.30/0.54 => ( ! [A5: $o,B5: $o] : % 0.30/0.54 ( ( X3 % 0.30/0.54 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.54 => ( Y % 0.30/0.54 = ( ~ ( ( ( Xa2 % 0.30/0.54 = ( zero_zero @ nat ) ) % 0.30/0.54 => A5 ) % 0.30/0.54 & ( ( Xa2 % 0.30/0.54 != ( zero_zero @ nat ) ) % 0.30/0.54 => ( ( ( Xa2 % 0.30/0.54 = ( one_one @ nat ) ) % 0.30/0.54 => B5 ) % 0.30/0.54 & ( Xa2 % 0.30/0.54 = ( one_one @ nat ) ) ) ) ) ) ) ) % 0.30/0.54 => ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 0.30/0.54 => Y ) % 0.30/0.54 => ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) ) % 0.30/0.54 => Y ) % 0.30/0.54 => ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) ) % 0.30/0.54 => Y ) % 0.30/0.54 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT] : % 0.30/0.54 ( ? [Summary2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) % 0.30/0.54 => ( Y % 0.30/0.54 = ( ~ ( ( Xa2 != Mi2 ) % 0.30/0.54 => ( ( Xa2 != Ma2 ) % 0.30/0.54 => ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 0.30/0.54 & ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 0.30/0.54 => ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 0.30/0.54 & ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 0.30/0.54 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.54 => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % vebt_member.elims(1) % 0.30/0.54 thf(fact_2334_set__vebt_H__def,axiom, % 0.30/0.54 ( vEBT_VEBT_set_vebt % 0.30/0.54 = ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % set_vebt'_def % 0.30/0.54 thf(fact_2335_finite__Collect__conjI,axiom, % 0.30/0.54 ! [A: $tType,P: A > $o,Q: A > $o] : % 0.30/0.54 ( ( ( finite_finite2 @ A @ ( collect @ A @ P ) ) % 0.30/0.54 | ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) % 0.30/0.54 => ( finite_finite2 @ A % 0.30/0.54 @ ( collect @ A % 0.30/0.54 @ ^ [X: A] : % 0.30/0.54 ( ( P @ X ) % 0.30/0.54 & ( Q @ X ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_Collect_conjI % 0.30/0.54 thf(fact_2336_finite__Collect__disjI,axiom, % 0.30/0.54 ! [A: $tType,P: A > $o,Q: A > $o] : % 0.30/0.54 ( ( finite_finite2 @ A % 0.30/0.54 @ ( collect @ A % 0.30/0.54 @ ^ [X: A] : % 0.30/0.54 ( ( P @ X ) % 0.30/0.54 | ( Q @ X ) ) ) ) % 0.30/0.54 = ( ( finite_finite2 @ A @ ( collect @ A @ P ) ) % 0.30/0.54 & ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_Collect_disjI % 0.30/0.54 thf(fact_2337_bit_Oxor__self,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_ri3973907225187159222ations @ A ) % 0.30/0.54 => ! [X3: A] : % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ A @ X3 @ X3 ) % 0.30/0.54 = ( zero_zero @ A ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit.xor_self % 0.30/0.54 thf(fact_2338_xor__self__eq,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.54 => ! [A2: A] : % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ A @ A2 @ A2 ) % 0.30/0.54 = ( zero_zero @ A ) ) ) ). % 0.30/0.54 % 0.30/0.54 % xor_self_eq % 0.30/0.54 thf(fact_2339_xor_Oleft__neutral,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.54 => ! [A2: A] : % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.54 = A2 ) ) ). % 0.30/0.54 % 0.30/0.54 % xor.left_neutral % 0.30/0.54 thf(fact_2340_xor_Oright__neutral,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.54 => ! [A2: A] : % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.54 = A2 ) ) ). % 0.30/0.54 % 0.30/0.54 % xor.right_neutral % 0.30/0.54 thf(fact_2341_finite__Collect__bounded__ex,axiom, % 0.30/0.54 ! [B: $tType,A: $tType,P: A > $o,Q: B > A > $o] : % 0.30/0.54 ( ( finite_finite2 @ A @ ( collect @ A @ P ) ) % 0.30/0.54 => ( ( finite_finite2 @ B % 0.30/0.54 @ ( collect @ B % 0.30/0.54 @ ^ [X: B] : % 0.30/0.54 ? [Y2: A] : % 0.30/0.54 ( ( P @ Y2 ) % 0.30/0.54 & ( Q @ X @ Y2 ) ) ) ) % 0.30/0.54 = ( ! [Y2: A] : % 0.30/0.54 ( ( P @ Y2 ) % 0.30/0.54 => ( finite_finite2 @ B % 0.30/0.54 @ ( collect @ B % 0.30/0.54 @ ^ [X: B] : ( Q @ X @ Y2 ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_Collect_bounded_ex % 0.30/0.54 thf(fact_2342_finite__Collect__subsets,axiom, % 0.30/0.54 ! [A: $tType,A3: set @ A] : % 0.30/0.54 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.54 => ( finite_finite2 @ ( set @ A ) % 0.30/0.54 @ ( collect @ ( set @ A ) % 0.30/0.54 @ ^ [B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_Collect_subsets % 0.30/0.54 thf(fact_2343_singleton__conv2,axiom, % 0.30/0.54 ! [A: $tType,A2: A] : % 0.30/0.54 ( ( collect @ A % 0.30/0.54 @ ( ^ [Y3: A,Z: A] : ( Y3 = Z ) % 0.30/0.54 @ A2 ) ) % 0.30/0.54 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % singleton_conv2 % 0.30/0.54 thf(fact_2344_singleton__conv,axiom, % 0.30/0.54 ! [A: $tType,A2: A] : % 0.30/0.54 ( ( collect @ A % 0.30/0.54 @ ^ [X: A] : ( X = A2 ) ) % 0.30/0.54 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % singleton_conv % 0.30/0.54 thf(fact_2345_finite__Collect__less__nat,axiom, % 0.30/0.54 ! [K: nat] : % 0.30/0.54 ( finite_finite2 @ nat % 0.30/0.54 @ ( collect @ nat % 0.30/0.54 @ ^ [N: nat] : ( ord_less @ nat @ N @ K ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_Collect_less_nat % 0.30/0.54 thf(fact_2346_finite__Collect__le__nat,axiom, % 0.30/0.54 ! [K: nat] : % 0.30/0.54 ( finite_finite2 @ nat % 0.30/0.54 @ ( collect @ nat % 0.30/0.54 @ ^ [N: nat] : ( ord_less_eq @ nat @ N @ K ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_Collect_le_nat % 0.30/0.54 thf(fact_2347_finite__interval__int1,axiom, % 0.30/0.54 ! [A2: int,B2: int] : % 0.30/0.54 ( finite_finite2 @ int % 0.30/0.54 @ ( collect @ int % 0.30/0.54 @ ^ [I3: int] : % 0.30/0.54 ( ( ord_less_eq @ int @ A2 @ I3 ) % 0.30/0.54 & ( ord_less_eq @ int @ I3 @ B2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_interval_int1 % 0.30/0.54 thf(fact_2348_finite__interval__int4,axiom, % 0.30/0.54 ! [A2: int,B2: int] : % 0.30/0.54 ( finite_finite2 @ int % 0.30/0.54 @ ( collect @ int % 0.30/0.54 @ ^ [I3: int] : % 0.30/0.54 ( ( ord_less @ int @ A2 @ I3 ) % 0.30/0.54 & ( ord_less @ int @ I3 @ B2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_interval_int4 % 0.30/0.54 thf(fact_2349_finite__interval__int2,axiom, % 0.30/0.54 ! [A2: int,B2: int] : % 0.30/0.54 ( finite_finite2 @ int % 0.30/0.54 @ ( collect @ int % 0.30/0.54 @ ^ [I3: int] : % 0.30/0.54 ( ( ord_less_eq @ int @ A2 @ I3 ) % 0.30/0.54 & ( ord_less @ int @ I3 @ B2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_interval_int2 % 0.30/0.54 thf(fact_2350_finite__interval__int3,axiom, % 0.30/0.54 ! [A2: int,B2: int] : % 0.30/0.54 ( finite_finite2 @ int % 0.30/0.54 @ ( collect @ int % 0.30/0.54 @ ^ [I3: int] : % 0.30/0.54 ( ( ord_less @ int @ A2 @ I3 ) % 0.30/0.54 & ( ord_less_eq @ int @ I3 @ B2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_interval_int3 % 0.30/0.54 thf(fact_2351_xor__nat__numerals_I4_J,axiom, % 0.30/0.54 ! [X3: num] : % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.54 = ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % xor_nat_numerals(4) % 0.30/0.54 thf(fact_2352_xor__nat__numerals_I3_J,axiom, % 0.30/0.54 ! [X3: num] : % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.54 = ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % xor_nat_numerals(3) % 0.30/0.54 thf(fact_2353_xor__nat__numerals_I2_J,axiom, % 0.30/0.54 ! [Y: num] : % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) % 0.30/0.54 = ( numeral_numeral @ nat @ ( bit0 @ Y ) ) ) ). % 0.30/0.54 % 0.30/0.54 % xor_nat_numerals(2) % 0.30/0.54 thf(fact_2354_xor__nat__numerals_I1_J,axiom, % 0.30/0.54 ! [Y: num] : % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) ) % 0.30/0.54 = ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ). % 0.30/0.54 % 0.30/0.54 % xor_nat_numerals(1) % 0.30/0.54 thf(fact_2355_xor__numerals_I6_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.54 => ! [X3: num,Y: num] : % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) ) % 0.30/0.54 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % xor_numerals(6) % 0.30/0.54 thf(fact_2356_xor__numerals_I4_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.54 => ! [X3: num,Y: num] : % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) % 0.30/0.54 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % xor_numerals(4) % 0.30/0.54 thf(fact_2357_empty__def,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bot_bot @ ( set @ A ) ) % 0.30/0.54 = ( collect @ A % 0.30/0.54 @ ^ [X: A] : $false ) ) ). % 0.30/0.54 % 0.30/0.54 % empty_def % 0.30/0.54 thf(fact_2358_Collect__conv__if2,axiom, % 0.30/0.54 ! [A: $tType,P: A > $o,A2: A] : % 0.30/0.54 ( ( ( P @ A2 ) % 0.30/0.54 => ( ( collect @ A % 0.30/0.54 @ ^ [X: A] : % 0.30/0.54 ( ( A2 = X ) % 0.30/0.54 & ( P @ X ) ) ) % 0.30/0.54 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.30/0.54 & ( ~ ( P @ A2 ) % 0.30/0.54 => ( ( collect @ A % 0.30/0.54 @ ^ [X: A] : % 0.30/0.54 ( ( A2 = X ) % 0.30/0.54 & ( P @ X ) ) ) % 0.30/0.54 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % Collect_conv_if2 % 0.30/0.54 thf(fact_2359_Collect__conv__if,axiom, % 0.30/0.54 ! [A: $tType,P: A > $o,A2: A] : % 0.30/0.54 ( ( ( P @ A2 ) % 0.30/0.54 => ( ( collect @ A % 0.30/0.54 @ ^ [X: A] : % 0.30/0.54 ( ( X = A2 ) % 0.30/0.54 & ( P @ X ) ) ) % 0.30/0.54 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.30/0.54 & ( ~ ( P @ A2 ) % 0.30/0.54 => ( ( collect @ A % 0.30/0.54 @ ^ [X: A] : % 0.30/0.54 ( ( X = A2 ) % 0.30/0.54 & ( P @ X ) ) ) % 0.30/0.54 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % Collect_conv_if % 0.30/0.54 thf(fact_2360_pred__subset__eq2,axiom, % 0.30/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] : % 0.30/0.54 ( ( ord_less_eq @ ( A > B > $o ) % 0.30/0.54 @ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R ) % 0.30/0.54 @ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S3 ) ) % 0.30/0.54 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ). % 0.30/0.54 % 0.30/0.54 % pred_subset_eq2 % 0.30/0.54 thf(fact_2361_strict__subset__divisors__dvd,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( comm_monoid_mult @ A ) % 0.30/0.54 => ! [A2: A,B2: A] : % 0.30/0.54 ( ( ord_less @ ( set @ A ) % 0.30/0.54 @ ( collect @ A % 0.30/0.54 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A2 ) ) % 0.30/0.54 @ ( collect @ A % 0.30/0.54 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) ) % 0.30/0.54 = ( ( dvd_dvd @ A @ A2 @ B2 ) % 0.30/0.54 & ~ ( dvd_dvd @ A @ B2 @ A2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % strict_subset_divisors_dvd % 0.30/0.54 thf(fact_2362_pred__equals__eq2,axiom, % 0.30/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] : % 0.30/0.54 ( ( ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R ) ) % 0.30/0.54 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S3 ) ) ) % 0.30/0.54 = ( R = S3 ) ) ). % 0.30/0.54 % 0.30/0.54 % pred_equals_eq2 % 0.30/0.54 thf(fact_2363_subset__divisors__dvd,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( comm_monoid_mult @ A ) % 0.30/0.54 => ! [A2: A,B2: A] : % 0.30/0.54 ( ( ord_less_eq @ ( set @ A ) % 0.30/0.54 @ ( collect @ A % 0.30/0.54 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A2 ) ) % 0.30/0.54 @ ( collect @ A % 0.30/0.54 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) ) % 0.30/0.54 = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % subset_divisors_dvd % 0.30/0.54 thf(fact_2364_less__set__def,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ord_less @ ( set @ A ) ) % 0.30/0.54 = ( ^ [A6: set @ A,B6: set @ A] : % 0.30/0.54 ( ord_less @ ( A > $o ) % 0.30/0.54 @ ^ [X: A] : ( member @ A @ X @ A6 ) % 0.30/0.54 @ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % less_set_def % 0.30/0.54 thf(fact_2365_Collect__subset,axiom, % 0.30/0.54 ! [A: $tType,A3: set @ A,P: A > $o] : % 0.30/0.54 ( ord_less_eq @ ( set @ A ) % 0.30/0.54 @ ( collect @ A % 0.30/0.54 @ ^ [X: A] : % 0.30/0.54 ( ( member @ A @ X @ A3 ) % 0.30/0.54 & ( P @ X ) ) ) % 0.30/0.54 @ A3 ) ). % 0.30/0.54 % 0.30/0.54 % Collect_subset % 0.30/0.54 thf(fact_2366_less__eq__set__def,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( ord_less_eq @ ( set @ A ) ) % 0.30/0.54 = ( ^ [A6: set @ A,B6: set @ A] : % 0.30/0.54 ( ord_less_eq @ ( A > $o ) % 0.30/0.54 @ ^ [X: A] : ( member @ A @ X @ A6 ) % 0.30/0.54 @ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % less_eq_set_def % 0.30/0.54 thf(fact_2367_prop__restrict,axiom, % 0.30/0.54 ! [A: $tType,X3: A,Z5: set @ A,X8: set @ A,P: A > $o] : % 0.30/0.54 ( ( member @ A @ X3 @ Z5 ) % 0.30/0.54 => ( ( ord_less_eq @ ( set @ A ) @ Z5 % 0.30/0.54 @ ( collect @ A % 0.30/0.54 @ ^ [X: A] : % 0.30/0.54 ( ( member @ A @ X @ X8 ) % 0.30/0.54 & ( P @ X ) ) ) ) % 0.30/0.54 => ( P @ X3 ) ) ) ). % 0.30/0.54 % 0.30/0.54 % prop_restrict % 0.30/0.54 thf(fact_2368_Collect__restrict,axiom, % 0.30/0.54 ! [A: $tType,X8: set @ A,P: A > $o] : % 0.30/0.54 ( ord_less_eq @ ( set @ A ) % 0.30/0.54 @ ( collect @ A % 0.30/0.54 @ ^ [X: A] : % 0.30/0.54 ( ( member @ A @ X @ X8 ) % 0.30/0.54 & ( P @ X ) ) ) % 0.30/0.54 @ X8 ) ). % 0.30/0.54 % 0.30/0.54 % Collect_restrict % 0.30/0.54 thf(fact_2369_pred__subset__eq,axiom, % 0.30/0.54 ! [A: $tType,R: set @ A,S3: set @ A] : % 0.30/0.54 ( ( ord_less_eq @ ( A > $o ) % 0.30/0.54 @ ^ [X: A] : ( member @ A @ X @ R ) % 0.30/0.54 @ ^ [X: A] : ( member @ A @ X @ S3 ) ) % 0.30/0.54 = ( ord_less_eq @ ( set @ A ) @ R @ S3 ) ) ). % 0.30/0.54 % 0.30/0.54 % pred_subset_eq % 0.30/0.54 thf(fact_2370_lambda__one,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( monoid_mult @ A ) % 0.30/0.54 => ( ( ^ [X: A] : X ) % 0.30/0.54 = ( times_times @ A @ ( one_one @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % lambda_one % 0.30/0.54 thf(fact_2371_lambda__zero,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( mult_zero @ A ) % 0.30/0.54 => ( ( ^ [H: A] : ( zero_zero @ A ) ) % 0.30/0.54 = ( times_times @ A @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % lambda_zero % 0.30/0.54 thf(fact_2372_uminus__set__def,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( uminus_uminus @ ( set @ A ) ) % 0.30/0.54 = ( ^ [A6: set @ A] : % 0.30/0.54 ( collect @ A % 0.30/0.54 @ ( uminus_uminus @ ( A > $o ) % 0.30/0.54 @ ^ [X: A] : ( member @ A @ X @ A6 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % uminus_set_def % 0.30/0.54 thf(fact_2373_Collect__neg__eq,axiom, % 0.30/0.54 ! [A: $tType,P: A > $o] : % 0.30/0.54 ( ( collect @ A % 0.30/0.54 @ ^ [X: A] : % 0.30/0.54 ~ ( P @ X ) ) % 0.30/0.54 = ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ). % 0.30/0.54 % 0.30/0.54 % Collect_neg_eq % 0.30/0.54 thf(fact_2374_Compl__eq,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( uminus_uminus @ ( set @ A ) ) % 0.30/0.54 = ( ^ [A6: set @ A] : % 0.30/0.54 ( collect @ A % 0.30/0.54 @ ^ [X: A] : % 0.30/0.54 ~ ( member @ A @ X @ A6 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % Compl_eq % 0.30/0.54 thf(fact_2375_set__diff__eq,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( minus_minus @ ( set @ A ) ) % 0.30/0.54 = ( ^ [A6: set @ A,B6: set @ A] : % 0.30/0.54 ( collect @ A % 0.30/0.54 @ ^ [X: A] : % 0.30/0.54 ( ( member @ A @ X @ A6 ) % 0.30/0.54 & ~ ( member @ A @ X @ B6 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % set_diff_eq % 0.30/0.54 thf(fact_2376_insert__compr,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( insert @ A ) % 0.30/0.54 = ( ^ [A4: A,B6: set @ A] : % 0.30/0.54 ( collect @ A % 0.30/0.54 @ ^ [X: A] : % 0.30/0.54 ( ( X = A4 ) % 0.30/0.54 | ( member @ A @ X @ B6 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % insert_compr % 0.30/0.54 thf(fact_2377_minus__set__def,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( minus_minus @ ( set @ A ) ) % 0.30/0.54 = ( ^ [A6: set @ A,B6: set @ A] : % 0.30/0.54 ( collect @ A % 0.30/0.54 @ ( minus_minus @ ( A > $o ) % 0.30/0.54 @ ^ [X: A] : ( member @ A @ X @ A6 ) % 0.30/0.54 @ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % minus_set_def % 0.30/0.54 thf(fact_2378_insert__Collect,axiom, % 0.30/0.54 ! [A: $tType,A2: A,P: A > $o] : % 0.30/0.54 ( ( insert @ A @ A2 @ ( collect @ A @ P ) ) % 0.30/0.54 = ( collect @ A % 0.30/0.54 @ ^ [U2: A] : % 0.30/0.54 ( ( U2 != A2 ) % 0.30/0.54 => ( P @ U2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % insert_Collect % 0.30/0.54 thf(fact_2379_not__finite__existsD,axiom, % 0.30/0.54 ! [A: $tType,P: A > $o] : % 0.30/0.54 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) ) % 0.30/0.54 => ? [X_1: A] : ( P @ X_1 ) ) ). % 0.30/0.54 % 0.30/0.54 % not_finite_existsD % 0.30/0.54 thf(fact_2380_pigeonhole__infinite__rel,axiom, % 0.30/0.54 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,R: A > B > $o] : % 0.30/0.54 ( ~ ( finite_finite2 @ A @ A3 ) % 0.30/0.54 => ( ( finite_finite2 @ B @ B3 ) % 0.30/0.54 => ( ! [X5: A] : % 0.30/0.54 ( ( member @ A @ X5 @ A3 ) % 0.30/0.54 => ? [Xa: B] : % 0.30/0.54 ( ( member @ B @ Xa @ B3 ) % 0.30/0.54 & ( R @ X5 @ Xa ) ) ) % 0.30/0.54 => ? [X5: B] : % 0.30/0.54 ( ( member @ B @ X5 @ B3 ) % 0.30/0.54 & ~ ( finite_finite2 @ A % 0.30/0.54 @ ( collect @ A % 0.30/0.54 @ ^ [A4: A] : % 0.30/0.54 ( ( member @ A @ A4 @ A3 ) % 0.30/0.54 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % pigeonhole_infinite_rel % 0.30/0.54 thf(fact_2381_finite__image__set2,axiom, % 0.30/0.54 ! [A: $tType,B: $tType,C: $tType,P: A > $o,Q: B > $o,F2: A > B > C] : % 0.30/0.54 ( ( finite_finite2 @ A @ ( collect @ A @ P ) ) % 0.30/0.54 => ( ( finite_finite2 @ B @ ( collect @ B @ Q ) ) % 0.30/0.54 => ( finite_finite2 @ C % 0.30/0.54 @ ( collect @ C % 0.30/0.54 @ ^ [Uu3: C] : % 0.30/0.54 ? [X: A,Y2: B] : % 0.30/0.54 ( ( Uu3 % 0.30/0.54 = ( F2 @ X @ Y2 ) ) % 0.30/0.54 & ( P @ X ) % 0.30/0.54 & ( Q @ Y2 ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_image_set2 % 0.30/0.54 thf(fact_2382_finite__image__set,axiom, % 0.30/0.54 ! [A: $tType,B: $tType,P: A > $o,F2: A > B] : % 0.30/0.54 ( ( finite_finite2 @ A @ ( collect @ A @ P ) ) % 0.30/0.54 => ( finite_finite2 @ B % 0.30/0.54 @ ( collect @ B % 0.30/0.54 @ ^ [Uu3: B] : % 0.30/0.54 ? [X: A] : % 0.30/0.54 ( ( Uu3 % 0.30/0.54 = ( F2 @ X ) ) % 0.30/0.54 & ( P @ X ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_image_set % 0.30/0.54 thf(fact_2383_bot__empty__eq2,axiom, % 0.30/0.54 ! [B: $tType,A: $tType] : % 0.30/0.54 ( ( bot_bot @ ( A > B > $o ) ) % 0.30/0.54 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bot_empty_eq2 % 0.30/0.54 thf(fact_2384_finite__M__bounded__by__nat,axiom, % 0.30/0.54 ! [P: nat > $o,I: nat] : % 0.30/0.54 ( finite_finite2 @ nat % 0.30/0.54 @ ( collect @ nat % 0.30/0.54 @ ^ [K3: nat] : % 0.30/0.54 ( ( P @ K3 ) % 0.30/0.54 & ( ord_less @ nat @ K3 @ I ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_M_bounded_by_nat % 0.30/0.54 thf(fact_2385_finite__less__ub,axiom, % 0.30/0.54 ! [F2: nat > nat,U: nat] : % 0.30/0.54 ( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F2 @ N3 ) ) % 0.30/0.54 => ( finite_finite2 @ nat % 0.30/0.54 @ ( collect @ nat % 0.30/0.54 @ ^ [N: nat] : ( ord_less_eq @ nat @ ( F2 @ N ) @ U ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_less_ub % 0.30/0.54 thf(fact_2386_numeral__code_I2_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( numeral @ A ) % 0.30/0.54 => ! [N2: num] : % 0.30/0.54 ( ( numeral_numeral @ A @ ( bit0 @ N2 ) ) % 0.30/0.54 = ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % numeral_code(2) % 0.30/0.54 thf(fact_2387_set__vebt__def,axiom, % 0.30/0.54 ( vEBT_set_vebt % 0.30/0.54 = ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % set_vebt_def % 0.30/0.54 thf(fact_2388_finite__divisors__int,axiom, % 0.30/0.54 ! [I: int] : % 0.30/0.54 ( ( I % 0.30/0.54 != ( zero_zero @ int ) ) % 0.30/0.54 => ( finite_finite2 @ int % 0.30/0.54 @ ( collect @ int % 0.30/0.54 @ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ I ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_divisors_int % 0.30/0.54 thf(fact_2389_numeral__code_I3_J,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( numeral @ A ) % 0.30/0.54 => ! [N2: num] : % 0.30/0.54 ( ( numeral_numeral @ A @ ( bit1 @ N2 ) ) % 0.30/0.54 = ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( one_one @ A ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % numeral_code(3) % 0.30/0.54 thf(fact_2390_finite__divisors__nat,axiom, % 0.30/0.54 ! [M: nat] : % 0.30/0.54 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.30/0.54 => ( finite_finite2 @ nat % 0.30/0.54 @ ( collect @ nat % 0.30/0.54 @ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_divisors_nat % 0.30/0.54 thf(fact_2391_set__conv__nth,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( set2 @ A ) % 0.30/0.54 = ( ^ [Xs2: list @ A] : % 0.30/0.54 ( collect @ A % 0.30/0.54 @ ^ [Uu3: A] : % 0.30/0.54 ? [I3: nat] : % 0.30/0.54 ( ( Uu3 % 0.30/0.54 = ( nth @ A @ Xs2 @ I3 ) ) % 0.30/0.54 & ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % set_conv_nth % 0.30/0.54 thf(fact_2392_finite__lists__length__eq,axiom, % 0.30/0.54 ! [A: $tType,A3: set @ A,N2: nat] : % 0.30/0.54 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.54 => ( finite_finite2 @ ( list @ A ) % 0.30/0.54 @ ( collect @ ( list @ A ) % 0.30/0.54 @ ^ [Xs2: list @ A] : % 0.30/0.54 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) % 0.30/0.54 & ( ( size_size @ ( list @ A ) @ Xs2 ) % 0.30/0.54 = N2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_lists_length_eq % 0.30/0.54 thf(fact_2393_finite__lists__length__le,axiom, % 0.30/0.54 ! [A: $tType,A3: set @ A,N2: nat] : % 0.30/0.54 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.54 => ( finite_finite2 @ ( list @ A ) % 0.30/0.54 @ ( collect @ ( list @ A ) % 0.30/0.54 @ ^ [Xs2: list @ A] : % 0.30/0.54 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) % 0.30/0.54 & ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % finite_lists_length_le % 0.30/0.54 thf(fact_2394_vebt__buildup_Osimps_I3_J,axiom, % 0.30/0.54 ! [Va: nat] : % 0.30/0.54 ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) ) % 0.30/0.54 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) ) % 0.30/0.54 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) % 0.30/0.54 & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) ) % 0.30/0.54 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) ) % 0.30/0.54 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % vebt_buildup.simps(3) % 0.30/0.54 thf(fact_2395_vebt__buildup_Oelims,axiom, % 0.30/0.54 ! [X3: nat,Y: vEBT_VEBT] : % 0.30/0.54 ( ( ( vEBT_vebt_buildup @ X3 ) % 0.30/0.54 = Y ) % 0.30/0.54 => ( ( ( X3 % 0.30/0.54 = ( zero_zero @ nat ) ) % 0.30/0.54 => ( Y % 0.30/0.54 != ( vEBT_Leaf @ $false @ $false ) ) ) % 0.30/0.54 => ( ( ( X3 % 0.30/0.54 = ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.54 => ( Y % 0.30/0.54 != ( vEBT_Leaf @ $false @ $false ) ) ) % 0.30/0.54 => ~ ! [Va2: nat] : % 0.30/0.54 ( ( X3 % 0.30/0.54 = ( suc @ ( suc @ Va2 ) ) ) % 0.30/0.54 => ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) ) % 0.30/0.54 => ( Y % 0.30/0.54 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) % 0.30/0.54 & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) ) % 0.30/0.54 => ( Y % 0.30/0.54 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % vebt_buildup.elims % 0.30/0.54 thf(fact_2396_VEBT__internal_Onaive__member_Osimps_I3_J,axiom, % 0.30/0.54 ! [Uy: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList2: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] : % 0.30/0.54 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList2 @ S ) @ X3 ) % 0.30/0.54 = ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) % 0.30/0.54 => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % VEBT_internal.naive_member.simps(3) % 0.30/0.54 thf(fact_2397_dbl__dec__def,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( neg_numeral @ A ) % 0.30/0.54 => ( ( neg_numeral_dbl_dec @ A ) % 0.30/0.54 = ( ^ [X: A] : ( minus_minus @ A @ ( plus_plus @ A @ X @ X ) @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % dbl_dec_def % 0.30/0.54 thf(fact_2398_VEBT__internal_Omembermima_Osimps_I5_J,axiom, % 0.30/0.54 ! [V2: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] : % 0.30/0.54 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ X3 ) % 0.30/0.54 = ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) % 0.30/0.54 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % VEBT_internal.membermima.simps(5) % 0.30/0.54 thf(fact_2399_vebt__member_Osimps_I5_J,axiom, % 0.30/0.54 ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : % 0.30/0.54 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 ) % 0.30/0.54 = ( ( X3 != Mi ) % 0.30/0.54 => ( ( X3 != Ma ) % 0.30/0.54 => ( ~ ( ord_less @ nat @ X3 @ Mi ) % 0.30/0.54 & ( ~ ( ord_less @ nat @ X3 @ Mi ) % 0.30/0.54 => ( ~ ( ord_less @ nat @ Ma @ X3 ) % 0.30/0.54 & ( ~ ( ord_less @ nat @ Ma @ X3 ) % 0.30/0.54 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) % 0.30/0.54 => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % vebt_member.simps(5) % 0.30/0.54 thf(fact_2400_VEBT__internal_Omembermima_Osimps_I4_J,axiom, % 0.30/0.54 ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] : % 0.30/0.54 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) @ X3 ) % 0.30/0.54 = ( ( X3 = Mi ) % 0.30/0.54 | ( X3 = Ma ) % 0.30/0.54 | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) % 0.30/0.54 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % VEBT_internal.membermima.simps(4) % 0.30/0.54 thf(fact_2401_VEBT__internal_Onaive__member_Oelims_I1_J,axiom, % 0.30/0.54 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 0.30/0.54 ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 ) % 0.30/0.54 = Y ) % 0.30/0.54 => ( ! [A5: $o,B5: $o] : % 0.30/0.54 ( ( X3 % 0.30/0.54 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.54 => ( Y % 0.30/0.54 = ( ~ ( ( ( Xa2 % 0.30/0.54 = ( zero_zero @ nat ) ) % 0.30/0.54 => A5 ) % 0.30/0.54 & ( ( Xa2 % 0.30/0.54 != ( zero_zero @ nat ) ) % 0.30/0.54 => ( ( ( Xa2 % 0.30/0.54 = ( one_one @ nat ) ) % 0.30/0.54 => B5 ) % 0.30/0.54 & ( Xa2 % 0.30/0.54 = ( one_one @ nat ) ) ) ) ) ) ) ) % 0.30/0.54 => ( ( ? [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) ) % 0.30/0.54 => Y ) % 0.30/0.54 => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT] : % 0.30/0.54 ( ? [S2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) ) % 0.30/0.54 => ( Y % 0.30/0.54 = ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.54 => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % VEBT_internal.naive_member.elims(1) % 0.30/0.54 thf(fact_2402_VEBT__internal_Onaive__member_Oelims_I2_J,axiom, % 0.30/0.54 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.30/0.54 ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 ) % 0.30/0.54 => ( ! [A5: $o,B5: $o] : % 0.30/0.54 ( ( X3 % 0.30/0.54 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.54 => ~ ( ( ( Xa2 % 0.30/0.54 = ( zero_zero @ nat ) ) % 0.30/0.54 => A5 ) % 0.30/0.54 & ( ( Xa2 % 0.30/0.54 != ( zero_zero @ nat ) ) % 0.30/0.54 => ( ( ( Xa2 % 0.30/0.54 = ( one_one @ nat ) ) % 0.30/0.54 => B5 ) % 0.30/0.54 & ( Xa2 % 0.30/0.54 = ( one_one @ nat ) ) ) ) ) ) % 0.30/0.54 => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT] : % 0.30/0.54 ( ? [S2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) ) % 0.30/0.54 => ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.54 => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % VEBT_internal.naive_member.elims(2) % 0.30/0.54 thf(fact_2403_VEBT__internal_Onaive__member_Oelims_I3_J,axiom, % 0.30/0.54 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.30/0.54 ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 ) % 0.30/0.54 => ( ! [A5: $o,B5: $o] : % 0.30/0.54 ( ( X3 % 0.30/0.54 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.54 => ( ( ( Xa2 % 0.30/0.54 = ( zero_zero @ nat ) ) % 0.30/0.54 => A5 ) % 0.30/0.54 & ( ( Xa2 % 0.30/0.54 != ( zero_zero @ nat ) ) % 0.30/0.54 => ( ( ( Xa2 % 0.30/0.54 = ( one_one @ nat ) ) % 0.30/0.54 => B5 ) % 0.30/0.54 & ( Xa2 % 0.30/0.54 = ( one_one @ nat ) ) ) ) ) ) % 0.30/0.54 => ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 != ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) ) % 0.30/0.54 => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT] : % 0.30/0.54 ( ? [S2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) ) % 0.30/0.54 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.54 => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % VEBT_internal.naive_member.elims(3) % 0.30/0.54 thf(fact_2404_VEBT__internal_Omembermima_Oelims_I2_J,axiom, % 0.30/0.54 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.30/0.54 ( ( vEBT_VEBT_membermima @ X3 @ Xa2 ) % 0.30/0.54 => ( ! [Mi2: nat,Ma2: nat] : % 0.30/0.54 ( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) ) % 0.30/0.54 => ~ ( ( Xa2 = Mi2 ) % 0.30/0.54 | ( Xa2 = Ma2 ) ) ) % 0.30/0.54 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT] : % 0.30/0.54 ( ? [Vc2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) ) % 0.30/0.54 => ~ ( ( Xa2 = Mi2 ) % 0.30/0.54 | ( Xa2 = Ma2 ) % 0.30/0.54 | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.54 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) % 0.30/0.54 => ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT] : % 0.30/0.54 ( ? [Vd: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) ) % 0.30/0.54 => ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.54 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % VEBT_internal.membermima.elims(2) % 0.30/0.54 thf(fact_2405_xor__nat__unfold,axiom, % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ nat ) % 0.30/0.54 = ( ^ [M2: nat,N: nat] : % 0.30/0.54 ( if @ nat % 0.30/0.54 @ ( M2 % 0.30/0.54 = ( zero_zero @ nat ) ) % 0.30/0.54 @ N % 0.30/0.54 @ ( if @ nat % 0.30/0.54 @ ( N % 0.30/0.54 = ( zero_zero @ nat ) ) % 0.30/0.54 @ M2 % 0.30/0.54 @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % xor_nat_unfold % 0.30/0.54 thf(fact_2406_xor__nat__rec,axiom, % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ nat ) % 0.30/0.54 = ( ^ [M2: nat,N: nat] : % 0.30/0.54 ( plus_plus @ nat % 0.30/0.54 @ ( zero_neq_one_of_bool @ nat % 0.30/0.54 @ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) % 0.30/0.54 != ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) % 0.30/0.54 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % xor_nat_rec % 0.30/0.54 thf(fact_2407_one__xor__eq,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.54 => ! [A2: A] : % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A2 ) % 0.30/0.54 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) % 0.30/0.54 @ ( zero_neq_one_of_bool @ A % 0.30/0.54 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % one_xor_eq % 0.30/0.54 thf(fact_2408_xor__one__eq,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_se359711467146920520ations @ A ) % 0.30/0.54 => ! [A2: A] : % 0.30/0.54 ( ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( one_one @ A ) ) % 0.30/0.54 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) % 0.30/0.54 @ ( zero_neq_one_of_bool @ A % 0.30/0.54 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % xor_one_eq % 0.30/0.54 thf(fact_2409_vebt__member_Oelims_I2_J,axiom, % 0.30/0.54 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.30/0.54 ( ( vEBT_vebt_member @ X3 @ Xa2 ) % 0.30/0.54 => ( ! [A5: $o,B5: $o] : % 0.30/0.54 ( ( X3 % 0.30/0.54 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.54 => ~ ( ( ( Xa2 % 0.30/0.54 = ( zero_zero @ nat ) ) % 0.30/0.54 => A5 ) % 0.30/0.54 & ( ( Xa2 % 0.30/0.54 != ( zero_zero @ nat ) ) % 0.30/0.54 => ( ( ( Xa2 % 0.30/0.54 = ( one_one @ nat ) ) % 0.30/0.54 => B5 ) % 0.30/0.54 & ( Xa2 % 0.30/0.54 = ( one_one @ nat ) ) ) ) ) ) % 0.30/0.54 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT] : % 0.30/0.54 ( ? [Summary2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) % 0.30/0.54 => ~ ( ( Xa2 != Mi2 ) % 0.30/0.54 => ( ( Xa2 != Ma2 ) % 0.30/0.54 => ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 0.30/0.54 & ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 0.30/0.54 => ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 0.30/0.54 & ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 0.30/0.54 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.54 => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % vebt_member.elims(2) % 0.30/0.54 thf(fact_2410_VEBT__internal_Omembermima_Oelims_I3_J,axiom, % 0.30/0.54 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.30/0.54 ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 ) % 0.30/0.54 => ( ! [Uu2: $o,Uv2: $o] : % 0.30/0.54 ( X3 % 0.30/0.54 != ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 0.30/0.54 => ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) ) % 0.30/0.54 => ( ! [Mi2: nat,Ma2: nat] : % 0.30/0.54 ( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) ) % 0.30/0.54 => ( ( Xa2 = Mi2 ) % 0.30/0.54 | ( Xa2 = Ma2 ) ) ) % 0.30/0.54 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT] : % 0.30/0.54 ( ? [Vc2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) ) % 0.30/0.54 => ( ( Xa2 = Mi2 ) % 0.30/0.54 | ( Xa2 = Ma2 ) % 0.30/0.54 | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.54 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) % 0.30/0.54 => ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT] : % 0.30/0.54 ( ? [Vd: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) ) % 0.30/0.54 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.54 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % VEBT_internal.membermima.elims(3) % 0.30/0.54 thf(fact_2411_VEBT__internal_Omembermima_Oelims_I1_J,axiom, % 0.30/0.54 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 0.30/0.54 ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 ) % 0.30/0.54 = Y ) % 0.30/0.54 => ( ( ? [Uu2: $o,Uv2: $o] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 0.30/0.54 => Y ) % 0.30/0.54 => ( ( ? [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) ) % 0.30/0.54 => Y ) % 0.30/0.54 => ( ! [Mi2: nat,Ma2: nat] : % 0.30/0.54 ( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) ) % 0.30/0.54 => ( Y % 0.30/0.54 = ( ~ ( ( Xa2 = Mi2 ) % 0.30/0.54 | ( Xa2 = Ma2 ) ) ) ) ) % 0.30/0.54 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT] : % 0.30/0.54 ( ? [Vc2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) ) % 0.30/0.54 => ( Y % 0.30/0.54 = ( ~ ( ( Xa2 = Mi2 ) % 0.30/0.54 | ( Xa2 = Ma2 ) % 0.30/0.54 | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.54 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) % 0.30/0.54 => ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT] : % 0.30/0.54 ( ? [Vd: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) ) % 0.30/0.54 => ( Y % 0.30/0.54 = ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.54 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % VEBT_internal.membermima.elims(1) % 0.30/0.54 thf(fact_2412_bit__horner__sum__bit__iff,axiom, % 0.30/0.54 ! [A: $tType] : % 0.30/0.54 ( ( bit_un5681908812861735899ations @ A ) % 0.30/0.54 => ! [Bs: list @ $o,N2: nat] : % 0.30/0.54 ( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) @ N2 ) % 0.30/0.54 = ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ $o ) @ Bs ) ) % 0.30/0.54 & ( nth @ $o @ Bs @ N2 ) ) ) ) ). % 0.30/0.54 % 0.30/0.54 % bit_horner_sum_bit_iff % 0.30/0.54 thf(fact_2413_vebt__member_Oelims_I3_J,axiom, % 0.30/0.54 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.30/0.54 ( ~ ( vEBT_vebt_member @ X3 @ Xa2 ) % 0.30/0.54 => ( ! [A5: $o,B5: $o] : % 0.30/0.54 ( ( X3 % 0.30/0.54 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.54 => ( ( ( Xa2 % 0.30/0.54 = ( zero_zero @ nat ) ) % 0.30/0.54 => A5 ) % 0.30/0.54 & ( ( Xa2 % 0.30/0.54 != ( zero_zero @ nat ) ) % 0.30/0.54 => ( ( ( Xa2 % 0.30/0.54 = ( one_one @ nat ) ) % 0.30/0.54 => B5 ) % 0.30/0.54 & ( Xa2 % 0.30/0.54 = ( one_one @ nat ) ) ) ) ) ) % 0.30/0.54 => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 0.30/0.54 => ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) ) % 0.30/0.54 => ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) ) % 0.30/0.54 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT] : % 0.30/0.54 ( ? [Summary2: vEBT_VEBT] : % 0.30/0.54 ( X3 % 0.30/0.54 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) % 0.30/0.54 => ( ( Xa2 != Mi2 ) % 0.30/0.54 => ( ( Xa2 != Ma2 ) % 0.30/0.54 => ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 0.30/0.54 & ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 0.30/0.54 => ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 0.30/0.54 & ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 0.30/0.54 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.54 => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.54 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % vebt_member.elims(3) % 0.30/0.55 thf(fact_2414_of__int__code__if,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ring_1 @ A ) % 0.30/0.55 => ( ( ring_1_of_int @ A ) % 0.30/0.55 = ( ^ [K3: int] : % 0.30/0.55 ( if @ A % 0.30/0.55 @ ( K3 % 0.30/0.55 = ( zero_zero @ int ) ) % 0.30/0.55 @ ( zero_zero @ A ) % 0.30/0.55 @ ( if @ A @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ ( uminus_uminus @ int @ K3 ) ) ) % 0.30/0.55 @ ( if @ A % 0.30/0.55 @ ( ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) % 0.30/0.55 = ( zero_zero @ int ) ) % 0.30/0.55 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) % 0.30/0.55 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ A ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_int_code_if % 0.30/0.55 thf(fact_2415_finite__nth__roots,axiom, % 0.30/0.55 ! [N2: nat,C2: complex] : % 0.30/0.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.30/0.55 => ( finite_finite2 @ complex % 0.30/0.55 @ ( collect @ complex % 0.30/0.55 @ ^ [Z6: complex] : % 0.30/0.55 ( ( power_power @ complex @ Z6 @ N2 ) % 0.30/0.55 = C2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % finite_nth_roots % 0.30/0.55 thf(fact_2416_finite__roots__unity,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ( real_V8999393235501362500lgebra @ A ) % 0.30/0.55 & ( idom @ A ) ) % 0.30/0.55 => ! [N2: nat] : % 0.30/0.55 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 ) % 0.30/0.55 => ( finite_finite2 @ A % 0.30/0.55 @ ( collect @ A % 0.30/0.55 @ ^ [Z6: A] : % 0.30/0.55 ( ( power_power @ A @ Z6 @ N2 ) % 0.30/0.55 = ( one_one @ A ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % finite_roots_unity % 0.30/0.55 thf(fact_2417_prod_Ofinite__Collect__op,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_mult @ A ) % 0.30/0.55 => ! [I6: set @ B,X3: B > A,Y: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B % 0.30/0.55 @ ( collect @ B % 0.30/0.55 @ ^ [I3: B] : % 0.30/0.55 ( ( member @ B @ I3 @ I6 ) % 0.30/0.55 & ( ( X3 @ I3 ) % 0.30/0.55 != ( one_one @ A ) ) ) ) ) % 0.30/0.55 => ( ( finite_finite2 @ B % 0.30/0.55 @ ( collect @ B % 0.30/0.55 @ ^ [I3: B] : % 0.30/0.55 ( ( member @ B @ I3 @ I6 ) % 0.30/0.55 & ( ( Y @ I3 ) % 0.30/0.55 != ( one_one @ A ) ) ) ) ) % 0.30/0.55 => ( finite_finite2 @ B % 0.30/0.55 @ ( collect @ B % 0.30/0.55 @ ^ [I3: B] : % 0.30/0.55 ( ( member @ B @ I3 @ I6 ) % 0.30/0.55 & ( ( times_times @ A @ ( X3 @ I3 ) @ ( Y @ I3 ) ) % 0.30/0.55 != ( one_one @ A ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % prod.finite_Collect_op % 0.30/0.55 thf(fact_2418_vebt__member_Opelims_I1_J,axiom, % 0.30/0.55 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 0.30/0.55 ( ( ( vEBT_vebt_member @ X3 @ Xa2 ) % 0.30/0.55 = Y ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 0.30/0.55 => ( ! [A5: $o,B5: $o] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.55 => ( ( Y % 0.30/0.55 = ( ( ( Xa2 % 0.30/0.55 = ( zero_zero @ nat ) ) % 0.30/0.55 => A5 ) % 0.30/0.55 & ( ( Xa2 % 0.30/0.55 != ( zero_zero @ nat ) ) % 0.30/0.55 => ( ( ( Xa2 % 0.30/0.55 = ( one_one @ nat ) ) % 0.30/0.55 => B5 ) % 0.30/0.55 & ( Xa2 % 0.30/0.55 = ( one_one @ nat ) ) ) ) ) ) % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) ) % 0.30/0.55 => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 0.30/0.55 => ( ~ Y % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) ) % 0.30/0.55 => ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) ) % 0.30/0.55 => ( ~ Y % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa2 ) ) ) ) % 0.30/0.55 => ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) ) % 0.30/0.55 => ( ~ Y % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) ) % 0.30/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) % 0.30/0.55 => ( ( Y % 0.30/0.55 = ( ( Xa2 != Mi2 ) % 0.30/0.55 => ( ( Xa2 != Ma2 ) % 0.30/0.55 => ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 0.30/0.55 & ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 0.30/0.55 => ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 0.30/0.55 & ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 0.30/0.55 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.55 => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.55 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % vebt_member.pelims(1) % 0.30/0.55 thf(fact_2419_vebt__member_Opelims_I3_J,axiom, % 0.30/0.55 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.30/0.55 ( ~ ( vEBT_vebt_member @ X3 @ Xa2 ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 0.30/0.55 => ( ! [A5: $o,B5: $o] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) % 0.30/0.55 => ( ( ( Xa2 % 0.30/0.55 = ( zero_zero @ nat ) ) % 0.30/0.55 => A5 ) % 0.30/0.55 & ( ( Xa2 % 0.30/0.55 != ( zero_zero @ nat ) ) % 0.30/0.55 => ( ( ( Xa2 % 0.30/0.55 = ( one_one @ nat ) ) % 0.30/0.55 => B5 ) % 0.30/0.55 & ( Xa2 % 0.30/0.55 = ( one_one @ nat ) ) ) ) ) ) ) % 0.30/0.55 => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) % 0.30/0.55 => ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) ) % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa2 ) ) ) % 0.30/0.55 => ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) ) % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) % 0.30/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) % 0.30/0.55 => ( ( Xa2 != Mi2 ) % 0.30/0.55 => ( ( Xa2 != Ma2 ) % 0.30/0.55 => ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 0.30/0.55 & ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 0.30/0.55 => ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 0.30/0.55 & ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 0.30/0.55 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.55 => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.55 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % vebt_member.pelims(3) % 0.30/0.55 thf(fact_2420_xor__negative__int__iff,axiom, % 0.30/0.55 ! [K: int,L: int] : % 0.30/0.55 ( ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) @ ( zero_zero @ int ) ) % 0.30/0.55 = ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 0.30/0.55 != ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % xor_negative_int_iff % 0.30/0.55 thf(fact_2421_bot2E,axiom, % 0.30/0.55 ! [A: $tType,B: $tType,X3: A,Y: B] : % 0.30/0.55 ~ ( bot_bot @ ( A > B > $o ) @ X3 @ Y ) ). % 0.30/0.55 % 0.30/0.55 % bot2E % 0.30/0.55 thf(fact_2422_XOR__upper,axiom, % 0.30/0.55 ! [X3: int,N2: nat,Y: int] : % 0.30/0.55 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 ) % 0.30/0.55 => ( ( ord_less @ int @ X3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.55 => ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.30/0.55 => ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X3 @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % XOR_upper % 0.30/0.55 thf(fact_2423_vebt__member_Opelims_I2_J,axiom, % 0.30/0.55 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.30/0.55 ( ( vEBT_vebt_member @ X3 @ Xa2 ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 0.30/0.55 => ( ! [A5: $o,B5: $o] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) % 0.30/0.55 => ~ ( ( ( Xa2 % 0.30/0.55 = ( zero_zero @ nat ) ) % 0.30/0.55 => A5 ) % 0.30/0.55 & ( ( Xa2 % 0.30/0.55 != ( zero_zero @ nat ) ) % 0.30/0.55 => ( ( ( Xa2 % 0.30/0.55 = ( one_one @ nat ) ) % 0.30/0.55 => B5 ) % 0.30/0.55 & ( Xa2 % 0.30/0.55 = ( one_one @ nat ) ) ) ) ) ) ) % 0.30/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) % 0.30/0.55 => ~ ( ( Xa2 != Mi2 ) % 0.30/0.55 => ( ( Xa2 != Ma2 ) % 0.30/0.55 => ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 0.30/0.55 & ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 0.30/0.55 => ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 0.30/0.55 & ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 0.30/0.55 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.55 => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.55 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % vebt_member.pelims(2) % 0.30/0.55 thf(fact_2424_sum_Ofinite__Collect__op,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [I6: set @ B,X3: B > A,Y: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B % 0.30/0.55 @ ( collect @ B % 0.30/0.55 @ ^ [I3: B] : % 0.30/0.55 ( ( member @ B @ I3 @ I6 ) % 0.30/0.55 & ( ( X3 @ I3 ) % 0.30/0.55 != ( zero_zero @ A ) ) ) ) ) % 0.30/0.55 => ( ( finite_finite2 @ B % 0.30/0.55 @ ( collect @ B % 0.30/0.55 @ ^ [I3: B] : % 0.30/0.55 ( ( member @ B @ I3 @ I6 ) % 0.30/0.55 & ( ( Y @ I3 ) % 0.30/0.55 != ( zero_zero @ A ) ) ) ) ) % 0.30/0.55 => ( finite_finite2 @ B % 0.30/0.55 @ ( collect @ B % 0.30/0.55 @ ^ [I3: B] : % 0.30/0.55 ( ( member @ B @ I3 @ I6 ) % 0.30/0.55 & ( ( plus_plus @ A @ ( X3 @ I3 ) @ ( Y @ I3 ) ) % 0.30/0.55 != ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.finite_Collect_op % 0.30/0.55 thf(fact_2425_VEBT__internal_Onaive__member_Opelims_I1_J,axiom, % 0.30/0.55 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 0.30/0.55 ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 ) % 0.30/0.55 = Y ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 0.30/0.55 => ( ! [A5: $o,B5: $o] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.55 => ( ( Y % 0.30/0.55 = ( ( ( Xa2 % 0.30/0.55 = ( zero_zero @ nat ) ) % 0.30/0.55 => A5 ) % 0.30/0.55 & ( ( Xa2 % 0.30/0.55 != ( zero_zero @ nat ) ) % 0.30/0.55 => ( ( ( Xa2 % 0.30/0.55 = ( one_one @ nat ) ) % 0.30/0.55 => B5 ) % 0.30/0.55 & ( Xa2 % 0.30/0.55 = ( one_one @ nat ) ) ) ) ) ) % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) ) % 0.30/0.55 => ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) ) % 0.30/0.55 => ( ~ Y % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa2 ) ) ) ) % 0.30/0.55 => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT,S2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) ) % 0.30/0.55 => ( ( Y % 0.30/0.55 = ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.55 => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.55 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % VEBT_internal.naive_member.pelims(1) % 0.30/0.55 thf(fact_2426_VEBT__internal_Onaive__member_Opelims_I2_J,axiom, % 0.30/0.55 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.30/0.55 ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 0.30/0.55 => ( ! [A5: $o,B5: $o] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) % 0.30/0.55 => ~ ( ( ( Xa2 % 0.30/0.55 = ( zero_zero @ nat ) ) % 0.30/0.55 => A5 ) % 0.30/0.55 & ( ( Xa2 % 0.30/0.55 != ( zero_zero @ nat ) ) % 0.30/0.55 => ( ( ( Xa2 % 0.30/0.55 = ( one_one @ nat ) ) % 0.30/0.55 => B5 ) % 0.30/0.55 & ( Xa2 % 0.30/0.55 = ( one_one @ nat ) ) ) ) ) ) ) % 0.30/0.55 => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT,S2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) @ Xa2 ) ) % 0.30/0.55 => ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.55 => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.55 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % VEBT_internal.naive_member.pelims(2) % 0.30/0.55 thf(fact_2427_VEBT__internal_Onaive__member_Opelims_I3_J,axiom, % 0.30/0.55 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.30/0.55 ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 0.30/0.55 => ( ! [A5: $o,B5: $o] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) % 0.30/0.55 => ( ( ( Xa2 % 0.30/0.55 = ( zero_zero @ nat ) ) % 0.30/0.55 => A5 ) % 0.30/0.55 & ( ( Xa2 % 0.30/0.55 != ( zero_zero @ nat ) ) % 0.30/0.55 => ( ( ( Xa2 % 0.30/0.55 = ( one_one @ nat ) ) % 0.30/0.55 => B5 ) % 0.30/0.55 & ( Xa2 % 0.30/0.55 = ( one_one @ nat ) ) ) ) ) ) ) % 0.30/0.55 => ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) ) % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa2 ) ) ) % 0.30/0.55 => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT,S2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) @ Xa2 ) ) % 0.30/0.55 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.55 => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.55 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % VEBT_internal.naive_member.pelims(3) % 0.30/0.55 thf(fact_2428_VEBT__internal_Omembermima_Opelims_I3_J,axiom, % 0.30/0.55 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.30/0.55 ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 0.30/0.55 => ( ! [Uu2: $o,Uv2: $o] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) % 0.30/0.55 => ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) ) % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa2 ) ) ) % 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa2 ) ) % 0.30/0.55 => ( ( Xa2 = Mi2 ) % 0.30/0.55 | ( Xa2 = Ma2 ) ) ) ) % 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) % 0.30/0.55 => ( ( Xa2 = Mi2 ) % 0.30/0.55 | ( Xa2 = Ma2 ) % 0.30/0.55 | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.55 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) % 0.30/0.55 => ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa2 ) ) % 0.30/0.55 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.55 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % VEBT_internal.membermima.pelims(3) % 0.30/0.55 thf(fact_2429_VEBT__internal_Omembermima_Opelims_I1_J,axiom, % 0.30/0.55 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 0.30/0.55 ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 ) % 0.30/0.55 = Y ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 0.30/0.55 => ( ! [Uu2: $o,Uv2: $o] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 0.30/0.55 => ( ~ Y % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) ) % 0.30/0.55 => ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) ) % 0.30/0.55 => ( ~ Y % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa2 ) ) ) ) % 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) ) % 0.30/0.55 => ( ( Y % 0.30/0.55 = ( ( Xa2 = Mi2 ) % 0.30/0.55 | ( Xa2 = Ma2 ) ) ) % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa2 ) ) ) ) % 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) ) % 0.30/0.55 => ( ( Y % 0.30/0.55 = ( ( Xa2 = Mi2 ) % 0.30/0.55 | ( Xa2 = Ma2 ) % 0.30/0.55 | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.55 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) ) % 0.30/0.55 => ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) ) % 0.30/0.55 => ( ( Y % 0.30/0.55 = ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.55 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) % 0.30/0.55 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % VEBT_internal.membermima.pelims(1) % 0.30/0.55 thf(fact_2430_VEBT__internal_Omembermima_Opelims_I2_J,axiom, % 0.30/0.55 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.30/0.55 ( ( vEBT_VEBT_membermima @ X3 @ Xa2 ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa2 ) ) % 0.30/0.55 => ~ ( ( Xa2 = Mi2 ) % 0.30/0.55 | ( Xa2 = Ma2 ) ) ) ) % 0.30/0.55 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) % 0.30/0.55 => ~ ( ( Xa2 = Mi2 ) % 0.30/0.55 | ( Xa2 = Ma2 ) % 0.30/0.55 | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.55 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) % 0.30/0.55 => ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) ) % 0.30/0.55 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa2 ) ) % 0.30/0.55 => ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 0.30/0.55 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.30/0.55 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % VEBT_internal.membermima.pelims(2) % 0.30/0.55 thf(fact_2431_arcosh__1,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ln @ A ) % 0.30/0.55 => ( ( arcosh @ A @ ( one_one @ A ) ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ). % 0.30/0.55 % 0.30/0.55 % arcosh_1 % 0.30/0.55 thf(fact_2432_arsinh__0,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ln @ A ) % 0.30/0.55 => ( ( arsinh @ A @ ( zero_zero @ A ) ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ). % 0.30/0.55 % 0.30/0.55 % arsinh_0 % 0.30/0.55 thf(fact_2433_artanh__0,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ( real_V3459762299906320749_field @ A ) % 0.30/0.55 & ( ln @ A ) ) % 0.30/0.55 => ( ( artanh @ A @ ( zero_zero @ A ) ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ). % 0.30/0.55 % 0.30/0.55 % artanh_0 % 0.30/0.55 thf(fact_2434_vebt__buildup_Opelims,axiom, % 0.30/0.55 ! [X3: nat,Y: vEBT_VEBT] : % 0.30/0.55 ( ( ( vEBT_vebt_buildup @ X3 ) % 0.30/0.55 = Y ) % 0.30/0.55 => ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X3 ) % 0.30/0.55 => ( ( ( X3 % 0.30/0.55 = ( zero_zero @ nat ) ) % 0.30/0.55 => ( ( Y % 0.30/0.55 = ( vEBT_Leaf @ $false @ $false ) ) % 0.30/0.55 => ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) ) % 0.30/0.55 => ( ( ( X3 % 0.30/0.55 = ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.55 => ( ( Y % 0.30/0.55 = ( vEBT_Leaf @ $false @ $false ) ) % 0.30/0.55 => ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) ) % 0.30/0.55 => ~ ! [Va2: nat] : % 0.30/0.55 ( ( X3 % 0.30/0.55 = ( suc @ ( suc @ Va2 ) ) ) % 0.30/0.55 => ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) ) % 0.30/0.55 => ( Y % 0.30/0.55 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) % 0.30/0.55 & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) ) % 0.30/0.55 => ( Y % 0.30/0.55 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) % 0.30/0.55 => ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % vebt_buildup.pelims % 0.30/0.55 thf(fact_2435_divmod__algorithm__code_I6_J,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.55 => ! [M: num,N2: num] : % 0.30/0.55 ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ N2 ) ) % 0.30/0.55 = ( product_case_prod @ A @ A @ ( product_prod @ A @ A ) % 0.30/0.55 @ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) ) % 0.30/0.55 @ ( unique8689654367752047608divmod @ A @ M @ N2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % divmod_algorithm_code(6) % 0.30/0.55 thf(fact_2436_artanh__def,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ( real_V3459762299906320749_field @ A ) % 0.30/0.55 & ( ln @ A ) ) % 0.30/0.55 => ( ( artanh @ A ) % 0.30/0.55 = ( ^ [X: A] : ( divide_divide @ A @ ( ln_ln @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % artanh_def % 0.30/0.55 thf(fact_2437_ln__inj__iff,axiom, % 0.30/0.55 ! [X3: real,Y: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 0.30/0.55 => ( ( ( ln_ln @ real @ X3 ) % 0.30/0.55 = ( ln_ln @ real @ Y ) ) % 0.30/0.55 = ( X3 = Y ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_inj_iff % 0.30/0.55 thf(fact_2438_ln__less__cancel__iff,axiom, % 0.30/0.55 ! [X3: real,Y: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 0.30/0.55 => ( ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y ) ) % 0.30/0.55 = ( ord_less @ real @ X3 @ Y ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_less_cancel_iff % 0.30/0.55 thf(fact_2439_case__prod__conv,axiom, % 0.30/0.55 ! [B: $tType,A: $tType,C: $tType,F2: B > C > A,A2: B,B2: C] : % 0.30/0.55 ( ( product_case_prod @ B @ C @ A @ F2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) % 0.30/0.55 = ( F2 @ A2 @ B2 ) ) ). % 0.30/0.55 % 0.30/0.55 % case_prod_conv % 0.30/0.55 thf(fact_2440_ln__le__cancel__iff,axiom, % 0.30/0.55 ! [X3: real,Y: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 0.30/0.55 => ( ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y ) ) % 0.30/0.55 = ( ord_less_eq @ real @ X3 @ Y ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_le_cancel_iff % 0.30/0.55 thf(fact_2441_ln__less__zero__iff,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ ( zero_zero @ real ) ) % 0.30/0.55 = ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_less_zero_iff % 0.30/0.55 thf(fact_2442_ln__gt__zero__iff,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) ) % 0.30/0.55 = ( ord_less @ real @ ( one_one @ real ) @ X3 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_gt_zero_iff % 0.30/0.55 thf(fact_2443_ln__eq__zero__iff,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ( ( ln_ln @ real @ X3 ) % 0.30/0.55 = ( zero_zero @ real ) ) % 0.30/0.55 = ( X3 % 0.30/0.55 = ( one_one @ real ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_eq_zero_iff % 0.30/0.55 thf(fact_2444_ln__one,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ln @ A ) % 0.30/0.55 => ( ( ln_ln @ A @ ( one_one @ A ) ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_one % 0.30/0.55 thf(fact_2445_ln__le__zero__iff,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( zero_zero @ real ) ) % 0.30/0.55 = ( ord_less_eq @ real @ X3 @ ( one_one @ real ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_le_zero_iff % 0.30/0.55 thf(fact_2446_ln__ge__zero__iff,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) ) % 0.30/0.55 = ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_ge_zero_iff % 0.30/0.55 thf(fact_2447_divmod__algorithm__code_I5_J,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.55 => ! [M: num,N2: num] : % 0.30/0.55 ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ N2 ) ) % 0.30/0.55 = ( product_case_prod @ A @ A @ ( product_prod @ A @ A ) % 0.30/0.55 @ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) ) % 0.30/0.55 @ ( unique8689654367752047608divmod @ A @ M @ N2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % divmod_algorithm_code(5) % 0.30/0.55 thf(fact_2448_prod_Ocase__distrib,axiom, % 0.30/0.55 ! [C: $tType,D: $tType,B: $tType,A: $tType,H2: C > D,F2: A > B > C,Prod: product_prod @ A @ B] : % 0.30/0.55 ( ( H2 @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) ) % 0.30/0.55 = ( product_case_prod @ A @ B @ D % 0.30/0.55 @ ^ [X15: A,X24: B] : ( H2 @ ( F2 @ X15 @ X24 ) ) % 0.30/0.55 @ Prod ) ) ). % 0.30/0.55 % 0.30/0.55 % prod.case_distrib % 0.30/0.55 thf(fact_2449_old_Oprod_Ocase,axiom, % 0.30/0.55 ! [A: $tType,C: $tType,B: $tType,F2: A > B > C,X1: A,X22: B] : % 0.30/0.55 ( ( product_case_prod @ A @ B @ C @ F2 @ ( product_Pair @ A @ B @ X1 @ X22 ) ) % 0.30/0.55 = ( F2 @ X1 @ X22 ) ) ). % 0.30/0.55 % 0.30/0.55 % old.prod.case % 0.30/0.55 thf(fact_2450_split__cong,axiom, % 0.30/0.55 ! [C: $tType,B: $tType,A: $tType,Q4: product_prod @ A @ B,F2: A > B > C,G: A > B > C,P4: product_prod @ A @ B] : % 0.30/0.55 ( ! [X5: A,Y4: B] : % 0.30/0.55 ( ( ( product_Pair @ A @ B @ X5 @ Y4 ) % 0.30/0.55 = Q4 ) % 0.30/0.55 => ( ( F2 @ X5 @ Y4 ) % 0.30/0.55 = ( G @ X5 @ Y4 ) ) ) % 0.30/0.55 => ( ( P4 = Q4 ) % 0.30/0.55 => ( ( product_case_prod @ A @ B @ C @ F2 @ P4 ) % 0.30/0.55 = ( product_case_prod @ A @ B @ C @ G @ Q4 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % split_cong % 0.30/0.55 thf(fact_2451_cond__case__prod__eta,axiom, % 0.30/0.55 ! [C: $tType,B: $tType,A: $tType,F2: A > B > C,G: ( product_prod @ A @ B ) > C] : % 0.30/0.55 ( ! [X5: A,Y4: B] : % 0.30/0.55 ( ( F2 @ X5 @ Y4 ) % 0.30/0.55 = ( G @ ( product_Pair @ A @ B @ X5 @ Y4 ) ) ) % 0.30/0.55 => ( ( product_case_prod @ A @ B @ C @ F2 ) % 0.30/0.55 = G ) ) ). % 0.30/0.55 % 0.30/0.55 % cond_case_prod_eta % 0.30/0.55 thf(fact_2452_case__prod__eta,axiom, % 0.30/0.55 ! [C: $tType,B: $tType,A: $tType,F2: ( product_prod @ A @ B ) > C] : % 0.30/0.55 ( ( product_case_prod @ A @ B @ C % 0.30/0.55 @ ^ [X: A,Y2: B] : ( F2 @ ( product_Pair @ A @ B @ X @ Y2 ) ) ) % 0.30/0.55 = F2 ) ). % 0.30/0.55 % 0.30/0.55 % case_prod_eta % 0.30/0.55 thf(fact_2453_case__prodE2,axiom, % 0.30/0.55 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z2: product_prod @ B @ C] : % 0.30/0.55 ( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) ) % 0.30/0.55 => ~ ! [X5: B,Y4: C] : % 0.30/0.55 ( ( Z2 % 0.30/0.55 = ( product_Pair @ B @ C @ X5 @ Y4 ) ) % 0.30/0.55 => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % case_prodE2 % 0.30/0.55 thf(fact_2454_ln__less__self,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ X3 ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_less_self % 0.30/0.55 thf(fact_2455_ln__bound,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ X3 ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_bound % 0.30/0.55 thf(fact_2456_ln__gt__zero__imp__gt__one,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) ) % 0.30/0.55 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ord_less @ real @ ( one_one @ real ) @ X3 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_gt_zero_imp_gt_one % 0.30/0.55 thf(fact_2457_ln__less__zero,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 0.30/0.55 => ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_less_zero % 0.30/0.55 thf(fact_2458_ln__gt__zero,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 0.30/0.55 => ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_gt_zero % 0.30/0.55 thf(fact_2459_ln__div,axiom, % 0.30/0.55 ! [X3: real,Y: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 0.30/0.55 => ( ( ln_ln @ real @ ( divide_divide @ real @ X3 @ Y ) ) % 0.30/0.55 = ( minus_minus @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_div % 0.30/0.55 thf(fact_2460_ln__2__less__1,axiom, % 0.30/0.55 ord_less @ real @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ). % 0.30/0.55 % 0.30/0.55 % ln_2_less_1 % 0.30/0.55 thf(fact_2461_ln__ge__zero__imp__ge__one,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) ) % 0.30/0.55 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_ge_zero_imp_ge_one % 0.30/0.55 thf(fact_2462_ln__diff__le,axiom, % 0.30/0.55 ! [X3: real,Y: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 0.30/0.55 => ( ord_less_eq @ real @ ( minus_minus @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ X3 @ Y ) @ Y ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_diff_le % 0.30/0.55 thf(fact_2463_ln__mult,axiom, % 0.30/0.55 ! [X3: real,Y: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 0.30/0.55 => ( ( ln_ln @ real @ ( times_times @ real @ X3 @ Y ) ) % 0.30/0.55 = ( plus_plus @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_mult % 0.30/0.55 thf(fact_2464_ln__eq__minus__one,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ( ( ln_ln @ real @ X3 ) % 0.30/0.55 = ( minus_minus @ real @ X3 @ ( one_one @ real ) ) ) % 0.30/0.55 => ( X3 % 0.30/0.55 = ( one_one @ real ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_eq_minus_one % 0.30/0.55 thf(fact_2465_ln__le__minus__one,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( minus_minus @ real @ X3 @ ( one_one @ real ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_le_minus_one % 0.30/0.55 thf(fact_2466_ln__add__one__self__le__self2,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 0.30/0.55 => ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) ) @ X3 ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_add_one_self_le_self2 % 0.30/0.55 thf(fact_2467_ln__one__minus__pos__upper__bound,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 0.30/0.55 => ( ord_less_eq @ real @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X3 ) ) @ ( uminus_uminus @ real @ X3 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % ln_one_minus_pos_upper_bound % 0.30/0.55 thf(fact_2468_divmod__step__nat__def,axiom, % 0.30/0.55 ( ( unique1321980374590559556d_step @ nat ) % 0.30/0.55 = ( ^ [L2: num] : % 0.30/0.55 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 0.30/0.55 @ ^ [Q5: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % divmod_step_nat_def % 0.30/0.55 thf(fact_2469_divmod__step__int__def,axiom, % 0.30/0.55 ( ( unique1321980374590559556d_step @ int ) % 0.30/0.55 = ( ^ [L2: num] : % 0.30/0.55 ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 0.30/0.55 @ ^ [Q5: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L2 ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % divmod_step_int_def % 0.30/0.55 thf(fact_2470_divmod__step__def,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( unique1627219031080169319umeral @ A ) % 0.30/0.55 => ( ( unique1321980374590559556d_step @ A ) % 0.30/0.55 = ( ^ [L2: num] : % 0.30/0.55 ( product_case_prod @ A @ A @ ( product_prod @ A @ A ) % 0.30/0.55 @ ^ [Q5: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % divmod_step_def % 0.30/0.55 thf(fact_2471_divmod__nat__if,axiom, % 0.30/0.55 ( divmod_nat % 0.30/0.55 = ( ^ [M2: nat,N: nat] : % 0.30/0.55 ( if @ ( product_prod @ nat @ nat ) % 0.30/0.55 @ ( ( N % 0.30/0.55 = ( zero_zero @ nat ) ) % 0.30/0.55 | ( ord_less @ nat @ M2 @ N ) ) % 0.30/0.55 @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M2 ) % 0.30/0.55 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 0.30/0.55 @ ^ [Q5: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q5 ) ) % 0.30/0.55 @ ( divmod_nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % divmod_nat_if % 0.30/0.55 thf(fact_2472_tanh__ln__real,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.55 => ( ( tanh @ real @ ( ln_ln @ real @ X3 ) ) % 0.30/0.55 = ( divide_divide @ real @ ( minus_minus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % tanh_ln_real % 0.30/0.55 thf(fact_2473_Sum__Icc__nat,axiom, % 0.30/0.55 ! [M: nat,N2: nat] : % 0.30/0.55 ( ( groups7311177749621191930dd_sum @ nat @ nat % 0.30/0.55 @ ^ [X: nat] : X % 0.30/0.55 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.30/0.55 = ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % Sum_Icc_nat % 0.30/0.55 thf(fact_2474_arith__series__nat,axiom, % 0.30/0.55 ! [A2: nat,D2: nat,N2: nat] : % 0.30/0.55 ( ( groups7311177749621191930dd_sum @ nat @ nat % 0.30/0.55 @ ^ [I3: nat] : ( plus_plus @ nat @ A2 @ ( times_times @ nat @ I3 @ D2 ) ) % 0.30/0.55 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.30/0.55 = ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ nat @ N2 @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % arith_series_nat % 0.30/0.55 thf(fact_2475_case__prod__Pair__iden,axiom, % 0.30/0.55 ! [B: $tType,A: $tType,P4: product_prod @ A @ B] : % 0.30/0.55 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P4 ) % 0.30/0.55 = P4 ) ). % 0.30/0.55 % 0.30/0.55 % case_prod_Pair_iden % 0.30/0.55 thf(fact_2476_round__unique_H,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.55 => ! [X3: A,N2: int] : % 0.30/0.55 ( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ ( ring_1_of_int @ A @ N2 ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) % 0.30/0.55 => ( ( archimedean_round @ A @ X3 ) % 0.30/0.55 = N2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % round_unique' % 0.30/0.55 thf(fact_2477_abs__abs,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( idom_abs_sgn @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) ) % 0.30/0.55 = ( abs_abs @ A @ A2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_abs % 0.30/0.55 thf(fact_2478_abs__idempotent,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) ) % 0.30/0.55 = ( abs_abs @ A @ A2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_idempotent % 0.30/0.55 thf(fact_2479_case__prodI,axiom, % 0.30/0.55 ! [A: $tType,B: $tType,F2: A > B > $o,A2: A,B2: B] : % 0.30/0.55 ( ( F2 @ A2 @ B2 ) % 0.30/0.55 => ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % case_prodI % 0.30/0.55 thf(fact_2480_case__prodI2,axiom, % 0.30/0.55 ! [B: $tType,A: $tType,P4: product_prod @ A @ B,C2: A > B > $o] : % 0.30/0.55 ( ! [A5: A,B5: B] : % 0.30/0.55 ( ( P4 % 0.30/0.55 = ( product_Pair @ A @ B @ A5 @ B5 ) ) % 0.30/0.55 => ( C2 @ A5 @ B5 ) ) % 0.30/0.55 => ( product_case_prod @ A @ B @ $o @ C2 @ P4 ) ) ). % 0.30/0.55 % 0.30/0.55 % case_prodI2 % 0.30/0.55 thf(fact_2481_mem__case__prodI,axiom, % 0.30/0.55 ! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A2: B,B2: C] : % 0.30/0.55 ( ( member @ A @ Z2 @ ( C2 @ A2 @ B2 ) ) % 0.30/0.55 => ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % mem_case_prodI % 0.30/0.55 thf(fact_2482_mem__case__prodI2,axiom, % 0.30/0.55 ! [C: $tType,B: $tType,A: $tType,P4: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] : % 0.30/0.55 ( ! [A5: A,B5: B] : % 0.30/0.55 ( ( P4 % 0.30/0.55 = ( product_Pair @ A @ B @ A5 @ B5 ) ) % 0.30/0.55 => ( member @ C @ Z2 @ ( C2 @ A5 @ B5 ) ) ) % 0.30/0.55 => ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P4 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % mem_case_prodI2 % 0.30/0.55 thf(fact_2483_case__prodI2_H,axiom, % 0.30/0.55 ! [A: $tType,B: $tType,C: $tType,P4: product_prod @ A @ B,C2: A > B > C > $o,X3: C] : % 0.30/0.55 ( ! [A5: A,B5: B] : % 0.30/0.55 ( ( ( product_Pair @ A @ B @ A5 @ B5 ) % 0.30/0.55 = P4 ) % 0.30/0.55 => ( C2 @ A5 @ B5 @ X3 ) ) % 0.30/0.55 => ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ X3 ) ) ). % 0.30/0.55 % 0.30/0.55 % case_prodI2' % 0.30/0.55 thf(fact_2484_abs__0__eq,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ( ( ( zero_zero @ A ) % 0.30/0.55 = ( abs_abs @ A @ A2 ) ) % 0.30/0.55 = ( A2 % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_0_eq % 0.30/0.55 thf(fact_2485_abs__eq__0,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ( ( ( abs_abs @ A @ A2 ) % 0.30/0.55 = ( zero_zero @ A ) ) % 0.30/0.55 = ( A2 % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_eq_0 % 0.30/0.55 thf(fact_2486_abs__zero,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ( ( abs_abs @ A @ ( zero_zero @ A ) ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_zero % 0.30/0.55 thf(fact_2487_abs__0,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( idom_abs_sgn @ A ) % 0.30/0.55 => ( ( abs_abs @ A @ ( zero_zero @ A ) ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_0 % 0.30/0.55 thf(fact_2488_abs__mult__self__eq,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ A2 ) ) % 0.30/0.55 = ( times_times @ A @ A2 @ A2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_mult_self_eq % 0.30/0.55 thf(fact_2489_abs__1,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( idom_abs_sgn @ A ) % 0.30/0.55 => ( ( abs_abs @ A @ ( one_one @ A ) ) % 0.30/0.55 = ( one_one @ A ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_1 % 0.30/0.55 thf(fact_2490_abs__add__abs,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : % 0.30/0.55 ( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) % 0.30/0.55 = ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_add_abs % 0.30/0.55 thf(fact_2491_abs__divide,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( field_abs_sgn @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : % 0.30/0.55 ( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 0.30/0.55 = ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_divide % 0.30/0.55 thf(fact_2492_abs__minus,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( idom_abs_sgn @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) ) % 0.30/0.55 = ( abs_abs @ A @ A2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_minus % 0.30/0.55 thf(fact_2493_abs__minus__cancel,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) ) % 0.30/0.55 = ( abs_abs @ A @ A2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_minus_cancel % 0.30/0.55 thf(fact_2494_dvd__abs__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [M: A,K: A] : % 0.30/0.55 ( ( dvd_dvd @ A @ M @ ( abs_abs @ A @ K ) ) % 0.30/0.55 = ( dvd_dvd @ A @ M @ K ) ) ) ). % 0.30/0.55 % 0.30/0.55 % dvd_abs_iff % 0.30/0.55 thf(fact_2495_abs__dvd__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [M: A,K: A] : % 0.30/0.55 ( ( dvd_dvd @ A @ ( abs_abs @ A @ M ) @ K ) % 0.30/0.55 = ( dvd_dvd @ A @ M @ K ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_dvd_iff % 0.30/0.55 thf(fact_2496_of__int__abs,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [X3: int] : % 0.30/0.55 ( ( ring_1_of_int @ A @ ( abs_abs @ int @ X3 ) ) % 0.30/0.55 = ( abs_abs @ A @ ( ring_1_of_int @ A @ X3 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_int_abs % 0.30/0.55 thf(fact_2497_abs__bool__eq,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [P: $o] : % 0.30/0.55 ( ( abs_abs @ A @ ( zero_neq_one_of_bool @ A @ P ) ) % 0.30/0.55 = ( zero_neq_one_of_bool @ A @ P ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_bool_eq % 0.30/0.55 thf(fact_2498_zdvd1__eq,axiom, % 0.30/0.55 ! [X3: int] : % 0.30/0.55 ( ( dvd_dvd @ int @ X3 @ ( one_one @ int ) ) % 0.30/0.55 = ( ( abs_abs @ int @ X3 ) % 0.30/0.55 = ( one_one @ int ) ) ) ). % 0.30/0.55 % 0.30/0.55 % zdvd1_eq % 0.30/0.55 thf(fact_2499_tanh__0,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ( real_Vector_banach @ A ) % 0.30/0.55 & ( real_V3459762299906320749_field @ A ) ) % 0.30/0.55 => ( ( tanh @ A @ ( zero_zero @ A ) ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ). % 0.30/0.55 % 0.30/0.55 % tanh_0 % 0.30/0.55 thf(fact_2500_tanh__real__less__iff,axiom, % 0.30/0.55 ! [X3: real,Y: real] : % 0.30/0.55 ( ( ord_less @ real @ ( tanh @ real @ X3 ) @ ( tanh @ real @ Y ) ) % 0.30/0.55 = ( ord_less @ real @ X3 @ Y ) ) ). % 0.30/0.55 % 0.30/0.55 % tanh_real_less_iff % 0.30/0.55 thf(fact_2501_sum_Oneutral__const,axiom, % 0.30/0.55 ! [B: $tType,A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [A3: set @ B] : % 0.30/0.55 ( ( groups7311177749621191930dd_sum @ B @ A % 0.30/0.55 @ ^ [Uu3: B] : ( zero_zero @ A ) % 0.30/0.55 @ A3 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.neutral_const % 0.30/0.55 thf(fact_2502_of__int__sum,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( ring_1 @ A ) % 0.30/0.55 => ! [F2: B > int,A3: set @ B] : % 0.30/0.55 ( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F2 @ A3 ) ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ B @ A % 0.30/0.55 @ ^ [X: B] : ( ring_1_of_int @ A @ ( F2 @ X ) ) % 0.30/0.55 @ A3 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_int_sum % 0.30/0.55 thf(fact_2503_abs__of__nonneg,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.55 => ( ( abs_abs @ A @ A2 ) % 0.30/0.55 = A2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_of_nonneg % 0.30/0.55 thf(fact_2504_abs__le__self__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 ) % 0.30/0.55 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_le_self_iff % 0.30/0.55 thf(fact_2505_abs__le__zero__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) ) % 0.30/0.55 = ( A2 % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_le_zero_iff % 0.30/0.55 thf(fact_2506_zero__less__abs__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) ) % 0.30/0.55 = ( A2 % 0.30/0.55 != ( zero_zero @ A ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % zero_less_abs_iff % 0.30/0.55 thf(fact_2507_sum_Oempty,axiom, % 0.30/0.55 ! [B: $tType,A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [G: B > A] : % 0.30/0.55 ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.empty % 0.30/0.55 thf(fact_2508_sum_Oinfinite,axiom, % 0.30/0.55 ! [B: $tType,A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [A3: set @ B,G: B > A] : % 0.30/0.55 ( ~ ( finite_finite2 @ B @ A3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.infinite % 0.30/0.55 thf(fact_2509_sum__eq__0__iff,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( canoni5634975068530333245id_add @ A ) % 0.30/0.55 => ! [F5: set @ B,F2: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ F5 ) % 0.30/0.55 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ F5 ) % 0.30/0.55 = ( zero_zero @ A ) ) % 0.30/0.55 = ( ! [X: B] : % 0.30/0.55 ( ( member @ B @ X @ F5 ) % 0.30/0.55 => ( ( F2 @ X ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_eq_0_iff % 0.30/0.55 thf(fact_2510_artanh__minus__real,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 0.30/0.55 => ( ( artanh @ real @ ( uminus_uminus @ real @ X3 ) ) % 0.30/0.55 = ( uminus_uminus @ real @ ( artanh @ real @ X3 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % artanh_minus_real % 0.30/0.55 thf(fact_2511_zabs__less__one__iff,axiom, % 0.30/0.55 ! [Z2: int] : % 0.30/0.55 ( ( ord_less @ int @ ( abs_abs @ int @ Z2 ) @ ( one_one @ int ) ) % 0.30/0.55 = ( Z2 % 0.30/0.55 = ( zero_zero @ int ) ) ) ). % 0.30/0.55 % 0.30/0.55 % zabs_less_one_iff % 0.30/0.55 thf(fact_2512_tanh__real__pos__iff,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X3 ) ) % 0.30/0.55 = ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ). % 0.30/0.55 % 0.30/0.55 % tanh_real_pos_iff % 0.30/0.55 thf(fact_2513_tanh__real__neg__iff,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( tanh @ real @ X3 ) @ ( zero_zero @ real ) ) % 0.30/0.55 = ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ). % 0.30/0.55 % 0.30/0.55 % tanh_real_neg_iff % 0.30/0.55 thf(fact_2514_sum_Odelta,axiom, % 0.30/0.55 ! [B: $tType,A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [S3: set @ B,A2: B,B2: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ S3 ) % 0.30/0.55 => ( ( ( member @ B @ A2 @ S3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A % 0.30/0.55 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) ) % 0.30/0.55 @ S3 ) % 0.30/0.55 = ( B2 @ A2 ) ) ) % 0.30/0.55 & ( ~ ( member @ B @ A2 @ S3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A % 0.30/0.55 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) ) % 0.30/0.55 @ S3 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.delta % 0.30/0.55 thf(fact_2515_sum_Odelta_H,axiom, % 0.30/0.55 ! [B: $tType,A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [S3: set @ B,A2: B,B2: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ S3 ) % 0.30/0.55 => ( ( ( member @ B @ A2 @ S3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A % 0.30/0.55 @ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) ) % 0.30/0.55 @ S3 ) % 0.30/0.55 = ( B2 @ A2 ) ) ) % 0.30/0.55 & ( ~ ( member @ B @ A2 @ S3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A % 0.30/0.55 @ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) ) % 0.30/0.55 @ S3 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.delta' % 0.30/0.55 thf(fact_2516_sum__abs,axiom, % 0.30/0.55 ! [B: $tType,A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ B ) % 0.30/0.55 => ! [F2: A > B,A3: set @ A] : % 0.30/0.55 ( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) ) % 0.30/0.55 @ ( groups7311177749621191930dd_sum @ A @ B % 0.30/0.55 @ ^ [I3: A] : ( abs_abs @ B @ ( F2 @ I3 ) ) % 0.30/0.55 @ A3 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_abs % 0.30/0.55 thf(fact_2517_divide__le__0__abs__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_field @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) ) % 0.30/0.55 = ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.55 | ( B2 % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % divide_le_0_abs_iff % 0.30/0.55 thf(fact_2518_zero__le__divide__abs__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_field @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) ) % 0.30/0.55 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.55 | ( B2 % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % zero_le_divide_abs_iff % 0.30/0.55 thf(fact_2519_abs__of__nonpos,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.55 => ( ( abs_abs @ A @ A2 ) % 0.30/0.55 = ( uminus_uminus @ A @ A2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_of_nonpos % 0.30/0.55 thf(fact_2520_sum_Oinsert,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [A3: set @ B,X3: B,G: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ A3 ) % 0.30/0.55 => ( ~ ( member @ B @ X3 @ A3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 0.30/0.55 = ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.insert % 0.30/0.55 thf(fact_2521_sum__abs__ge__zero,axiom, % 0.30/0.55 ! [B: $tType,A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ B ) % 0.30/0.55 => ! [F2: A > B,A3: set @ A] : % 0.30/0.55 ( ord_less_eq @ B @ ( zero_zero @ B ) % 0.30/0.55 @ ( groups7311177749621191930dd_sum @ A @ B % 0.30/0.55 @ ^ [I3: A] : ( abs_abs @ B @ ( F2 @ I3 ) ) % 0.30/0.55 @ A3 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_abs_ge_zero % 0.30/0.55 thf(fact_2522_zero__less__power__abs__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [A2: A,N2: nat] : % 0.30/0.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N2 ) ) % 0.30/0.55 = ( ( A2 % 0.30/0.55 != ( zero_zero @ A ) ) % 0.30/0.55 | ( N2 % 0.30/0.55 = ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % zero_less_power_abs_iff % 0.30/0.55 thf(fact_2523_sum_Ocl__ivl__Suc,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [N2: nat,M: nat,G: nat > A] : % 0.30/0.55 ( ( ( ord_less @ nat @ ( suc @ N2 ) @ M ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) ) % 0.30/0.55 = ( zero_zero @ A ) ) ) % 0.30/0.55 & ( ~ ( ord_less @ nat @ ( suc @ N2 ) @ M ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) ) % 0.30/0.55 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.cl_ivl_Suc % 0.30/0.55 thf(fact_2524_sum__zero__power,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( division_ring @ A ) % 0.30/0.55 => ! [A3: set @ nat,C2: nat > A] : % 0.30/0.55 ( ( ( ( finite_finite2 @ nat @ A3 ) % 0.30/0.55 & ( member @ nat @ ( zero_zero @ nat ) @ A3 ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.55 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) % 0.30/0.55 @ A3 ) % 0.30/0.55 = ( C2 @ ( zero_zero @ nat ) ) ) ) % 0.30/0.55 & ( ~ ( ( finite_finite2 @ nat @ A3 ) % 0.30/0.55 & ( member @ nat @ ( zero_zero @ nat ) @ A3 ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.55 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) % 0.30/0.55 @ A3 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_zero_power % 0.30/0.55 thf(fact_2525_sum__zero__power_H,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( field @ A ) % 0.30/0.55 => ! [A3: set @ nat,C2: nat > A,D2: nat > A] : % 0.30/0.55 ( ( ( ( finite_finite2 @ nat @ A3 ) % 0.30/0.55 & ( member @ nat @ ( zero_zero @ nat ) @ A3 ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.55 @ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D2 @ I3 ) ) % 0.30/0.55 @ A3 ) % 0.30/0.55 = ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) ) % 0.30/0.55 & ( ~ ( ( finite_finite2 @ nat @ A3 ) % 0.30/0.55 & ( member @ nat @ ( zero_zero @ nat ) @ A3 ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.55 @ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D2 @ I3 ) ) % 0.30/0.55 @ A3 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_zero_power' % 0.30/0.55 thf(fact_2526_sum_Onot__neutral__contains__not__neutral,axiom, % 0.30/0.55 ! [B: $tType,A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [G: B > A,A3: set @ B] : % 0.30/0.55 ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 0.30/0.55 != ( zero_zero @ A ) ) % 0.30/0.55 => ~ ! [A5: B] : % 0.30/0.55 ( ( member @ B @ A5 @ A3 ) % 0.30/0.55 => ( ( G @ A5 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.not_neutral_contains_not_neutral % 0.30/0.55 thf(fact_2527_sum_Oneutral,axiom, % 0.30/0.55 ! [B: $tType,A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [A3: set @ B,G: B > A] : % 0.30/0.55 ( ! [X5: B] : % 0.30/0.55 ( ( member @ B @ X5 @ A3 ) % 0.30/0.55 => ( ( G @ X5 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.neutral % 0.30/0.55 thf(fact_2528_abs__le__D1,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) % 0.30/0.55 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_le_D1 % 0.30/0.55 thf(fact_2529_abs__ge__self,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( abs_abs @ A @ A2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_ge_self % 0.30/0.55 thf(fact_2530_abs__eq__0__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( idom_abs_sgn @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ( ( ( abs_abs @ A @ A2 ) % 0.30/0.55 = ( zero_zero @ A ) ) % 0.30/0.55 = ( A2 % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_eq_0_iff % 0.30/0.55 thf(fact_2531_abs__mult,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( idom_abs_sgn @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : % 0.30/0.55 ( ( abs_abs @ A @ ( times_times @ A @ A2 @ B2 ) ) % 0.30/0.55 = ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_mult % 0.30/0.55 thf(fact_2532_abs__one,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ( ( abs_abs @ A @ ( one_one @ A ) ) % 0.30/0.55 = ( one_one @ A ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_one % 0.30/0.55 thf(fact_2533_abs__minus__commute,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : % 0.30/0.55 ( ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) % 0.30/0.55 = ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_minus_commute % 0.30/0.55 thf(fact_2534_abs__eq__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_ring @ A ) % 0.30/0.55 => ! [X3: A,Y: A] : % 0.30/0.55 ( ( ( abs_abs @ A @ X3 ) % 0.30/0.55 = ( abs_abs @ A @ Y ) ) % 0.30/0.55 = ( ( X3 = Y ) % 0.30/0.55 | ( X3 % 0.30/0.55 = ( uminus_uminus @ A @ Y ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_eq_iff % 0.30/0.55 thf(fact_2535_dvd__if__abs__eq,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [L: A,K: A] : % 0.30/0.55 ( ( ( abs_abs @ A @ L ) % 0.30/0.55 = ( abs_abs @ A @ K ) ) % 0.30/0.55 => ( dvd_dvd @ A @ L @ K ) ) ) ). % 0.30/0.55 % 0.30/0.55 % dvd_if_abs_eq % 0.30/0.55 thf(fact_2536_mem__case__prodE,axiom, % 0.30/0.55 ! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P4: product_prod @ B @ C] : % 0.30/0.55 ( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P4 ) ) % 0.30/0.55 => ~ ! [X5: B,Y4: C] : % 0.30/0.55 ( ( P4 % 0.30/0.55 = ( product_Pair @ B @ C @ X5 @ Y4 ) ) % 0.30/0.55 => ~ ( member @ A @ Z2 @ ( C2 @ X5 @ Y4 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % mem_case_prodE % 0.30/0.55 thf(fact_2537_zdvd__antisym__abs,axiom, % 0.30/0.55 ! [A2: int,B2: int] : % 0.30/0.55 ( ( dvd_dvd @ int @ A2 @ B2 ) % 0.30/0.55 => ( ( dvd_dvd @ int @ B2 @ A2 ) % 0.30/0.55 => ( ( abs_abs @ int @ A2 ) % 0.30/0.55 = ( abs_abs @ int @ B2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % zdvd_antisym_abs % 0.30/0.55 thf(fact_2538_case__prodD,axiom, % 0.30/0.55 ! [A: $tType,B: $tType,F2: A > B > $o,A2: A,B2: B] : % 0.30/0.55 ( ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) % 0.30/0.55 => ( F2 @ A2 @ B2 ) ) ). % 0.30/0.55 % 0.30/0.55 % case_prodD % 0.30/0.55 thf(fact_2539_case__prodE,axiom, % 0.30/0.55 ! [A: $tType,B: $tType,C2: A > B > $o,P4: product_prod @ A @ B] : % 0.30/0.55 ( ( product_case_prod @ A @ B @ $o @ C2 @ P4 ) % 0.30/0.55 => ~ ! [X5: A,Y4: B] : % 0.30/0.55 ( ( P4 % 0.30/0.55 = ( product_Pair @ A @ B @ X5 @ Y4 ) ) % 0.30/0.55 => ~ ( C2 @ X5 @ Y4 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % case_prodE % 0.30/0.55 thf(fact_2540_sum__mono,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.55 => ! [K5: set @ B,F2: B > A,G: B > A] : % 0.30/0.55 ( ! [I4: B] : % 0.30/0.55 ( ( member @ B @ I4 @ K5 ) % 0.30/0.55 => ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) ) % 0.30/0.55 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ K5 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_mono % 0.30/0.55 thf(fact_2541_sum_Odistrib,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [G: B > A,H2: B > A,A3: set @ B] : % 0.30/0.55 ( ( groups7311177749621191930dd_sum @ B @ A % 0.30/0.55 @ ^ [X: B] : ( plus_plus @ A @ ( G @ X ) @ ( H2 @ X ) ) % 0.30/0.55 @ A3 ) % 0.30/0.55 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ A3 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.distrib % 0.30/0.55 thf(fact_2542_sum_Oswap__restrict,axiom, % 0.30/0.55 ! [A: $tType,B: $tType,C: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [A3: set @ B,B3: set @ C,G: B > C > A,R: B > C > $o] : % 0.30/0.55 ( ( finite_finite2 @ B @ A3 ) % 0.30/0.55 => ( ( finite_finite2 @ C @ B3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A % 0.30/0.55 @ ^ [X: B] : % 0.30/0.55 ( groups7311177749621191930dd_sum @ C @ A @ ( G @ X ) % 0.30/0.55 @ ( collect @ C % 0.30/0.55 @ ^ [Y2: C] : % 0.30/0.55 ( ( member @ C @ Y2 @ B3 ) % 0.30/0.55 & ( R @ X @ Y2 ) ) ) ) % 0.30/0.55 @ A3 ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ C @ A % 0.30/0.55 @ ^ [Y2: C] : % 0.30/0.55 ( groups7311177749621191930dd_sum @ B @ A % 0.30/0.55 @ ^ [X: B] : ( G @ X @ Y2 ) % 0.30/0.55 @ ( collect @ B % 0.30/0.55 @ ^ [X: B] : % 0.30/0.55 ( ( member @ B @ X @ A3 ) % 0.30/0.55 & ( R @ X @ Y2 ) ) ) ) % 0.30/0.55 @ B3 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.swap_restrict % 0.30/0.55 thf(fact_2543_case__prodD_H,axiom, % 0.30/0.55 ! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A2: A,B2: B,C2: C] : % 0.30/0.55 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 ) % 0.30/0.55 => ( R @ A2 @ B2 @ C2 ) ) ). % 0.30/0.55 % 0.30/0.55 % case_prodD' % 0.30/0.55 thf(fact_2544_case__prodE_H,axiom, % 0.30/0.55 ! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P4: product_prod @ A @ B,Z2: C] : % 0.30/0.55 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ Z2 ) % 0.30/0.55 => ~ ! [X5: A,Y4: B] : % 0.30/0.55 ( ( P4 % 0.30/0.55 = ( product_Pair @ A @ B @ X5 @ Y4 ) ) % 0.30/0.55 => ~ ( C2 @ X5 @ Y4 @ Z2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % case_prodE' % 0.30/0.55 thf(fact_2545_sum__nonneg,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.55 => ! [A3: set @ B,F2: B > A] : % 0.30/0.55 ( ! [X5: B] : % 0.30/0.55 ( ( member @ B @ X5 @ A3 ) % 0.30/0.55 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) ) % 0.30/0.55 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_nonneg % 0.30/0.55 thf(fact_2546_sum__nonpos,axiom, % 0.30/0.55 ! [B: $tType,A: $tType] : % 0.30/0.55 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.55 => ! [A3: set @ B,F2: B > A] : % 0.30/0.55 ( ! [X5: B] : % 0.30/0.55 ( ( member @ B @ X5 @ A3 ) % 0.30/0.55 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( zero_zero @ A ) ) ) % 0.30/0.55 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_nonpos % 0.30/0.55 thf(fact_2547_sum__mono__inv,axiom, % 0.30/0.55 ! [A: $tType,I7: $tType] : % 0.30/0.55 ( ( ordere8940638589300402666id_add @ A ) % 0.30/0.55 => ! [F2: I7 > A,I6: set @ I7,G: I7 > A,I: I7] : % 0.30/0.55 ( ( ( groups7311177749621191930dd_sum @ I7 @ A @ F2 @ I6 ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ I7 @ A @ G @ I6 ) ) % 0.30/0.55 => ( ! [I4: I7] : % 0.30/0.55 ( ( member @ I7 @ I4 @ I6 ) % 0.30/0.55 => ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) ) % 0.30/0.55 => ( ( member @ I7 @ I @ I6 ) % 0.30/0.55 => ( ( finite_finite2 @ I7 @ I6 ) % 0.30/0.55 => ( ( F2 @ I ) % 0.30/0.55 = ( G @ I ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_mono_inv % 0.30/0.55 thf(fact_2548_abs__ge__zero,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_ge_zero % 0.30/0.55 thf(fact_2549_abs__not__less__zero,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ~ ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_not_less_zero % 0.30/0.55 thf(fact_2550_abs__of__pos,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.55 => ( ( abs_abs @ A @ A2 ) % 0.30/0.55 = A2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_of_pos % 0.30/0.55 thf(fact_2551_abs__triangle__ineq,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_triangle_ineq % 0.30/0.55 thf(fact_2552_abs__mult__less,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [A2: A,C2: A,B2: A,D2: A] : % 0.30/0.55 ( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ C2 ) % 0.30/0.55 => ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D2 ) % 0.30/0.55 => ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_mult_less % 0.30/0.55 thf(fact_2553_abs__triangle__ineq2,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_triangle_ineq2 % 0.30/0.55 thf(fact_2554_abs__triangle__ineq3,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_triangle_ineq3 % 0.30/0.55 thf(fact_2555_abs__triangle__ineq2__sym,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_triangle_ineq2_sym % 0.30/0.55 thf(fact_2556_nonzero__abs__divide,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_field @ A ) % 0.30/0.55 => ! [B2: A,A2: A] : % 0.30/0.55 ( ( B2 % 0.30/0.55 != ( zero_zero @ A ) ) % 0.30/0.55 => ( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 0.30/0.55 = ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % nonzero_abs_divide % 0.30/0.55 thf(fact_2557_abs__ge__minus__self,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( abs_abs @ A @ A2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_ge_minus_self % 0.30/0.55 thf(fact_2558_abs__le__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) % 0.30/0.55 = ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.55 & ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_le_iff % 0.30/0.55 thf(fact_2559_abs__le__D2,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) % 0.30/0.55 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_le_D2 % 0.30/0.55 thf(fact_2560_abs__leI,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.30/0.55 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 0.30/0.55 => ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_leI % 0.30/0.55 thf(fact_2561_abs__less__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : % 0.30/0.55 ( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ B2 ) % 0.30/0.55 = ( ( ord_less @ A @ A2 @ B2 ) % 0.30/0.55 & ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_less_iff % 0.30/0.55 thf(fact_2562_sum__cong__Suc,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [A3: set @ nat,F2: nat > A,G: nat > A] : % 0.30/0.55 ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 ) % 0.30/0.55 => ( ! [X5: nat] : % 0.30/0.55 ( ( member @ nat @ ( suc @ X5 ) @ A3 ) % 0.30/0.55 => ( ( F2 @ ( suc @ X5 ) ) % 0.30/0.55 = ( G @ ( suc @ X5 ) ) ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A3 ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ A3 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_cong_Suc % 0.30/0.55 thf(fact_2563_Collect__case__prod__mono,axiom, % 0.30/0.55 ! [B: $tType,A: $tType,A3: A > B > $o,B3: A > B > $o] : % 0.30/0.55 ( ( ord_less_eq @ ( A > B > $o ) @ A3 @ B3 ) % 0.30/0.55 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B3 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % Collect_case_prod_mono % 0.30/0.55 thf(fact_2564_abs__zmult__eq__1,axiom, % 0.30/0.55 ! [M: int,N2: int] : % 0.30/0.55 ( ( ( abs_abs @ int @ ( times_times @ int @ M @ N2 ) ) % 0.30/0.55 = ( one_one @ int ) ) % 0.30/0.55 => ( ( abs_abs @ int @ M ) % 0.30/0.55 = ( one_one @ int ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_zmult_eq_1 % 0.30/0.55 thf(fact_2565_sum_Ointer__filter,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [A3: set @ B,G: B > A,P: B > $o] : % 0.30/0.55 ( ( finite_finite2 @ B @ A3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G % 0.30/0.55 @ ( collect @ B % 0.30/0.55 @ ^ [X: B] : % 0.30/0.55 ( ( member @ B @ X @ A3 ) % 0.30/0.55 & ( P @ X ) ) ) ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ B @ A % 0.30/0.55 @ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( G @ X ) @ ( zero_zero @ A ) ) % 0.30/0.55 @ A3 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.inter_filter % 0.30/0.55 thf(fact_2566_sum__subtractf__nat,axiom, % 0.30/0.55 ! [A: $tType,A3: set @ A,G: A > nat,F2: A > nat] : % 0.30/0.55 ( ! [X5: A] : % 0.30/0.55 ( ( member @ A @ X5 @ A3 ) % 0.30/0.55 => ( ord_less_eq @ nat @ ( G @ X5 ) @ ( F2 @ X5 ) ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ A @ nat % 0.30/0.55 @ ^ [X: A] : ( minus_minus @ nat @ ( F2 @ X ) @ ( G @ X ) ) % 0.30/0.55 @ A3 ) % 0.30/0.55 = ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A3 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_subtractf_nat % 0.30/0.55 thf(fact_2567_convex__sum__bound__le,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( linordered_idom @ B ) % 0.30/0.55 => ! [I6: set @ A,X3: A > B,A2: A > B,B2: B,Delta: B] : % 0.30/0.55 ( ! [I4: A] : % 0.30/0.55 ( ( member @ A @ I4 @ I6 ) % 0.30/0.55 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X3 @ I4 ) ) ) % 0.30/0.55 => ( ( ( groups7311177749621191930dd_sum @ A @ B @ X3 @ I6 ) % 0.30/0.55 = ( one_one @ B ) ) % 0.30/0.55 => ( ! [I4: A] : % 0.30/0.55 ( ( member @ A @ I4 @ I6 ) % 0.30/0.55 => ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A2 @ I4 ) @ B2 ) ) @ Delta ) ) % 0.30/0.55 => ( ord_less_eq @ B % 0.30/0.55 @ ( abs_abs @ B % 0.30/0.55 @ ( minus_minus @ B % 0.30/0.55 @ ( groups7311177749621191930dd_sum @ A @ B % 0.30/0.55 @ ^ [I3: A] : ( times_times @ B @ ( A2 @ I3 ) @ ( X3 @ I3 ) ) % 0.30/0.55 @ I6 ) % 0.30/0.55 @ B2 ) ) % 0.30/0.55 @ Delta ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % convex_sum_bound_le % 0.30/0.55 thf(fact_2568_infinite__int__iff__unbounded__le,axiom, % 0.30/0.55 ! [S3: set @ int] : % 0.30/0.55 ( ( ~ ( finite_finite2 @ int @ S3 ) ) % 0.30/0.55 = ( ! [M2: int] : % 0.30/0.55 ? [N: int] : % 0.30/0.55 ( ( ord_less_eq @ int @ M2 @ ( abs_abs @ int @ N ) ) % 0.30/0.55 & ( member @ int @ N @ S3 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % infinite_int_iff_unbounded_le % 0.30/0.55 thf(fact_2569_infinite__int__iff__unbounded,axiom, % 0.30/0.55 ! [S3: set @ int] : % 0.30/0.55 ( ( ~ ( finite_finite2 @ int @ S3 ) ) % 0.30/0.55 = ( ! [M2: int] : % 0.30/0.55 ? [N: int] : % 0.30/0.55 ( ( ord_less @ int @ M2 @ ( abs_abs @ int @ N ) ) % 0.30/0.55 & ( member @ int @ N @ S3 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % infinite_int_iff_unbounded % 0.30/0.55 thf(fact_2570_sum_Oshift__bounds__cl__Suc__ivl,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [G: nat > A,M: nat,N2: nat] : % 0.30/0.55 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.55 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 0.30/0.55 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.shift_bounds_cl_Suc_ivl % 0.30/0.55 thf(fact_2571_sum_Oshift__bounds__cl__nat__ivl,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 0.30/0.55 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.55 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) ) % 0.30/0.55 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.shift_bounds_cl_nat_ivl % 0.30/0.55 thf(fact_2572_tanh__real__lt__1,axiom, % 0.30/0.55 ! [X3: real] : ( ord_less @ real @ ( tanh @ real @ X3 ) @ ( one_one @ real ) ) ). % 0.30/0.55 % 0.30/0.55 % tanh_real_lt_1 % 0.30/0.55 thf(fact_2573_sum__le__included,axiom, % 0.30/0.55 ! [B: $tType,A: $tType,C: $tType] : % 0.30/0.55 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.55 => ! [S: set @ B,T2: set @ C,G: C > A,I: C > B,F2: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ S ) % 0.30/0.55 => ( ( finite_finite2 @ C @ T2 ) % 0.30/0.55 => ( ! [X5: C] : % 0.30/0.55 ( ( member @ C @ X5 @ T2 ) % 0.30/0.55 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X5 ) ) ) % 0.30/0.55 => ( ! [X5: B] : % 0.30/0.55 ( ( member @ B @ X5 @ S ) % 0.30/0.55 => ? [Xa: C] : % 0.30/0.55 ( ( member @ C @ Xa @ T2 ) % 0.30/0.55 & ( ( I @ Xa ) % 0.30/0.55 = X5 ) % 0.30/0.55 & ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ Xa ) ) ) ) % 0.30/0.55 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_le_included % 0.30/0.55 thf(fact_2574_sum__nonneg__eq__0__iff,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.55 => ! [A3: set @ B,F2: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ A3 ) % 0.30/0.55 => ( ! [X5: B] : % 0.30/0.55 ( ( member @ B @ X5 @ A3 ) % 0.30/0.55 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) ) % 0.30/0.55 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) % 0.30/0.55 = ( zero_zero @ A ) ) % 0.30/0.55 = ( ! [X: B] : % 0.30/0.55 ( ( member @ B @ X @ A3 ) % 0.30/0.55 => ( ( F2 @ X ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_nonneg_eq_0_iff % 0.30/0.55 thf(fact_2575_sum__strict__mono__ex1,axiom, % 0.30/0.55 ! [A: $tType,I7: $tType] : % 0.30/0.55 ( ( ordere8940638589300402666id_add @ A ) % 0.30/0.55 => ! [A3: set @ I7,F2: I7 > A,G: I7 > A] : % 0.30/0.55 ( ( finite_finite2 @ I7 @ A3 ) % 0.30/0.55 => ( ! [X5: I7] : % 0.30/0.55 ( ( member @ I7 @ X5 @ A3 ) % 0.30/0.55 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 0.30/0.55 => ( ? [X2: I7] : % 0.30/0.55 ( ( member @ I7 @ X2 @ A3 ) % 0.30/0.55 & ( ord_less @ A @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) % 0.30/0.55 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I7 @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ I7 @ A @ G @ A3 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_strict_mono_ex1 % 0.30/0.55 thf(fact_2576_sum_Orelated,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [R: A > A > $o,S3: set @ B,H2: B > A,G: B > A] : % 0.30/0.55 ( ( R @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) % 0.30/0.55 => ( ! [X16: A,Y15: A,X23: A,Y23: A] : % 0.30/0.55 ( ( ( R @ X16 @ X23 ) % 0.30/0.55 & ( R @ Y15 @ Y23 ) ) % 0.30/0.55 => ( R @ ( plus_plus @ A @ X16 @ Y15 ) @ ( plus_plus @ A @ X23 @ Y23 ) ) ) % 0.30/0.55 => ( ( finite_finite2 @ B @ S3 ) % 0.30/0.55 => ( ! [X5: B] : % 0.30/0.55 ( ( member @ B @ X5 @ S3 ) % 0.30/0.55 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) ) % 0.30/0.55 => ( R @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.related % 0.30/0.55 thf(fact_2577_dense__eq0__I,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 & ( dense_linorder @ A ) ) % 0.30/0.55 => ! [X3: A] : % 0.30/0.55 ( ! [E: A] : % 0.30/0.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ E ) % 0.30/0.55 => ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ E ) ) % 0.30/0.55 => ( X3 % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % dense_eq0_I % 0.30/0.55 thf(fact_2578_abs__mult__pos,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [X3: A,Y: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.55 => ( ( times_times @ A @ ( abs_abs @ A @ Y ) @ X3 ) % 0.30/0.55 = ( abs_abs @ A @ ( times_times @ A @ Y @ X3 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_mult_pos % 0.30/0.55 thf(fact_2579_abs__eq__mult,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordered_ring_abs @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : % 0.30/0.55 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.55 | ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) % 0.30/0.55 & ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.55 | ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) % 0.30/0.55 => ( ( abs_abs @ A @ ( times_times @ A @ A2 @ B2 ) ) % 0.30/0.55 = ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_eq_mult % 0.30/0.55 thf(fact_2580_sum__strict__mono,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( strict7427464778891057005id_add @ A ) % 0.30/0.55 => ! [A3: set @ B,F2: B > A,G: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ A3 ) % 0.30/0.55 => ( ( A3 % 0.30/0.55 != ( bot_bot @ ( set @ B ) ) ) % 0.30/0.55 => ( ! [X5: B] : % 0.30/0.55 ( ( member @ B @ X5 @ A3 ) % 0.30/0.55 => ( ord_less @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 0.30/0.55 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_strict_mono % 0.30/0.55 thf(fact_2581_abs__minus__le__zero,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A2 ) ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_minus_le_zero % 0.30/0.55 thf(fact_2582_abs__eq__iff_H,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_ring @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : % 0.30/0.55 ( ( ( abs_abs @ A @ A2 ) % 0.30/0.55 = B2 ) % 0.30/0.55 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.30/0.55 & ( ( A2 = B2 ) % 0.30/0.55 | ( A2 % 0.30/0.55 = ( uminus_uminus @ A @ B2 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_eq_iff' % 0.30/0.55 thf(fact_2583_eq__abs__iff_H,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_ring @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : % 0.30/0.55 ( ( A2 % 0.30/0.55 = ( abs_abs @ A @ B2 ) ) % 0.30/0.55 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.30/0.55 & ( ( B2 = A2 ) % 0.30/0.55 | ( B2 % 0.30/0.55 = ( uminus_uminus @ A @ A2 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % eq_abs_iff' % 0.30/0.55 thf(fact_2584_zero__le__power__abs,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [A2: A,N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % zero_le_power_abs % 0.30/0.55 thf(fact_2585_abs__div__pos,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_field @ A ) % 0.30/0.55 => ! [Y: A,X3: A] : % 0.30/0.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.55 => ( ( divide_divide @ A @ ( abs_abs @ A @ X3 ) @ Y ) % 0.30/0.55 = ( abs_abs @ A @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_div_pos % 0.30/0.55 thf(fact_2586_abs__if__raw,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( abs_if @ A ) % 0.30/0.55 => ( ( abs_abs @ A ) % 0.30/0.55 = ( ^ [A4: A] : ( if @ A @ ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A4 ) @ A4 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_if_raw % 0.30/0.55 thf(fact_2587_abs__of__neg,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A] : % 0.30/0.55 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.30/0.55 => ( ( abs_abs @ A @ A2 ) % 0.30/0.55 = ( uminus_uminus @ A @ A2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_of_neg % 0.30/0.55 thf(fact_2588_abs__if,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( abs_if @ A ) % 0.30/0.55 => ( ( abs_abs @ A ) % 0.30/0.55 = ( ^ [A4: A] : ( if @ A @ ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A4 ) @ A4 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_if % 0.30/0.55 thf(fact_2589_abs__diff__triangle__ineq,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_diff_triangle_ineq % 0.30/0.55 thf(fact_2590_abs__triangle__ineq4,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ordere166539214618696060dd_abs @ A ) % 0.30/0.55 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_triangle_ineq4 % 0.30/0.55 thf(fact_2591_abs__diff__le__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [X3: A,A2: A,R2: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ R2 ) % 0.30/0.55 = ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X3 ) % 0.30/0.55 & ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_diff_le_iff % 0.30/0.55 thf(fact_2592_sum_Oinsert__if,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [A3: set @ B,X3: B,G: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ A3 ) % 0.30/0.55 => ( ( ( member @ B @ X3 @ A3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) % 0.30/0.55 & ( ~ ( member @ B @ X3 @ A3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 0.30/0.55 = ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.insert_if % 0.30/0.55 thf(fact_2593_abs__diff__less__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [X3: A,A2: A,R2: A] : % 0.30/0.55 ( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ R2 ) % 0.30/0.55 = ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X3 ) % 0.30/0.55 & ( ord_less @ A @ X3 @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_diff_less_iff % 0.30/0.55 thf(fact_2594_sum_Oreindex__bij__witness__not__neutral,axiom, % 0.30/0.55 ! [B: $tType,A: $tType,C: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [S6: set @ B,T6: set @ C,S3: set @ B,I: C > B,J: B > C,T4: set @ C,G: B > A,H2: C > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ S6 ) % 0.30/0.55 => ( ( finite_finite2 @ C @ T6 ) % 0.30/0.55 => ( ! [A5: B] : % 0.30/0.55 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) % 0.30/0.55 => ( ( I @ ( J @ A5 ) ) % 0.30/0.55 = A5 ) ) % 0.30/0.55 => ( ! [A5: B] : % 0.30/0.55 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) % 0.30/0.55 => ( member @ C @ ( J @ A5 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) ) % 0.30/0.55 => ( ! [B5: C] : % 0.30/0.55 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) % 0.30/0.55 => ( ( J @ ( I @ B5 ) ) % 0.30/0.55 = B5 ) ) % 0.30/0.55 => ( ! [B5: C] : % 0.30/0.55 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) % 0.30/0.55 => ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) ) % 0.30/0.55 => ( ! [A5: B] : % 0.30/0.55 ( ( member @ B @ A5 @ S6 ) % 0.30/0.55 => ( ( G @ A5 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) % 0.30/0.55 => ( ! [B5: C] : % 0.30/0.55 ( ( member @ C @ B5 @ T6 ) % 0.30/0.55 => ( ( H2 @ B5 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) % 0.30/0.55 => ( ! [A5: B] : % 0.30/0.55 ( ( member @ B @ A5 @ S3 ) % 0.30/0.55 => ( ( H2 @ ( J @ A5 ) ) % 0.30/0.55 = ( G @ A5 ) ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.reindex_bij_witness_not_neutral % 0.30/0.55 thf(fact_2595_sum__eq__Suc0__iff,axiom, % 0.30/0.55 ! [A: $tType,A3: set @ A,F2: A > nat] : % 0.30/0.55 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.55 => ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) % 0.30/0.55 = ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.55 = ( ? [X: A] : % 0.30/0.55 ( ( member @ A @ X @ A3 ) % 0.30/0.55 & ( ( F2 @ X ) % 0.30/0.55 = ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.55 & ! [Y2: A] : % 0.30/0.55 ( ( member @ A @ Y2 @ A3 ) % 0.30/0.55 => ( ( X != Y2 ) % 0.30/0.55 => ( ( F2 @ Y2 ) % 0.30/0.55 = ( zero_zero @ nat ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_eq_Suc0_iff % 0.30/0.55 thf(fact_2596_sum__SucD,axiom, % 0.30/0.55 ! [A: $tType,F2: A > nat,A3: set @ A,N2: nat] : % 0.30/0.55 ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) % 0.30/0.55 = ( suc @ N2 ) ) % 0.30/0.55 => ? [X5: A] : % 0.30/0.55 ( ( member @ A @ X5 @ A3 ) % 0.30/0.55 & ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X5 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_SucD % 0.30/0.55 thf(fact_2597_sum__eq__1__iff,axiom, % 0.30/0.55 ! [A: $tType,A3: set @ A,F2: A > nat] : % 0.30/0.55 ( ( finite_finite2 @ A @ A3 ) % 0.30/0.55 => ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) % 0.30/0.55 = ( one_one @ nat ) ) % 0.30/0.55 = ( ? [X: A] : % 0.30/0.55 ( ( member @ A @ X @ A3 ) % 0.30/0.55 & ( ( F2 @ X ) % 0.30/0.55 = ( one_one @ nat ) ) % 0.30/0.55 & ! [Y2: A] : % 0.30/0.55 ( ( member @ A @ Y2 @ A3 ) % 0.30/0.55 => ( ( X != Y2 ) % 0.30/0.55 => ( ( F2 @ Y2 ) % 0.30/0.55 = ( zero_zero @ nat ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_eq_1_iff % 0.30/0.55 thf(fact_2598_abs__real__def,axiom, % 0.30/0.55 ( ( abs_abs @ real ) % 0.30/0.55 = ( ^ [A4: real] : ( if @ real @ ( ord_less @ real @ A4 @ ( zero_zero @ real ) ) @ ( uminus_uminus @ real @ A4 ) @ A4 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_real_def % 0.30/0.55 thf(fact_2599_zabs__def,axiom, % 0.30/0.55 ( ( abs_abs @ int ) % 0.30/0.55 = ( ^ [I3: int] : ( if @ int @ ( ord_less @ int @ I3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I3 ) @ I3 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % zabs_def % 0.30/0.55 thf(fact_2600_sum__nonneg__leq__bound,axiom, % 0.30/0.55 ! [B: $tType,A: $tType] : % 0.30/0.55 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.55 => ! [S: set @ B,F2: B > A,B3: A,I: B] : % 0.30/0.55 ( ( finite_finite2 @ B @ S ) % 0.30/0.55 => ( ! [I4: B] : % 0.30/0.55 ( ( member @ B @ I4 @ S ) % 0.30/0.55 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) ) % 0.30/0.55 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S ) % 0.30/0.55 = B3 ) % 0.30/0.55 => ( ( member @ B @ I @ S ) % 0.30/0.55 => ( ord_less_eq @ A @ ( F2 @ I ) @ B3 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_nonneg_leq_bound % 0.30/0.55 thf(fact_2601_sum__nonneg__0,axiom, % 0.30/0.55 ! [B: $tType,A: $tType] : % 0.30/0.55 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.55 => ! [S: set @ B,F2: B > A,I: B] : % 0.30/0.55 ( ( finite_finite2 @ B @ S ) % 0.30/0.55 => ( ! [I4: B] : % 0.30/0.55 ( ( member @ B @ I4 @ S ) % 0.30/0.55 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) ) % 0.30/0.55 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S ) % 0.30/0.55 = ( zero_zero @ A ) ) % 0.30/0.55 => ( ( member @ B @ I @ S ) % 0.30/0.55 => ( ( F2 @ I ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_nonneg_0 % 0.30/0.55 thf(fact_2602_dvd__imp__le__int,axiom, % 0.30/0.55 ! [I: int,D2: int] : % 0.30/0.55 ( ( I % 0.30/0.55 != ( zero_zero @ int ) ) % 0.30/0.55 => ( ( dvd_dvd @ int @ D2 @ I ) % 0.30/0.55 => ( ord_less_eq @ int @ ( abs_abs @ int @ D2 ) @ ( abs_abs @ int @ I ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % dvd_imp_le_int % 0.30/0.55 thf(fact_2603_abs__mod__less,axiom, % 0.30/0.55 ! [L: int,K: int] : % 0.30/0.55 ( ( L % 0.30/0.55 != ( zero_zero @ int ) ) % 0.30/0.55 => ( ord_less @ int @ ( abs_abs @ int @ ( modulo_modulo @ int @ K @ L ) ) @ ( abs_abs @ int @ L ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_mod_less % 0.30/0.55 thf(fact_2604_sum_Osetdiff__irrelevant,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [A3: set @ B,G: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ A3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G % 0.30/0.55 @ ( minus_minus @ ( set @ B ) @ A3 % 0.30/0.55 @ ( collect @ B % 0.30/0.55 @ ^ [X: B] : % 0.30/0.55 ( ( G @ X ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.setdiff_irrelevant % 0.30/0.55 thf(fact_2605_sum__power__add,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ( monoid_mult @ A ) % 0.30/0.55 & ( comm_ring @ A ) ) % 0.30/0.55 => ! [X3: A,M: nat,I6: set @ nat] : % 0.30/0.55 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.55 @ ^ [I3: nat] : ( power_power @ A @ X3 @ ( plus_plus @ nat @ M @ I3 ) ) % 0.30/0.55 @ I6 ) % 0.30/0.55 = ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ I6 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_power_add % 0.30/0.55 thf(fact_2606_sum_OatLeastAtMost__rev,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [G: nat > A,N2: nat,M: nat] : % 0.30/0.55 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.55 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I3 ) ) % 0.30/0.55 @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.atLeastAtMost_rev % 0.30/0.55 thf(fact_2607_tanh__real__gt__neg1,axiom, % 0.30/0.55 ! [X3: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( tanh @ real @ X3 ) ) ). % 0.30/0.55 % 0.30/0.55 % tanh_real_gt_neg1 % 0.30/0.55 thf(fact_2608_sum__pos2,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.55 => ! [I6: set @ B,I: B,F2: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ I6 ) % 0.30/0.55 => ( ( member @ B @ I @ I6 ) % 0.30/0.55 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) ) % 0.30/0.55 => ( ! [I4: B] : % 0.30/0.55 ( ( member @ B @ I4 @ I6 ) % 0.30/0.55 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) ) % 0.30/0.55 => ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_pos2 % 0.30/0.55 thf(fact_2609_sum__pos,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.55 => ! [I6: set @ B,F2: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ I6 ) % 0.30/0.55 => ( ( I6 % 0.30/0.55 != ( bot_bot @ ( set @ B ) ) ) % 0.30/0.55 => ( ! [I4: B] : % 0.30/0.55 ( ( member @ B @ I4 @ I6 ) % 0.30/0.55 => ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) ) % 0.30/0.55 => ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_pos % 0.30/0.55 thf(fact_2610_abs__add__one__gt__zero,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [X3: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X3 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_add_one_gt_zero % 0.30/0.55 thf(fact_2611_sum_Omono__neutral__cong__right,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [T4: set @ B,S3: set @ B,G: B > A,H2: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ T4 ) % 0.30/0.55 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.30/0.55 => ( ! [X5: B] : % 0.30/0.55 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.30/0.55 => ( ( G @ X5 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) % 0.30/0.55 => ( ! [X5: B] : % 0.30/0.55 ( ( member @ B @ X5 @ S3 ) % 0.30/0.55 => ( ( G @ X5 ) % 0.30/0.55 = ( H2 @ X5 ) ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T4 ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.mono_neutral_cong_right % 0.30/0.55 thf(fact_2612_sum_Omono__neutral__cong__left,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [T4: set @ B,S3: set @ B,H2: B > A,G: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ T4 ) % 0.30/0.55 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.30/0.55 => ( ! [X5: B] : % 0.30/0.55 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.30/0.55 => ( ( H2 @ X5 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) % 0.30/0.55 => ( ! [X5: B] : % 0.30/0.55 ( ( member @ B @ X5 @ S3 ) % 0.30/0.55 => ( ( G @ X5 ) % 0.30/0.55 = ( H2 @ X5 ) ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.mono_neutral_cong_left % 0.30/0.55 thf(fact_2613_sum_Omono__neutral__right,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [T4: set @ B,S3: set @ B,G: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ T4 ) % 0.30/0.55 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.30/0.55 => ( ! [X5: B] : % 0.30/0.55 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.30/0.55 => ( ( G @ X5 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T4 ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.mono_neutral_right % 0.30/0.55 thf(fact_2614_sum_Omono__neutral__left,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [T4: set @ B,S3: set @ B,G: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ T4 ) % 0.30/0.55 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.30/0.55 => ( ! [X5: B] : % 0.30/0.55 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.30/0.55 => ( ( G @ X5 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ B @ A @ G @ T4 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.mono_neutral_left % 0.30/0.55 thf(fact_2615_sum_Osame__carrierI,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [C5: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ C5 ) % 0.30/0.55 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 ) % 0.30/0.55 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ C5 ) % 0.30/0.55 => ( ! [A5: B] : % 0.30/0.55 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) ) % 0.30/0.55 => ( ( G @ A5 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) % 0.30/0.55 => ( ! [B5: B] : % 0.30/0.55 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B3 ) ) % 0.30/0.55 => ( ( H2 @ B5 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) % 0.30/0.55 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B3 ) ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.same_carrierI % 0.30/0.55 thf(fact_2616_sum_Osame__carrier,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [C5: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ C5 ) % 0.30/0.55 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 ) % 0.30/0.55 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ C5 ) % 0.30/0.55 => ( ! [A5: B] : % 0.30/0.55 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) ) % 0.30/0.55 => ( ( G @ A5 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) % 0.30/0.55 => ( ! [B5: B] : % 0.30/0.55 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B3 ) ) % 0.30/0.55 => ( ( H2 @ B5 ) % 0.30/0.55 = ( zero_zero @ A ) ) ) % 0.30/0.55 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B3 ) ) % 0.30/0.55 = ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.same_carrier % 0.30/0.55 thf(fact_2617_sum_Osubset__diff,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [B3: set @ B,A3: set @ B,G: B > A] : % 0.30/0.55 ( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 ) % 0.30/0.55 => ( ( finite_finite2 @ B @ A3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 0.30/0.55 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B3 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.subset_diff % 0.30/0.55 thf(fact_2618_sum__diff,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( ab_group_add @ A ) % 0.30/0.55 => ! [A3: set @ B,B3: set @ B,F2: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ A3 ) % 0.30/0.55 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) % 0.30/0.55 = ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B3 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_diff % 0.30/0.55 thf(fact_2619_of__int__leD,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [N2: int,X3: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N2 ) ) @ X3 ) % 0.30/0.55 => ( ( N2 % 0.30/0.55 = ( zero_zero @ int ) ) % 0.30/0.55 | ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_int_leD % 0.30/0.55 thf(fact_2620_sum__diff__nat,axiom, % 0.30/0.55 ! [A: $tType,B3: set @ A,A3: set @ A,F2: A > nat] : % 0.30/0.55 ( ( finite_finite2 @ A @ B3 ) % 0.30/0.55 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 0.30/0.55 = ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ B3 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_diff_nat % 0.30/0.55 thf(fact_2621_of__int__lessD,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [N2: int,X3: A] : % 0.30/0.55 ( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N2 ) ) @ X3 ) % 0.30/0.55 => ( ( N2 % 0.30/0.55 = ( zero_zero @ int ) ) % 0.30/0.55 | ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_int_lessD % 0.30/0.55 thf(fact_2622_sum__diff1__nat,axiom, % 0.30/0.55 ! [A: $tType,A2: A,A3: set @ A,F2: A > nat] : % 0.30/0.55 ( ( ( member @ A @ A2 @ A3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.30/0.55 = ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( F2 @ A2 ) ) ) ) % 0.30/0.55 & ( ~ ( member @ A @ A2 @ A3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_diff1_nat % 0.30/0.55 thf(fact_2623_sum__shift__lb__Suc0__0,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [F2: nat > A,K: nat] : % 0.30/0.55 ( ( ( F2 @ ( zero_zero @ nat ) ) % 0.30/0.55 = ( zero_zero @ A ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_shift_lb_Suc0_0 % 0.30/0.55 thf(fact_2624_sum_OatLeast0__atMost__Suc,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [G: nat > A,N2: nat] : % 0.30/0.55 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 0.30/0.55 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.atLeast0_atMost_Suc % 0.30/0.55 thf(fact_2625_sum_OatLeast__Suc__atMost,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [M: nat,N2: nat,G: nat > A] : % 0.30/0.55 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.30/0.55 = ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.atLeast_Suc_atMost % 0.30/0.55 thf(fact_2626_sum_Onat__ivl__Suc_H,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [M: nat,N2: nat,G: nat > A] : % 0.30/0.55 ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) ) % 0.30/0.55 = ( plus_plus @ A @ ( G @ ( suc @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.nat_ivl_Suc' % 0.30/0.55 thf(fact_2627_round__diff__minimal,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.55 => ! [Z2: A,M: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z2 @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z2 ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z2 @ ( ring_1_of_int @ A @ M ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % round_diff_minimal % 0.30/0.55 thf(fact_2628_zdvd__mult__cancel1,axiom, % 0.30/0.55 ! [M: int,N2: int] : % 0.30/0.55 ( ( M % 0.30/0.55 != ( zero_zero @ int ) ) % 0.30/0.55 => ( ( dvd_dvd @ int @ ( times_times @ int @ M @ N2 ) @ M ) % 0.30/0.55 = ( ( abs_abs @ int @ N2 ) % 0.30/0.55 = ( one_one @ int ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % zdvd_mult_cancel1 % 0.30/0.55 thf(fact_2629_sum_OSuc__reindex__ivl,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [M: nat,N2: nat,G: nat > A] : % 0.30/0.55 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.55 => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) % 0.30/0.55 = ( plus_plus @ A @ ( G @ M ) % 0.30/0.55 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.55 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 0.30/0.55 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.Suc_reindex_ivl % 0.30/0.55 thf(fact_2630_sum__Suc__diff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ab_group_add @ A ) % 0.30/0.55 => ! [M: nat,N2: nat,F2: nat > A] : % 0.30/0.55 ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.55 @ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) % 0.30/0.55 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.30/0.55 = ( minus_minus @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ M ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_Suc_diff % 0.30/0.55 thf(fact_2631_fun__cong__unused__0,axiom, % 0.30/0.55 ! [A: $tType,B: $tType,C: $tType] : % 0.30/0.55 ( ( zero @ B ) % 0.30/0.55 => ! [F2: ( A > B ) > C,G: C] : % 0.30/0.55 ( ( F2 % 0.30/0.55 = ( ^ [X: A > B] : G ) ) % 0.30/0.55 => ( ( F2 % 0.30/0.55 @ ^ [X: A] : ( zero_zero @ B ) ) % 0.30/0.55 = G ) ) ) ). % 0.30/0.55 % 0.30/0.55 % fun_cong_unused_0 % 0.30/0.55 thf(fact_2632_sum__mono2,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( ordere6911136660526730532id_add @ A ) % 0.30/0.55 => ! [B3: set @ B,A3: set @ B,F2: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ B3 ) % 0.30/0.55 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 ) % 0.30/0.55 => ( ! [B5: B] : % 0.30/0.55 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ B3 @ A3 ) ) % 0.30/0.55 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ B5 ) ) ) % 0.30/0.55 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B3 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_mono2 % 0.30/0.55 thf(fact_2633_abs__le__square__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [X3: A,Y: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ ( abs_abs @ A @ Y ) ) % 0.30/0.55 = ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_le_square_iff % 0.30/0.55 thf(fact_2634_sum_Oremove,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [A3: set @ B,X3: B,G: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ A3 ) % 0.30/0.55 => ( ( member @ B @ X3 @ A3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 0.30/0.55 = ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.remove % 0.30/0.55 thf(fact_2635_sum_Oinsert__remove,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [A3: set @ B,G: B > A,X3: B] : % 0.30/0.55 ( ( finite_finite2 @ B @ A3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 0.30/0.55 = ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.insert_remove % 0.30/0.55 thf(fact_2636_sum__diff1,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( ab_group_add @ A ) % 0.30/0.55 => ! [A3: set @ B,A2: B,F2: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ A3 ) % 0.30/0.55 => ( ( ( member @ B @ A2 @ A3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) % 0.30/0.55 = ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( F2 @ A2 ) ) ) ) % 0.30/0.55 & ( ~ ( member @ B @ A2 @ A3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_diff1 % 0.30/0.55 thf(fact_2637_sum_Oub__add__nat,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [M: nat,N2: nat,G: nat > A,P4: nat] : % 0.30/0.55 ( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N2 @ P4 ) ) ) % 0.30/0.55 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ P4 ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.ub_add_nat % 0.30/0.55 thf(fact_2638_sum_Odelta__remove,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [S3: set @ B,A2: B,B2: B > A,C2: B > A] : % 0.30/0.55 ( ( finite_finite2 @ B @ S3 ) % 0.30/0.55 => ( ( ( member @ B @ A2 @ S3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A % 0.30/0.55 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) ) % 0.30/0.55 @ S3 ) % 0.30/0.55 = ( plus_plus @ A @ ( B2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) % 0.30/0.55 & ( ~ ( member @ B @ A2 @ S3 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A % 0.30/0.55 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) ) % 0.30/0.55 @ S3 ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.delta_remove % 0.30/0.55 thf(fact_2639_sum__count__set,axiom, % 0.30/0.55 ! [A: $tType,Xs: list @ A,X8: set @ A] : % 0.30/0.55 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X8 ) % 0.30/0.55 => ( ( finite_finite2 @ A @ X8 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs ) @ X8 ) % 0.30/0.55 = ( size_size @ ( list @ A ) @ Xs ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_count_set % 0.30/0.55 thf(fact_2640_sum__strict__mono2,axiom, % 0.30/0.55 ! [B: $tType,A: $tType] : % 0.30/0.55 ( ( ordere8940638589300402666id_add @ B ) % 0.30/0.55 => ! [B3: set @ A,A3: set @ A,B2: A,F2: A > B] : % 0.30/0.55 ( ( finite_finite2 @ A @ B3 ) % 0.30/0.55 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.30/0.55 => ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) % 0.30/0.55 => ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F2 @ B2 ) ) % 0.30/0.55 => ( ! [X5: A] : % 0.30/0.55 ( ( member @ A @ X5 @ B3 ) % 0.30/0.55 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X5 ) ) ) % 0.30/0.55 => ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ B3 ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_strict_mono2 % 0.30/0.55 thf(fact_2641_member__le__sum,axiom, % 0.30/0.55 ! [B: $tType,C: $tType] : % 0.30/0.55 ( ( ( ordere6911136660526730532id_add @ B ) % 0.30/0.55 & ( semiring_1 @ B ) ) % 0.30/0.55 => ! [I: C,A3: set @ C,F2: C > B] : % 0.30/0.55 ( ( member @ C @ I @ A3 ) % 0.30/0.55 => ( ! [X5: C] : % 0.30/0.55 ( ( member @ C @ X5 @ ( minus_minus @ ( set @ C ) @ A3 @ ( insert @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) ) % 0.30/0.55 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X5 ) ) ) % 0.30/0.55 => ( ( finite_finite2 @ C @ A3 ) % 0.30/0.55 => ( ord_less_eq @ B @ ( F2 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F2 @ A3 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % member_le_sum % 0.30/0.55 thf(fact_2642_power2__le__iff__abs__le,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [Y: A,X3: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 0.30/0.55 => ( ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.30/0.55 = ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ Y ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % power2_le_iff_abs_le % 0.30/0.55 thf(fact_2643_abs__sqrt__wlog,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [P: A > A > $o,X3: A] : % 0.30/0.55 ( ! [X5: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 ) % 0.30/0.55 => ( P @ X5 @ ( power_power @ A @ X5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.30/0.55 => ( P @ ( abs_abs @ A @ X3 ) @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_sqrt_wlog % 0.30/0.55 thf(fact_2644_abs__square__le__1,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [X3: A] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) % 0.30/0.55 = ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ ( one_one @ A ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_square_le_1 % 0.30/0.55 thf(fact_2645_abs__square__less__1,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [X3: A] : % 0.30/0.55 ( ( ord_less @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) % 0.30/0.55 = ( ord_less @ A @ ( abs_abs @ A @ X3 ) @ ( one_one @ A ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_square_less_1 % 0.30/0.55 thf(fact_2646_power__mono__even,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [N2: nat,A2: A,B2: A] : % 0.30/0.55 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.30/0.55 => ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) % 0.30/0.55 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % power_mono_even % 0.30/0.55 thf(fact_2647_nat__intermed__int__val,axiom, % 0.30/0.55 ! [M: nat,N2: nat,F2: nat > int,K: int] : % 0.30/0.55 ( ! [I4: nat] : % 0.30/0.55 ( ( ( ord_less_eq @ nat @ M @ I4 ) % 0.30/0.55 & ( ord_less @ nat @ I4 @ N2 ) ) % 0.30/0.55 => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I4 ) ) @ ( F2 @ I4 ) ) ) @ ( one_one @ int ) ) ) % 0.30/0.55 => ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.55 => ( ( ord_less_eq @ int @ ( F2 @ M ) @ K ) % 0.30/0.55 => ( ( ord_less_eq @ int @ K @ ( F2 @ N2 ) ) % 0.30/0.55 => ? [I4: nat] : % 0.30/0.55 ( ( ord_less_eq @ nat @ M @ I4 ) % 0.30/0.55 & ( ord_less_eq @ nat @ I4 @ N2 ) % 0.30/0.55 & ( ( F2 @ I4 ) % 0.30/0.55 = K ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % nat_intermed_int_val % 0.30/0.55 thf(fact_2648_incr__lemma,axiom, % 0.30/0.55 ! [D2: int,Z2: int,X3: int] : % 0.30/0.55 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 ) % 0.30/0.55 => ( ord_less @ int @ Z2 @ ( plus_plus @ int @ X3 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X3 @ Z2 ) ) @ ( one_one @ int ) ) @ D2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % incr_lemma % 0.30/0.55 thf(fact_2649_decr__lemma,axiom, % 0.30/0.55 ! [D2: int,X3: int,Z2: int] : % 0.30/0.55 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 ) % 0.30/0.55 => ( ord_less @ int @ ( minus_minus @ int @ X3 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X3 @ Z2 ) ) @ ( one_one @ int ) ) @ D2 ) ) @ Z2 ) ) ). % 0.30/0.55 % 0.30/0.55 % decr_lemma % 0.30/0.55 thf(fact_2650_sum__natinterval__diff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ab_group_add @ A ) % 0.30/0.55 => ! [M: nat,N2: nat,F2: nat > A] : % 0.30/0.55 ( ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.55 @ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) ) % 0.30/0.55 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.30/0.55 = ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) % 0.30/0.55 & ( ~ ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.55 @ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) ) % 0.30/0.55 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_natinterval_diff % 0.30/0.55 thf(fact_2651_sum__telescope_H_H,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ab_group_add @ A ) % 0.30/0.55 => ! [M: nat,N2: nat,F2: nat > A] : % 0.30/0.55 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.55 @ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) ) % 0.30/0.55 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) % 0.30/0.55 = ( minus_minus @ A @ ( F2 @ N2 ) @ ( F2 @ M ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_telescope'' % 0.30/0.55 thf(fact_2652_mask__eq__sum__exp,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( semiring_parity @ A ) % 0.30/0.55 => ! [N2: nat] : % 0.30/0.55 ( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 0.30/0.55 @ ( collect @ nat % 0.30/0.55 @ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N2 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % mask_eq_sum_exp % 0.30/0.55 thf(fact_2653_nat__ivt__aux,axiom, % 0.30/0.55 ! [N2: nat,F2: nat > int,K: int] : % 0.30/0.55 ( ! [I4: nat] : % 0.30/0.55 ( ( ord_less @ nat @ I4 @ N2 ) % 0.30/0.55 => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I4 ) ) @ ( F2 @ I4 ) ) ) @ ( one_one @ int ) ) ) % 0.30/0.55 => ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K ) % 0.30/0.55 => ( ( ord_less_eq @ int @ K @ ( F2 @ N2 ) ) % 0.30/0.55 => ? [I4: nat] : % 0.30/0.55 ( ( ord_less_eq @ nat @ I4 @ N2 ) % 0.30/0.55 & ( ( F2 @ I4 ) % 0.30/0.55 = K ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % nat_ivt_aux % 0.30/0.55 thf(fact_2654_divmod__nat__def,axiom, % 0.30/0.55 ( divmod_nat % 0.30/0.55 = ( ^ [M2: nat,N: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M2 @ N ) @ ( modulo_modulo @ nat @ M2 @ N ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % divmod_nat_def % 0.30/0.55 thf(fact_2655_sum__gp__multiplied,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ( monoid_mult @ A ) % 0.30/0.55 & ( comm_ring @ A ) ) % 0.30/0.55 => ! [M: nat,N2: nat,X3: A] : % 0.30/0.55 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.30/0.55 => ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) % 0.30/0.55 = ( minus_minus @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_gp_multiplied % 0.30/0.55 thf(fact_2656_sum_Oin__pairs,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( comm_monoid_add @ A ) % 0.30/0.55 => ! [G: nat > A,M: nat,N2: nat] : % 0.30/0.55 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.55 @ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) ) % 0.30/0.55 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum.in_pairs % 0.30/0.55 thf(fact_2657_nat0__intermed__int__val,axiom, % 0.30/0.55 ! [N2: nat,F2: nat > int,K: int] : % 0.30/0.55 ( ! [I4: nat] : % 0.30/0.55 ( ( ord_less @ nat @ I4 @ N2 ) % 0.30/0.55 => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) @ ( F2 @ I4 ) ) ) @ ( one_one @ int ) ) ) % 0.30/0.55 => ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K ) % 0.30/0.55 => ( ( ord_less_eq @ int @ K @ ( F2 @ N2 ) ) % 0.30/0.55 => ? [I4: nat] : % 0.30/0.55 ( ( ord_less_eq @ nat @ I4 @ N2 ) % 0.30/0.55 & ( ( F2 @ I4 ) % 0.30/0.55 = K ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % nat0_intermed_int_val % 0.30/0.55 thf(fact_2658_mask__eq__sum__exp__nat,axiom, % 0.30/0.55 ! [N2: nat] : % 0.30/0.55 ( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( suc @ ( zero_zero @ nat ) ) ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.30/0.55 @ ( collect @ nat % 0.30/0.55 @ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % mask_eq_sum_exp_nat % 0.30/0.55 thf(fact_2659_gauss__sum__nat,axiom, % 0.30/0.55 ! [N2: nat] : % 0.30/0.55 ( ( groups7311177749621191930dd_sum @ nat @ nat % 0.30/0.55 @ ^ [X: nat] : X % 0.30/0.55 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.30/0.55 = ( divide_divide @ nat @ ( times_times @ nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % gauss_sum_nat % 0.30/0.55 thf(fact_2660_of__int__round__abs__le,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.55 => ! [X3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) @ X3 ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_int_round_abs_le % 0.30/0.55 thf(fact_2661_listrel1p__def,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( listrel1p @ A ) % 0.30/0.55 = ( ^ [R5: A > A > $o,Xs2: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys3 ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % listrel1p_def % 0.30/0.55 thf(fact_2662_lemma__interval,axiom, % 0.30/0.55 ! [A2: real,X3: real,B2: real] : % 0.30/0.55 ( ( ord_less @ real @ A2 @ X3 ) % 0.30/0.55 => ( ( ord_less @ real @ X3 @ B2 ) % 0.30/0.55 => ? [D3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 0.30/0.55 & ! [Y5: real] : % 0.30/0.55 ( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y5 ) ) @ D3 ) % 0.30/0.55 => ( ( ord_less_eq @ real @ A2 @ Y5 ) % 0.30/0.55 & ( ord_less_eq @ real @ Y5 @ B2 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % lemma_interval % 0.30/0.55 thf(fact_2663_sum__gp,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ( division_ring @ A ) % 0.30/0.55 & ( comm_ring @ A ) ) % 0.30/0.55 => ! [N2: nat,M: nat,X3: A] : % 0.30/0.55 ( ( ( ord_less @ nat @ N2 @ M ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.30/0.55 = ( zero_zero @ A ) ) ) % 0.30/0.55 & ( ~ ( ord_less @ nat @ N2 @ M ) % 0.30/0.55 => ( ( ( X3 % 0.30/0.55 = ( one_one @ A ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.30/0.55 = ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ M ) ) ) ) % 0.30/0.55 & ( ( X3 % 0.30/0.55 != ( one_one @ A ) ) % 0.30/0.55 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.30/0.55 = ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % sum_gp % 0.30/0.55 thf(fact_2664_gauss__sum__from__Suc__0,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( euclid5411537665997757685th_nat @ A ) % 0.30/0.55 => ! [N2: nat] : % 0.30/0.55 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) % 0.30/0.55 = ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % gauss_sum_from_Suc_0 % 0.30/0.55 thf(fact_2665_lemma__interval__lt,axiom, % 0.30/0.55 ! [A2: real,X3: real,B2: real] : % 0.30/0.55 ( ( ord_less @ real @ A2 @ X3 ) % 0.30/0.55 => ( ( ord_less @ real @ X3 @ B2 ) % 0.30/0.55 => ? [D3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 0.30/0.55 & ! [Y5: real] : % 0.30/0.55 ( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y5 ) ) @ D3 ) % 0.30/0.55 => ( ( ord_less @ real @ A2 @ Y5 ) % 0.30/0.55 & ( ord_less @ real @ Y5 @ B2 ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % lemma_interval_lt % 0.30/0.55 thf(fact_2666_arctan__double,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 0.30/0.55 => ( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ X3 ) ) % 0.30/0.55 = ( arctan @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % arctan_double % 0.30/0.55 thf(fact_2667_of__nat__eq__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( semiring_char_0 @ A ) % 0.30/0.55 => ! [M: nat,N2: nat] : % 0.30/0.55 ( ( ( semiring_1_of_nat @ A @ M ) % 0.30/0.55 = ( semiring_1_of_nat @ A @ N2 ) ) % 0.30/0.55 = ( M = N2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_nat_eq_iff % 0.30/0.55 thf(fact_2668_split__part,axiom, % 0.30/0.55 ! [B: $tType,A: $tType,P: $o,Q: A > B > $o] : % 0.30/0.55 ( ( product_case_prod @ A @ B @ $o % 0.30/0.55 @ ^ [A4: A,B4: B] : % 0.30/0.55 ( P % 0.30/0.55 & ( Q @ A4 @ B4 ) ) ) % 0.30/0.55 = ( ^ [Ab: product_prod @ A @ B] : % 0.30/0.55 ( P % 0.30/0.55 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % split_part % 0.30/0.55 thf(fact_2669_int__eq__iff__numeral,axiom, % 0.30/0.55 ! [M: nat,V2: num] : % 0.30/0.55 ( ( ( semiring_1_of_nat @ int @ M ) % 0.30/0.55 = ( numeral_numeral @ int @ V2 ) ) % 0.30/0.55 = ( M % 0.30/0.55 = ( numeral_numeral @ nat @ V2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % int_eq_iff_numeral % 0.30/0.55 thf(fact_2670_abs__of__nat,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linordered_idom @ A ) % 0.30/0.55 => ! [N2: nat] : % 0.30/0.55 ( ( abs_abs @ A @ ( semiring_1_of_nat @ A @ N2 ) ) % 0.30/0.55 = ( semiring_1_of_nat @ A @ N2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % abs_of_nat % 0.30/0.55 thf(fact_2671_negative__eq__positive,axiom, % 0.30/0.55 ! [N2: nat,M: nat] : % 0.30/0.55 ( ( ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) % 0.30/0.55 = ( semiring_1_of_nat @ int @ M ) ) % 0.30/0.55 = ( ( N2 % 0.30/0.55 = ( zero_zero @ nat ) ) % 0.30/0.55 & ( M % 0.30/0.55 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % negative_eq_positive % 0.30/0.55 thf(fact_2672_of__int__of__nat__eq,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( ring_1 @ A ) % 0.30/0.55 => ! [N2: nat] : % 0.30/0.55 ( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N2 ) ) % 0.30/0.55 = ( semiring_1_of_nat @ A @ N2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_int_of_nat_eq % 0.30/0.55 thf(fact_2673_negative__zle,axiom, % 0.30/0.55 ! [N2: nat,M: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( semiring_1_of_nat @ int @ M ) ) ). % 0.30/0.55 % 0.30/0.55 % negative_zle % 0.30/0.55 thf(fact_2674_int__dvd__int__iff,axiom, % 0.30/0.55 ! [M: nat,N2: nat] : % 0.30/0.55 ( ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N2 ) ) % 0.30/0.55 = ( dvd_dvd @ nat @ M @ N2 ) ) ). % 0.30/0.55 % 0.30/0.55 % int_dvd_int_iff % 0.30/0.55 thf(fact_2675_of__nat__eq__0__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( semiring_char_0 @ A ) % 0.30/0.55 => ! [M: nat] : % 0.30/0.55 ( ( ( semiring_1_of_nat @ A @ M ) % 0.30/0.55 = ( zero_zero @ A ) ) % 0.30/0.55 = ( M % 0.30/0.55 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_nat_eq_0_iff % 0.30/0.55 thf(fact_2676_of__nat__0__eq__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( semiring_char_0 @ A ) % 0.30/0.55 => ! [N2: nat] : % 0.30/0.55 ( ( ( zero_zero @ A ) % 0.30/0.55 = ( semiring_1_of_nat @ A @ N2 ) ) % 0.30/0.55 = ( ( zero_zero @ nat ) % 0.30/0.55 = N2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_nat_0_eq_iff % 0.30/0.55 thf(fact_2677_of__nat__0,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( semiring_1 @ A ) % 0.30/0.55 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) ) % 0.30/0.55 = ( zero_zero @ A ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_nat_0 % 0.30/0.55 thf(fact_2678_of__nat__less__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linord181362715937106298miring @ A ) % 0.30/0.55 => ! [M: nat,N2: nat] : % 0.30/0.55 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) % 0.30/0.55 = ( ord_less @ nat @ M @ N2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_nat_less_iff % 0.30/0.55 thf(fact_2679_of__nat__le__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linord181362715937106298miring @ A ) % 0.30/0.55 => ! [M: nat,N2: nat] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) % 0.30/0.55 = ( ord_less_eq @ nat @ M @ N2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_nat_le_iff % 0.30/0.55 thf(fact_2680_of__nat__add,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( semiring_1 @ A ) % 0.30/0.55 => ! [M: nat,N2: nat] : % 0.30/0.55 ( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N2 ) ) % 0.30/0.55 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_nat_add % 0.30/0.55 thf(fact_2681_of__nat__mult,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( semiring_1 @ A ) % 0.30/0.55 => ! [M: nat,N2: nat] : % 0.30/0.55 ( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N2 ) ) % 0.30/0.55 = ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_nat_mult % 0.30/0.55 thf(fact_2682_of__nat__1,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( semiring_1 @ A ) % 0.30/0.55 => ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) ) % 0.30/0.55 = ( one_one @ A ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_nat_1 % 0.30/0.55 thf(fact_2683_of__nat__1__eq__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( semiring_char_0 @ A ) % 0.30/0.55 => ! [N2: nat] : % 0.30/0.55 ( ( ( one_one @ A ) % 0.30/0.55 = ( semiring_1_of_nat @ A @ N2 ) ) % 0.30/0.55 = ( N2 % 0.30/0.55 = ( one_one @ nat ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_nat_1_eq_iff % 0.30/0.55 thf(fact_2684_of__nat__eq__1__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( semiring_char_0 @ A ) % 0.30/0.55 => ! [N2: nat] : % 0.30/0.55 ( ( ( semiring_1_of_nat @ A @ N2 ) % 0.30/0.55 = ( one_one @ A ) ) % 0.30/0.55 = ( N2 % 0.30/0.55 = ( one_one @ nat ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_nat_eq_1_iff % 0.30/0.55 thf(fact_2685_negative__zless,axiom, % 0.30/0.55 ! [N2: nat,M: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) @ ( semiring_1_of_nat @ int @ M ) ) ). % 0.30/0.55 % 0.30/0.55 % negative_zless % 0.30/0.55 thf(fact_2686_arctan__less__zero__iff,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( arctan @ X3 ) @ ( zero_zero @ real ) ) % 0.30/0.55 = ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ). % 0.30/0.55 % 0.30/0.55 % arctan_less_zero_iff % 0.30/0.55 thf(fact_2687_zero__less__arctan__iff,axiom, % 0.30/0.55 ! [X3: real] : % 0.30/0.55 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( arctan @ X3 ) ) % 0.30/0.55 = ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ). % 0.30/0.55 % 0.30/0.55 % zero_less_arctan_iff % 0.30/0.55 thf(fact_2688_of__nat__of__bool,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( semiring_1 @ A ) % 0.30/0.55 => ! [P: $o] : % 0.30/0.55 ( ( semiring_1_of_nat @ A @ ( zero_neq_one_of_bool @ nat @ P ) ) % 0.30/0.55 = ( zero_neq_one_of_bool @ A @ P ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_nat_of_bool % 0.30/0.55 thf(fact_2689_of__nat__sum,axiom, % 0.30/0.55 ! [A: $tType,B: $tType] : % 0.30/0.55 ( ( semiring_1 @ A ) % 0.30/0.55 => ! [F2: B > nat,A3: set @ B] : % 0.30/0.55 ( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A3 ) ) % 0.30/0.55 = ( groups7311177749621191930dd_sum @ B @ A % 0.30/0.55 @ ^ [X: B] : ( semiring_1_of_nat @ A @ ( F2 @ X ) ) % 0.30/0.55 @ A3 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_nat_sum % 0.30/0.55 thf(fact_2690_of__nat__le__0__iff,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( linord181362715937106298miring @ A ) % 0.30/0.55 => ! [M: nat] : % 0.30/0.55 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) % 0.30/0.55 = ( M % 0.30/0.55 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_nat_le_0_iff % 0.30/0.55 thf(fact_2691_of__nat__Suc,axiom, % 0.30/0.55 ! [A: $tType] : % 0.30/0.55 ( ( semiring_1 @ A ) % 0.30/0.55 => ! [M: nat] : % 0.30/0.55 ( ( semiring_1_of_nat @ A @ ( suc @ M ) ) % 0.30/0.55 = ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) ) ) ). % 0.30/0.55 % 0.30/0.55 % of_nat_Suc % 0.30/0.55 thf(fact_2692_real__of__nat__less__numeral__iff,axiom, % 0.30/0.55 ! [N2: nat,W2: num] : % 0.30/0.55 ( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( numeral_numeral @ real @ W2 ) ) % 0.30/0.55 = ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ W2 ) ) ) ). % 0.30/0.55 % 0.30/0.55 % real_of_nat_less_numeral_iff % 0.30/0.55 thf(fact_2693_numeral__less__real__of__nat__iff,axiom, % 0.30/0.55 ! [W2: num,N2: nat] : % 0.30/0.55 ( ( ord_less @ real @ ( numeral_numeral @ real @ W2 ) @ ( semiring_1_of_nat @ real @ N2 ) ) % 0.30/0.56 = ( ord_less @ nat @ ( numeral_numeral @ nat @ W2 ) @ N2 ) ) ). % 0.30/0.56 % 0.30/0.56 % numeral_less_real_of_nat_iff % 0.30/0.56 thf(fact_2694_numeral__le__real__of__nat__iff,axiom, % 0.30/0.56 ! [N2: num,M: nat] : % 0.30/0.56 ( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N2 ) @ ( semiring_1_of_nat @ real @ M ) ) % 0.30/0.56 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N2 ) @ M ) ) ). % 0.30/0.56 % 0.30/0.56 % numeral_le_real_of_nat_iff % 0.30/0.56 thf(fact_2695_of__nat__0__less__iff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linord181362715937106298miring @ A ) % 0.30/0.56 => ! [N2: nat] : % 0.30/0.56 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) % 0.30/0.56 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_0_less_iff % 0.30/0.56 thf(fact_2696_of__nat__power__less__of__nat__cancel__iff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linordered_semidom @ A ) % 0.30/0.56 => ! [X3: nat,B2: nat,W2: nat] : % 0.30/0.56 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) ) % 0.30/0.56 = ( ord_less @ nat @ X3 @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_power_less_of_nat_cancel_iff % 0.30/0.56 thf(fact_2697_of__nat__less__of__nat__power__cancel__iff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linordered_semidom @ A ) % 0.30/0.56 => ! [B2: nat,W2: nat,X3: nat] : % 0.30/0.56 ( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X3 ) ) % 0.30/0.56 = ( ord_less @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X3 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_less_of_nat_power_cancel_iff % 0.30/0.56 thf(fact_2698_of__nat__power__le__of__nat__cancel__iff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linordered_semidom @ A ) % 0.30/0.56 => ! [X3: nat,B2: nat,W2: nat] : % 0.30/0.56 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) ) % 0.30/0.56 = ( ord_less_eq @ nat @ X3 @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_power_le_of_nat_cancel_iff % 0.30/0.56 thf(fact_2699_of__nat__le__of__nat__power__cancel__iff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linordered_semidom @ A ) % 0.30/0.56 => ! [B2: nat,W2: nat,X3: nat] : % 0.30/0.56 ( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X3 ) ) % 0.30/0.56 = ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X3 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_le_of_nat_power_cancel_iff % 0.30/0.56 thf(fact_2700_of__nat__zero__less__power__iff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linordered_semidom @ A ) % 0.30/0.56 => ! [X3: nat,N2: nat] : % 0.30/0.56 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X3 ) @ N2 ) ) % 0.30/0.56 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 ) % 0.30/0.56 | ( N2 % 0.30/0.56 = ( zero_zero @ nat ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_zero_less_power_iff % 0.30/0.56 thf(fact_2701_numeral__power__less__of__nat__cancel__iff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linordered_semidom @ A ) % 0.30/0.56 => ! [I: num,N2: nat,X3: nat] : % 0.30/0.56 ( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) @ ( semiring_1_of_nat @ A @ X3 ) ) % 0.30/0.56 = ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) @ X3 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % numeral_power_less_of_nat_cancel_iff % 0.30/0.56 thf(fact_2702_of__nat__less__numeral__power__cancel__iff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linordered_semidom @ A ) % 0.30/0.56 => ! [X3: nat,I: num,N2: nat] : % 0.30/0.56 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) ) % 0.30/0.56 = ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_less_numeral_power_cancel_iff % 0.30/0.56 thf(fact_2703_of__nat__le__numeral__power__cancel__iff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linordered_semidom @ A ) % 0.30/0.56 => ! [X3: nat,I: num,N2: nat] : % 0.30/0.56 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) ) % 0.30/0.56 = ( ord_less_eq @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_le_numeral_power_cancel_iff % 0.30/0.56 thf(fact_2704_numeral__power__le__of__nat__cancel__iff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linordered_semidom @ A ) % 0.30/0.56 => ! [I: num,N2: nat,X3: nat] : % 0.30/0.56 ( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) @ ( semiring_1_of_nat @ A @ X3 ) ) % 0.30/0.56 = ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) @ X3 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % numeral_power_le_of_nat_cancel_iff % 0.30/0.56 thf(fact_2705_prod_Odisc__eq__case,axiom, % 0.30/0.56 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] : % 0.30/0.56 ( product_case_prod @ A @ B @ $o % 0.30/0.56 @ ^ [Uu3: A,Uv3: B] : $true % 0.30/0.56 @ Prod ) ). % 0.30/0.56 % 0.30/0.56 % prod.disc_eq_case % 0.30/0.56 thf(fact_2706_int__sum,axiom, % 0.30/0.56 ! [B: $tType,F2: B > nat,A3: set @ B] : % 0.30/0.56 ( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A3 ) ) % 0.30/0.56 = ( groups7311177749621191930dd_sum @ B @ int % 0.30/0.56 @ ^ [X: B] : ( semiring_1_of_nat @ int @ ( F2 @ X ) ) % 0.30/0.56 @ A3 ) ) ). % 0.30/0.56 % 0.30/0.56 % int_sum % 0.30/0.56 thf(fact_2707_real__arch__simple,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( archim462609752435547400_field @ A ) % 0.30/0.56 => ! [X3: A] : % 0.30/0.56 ? [N3: nat] : ( ord_less_eq @ A @ X3 @ ( semiring_1_of_nat @ A @ N3 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % real_arch_simple % 0.30/0.56 thf(fact_2708_reals__Archimedean2,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( archim462609752435547400_field @ A ) % 0.30/0.56 => ! [X3: A] : % 0.30/0.56 ? [N3: nat] : ( ord_less @ A @ X3 @ ( semiring_1_of_nat @ A @ N3 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % reals_Archimedean2 % 0.30/0.56 thf(fact_2709_mult__of__nat__commute,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( semiring_1 @ A ) % 0.30/0.56 => ! [X3: nat,Y: A] : % 0.30/0.56 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ X3 ) @ Y ) % 0.30/0.56 = ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X3 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % mult_of_nat_commute % 0.30/0.56 thf(fact_2710_int__cases2,axiom, % 0.30/0.56 ! [Z2: int] : % 0.30/0.56 ( ! [N3: nat] : % 0.30/0.56 ( Z2 % 0.30/0.56 != ( semiring_1_of_nat @ int @ N3 ) ) % 0.30/0.56 => ~ ! [N3: nat] : % 0.30/0.56 ( Z2 % 0.30/0.56 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % int_cases2 % 0.30/0.56 thf(fact_2711_arctan__less__iff,axiom, % 0.30/0.56 ! [X3: real,Y: real] : % 0.30/0.56 ( ( ord_less @ real @ ( arctan @ X3 ) @ ( arctan @ Y ) ) % 0.30/0.56 = ( ord_less @ real @ X3 @ Y ) ) ). % 0.30/0.56 % 0.30/0.56 % arctan_less_iff % 0.30/0.56 thf(fact_2712_arctan__monotone,axiom, % 0.30/0.56 ! [X3: real,Y: real] : % 0.30/0.56 ( ( ord_less @ real @ X3 @ Y ) % 0.30/0.56 => ( ord_less @ real @ ( arctan @ X3 ) @ ( arctan @ Y ) ) ) ). % 0.30/0.56 % 0.30/0.56 % arctan_monotone % 0.30/0.56 thf(fact_2713_int__diff__cases,axiom, % 0.30/0.56 ! [Z2: int] : % 0.30/0.56 ~ ! [M3: nat,N3: nat] : % 0.30/0.56 ( Z2 % 0.30/0.56 != ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M3 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % int_diff_cases % 0.30/0.56 thf(fact_2714_of__nat__less__of__int__iff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linordered_idom @ A ) % 0.30/0.56 => ! [N2: nat,X3: int] : % 0.30/0.56 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( ring_1_of_int @ A @ X3 ) ) % 0.30/0.56 = ( ord_less @ int @ ( semiring_1_of_nat @ int @ N2 ) @ X3 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_less_of_int_iff % 0.30/0.56 thf(fact_2715_of__nat__0__le__iff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linord181362715937106298miring @ A ) % 0.30/0.56 => ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_0_le_iff % 0.30/0.56 thf(fact_2716_of__nat__less__0__iff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linord181362715937106298miring @ A ) % 0.30/0.56 => ! [M: nat] : % 0.30/0.56 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_less_0_iff % 0.30/0.56 thf(fact_2717_of__nat__neq__0,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( semiring_char_0 @ A ) % 0.30/0.56 => ! [N2: nat] : % 0.30/0.56 ( ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) % 0.30/0.56 != ( zero_zero @ A ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_neq_0 % 0.30/0.56 thf(fact_2718_of__nat__less__imp__less,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linord181362715937106298miring @ A ) % 0.30/0.56 => ! [M: nat,N2: nat] : % 0.30/0.56 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) % 0.30/0.56 => ( ord_less @ nat @ M @ N2 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_less_imp_less % 0.30/0.56 thf(fact_2719_less__imp__of__nat__less,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linord181362715937106298miring @ A ) % 0.30/0.56 => ! [M: nat,N2: nat] : % 0.30/0.56 ( ( ord_less @ nat @ M @ N2 ) % 0.30/0.56 => ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % less_imp_of_nat_less % 0.30/0.56 thf(fact_2720_of__nat__mono,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linord181362715937106298miring @ A ) % 0.30/0.56 => ! [I: nat,J: nat] : % 0.30/0.56 ( ( ord_less_eq @ nat @ I @ J ) % 0.30/0.56 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_mono % 0.30/0.56 thf(fact_2721_int__ops_I1_J,axiom, % 0.30/0.56 ( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) ) % 0.30/0.56 = ( zero_zero @ int ) ) ). % 0.30/0.56 % 0.30/0.56 % int_ops(1) % 0.30/0.56 thf(fact_2722_nat__int__comparison_I2_J,axiom, % 0.30/0.56 ( ( ord_less @ nat ) % 0.30/0.56 = ( ^ [A4: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % nat_int_comparison(2) % 0.30/0.56 thf(fact_2723_int__of__nat__induct,axiom, % 0.30/0.56 ! [P: int > $o,Z2: int] : % 0.30/0.56 ( ! [N3: nat] : ( P @ ( semiring_1_of_nat @ int @ N3 ) ) % 0.30/0.56 => ( ! [N3: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) % 0.30/0.56 => ( P @ Z2 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % int_of_nat_induct % 0.30/0.56 thf(fact_2724_int__cases,axiom, % 0.30/0.56 ! [Z2: int] : % 0.30/0.56 ( ! [N3: nat] : % 0.30/0.56 ( Z2 % 0.30/0.56 != ( semiring_1_of_nat @ int @ N3 ) ) % 0.30/0.56 => ~ ! [N3: nat] : % 0.30/0.56 ( Z2 % 0.30/0.56 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % int_cases % 0.30/0.56 thf(fact_2725_zle__int,axiom, % 0.30/0.56 ! [M: nat,N2: nat] : % 0.30/0.56 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N2 ) ) % 0.30/0.56 = ( ord_less_eq @ nat @ M @ N2 ) ) ). % 0.30/0.56 % 0.30/0.56 % zle_int % 0.30/0.56 thf(fact_2726_nat__int__comparison_I3_J,axiom, % 0.30/0.56 ( ( ord_less_eq @ nat ) % 0.30/0.56 = ( ^ [A4: nat,B4: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % nat_int_comparison(3) % 0.30/0.56 thf(fact_2727_zero__le__imp__eq__int,axiom, % 0.30/0.56 ! [K: int] : % 0.30/0.56 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.56 => ? [N3: nat] : % 0.30/0.56 ( K % 0.30/0.56 = ( semiring_1_of_nat @ int @ N3 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % zero_le_imp_eq_int % 0.30/0.56 thf(fact_2728_nonneg__int__cases,axiom, % 0.30/0.56 ! [K: int] : % 0.30/0.56 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.56 => ~ ! [N3: nat] : % 0.30/0.56 ( K % 0.30/0.56 != ( semiring_1_of_nat @ int @ N3 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % nonneg_int_cases % 0.30/0.56 thf(fact_2729_zadd__int__left,axiom, % 0.30/0.56 ! [M: nat,N2: nat,Z2: int] : % 0.30/0.56 ( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ Z2 ) ) % 0.30/0.56 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N2 ) ) @ Z2 ) ) ). % 0.30/0.56 % 0.30/0.56 % zadd_int_left % 0.30/0.56 thf(fact_2730_int__ops_I5_J,axiom, % 0.30/0.56 ! [A2: nat,B2: nat] : % 0.30/0.56 ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A2 @ B2 ) ) % 0.30/0.56 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % int_ops(5) % 0.30/0.56 thf(fact_2731_int__plus,axiom, % 0.30/0.56 ! [N2: nat,M: nat] : % 0.30/0.56 ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N2 @ M ) ) % 0.30/0.56 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( semiring_1_of_nat @ int @ M ) ) ) ). % 0.30/0.56 % 0.30/0.56 % int_plus % 0.30/0.56 thf(fact_2732_zle__iff__zadd,axiom, % 0.30/0.56 ( ( ord_less_eq @ int ) % 0.30/0.56 = ( ^ [W3: int,Z6: int] : % 0.30/0.56 ? [N: nat] : % 0.30/0.56 ( Z6 % 0.30/0.56 = ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ N ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % zle_iff_zadd % 0.30/0.56 thf(fact_2733_not__int__zless__negative,axiom, % 0.30/0.56 ! [N2: nat,M: nat] : % 0.30/0.56 ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) ) ). % 0.30/0.56 % 0.30/0.56 % not_int_zless_negative % 0.30/0.56 thf(fact_2734_nat__less__as__int,axiom, % 0.30/0.56 ( ( ord_less @ nat ) % 0.30/0.56 = ( ^ [A4: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % nat_less_as_int % 0.30/0.56 thf(fact_2735_nat__leq__as__int,axiom, % 0.30/0.56 ( ( ord_less_eq @ nat ) % 0.30/0.56 = ( ^ [A4: nat,B4: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % nat_leq_as_int % 0.30/0.56 thf(fact_2736_ex__less__of__nat__mult,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( archim462609752435547400_field @ A ) % 0.30/0.56 => ! [X3: A,Y: A] : % 0.30/0.56 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 0.30/0.56 => ? [N3: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ X3 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % ex_less_of_nat_mult % 0.30/0.56 thf(fact_2737_of__nat__diff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( semiring_1_cancel @ A ) % 0.30/0.56 => ! [N2: nat,M: nat] : % 0.30/0.56 ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.56 => ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N2 ) ) % 0.30/0.56 = ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_diff % 0.30/0.56 thf(fact_2738_reals__Archimedean3,axiom, % 0.30/0.56 ! [X3: real] : % 0.30/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.56 => ! [Y5: real] : % 0.30/0.56 ? [N3: nat] : ( ord_less @ real @ Y5 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ X3 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % reals_Archimedean3 % 0.30/0.56 thf(fact_2739_int__cases4,axiom, % 0.30/0.56 ! [M: int] : % 0.30/0.56 ( ! [N3: nat] : % 0.30/0.56 ( M % 0.30/0.56 != ( semiring_1_of_nat @ int @ N3 ) ) % 0.30/0.56 => ~ ! [N3: nat] : % 0.30/0.56 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 0.30/0.56 => ( M % 0.30/0.56 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % int_cases4 % 0.30/0.56 thf(fact_2740_int__zle__neg,axiom, % 0.30/0.56 ! [N2: nat,M: nat] : % 0.30/0.56 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) ) % 0.30/0.56 = ( ( N2 % 0.30/0.56 = ( zero_zero @ nat ) ) % 0.30/0.56 & ( M % 0.30/0.56 = ( zero_zero @ nat ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % int_zle_neg % 0.30/0.56 thf(fact_2741_int__ops_I4_J,axiom, % 0.30/0.56 ! [A2: nat] : % 0.30/0.56 ( ( semiring_1_of_nat @ int @ ( suc @ A2 ) ) % 0.30/0.56 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( one_one @ int ) ) ) ). % 0.30/0.56 % 0.30/0.56 % int_ops(4) % 0.30/0.56 thf(fact_2742_int__Suc,axiom, % 0.30/0.56 ! [N2: nat] : % 0.30/0.56 ( ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) % 0.30/0.56 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( one_one @ int ) ) ) ). % 0.30/0.56 % 0.30/0.56 % int_Suc % 0.30/0.56 thf(fact_2743_zless__iff__Suc__zadd,axiom, % 0.30/0.56 ( ( ord_less @ int ) % 0.30/0.56 = ( ^ [W3: int,Z6: int] : % 0.30/0.56 ? [N: nat] : % 0.30/0.56 ( Z6 % 0.30/0.56 = ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % zless_iff_Suc_zadd % 0.30/0.56 thf(fact_2744_negative__zle__0,axiom, % 0.30/0.56 ! [N2: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( zero_zero @ int ) ) ). % 0.30/0.56 % 0.30/0.56 % negative_zle_0 % 0.30/0.56 thf(fact_2745_nonpos__int__cases,axiom, % 0.30/0.56 ! [K: int] : % 0.30/0.56 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) ) % 0.30/0.56 => ~ ! [N3: nat] : % 0.30/0.56 ( K % 0.30/0.56 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % nonpos_int_cases % 0.30/0.56 thf(fact_2746_sum__roots__unity,axiom, % 0.30/0.56 ! [N2: nat] : % 0.30/0.56 ( ( ord_less @ nat @ ( one_one @ nat ) @ N2 ) % 0.30/0.56 => ( ( groups7311177749621191930dd_sum @ complex @ complex % 0.30/0.56 @ ^ [X: complex] : X % 0.30/0.56 @ ( collect @ complex % 0.30/0.56 @ ^ [Z6: complex] : % 0.30/0.56 ( ( power_power @ complex @ Z6 @ N2 ) % 0.30/0.56 = ( one_one @ complex ) ) ) ) % 0.30/0.56 = ( zero_zero @ complex ) ) ) ). % 0.30/0.56 % 0.30/0.56 % sum_roots_unity % 0.30/0.56 thf(fact_2747_sum__nth__roots,axiom, % 0.30/0.56 ! [N2: nat,C2: complex] : % 0.30/0.56 ( ( ord_less @ nat @ ( one_one @ nat ) @ N2 ) % 0.30/0.56 => ( ( groups7311177749621191930dd_sum @ complex @ complex % 0.30/0.56 @ ^ [X: complex] : X % 0.30/0.56 @ ( collect @ complex % 0.30/0.56 @ ^ [Z6: complex] : % 0.30/0.56 ( ( power_power @ complex @ Z6 @ N2 ) % 0.30/0.56 = C2 ) ) ) % 0.30/0.56 = ( zero_zero @ complex ) ) ) ). % 0.30/0.56 % 0.30/0.56 % sum_nth_roots % 0.30/0.56 thf(fact_2748_mod__mult2__eq_H,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( euclid5411537665997757685th_nat @ A ) % 0.30/0.56 => ! [A2: A,M: nat,N2: nat] : % 0.30/0.56 ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) % 0.30/0.56 = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) @ ( modulo_modulo @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % mod_mult2_eq' % 0.30/0.56 thf(fact_2749_pos__int__cases,axiom, % 0.30/0.56 ! [K: int] : % 0.30/0.56 ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.56 => ~ ! [N3: nat] : % 0.30/0.56 ( ( K % 0.30/0.56 = ( semiring_1_of_nat @ int @ N3 ) ) % 0.30/0.56 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % pos_int_cases % 0.30/0.56 thf(fact_2750_zero__less__imp__eq__int,axiom, % 0.30/0.56 ! [K: int] : % 0.30/0.56 ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 0.30/0.56 => ? [N3: nat] : % 0.30/0.56 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 0.30/0.56 & ( K % 0.30/0.56 = ( semiring_1_of_nat @ int @ N3 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % zero_less_imp_eq_int % 0.30/0.56 thf(fact_2751_int__cases3,axiom, % 0.30/0.56 ! [K: int] : % 0.30/0.56 ( ( K % 0.30/0.56 != ( zero_zero @ int ) ) % 0.30/0.56 => ( ! [N3: nat] : % 0.30/0.56 ( ( K % 0.30/0.56 = ( semiring_1_of_nat @ int @ N3 ) ) % 0.30/0.56 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) % 0.30/0.56 => ~ ! [N3: nat] : % 0.30/0.56 ( ( K % 0.30/0.56 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) % 0.30/0.56 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % int_cases3 % 0.30/0.56 thf(fact_2752_nat__less__real__le,axiom, % 0.30/0.56 ( ( ord_less @ nat ) % 0.30/0.56 = ( ^ [N: nat,M2: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % nat_less_real_le % 0.30/0.56 thf(fact_2753_nat__le__real__less,axiom, % 0.30/0.56 ( ( ord_less_eq @ nat ) % 0.30/0.56 = ( ^ [N: nat,M2: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( one_one @ real ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % nat_le_real_less % 0.30/0.56 thf(fact_2754_zmult__zless__mono2__lemma,axiom, % 0.30/0.56 ! [I: int,J: int,K: nat] : % 0.30/0.56 ( ( ord_less @ int @ I @ J ) % 0.30/0.56 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.30/0.56 => ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ I ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ J ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % zmult_zless_mono2_lemma % 0.30/0.56 thf(fact_2755_not__zle__0__negative,axiom, % 0.30/0.56 ! [N2: nat] : % 0.30/0.56 ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % not_zle_0_negative % 0.30/0.56 thf(fact_2756_negative__zless__0,axiom, % 0.30/0.56 ! [N2: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) @ ( zero_zero @ int ) ) ). % 0.30/0.56 % 0.30/0.56 % negative_zless_0 % 0.30/0.56 thf(fact_2757_negD,axiom, % 0.30/0.56 ! [X3: int] : % 0.30/0.56 ( ( ord_less @ int @ X3 @ ( zero_zero @ int ) ) % 0.30/0.56 => ? [N3: nat] : % 0.30/0.56 ( X3 % 0.30/0.56 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % negD % 0.30/0.56 thf(fact_2758_int__ops_I6_J,axiom, % 0.30/0.56 ! [A2: nat,B2: nat] : % 0.30/0.56 ( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) % 0.30/0.56 => ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) ) % 0.30/0.56 = ( zero_zero @ int ) ) ) % 0.30/0.56 & ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) % 0.30/0.56 => ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) ) % 0.30/0.56 = ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % int_ops(6) % 0.30/0.56 thf(fact_2759_nat__approx__posE,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.30/0.56 => ! [E3: A] : % 0.30/0.56 ( ( ord_less @ A @ ( zero_zero @ A ) @ E3 ) % 0.30/0.56 => ~ ! [N3: nat] : % 0.30/0.56 ~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ E3 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % nat_approx_posE % 0.30/0.56 thf(fact_2760_of__nat__less__two__power,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linordered_idom @ A ) % 0.30/0.56 => ! [N2: nat] : ( ord_less @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_less_two_power % 0.30/0.56 thf(fact_2761_inverse__of__nat__le,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linordered_field @ A ) % 0.30/0.56 => ! [N2: nat,M: nat] : % 0.30/0.56 ( ( ord_less_eq @ nat @ N2 @ M ) % 0.30/0.56 => ( ( N2 % 0.30/0.56 != ( zero_zero @ nat ) ) % 0.30/0.56 => ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % inverse_of_nat_le % 0.30/0.56 thf(fact_2762_real__archimedian__rdiv__eq__0,axiom, % 0.30/0.56 ! [X3: real,C2: real] : % 0.30/0.56 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.56 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 ) % 0.30/0.56 => ( ! [M3: nat] : % 0.30/0.56 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 ) % 0.30/0.56 => ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M3 ) @ X3 ) @ C2 ) ) % 0.30/0.56 => ( X3 % 0.30/0.56 = ( zero_zero @ real ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % real_archimedian_rdiv_eq_0 % 0.30/0.56 thf(fact_2763_neg__int__cases,axiom, % 0.30/0.56 ! [K: int] : % 0.30/0.56 ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 0.30/0.56 => ~ ! [N3: nat] : % 0.30/0.56 ( ( K % 0.30/0.56 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) % 0.30/0.56 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % neg_int_cases % 0.30/0.56 thf(fact_2764_zdiff__int__split,axiom, % 0.30/0.56 ! [P: int > $o,X3: nat,Y: nat] : % 0.30/0.56 ( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X3 @ Y ) ) ) % 0.30/0.56 = ( ( ( ord_less_eq @ nat @ Y @ X3 ) % 0.30/0.56 => ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X3 ) @ ( semiring_1_of_nat @ int @ Y ) ) ) ) % 0.30/0.56 & ( ( ord_less @ nat @ X3 @ Y ) % 0.30/0.56 => ( P @ ( zero_zero @ int ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % zdiff_int_split % 0.30/0.56 thf(fact_2765_ln__realpow,axiom, % 0.30/0.56 ! [X3: real,N2: nat] : % 0.30/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.56 => ( ( ln_ln @ real @ ( power_power @ real @ X3 @ N2 ) ) % 0.30/0.56 = ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( ln_ln @ real @ X3 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % ln_realpow % 0.30/0.56 thf(fact_2766_arctan__add,axiom, % 0.30/0.56 ! [X3: real,Y: real] : % 0.30/0.56 ( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 0.30/0.56 => ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) ) % 0.30/0.56 => ( ( plus_plus @ real @ ( arctan @ X3 ) @ ( arctan @ Y ) ) % 0.30/0.56 = ( arctan @ ( divide_divide @ real @ ( plus_plus @ real @ X3 @ Y ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( times_times @ real @ X3 @ Y ) ) ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % arctan_add % 0.30/0.56 thf(fact_2767_double__gauss__sum,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( comm_semiring_1 @ A ) % 0.30/0.56 => ! [N2: nat] : % 0.30/0.56 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) % 0.30/0.56 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % double_gauss_sum % 0.30/0.56 thf(fact_2768_double__arith__series,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( comm_semiring_1 @ A ) % 0.30/0.56 => ! [A2: A,D2: A,N2: nat] : % 0.30/0.56 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) % 0.30/0.56 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.56 @ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D2 ) ) % 0.30/0.56 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) % 0.30/0.56 = ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ D2 ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % double_arith_series % 0.30/0.56 thf(fact_2769_gauss__sum,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( euclid5411537665997757685th_nat @ A ) % 0.30/0.56 => ! [N2: nat] : % 0.30/0.56 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.30/0.56 = ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % gauss_sum % 0.30/0.56 thf(fact_2770_arith__series,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( euclid5411537665997757685th_nat @ A ) % 0.30/0.56 => ! [A2: A,D2: A,N2: nat] : % 0.30/0.56 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.56 @ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D2 ) ) % 0.30/0.56 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.30/0.56 = ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % arith_series % 0.30/0.56 thf(fact_2771_double__gauss__sum__from__Suc__0,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( comm_semiring_1 @ A ) % 0.30/0.56 => ! [N2: nat] : % 0.30/0.56 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) ) % 0.30/0.56 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % double_gauss_sum_from_Suc_0 % 0.30/0.56 thf(fact_2772_sum__gp__offset,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( ( division_ring @ A ) % 0.30/0.56 & ( comm_ring @ A ) ) % 0.30/0.56 => ! [X3: A,M: nat,N2: nat] : % 0.30/0.56 ( ( ( X3 % 0.30/0.56 = ( one_one @ A ) ) % 0.30/0.56 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N2 ) ) ) % 0.30/0.56 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) % 0.30/0.56 & ( ( X3 % 0.30/0.56 != ( one_one @ A ) ) % 0.30/0.56 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N2 ) ) ) % 0.30/0.56 = ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % sum_gp_offset % 0.30/0.56 thf(fact_2773_of__nat__code__if,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( semiring_1 @ A ) % 0.30/0.56 => ( ( semiring_1_of_nat @ A ) % 0.30/0.56 = ( ^ [N: nat] : % 0.30/0.56 ( if @ A % 0.30/0.56 @ ( N % 0.30/0.56 = ( zero_zero @ nat ) ) % 0.30/0.56 @ ( zero_zero @ A ) % 0.30/0.56 @ ( product_case_prod @ nat @ nat @ A % 0.30/0.56 @ ^ [M2: nat,Q5: nat] : % 0.30/0.56 ( if @ A % 0.30/0.56 @ ( Q5 % 0.30/0.56 = ( zero_zero @ nat ) ) % 0.30/0.56 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) ) % 0.30/0.56 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( one_one @ A ) ) ) % 0.30/0.56 @ ( divmod_nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % of_nat_code_if % 0.30/0.56 thf(fact_2774_monoseq__arctan__series,axiom, % 0.30/0.56 ! [X3: real] : % 0.30/0.56 ( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 0.30/0.56 => ( topological_monoseq @ real % 0.30/0.56 @ ^ [N: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % monoseq_arctan_series % 0.30/0.56 thf(fact_2775_lemma__termdiff3,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( real_V3459762299906320749_field @ A ) % 0.30/0.56 => ! [H2: A,Z2: A,K5: real,N2: nat] : % 0.30/0.56 ( ( H2 % 0.30/0.56 != ( zero_zero @ A ) ) % 0.30/0.56 => ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K5 ) % 0.30/0.56 => ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z2 @ H2 ) ) @ K5 ) % 0.30/0.56 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H2 ) @ N2 ) @ ( power_power @ A @ Z2 @ N2 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) @ ( times_times @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( semiring_1_of_nat @ real @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( power_power @ real @ K5 @ ( minus_minus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) ) ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % lemma_termdiff3 % 0.30/0.56 thf(fact_2776_ln__series,axiom, % 0.30/0.56 ! [X3: real] : % 0.30/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.30/0.56 => ( ( ord_less @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) % 0.30/0.56 => ( ( ln_ln @ real @ X3 ) % 0.30/0.56 = ( suminf @ real % 0.30/0.56 @ ^ [N: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X3 @ ( one_one @ real ) ) @ ( suc @ N ) ) ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % ln_series % 0.30/0.56 thf(fact_2777_lemma__termdiff2,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( field @ A ) % 0.30/0.56 => ! [H2: A,Z2: A,N2: nat] : % 0.30/0.56 ( ( H2 % 0.30/0.56 != ( zero_zero @ A ) ) % 0.30/0.56 => ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H2 ) @ N2 ) @ ( power_power @ A @ Z2 @ N2 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) % 0.30/0.56 = ( times_times @ A @ H2 % 0.30/0.56 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.56 @ ^ [P6: nat] : % 0.30/0.56 ( groups7311177749621191930dd_sum @ nat @ A % 0.30/0.56 @ ^ [Q5: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H2 ) @ Q5 ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ ( minus_minus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q5 ) ) ) % 0.30/0.56 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ P6 ) ) ) % 0.30/0.56 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % lemma_termdiff2 % 0.30/0.56 thf(fact_2778_arctan__series,axiom, % 0.30/0.56 ! [X3: real] : % 0.30/0.56 ( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 0.30/0.56 => ( ( arctan @ X3 ) % 0.30/0.56 = ( suminf @ real % 0.30/0.56 @ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % arctan_series % 0.30/0.56 thf(fact_2779_lessThan__iff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( ord @ A ) % 0.30/0.56 => ! [I: A,K: A] : % 0.30/0.56 ( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) ) % 0.30/0.56 = ( ord_less @ A @ I @ K ) ) ) ). % 0.30/0.56 % 0.30/0.56 % lessThan_iff % 0.30/0.56 thf(fact_2780_finite__lessThan,axiom, % 0.30/0.56 ! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_lessThan @ nat @ K ) ) ). % 0.30/0.56 % 0.30/0.56 % finite_lessThan % 0.30/0.56 thf(fact_2781_lessThan__subset__iff,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( linorder @ A ) % 0.30/0.56 => ! [X3: A,Y: A] : % 0.30/0.56 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X3 ) @ ( set_ord_lessThan @ A @ Y ) ) % 0.30/0.56 = ( ord_less_eq @ A @ X3 @ Y ) ) ) ). % 0.30/0.56 % 0.30/0.56 % lessThan_subset_iff % 0.30/0.56 thf(fact_2782_lessThan__0,axiom, % 0.30/0.56 ( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) ) % 0.30/0.56 = ( bot_bot @ ( set @ nat ) ) ) ). % 0.30/0.56 % 0.30/0.56 % lessThan_0 % 0.30/0.56 thf(fact_2783_single__Diff__lessThan,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( preorder @ A ) % 0.30/0.56 => ! [K: A] : % 0.30/0.56 ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K ) ) % 0.30/0.56 = ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % single_Diff_lessThan % 0.30/0.56 thf(fact_2784_sum_OlessThan__Suc,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( comm_monoid_add @ A ) % 0.30/0.56 => ! [G: nat > A,N2: nat] : % 0.30/0.56 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) % 0.30/0.56 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( G @ N2 ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % sum.lessThan_Suc % 0.30/0.56 thf(fact_2785_powser__zero,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( real_V2822296259951069270ebra_1 @ A ) % 0.30/0.56 => ! [F2: nat > A] : % 0.30/0.56 ( ( suminf @ A % 0.30/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) ) % 0.30/0.56 = ( F2 @ ( zero_zero @ nat ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % powser_zero % 0.30/0.56 thf(fact_2786_int__int__eq,axiom, % 0.30/0.56 ! [M: nat,N2: nat] : % 0.30/0.56 ( ( ( semiring_1_of_nat @ int @ M ) % 0.30/0.56 = ( semiring_1_of_nat @ int @ N2 ) ) % 0.30/0.56 = ( M = N2 ) ) ). % 0.30/0.56 % 0.30/0.56 % int_int_eq % 0.30/0.56 thf(fact_2787_lessThan__non__empty,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( no_bot @ A ) % 0.30/0.56 => ! [X3: A] : % 0.30/0.56 ( ( set_ord_lessThan @ A @ X3 ) % 0.30/0.56 != ( bot_bot @ ( set @ A ) ) ) ) ). % 0.30/0.56 % 0.30/0.56 % lessThan_non_empty % 0.30/0.56 thf(fact_2788_infinite__Iio,axiom, % 0.30/0.56 ! [A: $tType] : % 0.30/0.56 ( ( ( linorder @ A ) % 0.30/0.56 & ( no_bot @ A ) ) % 0.30/0.56 => ! [A2: A] : % 0.30/0.56 ~ ( finite_finite2 @ A @ ( set_ord_lessThan @ A @ A2 ) ) ) ). % 0.30/0.56 % 0.30/0.56 % infinite_Iio % 0.30/0.56 thf(fact_2789_lessThan__def,axiom, % 0.30/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ord @ A ) % 0.31/0.56 => ( ( set_ord_lessThan @ A ) % 0.31/0.56 = ( ^ [U2: A] : % 0.31/0.56 ( collect @ A % 0.31/0.56 @ ^ [X: A] : ( ord_less @ A @ X @ U2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % lessThan_def % 0.31/0.56 thf(fact_2790_Iio__eq__empty__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( linorder @ A ) % 0.31/0.56 & ( order_bot @ A ) ) % 0.31/0.56 => ! [N2: A] : % 0.31/0.56 ( ( ( set_ord_lessThan @ A @ N2 ) % 0.31/0.56 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.56 = ( N2 % 0.31/0.56 = ( bot_bot @ A ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % Iio_eq_empty_iff % 0.31/0.56 thf(fact_2791_lessThan__strict__subset__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( linorder @ A ) % 0.31/0.56 => ! [M: A,N2: A] : % 0.31/0.56 ( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N2 ) ) % 0.31/0.56 = ( ord_less @ A @ M @ N2 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % lessThan_strict_subset_iff % 0.31/0.56 thf(fact_2792_lessThan__Suc,axiom, % 0.31/0.56 ! [K: nat] : % 0.31/0.56 ( ( set_ord_lessThan @ nat @ ( suc @ K ) ) % 0.31/0.56 = ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ). % 0.31/0.56 % 0.31/0.56 % lessThan_Suc % 0.31/0.56 thf(fact_2793_lessThan__empty__iff,axiom, % 0.31/0.56 ! [N2: nat] : % 0.31/0.56 ( ( ( set_ord_lessThan @ nat @ N2 ) % 0.31/0.56 = ( bot_bot @ ( set @ nat ) ) ) % 0.31/0.56 = ( N2 % 0.31/0.56 = ( zero_zero @ nat ) ) ) ). % 0.31/0.56 % 0.31/0.56 % lessThan_empty_iff % 0.31/0.56 thf(fact_2794_finite__nat__iff__bounded,axiom, % 0.31/0.56 ( ( finite_finite2 @ nat ) % 0.31/0.56 = ( ^ [S7: set @ nat] : % 0.31/0.56 ? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S7 @ ( set_ord_lessThan @ nat @ K3 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % finite_nat_iff_bounded % 0.31/0.56 thf(fact_2795_finite__nat__bounded,axiom, % 0.31/0.56 ! [S3: set @ nat] : % 0.31/0.56 ( ( finite_finite2 @ nat @ S3 ) % 0.31/0.56 => ? [K2: nat] : ( ord_less_eq @ ( set @ nat ) @ S3 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % finite_nat_bounded % 0.31/0.56 thf(fact_2796_sum_Onat__diff__reindex,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( comm_monoid_add @ A ) % 0.31/0.56 => ! [G: nat > A,N2: nat] : % 0.31/0.56 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.56 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) ) % 0.31/0.56 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.56 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sum.nat_diff_reindex % 0.31/0.56 thf(fact_2797_sum__diff__distrib,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ord @ A ) % 0.31/0.56 => ! [Q: A > nat,P: A > nat,N2: A] : % 0.31/0.56 ( ! [X5: A] : ( ord_less_eq @ nat @ ( Q @ X5 ) @ ( P @ X5 ) ) % 0.31/0.56 => ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N2 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N2 ) ) ) % 0.31/0.56 = ( groups7311177749621191930dd_sum @ A @ nat % 0.31/0.56 @ ^ [X: A] : ( minus_minus @ nat @ ( P @ X ) @ ( Q @ X ) ) % 0.31/0.56 @ ( set_ord_lessThan @ A @ N2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sum_diff_distrib % 0.31/0.56 thf(fact_2798_lemma__NBseq__def2,axiom, % 0.31/0.56 ! [A: $tType,B: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ B ) % 0.31/0.56 => ! [X8: A > B] : % 0.31/0.56 ( ( ? [K6: real] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ K6 ) % 0.31/0.56 & ! [N: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ K6 ) ) ) % 0.31/0.56 = ( ? [N5: nat] : % 0.31/0.56 ! [N: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % lemma_NBseq_def2 % 0.31/0.56 thf(fact_2799_lemma__NBseq__def,axiom, % 0.31/0.56 ! [A: $tType,B: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ B ) % 0.31/0.56 => ! [X8: A > B] : % 0.31/0.56 ( ( ? [K6: real] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ K6 ) % 0.31/0.56 & ! [N: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ K6 ) ) ) % 0.31/0.56 = ( ? [N5: nat] : % 0.31/0.56 ! [N: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % lemma_NBseq_def % 0.31/0.56 thf(fact_2800_sum_OlessThan__Suc__shift,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( comm_monoid_add @ A ) % 0.31/0.56 => ! [G: nat > A,N2: nat] : % 0.31/0.56 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) % 0.31/0.56 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) % 0.31/0.56 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.56 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 0.31/0.56 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sum.lessThan_Suc_shift % 0.31/0.56 thf(fact_2801_sum__lessThan__telescope,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ab_group_add @ A ) % 0.31/0.56 => ! [F2: nat > A,M: nat] : % 0.31/0.56 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.56 @ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) ) % 0.31/0.56 @ ( set_ord_lessThan @ nat @ M ) ) % 0.31/0.56 = ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sum_lessThan_telescope % 0.31/0.56 thf(fact_2802_sum__lessThan__telescope_H,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ab_group_add @ A ) % 0.31/0.56 => ! [F2: nat > A,M: nat] : % 0.31/0.56 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.56 @ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) ) % 0.31/0.56 @ ( set_ord_lessThan @ nat @ M ) ) % 0.31/0.56 = ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ M ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sum_lessThan_telescope' % 0.31/0.56 thf(fact_2803_sum_OatLeast1__atMost__eq,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( comm_monoid_add @ A ) % 0.31/0.56 => ! [G: nat > A,N2: nat] : % 0.31/0.56 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) % 0.31/0.56 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.56 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) ) % 0.31/0.56 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sum.atLeast1_atMost_eq % 0.31/0.56 thf(fact_2804_lemma__termdiff1,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( monoid_mult @ A ) % 0.31/0.56 & ( comm_ring @ A ) ) % 0.31/0.56 => ! [Z2: A,H2: A,M: nat] : % 0.31/0.56 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.56 @ ^ [P6: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H2 ) @ ( minus_minus @ nat @ M @ P6 ) ) @ ( power_power @ A @ Z2 @ P6 ) ) @ ( power_power @ A @ Z2 @ M ) ) % 0.31/0.56 @ ( set_ord_lessThan @ nat @ M ) ) % 0.31/0.56 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.56 @ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ Z2 @ P6 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H2 ) @ ( minus_minus @ nat @ M @ P6 ) ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ M @ P6 ) ) ) ) % 0.31/0.56 @ ( set_ord_lessThan @ nat @ M ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % lemma_termdiff1 % 0.31/0.56 thf(fact_2805_diff__power__eq__sum,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( monoid_mult @ A ) % 0.31/0.56 & ( comm_ring @ A ) ) % 0.31/0.56 => ! [X3: A,N2: nat,Y: A] : % 0.31/0.56 ( ( minus_minus @ A @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) @ ( power_power @ A @ Y @ ( suc @ N2 ) ) ) % 0.31/0.56 = ( times_times @ A @ ( minus_minus @ A @ X3 @ Y ) % 0.31/0.56 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.56 @ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X3 @ P6 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N2 @ P6 ) ) ) % 0.31/0.56 @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % diff_power_eq_sum % 0.31/0.56 thf(fact_2806_power__diff__sumr2,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( monoid_mult @ A ) % 0.31/0.56 & ( comm_ring @ A ) ) % 0.31/0.56 => ! [X3: A,N2: nat,Y: A] : % 0.31/0.56 ( ( minus_minus @ A @ ( power_power @ A @ X3 @ N2 ) @ ( power_power @ A @ Y @ N2 ) ) % 0.31/0.56 = ( times_times @ A @ ( minus_minus @ A @ X3 @ Y ) % 0.31/0.56 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.56 @ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power @ A @ X3 @ I3 ) ) % 0.31/0.56 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % power_diff_sumr2 % 0.31/0.56 thf(fact_2807_real__sum__nat__ivl__bounded2,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( linordered_semidom @ A ) % 0.31/0.56 => ! [N2: nat,F2: nat > A,K5: A,K: nat] : % 0.31/0.56 ( ! [P7: nat] : % 0.31/0.56 ( ( ord_less @ nat @ P7 @ N2 ) % 0.31/0.56 => ( ord_less_eq @ A @ ( F2 @ P7 ) @ K5 ) ) % 0.31/0.56 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K5 ) % 0.31/0.56 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ K5 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % real_sum_nat_ivl_bounded2 % 0.31/0.56 thf(fact_2808_one__diff__power__eq_H,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( monoid_mult @ A ) % 0.31/0.56 & ( comm_ring @ A ) ) % 0.31/0.56 => ! [X3: A,N2: nat] : % 0.31/0.56 ( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ N2 ) ) % 0.31/0.56 = ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) % 0.31/0.56 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.56 @ ^ [I3: nat] : ( power_power @ A @ X3 @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) ) % 0.31/0.56 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % one_diff_power_eq' % 0.31/0.56 thf(fact_2809_sum__split__even__odd,axiom, % 0.31/0.56 ! [F2: nat > real,G: nat > real,N2: nat] : % 0.31/0.56 ( ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.56 @ ^ [I3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( F2 @ I3 ) @ ( G @ I3 ) ) % 0.31/0.56 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 0.31/0.56 = ( plus_plus @ real % 0.31/0.56 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.56 @ ^ [I3: nat] : ( F2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) % 0.31/0.56 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.56 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.56 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( one_one @ nat ) ) ) % 0.31/0.56 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sum_split_even_odd % 0.31/0.56 thf(fact_2810_norm__le__zero__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( zero_zero @ real ) ) % 0.31/0.56 = ( X3 % 0.31/0.56 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % norm_le_zero_iff % 0.31/0.56 thf(fact_2811_zero__less__norm__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) % 0.31/0.56 = ( X3 % 0.31/0.56 != ( zero_zero @ A ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % zero_less_norm_iff % 0.31/0.56 thf(fact_2812_suminf__geometric,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.56 => ! [C2: A] : % 0.31/0.56 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) ) % 0.31/0.56 => ( ( suminf @ A @ ( power_power @ A @ C2 ) ) % 0.31/0.56 = ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % suminf_geometric % 0.31/0.56 thf(fact_2813_norm__zero,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ( ( real_V7770717601297561774m_norm @ A @ ( zero_zero @ A ) ) % 0.31/0.56 = ( zero_zero @ real ) ) ) ). % 0.31/0.56 % 0.31/0.56 % norm_zero % 0.31/0.56 thf(fact_2814_norm__eq__zero,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ( real_V7770717601297561774m_norm @ A @ X3 ) % 0.31/0.56 = ( zero_zero @ real ) ) % 0.31/0.56 = ( X3 % 0.31/0.56 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % norm_eq_zero % 0.31/0.56 thf(fact_2815_suminf__zero,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( comm_monoid_add @ A ) % 0.31/0.56 & ( topological_t2_space @ A ) ) % 0.31/0.56 => ( ( suminf @ A % 0.31/0.56 @ ^ [N: nat] : ( zero_zero @ A ) ) % 0.31/0.56 = ( zero_zero @ A ) ) ) ). % 0.31/0.56 % 0.31/0.56 % suminf_zero % 0.31/0.56 thf(fact_2816_norm__not__less__zero,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ~ ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( zero_zero @ real ) ) ) ). % 0.31/0.56 % 0.31/0.56 % norm_not_less_zero % 0.31/0.56 thf(fact_2817_norm__uminus__minus,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [X3: A,Y: A] : % 0.31/0.56 ( ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) ) % 0.31/0.56 = ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % norm_uminus_minus % 0.31/0.56 thf(fact_2818_nonzero__norm__divide,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.56 => ! [B2: A,A2: A] : % 0.31/0.56 ( ( B2 % 0.31/0.56 != ( zero_zero @ A ) ) % 0.31/0.56 => ( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 0.31/0.56 = ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % nonzero_norm_divide % 0.31/0.56 thf(fact_2819_power__eq__imp__eq__norm,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.56 => ! [W2: A,N2: nat,Z2: A] : % 0.31/0.56 ( ( ( power_power @ A @ W2 @ N2 ) % 0.31/0.56 = ( power_power @ A @ Z2 @ N2 ) ) % 0.31/0.56 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.56 => ( ( real_V7770717601297561774m_norm @ A @ W2 ) % 0.31/0.56 = ( real_V7770717601297561774m_norm @ A @ Z2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % power_eq_imp_eq_norm % 0.31/0.56 thf(fact_2820_norm__mult__less,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V4412858255891104859lgebra @ A ) % 0.31/0.56 => ! [X3: A,R2: real,Y: A,S: real] : % 0.31/0.56 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ R2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S ) % 0.31/0.56 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X3 @ Y ) ) @ ( times_times @ real @ R2 @ S ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % norm_mult_less % 0.31/0.56 thf(fact_2821_norm__triangle__lt,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [X3: A,Y: A,E3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E3 ) % 0.31/0.56 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ E3 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % norm_triangle_lt % 0.31/0.56 thf(fact_2822_norm__add__less,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [X3: A,R2: real,Y: A,S: real] : % 0.31/0.56 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ R2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S ) % 0.31/0.56 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % norm_add_less % 0.31/0.56 thf(fact_2823_norm__triangle__mono,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [A2: A,R2: real,B2: A,S: real] : % 0.31/0.56 ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ R2 ) % 0.31/0.56 => ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ S ) % 0.31/0.56 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % norm_triangle_mono % 0.31/0.56 thf(fact_2824_norm__triangle__ineq,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [X3: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % norm_triangle_ineq % 0.31/0.56 thf(fact_2825_norm__triangle__le,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [X3: A,Y: A,E3: real] : % 0.31/0.56 ( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E3 ) % 0.31/0.56 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ E3 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % norm_triangle_le % 0.31/0.56 thf(fact_2826_norm__add__leD,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [A2: A,B2: A,C2: real] : % 0.31/0.56 ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ C2 ) % 0.31/0.56 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ C2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % norm_add_leD % 0.31/0.56 thf(fact_2827_norm__diff__triangle__less,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [X3: A,Y: A,E1: real,Z2: A,E22: real] : % 0.31/0.56 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ Y ) ) @ E1 ) % 0.31/0.56 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ Z2 ) ) @ E22 ) % 0.31/0.56 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ Z2 ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % norm_diff_triangle_less % 0.31/0.56 thf(fact_2828_norm__diff__ineq,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % norm_diff_ineq % 0.31/0.56 thf(fact_2829_suminf__finite,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( comm_monoid_add @ A ) % 0.31/0.56 & ( topological_t2_space @ A ) ) % 0.31/0.56 => ! [N7: set @ nat,F2: nat > A] : % 0.31/0.56 ( ( finite_finite2 @ nat @ N7 ) % 0.31/0.56 => ( ! [N3: nat] : % 0.31/0.56 ( ~ ( member @ nat @ N3 @ N7 ) % 0.31/0.56 => ( ( F2 @ N3 ) % 0.31/0.56 = ( zero_zero @ A ) ) ) % 0.31/0.56 => ( ( suminf @ A @ F2 ) % 0.31/0.56 = ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N7 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % suminf_finite % 0.31/0.56 thf(fact_2830_power__eq__1__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.56 => ! [W2: A,N2: nat] : % 0.31/0.56 ( ( ( power_power @ A @ W2 @ N2 ) % 0.31/0.56 = ( one_one @ A ) ) % 0.31/0.56 => ( ( ( real_V7770717601297561774m_norm @ A @ W2 ) % 0.31/0.56 = ( one_one @ real ) ) % 0.31/0.56 | ( N2 % 0.31/0.56 = ( zero_zero @ nat ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % power_eq_1_iff % 0.31/0.56 thf(fact_2831_norm__diff__triangle__ineq,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [A2: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % norm_diff_triangle_ineq % 0.31/0.56 thf(fact_2832_sum__bounds__lt__plus1,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( comm_monoid_add @ A ) % 0.31/0.56 => ! [F2: nat > A,Mm: nat] : % 0.31/0.56 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.56 @ ^ [K3: nat] : ( F2 @ ( suc @ K3 ) ) % 0.31/0.56 @ ( set_ord_lessThan @ nat @ Mm ) ) % 0.31/0.56 = ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sum_bounds_lt_plus1 % 0.31/0.56 thf(fact_2833_sumr__cos__zero__one,axiom, % 0.31/0.56 ! [N2: nat] : % 0.31/0.56 ( ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.56 @ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ ( zero_zero @ real ) @ M2 ) ) % 0.31/0.56 @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) % 0.31/0.56 = ( one_one @ real ) ) ). % 0.31/0.56 % 0.31/0.56 % sumr_cos_zero_one % 0.31/0.56 thf(fact_2834_pi__series,axiom, % 0.31/0.56 ( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) % 0.31/0.56 = ( suminf @ real % 0.31/0.56 @ ^ [K3: nat] : ( divide_divide @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % pi_series % 0.31/0.56 thf(fact_2835_summable__arctan__series,axiom, % 0.31/0.56 ! [X3: real] : % 0.31/0.56 ( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 0.31/0.56 => ( summable @ real % 0.31/0.56 @ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_arctan_series % 0.31/0.56 thf(fact_2836_ceiling__log__nat__eq__powr__iff,axiom, % 0.31/0.56 ! [B2: nat,K: nat,N2: nat] : % 0.31/0.56 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) % 0.31/0.56 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.31/0.56 => ( ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) ) % 0.31/0.56 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( one_one @ int ) ) ) % 0.31/0.56 = ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K ) % 0.31/0.56 & ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % ceiling_log_nat_eq_powr_iff % 0.31/0.56 thf(fact_2837_geometric__deriv__sums,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.56 => ! [Z2: A] : % 0.31/0.56 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( one_one @ real ) ) % 0.31/0.56 => ( sums @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( power_power @ A @ Z2 @ N ) ) % 0.31/0.56 @ ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( minus_minus @ A @ ( one_one @ A ) @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % geometric_deriv_sums % 0.31/0.56 thf(fact_2838_summable__zero,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( comm_monoid_add @ A ) % 0.31/0.56 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.56 => ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( zero_zero @ A ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_zero % 0.31/0.56 thf(fact_2839_summable__single,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( comm_monoid_add @ A ) % 0.31/0.56 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.56 => ! [I: nat,F2: nat > A] : % 0.31/0.56 ( summable @ A % 0.31/0.56 @ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_single % 0.31/0.56 thf(fact_2840_sums__zero,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( comm_monoid_add @ A ) % 0.31/0.56 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.56 => ( sums @ A % 0.31/0.56 @ ^ [N: nat] : ( zero_zero @ A ) % 0.31/0.56 @ ( zero_zero @ A ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_zero % 0.31/0.56 thf(fact_2841_summable__iff__shift,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [F2: nat > A,K: nat] : % 0.31/0.56 ( ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) ) % 0.31/0.56 = ( summable @ A @ F2 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_iff_shift % 0.31/0.56 thf(fact_2842_cos__coeff__0,axiom, % 0.31/0.56 ( ( cos_coeff @ ( zero_zero @ nat ) ) % 0.31/0.56 = ( one_one @ real ) ) ). % 0.31/0.56 % 0.31/0.56 % cos_coeff_0 % 0.31/0.56 thf(fact_2843_summable__cmult__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.56 => ! [C2: A,F2: nat > A] : % 0.31/0.56 ( ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) ) ) % 0.31/0.56 = ( ( C2 % 0.31/0.56 = ( zero_zero @ A ) ) % 0.31/0.56 | ( summable @ A @ F2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_cmult_iff % 0.31/0.56 thf(fact_2844_summable__divide__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.56 => ! [F2: nat > A,C2: A] : % 0.31/0.56 ( ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( divide_divide @ A @ ( F2 @ N ) @ C2 ) ) % 0.31/0.56 = ( ( C2 % 0.31/0.56 = ( zero_zero @ A ) ) % 0.31/0.56 | ( summable @ A @ F2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_divide_iff % 0.31/0.56 thf(fact_2845_summable__If__finite,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( comm_monoid_add @ A ) % 0.31/0.56 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.56 => ! [P: nat > $o,F2: nat > A] : % 0.31/0.56 ( ( finite_finite2 @ nat @ ( collect @ nat @ P ) ) % 0.31/0.56 => ( summable @ A % 0.31/0.56 @ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_If_finite % 0.31/0.56 thf(fact_2846_summable__If__finite__set,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( comm_monoid_add @ A ) % 0.31/0.56 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.56 => ! [A3: set @ nat,F2: nat > A] : % 0.31/0.56 ( ( finite_finite2 @ nat @ A3 ) % 0.31/0.56 => ( summable @ A % 0.31/0.56 @ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A3 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_If_finite_set % 0.31/0.56 thf(fact_2847_log__eq__one,axiom, % 0.31/0.56 ! [A2: real] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.56 => ( ( A2 % 0.31/0.56 != ( one_one @ real ) ) % 0.31/0.56 => ( ( log @ A2 @ A2 ) % 0.31/0.56 = ( one_one @ real ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_eq_one % 0.31/0.56 thf(fact_2848_log__less__cancel__iff,axiom, % 0.31/0.56 ! [A2: real,X3: real,Y: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 0.31/0.56 => ( ( ord_less @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y ) ) % 0.31/0.56 = ( ord_less @ real @ X3 @ Y ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_less_cancel_iff % 0.31/0.56 thf(fact_2849_log__less__one__cancel__iff,axiom, % 0.31/0.56 ! [A2: real,X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ ( log @ A2 @ X3 ) @ ( one_one @ real ) ) % 0.31/0.56 = ( ord_less @ real @ X3 @ A2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_less_one_cancel_iff % 0.31/0.56 thf(fact_2850_one__less__log__cancel__iff,axiom, % 0.31/0.56 ! [A2: real,X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ ( one_one @ real ) @ ( log @ A2 @ X3 ) ) % 0.31/0.56 = ( ord_less @ real @ A2 @ X3 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % one_less_log_cancel_iff % 0.31/0.56 thf(fact_2851_log__less__zero__cancel__iff,axiom, % 0.31/0.56 ! [A2: real,X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ ( log @ A2 @ X3 ) @ ( zero_zero @ real ) ) % 0.31/0.56 = ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_less_zero_cancel_iff % 0.31/0.56 thf(fact_2852_zero__less__log__cancel__iff,axiom, % 0.31/0.56 ! [A2: real,X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X3 ) ) % 0.31/0.56 = ( ord_less @ real @ ( one_one @ real ) @ X3 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % zero_less_log_cancel_iff % 0.31/0.56 thf(fact_2853_log__le__cancel__iff,axiom, % 0.31/0.56 ! [A2: real,X3: real,Y: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 0.31/0.56 => ( ( ord_less_eq @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y ) ) % 0.31/0.56 = ( ord_less_eq @ real @ X3 @ Y ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_le_cancel_iff % 0.31/0.56 thf(fact_2854_log__le__one__cancel__iff,axiom, % 0.31/0.56 ! [A2: real,X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less_eq @ real @ ( log @ A2 @ X3 ) @ ( one_one @ real ) ) % 0.31/0.56 = ( ord_less_eq @ real @ X3 @ A2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_le_one_cancel_iff % 0.31/0.56 thf(fact_2855_one__le__log__cancel__iff,axiom, % 0.31/0.56 ! [A2: real,X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( log @ A2 @ X3 ) ) % 0.31/0.56 = ( ord_less_eq @ real @ A2 @ X3 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % one_le_log_cancel_iff % 0.31/0.56 thf(fact_2856_log__le__zero__cancel__iff,axiom, % 0.31/0.56 ! [A2: real,X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less_eq @ real @ ( log @ A2 @ X3 ) @ ( zero_zero @ real ) ) % 0.31/0.56 = ( ord_less_eq @ real @ X3 @ ( one_one @ real ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_le_zero_cancel_iff % 0.31/0.56 thf(fact_2857_zero__le__log__cancel__iff,axiom, % 0.31/0.56 ! [A2: real,X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X3 ) ) % 0.31/0.56 = ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % zero_le_log_cancel_iff % 0.31/0.56 thf(fact_2858_powser__sums__zero__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.56 => ! [A2: nat > A,X3: A] : % 0.31/0.56 ( ( sums @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( A2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) % 0.31/0.56 @ X3 ) % 0.31/0.56 = ( ( A2 @ ( zero_zero @ nat ) ) % 0.31/0.56 = X3 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powser_sums_zero_iff % 0.31/0.56 thf(fact_2859_summable__geometric__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.56 => ! [C2: A] : % 0.31/0.56 ( ( summable @ A @ ( power_power @ A @ C2 ) ) % 0.31/0.56 = ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_geometric_iff % 0.31/0.56 thf(fact_2860_log__pow__cancel,axiom, % 0.31/0.56 ! [A2: real,B2: nat] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.56 => ( ( A2 % 0.31/0.56 != ( one_one @ real ) ) % 0.31/0.56 => ( ( log @ A2 @ ( power_power @ real @ A2 @ B2 ) ) % 0.31/0.56 = ( semiring_1_of_nat @ real @ B2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_pow_cancel % 0.31/0.56 thf(fact_2861_sums__0,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( comm_monoid_add @ A ) % 0.31/0.56 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.56 => ! [F2: nat > A] : % 0.31/0.56 ( ! [N3: nat] : % 0.31/0.56 ( ( F2 @ N3 ) % 0.31/0.56 = ( zero_zero @ A ) ) % 0.31/0.56 => ( sums @ A @ F2 @ ( zero_zero @ A ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_0 % 0.31/0.56 thf(fact_2862_sums__le,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( ordere6911136660526730532id_add @ A ) % 0.31/0.56 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.56 => ! [F2: nat > A,G: nat > A,S: A,T2: A] : % 0.31/0.56 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ N3 ) ) % 0.31/0.56 => ( ( sums @ A @ F2 @ S ) % 0.31/0.56 => ( ( sums @ A @ G @ T2 ) % 0.31/0.56 => ( ord_less_eq @ A @ S @ T2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_le % 0.31/0.56 thf(fact_2863_sums__single,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( comm_monoid_add @ A ) % 0.31/0.56 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.56 => ! [I: nat,F2: nat > A] : % 0.31/0.56 ( sums @ A % 0.31/0.56 @ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) % 0.31/0.56 @ ( F2 @ I ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_single % 0.31/0.56 thf(fact_2864_sums__add,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( topolo5987344860129210374id_add @ A ) % 0.31/0.56 & ( topological_t2_space @ A ) ) % 0.31/0.56 => ! [F2: nat > A,A2: A,G: nat > A,B2: A] : % 0.31/0.56 ( ( sums @ A @ F2 @ A2 ) % 0.31/0.56 => ( ( sums @ A @ G @ B2 ) % 0.31/0.56 => ( sums @ A % 0.31/0.56 @ ^ [N: nat] : ( plus_plus @ A @ ( F2 @ N ) @ ( G @ N ) ) % 0.31/0.56 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_add % 0.31/0.56 thf(fact_2865_summable__comparison__test,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_Vector_banach @ A ) % 0.31/0.56 => ! [F2: nat > A,G: nat > real] : % 0.31/0.56 ( ? [N8: nat] : % 0.31/0.56 ! [N3: nat] : % 0.31/0.56 ( ( ord_less_eq @ nat @ N8 @ N3 ) % 0.31/0.56 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) ) % 0.31/0.56 => ( ( summable @ real @ G ) % 0.31/0.56 => ( summable @ A @ F2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_comparison_test % 0.31/0.56 thf(fact_2866_summable__comparison__test_H,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_Vector_banach @ A ) % 0.31/0.56 => ! [G: nat > real,N7: nat,F2: nat > A] : % 0.31/0.56 ( ( summable @ real @ G ) % 0.31/0.56 => ( ! [N3: nat] : % 0.31/0.56 ( ( ord_less_eq @ nat @ N7 @ N3 ) % 0.31/0.56 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) ) % 0.31/0.56 => ( summable @ A @ F2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_comparison_test' % 0.31/0.56 thf(fact_2867_summable__const__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [C2: A] : % 0.31/0.56 ( ( summable @ A % 0.31/0.56 @ ^ [Uu3: nat] : C2 ) % 0.31/0.56 = ( C2 % 0.31/0.56 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_const_iff % 0.31/0.56 thf(fact_2868_summable__add,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( topolo5987344860129210374id_add @ A ) % 0.31/0.56 & ( topological_t2_space @ A ) ) % 0.31/0.56 => ! [F2: nat > A,G: nat > A] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ( summable @ A @ G ) % 0.31/0.56 => ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( plus_plus @ A @ ( F2 @ N ) @ ( G @ N ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_add % 0.31/0.56 thf(fact_2869_summable__Suc__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [F2: nat > A] : % 0.31/0.56 ( ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( F2 @ ( suc @ N ) ) ) % 0.31/0.56 = ( summable @ A @ F2 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_Suc_iff % 0.31/0.56 thf(fact_2870_summable__ignore__initial__segment,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [F2: nat > A,K: nat] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_ignore_initial_segment % 0.31/0.56 thf(fact_2871_powser__insidea,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.56 => ! [F2: nat > A,X3: A,Z2: A] : % 0.31/0.56 ( ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) % 0.31/0.56 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) % 0.31/0.56 => ( summable @ real % 0.31/0.56 @ ^ [N: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powser_insidea % 0.31/0.56 thf(fact_2872_suminf__le,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( ordere6911136660526730532id_add @ A ) % 0.31/0.56 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.56 => ! [F2: nat > A,G: nat > A] : % 0.31/0.56 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ N3 ) ) % 0.31/0.56 => ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ( summable @ A @ G ) % 0.31/0.56 => ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % suminf_le % 0.31/0.56 thf(fact_2873_summable__finite,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( comm_monoid_add @ A ) % 0.31/0.56 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.56 => ! [N7: set @ nat,F2: nat > A] : % 0.31/0.56 ( ( finite_finite2 @ nat @ N7 ) % 0.31/0.56 => ( ! [N3: nat] : % 0.31/0.56 ( ~ ( member @ nat @ N3 @ N7 ) % 0.31/0.56 => ( ( F2 @ N3 ) % 0.31/0.56 = ( zero_zero @ A ) ) ) % 0.31/0.56 => ( summable @ A @ F2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_finite % 0.31/0.56 thf(fact_2874_sums__mult__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( field @ A ) % 0.31/0.56 & ( real_V4412858255891104859lgebra @ A ) ) % 0.31/0.56 => ! [C2: A,F2: nat > A,D2: A] : % 0.31/0.56 ( ( C2 % 0.31/0.56 != ( zero_zero @ A ) ) % 0.31/0.56 => ( ( sums @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) ) % 0.31/0.56 @ ( times_times @ A @ C2 @ D2 ) ) % 0.31/0.56 = ( sums @ A @ F2 @ D2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_mult_iff % 0.31/0.56 thf(fact_2875_sums__mult2__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( field @ A ) % 0.31/0.56 & ( real_V4412858255891104859lgebra @ A ) ) % 0.31/0.56 => ! [C2: A,F2: nat > A,D2: A] : % 0.31/0.56 ( ( C2 % 0.31/0.56 != ( zero_zero @ A ) ) % 0.31/0.56 => ( ( sums @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ C2 ) % 0.31/0.56 @ ( times_times @ A @ D2 @ C2 ) ) % 0.31/0.56 = ( sums @ A @ F2 @ D2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_mult2_iff % 0.31/0.56 thf(fact_2876_summable__mult__D,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.56 => ! [C2: A,F2: nat > A] : % 0.31/0.56 ( ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) ) ) % 0.31/0.56 => ( ( C2 % 0.31/0.56 != ( zero_zero @ A ) ) % 0.31/0.56 => ( summable @ A @ F2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_mult_D % 0.31/0.56 thf(fact_2877_summable__zero__power,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( comm_ring_1 @ A ) % 0.31/0.56 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.56 => ( summable @ A @ ( power_power @ A @ ( zero_zero @ A ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_zero_power % 0.31/0.56 thf(fact_2878_pi__not__less__zero,axiom, % 0.31/0.56 ~ ( ord_less @ real @ pi @ ( zero_zero @ real ) ) ). % 0.31/0.56 % 0.31/0.56 % pi_not_less_zero % 0.31/0.56 thf(fact_2879_pi__gt__zero,axiom, % 0.31/0.56 ord_less @ real @ ( zero_zero @ real ) @ pi ). % 0.31/0.56 % 0.31/0.56 % pi_gt_zero % 0.31/0.56 thf(fact_2880_suminf__add,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( topolo5987344860129210374id_add @ A ) % 0.31/0.56 & ( topological_t2_space @ A ) ) % 0.31/0.56 => ! [F2: nat > A,G: nat > A] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ( summable @ A @ G ) % 0.31/0.56 => ( ( plus_plus @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) ) % 0.31/0.56 = ( suminf @ A % 0.31/0.56 @ ^ [N: nat] : ( plus_plus @ A @ ( F2 @ N ) @ ( G @ N ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % suminf_add % 0.31/0.56 thf(fact_2881_suminf__eq__zero__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( ordere6911136660526730532id_add @ A ) % 0.31/0.56 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.56 => ! [F2: nat > A] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) % 0.31/0.56 => ( ( ( suminf @ A @ F2 ) % 0.31/0.56 = ( zero_zero @ A ) ) % 0.31/0.56 = ( ! [N: nat] : % 0.31/0.56 ( ( F2 @ N ) % 0.31/0.56 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % suminf_eq_zero_iff % 0.31/0.56 thf(fact_2882_suminf__nonneg,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( ordere6911136660526730532id_add @ A ) % 0.31/0.56 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.56 => ! [F2: nat > A] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) % 0.31/0.56 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % suminf_nonneg % 0.31/0.56 thf(fact_2883_suminf__pos,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( ordere6911136660526730532id_add @ A ) % 0.31/0.56 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.56 => ! [F2: nat > A] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ! [N3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) % 0.31/0.56 => ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % suminf_pos % 0.31/0.56 thf(fact_2884_sums__mult__D,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.56 => ! [C2: A,F2: nat > A,A2: A] : % 0.31/0.56 ( ( sums @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) ) % 0.31/0.56 @ A2 ) % 0.31/0.56 => ( ( C2 % 0.31/0.56 != ( zero_zero @ A ) ) % 0.31/0.56 => ( sums @ A @ F2 @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_mult_D % 0.31/0.56 thf(fact_2885_sums__Suc__imp,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [F2: nat > A,S: A] : % 0.31/0.56 ( ( ( F2 @ ( zero_zero @ nat ) ) % 0.31/0.56 = ( zero_zero @ A ) ) % 0.31/0.56 => ( ( sums @ A % 0.31/0.56 @ ^ [N: nat] : ( F2 @ ( suc @ N ) ) % 0.31/0.56 @ S ) % 0.31/0.56 => ( sums @ A @ F2 @ S ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_Suc_imp % 0.31/0.56 thf(fact_2886_sums__Suc,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( topolo5987344860129210374id_add @ A ) % 0.31/0.56 & ( topological_t2_space @ A ) ) % 0.31/0.56 => ! [F2: nat > A,L: A] : % 0.31/0.56 ( ( sums @ A % 0.31/0.56 @ ^ [N: nat] : ( F2 @ ( suc @ N ) ) % 0.31/0.56 @ L ) % 0.31/0.56 => ( sums @ A @ F2 @ ( plus_plus @ A @ L @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_Suc % 0.31/0.56 thf(fact_2887_sums__Suc__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [F2: nat > A,S: A] : % 0.31/0.56 ( ( sums @ A % 0.31/0.56 @ ^ [N: nat] : ( F2 @ ( suc @ N ) ) % 0.31/0.56 @ S ) % 0.31/0.56 = ( sums @ A @ F2 @ ( plus_plus @ A @ S @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_Suc_iff % 0.31/0.56 thf(fact_2888_sums__zero__iff__shift,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [N2: nat,F2: nat > A,S: A] : % 0.31/0.56 ( ! [I4: nat] : % 0.31/0.56 ( ( ord_less @ nat @ I4 @ N2 ) % 0.31/0.56 => ( ( F2 @ I4 ) % 0.31/0.56 = ( zero_zero @ A ) ) ) % 0.31/0.56 => ( ( sums @ A % 0.31/0.56 @ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) ) % 0.31/0.56 @ S ) % 0.31/0.56 = ( sums @ A @ F2 @ S ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_zero_iff_shift % 0.31/0.56 thf(fact_2889_summable__0__powser,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.56 => ! [F2: nat > A] : % 0.31/0.56 ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_0_powser % 0.31/0.56 thf(fact_2890_summable__zero__power_H,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( ring_1 @ A ) % 0.31/0.56 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.56 => ! [F2: nat > A] : % 0.31/0.56 ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_zero_power' % 0.31/0.56 thf(fact_2891_summable__powser__split__head,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.56 => ! [F2: nat > A,Z2: A] : % 0.31/0.56 ( ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z2 @ N ) ) ) % 0.31/0.56 = ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_powser_split_head % 0.31/0.56 thf(fact_2892_powser__split__head_I3_J,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V8999393235501362500lgebra @ A ) ) % 0.31/0.56 => ! [F2: nat > A,Z2: A] : % 0.31/0.56 ( ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) % 0.31/0.56 => ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z2 @ N ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powser_split_head(3) % 0.31/0.56 thf(fact_2893_summable__powser__ignore__initial__segment,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.56 => ! [F2: nat > A,M: nat,Z2: A] : % 0.31/0.56 ( ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( plus_plus @ nat @ N @ M ) ) @ ( power_power @ A @ Z2 @ N ) ) ) % 0.31/0.56 = ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_powser_ignore_initial_segment % 0.31/0.56 thf(fact_2894_sums__finite,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( comm_monoid_add @ A ) % 0.31/0.56 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.56 => ! [N7: set @ nat,F2: nat > A] : % 0.31/0.56 ( ( finite_finite2 @ nat @ N7 ) % 0.31/0.56 => ( ! [N3: nat] : % 0.31/0.56 ( ~ ( member @ nat @ N3 @ N7 ) % 0.31/0.56 => ( ( F2 @ N3 ) % 0.31/0.56 = ( zero_zero @ A ) ) ) % 0.31/0.56 => ( sums @ A @ F2 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N7 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_finite % 0.31/0.56 thf(fact_2895_sums__If__finite,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( comm_monoid_add @ A ) % 0.31/0.56 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.56 => ! [P: nat > $o,F2: nat > A] : % 0.31/0.56 ( ( finite_finite2 @ nat @ ( collect @ nat @ P ) ) % 0.31/0.56 => ( sums @ A % 0.31/0.56 @ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) % 0.31/0.56 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( collect @ nat @ P ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_If_finite % 0.31/0.56 thf(fact_2896_sums__If__finite__set,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( comm_monoid_add @ A ) % 0.31/0.56 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.56 => ! [A3: set @ nat,F2: nat > A] : % 0.31/0.56 ( ( finite_finite2 @ nat @ A3 ) % 0.31/0.56 => ( sums @ A % 0.31/0.56 @ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A3 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) % 0.31/0.56 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A3 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_If_finite_set % 0.31/0.56 thf(fact_2897_summable__norm__comparison__test,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [F2: nat > A,G: nat > real] : % 0.31/0.56 ( ? [N8: nat] : % 0.31/0.56 ! [N3: nat] : % 0.31/0.56 ( ( ord_less_eq @ nat @ N8 @ N3 ) % 0.31/0.56 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) ) % 0.31/0.56 => ( ( summable @ real @ G ) % 0.31/0.56 => ( summable @ real % 0.31/0.56 @ ^ [N: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_norm_comparison_test % 0.31/0.56 thf(fact_2898_summable__rabs__comparison__test,axiom, % 0.31/0.56 ! [F2: nat > real,G: nat > real] : % 0.31/0.56 ( ? [N8: nat] : % 0.31/0.56 ! [N3: nat] : % 0.31/0.56 ( ( ord_less_eq @ nat @ N8 @ N3 ) % 0.31/0.56 => ( ord_less_eq @ real @ ( abs_abs @ real @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) ) % 0.31/0.56 => ( ( summable @ real @ G ) % 0.31/0.56 => ( summable @ real % 0.31/0.56 @ ^ [N: nat] : ( abs_abs @ real @ ( F2 @ N ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_rabs_comparison_test % 0.31/0.56 thf(fact_2899_suminf__pos__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( ordere6911136660526730532id_add @ A ) % 0.31/0.56 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.56 => ! [F2: nat > A] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) % 0.31/0.56 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) % 0.31/0.56 = ( ? [I3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % suminf_pos_iff % 0.31/0.56 thf(fact_2900_suminf__pos2,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( ordere6911136660526730532id_add @ A ) % 0.31/0.56 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.56 => ! [F2: nat > A,I: nat] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) % 0.31/0.56 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) ) % 0.31/0.56 => ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % suminf_pos2 % 0.31/0.56 thf(fact_2901_suminf__le__const,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( ordere6911136660526730532id_add @ A ) % 0.31/0.56 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.56 => ! [F2: nat > A,X3: A] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X3 ) % 0.31/0.56 => ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ X3 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % suminf_le_const % 0.31/0.56 thf(fact_2902_powser__sums__if,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( ring_1 @ A ) % 0.31/0.56 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.56 => ! [M: nat,Z2: A] : % 0.31/0.56 ( sums @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( if @ A @ ( N = M ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z2 @ N ) ) % 0.31/0.56 @ ( power_power @ A @ Z2 @ M ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powser_sums_if % 0.31/0.56 thf(fact_2903_powser__sums__zero,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.56 => ! [A2: nat > A] : % 0.31/0.56 ( sums @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( A2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) % 0.31/0.56 @ ( A2 @ ( zero_zero @ nat ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powser_sums_zero % 0.31/0.56 thf(fact_2904_log__base__change,axiom, % 0.31/0.56 ! [A2: real,B2: real,X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.56 => ( ( A2 % 0.31/0.56 != ( one_one @ real ) ) % 0.31/0.56 => ( ( log @ B2 @ X3 ) % 0.31/0.56 = ( divide_divide @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ B2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_base_change % 0.31/0.56 thf(fact_2905_log__of__power__eq,axiom, % 0.31/0.56 ! [M: nat,B2: real,N2: nat] : % 0.31/0.56 ( ( ( semiring_1_of_nat @ real @ M ) % 0.31/0.56 = ( power_power @ real @ B2 @ N2 ) ) % 0.31/0.56 => ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.56 => ( ( semiring_1_of_nat @ real @ N2 ) % 0.31/0.56 = ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_of_power_eq % 0.31/0.56 thf(fact_2906_less__log__of__power,axiom, % 0.31/0.56 ! [B2: real,N2: nat,M: real] : % 0.31/0.56 ( ( ord_less @ real @ ( power_power @ real @ B2 @ N2 ) @ M ) % 0.31/0.56 => ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.56 => ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ M ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % less_log_of_power % 0.31/0.56 thf(fact_2907_powser__inside,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V8999393235501362500lgebra @ A ) ) % 0.31/0.56 => ! [F2: nat > A,X3: A,Z2: A] : % 0.31/0.56 ( ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) % 0.31/0.56 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) % 0.31/0.56 => ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powser_inside % 0.31/0.56 thf(fact_2908_sums__iff__shift,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [F2: nat > A,N2: nat,S: A] : % 0.31/0.56 ( ( sums @ A % 0.31/0.56 @ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) ) % 0.31/0.56 @ S ) % 0.31/0.56 = ( sums @ A @ F2 @ ( plus_plus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_iff_shift % 0.31/0.56 thf(fact_2909_summableI__nonneg__bounded,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( condit6923001295902523014norder @ A ) % 0.31/0.56 & ( ordere6911136660526730532id_add @ A ) % 0.31/0.56 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.56 => ! [F2: nat > A,X3: A] : % 0.31/0.56 ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) % 0.31/0.56 => ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X3 ) % 0.31/0.56 => ( summable @ A @ F2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summableI_nonneg_bounded % 0.31/0.56 thf(fact_2910_sums__split__initial__segment,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [F2: nat > A,S: A,N2: nat] : % 0.31/0.56 ( ( sums @ A @ F2 @ S ) % 0.31/0.56 => ( sums @ A % 0.31/0.56 @ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) ) % 0.31/0.56 @ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_split_initial_segment % 0.31/0.56 thf(fact_2911_sums__iff__shift_H,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [F2: nat > A,N2: nat,S: A] : % 0.31/0.56 ( ( sums @ A % 0.31/0.56 @ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) ) % 0.31/0.56 @ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) % 0.31/0.56 = ( sums @ A @ F2 @ S ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_iff_shift' % 0.31/0.56 thf(fact_2912_complete__algebra__summable__geometric,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) ) % 0.31/0.56 => ( summable @ A @ ( power_power @ A @ X3 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % complete_algebra_summable_geometric % 0.31/0.56 thf(fact_2913_summable__geometric,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.56 => ! [C2: A] : % 0.31/0.56 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) ) % 0.31/0.56 => ( summable @ A @ ( power_power @ A @ C2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_geometric % 0.31/0.56 thf(fact_2914_sums__If__finite__set_H,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( topolo1287966508704411220up_add @ A ) % 0.31/0.56 & ( topological_t2_space @ A ) ) % 0.31/0.56 => ! [G: nat > A,S3: A,A3: set @ nat,S6: A,F2: nat > A] : % 0.31/0.56 ( ( sums @ A @ G @ S3 ) % 0.31/0.56 => ( ( finite_finite2 @ nat @ A3 ) % 0.31/0.56 => ( ( S6 % 0.31/0.56 = ( plus_plus @ A @ S3 % 0.31/0.56 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.56 @ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ N ) @ ( G @ N ) ) % 0.31/0.56 @ A3 ) ) ) % 0.31/0.56 => ( sums @ A % 0.31/0.56 @ ^ [N: nat] : ( if @ A @ ( member @ nat @ N @ A3 ) @ ( F2 @ N ) @ ( G @ N ) ) % 0.31/0.56 @ S6 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sums_If_finite_set' % 0.31/0.56 thf(fact_2915_suminf__split__head,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [F2: nat > A] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ( suminf @ A % 0.31/0.56 @ ^ [N: nat] : ( F2 @ ( suc @ N ) ) ) % 0.31/0.56 = ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % suminf_split_head % 0.31/0.56 thf(fact_2916_pi__less__4,axiom, % 0.31/0.56 ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % pi_less_4 % 0.31/0.56 thf(fact_2917_sum__le__suminf,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( ordere6911136660526730532id_add @ A ) % 0.31/0.56 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.56 => ! [F2: nat > A,I6: set @ nat] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ( finite_finite2 @ nat @ I6 ) % 0.31/0.56 => ( ! [N3: nat] : % 0.31/0.56 ( ( member @ nat @ N3 @ ( uminus_uminus @ ( set @ nat ) @ I6 ) ) % 0.31/0.56 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) ) % 0.31/0.56 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ I6 ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sum_le_suminf % 0.31/0.56 thf(fact_2918_log__mult,axiom, % 0.31/0.56 ! [A2: real,X3: real,Y: real] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.56 => ( ( A2 % 0.31/0.56 != ( one_one @ real ) ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 0.31/0.56 => ( ( log @ A2 @ ( times_times @ real @ X3 @ Y ) ) % 0.31/0.56 = ( plus_plus @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_mult % 0.31/0.56 thf(fact_2919_log__divide,axiom, % 0.31/0.56 ! [A2: real,X3: real,Y: real] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.56 => ( ( A2 % 0.31/0.56 != ( one_one @ real ) ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 0.31/0.56 => ( ( log @ A2 @ ( divide_divide @ real @ X3 @ Y ) ) % 0.31/0.56 = ( minus_minus @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_divide % 0.31/0.56 thf(fact_2920_le__log__of__power,axiom, % 0.31/0.56 ! [B2: real,N2: nat,M: real] : % 0.31/0.56 ( ( ord_less_eq @ real @ ( power_power @ real @ B2 @ N2 ) @ M ) % 0.31/0.56 => ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.56 => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ M ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % le_log_of_power % 0.31/0.56 thf(fact_2921_log__base__pow,axiom, % 0.31/0.56 ! [A2: real,N2: nat,X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.56 => ( ( log @ ( power_power @ real @ A2 @ N2 ) @ X3 ) % 0.31/0.56 = ( divide_divide @ real @ ( log @ A2 @ X3 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_base_pow % 0.31/0.56 thf(fact_2922_log__nat__power,axiom, % 0.31/0.56 ! [X3: real,B2: real,N2: nat] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( log @ B2 @ ( power_power @ real @ X3 @ N2 ) ) % 0.31/0.56 = ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ X3 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_nat_power % 0.31/0.56 thf(fact_2923_suminf__split__initial__segment,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [F2: nat > A,K: nat] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ( suminf @ A @ F2 ) % 0.31/0.56 = ( plus_plus @ A % 0.31/0.56 @ ( suminf @ A % 0.31/0.56 @ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) ) % 0.31/0.56 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % suminf_split_initial_segment % 0.31/0.56 thf(fact_2924_suminf__minus__initial__segment,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [F2: nat > A,K: nat] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ( suminf @ A % 0.31/0.56 @ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) ) % 0.31/0.56 = ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % suminf_minus_initial_segment % 0.31/0.56 thf(fact_2925_pi__half__less__two,axiom, % 0.31/0.56 ord_less @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ). % 0.31/0.56 % 0.31/0.56 % pi_half_less_two % 0.31/0.56 thf(fact_2926_sum__less__suminf,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( ordere8940638589300402666id_add @ A ) % 0.31/0.56 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.56 => ! [F2: nat > A,N2: nat] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ! [M3: nat] : % 0.31/0.56 ( ( ord_less_eq @ nat @ N2 @ M3 ) % 0.31/0.56 => ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ M3 ) ) ) % 0.31/0.56 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sum_less_suminf % 0.31/0.56 thf(fact_2927_powser__split__head_I1_J,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V8999393235501362500lgebra @ A ) ) % 0.31/0.56 => ! [F2: nat > A,Z2: A] : % 0.31/0.56 ( ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) % 0.31/0.56 => ( ( suminf @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) % 0.31/0.56 = ( plus_plus @ A @ ( F2 @ ( zero_zero @ nat ) ) % 0.31/0.56 @ ( times_times @ A % 0.31/0.56 @ ( suminf @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z2 @ N ) ) ) % 0.31/0.56 @ Z2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powser_split_head(1) % 0.31/0.56 thf(fact_2928_powser__split__head_I2_J,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V8999393235501362500lgebra @ A ) ) % 0.31/0.56 => ! [F2: nat > A,Z2: A] : % 0.31/0.56 ( ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) % 0.31/0.56 => ( ( times_times @ A % 0.31/0.56 @ ( suminf @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z2 @ N ) ) ) % 0.31/0.56 @ Z2 ) % 0.31/0.56 = ( minus_minus @ A % 0.31/0.56 @ ( suminf @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) % 0.31/0.56 @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powser_split_head(2) % 0.31/0.56 thf(fact_2929_log__of__power__less,axiom, % 0.31/0.56 ! [M: nat,B2: real,N2: nat] : % 0.31/0.56 ( ( ord_less @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N2 ) ) % 0.31/0.56 => ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.56 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.31/0.56 => ( ord_less @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_of_power_less % 0.31/0.56 thf(fact_2930_log__eq__div__ln__mult__log,axiom, % 0.31/0.56 ! [A2: real,B2: real,X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.56 => ( ( A2 % 0.31/0.56 != ( one_one @ real ) ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 0.31/0.56 => ( ( B2 % 0.31/0.56 != ( one_one @ real ) ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( log @ A2 @ X3 ) % 0.31/0.56 = ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( ln_ln @ real @ A2 ) ) @ ( log @ B2 @ X3 ) ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_eq_div_ln_mult_log % 0.31/0.56 thf(fact_2931_summable__partial__sum__bound,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_Vector_banach @ A ) % 0.31/0.56 => ! [F2: nat > A,E3: real] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 ) % 0.31/0.56 => ~ ! [N9: nat] : % 0.31/0.56 ~ ! [M6: nat] : % 0.31/0.56 ( ( ord_less_eq @ nat @ N9 @ M6 ) % 0.31/0.56 => ! [N6: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ M6 @ N6 ) ) ) @ E3 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_partial_sum_bound % 0.31/0.56 thf(fact_2932_suminf__exist__split,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [R2: real,F2: nat > A] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 ) % 0.31/0.56 => ( ( summable @ A @ F2 ) % 0.31/0.56 => ? [N9: nat] : % 0.31/0.56 ! [N6: nat] : % 0.31/0.56 ( ( ord_less_eq @ nat @ N9 @ N6 ) % 0.31/0.56 => ( ord_less @ real % 0.31/0.56 @ ( real_V7770717601297561774m_norm @ A % 0.31/0.56 @ ( suminf @ A % 0.31/0.56 @ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N6 ) ) ) ) % 0.31/0.56 @ R2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % suminf_exist_split % 0.31/0.56 thf(fact_2933_summable__power__series,axiom, % 0.31/0.56 ! [F2: nat > real,Z2: real] : % 0.31/0.56 ( ! [I4: nat] : ( ord_less_eq @ real @ ( F2 @ I4 ) @ ( one_one @ real ) ) % 0.31/0.56 => ( ! [I4: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ I4 ) ) % 0.31/0.56 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Z2 ) % 0.31/0.56 => ( ( ord_less @ real @ Z2 @ ( one_one @ real ) ) % 0.31/0.56 => ( summable @ real % 0.31/0.56 @ ^ [I3: nat] : ( times_times @ real @ ( F2 @ I3 ) @ ( power_power @ real @ Z2 @ I3 ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_power_series % 0.31/0.56 thf(fact_2934_Abel__lemma,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 => ! [R2: real,R0: real,A2: nat > A,M7: real] : % 0.31/0.56 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 ) % 0.31/0.56 => ( ( ord_less @ real @ R2 @ R0 ) % 0.31/0.56 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N3 ) ) @ ( power_power @ real @ R0 @ N3 ) ) @ M7 ) % 0.31/0.56 => ( summable @ real % 0.31/0.56 @ ^ [N: nat] : ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N ) ) @ ( power_power @ real @ R2 @ N ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % Abel_lemma % 0.31/0.56 thf(fact_2935_pi__half__gt__zero,axiom, % 0.31/0.56 ord_less @ real @ ( zero_zero @ real ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % pi_half_gt_zero % 0.31/0.56 thf(fact_2936_m2pi__less__pi,axiom, % 0.31/0.56 ord_less @ real @ ( uminus_uminus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) @ pi ). % 0.31/0.56 % 0.31/0.56 % m2pi_less_pi % 0.31/0.56 thf(fact_2937_summable__ratio__test,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_Vector_banach @ A ) % 0.31/0.56 => ! [C2: real,N7: nat,F2: nat > A] : % 0.31/0.56 ( ( ord_less @ real @ C2 @ ( one_one @ real ) ) % 0.31/0.56 => ( ! [N3: nat] : % 0.31/0.56 ( ( ord_less_eq @ nat @ N7 @ N3 ) % 0.31/0.56 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ ( suc @ N3 ) ) ) @ ( times_times @ real @ C2 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) ) ) ) % 0.31/0.56 => ( summable @ A @ F2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % summable_ratio_test % 0.31/0.56 thf(fact_2938_sum__less__suminf2,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( ordere8940638589300402666id_add @ A ) % 0.31/0.56 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.56 => ! [F2: nat > A,N2: nat,I: nat] : % 0.31/0.56 ( ( summable @ A @ F2 ) % 0.31/0.56 => ( ! [M3: nat] : % 0.31/0.56 ( ( ord_less_eq @ nat @ N2 @ M3 ) % 0.31/0.56 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ M3 ) ) ) % 0.31/0.56 => ( ( ord_less_eq @ nat @ N2 @ I ) % 0.31/0.56 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) ) % 0.31/0.56 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sum_less_suminf2 % 0.31/0.56 thf(fact_2939_arctan__ubound,axiom, % 0.31/0.56 ! [Y: real] : ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % arctan_ubound % 0.31/0.56 thf(fact_2940_geometric__sums,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.56 => ! [C2: A] : % 0.31/0.56 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) ) % 0.31/0.56 => ( sums @ A @ ( power_power @ A @ C2 ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % geometric_sums % 0.31/0.56 thf(fact_2941_power__half__series,axiom, % 0.31/0.56 ( sums @ real % 0.31/0.56 @ ^ [N: nat] : ( power_power @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( suc @ N ) ) % 0.31/0.56 @ ( one_one @ real ) ) ). % 0.31/0.56 % 0.31/0.56 % power_half_series % 0.31/0.56 thf(fact_2942_log__of__power__le,axiom, % 0.31/0.56 ! [M: nat,B2: real,N2: nat] : % 0.31/0.56 ( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N2 ) ) % 0.31/0.56 => ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.56 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.31/0.56 => ( ord_less_eq @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_of_power_le % 0.31/0.56 thf(fact_2943_minus__pi__half__less__zero,axiom, % 0.31/0.56 ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ real ) ). % 0.31/0.56 % 0.31/0.56 % minus_pi_half_less_zero % 0.31/0.56 thf(fact_2944_arctan__lbound,axiom, % 0.31/0.56 ! [Y: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) ) ). % 0.31/0.56 % 0.31/0.56 % arctan_lbound % 0.31/0.56 thf(fact_2945_arctan__bounded,axiom, % 0.31/0.56 ! [Y: real] : % 0.31/0.56 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) ) % 0.31/0.56 & ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % arctan_bounded % 0.31/0.56 thf(fact_2946_less__log2__of__power,axiom, % 0.31/0.56 ! [N2: nat,M: nat] : % 0.31/0.56 ( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M ) % 0.31/0.56 => ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % less_log2_of_power % 0.31/0.56 thf(fact_2947_le__log2__of__power,axiom, % 0.31/0.56 ! [N2: nat,M: nat] : % 0.31/0.56 ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M ) % 0.31/0.56 => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % le_log2_of_power % 0.31/0.56 thf(fact_2948_log2__of__power__less,axiom, % 0.31/0.56 ! [M: nat,N2: nat] : % 0.31/0.56 ( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.31/0.56 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.31/0.56 => ( ord_less @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log2_of_power_less % 0.31/0.56 thf(fact_2949_log2__of__power__le,axiom, % 0.31/0.56 ! [M: nat,N2: nat] : % 0.31/0.56 ( ( ord_less_eq @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.31/0.56 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.31/0.56 => ( ord_less_eq @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log2_of_power_le % 0.31/0.56 thf(fact_2950_sum__pos__lt__pair,axiom, % 0.31/0.56 ! [F2: nat > real,K: nat] : % 0.31/0.56 ( ( summable @ real @ F2 ) % 0.31/0.56 => ( ! [D3: nat] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( F2 @ ( plus_plus @ nat @ K @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D3 ) ) ) @ ( F2 @ ( plus_plus @ nat @ K @ ( plus_plus @ nat @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D3 ) @ ( one_one @ nat ) ) ) ) ) ) % 0.31/0.56 => ( ord_less @ real @ ( groups7311177749621191930dd_sum @ nat @ real @ F2 @ ( set_ord_lessThan @ nat @ K ) ) @ ( suminf @ real @ F2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sum_pos_lt_pair % 0.31/0.56 thf(fact_2951_ceiling__log__nat__eq__if,axiom, % 0.31/0.56 ! [B2: nat,N2: nat,K: nat] : % 0.31/0.56 ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K ) % 0.31/0.56 => ( ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) % 0.31/0.56 => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) % 0.31/0.56 => ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) ) % 0.31/0.56 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( one_one @ int ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % ceiling_log_nat_eq_if % 0.31/0.56 thf(fact_2952_ceiling__log2__div2,axiom, % 0.31/0.56 ! [N2: nat] : % 0.31/0.56 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.31/0.56 => ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) % 0.31/0.56 = ( plus_plus @ int @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) @ ( one_one @ int ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % ceiling_log2_div2 % 0.31/0.56 thf(fact_2953_sin__cos__npi,axiom, % 0.31/0.56 ! [N2: nat] : % 0.31/0.56 ( ( sin @ real @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.56 = ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) ) ). % 0.31/0.56 % 0.31/0.56 % sin_cos_npi % 0.31/0.56 thf(fact_2954_diffs__equiv,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.56 & ( ring_1 @ A ) ) % 0.31/0.56 => ! [C2: nat > A,X3: A] : % 0.31/0.56 ( ( summable @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) % 0.31/0.56 => ( sums @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( C2 @ N ) ) @ ( power_power @ A @ X3 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) % 0.31/0.56 @ ( suminf @ A % 0.31/0.56 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % diffs_equiv % 0.31/0.56 thf(fact_2955_cos__pi__eq__zero,axiom, % 0.31/0.56 ! [M: nat] : % 0.31/0.56 ( ( cos @ real @ ( divide_divide @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.56 = ( zero_zero @ real ) ) ). % 0.31/0.56 % 0.31/0.56 % cos_pi_eq_zero % 0.31/0.56 thf(fact_2956_ceiling__log__eq__powr__iff,axiom, % 0.31/0.56 ! [X3: real,B2: real,K: nat] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.56 => ( ( ( archimedean_ceiling @ real @ ( log @ B2 @ X3 ) ) % 0.31/0.56 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) ) ) % 0.31/0.56 = ( ( ord_less @ real @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ K ) ) @ X3 ) % 0.31/0.56 & ( ord_less_eq @ real @ X3 @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % ceiling_log_eq_powr_iff % 0.31/0.56 thf(fact_2957_floor__log__nat__eq__powr__iff,axiom, % 0.31/0.56 ! [B2: nat,K: nat,N2: nat] : % 0.31/0.56 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) % 0.31/0.56 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.31/0.56 => ( ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) ) % 0.31/0.56 = ( semiring_1_of_nat @ int @ N2 ) ) % 0.31/0.56 = ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K ) % 0.31/0.56 & ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_log_nat_eq_powr_iff % 0.31/0.56 thf(fact_2958_sin__zero,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.56 => ( ( sin @ A @ ( zero_zero @ A ) ) % 0.31/0.56 = ( zero_zero @ A ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sin_zero % 0.31/0.56 thf(fact_2959_powr__eq__0__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ln @ A ) % 0.31/0.56 => ! [W2: A,Z2: A] : % 0.31/0.56 ( ( ( powr @ A @ W2 @ Z2 ) % 0.31/0.56 = ( zero_zero @ A ) ) % 0.31/0.56 = ( W2 % 0.31/0.56 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_eq_0_iff % 0.31/0.56 thf(fact_2960_powr__0,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ln @ A ) % 0.31/0.56 => ! [Z2: A] : % 0.31/0.56 ( ( powr @ A @ ( zero_zero @ A ) @ Z2 ) % 0.31/0.56 = ( zero_zero @ A ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_0 % 0.31/0.56 thf(fact_2961_cos__zero,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.56 => ( ( cos @ A @ ( zero_zero @ A ) ) % 0.31/0.56 = ( one_one @ A ) ) ) ). % 0.31/0.56 % 0.31/0.56 % cos_zero % 0.31/0.56 thf(fact_2962_powr__zero__eq__one,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ln @ A ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ( X3 % 0.31/0.56 = ( zero_zero @ A ) ) % 0.31/0.56 => ( ( powr @ A @ X3 @ ( zero_zero @ A ) ) % 0.31/0.56 = ( zero_zero @ A ) ) ) % 0.31/0.56 & ( ( X3 % 0.31/0.56 != ( zero_zero @ A ) ) % 0.31/0.56 => ( ( powr @ A @ X3 @ ( zero_zero @ A ) ) % 0.31/0.56 = ( one_one @ A ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_zero_eq_one % 0.31/0.56 thf(fact_2963_floor__zero,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) ) % 0.31/0.56 = ( zero_zero @ int ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_zero % 0.31/0.56 thf(fact_2964_powr__gt__zero,axiom, % 0.31/0.56 ! [X3: real,A2: real] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( powr @ real @ X3 @ A2 ) ) % 0.31/0.56 = ( X3 % 0.31/0.56 != ( zero_zero @ real ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_gt_zero % 0.31/0.56 thf(fact_2965_powr__less__cancel__iff,axiom, % 0.31/0.56 ! [X3: real,A2: real,B2: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) ) % 0.31/0.56 = ( ord_less @ real @ A2 @ B2 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_less_cancel_iff % 0.31/0.56 thf(fact_2966_powr__eq__one__iff,axiom, % 0.31/0.56 ! [A2: real,X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 0.31/0.56 => ( ( ( powr @ real @ A2 @ X3 ) % 0.31/0.56 = ( one_one @ real ) ) % 0.31/0.56 = ( X3 % 0.31/0.56 = ( zero_zero @ real ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_eq_one_iff % 0.31/0.56 thf(fact_2967_powr__le__cancel__iff,axiom, % 0.31/0.56 ! [X3: real,A2: real,B2: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less_eq @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) ) % 0.31/0.56 = ( ord_less_eq @ real @ A2 @ B2 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_le_cancel_iff % 0.31/0.56 thf(fact_2968_zero__le__floor,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 0.31/0.56 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % zero_le_floor % 0.31/0.56 thf(fact_2969_sin__cos__squared__add3,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ X3 ) ) @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ X3 ) ) ) % 0.31/0.56 = ( one_one @ A ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sin_cos_squared_add3 % 0.31/0.56 thf(fact_2970_floor__less__zero,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( zero_zero @ int ) ) % 0.31/0.56 = ( ord_less @ A @ X3 @ ( zero_zero @ A ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_less_zero % 0.31/0.56 thf(fact_2971_numeral__le__floor,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [V2: num,X3: A] : % 0.31/0.56 ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 0.31/0.56 = ( ord_less_eq @ A @ ( numeral_numeral @ A @ V2 ) @ X3 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % numeral_le_floor % 0.31/0.56 thf(fact_2972_zero__less__floor,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 0.31/0.56 = ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % zero_less_floor % 0.31/0.56 thf(fact_2973_floor__le__zero,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( zero_zero @ int ) ) % 0.31/0.56 = ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_le_zero % 0.31/0.56 thf(fact_2974_floor__less__numeral,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A,V2: num] : % 0.31/0.56 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) % 0.31/0.56 = ( ord_less @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_less_numeral % 0.31/0.56 thf(fact_2975_one__le__floor,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 0.31/0.56 = ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % one_le_floor % 0.31/0.56 thf(fact_2976_floor__less__one,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) % 0.31/0.56 = ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_less_one % 0.31/0.56 thf(fact_2977_powr__log__cancel,axiom, % 0.31/0.56 ! [A2: real,X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.56 => ( ( A2 % 0.31/0.56 != ( one_one @ real ) ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( powr @ real @ A2 @ ( log @ A2 @ X3 ) ) % 0.31/0.56 = X3 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_log_cancel % 0.31/0.56 thf(fact_2978_log__powr__cancel,axiom, % 0.31/0.56 ! [A2: real,Y: real] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.56 => ( ( A2 % 0.31/0.56 != ( one_one @ real ) ) % 0.31/0.56 => ( ( log @ A2 @ ( powr @ real @ A2 @ Y ) ) % 0.31/0.56 = Y ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_powr_cancel % 0.31/0.56 thf(fact_2979_numeral__less__floor,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [V2: num,X3: A] : % 0.31/0.56 ( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 0.31/0.56 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % numeral_less_floor % 0.31/0.56 thf(fact_2980_floor__le__numeral,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A,V2: num] : % 0.31/0.56 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) % 0.31/0.56 = ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_le_numeral % 0.31/0.56 thf(fact_2981_one__less__floor,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 0.31/0.56 = ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % one_less_floor % 0.31/0.56 thf(fact_2982_floor__le__one,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) % 0.31/0.56 = ( ord_less @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_le_one % 0.31/0.56 thf(fact_2983_neg__numeral__le__floor,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [V2: num,X3: A] : % 0.31/0.56 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 0.31/0.56 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X3 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % neg_numeral_le_floor % 0.31/0.56 thf(fact_2984_floor__less__neg__numeral,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A,V2: num] : % 0.31/0.56 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) % 0.31/0.56 = ( ord_less @ A @ X3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_less_neg_numeral % 0.31/0.56 thf(fact_2985_sin__cos__squared__add2,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( plus_plus @ A @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.31/0.56 = ( one_one @ A ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sin_cos_squared_add2 % 0.31/0.56 thf(fact_2986_sin__cos__squared__add,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( plus_plus @ A @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.31/0.56 = ( one_one @ A ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sin_cos_squared_add % 0.31/0.56 thf(fact_2987_neg__numeral__less__floor,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [V2: num,X3: A] : % 0.31/0.56 ( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 0.31/0.56 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X3 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % neg_numeral_less_floor % 0.31/0.56 thf(fact_2988_floor__le__neg__numeral,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A,V2: num] : % 0.31/0.56 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) % 0.31/0.56 = ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_le_neg_numeral % 0.31/0.56 thf(fact_2989_cos__one__sin__zero,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ( cos @ A @ X3 ) % 0.31/0.56 = ( one_one @ A ) ) % 0.31/0.56 => ( ( sin @ A @ X3 ) % 0.31/0.56 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % cos_one_sin_zero % 0.31/0.56 thf(fact_2990_sin__add,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.56 => ! [X3: A,Y: A] : % 0.31/0.56 ( ( sin @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 0.31/0.56 = ( plus_plus @ A @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sin_add % 0.31/0.56 thf(fact_2991_cos__diff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.56 => ! [X3: A,Y: A] : % 0.31/0.56 ( ( cos @ A @ ( minus_minus @ A @ X3 @ Y ) ) % 0.31/0.56 = ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % cos_diff % 0.31/0.56 thf(fact_2992_cos__add,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.56 => ! [X3: A,Y: A] : % 0.31/0.56 ( ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 0.31/0.56 = ( minus_minus @ A @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % cos_add % 0.31/0.56 thf(fact_2993_sin__zero__norm__cos__one,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ( sin @ A @ X3 ) % 0.31/0.56 = ( zero_zero @ A ) ) % 0.31/0.56 => ( ( real_V7770717601297561774m_norm @ A @ ( cos @ A @ X3 ) ) % 0.31/0.56 = ( one_one @ real ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sin_zero_norm_cos_one % 0.31/0.56 thf(fact_2994_sincos__principal__value,axiom, % 0.31/0.56 ! [X3: real] : % 0.31/0.56 ? [Y4: real] : % 0.31/0.56 ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ Y4 ) % 0.31/0.56 & ( ord_less_eq @ real @ Y4 @ pi ) % 0.31/0.56 & ( ( sin @ real @ Y4 ) % 0.31/0.56 = ( sin @ real @ X3 ) ) % 0.31/0.56 & ( ( cos @ real @ Y4 ) % 0.31/0.56 = ( cos @ real @ X3 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sincos_principal_value % 0.31/0.56 thf(fact_2995_floor__mono,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A,Y: A] : % 0.31/0.56 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.56 => ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_mono % 0.31/0.56 thf(fact_2996_of__int__floor__le,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) @ X3 ) ) ). % 0.31/0.56 % 0.31/0.56 % of_int_floor_le % 0.31/0.56 thf(fact_2997_floor__less__cancel,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A,Y: A] : % 0.31/0.56 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) % 0.31/0.56 => ( ord_less @ A @ X3 @ Y ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_less_cancel % 0.31/0.56 thf(fact_2998_powr__non__neg,axiom, % 0.31/0.56 ! [A2: real,X3: real] : % 0.31/0.56 ~ ( ord_less @ real @ ( powr @ real @ A2 @ X3 ) @ ( zero_zero @ real ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_non_neg % 0.31/0.56 thf(fact_2999_powr__less__mono2__neg,axiom, % 0.31/0.56 ! [A2: real,X3: real,Y: real] : % 0.31/0.56 ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ X3 @ Y ) % 0.31/0.56 => ( ord_less @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X3 @ A2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_less_mono2_neg % 0.31/0.56 thf(fact_3000_powr__less__cancel,axiom, % 0.31/0.56 ! [X3: real,A2: real,B2: real] : % 0.31/0.56 ( ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) ) % 0.31/0.56 => ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 0.31/0.56 => ( ord_less @ real @ A2 @ B2 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_less_cancel % 0.31/0.56 thf(fact_3001_powr__less__mono,axiom, % 0.31/0.56 ! [A2: real,B2: real,X3: real] : % 0.31/0.56 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 0.31/0.56 => ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_less_mono % 0.31/0.56 thf(fact_3002_le__floor__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [Z2: int,X3: A] : % 0.31/0.56 ( ( ord_less_eq @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 0.31/0.56 = ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % le_floor_iff % 0.31/0.56 thf(fact_3003_floor__less__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A,Z2: int] : % 0.31/0.56 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z2 ) % 0.31/0.56 = ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_less_iff % 0.31/0.56 thf(fact_3004_powr__mono2_H,axiom, % 0.31/0.56 ! [A2: real,X3: real,Y: real] : % 0.31/0.56 ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less_eq @ real @ X3 @ Y ) % 0.31/0.56 => ( ord_less_eq @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X3 @ A2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_mono2' % 0.31/0.56 thf(fact_3005_powr__less__mono2,axiom, % 0.31/0.56 ! [A2: real,X3: real,Y: real] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.56 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ X3 @ Y ) % 0.31/0.56 => ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_less_mono2 % 0.31/0.56 thf(fact_3006_le__floor__add,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A,Y: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % le_floor_add % 0.31/0.56 thf(fact_3007_gr__one__powr,axiom, % 0.31/0.56 ! [X3: real,Y: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 0.31/0.56 => ( ord_less @ real @ ( one_one @ real ) @ ( powr @ real @ X3 @ Y ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % gr_one_powr % 0.31/0.56 thf(fact_3008_powr__inj,axiom, % 0.31/0.56 ! [A2: real,X3: real,Y: real] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.56 => ( ( A2 % 0.31/0.56 != ( one_one @ real ) ) % 0.31/0.56 => ( ( ( powr @ real @ A2 @ X3 ) % 0.31/0.56 = ( powr @ real @ A2 @ Y ) ) % 0.31/0.56 = ( X3 = Y ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_inj % 0.31/0.56 thf(fact_3009_int__add__floor,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [Z2: int,X3: A] : % 0.31/0.56 ( ( plus_plus @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 0.31/0.56 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % int_add_floor % 0.31/0.56 thf(fact_3010_floor__add__int,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A,Z2: int] : % 0.31/0.56 ( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z2 ) % 0.31/0.56 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_add_int % 0.31/0.56 thf(fact_3011_sin__gt__zero,axiom, % 0.31/0.56 ! [X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ X3 @ pi ) % 0.31/0.56 => ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sin_gt_zero % 0.31/0.56 thf(fact_3012_sin__times__sin,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.56 => ! [W2: A,Z2: A] : % 0.31/0.56 ( ( times_times @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z2 ) ) % 0.31/0.56 = ( divide_divide @ A @ ( minus_minus @ A @ ( cos @ A @ ( minus_minus @ A @ W2 @ Z2 ) ) @ ( cos @ A @ ( plus_plus @ A @ W2 @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sin_times_sin % 0.31/0.56 thf(fact_3013_sin__times__cos,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.56 => ! [W2: A,Z2: A] : % 0.31/0.56 ( ( times_times @ A @ ( sin @ A @ W2 ) @ ( cos @ A @ Z2 ) ) % 0.31/0.56 = ( divide_divide @ A @ ( plus_plus @ A @ ( sin @ A @ ( plus_plus @ A @ W2 @ Z2 ) ) @ ( sin @ A @ ( minus_minus @ A @ W2 @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sin_times_cos % 0.31/0.56 thf(fact_3014_cos__times__sin,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.56 => ! [W2: A,Z2: A] : % 0.31/0.56 ( ( times_times @ A @ ( cos @ A @ W2 ) @ ( sin @ A @ Z2 ) ) % 0.31/0.56 = ( divide_divide @ A @ ( minus_minus @ A @ ( sin @ A @ ( plus_plus @ A @ W2 @ Z2 ) ) @ ( sin @ A @ ( minus_minus @ A @ W2 @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % cos_times_sin % 0.31/0.56 thf(fact_3015_sin__plus__sin,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.56 => ! [W2: A,Z2: A] : % 0.31/0.56 ( ( plus_plus @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z2 ) ) % 0.31/0.56 = ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sin_plus_sin % 0.31/0.56 thf(fact_3016_sin__diff__sin,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.56 => ! [W2: A,Z2: A] : % 0.31/0.56 ( ( minus_minus @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z2 ) ) % 0.31/0.56 = ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sin_diff_sin % 0.31/0.56 thf(fact_3017_cos__diff__cos,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_Vector_banach @ A ) % 0.31/0.56 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.56 => ! [W2: A,Z2: A] : % 0.31/0.56 ( ( minus_minus @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z2 ) ) % 0.31/0.56 = ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ Z2 @ W2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % cos_diff_cos % 0.31/0.56 thf(fact_3018_powr__add,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ( real_V3459762299906320749_field @ A ) % 0.31/0.56 & ( ln @ A ) ) % 0.31/0.56 => ! [X3: A,A2: A,B2: A] : % 0.31/0.56 ( ( powr @ A @ X3 @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.31/0.56 = ( times_times @ A @ ( powr @ A @ X3 @ A2 ) @ ( powr @ A @ X3 @ B2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_add % 0.31/0.56 thf(fact_3019_one__add__floor,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) % 0.31/0.56 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % one_add_floor % 0.31/0.56 thf(fact_3020_floor__log__eq__powr__iff,axiom, % 0.31/0.56 ! [X3: real,B2: real,K: int] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.56 => ( ( ( archim6421214686448440834_floor @ real @ ( log @ B2 @ X3 ) ) % 0.31/0.56 = K ) % 0.31/0.56 = ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ K ) ) @ X3 ) % 0.31/0.56 & ( ord_less @ real @ X3 @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_log_eq_powr_iff % 0.31/0.56 thf(fact_3021_powr__realpow,axiom, % 0.31/0.56 ! [X3: real,N2: nat] : % 0.31/0.56 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( powr @ real @ X3 @ ( semiring_1_of_nat @ real @ N2 ) ) % 0.31/0.56 = ( power_power @ real @ X3 @ N2 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_realpow % 0.31/0.56 thf(fact_3022_less__log__iff,axiom, % 0.31/0.56 ! [B2: real,X3: real,Y: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ Y @ ( log @ B2 @ X3 ) ) % 0.31/0.56 = ( ord_less @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % less_log_iff % 0.31/0.56 thf(fact_3023_log__less__iff,axiom, % 0.31/0.56 ! [B2: real,X3: real,Y: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ ( log @ B2 @ X3 ) @ Y ) % 0.31/0.56 = ( ord_less @ real @ X3 @ ( powr @ real @ B2 @ Y ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % log_less_iff % 0.31/0.56 thf(fact_3024_less__powr__iff,axiom, % 0.31/0.56 ! [B2: real,X3: real,Y: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ X3 @ ( powr @ real @ B2 @ Y ) ) % 0.31/0.56 = ( ord_less @ real @ ( log @ B2 @ X3 ) @ Y ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % less_powr_iff % 0.31/0.56 thf(fact_3025_powr__less__iff,axiom, % 0.31/0.56 ! [B2: real,X3: real,Y: real] : % 0.31/0.56 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.56 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) % 0.31/0.56 = ( ord_less @ real @ Y @ ( log @ B2 @ X3 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % powr_less_iff % 0.31/0.56 thf(fact_3026_cos__mono__less__eq,axiom, % 0.31/0.56 ! [X3: real,Y: real] : % 0.31/0.56 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.56 => ( ( ord_less_eq @ real @ X3 @ pi ) % 0.31/0.56 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) % 0.31/0.56 => ( ( ord_less_eq @ real @ Y @ pi ) % 0.31/0.56 => ( ( ord_less @ real @ ( cos @ real @ X3 ) @ ( cos @ real @ Y ) ) % 0.31/0.56 = ( ord_less @ real @ Y @ X3 ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % cos_mono_less_eq % 0.31/0.56 thf(fact_3027_cos__monotone__0__pi,axiom, % 0.31/0.56 ! [Y: real,X3: real] : % 0.31/0.56 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) % 0.31/0.56 => ( ( ord_less @ real @ Y @ X3 ) % 0.31/0.56 => ( ( ord_less_eq @ real @ X3 @ pi ) % 0.31/0.56 => ( ord_less @ real @ ( cos @ real @ X3 ) @ ( cos @ real @ Y ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % cos_monotone_0_pi % 0.31/0.56 thf(fact_3028_sin__eq__0__pi,axiom, % 0.31/0.56 ! [X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ X3 @ pi ) % 0.31/0.56 => ( ( ( sin @ real @ X3 ) % 0.31/0.56 = ( zero_zero @ real ) ) % 0.31/0.56 => ( X3 % 0.31/0.56 = ( zero_zero @ real ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sin_eq_0_pi % 0.31/0.56 thf(fact_3029_real__of__int__floor__add__one__gt,axiom, % 0.31/0.56 ! [R2: real] : ( ord_less @ real @ R2 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R2 ) ) @ ( one_one @ real ) ) ) ). % 0.31/0.56 % 0.31/0.56 % real_of_int_floor_add_one_gt % 0.31/0.56 thf(fact_3030_floor__eq,axiom, % 0.31/0.56 ! [N2: int,X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( ring_1_of_int @ real @ N2 ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ real @ X3 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N2 ) @ ( one_one @ real ) ) ) % 0.31/0.56 => ( ( archim6421214686448440834_floor @ real @ X3 ) % 0.31/0.56 = N2 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_eq % 0.31/0.56 thf(fact_3031_sin__zero__pi__iff,axiom, % 0.31/0.56 ! [X3: real] : % 0.31/0.56 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ pi ) % 0.31/0.56 => ( ( ( sin @ real @ X3 ) % 0.31/0.56 = ( zero_zero @ real ) ) % 0.31/0.56 = ( X3 % 0.31/0.56 = ( zero_zero @ real ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sin_zero_pi_iff % 0.31/0.56 thf(fact_3032_real__of__int__floor__gt__diff__one,axiom, % 0.31/0.56 ! [R2: real] : ( ord_less @ real @ ( minus_minus @ real @ R2 @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R2 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % real_of_int_floor_gt_diff_one % 0.31/0.56 thf(fact_3033_diffs__def,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( ring_1 @ A ) % 0.31/0.56 => ( ( diffs @ A ) % 0.31/0.56 = ( ^ [C4: nat > A,N: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( C4 @ ( suc @ N ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % diffs_def % 0.31/0.56 thf(fact_3034_sin__expansion__lemma,axiom, % 0.31/0.56 ! [X3: real,M: nat] : % 0.31/0.56 ( ( sin @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.56 = ( cos @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % sin_expansion_lemma % 0.31/0.56 thf(fact_3035_floor__unique,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [Z2: int,X3: A] : % 0.31/0.56 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 ) % 0.31/0.56 => ( ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) % 0.31/0.56 => ( ( archim6421214686448440834_floor @ A @ X3 ) % 0.31/0.56 = Z2 ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_unique % 0.31/0.56 thf(fact_3036_floor__eq__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A,A2: int] : % 0.31/0.56 ( ( ( archim6421214686448440834_floor @ A @ X3 ) % 0.31/0.56 = A2 ) % 0.31/0.56 = ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ X3 ) % 0.31/0.56 & ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_eq_iff % 0.31/0.56 thf(fact_3037_floor__split,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [P: int > $o,T2: A] : % 0.31/0.56 ( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) ) % 0.31/0.56 = ( ! [I3: int] : % 0.31/0.56 ( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I3 ) @ T2 ) % 0.31/0.56 & ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) ) ) % 0.31/0.56 => ( P @ I3 ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_split % 0.31/0.56 thf(fact_3038_le__mult__floor,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [A2: A,B2: A] : % 0.31/0.56 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.56 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.31/0.56 => ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A2 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % le_mult_floor % 0.31/0.56 thf(fact_3039_less__floor__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [Z2: int,X3: A] : % 0.31/0.56 ( ( ord_less @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 0.31/0.56 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ). % 0.31/0.56 % 0.31/0.56 % less_floor_iff % 0.31/0.56 thf(fact_3040_floor__le__iff,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A,Z2: int] : % 0.31/0.56 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z2 ) % 0.31/0.56 = ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ). % 0.31/0.56 % 0.31/0.56 % floor_le_iff % 0.31/0.56 thf(fact_3041_floor__correct,axiom, % 0.31/0.56 ! [A: $tType] : % 0.31/0.56 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.56 => ! [X3: A] : % 0.31/0.56 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) @ X3 ) % 0.31/0.57 & ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % floor_correct % 0.31/0.57 thf(fact_3042_cos__expansion__lemma,axiom, % 0.31/0.57 ! [X3: real,M: nat] : % 0.31/0.57 ( ( cos @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.57 = ( uminus_uminus @ real @ ( sin @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % cos_expansion_lemma % 0.31/0.57 thf(fact_3043_powr__neg__one,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( powr @ real @ X3 @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) % 0.31/0.57 = ( divide_divide @ real @ ( one_one @ real ) @ X3 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % powr_neg_one % 0.31/0.57 thf(fact_3044_sin__gt__zero__02,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) % 0.31/0.57 => ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_gt_zero_02 % 0.31/0.57 thf(fact_3045_le__log__iff,axiom, % 0.31/0.57 ! [B2: real,X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.57 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less_eq @ real @ Y @ ( log @ B2 @ X3 ) ) % 0.31/0.57 = ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % le_log_iff % 0.31/0.57 thf(fact_3046_log__le__iff,axiom, % 0.31/0.57 ! [B2: real,X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.57 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less_eq @ real @ ( log @ B2 @ X3 ) @ Y ) % 0.31/0.57 = ( ord_less_eq @ real @ X3 @ ( powr @ real @ B2 @ Y ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % log_le_iff % 0.31/0.57 thf(fact_3047_le__powr__iff,axiom, % 0.31/0.57 ! [B2: real,X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.57 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less_eq @ real @ X3 @ ( powr @ real @ B2 @ Y ) ) % 0.31/0.57 = ( ord_less_eq @ real @ ( log @ B2 @ X3 ) @ Y ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % le_powr_iff % 0.31/0.57 thf(fact_3048_powr__le__iff,axiom, % 0.31/0.57 ! [B2: real,X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.57 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) % 0.31/0.57 = ( ord_less_eq @ real @ Y @ ( log @ B2 @ X3 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % powr_le_iff % 0.31/0.57 thf(fact_3049_cos__two__less__zero,axiom, % 0.31/0.57 ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ). % 0.31/0.57 % 0.31/0.57 % cos_two_less_zero % 0.31/0.57 thf(fact_3050_floor__eq2,axiom, % 0.31/0.57 ! [N2: int,X3: real] : % 0.31/0.57 ( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ N2 ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N2 ) @ ( one_one @ real ) ) ) % 0.31/0.57 => ( ( archim6421214686448440834_floor @ real @ X3 ) % 0.31/0.57 = N2 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % floor_eq2 % 0.31/0.57 thf(fact_3051_cos__monotone__minus__pi__0,axiom, % 0.31/0.57 ! [Y: real,X3: real] : % 0.31/0.57 ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y ) % 0.31/0.57 => ( ( ord_less @ real @ Y @ X3 ) % 0.31/0.57 => ( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) % 0.31/0.57 => ( ord_less @ real @ ( cos @ real @ Y ) @ ( cos @ real @ X3 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % cos_monotone_minus_pi_0 % 0.31/0.57 thf(fact_3052_floor__divide__lower,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.57 => ! [Q4: A,P4: A] : % 0.31/0.57 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 ) % 0.31/0.57 => ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % floor_divide_lower % 0.31/0.57 thf(fact_3053_sincos__total__2pi,axiom, % 0.31/0.57 ! [X3: real,Y: real] : % 0.31/0.57 ( ( ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 = ( one_one @ real ) ) % 0.31/0.57 => ~ ! [T7: real] : % 0.31/0.57 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 ) % 0.31/0.57 => ( ( ord_less @ real @ T7 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) % 0.31/0.57 => ( ( X3 % 0.31/0.57 = ( cos @ real @ T7 ) ) % 0.31/0.57 => ( Y % 0.31/0.57 != ( sin @ real @ T7 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sincos_total_2pi % 0.31/0.57 thf(fact_3054_ln__powr__bound,axiom, % 0.31/0.57 ! [X3: real,A2: real] : % 0.31/0.57 ( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.57 => ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( divide_divide @ real @ ( powr @ real @ X3 @ A2 ) @ A2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % ln_powr_bound % 0.31/0.57 thf(fact_3055_ln__powr__bound2,axiom, % 0.31/0.57 ! [X3: real,A2: real] : % 0.31/0.57 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.57 => ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X3 ) @ A2 ) @ ( times_times @ real @ ( powr @ real @ A2 @ A2 ) @ X3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % ln_powr_bound2 % 0.31/0.57 thf(fact_3056_log__add__eq__powr,axiom, % 0.31/0.57 ! [B2: real,X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 0.31/0.57 => ( ( B2 % 0.31/0.57 != ( one_one @ real ) ) % 0.31/0.57 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( plus_plus @ real @ ( log @ B2 @ X3 ) @ Y ) % 0.31/0.57 = ( log @ B2 @ ( times_times @ real @ X3 @ ( powr @ real @ B2 @ Y ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % log_add_eq_powr % 0.31/0.57 thf(fact_3057_add__log__eq__powr,axiom, % 0.31/0.57 ! [B2: real,X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 0.31/0.57 => ( ( B2 % 0.31/0.57 != ( one_one @ real ) ) % 0.31/0.57 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( plus_plus @ real @ Y @ ( log @ B2 @ X3 ) ) % 0.31/0.57 = ( log @ B2 @ ( times_times @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % add_log_eq_powr % 0.31/0.57 thf(fact_3058_minus__log__eq__powr,axiom, % 0.31/0.57 ! [B2: real,X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 0.31/0.57 => ( ( B2 % 0.31/0.57 != ( one_one @ real ) ) % 0.31/0.57 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( minus_minus @ real @ Y @ ( log @ B2 @ X3 ) ) % 0.31/0.57 = ( log @ B2 @ ( divide_divide @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % minus_log_eq_powr % 0.31/0.57 thf(fact_3059_sin__pi__divide__n__ge__0,axiom, % 0.31/0.57 ! [N2: nat] : % 0.31/0.57 ( ( N2 % 0.31/0.57 != ( zero_zero @ nat ) ) % 0.31/0.57 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_pi_divide_n_ge_0 % 0.31/0.57 thf(fact_3060_cos__times__cos,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ! [W2: A,Z2: A] : % 0.31/0.57 ( ( times_times @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z2 ) ) % 0.31/0.57 = ( divide_divide @ A @ ( plus_plus @ A @ ( cos @ A @ ( minus_minus @ A @ W2 @ Z2 ) ) @ ( cos @ A @ ( plus_plus @ A @ W2 @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % cos_times_cos % 0.31/0.57 thf(fact_3061_cos__plus__cos,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ! [W2: A,Z2: A] : % 0.31/0.57 ( ( plus_plus @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z2 ) ) % 0.31/0.57 = ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % cos_plus_cos % 0.31/0.57 thf(fact_3062_floor__divide__upper,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.57 => ! [Q4: A,P4: A] : % 0.31/0.57 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 ) % 0.31/0.57 => ( ord_less @ A @ P4 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % floor_divide_upper % 0.31/0.57 thf(fact_3063_round__def,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.57 => ( ( archimedean_round @ A ) % 0.31/0.57 = ( ^ [X: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % round_def % 0.31/0.57 thf(fact_3064_sin__gt__zero2,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_gt_zero2 % 0.31/0.57 thf(fact_3065_sin__lt__zero,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ pi @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) % 0.31/0.57 => ( ord_less @ real @ ( sin @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_lt_zero % 0.31/0.57 thf(fact_3066_cos__double__less__one,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) % 0.31/0.57 => ( ord_less @ real @ ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) ) @ ( one_one @ real ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % cos_double_less_one % 0.31/0.57 thf(fact_3067_cos__gt__zero,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % cos_gt_zero % 0.31/0.57 thf(fact_3068_log__minus__eq__powr,axiom, % 0.31/0.57 ! [B2: real,X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 0.31/0.57 => ( ( B2 % 0.31/0.57 != ( one_one @ real ) ) % 0.31/0.57 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( minus_minus @ real @ ( log @ B2 @ X3 ) @ Y ) % 0.31/0.57 = ( log @ B2 @ ( times_times @ real @ X3 @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ Y ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % log_minus_eq_powr % 0.31/0.57 thf(fact_3069_termdiff__converges,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ! [X3: A,K5: real,C2: nat > A] : % 0.31/0.57 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K5 ) % 0.31/0.57 => ( ! [X5: A] : % 0.31/0.57 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K5 ) % 0.31/0.57 => ( summable @ A % 0.31/0.57 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ X5 @ N ) ) ) ) % 0.31/0.57 => ( summable @ A % 0.31/0.57 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % termdiff_converges % 0.31/0.57 thf(fact_3070_powr__neg__numeral,axiom, % 0.31/0.57 ! [X3: real,N2: num] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( powr @ real @ X3 @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ N2 ) ) ) % 0.31/0.57 = ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % powr_neg_numeral % 0.31/0.57 thf(fact_3071_sin__le__zero,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less_eq @ real @ pi @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) % 0.31/0.57 => ( ord_less_eq @ real @ ( sin @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_le_zero % 0.31/0.57 thf(fact_3072_sin__less__zero,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) % 0.31/0.57 => ( ord_less @ real @ ( sin @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_less_zero % 0.31/0.57 thf(fact_3073_sin__mono__less__eq,axiom, % 0.31/0.57 ! [X3: real,Y: real] : % 0.31/0.57 ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 0.31/0.57 => ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y ) % 0.31/0.57 => ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ( ord_less @ real @ ( sin @ real @ X3 ) @ ( sin @ real @ Y ) ) % 0.31/0.57 = ( ord_less @ real @ X3 @ Y ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_mono_less_eq % 0.31/0.57 thf(fact_3074_sin__monotone__2pi,axiom, % 0.31/0.57 ! [Y: real,X3: real] : % 0.31/0.57 ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y ) % 0.31/0.57 => ( ( ord_less @ real @ Y @ X3 ) % 0.31/0.57 => ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ord_less @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X3 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_monotone_2pi % 0.31/0.57 thf(fact_3075_cos__gt__zero__pi,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % cos_gt_zero_pi % 0.31/0.57 thf(fact_3076_sin__pi__divide__n__gt__0,axiom, % 0.31/0.57 ! [N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.31/0.57 => ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_pi_divide_n_gt_0 % 0.31/0.57 thf(fact_3077_floor__log2__div2,axiom, % 0.31/0.57 ! [N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.31/0.57 => ( ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) % 0.31/0.57 = ( plus_plus @ int @ ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ int ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % floor_log2_div2 % 0.31/0.57 thf(fact_3078_floor__log__nat__eq__if,axiom, % 0.31/0.57 ! [B2: nat,N2: nat,K: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K ) % 0.31/0.57 => ( ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) % 0.31/0.57 => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) % 0.31/0.57 => ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) ) % 0.31/0.57 = ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % floor_log_nat_eq_if % 0.31/0.57 thf(fact_3079_Maclaurin__minus__cos__expansion,axiom, % 0.31/0.57 ! [N2: nat,X3: real] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) % 0.31/0.57 => ? [T7: real] : % 0.31/0.57 ( ( ord_less @ real @ X3 @ T7 ) % 0.31/0.57 & ( ord_less @ real @ T7 @ ( zero_zero @ real ) ) % 0.31/0.57 & ( ( cos @ real @ X3 ) % 0.31/0.57 = ( plus_plus @ real % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.57 @ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ X3 @ M2 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.57 @ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % Maclaurin_minus_cos_expansion % 0.31/0.57 thf(fact_3080_Maclaurin__cos__expansion2,axiom, % 0.31/0.57 ! [X3: real,N2: nat] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.57 => ? [T7: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ T7 ) % 0.31/0.57 & ( ord_less @ real @ T7 @ X3 ) % 0.31/0.57 & ( ( cos @ real @ X3 ) % 0.31/0.57 = ( plus_plus @ real % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.57 @ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ X3 @ M2 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.57 @ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % Maclaurin_cos_expansion2 % 0.31/0.57 thf(fact_3081_sin__paired,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( sums @ real % 0.31/0.57 @ ^ [N: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) % 0.31/0.57 @ ( sin @ real @ X3 ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_paired % 0.31/0.57 thf(fact_3082_tan__double,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ! [X3: A] : % 0.31/0.57 ( ( ( cos @ A @ X3 ) % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) ) % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( tan @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) ) % 0.31/0.57 = ( divide_divide @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( tan @ A @ X3 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_double % 0.31/0.57 thf(fact_3083_arcosh__def,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ln @ A ) % 0.31/0.57 => ( ( arcosh @ A ) % 0.31/0.57 = ( ^ [X: A] : ( ln_ln @ A @ ( plus_plus @ A @ X @ ( powr @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % arcosh_def % 0.31/0.57 thf(fact_3084_tan__zero,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ( ( tan @ A @ ( zero_zero @ A ) ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_zero % 0.31/0.57 thf(fact_3085_of__real__eq__0__iff,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V2191834092415804123ebra_1 @ A ) % 0.31/0.57 => ! [X3: real] : % 0.31/0.57 ( ( ( real_Vector_of_real @ A @ X3 ) % 0.31/0.57 = ( zero_zero @ A ) ) % 0.31/0.57 = ( X3 % 0.31/0.57 = ( zero_zero @ real ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % of_real_eq_0_iff % 0.31/0.57 thf(fact_3086_of__real__0,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V2191834092415804123ebra_1 @ A ) % 0.31/0.57 => ( ( real_Vector_of_real @ A @ ( zero_zero @ real ) ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % of_real_0 % 0.31/0.57 thf(fact_3087_fact__0,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( semiring_char_0 @ A ) % 0.31/0.57 => ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) ) % 0.31/0.57 = ( one_one @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_0 % 0.31/0.57 thf(fact_3088_of__real__add,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V2191834092415804123ebra_1 @ A ) % 0.31/0.57 => ! [X3: real,Y: real] : % 0.31/0.57 ( ( real_Vector_of_real @ A @ ( plus_plus @ real @ X3 @ Y ) ) % 0.31/0.57 = ( plus_plus @ A @ ( real_Vector_of_real @ A @ X3 ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % of_real_add % 0.31/0.57 thf(fact_3089_fact__Suc__0,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( semiring_char_0 @ A ) % 0.31/0.57 => ( ( semiring_char_0_fact @ A @ ( suc @ ( zero_zero @ nat ) ) ) % 0.31/0.57 = ( one_one @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_Suc_0 % 0.31/0.57 thf(fact_3090_fact__Suc,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( semiring_char_0 @ A ) % 0.31/0.57 => ! [N2: nat] : % 0.31/0.57 ( ( semiring_char_0_fact @ A @ ( suc @ N2 ) ) % 0.31/0.57 = ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_Suc % 0.31/0.57 thf(fact_3091_sin__of__real__pi,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.57 => ( ( sin @ A @ ( real_Vector_of_real @ A @ pi ) ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_of_real_pi % 0.31/0.57 thf(fact_3092_norm__of__real__add1,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.57 => ! [X3: real] : % 0.31/0.57 ( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X3 ) @ ( one_one @ A ) ) ) % 0.31/0.57 = ( abs_abs @ real @ ( plus_plus @ real @ X3 @ ( one_one @ real ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % norm_of_real_add1 % 0.31/0.57 thf(fact_3093_norm__of__real__addn,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.57 => ! [X3: real,B2: num] : % 0.31/0.57 ( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X3 ) @ ( numeral_numeral @ A @ B2 ) ) ) % 0.31/0.57 = ( abs_abs @ real @ ( plus_plus @ real @ X3 @ ( numeral_numeral @ real @ B2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % norm_of_real_addn % 0.31/0.57 thf(fact_3094_cos__of__real__pi__half,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V7773925162809079976_field @ A ) % 0.31/0.57 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.57 => ( ( cos @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % cos_of_real_pi_half % 0.31/0.57 thf(fact_3095_fact__nonzero,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( semiring_char_0 @ A ) % 0.31/0.57 & ( semiri3467727345109120633visors @ A ) ) % 0.31/0.57 => ! [N2: nat] : % 0.31/0.57 ( ( semiring_char_0_fact @ A @ N2 ) % 0.31/0.57 != ( zero_zero @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_nonzero % 0.31/0.57 thf(fact_3096_fact__ge__zero,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_ge_zero % 0.31/0.57 thf(fact_3097_fact__not__neg,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [N2: nat] : % 0.31/0.57 ~ ( ord_less @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( zero_zero @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_not_neg % 0.31/0.57 thf(fact_3098_fact__gt__zero,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [N2: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_gt_zero % 0.31/0.57 thf(fact_3099_fact__ge__1,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [N2: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_ge_1 % 0.31/0.57 thf(fact_3100_fact__mono,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [M: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.57 => ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_mono % 0.31/0.57 thf(fact_3101_fact__dvd,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [N2: nat,M: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ N2 @ M ) % 0.31/0.57 => ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ M ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_dvd % 0.31/0.57 thf(fact_3102_fact__less__mono,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [M: nat,N2: nat] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.31/0.57 => ( ( ord_less @ nat @ M @ N2 ) % 0.31/0.57 => ( ord_less @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_less_mono % 0.31/0.57 thf(fact_3103_fact__fact__dvd__fact,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [K: nat,N2: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N2 ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_fact_dvd_fact % 0.31/0.57 thf(fact_3104_fact__mod,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( linordered_semidom @ A ) % 0.31/0.57 & ( semidom_modulo @ A ) ) % 0.31/0.57 => ! [M: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.57 => ( ( modulo_modulo @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ M ) ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_mod % 0.31/0.57 thf(fact_3105_fact__le__power,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [N2: nat] : ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ N2 @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_le_power % 0.31/0.57 thf(fact_3106_norm__less__p1,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V2822296259951069270ebra_1 @ A ) % 0.31/0.57 => ! [X3: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) @ ( one_one @ A ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % norm_less_p1 % 0.31/0.57 thf(fact_3107_choose__dvd,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [K: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.57 => ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % choose_dvd % 0.31/0.57 thf(fact_3108_fact__num__eq__if,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( semiring_char_0 @ A ) % 0.31/0.57 => ( ( semiring_char_0_fact @ A ) % 0.31/0.57 = ( ^ [M2: nat] : % 0.31/0.57 ( if @ A % 0.31/0.57 @ ( M2 % 0.31/0.57 = ( zero_zero @ nat ) ) % 0.31/0.57 @ ( one_one @ A ) % 0.31/0.57 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_num_eq_if % 0.31/0.57 thf(fact_3109_fact__reduce,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( semiring_char_0 @ A ) % 0.31/0.57 => ! [N2: nat] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.57 => ( ( semiring_char_0_fact @ A @ N2 ) % 0.31/0.57 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_reduce % 0.31/0.57 thf(fact_3110_tan__gt__zero,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ord_less @ real @ ( zero_zero @ real ) @ ( tan @ real @ X3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_gt_zero % 0.31/0.57 thf(fact_3111_lemma__tan__total,axiom, % 0.31/0.57 ! [Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 0.31/0.57 => ? [X5: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X5 ) % 0.31/0.57 & ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 & ( ord_less @ real @ Y @ ( tan @ real @ X5 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % lemma_tan_total % 0.31/0.57 thf(fact_3112_lemma__tan__total1,axiom, % 0.31/0.57 ! [Y: real] : % 0.31/0.57 ? [X5: real] : % 0.31/0.57 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X5 ) % 0.31/0.57 & ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 & ( ( tan @ real @ X5 ) % 0.31/0.57 = Y ) ) ). % 0.31/0.57 % 0.31/0.57 % lemma_tan_total1 % 0.31/0.57 thf(fact_3113_tan__mono__lt__eq,axiom, % 0.31/0.57 ! [X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y ) % 0.31/0.57 => ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ( ord_less @ real @ ( tan @ real @ X3 ) @ ( tan @ real @ Y ) ) % 0.31/0.57 = ( ord_less @ real @ X3 @ Y ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_mono_lt_eq % 0.31/0.57 thf(fact_3114_tan__monotone_H,axiom, % 0.31/0.57 ! [Y: real,X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y ) % 0.31/0.57 => ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ( ord_less @ real @ Y @ X3 ) % 0.31/0.57 = ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X3 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_monotone' % 0.31/0.57 thf(fact_3115_tan__monotone,axiom, % 0.31/0.57 ! [Y: real,X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y ) % 0.31/0.57 => ( ( ord_less @ real @ Y @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X3 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_monotone % 0.31/0.57 thf(fact_3116_tan__total,axiom, % 0.31/0.57 ! [Y: real] : % 0.31/0.57 ? [X5: real] : % 0.31/0.57 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X5 ) % 0.31/0.57 & ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 & ( ( tan @ real @ X5 ) % 0.31/0.57 = Y ) % 0.31/0.57 & ! [Y5: real] : % 0.31/0.57 ( ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y5 ) % 0.31/0.57 & ( ord_less @ real @ Y5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 & ( ( tan @ real @ Y5 ) % 0.31/0.57 = Y ) ) % 0.31/0.57 => ( Y5 = X5 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_total % 0.31/0.57 thf(fact_3117_add__tan__eq,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ! [X3: A,Y: A] : % 0.31/0.57 ( ( ( cos @ A @ X3 ) % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( ( cos @ A @ Y ) % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( plus_plus @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) % 0.31/0.57 = ( divide_divide @ A @ ( sin @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % add_tan_eq % 0.31/0.57 thf(fact_3118_tan__total__pos,axiom, % 0.31/0.57 ! [Y: real] : % 0.31/0.57 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) % 0.31/0.57 => ? [X5: real] : % 0.31/0.57 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X5 ) % 0.31/0.57 & ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 & ( ( tan @ real @ X5 ) % 0.31/0.57 = Y ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_total_pos % 0.31/0.57 thf(fact_3119_tan__pos__pi2__le,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tan @ real @ X3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_pos_pi2_le % 0.31/0.57 thf(fact_3120_tan__less__zero,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) % 0.31/0.57 => ( ord_less @ real @ ( tan @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_less_zero % 0.31/0.57 thf(fact_3121_tan__mono__le__eq,axiom, % 0.31/0.57 ! [X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y ) % 0.31/0.57 => ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ( ord_less_eq @ real @ ( tan @ real @ X3 ) @ ( tan @ real @ Y ) ) % 0.31/0.57 = ( ord_less_eq @ real @ X3 @ Y ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_mono_le_eq % 0.31/0.57 thf(fact_3122_tan__mono__le,axiom, % 0.31/0.57 ! [X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 0.31/0.57 => ( ( ord_less_eq @ real @ X3 @ Y ) % 0.31/0.57 => ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ord_less_eq @ real @ ( tan @ real @ X3 ) @ ( tan @ real @ Y ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_mono_le % 0.31/0.57 thf(fact_3123_tan__bound__pi2,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.57 => ( ord_less @ real @ ( abs_abs @ real @ ( tan @ real @ X3 ) ) @ ( one_one @ real ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_bound_pi2 % 0.31/0.57 thf(fact_3124_arctan,axiom, % 0.31/0.57 ! [Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) ) % 0.31/0.57 & ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 & ( ( tan @ real @ ( arctan @ Y ) ) % 0.31/0.57 = Y ) ) ). % 0.31/0.57 % 0.31/0.57 % arctan % 0.31/0.57 thf(fact_3125_arctan__tan,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ( arctan @ ( tan @ real @ X3 ) ) % 0.31/0.57 = X3 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % arctan_tan % 0.31/0.57 thf(fact_3126_arctan__unique,axiom, % 0.31/0.57 ! [X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ( ( tan @ real @ X3 ) % 0.31/0.57 = Y ) % 0.31/0.57 => ( ( arctan @ Y ) % 0.31/0.57 = X3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % arctan_unique % 0.31/0.57 thf(fact_3127_Maclaurin__zero,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( zero @ A ) % 0.31/0.57 => ! [X3: real,N2: nat,Diff: nat > A > real] : % 0.31/0.57 ( ( X3 % 0.31/0.57 = ( zero_zero @ real ) ) % 0.31/0.57 => ( ( N2 % 0.31/0.57 != ( zero_zero @ nat ) ) % 0.31/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.57 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X3 @ M2 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.57 = ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % Maclaurin_zero % 0.31/0.57 thf(fact_3128_Maclaurin__lemma,axiom, % 0.31/0.57 ! [H2: real,F2: real > real,J: nat > real,N2: nat] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ H2 ) % 0.31/0.57 => ? [B10: real] : % 0.31/0.57 ( ( F2 @ H2 ) % 0.31/0.57 = ( plus_plus @ real % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.57 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J @ M2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.57 @ ( times_times @ real @ B10 @ ( divide_divide @ real @ ( power_power @ real @ H2 @ N2 ) @ ( semiring_char_0_fact @ real @ N2 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % Maclaurin_lemma % 0.31/0.57 thf(fact_3129_tan__add,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ! [X3: A,Y: A] : % 0.31/0.57 ( ( ( cos @ A @ X3 ) % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( ( cos @ A @ Y ) % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( tan @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 0.31/0.57 = ( divide_divide @ A @ ( plus_plus @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_add % 0.31/0.57 thf(fact_3130_tan__diff,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ! [X3: A,Y: A] : % 0.31/0.57 ( ( ( cos @ A @ X3 ) % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( ( cos @ A @ Y ) % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( ( cos @ A @ ( minus_minus @ A @ X3 @ Y ) ) % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( tan @ A @ ( minus_minus @ A @ X3 @ Y ) ) % 0.31/0.57 = ( divide_divide @ A @ ( minus_minus @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_diff % 0.31/0.57 thf(fact_3131_lemma__tan__add1,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ! [X3: A,Y: A] : % 0.31/0.57 ( ( ( cos @ A @ X3 ) % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( ( cos @ A @ Y ) % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) ) % 0.31/0.57 = ( divide_divide @ A @ ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % lemma_tan_add1 % 0.31/0.57 thf(fact_3132_minus__sin__cos__eq,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ! [X3: A] : % 0.31/0.57 ( ( uminus_uminus @ A @ ( sin @ A @ X3 ) ) % 0.31/0.57 = ( cos @ A @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % minus_sin_cos_eq % 0.31/0.57 thf(fact_3133_tan__total__pi4,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 0.31/0.57 => ? [Z3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) @ Z3 ) % 0.31/0.57 & ( ord_less @ real @ Z3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.57 & ( ( tan @ real @ Z3 ) % 0.31/0.57 = X3 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_total_pi4 % 0.31/0.57 thf(fact_3134_tan__half,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ( ( tan @ A ) % 0.31/0.57 = ( ^ [X: A] : ( divide_divide @ A @ ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) @ ( plus_plus @ A @ ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) @ ( one_one @ A ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tan_half % 0.31/0.57 thf(fact_3135_arsinh__def,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ln @ A ) % 0.31/0.57 => ( ( arsinh @ A ) % 0.31/0.57 = ( ^ [X: A] : ( ln_ln @ A @ ( plus_plus @ A @ X @ ( powr @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % arsinh_def % 0.31/0.57 thf(fact_3136_Maclaurin__sin__expansion3,axiom, % 0.31/0.57 ! [N2: nat,X3: real] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.57 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ? [T7: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ T7 ) % 0.31/0.57 & ( ord_less @ real @ T7 @ X3 ) % 0.31/0.57 & ( ( sin @ real @ X3 ) % 0.31/0.57 = ( plus_plus @ real % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.57 @ ^ [M2: nat] : ( times_times @ real @ ( sin_coeff @ M2 ) @ ( power_power @ real @ X3 @ M2 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.57 @ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % Maclaurin_sin_expansion3 % 0.31/0.57 thf(fact_3137_Maclaurin__sin__expansion4,axiom, % 0.31/0.57 ! [X3: real,N2: nat] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ? [T7: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ T7 ) % 0.31/0.57 & ( ord_less_eq @ real @ T7 @ X3 ) % 0.31/0.57 & ( ( sin @ real @ X3 ) % 0.31/0.57 = ( plus_plus @ real % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.57 @ ^ [M2: nat] : ( times_times @ real @ ( sin_coeff @ M2 ) @ ( power_power @ real @ X3 @ M2 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.57 @ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % Maclaurin_sin_expansion4 % 0.31/0.57 thf(fact_3138_sin__coeff__def,axiom, % 0.31/0.57 ( sin_coeff % 0.31/0.57 = ( ^ [N: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_coeff_def % 0.31/0.57 thf(fact_3139_Maclaurin__exp__lt,axiom, % 0.31/0.57 ! [X3: real,N2: nat] : % 0.31/0.57 ( ( X3 % 0.31/0.57 != ( zero_zero @ real ) ) % 0.31/0.57 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.57 => ? [T7: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T7 ) ) % 0.31/0.57 & ( ord_less @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) ) % 0.31/0.57 & ( ( exp @ real @ X3 ) % 0.31/0.57 = ( plus_plus @ real % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.57 @ ^ [M2: nat] : ( divide_divide @ real @ ( power_power @ real @ X3 @ M2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.57 @ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % Maclaurin_exp_lt % 0.31/0.57 thf(fact_3140_round__altdef,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.57 => ( ( archimedean_round @ A ) % 0.31/0.57 = ( ^ [X: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X ) ) @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % round_altdef % 0.31/0.57 thf(fact_3141_exp__less__mono,axiom, % 0.31/0.57 ! [X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ X3 @ Y ) % 0.31/0.57 => ( ord_less @ real @ ( exp @ real @ X3 ) @ ( exp @ real @ Y ) ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_less_mono % 0.31/0.57 thf(fact_3142_exp__less__cancel__iff,axiom, % 0.31/0.57 ! [X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( exp @ real @ X3 ) @ ( exp @ real @ Y ) ) % 0.31/0.57 = ( ord_less @ real @ X3 @ Y ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_less_cancel_iff % 0.31/0.57 thf(fact_3143_exp__zero,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.57 => ( ( exp @ A @ ( zero_zero @ A ) ) % 0.31/0.57 = ( one_one @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_zero % 0.31/0.57 thf(fact_3144_frac__of__int,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.57 => ! [Z2: int] : % 0.31/0.57 ( ( archimedean_frac @ A @ ( ring_1_of_int @ A @ Z2 ) ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % frac_of_int % 0.31/0.57 thf(fact_3145_sin__coeff__0,axiom, % 0.31/0.57 ( ( sin_coeff @ ( zero_zero @ nat ) ) % 0.31/0.57 = ( zero_zero @ real ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_coeff_0 % 0.31/0.57 thf(fact_3146_one__less__exp__iff,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X3 ) ) % 0.31/0.57 = ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ). % 0.31/0.57 % 0.31/0.57 % one_less_exp_iff % 0.31/0.57 thf(fact_3147_exp__less__one__iff,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( exp @ real @ X3 ) @ ( one_one @ real ) ) % 0.31/0.57 = ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_less_one_iff % 0.31/0.57 thf(fact_3148_exp__ln__iff,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ( exp @ real @ ( ln_ln @ real @ X3 ) ) % 0.31/0.57 = X3 ) % 0.31/0.57 = ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_ln_iff % 0.31/0.57 thf(fact_3149_exp__ln,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( exp @ real @ ( ln_ln @ real @ X3 ) ) % 0.31/0.57 = X3 ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_ln % 0.31/0.57 thf(fact_3150_fact__mono__nat,axiom, % 0.31/0.57 ! [M: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.57 => ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_mono_nat % 0.31/0.57 thf(fact_3151_fact__ge__self,axiom, % 0.31/0.57 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( semiring_char_0_fact @ nat @ N2 ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_ge_self % 0.31/0.57 thf(fact_3152_exp__less__cancel,axiom, % 0.31/0.57 ! [X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( exp @ real @ X3 ) @ ( exp @ real @ Y ) ) % 0.31/0.57 => ( ord_less @ real @ X3 @ Y ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_less_cancel % 0.31/0.57 thf(fact_3153_exp__not__eq__zero,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.57 => ! [X3: A] : % 0.31/0.57 ( ( exp @ A @ X3 ) % 0.31/0.57 != ( zero_zero @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_not_eq_zero % 0.31/0.57 thf(fact_3154_fact__less__mono__nat,axiom, % 0.31/0.57 ! [M: nat,N2: nat] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.31/0.57 => ( ( ord_less @ nat @ M @ N2 ) % 0.31/0.57 => ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_less_mono_nat % 0.31/0.57 thf(fact_3155_exp__total,axiom, % 0.31/0.57 ! [Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 0.31/0.57 => ? [X5: real] : % 0.31/0.57 ( ( exp @ real @ X5 ) % 0.31/0.57 = Y ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_total % 0.31/0.57 thf(fact_3156_exp__gt__zero,axiom, % 0.31/0.57 ! [X3: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( exp @ real @ X3 ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_gt_zero % 0.31/0.57 thf(fact_3157_not__exp__less__zero,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ~ ( ord_less @ real @ ( exp @ real @ X3 ) @ ( zero_zero @ real ) ) ). % 0.31/0.57 % 0.31/0.57 % not_exp_less_zero % 0.31/0.57 thf(fact_3158_mult__exp__exp,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ! [X3: A,Y: A] : % 0.31/0.57 ( ( times_times @ A @ ( exp @ A @ X3 ) @ ( exp @ A @ Y ) ) % 0.31/0.57 = ( exp @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % mult_exp_exp % 0.31/0.57 thf(fact_3159_exp__add__commuting,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.57 => ! [X3: A,Y: A] : % 0.31/0.57 ( ( ( times_times @ A @ X3 @ Y ) % 0.31/0.57 = ( times_times @ A @ Y @ X3 ) ) % 0.31/0.57 => ( ( exp @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 0.31/0.57 = ( times_times @ A @ ( exp @ A @ X3 ) @ ( exp @ A @ Y ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_add_commuting % 0.31/0.57 thf(fact_3160_frac__ge__0,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.57 => ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X3 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % frac_ge_0 % 0.31/0.57 thf(fact_3161_frac__lt__1,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.57 => ! [X3: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X3 ) @ ( one_one @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % frac_lt_1 % 0.31/0.57 thf(fact_3162_frac__1__eq,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.57 => ! [X3: A] : % 0.31/0.57 ( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) ) % 0.31/0.57 = ( archimedean_frac @ A @ X3 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % frac_1_eq % 0.31/0.57 thf(fact_3163_fact__ge__Suc__0__nat,axiom, % 0.31/0.57 ! [N2: nat] : ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_ge_Suc_0_nat % 0.31/0.57 thf(fact_3164_exp__gt__one,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X3 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_gt_one % 0.31/0.57 thf(fact_3165_dvd__fact,axiom, % 0.31/0.57 ! [M: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M ) % 0.31/0.57 => ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.57 => ( dvd_dvd @ nat @ M @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % dvd_fact % 0.31/0.57 thf(fact_3166_fact__diff__Suc,axiom, % 0.31/0.57 ! [N2: nat,M: nat] : % 0.31/0.57 ( ( ord_less @ nat @ N2 @ ( suc @ M ) ) % 0.31/0.57 => ( ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N2 ) ) % 0.31/0.57 = ( times_times @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_diff_Suc % 0.31/0.57 thf(fact_3167_ln__ge__iff,axiom, % 0.31/0.57 ! [X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less_eq @ real @ Y @ ( ln_ln @ real @ X3 ) ) % 0.31/0.57 = ( ord_less_eq @ real @ ( exp @ real @ Y ) @ X3 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % ln_ge_iff % 0.31/0.57 thf(fact_3168_fact__div__fact__le__pow,axiom, % 0.31/0.57 ! [R2: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ R2 @ N2 ) % 0.31/0.57 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ R2 ) ) ) @ ( power_power @ nat @ N2 @ R2 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_div_fact_le_pow % 0.31/0.57 thf(fact_3169_powr__def,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ln @ A ) % 0.31/0.57 => ( ( powr @ A ) % 0.31/0.57 = ( ^ [X: A,A4: A] : % 0.31/0.57 ( if @ A % 0.31/0.57 @ ( X % 0.31/0.57 = ( zero_zero @ A ) ) % 0.31/0.57 @ ( zero_zero @ A ) % 0.31/0.57 @ ( exp @ A @ ( times_times @ A @ A4 @ ( ln_ln @ A @ X ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % powr_def % 0.31/0.57 thf(fact_3170_exp__divide__power__eq,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ! [N2: nat,X3: A] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.57 => ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X3 @ ( semiring_1_of_nat @ A @ N2 ) ) ) @ N2 ) % 0.31/0.57 = ( exp @ A @ X3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_divide_power_eq % 0.31/0.57 thf(fact_3171_frac__eq,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.57 => ! [X3: A] : % 0.31/0.57 ( ( ( archimedean_frac @ A @ X3 ) % 0.31/0.57 = X3 ) % 0.31/0.57 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.57 & ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % frac_eq % 0.31/0.57 thf(fact_3172_frac__add,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.57 => ! [X3: A,Y: A] : % 0.31/0.57 ( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) % 0.31/0.57 => ( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 0.31/0.57 = ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) ) ) % 0.31/0.57 & ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) % 0.31/0.57 => ( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 0.31/0.57 = ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % frac_add % 0.31/0.57 thf(fact_3173_tanh__altdef,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ( ( tanh @ A ) % 0.31/0.57 = ( ^ [X: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % tanh_altdef % 0.31/0.57 thf(fact_3174_sin__coeff__Suc,axiom, % 0.31/0.57 ! [N2: nat] : % 0.31/0.57 ( ( sin_coeff @ ( suc @ N2 ) ) % 0.31/0.57 = ( divide_divide @ real @ ( cos_coeff @ N2 ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_coeff_Suc % 0.31/0.57 thf(fact_3175_floor__add,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.57 => ! [X3: A,Y: A] : % 0.31/0.57 ( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) % 0.31/0.57 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 0.31/0.57 = ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) % 0.31/0.57 & ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) % 0.31/0.57 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 0.31/0.57 = ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % floor_add % 0.31/0.57 thf(fact_3176_cos__coeff__Suc,axiom, % 0.31/0.57 ! [N2: nat] : % 0.31/0.57 ( ( cos_coeff @ ( suc @ N2 ) ) % 0.31/0.57 = ( divide_divide @ real @ ( uminus_uminus @ real @ ( sin_coeff @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % cos_coeff_Suc % 0.31/0.57 thf(fact_3177_exp__ge__one__plus__x__over__n__power__n,axiom, % 0.31/0.57 ! [N2: nat,X3: real] : % 0.31/0.57 ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N2 ) ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.57 => ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X3 @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ N2 ) @ ( exp @ real @ X3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_ge_one_plus_x_over_n_power_n % 0.31/0.57 thf(fact_3178_exp__ge__one__minus__x__over__n__power__n,axiom, % 0.31/0.57 ! [X3: real,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ real @ X3 @ ( semiring_1_of_nat @ real @ N2 ) ) % 0.31/0.57 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.57 => ( ord_less_eq @ real @ ( power_power @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X3 @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ N2 ) @ ( exp @ real @ ( uminus_uminus @ real @ X3 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_ge_one_minus_x_over_n_power_n % 0.31/0.57 thf(fact_3179_log__base__10__eq2,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 0.31/0.57 = ( times_times @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % log_base_10_eq2 % 0.31/0.57 thf(fact_3180_log__base__10__eq1,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 0.31/0.57 = ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( ln_ln @ real @ X3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % log_base_10_eq1 % 0.31/0.57 thf(fact_3181_sin__tan,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ( sin @ real @ X3 ) % 0.31/0.57 = ( divide_divide @ real @ ( tan @ real @ X3 ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_tan % 0.31/0.57 thf(fact_3182_cos__tan,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ( cos @ real @ X3 ) % 0.31/0.57 = ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % cos_tan % 0.31/0.57 thf(fact_3183_monoseq__def,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( order @ A ) % 0.31/0.57 => ( ( topological_monoseq @ A ) % 0.31/0.57 = ( ^ [X6: nat > A] : % 0.31/0.57 ( ! [M2: nat,N: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ M2 @ N ) % 0.31/0.57 => ( ord_less_eq @ A @ ( X6 @ M2 ) @ ( X6 @ N ) ) ) % 0.31/0.57 | ! [M2: nat,N: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ M2 @ N ) % 0.31/0.57 => ( ord_less_eq @ A @ ( X6 @ N ) @ ( X6 @ M2 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % monoseq_def % 0.31/0.57 thf(fact_3184_monoI2,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( order @ A ) % 0.31/0.57 => ! [X8: nat > A] : % 0.31/0.57 ( ! [M3: nat,N3: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ M3 @ N3 ) % 0.31/0.57 => ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ M3 ) ) ) % 0.31/0.57 => ( topological_monoseq @ A @ X8 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % monoI2 % 0.31/0.57 thf(fact_3185_monoI1,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( order @ A ) % 0.31/0.57 => ! [X8: nat > A] : % 0.31/0.57 ( ! [M3: nat,N3: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ M3 @ N3 ) % 0.31/0.57 => ( ord_less_eq @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) % 0.31/0.57 => ( topological_monoseq @ A @ X8 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % monoI1 % 0.31/0.57 thf(fact_3186_real__sqrt__less__iff,axiom, % 0.31/0.57 ! [X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( sqrt @ X3 ) @ ( sqrt @ Y ) ) % 0.31/0.57 = ( ord_less @ real @ X3 @ Y ) ) ). % 0.31/0.57 % 0.31/0.57 % real_sqrt_less_iff % 0.31/0.57 thf(fact_3187_real__sqrt__lt__0__iff,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( sqrt @ X3 ) @ ( zero_zero @ real ) ) % 0.31/0.57 = ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ). % 0.31/0.57 % 0.31/0.57 % real_sqrt_lt_0_iff % 0.31/0.57 thf(fact_3188_real__sqrt__gt__0__iff,axiom, % 0.31/0.57 ! [Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ Y ) ) % 0.31/0.57 = ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ). % 0.31/0.57 % 0.31/0.57 % real_sqrt_gt_0_iff % 0.31/0.57 thf(fact_3189_real__sqrt__lt__1__iff,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( sqrt @ X3 ) @ ( one_one @ real ) ) % 0.31/0.57 = ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ). % 0.31/0.57 % 0.31/0.57 % real_sqrt_lt_1_iff % 0.31/0.57 thf(fact_3190_real__sqrt__gt__1__iff,axiom, % 0.31/0.57 ! [Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( one_one @ real ) @ ( sqrt @ Y ) ) % 0.31/0.57 = ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ). % 0.31/0.57 % 0.31/0.57 % real_sqrt_gt_1_iff % 0.31/0.57 thf(fact_3191_real__sqrt__less__mono,axiom, % 0.31/0.57 ! [X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ X3 @ Y ) % 0.31/0.57 => ( ord_less @ real @ ( sqrt @ X3 ) @ ( sqrt @ Y ) ) ) ). % 0.31/0.57 % 0.31/0.57 % real_sqrt_less_mono % 0.31/0.57 thf(fact_3192_real__sqrt__gt__zero,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ X3 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % real_sqrt_gt_zero % 0.31/0.57 thf(fact_3193_sqrt2__less__2,axiom, % 0.31/0.57 ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ). % 0.31/0.57 % 0.31/0.57 % sqrt2_less_2 % 0.31/0.57 thf(fact_3194_real__less__rsqrt,axiom, % 0.31/0.57 ! [X3: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) % 0.31/0.57 => ( ord_less @ real @ X3 @ ( sqrt @ Y ) ) ) ). % 0.31/0.57 % 0.31/0.57 % real_less_rsqrt % 0.31/0.57 thf(fact_3195_lemma__real__divide__sqrt__less,axiom, % 0.31/0.57 ! [U: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ U ) % 0.31/0.57 => ( ord_less @ real @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ U ) ) ). % 0.31/0.57 % 0.31/0.57 % lemma_real_divide_sqrt_less % 0.31/0.57 thf(fact_3196_real__less__lsqrt,axiom, % 0.31/0.57 ! [X3: real,Y: real] : % 0.31/0.57 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ord_less @ real @ ( sqrt @ X3 ) @ Y ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % real_less_lsqrt % 0.31/0.57 thf(fact_3197_ln__sqrt,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( ln_ln @ real @ ( sqrt @ X3 ) ) % 0.31/0.57 = ( divide_divide @ real @ ( ln_ln @ real @ X3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % ln_sqrt % 0.31/0.57 thf(fact_3198_arsinh__real__aux,axiom, % 0.31/0.57 ! [X3: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X3 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % arsinh_real_aux % 0.31/0.57 thf(fact_3199_real__sqrt__sum__squares__less,axiom, % 0.31/0.57 ! [X3: real,U: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.57 => ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.57 => ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ). % 0.31/0.57 % 0.31/0.57 % real_sqrt_sum_squares_less % 0.31/0.57 thf(fact_3200_sqrt__sum__squares__half__less,axiom, % 0.31/0.57 ! [X3: real,U: real,Y: real] : % 0.31/0.57 ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ( ord_less @ real @ Y @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) % 0.31/0.57 => ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sqrt_sum_squares_half_less % 0.31/0.57 thf(fact_3201_monoseq__Suc,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( order @ A ) % 0.31/0.57 => ( ( topological_monoseq @ A ) % 0.31/0.57 = ( ^ [X6: nat > A] : % 0.31/0.57 ( ! [N: nat] : ( ord_less_eq @ A @ ( X6 @ N ) @ ( X6 @ ( suc @ N ) ) ) % 0.31/0.57 | ! [N: nat] : ( ord_less_eq @ A @ ( X6 @ ( suc @ N ) ) @ ( X6 @ N ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % monoseq_Suc % 0.31/0.57 thf(fact_3202_mono__SucI2,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( order @ A ) % 0.31/0.57 => ! [X8: nat > A] : % 0.31/0.57 ( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) ) % 0.31/0.57 => ( topological_monoseq @ A @ X8 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % mono_SucI2 % 0.31/0.57 thf(fact_3203_mono__SucI1,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( order @ A ) % 0.31/0.57 => ! [X8: nat > A] : % 0.31/0.57 ( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) ) % 0.31/0.57 => ( topological_monoseq @ A @ X8 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % mono_SucI1 % 0.31/0.57 thf(fact_3204_pochhammer__double,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [Z2: A,N2: nat] : % 0.31/0.57 ( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.31/0.57 = ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_double % 0.31/0.57 thf(fact_3205_of__nat__code,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( semiring_1 @ A ) % 0.31/0.57 => ( ( semiring_1_of_nat @ A ) % 0.31/0.57 = ( ^ [N: nat] : % 0.31/0.57 ( semiri8178284476397505188at_aux @ A % 0.31/0.57 @ ^ [I3: A] : ( plus_plus @ A @ I3 @ ( one_one @ A ) ) % 0.31/0.57 @ N % 0.31/0.57 @ ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % of_nat_code % 0.31/0.57 thf(fact_3206_gchoose__row__sum__weighted,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [R2: A,M: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ R2 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) ) % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M ) ) % 0.31/0.57 = ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R2 @ ( suc @ M ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gchoose_row_sum_weighted % 0.31/0.57 thf(fact_3207_central__binomial__lower__bound,axiom, % 0.31/0.57 ! [N2: nat] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.57 => ( ord_less_eq @ real @ ( divide_divide @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ N2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ ( semiring_1_of_nat @ real @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % central_binomial_lower_bound % 0.31/0.57 thf(fact_3208_complex__unimodular__polar,axiom, % 0.31/0.57 ! [Z2: complex] : % 0.31/0.57 ( ( ( real_V7770717601297561774m_norm @ complex @ Z2 ) % 0.31/0.57 = ( one_one @ real ) ) % 0.31/0.57 => ~ ! [T7: real] : % 0.31/0.57 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 ) % 0.31/0.57 => ( ( ord_less @ real @ T7 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) % 0.31/0.57 => ( Z2 % 0.31/0.57 != ( complex2 @ ( cos @ real @ T7 ) @ ( sin @ real @ T7 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % complex_unimodular_polar % 0.31/0.57 thf(fact_3209_binomial__Suc__n,axiom, % 0.31/0.57 ! [N2: nat] : % 0.31/0.57 ( ( binomial @ ( suc @ N2 ) @ N2 ) % 0.31/0.57 = ( suc @ N2 ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_Suc_n % 0.31/0.57 thf(fact_3210_gbinomial__0_I2_J,axiom, % 0.31/0.57 ! [B: $tType] : % 0.31/0.57 ( ( ( semiring_char_0 @ B ) % 0.31/0.57 & ( semidom_divide @ B ) ) % 0.31/0.57 => ! [K: nat] : % 0.31/0.57 ( ( gbinomial @ B @ ( zero_zero @ B ) @ ( suc @ K ) ) % 0.31/0.57 = ( zero_zero @ B ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_0(2) % 0.31/0.57 thf(fact_3211_binomial__1,axiom, % 0.31/0.57 ! [N2: nat] : % 0.31/0.57 ( ( binomial @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 0.31/0.57 = N2 ) ). % 0.31/0.57 % 0.31/0.57 % binomial_1 % 0.31/0.57 thf(fact_3212_binomial__0__Suc,axiom, % 0.31/0.57 ! [K: nat] : % 0.31/0.57 ( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) ) % 0.31/0.57 = ( zero_zero @ nat ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_0_Suc % 0.31/0.57 thf(fact_3213_binomial__eq__0__iff,axiom, % 0.31/0.57 ! [N2: nat,K: nat] : % 0.31/0.57 ( ( ( binomial @ N2 @ K ) % 0.31/0.57 = ( zero_zero @ nat ) ) % 0.31/0.57 = ( ord_less @ nat @ N2 @ K ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_eq_0_iff % 0.31/0.57 thf(fact_3214_gbinomial__0_I1_J,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( semiring_char_0 @ A ) % 0.31/0.57 & ( semidom_divide @ A ) ) % 0.31/0.57 => ! [A2: A] : % 0.31/0.57 ( ( gbinomial @ A @ A2 @ ( zero_zero @ nat ) ) % 0.31/0.57 = ( one_one @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_0(1) % 0.31/0.57 thf(fact_3215_binomial__Suc__Suc,axiom, % 0.31/0.57 ! [N2: nat,K: nat] : % 0.31/0.57 ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) % 0.31/0.57 = ( plus_plus @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_Suc_Suc % 0.31/0.57 thf(fact_3216_pochhammer__0,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [A2: A] : % 0.31/0.57 ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( zero_zero @ nat ) ) % 0.31/0.57 = ( one_one @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_0 % 0.31/0.57 thf(fact_3217_gbinomial__Suc0,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( semiring_char_0 @ A ) % 0.31/0.57 & ( semidom_divide @ A ) ) % 0.31/0.57 => ! [A2: A] : % 0.31/0.57 ( ( gbinomial @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) ) % 0.31/0.57 = A2 ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_Suc0 % 0.31/0.57 thf(fact_3218_pochhammer__Suc0,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [A2: A] : % 0.31/0.57 ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) ) % 0.31/0.57 = A2 ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_Suc0 % 0.31/0.57 thf(fact_3219_binomial__n__0,axiom, % 0.31/0.57 ! [N2: nat] : % 0.31/0.57 ( ( binomial @ N2 @ ( zero_zero @ nat ) ) % 0.31/0.57 = ( one_one @ nat ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_n_0 % 0.31/0.57 thf(fact_3220_zero__less__binomial__iff,axiom, % 0.31/0.57 ! [N2: nat,K: nat] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N2 @ K ) ) % 0.31/0.57 = ( ord_less_eq @ nat @ K @ N2 ) ) ). % 0.31/0.57 % 0.31/0.57 % zero_less_binomial_iff % 0.31/0.57 thf(fact_3221_binomial__eq__0,axiom, % 0.31/0.57 ! [N2: nat,K: nat] : % 0.31/0.57 ( ( ord_less @ nat @ N2 @ K ) % 0.31/0.57 => ( ( binomial @ N2 @ K ) % 0.31/0.57 = ( zero_zero @ nat ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_eq_0 % 0.31/0.57 thf(fact_3222_Suc__times__binomial,axiom, % 0.31/0.57 ! [K: nat,N2: nat] : % 0.31/0.57 ( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) ) % 0.31/0.57 = ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ). % 0.31/0.57 % 0.31/0.57 % Suc_times_binomial % 0.31/0.57 thf(fact_3223_Suc__times__binomial__eq,axiom, % 0.31/0.57 ! [N2: nat,K: nat] : % 0.31/0.57 ( ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) % 0.31/0.57 = ( times_times @ nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ). % 0.31/0.57 % 0.31/0.57 % Suc_times_binomial_eq % 0.31/0.57 thf(fact_3224_binomial__symmetric,axiom, % 0.31/0.57 ! [K: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.57 => ( ( binomial @ N2 @ K ) % 0.31/0.57 = ( binomial @ N2 @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_symmetric % 0.31/0.57 thf(fact_3225_choose__mult__lemma,axiom, % 0.31/0.57 ! [M: nat,R2: nat,K: nat] : % 0.31/0.57 ( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R2 ) @ K ) @ ( plus_plus @ nat @ M @ K ) ) @ ( binomial @ ( plus_plus @ nat @ M @ K ) @ K ) ) % 0.31/0.57 = ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus @ nat @ M @ R2 ) @ M ) ) ) ). % 0.31/0.57 % 0.31/0.57 % choose_mult_lemma % 0.31/0.57 thf(fact_3226_binomial__le__pow,axiom, % 0.31/0.57 ! [R2: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ R2 @ N2 ) % 0.31/0.57 => ( ord_less_eq @ nat @ ( binomial @ N2 @ R2 ) @ ( power_power @ nat @ N2 @ R2 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_le_pow % 0.31/0.57 thf(fact_3227_pochhammer__pos,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [X3: A,N2: nat] : % 0.31/0.57 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.57 => ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X3 @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_pos % 0.31/0.57 thf(fact_3228_pochhammer__neq__0__mono,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [A2: A,M: nat,N2: nat] : % 0.31/0.57 ( ( ( comm_s3205402744901411588hammer @ A @ A2 @ M ) % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( ord_less_eq @ nat @ N2 @ M ) % 0.31/0.57 => ( ( comm_s3205402744901411588hammer @ A @ A2 @ N2 ) % 0.31/0.57 != ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_neq_0_mono % 0.31/0.57 thf(fact_3229_pochhammer__eq__0__mono,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [A2: A,N2: nat,M: nat] : % 0.31/0.57 ( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N2 ) % 0.31/0.57 = ( zero_zero @ A ) ) % 0.31/0.57 => ( ( ord_less_eq @ nat @ N2 @ M ) % 0.31/0.57 => ( ( comm_s3205402744901411588hammer @ A @ A2 @ M ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_eq_0_mono % 0.31/0.57 thf(fact_3230_zero__less__binomial,axiom, % 0.31/0.57 ! [K: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.57 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N2 @ K ) ) ) ). % 0.31/0.57 % 0.31/0.57 % zero_less_binomial % 0.31/0.57 thf(fact_3231_Suc__times__binomial__add,axiom, % 0.31/0.57 ! [A2: nat,B2: nat] : % 0.31/0.57 ( ( times_times @ nat @ ( suc @ A2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( suc @ A2 ) ) ) % 0.31/0.57 = ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ A2 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % Suc_times_binomial_add % 0.31/0.57 thf(fact_3232_choose__mult,axiom, % 0.31/0.57 ! [K: nat,M: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ K @ M ) % 0.31/0.57 => ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.57 => ( ( times_times @ nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) ) % 0.31/0.57 = ( times_times @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus @ nat @ N2 @ K ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % choose_mult % 0.31/0.57 thf(fact_3233_binomial__Suc__Suc__eq__times,axiom, % 0.31/0.57 ! [N2: nat,K: nat] : % 0.31/0.57 ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) % 0.31/0.57 = ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_Suc_Suc_eq_times % 0.31/0.57 thf(fact_3234_gbinomial__pochhammer_H,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ( ( gbinomial @ A ) % 0.31/0.57 = ( ^ [A4: A,K3: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_pochhammer' % 0.31/0.57 thf(fact_3235_gbinomial__Suc__Suc,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [A2: A,K: nat] : % 0.31/0.57 ( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) % 0.31/0.57 = ( plus_plus @ A @ ( gbinomial @ A @ A2 @ K ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_Suc_Suc % 0.31/0.57 thf(fact_3236_gbinomial__of__nat__symmetric,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [K: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.57 => ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N2 ) @ K ) % 0.31/0.57 = ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_of_nat_symmetric % 0.31/0.57 thf(fact_3237_pochhammer__nonneg,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [X3: A,N2: nat] : % 0.31/0.57 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.57 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X3 @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_nonneg % 0.31/0.57 thf(fact_3238_of__nat__aux_Osimps_I2_J,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( semiring_1 @ A ) % 0.31/0.57 => ! [Inc: A > A,N2: nat,I: A] : % 0.31/0.57 ( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N2 ) @ I ) % 0.31/0.57 = ( semiri8178284476397505188at_aux @ A @ Inc @ N2 @ ( Inc @ I ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % of_nat_aux.simps(2) % 0.31/0.57 thf(fact_3239_of__nat__aux_Osimps_I1_J,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( semiring_1 @ A ) % 0.31/0.57 => ! [Inc: A > A,I: A] : % 0.31/0.57 ( ( semiri8178284476397505188at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I ) % 0.31/0.57 = I ) ) ). % 0.31/0.57 % 0.31/0.57 % of_nat_aux.simps(1) % 0.31/0.57 thf(fact_3240_pochhammer__0__left,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [N2: nat] : % 0.31/0.57 ( ( ( N2 % 0.31/0.57 = ( zero_zero @ nat ) ) % 0.31/0.57 => ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N2 ) % 0.31/0.57 = ( one_one @ A ) ) ) % 0.31/0.57 & ( ( N2 % 0.31/0.57 != ( zero_zero @ nat ) ) % 0.31/0.57 => ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N2 ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_0_left % 0.31/0.57 thf(fact_3241_binomial__absorption,axiom, % 0.31/0.57 ! [K: nat,N2: nat] : % 0.31/0.57 ( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) % 0.31/0.57 = ( times_times @ nat @ N2 @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_absorption % 0.31/0.57 thf(fact_3242_gbinomial__addition__formula,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [A2: A,K: nat] : % 0.31/0.57 ( ( gbinomial @ A @ A2 @ ( suc @ K ) ) % 0.31/0.57 = ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_addition_formula % 0.31/0.57 thf(fact_3243_gbinomial__ge__n__over__k__pow__k,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( linordered_field @ A ) % 0.31/0.57 => ! [K: nat,A2: A] : % 0.31/0.57 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K ) @ A2 ) % 0.31/0.57 => ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_ge_n_over_k_pow_k % 0.31/0.57 thf(fact_3244_gbinomial__mult__1_H,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [A2: A,K: nat] : % 0.31/0.57 ( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ A2 ) % 0.31/0.57 = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_mult_1' % 0.31/0.57 thf(fact_3245_gbinomial__mult__1,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [A2: A,K: nat] : % 0.31/0.57 ( ( times_times @ A @ A2 @ ( gbinomial @ A @ A2 @ K ) ) % 0.31/0.57 = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_mult_1 % 0.31/0.57 thf(fact_3246_binomial__fact__lemma,axiom, % 0.31/0.57 ! [K: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.57 => ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) ) % 0.31/0.57 = ( semiring_char_0_fact @ nat @ N2 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_fact_lemma % 0.31/0.57 thf(fact_3247_pochhammer__rec,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [A2: A,N2: nat] : % 0.31/0.57 ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) ) % 0.31/0.57 = ( times_times @ A @ A2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_rec % 0.31/0.57 thf(fact_3248_pochhammer__rec_H,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [Z2: A,N2: nat] : % 0.31/0.57 ( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N2 ) ) % 0.31/0.57 = ( times_times @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_rec' % 0.31/0.57 thf(fact_3249_pochhammer__Suc,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [A2: A,N2: nat] : % 0.31/0.57 ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) ) % 0.31/0.57 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A2 @ N2 ) @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_Suc % 0.31/0.57 thf(fact_3250_pochhammer__eq__0__iff,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [A2: A,N2: nat] : % 0.31/0.57 ( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N2 ) % 0.31/0.57 = ( zero_zero @ A ) ) % 0.31/0.57 = ( ? [K3: nat] : % 0.31/0.57 ( ( ord_less @ nat @ K3 @ N2 ) % 0.31/0.57 & ( A2 % 0.31/0.57 = ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_eq_0_iff % 0.31/0.57 thf(fact_3251_pochhammer__of__nat__eq__0__iff,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( ring_char_0 @ A ) % 0.31/0.57 & ( idom @ A ) ) % 0.31/0.57 => ! [N2: nat,K: nat] : % 0.31/0.57 ( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K ) % 0.31/0.57 = ( zero_zero @ A ) ) % 0.31/0.57 = ( ord_less @ nat @ N2 @ K ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_of_nat_eq_0_iff % 0.31/0.57 thf(fact_3252_pochhammer__of__nat__eq__0__lemma,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( idom @ A ) % 0.31/0.57 => ! [N2: nat,K: nat] : % 0.31/0.57 ( ( ord_less @ nat @ N2 @ K ) % 0.31/0.57 => ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_of_nat_eq_0_lemma % 0.31/0.57 thf(fact_3253_pochhammer__of__nat__eq__0__lemma_H,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( ring_char_0 @ A ) % 0.31/0.57 & ( idom @ A ) ) % 0.31/0.57 => ! [K: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.57 => ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K ) % 0.31/0.57 != ( zero_zero @ A ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_of_nat_eq_0_lemma' % 0.31/0.57 thf(fact_3254_pochhammer__product_H,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [Z2: A,N2: nat,M: nat] : % 0.31/0.57 ( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( plus_plus @ nat @ N2 @ M ) ) % 0.31/0.57 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N2 ) ) @ M ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_product' % 0.31/0.57 thf(fact_3255_binomial__ge__n__over__k__pow__k,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( linordered_field @ A ) % 0.31/0.57 => ! [K: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.57 => ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_ge_n_over_k_pow_k % 0.31/0.57 thf(fact_3256_binomial__maximum_H,axiom, % 0.31/0.57 ! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ N2 ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_maximum' % 0.31/0.57 thf(fact_3257_binomial__mono,axiom, % 0.31/0.57 ! [K: nat,K7: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ K @ K7 ) % 0.31/0.57 => ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N2 ) % 0.31/0.57 => ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K7 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_mono % 0.31/0.57 thf(fact_3258_binomial__antimono,axiom, % 0.31/0.57 ! [K: nat,K7: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ K @ K7 ) % 0.31/0.57 => ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K ) % 0.31/0.57 => ( ( ord_less_eq @ nat @ K7 @ N2 ) % 0.31/0.57 => ( ord_less_eq @ nat @ ( binomial @ N2 @ K7 ) @ ( binomial @ N2 @ K ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_antimono % 0.31/0.57 thf(fact_3259_binomial__maximum,axiom, % 0.31/0.57 ! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_maximum % 0.31/0.57 thf(fact_3260_binomial__le__pow2,axiom, % 0.31/0.57 ! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_le_pow2 % 0.31/0.57 thf(fact_3261_choose__reduce__nat,axiom, % 0.31/0.57 ! [N2: nat,K: nat] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.57 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.31/0.57 => ( ( binomial @ N2 @ K ) % 0.31/0.57 = ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % choose_reduce_nat % 0.31/0.57 thf(fact_3262_times__binomial__minus1__eq,axiom, % 0.31/0.57 ! [K: nat,N2: nat] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.31/0.57 => ( ( times_times @ nat @ K @ ( binomial @ N2 @ K ) ) % 0.31/0.57 = ( times_times @ nat @ N2 @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % times_binomial_minus1_eq % 0.31/0.57 thf(fact_3263_Suc__times__gbinomial,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [K: nat,A2: A] : % 0.31/0.57 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) ) % 0.31/0.57 = ( times_times @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % Suc_times_gbinomial % 0.31/0.57 thf(fact_3264_gbinomial__absorption,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [K: nat,A2: A] : % 0.31/0.57 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) % 0.31/0.57 = ( times_times @ A @ A2 @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_absorption % 0.31/0.57 thf(fact_3265_binomial__altdef__nat,axiom, % 0.31/0.57 ! [K: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.57 => ( ( binomial @ N2 @ K ) % 0.31/0.57 = ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N2 ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_altdef_nat % 0.31/0.57 thf(fact_3266_gbinomial__trinomial__revision,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [K: nat,M: nat,A2: A] : % 0.31/0.57 ( ( ord_less_eq @ nat @ K @ M ) % 0.31/0.57 => ( ( times_times @ A @ ( gbinomial @ A @ A2 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) ) % 0.31/0.57 = ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_trinomial_revision % 0.31/0.57 thf(fact_3267_pochhammer__product,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [M: nat,N2: nat,Z2: A] : % 0.31/0.57 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.57 => ( ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) % 0.31/0.57 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_product % 0.31/0.57 thf(fact_3268_binomial__strict__antimono,axiom, % 0.31/0.57 ! [K: nat,K7: nat,N2: nat] : % 0.31/0.57 ( ( ord_less @ nat @ K @ K7 ) % 0.31/0.57 => ( ( ord_less_eq @ nat @ N2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) ) % 0.31/0.57 => ( ( ord_less_eq @ nat @ K7 @ N2 ) % 0.31/0.57 => ( ord_less @ nat @ ( binomial @ N2 @ K7 ) @ ( binomial @ N2 @ K ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_strict_antimono % 0.31/0.57 thf(fact_3269_binomial__strict__mono,axiom, % 0.31/0.57 ! [K: nat,K7: nat,N2: nat] : % 0.31/0.57 ( ( ord_less @ nat @ K @ K7 ) % 0.31/0.57 => ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N2 ) % 0.31/0.57 => ( ord_less @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K7 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_strict_mono % 0.31/0.57 thf(fact_3270_binomial__less__binomial__Suc,axiom, % 0.31/0.57 ! [K: nat,N2: nat] : % 0.31/0.57 ( ( ord_less @ nat @ K @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 => ( ord_less @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_less_binomial_Suc % 0.31/0.57 thf(fact_3271_central__binomial__odd,axiom, % 0.31/0.57 ! [N2: nat] : % 0.31/0.57 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.31/0.57 => ( ( binomial @ N2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.57 = ( binomial @ N2 @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % central_binomial_odd % 0.31/0.57 thf(fact_3272_binomial__addition__formula,axiom, % 0.31/0.57 ! [N2: nat,K: nat] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.57 => ( ( binomial @ N2 @ ( suc @ K ) ) % 0.31/0.57 = ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_addition_formula % 0.31/0.57 thf(fact_3273_binomial__fact,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [K: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.57 => ( ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) % 0.31/0.57 = ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_fact % 0.31/0.57 thf(fact_3274_fact__binomial,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [K: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.57 => ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) ) % 0.31/0.57 = ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_binomial % 0.31/0.57 thf(fact_3275_gbinomial__rec,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [A2: A,K: nat] : % 0.31/0.57 ( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) % 0.31/0.57 = ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_rec % 0.31/0.57 thf(fact_3276_gbinomial__factors,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [A2: A,K: nat] : % 0.31/0.57 ( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) % 0.31/0.57 = ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_factors % 0.31/0.57 thf(fact_3277_pochhammer__absorb__comp,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_ring_1 @ A ) % 0.31/0.57 => ! [R2: A,K: nat] : % 0.31/0.57 ( ( times_times @ A @ ( minus_minus @ A @ R2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R2 ) @ K ) ) % 0.31/0.57 = ( times_times @ A @ R2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R2 ) @ ( one_one @ A ) ) @ K ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_absorb_comp % 0.31/0.57 thf(fact_3278_gbinomial__minus,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [A2: A,K: nat] : % 0.31/0.57 ( ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K ) % 0.31/0.57 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_minus % 0.31/0.57 thf(fact_3279_gbinomial__reduce__nat,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [K: nat,A2: A] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.31/0.57 => ( ( gbinomial @ A @ A2 @ K ) % 0.31/0.57 = ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_reduce_nat % 0.31/0.57 thf(fact_3280_pochhammer__minus,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_ring_1 @ A ) % 0.31/0.57 => ! [B2: A,K: nat] : % 0.31/0.57 ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) % 0.31/0.57 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_minus % 0.31/0.57 thf(fact_3281_pochhammer__minus_H,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_ring_1 @ A ) % 0.31/0.57 => ! [B2: A,K: nat] : % 0.31/0.57 ( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) % 0.31/0.57 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_minus' % 0.31/0.57 thf(fact_3282_gbinomial__sum__up__index,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [K: nat,N2: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K ) % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.31/0.57 = ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_sum_up_index % 0.31/0.57 thf(fact_3283_gbinomial__absorption_H,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [K: nat,A2: A] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.31/0.57 => ( ( gbinomial @ A @ A2 @ K ) % 0.31/0.57 = ( times_times @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_absorption' % 0.31/0.57 thf(fact_3284_binomial__code,axiom, % 0.31/0.57 ( binomial % 0.31/0.57 = ( ^ [N: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N @ ( minus_minus @ nat @ N @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N @ K3 ) @ ( one_one @ nat ) ) @ N @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_code % 0.31/0.57 thf(fact_3285_gbinomial__code,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ( ( gbinomial @ A ) % 0.31/0.57 = ( ^ [A4: A,K3: nat] : % 0.31/0.57 ( if @ A % 0.31/0.57 @ ( K3 % 0.31/0.57 = ( zero_zero @ nat ) ) % 0.31/0.57 @ ( one_one @ A ) % 0.31/0.57 @ ( divide_divide @ A % 0.31/0.57 @ ( set_fo6178422350223883121st_nat @ A % 0.31/0.57 @ ^ [L2: nat] : ( times_times @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ L2 ) ) ) % 0.31/0.57 @ ( zero_zero @ nat ) % 0.31/0.57 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) % 0.31/0.57 @ ( one_one @ A ) ) % 0.31/0.57 @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_code % 0.31/0.57 thf(fact_3286_pochhammer__times__pochhammer__half,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [Z2: A,N2: nat] : % 0.31/0.57 ( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N2 ) ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_times_pochhammer_half % 0.31/0.57 thf(fact_3287_pochhammer__code,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ( ( comm_s3205402744901411588hammer @ A ) % 0.31/0.57 = ( ^ [A4: A,N: nat] : % 0.31/0.57 ( if @ A % 0.31/0.57 @ ( N % 0.31/0.57 = ( zero_zero @ nat ) ) % 0.31/0.57 @ ( one_one @ A ) % 0.31/0.57 @ ( set_fo6178422350223883121st_nat @ A % 0.31/0.57 @ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ O ) ) ) % 0.31/0.57 @ ( zero_zero @ nat ) % 0.31/0.57 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) % 0.31/0.57 @ ( one_one @ A ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_code % 0.31/0.57 thf(fact_3288_gbinomial__partial__row__sum,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [A2: A,M: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ M ) ) % 0.31/0.57 = ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A2 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_partial_row_sum % 0.31/0.57 thf(fact_3289_atMost__iff,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ord @ A ) % 0.31/0.57 => ! [I: A,K: A] : % 0.31/0.57 ( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) ) % 0.31/0.57 = ( ord_less_eq @ A @ I @ K ) ) ) ). % 0.31/0.57 % 0.31/0.57 % atMost_iff % 0.31/0.57 thf(fact_3290_finite__atMost,axiom, % 0.31/0.57 ! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_atMost @ nat @ K ) ) ). % 0.31/0.57 % 0.31/0.57 % finite_atMost % 0.31/0.57 thf(fact_3291_of__nat__prod,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [F2: B > nat,A3: set @ B] : % 0.31/0.57 ( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A3 ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ A % 0.31/0.57 @ ^ [X: B] : ( semiring_1_of_nat @ A @ ( F2 @ X ) ) % 0.31/0.57 @ A3 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % of_nat_prod % 0.31/0.57 thf(fact_3292_of__int__prod,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_ring_1 @ A ) % 0.31/0.57 => ! [F2: B > int,A3: set @ B] : % 0.31/0.57 ( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F2 @ A3 ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ A % 0.31/0.57 @ ^ [X: B] : ( ring_1_of_int @ A @ ( F2 @ X ) ) % 0.31/0.57 @ A3 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % of_int_prod % 0.31/0.57 thf(fact_3293_prod__zero__iff,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( semidom @ A ) % 0.31/0.57 => ! [A3: set @ B,F2: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) % 0.31/0.57 = ( zero_zero @ A ) ) % 0.31/0.57 = ( ? [X: B] : % 0.31/0.57 ( ( member @ B @ X @ A3 ) % 0.31/0.57 & ( ( F2 @ X ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod_zero_iff % 0.31/0.57 thf(fact_3294_prod_Oempty,axiom, % 0.31/0.57 ! [B: $tType,A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [G: B > A] : % 0.31/0.57 ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) ) % 0.31/0.57 = ( one_one @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.empty % 0.31/0.57 thf(fact_3295_prod_Oinfinite,axiom, % 0.31/0.57 ! [B: $tType,A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [A3: set @ B,G: B > A] : % 0.31/0.57 ( ~ ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) % 0.31/0.57 = ( one_one @ A ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.infinite % 0.31/0.57 thf(fact_3296_atMost__subset__iff,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( preorder @ A ) % 0.31/0.57 => ! [X3: A,Y: A] : % 0.31/0.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X3 ) @ ( set_ord_atMost @ A @ Y ) ) % 0.31/0.57 = ( ord_less_eq @ A @ X3 @ Y ) ) ) ). % 0.31/0.57 % 0.31/0.57 % atMost_subset_iff % 0.31/0.57 thf(fact_3297_dvd__prod__eqI,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [A3: set @ B,A2: B,B2: A,F2: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ( member @ B @ A2 @ A3 ) % 0.31/0.57 => ( ( B2 % 0.31/0.57 = ( F2 @ A2 ) ) % 0.31/0.57 => ( dvd_dvd @ A @ B2 @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % dvd_prod_eqI % 0.31/0.57 thf(fact_3298_dvd__prodI,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [A3: set @ B,A2: B,F2: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ( member @ B @ A2 @ A3 ) % 0.31/0.57 => ( dvd_dvd @ A @ ( F2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % dvd_prodI % 0.31/0.57 thf(fact_3299_prod_Odelta,axiom, % 0.31/0.57 ! [B: $tType,A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [S3: set @ B,A2: B,B2: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ S3 ) % 0.31/0.57 => ( ( ( member @ B @ A2 @ S3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A % 0.31/0.57 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) ) % 0.31/0.57 @ S3 ) % 0.31/0.57 = ( B2 @ A2 ) ) ) % 0.31/0.57 & ( ~ ( member @ B @ A2 @ S3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A % 0.31/0.57 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) ) % 0.31/0.57 @ S3 ) % 0.31/0.57 = ( one_one @ A ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.delta % 0.31/0.57 thf(fact_3300_prod_Odelta_H,axiom, % 0.31/0.57 ! [B: $tType,A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [S3: set @ B,A2: B,B2: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ S3 ) % 0.31/0.57 => ( ( ( member @ B @ A2 @ S3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A % 0.31/0.57 @ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) ) % 0.31/0.57 @ S3 ) % 0.31/0.57 = ( B2 @ A2 ) ) ) % 0.31/0.57 & ( ~ ( member @ B @ A2 @ S3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A % 0.31/0.57 @ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) ) % 0.31/0.57 @ S3 ) % 0.31/0.57 = ( one_one @ A ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.delta' % 0.31/0.57 thf(fact_3301_prod_Oinsert,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [A3: set @ B,X3: B,G: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ~ ( member @ B @ X3 @ A3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 0.31/0.57 = ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.insert % 0.31/0.57 thf(fact_3302_Icc__subset__Iic__iff,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( preorder @ A ) % 0.31/0.57 => ! [L: A,H2: A,H3: A] : % 0.31/0.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atMost @ A @ H3 ) ) % 0.31/0.57 = ( ~ ( ord_less_eq @ A @ L @ H2 ) % 0.31/0.57 | ( ord_less_eq @ A @ H2 @ H3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % Icc_subset_Iic_iff % 0.31/0.57 thf(fact_3303_sum_OatMost__Suc,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_add @ A ) % 0.31/0.57 => ! [G: nat > A,N2: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) ) % 0.31/0.57 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum.atMost_Suc % 0.31/0.57 thf(fact_3304_prod_OlessThan__Suc,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [G: nat > A,N2: nat] : % 0.31/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) % 0.31/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( G @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.lessThan_Suc % 0.31/0.57 thf(fact_3305_prod_OatMost__Suc,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [G: nat > A,N2: nat] : % 0.31/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) ) % 0.31/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.atMost_Suc % 0.31/0.57 thf(fact_3306_atMost__0,axiom, % 0.31/0.57 ( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) ) % 0.31/0.57 = ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % atMost_0 % 0.31/0.57 thf(fact_3307_prod_Ocl__ivl__Suc,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [N2: nat,M: nat,G: nat > A] : % 0.31/0.57 ( ( ( ord_less @ nat @ ( suc @ N2 ) @ M ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) ) % 0.31/0.57 = ( one_one @ A ) ) ) % 0.31/0.57 & ( ~ ( ord_less @ nat @ ( suc @ N2 ) @ M ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) ) % 0.31/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.cl_ivl_Suc % 0.31/0.57 thf(fact_3308_not__empty__eq__Iic__eq__empty,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( preorder @ A ) % 0.31/0.57 => ! [H2: A] : % 0.31/0.57 ( ( bot_bot @ ( set @ A ) ) % 0.31/0.57 != ( set_ord_atMost @ A @ H2 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % not_empty_eq_Iic_eq_empty % 0.31/0.57 thf(fact_3309_infinite__Iic,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( linorder @ A ) % 0.31/0.57 & ( no_bot @ A ) ) % 0.31/0.57 => ! [A2: A] : % 0.31/0.57 ~ ( finite_finite2 @ A @ ( set_ord_atMost @ A @ A2 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % infinite_Iic % 0.31/0.57 thf(fact_3310_prod_Oswap__restrict,axiom, % 0.31/0.57 ! [A: $tType,B: $tType,C: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [A3: set @ B,B3: set @ C,G: B > C > A,R: B > C > $o] : % 0.31/0.57 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ( finite_finite2 @ C @ B3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A % 0.31/0.57 @ ^ [X: B] : % 0.31/0.57 ( groups7121269368397514597t_prod @ C @ A @ ( G @ X ) % 0.31/0.57 @ ( collect @ C % 0.31/0.57 @ ^ [Y2: C] : % 0.31/0.57 ( ( member @ C @ Y2 @ B3 ) % 0.31/0.57 & ( R @ X @ Y2 ) ) ) ) % 0.31/0.57 @ A3 ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ C @ A % 0.31/0.57 @ ^ [Y2: C] : % 0.31/0.57 ( groups7121269368397514597t_prod @ B @ A % 0.31/0.57 @ ^ [X: B] : ( G @ X @ Y2 ) % 0.31/0.57 @ ( collect @ B % 0.31/0.57 @ ^ [X: B] : % 0.31/0.57 ( ( member @ B @ X @ A3 ) % 0.31/0.57 & ( R @ X @ Y2 ) ) ) ) % 0.31/0.57 @ B3 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.swap_restrict % 0.31/0.57 thf(fact_3311_prod_OatMost__Suc__shift,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [G: nat > A,N2: nat] : % 0.31/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) ) % 0.31/0.57 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) % 0.31/0.57 @ ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.atMost_Suc_shift % 0.31/0.57 thf(fact_3312_atMost__def,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ord @ A ) % 0.31/0.57 => ( ( set_ord_atMost @ A ) % 0.31/0.57 = ( ^ [U2: A] : % 0.31/0.57 ( collect @ A % 0.31/0.57 @ ^ [X: A] : ( ord_less_eq @ A @ X @ U2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % atMost_def % 0.31/0.57 thf(fact_3313_prod_Onested__swap_H,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [A2: nat > nat > A,N2: nat] : % 0.31/0.57 ( ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [J3: nat] : % 0.31/0.57 ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( A2 @ I3 @ J3 ) % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.nested_swap' % 0.31/0.57 thf(fact_3314_prod__mono,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [A3: set @ B,F2: B > A,G: B > A] : % 0.31/0.57 ( ! [I4: B] : % 0.31/0.57 ( ( member @ B @ I4 @ A3 ) % 0.31/0.57 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) % 0.31/0.57 & ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) ) ) % 0.31/0.57 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod_mono % 0.31/0.57 thf(fact_3315_prod__nonneg,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [A3: set @ B,F2: B > A] : % 0.31/0.57 ( ! [X5: B] : % 0.31/0.57 ( ( member @ B @ X5 @ A3 ) % 0.31/0.57 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) ) % 0.31/0.57 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod_nonneg % 0.31/0.57 thf(fact_3316_prod__pos,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [A3: set @ B,F2: B > A] : % 0.31/0.57 ( ! [X5: B] : % 0.31/0.57 ( ( member @ B @ X5 @ A3 ) % 0.31/0.57 => ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) ) % 0.31/0.57 => ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod_pos % 0.31/0.57 thf(fact_3317_prod__ge__1,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( linord181362715937106298miring @ A ) % 0.31/0.57 => ! [A3: set @ B,F2: B > A] : % 0.31/0.57 ( ! [X5: B] : % 0.31/0.57 ( ( member @ B @ X5 @ A3 ) % 0.31/0.57 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( F2 @ X5 ) ) ) % 0.31/0.57 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod_ge_1 % 0.31/0.57 thf(fact_3318_prod__zero,axiom, % 0.31/0.57 ! [B: $tType,A: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [A3: set @ B,F2: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ? [X2: B] : % 0.31/0.57 ( ( member @ B @ X2 @ A3 ) % 0.31/0.57 & ( ( F2 @ X2 ) % 0.31/0.57 = ( zero_zero @ A ) ) ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod_zero % 0.31/0.57 thf(fact_3319_prod_OatMost__shift,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [G: nat > A,N2: nat] : % 0.31/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) % 0.31/0.57 @ ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.atMost_shift % 0.31/0.57 thf(fact_3320_atMost__atLeast0,axiom, % 0.31/0.57 ( ( set_ord_atMost @ nat ) % 0.31/0.57 = ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ). % 0.31/0.57 % 0.31/0.57 % atMost_atLeast0 % 0.31/0.57 thf(fact_3321_lessThan__Suc__atMost,axiom, % 0.31/0.57 ! [K: nat] : % 0.31/0.57 ( ( set_ord_lessThan @ nat @ ( suc @ K ) ) % 0.31/0.57 = ( set_ord_atMost @ nat @ K ) ) ). % 0.31/0.57 % 0.31/0.57 % lessThan_Suc_atMost % 0.31/0.57 thf(fact_3322_atMost__Suc,axiom, % 0.31/0.57 ! [K: nat] : % 0.31/0.57 ( ( set_ord_atMost @ nat @ ( suc @ K ) ) % 0.31/0.57 = ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ). % 0.31/0.57 % 0.31/0.57 % atMost_Suc % 0.31/0.57 thf(fact_3323_prod_Ointer__filter,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [A3: set @ B,G: B > A,P: B > $o] : % 0.31/0.57 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G % 0.31/0.57 @ ( collect @ B % 0.31/0.57 @ ^ [X: B] : % 0.31/0.57 ( ( member @ B @ X @ A3 ) % 0.31/0.57 & ( P @ X ) ) ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ A % 0.31/0.57 @ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( G @ X ) @ ( one_one @ A ) ) % 0.31/0.57 @ A3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.inter_filter % 0.31/0.57 thf(fact_3324_not__Iic__le__Icc,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( no_bot @ A ) % 0.31/0.57 => ! [H2: A,L3: A,H3: A] : % 0.31/0.57 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % not_Iic_le_Icc % 0.31/0.57 thf(fact_3325_prod_Oshift__bounds__cl__Suc__ivl,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [G: nat > A,M: nat,N2: nat] : % 0.31/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.shift_bounds_cl_Suc_ivl % 0.31/0.57 thf(fact_3326_prod_Oshift__bounds__cl__nat__ivl,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 0.31/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) ) % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.shift_bounds_cl_nat_ivl % 0.31/0.57 thf(fact_3327_finite__nat__iff__bounded__le,axiom, % 0.31/0.57 ( ( finite_finite2 @ nat ) % 0.31/0.57 = ( ^ [S7: set @ nat] : % 0.31/0.57 ? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S7 @ ( set_ord_atMost @ nat @ K3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % finite_nat_iff_bounded_le % 0.31/0.57 thf(fact_3328_prod__le__1,axiom, % 0.31/0.57 ! [B: $tType,A: $tType] : % 0.31/0.57 ( ( linord181362715937106298miring @ A ) % 0.31/0.57 => ! [A3: set @ B,F2: B > A] : % 0.31/0.57 ( ! [X5: B] : % 0.31/0.57 ( ( member @ B @ X5 @ A3 ) % 0.31/0.57 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) % 0.31/0.57 & ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( one_one @ A ) ) ) ) % 0.31/0.57 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( one_one @ A ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod_le_1 % 0.31/0.57 thf(fact_3329_prod_Orelated,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [R: A > A > $o,S3: set @ B,H2: B > A,G: B > A] : % 0.31/0.57 ( ( R @ ( one_one @ A ) @ ( one_one @ A ) ) % 0.31/0.57 => ( ! [X16: A,Y15: A,X23: A,Y23: A] : % 0.31/0.57 ( ( ( R @ X16 @ X23 ) % 0.31/0.57 & ( R @ Y15 @ Y23 ) ) % 0.31/0.57 => ( R @ ( times_times @ A @ X16 @ Y15 ) @ ( times_times @ A @ X23 @ Y23 ) ) ) % 0.31/0.57 => ( ( finite_finite2 @ B @ S3 ) % 0.31/0.57 => ( ! [X5: B] : % 0.31/0.57 ( ( member @ B @ X5 @ S3 ) % 0.31/0.57 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) ) % 0.31/0.57 => ( R @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.related % 0.31/0.57 thf(fact_3330_prod_Oinsert__if,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [A3: set @ B,X3: B,G: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ( ( member @ B @ X3 @ A3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) % 0.31/0.57 & ( ~ ( member @ B @ X3 @ A3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 0.31/0.57 = ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.insert_if % 0.31/0.57 thf(fact_3331_prod__dvd__prod__subset,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [B3: set @ B,A3: set @ B,F2: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ B3 ) % 0.31/0.57 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 ) % 0.31/0.57 => ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B3 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod_dvd_prod_subset % 0.31/0.57 thf(fact_3332_prod__dvd__prod__subset2,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [B3: set @ B,A3: set @ B,F2: B > A,G: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ B3 ) % 0.31/0.57 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 ) % 0.31/0.57 => ( ! [A5: B] : % 0.31/0.57 ( ( member @ B @ A5 @ A3 ) % 0.31/0.57 => ( dvd_dvd @ A @ ( F2 @ A5 ) @ ( G @ A5 ) ) ) % 0.31/0.57 => ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B3 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod_dvd_prod_subset2 % 0.31/0.57 thf(fact_3333_prod_Oreindex__bij__witness__not__neutral,axiom, % 0.31/0.57 ! [B: $tType,A: $tType,C: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [S6: set @ B,T6: set @ C,S3: set @ B,I: C > B,J: B > C,T4: set @ C,G: B > A,H2: C > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ S6 ) % 0.31/0.57 => ( ( finite_finite2 @ C @ T6 ) % 0.31/0.57 => ( ! [A5: B] : % 0.31/0.57 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) % 0.31/0.57 => ( ( I @ ( J @ A5 ) ) % 0.31/0.57 = A5 ) ) % 0.31/0.57 => ( ! [A5: B] : % 0.31/0.57 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) % 0.31/0.57 => ( member @ C @ ( J @ A5 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) ) % 0.31/0.57 => ( ! [B5: C] : % 0.31/0.57 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) % 0.31/0.57 => ( ( J @ ( I @ B5 ) ) % 0.31/0.57 = B5 ) ) % 0.31/0.57 => ( ! [B5: C] : % 0.31/0.57 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) % 0.31/0.57 => ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) ) % 0.31/0.57 => ( ! [A5: B] : % 0.31/0.57 ( ( member @ B @ A5 @ S6 ) % 0.31/0.57 => ( ( G @ A5 ) % 0.31/0.57 = ( one_one @ A ) ) ) % 0.31/0.57 => ( ! [B5: C] : % 0.31/0.57 ( ( member @ C @ B5 @ T6 ) % 0.31/0.57 => ( ( H2 @ B5 ) % 0.31/0.57 = ( one_one @ A ) ) ) % 0.31/0.57 => ( ! [A5: B] : % 0.31/0.57 ( ( member @ B @ A5 @ S3 ) % 0.31/0.57 => ( ( H2 @ ( J @ A5 ) ) % 0.31/0.57 = ( G @ A5 ) ) ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.reindex_bij_witness_not_neutral % 0.31/0.57 thf(fact_3334_prod_Oin__pairs__0,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [G: nat > A,N2: nat] : % 0.31/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.in_pairs_0 % 0.31/0.57 thf(fact_3335_prod_Osetdiff__irrelevant,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [A3: set @ B,G: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G % 0.31/0.57 @ ( minus_minus @ ( set @ B ) @ A3 % 0.31/0.57 @ ( collect @ B % 0.31/0.57 @ ^ [X: B] : % 0.31/0.57 ( ( G @ X ) % 0.31/0.57 = ( one_one @ A ) ) ) ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.setdiff_irrelevant % 0.31/0.57 thf(fact_3336_exp__sum,axiom, % 0.31/0.57 ! [B: $tType,A: $tType] : % 0.31/0.57 ( ( ( comm_monoid_mult @ B ) % 0.31/0.57 & ( real_Vector_banach @ B ) % 0.31/0.57 & ( real_V2822296259951069270ebra_1 @ B ) ) % 0.31/0.57 => ! [I6: set @ A,F2: A > B] : % 0.31/0.57 ( ( finite_finite2 @ A @ I6 ) % 0.31/0.57 => ( ( exp @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ I6 ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ A @ B % 0.31/0.57 @ ^ [X: A] : ( exp @ B @ ( F2 @ X ) ) % 0.31/0.57 @ I6 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % exp_sum % 0.31/0.57 thf(fact_3337_prod_Onat__diff__reindex,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [G: nat > A,N2: nat] : % 0.31/0.57 ( ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.nat_diff_reindex % 0.31/0.57 thf(fact_3338_Iic__subset__Iio__iff,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( linorder @ A ) % 0.31/0.57 => ! [A2: A,B2: A] : % 0.31/0.57 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A2 ) @ ( set_ord_lessThan @ A @ B2 ) ) % 0.31/0.57 = ( ord_less @ A @ A2 @ B2 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % Iic_subset_Iio_iff % 0.31/0.57 thf(fact_3339_prod_OatLeastAtMost__rev,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [G: nat > A,N2: nat,M: nat] : % 0.31/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I3 ) ) % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.atLeastAtMost_rev % 0.31/0.57 thf(fact_3340_prod_Ozero__middle,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [P4: nat,K: nat,G: nat > A,H2: nat > A] : % 0.31/0.57 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 ) % 0.31/0.57 => ( ( ord_less_eq @ nat @ K @ P4 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ P4 ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.zero_middle % 0.31/0.57 thf(fact_3341_less__1__prod2,axiom, % 0.31/0.57 ! [B: $tType,A: $tType] : % 0.31/0.57 ( ( linordered_idom @ B ) % 0.31/0.57 => ! [I6: set @ A,I: A,F2: A > B] : % 0.31/0.57 ( ( finite_finite2 @ A @ I6 ) % 0.31/0.57 => ( ( member @ A @ I @ I6 ) % 0.31/0.57 => ( ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I ) ) % 0.31/0.57 => ( ! [I4: A] : % 0.31/0.57 ( ( member @ A @ I4 @ I6 ) % 0.31/0.57 => ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ I4 ) ) ) % 0.31/0.57 => ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % less_1_prod2 % 0.31/0.57 thf(fact_3342_sum__choose__upper,axiom, % 0.31/0.57 ! [M: nat,N2: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ nat % 0.31/0.57 @ ^ [K3: nat] : ( binomial @ K3 @ M ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum_choose_upper % 0.31/0.57 thf(fact_3343_less__1__prod,axiom, % 0.31/0.57 ! [B: $tType,A: $tType] : % 0.31/0.57 ( ( linordered_idom @ B ) % 0.31/0.57 => ! [I6: set @ A,F2: A > B] : % 0.31/0.57 ( ( finite_finite2 @ A @ I6 ) % 0.31/0.57 => ( ( I6 % 0.31/0.57 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.57 => ( ! [I4: A] : % 0.31/0.57 ( ( member @ A @ I4 @ I6 ) % 0.31/0.57 => ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I4 ) ) ) % 0.31/0.57 => ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % less_1_prod % 0.31/0.57 thf(fact_3344_prod_Osubset__diff,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [B3: set @ B,A3: set @ B,G: B > A] : % 0.31/0.57 ( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 ) % 0.31/0.57 => ( ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) % 0.31/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B3 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.subset_diff % 0.31/0.57 thf(fact_3345_prod_Omono__neutral__cong__right,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [T4: set @ B,S3: set @ B,G: B > A,H2: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ T4 ) % 0.31/0.57 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.31/0.57 => ( ! [X5: B] : % 0.31/0.57 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.31/0.57 => ( ( G @ X5 ) % 0.31/0.57 = ( one_one @ A ) ) ) % 0.31/0.57 => ( ! [X5: B] : % 0.31/0.57 ( ( member @ B @ X5 @ S3 ) % 0.31/0.57 => ( ( G @ X5 ) % 0.31/0.57 = ( H2 @ X5 ) ) ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ T4 ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.mono_neutral_cong_right % 0.31/0.57 thf(fact_3346_prod_Omono__neutral__cong__left,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [T4: set @ B,S3: set @ B,H2: B > A,G: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ T4 ) % 0.31/0.57 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.31/0.57 => ( ! [X5: B] : % 0.31/0.57 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.31/0.57 => ( ( H2 @ X5 ) % 0.31/0.57 = ( one_one @ A ) ) ) % 0.31/0.57 => ( ! [X5: B] : % 0.31/0.57 ( ( member @ B @ X5 @ S3 ) % 0.31/0.57 => ( ( G @ X5 ) % 0.31/0.57 = ( H2 @ X5 ) ) ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.mono_neutral_cong_left % 0.31/0.57 thf(fact_3347_prod_Omono__neutral__right,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [T4: set @ B,S3: set @ B,G: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ T4 ) % 0.31/0.57 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.31/0.57 => ( ! [X5: B] : % 0.31/0.57 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.31/0.57 => ( ( G @ X5 ) % 0.31/0.57 = ( one_one @ A ) ) ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ T4 ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.mono_neutral_right % 0.31/0.57 thf(fact_3348_prod_Omono__neutral__left,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [T4: set @ B,S3: set @ B,G: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ T4 ) % 0.31/0.57 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.31/0.57 => ( ! [X5: B] : % 0.31/0.57 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.31/0.57 => ( ( G @ X5 ) % 0.31/0.57 = ( one_one @ A ) ) ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ A @ G @ T4 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.mono_neutral_left % 0.31/0.57 thf(fact_3349_prod_Osame__carrierI,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [C5: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ C5 ) % 0.31/0.57 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 ) % 0.31/0.57 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ C5 ) % 0.31/0.57 => ( ! [A5: B] : % 0.31/0.57 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) ) % 0.31/0.57 => ( ( G @ A5 ) % 0.31/0.57 = ( one_one @ A ) ) ) % 0.31/0.57 => ( ! [B5: B] : % 0.31/0.57 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B3 ) ) % 0.31/0.57 => ( ( H2 @ B5 ) % 0.31/0.57 = ( one_one @ A ) ) ) % 0.31/0.57 => ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ B3 ) ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.same_carrierI % 0.31/0.57 thf(fact_3350_prod_Osame__carrier,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [C5: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ C5 ) % 0.31/0.57 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 ) % 0.31/0.57 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ C5 ) % 0.31/0.57 => ( ! [A5: B] : % 0.31/0.57 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) ) % 0.31/0.57 => ( ( G @ A5 ) % 0.31/0.57 = ( one_one @ A ) ) ) % 0.31/0.57 => ( ! [B5: B] : % 0.31/0.57 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B3 ) ) % 0.31/0.57 => ( ( H2 @ B5 ) % 0.31/0.57 = ( one_one @ A ) ) ) % 0.31/0.57 => ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ B3 ) ) % 0.31/0.57 = ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.same_carrier % 0.31/0.57 thf(fact_3351_prod_OatLeast0__atMost__Suc,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [G: nat > A,N2: nat] : % 0.31/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 0.31/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.atLeast0_atMost_Suc % 0.31/0.57 thf(fact_3352_prod_OatLeast__Suc__atMost,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [M: nat,N2: nat,G: nat > A] : % 0.31/0.57 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.31/0.57 = ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.atLeast_Suc_atMost % 0.31/0.57 thf(fact_3353_prod_Onat__ivl__Suc_H,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [M: nat,N2: nat,G: nat > A] : % 0.31/0.57 ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) ) % 0.31/0.57 = ( times_times @ A @ ( G @ ( suc @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.nat_ivl_Suc' % 0.31/0.57 thf(fact_3354_prod_OlessThan__Suc__shift,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [G: nat > A,N2: nat] : % 0.31/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) % 0.31/0.57 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) % 0.31/0.57 @ ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.lessThan_Suc_shift % 0.31/0.57 thf(fact_3355_prod_OSuc__reindex__ivl,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [M: nat,N2: nat,G: nat > A] : % 0.31/0.57 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.57 => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) % 0.31/0.57 = ( times_times @ A @ ( G @ M ) % 0.31/0.57 @ ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.Suc_reindex_ivl % 0.31/0.57 thf(fact_3356_sum_OatMost__Suc__shift,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_add @ A ) % 0.31/0.57 => ! [G: nat > A,N2: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) ) % 0.31/0.57 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum.atMost_Suc_shift % 0.31/0.57 thf(fact_3357_sum__telescope,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ab_group_add @ A ) % 0.31/0.57 => ! [F2: nat > A,I: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ I ) ) % 0.31/0.57 = ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ ( suc @ I ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum_telescope % 0.31/0.57 thf(fact_3358_polyfun__eq__coeffs,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.57 & ( idom @ A ) ) % 0.31/0.57 => ! [C2: nat > A,N2: nat,D2: nat > A] : % 0.31/0.57 ( ( ! [X: A] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( D2 @ I3 ) @ ( power_power @ A @ X @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) % 0.31/0.57 = ( ! [I3: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ I3 @ N2 ) % 0.31/0.57 => ( ( C2 @ I3 ) % 0.31/0.57 = ( D2 @ I3 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % polyfun_eq_coeffs % 0.31/0.57 thf(fact_3359_prod_OatLeast1__atMost__eq,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [G: nat > A,N2: nat] : % 0.31/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.atLeast1_atMost_eq % 0.31/0.57 thf(fact_3360_bounded__imp__summable,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( condit6923001295902523014norder @ A ) % 0.31/0.57 & ( linord2810124833399127020strict @ A ) % 0.31/0.57 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.57 => ! [A2: nat > A,B3: A] : % 0.31/0.57 ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A2 @ N3 ) ) % 0.31/0.57 => ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_ord_atMost @ nat @ N3 ) ) @ B3 ) % 0.31/0.57 => ( summable @ A @ A2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % bounded_imp_summable % 0.31/0.57 thf(fact_3361_sum_Onested__swap_H,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_add @ A ) % 0.31/0.57 => ! [A2: nat > nat > A,N2: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [J3: nat] : % 0.31/0.57 ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( A2 @ I3 @ J3 ) % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum.nested_swap' % 0.31/0.57 thf(fact_3362_prod__mono__strict,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( linordered_semidom @ A ) % 0.31/0.57 => ! [A3: set @ B,F2: B > A,G: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ! [I4: B] : % 0.31/0.57 ( ( member @ B @ I4 @ A3 ) % 0.31/0.57 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) % 0.31/0.57 & ( ord_less @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) ) ) % 0.31/0.57 => ( ( A3 % 0.31/0.57 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.57 => ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod_mono_strict % 0.31/0.57 thf(fact_3363_even__prod__iff,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( semiring_parity @ A ) % 0.31/0.57 => ! [A3: set @ B,F2: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) % 0.31/0.57 = ( ? [X: B] : % 0.31/0.57 ( ( member @ B @ X @ A3 ) % 0.31/0.57 & ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % even_prod_iff % 0.31/0.57 thf(fact_3364_sum__choose__lower,axiom, % 0.31/0.57 ! [R2: nat,N2: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ nat % 0.31/0.57 @ ^ [K3: nat] : ( binomial @ ( plus_plus @ nat @ R2 @ K3 ) @ K3 ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( binomial @ ( suc @ ( plus_plus @ nat @ R2 @ N2 ) ) @ N2 ) ) ). % 0.31/0.57 % 0.31/0.57 % sum_choose_lower % 0.31/0.57 thf(fact_3365_prod_Oremove,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [A3: set @ B,X3: B,G: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ( member @ B @ X3 @ A3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) % 0.31/0.57 = ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.remove % 0.31/0.57 thf(fact_3366_prod_Oinsert__remove,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [A3: set @ B,G: B > A,X3: B] : % 0.31/0.57 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 0.31/0.57 = ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.insert_remove % 0.31/0.57 thf(fact_3367_choose__rising__sum_I1_J,axiom, % 0.31/0.57 ! [N2: nat,M: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ nat % 0.31/0.57 @ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N2 @ J3 ) @ N2 ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ M ) ) % 0.31/0.57 = ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % choose_rising_sum(1) % 0.31/0.57 thf(fact_3368_choose__rising__sum_I2_J,axiom, % 0.31/0.57 ! [N2: nat,M: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ nat % 0.31/0.57 @ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N2 @ J3 ) @ N2 ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ M ) ) % 0.31/0.57 = ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ ( one_one @ nat ) ) @ M ) ) ). % 0.31/0.57 % 0.31/0.57 % choose_rising_sum(2) % 0.31/0.57 thf(fact_3369_prod_Oub__add__nat,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [M: nat,N2: nat,G: nat > A,P4: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N2 @ P4 ) ) ) % 0.31/0.57 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ P4 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.ub_add_nat % 0.31/0.57 thf(fact_3370_prod_Odelta__remove,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [S3: set @ B,A2: B,B2: B > A,C2: B > A] : % 0.31/0.57 ( ( finite_finite2 @ B @ S3 ) % 0.31/0.57 => ( ( ( member @ B @ A2 @ S3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A % 0.31/0.57 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) ) % 0.31/0.57 @ S3 ) % 0.31/0.57 = ( times_times @ A @ ( B2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) % 0.31/0.57 & ( ~ ( member @ B @ A2 @ S3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A % 0.31/0.57 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) ) % 0.31/0.57 @ S3 ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.delta_remove % 0.31/0.57 thf(fact_3371_fold__atLeastAtMost__nat_Osimps,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( set_fo6178422350223883121st_nat @ A ) % 0.31/0.57 = ( ^ [F3: nat > A > A,A4: nat,B4: nat,Acc2: A] : ( if @ A @ ( ord_less @ nat @ B4 @ A4 ) @ Acc2 @ ( set_fo6178422350223883121st_nat @ A @ F3 @ ( plus_plus @ nat @ A4 @ ( one_one @ nat ) ) @ B4 @ ( F3 @ A4 @ Acc2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fold_atLeastAtMost_nat.simps % 0.31/0.57 thf(fact_3372_fold__atLeastAtMost__nat_Oelims,axiom, % 0.31/0.57 ! [A: $tType,X3: nat > A > A,Xa2: nat,Xb: nat,Xc: A,Y: A] : % 0.31/0.57 ( ( ( set_fo6178422350223883121st_nat @ A @ X3 @ Xa2 @ Xb @ Xc ) % 0.31/0.57 = Y ) % 0.31/0.57 => ( ( ( ord_less @ nat @ Xb @ Xa2 ) % 0.31/0.57 => ( Y = Xc ) ) % 0.31/0.57 & ( ~ ( ord_less @ nat @ Xb @ Xa2 ) % 0.31/0.57 => ( Y % 0.31/0.57 = ( set_fo6178422350223883121st_nat @ A @ X3 @ ( plus_plus @ nat @ Xa2 @ ( one_one @ nat ) ) @ Xb @ ( X3 @ Xa2 @ Xc ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fold_atLeastAtMost_nat.elims % 0.31/0.57 thf(fact_3373_polyfun__eq__0,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.57 & ( idom @ A ) ) % 0.31/0.57 => ! [C2: nat > A,N2: nat] : % 0.31/0.57 ( ( ! [X: A] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( zero_zero @ A ) ) ) % 0.31/0.57 = ( ! [I3: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ I3 @ N2 ) % 0.31/0.57 => ( ( C2 @ I3 ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % polyfun_eq_0 % 0.31/0.57 thf(fact_3374_zero__polynom__imp__zero__coeffs,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( ab_semigroup_mult @ A ) % 0.31/0.57 & ( real_V8999393235501362500lgebra @ A ) ) % 0.31/0.57 => ! [C2: nat > A,N2: nat,K: nat] : % 0.31/0.57 ( ! [W: A] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ W @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( zero_zero @ A ) ) % 0.31/0.57 => ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.57 => ( ( C2 @ K ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % zero_polynom_imp_zero_coeffs % 0.31/0.57 thf(fact_3375_sum_OatMost__shift,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_add @ A ) % 0.31/0.57 => ! [G: nat > A,N2: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum.atMost_shift % 0.31/0.57 thf(fact_3376_sum__up__index__split,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_add @ A ) % 0.31/0.57 => ! [F2: nat > A,M: nat,N2: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N2 ) ) ) % 0.31/0.57 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ M ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum_up_index_split % 0.31/0.57 thf(fact_3377_atLeast1__atMost__eq__remove0,axiom, % 0.31/0.57 ! [N2: nat] : % 0.31/0.57 ( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 0.31/0.57 = ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N2 ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % atLeast1_atMost_eq_remove0 % 0.31/0.57 thf(fact_3378_gbinomial__parallel__sum,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [A2: A,N2: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K3 ) ) @ K3 ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( one_one @ A ) ) @ N2 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_parallel_sum % 0.31/0.57 thf(fact_3379_sum_Otriangle__reindex__eq,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_add @ A ) % 0.31/0.57 => ! [G: nat > nat > A,N2: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G ) % 0.31/0.57 @ ( collect @ ( product_prod @ nat @ nat ) % 0.31/0.57 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.57 @ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N2 ) ) ) ) % 0.31/0.57 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : % 0.31/0.57 ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ K3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum.triangle_reindex_eq % 0.31/0.57 thf(fact_3380_fact__eq__fact__times,axiom, % 0.31/0.57 ! [N2: nat,M: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ N2 @ M ) % 0.31/0.57 => ( ( semiring_char_0_fact @ nat @ M ) % 0.31/0.57 = ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N2 ) % 0.31/0.57 @ ( groups7121269368397514597t_prod @ nat @ nat % 0.31/0.57 @ ^ [X: nat] : X % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_eq_fact_times % 0.31/0.57 thf(fact_3381_prod__mono2,axiom, % 0.31/0.57 ! [B: $tType,A: $tType] : % 0.31/0.57 ( ( linordered_idom @ B ) % 0.31/0.57 => ! [B3: set @ A,A3: set @ A,F2: A > B] : % 0.31/0.57 ( ( finite_finite2 @ A @ B3 ) % 0.31/0.57 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.57 => ( ! [B5: A] : % 0.31/0.57 ( ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) % 0.31/0.57 => ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ B5 ) ) ) % 0.31/0.57 => ( ! [A5: A] : % 0.31/0.57 ( ( member @ A @ A5 @ A3 ) % 0.31/0.57 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ A5 ) ) ) % 0.31/0.57 => ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ B3 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod_mono2 % 0.31/0.57 thf(fact_3382_prod__diff1,axiom, % 0.31/0.57 ! [A: $tType,B: $tType] : % 0.31/0.57 ( ( semidom_divide @ A ) % 0.31/0.57 => ! [A3: set @ B,F2: B > A,A2: B] : % 0.31/0.57 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.57 => ( ( ( F2 @ A2 ) % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( ( member @ B @ A2 @ A3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) % 0.31/0.57 = ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( F2 @ A2 ) ) ) ) % 0.31/0.57 & ( ~ ( member @ B @ A2 @ A3 ) % 0.31/0.57 => ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod_diff1 % 0.31/0.57 thf(fact_3383_sum__choose__diagonal,axiom, % 0.31/0.57 ! [M: nat,N2: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ nat % 0.31/0.57 @ ^ [K3: nat] : ( binomial @ ( minus_minus @ nat @ N2 @ K3 ) @ ( minus_minus @ nat @ M @ K3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ M ) ) % 0.31/0.57 = ( binomial @ ( suc @ N2 ) @ M ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum_choose_diagonal % 0.31/0.57 thf(fact_3384_vandermonde,axiom, % 0.31/0.57 ! [M: nat,N2: nat,R2: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ nat % 0.31/0.57 @ ^ [K3: nat] : ( times_times @ nat @ ( binomial @ M @ K3 ) @ ( binomial @ N2 @ ( minus_minus @ nat @ R2 @ K3 ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ R2 ) ) % 0.31/0.57 = ( binomial @ ( plus_plus @ nat @ M @ N2 ) @ R2 ) ) ). % 0.31/0.57 % 0.31/0.57 % vandermonde % 0.31/0.57 thf(fact_3385_sum__gp__basic,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( monoid_mult @ A ) % 0.31/0.57 & ( comm_ring @ A ) ) % 0.31/0.57 => ! [X3: A,N2: nat] : % 0.31/0.57 ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N2 ) ) ) % 0.31/0.57 = ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum_gp_basic % 0.31/0.57 thf(fact_3386_polyfun__finite__roots,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.57 & ( idom @ A ) ) % 0.31/0.57 => ! [C2: nat > A,N2: nat] : % 0.31/0.57 ( ( finite_finite2 @ A % 0.31/0.57 @ ( collect @ A % 0.31/0.57 @ ^ [X: A] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ) % 0.31/0.57 = ( ? [I3: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ I3 @ N2 ) % 0.31/0.57 & ( ( C2 @ I3 ) % 0.31/0.57 != ( zero_zero @ A ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % polyfun_finite_roots % 0.31/0.57 thf(fact_3387_polyfun__roots__finite,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.57 & ( idom @ A ) ) % 0.31/0.57 => ! [C2: nat > A,K: nat,N2: nat] : % 0.31/0.57 ( ( ( C2 @ K ) % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.57 => ( finite_finite2 @ A % 0.31/0.57 @ ( collect @ A % 0.31/0.57 @ ^ [Z6: A] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % polyfun_roots_finite % 0.31/0.57 thf(fact_3388_pochhammer__Suc__prod,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [A2: A,N2: nat] : % 0.31/0.57 ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) ) % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_Suc_prod % 0.31/0.57 thf(fact_3389_polyfun__linear__factor__root,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( idom @ A ) % 0.31/0.57 => ! [C2: nat > A,A2: A,N2: nat] : % 0.31/0.57 ( ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ A2 @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( zero_zero @ A ) ) % 0.31/0.57 => ~ ! [B5: nat > A] : % 0.31/0.57 ~ ! [Z4: A] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 ) % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( B5 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % polyfun_linear_factor_root % 0.31/0.57 thf(fact_3390_polyfun__linear__factor,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( idom @ A ) % 0.31/0.57 => ! [C2: nat > A,N2: nat,A2: A] : % 0.31/0.57 ? [B5: nat > A] : % 0.31/0.57 ! [Z4: A] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( plus_plus @ A % 0.31/0.57 @ ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 ) % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( B5 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) ) % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ A2 @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % polyfun_linear_factor % 0.31/0.57 thf(fact_3391_pochhammer__prod__rev,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ( ( comm_s3205402744901411588hammer @ A ) % 0.31/0.57 = ( ^ [A4: A,N: nat] : % 0.31/0.57 ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I3 ) ) ) % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_prod_rev % 0.31/0.57 thf(fact_3392_sum__power__shift,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( monoid_mult @ A ) % 0.31/0.57 & ( comm_ring @ A ) ) % 0.31/0.57 => ! [M: nat,N2: nat,X3: A] : % 0.31/0.57 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.31/0.57 = ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum_power_shift % 0.31/0.57 thf(fact_3393_sum_Otriangle__reindex,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_add @ A ) % 0.31/0.57 => ! [G: nat > nat > A,N2: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G ) % 0.31/0.57 @ ( collect @ ( product_prod @ nat @ nat ) % 0.31/0.57 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.57 @ ^ [I3: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N2 ) ) ) ) % 0.31/0.57 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : % 0.31/0.57 ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ K3 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum.triangle_reindex % 0.31/0.57 thf(fact_3394_fact__div__fact,axiom, % 0.31/0.57 ! [N2: nat,M: nat] : % 0.31/0.57 ( ( ord_less_eq @ nat @ N2 @ M ) % 0.31/0.57 => ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N2 ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ nat @ nat % 0.31/0.57 @ ^ [X: nat] : X % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ M ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % fact_div_fact % 0.31/0.57 thf(fact_3395_binomial,axiom, % 0.31/0.57 ! [A2: nat,B2: nat,N2: nat] : % 0.31/0.57 ( ( power_power @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ N2 ) % 0.31/0.57 = ( groups7311177749621191930dd_sum @ nat @ nat % 0.31/0.57 @ ^ [K3: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power @ nat @ A2 @ K3 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N2 @ K3 ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial % 0.31/0.57 thf(fact_3396_prod_Oin__pairs,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_mult @ A ) % 0.31/0.57 => ! [G: nat > A,M: nat,N2: nat] : % 0.31/0.57 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) ) % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod.in_pairs % 0.31/0.57 thf(fact_3397_sum__atLeastAtMost__code,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_add @ A ) % 0.31/0.57 => ! [F2: nat > A,A2: nat,B2: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) ) % 0.31/0.57 = ( set_fo6178422350223883121st_nat @ A % 0.31/0.57 @ ^ [A4: nat] : ( plus_plus @ A @ ( F2 @ A4 ) ) % 0.31/0.57 @ A2 % 0.31/0.57 @ B2 % 0.31/0.57 @ ( zero_zero @ A ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum_atLeastAtMost_code % 0.31/0.57 thf(fact_3398_sum_Oin__pairs__0,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_add @ A ) % 0.31/0.57 => ! [G: nat > A,N2: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) % 0.31/0.57 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum.in_pairs_0 % 0.31/0.57 thf(fact_3399_polynomial__product,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( idom @ A ) % 0.31/0.57 => ! [M: nat,A2: nat > A,N2: nat,B2: nat > A,X3: A] : % 0.31/0.57 ( ! [I4: nat] : % 0.31/0.57 ( ( ord_less @ nat @ M @ I4 ) % 0.31/0.57 => ( ( A2 @ I4 ) % 0.31/0.57 = ( zero_zero @ A ) ) ) % 0.31/0.57 => ( ! [J2: nat] : % 0.31/0.57 ( ( ord_less @ nat @ N2 @ J2 ) % 0.31/0.57 => ( ( B2 @ J2 ) % 0.31/0.57 = ( zero_zero @ A ) ) ) % 0.31/0.57 => ( ( times_times @ A % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X3 @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ M ) ) % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [J3: nat] : ( times_times @ A @ ( B2 @ J3 ) @ ( power_power @ A @ X3 @ J3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 0.31/0.57 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [R5: nat] : % 0.31/0.57 ( times_times @ A % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : ( times_times @ A @ ( A2 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ R5 ) ) % 0.31/0.57 @ ( power_power @ A @ X3 @ R5 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % polynomial_product % 0.31/0.57 thf(fact_3400_pochhammer__Suc__prod__rev,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [A2: A,N2: nat] : % 0.31/0.57 ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I3 ) ) ) % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_Suc_prod_rev % 0.31/0.57 thf(fact_3401_polyfun__eq__const,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.57 & ( idom @ A ) ) % 0.31/0.57 => ! [C2: nat > A,N2: nat,K: A] : % 0.31/0.57 ( ( ! [X: A] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = K ) ) % 0.31/0.57 = ( ( ( C2 @ ( zero_zero @ nat ) ) % 0.31/0.57 = K ) % 0.31/0.57 & ! [X: nat] : % 0.31/0.57 ( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) % 0.31/0.57 => ( ( C2 @ X ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % polyfun_eq_const % 0.31/0.57 thf(fact_3402_binomial__ring,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_semiring_1 @ A ) % 0.31/0.57 => ! [A2: A,B2: A,N2: nat] : % 0.31/0.57 ( ( power_power @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N2 ) % 0.31/0.57 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K3 ) ) @ ( power_power @ A @ A2 @ K3 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N2 @ K3 ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_ring % 0.31/0.57 thf(fact_3403_pochhammer__binomial__sum,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_ring_1 @ A ) % 0.31/0.57 => ! [A2: A,B2: A,N2: nat] : % 0.31/0.57 ( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N2 ) % 0.31/0.57 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ A2 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N2 @ K3 ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % pochhammer_binomial_sum % 0.31/0.57 thf(fact_3404_polynomial__product__nat,axiom, % 0.31/0.57 ! [M: nat,A2: nat > nat,N2: nat,B2: nat > nat,X3: nat] : % 0.31/0.57 ( ! [I4: nat] : % 0.31/0.57 ( ( ord_less @ nat @ M @ I4 ) % 0.31/0.57 => ( ( A2 @ I4 ) % 0.31/0.57 = ( zero_zero @ nat ) ) ) % 0.31/0.57 => ( ! [J2: nat] : % 0.31/0.57 ( ( ord_less @ nat @ N2 @ J2 ) % 0.31/0.57 => ( ( B2 @ J2 ) % 0.31/0.57 = ( zero_zero @ nat ) ) ) % 0.31/0.57 => ( ( times_times @ nat % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ nat % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ nat @ ( A2 @ I3 ) @ ( power_power @ nat @ X3 @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ M ) ) % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ nat % 0.31/0.57 @ ^ [J3: nat] : ( times_times @ nat @ ( B2 @ J3 ) @ ( power_power @ nat @ X3 @ J3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 0.31/0.57 = ( groups7311177749621191930dd_sum @ nat @ nat % 0.31/0.57 @ ^ [R5: nat] : % 0.31/0.57 ( times_times @ nat % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ nat % 0.31/0.57 @ ^ [K3: nat] : ( times_times @ nat @ ( A2 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ R5 ) ) % 0.31/0.57 @ ( power_power @ nat @ X3 @ R5 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % polynomial_product_nat % 0.31/0.57 thf(fact_3405_gbinomial__Suc,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( semiring_char_0 @ A ) % 0.31/0.57 & ( semidom_divide @ A ) ) % 0.31/0.57 => ! [A2: A,K: nat] : % 0.31/0.57 ( ( gbinomial @ A @ A2 @ ( suc @ K ) ) % 0.31/0.57 = ( divide_divide @ A % 0.31/0.57 @ ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) ) % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) % 0.31/0.57 @ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_Suc % 0.31/0.57 thf(fact_3406_sum_Ozero__middle,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_monoid_add @ A ) % 0.31/0.57 => ! [P4: nat,K: nat,G: nat > A,H2: nat > A] : % 0.31/0.57 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 ) % 0.31/0.57 => ( ( ord_less_eq @ nat @ K @ P4 ) % 0.31/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ P4 ) ) % 0.31/0.57 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum.zero_middle % 0.31/0.57 thf(fact_3407_gbinomial__partial__sum__poly,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [M: nat,A2: A,X3: A,Y: A] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K3 ) @ ( power_power @ A @ X3 @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K3 ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ M ) ) % 0.31/0.57 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( minus_minus @ nat @ M @ K3 ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ M ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_partial_sum_poly % 0.31/0.57 thf(fact_3408_root__polyfun,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( idom @ A ) % 0.31/0.57 => ! [N2: nat,Z2: A,A2: A] : % 0.31/0.57 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 ) % 0.31/0.57 => ( ( ( power_power @ A @ Z2 @ N2 ) % 0.31/0.57 = A2 ) % 0.31/0.57 = ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : % 0.31/0.57 ( times_times @ A % 0.31/0.57 @ ( if @ A % 0.31/0.57 @ ( I3 % 0.31/0.57 = ( zero_zero @ nat ) ) % 0.31/0.57 @ ( uminus_uminus @ A @ A2 ) % 0.31/0.57 @ ( if @ A @ ( I3 = N2 ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) % 0.31/0.57 @ ( power_power @ A @ Z2 @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % root_polyfun % 0.31/0.57 thf(fact_3409_sum__gp0,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( division_ring @ A ) % 0.31/0.57 & ( comm_ring @ A ) ) % 0.31/0.57 => ! [X3: A,N2: nat] : % 0.31/0.57 ( ( ( X3 % 0.31/0.57 = ( one_one @ A ) ) % 0.31/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) % 0.31/0.57 & ( ( X3 % 0.31/0.57 != ( one_one @ A ) ) % 0.31/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sum_gp0 % 0.31/0.57 thf(fact_3410_choose__alternating__linear__sum,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_ring_1 @ A ) % 0.31/0.57 => ! [N2: nat] : % 0.31/0.57 ( ( N2 % 0.31/0.57 != ( one_one @ nat ) ) % 0.31/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % choose_alternating_linear_sum % 0.31/0.57 thf(fact_3411_gbinomial__sum__nat__pow2,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [M: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K3 ) ) @ K3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ M ) ) % 0.31/0.57 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_sum_nat_pow2 % 0.31/0.57 thf(fact_3412_gbinomial__partial__sum__poly__xpos,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [M: nat,A2: A,X3: A,Y: A] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K3 ) @ ( power_power @ A @ X3 @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K3 ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ M ) ) % 0.31/0.57 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A2 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X3 @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( minus_minus @ nat @ M @ K3 ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ M ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_partial_sum_poly_xpos % 0.31/0.57 thf(fact_3413_polyfun__diff__alt,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( idom @ A ) % 0.31/0.57 => ! [N2: nat,A2: nat > A,X3: A,Y: A] : % 0.31/0.57 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 ) % 0.31/0.57 => ( ( minus_minus @ A % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X3 @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 0.31/0.57 = ( times_times @ A @ ( minus_minus @ A @ X3 @ Y ) % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [J3: nat] : % 0.31/0.57 ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( A2 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J3 @ K3 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y @ K3 ) ) @ ( power_power @ A @ X3 @ J3 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ J3 ) ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % polyfun_diff_alt % 0.31/0.57 thf(fact_3414_binomial__r__part__sum,axiom, % 0.31/0.57 ! [M: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M ) ) % 0.31/0.57 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ). % 0.31/0.57 % 0.31/0.57 % binomial_r_part_sum % 0.31/0.57 thf(fact_3415_choose__alternating__sum,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_ring_1 @ A ) % 0.31/0.57 => ! [N2: nat] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.57 => ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % choose_alternating_sum % 0.31/0.57 thf(fact_3416_polyfun__extremal__lemma,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.57 => ! [E3: real,C2: nat > A,N2: nat] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 ) % 0.31/0.57 => ? [M8: real] : % 0.31/0.57 ! [Z4: A] : % 0.31/0.57 ( ( ord_less_eq @ real @ M8 @ ( real_V7770717601297561774m_norm @ A @ Z4 ) ) % 0.31/0.57 => ( ord_less_eq @ real % 0.31/0.57 @ ( real_V7770717601297561774m_norm @ A % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 0.31/0.57 @ ( times_times @ real @ E3 @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z4 ) @ ( suc @ N2 ) ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % polyfun_extremal_lemma % 0.31/0.57 thf(fact_3417_polyfun__diff,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( idom @ A ) % 0.31/0.57 => ! [N2: nat,A2: nat > A,X3: A,Y: A] : % 0.31/0.57 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 ) % 0.31/0.57 => ( ( minus_minus @ A % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X3 @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y @ I3 ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 0.31/0.57 = ( times_times @ A @ ( minus_minus @ A @ X3 @ Y ) % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [J3: nat] : % 0.31/0.57 ( times_times @ A % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ J3 ) @ ( one_one @ nat ) ) ) ) % 0.31/0.57 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) ) % 0.31/0.57 @ ( power_power @ A @ X3 @ J3 ) ) % 0.31/0.57 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % polyfun_diff % 0.31/0.57 thf(fact_3418_gbinomial__r__part__sum,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( field_char_0 @ A ) % 0.31/0.57 => ! [M: nat] : % 0.31/0.57 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M ) ) % 0.31/0.57 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % gbinomial_r_part_sum % 0.31/0.57 thf(fact_3419_choose__odd__sum,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_ring_1 @ A ) % 0.31/0.57 => ! [N2: nat] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.57 => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : % 0.31/0.57 ( if @ A % 0.31/0.57 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) % 0.31/0.57 @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) ) % 0.31/0.57 @ ( zero_zero @ A ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 0.31/0.57 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % choose_odd_sum % 0.31/0.57 thf(fact_3420_choose__even__sum,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( comm_ring_1 @ A ) % 0.31/0.57 => ! [N2: nat] : % 0.31/0.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.57 => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) % 0.31/0.57 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [I3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) ) @ ( zero_zero @ A ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 0.31/0.57 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % choose_even_sum % 0.31/0.57 thf(fact_3421_sin__x__sin__y,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ! [X3: A,Y: A] : % 0.31/0.57 ( sums @ A % 0.31/0.57 @ ^ [P6: nat] : % 0.31/0.57 ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [N: nat] : % 0.31/0.57 ( if @ A % 0.31/0.57 @ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 ) % 0.31/0.57 & ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) % 0.31/0.57 @ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) ) @ ( power_power @ A @ X3 @ N ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N ) ) ) % 0.31/0.57 @ ( zero_zero @ A ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ P6 ) ) % 0.31/0.57 @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sin_x_sin_y % 0.31/0.57 thf(fact_3422_sums__cos__x__plus__y,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ! [X3: A,Y: A] : % 0.31/0.57 ( sums @ A % 0.31/0.57 @ ^ [P6: nat] : % 0.31/0.57 ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [N: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 ) @ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X3 @ N ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N ) ) ) @ ( zero_zero @ A ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ P6 ) ) % 0.31/0.57 @ ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % sums_cos_x_plus_y % 0.31/0.57 thf(fact_3423_cos__x__cos__y,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ! [X3: A,Y: A] : % 0.31/0.57 ( sums @ A % 0.31/0.57 @ ^ [P6: nat] : % 0.31/0.57 ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.57 @ ^ [N: nat] : % 0.31/0.57 ( if @ A % 0.31/0.57 @ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 ) % 0.31/0.57 & ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) % 0.31/0.57 @ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X3 @ N ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N ) ) ) % 0.31/0.57 @ ( zero_zero @ A ) ) % 0.31/0.57 @ ( set_ord_atMost @ nat @ P6 ) ) % 0.31/0.57 @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % cos_x_cos_y % 0.31/0.57 thf(fact_3424_cot__less__zero,axiom, % 0.31/0.57 ! [X3: real] : % 0.31/0.57 ( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 ) % 0.31/0.57 => ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) % 0.31/0.57 => ( ord_less @ real @ ( cot @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % cot_less_zero % 0.31/0.57 thf(fact_3425_int__ge__less__than2__def,axiom, % 0.31/0.57 ( int_ge_less_than2 % 0.31/0.57 = ( ^ [D5: int] : % 0.31/0.57 ( collect @ ( product_prod @ int @ int ) % 0.31/0.57 @ ( product_case_prod @ int @ int @ $o % 0.31/0.57 @ ^ [Z7: int,Z6: int] : % 0.31/0.57 ( ( ord_less_eq @ int @ D5 @ Z6 ) % 0.31/0.57 & ( ord_less @ int @ Z7 @ Z6 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % int_ge_less_than2_def % 0.31/0.57 thf(fact_3426_of__nat__id,axiom, % 0.31/0.57 ( ( semiring_1_of_nat @ nat ) % 0.31/0.57 = ( ^ [N: nat] : N ) ) ). % 0.31/0.57 % 0.31/0.57 % of_nat_id % 0.31/0.57 thf(fact_3427_scaleR__cancel__right,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.57 => ! [A2: real,X3: A,B2: real] : % 0.31/0.57 ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) % 0.31/0.57 = ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) % 0.31/0.57 = ( ( A2 = B2 ) % 0.31/0.57 | ( X3 % 0.31/0.57 = ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_cancel_right % 0.31/0.57 thf(fact_3428_scaleR__zero__right,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.57 => ! [A2: real] : % 0.31/0.57 ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( zero_zero @ A ) ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_zero_right % 0.31/0.57 thf(fact_3429_cot__zero,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( ( real_Vector_banach @ A ) % 0.31/0.57 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.57 => ( ( cot @ A @ ( zero_zero @ A ) ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % cot_zero % 0.31/0.57 thf(fact_3430_scaleR__zero__left,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.57 => ! [X3: A] : % 0.31/0.57 ( ( real_V8093663219630862766scaleR @ A @ ( zero_zero @ real ) @ X3 ) % 0.31/0.57 = ( zero_zero @ A ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_zero_left % 0.31/0.57 thf(fact_3431_scaleR__eq__0__iff,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.57 => ! [A2: real,X3: A] : % 0.31/0.57 ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) % 0.31/0.57 = ( zero_zero @ A ) ) % 0.31/0.57 = ( ( A2 % 0.31/0.57 = ( zero_zero @ real ) ) % 0.31/0.57 | ( X3 % 0.31/0.57 = ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_eq_0_iff % 0.31/0.57 thf(fact_3432_scaleR__eq__iff,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.57 => ! [B2: A,U: real,A2: A] : % 0.31/0.57 ( ( ( plus_plus @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ U @ A2 ) ) % 0.31/0.57 = ( plus_plus @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ U @ B2 ) ) ) % 0.31/0.57 = ( ( A2 = B2 ) % 0.31/0.57 | ( U % 0.31/0.57 = ( one_one @ real ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_eq_iff % 0.31/0.57 thf(fact_3433_prod__eq__1__iff,axiom, % 0.31/0.57 ! [A: $tType,A3: set @ A,F2: A > nat] : % 0.31/0.57 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.57 => ( ( ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A3 ) % 0.31/0.57 = ( one_one @ nat ) ) % 0.31/0.57 = ( ! [X: A] : % 0.31/0.57 ( ( member @ A @ X @ A3 ) % 0.31/0.57 => ( ( F2 @ X ) % 0.31/0.57 = ( one_one @ nat ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod_eq_1_iff % 0.31/0.57 thf(fact_3434_scaleR__collapse,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.57 => ! [U: real,A2: A] : % 0.31/0.57 ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ U @ A2 ) ) % 0.31/0.57 = A2 ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_collapse % 0.31/0.57 thf(fact_3435_prod__pos__nat__iff,axiom, % 0.31/0.57 ! [A: $tType,A3: set @ A,F2: A > nat] : % 0.31/0.57 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.57 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A3 ) ) % 0.31/0.57 = ( ! [X: A] : % 0.31/0.57 ( ( member @ A @ X @ A3 ) % 0.31/0.57 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % prod_pos_nat_iff % 0.31/0.57 thf(fact_3436_scaleR__half__double,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.57 => ! [A2: A] : % 0.31/0.57 ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ A2 @ A2 ) ) % 0.31/0.57 = A2 ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_half_double % 0.31/0.57 thf(fact_3437_int__prod,axiom, % 0.31/0.57 ! [B: $tType,F2: B > nat,A3: set @ B] : % 0.31/0.57 ( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A3 ) ) % 0.31/0.57 = ( groups7121269368397514597t_prod @ B @ int % 0.31/0.57 @ ^ [X: B] : ( semiring_1_of_nat @ int @ ( F2 @ X ) ) % 0.31/0.57 @ A3 ) ) ). % 0.31/0.57 % 0.31/0.57 % int_prod % 0.31/0.57 thf(fact_3438_scaleR__right__distrib,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.57 => ! [A2: real,X3: A,Y: A] : % 0.31/0.57 ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( plus_plus @ A @ X3 @ Y ) ) % 0.31/0.57 = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_right_distrib % 0.31/0.57 thf(fact_3439_scaleR__right__imp__eq,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.57 => ! [X3: A,A2: real,B2: real] : % 0.31/0.57 ( ( X3 % 0.31/0.57 != ( zero_zero @ A ) ) % 0.31/0.57 => ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) % 0.31/0.57 = ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) % 0.31/0.57 => ( A2 = B2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_right_imp_eq % 0.31/0.57 thf(fact_3440_scaleR__left__distrib,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.57 => ! [A2: real,B2: real,X3: A] : % 0.31/0.57 ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X3 ) % 0.31/0.57 = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_left_distrib % 0.31/0.57 thf(fact_3441_scaleR__left_Oadd,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.57 => ! [X3: real,Y: real,Xa2: A] : % 0.31/0.57 ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ X3 @ Y ) @ Xa2 ) % 0.31/0.57 = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ X3 @ Xa2 ) @ ( real_V8093663219630862766scaleR @ A @ Y @ Xa2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_left.add % 0.31/0.57 thf(fact_3442_scaleR__right__mono__neg,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.57 => ! [B2: real,A2: real,C2: A] : % 0.31/0.57 ( ( ord_less_eq @ real @ B2 @ A2 ) % 0.31/0.57 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 0.31/0.57 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ C2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_right_mono_neg % 0.31/0.57 thf(fact_3443_scaleR__right__mono,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.57 => ! [A2: real,B2: real,X3: A] : % 0.31/0.57 ( ( ord_less_eq @ real @ A2 @ B2 ) % 0.31/0.57 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.57 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_right_mono % 0.31/0.57 thf(fact_3444_scaleR__le__cancel__left,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.57 => ! [C2: real,A2: A,B2: A] : % 0.31/0.57 ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) ) % 0.31/0.57 = ( ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.57 => ( ord_less_eq @ A @ A2 @ B2 ) ) % 0.31/0.57 & ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 0.31/0.57 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_le_cancel_left % 0.31/0.57 thf(fact_3445_scaleR__le__cancel__left__neg,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.57 => ! [C2: real,A2: A,B2: A] : % 0.31/0.57 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 0.31/0.57 => ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) ) % 0.31/0.57 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_le_cancel_left_neg % 0.31/0.57 thf(fact_3446_scaleR__le__cancel__left__pos,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.57 => ! [C2: real,A2: A,B2: A] : % 0.31/0.57 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.57 => ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) ) % 0.31/0.57 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_le_cancel_left_pos % 0.31/0.57 thf(fact_3447_scaleR__left__mono__neg,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.57 => ! [B2: A,A2: A,C2: real] : % 0.31/0.57 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.57 => ( ( ord_less_eq @ real @ C2 @ ( zero_zero @ real ) ) % 0.31/0.57 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_left_mono_neg % 0.31/0.57 thf(fact_3448_scaleR__left__mono,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.57 => ! [X3: A,Y: A,A2: real] : % 0.31/0.57 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.57 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.57 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_left_mono % 0.31/0.57 thf(fact_3449_eq__vector__fraction__iff,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.57 => ! [X3: A,U: real,V2: real,A2: A] : % 0.31/0.57 ( ( X3 % 0.31/0.57 = ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V2 ) @ A2 ) ) % 0.31/0.57 = ( ( ( V2 % 0.31/0.57 = ( zero_zero @ real ) ) % 0.31/0.57 => ( X3 % 0.31/0.57 = ( zero_zero @ A ) ) ) % 0.31/0.57 & ( ( V2 % 0.31/0.57 != ( zero_zero @ real ) ) % 0.31/0.57 => ( ( real_V8093663219630862766scaleR @ A @ V2 @ X3 ) % 0.31/0.57 = ( real_V8093663219630862766scaleR @ A @ U @ A2 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % eq_vector_fraction_iff % 0.31/0.57 thf(fact_3450_vector__fraction__eq__iff,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.57 => ! [U: real,V2: real,A2: A,X3: A] : % 0.31/0.57 ( ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V2 ) @ A2 ) % 0.31/0.57 = X3 ) % 0.31/0.57 = ( ( ( V2 % 0.31/0.57 = ( zero_zero @ real ) ) % 0.31/0.57 => ( X3 % 0.31/0.57 = ( zero_zero @ A ) ) ) % 0.31/0.57 & ( ( V2 % 0.31/0.57 != ( zero_zero @ real ) ) % 0.31/0.57 => ( ( real_V8093663219630862766scaleR @ A @ U @ A2 ) % 0.31/0.57 = ( real_V8093663219630862766scaleR @ A @ V2 @ X3 ) ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % vector_fraction_eq_iff % 0.31/0.57 thf(fact_3451_Real__Vector__Spaces_Ole__add__iff1,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.57 => ! [A2: real,E3: A,C2: A,B2: real,D2: A] : % 0.31/0.57 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E3 ) @ D2 ) ) % 0.31/0.57 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ E3 ) @ C2 ) @ D2 ) ) ) ). % 0.31/0.57 % 0.31/0.57 % Real_Vector_Spaces.le_add_iff1 % 0.31/0.57 thf(fact_3452_Real__Vector__Spaces_Ole__add__iff2,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.57 => ! [A2: real,E3: A,C2: A,B2: real,D2: A] : % 0.31/0.57 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E3 ) @ D2 ) ) % 0.31/0.57 = ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % Real_Vector_Spaces.le_add_iff2 % 0.31/0.57 thf(fact_3453_scaleR__le__0__iff,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.57 => ! [A2: real,B2: A] : % 0.31/0.57 ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) % 0.31/0.57 = ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.57 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) % 0.31/0.57 | ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) ) % 0.31/0.57 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) % 0.31/0.57 | ( A2 % 0.31/0.57 = ( zero_zero @ real ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_le_0_iff % 0.31/0.57 thf(fact_3454_zero__le__scaleR__iff,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.57 => ! [A2: real,B2: A] : % 0.31/0.57 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) % 0.31/0.57 = ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.57 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) % 0.31/0.57 | ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) ) % 0.31/0.57 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) % 0.31/0.57 | ( A2 % 0.31/0.57 = ( zero_zero @ real ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % zero_le_scaleR_iff % 0.31/0.57 thf(fact_3455_scaleR__nonpos__nonpos,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.57 => ! [A2: real,B2: A] : % 0.31/0.57 ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) ) % 0.31/0.57 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) % 0.31/0.57 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_nonpos_nonpos % 0.31/0.57 thf(fact_3456_scaleR__nonpos__nonneg,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.57 => ! [A2: real,X3: A] : % 0.31/0.57 ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) ) % 0.31/0.57 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.57 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.57 % 0.31/0.57 % scaleR_nonpos_nonneg % 0.31/0.57 thf(fact_3457_scaleR__nonneg__nonpos,axiom, % 0.31/0.57 ! [A: $tType] : % 0.31/0.57 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.57 => ! [A2: real,X3: A] : % 0.31/0.57 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.58 => ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 0.31/0.58 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % scaleR_nonneg_nonpos % 0.31/0.58 thf(fact_3458_scaleR__nonneg__nonneg,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [A2: real,X3: A] : % 0.31/0.58 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.58 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.58 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % scaleR_nonneg_nonneg % 0.31/0.58 thf(fact_3459_split__scaleR__pos__le,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [A2: real,B2: A] : % 0.31/0.58 ( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.58 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) % 0.31/0.58 | ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) ) % 0.31/0.58 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) % 0.31/0.58 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % split_scaleR_pos_le % 0.31/0.58 thf(fact_3460_split__scaleR__neg__le,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [A2: real,X3: A] : % 0.31/0.58 ( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.58 & ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) ) % 0.31/0.58 | ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) ) % 0.31/0.58 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ) % 0.31/0.58 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( zero_zero @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % split_scaleR_neg_le % 0.31/0.58 thf(fact_3461_scaleR__mono_H,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [A2: real,B2: real,C2: A,D2: A] : % 0.31/0.58 ( ( ord_less_eq @ real @ A2 @ B2 ) % 0.31/0.58 => ( ( ord_less_eq @ A @ C2 @ D2 ) % 0.31/0.58 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.58 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 0.31/0.58 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ D2 ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % scaleR_mono' % 0.31/0.58 thf(fact_3462_scaleR__mono,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [A2: real,B2: real,X3: A,Y: A] : % 0.31/0.58 ( ( ord_less_eq @ real @ A2 @ B2 ) % 0.31/0.58 => ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.58 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B2 ) % 0.31/0.58 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.58 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ Y ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % scaleR_mono % 0.31/0.58 thf(fact_3463_scaleR__left__le__one__le,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [X3: A,A2: real] : % 0.31/0.58 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.58 => ( ( ord_less_eq @ real @ A2 @ ( one_one @ real ) ) % 0.31/0.58 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ X3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % scaleR_left_le_one_le % 0.31/0.58 thf(fact_3464_scaleR__2,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) % 0.31/0.58 = ( plus_plus @ A @ X3 @ X3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % scaleR_2 % 0.31/0.58 thf(fact_3465_prod__int__plus__eq,axiom, % 0.31/0.58 ! [I: nat,J: nat] : % 0.31/0.58 ( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ int @ int % 0.31/0.58 @ ^ [X: int] : X % 0.31/0.58 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod_int_plus_eq % 0.31/0.58 thf(fact_3466_ln__prod,axiom, % 0.31/0.58 ! [A: $tType,I6: set @ A,F2: A > real] : % 0.31/0.58 ( ( finite_finite2 @ A @ I6 ) % 0.31/0.58 => ( ! [I4: A] : % 0.31/0.58 ( ( member @ A @ I4 @ I6 ) % 0.31/0.58 => ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ I4 ) ) ) % 0.31/0.58 => ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F2 @ I6 ) ) % 0.31/0.58 = ( groups7311177749621191930dd_sum @ A @ real % 0.31/0.58 @ ^ [X: A] : ( ln_ln @ real @ ( F2 @ X ) ) % 0.31/0.58 @ I6 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % ln_prod % 0.31/0.58 thf(fact_3467_prod_Otriangle__reindex__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [G: nat > nat > A,N2: nat] : % 0.31/0.58 ( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G ) % 0.31/0.58 @ ( collect @ ( product_prod @ nat @ nat ) % 0.31/0.58 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.58 @ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N2 ) ) ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [K3: nat] : % 0.31/0.58 ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) ) % 0.31/0.58 @ ( set_ord_atMost @ nat @ K3 ) ) % 0.31/0.58 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.triangle_reindex_eq % 0.31/0.58 thf(fact_3468_prod_Otriangle__reindex,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [G: nat > nat > A,N2: nat] : % 0.31/0.58 ( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G ) % 0.31/0.58 @ ( collect @ ( product_prod @ nat @ nat ) % 0.31/0.58 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.58 @ ^ [I3: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N2 ) ) ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [K3: nat] : % 0.31/0.58 ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) ) % 0.31/0.58 @ ( set_ord_atMost @ nat @ K3 ) ) % 0.31/0.58 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.triangle_reindex % 0.31/0.58 thf(fact_3469_cot__gt__zero,axiom, % 0.31/0.58 ! [X3: real] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.58 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.58 => ( ord_less @ real @ ( zero_zero @ real ) @ ( cot @ real @ X3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % cot_gt_zero % 0.31/0.58 thf(fact_3470_int__ge__less__than__def,axiom, % 0.31/0.58 ( int_ge_less_than % 0.31/0.58 = ( ^ [D5: int] : % 0.31/0.58 ( collect @ ( product_prod @ int @ int ) % 0.31/0.58 @ ( product_case_prod @ int @ int @ $o % 0.31/0.58 @ ^ [Z7: int,Z6: int] : % 0.31/0.58 ( ( ord_less_eq @ int @ D5 @ Z7 ) % 0.31/0.58 & ( ord_less @ int @ Z7 @ Z6 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % int_ge_less_than_def % 0.31/0.58 thf(fact_3471_exp__first__two__terms,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.58 => ( ( exp @ A ) % 0.31/0.58 = ( ^ [X: A] : % 0.31/0.58 ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X ) % 0.31/0.58 @ ( suminf @ A % 0.31/0.58 @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ A @ X @ ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % exp_first_two_terms % 0.31/0.58 thf(fact_3472_arctan__def,axiom, % 0.31/0.58 ( arctan % 0.31/0.58 = ( ^ [Y2: real] : % 0.31/0.58 ( the @ real % 0.31/0.58 @ ^ [X: real] : % 0.31/0.58 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X ) % 0.31/0.58 & ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.58 & ( ( tan @ real @ X ) % 0.31/0.58 = Y2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % arctan_def % 0.31/0.58 thf(fact_3473_modulo__int__unfold,axiom, % 0.31/0.58 ! [L: int,K: int,N2: nat,M: nat] : % 0.31/0.58 ( ( ( ( ( sgn_sgn @ int @ L ) % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 | ( ( sgn_sgn @ int @ K ) % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 | ( N2 % 0.31/0.58 = ( zero_zero @ nat ) ) ) % 0.31/0.58 => ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) % 0.31/0.58 = ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) ) ) % 0.31/0.58 & ( ~ ( ( ( sgn_sgn @ int @ L ) % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 | ( ( sgn_sgn @ int @ K ) % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 | ( N2 % 0.31/0.58 = ( zero_zero @ nat ) ) ) % 0.31/0.58 => ( ( ( ( sgn_sgn @ int @ K ) % 0.31/0.58 = ( sgn_sgn @ int @ L ) ) % 0.31/0.58 => ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) % 0.31/0.58 = ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) ) % 0.31/0.58 & ( ( ( sgn_sgn @ int @ K ) % 0.31/0.58 != ( sgn_sgn @ int @ L ) ) % 0.31/0.58 => ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) % 0.31/0.58 = ( times_times @ int @ ( sgn_sgn @ int @ L ) % 0.31/0.58 @ ( minus_minus @ int % 0.31/0.58 @ ( semiring_1_of_nat @ int % 0.31/0.58 @ ( times_times @ nat @ N2 % 0.31/0.58 @ ( zero_neq_one_of_bool @ nat % 0.31/0.58 @ ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ) % 0.31/0.58 @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % modulo_int_unfold % 0.31/0.58 thf(fact_3474_powr__int,axiom, % 0.31/0.58 ! [X3: real,I: int] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.58 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I ) % 0.31/0.58 => ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ I ) ) % 0.31/0.58 = ( power_power @ real @ X3 @ ( nat2 @ I ) ) ) ) % 0.31/0.58 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I ) % 0.31/0.58 => ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ I ) ) % 0.31/0.58 = ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( nat2 @ ( uminus_uminus @ int @ I ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % powr_int % 0.31/0.58 thf(fact_3475_inverse__inverse__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) ) % 0.31/0.58 = A2 ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_inverse_eq % 0.31/0.58 thf(fact_3476_inverse__eq__iff__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ( inverse_inverse @ A @ A2 ) % 0.31/0.58 = ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 = ( A2 = B2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_eq_iff_eq % 0.31/0.58 thf(fact_3477_sgn__sgn,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( idom_abs_sgn @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) ) % 0.31/0.58 = ( sgn_sgn @ A @ A2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_sgn % 0.31/0.58 thf(fact_3478_inverse__zero,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) ) % 0.31/0.58 = ( zero_zero @ A ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_zero % 0.31/0.58 thf(fact_3479_inverse__nonzero__iff__nonzero,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ( inverse_inverse @ A @ A2 ) % 0.31/0.58 = ( zero_zero @ A ) ) % 0.31/0.58 = ( A2 % 0.31/0.58 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_nonzero_iff_nonzero % 0.31/0.58 thf(fact_3480_inverse__mult__distrib,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) ) % 0.31/0.58 = ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_mult_distrib % 0.31/0.58 thf(fact_3481_inverse__1,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ( ( inverse_inverse @ A @ ( one_one @ A ) ) % 0.31/0.58 = ( one_one @ A ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_1 % 0.31/0.58 thf(fact_3482_inverse__eq__1__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( field @ A ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( ( inverse_inverse @ A @ X3 ) % 0.31/0.58 = ( one_one @ A ) ) % 0.31/0.58 = ( X3 % 0.31/0.58 = ( one_one @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_eq_1_iff % 0.31/0.58 thf(fact_3483_inverse__divide,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( inverse_inverse @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 0.31/0.58 = ( divide_divide @ A @ B2 @ A2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_divide % 0.31/0.58 thf(fact_3484_inverse__minus__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A2 ) ) % 0.31/0.58 = ( uminus_uminus @ A @ ( inverse_inverse @ A @ A2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_minus_eq % 0.31/0.58 thf(fact_3485_sgn__0,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( idom_abs_sgn @ A ) % 0.31/0.58 => ( ( sgn_sgn @ A @ ( zero_zero @ A ) ) % 0.31/0.58 = ( zero_zero @ A ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_0 % 0.31/0.58 thf(fact_3486_sgn__zero,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V822414075346904944vector @ A ) % 0.31/0.58 => ( ( sgn_sgn @ A @ ( zero_zero @ A ) ) % 0.31/0.58 = ( zero_zero @ A ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_zero % 0.31/0.58 thf(fact_3487_sgn__1,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( idom_abs_sgn @ A ) % 0.31/0.58 => ( ( sgn_sgn @ A @ ( one_one @ A ) ) % 0.31/0.58 = ( one_one @ A ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_1 % 0.31/0.58 thf(fact_3488_sgn__divide,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( field_abs_sgn @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( sgn_sgn @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 0.31/0.58 = ( divide_divide @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_divide % 0.31/0.58 thf(fact_3489_idom__abs__sgn__class_Osgn__minus,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( idom_abs_sgn @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ A2 ) ) % 0.31/0.58 = ( uminus_uminus @ A @ ( sgn_sgn @ A @ A2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % idom_abs_sgn_class.sgn_minus % 0.31/0.58 thf(fact_3490_abs__inverse,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( field_abs_sgn @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( abs_abs @ A @ ( inverse_inverse @ A @ A2 ) ) % 0.31/0.58 = ( inverse_inverse @ A @ ( abs_abs @ A @ A2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % abs_inverse % 0.31/0.58 thf(fact_3491_inverse__sgn,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( inverse_inverse @ A @ ( sgn_sgn @ A @ A2 ) ) % 0.31/0.58 = ( sgn_sgn @ A @ A2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_sgn % 0.31/0.58 thf(fact_3492_sgn__inverse,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( field_abs_sgn @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( sgn_sgn @ A @ ( inverse_inverse @ A @ A2 ) ) % 0.31/0.58 = ( inverse_inverse @ A @ ( sgn_sgn @ A @ A2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_inverse % 0.31/0.58 thf(fact_3493_nat__int,axiom, % 0.31/0.58 ! [N2: nat] : % 0.31/0.58 ( ( nat2 @ ( semiring_1_of_nat @ int @ N2 ) ) % 0.31/0.58 = N2 ) ). % 0.31/0.58 % 0.31/0.58 % nat_int % 0.31/0.58 thf(fact_3494_inverse__nonnegative__iff__nonnegative,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) % 0.31/0.58 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_nonnegative_iff_nonnegative % 0.31/0.58 thf(fact_3495_inverse__nonpositive__iff__nonpositive,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) % 0.31/0.58 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_nonpositive_iff_nonpositive % 0.31/0.58 thf(fact_3496_inverse__positive__iff__positive,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) % 0.31/0.58 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_positive_iff_positive % 0.31/0.58 thf(fact_3497_inverse__negative__iff__negative,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) % 0.31/0.58 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_negative_iff_negative % 0.31/0.58 thf(fact_3498_inverse__less__iff__less__neg,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.31/0.58 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 0.31/0.58 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_less_iff_less_neg % 0.31/0.58 thf(fact_3499_inverse__less__iff__less,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.31/0.58 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_less_iff_less % 0.31/0.58 thf(fact_3500_sgn__greater,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) ) % 0.31/0.58 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_greater % 0.31/0.58 thf(fact_3501_sgn__less,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) ) % 0.31/0.58 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_less % 0.31/0.58 thf(fact_3502_divide__sgn,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( divide_divide @ A @ A2 @ ( sgn_sgn @ A @ B2 ) ) % 0.31/0.58 = ( times_times @ A @ A2 @ ( sgn_sgn @ A @ B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % divide_sgn % 0.31/0.58 thf(fact_3503_nat__numeral,axiom, % 0.31/0.58 ! [K: num] : % 0.31/0.58 ( ( nat2 @ ( numeral_numeral @ int @ K ) ) % 0.31/0.58 = ( numeral_numeral @ nat @ K ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_numeral % 0.31/0.58 thf(fact_3504_nat__of__bool,axiom, % 0.31/0.58 ! [P: $o] : % 0.31/0.58 ( ( nat2 @ ( zero_neq_one_of_bool @ int @ P ) ) % 0.31/0.58 = ( zero_neq_one_of_bool @ nat @ P ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_of_bool % 0.31/0.58 thf(fact_3505_inverse__le__iff__le__neg,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.31/0.58 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 0.31/0.58 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_le_iff_le_neg % 0.31/0.58 thf(fact_3506_inverse__le__iff__le,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 0.31/0.58 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_le_iff_le % 0.31/0.58 thf(fact_3507_right__inverse,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( times_times @ A @ A2 @ ( inverse_inverse @ A @ A2 ) ) % 0.31/0.58 = ( one_one @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % right_inverse % 0.31/0.58 thf(fact_3508_left__inverse,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 ) % 0.31/0.58 = ( one_one @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % left_inverse % 0.31/0.58 thf(fact_3509_sgn__pos,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.58 => ( ( sgn_sgn @ A @ A2 ) % 0.31/0.58 = ( one_one @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_pos % 0.31/0.58 thf(fact_3510_abs__sgn__eq__1,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) ) % 0.31/0.58 = ( one_one @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % abs_sgn_eq_1 % 0.31/0.58 thf(fact_3511_nat__1,axiom, % 0.31/0.58 ( ( nat2 @ ( one_one @ int ) ) % 0.31/0.58 = ( suc @ ( zero_zero @ nat ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_1 % 0.31/0.58 thf(fact_3512_sgn__mult__self__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ A2 ) ) % 0.31/0.58 = ( zero_neq_one_of_bool @ A % 0.31/0.58 @ ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_mult_self_eq % 0.31/0.58 thf(fact_3513_nat__0__iff,axiom, % 0.31/0.58 ! [I: int] : % 0.31/0.58 ( ( ( nat2 @ I ) % 0.31/0.58 = ( zero_zero @ nat ) ) % 0.31/0.58 = ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_0_iff % 0.31/0.58 thf(fact_3514_nat__le__0,axiom, % 0.31/0.58 ! [Z2: int] : % 0.31/0.58 ( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) ) % 0.31/0.58 => ( ( nat2 @ Z2 ) % 0.31/0.58 = ( zero_zero @ nat ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_le_0 % 0.31/0.58 thf(fact_3515_zless__nat__conj,axiom, % 0.31/0.58 ! [W2: int,Z2: int] : % 0.31/0.58 ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) ) % 0.31/0.58 = ( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) % 0.31/0.58 & ( ord_less @ int @ W2 @ Z2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % zless_nat_conj % 0.31/0.58 thf(fact_3516_nat__neg__numeral,axiom, % 0.31/0.58 ! [K: num] : % 0.31/0.58 ( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) % 0.31/0.58 = ( zero_zero @ nat ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_neg_numeral % 0.31/0.58 thf(fact_3517_sgn__abs,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( idom_abs_sgn @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) ) % 0.31/0.58 = ( zero_neq_one_of_bool @ A % 0.31/0.58 @ ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_abs % 0.31/0.58 thf(fact_3518_idom__abs__sgn__class_Oabs__sgn,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( idom_abs_sgn @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( sgn_sgn @ A @ ( abs_abs @ A @ A2 ) ) % 0.31/0.58 = ( zero_neq_one_of_bool @ A % 0.31/0.58 @ ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % idom_abs_sgn_class.abs_sgn % 0.31/0.58 thf(fact_3519_nat__zminus__int,axiom, % 0.31/0.58 ! [N2: nat] : % 0.31/0.58 ( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) % 0.31/0.58 = ( zero_zero @ nat ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_zminus_int % 0.31/0.58 thf(fact_3520_int__nat__eq,axiom, % 0.31/0.58 ! [Z2: int] : % 0.31/0.58 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 0.31/0.58 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) ) % 0.31/0.58 = Z2 ) ) % 0.31/0.58 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 0.31/0.58 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) ) % 0.31/0.58 = ( zero_zero @ int ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % int_nat_eq % 0.31/0.58 thf(fact_3521_sgn__mult__dvd__iff,axiom, % 0.31/0.58 ! [R2: int,L: int,K: int] : % 0.31/0.58 ( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ L ) @ K ) % 0.31/0.58 = ( ( dvd_dvd @ int @ L @ K ) % 0.31/0.58 & ( ( R2 % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 => ( K % 0.31/0.58 = ( zero_zero @ int ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_mult_dvd_iff % 0.31/0.58 thf(fact_3522_mult__sgn__dvd__iff,axiom, % 0.31/0.58 ! [L: int,R2: int,K: int] : % 0.31/0.58 ( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R2 ) ) @ K ) % 0.31/0.58 = ( ( dvd_dvd @ int @ L @ K ) % 0.31/0.58 & ( ( R2 % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 => ( K % 0.31/0.58 = ( zero_zero @ int ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % mult_sgn_dvd_iff % 0.31/0.58 thf(fact_3523_dvd__sgn__mult__iff,axiom, % 0.31/0.58 ! [L: int,R2: int,K: int] : % 0.31/0.58 ( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ K ) ) % 0.31/0.58 = ( ( dvd_dvd @ int @ L @ K ) % 0.31/0.58 | ( R2 % 0.31/0.58 = ( zero_zero @ int ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % dvd_sgn_mult_iff % 0.31/0.58 thf(fact_3524_dvd__mult__sgn__iff,axiom, % 0.31/0.58 ! [L: int,K: int,R2: int] : % 0.31/0.58 ( ( dvd_dvd @ int @ L @ ( times_times @ int @ K @ ( sgn_sgn @ int @ R2 ) ) ) % 0.31/0.58 = ( ( dvd_dvd @ int @ L @ K ) % 0.31/0.58 | ( R2 % 0.31/0.58 = ( zero_zero @ int ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % dvd_mult_sgn_iff % 0.31/0.58 thf(fact_3525_sgn__neg,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.31/0.58 => ( ( sgn_sgn @ A @ A2 ) % 0.31/0.58 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_neg % 0.31/0.58 thf(fact_3526_zero__less__nat__eq,axiom, % 0.31/0.58 ! [Z2: int] : % 0.31/0.58 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z2 ) ) % 0.31/0.58 = ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ). % 0.31/0.58 % 0.31/0.58 % zero_less_nat_eq % 0.31/0.58 thf(fact_3527_of__nat__nat,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ring_1 @ A ) % 0.31/0.58 => ! [Z2: int] : % 0.31/0.58 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 0.31/0.58 => ( ( semiring_1_of_nat @ A @ ( nat2 @ Z2 ) ) % 0.31/0.58 = ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % of_nat_nat % 0.31/0.58 thf(fact_3528_sgn__of__nat,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [N2: nat] : % 0.31/0.58 ( ( sgn_sgn @ A @ ( semiring_1_of_nat @ A @ N2 ) ) % 0.31/0.58 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_of_nat % 0.31/0.58 thf(fact_3529_diff__nat__numeral,axiom, % 0.31/0.58 ! [V2: num,V4: num] : % 0.31/0.58 ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( numeral_numeral @ nat @ V4 ) ) % 0.31/0.58 = ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ V4 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % diff_nat_numeral % 0.31/0.58 thf(fact_3530_numeral__power__eq__nat__cancel__iff,axiom, % 0.31/0.58 ! [X3: num,N2: nat,Y: int] : % 0.31/0.58 ( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) % 0.31/0.58 = ( nat2 @ Y ) ) % 0.31/0.58 = ( ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) % 0.31/0.58 = Y ) ) ). % 0.31/0.58 % 0.31/0.58 % numeral_power_eq_nat_cancel_iff % 0.31/0.58 thf(fact_3531_nat__eq__numeral__power__cancel__iff,axiom, % 0.31/0.58 ! [Y: int,X3: num,N2: nat] : % 0.31/0.58 ( ( ( nat2 @ Y ) % 0.31/0.58 = ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) ) % 0.31/0.58 = ( Y % 0.31/0.58 = ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_eq_numeral_power_cancel_iff % 0.31/0.58 thf(fact_3532_nat__abs__dvd__iff,axiom, % 0.31/0.58 ! [K: int,N2: nat] : % 0.31/0.58 ( ( dvd_dvd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N2 ) % 0.31/0.58 = ( dvd_dvd @ int @ K @ ( semiring_1_of_nat @ int @ N2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_abs_dvd_iff % 0.31/0.58 thf(fact_3533_dvd__nat__abs__iff,axiom, % 0.31/0.58 ! [N2: nat,K: int] : % 0.31/0.58 ( ( dvd_dvd @ nat @ N2 @ ( nat2 @ ( abs_abs @ int @ K ) ) ) % 0.31/0.58 = ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ). % 0.31/0.58 % 0.31/0.58 % dvd_nat_abs_iff % 0.31/0.58 thf(fact_3534_nat__ceiling__le__eq,axiom, % 0.31/0.58 ! [X3: real,A2: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X3 ) ) @ A2 ) % 0.31/0.58 = ( ord_less_eq @ real @ X3 @ ( semiring_1_of_nat @ real @ A2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_ceiling_le_eq % 0.31/0.58 thf(fact_3535_one__less__nat__eq,axiom, % 0.31/0.58 ! [Z2: int] : % 0.31/0.58 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z2 ) ) % 0.31/0.58 = ( ord_less @ int @ ( one_one @ int ) @ Z2 ) ) ). % 0.31/0.58 % 0.31/0.58 % one_less_nat_eq % 0.31/0.58 thf(fact_3536_nat__numeral__diff__1,axiom, % 0.31/0.58 ! [V2: num] : % 0.31/0.58 ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) % 0.31/0.58 = ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( one_one @ int ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_numeral_diff_1 % 0.31/0.58 thf(fact_3537_nat__less__numeral__power__cancel__iff,axiom, % 0.31/0.58 ! [A2: int,X3: num,N2: nat] : % 0.31/0.58 ( ( ord_less @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) ) % 0.31/0.58 = ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_less_numeral_power_cancel_iff % 0.31/0.58 thf(fact_3538_numeral__power__less__nat__cancel__iff,axiom, % 0.31/0.58 ! [X3: num,N2: nat,A2: int] : % 0.31/0.58 ( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) @ ( nat2 @ A2 ) ) % 0.31/0.58 = ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) @ A2 ) ) ). % 0.31/0.58 % 0.31/0.58 % numeral_power_less_nat_cancel_iff % 0.31/0.58 thf(fact_3539_nat__le__numeral__power__cancel__iff,axiom, % 0.31/0.58 ! [A2: int,X3: num,N2: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) ) % 0.31/0.58 = ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_le_numeral_power_cancel_iff % 0.31/0.58 thf(fact_3540_numeral__power__le__nat__cancel__iff,axiom, % 0.31/0.58 ! [X3: num,N2: nat,A2: int] : % 0.31/0.58 ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) @ ( nat2 @ A2 ) ) % 0.31/0.58 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) @ A2 ) ) ). % 0.31/0.58 % 0.31/0.58 % numeral_power_le_nat_cancel_iff % 0.31/0.58 thf(fact_3541_nonzero__norm__inverse,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ A2 ) ) % 0.31/0.58 = ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nonzero_norm_inverse % 0.31/0.58 thf(fact_3542_sgn__zero__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V822414075346904944vector @ A ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( ( sgn_sgn @ A @ X3 ) % 0.31/0.58 = ( zero_zero @ A ) ) % 0.31/0.58 = ( X3 % 0.31/0.58 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_zero_iff % 0.31/0.58 thf(fact_3543_sgn__eq__0__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( idom_abs_sgn @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ( sgn_sgn @ A @ A2 ) % 0.31/0.58 = ( zero_zero @ A ) ) % 0.31/0.58 = ( A2 % 0.31/0.58 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_eq_0_iff % 0.31/0.58 thf(fact_3544_sgn__0__0,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ( sgn_sgn @ A @ A2 ) % 0.31/0.58 = ( zero_zero @ A ) ) % 0.31/0.58 = ( A2 % 0.31/0.58 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_0_0 % 0.31/0.58 thf(fact_3545_sgn__mult,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( idom_abs_sgn @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( sgn_sgn @ A @ ( times_times @ A @ A2 @ B2 ) ) % 0.31/0.58 = ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_mult % 0.31/0.58 thf(fact_3546_same__sgn__sgn__add,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [B2: A,A2: A] : % 0.31/0.58 ( ( ( sgn_sgn @ A @ B2 ) % 0.31/0.58 = ( sgn_sgn @ A @ A2 ) ) % 0.31/0.58 => ( ( sgn_sgn @ A @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.31/0.58 = ( sgn_sgn @ A @ A2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % same_sgn_sgn_add % 0.31/0.58 thf(fact_3547_mult__commute__imp__mult__inverse__commute,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [Y: A,X3: A] : % 0.31/0.58 ( ( ( times_times @ A @ Y @ X3 ) % 0.31/0.58 = ( times_times @ A @ X3 @ Y ) ) % 0.31/0.58 => ( ( times_times @ A @ ( inverse_inverse @ A @ Y ) @ X3 ) % 0.31/0.58 = ( times_times @ A @ X3 @ ( inverse_inverse @ A @ Y ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % mult_commute_imp_mult_inverse_commute % 0.31/0.58 thf(fact_3548_inverse__eq__imp__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ( inverse_inverse @ A @ A2 ) % 0.31/0.58 = ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 => ( A2 = B2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_eq_imp_eq % 0.31/0.58 thf(fact_3549_nonzero__imp__inverse__nonzero,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( inverse_inverse @ A @ A2 ) % 0.31/0.58 != ( zero_zero @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nonzero_imp_inverse_nonzero % 0.31/0.58 thf(fact_3550_nonzero__inverse__inverse__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) ) % 0.31/0.58 = A2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nonzero_inverse_inverse_eq % 0.31/0.58 thf(fact_3551_nonzero__inverse__eq__imp__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ( inverse_inverse @ A @ A2 ) % 0.31/0.58 = ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 => ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( B2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( A2 = B2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nonzero_inverse_eq_imp_eq % 0.31/0.58 thf(fact_3552_inverse__zero__imp__zero,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ( inverse_inverse @ A @ A2 ) % 0.31/0.58 = ( zero_zero @ A ) ) % 0.31/0.58 => ( A2 % 0.31/0.58 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_zero_imp_zero % 0.31/0.58 thf(fact_3553_field__class_Ofield__inverse__zero,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( field @ A ) % 0.31/0.58 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) ) % 0.31/0.58 = ( zero_zero @ A ) ) ) ). % 0.31/0.58 % 0.31/0.58 % field_class.field_inverse_zero % 0.31/0.58 thf(fact_3554_nonzero__inverse__scaleR__distrib,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5047593784448816457lgebra @ A ) % 0.31/0.58 => ! [A2: real,X3: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ real ) ) % 0.31/0.58 => ( ( X3 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( inverse_inverse @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) ) % 0.31/0.58 = ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ A2 ) @ ( inverse_inverse @ A @ X3 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nonzero_inverse_scaleR_distrib % 0.31/0.58 thf(fact_3555_norm__inverse__le__norm,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.58 => ! [R2: real,X3: A] : % 0.31/0.58 ( ( ord_less_eq @ real @ R2 @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) % 0.31/0.58 => ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 ) % 0.31/0.58 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ X3 ) ) @ ( inverse_inverse @ real @ R2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % norm_inverse_le_norm % 0.31/0.58 thf(fact_3556_positive__imp__inverse__positive,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.58 => ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % positive_imp_inverse_positive % 0.31/0.58 thf(fact_3557_negative__imp__inverse__negative,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 0.31/0.58 => ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % negative_imp_inverse_negative % 0.31/0.58 thf(fact_3558_inverse__positive__imp__positive,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) % 0.31/0.58 => ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_positive_imp_positive % 0.31/0.58 thf(fact_3559_inverse__negative__imp__negative,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) % 0.31/0.58 => ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_negative_imp_negative % 0.31/0.58 thf(fact_3560_less__imp__inverse__less__neg,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.58 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 0.31/0.58 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % less_imp_inverse_less_neg % 0.31/0.58 thf(fact_3561_inverse__less__imp__less__neg,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 0.31/0.58 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_less_imp_less_neg % 0.31/0.58 thf(fact_3562_less__imp__inverse__less,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.58 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % less_imp_inverse_less % 0.31/0.58 thf(fact_3563_inverse__less__imp__less,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.58 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_less_imp_less % 0.31/0.58 thf(fact_3564_nonzero__inverse__mult__distrib,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( B2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) ) % 0.31/0.58 = ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nonzero_inverse_mult_distrib % 0.31/0.58 thf(fact_3565_nonzero__inverse__minus__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A2 ) ) % 0.31/0.58 = ( uminus_uminus @ A @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nonzero_inverse_minus_eq % 0.31/0.58 thf(fact_3566_inverse__unique,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ( times_times @ A @ A2 @ B2 ) % 0.31/0.58 = ( one_one @ A ) ) % 0.31/0.58 => ( ( inverse_inverse @ A @ A2 ) % 0.31/0.58 = B2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_unique % 0.31/0.58 thf(fact_3567_divide__inverse__commute,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( field @ A ) % 0.31/0.58 => ( ( divide_divide @ A ) % 0.31/0.58 = ( ^ [A4: A,B4: A] : ( times_times @ A @ ( inverse_inverse @ A @ B4 ) @ A4 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % divide_inverse_commute % 0.31/0.58 thf(fact_3568_divide__inverse,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ( ( divide_divide @ A ) % 0.31/0.58 = ( ^ [A4: A,B4: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B4 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % divide_inverse % 0.31/0.58 thf(fact_3569_field__class_Ofield__divide__inverse,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( field @ A ) % 0.31/0.58 => ( ( divide_divide @ A ) % 0.31/0.58 = ( ^ [A4: A,B4: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B4 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % field_class.field_divide_inverse % 0.31/0.58 thf(fact_3570_inverse__eq__divide,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ( ( inverse_inverse @ A ) % 0.31/0.58 = ( divide_divide @ A @ ( one_one @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_eq_divide % 0.31/0.58 thf(fact_3571_power__mult__power__inverse__commute,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [X3: A,M: nat,N2: nat] : % 0.31/0.58 ( ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ N2 ) ) % 0.31/0.58 = ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ N2 ) @ ( power_power @ A @ X3 @ M ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % power_mult_power_inverse_commute % 0.31/0.58 thf(fact_3572_power__mult__inverse__distrib,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [X3: A,M: nat] : % 0.31/0.58 ( ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( inverse_inverse @ A @ X3 ) ) % 0.31/0.58 = ( times_times @ A @ ( inverse_inverse @ A @ X3 ) @ ( power_power @ A @ X3 @ M ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % power_mult_inverse_distrib % 0.31/0.58 thf(fact_3573_mult__inverse__of__nat__commute,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [Xa2: nat,X3: A] : % 0.31/0.58 ( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa2 ) ) @ X3 ) % 0.31/0.58 = ( times_times @ A @ X3 @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % mult_inverse_of_nat_commute % 0.31/0.58 thf(fact_3574_nonzero__abs__inverse,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( abs_abs @ A @ ( inverse_inverse @ A @ A2 ) ) % 0.31/0.58 = ( inverse_inverse @ A @ ( abs_abs @ A @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nonzero_abs_inverse % 0.31/0.58 thf(fact_3575_sgn__not__eq__imp,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [B2: A,A2: A] : % 0.31/0.58 ( ( ( sgn_sgn @ A @ B2 ) % 0.31/0.58 != ( sgn_sgn @ A @ A2 ) ) % 0.31/0.58 => ( ( ( sgn_sgn @ A @ A2 ) % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( ( sgn_sgn @ A @ B2 ) % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( sgn_sgn @ A @ A2 ) % 0.31/0.58 = ( uminus_uminus @ A @ ( sgn_sgn @ A @ B2 ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_not_eq_imp % 0.31/0.58 thf(fact_3576_mult__inverse__of__int__commute,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [Xa2: int,X3: A] : % 0.31/0.58 ( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa2 ) ) @ X3 ) % 0.31/0.58 = ( times_times @ A @ X3 @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % mult_inverse_of_int_commute % 0.31/0.58 thf(fact_3577_sgn__minus__1,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( idom_abs_sgn @ A ) % 0.31/0.58 => ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.31/0.58 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_minus_1 % 0.31/0.58 thf(fact_3578_nat__zero__as__int,axiom, % 0.31/0.58 ( ( zero_zero @ nat ) % 0.31/0.58 = ( nat2 @ ( zero_zero @ int ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_zero_as_int % 0.31/0.58 thf(fact_3579_mult__sgn__abs,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( times_times @ A @ ( sgn_sgn @ A @ X3 ) @ ( abs_abs @ A @ X3 ) ) % 0.31/0.58 = X3 ) ) ). % 0.31/0.58 % 0.31/0.58 % mult_sgn_abs % 0.31/0.58 thf(fact_3580_sgn__mult__abs,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( idom_abs_sgn @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( abs_abs @ A @ A2 ) ) % 0.31/0.58 = A2 ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_mult_abs % 0.31/0.58 thf(fact_3581_abs__mult__sgn,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( idom_abs_sgn @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( sgn_sgn @ A @ A2 ) ) % 0.31/0.58 = A2 ) ) ). % 0.31/0.58 % 0.31/0.58 % abs_mult_sgn % 0.31/0.58 thf(fact_3582_linordered__idom__class_Oabs__sgn,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ( ( abs_abs @ A ) % 0.31/0.58 = ( ^ [K3: A] : ( times_times @ A @ K3 @ ( sgn_sgn @ A @ K3 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % linordered_idom_class.abs_sgn % 0.31/0.58 thf(fact_3583_nat__mono,axiom, % 0.31/0.58 ! [X3: int,Y: int] : % 0.31/0.58 ( ( ord_less_eq @ int @ X3 @ Y ) % 0.31/0.58 => ( ord_less_eq @ nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_mono % 0.31/0.58 thf(fact_3584_int__sgnE,axiom, % 0.31/0.58 ! [K: int] : % 0.31/0.58 ~ ! [N3: nat,L4: int] : % 0.31/0.58 ( K % 0.31/0.58 != ( times_times @ int @ ( sgn_sgn @ int @ L4 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % int_sgnE % 0.31/0.58 thf(fact_3585_same__sgn__abs__add,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [B2: A,A2: A] : % 0.31/0.58 ( ( ( sgn_sgn @ A @ B2 ) % 0.31/0.58 = ( sgn_sgn @ A @ A2 ) ) % 0.31/0.58 => ( ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.31/0.58 = ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % same_sgn_abs_add % 0.31/0.58 thf(fact_3586_ex__nat,axiom, % 0.31/0.58 ( ( ^ [P2: nat > $o] : % 0.31/0.58 ? [X7: nat] : ( P2 @ X7 ) ) % 0.31/0.58 = ( ^ [P3: nat > $o] : % 0.31/0.58 ? [X: int] : % 0.31/0.58 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X ) % 0.31/0.58 & ( P3 @ ( nat2 @ X ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % ex_nat % 0.31/0.58 thf(fact_3587_all__nat,axiom, % 0.31/0.58 ( ( ^ [P2: nat > $o] : % 0.31/0.58 ! [X7: nat] : ( P2 @ X7 ) ) % 0.31/0.58 = ( ^ [P3: nat > $o] : % 0.31/0.58 ! [X: int] : % 0.31/0.58 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X ) % 0.31/0.58 => ( P3 @ ( nat2 @ X ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % all_nat % 0.31/0.58 thf(fact_3588_eq__nat__nat__iff,axiom, % 0.31/0.58 ! [Z2: int,Z8: int] : % 0.31/0.58 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 0.31/0.58 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 ) % 0.31/0.58 => ( ( ( nat2 @ Z2 ) % 0.31/0.58 = ( nat2 @ Z8 ) ) % 0.31/0.58 = ( Z2 = Z8 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % eq_nat_nat_iff % 0.31/0.58 thf(fact_3589_le__imp__inverse__le__neg,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.58 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 0.31/0.58 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % le_imp_inverse_le_neg % 0.31/0.58 thf(fact_3590_inverse__le__imp__le__neg,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 0.31/0.58 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_le_imp_le_neg % 0.31/0.58 thf(fact_3591_le__imp__inverse__le,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.58 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % le_imp_inverse_le % 0.31/0.58 thf(fact_3592_inverse__le__imp__le,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.58 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_le_imp_le % 0.31/0.58 thf(fact_3593_inverse__le__1__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X3 ) @ ( one_one @ A ) ) % 0.31/0.58 = ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 0.31/0.58 | ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_le_1_iff % 0.31/0.58 thf(fact_3594_one__less__inverse__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X3 ) ) % 0.31/0.58 = ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.58 & ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % one_less_inverse_iff % 0.31/0.58 thf(fact_3595_one__less__inverse,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.58 => ( ( ord_less @ A @ A2 @ ( one_one @ A ) ) % 0.31/0.58 => ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % one_less_inverse % 0.31/0.58 thf(fact_3596_field__class_Ofield__inverse,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( field @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 ) % 0.31/0.58 = ( one_one @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % field_class.field_inverse % 0.31/0.58 thf(fact_3597_division__ring__inverse__add,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( B2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 = ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % division_ring_inverse_add % 0.31/0.58 thf(fact_3598_inverse__add,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( B2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 = ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_add % 0.31/0.58 thf(fact_3599_division__ring__inverse__diff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( B2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( minus_minus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 = ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( minus_minus @ A @ B2 @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % division_ring_inverse_diff % 0.31/0.58 thf(fact_3600_nonzero__inverse__eq__divide,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( inverse_inverse @ A @ A2 ) % 0.31/0.58 = ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nonzero_inverse_eq_divide % 0.31/0.58 thf(fact_3601_sgn__1__pos,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ( sgn_sgn @ A @ A2 ) % 0.31/0.58 = ( one_one @ A ) ) % 0.31/0.58 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_1_pos % 0.31/0.58 thf(fact_3602_abs__sgn__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ( A2 % 0.31/0.58 = ( zero_zero @ A ) ) % 0.31/0.58 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) ) % 0.31/0.58 = ( zero_zero @ A ) ) ) % 0.31/0.58 & ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) ) % 0.31/0.58 = ( one_one @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % abs_sgn_eq % 0.31/0.58 thf(fact_3603_nat__mono__iff,axiom, % 0.31/0.58 ! [Z2: int,W2: int] : % 0.31/0.58 ( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) % 0.31/0.58 => ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) ) % 0.31/0.58 = ( ord_less @ int @ W2 @ Z2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_mono_iff % 0.31/0.58 thf(fact_3604_of__nat__ceiling,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.58 => ! [R2: A] : ( ord_less_eq @ A @ R2 @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % of_nat_ceiling % 0.31/0.58 thf(fact_3605_zless__nat__eq__int__zless,axiom, % 0.31/0.58 ! [M: nat,Z2: int] : % 0.31/0.58 ( ( ord_less @ nat @ M @ ( nat2 @ Z2 ) ) % 0.31/0.58 = ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z2 ) ) ). % 0.31/0.58 % 0.31/0.58 % zless_nat_eq_int_zless % 0.31/0.58 thf(fact_3606_nat__le__iff,axiom, % 0.31/0.58 ! [X3: int,N2: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ ( nat2 @ X3 ) @ N2 ) % 0.31/0.58 = ( ord_less_eq @ int @ X3 @ ( semiring_1_of_nat @ int @ N2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_le_iff % 0.31/0.58 thf(fact_3607_nat__0__le,axiom, % 0.31/0.58 ! [Z2: int] : % 0.31/0.58 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 0.31/0.58 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) ) % 0.31/0.58 = Z2 ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_0_le % 0.31/0.58 thf(fact_3608_int__eq__iff,axiom, % 0.31/0.58 ! [M: nat,Z2: int] : % 0.31/0.58 ( ( ( semiring_1_of_nat @ int @ M ) % 0.31/0.58 = Z2 ) % 0.31/0.58 = ( ( M % 0.31/0.58 = ( nat2 @ Z2 ) ) % 0.31/0.58 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % int_eq_iff % 0.31/0.58 thf(fact_3609_nat__int__add,axiom, % 0.31/0.58 ! [A2: nat,B2: nat] : % 0.31/0.58 ( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) % 0.31/0.58 = ( plus_plus @ nat @ A2 @ B2 ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_int_add % 0.31/0.58 thf(fact_3610_nat__abs__mult__distrib,axiom, % 0.31/0.58 ! [W2: int,Z2: int] : % 0.31/0.58 ( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W2 @ Z2 ) ) ) % 0.31/0.58 = ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W2 ) ) @ ( nat2 @ ( abs_abs @ int @ Z2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_abs_mult_distrib % 0.31/0.58 thf(fact_3611_nat__plus__as__int,axiom, % 0.31/0.58 ( ( plus_plus @ nat ) % 0.31/0.58 = ( ^ [A4: nat,B4: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_plus_as_int % 0.31/0.58 thf(fact_3612_inverse__less__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 0.31/0.58 => ( ord_less @ A @ B2 @ A2 ) ) % 0.31/0.58 & ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) % 0.31/0.58 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_less_iff % 0.31/0.58 thf(fact_3613_inverse__le__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 0.31/0.58 => ( ord_less_eq @ A @ B2 @ A2 ) ) % 0.31/0.58 & ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) % 0.31/0.58 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_le_iff % 0.31/0.58 thf(fact_3614_one__le__inverse__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X3 ) ) % 0.31/0.58 = ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.58 & ( ord_less_eq @ A @ X3 @ ( one_one @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % one_le_inverse_iff % 0.31/0.58 thf(fact_3615_inverse__less__1__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( ord_less @ A @ ( inverse_inverse @ A @ X3 ) @ ( one_one @ A ) ) % 0.31/0.58 = ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 0.31/0.58 | ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_less_1_iff % 0.31/0.58 thf(fact_3616_one__le__inverse,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.58 => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) % 0.31/0.58 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % one_le_inverse % 0.31/0.58 thf(fact_3617_inverse__diff__inverse,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( A2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( B2 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( minus_minus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 0.31/0.58 = ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % inverse_diff_inverse % 0.31/0.58 thf(fact_3618_reals__Archimedean,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( archim462609752435547400_field @ A ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.58 => ? [N3: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ X3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % reals_Archimedean % 0.31/0.58 thf(fact_3619_sgn__if,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ( ( sgn_sgn @ A ) % 0.31/0.58 = ( ^ [X: A] : % 0.31/0.58 ( if @ A % 0.31/0.58 @ ( X % 0.31/0.58 = ( zero_zero @ A ) ) % 0.31/0.58 @ ( zero_zero @ A ) % 0.31/0.58 @ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_if % 0.31/0.58 thf(fact_3620_sgn__1__neg,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [A2: A] : % 0.31/0.58 ( ( ( sgn_sgn @ A @ A2 ) % 0.31/0.58 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.31/0.58 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_1_neg % 0.31/0.58 thf(fact_3621_of__nat__floor,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.58 => ! [R2: A] : % 0.31/0.58 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 ) % 0.31/0.58 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archim6421214686448440834_floor @ A @ R2 ) ) ) @ R2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % of_nat_floor % 0.31/0.58 thf(fact_3622_zsgn__def,axiom, % 0.31/0.58 ( ( sgn_sgn @ int ) % 0.31/0.58 = ( ^ [I3: int] : % 0.31/0.58 ( if @ int % 0.31/0.58 @ ( I3 % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 @ ( zero_zero @ int ) % 0.31/0.58 @ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I3 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % zsgn_def % 0.31/0.58 thf(fact_3623_powr__real__of__int,axiom, % 0.31/0.58 ! [X3: real,N2: int] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.58 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) % 0.31/0.58 => ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ N2 ) ) % 0.31/0.58 = ( power_power @ real @ X3 @ ( nat2 @ N2 ) ) ) ) % 0.31/0.58 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) % 0.31/0.58 => ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ N2 ) ) % 0.31/0.58 = ( inverse_inverse @ real @ ( power_power @ real @ X3 @ ( nat2 @ ( uminus_uminus @ int @ N2 ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % powr_real_of_int % 0.31/0.58 thf(fact_3624_real__vector__eq__affinity,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.58 => ! [M: real,Y: A,X3: A,C2: A] : % 0.31/0.58 ( ( M % 0.31/0.58 != ( zero_zero @ real ) ) % 0.31/0.58 => ( ( Y % 0.31/0.58 = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X3 ) @ C2 ) ) % 0.31/0.58 = ( ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) ) % 0.31/0.58 = X3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % real_vector_eq_affinity % 0.31/0.58 thf(fact_3625_real__vector__affinity__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.58 => ! [M: real,X3: A,C2: A,Y: A] : % 0.31/0.58 ( ( M % 0.31/0.58 != ( zero_zero @ real ) ) % 0.31/0.58 => ( ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X3 ) @ C2 ) % 0.31/0.58 = Y ) % 0.31/0.58 = ( X3 % 0.31/0.58 = ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % real_vector_affinity_eq % 0.31/0.58 thf(fact_3626_pos__divideR__le__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,B2: A,A2: A] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.58 => ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 ) % 0.31/0.58 = ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % pos_divideR_le_eq % 0.31/0.58 thf(fact_3627_pos__le__divideR__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.58 => ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) % 0.31/0.58 = ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % pos_le_divideR_eq % 0.31/0.58 thf(fact_3628_neg__divideR__le__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,B2: A,A2: A] : % 0.31/0.58 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 0.31/0.58 => ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 ) % 0.31/0.58 = ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % neg_divideR_le_eq % 0.31/0.58 thf(fact_3629_neg__le__divideR__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 0.31/0.58 => ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) % 0.31/0.58 = ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % neg_le_divideR_eq % 0.31/0.58 thf(fact_3630_pos__divideR__less__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,B2: A,A2: A] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.58 => ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 ) % 0.31/0.58 = ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % pos_divideR_less_eq % 0.31/0.58 thf(fact_3631_pos__less__divideR__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.58 => ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) % 0.31/0.58 = ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % pos_less_divideR_eq % 0.31/0.58 thf(fact_3632_neg__divideR__less__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,B2: A,A2: A] : % 0.31/0.58 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 0.31/0.58 => ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 ) % 0.31/0.58 = ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % neg_divideR_less_eq % 0.31/0.58 thf(fact_3633_neg__less__divideR__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 0.31/0.58 => ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) % 0.31/0.58 = ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % neg_less_divideR_eq % 0.31/0.58 thf(fact_3634_nat__less__eq__zless,axiom, % 0.31/0.58 ! [W2: int,Z2: int] : % 0.31/0.58 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 ) % 0.31/0.58 => ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) ) % 0.31/0.58 = ( ord_less @ int @ W2 @ Z2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_less_eq_zless % 0.31/0.58 thf(fact_3635_nat__le__eq__zle,axiom, % 0.31/0.58 ! [W2: int,Z2: int] : % 0.31/0.58 ( ( ( ord_less @ int @ ( zero_zero @ int ) @ W2 ) % 0.31/0.58 | ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) ) % 0.31/0.58 = ( ord_less_eq @ int @ W2 @ Z2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_le_eq_zle % 0.31/0.58 thf(fact_3636_norm__sgn,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V822414075346904944vector @ A ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( ( X3 % 0.31/0.58 = ( zero_zero @ A ) ) % 0.31/0.58 => ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X3 ) ) % 0.31/0.58 = ( zero_zero @ real ) ) ) % 0.31/0.58 & ( ( X3 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X3 ) ) % 0.31/0.58 = ( one_one @ real ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % norm_sgn % 0.31/0.58 thf(fact_3637_nat__eq__iff,axiom, % 0.31/0.58 ! [W2: int,M: nat] : % 0.31/0.58 ( ( ( nat2 @ W2 ) % 0.31/0.58 = M ) % 0.31/0.58 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 ) % 0.31/0.58 => ( W2 % 0.31/0.58 = ( semiring_1_of_nat @ int @ M ) ) ) % 0.31/0.58 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 ) % 0.31/0.58 => ( M % 0.31/0.58 = ( zero_zero @ nat ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_eq_iff % 0.31/0.58 thf(fact_3638_nat__eq__iff2,axiom, % 0.31/0.58 ! [M: nat,W2: int] : % 0.31/0.58 ( ( M % 0.31/0.58 = ( nat2 @ W2 ) ) % 0.31/0.58 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 ) % 0.31/0.58 => ( W2 % 0.31/0.58 = ( semiring_1_of_nat @ int @ M ) ) ) % 0.31/0.58 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 ) % 0.31/0.58 => ( M % 0.31/0.58 = ( zero_zero @ nat ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_eq_iff2 % 0.31/0.58 thf(fact_3639_split__nat,axiom, % 0.31/0.58 ! [P: nat > $o,I: int] : % 0.31/0.58 ( ( P @ ( nat2 @ I ) ) % 0.31/0.58 = ( ! [N: nat] : % 0.31/0.58 ( ( I % 0.31/0.58 = ( semiring_1_of_nat @ int @ N ) ) % 0.31/0.58 => ( P @ N ) ) % 0.31/0.58 & ( ( ord_less @ int @ I @ ( zero_zero @ int ) ) % 0.31/0.58 => ( P @ ( zero_zero @ nat ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % split_nat % 0.31/0.58 thf(fact_3640_le__mult__nat__floor,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( nat2 @ ( archim6421214686448440834_floor @ A @ A2 ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ B2 ) ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % le_mult_nat_floor % 0.31/0.58 thf(fact_3641_le__nat__iff,axiom, % 0.31/0.58 ! [K: int,N2: nat] : % 0.31/0.58 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ N2 @ ( nat2 @ K ) ) % 0.31/0.58 = ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ) ). % 0.31/0.58 % 0.31/0.58 % le_nat_iff % 0.31/0.58 thf(fact_3642_nat__add__distrib,axiom, % 0.31/0.58 ! [Z2: int,Z8: int] : % 0.31/0.58 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 0.31/0.58 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 ) % 0.31/0.58 => ( ( nat2 @ ( plus_plus @ int @ Z2 @ Z8 ) ) % 0.31/0.58 = ( plus_plus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_add_distrib % 0.31/0.58 thf(fact_3643_nat__mult__distrib,axiom, % 0.31/0.58 ! [Z2: int,Z8: int] : % 0.31/0.58 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 0.31/0.58 => ( ( nat2 @ ( times_times @ int @ Z2 @ Z8 ) ) % 0.31/0.58 = ( times_times @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_mult_distrib % 0.31/0.58 thf(fact_3644_Suc__as__int,axiom, % 0.31/0.58 ( suc % 0.31/0.58 = ( ^ [A4: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( one_one @ int ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Suc_as_int % 0.31/0.58 thf(fact_3645_forall__pos__mono__1,axiom, % 0.31/0.58 ! [P: real > $o,E3: real] : % 0.31/0.58 ( ! [D3: real,E: real] : % 0.31/0.58 ( ( ord_less @ real @ D3 @ E ) % 0.31/0.58 => ( ( P @ D3 ) % 0.31/0.58 => ( P @ E ) ) ) % 0.31/0.58 => ( ! [N3: nat] : ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) ) % 0.31/0.58 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 ) % 0.31/0.58 => ( P @ E3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % forall_pos_mono_1 % 0.31/0.58 thf(fact_3646_forall__pos__mono,axiom, % 0.31/0.58 ! [P: real > $o,E3: real] : % 0.31/0.58 ( ! [D3: real,E: real] : % 0.31/0.58 ( ( ord_less @ real @ D3 @ E ) % 0.31/0.58 => ( ( P @ D3 ) % 0.31/0.58 => ( P @ E ) ) ) % 0.31/0.58 => ( ! [N3: nat] : % 0.31/0.58 ( ( N3 % 0.31/0.58 != ( zero_zero @ nat ) ) % 0.31/0.58 => ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) % 0.31/0.58 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 ) % 0.31/0.58 => ( P @ E3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % forall_pos_mono % 0.31/0.58 thf(fact_3647_real__arch__inverse,axiom, % 0.31/0.58 ! [E3: real] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 ) % 0.31/0.58 = ( ? [N: nat] : % 0.31/0.58 ( ( N % 0.31/0.58 != ( zero_zero @ nat ) ) % 0.31/0.58 & ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N ) ) ) % 0.31/0.58 & ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N ) ) @ E3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % real_arch_inverse % 0.31/0.58 thf(fact_3648_nat__diff__distrib,axiom, % 0.31/0.58 ! [Z8: int,Z2: int] : % 0.31/0.58 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 ) % 0.31/0.58 => ( ( ord_less_eq @ int @ Z8 @ Z2 ) % 0.31/0.58 => ( ( nat2 @ ( minus_minus @ int @ Z2 @ Z8 ) ) % 0.31/0.58 = ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_diff_distrib % 0.31/0.58 thf(fact_3649_nat__diff__distrib_H,axiom, % 0.31/0.58 ! [X3: int,Y: int] : % 0.31/0.58 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 ) % 0.31/0.58 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y ) % 0.31/0.58 => ( ( nat2 @ ( minus_minus @ int @ X3 @ Y ) ) % 0.31/0.58 = ( minus_minus @ nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_diff_distrib' % 0.31/0.58 thf(fact_3650_nat__abs__triangle__ineq,axiom, % 0.31/0.58 ! [K: int,L: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K @ L ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_abs_triangle_ineq % 0.31/0.58 thf(fact_3651_nat__floor__neg,axiom, % 0.31/0.58 ! [X3: real] : % 0.31/0.58 ( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) % 0.31/0.58 => ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X3 ) ) % 0.31/0.58 = ( zero_zero @ nat ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_floor_neg % 0.31/0.58 thf(fact_3652_nat__power__eq,axiom, % 0.31/0.58 ! [Z2: int,N2: nat] : % 0.31/0.58 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 0.31/0.58 => ( ( nat2 @ ( power_power @ int @ Z2 @ N2 ) ) % 0.31/0.58 = ( power_power @ nat @ ( nat2 @ Z2 ) @ N2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_power_eq % 0.31/0.58 thf(fact_3653_ln__inverse,axiom, % 0.31/0.58 ! [X3: real] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.58 => ( ( ln_ln @ real @ ( inverse_inverse @ real @ X3 ) ) % 0.31/0.58 = ( uminus_uminus @ real @ ( ln_ln @ real @ X3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % ln_inverse % 0.31/0.58 thf(fact_3654_floor__eq3,axiom, % 0.31/0.58 ! [N2: nat,X3: real] : % 0.31/0.58 ( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X3 ) % 0.31/0.58 => ( ( ord_less @ real @ X3 @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) % 0.31/0.58 => ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X3 ) ) % 0.31/0.58 = N2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % floor_eq3 % 0.31/0.58 thf(fact_3655_le__nat__floor,axiom, % 0.31/0.58 ! [X3: nat,A2: real] : % 0.31/0.58 ( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X3 ) @ A2 ) % 0.31/0.58 => ( ord_less_eq @ nat @ X3 @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % le_nat_floor % 0.31/0.58 thf(fact_3656_the__elem__def,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( the_elem @ A ) % 0.31/0.58 = ( ^ [X6: set @ A] : % 0.31/0.58 ( the @ A % 0.31/0.58 @ ^ [X: A] : % 0.31/0.58 ( X6 % 0.31/0.58 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % the_elem_def % 0.31/0.58 thf(fact_3657_divide__int__def,axiom, % 0.31/0.58 ( ( divide_divide @ int ) % 0.31/0.58 = ( ^ [K3: int,L2: int] : % 0.31/0.58 ( if @ int % 0.31/0.58 @ ( L2 % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 @ ( zero_zero @ int ) % 0.31/0.58 @ ( if @ int % 0.31/0.58 @ ( ( sgn_sgn @ int @ K3 ) % 0.31/0.58 = ( sgn_sgn @ int @ L2 ) ) % 0.31/0.58 @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) % 0.31/0.58 @ ( uminus_uminus @ int % 0.31/0.58 @ ( semiring_1_of_nat @ int % 0.31/0.58 @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) % 0.31/0.58 @ ( zero_neq_one_of_bool @ nat % 0.31/0.58 @ ~ ( dvd_dvd @ int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % divide_int_def % 0.31/0.58 thf(fact_3658_ex__inverse__of__nat__less,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( archim462609752435547400_field @ A ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.58 => ? [N3: nat] : % 0.31/0.58 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 0.31/0.58 & ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N3 ) ) @ X3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % ex_inverse_of_nat_less % 0.31/0.58 thf(fact_3659_power__diff__conv__inverse,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( division_ring @ A ) % 0.31/0.58 => ! [X3: A,M: nat,N2: nat] : % 0.31/0.58 ( ( X3 % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.58 => ( ( power_power @ A @ X3 @ ( minus_minus @ nat @ N2 @ M ) ) % 0.31/0.58 = ( times_times @ A @ ( power_power @ A @ X3 @ N2 ) @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ M ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % power_diff_conv_inverse % 0.31/0.58 thf(fact_3660_nat__2,axiom, % 0.31/0.58 ( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) % 0.31/0.58 = ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_2 % 0.31/0.58 thf(fact_3661_pos__le__minus__divideR__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.58 => ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) ) % 0.31/0.58 = ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % pos_le_minus_divideR_eq % 0.31/0.58 thf(fact_3662_pos__minus__divideR__le__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,B2: A,A2: A] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.58 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 ) % 0.31/0.58 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % pos_minus_divideR_le_eq % 0.31/0.58 thf(fact_3663_neg__le__minus__divideR__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 0.31/0.58 => ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) ) % 0.31/0.58 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % neg_le_minus_divideR_eq % 0.31/0.58 thf(fact_3664_neg__minus__divideR__le__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,B2: A,A2: A] : % 0.31/0.58 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 0.31/0.58 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 ) % 0.31/0.58 = ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % neg_minus_divideR_le_eq % 0.31/0.58 thf(fact_3665_pos__less__minus__divideR__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.58 => ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) ) % 0.31/0.58 = ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % pos_less_minus_divideR_eq % 0.31/0.58 thf(fact_3666_pos__minus__divideR__less__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,B2: A,A2: A] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.58 => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 ) % 0.31/0.58 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % pos_minus_divideR_less_eq % 0.31/0.58 thf(fact_3667_neg__less__minus__divideR__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 0.31/0.58 => ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) ) % 0.31/0.58 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % neg_less_minus_divideR_eq % 0.31/0.58 thf(fact_3668_neg__minus__divideR__less__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.58 => ! [C2: real,B2: A,A2: A] : % 0.31/0.58 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 0.31/0.58 => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 ) % 0.31/0.58 = ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % neg_minus_divideR_less_eq % 0.31/0.58 thf(fact_3669_Suc__nat__eq__nat__zadd1,axiom, % 0.31/0.58 ! [Z2: int] : % 0.31/0.58 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 0.31/0.58 => ( ( suc @ ( nat2 @ Z2 ) ) % 0.31/0.58 = ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Suc_nat_eq_nat_zadd1 % 0.31/0.58 thf(fact_3670_nat__less__iff,axiom, % 0.31/0.58 ! [W2: int,M: nat] : % 0.31/0.58 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 ) % 0.31/0.58 => ( ( ord_less @ nat @ ( nat2 @ W2 ) @ M ) % 0.31/0.58 = ( ord_less @ int @ W2 @ ( semiring_1_of_nat @ int @ M ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_less_iff % 0.31/0.58 thf(fact_3671_nat__mult__distrib__neg,axiom, % 0.31/0.58 ! [Z2: int,Z8: int] : % 0.31/0.58 ( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) ) % 0.31/0.58 => ( ( nat2 @ ( times_times @ int @ Z2 @ Z8 ) ) % 0.31/0.58 = ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z8 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_mult_distrib_neg % 0.31/0.58 thf(fact_3672_nat__abs__int__diff,axiom, % 0.31/0.58 ! [A2: nat,B2: nat] : % 0.31/0.58 ( ( ( ord_less_eq @ nat @ A2 @ B2 ) % 0.31/0.58 => ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) % 0.31/0.58 = ( minus_minus @ nat @ B2 @ A2 ) ) ) % 0.31/0.58 & ( ~ ( ord_less_eq @ nat @ A2 @ B2 ) % 0.31/0.58 => ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) % 0.31/0.58 = ( minus_minus @ nat @ A2 @ B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_abs_int_diff % 0.31/0.58 thf(fact_3673_log__inverse,axiom, % 0.31/0.58 ! [A2: real,X3: real] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.58 => ( ( A2 % 0.31/0.58 != ( one_one @ real ) ) % 0.31/0.58 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.58 => ( ( log @ A2 @ ( inverse_inverse @ real @ X3 ) ) % 0.31/0.58 = ( uminus_uminus @ real @ ( log @ A2 @ X3 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % log_inverse % 0.31/0.58 thf(fact_3674_floor__eq4,axiom, % 0.31/0.58 ! [N2: nat,X3: real] : % 0.31/0.58 ( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X3 ) % 0.31/0.58 => ( ( ord_less @ real @ X3 @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) % 0.31/0.58 => ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X3 ) ) % 0.31/0.58 = N2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % floor_eq4 % 0.31/0.58 thf(fact_3675_diff__nat__eq__if,axiom, % 0.31/0.58 ! [Z8: int,Z2: int] : % 0.31/0.58 ( ( ( ord_less @ int @ Z8 @ ( zero_zero @ int ) ) % 0.31/0.58 => ( ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) % 0.31/0.58 = ( nat2 @ Z2 ) ) ) % 0.31/0.58 & ( ~ ( ord_less @ int @ Z8 @ ( zero_zero @ int ) ) % 0.31/0.58 => ( ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) % 0.31/0.58 = ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z2 @ Z8 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z2 @ Z8 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % diff_nat_eq_if % 0.31/0.58 thf(fact_3676_of__int__of__nat,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ring_1 @ A ) % 0.31/0.58 => ( ( ring_1_of_int @ A ) % 0.31/0.58 = ( ^ [K3: int] : ( if @ A @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K3 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K3 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % of_int_of_nat % 0.31/0.58 thf(fact_3677_nat__dvd__iff,axiom, % 0.31/0.58 ! [Z2: int,M: nat] : % 0.31/0.58 ( ( dvd_dvd @ nat @ ( nat2 @ Z2 ) @ M ) % 0.31/0.58 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 0.31/0.58 => ( dvd_dvd @ int @ Z2 @ ( semiring_1_of_nat @ int @ M ) ) ) % 0.31/0.58 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 0.31/0.58 => ( M % 0.31/0.58 = ( zero_zero @ nat ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_dvd_iff % 0.31/0.58 thf(fact_3678_eucl__rel__int__remainderI,axiom, % 0.31/0.58 ! [R2: int,L: int,K: int,Q4: int] : % 0.31/0.58 ( ( ( sgn_sgn @ int @ R2 ) % 0.31/0.58 = ( sgn_sgn @ int @ L ) ) % 0.31/0.58 => ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ L ) ) % 0.31/0.58 => ( ( K % 0.31/0.58 = ( plus_plus @ int @ ( times_times @ int @ Q4 @ L ) @ R2 ) ) % 0.31/0.58 => ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % eucl_rel_int_remainderI % 0.31/0.58 thf(fact_3679_plus__inverse__ge__2,axiom, % 0.31/0.58 ! [X3: real] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.58 => ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ X3 @ ( inverse_inverse @ real @ X3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % plus_inverse_ge_2 % 0.31/0.58 thf(fact_3680_real__inv__sqrt__pow2,axiom, % 0.31/0.58 ! [X3: real] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.58 => ( ( power_power @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.31/0.58 = ( inverse_inverse @ real @ X3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % real_inv_sqrt_pow2 % 0.31/0.58 thf(fact_3681_eucl__rel__int_Ocases,axiom, % 0.31/0.58 ! [A12: int,A23: int,A32: product_prod @ int @ int] : % 0.31/0.58 ( ( eucl_rel_int @ A12 @ A23 @ A32 ) % 0.31/0.58 => ( ( ( A23 % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 => ( A32 % 0.31/0.58 != ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A12 ) ) ) % 0.31/0.58 => ( ! [Q2: int] : % 0.31/0.58 ( ( A32 % 0.31/0.58 = ( product_Pair @ int @ int @ Q2 @ ( zero_zero @ int ) ) ) % 0.31/0.58 => ( ( A23 % 0.31/0.58 != ( zero_zero @ int ) ) % 0.31/0.58 => ( A12 % 0.31/0.58 != ( times_times @ int @ Q2 @ A23 ) ) ) ) % 0.31/0.58 => ~ ! [R4: int,Q2: int] : % 0.31/0.58 ( ( A32 % 0.31/0.58 = ( product_Pair @ int @ int @ Q2 @ R4 ) ) % 0.31/0.58 => ( ( ( sgn_sgn @ int @ R4 ) % 0.31/0.58 = ( sgn_sgn @ int @ A23 ) ) % 0.31/0.58 => ( ( ord_less @ int @ ( abs_abs @ int @ R4 ) @ ( abs_abs @ int @ A23 ) ) % 0.31/0.58 => ( A12 % 0.31/0.58 != ( plus_plus @ int @ ( times_times @ int @ Q2 @ A23 ) @ R4 ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % eucl_rel_int.cases % 0.31/0.58 thf(fact_3682_eucl__rel__int_Osimps,axiom, % 0.31/0.58 ( eucl_rel_int % 0.31/0.58 = ( ^ [A1: int,A22: int,A33: product_prod @ int @ int] : % 0.31/0.58 ( ? [K3: int] : % 0.31/0.58 ( ( A1 = K3 ) % 0.31/0.58 & ( A22 % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 & ( A33 % 0.31/0.58 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) ) % 0.31/0.58 | ? [L2: int,K3: int,Q5: int] : % 0.31/0.58 ( ( A1 = K3 ) % 0.31/0.58 & ( A22 = L2 ) % 0.31/0.58 & ( A33 % 0.31/0.58 = ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) ) % 0.31/0.58 & ( L2 % 0.31/0.58 != ( zero_zero @ int ) ) % 0.31/0.58 & ( K3 % 0.31/0.58 = ( times_times @ int @ Q5 @ L2 ) ) ) % 0.31/0.58 | ? [R5: int,L2: int,K3: int,Q5: int] : % 0.31/0.58 ( ( A1 = K3 ) % 0.31/0.58 & ( A22 = L2 ) % 0.31/0.58 & ( A33 % 0.31/0.58 = ( product_Pair @ int @ int @ Q5 @ R5 ) ) % 0.31/0.58 & ( ( sgn_sgn @ int @ R5 ) % 0.31/0.58 = ( sgn_sgn @ int @ L2 ) ) % 0.31/0.58 & ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L2 ) ) % 0.31/0.58 & ( K3 % 0.31/0.58 = ( plus_plus @ int @ ( times_times @ int @ Q5 @ L2 ) @ R5 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % eucl_rel_int.simps % 0.31/0.58 thf(fact_3683_exp__series__add__commuting,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.58 => ! [X3: A,Y: A,N2: nat] : % 0.31/0.58 ( ( ( times_times @ A @ X3 @ Y ) % 0.31/0.58 = ( times_times @ A @ Y @ X3 ) ) % 0.31/0.58 => ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ N2 ) ) % 0.31/0.58 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I3 ) ) @ ( power_power @ A @ X3 @ I3 ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N2 @ I3 ) ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N2 @ I3 ) ) ) ) % 0.31/0.58 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % exp_series_add_commuting % 0.31/0.58 thf(fact_3684_exp__first__term,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.58 => ( ( exp @ A ) % 0.31/0.58 = ( ^ [X: A] : % 0.31/0.58 ( plus_plus @ A @ ( one_one @ A ) % 0.31/0.58 @ ( suminf @ A % 0.31/0.58 @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N ) ) ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % exp_first_term % 0.31/0.58 thf(fact_3685_tan__sec,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( ( cos @ A @ X3 ) % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.31/0.58 = ( power_power @ A @ ( inverse_inverse @ A @ ( cos @ A @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % tan_sec % 0.31/0.58 thf(fact_3686_exp__first__terms,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.58 => ! [K: nat] : % 0.31/0.58 ( ( exp @ A ) % 0.31/0.58 = ( ^ [X: A] : % 0.31/0.58 ( plus_plus @ A % 0.31/0.58 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X @ N ) ) % 0.31/0.58 @ ( set_ord_lessThan @ nat @ K ) ) % 0.31/0.58 @ ( suminf @ A % 0.31/0.58 @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N @ K ) ) ) @ ( power_power @ A @ X @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % exp_first_terms % 0.31/0.58 thf(fact_3687_divide__int__unfold,axiom, % 0.31/0.58 ! [L: int,K: int,N2: nat,M: nat] : % 0.31/0.58 ( ( ( ( ( sgn_sgn @ int @ L ) % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 | ( ( sgn_sgn @ int @ K ) % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 | ( N2 % 0.31/0.58 = ( zero_zero @ nat ) ) ) % 0.31/0.58 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) % 0.31/0.58 = ( zero_zero @ int ) ) ) % 0.31/0.58 & ( ~ ( ( ( sgn_sgn @ int @ L ) % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 | ( ( sgn_sgn @ int @ K ) % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 | ( N2 % 0.31/0.58 = ( zero_zero @ nat ) ) ) % 0.31/0.58 => ( ( ( ( sgn_sgn @ int @ K ) % 0.31/0.58 = ( sgn_sgn @ int @ L ) ) % 0.31/0.58 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) % 0.31/0.58 = ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N2 ) ) ) ) % 0.31/0.58 & ( ( ( sgn_sgn @ int @ K ) % 0.31/0.58 != ( sgn_sgn @ int @ L ) ) % 0.31/0.58 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) % 0.31/0.58 = ( uminus_uminus @ int % 0.31/0.58 @ ( semiring_1_of_nat @ int % 0.31/0.58 @ ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N2 ) % 0.31/0.58 @ ( zero_neq_one_of_bool @ nat % 0.31/0.58 @ ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % divide_int_unfold % 0.31/0.58 thf(fact_3688_old_Orec__prod__def,axiom, % 0.31/0.58 ! [T: $tType,B: $tType,A: $tType] : % 0.31/0.58 ( ( product_rec_prod @ A @ B @ T ) % 0.31/0.58 = ( ^ [F12: A > B > T,X: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % old.rec_prod_def % 0.31/0.58 thf(fact_3689_sinh__converges,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( sums @ A % 0.31/0.58 @ ^ [N: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X3 @ N ) ) ) % 0.31/0.58 @ ( sinh @ A @ X3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sinh_converges % 0.31/0.58 thf(fact_3690_cosh__converges,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( sums @ A % 0.31/0.58 @ ^ [N: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X3 @ N ) ) @ ( zero_zero @ A ) ) % 0.31/0.58 @ ( cosh @ A @ X3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % cosh_converges % 0.31/0.58 thf(fact_3691_sinh__ln__real,axiom, % 0.31/0.58 ! [X3: real] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.58 => ( ( sinh @ real @ ( ln_ln @ real @ X3 ) ) % 0.31/0.58 = ( divide_divide @ real @ ( minus_minus @ real @ X3 @ ( inverse_inverse @ real @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sinh_ln_real % 0.31/0.58 thf(fact_3692_arcsin__lt__bounded,axiom, % 0.31/0.58 ! [Y: real] : % 0.31/0.58 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y ) % 0.31/0.58 => ( ( ord_less @ real @ Y @ ( one_one @ real ) ) % 0.31/0.58 => ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) ) % 0.31/0.58 & ( ord_less @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % arcsin_lt_bounded % 0.31/0.58 thf(fact_3693_sinh__0,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.58 => ( ( sinh @ A @ ( zero_zero @ A ) ) % 0.31/0.58 = ( zero_zero @ A ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sinh_0 % 0.31/0.58 thf(fact_3694_sinh__real__less__iff,axiom, % 0.31/0.58 ! [X3: real,Y: real] : % 0.31/0.58 ( ( ord_less @ real @ ( sinh @ real @ X3 ) @ ( sinh @ real @ Y ) ) % 0.31/0.58 = ( ord_less @ real @ X3 @ Y ) ) ). % 0.31/0.58 % 0.31/0.58 % sinh_real_less_iff % 0.31/0.58 thf(fact_3695_The__split__eq,axiom, % 0.31/0.58 ! [A: $tType,B: $tType,X3: A,Y: B] : % 0.31/0.58 ( ( the @ ( product_prod @ A @ B ) % 0.31/0.58 @ ( product_case_prod @ A @ B @ $o % 0.31/0.58 @ ^ [X9: A,Y6: B] : % 0.31/0.58 ( ( X3 = X9 ) % 0.31/0.58 & ( Y = Y6 ) ) ) ) % 0.31/0.58 = ( product_Pair @ A @ B @ X3 @ Y ) ) ). % 0.31/0.58 % 0.31/0.58 % The_split_eq % 0.31/0.58 thf(fact_3696_cosh__0,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.58 => ( ( cosh @ A @ ( zero_zero @ A ) ) % 0.31/0.58 = ( one_one @ A ) ) ) ). % 0.31/0.58 % 0.31/0.58 % cosh_0 % 0.31/0.58 thf(fact_3697_sinh__real__neg__iff,axiom, % 0.31/0.58 ! [X3: real] : % 0.31/0.58 ( ( ord_less @ real @ ( sinh @ real @ X3 ) @ ( zero_zero @ real ) ) % 0.31/0.58 = ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sinh_real_neg_iff % 0.31/0.58 thf(fact_3698_sinh__real__pos__iff,axiom, % 0.31/0.58 ! [X3: real] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X3 ) ) % 0.31/0.58 = ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ). % 0.31/0.58 % 0.31/0.58 % sinh_real_pos_iff % 0.31/0.58 thf(fact_3699_sinh__plus__cosh,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( plus_plus @ A @ ( sinh @ A @ X3 ) @ ( cosh @ A @ X3 ) ) % 0.31/0.58 = ( exp @ A @ X3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sinh_plus_cosh % 0.31/0.58 thf(fact_3700_cosh__plus__sinh,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( plus_plus @ A @ ( cosh @ A @ X3 ) @ ( sinh @ A @ X3 ) ) % 0.31/0.58 = ( exp @ A @ X3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % cosh_plus_sinh % 0.31/0.58 thf(fact_3701_sinh__less__cosh__real,axiom, % 0.31/0.58 ! [X3: real] : ( ord_less @ real @ ( sinh @ real @ X3 ) @ ( cosh @ real @ X3 ) ) ). % 0.31/0.58 % 0.31/0.58 % sinh_less_cosh_real % 0.31/0.58 thf(fact_3702_sinh__add,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.58 => ! [X3: A,Y: A] : % 0.31/0.58 ( ( sinh @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 0.31/0.58 = ( plus_plus @ A @ ( times_times @ A @ ( sinh @ A @ X3 ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( cosh @ A @ X3 ) @ ( sinh @ A @ Y ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sinh_add % 0.31/0.58 thf(fact_3703_cosh__add,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.58 => ! [X3: A,Y: A] : % 0.31/0.58 ( ( cosh @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 0.31/0.58 = ( plus_plus @ A @ ( times_times @ A @ ( cosh @ A @ X3 ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( sinh @ A @ X3 ) @ ( sinh @ A @ Y ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % cosh_add % 0.31/0.58 thf(fact_3704_cosh__real__pos,axiom, % 0.31/0.58 ! [X3: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X3 ) ) ). % 0.31/0.58 % 0.31/0.58 % cosh_real_pos % 0.31/0.58 thf(fact_3705_cosh__real__strict__mono,axiom, % 0.31/0.58 ! [X3: real,Y: real] : % 0.31/0.58 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.58 => ( ( ord_less @ real @ X3 @ Y ) % 0.31/0.58 => ( ord_less @ real @ ( cosh @ real @ X3 ) @ ( cosh @ real @ Y ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % cosh_real_strict_mono % 0.31/0.58 thf(fact_3706_cosh__real__nonneg__less__iff,axiom, % 0.31/0.58 ! [X3: real,Y: real] : % 0.31/0.58 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.58 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) % 0.31/0.58 => ( ( ord_less @ real @ ( cosh @ real @ X3 ) @ ( cosh @ real @ Y ) ) % 0.31/0.58 = ( ord_less @ real @ X3 @ Y ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % cosh_real_nonneg_less_iff % 0.31/0.58 thf(fact_3707_cosh__real__nonpos__less__iff,axiom, % 0.31/0.58 ! [X3: real,Y: real] : % 0.31/0.58 ( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) % 0.31/0.58 => ( ( ord_less_eq @ real @ Y @ ( zero_zero @ real ) ) % 0.31/0.58 => ( ( ord_less @ real @ ( cosh @ real @ X3 ) @ ( cosh @ real @ Y ) ) % 0.31/0.58 = ( ord_less @ real @ Y @ X3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % cosh_real_nonpos_less_iff % 0.31/0.58 thf(fact_3708_cosh__square__eq,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( power_power @ A @ ( cosh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.31/0.58 = ( plus_plus @ A @ ( power_power @ A @ ( sinh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % cosh_square_eq % 0.31/0.58 thf(fact_3709_cosh__double,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( cosh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) ) % 0.31/0.58 = ( plus_plus @ A @ ( power_power @ A @ ( cosh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % cosh_double % 0.31/0.58 thf(fact_3710_sgn__real__def,axiom, % 0.31/0.58 ( ( sgn_sgn @ real ) % 0.31/0.58 = ( ^ [A4: real] : % 0.31/0.58 ( if @ real % 0.31/0.58 @ ( A4 % 0.31/0.58 = ( zero_zero @ real ) ) % 0.31/0.58 @ ( zero_zero @ real ) % 0.31/0.58 @ ( if @ real @ ( ord_less @ real @ ( zero_zero @ real ) @ A4 ) @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_real_def % 0.31/0.58 thf(fact_3711_arcsin__less__arcsin,axiom, % 0.31/0.58 ! [X3: real,Y: real] : % 0.31/0.58 ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 0.31/0.58 => ( ( ord_less @ real @ X3 @ Y ) % 0.31/0.58 => ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) ) % 0.31/0.58 => ( ord_less @ real @ ( arcsin @ X3 ) @ ( arcsin @ Y ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % arcsin_less_arcsin % 0.31/0.58 thf(fact_3712_arcsin__less__mono,axiom, % 0.31/0.58 ! [X3: real,Y: real] : % 0.31/0.58 ( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 0.31/0.58 => ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) ) % 0.31/0.58 => ( ( ord_less @ real @ ( arcsin @ X3 ) @ ( arcsin @ Y ) ) % 0.31/0.58 = ( ord_less @ real @ X3 @ Y ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % arcsin_less_mono % 0.31/0.58 thf(fact_3713_sgn__power__injE,axiom, % 0.31/0.58 ! [A2: real,N2: nat,X3: real,B2: real] : % 0.31/0.58 ( ( ( times_times @ real @ ( sgn_sgn @ real @ A2 ) @ ( power_power @ real @ ( abs_abs @ real @ A2 ) @ N2 ) ) % 0.31/0.58 = X3 ) % 0.31/0.58 => ( ( X3 % 0.31/0.58 = ( times_times @ real @ ( sgn_sgn @ real @ B2 ) @ ( power_power @ real @ ( abs_abs @ real @ B2 ) @ N2 ) ) ) % 0.31/0.58 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.58 => ( A2 = B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_power_injE % 0.31/0.58 thf(fact_3714_cos__arcsin__nonzero,axiom, % 0.31/0.58 ! [X3: real] : % 0.31/0.58 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 0.31/0.58 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 0.31/0.58 => ( ( cos @ real @ ( arcsin @ X3 ) ) % 0.31/0.58 != ( zero_zero @ real ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % cos_arcsin_nonzero % 0.31/0.58 thf(fact_3715_tanh__add,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.58 => ! [X3: A,Y: A] : % 0.31/0.58 ( ( ( cosh @ A @ X3 ) % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( ( cosh @ A @ Y ) % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( tanh @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 0.31/0.58 = ( divide_divide @ A @ ( plus_plus @ A @ ( tanh @ A @ X3 ) @ ( tanh @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tanh @ A @ X3 ) @ ( tanh @ A @ Y ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % tanh_add % 0.31/0.58 thf(fact_3716_sinh__zero__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( ( sinh @ A @ X3 ) % 0.31/0.58 = ( zero_zero @ A ) ) % 0.31/0.58 = ( member @ A @ ( exp @ A @ X3 ) @ ( insert @ A @ ( one_one @ A ) @ ( insert @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sinh_zero_iff % 0.31/0.58 thf(fact_3717_floor__real__def,axiom, % 0.31/0.58 ( ( archim6421214686448440834_floor @ real ) % 0.31/0.58 = ( ^ [X: real] : % 0.31/0.58 ( the @ int % 0.31/0.58 @ ^ [Z6: int] : % 0.31/0.58 ( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ Z6 ) @ X ) % 0.31/0.58 & ( ord_less @ real @ X @ ( ring_1_of_int @ real @ ( plus_plus @ int @ Z6 @ ( one_one @ int ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % floor_real_def % 0.31/0.58 thf(fact_3718_cosh__field__def,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.58 => ( ( cosh @ A ) % 0.31/0.58 = ( ^ [Z6: A] : ( divide_divide @ A @ ( plus_plus @ A @ ( exp @ A @ Z6 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z6 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % cosh_field_def % 0.31/0.58 thf(fact_3719_cosh__zero__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.58 => ! [X3: A] : % 0.31/0.58 ( ( ( cosh @ A @ X3 ) % 0.31/0.58 = ( zero_zero @ A ) ) % 0.31/0.58 = ( ( power_power @ A @ ( exp @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.31/0.58 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % cosh_zero_iff % 0.31/0.58 thf(fact_3720_cosh__def,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_Vector_banach @ A ) % 0.31/0.58 & ( real_V2822296259951069270ebra_1 @ A ) ) % 0.31/0.58 => ( ( cosh @ A ) % 0.31/0.58 = ( ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % cosh_def % 0.31/0.58 thf(fact_3721_cosh__ln__real,axiom, % 0.31/0.58 ! [X3: real] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.58 => ( ( cosh @ real @ ( ln_ln @ real @ X3 ) ) % 0.31/0.58 = ( divide_divide @ real @ ( plus_plus @ real @ X3 @ ( inverse_inverse @ real @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % cosh_ln_real % 0.31/0.58 thf(fact_3722_floor__rat__def,axiom, % 0.31/0.58 ( ( archim6421214686448440834_floor @ rat ) % 0.31/0.58 = ( ^ [X: rat] : % 0.31/0.58 ( the @ int % 0.31/0.58 @ ^ [Z6: int] : % 0.31/0.58 ( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z6 ) @ X ) % 0.31/0.58 & ( ord_less @ rat @ X @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z6 @ ( one_one @ int ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % floor_rat_def % 0.31/0.58 thf(fact_3723_sum__diff1_H__aux,axiom, % 0.31/0.58 ! [B: $tType,A: $tType] : % 0.31/0.58 ( ( ab_group_add @ B ) % 0.31/0.58 => ! [F5: set @ A,I6: set @ A,F2: A > B,I: A] : % 0.31/0.58 ( ( finite_finite2 @ A @ F5 ) % 0.31/0.58 => ( ( ord_less_eq @ ( set @ A ) % 0.31/0.58 @ ( collect @ A % 0.31/0.58 @ ^ [I3: A] : % 0.31/0.58 ( ( member @ A @ I3 @ I6 ) % 0.31/0.58 & ( ( F2 @ I3 ) % 0.31/0.58 != ( zero_zero @ B ) ) ) ) % 0.31/0.58 @ F5 ) % 0.31/0.58 => ( ( ( member @ A @ I @ I6 ) % 0.31/0.58 => ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.58 = ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) @ ( F2 @ I ) ) ) ) % 0.31/0.58 & ( ~ ( member @ A @ I @ I6 ) % 0.31/0.58 => ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.58 = ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum_diff1'_aux % 0.31/0.58 thf(fact_3724_VEBT_Osize_I3_J,axiom, % 0.31/0.58 ! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] : % 0.31/0.58 ( ( size_size @ vEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) ) % 0.31/0.58 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_list @ vEBT_VEBT @ ( size_size @ vEBT_VEBT ) @ X13 ) @ ( size_size @ vEBT_VEBT @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % VEBT.size(3) % 0.31/0.58 thf(fact_3725_sum__power2,axiom, % 0.31/0.58 ! [K: nat] : % 0.31/0.58 ( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) % 0.31/0.58 = ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ ( one_one @ nat ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum_power2 % 0.31/0.58 thf(fact_3726_finite__atLeastLessThan,axiom, % 0.31/0.58 ! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) ) ). % 0.31/0.58 % 0.31/0.58 % finite_atLeastLessThan % 0.31/0.58 thf(fact_3727_atLeastLessThan__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ord @ A ) % 0.31/0.58 => ! [I: A,L: A,U: A] : % 0.31/0.58 ( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) ) % 0.31/0.58 = ( ( ord_less_eq @ A @ L @ I ) % 0.31/0.58 & ( ord_less @ A @ I @ U ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastLessThan_iff % 0.31/0.58 thf(fact_3728_atLeastLessThan__empty,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( order @ A ) % 0.31/0.58 => ! [B2: A,A2: A] : % 0.31/0.58 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.58 => ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) % 0.31/0.58 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastLessThan_empty % 0.31/0.58 thf(fact_3729_ivl__subset,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linorder @ A ) % 0.31/0.58 => ! [I: A,J: A,M: A,N2: A] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M @ N2 ) ) % 0.31/0.58 = ( ( ord_less_eq @ A @ J @ I ) % 0.31/0.58 | ( ( ord_less_eq @ A @ M @ I ) % 0.31/0.58 & ( ord_less_eq @ A @ J @ N2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % ivl_subset % 0.31/0.58 thf(fact_3730_atLeastLessThan__empty__iff2,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( preorder @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ( bot_bot @ ( set @ A ) ) % 0.31/0.58 = ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) % 0.31/0.58 = ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastLessThan_empty_iff2 % 0.31/0.58 thf(fact_3731_atLeastLessThan__empty__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( preorder @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) % 0.31/0.58 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.58 = ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastLessThan_empty_iff % 0.31/0.58 thf(fact_3732_infinite__Ico__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( dense_linorder @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) ) % 0.31/0.58 = ( ord_less @ A @ A2 @ B2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % infinite_Ico_iff % 0.31/0.58 thf(fact_3733_ivl__diff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linorder @ A ) % 0.31/0.58 => ! [I: A,N2: A,M: A] : % 0.31/0.58 ( ( ord_less_eq @ A @ I @ N2 ) % 0.31/0.58 => ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M ) @ ( set_or7035219750837199246ssThan @ A @ I @ N2 ) ) % 0.31/0.58 = ( set_or7035219750837199246ssThan @ A @ N2 @ M ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % ivl_diff % 0.31/0.58 thf(fact_3734_sum_Oempty_H,axiom, % 0.31/0.58 ! [B: $tType,A: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [P4: B > A] : % 0.31/0.58 ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) ) % 0.31/0.58 = ( zero_zero @ A ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.empty' % 0.31/0.58 thf(fact_3735_sum_Oeq__sum,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [I6: set @ B,P4: B > A] : % 0.31/0.58 ( ( finite_finite2 @ B @ I6 ) % 0.31/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I6 ) % 0.31/0.58 = ( groups7311177749621191930dd_sum @ B @ A @ P4 @ I6 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.eq_sum % 0.31/0.58 thf(fact_3736_atLeastLessThan__singleton,axiom, % 0.31/0.58 ! [M: nat] : % 0.31/0.58 ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) ) % 0.31/0.58 = ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastLessThan_singleton % 0.31/0.58 thf(fact_3737_sum_Oinsert_H,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [I6: set @ B,P4: B > A,I: B] : % 0.31/0.58 ( ( finite_finite2 @ B % 0.31/0.58 @ ( collect @ B % 0.31/0.58 @ ^ [X: B] : % 0.31/0.58 ( ( member @ B @ X @ I6 ) % 0.31/0.58 & ( ( P4 @ X ) % 0.31/0.58 != ( zero_zero @ A ) ) ) ) ) % 0.31/0.58 => ( ( ( member @ B @ I @ I6 ) % 0.31/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) ) % 0.31/0.58 = ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I6 ) ) ) % 0.31/0.58 & ( ~ ( member @ B @ I @ I6 ) % 0.31/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) ) % 0.31/0.58 = ( plus_plus @ A @ ( P4 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I6 ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.insert' % 0.31/0.58 thf(fact_3738_sum_Oop__ivl__Suc,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [N2: nat,M: nat,G: nat > A] : % 0.31/0.58 ( ( ( ord_less @ nat @ N2 @ M ) % 0.31/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) ) % 0.31/0.58 = ( zero_zero @ A ) ) ) % 0.31/0.58 & ( ~ ( ord_less @ nat @ N2 @ M ) % 0.31/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) ) % 0.31/0.58 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.op_ivl_Suc % 0.31/0.58 thf(fact_3739_prod_Oop__ivl__Suc,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [N2: nat,M: nat,G: nat > A] : % 0.31/0.58 ( ( ( ord_less @ nat @ N2 @ M ) % 0.31/0.58 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) ) % 0.31/0.58 = ( one_one @ A ) ) ) % 0.31/0.58 & ( ~ ( ord_less @ nat @ N2 @ M ) % 0.31/0.58 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) ) % 0.31/0.58 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.op_ivl_Suc % 0.31/0.58 thf(fact_3740_sgn__rat__def,axiom, % 0.31/0.58 ( ( sgn_sgn @ rat ) % 0.31/0.58 = ( ^ [A4: rat] : % 0.31/0.58 ( if @ rat % 0.31/0.58 @ ( A4 % 0.31/0.58 = ( zero_zero @ rat ) ) % 0.31/0.58 @ ( zero_zero @ rat ) % 0.31/0.58 @ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A4 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_rat_def % 0.31/0.58 thf(fact_3741_abs__rat__def,axiom, % 0.31/0.58 ( ( abs_abs @ rat ) % 0.31/0.58 = ( ^ [A4: rat] : ( if @ rat @ ( ord_less @ rat @ A4 @ ( zero_zero @ rat ) ) @ ( uminus_uminus @ rat @ A4 ) @ A4 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % abs_rat_def % 0.31/0.58 thf(fact_3742_atLeastLessThan__eq__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linorder @ A ) % 0.31/0.58 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.58 ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.58 => ( ( ord_less @ A @ C2 @ D2 ) % 0.31/0.58 => ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) % 0.31/0.58 = ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) ) % 0.31/0.58 = ( ( A2 = C2 ) % 0.31/0.58 & ( B2 = D2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastLessThan_eq_iff % 0.31/0.58 thf(fact_3743_atLeastLessThan__inj_I1_J,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linorder @ A ) % 0.31/0.58 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.58 ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) % 0.31/0.58 = ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) ) % 0.31/0.58 => ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.58 => ( ( ord_less @ A @ C2 @ D2 ) % 0.31/0.58 => ( A2 = C2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastLessThan_inj(1) % 0.31/0.58 thf(fact_3744_atLeastLessThan__inj_I2_J,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linorder @ A ) % 0.31/0.58 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.58 ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) % 0.31/0.58 = ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) ) % 0.31/0.58 => ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.58 => ( ( ord_less @ A @ C2 @ D2 ) % 0.31/0.58 => ( B2 = D2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastLessThan_inj(2) % 0.31/0.58 thf(fact_3745_less__eq__rat__def,axiom, % 0.31/0.58 ( ( ord_less_eq @ rat ) % 0.31/0.58 = ( ^ [X: rat,Y2: rat] : % 0.31/0.58 ( ( ord_less @ rat @ X @ Y2 ) % 0.31/0.58 | ( X = Y2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % less_eq_rat_def % 0.31/0.58 thf(fact_3746_obtain__pos__sum,axiom, % 0.31/0.58 ! [R2: rat] : % 0.31/0.58 ( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 ) % 0.31/0.58 => ~ ! [S2: rat] : % 0.31/0.58 ( ( ord_less @ rat @ ( zero_zero @ rat ) @ S2 ) % 0.31/0.58 => ! [T7: rat] : % 0.31/0.58 ( ( ord_less @ rat @ ( zero_zero @ rat ) @ T7 ) % 0.31/0.58 => ( R2 % 0.31/0.58 != ( plus_plus @ rat @ S2 @ T7 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % obtain_pos_sum % 0.31/0.58 thf(fact_3747_sum_Onon__neutral_H,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [G: B > A,I6: set @ B] : % 0.31/0.58 ( ( groups1027152243600224163dd_sum @ B @ A @ G % 0.31/0.58 @ ( collect @ B % 0.31/0.58 @ ^ [X: B] : % 0.31/0.58 ( ( member @ B @ X @ I6 ) % 0.31/0.58 & ( ( G @ X ) % 0.31/0.58 != ( zero_zero @ A ) ) ) ) ) % 0.31/0.58 = ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.non_neutral' % 0.31/0.58 thf(fact_3748_atLeastLessThan__subset__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linorder @ A ) % 0.31/0.58 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) ) % 0.31/0.58 => ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.58 | ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.58 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastLessThan_subset_iff % 0.31/0.58 thf(fact_3749_infinite__Ico,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( dense_linorder @ A ) % 0.31/0.58 => ! [A2: A,B2: A] : % 0.31/0.58 ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.58 => ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % infinite_Ico % 0.31/0.58 thf(fact_3750_ex__nat__less__eq,axiom, % 0.31/0.58 ! [N2: nat,P: nat > $o] : % 0.31/0.58 ( ( ? [M2: nat] : % 0.31/0.58 ( ( ord_less @ nat @ M2 @ N2 ) % 0.31/0.58 & ( P @ M2 ) ) ) % 0.31/0.58 = ( ? [X: nat] : % 0.31/0.58 ( ( member @ nat @ X @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.31/0.58 & ( P @ X ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % ex_nat_less_eq % 0.31/0.58 thf(fact_3751_all__nat__less__eq,axiom, % 0.31/0.58 ! [N2: nat,P: nat > $o] : % 0.31/0.58 ( ( ! [M2: nat] : % 0.31/0.58 ( ( ord_less @ nat @ M2 @ N2 ) % 0.31/0.58 => ( P @ M2 ) ) ) % 0.31/0.58 = ( ! [X: nat] : % 0.31/0.58 ( ( member @ nat @ X @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.31/0.58 => ( P @ X ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % all_nat_less_eq % 0.31/0.58 thf(fact_3752_atLeastLessThanSuc__atLeastAtMost,axiom, % 0.31/0.58 ! [L: nat,U: nat] : % 0.31/0.58 ( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) ) % 0.31/0.58 = ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastLessThanSuc_atLeastAtMost % 0.31/0.58 thf(fact_3753_lessThan__atLeast0,axiom, % 0.31/0.58 ( ( set_ord_lessThan @ nat ) % 0.31/0.58 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ). % 0.31/0.58 % 0.31/0.58 % lessThan_atLeast0 % 0.31/0.58 thf(fact_3754_atLeastLessThan0,axiom, % 0.31/0.58 ! [M: nat] : % 0.31/0.58 ( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) ) % 0.31/0.58 = ( bot_bot @ ( set @ nat ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastLessThan0 % 0.31/0.58 thf(fact_3755_sum_Oshift__bounds__Suc__ivl,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [G: nat > A,M: nat,N2: nat] : % 0.31/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 0.31/0.58 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.shift_bounds_Suc_ivl % 0.31/0.58 thf(fact_3756_sum_Oshift__bounds__nat__ivl,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 0.31/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 0.31/0.58 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.shift_bounds_nat_ivl % 0.31/0.58 thf(fact_3757_prod_Oshift__bounds__Suc__ivl,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [G: nat > A,M: nat,N2: nat] : % 0.31/0.58 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.shift_bounds_Suc_ivl % 0.31/0.58 thf(fact_3758_prod_Oshift__bounds__nat__ivl,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 0.31/0.58 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.shift_bounds_nat_ivl % 0.31/0.58 thf(fact_3759_sum_Odistrib__triv_H,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [I6: set @ B,G: B > A,H2: B > A] : % 0.31/0.58 ( ( finite_finite2 @ B @ I6 ) % 0.31/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A % 0.31/0.58 @ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H2 @ I3 ) ) % 0.31/0.58 @ I6 ) % 0.31/0.58 = ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.distrib_triv' % 0.31/0.58 thf(fact_3760_sum_Oivl__cong,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( ( ord @ B ) % 0.31/0.58 & ( comm_monoid_add @ A ) ) % 0.31/0.58 => ! [A2: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] : % 0.31/0.58 ( ( A2 = C2 ) % 0.31/0.58 => ( ( B2 = D2 ) % 0.31/0.58 => ( ! [X5: B] : % 0.31/0.58 ( ( ord_less_eq @ B @ C2 @ X5 ) % 0.31/0.58 => ( ( ord_less @ B @ X5 @ D2 ) % 0.31/0.58 => ( ( G @ X5 ) % 0.31/0.58 = ( H2 @ X5 ) ) ) ) % 0.31/0.58 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) ) % 0.31/0.58 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.ivl_cong % 0.31/0.58 thf(fact_3761_prod_Oivl__cong,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( ( ord @ B ) % 0.31/0.58 & ( comm_monoid_mult @ A ) ) % 0.31/0.58 => ! [A2: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] : % 0.31/0.58 ( ( A2 = C2 ) % 0.31/0.58 => ( ( B2 = D2 ) % 0.31/0.58 => ( ! [X5: B] : % 0.31/0.58 ( ( ord_less_eq @ B @ C2 @ X5 ) % 0.31/0.58 => ( ( ord_less @ B @ X5 @ D2 ) % 0.31/0.58 => ( ( G @ X5 ) % 0.31/0.58 = ( H2 @ X5 ) ) ) ) % 0.31/0.58 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.ivl_cong % 0.31/0.58 thf(fact_3762_sum_OatLeastLessThan__concat,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [M: nat,N2: nat,P4: nat,G: nat > A] : % 0.31/0.58 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ N2 @ P4 ) % 0.31/0.58 => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P4 ) ) ) % 0.31/0.58 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.atLeastLessThan_concat % 0.31/0.58 thf(fact_3763_sum__diff__nat__ivl,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ab_group_add @ A ) % 0.31/0.58 => ! [M: nat,N2: nat,P4: nat,F2: nat > A] : % 0.31/0.58 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ N2 @ P4 ) % 0.31/0.58 => ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) % 0.31/0.58 = ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P4 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum_diff_nat_ivl % 0.31/0.58 thf(fact_3764_size__list__estimation,axiom, % 0.31/0.58 ! [A: $tType,X3: A,Xs: list @ A,Y: nat,F2: A > nat] : % 0.31/0.58 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.58 => ( ( ord_less @ nat @ Y @ ( F2 @ X3 ) ) % 0.31/0.58 => ( ord_less @ nat @ Y @ ( size_list @ A @ F2 @ Xs ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % size_list_estimation % 0.31/0.58 thf(fact_3765_size__list__pointwise,axiom, % 0.31/0.58 ! [A: $tType,Xs: list @ A,F2: A > nat,G: A > nat] : % 0.31/0.58 ( ! [X5: A] : % 0.31/0.58 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.58 => ( ord_less_eq @ nat @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 0.31/0.58 => ( ord_less_eq @ nat @ ( size_list @ A @ F2 @ Xs ) @ ( size_list @ A @ G @ Xs ) ) ) ). % 0.31/0.58 % 0.31/0.58 % size_list_pointwise % 0.31/0.58 thf(fact_3766_size__list__estimation_H,axiom, % 0.31/0.58 ! [A: $tType,X3: A,Xs: list @ A,Y: nat,F2: A > nat] : % 0.31/0.58 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ Y @ ( F2 @ X3 ) ) % 0.31/0.58 => ( ord_less_eq @ nat @ Y @ ( size_list @ A @ F2 @ Xs ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % size_list_estimation' % 0.31/0.58 thf(fact_3767_prod_OatLeastLessThan__concat,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [M: nat,N2: nat,P4: nat,G: nat > A] : % 0.31/0.58 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ N2 @ P4 ) % 0.31/0.58 => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P4 ) ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.atLeastLessThan_concat % 0.31/0.58 thf(fact_3768_atLeast0__lessThan__Suc,axiom, % 0.31/0.58 ! [N2: nat] : % 0.31/0.58 ( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) % 0.31/0.58 = ( insert @ nat @ N2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeast0_lessThan_Suc % 0.31/0.58 thf(fact_3769_subset__eq__atLeast0__lessThan__finite,axiom, % 0.31/0.58 ! [N7: set @ nat,N2: nat] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.31/0.58 => ( finite_finite2 @ nat @ N7 ) ) ). % 0.31/0.58 % 0.31/0.58 % subset_eq_atLeast0_lessThan_finite % 0.31/0.58 thf(fact_3770_sum_Omono__neutral__left_H,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [S3: set @ B,T4: set @ B,G: B > A] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.31/0.58 => ( ! [X5: B] : % 0.31/0.58 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.31/0.58 => ( ( G @ X5 ) % 0.31/0.58 = ( zero_zero @ A ) ) ) % 0.31/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 ) % 0.31/0.58 = ( groups1027152243600224163dd_sum @ B @ A @ G @ T4 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.mono_neutral_left' % 0.31/0.58 thf(fact_3771_sum_Omono__neutral__right_H,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [S3: set @ B,T4: set @ B,G: B > A] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.31/0.58 => ( ! [X5: B] : % 0.31/0.58 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.31/0.58 => ( ( G @ X5 ) % 0.31/0.58 = ( zero_zero @ A ) ) ) % 0.31/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T4 ) % 0.31/0.58 = ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.mono_neutral_right' % 0.31/0.58 thf(fact_3772_sum_Omono__neutral__cong__left_H,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [S3: set @ B,T4: set @ B,H2: B > A,G: B > A] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.31/0.58 => ( ! [I4: B] : % 0.31/0.58 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.31/0.58 => ( ( H2 @ I4 ) % 0.31/0.58 = ( zero_zero @ A ) ) ) % 0.31/0.58 => ( ! [X5: B] : % 0.31/0.58 ( ( member @ B @ X5 @ S3 ) % 0.31/0.58 => ( ( G @ X5 ) % 0.31/0.58 = ( H2 @ X5 ) ) ) % 0.31/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 ) % 0.31/0.58 = ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.mono_neutral_cong_left' % 0.31/0.58 thf(fact_3773_sum_Omono__neutral__cong__right_H,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [S3: set @ B,T4: set @ B,G: B > A,H2: B > A] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.31/0.58 => ( ! [X5: B] : % 0.31/0.58 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.31/0.58 => ( ( G @ X5 ) % 0.31/0.58 = ( zero_zero @ A ) ) ) % 0.31/0.58 => ( ! [X5: B] : % 0.31/0.58 ( ( member @ B @ X5 @ S3 ) % 0.31/0.58 => ( ( G @ X5 ) % 0.31/0.58 = ( H2 @ X5 ) ) ) % 0.31/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T4 ) % 0.31/0.58 = ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.mono_neutral_cong_right' % 0.31/0.58 thf(fact_3774_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( dense_linorder @ A ) % 0.31/0.58 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) ) % 0.31/0.58 = ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.58 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.58 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastLessThan_subseteq_atLeastAtMost_iff % 0.31/0.58 thf(fact_3775_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( preorder @ A ) % 0.31/0.58 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) ) % 0.31/0.58 = ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.58 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.58 & ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastAtMost_subseteq_atLeastLessThan_iff % 0.31/0.58 thf(fact_3776_sum__shift__lb__Suc0__0__upt,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [F2: nat > A,K: nat] : % 0.31/0.58 ( ( ( F2 @ ( zero_zero @ nat ) ) % 0.31/0.58 = ( zero_zero @ A ) ) % 0.31/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ) % 0.31/0.58 = ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum_shift_lb_Suc0_0_upt % 0.31/0.58 thf(fact_3777_sum_OatLeast0__lessThan__Suc,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [G: nat > A,N2: nat] : % 0.31/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 0.31/0.58 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ N2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.atLeast0_lessThan_Suc % 0.31/0.58 thf(fact_3778_sum_OatLeast__Suc__lessThan,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [M: nat,N2: nat,G: nat > A] : % 0.31/0.58 ( ( ord_less @ nat @ M @ N2 ) % 0.31/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) % 0.31/0.58 = ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.atLeast_Suc_lessThan % 0.31/0.58 thf(fact_3779_sum_OatLeastLessThan__Suc,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [A2: nat,B2: nat,G: nat > A] : % 0.31/0.58 ( ( ord_less_eq @ nat @ A2 @ B2 ) % 0.31/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) ) % 0.31/0.58 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.atLeastLessThan_Suc % 0.31/0.58 thf(fact_3780_sum_Odistrib_H,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [I6: set @ B,G: B > A,H2: B > A] : % 0.31/0.58 ( ( finite_finite2 @ B % 0.31/0.58 @ ( collect @ B % 0.31/0.58 @ ^ [X: B] : % 0.31/0.58 ( ( member @ B @ X @ I6 ) % 0.31/0.58 & ( ( G @ X ) % 0.31/0.58 != ( zero_zero @ A ) ) ) ) ) % 0.31/0.58 => ( ( finite_finite2 @ B % 0.31/0.58 @ ( collect @ B % 0.31/0.58 @ ^ [X: B] : % 0.31/0.58 ( ( member @ B @ X @ I6 ) % 0.31/0.58 & ( ( H2 @ X ) % 0.31/0.58 != ( zero_zero @ A ) ) ) ) ) % 0.31/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A % 0.31/0.58 @ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H2 @ I3 ) ) % 0.31/0.58 @ I6 ) % 0.31/0.58 = ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.distrib' % 0.31/0.58 thf(fact_3781_atLeastLessThan__eq__atLeastAtMost__diff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( order @ A ) % 0.31/0.58 => ( ( set_or7035219750837199246ssThan @ A ) % 0.31/0.58 = ( ^ [A4: A,B4: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A4 @ B4 ) @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastLessThan_eq_atLeastAtMost_diff % 0.31/0.58 thf(fact_3782_prod_OatLeast0__lessThan__Suc,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [G: nat > A,N2: nat] : % 0.31/0.58 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 0.31/0.58 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ N2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.atLeast0_lessThan_Suc % 0.31/0.58 thf(fact_3783_prod_OatLeast__Suc__lessThan,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [M: nat,N2: nat,G: nat > A] : % 0.31/0.58 ( ( ord_less @ nat @ M @ N2 ) % 0.31/0.58 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) % 0.31/0.58 = ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.atLeast_Suc_lessThan % 0.31/0.58 thf(fact_3784_prod_OatLeastLessThan__Suc,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [A2: nat,B2: nat,G: nat > A] : % 0.31/0.58 ( ( ord_less_eq @ nat @ A2 @ B2 ) % 0.31/0.58 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) ) % 0.31/0.58 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.atLeastLessThan_Suc % 0.31/0.58 thf(fact_3785_sum_OG__def,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ( ( groups1027152243600224163dd_sum @ B @ A ) % 0.31/0.58 = ( ^ [P6: B > A,I8: set @ B] : % 0.31/0.58 ( if @ A % 0.31/0.58 @ ( finite_finite2 @ B % 0.31/0.58 @ ( collect @ B % 0.31/0.58 @ ^ [X: B] : % 0.31/0.58 ( ( member @ B @ X @ I8 ) % 0.31/0.58 & ( ( P6 @ X ) % 0.31/0.58 != ( zero_zero @ A ) ) ) ) ) % 0.31/0.58 @ ( groups7311177749621191930dd_sum @ B @ A @ P6 % 0.31/0.58 @ ( collect @ B % 0.31/0.58 @ ^ [X: B] : % 0.31/0.58 ( ( member @ B @ X @ I8 ) % 0.31/0.58 & ( ( P6 @ X ) % 0.31/0.58 != ( zero_zero @ A ) ) ) ) ) % 0.31/0.58 @ ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.G_def % 0.31/0.58 thf(fact_3786_sum_Olast__plus,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [M: nat,N2: nat,G: nat > A] : % 0.31/0.58 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.31/0.58 = ( plus_plus @ A @ ( G @ N2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.last_plus % 0.31/0.58 thf(fact_3787_prod_Olast__plus,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [M: nat,N2: nat,G: nat > A] : % 0.31/0.58 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.58 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.31/0.58 = ( times_times @ A @ ( G @ N2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.last_plus % 0.31/0.58 thf(fact_3788_sum__Suc__diff_H,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ab_group_add @ A ) % 0.31/0.58 => ! [M: nat,N2: nat,F2: nat > A] : % 0.31/0.58 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) % 0.31/0.58 = ( minus_minus @ A @ ( F2 @ N2 ) @ ( F2 @ M ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum_Suc_diff' % 0.31/0.58 thf(fact_3789_atLeastLessThanSuc,axiom, % 0.31/0.58 ! [M: nat,N2: nat] : % 0.31/0.58 ( ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.58 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) % 0.31/0.58 = ( insert @ nat @ N2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) % 0.31/0.58 & ( ~ ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.58 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) % 0.31/0.58 = ( bot_bot @ ( set @ nat ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastLessThanSuc % 0.31/0.58 thf(fact_3790_sum_OatLeastLessThan__rev,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [G: nat > A,N2: nat,M: nat] : % 0.31/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) % 0.31/0.58 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( suc @ I3 ) ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.atLeastLessThan_rev % 0.31/0.58 thf(fact_3791_sum_Onested__swap,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [A2: nat > nat > A,N2: nat] : % 0.31/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) ) % 0.31/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.31/0.58 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [J3: nat] : % 0.31/0.58 ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( A2 @ I3 @ J3 ) % 0.31/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.nested_swap % 0.31/0.58 thf(fact_3792_prod_OatLeastLessThan__rev,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [G: nat > A,N2: nat,M: nat] : % 0.31/0.58 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( suc @ I3 ) ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.atLeastLessThan_rev % 0.31/0.58 thf(fact_3793_prod_Onested__swap,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [A2: nat > nat > A,N2: nat] : % 0.31/0.58 ( ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) ) % 0.31/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [J3: nat] : % 0.31/0.58 ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( A2 @ I3 @ J3 ) % 0.31/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.nested_swap % 0.31/0.58 thf(fact_3794_sum_Onat__group,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [G: nat > A,K: nat,N2: nat] : % 0.31/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [M2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) ) % 0.31/0.58 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.58 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N2 @ K ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.nat_group % 0.31/0.58 thf(fact_3795_prod_Onat__group,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [G: nat > A,K: nat,N2: nat] : % 0.31/0.58 ( ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [M2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) ) % 0.31/0.58 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N2 @ K ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.nat_group % 0.31/0.58 thf(fact_3796_prod__Suc__Suc__fact,axiom, % 0.31/0.58 ! [N2: nat] : % 0.31/0.58 ( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) % 0.31/0.58 = ( semiring_char_0_fact @ nat @ N2 ) ) ). % 0.31/0.58 % 0.31/0.58 % prod_Suc_Suc_fact % 0.31/0.58 thf(fact_3797_prod__Suc__fact,axiom, % 0.31/0.58 ! [N2: nat] : % 0.31/0.58 ( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.31/0.58 = ( semiring_char_0_fact @ nat @ N2 ) ) ). % 0.31/0.58 % 0.31/0.58 % prod_Suc_fact % 0.31/0.58 thf(fact_3798_sum_Ohead__if,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [N2: nat,M: nat,G: nat > A] : % 0.31/0.58 ( ( ( ord_less @ nat @ N2 @ M ) % 0.31/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.31/0.58 = ( zero_zero @ A ) ) ) % 0.31/0.58 & ( ~ ( ord_less @ nat @ N2 @ M ) % 0.31/0.58 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.31/0.58 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.head_if % 0.31/0.58 thf(fact_3799_prod_Ohead__if,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [N2: nat,M: nat,G: nat > A] : % 0.31/0.58 ( ( ( ord_less @ nat @ N2 @ M ) % 0.31/0.58 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.31/0.58 = ( one_one @ A ) ) ) % 0.31/0.58 & ( ~ ( ord_less @ nat @ N2 @ M ) % 0.31/0.58 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.31/0.58 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.head_if % 0.31/0.58 thf(fact_3800_fact__prod__Suc,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( semiring_char_0 @ A ) % 0.31/0.58 => ( ( semiring_char_0_fact @ A ) % 0.31/0.58 = ( ^ [N: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % fact_prod_Suc % 0.31/0.58 thf(fact_3801_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_add @ A ) % 0.31/0.58 => ! [G: nat > A,N2: nat,M: nat] : % 0.31/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) % 0.31/0.58 = ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I3 ) ) % 0.31/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum.atLeastLessThan_rev_at_least_Suc_atMost % 0.31/0.58 thf(fact_3802_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [G: nat > A,N2: nat,M: nat] : % 0.31/0.58 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I3 ) ) % 0.31/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod.atLeastLessThan_rev_at_least_Suc_atMost % 0.31/0.58 thf(fact_3803_pochhammer__prod,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( comm_semiring_1 @ A ) % 0.31/0.58 => ( ( comm_s3205402744901411588hammer @ A ) % 0.31/0.58 = ( ^ [A4: A,N: nat] : % 0.31/0.58 ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ I3 ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % pochhammer_prod % 0.31/0.58 thf(fact_3804_atLeastLessThan__nat__numeral,axiom, % 0.31/0.58 ! [M: nat,K: num] : % 0.31/0.58 ( ( ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) ) % 0.31/0.58 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) ) % 0.31/0.58 = ( insert @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M @ ( pred_numeral @ K ) ) ) ) ) % 0.31/0.58 & ( ~ ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) ) % 0.31/0.58 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) ) % 0.31/0.58 = ( bot_bot @ ( set @ nat ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeastLessThan_nat_numeral % 0.31/0.58 thf(fact_3805_fact__prod__rev,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( semiring_char_0 @ A ) % 0.31/0.58 => ( ( semiring_char_0_fact @ A ) % 0.31/0.58 = ( ^ [N: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % fact_prod_rev % 0.31/0.58 thf(fact_3806_summable__Cauchy,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_Vector_banach @ A ) % 0.31/0.58 => ( ( summable @ A ) % 0.31/0.58 = ( ^ [F3: nat > A] : % 0.31/0.58 ! [E4: real] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 ) % 0.31/0.58 => ? [N5: nat] : % 0.31/0.58 ! [M2: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ N5 @ M2 ) % 0.31/0.58 => ! [N: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) @ E4 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % summable_Cauchy % 0.31/0.58 thf(fact_3807_sums__group,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( comm_monoid_add @ A ) % 0.31/0.58 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.58 => ! [F2: nat > A,S: A,K: nat] : % 0.31/0.58 ( ( sums @ A @ F2 @ S ) % 0.31/0.58 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.31/0.58 => ( sums @ A % 0.31/0.58 @ ^ [N: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K ) @ K ) ) ) % 0.31/0.58 @ S ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sums_group % 0.31/0.58 thf(fact_3808_atLeast1__lessThan__eq__remove0,axiom, % 0.31/0.58 ! [N2: nat] : % 0.31/0.58 ( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 0.31/0.58 = ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N2 ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % atLeast1_lessThan_eq_remove0 % 0.31/0.58 thf(fact_3809_fact__split,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( semiring_char_0 @ A ) % 0.31/0.58 => ! [K: nat,N2: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.58 => ( ( semiring_char_0_fact @ A @ N2 ) % 0.31/0.58 = ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N2 @ K ) @ N2 ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % fact_split % 0.31/0.58 thf(fact_3810_binomial__altdef__of__nat,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( field_char_0 @ A ) % 0.31/0.58 => ! [K: nat,N2: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.58 => ( ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I3 ) ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % binomial_altdef_of_nat % 0.31/0.58 thf(fact_3811_gbinomial__altdef__of__nat,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( field_char_0 @ A ) % 0.31/0.58 => ( ( gbinomial @ A ) % 0.31/0.58 = ( ^ [A4: A,K3: nat] : % 0.31/0.58 ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I3 ) ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % gbinomial_altdef_of_nat % 0.31/0.58 thf(fact_3812_gbinomial__mult__fact_H,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( field_char_0 @ A ) % 0.31/0.58 => ! [A2: A,K: nat] : % 0.31/0.58 ( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( semiring_char_0_fact @ A @ K ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % gbinomial_mult_fact' % 0.31/0.58 thf(fact_3813_gbinomial__mult__fact,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( field_char_0 @ A ) % 0.31/0.58 => ! [K: nat,A2: A] : % 0.31/0.58 ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % gbinomial_mult_fact % 0.31/0.58 thf(fact_3814_gbinomial__prod__rev,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( semiring_char_0 @ A ) % 0.31/0.58 & ( semidom_divide @ A ) ) % 0.31/0.58 => ( ( gbinomial @ A ) % 0.31/0.58 = ( ^ [A4: A,K3: nat] : % 0.31/0.58 ( divide_divide @ A % 0.31/0.58 @ ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I3 ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) % 0.31/0.58 @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % gbinomial_prod_rev % 0.31/0.58 thf(fact_3815_sum__diff1_H,axiom, % 0.31/0.58 ! [B: $tType,A: $tType] : % 0.31/0.58 ( ( ab_group_add @ B ) % 0.31/0.58 => ! [I6: set @ A,F2: A > B,I: A] : % 0.31/0.58 ( ( finite_finite2 @ A % 0.31/0.58 @ ( collect @ A % 0.31/0.58 @ ^ [I3: A] : % 0.31/0.58 ( ( member @ A @ I3 @ I6 ) % 0.31/0.58 & ( ( F2 @ I3 ) % 0.31/0.58 != ( zero_zero @ B ) ) ) ) ) % 0.31/0.58 => ( ( ( member @ A @ I @ I6 ) % 0.31/0.58 => ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.58 = ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) @ ( F2 @ I ) ) ) ) % 0.31/0.58 & ( ~ ( member @ A @ I @ I6 ) % 0.31/0.58 => ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.58 = ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum_diff1' % 0.31/0.58 thf(fact_3816_horner__sum__eq__sum,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( comm_semiring_1 @ A ) % 0.31/0.58 => ( ( groups4207007520872428315er_sum @ B @ A ) % 0.31/0.58 = ( ^ [F3: B > A,A4: A,Xs2: list @ B] : % 0.31/0.58 ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [N: nat] : ( times_times @ A @ ( F3 @ ( nth @ B @ Xs2 @ N ) ) @ ( power_power @ A @ A4 @ N ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % horner_sum_eq_sum % 0.31/0.58 thf(fact_3817_Chebyshev__sum__upper,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linordered_idom @ A ) % 0.31/0.58 => ! [N2: nat,A2: nat > A,B2: nat > A] : % 0.31/0.58 ( ! [I4: nat,J2: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ I4 @ J2 ) % 0.31/0.58 => ( ( ord_less @ nat @ J2 @ N2 ) % 0.31/0.58 => ( ord_less_eq @ A @ ( A2 @ I4 ) @ ( A2 @ J2 ) ) ) ) % 0.31/0.58 => ( ! [I4: nat,J2: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ I4 @ J2 ) % 0.31/0.58 => ( ( ord_less @ nat @ J2 @ N2 ) % 0.31/0.58 => ( ord_less_eq @ A @ ( B2 @ J2 ) @ ( B2 @ I4 ) ) ) ) % 0.31/0.58 => ( ord_less_eq @ A % 0.31/0.58 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) % 0.31/0.58 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [K3: nat] : ( times_times @ A @ ( A2 @ K3 ) @ ( B2 @ K3 ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) % 0.31/0.58 @ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Chebyshev_sum_upper % 0.31/0.58 thf(fact_3818_Chebyshev__sum__upper__nat,axiom, % 0.31/0.58 ! [N2: nat,A2: nat > nat,B2: nat > nat] : % 0.31/0.58 ( ! [I4: nat,J2: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ I4 @ J2 ) % 0.31/0.58 => ( ( ord_less @ nat @ J2 @ N2 ) % 0.31/0.58 => ( ord_less_eq @ nat @ ( A2 @ I4 ) @ ( A2 @ J2 ) ) ) ) % 0.31/0.58 => ( ! [I4: nat,J2: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ I4 @ J2 ) % 0.31/0.58 => ( ( ord_less @ nat @ J2 @ N2 ) % 0.31/0.58 => ( ord_less_eq @ nat @ ( B2 @ J2 ) @ ( B2 @ I4 ) ) ) ) % 0.31/0.58 => ( ord_less_eq @ nat % 0.31/0.58 @ ( times_times @ nat @ N2 % 0.31/0.58 @ ( groups7311177749621191930dd_sum @ nat @ nat % 0.31/0.58 @ ^ [I3: nat] : ( times_times @ nat @ ( A2 @ I3 ) @ ( B2 @ I3 ) ) % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) % 0.31/0.58 @ ( times_times @ nat @ ( groups7311177749621191930dd_sum @ nat @ nat @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ nat @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Chebyshev_sum_upper_nat % 0.31/0.58 thf(fact_3819_VEBT_Osize__gen_I1_J,axiom, % 0.31/0.58 ! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] : % 0.31/0.58 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) ) % 0.31/0.58 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_list @ vEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % VEBT.size_gen(1) % 0.31/0.58 thf(fact_3820_finite__atLeastLessThan__int,axiom, % 0.31/0.58 ! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ). % 0.31/0.58 % 0.31/0.58 % finite_atLeastLessThan_int % 0.31/0.58 thf(fact_3821_finite__atLeastZeroLessThan__int,axiom, % 0.31/0.58 ! [U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ). % 0.31/0.58 % 0.31/0.58 % finite_atLeastZeroLessThan_int % 0.31/0.58 thf(fact_3822_VEBT_Osize__gen_I2_J,axiom, % 0.31/0.58 ! [X21: $o,X222: $o] : % 0.31/0.58 ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) ) % 0.31/0.58 = ( zero_zero @ nat ) ) ). % 0.31/0.58 % 0.31/0.58 % VEBT.size_gen(2) % 0.31/0.58 thf(fact_3823_rat__inverse__code,axiom, % 0.31/0.58 ! [P4: rat] : % 0.31/0.58 ( ( quotient_of @ ( inverse_inverse @ rat @ P4 ) ) % 0.31/0.58 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 0.31/0.58 @ ^ [A4: int,B4: int] : % 0.31/0.58 ( if @ ( product_prod @ int @ int ) % 0.31/0.58 @ ( A4 % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) % 0.31/0.58 @ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A4 ) @ B4 ) @ ( abs_abs @ int @ A4 ) ) ) % 0.31/0.58 @ ( quotient_of @ P4 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % rat_inverse_code % 0.31/0.58 thf(fact_3824_normalize__negative,axiom, % 0.31/0.58 ! [Q4: int,P4: int] : % 0.31/0.58 ( ( ord_less @ int @ Q4 @ ( zero_zero @ int ) ) % 0.31/0.58 => ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q4 ) ) % 0.31/0.58 = ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P4 ) @ ( uminus_uminus @ int @ Q4 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % normalize_negative % 0.31/0.58 thf(fact_3825_Cauchy__iff2,axiom, % 0.31/0.58 ( ( topolo3814608138187158403Cauchy @ real ) % 0.31/0.58 = ( ^ [X6: nat > real] : % 0.31/0.58 ! [J3: nat] : % 0.31/0.58 ? [M9: nat] : % 0.31/0.58 ! [M2: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ M9 @ M2 ) % 0.31/0.58 => ! [N: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ M9 @ N ) % 0.31/0.58 => ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X6 @ M2 ) @ ( X6 @ N ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Cauchy_iff2 % 0.31/0.58 thf(fact_3826_vebt__mint_Opelims,axiom, % 0.31/0.58 ! [X3: vEBT_VEBT,Y: option @ nat] : % 0.31/0.58 ( ( ( vEBT_vebt_mint @ X3 ) % 0.31/0.58 = Y ) % 0.31/0.58 => ( ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ X3 ) % 0.31/0.58 => ( ! [A5: $o,B5: $o] : % 0.31/0.58 ( ( X3 % 0.31/0.58 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.31/0.58 => ( ( ( A5 % 0.31/0.58 => ( Y % 0.31/0.58 = ( some @ nat @ ( zero_zero @ nat ) ) ) ) % 0.31/0.58 & ( ~ A5 % 0.31/0.58 => ( ( B5 % 0.31/0.58 => ( Y % 0.31/0.58 = ( some @ nat @ ( one_one @ nat ) ) ) ) % 0.31/0.58 & ( ~ B5 % 0.31/0.58 => ( Y % 0.31/0.58 = ( none @ nat ) ) ) ) ) ) % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) ) % 0.31/0.58 => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 0.31/0.58 ( ( X3 % 0.31/0.58 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 0.31/0.58 => ( ( Y % 0.31/0.58 = ( none @ nat ) ) % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) ) % 0.31/0.58 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 0.31/0.58 ( ( X3 % 0.31/0.58 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) % 0.31/0.58 => ( ( Y % 0.31/0.58 = ( some @ nat @ Mi2 ) ) % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % vebt_mint.pelims % 0.31/0.58 thf(fact_3827_quotient__of__number_I3_J,axiom, % 0.31/0.58 ! [K: num] : % 0.31/0.58 ( ( quotient_of @ ( numeral_numeral @ rat @ K ) ) % 0.31/0.58 = ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) ) ). % 0.31/0.58 % 0.31/0.58 % quotient_of_number(3) % 0.31/0.58 thf(fact_3828_normalize__denom__zero,axiom, % 0.31/0.58 ! [P4: int] : % 0.31/0.58 ( ( normalize @ ( product_Pair @ int @ int @ P4 @ ( zero_zero @ int ) ) ) % 0.31/0.58 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ). % 0.31/0.58 % 0.31/0.58 % normalize_denom_zero % 0.31/0.58 thf(fact_3829_rat__one__code,axiom, % 0.31/0.58 ( ( quotient_of @ ( one_one @ rat ) ) % 0.31/0.58 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ). % 0.31/0.58 % 0.31/0.58 % rat_one_code % 0.31/0.58 thf(fact_3830_rat__zero__code,axiom, % 0.31/0.58 ( ( quotient_of @ ( zero_zero @ rat ) ) % 0.31/0.58 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ). % 0.31/0.58 % 0.31/0.58 % rat_zero_code % 0.31/0.58 thf(fact_3831_quotient__of__number_I5_J,axiom, % 0.31/0.58 ! [K: num] : % 0.31/0.58 ( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) ) ) % 0.31/0.58 = ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ). % 0.31/0.58 % 0.31/0.58 % quotient_of_number(5) % 0.31/0.58 thf(fact_3832_quotient__of__number_I4_J,axiom, % 0.31/0.58 ( ( quotient_of @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) % 0.31/0.58 = ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ). % 0.31/0.58 % 0.31/0.58 % quotient_of_number(4) % 0.31/0.58 thf(fact_3833_rat__divide__code,axiom, % 0.31/0.58 ! [P4: rat,Q4: rat] : % 0.31/0.58 ( ( quotient_of @ ( divide_divide @ rat @ P4 @ Q4 ) ) % 0.31/0.58 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 0.31/0.58 @ ^ [A4: int,C4: int] : % 0.31/0.58 ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 0.31/0.58 @ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A4 @ D5 ) @ ( times_times @ int @ C4 @ B4 ) ) ) % 0.31/0.58 @ ( quotient_of @ Q4 ) ) % 0.31/0.58 @ ( quotient_of @ P4 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % rat_divide_code % 0.31/0.58 thf(fact_3834_rat__times__code,axiom, % 0.31/0.58 ! [P4: rat,Q4: rat] : % 0.31/0.58 ( ( quotient_of @ ( times_times @ rat @ P4 @ Q4 ) ) % 0.31/0.58 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 0.31/0.58 @ ^ [A4: int,C4: int] : % 0.31/0.58 ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 0.31/0.58 @ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A4 @ B4 ) @ ( times_times @ int @ C4 @ D5 ) ) ) % 0.31/0.58 @ ( quotient_of @ Q4 ) ) % 0.31/0.58 @ ( quotient_of @ P4 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % rat_times_code % 0.31/0.58 thf(fact_3835_quotient__of__div,axiom, % 0.31/0.58 ! [R2: rat,N2: int,D2: int] : % 0.31/0.58 ( ( ( quotient_of @ R2 ) % 0.31/0.58 = ( product_Pair @ int @ int @ N2 @ D2 ) ) % 0.31/0.58 => ( R2 % 0.31/0.58 = ( divide_divide @ rat @ ( ring_1_of_int @ rat @ N2 ) @ ( ring_1_of_int @ rat @ D2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % quotient_of_div % 0.31/0.58 thf(fact_3836_rat__minus__code,axiom, % 0.31/0.58 ! [P4: rat,Q4: rat] : % 0.31/0.58 ( ( quotient_of @ ( minus_minus @ rat @ P4 @ Q4 ) ) % 0.31/0.58 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 0.31/0.58 @ ^ [A4: int,C4: int] : % 0.31/0.58 ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 0.31/0.58 @ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A4 @ D5 ) @ ( times_times @ int @ B4 @ C4 ) ) @ ( times_times @ int @ C4 @ D5 ) ) ) % 0.31/0.58 @ ( quotient_of @ Q4 ) ) % 0.31/0.58 @ ( quotient_of @ P4 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % rat_minus_code % 0.31/0.58 thf(fact_3837_quotient__of__denom__pos,axiom, % 0.31/0.58 ! [R2: rat,P4: int,Q4: int] : % 0.31/0.58 ( ( ( quotient_of @ R2 ) % 0.31/0.58 = ( product_Pair @ int @ int @ P4 @ Q4 ) ) % 0.31/0.58 => ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ). % 0.31/0.58 % 0.31/0.58 % quotient_of_denom_pos % 0.31/0.58 thf(fact_3838_rat__plus__code,axiom, % 0.31/0.58 ! [P4: rat,Q4: rat] : % 0.31/0.58 ( ( quotient_of @ ( plus_plus @ rat @ P4 @ Q4 ) ) % 0.31/0.58 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 0.31/0.58 @ ^ [A4: int,C4: int] : % 0.31/0.58 ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 0.31/0.58 @ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A4 @ D5 ) @ ( times_times @ int @ B4 @ C4 ) ) @ ( times_times @ int @ C4 @ D5 ) ) ) % 0.31/0.58 @ ( quotient_of @ Q4 ) ) % 0.31/0.58 @ ( quotient_of @ P4 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % rat_plus_code % 0.31/0.58 thf(fact_3839_normalize__denom__pos,axiom, % 0.31/0.58 ! [R2: product_prod @ int @ int,P4: int,Q4: int] : % 0.31/0.58 ( ( ( normalize @ R2 ) % 0.31/0.58 = ( product_Pair @ int @ int @ P4 @ Q4 ) ) % 0.31/0.58 => ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ). % 0.31/0.58 % 0.31/0.58 % normalize_denom_pos % 0.31/0.58 thf(fact_3840_normalize__crossproduct,axiom, % 0.31/0.58 ! [Q4: int,S: int,P4: int,R2: int] : % 0.31/0.58 ( ( Q4 % 0.31/0.58 != ( zero_zero @ int ) ) % 0.31/0.58 => ( ( S % 0.31/0.58 != ( zero_zero @ int ) ) % 0.31/0.58 => ( ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q4 ) ) % 0.31/0.58 = ( normalize @ ( product_Pair @ int @ int @ R2 @ S ) ) ) % 0.31/0.58 => ( ( times_times @ int @ P4 @ S ) % 0.31/0.58 = ( times_times @ int @ R2 @ Q4 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % normalize_crossproduct % 0.31/0.58 thf(fact_3841_rat__uminus__code,axiom, % 0.31/0.58 ! [P4: rat] : % 0.31/0.58 ( ( quotient_of @ ( uminus_uminus @ rat @ P4 ) ) % 0.31/0.58 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 0.31/0.58 @ ^ [A4: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A4 ) ) % 0.31/0.58 @ ( quotient_of @ P4 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % rat_uminus_code % 0.31/0.58 thf(fact_3842_rat__less__code,axiom, % 0.31/0.58 ( ( ord_less @ rat ) % 0.31/0.58 = ( ^ [P6: rat,Q5: rat] : % 0.31/0.58 ( product_case_prod @ int @ int @ $o % 0.31/0.58 @ ^ [A4: int,C4: int] : % 0.31/0.58 ( product_case_prod @ int @ int @ $o % 0.31/0.58 @ ^ [B4: int,D5: int] : ( ord_less @ int @ ( times_times @ int @ A4 @ D5 ) @ ( times_times @ int @ C4 @ B4 ) ) % 0.31/0.58 @ ( quotient_of @ Q5 ) ) % 0.31/0.58 @ ( quotient_of @ P6 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % rat_less_code % 0.31/0.58 thf(fact_3843_rat__abs__code,axiom, % 0.31/0.58 ! [P4: rat] : % 0.31/0.58 ( ( quotient_of @ ( abs_abs @ rat @ P4 ) ) % 0.31/0.58 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 0.31/0.58 @ ^ [A4: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A4 ) ) % 0.31/0.58 @ ( quotient_of @ P4 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % rat_abs_code % 0.31/0.58 thf(fact_3844_Cauchy__iff,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V822414075346904944vector @ A ) % 0.31/0.58 => ( ( topolo3814608138187158403Cauchy @ A ) % 0.31/0.58 = ( ^ [X6: nat > A] : % 0.31/0.58 ! [E4: real] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 ) % 0.31/0.58 => ? [M9: nat] : % 0.31/0.58 ! [M2: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ M9 @ M2 ) % 0.31/0.58 => ! [N: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ M9 @ N ) % 0.31/0.58 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ M2 ) @ ( X6 @ N ) ) ) @ E4 ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Cauchy_iff % 0.31/0.58 thf(fact_3845_CauchyI,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V822414075346904944vector @ A ) % 0.31/0.58 => ! [X8: nat > A] : % 0.31/0.58 ( ! [E: real] : % 0.31/0.58 ( ( ord_less @ real @ ( zero_zero @ real ) @ E ) % 0.31/0.58 => ? [M10: nat] : % 0.31/0.58 ! [M3: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ M10 @ M3 ) % 0.31/0.58 => ! [N3: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ M10 @ N3 ) % 0.31/0.58 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) @ E ) ) ) ) % 0.31/0.58 => ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % CauchyI % 0.31/0.58 thf(fact_3846_CauchyD,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( real_V822414075346904944vector @ A ) % 0.31/0.58 => ! [X8: nat > A,E3: real] : % 0.31/0.58 ( ( topolo3814608138187158403Cauchy @ A @ X8 ) % 0.31/0.58 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 ) % 0.31/0.58 => ? [M8: nat] : % 0.31/0.58 ! [M6: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ M8 @ M6 ) % 0.31/0.58 => ! [N6: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ M8 @ N6 ) % 0.31/0.58 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M6 ) @ ( X8 @ N6 ) ) ) @ E3 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % CauchyD % 0.31/0.58 thf(fact_3847_vebt__maxt_Opelims,axiom, % 0.31/0.58 ! [X3: vEBT_VEBT,Y: option @ nat] : % 0.31/0.58 ( ( ( vEBT_vebt_maxt @ X3 ) % 0.31/0.58 = Y ) % 0.31/0.58 => ( ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ X3 ) % 0.31/0.58 => ( ! [A5: $o,B5: $o] : % 0.31/0.58 ( ( X3 % 0.31/0.58 = ( vEBT_Leaf @ A5 @ B5 ) ) % 0.31/0.58 => ( ( ( B5 % 0.31/0.58 => ( Y % 0.31/0.58 = ( some @ nat @ ( one_one @ nat ) ) ) ) % 0.31/0.58 & ( ~ B5 % 0.31/0.58 => ( ( A5 % 0.31/0.58 => ( Y % 0.31/0.58 = ( some @ nat @ ( zero_zero @ nat ) ) ) ) % 0.31/0.58 & ( ~ A5 % 0.31/0.58 => ( Y % 0.31/0.58 = ( none @ nat ) ) ) ) ) ) % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) ) % 0.31/0.58 => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 0.31/0.58 ( ( X3 % 0.31/0.58 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 0.31/0.58 => ( ( Y % 0.31/0.58 = ( none @ nat ) ) % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) ) % 0.31/0.58 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 0.31/0.58 ( ( X3 % 0.31/0.58 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) % 0.31/0.58 => ( ( Y % 0.31/0.58 = ( some @ nat @ Ma2 ) ) % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % vebt_maxt.pelims % 0.31/0.58 thf(fact_3848_quotient__of__int,axiom, % 0.31/0.58 ! [A2: int] : % 0.31/0.58 ( ( quotient_of @ ( of_int @ A2 ) ) % 0.31/0.58 = ( product_Pair @ int @ int @ A2 @ ( one_one @ int ) ) ) ). % 0.31/0.58 % 0.31/0.58 % quotient_of_int % 0.31/0.58 thf(fact_3849_VEBT__internal_OminNull_Opelims_I1_J,axiom, % 0.31/0.58 ! [X3: vEBT_VEBT,Y: $o] : % 0.31/0.58 ( ( ( vEBT_VEBT_minNull @ X3 ) % 0.31/0.58 = Y ) % 0.31/0.58 => ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 ) % 0.31/0.58 => ( ( ( X3 % 0.31/0.58 = ( vEBT_Leaf @ $false @ $false ) ) % 0.31/0.58 => ( Y % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) ) % 0.31/0.58 => ( ! [Uv2: $o] : % 0.31/0.58 ( ( X3 % 0.31/0.58 = ( vEBT_Leaf @ $true @ Uv2 ) ) % 0.31/0.58 => ( ~ Y % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) ) % 0.31/0.58 => ( ! [Uu2: $o] : % 0.31/0.58 ( ( X3 % 0.31/0.58 = ( vEBT_Leaf @ Uu2 @ $true ) ) % 0.31/0.58 => ( ~ Y % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) ) % 0.31/0.58 => ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 0.31/0.58 ( ( X3 % 0.31/0.58 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) % 0.31/0.58 => ( Y % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) % 0.31/0.58 => ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 0.31/0.58 ( ( X3 % 0.31/0.58 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) % 0.31/0.58 => ( ~ Y % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % VEBT_internal.minNull.pelims(1) % 0.31/0.58 thf(fact_3850_VEBT__internal_OminNull_Opelims_I2_J,axiom, % 0.31/0.58 ! [X3: vEBT_VEBT] : % 0.31/0.58 ( ( vEBT_VEBT_minNull @ X3 ) % 0.31/0.58 => ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 ) % 0.31/0.58 => ( ( ( X3 % 0.31/0.58 = ( vEBT_Leaf @ $false @ $false ) ) % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) % 0.31/0.58 => ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 0.31/0.58 ( ( X3 % 0.31/0.58 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % VEBT_internal.minNull.pelims(2) % 0.31/0.58 thf(fact_3851_VEBT__internal_OminNull_Opelims_I3_J,axiom, % 0.31/0.58 ! [X3: vEBT_VEBT] : % 0.31/0.58 ( ~ ( vEBT_VEBT_minNull @ X3 ) % 0.31/0.58 => ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 ) % 0.31/0.58 => ( ! [Uv2: $o] : % 0.31/0.58 ( ( X3 % 0.31/0.58 = ( vEBT_Leaf @ $true @ Uv2 ) ) % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) % 0.31/0.58 => ( ! [Uu2: $o] : % 0.31/0.58 ( ( X3 % 0.31/0.58 = ( vEBT_Leaf @ Uu2 @ $true ) ) % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) % 0.31/0.58 => ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 0.31/0.58 ( ( X3 % 0.31/0.58 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) % 0.31/0.58 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % VEBT_internal.minNull.pelims(3) % 0.31/0.58 thf(fact_3852_Frct__code__post_I5_J,axiom, % 0.31/0.58 ! [K: num] : % 0.31/0.58 ( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K ) ) ) % 0.31/0.58 = ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Frct_code_post(5) % 0.31/0.58 thf(fact_3853_is__singleton__the__elem,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( is_singleton @ A ) % 0.31/0.58 = ( ^ [A6: set @ A] : % 0.31/0.58 ( A6 % 0.31/0.58 = ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % is_singleton_the_elem % 0.31/0.58 thf(fact_3854_length__subseqs,axiom, % 0.31/0.58 ! [A: $tType,Xs: list @ A] : % 0.31/0.58 ( ( size_size @ ( list @ ( list @ A ) ) @ ( subseqs @ A @ Xs ) ) % 0.31/0.58 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ). % 0.31/0.58 % 0.31/0.58 % length_subseqs % 0.31/0.58 thf(fact_3855_divmod__step__integer__def,axiom, % 0.31/0.58 ( ( unique1321980374590559556d_step @ code_integer ) % 0.31/0.58 = ( ^ [L2: num] : % 0.31/0.58 ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer ) % 0.31/0.58 @ ^ [Q5: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L2 ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L2 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % divmod_step_integer_def % 0.31/0.58 thf(fact_3856_is__singletonI,axiom, % 0.31/0.58 ! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % is_singletonI % 0.31/0.58 thf(fact_3857_divmod__integer_H__def,axiom, % 0.31/0.58 ( ( unique8689654367752047608divmod @ code_integer ) % 0.31/0.58 = ( ^ [M2: num,N: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % divmod_integer'_def % 0.31/0.58 thf(fact_3858_sgn__integer__code,axiom, % 0.31/0.58 ( ( sgn_sgn @ code_integer ) % 0.31/0.58 = ( ^ [K3: code_integer] : % 0.31/0.58 ( if @ code_integer % 0.31/0.58 @ ( K3 % 0.31/0.58 = ( zero_zero @ code_integer ) ) % 0.31/0.58 @ ( zero_zero @ code_integer ) % 0.31/0.58 @ ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sgn_integer_code % 0.31/0.58 thf(fact_3859_exhaustive__integer_H_Ocases,axiom, % 0.31/0.58 ! [X3: product_prod @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] : % 0.31/0.58 ~ ! [F4: code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: code_integer,I4: code_integer] : % 0.31/0.58 ( X3 % 0.31/0.58 != ( product_Pair @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D3 @ I4 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % exhaustive_integer'.cases % 0.31/0.58 thf(fact_3860_full__exhaustive__integer_H_Ocases,axiom, % 0.31/0.58 ! [X3: product_prod @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] : % 0.31/0.58 ~ ! [F4: ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: code_integer,I4: code_integer] : % 0.31/0.58 ( X3 % 0.31/0.58 != ( product_Pair @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D3 @ I4 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % full_exhaustive_integer'.cases % 0.31/0.58 thf(fact_3861_subseqs__refl,axiom, % 0.31/0.58 ! [A: $tType,Xs: list @ A] : ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ). % 0.31/0.58 % 0.31/0.58 % subseqs_refl % 0.31/0.58 thf(fact_3862_is__singletonI_H,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ( A3 % 0.31/0.58 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.58 => ( ! [X5: A,Y4: A] : % 0.31/0.58 ( ( member @ A @ X5 @ A3 ) % 0.31/0.58 => ( ( member @ A @ Y4 @ A3 ) % 0.31/0.58 => ( X5 = Y4 ) ) ) % 0.31/0.58 => ( is_singleton @ A @ A3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % is_singletonI' % 0.31/0.58 thf(fact_3863_zero__natural_Orsp,axiom, % 0.31/0.58 ( ( zero_zero @ nat ) % 0.31/0.58 = ( zero_zero @ nat ) ) ). % 0.31/0.58 % 0.31/0.58 % zero_natural.rsp % 0.31/0.58 thf(fact_3864_is__singleton__def,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( is_singleton @ A ) % 0.31/0.58 = ( ^ [A6: set @ A] : % 0.31/0.58 ? [X: A] : % 0.31/0.58 ( A6 % 0.31/0.58 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % is_singleton_def % 0.31/0.58 thf(fact_3865_is__singletonE,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ( is_singleton @ A @ A3 ) % 0.31/0.58 => ~ ! [X5: A] : % 0.31/0.58 ( A3 % 0.31/0.58 != ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % is_singletonE % 0.31/0.58 thf(fact_3866_Frct__code__post_I1_J,axiom, % 0.31/0.58 ! [A2: int] : % 0.31/0.58 ( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A2 ) ) % 0.31/0.58 = ( zero_zero @ rat ) ) ). % 0.31/0.58 % 0.31/0.58 % Frct_code_post(1) % 0.31/0.58 thf(fact_3867_Frct__code__post_I2_J,axiom, % 0.31/0.58 ! [A2: int] : % 0.31/0.58 ( ( frct @ ( product_Pair @ int @ int @ A2 @ ( zero_zero @ int ) ) ) % 0.31/0.58 = ( zero_zero @ rat ) ) ). % 0.31/0.58 % 0.31/0.58 % Frct_code_post(2) % 0.31/0.58 thf(fact_3868_Frct__code__post_I8_J,axiom, % 0.31/0.58 ! [A2: int,B2: int] : % 0.31/0.58 ( ( frct @ ( product_Pair @ int @ int @ A2 @ ( uminus_uminus @ int @ B2 ) ) ) % 0.31/0.58 = ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A2 @ B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Frct_code_post(8) % 0.31/0.58 thf(fact_3869_Frct__code__post_I7_J,axiom, % 0.31/0.58 ! [A2: int,B2: int] : % 0.31/0.58 ( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A2 ) @ B2 ) ) % 0.31/0.58 = ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A2 @ B2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Frct_code_post(7) % 0.31/0.58 thf(fact_3870_Frct__code__post_I3_J,axiom, % 0.31/0.58 ( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) % 0.31/0.58 = ( one_one @ rat ) ) ). % 0.31/0.58 % 0.31/0.58 % Frct_code_post(3) % 0.31/0.58 thf(fact_3871_Frct__code__post_I4_J,axiom, % 0.31/0.58 ! [K: num] : % 0.31/0.58 ( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) ) % 0.31/0.58 = ( numeral_numeral @ rat @ K ) ) ). % 0.31/0.58 % 0.31/0.58 % Frct_code_post(4) % 0.31/0.58 thf(fact_3872_Frct__code__post_I6_J,axiom, % 0.31/0.58 ! [K: num,L: num] : % 0.31/0.58 ( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( numeral_numeral @ int @ L ) ) ) % 0.31/0.58 = ( divide_divide @ rat @ ( numeral_numeral @ rat @ K ) @ ( numeral_numeral @ rat @ L ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Frct_code_post(6) % 0.31/0.58 thf(fact_3873_integer__of__int__code,axiom, % 0.31/0.58 ( code_integer_of_int % 0.31/0.58 = ( ^ [K3: int] : % 0.31/0.58 ( if @ code_integer @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K3 ) ) ) % 0.31/0.58 @ ( if @ code_integer % 0.31/0.58 @ ( K3 % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 @ ( zero_zero @ code_integer ) % 0.31/0.58 @ ( if @ code_integer % 0.31/0.58 @ ( ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) % 0.31/0.58 = ( zero_zero @ int ) ) % 0.31/0.58 @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.58 @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % integer_of_int_code % 0.31/0.58 thf(fact_3874_length__mul__elem,axiom, % 0.31/0.58 ! [A: $tType,Xs: list @ ( list @ A ),N2: nat] : % 0.31/0.58 ( ! [X5: list @ A] : % 0.31/0.58 ( ( member @ ( list @ A ) @ X5 @ ( set2 @ ( list @ A ) @ Xs ) ) % 0.31/0.58 => ( ( size_size @ ( list @ A ) @ X5 ) % 0.31/0.58 = N2 ) ) % 0.31/0.58 => ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs ) ) % 0.31/0.58 = ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) @ N2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % length_mul_elem % 0.31/0.58 thf(fact_3875_even__sum__iff,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( semiring_parity @ A ) % 0.31/0.58 => ! [A3: set @ B,F2: B > A] : % 0.31/0.58 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.58 => ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) % 0.31/0.58 = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) % 0.31/0.58 @ ( finite_card @ B % 0.31/0.58 @ ( collect @ B % 0.31/0.58 @ ^ [A4: B] : % 0.31/0.58 ( ( member @ B @ A4 @ A3 ) % 0.31/0.58 & ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ A4 ) ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % even_sum_iff % 0.31/0.58 thf(fact_3876_pair__leqI2,axiom, % 0.31/0.58 ! [A2: nat,B2: nat,S: nat,T2: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ A2 @ B2 ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ S @ T2 ) % 0.31/0.58 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ) ). % 0.31/0.58 % 0.31/0.58 % pair_leqI2 % 0.31/0.58 thf(fact_3877_card__Collect__less__nat,axiom, % 0.31/0.58 ! [N2: nat] : % 0.31/0.58 ( ( finite_card @ nat % 0.31/0.58 @ ( collect @ nat % 0.31/0.58 @ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) ) % 0.31/0.58 = N2 ) ). % 0.31/0.58 % 0.31/0.58 % card_Collect_less_nat % 0.31/0.58 thf(fact_3878_card__atMost,axiom, % 0.31/0.58 ! [U: nat] : % 0.31/0.58 ( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) ) % 0.31/0.58 = ( suc @ U ) ) ). % 0.31/0.58 % 0.31/0.58 % card_atMost % 0.31/0.58 thf(fact_3879_card__Collect__le__nat,axiom, % 0.31/0.58 ! [N2: nat] : % 0.31/0.58 ( ( finite_card @ nat % 0.31/0.58 @ ( collect @ nat % 0.31/0.58 @ ^ [I3: nat] : ( ord_less_eq @ nat @ I3 @ N2 ) ) ) % 0.31/0.58 = ( suc @ N2 ) ) ). % 0.31/0.58 % 0.31/0.58 % card_Collect_le_nat % 0.31/0.58 thf(fact_3880_card_Oempty,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.58 = ( zero_zero @ nat ) ) ). % 0.31/0.58 % 0.31/0.58 % card.empty % 0.31/0.58 thf(fact_3881_card_Oinfinite,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ~ ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( finite_card @ A @ A3 ) % 0.31/0.58 = ( zero_zero @ nat ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card.infinite % 0.31/0.58 thf(fact_3882_card__atLeastAtMost,axiom, % 0.31/0.58 ! [L: nat,U: nat] : % 0.31/0.58 ( ( finite_card @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) % 0.31/0.58 = ( minus_minus @ nat @ ( suc @ U ) @ L ) ) ). % 0.31/0.58 % 0.31/0.58 % card_atLeastAtMost % 0.31/0.58 thf(fact_3883_card__0__eq,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( ( finite_card @ A @ A3 ) % 0.31/0.58 = ( zero_zero @ nat ) ) % 0.31/0.58 = ( A3 % 0.31/0.58 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_0_eq % 0.31/0.58 thf(fact_3884_card__insert__disjoint,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,X3: A] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ~ ( member @ A @ X3 @ A3 ) % 0.31/0.58 => ( ( finite_card @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.58 = ( suc @ ( finite_card @ A @ A3 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_insert_disjoint % 0.31/0.58 thf(fact_3885_abs__integer__code,axiom, % 0.31/0.58 ( ( abs_abs @ code_integer ) % 0.31/0.58 = ( ^ [K3: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K3 ) @ K3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % abs_integer_code % 0.31/0.58 thf(fact_3886_less__integer__code_I1_J,axiom, % 0.31/0.58 ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ). % 0.31/0.58 % 0.31/0.58 % less_integer_code(1) % 0.31/0.58 thf(fact_3887_less__integer_Oabs__eq,axiom, % 0.31/0.58 ! [Xa2: int,X3: int] : % 0.31/0.58 ( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) ) % 0.31/0.58 = ( ord_less @ int @ Xa2 @ X3 ) ) ). % 0.31/0.58 % 0.31/0.58 % less_integer.abs_eq % 0.31/0.58 thf(fact_3888_n__subsets,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,K: nat] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( finite_card @ ( set @ A ) % 0.31/0.58 @ ( collect @ ( set @ A ) % 0.31/0.58 @ ^ [B6: set @ A] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) % 0.31/0.58 & ( ( finite_card @ A @ B6 ) % 0.31/0.58 = K ) ) ) ) % 0.31/0.58 = ( binomial @ ( finite_card @ A @ A3 ) @ K ) ) ) ). % 0.31/0.58 % 0.31/0.58 % n_subsets % 0.31/0.58 thf(fact_3889_card__subset__eq,axiom, % 0.31/0.58 ! [A: $tType,B3: set @ A,A3: set @ A] : % 0.31/0.58 ( ( finite_finite2 @ A @ B3 ) % 0.31/0.58 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.58 => ( ( ( finite_card @ A @ A3 ) % 0.31/0.58 = ( finite_card @ A @ B3 ) ) % 0.31/0.58 => ( A3 = B3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_subset_eq % 0.31/0.58 thf(fact_3890_infinite__arbitrarily__large,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,N2: nat] : % 0.31/0.58 ( ~ ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ? [B10: set @ A] : % 0.31/0.58 ( ( finite_finite2 @ A @ B10 ) % 0.31/0.58 & ( ( finite_card @ A @ B10 ) % 0.31/0.58 = N2 ) % 0.31/0.58 & ( ord_less_eq @ ( set @ A ) @ B10 @ A3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % infinite_arbitrarily_large % 0.31/0.58 thf(fact_3891_card__le__if__inj__on__rel,axiom, % 0.31/0.58 ! [B: $tType,A: $tType,B3: set @ A,A3: set @ B,R2: B > A > $o] : % 0.31/0.58 ( ( finite_finite2 @ A @ B3 ) % 0.31/0.58 => ( ! [A5: B] : % 0.31/0.58 ( ( member @ B @ A5 @ A3 ) % 0.31/0.58 => ? [B11: A] : % 0.31/0.58 ( ( member @ A @ B11 @ B3 ) % 0.31/0.58 & ( R2 @ A5 @ B11 ) ) ) % 0.31/0.58 => ( ! [A13: B,A24: B,B5: A] : % 0.31/0.58 ( ( member @ B @ A13 @ A3 ) % 0.31/0.58 => ( ( member @ B @ A24 @ A3 ) % 0.31/0.58 => ( ( member @ A @ B5 @ B3 ) % 0.31/0.58 => ( ( R2 @ A13 @ B5 ) % 0.31/0.58 => ( ( R2 @ A24 @ B5 ) % 0.31/0.58 => ( A13 = A24 ) ) ) ) ) ) % 0.31/0.58 => ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_le_if_inj_on_rel % 0.31/0.58 thf(fact_3892_card__insert__le,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,X3: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ ( insert @ A @ X3 @ A3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_insert_le % 0.31/0.58 thf(fact_3893_sum__multicount__gen,axiom, % 0.31/0.58 ! [A: $tType,B: $tType,S: set @ A,T2: set @ B,R: A > B > $o,K: B > nat] : % 0.31/0.58 ( ( finite_finite2 @ A @ S ) % 0.31/0.58 => ( ( finite_finite2 @ B @ T2 ) % 0.31/0.58 => ( ! [X5: B] : % 0.31/0.58 ( ( member @ B @ X5 @ T2 ) % 0.31/0.58 => ( ( finite_card @ A % 0.31/0.58 @ ( collect @ A % 0.31/0.58 @ ^ [I3: A] : % 0.31/0.58 ( ( member @ A @ I3 @ S ) % 0.31/0.58 & ( R @ I3 @ X5 ) ) ) ) % 0.31/0.58 = ( K @ X5 ) ) ) % 0.31/0.58 => ( ( groups7311177749621191930dd_sum @ A @ nat % 0.31/0.58 @ ^ [I3: A] : % 0.31/0.58 ( finite_card @ B % 0.31/0.58 @ ( collect @ B % 0.31/0.58 @ ^ [J3: B] : % 0.31/0.58 ( ( member @ B @ J3 @ T2 ) % 0.31/0.58 & ( R @ I3 @ J3 ) ) ) ) % 0.31/0.58 @ S ) % 0.31/0.58 = ( groups7311177749621191930dd_sum @ B @ nat @ K @ T2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum_multicount_gen % 0.31/0.58 thf(fact_3894_card__lists__length__eq,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,N2: nat] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( finite_card @ ( list @ A ) % 0.31/0.58 @ ( collect @ ( list @ A ) % 0.31/0.58 @ ^ [Xs2: list @ A] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) % 0.31/0.58 & ( ( size_size @ ( list @ A ) @ Xs2 ) % 0.31/0.58 = N2 ) ) ) ) % 0.31/0.58 = ( power_power @ nat @ ( finite_card @ A @ A3 ) @ N2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_lists_length_eq % 0.31/0.58 thf(fact_3895_card__eq__0__iff,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ( ( finite_card @ A @ A3 ) % 0.31/0.58 = ( zero_zero @ nat ) ) % 0.31/0.58 = ( ( A3 % 0.31/0.58 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.58 | ~ ( finite_finite2 @ A @ A3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_eq_0_iff % 0.31/0.58 thf(fact_3896_card__ge__0__finite,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) ) % 0.31/0.58 => ( finite_finite2 @ A @ A3 ) ) ). % 0.31/0.58 % 0.31/0.58 % card_ge_0_finite % 0.31/0.58 thf(fact_3897_card__insert__if,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,X3: A] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( ( member @ A @ X3 @ A3 ) % 0.31/0.58 => ( ( finite_card @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.58 = ( finite_card @ A @ A3 ) ) ) % 0.31/0.58 & ( ~ ( member @ A @ X3 @ A3 ) % 0.31/0.58 => ( ( finite_card @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.58 = ( suc @ ( finite_card @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_insert_if % 0.31/0.58 thf(fact_3898_card__Suc__eq__finite,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,K: nat] : % 0.31/0.58 ( ( ( finite_card @ A @ A3 ) % 0.31/0.58 = ( suc @ K ) ) % 0.31/0.58 = ( ? [B4: A,B6: set @ A] : % 0.31/0.58 ( ( A3 % 0.31/0.58 = ( insert @ A @ B4 @ B6 ) ) % 0.31/0.58 & ~ ( member @ A @ B4 @ B6 ) % 0.31/0.58 & ( ( finite_card @ A @ B6 ) % 0.31/0.58 = K ) % 0.31/0.58 & ( finite_finite2 @ A @ B6 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_Suc_eq_finite % 0.31/0.58 thf(fact_3899_finite__if__finite__subsets__card__bdd,axiom, % 0.31/0.58 ! [A: $tType,F5: set @ A,C5: nat] : % 0.31/0.58 ( ! [G5: set @ A] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ A ) @ G5 @ F5 ) % 0.31/0.58 => ( ( finite_finite2 @ A @ G5 ) % 0.31/0.58 => ( ord_less_eq @ nat @ ( finite_card @ A @ G5 ) @ C5 ) ) ) % 0.31/0.58 => ( ( finite_finite2 @ A @ F5 ) % 0.31/0.58 & ( ord_less_eq @ nat @ ( finite_card @ A @ F5 ) @ C5 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % finite_if_finite_subsets_card_bdd % 0.31/0.58 thf(fact_3900_obtain__subset__with__card__n,axiom, % 0.31/0.58 ! [A: $tType,N2: nat,S3: set @ A] : % 0.31/0.58 ( ( ord_less_eq @ nat @ N2 @ ( finite_card @ A @ S3 ) ) % 0.31/0.58 => ~ ! [T5: set @ A] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ A ) @ T5 @ S3 ) % 0.31/0.58 => ( ( ( finite_card @ A @ T5 ) % 0.31/0.58 = N2 ) % 0.31/0.58 => ~ ( finite_finite2 @ A @ T5 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % obtain_subset_with_card_n % 0.31/0.58 thf(fact_3901_card__seteq,axiom, % 0.31/0.58 ! [A: $tType,B3: set @ A,A3: set @ A] : % 0.31/0.58 ( ( finite_finite2 @ A @ B3 ) % 0.31/0.58 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ B3 ) @ ( finite_card @ A @ A3 ) ) % 0.31/0.58 => ( A3 = B3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_seteq % 0.31/0.58 thf(fact_3902_card__mono,axiom, % 0.31/0.58 ! [A: $tType,B3: set @ A,A3: set @ A] : % 0.31/0.58 ( ( finite_finite2 @ A @ B3 ) % 0.31/0.58 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.58 => ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_mono % 0.31/0.58 thf(fact_3903_card__less__sym__Diff,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.58 => ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) % 0.31/0.58 => ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_less_sym_Diff % 0.31/0.58 thf(fact_3904_card__le__sym__Diff,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) % 0.31/0.58 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_le_sym_Diff % 0.31/0.58 thf(fact_3905_card__length,axiom, % 0.31/0.58 ! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( set2 @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ). % 0.31/0.58 % 0.31/0.58 % card_length % 0.31/0.58 thf(fact_3906_card__1__singletonE,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ( ( finite_card @ A @ A3 ) % 0.31/0.58 = ( one_one @ nat ) ) % 0.31/0.58 => ~ ! [X5: A] : % 0.31/0.58 ( A3 % 0.31/0.58 != ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_1_singletonE % 0.31/0.58 thf(fact_3907_psubset__card__mono,axiom, % 0.31/0.58 ! [A: $tType,B3: set @ A,A3: set @ A] : % 0.31/0.58 ( ( finite_finite2 @ A @ B3 ) % 0.31/0.58 => ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.58 => ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % psubset_card_mono % 0.31/0.58 thf(fact_3908_card__less,axiom, % 0.31/0.58 ! [M7: set @ nat,I: nat] : % 0.31/0.58 ( ( member @ nat @ ( zero_zero @ nat ) @ M7 ) % 0.31/0.58 => ( ( finite_card @ nat % 0.31/0.58 @ ( collect @ nat % 0.31/0.58 @ ^ [K3: nat] : % 0.31/0.58 ( ( member @ nat @ K3 @ M7 ) % 0.31/0.58 & ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) % 0.31/0.58 != ( zero_zero @ nat ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_less % 0.31/0.58 thf(fact_3909_card__less__Suc,axiom, % 0.31/0.58 ! [M7: set @ nat,I: nat] : % 0.31/0.58 ( ( member @ nat @ ( zero_zero @ nat ) @ M7 ) % 0.31/0.58 => ( ( suc % 0.31/0.58 @ ( finite_card @ nat % 0.31/0.58 @ ( collect @ nat % 0.31/0.58 @ ^ [K3: nat] : % 0.31/0.58 ( ( member @ nat @ ( suc @ K3 ) @ M7 ) % 0.31/0.58 & ( ord_less @ nat @ K3 @ I ) ) ) ) ) % 0.31/0.58 = ( finite_card @ nat % 0.31/0.58 @ ( collect @ nat % 0.31/0.58 @ ^ [K3: nat] : % 0.31/0.58 ( ( member @ nat @ K3 @ M7 ) % 0.31/0.58 & ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_less_Suc % 0.31/0.58 thf(fact_3910_card__less__Suc2,axiom, % 0.31/0.58 ! [M7: set @ nat,I: nat] : % 0.31/0.58 ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 ) % 0.31/0.58 => ( ( finite_card @ nat % 0.31/0.58 @ ( collect @ nat % 0.31/0.58 @ ^ [K3: nat] : % 0.31/0.58 ( ( member @ nat @ ( suc @ K3 ) @ M7 ) % 0.31/0.58 & ( ord_less @ nat @ K3 @ I ) ) ) ) % 0.31/0.58 = ( finite_card @ nat % 0.31/0.58 @ ( collect @ nat % 0.31/0.58 @ ^ [K3: nat] : % 0.31/0.58 ( ( member @ nat @ K3 @ M7 ) % 0.31/0.58 & ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_less_Suc2 % 0.31/0.58 thf(fact_3911_sum__Suc,axiom, % 0.31/0.58 ! [A: $tType,F2: A > nat,A3: set @ A] : % 0.31/0.58 ( ( groups7311177749621191930dd_sum @ A @ nat % 0.31/0.58 @ ^ [X: A] : ( suc @ ( F2 @ X ) ) % 0.31/0.58 @ A3 ) % 0.31/0.58 = ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( finite_card @ A @ A3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum_Suc % 0.31/0.58 thf(fact_3912_subset__card__intvl__is__intvl,axiom, % 0.31/0.58 ! [A3: set @ nat,K: nat] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ nat ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A3 ) ) ) ) % 0.31/0.58 => ( A3 % 0.31/0.58 = ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A3 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % subset_card_intvl_is_intvl % 0.31/0.58 thf(fact_3913_sum__multicount,axiom, % 0.31/0.58 ! [A: $tType,B: $tType,S3: set @ A,T4: set @ B,R: A > B > $o,K: nat] : % 0.31/0.58 ( ( finite_finite2 @ A @ S3 ) % 0.31/0.58 => ( ( finite_finite2 @ B @ T4 ) % 0.31/0.58 => ( ! [X5: B] : % 0.31/0.58 ( ( member @ B @ X5 @ T4 ) % 0.31/0.58 => ( ( finite_card @ A % 0.31/0.58 @ ( collect @ A % 0.31/0.58 @ ^ [I3: A] : % 0.31/0.58 ( ( member @ A @ I3 @ S3 ) % 0.31/0.58 & ( R @ I3 @ X5 ) ) ) ) % 0.31/0.58 = K ) ) % 0.31/0.58 => ( ( groups7311177749621191930dd_sum @ A @ nat % 0.31/0.58 @ ^ [I3: A] : % 0.31/0.58 ( finite_card @ B % 0.31/0.58 @ ( collect @ B % 0.31/0.58 @ ^ [J3: B] : % 0.31/0.58 ( ( member @ B @ J3 @ T4 ) % 0.31/0.58 & ( R @ I3 @ J3 ) ) ) ) % 0.31/0.58 @ S3 ) % 0.31/0.58 = ( times_times @ nat @ K @ ( finite_card @ B @ T4 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum_multicount % 0.31/0.58 thf(fact_3914_sum__bounded__below,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( ( ordere6911136660526730532id_add @ A ) % 0.31/0.58 & ( semiring_1 @ A ) ) % 0.31/0.58 => ! [A3: set @ B,K5: A,F2: B > A] : % 0.31/0.58 ( ! [I4: B] : % 0.31/0.58 ( ( member @ B @ I4 @ A3 ) % 0.31/0.58 => ( ord_less_eq @ A @ K5 @ ( F2 @ I4 ) ) ) % 0.31/0.58 => ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum_bounded_below % 0.31/0.58 thf(fact_3915_sum__bounded__above,axiom, % 0.31/0.58 ! [B: $tType,A: $tType] : % 0.31/0.58 ( ( ( ordere6911136660526730532id_add @ A ) % 0.31/0.58 & ( semiring_1 @ A ) ) % 0.31/0.58 => ! [A3: set @ B,F2: B > A,K5: A] : % 0.31/0.58 ( ! [I4: B] : % 0.31/0.58 ( ( member @ B @ I4 @ A3 ) % 0.31/0.58 => ( ord_less_eq @ A @ ( F2 @ I4 ) @ K5 ) ) % 0.31/0.58 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K5 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum_bounded_above % 0.31/0.58 thf(fact_3916_card__gt__0__iff,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) ) % 0.31/0.58 = ( ( A3 % 0.31/0.58 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.58 & ( finite_finite2 @ A @ A3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_gt_0_iff % 0.31/0.58 thf(fact_3917_card__Suc__eq,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,K: nat] : % 0.31/0.58 ( ( ( finite_card @ A @ A3 ) % 0.31/0.58 = ( suc @ K ) ) % 0.31/0.58 = ( ? [B4: A,B6: set @ A] : % 0.31/0.58 ( ( A3 % 0.31/0.58 = ( insert @ A @ B4 @ B6 ) ) % 0.31/0.58 & ~ ( member @ A @ B4 @ B6 ) % 0.31/0.58 & ( ( finite_card @ A @ B6 ) % 0.31/0.58 = K ) % 0.31/0.58 & ( ( K % 0.31/0.58 = ( zero_zero @ nat ) ) % 0.31/0.58 => ( B6 % 0.31/0.58 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_Suc_eq % 0.31/0.58 thf(fact_3918_card__eq__SucD,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,K: nat] : % 0.31/0.58 ( ( ( finite_card @ A @ A3 ) % 0.31/0.58 = ( suc @ K ) ) % 0.31/0.58 => ? [B5: A,B10: set @ A] : % 0.31/0.58 ( ( A3 % 0.31/0.58 = ( insert @ A @ B5 @ B10 ) ) % 0.31/0.58 & ~ ( member @ A @ B5 @ B10 ) % 0.31/0.58 & ( ( finite_card @ A @ B10 ) % 0.31/0.58 = K ) % 0.31/0.58 & ( ( K % 0.31/0.58 = ( zero_zero @ nat ) ) % 0.31/0.58 => ( B10 % 0.31/0.58 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_eq_SucD % 0.31/0.58 thf(fact_3919_card__1__singleton__iff,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ( ( finite_card @ A @ A3 ) % 0.31/0.58 = ( suc @ ( zero_zero @ nat ) ) ) % 0.31/0.58 = ( ? [X: A] : % 0.31/0.58 ( A3 % 0.31/0.58 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_1_singleton_iff % 0.31/0.58 thf(fact_3920_card__le__Suc0__iff__eq,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( suc @ ( zero_zero @ nat ) ) ) % 0.31/0.58 = ( ! [X: A] : % 0.31/0.58 ( ( member @ A @ X @ A3 ) % 0.31/0.58 => ! [Y2: A] : % 0.31/0.58 ( ( member @ A @ Y2 @ A3 ) % 0.31/0.58 => ( X = Y2 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_le_Suc0_iff_eq % 0.31/0.58 thf(fact_3921_card__le__Suc__iff,axiom, % 0.31/0.58 ! [A: $tType,N2: nat,A3: set @ A] : % 0.31/0.58 ( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( finite_card @ A @ A3 ) ) % 0.31/0.58 = ( ? [A4: A,B6: set @ A] : % 0.31/0.58 ( ( A3 % 0.31/0.58 = ( insert @ A @ A4 @ B6 ) ) % 0.31/0.58 & ~ ( member @ A @ A4 @ B6 ) % 0.31/0.58 & ( ord_less_eq @ nat @ N2 @ ( finite_card @ A @ B6 ) ) % 0.31/0.58 & ( finite_finite2 @ A @ B6 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_le_Suc_iff % 0.31/0.58 thf(fact_3922_card__Diff1__le,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,X3: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ). % 0.31/0.58 % 0.31/0.58 % card_Diff1_le % 0.31/0.58 thf(fact_3923_card__Diff__subset,axiom, % 0.31/0.58 ! [A: $tType,B3: set @ A,A3: set @ A] : % 0.31/0.58 ( ( finite_finite2 @ A @ B3 ) % 0.31/0.58 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.58 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.58 = ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_Diff_subset % 0.31/0.58 thf(fact_3924_card__psubset,axiom, % 0.31/0.58 ! [A: $tType,B3: set @ A,A3: set @ A] : % 0.31/0.58 ( ( finite_finite2 @ A @ B3 ) % 0.31/0.58 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.58 => ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) % 0.31/0.58 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_psubset % 0.31/0.58 thf(fact_3925_diff__card__le__card__Diff,axiom, % 0.31/0.58 ! [A: $tType,B3: set @ A,A3: set @ A] : % 0.31/0.58 ( ( finite_finite2 @ A @ B3 ) % 0.31/0.58 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % diff_card_le_card_Diff % 0.31/0.58 thf(fact_3926_card__lists__length__le,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,N2: nat] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( finite_card @ ( list @ A ) % 0.31/0.58 @ ( collect @ ( list @ A ) % 0.31/0.58 @ ^ [Xs2: list @ A] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) % 0.31/0.58 & ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N2 ) ) ) ) % 0.31/0.58 = ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A3 ) ) @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_lists_length_le % 0.31/0.58 thf(fact_3927_card__roots__unity,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.58 & ( idom @ A ) ) % 0.31/0.58 => ! [N2: nat] : % 0.31/0.58 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 ) % 0.31/0.58 => ( ord_less_eq @ nat % 0.31/0.58 @ ( finite_card @ A % 0.31/0.58 @ ( collect @ A % 0.31/0.58 @ ^ [Z6: A] : % 0.31/0.58 ( ( power_power @ A @ Z6 @ N2 ) % 0.31/0.58 = ( one_one @ A ) ) ) ) % 0.31/0.58 @ N2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_roots_unity % 0.31/0.58 thf(fact_3928_subset__eq__atLeast0__lessThan__card,axiom, % 0.31/0.58 ! [N7: set @ nat,N2: nat] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 0.31/0.58 => ( ord_less_eq @ nat @ ( finite_card @ nat @ N7 ) @ N2 ) ) ). % 0.31/0.58 % 0.31/0.58 % subset_eq_atLeast0_lessThan_card % 0.31/0.58 thf(fact_3929_card__sum__le__nat__sum,axiom, % 0.31/0.58 ! [S3: set @ nat] : % 0.31/0.58 ( ord_less_eq @ nat % 0.31/0.58 @ ( groups7311177749621191930dd_sum @ nat @ nat % 0.31/0.58 @ ^ [X: nat] : X % 0.31/0.58 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S3 ) ) ) % 0.31/0.58 @ ( groups7311177749621191930dd_sum @ nat @ nat % 0.31/0.58 @ ^ [X: nat] : X % 0.31/0.58 @ S3 ) ) ). % 0.31/0.58 % 0.31/0.58 % card_sum_le_nat_sum % 0.31/0.58 thf(fact_3930_card__nth__roots,axiom, % 0.31/0.58 ! [C2: complex,N2: nat] : % 0.31/0.58 ( ( C2 % 0.31/0.58 != ( zero_zero @ complex ) ) % 0.31/0.58 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.58 => ( ( finite_card @ complex % 0.31/0.58 @ ( collect @ complex % 0.31/0.58 @ ^ [Z6: complex] : % 0.31/0.58 ( ( power_power @ complex @ Z6 @ N2 ) % 0.31/0.58 = C2 ) ) ) % 0.31/0.58 = N2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_nth_roots % 0.31/0.58 thf(fact_3931_card__roots__unity__eq,axiom, % 0.31/0.58 ! [N2: nat] : % 0.31/0.58 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.58 => ( ( finite_card @ complex % 0.31/0.58 @ ( collect @ complex % 0.31/0.58 @ ^ [Z6: complex] : % 0.31/0.58 ( ( power_power @ complex @ Z6 @ N2 ) % 0.31/0.58 = ( one_one @ complex ) ) ) ) % 0.31/0.58 = N2 ) ) ). % 0.31/0.58 % 0.31/0.58 % card_roots_unity_eq % 0.31/0.58 thf(fact_3932_card__2__iff,axiom, % 0.31/0.58 ! [A: $tType,S3: set @ A] : % 0.31/0.58 ( ( ( finite_card @ A @ S3 ) % 0.31/0.58 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 0.31/0.58 = ( ? [X: A,Y2: A] : % 0.31/0.58 ( ( S3 % 0.31/0.58 = ( insert @ A @ X @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.58 & ( X != Y2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_2_iff % 0.31/0.58 thf(fact_3933_card__3__iff,axiom, % 0.31/0.58 ! [A: $tType,S3: set @ A] : % 0.31/0.58 ( ( ( finite_card @ A @ S3 ) % 0.31/0.58 = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) % 0.31/0.58 = ( ? [X: A,Y2: A,Z6: A] : % 0.31/0.58 ( ( S3 % 0.31/0.58 = ( insert @ A @ X @ ( insert @ A @ Y2 @ ( insert @ A @ Z6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) % 0.31/0.58 & ( X != Y2 ) % 0.31/0.58 & ( Y2 != Z6 ) % 0.31/0.58 & ( X != Z6 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_3_iff % 0.31/0.58 thf(fact_3934_odd__card__imp__not__empty,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A3 ) ) % 0.31/0.58 => ( A3 % 0.31/0.58 != ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % odd_card_imp_not_empty % 0.31/0.58 thf(fact_3935_card__Suc__Diff1,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,X3: A] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.58 => ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) % 0.31/0.58 = ( finite_card @ A @ A3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_Suc_Diff1 % 0.31/0.58 thf(fact_3936_card_Oinsert__remove,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,X3: A] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( finite_card @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.58 = ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card.insert_remove % 0.31/0.58 thf(fact_3937_card_Oremove,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,X3: A] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.58 => ( ( finite_card @ A @ A3 ) % 0.31/0.58 = ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card.remove % 0.31/0.58 thf(fact_3938_card__Diff1__less,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,X3: A] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.58 => ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_Diff1_less % 0.31/0.58 thf(fact_3939_card__Diff2__less,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,X3: A,Y: A] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.58 => ( ( member @ A @ Y @ A3 ) % 0.31/0.58 => ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_Diff2_less % 0.31/0.58 thf(fact_3940_card__Diff1__less__iff,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,X3: A] : % 0.31/0.58 ( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) % 0.31/0.58 = ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 & ( member @ A @ X3 @ A3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_Diff1_less_iff % 0.31/0.58 thf(fact_3941_card__Diff__singleton__if,axiom, % 0.31/0.58 ! [A: $tType,X3: A,A3: set @ A] : % 0.31/0.58 ( ( ( member @ A @ X3 @ A3 ) % 0.31/0.58 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.58 = ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) ) % 0.31/0.58 & ( ~ ( member @ A @ X3 @ A3 ) % 0.31/0.58 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.58 = ( finite_card @ A @ A3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_Diff_singleton_if % 0.31/0.58 thf(fact_3942_card__Diff__singleton,axiom, % 0.31/0.58 ! [A: $tType,X3: A,A3: set @ A] : % 0.31/0.58 ( ( member @ A @ X3 @ A3 ) % 0.31/0.58 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.58 = ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_Diff_singleton % 0.31/0.58 thf(fact_3943_prod__le__power,axiom, % 0.31/0.58 ! [B: $tType,A: $tType] : % 0.31/0.58 ( ( linordered_semidom @ A ) % 0.31/0.58 => ! [A3: set @ B,F2: B > A,N2: A,K: nat] : % 0.31/0.58 ( ! [I4: B] : % 0.31/0.58 ( ( member @ B @ I4 @ A3 ) % 0.31/0.58 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) % 0.31/0.58 & ( ord_less_eq @ A @ ( F2 @ I4 ) @ N2 ) ) ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ K ) % 0.31/0.58 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ N2 ) % 0.31/0.58 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( power_power @ A @ N2 @ K ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod_le_power % 0.31/0.58 thf(fact_3944_sum__bounded__above__strict,axiom, % 0.31/0.58 ! [B: $tType,A: $tType] : % 0.31/0.58 ( ( ( ordere8940638589300402666id_add @ A ) % 0.31/0.58 & ( semiring_1 @ A ) ) % 0.31/0.58 => ! [A3: set @ B,F2: B > A,K5: A] : % 0.31/0.58 ( ! [I4: B] : % 0.31/0.58 ( ( member @ B @ I4 @ A3 ) % 0.31/0.58 => ( ord_less @ A @ ( F2 @ I4 ) @ K5 ) ) % 0.31/0.58 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A3 ) ) % 0.31/0.58 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K5 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum_bounded_above_strict % 0.31/0.58 thf(fact_3945_sum__bounded__above__divide,axiom, % 0.31/0.58 ! [B: $tType,A: $tType] : % 0.31/0.58 ( ( linordered_field @ A ) % 0.31/0.58 => ! [A3: set @ B,F2: B > A,K5: A] : % 0.31/0.58 ( ! [I4: B] : % 0.31/0.58 ( ( member @ B @ I4 @ A3 ) % 0.31/0.58 => ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( divide_divide @ A @ K5 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) ) ) ) % 0.31/0.58 => ( ( finite_finite2 @ B @ A3 ) % 0.31/0.58 => ( ( A3 % 0.31/0.58 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.58 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ K5 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sum_bounded_above_divide % 0.31/0.58 thf(fact_3946_card__insert__le__m1,axiom, % 0.31/0.58 ! [A: $tType,N2: nat,Y: set @ A,X3: A] : % 0.31/0.58 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) % 0.31/0.58 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X3 @ Y ) ) @ N2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_insert_le_m1 % 0.31/0.58 thf(fact_3947_polyfun__roots__card,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.58 & ( idom @ A ) ) % 0.31/0.58 => ! [C2: nat > A,K: nat,N2: nat] : % 0.31/0.58 ( ( ( C2 @ K ) % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.58 => ( ord_less_eq @ nat % 0.31/0.58 @ ( finite_card @ A % 0.31/0.58 @ ( collect @ A % 0.31/0.58 @ ^ [Z6: A] : % 0.31/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) ) % 0.31/0.58 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.58 = ( zero_zero @ A ) ) ) ) % 0.31/0.58 @ N2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % polyfun_roots_card % 0.31/0.58 thf(fact_3948_prod__gen__delta,axiom, % 0.31/0.58 ! [A: $tType,B: $tType] : % 0.31/0.58 ( ( comm_monoid_mult @ A ) % 0.31/0.58 => ! [S3: set @ B,A2: B,B2: B > A,C2: A] : % 0.31/0.58 ( ( finite_finite2 @ B @ S3 ) % 0.31/0.58 => ( ( ( member @ B @ A2 @ S3 ) % 0.31/0.58 => ( ( groups7121269368397514597t_prod @ B @ A % 0.31/0.58 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ C2 ) % 0.31/0.58 @ S3 ) % 0.31/0.58 = ( times_times @ A @ ( B2 @ A2 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S3 ) @ ( one_one @ nat ) ) ) ) ) ) % 0.31/0.58 & ( ~ ( member @ B @ A2 @ S3 ) % 0.31/0.58 => ( ( groups7121269368397514597t_prod @ B @ A % 0.31/0.58 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ C2 ) % 0.31/0.58 @ S3 ) % 0.31/0.58 = ( power_power @ A @ C2 @ ( finite_card @ B @ S3 ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % prod_gen_delta % 0.31/0.58 thf(fact_3949_polyfun__rootbound,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.58 & ( idom @ A ) ) % 0.31/0.58 => ! [C2: nat > A,K: nat,N2: nat] : % 0.31/0.58 ( ( ( C2 @ K ) % 0.31/0.58 != ( zero_zero @ A ) ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.58 => ( ( finite_finite2 @ A % 0.31/0.58 @ ( collect @ A % 0.31/0.58 @ ^ [Z6: A] : % 0.31/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) ) % 0.31/0.58 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.58 = ( zero_zero @ A ) ) ) ) % 0.31/0.58 & ( ord_less_eq @ nat % 0.31/0.58 @ ( finite_card @ A % 0.31/0.58 @ ( collect @ A % 0.31/0.58 @ ^ [Z6: A] : % 0.31/0.58 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.58 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) ) % 0.31/0.58 @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.58 = ( zero_zero @ A ) ) ) ) % 0.31/0.58 @ N2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % polyfun_rootbound % 0.31/0.58 thf(fact_3950_pair__leqI1,axiom, % 0.31/0.58 ! [A2: nat,B2: nat,S: nat,T2: nat] : % 0.31/0.58 ( ( ord_less @ nat @ A2 @ B2 ) % 0.31/0.58 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ). % 0.31/0.58 % 0.31/0.58 % pair_leqI1 % 0.31/0.58 thf(fact_3951_card__lists__distinct__length__eq,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,K: nat] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A3 ) ) % 0.31/0.58 => ( ( finite_card @ ( list @ A ) % 0.31/0.58 @ ( collect @ ( list @ A ) % 0.31/0.58 @ ^ [Xs2: list @ A] : % 0.31/0.58 ( ( ( size_size @ ( list @ A ) @ Xs2 ) % 0.31/0.58 = K ) % 0.31/0.58 & ( distinct @ A @ Xs2 ) % 0.31/0.58 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) ) ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ nat @ nat % 0.31/0.58 @ ^ [X: nat] : X % 0.31/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_lists_distinct_length_eq % 0.31/0.58 thf(fact_3952_card__lists__distinct__length__eq_H,axiom, % 0.31/0.58 ! [A: $tType,K: nat,A3: set @ A] : % 0.31/0.58 ( ( ord_less @ nat @ K @ ( finite_card @ A @ A3 ) ) % 0.31/0.58 => ( ( finite_card @ ( list @ A ) % 0.31/0.58 @ ( collect @ ( list @ A ) % 0.31/0.58 @ ^ [Xs2: list @ A] : % 0.31/0.58 ( ( ( size_size @ ( list @ A ) @ Xs2 ) % 0.31/0.58 = K ) % 0.31/0.58 & ( distinct @ A @ Xs2 ) % 0.31/0.58 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) ) ) ) % 0.31/0.58 = ( groups7121269368397514597t_prod @ nat @ nat % 0.31/0.58 @ ^ [X: nat] : X % 0.31/0.58 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_lists_distinct_length_eq' % 0.31/0.58 thf(fact_3953_set__n__lists,axiom, % 0.31/0.58 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.58 ( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N2 @ Xs ) ) % 0.31/0.58 = ( collect @ ( list @ A ) % 0.31/0.58 @ ^ [Ys3: list @ A] : % 0.31/0.58 ( ( ( size_size @ ( list @ A ) @ Ys3 ) % 0.31/0.58 = N2 ) % 0.31/0.58 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Xs ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % set_n_lists % 0.31/0.58 thf(fact_3954_int__of__integer__code,axiom, % 0.31/0.58 ( code_int_of_integer % 0.31/0.58 = ( ^ [K3: code_integer] : % 0.31/0.58 ( if @ int @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K3 ) ) ) % 0.31/0.58 @ ( if @ int % 0.31/0.58 @ ( K3 % 0.31/0.58 = ( zero_zero @ code_integer ) ) % 0.31/0.58 @ ( zero_zero @ int ) % 0.31/0.58 @ ( product_case_prod @ code_integer @ code_integer @ int % 0.31/0.58 @ ^ [L2: code_integer,J3: code_integer] : % 0.31/0.58 ( if @ int % 0.31/0.58 @ ( J3 % 0.31/0.58 = ( zero_zero @ code_integer ) ) % 0.31/0.58 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) ) % 0.31/0.58 @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) ) @ ( one_one @ int ) ) ) % 0.31/0.58 @ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % int_of_integer_code % 0.31/0.58 thf(fact_3955_distinct__swap,axiom, % 0.31/0.58 ! [A: $tType,I: nat,Xs: list @ A,J: nat] : % 0.31/0.58 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.58 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.58 => ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) ) % 0.31/0.58 = ( distinct @ A @ Xs ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % distinct_swap % 0.31/0.58 thf(fact_3956_finite__lists__distinct__length__eq,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,N2: nat] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( finite_finite2 @ ( list @ A ) % 0.31/0.58 @ ( collect @ ( list @ A ) % 0.31/0.58 @ ^ [Xs2: list @ A] : % 0.31/0.58 ( ( ( size_size @ ( list @ A ) @ Xs2 ) % 0.31/0.58 = N2 ) % 0.31/0.58 & ( distinct @ A @ Xs2 ) % 0.31/0.58 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % finite_lists_distinct_length_eq % 0.31/0.58 thf(fact_3957_distinct__enumerate,axiom, % 0.31/0.58 ! [A: $tType,N2: nat,Xs: list @ A] : ( distinct @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) ) ). % 0.31/0.58 % 0.31/0.58 % distinct_enumerate % 0.31/0.58 thf(fact_3958_distinct__product,axiom, % 0.31/0.58 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 0.31/0.58 ( ( distinct @ A @ Xs ) % 0.31/0.58 => ( ( distinct @ B @ Ys ) % 0.31/0.58 => ( distinct @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % distinct_product % 0.31/0.58 thf(fact_3959_sorted__list__of__set_Odistinct__if__distinct__map,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( linorder @ A ) % 0.31/0.58 => ! [Xs: list @ A] : % 0.31/0.58 ( ( distinct @ A @ Xs ) % 0.31/0.58 => ( distinct @ A @ Xs ) ) ) ). % 0.31/0.58 % 0.31/0.58 % sorted_list_of_set.distinct_if_distinct_map % 0.31/0.58 thf(fact_3960_distinct__zipI1,axiom, % 0.31/0.58 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 0.31/0.58 ( ( distinct @ A @ Xs ) % 0.31/0.58 => ( distinct @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ). % 0.31/0.58 % 0.31/0.58 % distinct_zipI1 % 0.31/0.58 thf(fact_3961_distinct__zipI2,axiom, % 0.31/0.58 ! [B: $tType,A: $tType,Ys: list @ A,Xs: list @ B] : % 0.31/0.58 ( ( distinct @ A @ Ys ) % 0.31/0.58 => ( distinct @ ( product_prod @ B @ A ) @ ( zip @ B @ A @ Xs @ Ys ) ) ) ). % 0.31/0.58 % 0.31/0.58 % distinct_zipI2 % 0.31/0.58 thf(fact_3962_finite__distinct__list,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ? [Xs3: list @ A] : % 0.31/0.58 ( ( ( set2 @ A @ Xs3 ) % 0.31/0.58 = A3 ) % 0.31/0.58 & ( distinct @ A @ Xs3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % finite_distinct_list % 0.31/0.58 thf(fact_3963_less__integer_Orep__eq,axiom, % 0.31/0.58 ( ( ord_less @ code_integer ) % 0.31/0.58 = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % less_integer.rep_eq % 0.31/0.58 thf(fact_3964_integer__less__iff,axiom, % 0.31/0.58 ( ( ord_less @ code_integer ) % 0.31/0.58 = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % integer_less_iff % 0.31/0.58 thf(fact_3965_subseqs__distinctD,axiom, % 0.31/0.58 ! [A: $tType,Ys: list @ A,Xs: list @ A] : % 0.31/0.58 ( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) % 0.31/0.58 => ( ( distinct @ A @ Xs ) % 0.31/0.58 => ( distinct @ A @ Ys ) ) ) ). % 0.31/0.58 % 0.31/0.58 % subseqs_distinctD % 0.31/0.58 thf(fact_3966_distinct__card,axiom, % 0.31/0.58 ! [A: $tType,Xs: list @ A] : % 0.31/0.58 ( ( distinct @ A @ Xs ) % 0.31/0.58 => ( ( finite_card @ A @ ( set2 @ A @ Xs ) ) % 0.31/0.58 = ( size_size @ ( list @ A ) @ Xs ) ) ) ). % 0.31/0.58 % 0.31/0.58 % distinct_card % 0.31/0.58 thf(fact_3967_card__distinct,axiom, % 0.31/0.58 ! [A: $tType,Xs: list @ A] : % 0.31/0.58 ( ( ( finite_card @ A @ ( set2 @ A @ Xs ) ) % 0.31/0.58 = ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.58 => ( distinct @ A @ Xs ) ) ). % 0.31/0.58 % 0.31/0.58 % card_distinct % 0.31/0.58 thf(fact_3968_distinct__conv__nth,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( distinct @ A ) % 0.31/0.58 = ( ^ [Xs2: list @ A] : % 0.31/0.58 ! [I3: nat] : % 0.31/0.58 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) % 0.31/0.58 => ! [J3: nat] : % 0.31/0.58 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) % 0.31/0.58 => ( ( I3 != J3 ) % 0.31/0.58 => ( ( nth @ A @ Xs2 @ I3 ) % 0.31/0.58 != ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % distinct_conv_nth % 0.31/0.58 thf(fact_3969_nth__eq__iff__index__eq,axiom, % 0.31/0.58 ! [A: $tType,Xs: list @ A,I: nat,J: nat] : % 0.31/0.58 ( ( distinct @ A @ Xs ) % 0.31/0.58 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.58 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.58 => ( ( ( nth @ A @ Xs @ I ) % 0.31/0.58 = ( nth @ A @ Xs @ J ) ) % 0.31/0.58 = ( I = J ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nth_eq_iff_index_eq % 0.31/0.58 thf(fact_3970_distinct__Ex1,axiom, % 0.31/0.58 ! [A: $tType,Xs: list @ A,X3: A] : % 0.31/0.58 ( ( distinct @ A @ Xs ) % 0.31/0.58 => ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.58 => ? [X5: nat] : % 0.31/0.58 ( ( ord_less @ nat @ X5 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.58 & ( ( nth @ A @ Xs @ X5 ) % 0.31/0.58 = X3 ) % 0.31/0.58 & ! [Y5: nat] : % 0.31/0.58 ( ( ( ord_less @ nat @ Y5 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.58 & ( ( nth @ A @ Xs @ Y5 ) % 0.31/0.58 = X3 ) ) % 0.31/0.58 => ( Y5 = X5 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % distinct_Ex1 % 0.31/0.58 thf(fact_3971_length__n__lists__elem,axiom, % 0.31/0.58 ! [A: $tType,Ys: list @ A,N2: nat,Xs: list @ A] : % 0.31/0.58 ( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N2 @ Xs ) ) ) % 0.31/0.58 => ( ( size_size @ ( list @ A ) @ Ys ) % 0.31/0.58 = N2 ) ) ). % 0.31/0.58 % 0.31/0.58 % length_n_lists_elem % 0.31/0.58 thf(fact_3972_distinct__list__update,axiom, % 0.31/0.58 ! [A: $tType,Xs: list @ A,A2: A,I: nat] : % 0.31/0.58 ( ( distinct @ A @ Xs ) % 0.31/0.58 => ( ~ ( member @ A @ A2 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ I ) @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.58 => ( distinct @ A @ ( list_update @ A @ Xs @ I @ A2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % distinct_list_update % 0.31/0.58 thf(fact_3973_length__n__lists,axiom, % 0.31/0.58 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.58 ( ( size_size @ ( list @ ( list @ A ) ) @ ( n_lists @ A @ N2 @ Xs ) ) % 0.31/0.58 = ( power_power @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ). % 0.31/0.58 % 0.31/0.58 % length_n_lists % 0.31/0.58 thf(fact_3974_set__update__distinct,axiom, % 0.31/0.58 ! [A: $tType,Xs: list @ A,N2: nat,X3: A] : % 0.31/0.58 ( ( distinct @ A @ Xs ) % 0.31/0.58 => ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.58 => ( ( set2 @ A @ ( list_update @ A @ Xs @ N2 @ X3 ) ) % 0.31/0.58 = ( insert @ A @ X3 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ N2 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % set_update_distinct % 0.31/0.58 thf(fact_3975_divmod__integer__def,axiom, % 0.31/0.58 ( code_divmod_integer % 0.31/0.58 = ( ^ [K3: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K3 @ L2 ) @ ( modulo_modulo @ code_integer @ K3 @ L2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % divmod_integer_def % 0.31/0.58 thf(fact_3976_nat__of__integer__code,axiom, % 0.31/0.58 ( code_nat_of_integer % 0.31/0.58 = ( ^ [K3: code_integer] : % 0.31/0.58 ( if @ nat @ ( ord_less_eq @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat ) % 0.31/0.58 @ ( product_case_prod @ code_integer @ code_integer @ nat % 0.31/0.58 @ ^ [L2: code_integer,J3: code_integer] : % 0.31/0.58 ( if @ nat % 0.31/0.58 @ ( J3 % 0.31/0.58 = ( zero_zero @ code_integer ) ) % 0.31/0.58 @ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) % 0.31/0.58 @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( one_one @ nat ) ) ) % 0.31/0.58 @ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_of_integer_code % 0.31/0.58 thf(fact_3977_distinct__union,axiom, % 0.31/0.58 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.58 ( ( distinct @ A @ ( union @ A @ Xs @ Ys ) ) % 0.31/0.58 = ( distinct @ A @ Ys ) ) ). % 0.31/0.58 % 0.31/0.58 % distinct_union % 0.31/0.58 thf(fact_3978_bit__cut__integer__def,axiom, % 0.31/0.58 ( code_bit_cut_integer % 0.31/0.58 = ( ^ [K3: code_integer] : % 0.31/0.58 ( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) % 0.31/0.58 @ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % bit_cut_integer_def % 0.31/0.58 thf(fact_3979_divmod__abs__def,axiom, % 0.31/0.58 ( code_divmod_abs % 0.31/0.58 = ( ^ [K3: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % divmod_abs_def % 0.31/0.58 thf(fact_3980_nat__of__integer__non__positive,axiom, % 0.31/0.58 ! [K: code_integer] : % 0.31/0.58 ( ( ord_less_eq @ code_integer @ K @ ( zero_zero @ code_integer ) ) % 0.31/0.58 => ( ( code_nat_of_integer @ K ) % 0.31/0.58 = ( zero_zero @ nat ) ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_of_integer_non_positive % 0.31/0.58 thf(fact_3981_nat__of__integer__code__post_I1_J,axiom, % 0.31/0.58 ( ( code_nat_of_integer @ ( zero_zero @ code_integer ) ) % 0.31/0.58 = ( zero_zero @ nat ) ) ). % 0.31/0.58 % 0.31/0.58 % nat_of_integer_code_post(1) % 0.31/0.58 thf(fact_3982_divmod__abs__code_I6_J,axiom, % 0.31/0.58 ! [J: code_integer] : % 0.31/0.58 ( ( code_divmod_abs @ ( zero_zero @ code_integer ) @ J ) % 0.31/0.58 = ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ) ). % 0.31/0.58 % 0.31/0.58 % divmod_abs_code(6) % 0.31/0.58 thf(fact_3983_divmod__abs__code_I5_J,axiom, % 0.31/0.58 ! [J: code_integer] : % 0.31/0.58 ( ( code_divmod_abs @ J @ ( zero_zero @ code_integer ) ) % 0.31/0.58 = ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( abs_abs @ code_integer @ J ) ) ) ). % 0.31/0.58 % 0.31/0.58 % divmod_abs_code(5) % 0.31/0.58 thf(fact_3984_bit__cut__integer__code,axiom, % 0.31/0.58 ( code_bit_cut_integer % 0.31/0.58 = ( ^ [K3: code_integer] : % 0.31/0.58 ( if @ ( product_prod @ code_integer @ $o ) % 0.31/0.58 @ ( K3 % 0.31/0.58 = ( zero_zero @ code_integer ) ) % 0.31/0.58 @ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false ) % 0.31/0.58 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o ) % 0.31/0.58 @ ^ [R5: code_integer,S8: code_integer] : % 0.31/0.58 ( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ R5 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ S8 ) ) % 0.31/0.58 @ ( S8 % 0.31/0.58 = ( one_one @ code_integer ) ) ) % 0.31/0.58 @ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % bit_cut_integer_code % 0.31/0.58 thf(fact_3985_divmod__integer__code,axiom, % 0.31/0.58 ( code_divmod_integer % 0.31/0.58 = ( ^ [K3: code_integer,L2: code_integer] : % 0.31/0.58 ( if @ ( product_prod @ code_integer @ code_integer ) % 0.31/0.58 @ ( K3 % 0.31/0.58 = ( zero_zero @ code_integer ) ) % 0.31/0.58 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) % 0.31/0.58 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L2 ) % 0.31/0.58 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ ( code_divmod_abs @ K3 @ L2 ) % 0.31/0.58 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer ) % 0.31/0.58 @ ^ [R5: code_integer,S8: code_integer] : % 0.31/0.58 ( if @ ( product_prod @ code_integer @ code_integer ) % 0.31/0.58 @ ( S8 % 0.31/0.58 = ( zero_zero @ code_integer ) ) % 0.31/0.58 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) ) % 0.31/0.58 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L2 @ S8 ) ) ) % 0.31/0.58 @ ( code_divmod_abs @ K3 @ L2 ) ) ) % 0.31/0.58 @ ( if @ ( product_prod @ code_integer @ code_integer ) % 0.31/0.58 @ ( L2 % 0.31/0.58 = ( zero_zero @ code_integer ) ) % 0.31/0.58 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) % 0.31/0.58 @ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer ) % 0.31/0.58 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K3 @ L2 ) % 0.31/0.58 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer ) % 0.31/0.58 @ ^ [R5: code_integer,S8: code_integer] : % 0.31/0.58 ( if @ ( product_prod @ code_integer @ code_integer ) % 0.31/0.58 @ ( S8 % 0.31/0.58 = ( zero_zero @ code_integer ) ) % 0.31/0.58 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) ) % 0.31/0.58 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L2 ) @ S8 ) ) ) % 0.31/0.58 @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % divmod_integer_code % 0.31/0.58 thf(fact_3986_card__Pow,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.58 => ( ( finite_card @ ( set @ A ) @ ( pow2 @ A @ A3 ) ) % 0.31/0.58 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A3 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % card_Pow % 0.31/0.58 thf(fact_3987_rec__nat__add__eq__if,axiom, % 0.31/0.58 ! [A: $tType,A2: A,F2: nat > A > A,V2: num,N2: nat] : % 0.31/0.58 ( ( rec_nat @ A @ A2 @ F2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N2 ) ) % 0.31/0.58 = ( F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N2 ) @ ( rec_nat @ A @ A2 @ F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N2 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % rec_nat_add_eq_if % 0.31/0.58 thf(fact_3988_case__nat__add__eq__if,axiom, % 0.31/0.58 ! [A: $tType,A2: A,F2: nat > A,V2: num,N2: nat] : % 0.31/0.58 ( ( case_nat @ A @ A2 @ F2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N2 ) ) % 0.31/0.58 = ( F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N2 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % case_nat_add_eq_if % 0.31/0.58 thf(fact_3989_Pow__empty,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( pow2 @ A @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.58 = ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Pow_empty % 0.31/0.58 thf(fact_3990_Pow__singleton__iff,axiom, % 0.31/0.58 ! [A: $tType,X8: set @ A,Y7: set @ A] : % 0.31/0.58 ( ( ( pow2 @ A @ X8 ) % 0.31/0.58 = ( insert @ ( set @ A ) @ Y7 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) % 0.31/0.58 = ( ( X8 % 0.31/0.58 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.58 & ( Y7 % 0.31/0.58 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Pow_singleton_iff % 0.31/0.58 thf(fact_3991_PowI,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.58 => ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % PowI % 0.31/0.58 thf(fact_3992_Pow__iff,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.58 ( ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B3 ) ) % 0.31/0.58 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ). % 0.31/0.58 % 0.31/0.58 % Pow_iff % 0.31/0.58 thf(fact_3993_apsnd__conv,axiom, % 0.31/0.58 ! [A: $tType,B: $tType,C: $tType,F2: C > B,X3: A,Y: C] : % 0.31/0.58 ( ( product_apsnd @ C @ B @ A @ F2 @ ( product_Pair @ A @ C @ X3 @ Y ) ) % 0.31/0.58 = ( product_Pair @ A @ B @ X3 @ ( F2 @ Y ) ) ) ). % 0.31/0.58 % 0.31/0.58 % apsnd_conv % 0.31/0.58 thf(fact_3994_old_Onat_Osimps_I7_J,axiom, % 0.31/0.58 ! [T: $tType,F1: T,F22: nat > T > T,Nat: nat] : % 0.31/0.58 ( ( rec_nat @ T @ F1 @ F22 @ ( suc @ Nat ) ) % 0.31/0.58 = ( F22 @ Nat @ ( rec_nat @ T @ F1 @ F22 @ Nat ) ) ) ). % 0.31/0.58 % 0.31/0.58 % old.nat.simps(7) % 0.31/0.58 thf(fact_3995_old_Onat_Osimps_I6_J,axiom, % 0.31/0.58 ! [T: $tType,F1: T,F22: nat > T > T] : % 0.31/0.58 ( ( rec_nat @ T @ F1 @ F22 @ ( zero_zero @ nat ) ) % 0.31/0.58 = F1 ) ). % 0.31/0.58 % 0.31/0.58 % old.nat.simps(6) % 0.31/0.58 thf(fact_3996_finite__Pow__iff,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ( finite_finite2 @ ( set @ A ) @ ( pow2 @ A @ A3 ) ) % 0.31/0.58 = ( finite_finite2 @ A @ A3 ) ) ). % 0.31/0.58 % 0.31/0.58 % finite_Pow_iff % 0.31/0.58 thf(fact_3997_Pow__bottom,axiom, % 0.31/0.58 ! [A: $tType,B3: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow2 @ A @ B3 ) ) ). % 0.31/0.58 % 0.31/0.58 % Pow_bottom % 0.31/0.58 thf(fact_3998_nat_Ocase__distrib,axiom, % 0.31/0.58 ! [B: $tType,A: $tType,H2: A > B,F1: A,F22: nat > A,Nat: nat] : % 0.31/0.58 ( ( H2 @ ( case_nat @ A @ F1 @ F22 @ Nat ) ) % 0.31/0.58 = ( case_nat @ B @ ( H2 @ F1 ) % 0.31/0.58 @ ^ [X: nat] : ( H2 @ ( F22 @ X ) ) % 0.31/0.58 @ Nat ) ) ). % 0.31/0.58 % 0.31/0.58 % nat.case_distrib % 0.31/0.58 thf(fact_3999_Pow__not__empty,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ( pow2 @ A @ A3 ) % 0.31/0.58 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Pow_not_empty % 0.31/0.58 thf(fact_4000_Pow__top,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ A3 ) ) ). % 0.31/0.58 % 0.31/0.58 % Pow_top % 0.31/0.58 thf(fact_4001_Pow__mono,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.58 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.58 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B3 ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Pow_mono % 0.31/0.58 thf(fact_4002_PowD,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.58 ( ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B3 ) ) % 0.31/0.58 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ). % 0.31/0.58 % 0.31/0.58 % PowD % 0.31/0.58 thf(fact_4003_Pow__def,axiom, % 0.31/0.58 ! [A: $tType] : % 0.31/0.58 ( ( pow2 @ A ) % 0.31/0.58 = ( ^ [A6: set @ A] : % 0.31/0.58 ( collect @ ( set @ A ) % 0.31/0.58 @ ^ [B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Pow_def % 0.31/0.58 thf(fact_4004_Pow__Compl,axiom, % 0.31/0.58 ! [A: $tType,A3: set @ A] : % 0.31/0.58 ( ( pow2 @ A @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) % 0.31/0.58 = ( collect @ ( set @ A ) % 0.31/0.58 @ ^ [Uu3: set @ A] : % 0.31/0.58 ? [B6: set @ A] : % 0.31/0.58 ( ( Uu3 % 0.31/0.58 = ( uminus_uminus @ ( set @ A ) @ B6 ) ) % 0.31/0.58 & ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B6 ) ) ) ) ) ). % 0.31/0.58 % 0.31/0.58 % Pow_Compl % 0.31/0.58 thf(fact_4005_old_Onat_Osimps_I5_J,axiom, % 0.31/0.58 ! [A: $tType,F1: A,F22: nat > A,X22: nat] : % 0.31/0.58 ( ( case_nat @ A @ F1 @ F22 @ ( suc @ X22 ) ) % 0.31/0.58 = ( F22 @ X22 ) ) ). % 0.31/0.58 % 0.31/0.58 % old.nat.simps(5) % 0.31/0.58 thf(fact_4006_old_Onat_Osimps_I4_J,axiom, % 0.31/0.58 ! [A: $tType,F1: A,F22: nat > A] : % 0.31/0.59 ( ( case_nat @ A @ F1 @ F22 @ ( zero_zero @ nat ) ) % 0.31/0.59 = F1 ) ). % 0.31/0.59 % 0.31/0.59 % old.nat.simps(4) % 0.31/0.59 thf(fact_4007_nat_Odisc__eq__case_I2_J,axiom, % 0.31/0.59 ! [Nat: nat] : % 0.31/0.59 ( ( Nat % 0.31/0.59 != ( zero_zero @ nat ) ) % 0.31/0.59 = ( case_nat @ $o @ $false % 0.31/0.59 @ ^ [Uu3: nat] : $true % 0.31/0.59 @ Nat ) ) ). % 0.31/0.59 % 0.31/0.59 % nat.disc_eq_case(2) % 0.31/0.59 thf(fact_4008_nat_Odisc__eq__case_I1_J,axiom, % 0.31/0.59 ! [Nat: nat] : % 0.31/0.59 ( ( Nat % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 = ( case_nat @ $o @ $true % 0.31/0.59 @ ^ [Uu3: nat] : $false % 0.31/0.59 @ Nat ) ) ). % 0.31/0.59 % 0.31/0.59 % nat.disc_eq_case(1) % 0.31/0.59 thf(fact_4009_less__eq__nat_Osimps_I2_J,axiom, % 0.31/0.59 ! [M: nat,N2: nat] : % 0.31/0.59 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 ) % 0.31/0.59 = ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M ) @ N2 ) ) ). % 0.31/0.59 % 0.31/0.59 % less_eq_nat.simps(2) % 0.31/0.59 thf(fact_4010_diff__Suc,axiom, % 0.31/0.59 ! [M: nat,N2: nat] : % 0.31/0.59 ( ( minus_minus @ nat @ M @ ( suc @ N2 ) ) % 0.31/0.59 = ( case_nat @ nat @ ( zero_zero @ nat ) % 0.31/0.59 @ ^ [K3: nat] : K3 % 0.31/0.59 @ ( minus_minus @ nat @ M @ N2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % diff_Suc % 0.31/0.59 thf(fact_4011_Nitpick_Ocase__nat__unfold,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( case_nat @ A ) % 0.31/0.59 = ( ^ [X: A,F3: nat > A,N: nat] : % 0.31/0.59 ( if @ A % 0.31/0.59 @ ( N % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 @ X % 0.31/0.59 @ ( F3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Nitpick.case_nat_unfold % 0.31/0.59 thf(fact_4012_binomial__def,axiom, % 0.31/0.59 ( binomial % 0.31/0.59 = ( ^ [N: nat,K3: nat] : % 0.31/0.59 ( finite_card @ ( set @ nat ) % 0.31/0.59 @ ( collect @ ( set @ nat ) % 0.31/0.59 @ ^ [K6: set @ nat] : % 0.31/0.59 ( ( member @ ( set @ nat ) @ K6 @ ( pow2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) % 0.31/0.59 & ( ( finite_card @ nat @ K6 ) % 0.31/0.59 = K3 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % binomial_def % 0.31/0.59 thf(fact_4013_old_Orec__nat__def,axiom, % 0.31/0.59 ! [T: $tType] : % 0.31/0.59 ( ( rec_nat @ T ) % 0.31/0.59 = ( ^ [F12: T,F23: nat > T > T,X: nat] : ( the @ T @ ( rec_set_nat @ T @ F12 @ F23 @ X ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % old.rec_nat_def % 0.31/0.59 thf(fact_4014_rec__nat__0__imp,axiom, % 0.31/0.59 ! [A: $tType,F2: nat > A,F1: A,F22: nat > A > A] : % 0.31/0.59 ( ( F2 % 0.31/0.59 = ( rec_nat @ A @ F1 @ F22 ) ) % 0.31/0.59 => ( ( F2 @ ( zero_zero @ nat ) ) % 0.31/0.59 = F1 ) ) ). % 0.31/0.59 % 0.31/0.59 % rec_nat_0_imp % 0.31/0.59 thf(fact_4015_rec__nat__Suc__imp,axiom, % 0.31/0.59 ! [A: $tType,F2: nat > A,F1: A,F22: nat > A > A,N2: nat] : % 0.31/0.59 ( ( F2 % 0.31/0.59 = ( rec_nat @ A @ F1 @ F22 ) ) % 0.31/0.59 => ( ( F2 @ ( suc @ N2 ) ) % 0.31/0.59 = ( F22 @ N2 @ ( F2 @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % rec_nat_Suc_imp % 0.31/0.59 thf(fact_4016_nat_Osplit__sels_I1_J,axiom, % 0.31/0.59 ! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] : % 0.31/0.59 ( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) ) % 0.31/0.59 = ( ( ( Nat % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 => ( P @ F1 ) ) % 0.31/0.59 & ( ( Nat % 0.31/0.59 = ( suc @ ( pred @ Nat ) ) ) % 0.31/0.59 => ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % nat.split_sels(1) % 0.31/0.59 thf(fact_4017_pred__def,axiom, % 0.31/0.59 ( pred % 0.31/0.59 = ( case_nat @ nat @ ( zero_zero @ nat ) % 0.31/0.59 @ ^ [X24: nat] : X24 ) ) ). % 0.31/0.59 % 0.31/0.59 % pred_def % 0.31/0.59 thf(fact_4018_subset__Collect__iff,axiom, % 0.31/0.59 ! [A: $tType,B3: set @ A,A3: set @ A,P: A > $o] : % 0.31/0.59 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.59 => ( ( ord_less_eq @ ( set @ A ) @ B3 % 0.31/0.59 @ ( collect @ A % 0.31/0.59 @ ^ [X: A] : % 0.31/0.59 ( ( member @ A @ X @ A3 ) % 0.31/0.59 & ( P @ X ) ) ) ) % 0.31/0.59 = ( ! [X: A] : % 0.31/0.59 ( ( member @ A @ X @ B3 ) % 0.31/0.59 => ( P @ X ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % subset_Collect_iff % 0.31/0.59 thf(fact_4019_subset__CollectI,axiom, % 0.31/0.59 ! [A: $tType,B3: set @ A,A3: set @ A,Q: A > $o,P: A > $o] : % 0.31/0.59 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.59 => ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ B3 ) % 0.31/0.59 => ( ( Q @ X5 ) % 0.31/0.59 => ( P @ X5 ) ) ) % 0.31/0.59 => ( ord_less_eq @ ( set @ A ) % 0.31/0.59 @ ( collect @ A % 0.31/0.59 @ ^ [X: A] : % 0.31/0.59 ( ( member @ A @ X @ B3 ) % 0.31/0.59 & ( Q @ X ) ) ) % 0.31/0.59 @ ( collect @ A % 0.31/0.59 @ ^ [X: A] : % 0.31/0.59 ( ( member @ A @ X @ A3 ) % 0.31/0.59 & ( P @ X ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % subset_CollectI % 0.31/0.59 thf(fact_4020_nat_Osplit__sels_I2_J,axiom, % 0.31/0.59 ! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] : % 0.31/0.59 ( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) ) % 0.31/0.59 = ( ~ ( ( ( Nat % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 & ~ ( P @ F1 ) ) % 0.31/0.59 | ( ( Nat % 0.31/0.59 = ( suc @ ( pred @ Nat ) ) ) % 0.31/0.59 & ~ ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % nat.split_sels(2) % 0.31/0.59 thf(fact_4021_signed__take__bit__code,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.59 => ( ( bit_ri4674362597316999326ke_bit @ A ) % 0.31/0.59 = ( ^ [N: nat,A4: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A4 ) @ N ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A4 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A4 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % signed_take_bit_code % 0.31/0.59 thf(fact_4022_wmin__insertI,axiom, % 0.31/0.59 ! [X3: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] : % 0.31/0.59 ( ( member @ ( product_prod @ nat @ nat ) @ X3 @ XS ) % 0.31/0.59 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_leq ) % 0.31/0.59 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_min_weak ) % 0.31/0.59 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ ( insert @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % wmin_insertI % 0.31/0.59 thf(fact_4023_wmax__insertI,axiom, % 0.31/0.59 ! [Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X3: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat )] : % 0.31/0.59 ( ( member @ ( product_prod @ nat @ nat ) @ Y @ YS ) % 0.31/0.59 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_leq ) % 0.31/0.59 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_max_weak ) % 0.31/0.59 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X3 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % wmax_insertI % 0.31/0.59 thf(fact_4024_bezw__0,axiom, % 0.31/0.59 ! [X3: nat] : % 0.31/0.59 ( ( bezw @ X3 @ ( zero_zero @ nat ) ) % 0.31/0.59 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ). % 0.31/0.59 % 0.31/0.59 % bezw_0 % 0.31/0.59 thf(fact_4025_push__bit__negative__int__iff,axiom, % 0.31/0.59 ! [N2: nat,K: int] : % 0.31/0.59 ( ( ord_less @ int @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) % 0.31/0.59 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ). % 0.31/0.59 % 0.31/0.59 % push_bit_negative_int_iff % 0.31/0.59 thf(fact_4026_push__bit__eq__0__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_un5681908812861735899ations @ A ) % 0.31/0.59 => ! [N2: nat,A2: A] : % 0.31/0.59 ( ( ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) % 0.31/0.59 = ( zero_zero @ A ) ) % 0.31/0.59 = ( A2 % 0.31/0.59 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % push_bit_eq_0_iff % 0.31/0.59 thf(fact_4027_push__bit__of__0,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [N2: nat] : % 0.31/0.59 ( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( zero_zero @ A ) ) % 0.31/0.59 = ( zero_zero @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % push_bit_of_0 % 0.31/0.59 thf(fact_4028_push__bit__push__bit,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [M: nat,N2: nat,A2: A] : % 0.31/0.59 ( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) ) % 0.31/0.59 = ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ A2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % push_bit_push_bit % 0.31/0.59 thf(fact_4029_push__bit__Suc__numeral,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [N2: nat,K: num] : % 0.31/0.59 ( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ K ) ) % 0.31/0.59 = ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % push_bit_Suc_numeral % 0.31/0.59 thf(fact_4030_push__bit__Suc__minus__numeral,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.59 => ! [N2: nat,K: num] : % 0.31/0.59 ( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) % 0.31/0.59 = ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % push_bit_Suc_minus_numeral % 0.31/0.59 thf(fact_4031_push__bit__Suc,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [N2: nat,A2: A] : % 0.31/0.59 ( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ A2 ) % 0.31/0.59 = ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % push_bit_Suc % 0.31/0.59 thf(fact_4032_push__bit__of__Suc__0,axiom, % 0.31/0.59 ! [N2: nat] : % 0.31/0.59 ( ( bit_se4730199178511100633sh_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 0.31/0.59 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 0.31/0.59 % 0.31/0.59 % push_bit_of_Suc_0 % 0.31/0.59 thf(fact_4033_even__push__bit__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [N2: nat,A2: A] : % 0.31/0.59 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) ) % 0.31/0.59 = ( ( N2 % 0.31/0.59 != ( zero_zero @ nat ) ) % 0.31/0.59 | ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % even_push_bit_iff % 0.31/0.59 thf(fact_4034_wmin__emptyI,axiom, % 0.31/0.59 ! [X8: set @ ( product_prod @ nat @ nat )] : ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X8 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ). % 0.31/0.59 % 0.31/0.59 % wmin_emptyI % 0.31/0.59 thf(fact_4035_wmax__emptyI,axiom, % 0.31/0.59 ! [X8: set @ ( product_prod @ nat @ nat )] : % 0.31/0.59 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X8 ) % 0.31/0.59 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ X8 ) @ fun_max_weak ) ) ). % 0.31/0.59 % 0.31/0.59 % wmax_emptyI % 0.31/0.59 thf(fact_4036_push__bit__add,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [N2: nat,A2: A,B2: A] : % 0.31/0.59 ( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.31/0.59 = ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ B2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % push_bit_add % 0.31/0.59 thf(fact_4037_push__bit__take__bit,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [M: nat,N2: nat,A2: A] : % 0.31/0.59 ( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) % 0.31/0.59 = ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ ( bit_se4730199178511100633sh_bit @ A @ M @ A2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % push_bit_take_bit % 0.31/0.59 thf(fact_4038_bit__push__bit__iff__int,axiom, % 0.31/0.59 ! [M: nat,K: int,N2: nat] : % 0.31/0.59 ( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M @ K ) @ N2 ) % 0.31/0.59 = ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.59 & ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % bit_push_bit_iff_int % 0.31/0.59 thf(fact_4039_bit__push__bit__iff__nat,axiom, % 0.31/0.59 ! [M: nat,Q4: nat,N2: nat] : % 0.31/0.59 ( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M @ Q4 ) @ N2 ) % 0.31/0.59 = ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.59 & ( bit_se5641148757651400278ts_bit @ nat @ Q4 @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % bit_push_bit_iff_nat % 0.31/0.59 thf(fact_4040_bit__iff__and__push__bit__not__eq__0,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ( ( bit_se5641148757651400278ts_bit @ A ) % 0.31/0.59 = ( ^ [A4: A,N: nat] : % 0.31/0.59 ( ( bit_se5824344872417868541ns_and @ A @ A4 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) ) % 0.31/0.59 != ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % bit_iff_and_push_bit_not_eq_0 % 0.31/0.59 thf(fact_4041_take__bit__sum,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_un5681908812861735899ations @ A ) % 0.31/0.59 => ( ( bit_se2584673776208193580ke_bit @ A ) % 0.31/0.59 = ( ^ [N: nat,A4: A] : % 0.31/0.59 ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.59 @ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A4 @ K3 ) ) ) % 0.31/0.59 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % take_bit_sum % 0.31/0.59 thf(fact_4042_smax__insertI,axiom, % 0.31/0.59 ! [Y: product_prod @ nat @ nat,Y7: set @ ( product_prod @ nat @ nat ),X3: product_prod @ nat @ nat,X8: set @ ( product_prod @ nat @ nat )] : % 0.31/0.59 ( ( member @ ( product_prod @ nat @ nat ) @ Y @ Y7 ) % 0.31/0.59 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_less ) % 0.31/0.59 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X8 @ Y7 ) @ fun_max_strict ) % 0.31/0.59 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X3 @ X8 ) @ Y7 ) @ fun_max_strict ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % smax_insertI % 0.31/0.59 thf(fact_4043_smin__insertI,axiom, % 0.31/0.59 ! [X3: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] : % 0.31/0.59 ( ( member @ ( product_prod @ nat @ nat ) @ X3 @ XS ) % 0.31/0.59 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_less ) % 0.31/0.59 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_min_strict ) % 0.31/0.59 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ ( insert @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % smin_insertI % 0.31/0.59 thf(fact_4044_prod__decode__aux_Osimps,axiom, % 0.31/0.59 ( nat_prod_decode_aux % 0.31/0.59 = ( ^ [K3: nat,M2: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M2 @ K3 ) @ ( product_Pair @ nat @ nat @ M2 @ ( minus_minus @ nat @ K3 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus @ nat @ M2 @ ( suc @ K3 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod_decode_aux.simps % 0.31/0.59 thf(fact_4045_prod__decode__aux_Oelims,axiom, % 0.31/0.59 ! [X3: nat,Xa2: nat,Y: product_prod @ nat @ nat] : % 0.31/0.59 ( ( ( nat_prod_decode_aux @ X3 @ Xa2 ) % 0.31/0.59 = Y ) % 0.31/0.59 => ( ( ( ord_less_eq @ nat @ Xa2 @ X3 ) % 0.31/0.59 => ( Y % 0.31/0.59 = ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X3 @ Xa2 ) ) ) ) % 0.31/0.59 & ( ~ ( ord_less_eq @ nat @ Xa2 @ X3 ) % 0.31/0.59 => ( Y % 0.31/0.59 = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod_decode_aux.elims % 0.31/0.59 thf(fact_4046_smax__emptyI,axiom, % 0.31/0.59 ! [Y7: set @ ( product_prod @ nat @ nat )] : % 0.31/0.59 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y7 ) % 0.31/0.59 => ( ( Y7 % 0.31/0.59 != ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) % 0.31/0.59 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ Y7 ) @ fun_max_strict ) ) ) ). % 0.31/0.59 % 0.31/0.59 % smax_emptyI % 0.31/0.59 thf(fact_4047_smin__emptyI,axiom, % 0.31/0.59 ! [X8: set @ ( product_prod @ nat @ nat )] : % 0.31/0.59 ( ( X8 % 0.31/0.59 != ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) % 0.31/0.59 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X8 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ). % 0.31/0.59 % 0.31/0.59 % smin_emptyI % 0.31/0.59 thf(fact_4048_Suc__0__div__numeral,axiom, % 0.31/0.59 ! [K: num] : % 0.31/0.59 ( ( divide_divide @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) ) % 0.31/0.59 = ( product_fst @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Suc_0_div_numeral % 0.31/0.59 thf(fact_4049_Suc__0__mod__numeral,axiom, % 0.31/0.59 ! [K: num] : % 0.31/0.59 ( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) ) % 0.31/0.59 = ( product_snd @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Suc_0_mod_numeral % 0.31/0.59 thf(fact_4050_finite__enumerate,axiom, % 0.31/0.59 ! [S3: set @ nat] : % 0.31/0.59 ( ( finite_finite2 @ nat @ S3 ) % 0.31/0.59 => ? [R4: nat > nat] : % 0.31/0.59 ( ( strict_mono_on @ nat @ nat @ R4 @ ( set_ord_lessThan @ nat @ ( finite_card @ nat @ S3 ) ) ) % 0.31/0.59 & ! [N6: nat] : % 0.31/0.59 ( ( ord_less @ nat @ N6 @ ( finite_card @ nat @ S3 ) ) % 0.31/0.59 => ( member @ nat @ ( R4 @ N6 ) @ S3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % finite_enumerate % 0.31/0.59 thf(fact_4051_card__UNION,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ ( set @ A )] : % 0.31/0.59 ( ( finite_finite2 @ ( set @ A ) @ A3 ) % 0.31/0.59 => ( ! [X5: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ X5 @ A3 ) % 0.31/0.59 => ( finite_finite2 @ A @ X5 ) ) % 0.31/0.59 => ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) % 0.31/0.59 = ( nat2 % 0.31/0.59 @ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int % 0.31/0.59 @ ^ [I8: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I8 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I8 ) ) ) ) % 0.31/0.59 @ ( collect @ ( set @ ( set @ A ) ) % 0.31/0.59 @ ^ [I8: set @ ( set @ A )] : % 0.31/0.59 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I8 @ A3 ) % 0.31/0.59 & ( I8 % 0.31/0.59 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % card_UNION % 0.31/0.59 thf(fact_4052_finite__Inter,axiom, % 0.31/0.59 ! [A: $tType,M7: set @ ( set @ A )] : % 0.31/0.59 ( ? [X2: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ X2 @ M7 ) % 0.31/0.59 & ( finite_finite2 @ A @ X2 ) ) % 0.31/0.59 => ( finite_finite2 @ A @ ( complete_Inf_Inf @ ( set @ A ) @ M7 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % finite_Inter % 0.31/0.59 thf(fact_4053_fst__apsnd,axiom, % 0.31/0.59 ! [B: $tType,C: $tType,A: $tType,F2: C > B,X3: product_prod @ A @ C] : % 0.31/0.59 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F2 @ X3 ) ) % 0.31/0.59 = ( product_fst @ A @ C @ X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % fst_apsnd % 0.31/0.59 thf(fact_4054_apsnd__eq__conv,axiom, % 0.31/0.59 ! [B: $tType,C: $tType,A: $tType,F2: C > B,X3: product_prod @ A @ C,G: C > B] : % 0.31/0.59 ( ( ( product_apsnd @ C @ B @ A @ F2 @ X3 ) % 0.31/0.59 = ( product_apsnd @ C @ B @ A @ G @ X3 ) ) % 0.31/0.59 = ( ( F2 @ ( product_snd @ A @ C @ X3 ) ) % 0.31/0.59 = ( G @ ( product_snd @ A @ C @ X3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % apsnd_eq_conv % 0.31/0.59 thf(fact_4055_snd__apsnd,axiom, % 0.31/0.59 ! [A: $tType,C: $tType,B: $tType,F2: C > A,X3: product_prod @ B @ C] : % 0.31/0.59 ( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F2 @ X3 ) ) % 0.31/0.59 = ( F2 @ ( product_snd @ B @ C @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % snd_apsnd % 0.31/0.59 thf(fact_4056_cSup__atLeastAtMost,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit1219197933456340205attice @ A ) % 0.31/0.59 => ! [Y: A,X3: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ Y @ X3 ) % 0.31/0.59 => ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X3 ) ) % 0.31/0.59 = X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cSup_atLeastAtMost % 0.31/0.59 thf(fact_4057_Sup__atLeastAtMost,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [X3: A,Y: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.59 => ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 0.31/0.59 = Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_atLeastAtMost % 0.31/0.59 thf(fact_4058_cSup__singleton,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit1219197933456340205attice @ A ) % 0.31/0.59 => ! [X3: A] : % 0.31/0.59 ( ( complete_Sup_Sup @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.59 = X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % cSup_singleton % 0.31/0.59 thf(fact_4059_cInf__atLeastAtMost,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit1219197933456340205attice @ A ) % 0.31/0.59 => ! [Y: A,X3: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ Y @ X3 ) % 0.31/0.59 => ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X3 ) ) % 0.31/0.59 = Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cInf_atLeastAtMost % 0.31/0.59 thf(fact_4060_Inf__atLeastAtMost,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [X3: A,Y: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.59 => ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 0.31/0.59 = X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inf_atLeastAtMost % 0.31/0.59 thf(fact_4061_cInf__singleton,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit1219197933456340205attice @ A ) % 0.31/0.59 => ! [X3: A] : % 0.31/0.59 ( ( complete_Inf_Inf @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.59 = X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % cInf_singleton % 0.31/0.59 thf(fact_4062_Sup__atLeastLessThan,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 & ( dense_linorder @ A ) ) % 0.31/0.59 => ! [X3: A,Y: A] : % 0.31/0.59 ( ( ord_less @ A @ X3 @ Y ) % 0.31/0.59 => ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X3 @ Y ) ) % 0.31/0.59 = Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_atLeastLessThan % 0.31/0.59 thf(fact_4063_cSup__atLeastLessThan,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ( condit6923001295902523014norder @ A ) % 0.31/0.59 & ( dense_linorder @ A ) ) % 0.31/0.59 => ! [Y: A,X3: A] : % 0.31/0.59 ( ( ord_less @ A @ Y @ X3 ) % 0.31/0.59 => ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X3 ) ) % 0.31/0.59 = X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cSup_atLeastLessThan % 0.31/0.59 thf(fact_4064_Inf__atLeastLessThan,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [X3: A,Y: A] : % 0.31/0.59 ( ( ord_less @ A @ X3 @ Y ) % 0.31/0.59 => ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X3 @ Y ) ) % 0.31/0.59 = X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inf_atLeastLessThan % 0.31/0.59 thf(fact_4065_cInf__atLeastLessThan,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit1219197933456340205attice @ A ) % 0.31/0.59 => ! [Y: A,X3: A] : % 0.31/0.59 ( ( ord_less @ A @ Y @ X3 ) % 0.31/0.59 => ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X3 ) ) % 0.31/0.59 = Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cInf_atLeastLessThan % 0.31/0.59 thf(fact_4066_Inf__atMost,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [X3: A] : % 0.31/0.59 ( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X3 ) ) % 0.31/0.59 = ( bot_bot @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inf_atMost % 0.31/0.59 thf(fact_4067_prod_Ocollapse,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] : % 0.31/0.59 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) % 0.31/0.59 = Prod ) ). % 0.31/0.59 % 0.31/0.59 % prod.collapse % 0.31/0.59 thf(fact_4068_finite__Union,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ ( set @ A )] : % 0.31/0.59 ( ( finite_finite2 @ ( set @ A ) @ A3 ) % 0.31/0.59 => ( ! [M8: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ M8 @ A3 ) % 0.31/0.59 => ( finite_finite2 @ A @ M8 ) ) % 0.31/0.59 => ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % finite_Union % 0.31/0.59 thf(fact_4069_split__comp__eq,axiom, % 0.31/0.59 ! [A: $tType,C: $tType,B: $tType,D: $tType,F2: A > B > C,G: D > A] : % 0.31/0.59 ( ( ^ [U2: product_prod @ D @ B] : ( F2 @ ( G @ ( product_fst @ D @ B @ U2 ) ) @ ( product_snd @ D @ B @ U2 ) ) ) % 0.31/0.59 = ( product_case_prod @ D @ B @ C % 0.31/0.59 @ ^ [X: D] : ( F2 @ ( G @ X ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % split_comp_eq % 0.31/0.59 thf(fact_4070_case__prod__beta_H,axiom, % 0.31/0.59 ! [C: $tType,B: $tType,A: $tType] : % 0.31/0.59 ( ( product_case_prod @ A @ B @ C ) % 0.31/0.59 = ( ^ [F3: A > B > C,X: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % case_prod_beta' % 0.31/0.59 thf(fact_4071_case__prod__unfold,axiom, % 0.31/0.59 ! [C: $tType,B: $tType,A: $tType] : % 0.31/0.59 ( ( product_case_prod @ A @ B @ C ) % 0.31/0.59 = ( ^ [C4: A > B > C,P6: product_prod @ A @ B] : ( C4 @ ( product_fst @ A @ B @ P6 ) @ ( product_snd @ A @ B @ P6 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % case_prod_unfold % 0.31/0.59 thf(fact_4072_The__case__prod,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,P: A > B > $o] : % 0.31/0.59 ( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) % 0.31/0.59 = ( the @ ( product_prod @ A @ B ) % 0.31/0.59 @ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % The_case_prod % 0.31/0.59 thf(fact_4073_cSup__eq__maximum,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit1219197933456340205attice @ A ) % 0.31/0.59 => ! [Z2: A,X8: set @ A] : % 0.31/0.59 ( ( member @ A @ Z2 @ X8 ) % 0.31/0.59 => ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ X8 ) % 0.31/0.59 => ( ord_less_eq @ A @ X5 @ Z2 ) ) % 0.31/0.59 => ( ( complete_Sup_Sup @ A @ X8 ) % 0.31/0.59 = Z2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cSup_eq_maximum % 0.31/0.59 thf(fact_4074_cSup__eq,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ( condit1219197933456340205attice @ A ) % 0.31/0.59 & ( no_bot @ A ) ) % 0.31/0.59 => ! [X8: set @ A,A2: A] : % 0.31/0.59 ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ X8 ) % 0.31/0.59 => ( ord_less_eq @ A @ X5 @ A2 ) ) % 0.31/0.59 => ( ! [Y4: A] : % 0.31/0.59 ( ! [X2: A] : % 0.31/0.59 ( ( member @ A @ X2 @ X8 ) % 0.31/0.59 => ( ord_less_eq @ A @ X2 @ Y4 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ A2 @ Y4 ) ) % 0.31/0.59 => ( ( complete_Sup_Sup @ A @ X8 ) % 0.31/0.59 = A2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cSup_eq % 0.31/0.59 thf(fact_4075_cInf__eq__minimum,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit1219197933456340205attice @ A ) % 0.31/0.59 => ! [Z2: A,X8: set @ A] : % 0.31/0.59 ( ( member @ A @ Z2 @ X8 ) % 0.31/0.59 => ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ X8 ) % 0.31/0.59 => ( ord_less_eq @ A @ Z2 @ X5 ) ) % 0.31/0.59 => ( ( complete_Inf_Inf @ A @ X8 ) % 0.31/0.59 = Z2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cInf_eq_minimum % 0.31/0.59 thf(fact_4076_cInf__eq,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ( condit1219197933456340205attice @ A ) % 0.31/0.59 & ( no_top @ A ) ) % 0.31/0.59 => ! [X8: set @ A,A2: A] : % 0.31/0.59 ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ X8 ) % 0.31/0.59 => ( ord_less_eq @ A @ A2 @ X5 ) ) % 0.31/0.59 => ( ! [Y4: A] : % 0.31/0.59 ( ! [X2: A] : % 0.31/0.59 ( ( member @ A @ X2 @ X8 ) % 0.31/0.59 => ( ord_less_eq @ A @ Y4 @ X2 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ Y4 @ A2 ) ) % 0.31/0.59 => ( ( complete_Inf_Inf @ A @ X8 ) % 0.31/0.59 = A2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cInf_eq % 0.31/0.59 thf(fact_4077_snd__def,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( product_snd @ A @ B ) % 0.31/0.59 = ( product_case_prod @ A @ B @ B % 0.31/0.59 @ ^ [X15: A,X24: B] : X24 ) ) ). % 0.31/0.59 % 0.31/0.59 % snd_def % 0.31/0.59 thf(fact_4078_fst__def,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( product_fst @ A @ B ) % 0.31/0.59 = ( product_case_prod @ A @ B @ A % 0.31/0.59 @ ^ [X15: A,X24: B] : X15 ) ) ). % 0.31/0.59 % 0.31/0.59 % fst_def % 0.31/0.59 thf(fact_4079_prod__eq__iff,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( ^ [Y3: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y3 = Z ) ) % 0.31/0.59 = ( ^ [S8: product_prod @ A @ B,T3: product_prod @ A @ B] : % 0.31/0.59 ( ( ( product_fst @ A @ B @ S8 ) % 0.31/0.59 = ( product_fst @ A @ B @ T3 ) ) % 0.31/0.59 & ( ( product_snd @ A @ B @ S8 ) % 0.31/0.59 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod_eq_iff % 0.31/0.59 thf(fact_4080_prod__eqI,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,P4: product_prod @ A @ B,Q4: product_prod @ A @ B] : % 0.31/0.59 ( ( ( product_fst @ A @ B @ P4 ) % 0.31/0.59 = ( product_fst @ A @ B @ Q4 ) ) % 0.31/0.59 => ( ( ( product_snd @ A @ B @ P4 ) % 0.31/0.59 = ( product_snd @ A @ B @ Q4 ) ) % 0.31/0.59 => ( P4 = Q4 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod_eqI % 0.31/0.59 thf(fact_4081_prod_Oexpand,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] : % 0.31/0.59 ( ( ( ( product_fst @ A @ B @ Prod ) % 0.31/0.59 = ( product_fst @ A @ B @ Prod2 ) ) % 0.31/0.59 & ( ( product_snd @ A @ B @ Prod ) % 0.31/0.59 = ( product_snd @ A @ B @ Prod2 ) ) ) % 0.31/0.59 => ( Prod = Prod2 ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.expand % 0.31/0.59 thf(fact_4082_split__beta,axiom, % 0.31/0.59 ! [C: $tType,B: $tType,A: $tType] : % 0.31/0.59 ( ( product_case_prod @ A @ B @ C ) % 0.31/0.59 = ( ^ [F3: A > B > C,Prod3: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % split_beta % 0.31/0.59 thf(fact_4083_case__prod__beta,axiom, % 0.31/0.59 ! [A: $tType,C: $tType,B: $tType] : % 0.31/0.59 ( ( product_case_prod @ B @ C @ A ) % 0.31/0.59 = ( ^ [F3: B > C > A,P6: product_prod @ B @ C] : ( F3 @ ( product_fst @ B @ C @ P6 ) @ ( product_snd @ B @ C @ P6 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % case_prod_beta % 0.31/0.59 thf(fact_4084_Product__Type_OCollect__case__prodD,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,X3: product_prod @ A @ B,A3: A > B > $o] : % 0.31/0.59 ( ( member @ ( product_prod @ A @ B ) @ X3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) ) % 0.31/0.59 => ( A3 @ ( product_fst @ A @ B @ X3 ) @ ( product_snd @ A @ B @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Product_Type.Collect_case_prodD % 0.31/0.59 thf(fact_4085_prod_Osplit__sel__asm,axiom, % 0.31/0.59 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F2: A > B > C,Prod: product_prod @ A @ B] : % 0.31/0.59 ( ( P @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) ) % 0.31/0.59 = ( ~ ( ( Prod % 0.31/0.59 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) % 0.31/0.59 & ~ ( P @ ( F2 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.split_sel_asm % 0.31/0.59 thf(fact_4086_prod_Osplit__sel,axiom, % 0.31/0.59 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F2: A > B > C,Prod: product_prod @ A @ B] : % 0.31/0.59 ( ( P @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) ) % 0.31/0.59 = ( ( Prod % 0.31/0.59 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) % 0.31/0.59 => ( P @ ( F2 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.split_sel % 0.31/0.59 thf(fact_4087_fst__conv,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,X1: A,X22: B] : % 0.31/0.59 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) ) % 0.31/0.59 = X1 ) ). % 0.31/0.59 % 0.31/0.59 % fst_conv % 0.31/0.59 thf(fact_4088_fst__eqD,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,X3: A,Y: B,A2: A] : % 0.31/0.59 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y ) ) % 0.31/0.59 = A2 ) % 0.31/0.59 => ( X3 = A2 ) ) ). % 0.31/0.59 % 0.31/0.59 % fst_eqD % 0.31/0.59 thf(fact_4089_prod_Oexhaust__sel,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] : % 0.31/0.59 ( Prod % 0.31/0.59 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.exhaust_sel % 0.31/0.59 thf(fact_4090_surjective__pairing,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] : % 0.31/0.59 ( T2 % 0.31/0.59 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % surjective_pairing % 0.31/0.59 thf(fact_4091_snd__conv,axiom, % 0.31/0.59 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] : % 0.31/0.59 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) ) % 0.31/0.59 = X22 ) ). % 0.31/0.59 % 0.31/0.59 % snd_conv % 0.31/0.59 thf(fact_4092_snd__eqD,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,X3: B,Y: A,A2: A] : % 0.31/0.59 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) ) % 0.31/0.59 = A2 ) % 0.31/0.59 => ( Y = A2 ) ) ). % 0.31/0.59 % 0.31/0.59 % snd_eqD % 0.31/0.59 thf(fact_4093_BNF__Greatest__Fixpoint_Osubst__Pair,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,P: A > B > $o,X3: A,Y: B,A2: product_prod @ A @ B] : % 0.31/0.59 ( ( P @ X3 @ Y ) % 0.31/0.59 => ( ( A2 % 0.31/0.59 = ( product_Pair @ A @ B @ X3 @ Y ) ) % 0.31/0.59 => ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % BNF_Greatest_Fixpoint.subst_Pair % 0.31/0.59 thf(fact_4094_cInf__eq__non__empty,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit1219197933456340205attice @ A ) % 0.31/0.59 => ! [X8: set @ A,A2: A] : % 0.31/0.59 ( ( X8 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ X8 ) % 0.31/0.59 => ( ord_less_eq @ A @ A2 @ X5 ) ) % 0.31/0.59 => ( ! [Y4: A] : % 0.31/0.59 ( ! [X2: A] : % 0.31/0.59 ( ( member @ A @ X2 @ X8 ) % 0.31/0.59 => ( ord_less_eq @ A @ Y4 @ X2 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ Y4 @ A2 ) ) % 0.31/0.59 => ( ( complete_Inf_Inf @ A @ X8 ) % 0.31/0.59 = A2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cInf_eq_non_empty % 0.31/0.59 thf(fact_4095_cInf__greatest,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit1219197933456340205attice @ A ) % 0.31/0.59 => ! [X8: set @ A,Z2: A] : % 0.31/0.59 ( ( X8 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ X8 ) % 0.31/0.59 => ( ord_less_eq @ A @ Z2 @ X5 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cInf_greatest % 0.31/0.59 thf(fact_4096_cInf__le__finite,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit1219197933456340205attice @ A ) % 0.31/0.59 => ! [X8: set @ A,X3: A] : % 0.31/0.59 ( ( finite_finite2 @ A @ X8 ) % 0.31/0.59 => ( ( member @ A @ X3 @ X8 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cInf_le_finite % 0.31/0.59 thf(fact_4097_cInf__lessD,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit6923001295902523014norder @ A ) % 0.31/0.59 => ! [X8: set @ A,Z2: A] : % 0.31/0.59 ( ( X8 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Z2 ) % 0.31/0.59 => ? [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ X8 ) % 0.31/0.59 & ( ord_less @ A @ X5 @ Z2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cInf_lessD % 0.31/0.59 thf(fact_4098_finite__imp__less__Inf,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit6923001295902523014norder @ A ) % 0.31/0.59 => ! [X8: set @ A,X3: A,A2: A] : % 0.31/0.59 ( ( finite_finite2 @ A @ X8 ) % 0.31/0.59 => ( ( member @ A @ X3 @ X8 ) % 0.31/0.59 => ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ X8 ) % 0.31/0.59 => ( ord_less @ A @ A2 @ X5 ) ) % 0.31/0.59 => ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % finite_imp_less_Inf % 0.31/0.59 thf(fact_4099_cSup__eq__non__empty,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit1219197933456340205attice @ A ) % 0.31/0.59 => ! [X8: set @ A,A2: A] : % 0.31/0.59 ( ( X8 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ X8 ) % 0.31/0.59 => ( ord_less_eq @ A @ X5 @ A2 ) ) % 0.31/0.59 => ( ! [Y4: A] : % 0.31/0.59 ( ! [X2: A] : % 0.31/0.59 ( ( member @ A @ X2 @ X8 ) % 0.31/0.59 => ( ord_less_eq @ A @ X2 @ Y4 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ A2 @ Y4 ) ) % 0.31/0.59 => ( ( complete_Sup_Sup @ A @ X8 ) % 0.31/0.59 = A2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cSup_eq_non_empty % 0.31/0.59 thf(fact_4100_cSup__least,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit1219197933456340205attice @ A ) % 0.31/0.59 => ! [X8: set @ A,Z2: A] : % 0.31/0.59 ( ( X8 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ X8 ) % 0.31/0.59 => ( ord_less_eq @ A @ X5 @ Z2 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X8 ) @ Z2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cSup_least % 0.31/0.59 thf(fact_4101_le__cSup__finite,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit1219197933456340205attice @ A ) % 0.31/0.59 => ! [X8: set @ A,X3: A] : % 0.31/0.59 ( ( finite_finite2 @ A @ X8 ) % 0.31/0.59 => ( ( member @ A @ X3 @ X8 ) % 0.31/0.59 => ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % le_cSup_finite % 0.31/0.59 thf(fact_4102_update__zip,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,I: nat,Xy2: product_prod @ A @ B] : % 0.31/0.59 ( ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I @ Xy2 ) % 0.31/0.59 = ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ ( product_fst @ A @ B @ Xy2 ) ) @ ( list_update @ B @ Ys @ I @ ( product_snd @ A @ B @ Xy2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % update_zip % 0.31/0.59 thf(fact_4103_less__cSupE,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit6923001295902523014norder @ A ) % 0.31/0.59 => ! [Y: A,X8: set @ A] : % 0.31/0.59 ( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X8 ) ) % 0.31/0.59 => ( ( X8 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ~ ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ X8 ) % 0.31/0.59 => ~ ( ord_less @ A @ Y @ X5 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % less_cSupE % 0.31/0.59 thf(fact_4104_less__cSupD,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit6923001295902523014norder @ A ) % 0.31/0.59 => ! [X8: set @ A,Z2: A] : % 0.31/0.59 ( ( X8 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ( ord_less @ A @ Z2 @ ( complete_Sup_Sup @ A @ X8 ) ) % 0.31/0.59 => ? [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ X8 ) % 0.31/0.59 & ( ord_less @ A @ Z2 @ X5 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % less_cSupD % 0.31/0.59 thf(fact_4105_finite__imp__Sup__less,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit6923001295902523014norder @ A ) % 0.31/0.59 => ! [X8: set @ A,X3: A,A2: A] : % 0.31/0.59 ( ( finite_finite2 @ A @ X8 ) % 0.31/0.59 => ( ( member @ A @ X3 @ X8 ) % 0.31/0.59 => ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ X8 ) % 0.31/0.59 => ( ord_less @ A @ X5 @ A2 ) ) % 0.31/0.59 => ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % finite_imp_Sup_less % 0.31/0.59 thf(fact_4106_card__Union__le__sum__card,axiom, % 0.31/0.59 ! [A: $tType,U3: set @ ( set @ A )] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U3 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U3 ) ) ). % 0.31/0.59 % 0.31/0.59 % card_Union_le_sum_card % 0.31/0.59 thf(fact_4107_finite__UnionD,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ ( set @ A )] : % 0.31/0.59 ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) % 0.31/0.59 => ( finite_finite2 @ ( set @ A ) @ A3 ) ) ). % 0.31/0.59 % 0.31/0.59 % finite_UnionD % 0.31/0.59 thf(fact_4108_in__set__zip,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,P4: product_prod @ A @ B,Xs: list @ A,Ys: list @ B] : % 0.31/0.59 ( ( member @ ( product_prod @ A @ B ) @ P4 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) % 0.31/0.59 = ( ? [N: nat] : % 0.31/0.59 ( ( ( nth @ A @ Xs @ N ) % 0.31/0.59 = ( product_fst @ A @ B @ P4 ) ) % 0.31/0.59 & ( ( nth @ B @ Ys @ N ) % 0.31/0.59 = ( product_snd @ A @ B @ P4 ) ) % 0.31/0.59 & ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.59 & ( ord_less @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % in_set_zip % 0.31/0.59 thf(fact_4109_divides__aux__def,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( unique1627219031080169319umeral @ A ) % 0.31/0.59 => ( ( unique5940410009612947441es_aux @ A ) % 0.31/0.59 = ( ^ [Qr: product_prod @ A @ A] : % 0.31/0.59 ( ( product_snd @ A @ A @ Qr ) % 0.31/0.59 = ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % divides_aux_def % 0.31/0.59 thf(fact_4110_finite__less__Inf__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit6923001295902523014norder @ A ) % 0.31/0.59 => ! [X8: set @ A,A2: A] : % 0.31/0.59 ( ( finite_finite2 @ A @ X8 ) % 0.31/0.59 => ( ( X8 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) ) % 0.31/0.59 = ( ! [X: A] : % 0.31/0.59 ( ( member @ A @ X @ X8 ) % 0.31/0.59 => ( ord_less @ A @ A2 @ X ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % finite_less_Inf_iff % 0.31/0.59 thf(fact_4111_finite__Sup__less__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit6923001295902523014norder @ A ) % 0.31/0.59 => ! [X8: set @ A,A2: A] : % 0.31/0.59 ( ( finite_finite2 @ A @ X8 ) % 0.31/0.59 => ( ( X8 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A2 ) % 0.31/0.59 = ( ! [X: A] : % 0.31/0.59 ( ( member @ A @ X @ X8 ) % 0.31/0.59 => ( ord_less @ A @ X @ A2 ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % finite_Sup_less_iff % 0.31/0.59 thf(fact_4112_cInf__abs__ge,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ( condit6923001295902523014norder @ A ) % 0.31/0.59 & ( linordered_idom @ A ) ) % 0.31/0.59 => ! [S3: set @ A,A2: A] : % 0.31/0.59 ( ( S3 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ S3 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ X5 ) @ A2 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S3 ) ) @ A2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cInf_abs_ge % 0.31/0.59 thf(fact_4113_cSup__abs__le,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ( condit6923001295902523014norder @ A ) % 0.31/0.59 & ( linordered_idom @ A ) ) % 0.31/0.59 => ! [S3: set @ A,A2: A] : % 0.31/0.59 ( ( S3 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ S3 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ X5 ) @ A2 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S3 ) ) @ A2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cSup_abs_le % 0.31/0.59 thf(fact_4114_card__Union__le__sum__card__weak,axiom, % 0.31/0.59 ! [A: $tType,U3: set @ ( set @ A )] : % 0.31/0.59 ( ! [X5: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ X5 @ U3 ) % 0.31/0.59 => ( finite_finite2 @ A @ X5 ) ) % 0.31/0.59 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U3 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % card_Union_le_sum_card_weak % 0.31/0.59 thf(fact_4115_in__set__enumerate__eq,axiom, % 0.31/0.59 ! [A: $tType,P4: product_prod @ nat @ A,N2: nat,Xs: list @ A] : % 0.31/0.59 ( ( member @ ( product_prod @ nat @ A ) @ P4 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) ) ) % 0.31/0.59 = ( ( ord_less_eq @ nat @ N2 @ ( product_fst @ nat @ A @ P4 ) ) % 0.31/0.59 & ( ord_less @ nat @ ( product_fst @ nat @ A @ P4 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) % 0.31/0.59 & ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P4 ) @ N2 ) ) % 0.31/0.59 = ( product_snd @ nat @ A @ P4 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % in_set_enumerate_eq % 0.31/0.59 thf(fact_4116_cInf__asclose,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ( condit6923001295902523014norder @ A ) % 0.31/0.59 & ( linordered_idom @ A ) ) % 0.31/0.59 => ! [S3: set @ A,L: A,E3: A] : % 0.31/0.59 ( ( S3 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ S3 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X5 @ L ) ) @ E3 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S3 ) @ L ) ) @ E3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cInf_asclose % 0.31/0.59 thf(fact_4117_cSup__asclose,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ( condit6923001295902523014norder @ A ) % 0.31/0.59 & ( linordered_idom @ A ) ) % 0.31/0.59 => ! [S3: set @ A,L: A,E3: A] : % 0.31/0.59 ( ( S3 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ S3 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X5 @ L ) ) @ E3 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S3 ) @ L ) ) @ E3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % cSup_asclose % 0.31/0.59 thf(fact_4118_finite__subset__Union,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B12: set @ ( set @ A )] : % 0.31/0.59 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.59 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ B12 ) ) % 0.31/0.59 => ~ ! [F7: set @ ( set @ A )] : % 0.31/0.59 ( ( finite_finite2 @ ( set @ A ) @ F7 ) % 0.31/0.59 => ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F7 @ B12 ) % 0.31/0.59 => ~ ( ord_less_eq @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ F7 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % finite_subset_Union % 0.31/0.59 thf(fact_4119_ccpo__Sup__singleton,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple9053668089753744459l_ccpo @ A ) % 0.31/0.59 => ! [X3: A] : % 0.31/0.59 ( ( complete_Sup_Sup @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.59 = X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % ccpo_Sup_singleton % 0.31/0.59 thf(fact_4120_Sup__empty,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 = ( bot_bot @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_empty % 0.31/0.59 thf(fact_4121_ccSup__empty,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( counta3822494911875563373attice @ A ) % 0.31/0.59 => ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 = ( bot_bot @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % ccSup_empty % 0.31/0.59 thf(fact_4122_Inf__eq__bot__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple5582772986160207858norder @ A ) % 0.31/0.59 => ! [A3: set @ A] : % 0.31/0.59 ( ( ( complete_Inf_Inf @ A @ A3 ) % 0.31/0.59 = ( bot_bot @ A ) ) % 0.31/0.59 = ( ! [X: A] : % 0.31/0.59 ( ( ord_less @ A @ ( bot_bot @ A ) @ X ) % 0.31/0.59 => ? [Y2: A] : % 0.31/0.59 ( ( member @ A @ Y2 @ A3 ) % 0.31/0.59 & ( ord_less @ A @ Y2 @ X ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inf_eq_bot_iff % 0.31/0.59 thf(fact_4123_Sup__bot__conv_I1_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [A3: set @ A] : % 0.31/0.59 ( ( ( complete_Sup_Sup @ A @ A3 ) % 0.31/0.59 = ( bot_bot @ A ) ) % 0.31/0.59 = ( ! [X: A] : % 0.31/0.59 ( ( member @ A @ X @ A3 ) % 0.31/0.59 => ( X % 0.31/0.59 = ( bot_bot @ A ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_bot_conv(1) % 0.31/0.59 thf(fact_4124_Sup__bot__conv_I2_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [A3: set @ A] : % 0.31/0.59 ( ( ( bot_bot @ A ) % 0.31/0.59 = ( complete_Sup_Sup @ A @ A3 ) ) % 0.31/0.59 = ( ! [X: A] : % 0.31/0.59 ( ( member @ A @ X @ A3 ) % 0.31/0.59 => ( X % 0.31/0.59 = ( bot_bot @ A ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_bot_conv(2) % 0.31/0.59 thf(fact_4125_Sup__nat__empty,axiom, % 0.31/0.59 ( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) ) % 0.31/0.59 = ( zero_zero @ nat ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_nat_empty % 0.31/0.59 thf(fact_4126_Inf__nat__def1,axiom, % 0.31/0.59 ! [K5: set @ nat] : % 0.31/0.59 ( ( K5 % 0.31/0.59 != ( bot_bot @ ( set @ nat ) ) ) % 0.31/0.59 => ( member @ nat @ ( complete_Inf_Inf @ nat @ K5 ) @ K5 ) ) ). % 0.31/0.59 % 0.31/0.59 % Inf_nat_def1 % 0.31/0.59 thf(fact_4127_quotient__of__denom__pos_H,axiom, % 0.31/0.59 ! [R2: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ ( quotient_of @ R2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % quotient_of_denom_pos' % 0.31/0.59 thf(fact_4128_bezw__non__0,axiom, % 0.31/0.59 ! [Y: nat,X3: nat] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y ) % 0.31/0.59 => ( ( bezw @ X3 @ Y ) % 0.31/0.59 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Y ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % bezw_non_0 % 0.31/0.59 thf(fact_4129_bezw_Oelims,axiom, % 0.31/0.59 ! [X3: nat,Xa2: nat,Y: product_prod @ int @ int] : % 0.31/0.59 ( ( ( bezw @ X3 @ Xa2 ) % 0.31/0.59 = Y ) % 0.31/0.59 => ( ( ( Xa2 % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 => ( Y % 0.31/0.59 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ) % 0.31/0.59 & ( ( Xa2 % 0.31/0.59 != ( zero_zero @ nat ) ) % 0.31/0.59 => ( Y % 0.31/0.59 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Xa2 ) ) ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % bezw.elims % 0.31/0.59 thf(fact_4130_bezw_Osimps,axiom, % 0.31/0.59 ( bezw % 0.31/0.59 = ( ^ [X: nat,Y2: nat] : % 0.31/0.59 ( if @ ( product_prod @ int @ int ) % 0.31/0.59 @ ( Y2 % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) % 0.31/0.59 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y2 ) ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % bezw.simps % 0.31/0.59 thf(fact_4131_rat__sgn__code,axiom, % 0.31/0.59 ! [P4: rat] : % 0.31/0.59 ( ( quotient_of @ ( sgn_sgn @ rat @ P4 ) ) % 0.31/0.59 = ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P4 ) ) ) @ ( one_one @ int ) ) ) ). % 0.31/0.59 % 0.31/0.59 % rat_sgn_code % 0.31/0.59 thf(fact_4132_Sup__upper2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [U: A,A3: set @ A,V2: A] : % 0.31/0.59 ( ( member @ A @ U @ A3 ) % 0.31/0.59 => ( ( ord_less_eq @ A @ V2 @ U ) % 0.31/0.59 => ( ord_less_eq @ A @ V2 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_upper2 % 0.31/0.59 thf(fact_4133_Sup__le__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [A3: set @ A,B2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ B2 ) % 0.31/0.59 = ( ! [X: A] : % 0.31/0.59 ( ( member @ A @ X @ A3 ) % 0.31/0.59 => ( ord_less_eq @ A @ X @ B2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_le_iff % 0.31/0.59 thf(fact_4134_Sup__upper,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [X3: A,A3: set @ A] : % 0.31/0.59 ( ( member @ A @ X3 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_upper % 0.31/0.59 thf(fact_4135_Sup__least,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [A3: set @ A,Z2: A] : % 0.31/0.59 ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ A @ X5 @ Z2 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ Z2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_least % 0.31/0.59 thf(fact_4136_Sup__mono,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ! [A5: A] : % 0.31/0.59 ( ( member @ A @ A5 @ A3 ) % 0.31/0.59 => ? [X2: A] : % 0.31/0.59 ( ( member @ A @ X2 @ B3 ) % 0.31/0.59 & ( ord_less_eq @ A @ A5 @ X2 ) ) ) % 0.31/0.59 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_mono % 0.31/0.59 thf(fact_4137_Sup__eqI,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [A3: set @ A,X3: A] : % 0.31/0.59 ( ! [Y4: A] : % 0.31/0.59 ( ( member @ A @ Y4 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ A @ Y4 @ X3 ) ) % 0.31/0.59 => ( ! [Y4: A] : % 0.31/0.59 ( ! [Z4: A] : % 0.31/0.59 ( ( member @ A @ Z4 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ A @ Z4 @ Y4 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ X3 @ Y4 ) ) % 0.31/0.59 => ( ( complete_Sup_Sup @ A @ A3 ) % 0.31/0.59 = X3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_eqI % 0.31/0.59 thf(fact_4138_less__Sup__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple5582772986160207858norder @ A ) % 0.31/0.59 => ! [A2: A,S3: set @ A] : % 0.31/0.59 ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ S3 ) ) % 0.31/0.59 = ( ? [X: A] : % 0.31/0.59 ( ( member @ A @ X @ S3 ) % 0.31/0.59 & ( ord_less @ A @ A2 @ X ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % less_Sup_iff % 0.31/0.59 thf(fact_4139_Inf__greatest,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [A3: set @ A,Z2: A] : % 0.31/0.59 ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ A @ Z2 @ X5 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inf_greatest % 0.31/0.59 thf(fact_4140_le__Inf__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [B2: A,A3: set @ A] : % 0.31/0.59 ( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A3 ) ) % 0.31/0.59 = ( ! [X: A] : % 0.31/0.59 ( ( member @ A @ X @ A3 ) % 0.31/0.59 => ( ord_less_eq @ A @ B2 @ X ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % le_Inf_iff % 0.31/0.59 thf(fact_4141_Inf__lower2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [U: A,A3: set @ A,V2: A] : % 0.31/0.59 ( ( member @ A @ U @ A3 ) % 0.31/0.59 => ( ( ord_less_eq @ A @ U @ V2 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ V2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inf_lower2 % 0.31/0.59 thf(fact_4142_Inf__lower,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [X3: A,A3: set @ A] : % 0.31/0.59 ( ( member @ A @ X3 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inf_lower % 0.31/0.59 thf(fact_4143_Inf__mono,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [B3: set @ A,A3: set @ A] : % 0.31/0.59 ( ! [B5: A] : % 0.31/0.59 ( ( member @ A @ B5 @ B3 ) % 0.31/0.59 => ? [X2: A] : % 0.31/0.59 ( ( member @ A @ X2 @ A3 ) % 0.31/0.59 & ( ord_less_eq @ A @ X2 @ B5 ) ) ) % 0.31/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inf_mono % 0.31/0.59 thf(fact_4144_Inf__eqI,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [A3: set @ A,X3: A] : % 0.31/0.59 ( ! [I4: A] : % 0.31/0.59 ( ( member @ A @ I4 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ A @ X3 @ I4 ) ) % 0.31/0.59 => ( ! [Y4: A] : % 0.31/0.59 ( ! [I5: A] : % 0.31/0.59 ( ( member @ A @ I5 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ A @ Y4 @ I5 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ Y4 @ X3 ) ) % 0.31/0.59 => ( ( complete_Inf_Inf @ A @ A3 ) % 0.31/0.59 = X3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inf_eqI % 0.31/0.59 thf(fact_4145_Inf__less__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple5582772986160207858norder @ A ) % 0.31/0.59 => ! [S3: set @ A,A2: A] : % 0.31/0.59 ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S3 ) @ A2 ) % 0.31/0.59 = ( ? [X: A] : % 0.31/0.59 ( ( member @ A @ X @ S3 ) % 0.31/0.59 & ( ord_less @ A @ X @ A2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inf_less_iff % 0.31/0.59 thf(fact_4146_Union__empty__conv,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ ( set @ A )] : % 0.31/0.59 ( ( ( complete_Sup_Sup @ ( set @ A ) @ A3 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 = ( ! [X: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ X @ A3 ) % 0.31/0.59 => ( X % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Union_empty_conv % 0.31/0.59 thf(fact_4147_empty__Union__conv,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ ( set @ A )] : % 0.31/0.59 ( ( ( bot_bot @ ( set @ A ) ) % 0.31/0.59 = ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) % 0.31/0.59 = ( ! [X: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ X @ A3 ) % 0.31/0.59 => ( X % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % empty_Union_conv % 0.31/0.59 thf(fact_4148_Union__least,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ ( set @ A ),C5: set @ A] : % 0.31/0.59 ( ! [X10: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ X10 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ ( set @ A ) @ X10 @ C5 ) ) % 0.31/0.59 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ C5 ) ) ). % 0.31/0.59 % 0.31/0.59 % Union_least % 0.31/0.59 thf(fact_4149_Union__upper,axiom, % 0.31/0.59 ! [A: $tType,B3: set @ A,A3: set @ ( set @ A )] : % 0.31/0.59 ( ( member @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ ( set @ A ) @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Union_upper % 0.31/0.59 thf(fact_4150_Union__subsetI,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] : % 0.31/0.59 ( ! [X5: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ X5 @ A3 ) % 0.31/0.59 => ? [Y5: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ Y5 @ B3 ) % 0.31/0.59 & ( ord_less_eq @ ( set @ A ) @ X5 @ Y5 ) ) ) % 0.31/0.59 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Union_subsetI % 0.31/0.59 thf(fact_4151_Inter__lower,axiom, % 0.31/0.59 ! [A: $tType,B3: set @ A,A3: set @ ( set @ A )] : % 0.31/0.59 ( ( member @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ B3 ) ) ). % 0.31/0.59 % 0.31/0.59 % Inter_lower % 0.31/0.59 thf(fact_4152_Inter__greatest,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ ( set @ A ),C5: set @ A] : % 0.31/0.59 ( ! [X10: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ X10 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ ( set @ A ) @ C5 @ X10 ) ) % 0.31/0.59 => ( ord_less_eq @ ( set @ A ) @ C5 @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inter_greatest % 0.31/0.59 thf(fact_4153_Union__SetCompr__eq,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,F2: B > ( set @ A ),P: B > $o] : % 0.31/0.59 ( ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.59 @ ( collect @ ( set @ A ) % 0.31/0.59 @ ^ [Uu3: set @ A] : % 0.31/0.59 ? [X: B] : % 0.31/0.59 ( ( Uu3 % 0.31/0.59 = ( F2 @ X ) ) % 0.31/0.59 & ( P @ X ) ) ) ) % 0.31/0.59 = ( collect @ A % 0.31/0.59 @ ^ [A4: A] : % 0.31/0.59 ? [X: B] : % 0.31/0.59 ( ( P @ X ) % 0.31/0.59 & ( member @ A @ A4 @ ( F2 @ X ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Union_SetCompr_eq % 0.31/0.59 thf(fact_4154_le__Sup__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple5582772986160207858norder @ A ) % 0.31/0.59 => ! [X3: A,A3: set @ A] : % 0.31/0.59 ( ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A3 ) ) % 0.31/0.59 = ( ! [Y2: A] : % 0.31/0.59 ( ( ord_less @ A @ Y2 @ X3 ) % 0.31/0.59 => ? [X: A] : % 0.31/0.59 ( ( member @ A @ X @ A3 ) % 0.31/0.59 & ( ord_less @ A @ Y2 @ X ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % le_Sup_iff % 0.31/0.59 thf(fact_4155_Inf__le__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple5582772986160207858norder @ A ) % 0.31/0.59 => ! [A3: set @ A,X3: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ X3 ) % 0.31/0.59 = ( ! [Y2: A] : % 0.31/0.59 ( ( ord_less @ A @ X3 @ Y2 ) % 0.31/0.59 => ? [X: A] : % 0.31/0.59 ( ( member @ A @ X @ A3 ) % 0.31/0.59 & ( ord_less @ A @ X @ Y2 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inf_le_iff % 0.31/0.59 thf(fact_4156_less__eq__Sup,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [A3: set @ A,U: A] : % 0.31/0.59 ( ! [V3: A] : % 0.31/0.59 ( ( member @ A @ V3 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ A @ U @ V3 ) ) % 0.31/0.59 => ( ( A3 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % less_eq_Sup % 0.31/0.59 thf(fact_4157_Sup__subset__mono,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_subset_mono % 0.31/0.59 thf(fact_4158_Inf__less__eq,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [A3: set @ A,U: A] : % 0.31/0.59 ( ! [V3: A] : % 0.31/0.59 ( ( member @ A @ V3 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ A @ V3 @ U ) ) % 0.31/0.59 => ( ( A3 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ U ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inf_less_eq % 0.31/0.59 thf(fact_4159_Inf__superset__mono,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [B3: set @ A,A3: set @ A] : % 0.31/0.59 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inf_superset_mono % 0.31/0.59 thf(fact_4160_Union__mono,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] : % 0.31/0.59 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A3 @ B3 ) % 0.31/0.59 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Union_mono % 0.31/0.59 thf(fact_4161_Inter__anti__mono,axiom, % 0.31/0.59 ! [A: $tType,B3: set @ ( set @ A ),A3: set @ ( set @ A )] : % 0.31/0.59 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B3 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inter_anti_mono % 0.31/0.59 thf(fact_4162_Union__empty,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Union_empty % 0.31/0.59 thf(fact_4163_Inter__subset,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ ( set @ A ),B3: set @ A] : % 0.31/0.59 ( ! [X10: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ X10 @ A3 ) % 0.31/0.59 => ( ord_less_eq @ ( set @ A ) @ X10 @ B3 ) ) % 0.31/0.59 => ( ( A3 % 0.31/0.59 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 0.31/0.59 => ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ B3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inter_subset % 0.31/0.59 thf(fact_4164_Inf__le__Sup,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [A3: set @ A] : % 0.31/0.59 ( ( A3 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Inf_le_Sup % 0.31/0.59 thf(fact_4165_size__prod__simp,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( basic_BNF_size_prod @ A @ B ) % 0.31/0.59 = ( ^ [F3: A > nat,G2: B > nat,P6: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F3 @ ( product_fst @ A @ B @ P6 ) ) @ ( G2 @ ( product_snd @ A @ B @ P6 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % size_prod_simp % 0.31/0.59 thf(fact_4166_conjI__realizer,axiom, % 0.31/0.59 ! [A: $tType,B: $tType,P: A > $o,P4: A,Q: B > $o,Q4: B] : % 0.31/0.59 ( ( P @ P4 ) % 0.31/0.59 => ( ( Q @ Q4 ) % 0.31/0.59 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P4 @ Q4 ) ) ) % 0.31/0.59 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P4 @ Q4 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % conjI_realizer % 0.31/0.59 thf(fact_4167_exI__realizer,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X3: B] : % 0.31/0.59 ( ( P @ Y @ X3 ) % 0.31/0.59 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % exI_realizer % 0.31/0.59 thf(fact_4168_bezw_Opelims,axiom, % 0.31/0.59 ! [X3: nat,Xa2: nat,Y: product_prod @ int @ int] : % 0.31/0.59 ( ( ( bezw @ X3 @ Xa2 ) % 0.31/0.59 = Y ) % 0.31/0.59 => ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) % 0.31/0.59 => ~ ( ( ( ( Xa2 % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 => ( Y % 0.31/0.59 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ) % 0.31/0.59 & ( ( Xa2 % 0.31/0.59 != ( zero_zero @ nat ) ) % 0.31/0.59 => ( Y % 0.31/0.59 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Xa2 ) ) ) ) ) ) ) ) % 0.31/0.59 => ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % bezw.pelims % 0.31/0.59 thf(fact_4169_prod__decode__aux_Opelims,axiom, % 0.31/0.59 ! [X3: nat,Xa2: nat,Y: product_prod @ nat @ nat] : % 0.31/0.59 ( ( ( nat_prod_decode_aux @ X3 @ Xa2 ) % 0.31/0.59 = Y ) % 0.31/0.59 => ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) % 0.31/0.59 => ~ ( ( ( ( ord_less_eq @ nat @ Xa2 @ X3 ) % 0.31/0.59 => ( Y % 0.31/0.59 = ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X3 @ Xa2 ) ) ) ) % 0.31/0.59 & ( ~ ( ord_less_eq @ nat @ Xa2 @ X3 ) % 0.31/0.59 => ( Y % 0.31/0.59 = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) % 0.31/0.59 => ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod_decode_aux.pelims % 0.31/0.59 thf(fact_4170_normalize__def,axiom, % 0.31/0.59 ( normalize % 0.31/0.59 = ( ^ [P6: product_prod @ int @ int] : % 0.31/0.59 ( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P6 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) % 0.31/0.59 @ ( if @ ( product_prod @ int @ int ) % 0.31/0.59 @ ( ( product_snd @ int @ int @ P6 ) % 0.31/0.59 = ( zero_zero @ int ) ) % 0.31/0.59 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) % 0.31/0.59 @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % normalize_def % 0.31/0.59 thf(fact_4171_strict__mono__on__def,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( ( ord @ A ) % 0.31/0.59 & ( ord @ B ) ) % 0.31/0.59 => ( ( strict_mono_on @ A @ B ) % 0.31/0.59 = ( ^ [F3: A > B,A6: set @ A] : % 0.31/0.59 ! [R5: A,S8: A] : % 0.31/0.59 ( ( ( member @ A @ R5 @ A6 ) % 0.31/0.59 & ( member @ A @ S8 @ A6 ) % 0.31/0.59 & ( ord_less @ A @ R5 @ S8 ) ) % 0.31/0.59 => ( ord_less @ B @ ( F3 @ R5 ) @ ( F3 @ S8 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % strict_mono_on_def % 0.31/0.59 thf(fact_4172_strict__mono__onI,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( ( ord @ A ) % 0.31/0.59 & ( ord @ B ) ) % 0.31/0.59 => ! [A3: set @ A,F2: A > B] : % 0.31/0.59 ( ! [R4: A,S2: A] : % 0.31/0.59 ( ( member @ A @ R4 @ A3 ) % 0.31/0.59 => ( ( member @ A @ S2 @ A3 ) % 0.31/0.59 => ( ( ord_less @ A @ R4 @ S2 ) % 0.31/0.59 => ( ord_less @ B @ ( F2 @ R4 ) @ ( F2 @ S2 ) ) ) ) ) % 0.31/0.59 => ( strict_mono_on @ A @ B @ F2 @ A3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % strict_mono_onI % 0.31/0.59 thf(fact_4173_gcd__left__idem,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : % 0.31/0.59 ( ( gcd_gcd @ A @ A2 @ ( gcd_gcd @ A @ A2 @ B2 ) ) % 0.31/0.59 = ( gcd_gcd @ A @ A2 @ B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_left_idem % 0.31/0.59 thf(fact_4174_gcd__right__idem,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : % 0.31/0.59 ( ( gcd_gcd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ B2 ) % 0.31/0.59 = ( gcd_gcd @ A @ A2 @ B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_right_idem % 0.31/0.59 thf(fact_4175_gcd__eq__0__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : % 0.31/0.59 ( ( ( gcd_gcd @ A @ A2 @ B2 ) % 0.31/0.59 = ( zero_zero @ A ) ) % 0.31/0.59 = ( ( A2 % 0.31/0.59 = ( zero_zero @ A ) ) % 0.31/0.59 & ( B2 % 0.31/0.59 = ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_eq_0_iff % 0.31/0.59 thf(fact_4176_gcd_Obottom__right__bottom,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A] : % 0.31/0.59 ( ( gcd_gcd @ A @ A2 @ ( one_one @ A ) ) % 0.31/0.59 = ( one_one @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd.bottom_right_bottom % 0.31/0.59 thf(fact_4177_gcd_Obottom__left__bottom,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A] : % 0.31/0.59 ( ( gcd_gcd @ A @ ( one_one @ A ) @ A2 ) % 0.31/0.59 = ( one_one @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd.bottom_left_bottom % 0.31/0.59 thf(fact_4178_gcd__add1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [M: A,N2: A] : % 0.31/0.59 ( ( gcd_gcd @ A @ ( plus_plus @ A @ M @ N2 ) @ N2 ) % 0.31/0.59 = ( gcd_gcd @ A @ M @ N2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_add1 % 0.31/0.59 thf(fact_4179_gcd__add2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [M: A,N2: A] : % 0.31/0.59 ( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ M @ N2 ) ) % 0.31/0.59 = ( gcd_gcd @ A @ M @ N2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_add2 % 0.31/0.59 thf(fact_4180_gcd__neg2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : % 0.31/0.59 ( ( gcd_gcd @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) % 0.31/0.59 = ( gcd_gcd @ A @ A2 @ B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_neg2 % 0.31/0.59 thf(fact_4181_gcd__neg1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : % 0.31/0.59 ( ( gcd_gcd @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 0.31/0.59 = ( gcd_gcd @ A @ A2 @ B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_neg1 % 0.31/0.59 thf(fact_4182_gcd__exp,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiri6843258321239162965malize @ A ) % 0.31/0.59 => ! [A2: A,N2: nat,B2: A] : % 0.31/0.59 ( ( gcd_gcd @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) % 0.31/0.59 = ( power_power @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ N2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_exp % 0.31/0.59 thf(fact_4183_gcd__dvd1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ A2 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_dvd1 % 0.31/0.59 thf(fact_4184_gcd__dvd2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ B2 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_dvd2 % 0.31/0.59 thf(fact_4185_gcd__greatest__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A,C2: A] : % 0.31/0.59 ( ( dvd_dvd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) % 0.31/0.59 = ( ( dvd_dvd @ A @ A2 @ B2 ) % 0.31/0.59 & ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_greatest_iff % 0.31/0.59 thf(fact_4186_gcd__1__int,axiom, % 0.31/0.59 ! [M: int] : % 0.31/0.59 ( ( gcd_gcd @ int @ M @ ( one_one @ int ) ) % 0.31/0.59 = ( one_one @ int ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_1_int % 0.31/0.59 thf(fact_4187_gcd__neg2__int,axiom, % 0.31/0.59 ! [X3: int,Y: int] : % 0.31/0.59 ( ( gcd_gcd @ int @ X3 @ ( uminus_uminus @ int @ Y ) ) % 0.31/0.59 = ( gcd_gcd @ int @ X3 @ Y ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_neg2_int % 0.31/0.59 thf(fact_4188_gcd__neg1__int,axiom, % 0.31/0.59 ! [X3: int,Y: int] : % 0.31/0.59 ( ( gcd_gcd @ int @ ( uminus_uminus @ int @ X3 ) @ Y ) % 0.31/0.59 = ( gcd_gcd @ int @ X3 @ Y ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_neg1_int % 0.31/0.59 thf(fact_4189_abs__gcd__int,axiom, % 0.31/0.59 ! [X3: int,Y: int] : % 0.31/0.59 ( ( abs_abs @ int @ ( gcd_gcd @ int @ X3 @ Y ) ) % 0.31/0.59 = ( gcd_gcd @ int @ X3 @ Y ) ) ). % 0.31/0.59 % 0.31/0.59 % abs_gcd_int % 0.31/0.59 thf(fact_4190_gcd__abs1__int,axiom, % 0.31/0.59 ! [X3: int,Y: int] : % 0.31/0.59 ( ( gcd_gcd @ int @ ( abs_abs @ int @ X3 ) @ Y ) % 0.31/0.59 = ( gcd_gcd @ int @ X3 @ Y ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_abs1_int % 0.31/0.59 thf(fact_4191_gcd__abs2__int,axiom, % 0.31/0.59 ! [X3: int,Y: int] : % 0.31/0.59 ( ( gcd_gcd @ int @ X3 @ ( abs_abs @ int @ Y ) ) % 0.31/0.59 = ( gcd_gcd @ int @ X3 @ Y ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_abs2_int % 0.31/0.59 thf(fact_4192_gcd__neg__numeral__1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ring_gcd @ A ) % 0.31/0.59 => ! [N2: num,A2: A] : % 0.31/0.59 ( ( gcd_gcd @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ A2 ) % 0.31/0.59 = ( gcd_gcd @ A @ ( numeral_numeral @ A @ N2 ) @ A2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_neg_numeral_1 % 0.31/0.59 thf(fact_4193_gcd__neg__numeral__2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ring_gcd @ A ) % 0.31/0.59 => ! [A2: A,N2: num] : % 0.31/0.59 ( ( gcd_gcd @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) % 0.31/0.59 = ( gcd_gcd @ A @ A2 @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_neg_numeral_2 % 0.31/0.59 thf(fact_4194_is__unit__gcd__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : % 0.31/0.59 ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( one_one @ A ) ) % 0.31/0.59 = ( ( gcd_gcd @ A @ A2 @ B2 ) % 0.31/0.59 = ( one_one @ A ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % is_unit_gcd_iff % 0.31/0.59 thf(fact_4195_gcd__pos__int,axiom, % 0.31/0.59 ! [M: int,N2: int] : % 0.31/0.59 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ M @ N2 ) ) % 0.31/0.59 = ( ( M % 0.31/0.59 != ( zero_zero @ int ) ) % 0.31/0.59 | ( N2 % 0.31/0.59 != ( zero_zero @ int ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_pos_int % 0.31/0.59 thf(fact_4196_gcd__neg__numeral__1__int,axiom, % 0.31/0.59 ! [N2: num,X3: int] : % 0.31/0.59 ( ( gcd_gcd @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) @ X3 ) % 0.31/0.59 = ( gcd_gcd @ int @ ( numeral_numeral @ int @ N2 ) @ X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_neg_numeral_1_int % 0.31/0.59 thf(fact_4197_gcd__neg__numeral__2__int,axiom, % 0.31/0.59 ! [X3: int,N2: num] : % 0.31/0.59 ( ( gcd_gcd @ int @ X3 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) % 0.31/0.59 = ( gcd_gcd @ int @ X3 @ ( numeral_numeral @ int @ N2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_neg_numeral_2_int % 0.31/0.59 thf(fact_4198_gcd__0__int,axiom, % 0.31/0.59 ! [X3: int] : % 0.31/0.59 ( ( gcd_gcd @ int @ X3 @ ( zero_zero @ int ) ) % 0.31/0.59 = ( abs_abs @ int @ X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_0_int % 0.31/0.59 thf(fact_4199_gcd__0__left__int,axiom, % 0.31/0.59 ! [X3: int] : % 0.31/0.59 ( ( gcd_gcd @ int @ ( zero_zero @ int ) @ X3 ) % 0.31/0.59 = ( abs_abs @ int @ X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_0_left_int % 0.31/0.59 thf(fact_4200_gcd__proj1__if__dvd__int,axiom, % 0.31/0.59 ! [X3: int,Y: int] : % 0.31/0.59 ( ( dvd_dvd @ int @ X3 @ Y ) % 0.31/0.59 => ( ( gcd_gcd @ int @ X3 @ Y ) % 0.31/0.59 = ( abs_abs @ int @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_proj1_if_dvd_int % 0.31/0.59 thf(fact_4201_gcd__proj2__if__dvd__int,axiom, % 0.31/0.59 ! [Y: int,X3: int] : % 0.31/0.59 ( ( dvd_dvd @ int @ Y @ X3 ) % 0.31/0.59 => ( ( gcd_gcd @ int @ X3 @ Y ) % 0.31/0.59 = ( abs_abs @ int @ Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_proj2_if_dvd_int % 0.31/0.59 thf(fact_4202_gcd__idem__int,axiom, % 0.31/0.59 ! [X3: int] : % 0.31/0.59 ( ( gcd_gcd @ int @ X3 @ X3 ) % 0.31/0.59 = ( abs_abs @ int @ X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_idem_int % 0.31/0.59 thf(fact_4203_gcd__diff1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ring_gcd @ A ) % 0.31/0.59 => ! [M: A,N2: A] : % 0.31/0.59 ( ( gcd_gcd @ A @ ( minus_minus @ A @ M @ N2 ) @ N2 ) % 0.31/0.59 = ( gcd_gcd @ A @ M @ N2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_diff1 % 0.31/0.59 thf(fact_4204_gcd__diff2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ring_gcd @ A ) % 0.31/0.59 => ! [N2: A,M: A] : % 0.31/0.59 ( ( gcd_gcd @ A @ ( minus_minus @ A @ N2 @ M ) @ N2 ) % 0.31/0.59 = ( gcd_gcd @ A @ M @ N2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_diff2 % 0.31/0.59 thf(fact_4205_gcd_Oassoc,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A,C2: A] : % 0.31/0.59 ( ( gcd_gcd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ C2 ) % 0.31/0.59 = ( gcd_gcd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd.assoc % 0.31/0.59 thf(fact_4206_gcd_Ocommute,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ( ( gcd_gcd @ A ) % 0.31/0.59 = ( ^ [A4: A,B4: A] : ( gcd_gcd @ A @ B4 @ A4 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd.commute % 0.31/0.59 thf(fact_4207_gcd_Oleft__commute,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [B2: A,A2: A,C2: A] : % 0.31/0.59 ( ( gcd_gcd @ A @ B2 @ ( gcd_gcd @ A @ A2 @ C2 ) ) % 0.31/0.59 = ( gcd_gcd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd.left_commute % 0.31/0.59 thf(fact_4208_gcd__red__int,axiom, % 0.31/0.59 ( ( gcd_gcd @ int ) % 0.31/0.59 = ( ^ [X: int,Y2: int] : ( gcd_gcd @ int @ Y2 @ ( modulo_modulo @ int @ X @ Y2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_red_int % 0.31/0.59 thf(fact_4209_gcd__mono,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,C2: A,B2: A,D2: A] : % 0.31/0.59 ( ( dvd_dvd @ A @ A2 @ C2 ) % 0.31/0.59 => ( ( dvd_dvd @ A @ B2 @ D2 ) % 0.31/0.59 => ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( gcd_gcd @ A @ C2 @ D2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_mono % 0.31/0.59 thf(fact_4210_dvd__gcdD1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A,C2: A] : % 0.31/0.59 ( ( dvd_dvd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) % 0.31/0.59 => ( dvd_dvd @ A @ A2 @ B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % dvd_gcdD1 % 0.31/0.59 thf(fact_4211_dvd__gcdD2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A,C2: A] : % 0.31/0.59 ( ( dvd_dvd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) % 0.31/0.59 => ( dvd_dvd @ A @ A2 @ C2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % dvd_gcdD2 % 0.31/0.59 thf(fact_4212_gcd__dvdI1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,C2: A,B2: A] : % 0.31/0.59 ( ( dvd_dvd @ A @ A2 @ C2 ) % 0.31/0.59 => ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_dvdI1 % 0.31/0.59 thf(fact_4213_gcd__dvdI2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [B2: A,C2: A,A2: A] : % 0.31/0.59 ( ( dvd_dvd @ A @ B2 @ C2 ) % 0.31/0.59 => ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_dvdI2 % 0.31/0.59 thf(fact_4214_gcd__greatest,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [C2: A,A2: A,B2: A] : % 0.31/0.59 ( ( dvd_dvd @ A @ C2 @ A2 ) % 0.31/0.59 => ( ( dvd_dvd @ A @ C2 @ B2 ) % 0.31/0.59 => ( dvd_dvd @ A @ C2 @ ( gcd_gcd @ A @ A2 @ B2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_greatest % 0.31/0.59 thf(fact_4215_gcd__dvd__antisym,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.59 ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( gcd_gcd @ A @ C2 @ D2 ) ) % 0.31/0.59 => ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ C2 @ D2 ) @ ( gcd_gcd @ A @ A2 @ B2 ) ) % 0.31/0.59 => ( ( gcd_gcd @ A @ A2 @ B2 ) % 0.31/0.59 = ( gcd_gcd @ A @ C2 @ D2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_dvd_antisym % 0.31/0.59 thf(fact_4216_gcd__add__mult,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [M: A,K: A,N2: A] : % 0.31/0.59 ( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ ( times_times @ A @ K @ M ) @ N2 ) ) % 0.31/0.59 = ( gcd_gcd @ A @ M @ N2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_add_mult % 0.31/0.59 thf(fact_4217_gcd__dvd__prod,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A,K: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( times_times @ A @ K @ B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_dvd_prod % 0.31/0.59 thf(fact_4218_gcd__ge__0__int,axiom, % 0.31/0.59 ! [X3: int,Y: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ X3 @ Y ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_ge_0_int % 0.31/0.59 thf(fact_4219_bezout__int,axiom, % 0.31/0.59 ! [X3: int,Y: int] : % 0.31/0.59 ? [U4: int,V3: int] : % 0.31/0.59 ( ( plus_plus @ int @ ( times_times @ int @ U4 @ X3 ) @ ( times_times @ int @ V3 @ Y ) ) % 0.31/0.59 = ( gcd_gcd @ int @ X3 @ Y ) ) ). % 0.31/0.59 % 0.31/0.59 % bezout_int % 0.31/0.59 thf(fact_4220_gcd__mult__distrib__int,axiom, % 0.31/0.59 ! [K: int,M: int,N2: int] : % 0.31/0.59 ( ( times_times @ int @ ( abs_abs @ int @ K ) @ ( gcd_gcd @ int @ M @ N2 ) ) % 0.31/0.59 = ( gcd_gcd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_mult_distrib_int % 0.31/0.59 thf(fact_4221_gcd__mult__unit2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A,C2: A] : % 0.31/0.59 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.31/0.59 => ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) ) % 0.31/0.59 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_mult_unit2 % 0.31/0.59 thf(fact_4222_gcd__mult__unit1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A,C2: A] : % 0.31/0.59 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.31/0.59 => ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) % 0.31/0.59 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_mult_unit1 % 0.31/0.59 thf(fact_4223_gcd__div__unit2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A,C2: A] : % 0.31/0.59 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.31/0.59 => ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C2 @ A2 ) ) % 0.31/0.59 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_div_unit2 % 0.31/0.59 thf(fact_4224_gcd__div__unit1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semiring_gcd @ A ) % 0.31/0.59 => ! [A2: A,B2: A,C2: A] : % 0.31/0.59 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.31/0.59 => ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A2 ) @ C2 ) % 0.31/0.59 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_div_unit1 % 0.31/0.59 thf(fact_4225_gcd__le2__int,axiom, % 0.31/0.59 ! [B2: int,A2: int] : % 0.31/0.59 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.31/0.59 => ( ord_less_eq @ int @ ( gcd_gcd @ int @ A2 @ B2 ) @ B2 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_le2_int % 0.31/0.59 thf(fact_4226_gcd__le1__int,axiom, % 0.31/0.59 ! [A2: int,B2: int] : % 0.31/0.59 ( ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) % 0.31/0.59 => ( ord_less_eq @ int @ ( gcd_gcd @ int @ A2 @ B2 ) @ A2 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_le1_int % 0.31/0.59 thf(fact_4227_gcd__cases__int,axiom, % 0.31/0.59 ! [X3: int,Y: int,P: int > $o] : % 0.31/0.59 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 ) % 0.31/0.59 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y ) % 0.31/0.59 => ( P @ ( gcd_gcd @ int @ X3 @ Y ) ) ) ) % 0.31/0.59 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 ) % 0.31/0.59 => ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) ) % 0.31/0.59 => ( P @ ( gcd_gcd @ int @ X3 @ ( uminus_uminus @ int @ Y ) ) ) ) ) % 0.31/0.59 => ( ( ( ord_less_eq @ int @ X3 @ ( zero_zero @ int ) ) % 0.31/0.59 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y ) % 0.31/0.59 => ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X3 ) @ Y ) ) ) ) % 0.31/0.59 => ( ( ( ord_less_eq @ int @ X3 @ ( zero_zero @ int ) ) % 0.31/0.59 => ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) ) % 0.31/0.59 => ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X3 ) @ ( uminus_uminus @ int @ Y ) ) ) ) ) % 0.31/0.59 => ( P @ ( gcd_gcd @ int @ X3 @ Y ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_cases_int % 0.31/0.59 thf(fact_4228_gcd__unique__int,axiom, % 0.31/0.59 ! [D2: int,A2: int,B2: int] : % 0.31/0.59 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 ) % 0.31/0.59 & ( dvd_dvd @ int @ D2 @ A2 ) % 0.31/0.59 & ( dvd_dvd @ int @ D2 @ B2 ) % 0.31/0.59 & ! [E4: int] : % 0.31/0.59 ( ( ( dvd_dvd @ int @ E4 @ A2 ) % 0.31/0.59 & ( dvd_dvd @ int @ E4 @ B2 ) ) % 0.31/0.59 => ( dvd_dvd @ int @ E4 @ D2 ) ) ) % 0.31/0.59 = ( D2 % 0.31/0.59 = ( gcd_gcd @ int @ A2 @ B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_unique_int % 0.31/0.59 thf(fact_4229_gcd__non__0__int,axiom, % 0.31/0.59 ! [Y: int,X3: int] : % 0.31/0.59 ( ( ord_less @ int @ ( zero_zero @ int ) @ Y ) % 0.31/0.59 => ( ( gcd_gcd @ int @ X3 @ Y ) % 0.31/0.59 = ( gcd_gcd @ int @ Y @ ( modulo_modulo @ int @ X3 @ Y ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_non_0_int % 0.31/0.59 thf(fact_4230_gcd__code__int,axiom, % 0.31/0.59 ( ( gcd_gcd @ int ) % 0.31/0.59 = ( ^ [K3: int,L2: int] : % 0.31/0.59 ( abs_abs @ int % 0.31/0.59 @ ( if @ int % 0.31/0.59 @ ( L2 % 0.31/0.59 = ( zero_zero @ int ) ) % 0.31/0.59 @ K3 % 0.31/0.59 @ ( gcd_gcd @ int @ L2 @ ( modulo_modulo @ int @ ( abs_abs @ int @ K3 ) @ ( abs_abs @ int @ L2 ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_code_int % 0.31/0.59 thf(fact_4231_strict__mono__on__leD,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( ( linorder @ A ) % 0.31/0.59 & ( preorder @ B ) ) % 0.31/0.59 => ! [F2: A > B,A3: set @ A,X3: A,Y: A] : % 0.31/0.59 ( ( strict_mono_on @ A @ B @ F2 @ A3 ) % 0.31/0.59 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.59 => ( ( member @ A @ Y @ A3 ) % 0.31/0.59 => ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.59 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % strict_mono_on_leD % 0.31/0.59 thf(fact_4232_strict__mono__onD,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( ( ord @ A ) % 0.31/0.59 & ( ord @ B ) ) % 0.31/0.59 => ! [F2: A > B,A3: set @ A,R2: A,S: A] : % 0.31/0.59 ( ( strict_mono_on @ A @ B @ F2 @ A3 ) % 0.31/0.59 => ( ( member @ A @ R2 @ A3 ) % 0.31/0.59 => ( ( member @ A @ S @ A3 ) % 0.31/0.59 => ( ( ord_less @ A @ R2 @ S ) % 0.31/0.59 => ( ord_less @ B @ ( F2 @ R2 ) @ ( F2 @ S ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % strict_mono_onD % 0.31/0.59 thf(fact_4233_card__partition,axiom, % 0.31/0.59 ! [A: $tType,C5: set @ ( set @ A ),K: nat] : % 0.31/0.59 ( ( finite_finite2 @ ( set @ A ) @ C5 ) % 0.31/0.59 => ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) % 0.31/0.59 => ( ! [C3: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ C3 @ C5 ) % 0.31/0.59 => ( ( finite_card @ A @ C3 ) % 0.31/0.59 = K ) ) % 0.31/0.59 => ( ! [C1: set @ A,C22: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ C1 @ C5 ) % 0.31/0.59 => ( ( member @ ( set @ A ) @ C22 @ C5 ) % 0.31/0.59 => ( ( C1 != C22 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ C1 @ C22 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ) ) % 0.31/0.59 => ( ( times_times @ nat @ K @ ( finite_card @ ( set @ A ) @ C5 ) ) % 0.31/0.59 = ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % card_partition % 0.31/0.59 thf(fact_4234_or__numerals_I4_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_un5681908812861735899ations @ A ) % 0.31/0.59 => ! [X3: num,Y: num] : % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) % 0.31/0.59 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % or_numerals(4) % 0.31/0.59 thf(fact_4235_or__numerals_I6_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_un5681908812861735899ations @ A ) % 0.31/0.59 => ! [X3: num,Y: num] : % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) ) % 0.31/0.59 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % or_numerals(6) % 0.31/0.59 thf(fact_4236_or__numerals_I7_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_un5681908812861735899ations @ A ) % 0.31/0.59 => ! [X3: num,Y: num] : % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) % 0.31/0.59 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % or_numerals(7) % 0.31/0.59 thf(fact_4237_gcd__nat_Oeq__neutr__iff,axiom, % 0.31/0.59 ! [A2: nat,B2: nat] : % 0.31/0.59 ( ( ( gcd_gcd @ nat @ A2 @ B2 ) % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 = ( ( A2 % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 & ( B2 % 0.31/0.59 = ( zero_zero @ nat ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.eq_neutr_iff % 0.31/0.59 thf(fact_4238_gcd__nat_Oleft__neutral,axiom, % 0.31/0.59 ! [A2: nat] : % 0.31/0.59 ( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ A2 ) % 0.31/0.59 = A2 ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.left_neutral % 0.31/0.59 thf(fact_4239_gcd__nat_Oneutr__eq__iff,axiom, % 0.31/0.59 ! [A2: nat,B2: nat] : % 0.31/0.59 ( ( ( zero_zero @ nat ) % 0.31/0.59 = ( gcd_gcd @ nat @ A2 @ B2 ) ) % 0.31/0.59 = ( ( A2 % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 & ( B2 % 0.31/0.59 = ( zero_zero @ nat ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.neutr_eq_iff % 0.31/0.59 thf(fact_4240_gcd__nat_Oright__neutral,axiom, % 0.31/0.59 ! [A2: nat] : % 0.31/0.59 ( ( gcd_gcd @ nat @ A2 @ ( zero_zero @ nat ) ) % 0.31/0.59 = A2 ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.right_neutral % 0.31/0.59 thf(fact_4241_gcd__0__nat,axiom, % 0.31/0.59 ! [X3: nat] : % 0.31/0.59 ( ( gcd_gcd @ nat @ X3 @ ( zero_zero @ nat ) ) % 0.31/0.59 = X3 ) ). % 0.31/0.59 % 0.31/0.59 % gcd_0_nat % 0.31/0.59 thf(fact_4242_gcd__0__left__nat,axiom, % 0.31/0.59 ! [X3: nat] : % 0.31/0.59 ( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ X3 ) % 0.31/0.59 = X3 ) ). % 0.31/0.59 % 0.31/0.59 % gcd_0_left_nat % 0.31/0.59 thf(fact_4243_Int__iff,axiom, % 0.31/0.59 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.59 = ( ( member @ A @ C2 @ A3 ) % 0.31/0.59 & ( member @ A @ C2 @ B3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_iff % 0.31/0.59 thf(fact_4244_IntI,axiom, % 0.31/0.59 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( member @ A @ C2 @ A3 ) % 0.31/0.59 => ( ( member @ A @ C2 @ B3 ) % 0.31/0.59 => ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % IntI % 0.31/0.59 thf(fact_4245_gcd__1__nat,axiom, % 0.31/0.59 ! [M: nat] : % 0.31/0.59 ( ( gcd_gcd @ nat @ M @ ( one_one @ nat ) ) % 0.31/0.59 = ( one_one @ nat ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_1_nat % 0.31/0.59 thf(fact_4246_gcd__proj2__if__dvd__nat,axiom, % 0.31/0.59 ! [Y: nat,X3: nat] : % 0.31/0.59 ( ( dvd_dvd @ nat @ Y @ X3 ) % 0.31/0.59 => ( ( gcd_gcd @ nat @ X3 @ Y ) % 0.31/0.59 = Y ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_proj2_if_dvd_nat % 0.31/0.59 thf(fact_4247_gcd__proj1__if__dvd__nat,axiom, % 0.31/0.59 ! [X3: nat,Y: nat] : % 0.31/0.59 ( ( dvd_dvd @ nat @ X3 @ Y ) % 0.31/0.59 => ( ( gcd_gcd @ nat @ X3 @ Y ) % 0.31/0.59 = X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_proj1_if_dvd_nat % 0.31/0.59 thf(fact_4248_gcd__nat_Obounded__iff,axiom, % 0.31/0.59 ! [A2: nat,B2: nat,C2: nat] : % 0.31/0.59 ( ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) ) % 0.31/0.59 = ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.31/0.59 & ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.bounded_iff % 0.31/0.59 thf(fact_4249_gcd__nat_Oabsorb2,axiom, % 0.31/0.59 ! [B2: nat,A2: nat] : % 0.31/0.59 ( ( dvd_dvd @ nat @ B2 @ A2 ) % 0.31/0.59 => ( ( gcd_gcd @ nat @ A2 @ B2 ) % 0.31/0.59 = B2 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.absorb2 % 0.31/0.59 thf(fact_4250_gcd__nat_Oabsorb1,axiom, % 0.31/0.59 ! [A2: nat,B2: nat] : % 0.31/0.59 ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.31/0.59 => ( ( gcd_gcd @ nat @ A2 @ B2 ) % 0.31/0.59 = A2 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.absorb1 % 0.31/0.59 thf(fact_4251_gcd__Suc__0,axiom, % 0.31/0.59 ! [M: nat] : % 0.31/0.59 ( ( gcd_gcd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) % 0.31/0.59 = ( suc @ ( zero_zero @ nat ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_Suc_0 % 0.31/0.59 thf(fact_4252_gcd__pos__nat,axiom, % 0.31/0.59 ! [M: nat,N2: nat] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_gcd @ nat @ M @ N2 ) ) % 0.31/0.59 = ( ( M % 0.31/0.59 != ( zero_zero @ nat ) ) % 0.31/0.59 | ( N2 % 0.31/0.59 != ( zero_zero @ nat ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_pos_nat % 0.31/0.59 thf(fact_4253_boolean__algebra_Oconj__zero__left,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.59 => ! [X3: A] : % 0.31/0.59 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X3 ) % 0.31/0.59 = ( bot_bot @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % boolean_algebra.conj_zero_left % 0.31/0.59 thf(fact_4254_boolean__algebra_Oconj__zero__right,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.59 => ! [X3: A] : % 0.31/0.59 ( ( inf_inf @ A @ X3 @ ( bot_bot @ A ) ) % 0.31/0.59 = ( bot_bot @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % boolean_algebra.conj_zero_right % 0.31/0.59 thf(fact_4255_finite__Int,axiom, % 0.31/0.59 ! [A: $tType,F5: set @ A,G6: set @ A] : % 0.31/0.59 ( ( ( finite_finite2 @ A @ F5 ) % 0.31/0.59 | ( finite_finite2 @ A @ G6 ) ) % 0.31/0.59 => ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F5 @ G6 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % finite_Int % 0.31/0.59 thf(fact_4256_Int__subset__iff,axiom, % 0.31/0.59 ! [A: $tType,C5: set @ A,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( ord_less_eq @ ( set @ A ) @ C5 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.59 = ( ( ord_less_eq @ ( set @ A ) @ C5 @ A3 ) % 0.31/0.59 & ( ord_less_eq @ ( set @ A ) @ C5 @ B3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_subset_iff % 0.31/0.59 thf(fact_4257_Int__insert__left__if0,axiom, % 0.31/0.59 ! [A: $tType,A2: A,C5: set @ A,B3: set @ A] : % 0.31/0.59 ( ~ ( member @ A @ A2 @ C5 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C5 ) % 0.31/0.59 = ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_insert_left_if0 % 0.31/0.59 thf(fact_4258_Int__insert__left__if1,axiom, % 0.31/0.59 ! [A: $tType,A2: A,C5: set @ A,B3: set @ A] : % 0.31/0.59 ( ( member @ A @ A2 @ C5 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C5 ) % 0.31/0.59 = ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_insert_left_if1 % 0.31/0.59 thf(fact_4259_insert__inter__insert,axiom, % 0.31/0.59 ! [A: $tType,A2: A,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ ( insert @ A @ A2 @ B3 ) ) % 0.31/0.59 = ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % insert_inter_insert % 0.31/0.59 thf(fact_4260_Int__insert__right__if0,axiom, % 0.31/0.59 ! [A: $tType,A2: A,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ~ ( member @ A @ A2 @ A3 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) % 0.31/0.59 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_insert_right_if0 % 0.31/0.59 thf(fact_4261_Int__insert__right__if1,axiom, % 0.31/0.59 ! [A: $tType,A2: A,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( member @ A @ A2 @ A3 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) % 0.31/0.59 = ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_insert_right_if1 % 0.31/0.59 thf(fact_4262_or_Oright__neutral,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [A2: A] : % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( zero_zero @ A ) ) % 0.31/0.59 = A2 ) ) ). % 0.31/0.59 % 0.31/0.59 % or.right_neutral % 0.31/0.59 thf(fact_4263_or_Oleft__neutral,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [A2: A] : % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.59 = A2 ) ) ). % 0.31/0.59 % 0.31/0.59 % or.left_neutral % 0.31/0.59 thf(fact_4264_or__negative__int__iff,axiom, % 0.31/0.59 ! [K: int,L: int] : % 0.31/0.59 ( ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) @ ( zero_zero @ int ) ) % 0.31/0.59 = ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 0.31/0.59 | ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % or_negative_int_iff % 0.31/0.59 thf(fact_4265_Pow__Int__eq,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( pow2 @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.59 = ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Pow_Int_eq % 0.31/0.59 thf(fact_4266_gcd__int__int__eq,axiom, % 0.31/0.59 ! [M: nat,N2: nat] : % 0.31/0.59 ( ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N2 ) ) % 0.31/0.59 = ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ M @ N2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_int_int_eq % 0.31/0.59 thf(fact_4267_boolean__algebra_Oconj__cancel__right,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.59 => ! [X3: A] : % 0.31/0.59 ( ( inf_inf @ A @ X3 @ ( uminus_uminus @ A @ X3 ) ) % 0.31/0.59 = ( bot_bot @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % boolean_algebra.conj_cancel_right % 0.31/0.59 thf(fact_4268_boolean__algebra_Oconj__cancel__left,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.59 => ! [X3: A] : % 0.31/0.59 ( ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ X3 ) % 0.31/0.59 = ( bot_bot @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % boolean_algebra.conj_cancel_left % 0.31/0.59 thf(fact_4269_inf__compl__bot__right,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.59 => ! [X3: A,Y: A] : % 0.31/0.59 ( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y @ ( uminus_uminus @ A @ X3 ) ) ) % 0.31/0.59 = ( bot_bot @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_compl_bot_right % 0.31/0.59 thf(fact_4270_inf__compl__bot__left2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.59 => ! [X3: A,Y: A] : % 0.31/0.59 ( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) ) % 0.31/0.59 = ( bot_bot @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_compl_bot_left2 % 0.31/0.59 thf(fact_4271_inf__compl__bot__left1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.59 => ! [X3: A,Y: A] : % 0.31/0.59 ( ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ ( inf_inf @ A @ X3 @ Y ) ) % 0.31/0.59 = ( bot_bot @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_compl_bot_left1 % 0.31/0.59 thf(fact_4272_insert__disjoint_I1_J,axiom, % 0.31/0.59 ! [A: $tType,A2: A,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ B3 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 = ( ~ ( member @ A @ A2 @ B3 ) % 0.31/0.59 & ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % insert_disjoint(1) % 0.31/0.59 thf(fact_4273_insert__disjoint_I2_J,axiom, % 0.31/0.59 ! [A: $tType,A2: A,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( ( bot_bot @ ( set @ A ) ) % 0.31/0.59 = ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ B3 ) ) % 0.31/0.59 = ( ~ ( member @ A @ A2 @ B3 ) % 0.31/0.59 & ( ( bot_bot @ ( set @ A ) ) % 0.31/0.59 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % insert_disjoint(2) % 0.31/0.59 thf(fact_4274_disjoint__insert_I1_J,axiom, % 0.31/0.59 ! [A: $tType,B3: set @ A,A2: A,A3: set @ A] : % 0.31/0.59 ( ( ( inf_inf @ ( set @ A ) @ B3 @ ( insert @ A @ A2 @ A3 ) ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 = ( ~ ( member @ A @ A2 @ B3 ) % 0.31/0.59 & ( ( inf_inf @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % disjoint_insert(1) % 0.31/0.59 thf(fact_4275_disjoint__insert_I2_J,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B2: A,B3: set @ A] : % 0.31/0.59 ( ( ( bot_bot @ ( set @ A ) ) % 0.31/0.59 = ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B3 ) ) ) % 0.31/0.59 = ( ~ ( member @ A @ B2 @ A3 ) % 0.31/0.59 & ( ( bot_bot @ ( set @ A ) ) % 0.31/0.59 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % disjoint_insert(2) % 0.31/0.59 thf(fact_4276_Diff__disjoint,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Diff_disjoint % 0.31/0.59 thf(fact_4277_Compl__disjoint2,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ A3 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Compl_disjoint2 % 0.31/0.59 thf(fact_4278_Compl__disjoint,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Compl_disjoint % 0.31/0.59 thf(fact_4279_Diff__Compl,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( minus_minus @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) % 0.31/0.59 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ). % 0.31/0.59 % 0.31/0.59 % Diff_Compl % 0.31/0.59 thf(fact_4280_or__nat__numerals_I4_J,axiom, % 0.31/0.59 ! [X3: num] : % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) % 0.31/0.59 = ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % or_nat_numerals(4) % 0.31/0.59 thf(fact_4281_or__nat__numerals_I2_J,axiom, % 0.31/0.59 ! [Y: num] : % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) % 0.31/0.59 = ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % or_nat_numerals(2) % 0.31/0.59 thf(fact_4282_gcd__nat__abs__right__eq,axiom, % 0.31/0.59 ! [N2: nat,K: int] : % 0.31/0.59 ( ( gcd_gcd @ nat @ N2 @ ( nat2 @ ( abs_abs @ int @ K ) ) ) % 0.31/0.59 = ( nat2 @ ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat_abs_right_eq % 0.31/0.59 thf(fact_4283_gcd__nat__abs__left__eq,axiom, % 0.31/0.59 ! [K: int,N2: nat] : % 0.31/0.59 ( ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N2 ) % 0.31/0.59 = ( nat2 @ ( gcd_gcd @ int @ K @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat_abs_left_eq % 0.31/0.59 thf(fact_4284_or__nat__numerals_I3_J,axiom, % 0.31/0.59 ! [X3: num] : % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) % 0.31/0.59 = ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % or_nat_numerals(3) % 0.31/0.59 thf(fact_4285_or__nat__numerals_I1_J,axiom, % 0.31/0.59 ! [Y: num] : % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) ) % 0.31/0.59 = ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % or_nat_numerals(1) % 0.31/0.59 thf(fact_4286_sum__of__bool__mult__eq,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( semiring_1 @ A ) % 0.31/0.59 => ! [A3: set @ B,P: B > $o,F2: B > A] : % 0.31/0.59 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.59 => ( ( groups7311177749621191930dd_sum @ B @ A % 0.31/0.59 @ ^ [X: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X ) ) @ ( F2 @ X ) ) % 0.31/0.59 @ A3 ) % 0.31/0.59 = ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum_of_bool_mult_eq % 0.31/0.59 thf(fact_4287_sum__mult__of__bool__eq,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( semiring_1 @ A ) % 0.31/0.59 => ! [A3: set @ B,F2: B > A,P: B > $o] : % 0.31/0.59 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.59 => ( ( groups7311177749621191930dd_sum @ B @ A % 0.31/0.59 @ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ ( zero_neq_one_of_bool @ A @ ( P @ X ) ) ) % 0.31/0.59 @ A3 ) % 0.31/0.59 = ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum_mult_of_bool_eq % 0.31/0.59 thf(fact_4288_sum__of__bool__eq,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( semiring_1 @ A ) % 0.31/0.59 => ! [A3: set @ B,P: B > $o] : % 0.31/0.59 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.59 => ( ( finite_finite2 @ B @ A3 ) % 0.31/0.59 => ( ( groups7311177749621191930dd_sum @ B @ A % 0.31/0.59 @ ^ [X: B] : ( zero_neq_one_of_bool @ A @ ( P @ X ) ) % 0.31/0.59 @ A3 ) % 0.31/0.59 = ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum_of_bool_eq % 0.31/0.59 thf(fact_4289_gcd__mult__distrib__nat,axiom, % 0.31/0.59 ! [K: nat,M: nat,N2: nat] : % 0.31/0.59 ( ( times_times @ nat @ K @ ( gcd_gcd @ nat @ M @ N2 ) ) % 0.31/0.59 = ( gcd_gcd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_mult_distrib_nat % 0.31/0.59 thf(fact_4290_gcd__red__nat,axiom, % 0.31/0.59 ( ( gcd_gcd @ nat ) % 0.31/0.59 = ( ^ [X: nat,Y2: nat] : ( gcd_gcd @ nat @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_red_nat % 0.31/0.59 thf(fact_4291_gcd__unique__nat,axiom, % 0.31/0.59 ! [D2: nat,A2: nat,B2: nat] : % 0.31/0.59 ( ( ( dvd_dvd @ nat @ D2 @ A2 ) % 0.31/0.59 & ( dvd_dvd @ nat @ D2 @ B2 ) % 0.31/0.59 & ! [E4: nat] : % 0.31/0.59 ( ( ( dvd_dvd @ nat @ E4 @ A2 ) % 0.31/0.59 & ( dvd_dvd @ nat @ E4 @ B2 ) ) % 0.31/0.59 => ( dvd_dvd @ nat @ E4 @ D2 ) ) ) % 0.31/0.59 = ( D2 % 0.31/0.59 = ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_unique_nat % 0.31/0.59 thf(fact_4292_gcd__nat_Ostrict__coboundedI2,axiom, % 0.31/0.59 ! [B2: nat,C2: nat,A2: nat] : % 0.31/0.59 ( ( ( dvd_dvd @ nat @ B2 @ C2 ) % 0.31/0.59 & ( B2 != C2 ) ) % 0.31/0.59 => ( ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 ) % 0.31/0.59 & ( ( gcd_gcd @ nat @ A2 @ B2 ) % 0.31/0.59 != C2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.strict_coboundedI2 % 0.31/0.59 thf(fact_4293_gcd__nat_Ostrict__coboundedI1,axiom, % 0.31/0.59 ! [A2: nat,C2: nat,B2: nat] : % 0.31/0.59 ( ( ( dvd_dvd @ nat @ A2 @ C2 ) % 0.31/0.59 & ( A2 != C2 ) ) % 0.31/0.59 => ( ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 ) % 0.31/0.59 & ( ( gcd_gcd @ nat @ A2 @ B2 ) % 0.31/0.59 != C2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.strict_coboundedI1 % 0.31/0.59 thf(fact_4294_gcd__nat_Ostrict__order__iff,axiom, % 0.31/0.59 ! [A2: nat,B2: nat] : % 0.31/0.59 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.31/0.59 & ( A2 != B2 ) ) % 0.31/0.59 = ( ( A2 % 0.31/0.59 = ( gcd_gcd @ nat @ A2 @ B2 ) ) % 0.31/0.59 & ( A2 != B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.strict_order_iff % 0.31/0.59 thf(fact_4295_gcd__nat_Ostrict__boundedE,axiom, % 0.31/0.59 ! [A2: nat,B2: nat,C2: nat] : % 0.31/0.59 ( ( ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) ) % 0.31/0.59 & ( A2 % 0.31/0.59 != ( gcd_gcd @ nat @ B2 @ C2 ) ) ) % 0.31/0.59 => ~ ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.31/0.59 & ( A2 != B2 ) ) % 0.31/0.59 => ~ ( ( dvd_dvd @ nat @ A2 @ C2 ) % 0.31/0.59 & ( A2 != C2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.strict_boundedE % 0.31/0.59 thf(fact_4296_gcd__nat_OcoboundedI2,axiom, % 0.31/0.59 ! [B2: nat,C2: nat,A2: nat] : % 0.31/0.59 ( ( dvd_dvd @ nat @ B2 @ C2 ) % 0.31/0.59 => ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.coboundedI2 % 0.31/0.59 thf(fact_4297_gcd__nat_OcoboundedI1,axiom, % 0.31/0.59 ! [A2: nat,C2: nat,B2: nat] : % 0.31/0.59 ( ( dvd_dvd @ nat @ A2 @ C2 ) % 0.31/0.59 => ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.coboundedI1 % 0.31/0.59 thf(fact_4298_gcd__nat_Oabsorb__iff2,axiom, % 0.31/0.59 ( ( dvd_dvd @ nat ) % 0.31/0.59 = ( ^ [B4: nat,A4: nat] : % 0.31/0.59 ( ( gcd_gcd @ nat @ A4 @ B4 ) % 0.31/0.59 = B4 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.absorb_iff2 % 0.31/0.59 thf(fact_4299_gcd__nat_Oabsorb__iff1,axiom, % 0.31/0.59 ( ( dvd_dvd @ nat ) % 0.31/0.59 = ( ^ [A4: nat,B4: nat] : % 0.31/0.59 ( ( gcd_gcd @ nat @ A4 @ B4 ) % 0.31/0.59 = A4 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.absorb_iff1 % 0.31/0.59 thf(fact_4300_gcd__nat_Ocobounded2,axiom, % 0.31/0.59 ! [A2: nat,B2: nat] : ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.cobounded2 % 0.31/0.59 thf(fact_4301_gcd__nat_Ocobounded1,axiom, % 0.31/0.59 ! [A2: nat,B2: nat] : ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ A2 ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.cobounded1 % 0.31/0.59 thf(fact_4302_gcd__nat_Oorder__iff,axiom, % 0.31/0.59 ( ( dvd_dvd @ nat ) % 0.31/0.59 = ( ^ [A4: nat,B4: nat] : % 0.31/0.59 ( A4 % 0.31/0.59 = ( gcd_gcd @ nat @ A4 @ B4 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.order_iff % 0.31/0.59 thf(fact_4303_gcd__nat_OboundedI,axiom, % 0.31/0.59 ! [A2: nat,B2: nat,C2: nat] : % 0.31/0.59 ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.31/0.59 => ( ( dvd_dvd @ nat @ A2 @ C2 ) % 0.31/0.59 => ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.boundedI % 0.31/0.59 thf(fact_4304_gcd__nat_OboundedE,axiom, % 0.31/0.59 ! [A2: nat,B2: nat,C2: nat] : % 0.31/0.59 ( ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) ) % 0.31/0.59 => ~ ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.31/0.59 => ~ ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.boundedE % 0.31/0.59 thf(fact_4305_gcd__nat_Oabsorb4,axiom, % 0.31/0.59 ! [B2: nat,A2: nat] : % 0.31/0.59 ( ( ( dvd_dvd @ nat @ B2 @ A2 ) % 0.31/0.59 & ( B2 != A2 ) ) % 0.31/0.59 => ( ( gcd_gcd @ nat @ A2 @ B2 ) % 0.31/0.59 = B2 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.absorb4 % 0.31/0.59 thf(fact_4306_gcd__nat_Oabsorb3,axiom, % 0.31/0.59 ! [A2: nat,B2: nat] : % 0.31/0.59 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.31/0.59 & ( A2 != B2 ) ) % 0.31/0.59 => ( ( gcd_gcd @ nat @ A2 @ B2 ) % 0.31/0.59 = A2 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.absorb3 % 0.31/0.59 thf(fact_4307_gcd__nat_OorderI,axiom, % 0.31/0.59 ! [A2: nat,B2: nat] : % 0.31/0.59 ( ( A2 % 0.31/0.59 = ( gcd_gcd @ nat @ A2 @ B2 ) ) % 0.31/0.59 => ( dvd_dvd @ nat @ A2 @ B2 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.orderI % 0.31/0.59 thf(fact_4308_gcd__nat_OorderE,axiom, % 0.31/0.59 ! [A2: nat,B2: nat] : % 0.31/0.59 ( ( dvd_dvd @ nat @ A2 @ B2 ) % 0.31/0.59 => ( A2 % 0.31/0.59 = ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.orderE % 0.31/0.59 thf(fact_4309_gcd__nat_Omono,axiom, % 0.31/0.59 ! [A2: nat,C2: nat,B2: nat,D2: nat] : % 0.31/0.59 ( ( dvd_dvd @ nat @ A2 @ C2 ) % 0.31/0.59 => ( ( dvd_dvd @ nat @ B2 @ D2 ) % 0.31/0.59 => ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ ( gcd_gcd @ nat @ C2 @ D2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.mono % 0.31/0.59 thf(fact_4310_disjoint__iff__not__equal,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 = ( ! [X: A] : % 0.31/0.59 ( ( member @ A @ X @ A3 ) % 0.31/0.59 => ! [Y2: A] : % 0.31/0.59 ( ( member @ A @ Y2 @ B3 ) % 0.31/0.59 => ( X != Y2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % disjoint_iff_not_equal % 0.31/0.59 thf(fact_4311_Int__empty__right,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_empty_right % 0.31/0.59 thf(fact_4312_Int__empty__left,axiom, % 0.31/0.59 ! [A: $tType,B3: set @ A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_empty_left % 0.31/0.59 thf(fact_4313_disjoint__iff,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 = ( ! [X: A] : % 0.31/0.59 ( ( member @ A @ X @ A3 ) % 0.31/0.59 => ~ ( member @ A @ X @ B3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % disjoint_iff % 0.31/0.59 thf(fact_4314_Int__emptyI,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ A3 ) % 0.31/0.59 => ~ ( member @ A @ X5 @ B3 ) ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_emptyI % 0.31/0.59 thf(fact_4315_Int__mono,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,C5: set @ A,B3: set @ A,D6: set @ A] : % 0.31/0.59 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) % 0.31/0.59 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ D6 ) % 0.31/0.59 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( inf_inf @ ( set @ A ) @ C5 @ D6 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_mono % 0.31/0.59 thf(fact_4316_Int__lower1,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ A3 ) ). % 0.31/0.59 % 0.31/0.59 % Int_lower1 % 0.31/0.59 thf(fact_4317_Int__lower2,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ B3 ) ). % 0.31/0.59 % 0.31/0.59 % Int_lower2 % 0.31/0.59 thf(fact_4318_Int__absorb1,axiom, % 0.31/0.59 ! [A: $tType,B3: set @ A,A3: set @ A] : % 0.31/0.59 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.59 = B3 ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_absorb1 % 0.31/0.59 thf(fact_4319_Int__absorb2,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.59 = A3 ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_absorb2 % 0.31/0.59 thf(fact_4320_Int__greatest,axiom, % 0.31/0.59 ! [A: $tType,C5: set @ A,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( ord_less_eq @ ( set @ A ) @ C5 @ A3 ) % 0.31/0.59 => ( ( ord_less_eq @ ( set @ A ) @ C5 @ B3 ) % 0.31/0.59 => ( ord_less_eq @ ( set @ A ) @ C5 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_greatest % 0.31/0.59 thf(fact_4321_Int__Collect__mono,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o,Q: A > $o] : % 0.31/0.59 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.59 => ( ! [X5: A] : % 0.31/0.59 ( ( member @ A @ X5 @ A3 ) % 0.31/0.59 => ( ( P @ X5 ) % 0.31/0.59 => ( Q @ X5 ) ) ) % 0.31/0.59 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B3 @ ( collect @ A @ Q ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_Collect_mono % 0.31/0.59 thf(fact_4322_Collect__conj__eq,axiom, % 0.31/0.59 ! [A: $tType,P: A > $o,Q: A > $o] : % 0.31/0.59 ( ( collect @ A % 0.31/0.59 @ ^ [X: A] : % 0.31/0.59 ( ( P @ X ) % 0.31/0.59 & ( Q @ X ) ) ) % 0.31/0.59 = ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Collect_conj_eq % 0.31/0.59 thf(fact_4323_Int__Collect,axiom, % 0.31/0.59 ! [A: $tType,X3: A,A3: set @ A,P: A > $o] : % 0.31/0.59 ( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) ) % 0.31/0.59 = ( ( member @ A @ X3 @ A3 ) % 0.31/0.59 & ( P @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_Collect % 0.31/0.59 thf(fact_4324_Int__def,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) ) % 0.31/0.59 = ( ^ [A6: set @ A,B6: set @ A] : % 0.31/0.59 ( collect @ A % 0.31/0.59 @ ^ [X: A] : % 0.31/0.59 ( ( member @ A @ X @ A6 ) % 0.31/0.59 & ( member @ A @ X @ B6 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_def % 0.31/0.59 thf(fact_4325_or__eq__0__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : % 0.31/0.59 ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) % 0.31/0.59 = ( zero_zero @ A ) ) % 0.31/0.59 = ( ( A2 % 0.31/0.59 = ( zero_zero @ A ) ) % 0.31/0.59 & ( B2 % 0.31/0.59 = ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % or_eq_0_iff % 0.31/0.59 thf(fact_4326_bit_Odisj__zero__right,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.59 => ! [X3: A] : % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ A @ X3 @ ( zero_zero @ A ) ) % 0.31/0.59 = X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % bit.disj_zero_right % 0.31/0.59 thf(fact_4327_Int__left__commute,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) % 0.31/0.59 = ( inf_inf @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ A ) @ A3 @ C5 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_left_commute % 0.31/0.59 thf(fact_4328_Int__left__absorb,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.59 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_left_absorb % 0.31/0.59 thf(fact_4329_Int__commute,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) ) % 0.31/0.59 = ( ^ [A6: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ B6 @ A6 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_commute % 0.31/0.59 thf(fact_4330_Int__absorb,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ A3 @ A3 ) % 0.31/0.59 = A3 ) ). % 0.31/0.59 % 0.31/0.59 % Int_absorb % 0.31/0.59 thf(fact_4331_Int__assoc,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C5 ) % 0.31/0.59 = ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_assoc % 0.31/0.59 thf(fact_4332_IntD2,axiom, % 0.31/0.59 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.59 => ( member @ A @ C2 @ B3 ) ) ). % 0.31/0.59 % 0.31/0.59 % IntD2 % 0.31/0.59 thf(fact_4333_IntD1,axiom, % 0.31/0.59 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.59 => ( member @ A @ C2 @ A3 ) ) ). % 0.31/0.59 % 0.31/0.59 % IntD1 % 0.31/0.59 thf(fact_4334_IntE,axiom, % 0.31/0.59 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.59 => ~ ( ( member @ A @ C2 @ A3 ) % 0.31/0.59 => ~ ( member @ A @ C2 @ B3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % IntE % 0.31/0.59 thf(fact_4335_Diff__Int__distrib2,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ C5 ) % 0.31/0.59 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C5 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Diff_Int_distrib2 % 0.31/0.59 thf(fact_4336_Diff__Int__distrib,axiom, % 0.31/0.59 ! [A: $tType,C5: set @ A,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ C5 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.59 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C5 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C5 @ B3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Diff_Int_distrib % 0.31/0.59 thf(fact_4337_Diff__Diff__Int,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( minus_minus @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.59 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ). % 0.31/0.59 % 0.31/0.59 % Diff_Diff_Int % 0.31/0.59 thf(fact_4338_Diff__Int2,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,C5: set @ A,B3: set @ A] : % 0.31/0.59 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C5 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) % 0.31/0.59 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C5 ) @ B3 ) ) ). % 0.31/0.59 % 0.31/0.59 % Diff_Int2 % 0.31/0.59 thf(fact_4339_Int__Diff,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.31/0.59 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C5 ) % 0.31/0.59 = ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ C5 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_Diff % 0.31/0.59 thf(fact_4340_Int__insert__left,axiom, % 0.31/0.59 ! [A: $tType,A2: A,C5: set @ A,B3: set @ A] : % 0.31/0.59 ( ( ( member @ A @ A2 @ C5 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C5 ) % 0.31/0.59 = ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ) % 0.31/0.59 & ( ~ ( member @ A @ A2 @ C5 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C5 ) % 0.31/0.59 = ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_insert_left % 0.31/0.59 thf(fact_4341_Int__insert__right,axiom, % 0.31/0.59 ! [A: $tType,A2: A,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( ( member @ A @ A2 @ A3 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) % 0.31/0.59 = ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) % 0.31/0.59 & ( ~ ( member @ A @ A2 @ A3 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) % 0.31/0.59 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_insert_right % 0.31/0.59 thf(fact_4342_Union__Int__subset,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A3 @ B3 ) ) @ ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Union_Int_subset % 0.31/0.59 thf(fact_4343_Sup__inter__less__eq,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [A3: set @ A,B3: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_inter_less_eq % 0.31/0.59 thf(fact_4344_inf__cancel__left2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.59 => ! [X3: A,A2: A,B2: A] : % 0.31/0.59 ( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ A2 ) @ ( inf_inf @ A @ X3 @ B2 ) ) % 0.31/0.59 = ( bot_bot @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_cancel_left2 % 0.31/0.59 thf(fact_4345_inf__cancel__left1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.59 => ! [X3: A,A2: A,B2: A] : % 0.31/0.59 ( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ A2 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ B2 ) ) % 0.31/0.59 = ( bot_bot @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_cancel_left1 % 0.31/0.59 thf(fact_4346_gcd__le1__nat,axiom, % 0.31/0.59 ! [A2: nat,B2: nat] : % 0.31/0.59 ( ( A2 % 0.31/0.59 != ( zero_zero @ nat ) ) % 0.31/0.59 => ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ A2 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_le1_nat % 0.31/0.59 thf(fact_4347_gcd__le2__nat,axiom, % 0.31/0.59 ! [B2: nat,A2: nat] : % 0.31/0.59 ( ( B2 % 0.31/0.59 != ( zero_zero @ nat ) ) % 0.31/0.59 => ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_le2_nat % 0.31/0.59 thf(fact_4348_Sup__inf__eq__bot__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple592849572758109894attice @ A ) % 0.31/0.59 => ! [B3: set @ A,A2: A] : % 0.31/0.59 ( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B3 ) @ A2 ) % 0.31/0.59 = ( bot_bot @ A ) ) % 0.31/0.59 = ( ! [X: A] : % 0.31/0.59 ( ( member @ A @ X @ B3 ) % 0.31/0.59 => ( ( inf_inf @ A @ X @ A2 ) % 0.31/0.59 = ( bot_bot @ A ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_inf_eq_bot_iff % 0.31/0.59 thf(fact_4349_disjunctive__add,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : % 0.31/0.59 ( ! [N3: nat] : % 0.31/0.59 ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 ) % 0.31/0.59 | ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) ) % 0.31/0.59 => ( ( plus_plus @ A @ A2 @ B2 ) % 0.31/0.59 = ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % disjunctive_add % 0.31/0.59 thf(fact_4350_gcd__diff1__nat,axiom, % 0.31/0.59 ! [N2: nat,M: nat] : % 0.31/0.59 ( ( ord_less_eq @ nat @ N2 @ M ) % 0.31/0.59 => ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) % 0.31/0.59 = ( gcd_gcd @ nat @ M @ N2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_diff1_nat % 0.31/0.59 thf(fact_4351_gcd__diff2__nat,axiom, % 0.31/0.59 ! [M: nat,N2: nat] : % 0.31/0.59 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.59 => ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ N2 @ M ) @ N2 ) % 0.31/0.59 = ( gcd_gcd @ nat @ M @ N2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_diff2_nat % 0.31/0.59 thf(fact_4352_gcd__nat_Oelims,axiom, % 0.31/0.59 ! [X3: nat,Xa2: nat,Y: nat] : % 0.31/0.59 ( ( ( gcd_gcd @ nat @ X3 @ Xa2 ) % 0.31/0.59 = Y ) % 0.31/0.59 => ( ( ( Xa2 % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 => ( Y = X3 ) ) % 0.31/0.59 & ( ( Xa2 % 0.31/0.59 != ( zero_zero @ nat ) ) % 0.31/0.59 => ( Y % 0.31/0.59 = ( gcd_gcd @ nat @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.elims % 0.31/0.59 thf(fact_4353_gcd__nat_Osimps,axiom, % 0.31/0.59 ( ( gcd_gcd @ nat ) % 0.31/0.59 = ( ^ [X: nat,Y2: nat] : % 0.31/0.59 ( if @ nat % 0.31/0.59 @ ( Y2 % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 @ X % 0.31/0.59 @ ( gcd_gcd @ nat @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.simps % 0.31/0.59 thf(fact_4354_gcd__non__0__nat,axiom, % 0.31/0.59 ! [Y: nat,X3: nat] : % 0.31/0.59 ( ( Y % 0.31/0.59 != ( zero_zero @ nat ) ) % 0.31/0.59 => ( ( gcd_gcd @ nat @ X3 @ Y ) % 0.31/0.59 = ( gcd_gcd @ nat @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_non_0_nat % 0.31/0.59 thf(fact_4355_Int__Diff__disjoint,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Int_Diff_disjoint % 0.31/0.59 thf(fact_4356_Diff__triv,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ( minus_minus @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.59 = A3 ) ) ). % 0.31/0.59 % 0.31/0.59 % Diff_triv % 0.31/0.59 thf(fact_4357_Union__disjoint,axiom, % 0.31/0.59 ! [A: $tType,C5: set @ ( set @ A ),A3: set @ A] : % 0.31/0.59 ( ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) @ A3 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 = ( ! [X: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ X @ C5 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ X @ A3 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Union_disjoint % 0.31/0.59 thf(fact_4358_ivl__disj__int__two_I3_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( order @ A ) % 0.31/0.59 => ! [L: A,M: A,U: A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % ivl_disj_int_two(3) % 0.31/0.59 thf(fact_4359_gcd__integer_Orep__eq,axiom, % 0.31/0.59 ! [X3: code_integer,Xa2: code_integer] : % 0.31/0.59 ( ( code_int_of_integer @ ( gcd_gcd @ code_integer @ X3 @ Xa2 ) ) % 0.31/0.59 = ( gcd_gcd @ int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_integer.rep_eq % 0.31/0.59 thf(fact_4360_gcd__integer_Oabs__eq,axiom, % 0.31/0.59 ! [Xa2: int,X3: int] : % 0.31/0.59 ( ( gcd_gcd @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) ) % 0.31/0.59 = ( code_integer_of_int @ ( gcd_gcd @ int @ Xa2 @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_integer.abs_eq % 0.31/0.59 thf(fact_4361_Diff__eq,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( minus_minus @ ( set @ A ) ) % 0.31/0.59 = ( ^ [A6: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ B6 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Diff_eq % 0.31/0.59 thf(fact_4362_inf__shunt,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.59 => ! [X3: A,Y: A] : % 0.31/0.59 ( ( ( inf_inf @ A @ X3 @ Y ) % 0.31/0.59 = ( bot_bot @ A ) ) % 0.31/0.59 = ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_shunt % 0.31/0.59 thf(fact_4363_finite__Inf__in,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comple6319245703460814977attice @ A ) % 0.31/0.59 => ! [A3: set @ A] : % 0.31/0.59 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.59 => ( ( A3 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ! [X5: A,Y4: A] : % 0.31/0.59 ( ( member @ A @ X5 @ A3 ) % 0.31/0.59 => ( ( member @ A @ Y4 @ A3 ) % 0.31/0.59 => ( member @ A @ ( inf_inf @ A @ X5 @ Y4 ) @ A3 ) ) ) % 0.31/0.59 => ( member @ A @ ( complete_Inf_Inf @ A @ A3 ) @ A3 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % finite_Inf_in % 0.31/0.59 thf(fact_4364_bezout__nat,axiom, % 0.31/0.59 ! [A2: nat,B2: nat] : % 0.31/0.59 ( ( A2 % 0.31/0.59 != ( zero_zero @ nat ) ) % 0.31/0.59 => ? [X5: nat,Y4: nat] : % 0.31/0.59 ( ( times_times @ nat @ A2 @ X5 ) % 0.31/0.59 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % bezout_nat % 0.31/0.59 thf(fact_4365_bezout__gcd__nat_H,axiom, % 0.31/0.59 ! [B2: nat,A2: nat] : % 0.31/0.59 ? [X5: nat,Y4: nat] : % 0.31/0.59 ( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( times_times @ nat @ A2 @ X5 ) ) % 0.31/0.59 & ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X5 ) @ ( times_times @ nat @ B2 @ Y4 ) ) % 0.31/0.59 = ( gcd_gcd @ nat @ A2 @ B2 ) ) ) % 0.31/0.59 | ( ( ord_less_eq @ nat @ ( times_times @ nat @ A2 @ Y4 ) @ ( times_times @ nat @ B2 @ X5 ) ) % 0.31/0.59 & ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X5 ) @ ( times_times @ nat @ A2 @ Y4 ) ) % 0.31/0.59 = ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % bezout_gcd_nat' % 0.31/0.59 thf(fact_4366_ivl__disj__int__two_I7_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( order @ A ) % 0.31/0.59 => ! [L: A,M: A,U: A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % ivl_disj_int_two(7) % 0.31/0.59 thf(fact_4367_ivl__disj__int__one_I4_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( order @ A ) % 0.31/0.59 => ! [L: A,U: A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % ivl_disj_int_one(4) % 0.31/0.59 thf(fact_4368_ivl__disj__int__one_I2_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( order @ A ) % 0.31/0.59 => ! [L: A,U: A] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % ivl_disj_int_one(2) % 0.31/0.59 thf(fact_4369_disjoint__eq__subset__Compl,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 = ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % disjoint_eq_subset_Compl % 0.31/0.59 thf(fact_4370_insert__partition,axiom, % 0.31/0.59 ! [A: $tType,X3: set @ A,F5: set @ ( set @ A )] : % 0.31/0.59 ( ~ ( member @ ( set @ A ) @ X3 @ F5 ) % 0.31/0.59 => ( ! [X5: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ X5 @ ( insert @ ( set @ A ) @ X3 @ F5 ) ) % 0.31/0.59 => ! [Xa3: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ Xa3 @ ( insert @ ( set @ A ) @ X3 @ F5 ) ) % 0.31/0.59 => ( ( X5 != Xa3 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ X5 @ Xa3 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ) ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ X3 @ ( complete_Sup_Sup @ ( set @ A ) @ F5 ) ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % insert_partition % 0.31/0.59 thf(fact_4371_sum_Ointer__restrict,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [A3: set @ B,G: B > A,B3: set @ B] : % 0.31/0.59 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.59 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.59 = ( groups7311177749621191930dd_sum @ B @ A % 0.31/0.59 @ ^ [X: B] : ( if @ A @ ( member @ B @ X @ B3 ) @ ( G @ X ) @ ( zero_zero @ A ) ) % 0.31/0.59 @ A3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.inter_restrict % 0.31/0.59 thf(fact_4372_prod_Ointer__restrict,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [A3: set @ B,G: B > A,B3: set @ B] : % 0.31/0.59 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.59 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.59 = ( groups7121269368397514597t_prod @ B @ A % 0.31/0.59 @ ^ [X: B] : ( if @ A @ ( member @ B @ X @ B3 ) @ ( G @ X ) @ ( one_one @ A ) ) % 0.31/0.59 @ A3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.inter_restrict % 0.31/0.59 thf(fact_4373_gcd__code__integer,axiom, % 0.31/0.59 ( ( gcd_gcd @ code_integer ) % 0.31/0.59 = ( ^ [K3: code_integer,L2: code_integer] : % 0.31/0.59 ( abs_abs @ code_integer % 0.31/0.59 @ ( if @ code_integer % 0.31/0.59 @ ( L2 % 0.31/0.59 = ( zero_zero @ code_integer ) ) % 0.31/0.59 @ K3 % 0.31/0.59 @ ( gcd_gcd @ code_integer @ L2 @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_code_integer % 0.31/0.59 thf(fact_4374_bit_Ocomplement__unique,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.59 => ! [A2: A,X3: A,Y: A] : % 0.31/0.59 ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ X3 ) % 0.31/0.59 = ( zero_zero @ A ) ) % 0.31/0.59 => ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ X3 ) % 0.31/0.59 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.31/0.59 => ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ Y ) % 0.31/0.59 = ( zero_zero @ A ) ) % 0.31/0.59 => ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ Y ) % 0.31/0.59 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.31/0.59 => ( X3 = Y ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % bit.complement_unique % 0.31/0.59 thf(fact_4375_sum_Omono__neutral__cong,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [T4: set @ B,S3: set @ B,H2: B > A,G: B > A] : % 0.31/0.59 ( ( finite_finite2 @ B @ T4 ) % 0.31/0.59 => ( ( finite_finite2 @ B @ S3 ) % 0.31/0.59 => ( ! [I4: B] : % 0.31/0.59 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.31/0.59 => ( ( H2 @ I4 ) % 0.31/0.59 = ( zero_zero @ A ) ) ) % 0.31/0.59 => ( ! [I4: B] : % 0.31/0.59 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ S3 @ T4 ) ) % 0.31/0.59 => ( ( G @ I4 ) % 0.31/0.59 = ( zero_zero @ A ) ) ) % 0.31/0.59 => ( ! [X5: B] : % 0.31/0.59 ( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ S3 @ T4 ) ) % 0.31/0.59 => ( ( G @ X5 ) % 0.31/0.59 = ( H2 @ X5 ) ) ) % 0.31/0.59 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) % 0.31/0.59 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.mono_neutral_cong % 0.31/0.59 thf(fact_4376_Iio__Int__singleton,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( order @ A ) % 0.31/0.59 => ! [X3: A,K: A] : % 0.31/0.59 ( ( ( ord_less @ A @ X3 @ K ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.59 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.59 & ( ~ ( ord_less @ A @ X3 @ K ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Iio_Int_singleton % 0.31/0.59 thf(fact_4377_sum_OInt__Diff,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [A3: set @ B,G: B > A,B3: set @ B] : % 0.31/0.59 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.59 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 0.31/0.59 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.Int_Diff % 0.31/0.59 thf(fact_4378_prod_OInt__Diff,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [A3: set @ B,G: B > A,B3: set @ B] : % 0.31/0.59 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.59 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) % 0.31/0.59 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.Int_Diff % 0.31/0.59 thf(fact_4379_prod_Omono__neutral__cong,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [T4: set @ B,S3: set @ B,H2: B > A,G: B > A] : % 0.31/0.59 ( ( finite_finite2 @ B @ T4 ) % 0.31/0.59 => ( ( finite_finite2 @ B @ S3 ) % 0.31/0.59 => ( ! [I4: B] : % 0.31/0.59 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.31/0.59 => ( ( H2 @ I4 ) % 0.31/0.59 = ( one_one @ A ) ) ) % 0.31/0.59 => ( ! [I4: B] : % 0.31/0.59 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ S3 @ T4 ) ) % 0.31/0.59 => ( ( G @ I4 ) % 0.31/0.59 = ( one_one @ A ) ) ) % 0.31/0.59 => ( ! [X5: B] : % 0.31/0.59 ( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ S3 @ T4 ) ) % 0.31/0.59 => ( ( G @ X5 ) % 0.31/0.59 = ( H2 @ X5 ) ) ) % 0.31/0.59 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) % 0.31/0.59 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.mono_neutral_cong % 0.31/0.59 thf(fact_4380_card__Diff__subset__Int,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.59 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.59 = ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % card_Diff_subset_Int % 0.31/0.59 thf(fact_4381_gcd__int__def,axiom, % 0.31/0.59 ( ( gcd_gcd @ int ) % 0.31/0.59 = ( ^ [X: int,Y2: int] : ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ X ) ) @ ( nat2 @ ( abs_abs @ int @ Y2 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_int_def % 0.31/0.59 thf(fact_4382_sum_OIf__cases,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [A3: set @ B,P: B > $o,H2: B > A,G: B > A] : % 0.31/0.59 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.59 => ( ( groups7311177749621191930dd_sum @ B @ A % 0.31/0.59 @ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( H2 @ X ) @ ( G @ X ) ) % 0.31/0.59 @ A3 ) % 0.31/0.59 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.If_cases % 0.31/0.59 thf(fact_4383_prod_OIf__cases,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [A3: set @ B,P: B > $o,H2: B > A,G: B > A] : % 0.31/0.59 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.59 => ( ( groups7121269368397514597t_prod @ B @ A % 0.31/0.59 @ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( H2 @ X ) @ ( G @ X ) ) % 0.31/0.59 @ A3 ) % 0.31/0.59 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.If_cases % 0.31/0.59 thf(fact_4384_dvd__partition,axiom, % 0.31/0.59 ! [A: $tType,C5: set @ ( set @ A ),K: nat] : % 0.31/0.59 ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) % 0.31/0.59 => ( ! [X5: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ X5 @ C5 ) % 0.31/0.59 => ( dvd_dvd @ nat @ K @ ( finite_card @ A @ X5 ) ) ) % 0.31/0.59 => ( ! [X5: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ X5 @ C5 ) % 0.31/0.59 => ! [Xa3: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ Xa3 @ C5 ) % 0.31/0.59 => ( ( X5 != Xa3 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ X5 @ Xa3 ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ) ) % 0.31/0.59 => ( dvd_dvd @ nat @ K @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % dvd_partition % 0.31/0.59 thf(fact_4385_sum__div__partition,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( euclid4440199948858584721cancel @ A ) % 0.31/0.59 => ! [A3: set @ B,F2: B > A,B2: A] : % 0.31/0.59 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.59 => ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ B2 ) % 0.31/0.59 = ( plus_plus @ A % 0.31/0.59 @ ( groups7311177749621191930dd_sum @ B @ A % 0.31/0.59 @ ^ [A4: B] : ( divide_divide @ A @ ( F2 @ A4 ) @ B2 ) % 0.31/0.59 @ ( inf_inf @ ( set @ B ) @ A3 % 0.31/0.59 @ ( collect @ B % 0.31/0.59 @ ^ [A4: B] : ( dvd_dvd @ A @ B2 @ ( F2 @ A4 ) ) ) ) ) % 0.31/0.59 @ ( divide_divide @ A % 0.31/0.59 @ ( groups7311177749621191930dd_sum @ B @ A @ F2 % 0.31/0.59 @ ( inf_inf @ ( set @ B ) @ A3 % 0.31/0.59 @ ( collect @ B % 0.31/0.59 @ ^ [A4: B] : % 0.31/0.59 ~ ( dvd_dvd @ A @ B2 @ ( F2 @ A4 ) ) ) ) ) % 0.31/0.59 @ B2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum_div_partition % 0.31/0.59 thf(fact_4386_distinct__concat,axiom, % 0.31/0.59 ! [A: $tType,Xs: list @ ( list @ A )] : % 0.31/0.59 ( ( distinct @ ( list @ A ) @ Xs ) % 0.31/0.59 => ( ! [Ys4: list @ A] : % 0.31/0.59 ( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) ) % 0.31/0.59 => ( distinct @ A @ Ys4 ) ) % 0.31/0.59 => ( ! [Ys4: list @ A,Zs: list @ A] : % 0.31/0.59 ( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) ) % 0.31/0.59 => ( ( member @ ( list @ A ) @ Zs @ ( set2 @ ( list @ A ) @ Xs ) ) % 0.31/0.59 => ( ( Ys4 != Zs ) % 0.31/0.59 => ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys4 ) @ ( set2 @ A @ Zs ) ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) ) ) ) % 0.31/0.59 => ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % distinct_concat % 0.31/0.59 thf(fact_4387_mask__Suc__exp,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [N2: nat] : % 0.31/0.59 ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N2 ) ) % 0.31/0.59 = ( bit_se1065995026697491101ons_or @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % mask_Suc_exp % 0.31/0.59 thf(fact_4388_bezw__aux,axiom, % 0.31/0.59 ! [X3: nat,Y: nat] : % 0.31/0.59 ( ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ X3 @ Y ) ) % 0.31/0.59 = ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ ( bezw @ X3 @ Y ) ) @ ( semiring_1_of_nat @ int @ X3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ X3 @ Y ) ) @ ( semiring_1_of_nat @ int @ Y ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % bezw_aux % 0.31/0.59 thf(fact_4389_one__or__eq,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [A2: A] : % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A2 ) % 0.31/0.59 = ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % one_or_eq % 0.31/0.59 thf(fact_4390_or__one__eq,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [A2: A] : % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( one_one @ A ) ) % 0.31/0.59 = ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % or_one_eq % 0.31/0.59 thf(fact_4391_mask__Suc__double,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [N2: nat] : % 0.31/0.59 ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N2 ) ) % 0.31/0.59 = ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % mask_Suc_double % 0.31/0.59 thf(fact_4392_OR__upper,axiom, % 0.31/0.59 ! [X3: int,N2: nat,Y: int] : % 0.31/0.59 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 ) % 0.31/0.59 => ( ( ord_less @ int @ X3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.31/0.59 => ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) % 0.31/0.59 => ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X3 @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % OR_upper % 0.31/0.59 thf(fact_4393_or__Suc__0__eq,axiom, % 0.31/0.59 ! [N2: nat] : % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 0.31/0.59 = ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % or_Suc_0_eq % 0.31/0.59 thf(fact_4394_Suc__0__or__eq,axiom, % 0.31/0.59 ! [N2: nat] : % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 0.31/0.59 = ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Suc_0_or_eq % 0.31/0.59 thf(fact_4395_or__nat__rec,axiom, % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ nat ) % 0.31/0.59 = ( ^ [M2: nat,N: nat] : % 0.31/0.59 ( plus_plus @ nat % 0.31/0.59 @ ( zero_neq_one_of_bool @ nat % 0.31/0.59 @ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) % 0.31/0.59 | ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) % 0.31/0.59 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % or_nat_rec % 0.31/0.59 thf(fact_4396_gcd__nat_Opelims,axiom, % 0.31/0.59 ! [X3: nat,Xa2: nat,Y: nat] : % 0.31/0.59 ( ( ( gcd_gcd @ nat @ X3 @ Xa2 ) % 0.31/0.59 = Y ) % 0.31/0.59 => ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) % 0.31/0.59 => ~ ( ( ( ( Xa2 % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 => ( Y = X3 ) ) % 0.31/0.59 & ( ( Xa2 % 0.31/0.59 != ( zero_zero @ nat ) ) % 0.31/0.59 => ( Y % 0.31/0.59 = ( gcd_gcd @ nat @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) ) ) % 0.31/0.59 => ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.pelims % 0.31/0.59 thf(fact_4397_inf__bot__left,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bounded_lattice_bot @ A ) % 0.31/0.59 => ! [X3: A] : % 0.31/0.59 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X3 ) % 0.31/0.59 = ( bot_bot @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_bot_left % 0.31/0.59 thf(fact_4398_inf__bot__right,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bounded_lattice_bot @ A ) % 0.31/0.59 => ! [X3: A] : % 0.31/0.59 ( ( inf_inf @ A @ X3 @ ( bot_bot @ A ) ) % 0.31/0.59 = ( bot_bot @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_bot_right % 0.31/0.59 thf(fact_4399_inf_Obounded__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [A2: A,B2: A,C2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) % 0.31/0.59 = ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.59 & ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.bounded_iff % 0.31/0.59 thf(fact_4400_gcd__nat_Oidem,axiom, % 0.31/0.59 ! [A2: nat] : % 0.31/0.59 ( ( gcd_gcd @ nat @ A2 @ A2 ) % 0.31/0.59 = A2 ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.idem % 0.31/0.59 thf(fact_4401_gcd__nat_Oleft__idem,axiom, % 0.31/0.59 ! [A2: nat,B2: nat] : % 0.31/0.59 ( ( gcd_gcd @ nat @ A2 @ ( gcd_gcd @ nat @ A2 @ B2 ) ) % 0.31/0.59 = ( gcd_gcd @ nat @ A2 @ B2 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.left_idem % 0.31/0.59 thf(fact_4402_gcd__nat_Oright__idem,axiom, % 0.31/0.59 ! [A2: nat,B2: nat] : % 0.31/0.59 ( ( gcd_gcd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 ) % 0.31/0.59 = ( gcd_gcd @ nat @ A2 @ B2 ) ) ). % 0.31/0.59 % 0.31/0.59 % gcd_nat.right_idem % 0.31/0.59 thf(fact_4403_gcd__idem__nat,axiom, % 0.31/0.59 ! [X3: nat] : % 0.31/0.59 ( ( gcd_gcd @ nat @ X3 @ X3 ) % 0.31/0.59 = X3 ) ). % 0.31/0.59 % 0.31/0.59 % gcd_idem_nat % 0.31/0.59 thf(fact_4404_le__inf__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [X3: A,Y: A,Z2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) ) % 0.31/0.59 = ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.59 & ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % le_inf_iff % 0.31/0.59 thf(fact_4405_inf__set__def,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( inf_inf @ ( set @ A ) ) % 0.31/0.59 = ( ^ [A6: set @ A,B6: set @ A] : % 0.31/0.59 ( collect @ A % 0.31/0.59 @ ( inf_inf @ ( A > $o ) % 0.31/0.59 @ ^ [X: A] : ( member @ A @ X @ A6 ) % 0.31/0.59 @ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_set_def % 0.31/0.59 thf(fact_4406_inf__Int__eq2,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] : % 0.31/0.59 ( ( inf_inf @ ( A > B > $o ) % 0.31/0.59 @ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R ) % 0.31/0.59 @ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S3 ) ) % 0.31/0.59 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_Int_eq2 % 0.31/0.59 thf(fact_4407_inf_OcoboundedI2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [B2: A,C2: A,A2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ B2 @ C2 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.coboundedI2 % 0.31/0.59 thf(fact_4408_inf_OcoboundedI1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [A2: A,C2: A,B2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ A2 @ C2 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.coboundedI1 % 0.31/0.59 thf(fact_4409_inf_Oabsorb__iff2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ( ( ord_less_eq @ A ) % 0.31/0.59 = ( ^ [B4: A,A4: A] : % 0.31/0.59 ( ( inf_inf @ A @ A4 @ B4 ) % 0.31/0.59 = B4 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.absorb_iff2 % 0.31/0.59 thf(fact_4410_inf_Oabsorb__iff1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ( ( ord_less_eq @ A ) % 0.31/0.59 = ( ^ [A4: A,B4: A] : % 0.31/0.59 ( ( inf_inf @ A @ A4 @ B4 ) % 0.31/0.59 = A4 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.absorb_iff1 % 0.31/0.59 thf(fact_4411_inf_Ocobounded2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.cobounded2 % 0.31/0.59 thf(fact_4412_inf_Ocobounded1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ A2 ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.cobounded1 % 0.31/0.59 thf(fact_4413_inf_Oorder__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ( ( ord_less_eq @ A ) % 0.31/0.59 = ( ^ [A4: A,B4: A] : % 0.31/0.59 ( A4 % 0.31/0.59 = ( inf_inf @ A @ A4 @ B4 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.order_iff % 0.31/0.59 thf(fact_4414_inf__greatest,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [X3: A,Y: A,Z2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.59 => ( ( ord_less_eq @ A @ X3 @ Z2 ) % 0.31/0.59 => ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_greatest % 0.31/0.59 thf(fact_4415_inf_OboundedI,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [A2: A,B2: A,C2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.59 => ( ( ord_less_eq @ A @ A2 @ C2 ) % 0.31/0.59 => ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.boundedI % 0.31/0.59 thf(fact_4416_inf_OboundedE,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [A2: A,B2: A,C2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) % 0.31/0.59 => ~ ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.59 => ~ ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.boundedE % 0.31/0.59 thf(fact_4417_inf__absorb2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [Y: A,X3: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ Y @ X3 ) % 0.31/0.59 => ( ( inf_inf @ A @ X3 @ Y ) % 0.31/0.59 = Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_absorb2 % 0.31/0.59 thf(fact_4418_inf__absorb1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [X3: A,Y: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.59 => ( ( inf_inf @ A @ X3 @ Y ) % 0.31/0.59 = X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_absorb1 % 0.31/0.59 thf(fact_4419_inf_Oabsorb2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [B2: A,A2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.59 => ( ( inf_inf @ A @ A2 @ B2 ) % 0.31/0.59 = B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.absorb2 % 0.31/0.59 thf(fact_4420_inf_Oabsorb1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.59 => ( ( inf_inf @ A @ A2 @ B2 ) % 0.31/0.59 = A2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.absorb1 % 0.31/0.59 thf(fact_4421_le__iff__inf,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ( ( ord_less_eq @ A ) % 0.31/0.59 = ( ^ [X: A,Y2: A] : % 0.31/0.59 ( ( inf_inf @ A @ X @ Y2 ) % 0.31/0.59 = X ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % le_iff_inf % 0.31/0.59 thf(fact_4422_inf__unique,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [F2: A > A > A,X3: A,Y: A] : % 0.31/0.59 ( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ ( F2 @ X5 @ Y4 ) @ X5 ) % 0.31/0.59 => ( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ ( F2 @ X5 @ Y4 ) @ Y4 ) % 0.31/0.59 => ( ! [X5: A,Y4: A,Z3: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ X5 @ Y4 ) % 0.31/0.59 => ( ( ord_less_eq @ A @ X5 @ Z3 ) % 0.31/0.59 => ( ord_less_eq @ A @ X5 @ ( F2 @ Y4 @ Z3 ) ) ) ) % 0.31/0.59 => ( ( inf_inf @ A @ X3 @ Y ) % 0.31/0.59 = ( F2 @ X3 @ Y ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_unique % 0.31/0.59 thf(fact_4423_inf_OorderI,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : % 0.31/0.59 ( ( A2 % 0.31/0.59 = ( inf_inf @ A @ A2 @ B2 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.orderI % 0.31/0.59 thf(fact_4424_inf_OorderE,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.59 => ( A2 % 0.31/0.59 = ( inf_inf @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.orderE % 0.31/0.59 thf(fact_4425_le__infI2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [B2: A,X3: A,A2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ B2 @ X3 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % le_infI2 % 0.31/0.59 thf(fact_4426_le__infI1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [A2: A,X3: A,B2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ A2 @ X3 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % le_infI1 % 0.31/0.59 thf(fact_4427_inf__mono,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [A2: A,C2: A,B2: A,D2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ A2 @ C2 ) % 0.31/0.59 => ( ( ord_less_eq @ A @ B2 @ D2 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ ( inf_inf @ A @ C2 @ D2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_mono % 0.31/0.59 thf(fact_4428_le__infI,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [X3: A,A2: A,B2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ X3 @ A2 ) % 0.31/0.59 => ( ( ord_less_eq @ A @ X3 @ B2 ) % 0.31/0.59 => ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % le_infI % 0.31/0.59 thf(fact_4429_le__infE,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [X3: A,A2: A,B2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A2 @ B2 ) ) % 0.31/0.59 => ~ ( ( ord_less_eq @ A @ X3 @ A2 ) % 0.31/0.59 => ~ ( ord_less_eq @ A @ X3 @ B2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % le_infE % 0.31/0.59 thf(fact_4430_inf__le2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ Y ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_le2 % 0.31/0.59 thf(fact_4431_inf__le1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_le1 % 0.31/0.59 thf(fact_4432_inf__sup__ord_I1_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( lattice @ A ) % 0.31/0.59 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_sup_ord(1) % 0.31/0.59 thf(fact_4433_inf__sup__ord_I2_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( lattice @ A ) % 0.31/0.59 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ Y ) ) ). % 0.31/0.59 % 0.31/0.59 % inf_sup_ord(2) % 0.31/0.59 thf(fact_4434_less__infI1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [A2: A,X3: A,B2: A] : % 0.31/0.59 ( ( ord_less @ A @ A2 @ X3 ) % 0.31/0.59 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % less_infI1 % 0.31/0.59 thf(fact_4435_less__infI2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [B2: A,X3: A,A2: A] : % 0.31/0.59 ( ( ord_less @ A @ B2 @ X3 ) % 0.31/0.59 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % less_infI2 % 0.31/0.59 thf(fact_4436_inf_Oabsorb3,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : % 0.31/0.59 ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.59 => ( ( inf_inf @ A @ A2 @ B2 ) % 0.31/0.59 = A2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.absorb3 % 0.31/0.59 thf(fact_4437_inf_Oabsorb4,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [B2: A,A2: A] : % 0.31/0.59 ( ( ord_less @ A @ B2 @ A2 ) % 0.31/0.59 => ( ( inf_inf @ A @ A2 @ B2 ) % 0.31/0.59 = B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.absorb4 % 0.31/0.59 thf(fact_4438_inf_Ostrict__boundedE,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [A2: A,B2: A,C2: A] : % 0.31/0.59 ( ( ord_less @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) % 0.31/0.59 => ~ ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.59 => ~ ( ord_less @ A @ A2 @ C2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.strict_boundedE % 0.31/0.59 thf(fact_4439_inf_Ostrict__order__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ( ( ord_less @ A ) % 0.31/0.59 = ( ^ [A4: A,B4: A] : % 0.31/0.59 ( ( A4 % 0.31/0.59 = ( inf_inf @ A @ A4 @ B4 ) ) % 0.31/0.59 & ( A4 != B4 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.strict_order_iff % 0.31/0.59 thf(fact_4440_inf_Ostrict__coboundedI1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [A2: A,C2: A,B2: A] : % 0.31/0.59 ( ( ord_less @ A @ A2 @ C2 ) % 0.31/0.59 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.strict_coboundedI1 % 0.31/0.59 thf(fact_4441_inf_Ostrict__coboundedI2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( semilattice_inf @ A ) % 0.31/0.59 => ! [B2: A,C2: A,A2: A] : % 0.31/0.59 ( ( ord_less @ A @ B2 @ C2 ) % 0.31/0.59 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % inf.strict_coboundedI2 % 0.31/0.59 thf(fact_4442_finite__inf__Sup,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( finite8700451911770168679attice @ A ) % 0.31/0.59 => ! [A2: A,A3: set @ A] : % 0.31/0.59 ( ( inf_inf @ A @ A2 @ ( complete_Sup_Sup @ A @ A3 ) ) % 0.31/0.59 = ( complete_Sup_Sup @ A % 0.31/0.59 @ ( collect @ A % 0.31/0.59 @ ^ [Uu3: A] : % 0.31/0.59 ? [B4: A] : % 0.31/0.59 ( ( Uu3 % 0.31/0.59 = ( inf_inf @ A @ A2 @ B4 ) ) % 0.31/0.59 & ( member @ A @ B4 @ A3 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % finite_inf_Sup % 0.31/0.59 thf(fact_4443_or__nat__unfold,axiom, % 0.31/0.59 ( ( bit_se1065995026697491101ons_or @ nat ) % 0.31/0.59 = ( ^ [M2: nat,N: nat] : % 0.31/0.59 ( if @ nat % 0.31/0.59 @ ( M2 % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 @ N % 0.31/0.59 @ ( if @ nat % 0.31/0.59 @ ( N % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 @ M2 % 0.31/0.59 @ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % or_nat_unfold % 0.31/0.59 thf(fact_4444_card__disjoint__shuffles,axiom, % 0.31/0.59 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.59 ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) % 0.31/0.59 = ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % card_disjoint_shuffles % 0.31/0.59 thf(fact_4445_distinct__product__lists,axiom, % 0.31/0.59 ! [A: $tType,Xss: list @ ( list @ A )] : % 0.31/0.59 ( ! [X5: list @ A] : % 0.31/0.59 ( ( member @ ( list @ A ) @ X5 @ ( set2 @ ( list @ A ) @ Xss ) ) % 0.31/0.59 => ( distinct @ A @ X5 ) ) % 0.31/0.59 => ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) ) ). % 0.31/0.59 % 0.31/0.59 % distinct_product_lists % 0.31/0.59 thf(fact_4446_max_Oabsorb1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [B2: A,A2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.59 => ( ( ord_max @ A @ A2 @ B2 ) % 0.31/0.59 = A2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.absorb1 % 0.31/0.59 thf(fact_4447_max_Oabsorb2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.59 => ( ( ord_max @ A @ A2 @ B2 ) % 0.31/0.59 = B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.absorb2 % 0.31/0.59 thf(fact_4448_max_Obounded__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [B2: A,C2: A,A2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 ) % 0.31/0.59 = ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.59 & ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.bounded_iff % 0.31/0.59 thf(fact_4449_max_Oabsorb3,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [B2: A,A2: A] : % 0.31/0.59 ( ( ord_less @ A @ B2 @ A2 ) % 0.31/0.59 => ( ( ord_max @ A @ A2 @ B2 ) % 0.31/0.59 = A2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.absorb3 % 0.31/0.59 thf(fact_4450_max_Oabsorb4,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : % 0.31/0.59 ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.59 => ( ( ord_max @ A @ A2 @ B2 ) % 0.31/0.59 = B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.absorb4 % 0.31/0.59 thf(fact_4451_max__less__iff__conj,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [X3: A,Y: A,Z2: A] : % 0.31/0.59 ( ( ord_less @ A @ ( ord_max @ A @ X3 @ Y ) @ Z2 ) % 0.31/0.59 = ( ( ord_less @ A @ X3 @ Z2 ) % 0.31/0.59 & ( ord_less @ A @ Y @ Z2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_less_iff_conj % 0.31/0.59 thf(fact_4452_max__bot2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( order_bot @ A ) % 0.31/0.59 => ! [X3: A] : % 0.31/0.59 ( ( ord_max @ A @ X3 @ ( bot_bot @ A ) ) % 0.31/0.59 = X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % max_bot2 % 0.31/0.59 thf(fact_4453_max__bot,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( order_bot @ A ) % 0.31/0.59 => ! [X3: A] : % 0.31/0.59 ( ( ord_max @ A @ ( bot_bot @ A ) @ X3 ) % 0.31/0.59 = X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % max_bot % 0.31/0.59 thf(fact_4454_of__bool__or__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linordered_semidom @ A ) % 0.31/0.59 => ! [P: $o,Q: $o] : % 0.31/0.59 ( ( zero_neq_one_of_bool @ A % 0.31/0.59 @ ( P % 0.31/0.59 | Q ) ) % 0.31/0.59 = ( ord_max @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % of_bool_or_iff % 0.31/0.59 thf(fact_4455_max__Suc__Suc,axiom, % 0.31/0.59 ! [M: nat,N2: nat] : % 0.31/0.59 ( ( ord_max @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) % 0.31/0.59 = ( suc @ ( ord_max @ nat @ M @ N2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_Suc_Suc % 0.31/0.59 thf(fact_4456_max__0R,axiom, % 0.31/0.59 ! [N2: nat] : % 0.31/0.59 ( ( ord_max @ nat @ N2 @ ( zero_zero @ nat ) ) % 0.31/0.59 = N2 ) ). % 0.31/0.59 % 0.31/0.59 % max_0R % 0.31/0.59 thf(fact_4457_max__0L,axiom, % 0.31/0.59 ! [N2: nat] : % 0.31/0.59 ( ( ord_max @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 = N2 ) ). % 0.31/0.59 % 0.31/0.59 % max_0L % 0.31/0.59 thf(fact_4458_max__nat_Oright__neutral,axiom, % 0.31/0.59 ! [A2: nat] : % 0.31/0.59 ( ( ord_max @ nat @ A2 @ ( zero_zero @ nat ) ) % 0.31/0.59 = A2 ) ). % 0.31/0.59 % 0.31/0.59 % max_nat.right_neutral % 0.31/0.59 thf(fact_4459_max__nat_Oneutr__eq__iff,axiom, % 0.31/0.59 ! [A2: nat,B2: nat] : % 0.31/0.59 ( ( ( zero_zero @ nat ) % 0.31/0.59 = ( ord_max @ nat @ A2 @ B2 ) ) % 0.31/0.59 = ( ( A2 % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 & ( B2 % 0.31/0.59 = ( zero_zero @ nat ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_nat.neutr_eq_iff % 0.31/0.59 thf(fact_4460_max__nat_Oleft__neutral,axiom, % 0.31/0.59 ! [A2: nat] : % 0.31/0.59 ( ( ord_max @ nat @ ( zero_zero @ nat ) @ A2 ) % 0.31/0.59 = A2 ) ). % 0.31/0.59 % 0.31/0.59 % max_nat.left_neutral % 0.31/0.59 thf(fact_4461_max__nat_Oeq__neutr__iff,axiom, % 0.31/0.59 ! [A2: nat,B2: nat] : % 0.31/0.59 ( ( ( ord_max @ nat @ A2 @ B2 ) % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 = ( ( A2 % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 & ( B2 % 0.31/0.59 = ( zero_zero @ nat ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_nat.eq_neutr_iff % 0.31/0.59 thf(fact_4462_finite__shuffles,axiom, % 0.31/0.59 ! [A: $tType,Xs: list @ A,Ys: list @ A] : ( finite_finite2 @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ). % 0.31/0.59 % 0.31/0.59 % finite_shuffles % 0.31/0.59 thf(fact_4463_max__number__of_I1_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ( numeral @ A ) % 0.31/0.59 & ( ord @ A ) ) % 0.31/0.59 => ! [U: num,V2: num] : % 0.31/0.59 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.59 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.59 = ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.59 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.59 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.59 = ( numeral_numeral @ A @ U ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_number_of(1) % 0.31/0.59 thf(fact_4464_max__0__1_I4_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linord181362715937106298miring @ A ) % 0.31/0.59 => ! [X3: num] : % 0.31/0.59 ( ( ord_max @ A @ ( numeral_numeral @ A @ X3 ) @ ( zero_zero @ A ) ) % 0.31/0.59 = ( numeral_numeral @ A @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_0_1(4) % 0.31/0.59 thf(fact_4465_max__0__1_I3_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linord181362715937106298miring @ A ) % 0.31/0.59 => ! [X3: num] : % 0.31/0.59 ( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X3 ) ) % 0.31/0.59 = ( numeral_numeral @ A @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_0_1(3) % 0.31/0.59 thf(fact_4466_max__0__1_I1_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linord181362715937106298miring @ A ) % 0.31/0.59 => ( ( ord_max @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) % 0.31/0.59 = ( one_one @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_0_1(1) % 0.31/0.59 thf(fact_4467_max__0__1_I2_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linord181362715937106298miring @ A ) % 0.31/0.59 => ( ( ord_max @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) % 0.31/0.59 = ( one_one @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_0_1(2) % 0.31/0.59 thf(fact_4468_max__number__of_I2_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ( uminus @ A ) % 0.31/0.59 & ( numeral @ A ) % 0.31/0.59 & ( ord @ A ) ) % 0.31/0.59 => ! [U: num,V2: num] : % 0.31/0.59 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.59 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.59 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) % 0.31/0.59 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.59 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.59 = ( numeral_numeral @ A @ U ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_number_of(2) % 0.31/0.59 thf(fact_4469_max__number__of_I3_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ( uminus @ A ) % 0.31/0.59 & ( numeral @ A ) % 0.31/0.59 & ( ord @ A ) ) % 0.31/0.59 => ! [U: num,V2: num] : % 0.31/0.59 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.59 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.59 = ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.59 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.59 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.59 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_number_of(3) % 0.31/0.59 thf(fact_4470_max__number__of_I4_J,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ( uminus @ A ) % 0.31/0.59 & ( numeral @ A ) % 0.31/0.59 & ( ord @ A ) ) % 0.31/0.59 => ! [U: num,V2: num] : % 0.31/0.59 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.59 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.59 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) % 0.31/0.59 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.59 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.59 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_number_of(4) % 0.31/0.59 thf(fact_4471_max__Suc__numeral,axiom, % 0.31/0.59 ! [N2: nat,K: num] : % 0.31/0.59 ( ( ord_max @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) ) % 0.31/0.59 = ( suc @ ( ord_max @ nat @ N2 @ ( pred_numeral @ K ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_Suc_numeral % 0.31/0.59 thf(fact_4472_max__numeral__Suc,axiom, % 0.31/0.59 ! [K: num,N2: nat] : % 0.31/0.59 ( ( ord_max @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) ) % 0.31/0.59 = ( suc @ ( ord_max @ nat @ ( pred_numeral @ K ) @ N2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_numeral_Suc % 0.31/0.59 thf(fact_4473_max_Ostrict__coboundedI2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [C2: A,B2: A,A2: A] : % 0.31/0.59 ( ( ord_less @ A @ C2 @ B2 ) % 0.31/0.59 => ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.strict_coboundedI2 % 0.31/0.59 thf(fact_4474_max_Ostrict__coboundedI1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [C2: A,A2: A,B2: A] : % 0.31/0.59 ( ( ord_less @ A @ C2 @ A2 ) % 0.31/0.59 => ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.strict_coboundedI1 % 0.31/0.59 thf(fact_4475_max_Ostrict__order__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ( ( ord_less @ A ) % 0.31/0.59 = ( ^ [B4: A,A4: A] : % 0.31/0.59 ( ( A4 % 0.31/0.59 = ( ord_max @ A @ A4 @ B4 ) ) % 0.31/0.59 & ( A4 != B4 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.strict_order_iff % 0.31/0.59 thf(fact_4476_max_Ostrict__boundedE,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [B2: A,C2: A,A2: A] : % 0.31/0.59 ( ( ord_less @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 ) % 0.31/0.59 => ~ ( ( ord_less @ A @ B2 @ A2 ) % 0.31/0.59 => ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.strict_boundedE % 0.31/0.59 thf(fact_4477_less__max__iff__disj,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [Z2: A,X3: A,Y: A] : % 0.31/0.59 ( ( ord_less @ A @ Z2 @ ( ord_max @ A @ X3 @ Y ) ) % 0.31/0.59 = ( ( ord_less @ A @ Z2 @ X3 ) % 0.31/0.59 | ( ord_less @ A @ Z2 @ Y ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % less_max_iff_disj % 0.31/0.59 thf(fact_4478_max_Omono,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [C2: A,A2: A,D2: A,B2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.59 => ( ( ord_less_eq @ A @ D2 @ B2 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D2 ) @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.mono % 0.31/0.59 thf(fact_4479_max_OorderE,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [B2: A,A2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.59 => ( A2 % 0.31/0.59 = ( ord_max @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.orderE % 0.31/0.59 thf(fact_4480_max_OorderI,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : % 0.31/0.59 ( ( A2 % 0.31/0.59 = ( ord_max @ A @ A2 @ B2 ) ) % 0.31/0.59 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.orderI % 0.31/0.59 thf(fact_4481_max_OboundedE,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [B2: A,C2: A,A2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 ) % 0.31/0.59 => ~ ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.59 => ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.boundedE % 0.31/0.59 thf(fact_4482_max_OboundedI,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [B2: A,A2: A,C2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.59 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.59 => ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.boundedI % 0.31/0.59 thf(fact_4483_max_Oorder__iff,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ( ( ord_less_eq @ A ) % 0.31/0.59 = ( ^ [B4: A,A4: A] : % 0.31/0.59 ( A4 % 0.31/0.59 = ( ord_max @ A @ A4 @ B4 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.order_iff % 0.31/0.59 thf(fact_4484_max_Ocobounded1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.cobounded1 % 0.31/0.59 thf(fact_4485_max_Ocobounded2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.cobounded2 % 0.31/0.59 thf(fact_4486_le__max__iff__disj,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [Z2: A,X3: A,Y: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ Z2 @ ( ord_max @ A @ X3 @ Y ) ) % 0.31/0.59 = ( ( ord_less_eq @ A @ Z2 @ X3 ) % 0.31/0.59 | ( ord_less_eq @ A @ Z2 @ Y ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % le_max_iff_disj % 0.31/0.59 thf(fact_4487_max_Oabsorb__iff1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ( ( ord_less_eq @ A ) % 0.31/0.59 = ( ^ [B4: A,A4: A] : % 0.31/0.59 ( ( ord_max @ A @ A4 @ B4 ) % 0.31/0.59 = A4 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.absorb_iff1 % 0.31/0.59 thf(fact_4488_max_Oabsorb__iff2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ( ( ord_less_eq @ A ) % 0.31/0.59 = ( ^ [A4: A,B4: A] : % 0.31/0.59 ( ( ord_max @ A @ A4 @ B4 ) % 0.31/0.59 = B4 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.absorb_iff2 % 0.31/0.59 thf(fact_4489_max_OcoboundedI1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [C2: A,A2: A,B2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.59 => ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.coboundedI1 % 0.31/0.59 thf(fact_4490_max_OcoboundedI2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linorder @ A ) % 0.31/0.59 => ! [C2: A,B2: A,A2: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ C2 @ B2 ) % 0.31/0.59 => ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max.coboundedI2 % 0.31/0.59 thf(fact_4491_max__def__raw,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ord @ A ) % 0.31/0.59 => ( ( ord_max @ A ) % 0.31/0.59 = ( ^ [A4: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B4 ) @ B4 @ A4 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_def_raw % 0.31/0.59 thf(fact_4492_max__def,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ord @ A ) % 0.31/0.59 => ( ( ord_max @ A ) % 0.31/0.59 = ( ^ [A4: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B4 ) @ B4 @ A4 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_def % 0.31/0.59 thf(fact_4493_max__absorb1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( order @ A ) % 0.31/0.59 => ! [Y: A,X3: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ Y @ X3 ) % 0.31/0.59 => ( ( ord_max @ A @ X3 @ Y ) % 0.31/0.59 = X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_absorb1 % 0.31/0.59 thf(fact_4494_max__absorb2,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ord @ A ) % 0.31/0.59 => ! [X3: A,Y: A] : % 0.31/0.59 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.59 => ( ( ord_max @ A @ X3 @ Y ) % 0.31/0.59 = Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_absorb2 % 0.31/0.59 thf(fact_4495_of__nat__max,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linord181362715937106298miring @ A ) % 0.31/0.59 => ! [X3: nat,Y: nat] : % 0.31/0.59 ( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X3 @ Y ) ) % 0.31/0.59 = ( ord_max @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % of_nat_max % 0.31/0.59 thf(fact_4496_shuffles__commutes,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( shuffles @ A ) % 0.31/0.59 = ( ^ [Xs2: list @ A,Ys3: list @ A] : ( shuffles @ A @ Ys3 @ Xs2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % shuffles_commutes % 0.31/0.59 thf(fact_4497_of__int__max,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( linordered_idom @ A ) % 0.31/0.59 => ! [X3: int,Y: int] : % 0.31/0.59 ( ( ring_1_of_int @ A @ ( ord_max @ int @ X3 @ Y ) ) % 0.31/0.59 = ( ord_max @ A @ ( ring_1_of_int @ A @ X3 ) @ ( ring_1_of_int @ A @ Y ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % of_int_max % 0.31/0.59 thf(fact_4498_nat__add__max__right,axiom, % 0.31/0.59 ! [M: nat,N2: nat,Q4: nat] : % 0.31/0.59 ( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N2 @ Q4 ) ) % 0.31/0.59 = ( ord_max @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( plus_plus @ nat @ M @ Q4 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % nat_add_max_right % 0.31/0.59 thf(fact_4499_nat__add__max__left,axiom, % 0.31/0.59 ! [M: nat,N2: nat,Q4: nat] : % 0.31/0.59 ( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N2 ) @ Q4 ) % 0.31/0.59 = ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q4 ) @ ( plus_plus @ nat @ N2 @ Q4 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % nat_add_max_left % 0.31/0.59 thf(fact_4500_max__add__distrib__right,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ordere2412721322843649153imp_le @ A ) % 0.31/0.59 => ! [X3: A,Y: A,Z2: A] : % 0.31/0.59 ( ( plus_plus @ A @ X3 @ ( ord_max @ A @ Y @ Z2 ) ) % 0.31/0.59 = ( ord_max @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( plus_plus @ A @ X3 @ Z2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_add_distrib_right % 0.31/0.59 thf(fact_4501_max__add__distrib__left,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ordere2412721322843649153imp_le @ A ) % 0.31/0.59 => ! [X3: A,Y: A,Z2: A] : % 0.31/0.59 ( ( plus_plus @ A @ ( ord_max @ A @ X3 @ Y ) @ Z2 ) % 0.31/0.59 = ( ord_max @ A @ ( plus_plus @ A @ X3 @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_add_distrib_left % 0.31/0.59 thf(fact_4502_max__diff__distrib__left,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( ordered_ab_group_add @ A ) % 0.31/0.59 => ! [X3: A,Y: A,Z2: A] : % 0.31/0.59 ( ( minus_minus @ A @ ( ord_max @ A @ X3 @ Y ) @ Z2 ) % 0.31/0.59 = ( ord_max @ A @ ( minus_minus @ A @ X3 @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % max_diff_distrib_left % 0.31/0.59 thf(fact_4503_nat__mult__max__left,axiom, % 0.31/0.59 ! [M: nat,N2: nat,Q4: nat] : % 0.31/0.59 ( ( times_times @ nat @ ( ord_max @ nat @ M @ N2 ) @ Q4 ) % 0.31/0.59 = ( ord_max @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N2 @ Q4 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % nat_mult_max_left % 0.31/0.59 thf(fact_4504_nat__mult__max__right,axiom, % 0.31/0.59 ! [M: nat,N2: nat,Q4: nat] : % 0.31/0.59 ( ( times_times @ nat @ M @ ( ord_max @ nat @ N2 @ Q4 ) ) % 0.31/0.59 = ( ord_max @ nat @ ( times_times @ nat @ M @ N2 ) @ ( times_times @ nat @ M @ Q4 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % nat_mult_max_right % 0.31/0.59 thf(fact_4505_nat__minus__add__max,axiom, % 0.31/0.59 ! [N2: nat,M: nat] : % 0.31/0.59 ( ( plus_plus @ nat @ ( minus_minus @ nat @ N2 @ M ) @ M ) % 0.31/0.59 = ( ord_max @ nat @ N2 @ M ) ) ). % 0.31/0.59 % 0.31/0.59 % nat_minus_add_max % 0.31/0.59 thf(fact_4506_length__shuffles,axiom, % 0.31/0.59 ! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys: list @ A] : % 0.31/0.59 ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) ) % 0.31/0.59 => ( ( size_size @ ( list @ A ) @ Zs2 ) % 0.31/0.59 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % length_shuffles % 0.31/0.59 thf(fact_4507_max__Suc2,axiom, % 0.31/0.59 ! [M: nat,N2: nat] : % 0.31/0.59 ( ( ord_max @ nat @ M @ ( suc @ N2 ) ) % 0.31/0.59 = ( case_nat @ nat @ ( suc @ N2 ) % 0.31/0.59 @ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ M4 @ N2 ) ) % 0.31/0.59 @ M ) ) ). % 0.31/0.59 % 0.31/0.59 % max_Suc2 % 0.31/0.59 thf(fact_4508_max__Suc1,axiom, % 0.31/0.59 ! [N2: nat,M: nat] : % 0.31/0.59 ( ( ord_max @ nat @ ( suc @ N2 ) @ M ) % 0.31/0.59 = ( case_nat @ nat @ ( suc @ N2 ) % 0.31/0.59 @ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ N2 @ M4 ) ) % 0.31/0.59 @ M ) ) ). % 0.31/0.59 % 0.31/0.59 % max_Suc1 % 0.31/0.59 thf(fact_4509_Sup__insert__finite,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( condit6923001295902523014norder @ A ) % 0.31/0.59 => ! [S3: set @ A,X3: A] : % 0.31/0.59 ( ( finite_finite2 @ A @ S3 ) % 0.31/0.59 => ( ( ( S3 % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ( complete_Sup_Sup @ A @ ( insert @ A @ X3 @ S3 ) ) % 0.31/0.59 = X3 ) ) % 0.31/0.59 & ( ( S3 % 0.31/0.59 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ( complete_Sup_Sup @ A @ ( insert @ A @ X3 @ S3 ) ) % 0.31/0.59 = ( ord_max @ A @ X3 @ ( complete_Sup_Sup @ A @ S3 ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % Sup_insert_finite % 0.31/0.59 thf(fact_4510_distinct__disjoint__shuffles,axiom, % 0.31/0.59 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] : % 0.31/0.59 ( ( distinct @ A @ Xs ) % 0.31/0.59 => ( ( distinct @ A @ Ys ) % 0.31/0.59 => ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) % 0.31/0.59 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.59 => ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) ) % 0.31/0.59 => ( distinct @ A @ Zs2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % distinct_disjoint_shuffles % 0.31/0.59 thf(fact_4511_in__set__product__lists__length,axiom, % 0.31/0.59 ! [A: $tType,Xs: list @ A,Xss: list @ ( list @ A )] : % 0.31/0.59 ( ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) ) % 0.31/0.59 => ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.59 = ( size_size @ ( list @ ( list @ A ) ) @ Xss ) ) ) ). % 0.31/0.59 % 0.31/0.59 % in_set_product_lists_length % 0.31/0.59 thf(fact_4512_mlex__eq,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( mlex_prod @ A ) % 0.31/0.59 = ( ^ [F3: A > nat,R6: set @ ( product_prod @ A @ A )] : % 0.31/0.59 ( collect @ ( product_prod @ A @ A ) % 0.31/0.59 @ ( product_case_prod @ A @ A @ $o % 0.31/0.59 @ ^ [X: A,Y2: A] : % 0.31/0.59 ( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y2 ) ) % 0.31/0.59 | ( ( ord_less_eq @ nat @ ( F3 @ X ) @ ( F3 @ Y2 ) ) % 0.31/0.59 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R6 ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % mlex_eq % 0.31/0.59 thf(fact_4513_divmod__integer__eq__cases,axiom, % 0.31/0.59 ( code_divmod_integer % 0.31/0.59 = ( ^ [K3: code_integer,L2: code_integer] : % 0.31/0.59 ( if @ ( product_prod @ code_integer @ code_integer ) % 0.31/0.59 @ ( K3 % 0.31/0.59 = ( zero_zero @ code_integer ) ) % 0.31/0.59 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) % 0.31/0.59 @ ( if @ ( product_prod @ code_integer @ code_integer ) % 0.31/0.59 @ ( L2 % 0.31/0.59 = ( zero_zero @ code_integer ) ) % 0.31/0.59 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) % 0.31/0.59 @ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( product_apsnd @ code_integer @ code_integer @ code_integer ) @ ( times_times @ code_integer ) ) @ ( sgn_sgn @ code_integer ) @ L2 % 0.31/0.59 @ ( if @ ( product_prod @ code_integer @ code_integer ) % 0.31/0.59 @ ( ( sgn_sgn @ code_integer @ K3 ) % 0.31/0.59 = ( sgn_sgn @ code_integer @ L2 ) ) % 0.31/0.59 @ ( code_divmod_abs @ K3 @ L2 ) % 0.31/0.59 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer ) % 0.31/0.59 @ ^ [R5: code_integer,S8: code_integer] : % 0.31/0.59 ( if @ ( product_prod @ code_integer @ code_integer ) % 0.31/0.59 @ ( S8 % 0.31/0.59 = ( zero_zero @ code_integer ) ) % 0.31/0.59 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) ) % 0.31/0.59 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L2 ) @ S8 ) ) ) % 0.31/0.59 @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % divmod_integer_eq_cases % 0.31/0.59 thf(fact_4514_totally__bounded__Union,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( topolo7287701948861334536_space @ A ) % 0.31/0.59 => ! [M7: set @ ( set @ A )] : % 0.31/0.59 ( ( finite_finite2 @ ( set @ A ) @ M7 ) % 0.31/0.59 => ( ! [S5: set @ A] : % 0.31/0.59 ( ( member @ ( set @ A ) @ S5 @ M7 ) % 0.31/0.59 => ( topolo6688025880775521714ounded @ A @ S5 ) ) % 0.31/0.59 => ( topolo6688025880775521714ounded @ A @ ( complete_Sup_Sup @ ( set @ A ) @ M7 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % totally_bounded_Union % 0.31/0.59 thf(fact_4515_nth__rotate1,axiom, % 0.31/0.59 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.59 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.59 => ( ( nth @ A @ ( rotate1 @ A @ Xs ) @ N2 ) % 0.31/0.59 = ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( suc @ N2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % nth_rotate1 % 0.31/0.59 thf(fact_4516_set__rotate1,axiom, % 0.31/0.59 ! [A: $tType,Xs: list @ A] : % 0.31/0.59 ( ( set2 @ A @ ( rotate1 @ A @ Xs ) ) % 0.31/0.59 = ( set2 @ A @ Xs ) ) ). % 0.31/0.59 % 0.31/0.59 % set_rotate1 % 0.31/0.59 thf(fact_4517_length__rotate1,axiom, % 0.31/0.59 ! [A: $tType,Xs: list @ A] : % 0.31/0.59 ( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) ) % 0.31/0.59 = ( size_size @ ( list @ A ) @ Xs ) ) ). % 0.31/0.59 % 0.31/0.59 % length_rotate1 % 0.31/0.59 thf(fact_4518_distinct1__rotate,axiom, % 0.31/0.59 ! [A: $tType,Xs: list @ A] : % 0.31/0.59 ( ( distinct @ A @ ( rotate1 @ A @ Xs ) ) % 0.31/0.59 = ( distinct @ A @ Xs ) ) ). % 0.31/0.59 % 0.31/0.59 % distinct1_rotate % 0.31/0.59 thf(fact_4519_totally__bounded__empty,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( topolo7287701948861334536_space @ A ) % 0.31/0.59 => ( topolo6688025880775521714ounded @ A @ ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % totally_bounded_empty % 0.31/0.59 thf(fact_4520_rotate1__length01,axiom, % 0.31/0.59 ! [A: $tType,Xs: list @ A] : % 0.31/0.59 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) % 0.31/0.59 => ( ( rotate1 @ A @ Xs ) % 0.31/0.59 = Xs ) ) ). % 0.31/0.59 % 0.31/0.59 % rotate1_length01 % 0.31/0.59 thf(fact_4521_apsnd__compose,axiom, % 0.31/0.59 ! [C: $tType,B: $tType,D: $tType,A: $tType,F2: C > B,G: D > C,X3: product_prod @ A @ D] : % 0.31/0.59 ( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apsnd @ D @ C @ A @ G @ X3 ) ) % 0.31/0.59 = ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F2 @ G ) @ X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % apsnd_compose % 0.31/0.59 thf(fact_4522_totally__bounded__subset,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( topolo7287701948861334536_space @ A ) % 0.31/0.59 => ! [S3: set @ A,T4: set @ A] : % 0.31/0.59 ( ( topolo6688025880775521714ounded @ A @ S3 ) % 0.31/0.59 => ( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 ) % 0.31/0.59 => ( topolo6688025880775521714ounded @ A @ T4 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % totally_bounded_subset % 0.31/0.59 thf(fact_4523_sum__comp__morphism,axiom, % 0.31/0.59 ! [A: $tType,B: $tType,C: $tType] : % 0.31/0.59 ( ( ( comm_monoid_add @ B ) % 0.31/0.59 & ( comm_monoid_add @ A ) ) % 0.31/0.59 => ! [H2: B > A,G: C > B,A3: set @ C] : % 0.31/0.59 ( ( ( H2 @ ( zero_zero @ B ) ) % 0.31/0.59 = ( zero_zero @ A ) ) % 0.31/0.59 => ( ! [X5: B,Y4: B] : % 0.31/0.59 ( ( H2 @ ( plus_plus @ B @ X5 @ Y4 ) ) % 0.31/0.59 = ( plus_plus @ A @ ( H2 @ X5 ) @ ( H2 @ Y4 ) ) ) % 0.31/0.59 => ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H2 @ G ) @ A3 ) % 0.31/0.59 = ( H2 @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A3 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum_comp_morphism % 0.31/0.59 thf(fact_4524_case__prod__comp,axiom, % 0.31/0.59 ! [D: $tType,A: $tType,C: $tType,B: $tType,F2: D > C > A,G: B > D,X3: product_prod @ B @ C] : % 0.31/0.59 ( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F2 @ G ) @ X3 ) % 0.31/0.59 = ( F2 @ ( G @ ( product_fst @ B @ C @ X3 ) ) @ ( product_snd @ B @ C @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % case_prod_comp % 0.31/0.59 thf(fact_4525_mlex__leq,axiom, % 0.31/0.59 ! [A: $tType,F2: A > nat,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] : % 0.31/0.59 ( ( ord_less_eq @ nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) % 0.31/0.59 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R ) % 0.31/0.59 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( mlex_prod @ A @ F2 @ R ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % mlex_leq % 0.31/0.59 thf(fact_4526_mlex__less,axiom, % 0.31/0.59 ! [A: $tType,F2: A > nat,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] : % 0.31/0.59 ( ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) % 0.31/0.59 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( mlex_prod @ A @ F2 @ R ) ) ) ). % 0.31/0.59 % 0.31/0.59 % mlex_less % 0.31/0.59 thf(fact_4527_mlex__iff,axiom, % 0.31/0.59 ! [A: $tType,X3: A,Y: A,F2: A > nat,R: set @ ( product_prod @ A @ A )] : % 0.31/0.59 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( mlex_prod @ A @ F2 @ R ) ) % 0.31/0.59 = ( ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) % 0.31/0.59 | ( ( ( F2 @ X3 ) % 0.31/0.59 = ( F2 @ Y ) ) % 0.31/0.59 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % mlex_iff % 0.31/0.59 thf(fact_4528_set__remove1__eq,axiom, % 0.31/0.59 ! [A: $tType,Xs: list @ A,X3: A] : % 0.31/0.59 ( ( distinct @ A @ Xs ) % 0.31/0.59 => ( ( set2 @ A @ ( remove1 @ A @ X3 @ Xs ) ) % 0.31/0.59 = ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % set_remove1_eq % 0.31/0.59 thf(fact_4529_in__set__remove1,axiom, % 0.31/0.59 ! [A: $tType,A2: A,B2: A,Xs: list @ A] : % 0.31/0.59 ( ( A2 != B2 ) % 0.31/0.59 => ( ( member @ A @ A2 @ ( set2 @ A @ ( remove1 @ A @ B2 @ Xs ) ) ) % 0.31/0.59 = ( member @ A @ A2 @ ( set2 @ A @ Xs ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % in_set_remove1 % 0.31/0.59 thf(fact_4530_fst__comp__apsnd,axiom, % 0.31/0.59 ! [C: $tType,B: $tType,A: $tType,F2: B > C] : % 0.31/0.59 ( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 ) ) % 0.31/0.59 = ( product_fst @ A @ B ) ) ). % 0.31/0.59 % 0.31/0.59 % fst_comp_apsnd % 0.31/0.59 thf(fact_4531_snd__comp__apsnd,axiom, % 0.31/0.59 ! [C: $tType,B: $tType,A: $tType,F2: B > C] : % 0.31/0.59 ( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 ) ) % 0.31/0.59 = ( comp @ B @ C @ ( product_prod @ A @ B ) @ F2 @ ( product_snd @ A @ B ) ) ) ). % 0.31/0.59 % 0.31/0.59 % snd_comp_apsnd % 0.31/0.59 thf(fact_4532_fst__diag__fst,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A ) % 0.31/0.59 @ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B ) % 0.31/0.59 @ ^ [X: A] : ( product_Pair @ A @ A @ X @ X ) % 0.31/0.59 @ ( product_fst @ A @ B ) ) ) % 0.31/0.59 = ( product_fst @ A @ B ) ) ). % 0.31/0.59 % 0.31/0.59 % fst_diag_fst % 0.31/0.59 thf(fact_4533_snd__diag__snd,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B ) % 0.31/0.59 @ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B ) % 0.31/0.59 @ ^ [X: B] : ( product_Pair @ B @ B @ X @ X ) % 0.31/0.59 @ ( product_snd @ A @ B ) ) ) % 0.31/0.59 = ( product_snd @ A @ B ) ) ). % 0.31/0.59 % 0.31/0.59 % snd_diag_snd % 0.31/0.59 thf(fact_4534_card_Ocomp__fun__commute__on,axiom, % 0.31/0.59 ( ( comp @ nat @ nat @ nat @ suc @ suc ) % 0.31/0.59 = ( comp @ nat @ nat @ nat @ suc @ suc ) ) ). % 0.31/0.59 % 0.31/0.59 % card.comp_fun_commute_on % 0.31/0.59 thf(fact_4535_fst__diag__snd,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B ) % 0.31/0.59 @ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B ) % 0.31/0.59 @ ^ [X: B] : ( product_Pair @ B @ B @ X @ X ) % 0.31/0.59 @ ( product_snd @ A @ B ) ) ) % 0.31/0.59 = ( product_snd @ A @ B ) ) ). % 0.31/0.59 % 0.31/0.59 % fst_diag_snd % 0.31/0.59 thf(fact_4536_snd__diag__fst,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A ) % 0.31/0.59 @ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B ) % 0.31/0.59 @ ^ [X: A] : ( product_Pair @ A @ A @ X @ X ) % 0.31/0.59 @ ( product_fst @ A @ B ) ) ) % 0.31/0.59 = ( product_fst @ A @ B ) ) ). % 0.31/0.59 % 0.31/0.59 % snd_diag_fst % 0.31/0.59 thf(fact_4537_distinct__remove1,axiom, % 0.31/0.59 ! [A: $tType,Xs: list @ A,X3: A] : % 0.31/0.59 ( ( distinct @ A @ Xs ) % 0.31/0.59 => ( distinct @ A @ ( remove1 @ A @ X3 @ Xs ) ) ) ). % 0.31/0.59 % 0.31/0.59 % distinct_remove1 % 0.31/0.59 thf(fact_4538_notin__set__remove1,axiom, % 0.31/0.59 ! [A: $tType,X3: A,Xs: list @ A,Y: A] : % 0.31/0.59 ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.59 => ~ ( member @ A @ X3 @ ( set2 @ A @ ( remove1 @ A @ Y @ Xs ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % notin_set_remove1 % 0.31/0.59 thf(fact_4539_remove1__idem,axiom, % 0.31/0.59 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.59 ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.59 => ( ( remove1 @ A @ X3 @ Xs ) % 0.31/0.59 = Xs ) ) ). % 0.31/0.59 % 0.31/0.59 % remove1_idem % 0.31/0.59 thf(fact_4540_remove1__commute,axiom, % 0.31/0.59 ! [A: $tType,X3: A,Y: A,Zs2: list @ A] : % 0.31/0.59 ( ( remove1 @ A @ X3 @ ( remove1 @ A @ Y @ Zs2 ) ) % 0.31/0.59 = ( remove1 @ A @ Y @ ( remove1 @ A @ X3 @ Zs2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % remove1_commute % 0.31/0.59 thf(fact_4541_set__remove1__subset,axiom, % 0.31/0.59 ! [A: $tType,X3: A,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( remove1 @ A @ X3 @ Xs ) ) @ ( set2 @ A @ Xs ) ) ). % 0.31/0.59 % 0.31/0.59 % set_remove1_subset % 0.31/0.59 thf(fact_4542_sum_OatLeast__Suc__atMost__Suc__shift,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [G: nat > A,M: nat,N2: nat] : % 0.31/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 0.31/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.atLeast_Suc_atMost_Suc_shift % 0.31/0.59 thf(fact_4543_sum_OatLeast__Suc__lessThan__Suc__shift,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [G: nat > A,M: nat,N2: nat] : % 0.31/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 0.31/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.atLeast_Suc_lessThan_Suc_shift % 0.31/0.59 thf(fact_4544_sum_OatLeastAtMost__shift__bounds,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 0.31/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 0.31/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.atLeastAtMost_shift_bounds % 0.31/0.59 thf(fact_4545_sum_OatLeastLessThan__shift__bounds,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 0.31/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 0.31/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.atLeastLessThan_shift_bounds % 0.31/0.59 thf(fact_4546_prod_OatLeast__Suc__atMost__Suc__shift,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [G: nat > A,M: nat,N2: nat] : % 0.31/0.59 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 0.31/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.atLeast_Suc_atMost_Suc_shift % 0.31/0.59 thf(fact_4547_prod_OatLeast__Suc__lessThan__Suc__shift,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [G: nat > A,M: nat,N2: nat] : % 0.31/0.59 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 0.31/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.atLeast_Suc_lessThan_Suc_shift % 0.31/0.59 thf(fact_4548_prod_OatLeastAtMost__shift__bounds,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 0.31/0.59 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 0.31/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.atLeastAtMost_shift_bounds % 0.31/0.59 thf(fact_4549_prod_OatLeastLessThan__shift__bounds,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 0.31/0.59 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 0.31/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.atLeastLessThan_shift_bounds % 0.31/0.59 thf(fact_4550_prod_OUnion__comp,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [B3: set @ ( set @ B ),G: B > A] : % 0.31/0.59 ( ! [X5: set @ B] : % 0.31/0.59 ( ( member @ ( set @ B ) @ X5 @ B3 ) % 0.31/0.59 => ( finite_finite2 @ B @ X5 ) ) % 0.31/0.59 => ( ! [A14: set @ B] : % 0.31/0.59 ( ( member @ ( set @ B ) @ A14 @ B3 ) % 0.31/0.59 => ! [A25: set @ B] : % 0.31/0.59 ( ( member @ ( set @ B ) @ A25 @ B3 ) % 0.31/0.59 => ( ( A14 != A25 ) % 0.31/0.59 => ! [X5: B] : % 0.31/0.59 ( ( member @ B @ X5 @ A14 ) % 0.31/0.59 => ( ( member @ B @ X5 @ A25 ) % 0.31/0.59 => ( ( G @ X5 ) % 0.31/0.59 = ( one_one @ A ) ) ) ) ) ) ) % 0.31/0.59 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B3 ) ) % 0.31/0.59 = ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B3 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.Union_comp % 0.31/0.59 thf(fact_4551_prod_OUnion__disjoint,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [C5: set @ ( set @ B ),G: B > A] : % 0.31/0.59 ( ! [X5: set @ B] : % 0.31/0.59 ( ( member @ ( set @ B ) @ X5 @ C5 ) % 0.31/0.59 => ( finite_finite2 @ B @ X5 ) ) % 0.31/0.59 => ( ! [X5: set @ B] : % 0.31/0.59 ( ( member @ ( set @ B ) @ X5 @ C5 ) % 0.31/0.59 => ! [Xa3: set @ B] : % 0.31/0.59 ( ( member @ ( set @ B ) @ Xa3 @ C5 ) % 0.31/0.59 => ( ( X5 != Xa3 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ B ) @ X5 @ Xa3 ) % 0.31/0.59 = ( bot_bot @ ( set @ B ) ) ) ) ) ) % 0.31/0.59 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C5 ) ) % 0.31/0.59 = ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ C5 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.Union_disjoint % 0.31/0.59 thf(fact_4552_sum_OatLeast0__atMost__Suc__shift,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [G: nat > A,N2: nat] : % 0.31/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 0.31/0.59 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.atLeast0_atMost_Suc_shift % 0.31/0.59 thf(fact_4553_sum_OatLeast0__lessThan__Suc__shift,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [G: nat > A,N2: nat] : % 0.31/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 0.31/0.59 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.atLeast0_lessThan_Suc_shift % 0.31/0.59 thf(fact_4554_prod_OatLeast0__atMost__Suc__shift,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [G: nat > A,N2: nat] : % 0.31/0.59 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 0.31/0.59 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.atLeast0_atMost_Suc_shift % 0.31/0.59 thf(fact_4555_prod_OatLeast0__lessThan__Suc__shift,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [G: nat > A,N2: nat] : % 0.31/0.59 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 0.31/0.59 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.atLeast0_lessThan_Suc_shift % 0.31/0.59 thf(fact_4556_sum_OatLeastLessThan__shift__0,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [G: nat > A,M: nat,N2: nat] : % 0.31/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) % 0.31/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.atLeastLessThan_shift_0 % 0.31/0.59 thf(fact_4557_prod_OatLeastLessThan__shift__0,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [G: nat > A,M: nat,N2: nat] : % 0.31/0.59 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) % 0.31/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.atLeastLessThan_shift_0 % 0.31/0.59 thf(fact_4558_length__remove1,axiom, % 0.31/0.59 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.59 ( ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.59 => ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X3 @ Xs ) ) % 0.31/0.59 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) % 0.31/0.59 & ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.59 => ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X3 @ Xs ) ) % 0.31/0.59 = ( size_size @ ( list @ A ) @ Xs ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % length_remove1 % 0.31/0.59 thf(fact_4559_sum_OUnion__comp,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [B3: set @ ( set @ B ),G: B > A] : % 0.31/0.59 ( ! [X5: set @ B] : % 0.31/0.59 ( ( member @ ( set @ B ) @ X5 @ B3 ) % 0.31/0.59 => ( finite_finite2 @ B @ X5 ) ) % 0.31/0.59 => ( ! [A14: set @ B] : % 0.31/0.59 ( ( member @ ( set @ B ) @ A14 @ B3 ) % 0.31/0.59 => ! [A25: set @ B] : % 0.31/0.59 ( ( member @ ( set @ B ) @ A25 @ B3 ) % 0.31/0.59 => ( ( A14 != A25 ) % 0.31/0.59 => ! [X5: B] : % 0.31/0.59 ( ( member @ B @ X5 @ A14 ) % 0.31/0.59 => ( ( member @ B @ X5 @ A25 ) % 0.31/0.59 => ( ( G @ X5 ) % 0.31/0.59 = ( zero_zero @ A ) ) ) ) ) ) ) % 0.31/0.59 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B3 ) ) % 0.31/0.59 = ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ B3 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.Union_comp % 0.31/0.59 thf(fact_4560_sum_OatLeast__atMost__pred__shift,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [G: nat > A,M: nat,N2: nat] : % 0.31/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.59 @ ( comp @ nat @ A @ nat @ G % 0.31/0.59 @ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) % 0.31/0.59 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 0.31/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.atLeast_atMost_pred_shift % 0.31/0.59 thf(fact_4561_sum_OatLeast__lessThan__pred__shift,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [G: nat > A,M: nat,N2: nat] : % 0.31/0.59 ( ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.59 @ ( comp @ nat @ A @ nat @ G % 0.31/0.59 @ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) % 0.31/0.59 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 0.31/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.atLeast_lessThan_pred_shift % 0.31/0.59 thf(fact_4562_prod_OatLeast__atMost__pred__shift,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [G: nat > A,M: nat,N2: nat] : % 0.31/0.59 ( ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.59 @ ( comp @ nat @ A @ nat @ G % 0.31/0.59 @ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) % 0.31/0.59 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 0.31/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.atLeast_atMost_pred_shift % 0.31/0.59 thf(fact_4563_prod_OatLeast__lessThan__pred__shift,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [G: nat > A,M: nat,N2: nat] : % 0.31/0.59 ( ( groups7121269368397514597t_prod @ nat @ A % 0.31/0.59 @ ( comp @ nat @ A @ nat @ G % 0.31/0.59 @ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) % 0.31/0.59 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 0.31/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.atLeast_lessThan_pred_shift % 0.31/0.59 thf(fact_4564_sum_OatLeastAtMost__shift__0,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [M: nat,N2: nat,G: nat > A] : % 0.31/0.59 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.59 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.31/0.59 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.atLeastAtMost_shift_0 % 0.31/0.59 thf(fact_4565_prod_OatLeastAtMost__shift__0,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( comm_monoid_mult @ A ) % 0.31/0.59 => ! [M: nat,N2: nat,G: nat > A] : % 0.31/0.59 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.59 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.31/0.59 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % prod.atLeastAtMost_shift_0 % 0.31/0.59 thf(fact_4566_sum_OUnion__disjoint,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( comm_monoid_add @ A ) % 0.31/0.59 => ! [C5: set @ ( set @ B ),G: B > A] : % 0.31/0.59 ( ! [X5: set @ B] : % 0.31/0.59 ( ( member @ ( set @ B ) @ X5 @ C5 ) % 0.31/0.59 => ( finite_finite2 @ B @ X5 ) ) % 0.31/0.59 => ( ! [X5: set @ B] : % 0.31/0.59 ( ( member @ ( set @ B ) @ X5 @ C5 ) % 0.31/0.59 => ! [Xa3: set @ B] : % 0.31/0.59 ( ( member @ ( set @ B ) @ Xa3 @ C5 ) % 0.31/0.59 => ( ( X5 != Xa3 ) % 0.31/0.59 => ( ( inf_inf @ ( set @ B ) @ X5 @ Xa3 ) % 0.31/0.59 = ( bot_bot @ ( set @ B ) ) ) ) ) ) % 0.31/0.59 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C5 ) ) % 0.31/0.59 = ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ C5 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sum.Union_disjoint % 0.31/0.59 thf(fact_4567_in__finite__psubset,axiom, % 0.31/0.59 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.59 ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A3 @ B3 ) @ ( finite_psubset @ A ) ) % 0.31/0.59 = ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.59 & ( finite_finite2 @ A @ B3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % in_finite_psubset % 0.31/0.59 thf(fact_4568_snd__fst__flip,axiom, % 0.31/0.59 ! [A: $tType,B: $tType] : % 0.31/0.59 ( ( product_snd @ B @ A ) % 0.31/0.59 = ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B ) % 0.31/0.59 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B ) % 0.31/0.59 @ ^ [X: B,Y2: A] : ( product_Pair @ A @ B @ Y2 @ X ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % snd_fst_flip % 0.31/0.59 thf(fact_4569_fst__snd__flip,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( product_fst @ A @ B ) % 0.31/0.59 = ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A ) % 0.31/0.59 @ ( product_case_prod @ A @ B @ ( product_prod @ B @ A ) % 0.31/0.59 @ ^ [X: A,Y2: B] : ( product_Pair @ B @ A @ Y2 @ X ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % fst_snd_flip % 0.31/0.59 thf(fact_4570_in__measure,axiom, % 0.31/0.59 ! [A: $tType,X3: A,Y: A,F2: A > nat] : % 0.31/0.59 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measure @ A @ F2 ) ) % 0.31/0.59 = ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % in_measure % 0.31/0.59 thf(fact_4571_fstI,axiom, % 0.31/0.59 ! [B: $tType,A: $tType,X3: product_prod @ A @ B,Y: A,Z2: B] : % 0.31/0.59 ( ( X3 % 0.31/0.59 = ( product_Pair @ A @ B @ Y @ Z2 ) ) % 0.31/0.59 => ( ( product_fst @ A @ B @ X3 ) % 0.31/0.59 = Y ) ) ). % 0.31/0.59 % 0.31/0.59 % fstI % 0.31/0.59 thf(fact_4572_sndI,axiom, % 0.31/0.59 ! [A: $tType,B: $tType,X3: product_prod @ A @ B,Y: A,Z2: B] : % 0.31/0.59 ( ( X3 % 0.31/0.59 = ( product_Pair @ A @ B @ Y @ Z2 ) ) % 0.31/0.59 => ( ( product_snd @ A @ B @ X3 ) % 0.31/0.59 = Z2 ) ) ). % 0.31/0.59 % 0.31/0.59 % sndI % 0.31/0.59 thf(fact_4573_finite__psubset__def,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( finite_psubset @ A ) % 0.31/0.59 = ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) % 0.31/0.59 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o % 0.31/0.59 @ ^ [A6: set @ A,B6: set @ A] : % 0.31/0.59 ( ( ord_less @ ( set @ A ) @ A6 @ B6 ) % 0.31/0.59 & ( finite_finite2 @ A @ B6 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % finite_psubset_def % 0.31/0.59 thf(fact_4574_drop__bit__rec,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ( ( bit_se4197421643247451524op_bit @ A ) % 0.31/0.59 = ( ^ [N: nat,A4: A] : % 0.31/0.59 ( if @ A % 0.31/0.59 @ ( N % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 @ A4 % 0.31/0.59 @ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % drop_bit_rec % 0.31/0.59 thf(fact_4575_root__powr__inverse,axiom, % 0.31/0.59 ! [N2: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.59 => ( ( root @ N2 @ X3 ) % 0.31/0.59 = ( powr @ real @ X3 @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % root_powr_inverse % 0.31/0.59 thf(fact_4576_image2__def,axiom, % 0.31/0.59 ! [A: $tType,B: $tType,C: $tType] : % 0.31/0.59 ( ( bNF_Greatest_image2 @ C @ A @ B ) % 0.31/0.59 = ( ^ [A6: set @ C,F3: C > A,G2: C > B] : % 0.31/0.59 ( collect @ ( product_prod @ A @ B ) % 0.31/0.59 @ ^ [Uu3: product_prod @ A @ B] : % 0.31/0.59 ? [A4: C] : % 0.31/0.59 ( ( Uu3 % 0.31/0.59 = ( product_Pair @ A @ B @ ( F3 @ A4 ) @ ( G2 @ A4 ) ) ) % 0.31/0.59 & ( member @ C @ A4 @ A6 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % image2_def % 0.31/0.59 thf(fact_4577_drop__bit__negative__int__iff,axiom, % 0.31/0.59 ! [N2: nat,K: int] : % 0.31/0.59 ( ( ord_less @ int @ ( bit_se4197421643247451524op_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) % 0.31/0.59 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ). % 0.31/0.59 % 0.31/0.59 % drop_bit_negative_int_iff % 0.31/0.59 thf(fact_4578_drop__bit__of__0,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [N2: nat] : % 0.31/0.59 ( ( bit_se4197421643247451524op_bit @ A @ N2 @ ( zero_zero @ A ) ) % 0.31/0.59 = ( zero_zero @ A ) ) ) ). % 0.31/0.59 % 0.31/0.59 % drop_bit_of_0 % 0.31/0.59 thf(fact_4579_drop__bit__drop__bit,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [M: nat,N2: nat,A2: A] : % 0.31/0.59 ( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) ) % 0.31/0.59 = ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ A2 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % drop_bit_drop_bit % 0.31/0.59 thf(fact_4580_real__root__Suc__0,axiom, % 0.31/0.59 ! [X3: real] : % 0.31/0.59 ( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X3 ) % 0.31/0.59 = X3 ) ). % 0.31/0.59 % 0.31/0.59 % real_root_Suc_0 % 0.31/0.59 thf(fact_4581_root__0,axiom, % 0.31/0.59 ! [X3: real] : % 0.31/0.59 ( ( root @ ( zero_zero @ nat ) @ X3 ) % 0.31/0.59 = ( zero_zero @ real ) ) ). % 0.31/0.59 % 0.31/0.59 % root_0 % 0.31/0.59 thf(fact_4582_real__root__eq__iff,axiom, % 0.31/0.59 ! [N2: nat,X3: real,Y: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ( root @ N2 @ X3 ) % 0.31/0.59 = ( root @ N2 @ Y ) ) % 0.31/0.59 = ( X3 = Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_eq_iff % 0.31/0.59 thf(fact_4583_drop__bit__of__bool,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [N2: nat,B2: $o] : % 0.31/0.59 ( ( bit_se4197421643247451524op_bit @ A @ N2 @ ( zero_neq_one_of_bool @ A @ B2 ) ) % 0.31/0.59 = ( zero_neq_one_of_bool @ A % 0.31/0.59 @ ( ( N2 % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 & B2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % drop_bit_of_bool % 0.31/0.59 thf(fact_4584_drop__bit__of__Suc__0,axiom, % 0.31/0.59 ! [N2: nat] : % 0.31/0.59 ( ( bit_se4197421643247451524op_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 0.31/0.59 = ( zero_neq_one_of_bool @ nat % 0.31/0.59 @ ( N2 % 0.31/0.59 = ( zero_zero @ nat ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % drop_bit_of_Suc_0 % 0.31/0.59 thf(fact_4585_drop__bit__Suc__minus__bit0,axiom, % 0.31/0.59 ! [N2: nat,K: num] : % 0.31/0.59 ( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) ) % 0.31/0.59 = ( bit_se4197421643247451524op_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % drop_bit_Suc_minus_bit0 % 0.31/0.59 thf(fact_4586_drop__bit__Suc__bit0,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_un5681908812861735899ations @ A ) % 0.31/0.59 => ! [N2: nat,K: num] : % 0.31/0.59 ( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) % 0.31/0.59 = ( bit_se4197421643247451524op_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % drop_bit_Suc_bit0 % 0.31/0.59 thf(fact_4587_drop__bit__Suc__bit1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_un5681908812861735899ations @ A ) % 0.31/0.59 => ! [N2: nat,K: num] : % 0.31/0.59 ( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) ) % 0.31/0.59 = ( bit_se4197421643247451524op_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % drop_bit_Suc_bit1 % 0.31/0.59 thf(fact_4588_real__root__eq__0__iff,axiom, % 0.31/0.59 ! [N2: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ( root @ N2 @ X3 ) % 0.31/0.59 = ( zero_zero @ real ) ) % 0.31/0.59 = ( X3 % 0.31/0.59 = ( zero_zero @ real ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_eq_0_iff % 0.31/0.59 thf(fact_4589_real__root__less__iff,axiom, % 0.31/0.59 ! [N2: nat,X3: real,Y: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less @ real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y ) ) % 0.31/0.59 = ( ord_less @ real @ X3 @ Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_less_iff % 0.31/0.59 thf(fact_4590_real__root__le__iff,axiom, % 0.31/0.59 ! [N2: nat,X3: real,Y: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less_eq @ real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y ) ) % 0.31/0.59 = ( ord_less_eq @ real @ X3 @ Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_le_iff % 0.31/0.59 thf(fact_4591_real__root__eq__1__iff,axiom, % 0.31/0.59 ! [N2: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ( root @ N2 @ X3 ) % 0.31/0.59 = ( one_one @ real ) ) % 0.31/0.59 = ( X3 % 0.31/0.59 = ( one_one @ real ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_eq_1_iff % 0.31/0.59 thf(fact_4592_real__root__one,axiom, % 0.31/0.59 ! [N2: nat] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( root @ N2 @ ( one_one @ real ) ) % 0.31/0.59 = ( one_one @ real ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_one % 0.31/0.59 thf(fact_4593_drop__bit__of__1,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [N2: nat] : % 0.31/0.59 ( ( bit_se4197421643247451524op_bit @ A @ N2 @ ( one_one @ A ) ) % 0.31/0.59 = ( zero_neq_one_of_bool @ A % 0.31/0.59 @ ( N2 % 0.31/0.59 = ( zero_zero @ nat ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % drop_bit_of_1 % 0.31/0.59 thf(fact_4594_real__root__lt__0__iff,axiom, % 0.31/0.59 ! [N2: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less @ real @ ( root @ N2 @ X3 ) @ ( zero_zero @ real ) ) % 0.31/0.59 = ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_lt_0_iff % 0.31/0.59 thf(fact_4595_real__root__gt__0__iff,axiom, % 0.31/0.59 ! [N2: nat,Y: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N2 @ Y ) ) % 0.31/0.59 = ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_gt_0_iff % 0.31/0.59 thf(fact_4596_real__root__le__0__iff,axiom, % 0.31/0.59 ! [N2: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less_eq @ real @ ( root @ N2 @ X3 ) @ ( zero_zero @ real ) ) % 0.31/0.59 = ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_le_0_iff % 0.31/0.59 thf(fact_4597_real__root__ge__0__iff,axiom, % 0.31/0.59 ! [N2: nat,Y: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N2 @ Y ) ) % 0.31/0.59 = ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_ge_0_iff % 0.31/0.59 thf(fact_4598_real__root__lt__1__iff,axiom, % 0.31/0.59 ! [N2: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less @ real @ ( root @ N2 @ X3 ) @ ( one_one @ real ) ) % 0.31/0.59 = ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_lt_1_iff % 0.31/0.59 thf(fact_4599_real__root__gt__1__iff,axiom, % 0.31/0.59 ! [N2: nat,Y: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N2 @ Y ) ) % 0.31/0.59 = ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_gt_1_iff % 0.31/0.59 thf(fact_4600_real__root__le__1__iff,axiom, % 0.31/0.59 ! [N2: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less_eq @ real @ ( root @ N2 @ X3 ) @ ( one_one @ real ) ) % 0.31/0.59 = ( ord_less_eq @ real @ X3 @ ( one_one @ real ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_le_1_iff % 0.31/0.59 thf(fact_4601_real__root__ge__1__iff,axiom, % 0.31/0.59 ! [N2: nat,Y: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N2 @ Y ) ) % 0.31/0.59 = ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_ge_1_iff % 0.31/0.59 thf(fact_4602_drop__bit__Suc__minus__bit1,axiom, % 0.31/0.59 ! [N2: nat,K: num] : % 0.31/0.59 ( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) ) % 0.31/0.59 = ( bit_se4197421643247451524op_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % drop_bit_Suc_minus_bit1 % 0.31/0.59 thf(fact_4603_real__root__pow__pos2,axiom, % 0.31/0.59 ! [N2: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.59 => ( ( power_power @ real @ ( root @ N2 @ X3 ) @ N2 ) % 0.31/0.59 = X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_pow_pos2 % 0.31/0.59 thf(fact_4604_take__bit__eq__self__iff__drop__bit__eq__0,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [N2: nat,A2: A] : % 0.31/0.59 ( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) % 0.31/0.59 = A2 ) % 0.31/0.59 = ( ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) % 0.31/0.59 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % take_bit_eq_self_iff_drop_bit_eq_0 % 0.31/0.59 thf(fact_4605_take__bit__drop__bit,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [M: nat,N2: nat,A2: A] : % 0.31/0.59 ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) ) % 0.31/0.59 = ( bit_se4197421643247451524op_bit @ A @ N2 @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ A2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % take_bit_drop_bit % 0.31/0.59 thf(fact_4606_bit__drop__bit__eq,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [N2: nat,A2: A] : % 0.31/0.59 ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) ) % 0.31/0.59 = ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( plus_plus @ nat @ N2 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % bit_drop_bit_eq % 0.31/0.59 thf(fact_4607_real__root__less__mono,axiom, % 0.31/0.59 ! [N2: nat,X3: real,Y: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less @ real @ X3 @ Y ) % 0.31/0.59 => ( ord_less @ real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_less_mono % 0.31/0.59 thf(fact_4608_real__root__le__mono,axiom, % 0.31/0.59 ! [N2: nat,X3: real,Y: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less_eq @ real @ X3 @ Y ) % 0.31/0.59 => ( ord_less_eq @ real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_le_mono % 0.31/0.59 thf(fact_4609_real__root__power,axiom, % 0.31/0.59 ! [N2: nat,X3: real,K: nat] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( root @ N2 @ ( power_power @ real @ X3 @ K ) ) % 0.31/0.59 = ( power_power @ real @ ( root @ N2 @ X3 ) @ K ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_power % 0.31/0.59 thf(fact_4610_real__root__abs,axiom, % 0.31/0.59 ! [N2: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( root @ N2 @ ( abs_abs @ real @ X3 ) ) % 0.31/0.59 = ( abs_abs @ real @ ( root @ N2 @ X3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_abs % 0.31/0.59 thf(fact_4611_sgn__root,axiom, % 0.31/0.59 ! [N2: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( sgn_sgn @ real @ ( root @ N2 @ X3 ) ) % 0.31/0.59 = ( sgn_sgn @ real @ X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % sgn_root % 0.31/0.59 thf(fact_4612_bits__ident,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [N2: nat,A2: A] : % 0.31/0.59 ( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) % 0.31/0.59 = A2 ) ) ). % 0.31/0.59 % 0.31/0.59 % bits_ident % 0.31/0.59 thf(fact_4613_image2__eqI,axiom, % 0.31/0.59 ! [A: $tType,C: $tType,B: $tType,B2: A,F2: B > A,X3: B,C2: C,G: B > C,A3: set @ B] : % 0.31/0.59 ( ( B2 % 0.31/0.59 = ( F2 @ X3 ) ) % 0.31/0.59 => ( ( C2 % 0.31/0.59 = ( G @ X3 ) ) % 0.31/0.59 => ( ( member @ B @ X3 @ A3 ) % 0.31/0.59 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A3 @ F2 @ G ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % image2_eqI % 0.31/0.59 thf(fact_4614_real__root__gt__zero,axiom, % 0.31/0.59 ! [N2: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.59 => ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N2 @ X3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_gt_zero % 0.31/0.59 thf(fact_4615_real__root__strict__decreasing,axiom, % 0.31/0.59 ! [N2: nat,N7: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less @ nat @ N2 @ N7 ) % 0.31/0.59 => ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 0.31/0.59 => ( ord_less @ real @ ( root @ N7 @ X3 ) @ ( root @ N2 @ X3 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_strict_decreasing % 0.31/0.59 thf(fact_4616_root__abs__power,axiom, % 0.31/0.59 ! [N2: nat,Y: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( abs_abs @ real @ ( root @ N2 @ ( power_power @ real @ Y @ N2 ) ) ) % 0.31/0.59 = ( abs_abs @ real @ Y ) ) ) ). % 0.31/0.59 % 0.31/0.59 % root_abs_power % 0.31/0.59 thf(fact_4617_real__root__pos__pos,axiom, % 0.31/0.59 ! [N2: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.59 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N2 @ X3 ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_pos_pos % 0.31/0.59 thf(fact_4618_real__root__strict__increasing,axiom, % 0.31/0.59 ! [N2: nat,N7: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less @ nat @ N2 @ N7 ) % 0.31/0.59 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.59 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 0.31/0.59 => ( ord_less @ real @ ( root @ N2 @ X3 ) @ ( root @ N7 @ X3 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_strict_increasing % 0.31/0.59 thf(fact_4619_real__root__decreasing,axiom, % 0.31/0.59 ! [N2: nat,N7: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less_eq @ nat @ N2 @ N7 ) % 0.31/0.59 => ( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) % 0.31/0.59 => ( ord_less_eq @ real @ ( root @ N7 @ X3 ) @ ( root @ N2 @ X3 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_decreasing % 0.31/0.59 thf(fact_4620_real__root__pow__pos,axiom, % 0.31/0.59 ! [N2: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.59 => ( ( power_power @ real @ ( root @ N2 @ X3 ) @ N2 ) % 0.31/0.59 = X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_pow_pos % 0.31/0.59 thf(fact_4621_real__root__pos__unique,axiom, % 0.31/0.59 ! [N2: nat,Y: real,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) % 0.31/0.59 => ( ( ( power_power @ real @ Y @ N2 ) % 0.31/0.59 = X3 ) % 0.31/0.59 => ( ( root @ N2 @ X3 ) % 0.31/0.59 = Y ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_pos_unique % 0.31/0.59 thf(fact_4622_real__root__power__cancel,axiom, % 0.31/0.59 ! [N2: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.59 => ( ( root @ N2 @ ( power_power @ real @ X3 @ N2 ) ) % 0.31/0.59 = X3 ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_power_cancel % 0.31/0.59 thf(fact_4623_drop__bit__Suc,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.59 => ! [N2: nat,A2: A] : % 0.31/0.59 ( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ A2 ) % 0.31/0.59 = ( bit_se4197421643247451524op_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % drop_bit_Suc % 0.31/0.59 thf(fact_4624_real__root__increasing,axiom, % 0.31/0.59 ! [N2: nat,N7: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less_eq @ nat @ N2 @ N7 ) % 0.31/0.59 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.59 => ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) ) % 0.31/0.59 => ( ord_less_eq @ real @ ( root @ N2 @ X3 ) @ ( root @ N7 @ X3 ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % real_root_increasing % 0.31/0.59 thf(fact_4625_root__sgn__power,axiom, % 0.31/0.59 ! [N2: nat,Y: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( root @ N2 @ ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N2 ) ) ) % 0.31/0.59 = Y ) ) ). % 0.31/0.59 % 0.31/0.59 % root_sgn_power % 0.31/0.59 thf(fact_4626_sgn__power__root,axiom, % 0.31/0.59 ! [N2: nat,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N2 @ X3 ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N2 @ X3 ) ) @ N2 ) ) % 0.31/0.59 = X3 ) ) ). % 0.31/0.59 % 0.31/0.59 % sgn_power_root % 0.31/0.59 thf(fact_4627_ln__root,axiom, % 0.31/0.59 ! [N2: nat,B2: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 0.31/0.59 => ( ( ln_ln @ real @ ( root @ N2 @ B2 ) ) % 0.31/0.59 = ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % ln_root % 0.31/0.59 thf(fact_4628_log__root,axiom, % 0.31/0.59 ! [N2: nat,A2: real,B2: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.59 => ( ( log @ B2 @ ( root @ N2 @ A2 ) ) % 0.31/0.59 = ( divide_divide @ real @ ( log @ B2 @ A2 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % log_root % 0.31/0.59 thf(fact_4629_log__base__root,axiom, % 0.31/0.59 ! [N2: nat,B2: real,X3: real] : % 0.31/0.59 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 0.31/0.59 => ( ( log @ ( root @ N2 @ B2 ) @ X3 ) % 0.31/0.59 = ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ X3 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % log_base_root % 0.31/0.59 thf(fact_4630_split__root,axiom, % 0.31/0.59 ! [P: real > $o,N2: nat,X3: real] : % 0.31/0.59 ( ( P @ ( root @ N2 @ X3 ) ) % 0.31/0.59 = ( ( ( N2 % 0.31/0.59 = ( zero_zero @ nat ) ) % 0.31/0.59 => ( P @ ( zero_zero @ real ) ) ) % 0.31/0.59 & ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.59 => ! [Y2: real] : % 0.31/0.59 ( ( ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N2 ) ) % 0.31/0.59 = X3 ) % 0.31/0.59 => ( P @ Y2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % split_root % 0.31/0.59 thf(fact_4631_in__lex__prod,axiom, % 0.31/0.59 ! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] : % 0.31/0.59 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ A7 @ B7 ) ) @ ( lex_prod @ A @ B @ R2 @ S ) ) % 0.31/0.59 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A7 ) @ R2 ) % 0.31/0.59 | ( ( A2 = A7 ) % 0.31/0.59 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B7 ) @ S ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % in_lex_prod % 0.31/0.59 thf(fact_4632_div__add__self1__no__field,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( ( euclid4440199948858584721cancel @ A ) % 0.31/0.59 & ( field @ B ) ) % 0.31/0.59 => ! [X3: B,B2: A,A2: A] : % 0.31/0.59 ( ( nO_MATCH @ B @ A @ X3 @ B2 ) % 0.31/0.59 => ( ( B2 % 0.31/0.59 != ( zero_zero @ A ) ) % 0.31/0.59 => ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 ) % 0.31/0.59 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % div_add_self1_no_field % 0.31/0.59 thf(fact_4633_div__add__self2__no__field,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( ( euclid4440199948858584721cancel @ A ) % 0.31/0.59 & ( field @ B ) ) % 0.31/0.59 => ! [X3: B,B2: A,A2: A] : % 0.31/0.59 ( ( nO_MATCH @ B @ A @ X3 @ B2 ) % 0.31/0.59 => ( ( B2 % 0.31/0.59 != ( zero_zero @ A ) ) % 0.31/0.59 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 ) % 0.31/0.59 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % div_add_self2_no_field % 0.31/0.59 thf(fact_4634_scale__right__distrib__NO__MATCH,axiom, % 0.31/0.59 ! [A: $tType] : % 0.31/0.59 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.59 => ! [X3: A,Y: A,A2: real] : % 0.31/0.59 ( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X3 @ Y ) @ A2 ) % 0.31/0.59 => ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( plus_plus @ A @ X3 @ Y ) ) % 0.31/0.59 = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % scale_right_distrib_NO_MATCH % 0.31/0.59 thf(fact_4635_distrib__right__NO__MATCH,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( semiring @ A ) % 0.31/0.59 => ! [X3: B,Y: B,C2: A,A2: A,B2: A] : % 0.31/0.59 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ C2 ) % 0.31/0.59 => ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 0.31/0.59 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % distrib_right_NO_MATCH % 0.31/0.59 thf(fact_4636_distrib__left__NO__MATCH,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( semiring @ A ) % 0.31/0.59 => ! [X3: B,Y: B,A2: A,B2: A,C2: A] : % 0.31/0.59 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ A2 ) % 0.31/0.59 => ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) % 0.31/0.59 = ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % distrib_left_NO_MATCH % 0.31/0.59 thf(fact_4637_left__diff__distrib__NO__MATCH,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( ring @ A ) % 0.31/0.59 => ! [X3: B,Y: B,C2: A,A2: A,B2: A] : % 0.31/0.59 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ C2 ) % 0.31/0.59 => ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 0.31/0.59 = ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % left_diff_distrib_NO_MATCH % 0.31/0.59 thf(fact_4638_right__diff__distrib__NO__MATCH,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( ring @ A ) % 0.31/0.59 => ! [X3: B,Y: B,A2: A,B2: A,C2: A] : % 0.31/0.59 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ A2 ) % 0.31/0.59 => ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) ) % 0.31/0.59 = ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % right_diff_distrib_NO_MATCH % 0.31/0.59 thf(fact_4639_lex__prod__def,axiom, % 0.31/0.59 ! [B: $tType,A: $tType] : % 0.31/0.59 ( ( lex_prod @ A @ B ) % 0.31/0.59 = ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] : % 0.31/0.59 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) % 0.31/0.59 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o % 0.31/0.59 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o ) % 0.31/0.59 @ ^ [A4: A,B4: B] : % 0.31/0.59 ( product_case_prod @ A @ B @ $o % 0.31/0.59 @ ^ [A9: A,B13: B] : % 0.31/0.59 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A9 ) @ Ra ) % 0.31/0.59 | ( ( A4 = A9 ) % 0.31/0.59 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B4 @ B13 ) @ Rb ) ) ) ) ) ) ) ) ) ). % 0.31/0.59 % 0.31/0.59 % lex_prod_def % 0.31/0.59 thf(fact_4640_scale__left__distrib__NO__MATCH,axiom, % 0.31/0.59 ! [C: $tType,A: $tType] : % 0.31/0.59 ( ( real_V4867850818363320053vector @ A ) % 0.31/0.59 => ! [X3: A,Y: A,C2: C,A2: real,B2: real] : % 0.31/0.59 ( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X3 @ Y ) @ C2 ) % 0.31/0.59 => ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X3 ) % 0.31/0.60 = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % scale_left_distrib_NO_MATCH % 0.31/0.60 thf(fact_4641_horner__sum__eq__sum__funpow,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comm_semiring_0 @ A ) % 0.31/0.60 => ( ( groups4207007520872428315er_sum @ B @ A ) % 0.31/0.60 = ( ^ [F3: B > A,A4: A,Xs2: list @ B] : % 0.31/0.60 ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.60 @ ^ [N: nat] : ( compow @ ( A > A ) @ N @ ( times_times @ A @ A4 ) @ ( F3 @ ( nth @ B @ Xs2 @ N ) ) ) % 0.31/0.60 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % horner_sum_eq_sum_funpow % 0.31/0.60 thf(fact_4642_set__zip,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] : % 0.31/0.60 ( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) % 0.31/0.60 = ( collect @ ( product_prod @ A @ B ) % 0.31/0.60 @ ^ [Uu3: product_prod @ A @ B] : % 0.31/0.60 ? [I3: nat] : % 0.31/0.60 ( ( Uu3 % 0.31/0.60 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I3 ) @ ( nth @ B @ Ys @ I3 ) ) ) % 0.31/0.60 & ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % set_zip % 0.31/0.60 thf(fact_4643_Fpow__Pow__finite,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( finite_Fpow @ A ) % 0.31/0.60 = ( ^ [A6: set @ A] : ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A6 ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Fpow_Pow_finite % 0.31/0.60 thf(fact_4644_Suc__funpow,axiom, % 0.31/0.60 ! [N2: nat] : % 0.31/0.60 ( ( compow @ ( nat > nat ) @ N2 @ suc ) % 0.31/0.60 = ( plus_plus @ nat @ N2 ) ) ). % 0.31/0.60 % 0.31/0.60 % Suc_funpow % 0.31/0.60 thf(fact_4645_min_Oabsorb1,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A2: A,B2: A] : % 0.31/0.60 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.60 => ( ( ord_min @ A @ A2 @ B2 ) % 0.31/0.60 = A2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.absorb1 % 0.31/0.60 thf(fact_4646_min_Oabsorb2,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [B2: A,A2: A] : % 0.31/0.60 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.60 => ( ( ord_min @ A @ A2 @ B2 ) % 0.31/0.60 = B2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.absorb2 % 0.31/0.60 thf(fact_4647_min_Obounded__iff,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A2: A,B2: A,C2: A] : % 0.31/0.60 ( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) ) % 0.31/0.60 = ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.60 & ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.bounded_iff % 0.31/0.60 thf(fact_4648_min_Oabsorb3,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A2: A,B2: A] : % 0.31/0.60 ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.60 => ( ( ord_min @ A @ A2 @ B2 ) % 0.31/0.60 = A2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.absorb3 % 0.31/0.60 thf(fact_4649_min_Oabsorb4,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [B2: A,A2: A] : % 0.31/0.60 ( ( ord_less @ A @ B2 @ A2 ) % 0.31/0.60 => ( ( ord_min @ A @ A2 @ B2 ) % 0.31/0.60 = B2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.absorb4 % 0.31/0.60 thf(fact_4650_min__less__iff__conj,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [Z2: A,X3: A,Y: A] : % 0.31/0.60 ( ( ord_less @ A @ Z2 @ ( ord_min @ A @ X3 @ Y ) ) % 0.31/0.60 = ( ( ord_less @ A @ Z2 @ X3 ) % 0.31/0.60 & ( ord_less @ A @ Z2 @ Y ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_less_iff_conj % 0.31/0.60 thf(fact_4651_min__bot2,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( order_bot @ A ) % 0.31/0.60 => ! [X3: A] : % 0.31/0.60 ( ( ord_min @ A @ X3 @ ( bot_bot @ A ) ) % 0.31/0.60 = ( bot_bot @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_bot2 % 0.31/0.60 thf(fact_4652_min__bot,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( order_bot @ A ) % 0.31/0.60 => ! [X3: A] : % 0.31/0.60 ( ( ord_min @ A @ ( bot_bot @ A ) @ X3 ) % 0.31/0.60 = ( bot_bot @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_bot % 0.31/0.60 thf(fact_4653_funpow__0,axiom, % 0.31/0.60 ! [A: $tType,F2: A > A,X3: A] : % 0.31/0.60 ( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F2 @ X3 ) % 0.31/0.60 = X3 ) ). % 0.31/0.60 % 0.31/0.60 % funpow_0 % 0.31/0.60 thf(fact_4654_max__min__same_I1_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [X3: A,Y: A] : % 0.31/0.60 ( ( ord_max @ A @ X3 @ ( ord_min @ A @ X3 @ Y ) ) % 0.31/0.60 = X3 ) ) ). % 0.31/0.60 % 0.31/0.60 % max_min_same(1) % 0.31/0.60 thf(fact_4655_max__min__same_I2_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [X3: A,Y: A] : % 0.31/0.60 ( ( ord_max @ A @ ( ord_min @ A @ X3 @ Y ) @ X3 ) % 0.31/0.60 = X3 ) ) ). % 0.31/0.60 % 0.31/0.60 % max_min_same(2) % 0.31/0.60 thf(fact_4656_max__min__same_I3_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [X3: A,Y: A] : % 0.31/0.60 ( ( ord_max @ A @ ( ord_min @ A @ X3 @ Y ) @ Y ) % 0.31/0.60 = Y ) ) ). % 0.31/0.60 % 0.31/0.60 % max_min_same(3) % 0.31/0.60 thf(fact_4657_max__min__same_I4_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [Y: A,X3: A] : % 0.31/0.60 ( ( ord_max @ A @ Y @ ( ord_min @ A @ X3 @ Y ) ) % 0.31/0.60 = Y ) ) ). % 0.31/0.60 % 0.31/0.60 % max_min_same(4) % 0.31/0.60 thf(fact_4658_min__Suc__Suc,axiom, % 0.31/0.60 ! [M: nat,N2: nat] : % 0.31/0.60 ( ( ord_min @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) % 0.31/0.60 = ( suc @ ( ord_min @ nat @ M @ N2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_Suc_Suc % 0.31/0.60 thf(fact_4659_min__0R,axiom, % 0.31/0.60 ! [N2: nat] : % 0.31/0.60 ( ( ord_min @ nat @ N2 @ ( zero_zero @ nat ) ) % 0.31/0.60 = ( zero_zero @ nat ) ) ). % 0.31/0.60 % 0.31/0.60 % min_0R % 0.31/0.60 thf(fact_4660_min__0L,axiom, % 0.31/0.60 ! [N2: nat] : % 0.31/0.60 ( ( ord_min @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.60 = ( zero_zero @ nat ) ) ). % 0.31/0.60 % 0.31/0.60 % min_0L % 0.31/0.60 thf(fact_4661_min__number__of_I1_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ( numeral @ A ) % 0.31/0.60 & ( ord @ A ) ) % 0.31/0.60 => ! [U: num,V2: num] : % 0.31/0.60 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.60 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.60 = ( numeral_numeral @ A @ U ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.60 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.60 = ( numeral_numeral @ A @ V2 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_number_of(1) % 0.31/0.60 thf(fact_4662_min__0__1_I4_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linord181362715937106298miring @ A ) % 0.31/0.60 => ! [X3: num] : % 0.31/0.60 ( ( ord_min @ A @ ( numeral_numeral @ A @ X3 ) @ ( zero_zero @ A ) ) % 0.31/0.60 = ( zero_zero @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_0_1(4) % 0.31/0.60 thf(fact_4663_min__0__1_I3_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linord181362715937106298miring @ A ) % 0.31/0.60 => ! [X3: num] : % 0.31/0.60 ( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X3 ) ) % 0.31/0.60 = ( zero_zero @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_0_1(3) % 0.31/0.60 thf(fact_4664_min__0__1_I1_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linord181362715937106298miring @ A ) % 0.31/0.60 => ( ( ord_min @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) % 0.31/0.60 = ( zero_zero @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_0_1(1) % 0.31/0.60 thf(fact_4665_min__0__1_I2_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linord181362715937106298miring @ A ) % 0.31/0.60 => ( ( ord_min @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) % 0.31/0.60 = ( zero_zero @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_0_1(2) % 0.31/0.60 thf(fact_4666_min__number__of_I2_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ( uminus @ A ) % 0.31/0.60 & ( numeral @ A ) % 0.31/0.60 & ( ord @ A ) ) % 0.31/0.60 => ! [U: num,V2: num] : % 0.31/0.60 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.60 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.60 = ( numeral_numeral @ A @ U ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.60 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.60 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_number_of(2) % 0.31/0.60 thf(fact_4667_min__number__of_I3_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ( uminus @ A ) % 0.31/0.60 & ( numeral @ A ) % 0.31/0.60 & ( ord @ A ) ) % 0.31/0.60 => ! [U: num,V2: num] : % 0.31/0.60 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.60 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.60 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.60 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 0.31/0.60 = ( numeral_numeral @ A @ V2 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_number_of(3) % 0.31/0.60 thf(fact_4668_min__number__of_I4_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ( uminus @ A ) % 0.31/0.60 & ( numeral @ A ) % 0.31/0.60 & ( ord @ A ) ) % 0.31/0.60 => ! [U: num,V2: num] : % 0.31/0.60 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.60 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.60 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.60 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 0.31/0.60 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_number_of(4) % 0.31/0.60 thf(fact_4669_min__Suc__numeral,axiom, % 0.31/0.60 ! [N2: nat,K: num] : % 0.31/0.60 ( ( ord_min @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) ) % 0.31/0.60 = ( suc @ ( ord_min @ nat @ N2 @ ( pred_numeral @ K ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_Suc_numeral % 0.31/0.60 thf(fact_4670_min__numeral__Suc,axiom, % 0.31/0.60 ! [K: num,N2: nat] : % 0.31/0.60 ( ( ord_min @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) ) % 0.31/0.60 = ( suc @ ( ord_min @ nat @ ( pred_numeral @ K ) @ N2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_numeral_Suc % 0.31/0.60 thf(fact_4671_zip__replicate,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,I: nat,X3: A,J: nat,Y: B] : % 0.31/0.60 ( ( zip @ A @ B @ ( replicate @ A @ I @ X3 ) @ ( replicate @ B @ J @ Y ) ) % 0.31/0.60 = ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I @ J ) @ ( product_Pair @ A @ B @ X3 @ Y ) ) ) ). % 0.31/0.60 % 0.31/0.60 % zip_replicate % 0.31/0.60 thf(fact_4672_length__zip,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 0.31/0.60 ( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) ) % 0.31/0.60 = ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ). % 0.31/0.60 % 0.31/0.60 % length_zip % 0.31/0.60 thf(fact_4673_comp__funpow,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,N2: nat,F2: A > A] : % 0.31/0.60 ( ( compow @ ( ( B > A ) > B > A ) @ N2 @ ( comp @ A @ A @ B @ F2 ) ) % 0.31/0.60 = ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N2 @ F2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % comp_funpow % 0.31/0.60 thf(fact_4674_min_Ostrict__coboundedI2,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [B2: A,C2: A,A2: A] : % 0.31/0.60 ( ( ord_less @ A @ B2 @ C2 ) % 0.31/0.60 => ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.strict_coboundedI2 % 0.31/0.60 thf(fact_4675_min_Ostrict__coboundedI1,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A2: A,C2: A,B2: A] : % 0.31/0.60 ( ( ord_less @ A @ A2 @ C2 ) % 0.31/0.60 => ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.strict_coboundedI1 % 0.31/0.60 thf(fact_4676_min_Ostrict__order__iff,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ( ( ord_less @ A ) % 0.31/0.60 = ( ^ [A4: A,B4: A] : % 0.31/0.60 ( ( A4 % 0.31/0.60 = ( ord_min @ A @ A4 @ B4 ) ) % 0.31/0.60 & ( A4 != B4 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.strict_order_iff % 0.31/0.60 thf(fact_4677_min_Ostrict__boundedE,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A2: A,B2: A,C2: A] : % 0.31/0.60 ( ( ord_less @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) ) % 0.31/0.60 => ~ ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.60 => ~ ( ord_less @ A @ A2 @ C2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.strict_boundedE % 0.31/0.60 thf(fact_4678_min__less__iff__disj,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [X3: A,Y: A,Z2: A] : % 0.31/0.60 ( ( ord_less @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 ) % 0.31/0.60 = ( ( ord_less @ A @ X3 @ Z2 ) % 0.31/0.60 | ( ord_less @ A @ Y @ Z2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_less_iff_disj % 0.31/0.60 thf(fact_4679_min_Omono,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A2: A,C2: A,B2: A,D2: A] : % 0.31/0.60 ( ( ord_less_eq @ A @ A2 @ C2 ) % 0.31/0.60 => ( ( ord_less_eq @ A @ B2 @ D2 ) % 0.31/0.60 => ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ ( ord_min @ A @ C2 @ D2 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.mono % 0.31/0.60 thf(fact_4680_min_OorderE,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A2: A,B2: A] : % 0.31/0.60 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.60 => ( A2 % 0.31/0.60 = ( ord_min @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.orderE % 0.31/0.60 thf(fact_4681_min_OorderI,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A2: A,B2: A] : % 0.31/0.60 ( ( A2 % 0.31/0.60 = ( ord_min @ A @ A2 @ B2 ) ) % 0.31/0.60 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.orderI % 0.31/0.60 thf(fact_4682_min_OboundedE,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A2: A,B2: A,C2: A] : % 0.31/0.60 ( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) ) % 0.31/0.60 => ~ ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.60 => ~ ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.boundedE % 0.31/0.60 thf(fact_4683_min_OboundedI,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A2: A,B2: A,C2: A] : % 0.31/0.60 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.60 => ( ( ord_less_eq @ A @ A2 @ C2 ) % 0.31/0.60 => ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.boundedI % 0.31/0.60 thf(fact_4684_min_Oorder__iff,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ( ( ord_less_eq @ A ) % 0.31/0.60 = ( ^ [A4: A,B4: A] : % 0.31/0.60 ( A4 % 0.31/0.60 = ( ord_min @ A @ A4 @ B4 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.order_iff % 0.31/0.60 thf(fact_4685_min_Ocobounded1,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ A2 ) ) ). % 0.31/0.60 % 0.31/0.60 % min.cobounded1 % 0.31/0.60 thf(fact_4686_min_Ocobounded2,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ B2 ) ) ). % 0.31/0.60 % 0.31/0.60 % min.cobounded2 % 0.31/0.60 thf(fact_4687_min_Oabsorb__iff1,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ( ( ord_less_eq @ A ) % 0.31/0.60 = ( ^ [A4: A,B4: A] : % 0.31/0.60 ( ( ord_min @ A @ A4 @ B4 ) % 0.31/0.60 = A4 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.absorb_iff1 % 0.31/0.60 thf(fact_4688_min_Oabsorb__iff2,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ( ( ord_less_eq @ A ) % 0.31/0.60 = ( ^ [B4: A,A4: A] : % 0.31/0.60 ( ( ord_min @ A @ A4 @ B4 ) % 0.31/0.60 = B4 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.absorb_iff2 % 0.31/0.60 thf(fact_4689_min_OcoboundedI1,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A2: A,C2: A,B2: A] : % 0.31/0.60 ( ( ord_less_eq @ A @ A2 @ C2 ) % 0.31/0.60 => ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.coboundedI1 % 0.31/0.60 thf(fact_4690_min_OcoboundedI2,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [B2: A,C2: A,A2: A] : % 0.31/0.60 ( ( ord_less_eq @ A @ B2 @ C2 ) % 0.31/0.60 => ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min.coboundedI2 % 0.31/0.60 thf(fact_4691_min__le__iff__disj,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [X3: A,Y: A,Z2: A] : % 0.31/0.60 ( ( ord_less_eq @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 ) % 0.31/0.60 = ( ( ord_less_eq @ A @ X3 @ Z2 ) % 0.31/0.60 | ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_le_iff_disj % 0.31/0.60 thf(fact_4692_min__def__raw,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ord @ A ) % 0.31/0.60 => ( ( ord_min @ A ) % 0.31/0.60 = ( ^ [A4: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B4 ) @ A4 @ B4 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_def_raw % 0.31/0.60 thf(fact_4693_min__def,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ord @ A ) % 0.31/0.60 => ( ( ord_min @ A ) % 0.31/0.60 = ( ^ [A4: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B4 ) @ A4 @ B4 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_def % 0.31/0.60 thf(fact_4694_min__absorb1,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ord @ A ) % 0.31/0.60 => ! [X3: A,Y: A] : % 0.31/0.60 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.60 => ( ( ord_min @ A @ X3 @ Y ) % 0.31/0.60 = X3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_absorb1 % 0.31/0.60 thf(fact_4695_min__absorb2,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( order @ A ) % 0.31/0.60 => ! [Y: A,X3: A] : % 0.31/0.60 ( ( ord_less_eq @ A @ Y @ X3 ) % 0.31/0.60 => ( ( ord_min @ A @ X3 @ Y ) % 0.31/0.60 = Y ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_absorb2 % 0.31/0.60 thf(fact_4696_of__nat__min,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linord181362715937106298miring @ A ) % 0.31/0.60 => ! [X3: nat,Y: nat] : % 0.31/0.60 ( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X3 @ Y ) ) % 0.31/0.60 = ( ord_min @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % of_nat_min % 0.31/0.60 thf(fact_4697_funpow__swap1,axiom, % 0.31/0.60 ! [A: $tType,F2: A > A,N2: nat,X3: A] : % 0.31/0.60 ( ( F2 @ ( compow @ ( A > A ) @ N2 @ F2 @ X3 ) ) % 0.31/0.60 = ( compow @ ( A > A ) @ N2 @ F2 @ ( F2 @ X3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % funpow_swap1 % 0.31/0.60 thf(fact_4698_of__int__min,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_idom @ A ) % 0.31/0.60 => ! [X3: int,Y: int] : % 0.31/0.60 ( ( ring_1_of_int @ A @ ( ord_min @ int @ X3 @ Y ) ) % 0.31/0.60 = ( ord_min @ A @ ( ring_1_of_int @ A @ X3 ) @ ( ring_1_of_int @ A @ Y ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % of_int_min % 0.31/0.60 thf(fact_4699_min__add__distrib__right,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ordere2412721322843649153imp_le @ A ) % 0.31/0.60 => ! [X3: A,Y: A,Z2: A] : % 0.31/0.60 ( ( plus_plus @ A @ X3 @ ( ord_min @ A @ Y @ Z2 ) ) % 0.31/0.60 = ( ord_min @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( plus_plus @ A @ X3 @ Z2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_add_distrib_right % 0.31/0.60 thf(fact_4700_min__add__distrib__left,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ordere2412721322843649153imp_le @ A ) % 0.31/0.60 => ! [X3: A,Y: A,Z2: A] : % 0.31/0.60 ( ( plus_plus @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 ) % 0.31/0.60 = ( ord_min @ A @ ( plus_plus @ A @ X3 @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_add_distrib_left % 0.31/0.60 thf(fact_4701_min__diff__distrib__left,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ordered_ab_group_add @ A ) % 0.31/0.60 => ! [X3: A,Y: A,Z2: A] : % 0.31/0.60 ( ( minus_minus @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 ) % 0.31/0.60 = ( ord_min @ A @ ( minus_minus @ A @ X3 @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_diff_distrib_left % 0.31/0.60 thf(fact_4702_min__diff,axiom, % 0.31/0.60 ! [M: nat,I: nat,N2: nat] : % 0.31/0.60 ( ( ord_min @ nat @ ( minus_minus @ nat @ M @ I ) @ ( minus_minus @ nat @ N2 @ I ) ) % 0.31/0.60 = ( minus_minus @ nat @ ( ord_min @ nat @ M @ N2 ) @ I ) ) ). % 0.31/0.60 % 0.31/0.60 % min_diff % 0.31/0.60 thf(fact_4703_funpow__mult,axiom, % 0.31/0.60 ! [A: $tType,N2: nat,M: nat,F2: A > A] : % 0.31/0.60 ( ( compow @ ( A > A ) @ N2 @ ( compow @ ( A > A ) @ M @ F2 ) ) % 0.31/0.60 = ( compow @ ( A > A ) @ ( times_times @ nat @ M @ N2 ) @ F2 ) ) ). % 0.31/0.60 % 0.31/0.60 % funpow_mult % 0.31/0.60 thf(fact_4704_nat__mult__min__left,axiom, % 0.31/0.60 ! [M: nat,N2: nat,Q4: nat] : % 0.31/0.60 ( ( times_times @ nat @ ( ord_min @ nat @ M @ N2 ) @ Q4 ) % 0.31/0.60 = ( ord_min @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N2 @ Q4 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % nat_mult_min_left % 0.31/0.60 thf(fact_4705_nat__mult__min__right,axiom, % 0.31/0.60 ! [M: nat,N2: nat,Q4: nat] : % 0.31/0.60 ( ( times_times @ nat @ M @ ( ord_min @ nat @ N2 @ Q4 ) ) % 0.31/0.60 = ( ord_min @ nat @ ( times_times @ nat @ M @ N2 ) @ ( times_times @ nat @ M @ Q4 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % nat_mult_min_right % 0.31/0.60 thf(fact_4706_inf__nat__def,axiom, % 0.31/0.60 ( ( inf_inf @ nat ) % 0.31/0.60 = ( ord_min @ nat ) ) ). % 0.31/0.60 % 0.31/0.60 % inf_nat_def % 0.31/0.60 thf(fact_4707_funpow__Suc__right,axiom, % 0.31/0.60 ! [A: $tType,N2: nat,F2: A > A] : % 0.31/0.60 ( ( compow @ ( A > A ) @ ( suc @ N2 ) @ F2 ) % 0.31/0.60 = ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ F2 ) ) ). % 0.31/0.60 % 0.31/0.60 % funpow_Suc_right % 0.31/0.60 thf(fact_4708_funpow_Osimps_I2_J,axiom, % 0.31/0.60 ! [A: $tType,N2: nat,F2: A > A] : % 0.31/0.60 ( ( compow @ ( A > A ) @ ( suc @ N2 ) @ F2 ) % 0.31/0.60 = ( comp @ A @ A @ A @ F2 @ ( compow @ ( A > A ) @ N2 @ F2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % funpow.simps(2) % 0.31/0.60 thf(fact_4709_funpow__add,axiom, % 0.31/0.60 ! [A: $tType,M: nat,N2: nat,F2: A > A] : % 0.31/0.60 ( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M @ N2 ) @ F2 ) % 0.31/0.60 = ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M @ F2 ) @ ( compow @ ( A > A ) @ N2 @ F2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % funpow_add % 0.31/0.60 thf(fact_4710_minus__min__eq__max,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linord5086331880401160121up_add @ A ) % 0.31/0.60 => ! [X3: A,Y: A] : % 0.31/0.60 ( ( uminus_uminus @ A @ ( ord_min @ A @ X3 @ Y ) ) % 0.31/0.60 = ( ord_max @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % minus_min_eq_max % 0.31/0.60 thf(fact_4711_minus__max__eq__min,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linord5086331880401160121up_add @ A ) % 0.31/0.60 => ! [X3: A,Y: A] : % 0.31/0.60 ( ( uminus_uminus @ A @ ( ord_max @ A @ X3 @ Y ) ) % 0.31/0.60 = ( ord_min @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % minus_max_eq_min % 0.31/0.60 thf(fact_4712_empty__in__Fpow,axiom, % 0.31/0.60 ! [A: $tType,A3: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) ) ). % 0.31/0.60 % 0.31/0.60 % empty_in_Fpow % 0.31/0.60 thf(fact_4713_Fpow__not__empty,axiom, % 0.31/0.60 ! [A: $tType,A3: set @ A] : % 0.31/0.60 ( ( finite_Fpow @ A @ A3 ) % 0.31/0.60 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Fpow_not_empty % 0.31/0.60 thf(fact_4714_max__mult__distrib__left,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_idom @ A ) % 0.31/0.60 => ! [P4: A,X3: A,Y: A] : % 0.31/0.60 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 0.31/0.60 => ( ( times_times @ A @ P4 @ ( ord_max @ A @ X3 @ Y ) ) % 0.31/0.60 = ( ord_max @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 0.31/0.60 => ( ( times_times @ A @ P4 @ ( ord_max @ A @ X3 @ Y ) ) % 0.31/0.60 = ( ord_min @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % max_mult_distrib_left % 0.31/0.60 thf(fact_4715_min__mult__distrib__left,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_idom @ A ) % 0.31/0.60 => ! [P4: A,X3: A,Y: A] : % 0.31/0.60 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 0.31/0.60 => ( ( times_times @ A @ P4 @ ( ord_min @ A @ X3 @ Y ) ) % 0.31/0.60 = ( ord_min @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 0.31/0.60 => ( ( times_times @ A @ P4 @ ( ord_min @ A @ X3 @ Y ) ) % 0.31/0.60 = ( ord_max @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_mult_distrib_left % 0.31/0.60 thf(fact_4716_max__mult__distrib__right,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_idom @ A ) % 0.31/0.60 => ! [P4: A,X3: A,Y: A] : % 0.31/0.60 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 0.31/0.60 => ( ( times_times @ A @ ( ord_max @ A @ X3 @ Y ) @ P4 ) % 0.31/0.60 = ( ord_max @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 0.31/0.60 => ( ( times_times @ A @ ( ord_max @ A @ X3 @ Y ) @ P4 ) % 0.31/0.60 = ( ord_min @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % max_mult_distrib_right % 0.31/0.60 thf(fact_4717_min__mult__distrib__right,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_idom @ A ) % 0.31/0.60 => ! [P4: A,X3: A,Y: A] : % 0.31/0.60 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 0.31/0.60 => ( ( times_times @ A @ ( ord_min @ A @ X3 @ Y ) @ P4 ) % 0.31/0.60 = ( ord_min @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 0.31/0.60 => ( ( times_times @ A @ ( ord_min @ A @ X3 @ Y ) @ P4 ) % 0.31/0.60 = ( ord_max @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_mult_distrib_right % 0.31/0.60 thf(fact_4718_of__nat__def,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_1 @ A ) % 0.31/0.60 => ( ( semiring_1_of_nat @ A ) % 0.31/0.60 = ( ^ [N: nat] : ( compow @ ( A > A ) @ N @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % of_nat_def % 0.31/0.60 thf(fact_4719_numeral__add__unfold__funpow,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_numeral @ A ) % 0.31/0.60 => ! [K: num,A2: A] : % 0.31/0.60 ( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A2 ) % 0.31/0.60 = ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % numeral_add_unfold_funpow % 0.31/0.60 thf(fact_4720_min__divide__distrib__right,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_field @ A ) % 0.31/0.60 => ! [P4: A,X3: A,Y: A] : % 0.31/0.60 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 0.31/0.60 => ( ( divide_divide @ A @ ( ord_min @ A @ X3 @ Y ) @ P4 ) % 0.31/0.60 = ( ord_min @ A @ ( divide_divide @ A @ X3 @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 0.31/0.60 => ( ( divide_divide @ A @ ( ord_min @ A @ X3 @ Y ) @ P4 ) % 0.31/0.60 = ( ord_max @ A @ ( divide_divide @ A @ X3 @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % min_divide_distrib_right % 0.31/0.60 thf(fact_4721_max__divide__distrib__right,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_field @ A ) % 0.31/0.60 => ! [P4: A,X3: A,Y: A] : % 0.31/0.60 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 0.31/0.60 => ( ( divide_divide @ A @ ( ord_max @ A @ X3 @ Y ) @ P4 ) % 0.31/0.60 = ( ord_max @ A @ ( divide_divide @ A @ X3 @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 0.31/0.60 => ( ( divide_divide @ A @ ( ord_max @ A @ X3 @ Y ) @ P4 ) % 0.31/0.60 = ( ord_min @ A @ ( divide_divide @ A @ X3 @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % max_divide_distrib_right % 0.31/0.60 thf(fact_4722_min__Suc2,axiom, % 0.31/0.60 ! [M: nat,N2: nat] : % 0.31/0.60 ( ( ord_min @ nat @ M @ ( suc @ N2 ) ) % 0.31/0.60 = ( case_nat @ nat @ ( zero_zero @ nat ) % 0.31/0.60 @ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ M4 @ N2 ) ) % 0.31/0.60 @ M ) ) ). % 0.31/0.60 % 0.31/0.60 % min_Suc2 % 0.31/0.60 thf(fact_4723_min__Suc1,axiom, % 0.31/0.60 ! [N2: nat,M: nat] : % 0.31/0.60 ( ( ord_min @ nat @ ( suc @ N2 ) @ M ) % 0.31/0.60 = ( case_nat @ nat @ ( zero_zero @ nat ) % 0.31/0.60 @ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ N2 @ M4 ) ) % 0.31/0.60 @ M ) ) ). % 0.31/0.60 % 0.31/0.60 % min_Suc1 % 0.31/0.60 thf(fact_4724_Inf__insert__finite,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( condit6923001295902523014norder @ A ) % 0.31/0.60 => ! [S3: set @ A,X3: A] : % 0.31/0.60 ( ( finite_finite2 @ A @ S3 ) % 0.31/0.60 => ( ( ( S3 % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 => ( ( complete_Inf_Inf @ A @ ( insert @ A @ X3 @ S3 ) ) % 0.31/0.60 = X3 ) ) % 0.31/0.60 & ( ( S3 % 0.31/0.60 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 => ( ( complete_Inf_Inf @ A @ ( insert @ A @ X3 @ S3 ) ) % 0.31/0.60 = ( ord_min @ A @ X3 @ ( complete_Inf_Inf @ A @ S3 ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Inf_insert_finite % 0.31/0.60 thf(fact_4725_Fpow__mono,axiom, % 0.31/0.60 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.60 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) @ ( finite_Fpow @ A @ B3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Fpow_mono % 0.31/0.60 thf(fact_4726_Fpow__def,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( finite_Fpow @ A ) % 0.31/0.60 = ( ^ [A6: set @ A] : % 0.31/0.60 ( collect @ ( set @ A ) % 0.31/0.60 @ ^ [X6: set @ A] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ X6 @ A6 ) % 0.31/0.60 & ( finite_finite2 @ A @ X6 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Fpow_def % 0.31/0.60 thf(fact_4727_numeral__unfold__funpow,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_1 @ A ) % 0.31/0.60 => ( ( numeral_numeral @ A ) % 0.31/0.60 = ( ^ [K3: num] : ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K3 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % numeral_unfold_funpow % 0.31/0.60 thf(fact_4728_relpowp__bot,axiom, % 0.31/0.60 ! [A: $tType,N2: nat] : % 0.31/0.60 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.60 => ( ( compow @ ( A > A > $o ) @ N2 @ ( bot_bot @ ( A > A > $o ) ) ) % 0.31/0.60 = ( bot_bot @ ( A > A > $o ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % relpowp_bot % 0.31/0.60 thf(fact_4729_relpowp__fun__conv,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( compow @ ( A > A > $o ) ) % 0.31/0.60 = ( ^ [N: nat,P3: A > A > $o,X: A,Y2: A] : % 0.31/0.60 ? [F3: nat > A] : % 0.31/0.60 ( ( ( F3 @ ( zero_zero @ nat ) ) % 0.31/0.60 = X ) % 0.31/0.60 & ( ( F3 @ N ) % 0.31/0.60 = Y2 ) % 0.31/0.60 & ! [I3: nat] : % 0.31/0.60 ( ( ord_less @ nat @ I3 @ N ) % 0.31/0.60 => ( P3 @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % relpowp_fun_conv % 0.31/0.60 thf(fact_4730_same__fst__def,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( same_fst @ A @ B ) % 0.31/0.60 = ( ^ [P3: A > $o,R6: A > ( set @ ( product_prod @ B @ B ) )] : % 0.31/0.60 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) % 0.31/0.60 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o % 0.31/0.60 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o ) % 0.31/0.60 @ ^ [X9: A,Y6: B] : % 0.31/0.60 ( product_case_prod @ A @ B @ $o % 0.31/0.60 @ ^ [X: A,Y2: B] : % 0.31/0.60 ( ( X9 = X ) % 0.31/0.60 & ( P3 @ X ) % 0.31/0.60 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y2 ) @ ( R6 @ X ) ) ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % same_fst_def % 0.31/0.60 thf(fact_4731_same__fstI,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,P: A > $o,X3: A,Y8: B,Y: B,R: A > ( set @ ( product_prod @ B @ B ) )] : % 0.31/0.60 ( ( P @ X3 ) % 0.31/0.60 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y ) @ ( R @ X3 ) ) % 0.31/0.60 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y8 ) @ ( product_Pair @ A @ B @ X3 @ Y ) ) @ ( same_fst @ A @ B @ P @ R ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % same_fstI % 0.31/0.60 thf(fact_4732_inf__int__def,axiom, % 0.31/0.60 ( ( inf_inf @ int ) % 0.31/0.60 = ( ord_min @ int ) ) ). % 0.31/0.60 % 0.31/0.60 % inf_int_def % 0.31/0.60 thf(fact_4733_relpowp__Suc__E,axiom, % 0.31/0.60 ! [A: $tType,N2: nat,P: A > A > $o,X3: A,Z2: A] : % 0.31/0.60 ( ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X3 @ Z2 ) % 0.31/0.60 => ~ ! [Y4: A] : % 0.31/0.60 ( ( compow @ ( A > A > $o ) @ N2 @ P @ X3 @ Y4 ) % 0.31/0.60 => ~ ( P @ Y4 @ Z2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % relpowp_Suc_E % 0.31/0.60 thf(fact_4734_relpowp__Suc__I,axiom, % 0.31/0.60 ! [A: $tType,N2: nat,P: A > A > $o,X3: A,Y: A,Z2: A] : % 0.31/0.60 ( ( compow @ ( A > A > $o ) @ N2 @ P @ X3 @ Y ) % 0.31/0.60 => ( ( P @ Y @ Z2 ) % 0.31/0.60 => ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X3 @ Z2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % relpowp_Suc_I % 0.31/0.60 thf(fact_4735_relpowp__Suc__D2,axiom, % 0.31/0.60 ! [A: $tType,N2: nat,P: A > A > $o,X3: A,Z2: A] : % 0.31/0.60 ( ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X3 @ Z2 ) % 0.31/0.60 => ? [Y4: A] : % 0.31/0.60 ( ( P @ X3 @ Y4 ) % 0.31/0.60 & ( compow @ ( A > A > $o ) @ N2 @ P @ Y4 @ Z2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % relpowp_Suc_D2 % 0.31/0.60 thf(fact_4736_relpowp__Suc__E2,axiom, % 0.31/0.60 ! [A: $tType,N2: nat,P: A > A > $o,X3: A,Z2: A] : % 0.31/0.60 ( ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X3 @ Z2 ) % 0.31/0.60 => ~ ! [Y4: A] : % 0.31/0.60 ( ( P @ X3 @ Y4 ) % 0.31/0.60 => ~ ( compow @ ( A > A > $o ) @ N2 @ P @ Y4 @ Z2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % relpowp_Suc_E2 % 0.31/0.60 thf(fact_4737_relpowp__Suc__I2,axiom, % 0.31/0.60 ! [A: $tType,P: A > A > $o,X3: A,Y: A,N2: nat,Z2: A] : % 0.31/0.60 ( ( P @ X3 @ Y ) % 0.31/0.60 => ( ( compow @ ( A > A > $o ) @ N2 @ P @ Y @ Z2 ) % 0.31/0.60 => ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X3 @ Z2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % relpowp_Suc_I2 % 0.31/0.60 thf(fact_4738_relpowp__0__I,axiom, % 0.31/0.60 ! [A: $tType,P: A > A > $o,X3: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X3 @ X3 ) ). % 0.31/0.60 % 0.31/0.60 % relpowp_0_I % 0.31/0.60 thf(fact_4739_relpowp__0__E,axiom, % 0.31/0.60 ! [A: $tType,P: A > A > $o,X3: A,Y: A] : % 0.31/0.60 ( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X3 @ Y ) % 0.31/0.60 => ( X3 = Y ) ) ). % 0.31/0.60 % 0.31/0.60 % relpowp_0_E % 0.31/0.60 thf(fact_4740_relpowp_Osimps_I1_J,axiom, % 0.31/0.60 ! [A: $tType,R: A > A > $o] : % 0.31/0.60 ( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R ) % 0.31/0.60 = ( ^ [Y3: A,Z: A] : ( Y3 = Z ) ) ) ). % 0.31/0.60 % 0.31/0.60 % relpowp.simps(1) % 0.31/0.60 thf(fact_4741_relpowp__E2,axiom, % 0.31/0.60 ! [A: $tType,N2: nat,P: A > A > $o,X3: A,Z2: A] : % 0.31/0.60 ( ( compow @ ( A > A > $o ) @ N2 @ P @ X3 @ Z2 ) % 0.31/0.60 => ( ( ( N2 % 0.31/0.60 = ( zero_zero @ nat ) ) % 0.31/0.60 => ( X3 != Z2 ) ) % 0.31/0.60 => ~ ! [Y4: A,M3: nat] : % 0.31/0.60 ( ( N2 % 0.31/0.60 = ( suc @ M3 ) ) % 0.31/0.60 => ( ( P @ X3 @ Y4 ) % 0.31/0.60 => ~ ( compow @ ( A > A > $o ) @ M3 @ P @ Y4 @ Z2 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % relpowp_E2 % 0.31/0.60 thf(fact_4742_relpowp__E,axiom, % 0.31/0.60 ! [A: $tType,N2: nat,P: A > A > $o,X3: A,Z2: A] : % 0.31/0.60 ( ( compow @ ( A > A > $o ) @ N2 @ P @ X3 @ Z2 ) % 0.31/0.60 => ( ( ( N2 % 0.31/0.60 = ( zero_zero @ nat ) ) % 0.31/0.60 => ( X3 != Z2 ) ) % 0.31/0.60 => ~ ! [Y4: A,M3: nat] : % 0.31/0.60 ( ( N2 % 0.31/0.60 = ( suc @ M3 ) ) % 0.31/0.60 => ( ( compow @ ( A > A > $o ) @ M3 @ P @ X3 @ Y4 ) % 0.31/0.60 => ~ ( P @ Y4 @ Z2 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % relpowp_E % 0.31/0.60 thf(fact_4743_Nat_Ofunpow__code__def,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( funpow @ A ) % 0.31/0.60 = ( compow @ ( A > A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Nat.funpow_code_def % 0.31/0.60 thf(fact_4744_max__nat_Osemilattice__neutr__order__axioms,axiom, % 0.31/0.60 ( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) % 0.31/0.60 @ ^ [X: nat,Y2: nat] : ( ord_less_eq @ nat @ Y2 @ X ) % 0.31/0.60 @ ^ [X: nat,Y2: nat] : ( ord_less @ nat @ Y2 @ X ) ) ). % 0.31/0.60 % 0.31/0.60 % max_nat.semilattice_neutr_order_axioms % 0.31/0.60 thf(fact_4745_apsnd__apfst,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,C: $tType,D: $tType,F2: C > B,G: D > A,X3: product_prod @ D @ C] : % 0.31/0.60 ( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apfst @ D @ A @ C @ G @ X3 ) ) % 0.31/0.60 = ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X3 ) ) @ ( F2 @ ( product_snd @ D @ C @ X3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % apsnd_apfst % 0.31/0.60 thf(fact_4746_apfst__conv,axiom, % 0.31/0.60 ! [C: $tType,A: $tType,B: $tType,F2: C > A,X3: C,Y: B] : % 0.31/0.60 ( ( product_apfst @ C @ A @ B @ F2 @ ( product_Pair @ C @ B @ X3 @ Y ) ) % 0.31/0.60 = ( product_Pair @ A @ B @ ( F2 @ X3 ) @ Y ) ) ). % 0.31/0.60 % 0.31/0.60 % apfst_conv % 0.31/0.60 thf(fact_4747_apfst__eq__conv,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,C: $tType,F2: C > A,X3: product_prod @ C @ B,G: C > A] : % 0.31/0.60 ( ( ( product_apfst @ C @ A @ B @ F2 @ X3 ) % 0.31/0.60 = ( product_apfst @ C @ A @ B @ G @ X3 ) ) % 0.31/0.60 = ( ( F2 @ ( product_fst @ C @ B @ X3 ) ) % 0.31/0.60 = ( G @ ( product_fst @ C @ B @ X3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % apfst_eq_conv % 0.31/0.60 thf(fact_4748_fst__apfst,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,C: $tType,F2: C > A,X3: product_prod @ C @ B] : % 0.31/0.60 ( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F2 @ X3 ) ) % 0.31/0.60 = ( F2 @ ( product_fst @ C @ B @ X3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % fst_apfst % 0.31/0.60 thf(fact_4749_snd__apfst,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,C: $tType,F2: C > B,X3: product_prod @ C @ A] : % 0.31/0.60 ( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F2 @ X3 ) ) % 0.31/0.60 = ( product_snd @ C @ A @ X3 ) ) ). % 0.31/0.60 % 0.31/0.60 % snd_apfst % 0.31/0.60 thf(fact_4750_snd__comp__apfst,axiom, % 0.31/0.60 ! [C: $tType,B: $tType,A: $tType,F2: A > C] : % 0.31/0.60 ( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 ) ) % 0.31/0.60 = ( product_snd @ A @ B ) ) ). % 0.31/0.60 % 0.31/0.60 % snd_comp_apfst % 0.31/0.60 thf(fact_4751_fst__comp__apfst,axiom, % 0.31/0.60 ! [C: $tType,B: $tType,A: $tType,F2: A > C] : % 0.31/0.60 ( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 ) ) % 0.31/0.60 = ( comp @ A @ C @ ( product_prod @ A @ B ) @ F2 @ ( product_fst @ A @ B ) ) ) ). % 0.31/0.60 % 0.31/0.60 % fst_comp_apfst % 0.31/0.60 thf(fact_4752_apfst__apsnd,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,D: $tType,C: $tType,F2: C > A,G: D > B,X3: product_prod @ C @ D] : % 0.31/0.60 ( ( product_apfst @ C @ A @ B @ F2 @ ( product_apsnd @ D @ B @ C @ G @ X3 ) ) % 0.31/0.60 = ( product_Pair @ A @ B @ ( F2 @ ( product_fst @ C @ D @ X3 ) ) @ ( G @ ( product_snd @ C @ D @ X3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % apfst_apsnd % 0.31/0.60 thf(fact_4753_apfst__compose,axiom, % 0.31/0.60 ! [C: $tType,A: $tType,B: $tType,D: $tType,F2: C > A,G: D > C,X3: product_prod @ D @ B] : % 0.31/0.60 ( ( product_apfst @ C @ A @ B @ F2 @ ( product_apfst @ D @ C @ B @ G @ X3 ) ) % 0.31/0.60 = ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F2 @ G ) @ X3 ) ) ). % 0.31/0.60 % 0.31/0.60 % apfst_compose % 0.31/0.60 thf(fact_4754_apsnd__apfst__commute,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,C: $tType,D: $tType,F2: C > B,G: D > A,P4: product_prod @ D @ C] : % 0.31/0.60 ( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apfst @ D @ A @ C @ G @ P4 ) ) % 0.31/0.60 = ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F2 @ P4 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % apsnd_apfst_commute % 0.31/0.60 thf(fact_4755_gcd__nat_Osemilattice__neutr__order__axioms,axiom, % 0.31/0.60 ( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat ) % 0.31/0.60 @ ^ [M2: nat,N: nat] : % 0.31/0.60 ( ( dvd_dvd @ nat @ M2 @ N ) % 0.31/0.60 & ( M2 != N ) ) ) ). % 0.31/0.60 % 0.31/0.60 % gcd_nat.semilattice_neutr_order_axioms % 0.31/0.60 thf(fact_4756_apfst__convE,axiom, % 0.31/0.60 ! [C: $tType,A: $tType,B: $tType,Q4: product_prod @ A @ B,F2: C > A,P4: product_prod @ C @ B] : % 0.31/0.60 ( ( Q4 % 0.31/0.60 = ( product_apfst @ C @ A @ B @ F2 @ P4 ) ) % 0.31/0.60 => ~ ! [X5: C,Y4: B] : % 0.31/0.60 ( ( P4 % 0.31/0.60 = ( product_Pair @ C @ B @ X5 @ Y4 ) ) % 0.31/0.60 => ( Q4 % 0.31/0.60 != ( product_Pair @ A @ B @ ( F2 @ X5 ) @ Y4 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % apfst_convE % 0.31/0.60 thf(fact_4757_set__removeAll,axiom, % 0.31/0.60 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.60 ( ( set2 @ A @ ( removeAll @ A @ X3 @ Xs ) ) % 0.31/0.60 = ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % set_removeAll % 0.31/0.60 thf(fact_4758_times__int_Oabs__eq,axiom, % 0.31/0.60 ! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] : % 0.31/0.60 ( ( times_times @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) ) % 0.31/0.60 = ( abs_Integ % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) % 0.31/0.60 @ ^ [X: nat,Y2: nat] : % 0.31/0.60 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y2 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V5 ) @ ( times_times @ nat @ Y2 @ U2 ) ) ) ) % 0.31/0.60 @ Xa2 % 0.31/0.60 @ X3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % times_int.abs_eq % 0.31/0.60 thf(fact_4759_removeAll__id,axiom, % 0.31/0.60 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.60 ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.60 => ( ( removeAll @ A @ X3 @ Xs ) % 0.31/0.60 = Xs ) ) ). % 0.31/0.60 % 0.31/0.60 % removeAll_id % 0.31/0.60 thf(fact_4760_distinct__removeAll,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A,X3: A] : % 0.31/0.60 ( ( distinct @ A @ Xs ) % 0.31/0.60 => ( distinct @ A @ ( removeAll @ A @ X3 @ Xs ) ) ) ). % 0.31/0.60 % 0.31/0.60 % distinct_removeAll % 0.31/0.60 thf(fact_4761_int_Oabs__induct,axiom, % 0.31/0.60 ! [P: int > $o,X3: int] : % 0.31/0.60 ( ! [Y4: product_prod @ nat @ nat] : ( P @ ( abs_Integ @ Y4 ) ) % 0.31/0.60 => ( P @ X3 ) ) ). % 0.31/0.60 % 0.31/0.60 % int.abs_induct % 0.31/0.60 thf(fact_4762_eq__Abs__Integ,axiom, % 0.31/0.60 ! [Z2: int] : % 0.31/0.60 ~ ! [X5: nat,Y4: nat] : % 0.31/0.60 ( Z2 % 0.31/0.60 != ( abs_Integ @ ( product_Pair @ nat @ nat @ X5 @ Y4 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % eq_Abs_Integ % 0.31/0.60 thf(fact_4763_nat_Oabs__eq,axiom, % 0.31/0.60 ! [X3: product_prod @ nat @ nat] : % 0.31/0.60 ( ( nat2 @ ( abs_Integ @ X3 ) ) % 0.31/0.60 = ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ X3 ) ) ). % 0.31/0.60 % 0.31/0.60 % nat.abs_eq % 0.31/0.60 thf(fact_4764_length__removeAll__less__eq,axiom, % 0.31/0.60 ! [A: $tType,X3: A,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ). % 0.31/0.60 % 0.31/0.60 % length_removeAll_less_eq % 0.31/0.60 thf(fact_4765_distinct__remove1__removeAll,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A,X3: A] : % 0.31/0.60 ( ( distinct @ A @ Xs ) % 0.31/0.60 => ( ( remove1 @ A @ X3 @ Xs ) % 0.31/0.60 = ( removeAll @ A @ X3 @ Xs ) ) ) ). % 0.31/0.60 % 0.31/0.60 % distinct_remove1_removeAll % 0.31/0.60 thf(fact_4766_zero__int__def,axiom, % 0.31/0.60 ( ( zero_zero @ int ) % 0.31/0.60 = ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % zero_int_def % 0.31/0.60 thf(fact_4767_int__def,axiom, % 0.31/0.60 ( ( semiring_1_of_nat @ int ) % 0.31/0.60 = ( ^ [N: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N @ ( zero_zero @ nat ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % int_def % 0.31/0.60 thf(fact_4768_length__removeAll__less,axiom, % 0.31/0.60 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.60 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.60 => ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ). % 0.31/0.60 % 0.31/0.60 % length_removeAll_less % 0.31/0.60 thf(fact_4769_uminus__int_Oabs__eq,axiom, % 0.31/0.60 ! [X3: product_prod @ nat @ nat] : % 0.31/0.60 ( ( uminus_uminus @ int @ ( abs_Integ @ X3 ) ) % 0.31/0.60 = ( abs_Integ % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ^ [X: nat,Y2: nat] : ( product_Pair @ nat @ nat @ Y2 @ X ) % 0.31/0.60 @ X3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % uminus_int.abs_eq % 0.31/0.60 thf(fact_4770_one__int__def,axiom, % 0.31/0.60 ( ( one_one @ int ) % 0.31/0.60 = ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % one_int_def % 0.31/0.60 thf(fact_4771_of__int_Oabs__eq,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ring_1 @ A ) % 0.31/0.60 => ! [X3: product_prod @ nat @ nat] : % 0.31/0.60 ( ( ring_1_of_int @ A @ ( abs_Integ @ X3 ) ) % 0.31/0.60 = ( product_case_prod @ nat @ nat @ A % 0.31/0.60 @ ^ [I3: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J3 ) ) % 0.31/0.60 @ X3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % of_int.abs_eq % 0.31/0.60 thf(fact_4772_less__int_Oabs__eq,axiom, % 0.31/0.60 ! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] : % 0.31/0.60 ( ( ord_less @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) ) % 0.31/0.60 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o ) % 0.31/0.60 @ ^ [X: nat,Y2: nat] : % 0.31/0.60 ( product_case_prod @ nat @ nat @ $o % 0.31/0.60 @ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y2 ) ) ) % 0.31/0.60 @ Xa2 % 0.31/0.60 @ X3 ) ) ). % 0.31/0.60 % 0.31/0.60 % less_int.abs_eq % 0.31/0.60 thf(fact_4773_less__eq__int_Oabs__eq,axiom, % 0.31/0.60 ! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] : % 0.31/0.60 ( ( ord_less_eq @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) ) % 0.31/0.60 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o ) % 0.31/0.60 @ ^ [X: nat,Y2: nat] : % 0.31/0.60 ( product_case_prod @ nat @ nat @ $o % 0.31/0.60 @ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y2 ) ) ) % 0.31/0.60 @ Xa2 % 0.31/0.60 @ X3 ) ) ). % 0.31/0.60 % 0.31/0.60 % less_eq_int.abs_eq % 0.31/0.60 thf(fact_4774_plus__int_Oabs__eq,axiom, % 0.31/0.60 ! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] : % 0.31/0.60 ( ( plus_plus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) ) % 0.31/0.60 = ( abs_Integ % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) % 0.31/0.60 @ ^ [X: nat,Y2: nat] : % 0.31/0.60 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y2 @ V5 ) ) ) % 0.31/0.60 @ Xa2 % 0.31/0.60 @ X3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % plus_int.abs_eq % 0.31/0.60 thf(fact_4775_minus__int_Oabs__eq,axiom, % 0.31/0.60 ! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] : % 0.31/0.60 ( ( minus_minus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) ) % 0.31/0.60 = ( abs_Integ % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) % 0.31/0.60 @ ^ [X: nat,Y2: nat] : % 0.31/0.60 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ Y2 @ U2 ) ) ) % 0.31/0.60 @ Xa2 % 0.31/0.60 @ X3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % minus_int.abs_eq % 0.31/0.60 thf(fact_4776_distinct__concat__iff,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ ( list @ A )] : % 0.31/0.60 ( ( distinct @ A @ ( concat @ A @ Xs ) ) % 0.31/0.60 = ( ( distinct @ ( list @ A ) @ ( removeAll @ ( list @ A ) @ ( nil @ A ) @ Xs ) ) % 0.31/0.60 & ! [Ys3: list @ A] : % 0.31/0.60 ( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) ) % 0.31/0.60 => ( distinct @ A @ Ys3 ) ) % 0.31/0.60 & ! [Ys3: list @ A,Zs3: list @ A] : % 0.31/0.60 ( ( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) ) % 0.31/0.60 & ( member @ ( list @ A ) @ Zs3 @ ( set2 @ ( list @ A ) @ Xs ) ) % 0.31/0.60 & ( Ys3 != Zs3 ) ) % 0.31/0.60 => ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Zs3 ) ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % distinct_concat_iff % 0.31/0.60 thf(fact_4777_max__rpair__set,axiom, % 0.31/0.60 fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_max_strict @ fun_max_weak ) ). % 0.31/0.60 % 0.31/0.60 % max_rpair_set % 0.31/0.60 thf(fact_4778_min__rpair__set,axiom, % 0.31/0.60 fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict @ fun_min_weak ) ). % 0.31/0.60 % 0.31/0.60 % min_rpair_set % 0.31/0.60 thf(fact_4779_zip__Nil,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,Ys: list @ B] : % 0.31/0.60 ( ( zip @ A @ B @ ( nil @ A ) @ Ys ) % 0.31/0.60 = ( nil @ ( product_prod @ A @ B ) ) ) ). % 0.31/0.60 % 0.31/0.60 % zip_Nil % 0.31/0.60 thf(fact_4780_Nil__eq__zip__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 0.31/0.60 ( ( ( nil @ ( product_prod @ A @ B ) ) % 0.31/0.60 = ( zip @ A @ B @ Xs @ Ys ) ) % 0.31/0.60 = ( ( Xs % 0.31/0.60 = ( nil @ A ) ) % 0.31/0.60 | ( Ys % 0.31/0.60 = ( nil @ B ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Nil_eq_zip_iff % 0.31/0.60 thf(fact_4781_zip__eq__Nil__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 0.31/0.60 ( ( ( zip @ A @ B @ Xs @ Ys ) % 0.31/0.60 = ( nil @ ( product_prod @ A @ B ) ) ) % 0.31/0.60 = ( ( Xs % 0.31/0.60 = ( nil @ A ) ) % 0.31/0.60 | ( Ys % 0.31/0.60 = ( nil @ B ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % zip_eq_Nil_iff % 0.31/0.60 thf(fact_4782_list__update__nonempty,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A,K: nat,X3: A] : % 0.31/0.60 ( ( ( list_update @ A @ Xs @ K @ X3 ) % 0.31/0.60 = ( nil @ A ) ) % 0.31/0.60 = ( Xs % 0.31/0.60 = ( nil @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % list_update_nonempty % 0.31/0.60 thf(fact_4783_concat__replicate__trivial,axiom, % 0.31/0.60 ! [A: $tType,I: nat] : % 0.31/0.60 ( ( concat @ A @ ( replicate @ ( list @ A ) @ I @ ( nil @ A ) ) ) % 0.31/0.60 = ( nil @ A ) ) ). % 0.31/0.60 % 0.31/0.60 % concat_replicate_trivial % 0.31/0.60 thf(fact_4784_Nil__in__shuffles,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.60 ( ( member @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) % 0.31/0.60 = ( ( Xs % 0.31/0.60 = ( nil @ A ) ) % 0.31/0.60 & ( Ys % 0.31/0.60 = ( nil @ A ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Nil_in_shuffles % 0.31/0.60 thf(fact_4785_enumerate__simps_I1_J,axiom, % 0.31/0.60 ! [A: $tType,N2: nat] : % 0.31/0.60 ( ( enumerate @ A @ N2 @ ( nil @ A ) ) % 0.31/0.60 = ( nil @ ( product_prod @ nat @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % enumerate_simps(1) % 0.31/0.60 thf(fact_4786_rotate1__is__Nil__conv,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A] : % 0.31/0.60 ( ( ( rotate1 @ A @ Xs ) % 0.31/0.60 = ( nil @ A ) ) % 0.31/0.60 = ( Xs % 0.31/0.60 = ( nil @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % rotate1_is_Nil_conv % 0.31/0.60 thf(fact_4787_set__empty,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A] : % 0.31/0.60 ( ( ( set2 @ A @ Xs ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 = ( Xs % 0.31/0.60 = ( nil @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % set_empty % 0.31/0.60 thf(fact_4788_set__empty2,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A] : % 0.31/0.60 ( ( ( bot_bot @ ( set @ A ) ) % 0.31/0.60 = ( set2 @ A @ Xs ) ) % 0.31/0.60 = ( Xs % 0.31/0.60 = ( nil @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % set_empty2 % 0.31/0.60 thf(fact_4789_length__0__conv,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A] : % 0.31/0.60 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.60 = ( zero_zero @ nat ) ) % 0.31/0.60 = ( Xs % 0.31/0.60 = ( nil @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % length_0_conv % 0.31/0.60 thf(fact_4790_empty__replicate,axiom, % 0.31/0.60 ! [A: $tType,N2: nat,X3: A] : % 0.31/0.60 ( ( ( nil @ A ) % 0.31/0.60 = ( replicate @ A @ N2 @ X3 ) ) % 0.31/0.60 = ( N2 % 0.31/0.60 = ( zero_zero @ nat ) ) ) ). % 0.31/0.60 % 0.31/0.60 % empty_replicate % 0.31/0.60 thf(fact_4791_replicate__empty,axiom, % 0.31/0.60 ! [A: $tType,N2: nat,X3: A] : % 0.31/0.60 ( ( ( replicate @ A @ N2 @ X3 ) % 0.31/0.60 = ( nil @ A ) ) % 0.31/0.60 = ( N2 % 0.31/0.60 = ( zero_zero @ nat ) ) ) ). % 0.31/0.60 % 0.31/0.60 % replicate_empty % 0.31/0.60 thf(fact_4792_horner__sum__simps_I1_J,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comm_semiring_0 @ A ) % 0.31/0.60 => ! [F2: B > A,A2: A] : % 0.31/0.60 ( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ ( nil @ B ) ) % 0.31/0.60 = ( zero_zero @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % horner_sum_simps(1) % 0.31/0.60 thf(fact_4793_concat__eq__Nil__conv,axiom, % 0.31/0.60 ! [A: $tType,Xss: list @ ( list @ A )] : % 0.31/0.60 ( ( ( concat @ A @ Xss ) % 0.31/0.60 = ( nil @ A ) ) % 0.31/0.60 = ( ! [X: list @ A] : % 0.31/0.60 ( ( member @ ( list @ A ) @ X @ ( set2 @ ( list @ A ) @ Xss ) ) % 0.31/0.60 => ( X % 0.31/0.60 = ( nil @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % concat_eq_Nil_conv % 0.31/0.60 thf(fact_4794_Nil__eq__concat__conv,axiom, % 0.31/0.60 ! [A: $tType,Xss: list @ ( list @ A )] : % 0.31/0.60 ( ( ( nil @ A ) % 0.31/0.60 = ( concat @ A @ Xss ) ) % 0.31/0.60 = ( ! [X: list @ A] : % 0.31/0.60 ( ( member @ ( list @ A ) @ X @ ( set2 @ ( list @ A ) @ Xss ) ) % 0.31/0.60 => ( X % 0.31/0.60 = ( nil @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Nil_eq_concat_conv % 0.31/0.60 thf(fact_4795_length__greater__0__conv,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A] : % 0.31/0.60 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.60 = ( Xs % 0.31/0.60 != ( nil @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % length_greater_0_conv % 0.31/0.60 thf(fact_4796_removeAll_Osimps_I1_J,axiom, % 0.31/0.60 ! [A: $tType,X3: A] : % 0.31/0.60 ( ( removeAll @ A @ X3 @ ( nil @ A ) ) % 0.31/0.60 = ( nil @ A ) ) ). % 0.31/0.60 % 0.31/0.60 % removeAll.simps(1) % 0.31/0.60 thf(fact_4797_rotate1_Osimps_I1_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( rotate1 @ A @ ( nil @ A ) ) % 0.31/0.60 = ( nil @ A ) ) ). % 0.31/0.60 % 0.31/0.60 % rotate1.simps(1) % 0.31/0.60 thf(fact_4798_distinct_Osimps_I1_J,axiom, % 0.31/0.60 ! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ). % 0.31/0.60 % 0.31/0.60 % distinct.simps(1) % 0.31/0.60 thf(fact_4799_concat_Osimps_I1_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( concat @ A @ ( nil @ ( list @ A ) ) ) % 0.31/0.60 = ( nil @ A ) ) ). % 0.31/0.60 % 0.31/0.60 % concat.simps(1) % 0.31/0.60 thf(fact_4800_list__update_Osimps_I1_J,axiom, % 0.31/0.60 ! [A: $tType,I: nat,V2: A] : % 0.31/0.60 ( ( list_update @ A @ ( nil @ A ) @ I @ V2 ) % 0.31/0.60 = ( nil @ A ) ) ). % 0.31/0.60 % 0.31/0.60 % list_update.simps(1) % 0.31/0.60 thf(fact_4801_list__update__code_I1_J,axiom, % 0.31/0.60 ! [A: $tType,I: nat,Y: A] : % 0.31/0.60 ( ( list_update @ A @ ( nil @ A ) @ I @ Y ) % 0.31/0.60 = ( nil @ A ) ) ). % 0.31/0.60 % 0.31/0.60 % list_update_code(1) % 0.31/0.60 thf(fact_4802_product_Osimps_I1_J,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,Uu: list @ B] : % 0.31/0.60 ( ( product @ A @ B @ ( nil @ A ) @ Uu ) % 0.31/0.60 = ( nil @ ( product_prod @ A @ B ) ) ) ). % 0.31/0.60 % 0.31/0.60 % product.simps(1) % 0.31/0.60 thf(fact_4803_zip_Osimps_I1_J,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,Xs: list @ A] : % 0.31/0.60 ( ( zip @ A @ B @ Xs @ ( nil @ B ) ) % 0.31/0.60 = ( nil @ ( product_prod @ A @ B ) ) ) ). % 0.31/0.60 % 0.31/0.60 % zip.simps(1) % 0.31/0.60 thf(fact_4804_Nil__in__shufflesI,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.60 ( ( Xs % 0.31/0.60 = ( nil @ A ) ) % 0.31/0.60 => ( ( Ys % 0.31/0.60 = ( nil @ A ) ) % 0.31/0.60 => ( member @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Nil_in_shufflesI % 0.31/0.60 thf(fact_4805_shuffles_Osimps_I1_J,axiom, % 0.31/0.60 ! [A: $tType,Ys: list @ A] : % 0.31/0.60 ( ( shuffles @ A @ ( nil @ A ) @ Ys ) % 0.31/0.60 = ( insert @ ( list @ A ) @ Ys @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % shuffles.simps(1) % 0.31/0.60 thf(fact_4806_shuffles_Osimps_I2_J,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A] : % 0.31/0.60 ( ( shuffles @ A @ Xs @ ( nil @ A ) ) % 0.31/0.60 = ( insert @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % shuffles.simps(2) % 0.31/0.60 thf(fact_4807_remove1_Osimps_I1_J,axiom, % 0.31/0.60 ! [A: $tType,X3: A] : % 0.31/0.60 ( ( remove1 @ A @ X3 @ ( nil @ A ) ) % 0.31/0.60 = ( nil @ A ) ) ). % 0.31/0.60 % 0.31/0.60 % remove1.simps(1) % 0.31/0.60 thf(fact_4808_gen__length__code_I1_J,axiom, % 0.31/0.60 ! [A: $tType,N2: nat] : % 0.31/0.60 ( ( gen_length @ A @ N2 @ ( nil @ A ) ) % 0.31/0.60 = N2 ) ). % 0.31/0.60 % 0.31/0.60 % gen_length_code(1) % 0.31/0.60 thf(fact_4809_empty__set,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( bot_bot @ ( set @ A ) ) % 0.31/0.60 = ( set2 @ A @ ( nil @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % empty_set % 0.31/0.60 thf(fact_4810_list_Osize_I3_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( size_size @ ( list @ A ) @ ( nil @ A ) ) % 0.31/0.60 = ( zero_zero @ nat ) ) ). % 0.31/0.60 % 0.31/0.60 % list.size(3) % 0.31/0.60 thf(fact_4811_replicate__0,axiom, % 0.31/0.60 ! [A: $tType,X3: A] : % 0.31/0.60 ( ( replicate @ A @ ( zero_zero @ nat ) @ X3 ) % 0.31/0.60 = ( nil @ A ) ) ). % 0.31/0.60 % 0.31/0.60 % replicate_0 % 0.31/0.60 thf(fact_4812_not__listrel1__Nil,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.60 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( listrel1 @ A @ R2 ) ) ). % 0.31/0.60 % 0.31/0.60 % not_listrel1_Nil % 0.31/0.60 thf(fact_4813_not__Nil__listrel1,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.60 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs ) @ ( listrel1 @ A @ R2 ) ) ). % 0.31/0.60 % 0.31/0.60 % not_Nil_listrel1 % 0.31/0.60 thf(fact_4814_list_Osize__gen_I1_J,axiom, % 0.31/0.60 ! [A: $tType,X3: A > nat] : % 0.31/0.60 ( ( size_list @ A @ X3 @ ( nil @ A ) ) % 0.31/0.60 = ( zero_zero @ nat ) ) ). % 0.31/0.60 % 0.31/0.60 % list.size_gen(1) % 0.31/0.60 thf(fact_4815_find_Osimps_I1_J,axiom, % 0.31/0.60 ! [A: $tType,Uu: A > $o] : % 0.31/0.60 ( ( find @ A @ Uu @ ( nil @ A ) ) % 0.31/0.60 = ( none @ A ) ) ). % 0.31/0.60 % 0.31/0.60 % find.simps(1) % 0.31/0.60 thf(fact_4816_count__list_Osimps_I1_J,axiom, % 0.31/0.60 ! [A: $tType,Y: A] : % 0.31/0.60 ( ( count_list @ A @ ( nil @ A ) @ Y ) % 0.31/0.60 = ( zero_zero @ nat ) ) ). % 0.31/0.60 % 0.31/0.60 % count_list.simps(1) % 0.31/0.60 thf(fact_4817_Pow__set_I1_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( pow2 @ A @ ( set2 @ A @ ( nil @ A ) ) ) % 0.31/0.60 = ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Pow_set(1) % 0.31/0.60 thf(fact_4818_less__eq__int_Orep__eq,axiom, % 0.31/0.60 ( ( ord_less_eq @ int ) % 0.31/0.60 = ( ^ [X: int,Xa4: int] : % 0.31/0.60 ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o ) % 0.31/0.60 @ ^ [Y2: nat,Z6: nat] : % 0.31/0.60 ( product_case_prod @ nat @ nat @ $o % 0.31/0.60 @ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) ) % 0.31/0.60 @ ( rep_Integ @ X ) % 0.31/0.60 @ ( rep_Integ @ Xa4 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % less_eq_int.rep_eq % 0.31/0.60 thf(fact_4819_less__int_Orep__eq,axiom, % 0.31/0.60 ( ( ord_less @ int ) % 0.31/0.60 = ( ^ [X: int,Xa4: int] : % 0.31/0.60 ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o ) % 0.31/0.60 @ ^ [Y2: nat,Z6: nat] : % 0.31/0.60 ( product_case_prod @ nat @ nat @ $o % 0.31/0.60 @ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) ) % 0.31/0.60 @ ( rep_Integ @ X ) % 0.31/0.60 @ ( rep_Integ @ Xa4 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % less_int.rep_eq % 0.31/0.60 thf(fact_4820_prod__encode__def,axiom, % 0.31/0.60 ( nat_prod_encode % 0.31/0.60 = ( product_case_prod @ nat @ nat @ nat % 0.31/0.60 @ ^ [M2: nat,N: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M2 @ N ) ) @ M2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % prod_encode_def % 0.31/0.60 thf(fact_4821_le__prod__encode__2,axiom, % 0.31/0.60 ! [B2: nat,A2: nat] : ( ord_less_eq @ nat @ B2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A2 @ B2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % le_prod_encode_2 % 0.31/0.60 thf(fact_4822_le__prod__encode__1,axiom, % 0.31/0.60 ! [A2: nat,B2: nat] : ( ord_less_eq @ nat @ A2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A2 @ B2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % le_prod_encode_1 % 0.31/0.60 thf(fact_4823_nat_Orep__eq,axiom, % 0.31/0.60 ( nat2 % 0.31/0.60 = ( ^ [X: int] : ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ ( rep_Integ @ X ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % nat.rep_eq % 0.31/0.60 thf(fact_4824_of__int_Orep__eq,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ring_1 @ A ) % 0.31/0.60 => ( ( ring_1_of_int @ A ) % 0.31/0.60 = ( ^ [X: int] : % 0.31/0.60 ( product_case_prod @ nat @ nat @ A % 0.31/0.60 @ ^ [I3: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J3 ) ) % 0.31/0.60 @ ( rep_Integ @ X ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % of_int.rep_eq % 0.31/0.60 thf(fact_4825_prod__encode__prod__decode__aux,axiom, % 0.31/0.60 ! [K: nat,M: nat] : % 0.31/0.60 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) ) % 0.31/0.60 = ( plus_plus @ nat @ ( nat_triangle @ K ) @ M ) ) ). % 0.31/0.60 % 0.31/0.60 % prod_encode_prod_decode_aux % 0.31/0.60 thf(fact_4826_listset_Osimps_I1_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( listset @ A @ ( nil @ ( set @ A ) ) ) % 0.31/0.60 = ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % listset.simps(1) % 0.31/0.60 thf(fact_4827_uminus__int__def,axiom, % 0.31/0.60 ( ( uminus_uminus @ int ) % 0.31/0.60 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ^ [X: nat,Y2: nat] : ( product_Pair @ nat @ nat @ Y2 @ X ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % uminus_int_def % 0.31/0.60 thf(fact_4828_eq__snd__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,B2: A,P4: product_prod @ B @ A] : % 0.31/0.60 ( ( B2 % 0.31/0.60 = ( product_snd @ B @ A @ P4 ) ) % 0.31/0.60 = ( ? [A4: B] : % 0.31/0.60 ( P4 % 0.31/0.60 = ( product_Pair @ B @ A @ A4 @ B2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % eq_snd_iff % 0.31/0.60 thf(fact_4829_prod_Osize__neq,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,X3: product_prod @ A @ B] : % 0.31/0.60 ( ( size_size @ ( product_prod @ A @ B ) @ X3 ) % 0.31/0.60 != ( zero_zero @ nat ) ) ). % 0.31/0.60 % 0.31/0.60 % prod.size_neq % 0.31/0.60 thf(fact_4830_sum_Osize__neq,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,X3: sum_sum @ A @ B] : % 0.31/0.60 ( ( size_size @ ( sum_sum @ A @ B ) @ X3 ) % 0.31/0.60 != ( zero_zero @ nat ) ) ). % 0.31/0.60 % 0.31/0.60 % sum.size_neq % 0.31/0.60 thf(fact_4831_eq__fst__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,A2: A,P4: product_prod @ A @ B] : % 0.31/0.60 ( ( A2 % 0.31/0.60 = ( product_fst @ A @ B @ P4 ) ) % 0.31/0.60 = ( ? [B4: B] : % 0.31/0.60 ( P4 % 0.31/0.60 = ( product_Pair @ A @ B @ A2 @ B4 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % eq_fst_iff % 0.31/0.60 thf(fact_4832_times__int__def,axiom, % 0.31/0.60 ( ( times_times @ int ) % 0.31/0.60 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ ) % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) % 0.31/0.60 @ ^ [X: nat,Y2: nat] : % 0.31/0.60 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y2 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V5 ) @ ( times_times @ nat @ Y2 @ U2 ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % times_int_def % 0.31/0.60 thf(fact_4833_minus__int__def,axiom, % 0.31/0.60 ( ( minus_minus @ int ) % 0.31/0.60 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ ) % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) % 0.31/0.60 @ ^ [X: nat,Y2: nat] : % 0.31/0.60 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ Y2 @ U2 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % minus_int_def % 0.31/0.60 thf(fact_4834_plus__int__def,axiom, % 0.31/0.60 ( ( plus_plus @ int ) % 0.31/0.60 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ ) % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) % 0.31/0.60 @ ^ [X: nat,Y2: nat] : % 0.31/0.60 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y2 @ V5 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % plus_int_def % 0.31/0.60 thf(fact_4835_Gcd__remove0__nat,axiom, % 0.31/0.60 ! [M7: set @ nat] : % 0.31/0.60 ( ( finite_finite2 @ nat @ M7 ) % 0.31/0.60 => ( ( gcd_Gcd @ nat @ M7 ) % 0.31/0.60 = ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M7 @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_remove0_nat % 0.31/0.60 thf(fact_4836_add__neg__numeral__special_I4_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( neg_numeral @ A ) % 0.31/0.60 => ! [N2: num] : % 0.31/0.60 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N2 ) ) % 0.31/0.60 = ( neg_numeral_sub @ A @ N2 @ one2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % add_neg_numeral_special(4) % 0.31/0.60 thf(fact_4837_add__neg__numeral__special_I3_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( neg_numeral @ A ) % 0.31/0.60 => ! [M: num] : % 0.31/0.60 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.31/0.60 = ( neg_numeral_sub @ A @ M @ one2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % add_neg_numeral_special(3) % 0.31/0.60 thf(fact_4838_sub__num__simps_I1_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( neg_numeral @ A ) % 0.31/0.60 => ( ( neg_numeral_sub @ A @ one2 @ one2 ) % 0.31/0.60 = ( zero_zero @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sub_num_simps(1) % 0.31/0.60 thf(fact_4839_Gcd__empty,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_Gcd @ A ) % 0.31/0.60 => ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 = ( zero_zero @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_empty % 0.31/0.60 thf(fact_4840_Gcd__insert,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_Gcd @ A ) % 0.31/0.60 => ! [A2: A,A3: set @ A] : % 0.31/0.60 ( ( gcd_Gcd @ A @ ( insert @ A @ A2 @ A3 ) ) % 0.31/0.60 = ( gcd_gcd @ A @ A2 @ ( gcd_Gcd @ A @ A3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_insert % 0.31/0.60 thf(fact_4841_semiring__norm_I167_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( neg_numeral @ A ) % 0.31/0.60 => ! [V2: num,W2: num,Y: A] : % 0.31/0.60 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) ) % 0.31/0.60 = ( plus_plus @ A @ ( neg_numeral_sub @ A @ W2 @ V2 ) @ Y ) ) ) ). % 0.31/0.60 % 0.31/0.60 % semiring_norm(167) % 0.31/0.60 thf(fact_4842_semiring__norm_I166_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( neg_numeral @ A ) % 0.31/0.60 => ! [V2: num,W2: num,Y: A] : % 0.31/0.60 ( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) ) % 0.31/0.60 = ( plus_plus @ A @ ( neg_numeral_sub @ A @ V2 @ W2 ) @ Y ) ) ) ). % 0.31/0.60 % 0.31/0.60 % semiring_norm(166) % 0.31/0.60 thf(fact_4843_add__neg__numeral__simps_I1_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( neg_numeral @ A ) % 0.31/0.60 => ! [M: num,N2: num] : % 0.31/0.60 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) % 0.31/0.60 = ( neg_numeral_sub @ A @ M @ N2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % add_neg_numeral_simps(1) % 0.31/0.60 thf(fact_4844_add__neg__numeral__simps_I2_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( neg_numeral @ A ) % 0.31/0.60 => ! [M: num,N2: num] : % 0.31/0.60 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) ) % 0.31/0.60 = ( neg_numeral_sub @ A @ N2 @ M ) ) ) ). % 0.31/0.60 % 0.31/0.60 % add_neg_numeral_simps(2) % 0.31/0.60 thf(fact_4845_Gcd__2,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_Gcd @ A ) % 0.31/0.60 => ! [A2: A,B2: A] : % 0.31/0.60 ( ( gcd_Gcd @ A @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.60 = ( gcd_gcd @ A @ A2 @ B2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_2 % 0.31/0.60 thf(fact_4846_Gcd__0__iff,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_Gcd @ A ) % 0.31/0.60 => ! [A3: set @ A] : % 0.31/0.60 ( ( ( gcd_Gcd @ A @ A3 ) % 0.31/0.60 = ( zero_zero @ A ) ) % 0.31/0.60 = ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_0_iff % 0.31/0.60 thf(fact_4847_add__neg__numeral__special_I1_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( neg_numeral @ A ) % 0.31/0.60 => ! [M: num] : % 0.31/0.60 ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) % 0.31/0.60 = ( neg_numeral_sub @ A @ one2 @ M ) ) ) ). % 0.31/0.60 % 0.31/0.60 % add_neg_numeral_special(1) % 0.31/0.60 thf(fact_4848_add__neg__numeral__special_I2_J,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( neg_numeral @ A ) % 0.31/0.60 => ! [M: num] : % 0.31/0.60 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) % 0.31/0.60 = ( neg_numeral_sub @ A @ one2 @ M ) ) ) ). % 0.31/0.60 % 0.31/0.60 % add_neg_numeral_special(2) % 0.31/0.60 thf(fact_4849_Gcd__dvd,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_Gcd @ A ) % 0.31/0.60 => ! [A2: A,A3: set @ A] : % 0.31/0.60 ( ( member @ A @ A2 @ A3 ) % 0.31/0.60 => ( dvd_dvd @ A @ ( gcd_Gcd @ A @ A3 ) @ A2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_dvd % 0.31/0.60 thf(fact_4850_dvd__GcdD,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_Gcd @ A ) % 0.31/0.60 => ! [X3: A,A3: set @ A,Y: A] : % 0.31/0.60 ( ( dvd_dvd @ A @ X3 @ ( gcd_Gcd @ A @ A3 ) ) % 0.31/0.60 => ( ( member @ A @ Y @ A3 ) % 0.31/0.60 => ( dvd_dvd @ A @ X3 @ Y ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % dvd_GcdD % 0.31/0.60 thf(fact_4851_dvd__Gcd__iff,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_Gcd @ A ) % 0.31/0.60 => ! [X3: A,A3: set @ A] : % 0.31/0.60 ( ( dvd_dvd @ A @ X3 @ ( gcd_Gcd @ A @ A3 ) ) % 0.31/0.60 = ( ! [X: A] : % 0.31/0.60 ( ( member @ A @ X @ A3 ) % 0.31/0.60 => ( dvd_dvd @ A @ X3 @ X ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % dvd_Gcd_iff % 0.31/0.60 thf(fact_4852_Gcd__greatest,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_Gcd @ A ) % 0.31/0.60 => ! [A3: set @ A,A2: A] : % 0.31/0.60 ( ! [B5: A] : % 0.31/0.60 ( ( member @ A @ B5 @ A3 ) % 0.31/0.60 => ( dvd_dvd @ A @ A2 @ B5 ) ) % 0.31/0.60 => ( dvd_dvd @ A @ A2 @ ( gcd_Gcd @ A @ A3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_greatest % 0.31/0.60 thf(fact_4853_Gcd__dvd__nat,axiom, % 0.31/0.60 ! [A2: nat,A3: set @ nat] : % 0.31/0.60 ( ( member @ nat @ A2 @ A3 ) % 0.31/0.60 => ( dvd_dvd @ nat @ ( gcd_Gcd @ nat @ A3 ) @ A2 ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_dvd_nat % 0.31/0.60 thf(fact_4854_Gcd__greatest__nat,axiom, % 0.31/0.60 ! [A3: set @ nat,A2: nat] : % 0.31/0.60 ( ! [B5: nat] : % 0.31/0.60 ( ( member @ nat @ B5 @ A3 ) % 0.31/0.60 => ( dvd_dvd @ nat @ A2 @ B5 ) ) % 0.31/0.60 => ( dvd_dvd @ nat @ A2 @ ( gcd_Gcd @ nat @ A3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_greatest_nat % 0.31/0.60 thf(fact_4855_Gcd__1,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_Gcd @ A ) % 0.31/0.60 => ! [A3: set @ A] : % 0.31/0.60 ( ( member @ A @ ( one_one @ A ) @ A3 ) % 0.31/0.60 => ( ( gcd_Gcd @ A @ A3 ) % 0.31/0.60 = ( one_one @ A ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_1 % 0.31/0.60 thf(fact_4856_Gcd__nat__eq__one,axiom, % 0.31/0.60 ! [N7: set @ nat] : % 0.31/0.60 ( ( member @ nat @ ( one_one @ nat ) @ N7 ) % 0.31/0.60 => ( ( gcd_Gcd @ nat @ N7 ) % 0.31/0.60 = ( one_one @ nat ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_nat_eq_one % 0.31/0.60 thf(fact_4857_Gcd__eq__1__I,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_Gcd @ A ) % 0.31/0.60 => ! [A2: A,A3: set @ A] : % 0.31/0.60 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 0.31/0.60 => ( ( member @ A @ A2 @ A3 ) % 0.31/0.60 => ( ( gcd_Gcd @ A @ A3 ) % 0.31/0.60 = ( one_one @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_eq_1_I % 0.31/0.60 thf(fact_4858_Gcd__in,axiom, % 0.31/0.60 ! [A3: set @ nat] : % 0.31/0.60 ( ! [A5: nat,B5: nat] : % 0.31/0.60 ( ( member @ nat @ A5 @ A3 ) % 0.31/0.60 => ( ( member @ nat @ B5 @ A3 ) % 0.31/0.60 => ( member @ nat @ ( gcd_gcd @ nat @ A5 @ B5 ) @ A3 ) ) ) % 0.31/0.60 => ( ( A3 % 0.31/0.60 != ( bot_bot @ ( set @ nat ) ) ) % 0.31/0.60 => ( member @ nat @ ( gcd_Gcd @ nat @ A3 ) @ A3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_in % 0.31/0.60 thf(fact_4859_sub__non__negative,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_idom @ A ) % 0.31/0.60 => ! [N2: num,M: num] : % 0.31/0.60 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N2 @ M ) ) % 0.31/0.60 = ( ord_less_eq @ num @ M @ N2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sub_non_negative % 0.31/0.60 thf(fact_4860_sub__non__positive,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_idom @ A ) % 0.31/0.60 => ! [N2: num,M: num] : % 0.31/0.60 ( ( ord_less_eq @ A @ ( neg_numeral_sub @ A @ N2 @ M ) @ ( zero_zero @ A ) ) % 0.31/0.60 = ( ord_less_eq @ num @ N2 @ M ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sub_non_positive % 0.31/0.60 thf(fact_4861_sub__positive,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_idom @ A ) % 0.31/0.60 => ! [N2: num,M: num] : % 0.31/0.60 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N2 @ M ) ) % 0.31/0.60 = ( ord_less @ num @ M @ N2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sub_positive % 0.31/0.60 thf(fact_4862_sub__negative,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_idom @ A ) % 0.31/0.60 => ! [N2: num,M: num] : % 0.31/0.60 ( ( ord_less @ A @ ( neg_numeral_sub @ A @ N2 @ M ) @ ( zero_zero @ A ) ) % 0.31/0.60 = ( ord_less @ num @ N2 @ M ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sub_negative % 0.31/0.60 thf(fact_4863_sub__BitM__One__eq,axiom, % 0.31/0.60 ! [N2: num] : % 0.31/0.60 ( ( neg_numeral_sub @ int @ ( bitM @ N2 ) @ one2 ) % 0.31/0.60 = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( neg_numeral_sub @ int @ N2 @ one2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sub_BitM_One_eq % 0.31/0.60 thf(fact_4864_semiring__char__def,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_1 @ A ) % 0.31/0.60 => ( ( semiri4206861660011772517g_char @ A ) % 0.31/0.60 = ( ^ [Uu4: itself @ A] : % 0.31/0.60 ( gcd_Gcd @ nat % 0.31/0.60 @ ( collect @ nat % 0.31/0.60 @ ^ [N: nat] : % 0.31/0.60 ( ( semiring_1_of_nat @ A @ N ) % 0.31/0.60 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % semiring_char_def % 0.31/0.60 thf(fact_4865_num__of__nat_Osimps_I2_J,axiom, % 0.31/0.60 ! [N2: nat] : % 0.31/0.60 ( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.60 => ( ( num_of_nat @ ( suc @ N2 ) ) % 0.31/0.60 = ( inc @ ( num_of_nat @ N2 ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.60 => ( ( num_of_nat @ ( suc @ N2 ) ) % 0.31/0.60 = one2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % num_of_nat.simps(2) % 0.31/0.60 thf(fact_4866_pred__nat__def,axiom, % 0.31/0.60 ( pred_nat % 0.31/0.60 = ( collect @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.60 @ ^ [M2: nat,N: nat] : % 0.31/0.60 ( N % 0.31/0.60 = ( suc @ M2 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % pred_nat_def % 0.31/0.60 thf(fact_4867_abs__Gcd__eq,axiom, % 0.31/0.60 ! [K5: set @ int] : % 0.31/0.60 ( ( abs_abs @ int @ ( gcd_Gcd @ int @ K5 ) ) % 0.31/0.60 = ( gcd_Gcd @ int @ K5 ) ) ). % 0.31/0.60 % 0.31/0.60 % abs_Gcd_eq % 0.31/0.60 thf(fact_4868_Gcd__dvd__int,axiom, % 0.31/0.60 ! [A2: int,A3: set @ int] : % 0.31/0.60 ( ( member @ int @ A2 @ A3 ) % 0.31/0.60 => ( dvd_dvd @ int @ ( gcd_Gcd @ int @ A3 ) @ A2 ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_dvd_int % 0.31/0.60 thf(fact_4869_Gcd__greatest__int,axiom, % 0.31/0.60 ! [A3: set @ int,A2: int] : % 0.31/0.60 ( ! [B5: int] : % 0.31/0.60 ( ( member @ int @ B5 @ A3 ) % 0.31/0.60 => ( dvd_dvd @ int @ A2 @ B5 ) ) % 0.31/0.60 => ( dvd_dvd @ int @ A2 @ ( gcd_Gcd @ int @ A3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_greatest_int % 0.31/0.60 thf(fact_4870_Gcd__int__greater__eq__0,axiom, % 0.31/0.60 ! [K5: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Gcd @ int @ K5 ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_int_greater_eq_0 % 0.31/0.60 thf(fact_4871_num__of__nat_Osimps_I1_J,axiom, % 0.31/0.60 ( ( num_of_nat @ ( zero_zero @ nat ) ) % 0.31/0.60 = one2 ) ). % 0.31/0.60 % 0.31/0.60 % num_of_nat.simps(1) % 0.31/0.60 thf(fact_4872_numeral__num__of__nat,axiom, % 0.31/0.60 ! [N2: nat] : % 0.31/0.60 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.60 => ( ( numeral_numeral @ nat @ ( num_of_nat @ N2 ) ) % 0.31/0.60 = N2 ) ) ). % 0.31/0.60 % 0.31/0.60 % numeral_num_of_nat % 0.31/0.60 thf(fact_4873_num__of__nat__One,axiom, % 0.31/0.60 ! [N2: nat] : % 0.31/0.60 ( ( ord_less_eq @ nat @ N2 @ ( one_one @ nat ) ) % 0.31/0.60 => ( ( num_of_nat @ N2 ) % 0.31/0.60 = one2 ) ) ). % 0.31/0.60 % 0.31/0.60 % num_of_nat_One % 0.31/0.60 thf(fact_4874_numeral__num__of__nat__unfold,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_1 @ A ) % 0.31/0.60 => ! [N2: nat] : % 0.31/0.60 ( ( ( N2 % 0.31/0.60 = ( zero_zero @ nat ) ) % 0.31/0.60 => ( ( numeral_numeral @ A @ ( num_of_nat @ N2 ) ) % 0.31/0.60 = ( one_one @ A ) ) ) % 0.31/0.60 & ( ( N2 % 0.31/0.60 != ( zero_zero @ nat ) ) % 0.31/0.60 => ( ( numeral_numeral @ A @ ( num_of_nat @ N2 ) ) % 0.31/0.60 = ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % numeral_num_of_nat_unfold % 0.31/0.60 thf(fact_4875_num__of__nat__double,axiom, % 0.31/0.60 ! [N2: nat] : % 0.31/0.60 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.60 => ( ( num_of_nat @ ( plus_plus @ nat @ N2 @ N2 ) ) % 0.31/0.60 = ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % num_of_nat_double % 0.31/0.60 thf(fact_4876_num__of__nat__plus__distrib,axiom, % 0.31/0.60 ! [M: nat,N2: nat] : % 0.31/0.60 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 0.31/0.60 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.60 => ( ( num_of_nat @ ( plus_plus @ nat @ M @ N2 ) ) % 0.31/0.60 = ( plus_plus @ num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % num_of_nat_plus_distrib % 0.31/0.60 thf(fact_4877_prod_Oinsert_H,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_mult @ A ) % 0.31/0.60 => ! [I6: set @ B,P4: B > A,I: B] : % 0.31/0.60 ( ( finite_finite2 @ B % 0.31/0.60 @ ( collect @ B % 0.31/0.60 @ ^ [X: B] : % 0.31/0.60 ( ( member @ B @ X @ I6 ) % 0.31/0.60 & ( ( P4 @ X ) % 0.31/0.60 != ( one_one @ A ) ) ) ) ) % 0.31/0.60 => ( ( ( member @ B @ I @ I6 ) % 0.31/0.60 => ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) ) % 0.31/0.60 = ( groups1962203154675924110t_prod @ B @ A @ P4 @ I6 ) ) ) % 0.31/0.60 & ( ~ ( member @ B @ I @ I6 ) % 0.31/0.60 => ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) ) % 0.31/0.60 = ( times_times @ A @ ( P4 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P4 @ I6 ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % prod.insert' % 0.31/0.60 thf(fact_4878_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A3: set @ A,X3: A] : % 0.31/0.60 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.60 => ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.60 = ( remove1 @ A @ X3 @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sorted_list_of_set.sorted_key_list_of_set_remove % 0.31/0.60 thf(fact_4879_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom, % 0.31/0.60 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.31/0.60 ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 ) % 0.31/0.60 => ( ( ? [Uu2: $o,Uv2: $o] : % 0.31/0.60 ( X3 % 0.31/0.60 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 0.31/0.60 => ( Xa2 % 0.31/0.60 = ( one_one @ nat ) ) ) % 0.31/0.60 => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 0.31/0.60 ( ( X3 % 0.31/0.60 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) ) % 0.31/0.60 => ( ( Deg2 = Xa2 ) % 0.31/0.60 & ! [X5: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.60 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.60 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 0.31/0.60 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.31/0.60 & ( case_option @ $o @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 ) % 0.31/0.60 & ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.60 @ ^ [Mi3: nat,Ma3: nat] : % 0.31/0.60 ( ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 0.31/0.60 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 0.31/0.60 & ! [I3: nat] : % 0.31/0.60 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.31/0.60 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) ) % 0.31/0.60 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) ) % 0.31/0.60 & ( ( Mi3 = Ma3 ) % 0.31/0.60 => ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.31/0.60 & ( ( Mi3 != Ma3 ) % 0.31/0.60 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 ) % 0.31/0.60 & ! [X: nat] : % 0.31/0.60 ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 0.31/0.60 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X ) % 0.31/0.60 => ( ( ord_less @ nat @ Mi3 @ X ) % 0.31/0.60 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) % 0.31/0.60 @ Mima ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % VEBT_internal.valid'.elims(3) % 0.31/0.60 thf(fact_4880_ball__empty,axiom, % 0.31/0.60 ! [A: $tType,P: A > $o,X2: A] : % 0.31/0.60 ( ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 => ( P @ X2 ) ) ). % 0.31/0.60 % 0.31/0.60 % ball_empty % 0.31/0.60 thf(fact_4881_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ( ( linord4507533701916653071of_set @ A @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 = ( nil @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sorted_list_of_set.sorted_key_list_of_set_empty % 0.31/0.60 thf(fact_4882_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A3: set @ A] : % 0.31/0.60 ( ~ ( finite_finite2 @ A @ A3 ) % 0.31/0.60 => ( ( linord4507533701916653071of_set @ A @ A3 ) % 0.31/0.60 = ( nil @ A ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sorted_list_of_set.fold_insort_key.infinite % 0.31/0.60 thf(fact_4883_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A3: set @ A] : % 0.31/0.60 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.60 => ( ( set2 @ A @ ( linord4507533701916653071of_set @ A @ A3 ) ) % 0.31/0.60 = A3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sorted_list_of_set.set_sorted_key_list_of_set % 0.31/0.60 thf(fact_4884_prod_Oempty_H,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comm_monoid_mult @ A ) % 0.31/0.60 => ! [P4: B > A] : % 0.31/0.60 ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 = ( one_one @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % prod.empty' % 0.31/0.60 thf(fact_4885_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A3: set @ A] : % 0.31/0.60 ( ( size_size @ ( list @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) ) % 0.31/0.60 = ( finite_card @ A @ A3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sorted_list_of_set.length_sorted_key_list_of_set % 0.31/0.60 thf(fact_4886_prod_Oeq__sum,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_mult @ A ) % 0.31/0.60 => ! [I6: set @ B,P4: B > A] : % 0.31/0.60 ( ( finite_finite2 @ B @ I6 ) % 0.31/0.60 => ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ I6 ) % 0.31/0.60 = ( groups7121269368397514597t_prod @ B @ A @ P4 @ I6 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % prod.eq_sum % 0.31/0.60 thf(fact_4887_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A3: set @ A] : % 0.31/0.60 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.60 => ( ( ( linord4507533701916653071of_set @ A @ A3 ) % 0.31/0.60 = ( nil @ A ) ) % 0.31/0.60 = ( A3 % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff % 0.31/0.60 thf(fact_4888_finite__set__of__finite__funs,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,D2: B] : % 0.31/0.60 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.60 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.60 => ( finite_finite2 @ ( A > B ) % 0.31/0.60 @ ( collect @ ( A > B ) % 0.31/0.60 @ ^ [F3: A > B] : % 0.31/0.60 ! [X: A] : % 0.31/0.60 ( ( ( member @ A @ X @ A3 ) % 0.31/0.60 => ( member @ B @ ( F3 @ X ) @ B3 ) ) % 0.31/0.60 & ( ~ ( member @ A @ X @ A3 ) % 0.31/0.60 => ( ( F3 @ X ) % 0.31/0.60 = D2 ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % finite_set_of_finite_funs % 0.31/0.60 thf(fact_4889_option_Osimps_I5_J,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,F1: B,F22: A > B,X22: A] : % 0.31/0.60 ( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X22 ) ) % 0.31/0.60 = ( F22 @ X22 ) ) ). % 0.31/0.60 % 0.31/0.60 % option.simps(5) % 0.31/0.60 thf(fact_4890_option_Osimps_I4_J,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F1: B,F22: A > B] : % 0.31/0.60 ( ( case_option @ B @ A @ F1 @ F22 @ ( none @ A ) ) % 0.31/0.60 = F1 ) ). % 0.31/0.60 % 0.31/0.60 % option.simps(4) % 0.31/0.60 thf(fact_4891_Ball__Collect,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ball @ A ) % 0.31/0.60 = ( ^ [A6: set @ A,P3: A > $o] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( collect @ A @ P3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Ball_Collect % 0.31/0.60 thf(fact_4892_option_Ocase__distrib,axiom, % 0.31/0.60 ! [C: $tType,B: $tType,A: $tType,H2: B > C,F1: B,F22: A > B,Option: option @ A] : % 0.31/0.60 ( ( H2 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) ) % 0.31/0.60 = ( case_option @ C @ A @ ( H2 @ F1 ) % 0.31/0.60 @ ^ [X: A] : ( H2 @ ( F22 @ X ) ) % 0.31/0.60 @ Option ) ) ). % 0.31/0.60 % 0.31/0.60 % option.case_distrib % 0.31/0.60 thf(fact_4893_Ball__def,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ball @ A ) % 0.31/0.60 = ( ^ [A6: set @ A,P3: A > $o] : % 0.31/0.60 ! [X: A] : % 0.31/0.60 ( ( member @ A @ X @ A6 ) % 0.31/0.60 => ( P3 @ X ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Ball_def % 0.31/0.60 thf(fact_4894_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.60 ( ( ( linord4507533701916653071of_set @ A @ A3 ) % 0.31/0.60 = ( linord4507533701916653071of_set @ A @ B3 ) ) % 0.31/0.60 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.60 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.60 => ( A3 = B3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sorted_list_of_set.sorted_key_list_of_set_inject % 0.31/0.60 thf(fact_4895_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A3: set @ A] : ( distinct @ A @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sorted_list_of_set.distinct_sorted_key_list_of_set % 0.31/0.60 thf(fact_4896_option_Odisc__eq__case_I2_J,axiom, % 0.31/0.60 ! [A: $tType,Option: option @ A] : % 0.31/0.60 ( ( Option % 0.31/0.60 != ( none @ A ) ) % 0.31/0.60 = ( case_option @ $o @ A @ $false % 0.31/0.60 @ ^ [Uu3: A] : $true % 0.31/0.60 @ Option ) ) ). % 0.31/0.60 % 0.31/0.60 % option.disc_eq_case(2) % 0.31/0.60 thf(fact_4897_option_Odisc__eq__case_I1_J,axiom, % 0.31/0.60 ! [A: $tType,Option: option @ A] : % 0.31/0.60 ( ( Option % 0.31/0.60 = ( none @ A ) ) % 0.31/0.60 = ( case_option @ $o @ A @ $true % 0.31/0.60 @ ^ [Uu3: A] : $false % 0.31/0.60 @ Option ) ) ). % 0.31/0.60 % 0.31/0.60 % option.disc_eq_case(1) % 0.31/0.60 thf(fact_4898_prod_Odistrib__triv_H,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_mult @ A ) % 0.31/0.60 => ! [I6: set @ B,G: B > A,H2: B > A] : % 0.31/0.60 ( ( finite_finite2 @ B @ I6 ) % 0.31/0.60 => ( ( groups1962203154675924110t_prod @ B @ A % 0.31/0.60 @ ^ [I3: B] : ( times_times @ A @ ( G @ I3 ) @ ( H2 @ I3 ) ) % 0.31/0.60 @ I6 ) % 0.31/0.60 = ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % prod.distrib_triv' % 0.31/0.60 thf(fact_4899_case__optionE,axiom, % 0.31/0.60 ! [A: $tType,P: $o,Q: A > $o,X3: option @ A] : % 0.31/0.60 ( ( case_option @ $o @ A @ P @ Q @ X3 ) % 0.31/0.60 => ( ( ( X3 % 0.31/0.60 = ( none @ A ) ) % 0.31/0.60 => ~ P ) % 0.31/0.60 => ~ ! [Y4: A] : % 0.31/0.60 ( ( X3 % 0.31/0.60 = ( some @ A @ Y4 ) ) % 0.31/0.60 => ~ ( Q @ Y4 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % case_optionE % 0.31/0.60 thf(fact_4900_Sup__eq__Inf,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ( ( complete_Sup_Sup @ A ) % 0.31/0.60 = ( ^ [A6: set @ A] : % 0.31/0.60 ( complete_Inf_Inf @ A % 0.31/0.60 @ ( collect @ A % 0.31/0.60 @ ^ [B4: A] : % 0.31/0.60 ! [X: A] : % 0.31/0.60 ( ( member @ A @ X @ A6 ) % 0.31/0.60 => ( ord_less_eq @ A @ X @ B4 ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Sup_eq_Inf % 0.31/0.60 thf(fact_4901_Inf__eq__Sup,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ( ( complete_Inf_Inf @ A ) % 0.31/0.60 = ( ^ [A6: set @ A] : % 0.31/0.60 ( complete_Sup_Sup @ A % 0.31/0.60 @ ( collect @ A % 0.31/0.60 @ ^ [B4: A] : % 0.31/0.60 ! [X: A] : % 0.31/0.60 ( ( member @ A @ X @ A6 ) % 0.31/0.60 => ( ord_less_eq @ A @ B4 @ X ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Inf_eq_Sup % 0.31/0.60 thf(fact_4902_option_Ocase__eq__if,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( case_option @ B @ A ) % 0.31/0.60 = ( ^ [F12: B,F23: A > B,Option3: option @ A] : % 0.31/0.60 ( if @ B % 0.31/0.60 @ ( Option3 % 0.31/0.60 = ( none @ A ) ) % 0.31/0.60 @ F12 % 0.31/0.60 @ ( F23 @ ( the2 @ A @ Option3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % option.case_eq_if % 0.31/0.60 thf(fact_4903_prod_Omono__neutral__cong__right_H,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_mult @ A ) % 0.31/0.60 => ! [S3: set @ B,T4: set @ B,G: B > A,H2: B > A] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.31/0.60 => ( ! [X5: B] : % 0.31/0.60 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.31/0.60 => ( ( G @ X5 ) % 0.31/0.60 = ( one_one @ A ) ) ) % 0.31/0.60 => ( ! [X5: B] : % 0.31/0.60 ( ( member @ B @ X5 @ S3 ) % 0.31/0.60 => ( ( G @ X5 ) % 0.31/0.60 = ( H2 @ X5 ) ) ) % 0.31/0.60 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ T4 ) % 0.31/0.60 = ( groups1962203154675924110t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % prod.mono_neutral_cong_right' % 0.31/0.60 thf(fact_4904_prod_Omono__neutral__cong__left_H,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_mult @ A ) % 0.31/0.60 => ! [S3: set @ B,T4: set @ B,H2: B > A,G: B > A] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.31/0.60 => ( ! [I4: B] : % 0.31/0.60 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.31/0.60 => ( ( H2 @ I4 ) % 0.31/0.60 = ( one_one @ A ) ) ) % 0.31/0.60 => ( ! [X5: B] : % 0.31/0.60 ( ( member @ B @ X5 @ S3 ) % 0.31/0.60 => ( ( G @ X5 ) % 0.31/0.60 = ( H2 @ X5 ) ) ) % 0.31/0.60 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 ) % 0.31/0.60 = ( groups1962203154675924110t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % prod.mono_neutral_cong_left' % 0.31/0.60 thf(fact_4905_prod_Omono__neutral__right_H,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_mult @ A ) % 0.31/0.60 => ! [S3: set @ B,T4: set @ B,G: B > A] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.31/0.60 => ( ! [X5: B] : % 0.31/0.60 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.31/0.60 => ( ( G @ X5 ) % 0.31/0.60 = ( one_one @ A ) ) ) % 0.31/0.60 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ T4 ) % 0.31/0.60 = ( groups1962203154675924110t_prod @ B @ A @ G @ S3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % prod.mono_neutral_right' % 0.31/0.60 thf(fact_4906_prod_Omono__neutral__left_H,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_mult @ A ) % 0.31/0.60 => ! [S3: set @ B,T4: set @ B,G: B > A] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 0.31/0.60 => ( ! [X5: B] : % 0.31/0.60 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 0.31/0.60 => ( ( G @ X5 ) % 0.31/0.60 = ( one_one @ A ) ) ) % 0.31/0.60 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 ) % 0.31/0.60 = ( groups1962203154675924110t_prod @ B @ A @ G @ T4 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % prod.mono_neutral_left' % 0.31/0.60 thf(fact_4907_prod_Odistrib_H,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_mult @ A ) % 0.31/0.60 => ! [I6: set @ B,G: B > A,H2: B > A] : % 0.31/0.60 ( ( finite_finite2 @ B % 0.31/0.60 @ ( collect @ B % 0.31/0.60 @ ^ [X: B] : % 0.31/0.60 ( ( member @ B @ X @ I6 ) % 0.31/0.60 & ( ( G @ X ) % 0.31/0.60 != ( one_one @ A ) ) ) ) ) % 0.31/0.60 => ( ( finite_finite2 @ B % 0.31/0.60 @ ( collect @ B % 0.31/0.60 @ ^ [X: B] : % 0.31/0.60 ( ( member @ B @ X @ I6 ) % 0.31/0.60 & ( ( H2 @ X ) % 0.31/0.60 != ( one_one @ A ) ) ) ) ) % 0.31/0.60 => ( ( groups1962203154675924110t_prod @ B @ A % 0.31/0.60 @ ^ [I3: B] : ( times_times @ A @ ( G @ I3 ) @ ( H2 @ I3 ) ) % 0.31/0.60 @ I6 ) % 0.31/0.60 = ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % prod.distrib' % 0.31/0.60 thf(fact_4908_prod_OG__def,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_mult @ A ) % 0.31/0.60 => ( ( groups1962203154675924110t_prod @ B @ A ) % 0.31/0.60 = ( ^ [P6: B > A,I8: set @ B] : % 0.31/0.60 ( if @ A % 0.31/0.60 @ ( finite_finite2 @ B % 0.31/0.60 @ ( collect @ B % 0.31/0.60 @ ^ [X: B] : % 0.31/0.60 ( ( member @ B @ X @ I8 ) % 0.31/0.60 & ( ( P6 @ X ) % 0.31/0.60 != ( one_one @ A ) ) ) ) ) % 0.31/0.60 @ ( groups7121269368397514597t_prod @ B @ A @ P6 % 0.31/0.60 @ ( collect @ B % 0.31/0.60 @ ^ [X: B] : % 0.31/0.60 ( ( member @ B @ X @ I8 ) % 0.31/0.60 & ( ( P6 @ X ) % 0.31/0.60 != ( one_one @ A ) ) ) ) ) % 0.31/0.60 @ ( one_one @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % prod.G_def % 0.31/0.60 thf(fact_4909_option_Osplit__sel__asm,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] : % 0.31/0.60 ( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) ) % 0.31/0.60 = ( ~ ( ( ( Option % 0.31/0.60 = ( none @ A ) ) % 0.31/0.60 & ~ ( P @ F1 ) ) % 0.31/0.60 | ( ( Option % 0.31/0.60 = ( some @ A @ ( the2 @ A @ Option ) ) ) % 0.31/0.60 & ~ ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % option.split_sel_asm % 0.31/0.60 thf(fact_4910_option_Osplit__sel,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] : % 0.31/0.60 ( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) ) % 0.31/0.60 = ( ( ( Option % 0.31/0.60 = ( none @ A ) ) % 0.31/0.60 => ( P @ F1 ) ) % 0.31/0.60 & ( ( Option % 0.31/0.60 = ( some @ A @ ( the2 @ A @ Option ) ) ) % 0.31/0.60 => ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % option.split_sel % 0.31/0.60 thf(fact_4911_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom, % 0.31/0.60 ! [Mima2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] : % 0.31/0.60 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 ) % 0.31/0.60 = ( ( Deg = Deg4 ) % 0.31/0.60 & ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 0.31/0.60 => ( vEBT_VEBT_valid @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.60 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.60 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) % 0.31/0.60 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.31/0.60 & ( case_option @ $o @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 ) % 0.31/0.60 & ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 0.31/0.60 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.60 @ ^ [Mi3: nat,Ma3: nat] : % 0.31/0.60 ( ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 0.31/0.60 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) % 0.31/0.60 & ! [I3: nat] : % 0.31/0.60 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.31/0.60 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) ) % 0.31/0.60 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) ) % 0.31/0.60 & ( ( Mi3 = Ma3 ) % 0.31/0.60 => ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 0.31/0.60 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.31/0.60 & ( ( Mi3 != Ma3 ) % 0.31/0.60 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 ) % 0.31/0.60 & ! [X: nat] : % 0.31/0.60 ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) % 0.31/0.60 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X ) % 0.31/0.60 => ( ( ord_less @ nat @ Mi3 @ X ) % 0.31/0.60 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) % 0.31/0.60 @ Mima2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % VEBT_internal.valid'.simps(2) % 0.31/0.60 thf(fact_4912_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom, % 0.31/0.60 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 0.31/0.60 ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 ) % 0.31/0.60 = Y ) % 0.31/0.60 => ( ( ? [Uu2: $o,Uv2: $o] : % 0.31/0.60 ( X3 % 0.31/0.60 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 0.31/0.60 => ( Y % 0.31/0.60 = ( Xa2 % 0.31/0.60 != ( one_one @ nat ) ) ) ) % 0.31/0.60 => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 0.31/0.60 ( ( X3 % 0.31/0.60 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) ) % 0.31/0.60 => ( Y % 0.31/0.60 = ( ~ ( ( Deg2 = Xa2 ) % 0.31/0.60 & ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ( vEBT_VEBT_valid @ X @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.60 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.60 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 0.31/0.60 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.31/0.60 & ( case_option @ $o @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 ) % 0.31/0.60 & ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.60 @ ^ [Mi3: nat,Ma3: nat] : % 0.31/0.60 ( ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 0.31/0.60 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 0.31/0.60 & ! [I3: nat] : % 0.31/0.60 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.31/0.60 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) ) % 0.31/0.60 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) ) % 0.31/0.60 & ( ( Mi3 = Ma3 ) % 0.31/0.60 => ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.31/0.60 & ( ( Mi3 != Ma3 ) % 0.31/0.60 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 ) % 0.31/0.60 & ! [X: nat] : % 0.31/0.60 ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 0.31/0.60 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X ) % 0.31/0.60 => ( ( ord_less @ nat @ Mi3 @ X ) % 0.31/0.60 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) % 0.31/0.60 @ Mima ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % VEBT_internal.valid'.elims(1) % 0.31/0.60 thf(fact_4913_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom, % 0.31/0.60 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.31/0.60 ( ( vEBT_VEBT_valid @ X3 @ Xa2 ) % 0.31/0.60 => ( ( ? [Uu2: $o,Uv2: $o] : % 0.31/0.60 ( X3 % 0.31/0.60 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 0.31/0.60 => ( Xa2 % 0.31/0.60 != ( one_one @ nat ) ) ) % 0.31/0.60 => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 0.31/0.60 ( ( X3 % 0.31/0.60 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) ) % 0.31/0.60 => ~ ( ( Deg2 = Xa2 ) % 0.31/0.60 & ! [X2: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.60 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.60 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 0.31/0.60 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.31/0.60 & ( case_option @ $o @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 ) % 0.31/0.60 & ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.60 @ ^ [Mi3: nat,Ma3: nat] : % 0.31/0.60 ( ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 0.31/0.60 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 0.31/0.60 & ! [I3: nat] : % 0.31/0.60 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.31/0.60 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) ) % 0.31/0.60 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) ) % 0.31/0.60 & ( ( Mi3 = Ma3 ) % 0.31/0.60 => ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.31/0.60 & ( ( Mi3 != Ma3 ) % 0.31/0.60 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 ) % 0.31/0.60 & ! [X: nat] : % 0.31/0.60 ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 0.31/0.60 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X ) % 0.31/0.60 => ( ( ord_less @ nat @ Mi3 @ X ) % 0.31/0.60 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) % 0.31/0.60 @ Mima ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % VEBT_internal.valid'.elims(2) % 0.31/0.60 thf(fact_4914_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom, % 0.31/0.60 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 0.31/0.60 ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 ) % 0.31/0.60 = Y ) % 0.31/0.60 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 0.31/0.60 => ( ! [Uu2: $o,Uv2: $o] : % 0.31/0.60 ( ( X3 % 0.31/0.60 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 0.31/0.60 => ( ( Y % 0.31/0.60 = ( Xa2 % 0.31/0.60 = ( one_one @ nat ) ) ) % 0.31/0.60 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) ) % 0.31/0.60 => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 0.31/0.60 ( ( X3 % 0.31/0.60 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) ) % 0.31/0.60 => ( ( Y % 0.31/0.60 = ( ( Deg2 = Xa2 ) % 0.31/0.60 & ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ( vEBT_VEBT_valid @ X @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.60 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.60 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 0.31/0.60 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.31/0.60 & ( case_option @ $o @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 ) % 0.31/0.60 & ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.60 @ ^ [Mi3: nat,Ma3: nat] : % 0.31/0.60 ( ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 0.31/0.60 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 0.31/0.60 & ! [I3: nat] : % 0.31/0.60 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.31/0.60 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) ) % 0.31/0.60 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) ) % 0.31/0.60 & ( ( Mi3 = Ma3 ) % 0.31/0.60 => ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.31/0.60 & ( ( Mi3 != Ma3 ) % 0.31/0.60 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 ) % 0.31/0.60 & ! [X: nat] : % 0.31/0.60 ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 0.31/0.60 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X ) % 0.31/0.60 => ( ( ord_less @ nat @ Mi3 @ X ) % 0.31/0.60 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) % 0.31/0.60 @ Mima ) ) ) % 0.31/0.60 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % VEBT_internal.valid'.pelims(1) % 0.31/0.60 thf(fact_4915_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom, % 0.31/0.60 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.31/0.60 ( ( vEBT_VEBT_valid @ X3 @ Xa2 ) % 0.31/0.60 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 0.31/0.60 => ( ! [Uu2: $o,Uv2: $o] : % 0.31/0.60 ( ( X3 % 0.31/0.60 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 0.31/0.60 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) % 0.31/0.60 => ( Xa2 % 0.31/0.60 != ( one_one @ nat ) ) ) ) % 0.31/0.60 => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 0.31/0.60 ( ( X3 % 0.31/0.60 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) ) % 0.31/0.60 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) % 0.31/0.60 => ~ ( ( Deg2 = Xa2 ) % 0.31/0.60 & ! [X2: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.60 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.60 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 0.31/0.60 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.31/0.60 & ( case_option @ $o @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 ) % 0.31/0.60 & ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.60 @ ^ [Mi3: nat,Ma3: nat] : % 0.31/0.60 ( ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 0.31/0.60 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 0.31/0.60 & ! [I3: nat] : % 0.31/0.60 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.31/0.60 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) ) % 0.31/0.60 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) ) % 0.31/0.60 & ( ( Mi3 = Ma3 ) % 0.31/0.60 => ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.31/0.60 & ( ( Mi3 != Ma3 ) % 0.31/0.60 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 ) % 0.31/0.60 & ! [X: nat] : % 0.31/0.60 ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 0.31/0.60 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X ) % 0.31/0.60 => ( ( ord_less @ nat @ Mi3 @ X ) % 0.31/0.60 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) % 0.31/0.60 @ Mima ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % VEBT_internal.valid'.pelims(2) % 0.31/0.60 thf(fact_4916_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom, % 0.31/0.60 ! [X3: vEBT_VEBT,Xa2: nat] : % 0.31/0.60 ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 ) % 0.31/0.60 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 0.31/0.60 => ( ! [Uu2: $o,Uv2: $o] : % 0.31/0.60 ( ( X3 % 0.31/0.60 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 0.31/0.60 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) % 0.31/0.60 => ( Xa2 % 0.31/0.60 = ( one_one @ nat ) ) ) ) % 0.31/0.60 => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 0.31/0.60 ( ( X3 % 0.31/0.60 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) ) % 0.31/0.60 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) % 0.31/0.60 => ( ( Deg2 = Xa2 ) % 0.31/0.60 & ! [X5: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.60 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.60 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 0.31/0.60 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.31/0.60 & ( case_option @ $o @ ( product_prod @ nat @ nat ) % 0.31/0.60 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 ) % 0.31/0.60 & ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.31/0.60 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.60 @ ^ [Mi3: nat,Ma3: nat] : % 0.31/0.60 ( ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 0.31/0.60 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 0.31/0.60 & ! [I3: nat] : % 0.31/0.60 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.31/0.60 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) ) % 0.31/0.60 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) ) % 0.31/0.60 & ( ( Mi3 = Ma3 ) % 0.31/0.60 => ! [X: vEBT_VEBT] : % 0.31/0.60 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 0.31/0.60 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 0.31/0.60 & ( ( Mi3 != Ma3 ) % 0.31/0.60 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 ) % 0.31/0.60 & ! [X: nat] : % 0.31/0.60 ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 0.31/0.60 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X ) % 0.31/0.60 => ( ( ord_less @ nat @ Mi3 @ X ) % 0.31/0.60 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) % 0.31/0.60 @ Mima ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % VEBT_internal.valid'.pelims(3) % 0.31/0.60 thf(fact_4917_Union__maximal__sets,axiom, % 0.31/0.60 ! [A: $tType,F8: set @ ( set @ A )] : % 0.31/0.60 ( ( finite_finite2 @ ( set @ A ) @ F8 ) % 0.31/0.60 => ( ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.60 @ ( collect @ ( set @ A ) % 0.31/0.60 @ ^ [T8: set @ A] : % 0.31/0.60 ( ( member @ ( set @ A ) @ T8 @ F8 ) % 0.31/0.60 & ! [X: set @ A] : % 0.31/0.60 ( ( member @ ( set @ A ) @ X @ F8 ) % 0.31/0.60 => ~ ( ord_less @ ( set @ A ) @ T8 @ X ) ) ) ) ) % 0.31/0.60 = ( complete_Sup_Sup @ ( set @ A ) @ F8 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Union_maximal_sets % 0.31/0.60 thf(fact_4918_concat__injective,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] : % 0.31/0.60 ( ( ( concat @ A @ Xs ) % 0.31/0.60 = ( concat @ A @ Ys ) ) % 0.31/0.60 => ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs ) % 0.31/0.60 = ( size_size @ ( list @ ( list @ A ) ) @ Ys ) ) % 0.31/0.60 => ( ! [X5: product_prod @ ( list @ A ) @ ( list @ A )] : % 0.31/0.60 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X5 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) ) % 0.31/0.60 => ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o % 0.31/0.60 @ ^ [Y2: list @ A,Z6: list @ A] : % 0.31/0.60 ( ( size_size @ ( list @ A ) @ Y2 ) % 0.31/0.60 = ( size_size @ ( list @ A ) @ Z6 ) ) % 0.31/0.60 @ X5 ) ) % 0.31/0.60 => ( Xs = Ys ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % concat_injective % 0.31/0.60 thf(fact_4919_concat__eq__concat__iff,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] : % 0.31/0.60 ( ! [X5: product_prod @ ( list @ A ) @ ( list @ A )] : % 0.31/0.60 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X5 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) ) % 0.31/0.60 => ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o % 0.31/0.60 @ ^ [Y2: list @ A,Z6: list @ A] : % 0.31/0.60 ( ( size_size @ ( list @ A ) @ Y2 ) % 0.31/0.60 = ( size_size @ ( list @ A ) @ Z6 ) ) % 0.31/0.60 @ X5 ) ) % 0.31/0.60 => ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs ) % 0.31/0.60 = ( size_size @ ( list @ ( list @ A ) ) @ Ys ) ) % 0.31/0.60 => ( ( ( concat @ A @ Xs ) % 0.31/0.60 = ( concat @ A @ Ys ) ) % 0.31/0.60 = ( Xs = Ys ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % concat_eq_concat_iff % 0.31/0.60 thf(fact_4920_list__eq__iff__zip__eq,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ^ [Y3: list @ A,Z: list @ A] : ( Y3 = Z ) ) % 0.31/0.60 = ( ^ [Xs2: list @ A,Ys3: list @ A] : % 0.31/0.60 ( ( ( size_size @ ( list @ A ) @ Xs2 ) % 0.31/0.60 = ( size_size @ ( list @ A ) @ Ys3 ) ) % 0.31/0.60 & ! [X: product_prod @ A @ A] : % 0.31/0.60 ( ( member @ ( product_prod @ A @ A ) @ X @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs2 @ Ys3 ) ) ) % 0.31/0.60 => ( product_case_prod @ A @ A @ $o % 0.31/0.60 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) % 0.31/0.60 @ X ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % list_eq_iff_zip_eq % 0.31/0.60 thf(fact_4921_Greatest__def,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( order @ A ) % 0.31/0.60 => ( ( order_Greatest @ A ) % 0.31/0.60 = ( ^ [P3: A > $o] : % 0.31/0.60 ( the @ A % 0.31/0.60 @ ^ [X: A] : % 0.31/0.60 ( ( P3 @ X ) % 0.31/0.60 & ! [Y2: A] : % 0.31/0.60 ( ( P3 @ Y2 ) % 0.31/0.60 => ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Greatest_def % 0.31/0.60 thf(fact_4922_ord_OLeast__def,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( least @ A ) % 0.31/0.60 = ( ^ [Less_eq: A > A > $o,P3: A > $o] : % 0.31/0.60 ( the @ A % 0.31/0.60 @ ^ [X: A] : % 0.31/0.60 ( ( P3 @ X ) % 0.31/0.60 & ! [Y2: A] : % 0.31/0.60 ( ( P3 @ Y2 ) % 0.31/0.60 => ( Less_eq @ X @ Y2 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % ord.Least_def % 0.31/0.60 thf(fact_4923_listrel__iff__zip,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] : % 0.31/0.60 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) ) % 0.31/0.60 = ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.60 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.60 & ! [X: product_prod @ A @ B] : % 0.31/0.60 ( ( member @ ( product_prod @ A @ B ) @ X @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) % 0.31/0.60 => ( product_case_prod @ A @ B @ $o % 0.31/0.60 @ ^ [Y2: A,Z6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y2 @ Z6 ) @ R2 ) % 0.31/0.60 @ X ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % listrel_iff_zip % 0.31/0.60 thf(fact_4924_ord_OLeast_Ocong,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( least @ A ) % 0.31/0.60 = ( least @ A ) ) ). % 0.31/0.60 % 0.31/0.60 % ord.Least.cong % 0.31/0.60 thf(fact_4925_listrel__Nil2,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ B )] : % 0.31/0.60 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( nil @ B ) ) @ ( listrel @ A @ B @ R2 ) ) % 0.31/0.60 => ( Xs % 0.31/0.60 = ( nil @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % listrel_Nil2 % 0.31/0.60 thf(fact_4926_listrel__Nil1,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,Xs: list @ B,R2: set @ ( product_prod @ A @ B )] : % 0.31/0.60 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs ) @ ( listrel @ A @ B @ R2 ) ) % 0.31/0.60 => ( Xs % 0.31/0.60 = ( nil @ B ) ) ) ). % 0.31/0.60 % 0.31/0.60 % listrel_Nil1 % 0.31/0.60 thf(fact_4927_listrel_ONil,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R2 ) ) ). % 0.31/0.60 % 0.31/0.60 % listrel.Nil % 0.31/0.60 thf(fact_4928_listrel__eq__len,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] : % 0.31/0.60 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) ) % 0.31/0.60 => ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.60 = ( size_size @ ( list @ B ) @ Ys ) ) ) ). % 0.31/0.60 % 0.31/0.60 % listrel_eq_len % 0.31/0.60 thf(fact_4929_GreatestI__nat,axiom, % 0.31/0.60 ! [P: nat > $o,K: nat,B2: nat] : % 0.31/0.60 ( ( P @ K ) % 0.31/0.60 => ( ! [Y4: nat] : % 0.31/0.60 ( ( P @ Y4 ) % 0.31/0.60 => ( ord_less_eq @ nat @ Y4 @ B2 ) ) % 0.31/0.60 => ( P @ ( order_Greatest @ nat @ P ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % GreatestI_nat % 0.31/0.60 thf(fact_4930_Greatest__le__nat,axiom, % 0.31/0.60 ! [P: nat > $o,K: nat,B2: nat] : % 0.31/0.60 ( ( P @ K ) % 0.31/0.60 => ( ! [Y4: nat] : % 0.31/0.60 ( ( P @ Y4 ) % 0.31/0.60 => ( ord_less_eq @ nat @ Y4 @ B2 ) ) % 0.31/0.60 => ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Greatest_le_nat % 0.31/0.60 thf(fact_4931_GreatestI__ex__nat,axiom, % 0.31/0.60 ! [P: nat > $o,B2: nat] : % 0.31/0.60 ( ? [X_12: nat] : ( P @ X_12 ) % 0.31/0.60 => ( ! [Y4: nat] : % 0.31/0.60 ( ( P @ Y4 ) % 0.31/0.60 => ( ord_less_eq @ nat @ Y4 @ B2 ) ) % 0.31/0.60 => ( P @ ( order_Greatest @ nat @ P ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % GreatestI_ex_nat % 0.31/0.60 thf(fact_4932_listrel__mono,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) % 0.31/0.60 => ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( listrel @ A @ B @ R2 ) @ ( listrel @ A @ B @ S ) ) ) ). % 0.31/0.60 % 0.31/0.60 % listrel_mono % 0.31/0.60 thf(fact_4933_GreatestI2__order,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( order @ A ) % 0.31/0.60 => ! [P: A > $o,X3: A,Q: A > $o] : % 0.31/0.60 ( ( P @ X3 ) % 0.31/0.60 => ( ! [Y4: A] : % 0.31/0.60 ( ( P @ Y4 ) % 0.31/0.60 => ( ord_less_eq @ A @ Y4 @ X3 ) ) % 0.31/0.60 => ( ! [X5: A] : % 0.31/0.60 ( ( P @ X5 ) % 0.31/0.60 => ( ! [Y5: A] : % 0.31/0.60 ( ( P @ Y5 ) % 0.31/0.60 => ( ord_less_eq @ A @ Y5 @ X5 ) ) % 0.31/0.60 => ( Q @ X5 ) ) ) % 0.31/0.60 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % GreatestI2_order % 0.31/0.60 thf(fact_4934_Greatest__equality,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( order @ A ) % 0.31/0.60 => ! [P: A > $o,X3: A] : % 0.31/0.60 ( ( P @ X3 ) % 0.31/0.60 => ( ! [Y4: A] : % 0.31/0.60 ( ( P @ Y4 ) % 0.31/0.60 => ( ord_less_eq @ A @ Y4 @ X3 ) ) % 0.31/0.60 => ( ( order_Greatest @ A @ P ) % 0.31/0.60 = X3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Greatest_equality % 0.31/0.60 thf(fact_4935_listrel__iff__nth,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] : % 0.31/0.60 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) ) % 0.31/0.60 = ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.60 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.60 & ! [N: nat] : % 0.31/0.60 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.60 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N ) @ ( nth @ B @ Ys @ N ) ) @ R2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % listrel_iff_nth % 0.31/0.60 thf(fact_4936_take__bit__numeral__numeral,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.60 => ! [M: num,N2: num] : % 0.31/0.60 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ A @ N2 ) ) % 0.31/0.60 = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % take_bit_numeral_numeral % 0.31/0.60 thf(fact_4937_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A3: set @ A,X3: A] : % 0.31/0.60 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.60 => ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.60 = ( linorder_insort_key @ A @ A % 0.31/0.60 @ ^ [X: A] : X % 0.31/0.60 @ X3 % 0.31/0.60 @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sorted_list_of_set.sorted_key_list_of_set_insert_remove % 0.31/0.60 thf(fact_4938_sorted__list__of__set__def,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ( ( linord4507533701916653071of_set @ A ) % 0.31/0.60 = ( linord144544945434240204of_set @ A @ A % 0.31/0.60 @ ^ [X: A] : X ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sorted_list_of_set_def % 0.31/0.60 thf(fact_4939_take__bit__num__simps_I1_J,axiom, % 0.31/0.60 ! [M: num] : % 0.31/0.60 ( ( bit_take_bit_num @ ( zero_zero @ nat ) @ M ) % 0.31/0.60 = ( none @ num ) ) ). % 0.31/0.60 % 0.31/0.60 % take_bit_num_simps(1) % 0.31/0.60 thf(fact_4940_remove1__insort__key,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [X3: B,F2: B > A,Xs: list @ B] : % 0.31/0.60 ( ( remove1 @ B @ X3 @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) % 0.31/0.60 = Xs ) ) ). % 0.31/0.60 % 0.31/0.60 % remove1_insort_key % 0.31/0.60 thf(fact_4941_length__insort,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [F2: B > A,X3: B,Xs: list @ B] : % 0.31/0.60 ( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) % 0.31/0.60 = ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % length_insort % 0.31/0.60 thf(fact_4942_take__bit__num__simps_I2_J,axiom, % 0.31/0.60 ! [N2: nat] : % 0.31/0.60 ( ( bit_take_bit_num @ ( suc @ N2 ) @ one2 ) % 0.31/0.60 = ( some @ num @ one2 ) ) ). % 0.31/0.60 % 0.31/0.60 % take_bit_num_simps(2) % 0.31/0.60 thf(fact_4943_take__bit__num__simps_I3_J,axiom, % 0.31/0.60 ! [N2: nat,M: num] : % 0.31/0.60 ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) ) % 0.31/0.60 = ( case_option @ ( option @ num ) @ num @ ( none @ num ) % 0.31/0.60 @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) ) % 0.31/0.60 @ ( bit_take_bit_num @ N2 @ M ) ) ) ). % 0.31/0.60 % 0.31/0.60 % take_bit_num_simps(3) % 0.31/0.60 thf(fact_4944_take__bit__num__simps_I4_J,axiom, % 0.31/0.60 ! [N2: nat,M: num] : % 0.31/0.60 ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) ) % 0.31/0.60 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % take_bit_num_simps(4) % 0.31/0.60 thf(fact_4945_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A3: set @ A,X3: A] : % 0.31/0.60 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.60 => ( ~ ( member @ A @ X3 @ A3 ) % 0.31/0.60 => ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.60 = ( linorder_insort_key @ A @ A % 0.31/0.60 @ ^ [X: A] : X % 0.31/0.60 @ X3 % 0.31/0.60 @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sorted_list_of_set.sorted_key_list_of_set_insert % 0.31/0.60 thf(fact_4946_take__bit__num__simps_I6_J,axiom, % 0.31/0.60 ! [R2: num,M: num] : % 0.31/0.60 ( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R2 ) @ ( bit0 @ M ) ) % 0.31/0.60 = ( case_option @ ( option @ num ) @ num @ ( none @ num ) % 0.31/0.60 @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) ) % 0.31/0.60 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ). % 0.31/0.60 % 0.31/0.60 % take_bit_num_simps(6) % 0.31/0.60 thf(fact_4947_insort__not__Nil,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [F2: B > A,A2: B,Xs: list @ B] : % 0.31/0.60 ( ( linorder_insort_key @ B @ A @ F2 @ A2 @ Xs ) % 0.31/0.60 != ( nil @ B ) ) ) ). % 0.31/0.60 % 0.31/0.60 % insort_not_Nil % 0.31/0.60 thf(fact_4948_insort__key__left__comm,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [F2: B > A,X3: B,Y: B,Xs: list @ B] : % 0.31/0.60 ( ( ( F2 @ X3 ) % 0.31/0.60 != ( F2 @ Y ) ) % 0.31/0.60 => ( ( linorder_insort_key @ B @ A @ F2 @ Y @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) % 0.31/0.60 = ( linorder_insort_key @ B @ A @ F2 @ X3 @ ( linorder_insort_key @ B @ A @ F2 @ Y @ Xs ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % insort_key_left_comm % 0.31/0.60 thf(fact_4949_insort__left__comm,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [X3: A,Y: A,Xs: list @ A] : % 0.31/0.60 ( ( linorder_insort_key @ A @ A % 0.31/0.60 @ ^ [X: A] : X % 0.31/0.60 @ X3 % 0.31/0.60 @ ( linorder_insort_key @ A @ A % 0.31/0.60 @ ^ [X: A] : X % 0.31/0.60 @ Y % 0.31/0.60 @ Xs ) ) % 0.31/0.60 = ( linorder_insort_key @ A @ A % 0.31/0.60 @ ^ [X: A] : X % 0.31/0.60 @ Y % 0.31/0.60 @ ( linorder_insort_key @ A @ A % 0.31/0.60 @ ^ [X: A] : X % 0.31/0.60 @ X3 % 0.31/0.60 @ Xs ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % insort_left_comm % 0.31/0.60 thf(fact_4950_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [Y: A,X3: A] : % 0.31/0.60 ( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) % 0.31/0.60 @ ( linorder_insort_key @ A @ A % 0.31/0.60 @ ^ [X: A] : X % 0.31/0.60 @ Y ) % 0.31/0.60 @ ( linorder_insort_key @ A @ A % 0.31/0.60 @ ^ [X: A] : X % 0.31/0.60 @ X3 ) ) % 0.31/0.60 = ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) % 0.31/0.60 @ ( linorder_insort_key @ A @ A % 0.31/0.60 @ ^ [X: A] : X % 0.31/0.60 @ X3 ) % 0.31/0.60 @ ( linorder_insort_key @ A @ A % 0.31/0.60 @ ^ [X: A] : X % 0.31/0.60 @ Y ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sorted_list_of_set.fold_insort_key.comp_fun_commute_on % 0.31/0.60 thf(fact_4951_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom, % 0.31/0.60 ! [N2: nat,M: num] : % 0.31/0.60 ( ( bit_take_bit_num @ N2 @ ( bit0 @ M ) ) % 0.31/0.60 = ( case_nat @ ( option @ num ) @ ( none @ num ) % 0.31/0.60 @ ^ [N: nat] : % 0.31/0.60 ( case_option @ ( option @ num ) @ num @ ( none @ num ) % 0.31/0.60 @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) ) % 0.31/0.60 @ ( bit_take_bit_num @ N @ M ) ) % 0.31/0.60 @ N2 ) ) ). % 0.31/0.60 % 0.31/0.60 % Code_Abstract_Nat.take_bit_num_code(2) % 0.31/0.60 thf(fact_4952_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom, % 0.31/0.60 ! [N2: nat] : % 0.31/0.60 ( ( bit_take_bit_num @ N2 @ one2 ) % 0.31/0.60 = ( case_nat @ ( option @ num ) @ ( none @ num ) % 0.31/0.60 @ ^ [N: nat] : ( some @ num @ one2 ) % 0.31/0.60 @ N2 ) ) ). % 0.31/0.60 % 0.31/0.60 % Code_Abstract_Nat.take_bit_num_code(1) % 0.31/0.60 thf(fact_4953_set__insort__key,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [F2: B > A,X3: B,Xs: list @ B] : % 0.31/0.60 ( ( set2 @ B @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) % 0.31/0.60 = ( insert @ B @ X3 @ ( set2 @ B @ Xs ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % set_insort_key % 0.31/0.60 thf(fact_4954_distinct__insort,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [F2: B > A,X3: B,Xs: list @ B] : % 0.31/0.60 ( ( distinct @ B @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) % 0.31/0.60 = ( ~ ( member @ B @ X3 @ ( set2 @ B @ Xs ) ) % 0.31/0.60 & ( distinct @ B @ Xs ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % distinct_insort % 0.31/0.60 thf(fact_4955_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom, % 0.31/0.60 ! [N2: nat,M: num] : % 0.31/0.60 ( ( bit_take_bit_num @ N2 @ ( bit1 @ M ) ) % 0.31/0.60 = ( case_nat @ ( option @ num ) @ ( none @ num ) % 0.31/0.60 @ ^ [N: nat] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) % 0.31/0.60 @ N2 ) ) ). % 0.31/0.60 % 0.31/0.60 % Code_Abstract_Nat.take_bit_num_code(3) % 0.31/0.60 thf(fact_4956_take__bit__num__eq__None__imp,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.60 => ! [M: nat,N2: num] : % 0.31/0.60 ( ( ( bit_take_bit_num @ M @ N2 ) % 0.31/0.60 = ( none @ num ) ) % 0.31/0.60 => ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N2 ) ) % 0.31/0.60 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % take_bit_num_eq_None_imp % 0.31/0.60 thf(fact_4957_sorted__list__of__set_Ofold__insort__key_Oremove,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linorder @ A ) % 0.31/0.60 => ! [A3: set @ A,X3: A] : % 0.31/0.60 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.60 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.60 => ( ( linord4507533701916653071of_set @ A @ A3 ) % 0.31/0.60 = ( linorder_insort_key @ A @ A % 0.31/0.60 @ ^ [X: A] : X % 0.31/0.60 @ X3 % 0.31/0.60 @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sorted_list_of_set.fold_insort_key.remove % 0.31/0.60 thf(fact_4958_take__bit__num__def,axiom, % 0.31/0.60 ( bit_take_bit_num % 0.31/0.60 = ( ^ [N: nat,M2: num] : % 0.31/0.60 ( if @ ( option @ num ) % 0.31/0.60 @ ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( numeral_numeral @ nat @ M2 ) ) % 0.31/0.60 = ( zero_zero @ nat ) ) % 0.31/0.60 @ ( none @ num ) % 0.31/0.60 @ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ ( numeral_numeral @ nat @ M2 ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % take_bit_num_def % 0.31/0.60 thf(fact_4959_Bit__Operations_Otake__bit__num__code,axiom, % 0.31/0.60 ( bit_take_bit_num % 0.31/0.60 = ( ^ [N: nat,M2: num] : % 0.31/0.60 ( product_case_prod @ nat @ num @ ( option @ num ) % 0.31/0.60 @ ^ [A4: nat,X: num] : % 0.31/0.60 ( case_nat @ ( option @ num ) @ ( none @ num ) % 0.31/0.60 @ ^ [O: nat] : % 0.31/0.60 ( case_num @ ( option @ num ) @ ( some @ num @ one2 ) % 0.31/0.60 @ ^ [P6: num] : % 0.31/0.60 ( case_option @ ( option @ num ) @ num @ ( none @ num ) % 0.31/0.60 @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) ) % 0.31/0.60 @ ( bit_take_bit_num @ O @ P6 ) ) % 0.31/0.60 @ ^ [P6: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) ) % 0.31/0.60 @ X ) % 0.31/0.60 @ A4 ) % 0.31/0.60 @ ( product_Pair @ nat @ num @ N @ M2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Bit_Operations.take_bit_num_code % 0.31/0.60 thf(fact_4960_set__nths,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A,I6: set @ nat] : % 0.31/0.60 ( ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) % 0.31/0.60 = ( collect @ A % 0.31/0.60 @ ^ [Uu3: A] : % 0.31/0.60 ? [I3: nat] : % 0.31/0.60 ( ( Uu3 % 0.31/0.60 = ( nth @ A @ Xs @ I3 ) ) % 0.31/0.60 & ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.60 & ( member @ nat @ I3 @ I6 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % set_nths % 0.31/0.60 thf(fact_4961_Rats__eq__int__div__nat,axiom, % 0.31/0.60 ( ( field_char_0_Rats @ real ) % 0.31/0.60 = ( collect @ real % 0.31/0.60 @ ^ [Uu3: real] : % 0.31/0.60 ? [I3: int,N: nat] : % 0.31/0.60 ( ( Uu3 % 0.31/0.60 = ( divide_divide @ real @ ( ring_1_of_int @ real @ I3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) % 0.31/0.60 & ( N % 0.31/0.60 != ( zero_zero @ nat ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Rats_eq_int_div_nat % 0.31/0.60 thf(fact_4962_nths__nil,axiom, % 0.31/0.60 ! [A: $tType,A3: set @ nat] : % 0.31/0.60 ( ( nths @ A @ ( nil @ A ) @ A3 ) % 0.31/0.60 = ( nil @ A ) ) ). % 0.31/0.60 % 0.31/0.60 % nths_nil % 0.31/0.60 thf(fact_4963_nths__empty,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A] : % 0.31/0.60 ( ( nths @ A @ Xs @ ( bot_bot @ ( set @ nat ) ) ) % 0.31/0.60 = ( nil @ A ) ) ). % 0.31/0.60 % 0.31/0.60 % nths_empty % 0.31/0.60 thf(fact_4964_Rats__infinite,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( field_char_0 @ A ) % 0.31/0.60 => ~ ( finite_finite2 @ A @ ( field_char_0_Rats @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Rats_infinite % 0.31/0.60 thf(fact_4965_distinct__nthsI,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A,I6: set @ nat] : % 0.31/0.60 ( ( distinct @ A @ Xs ) % 0.31/0.60 => ( distinct @ A @ ( nths @ A @ Xs @ I6 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % distinct_nthsI % 0.31/0.60 thf(fact_4966_Rats__add,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( field_char_0 @ A ) % 0.31/0.60 => ! [A2: A,B2: A] : % 0.31/0.60 ( ( member @ A @ A2 @ ( field_char_0_Rats @ A ) ) % 0.31/0.60 => ( ( member @ A @ B2 @ ( field_char_0_Rats @ A ) ) % 0.31/0.60 => ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( field_char_0_Rats @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Rats_add % 0.31/0.60 thf(fact_4967_Rats__dense__in__real,axiom, % 0.31/0.60 ! [X3: real,Y: real] : % 0.31/0.60 ( ( ord_less @ real @ X3 @ Y ) % 0.31/0.60 => ? [X5: real] : % 0.31/0.60 ( ( member @ real @ X5 @ ( field_char_0_Rats @ real ) ) % 0.31/0.60 & ( ord_less @ real @ X3 @ X5 ) % 0.31/0.60 & ( ord_less @ real @ X5 @ Y ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Rats_dense_in_real % 0.31/0.60 thf(fact_4968_Rats__no__bot__less,axiom, % 0.31/0.60 ! [X3: real] : % 0.31/0.60 ? [X5: real] : % 0.31/0.60 ( ( member @ real @ X5 @ ( field_char_0_Rats @ real ) ) % 0.31/0.60 & ( ord_less @ real @ X5 @ X3 ) ) ). % 0.31/0.60 % 0.31/0.60 % Rats_no_bot_less % 0.31/0.60 thf(fact_4969_Rats__0,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( field_char_0 @ A ) % 0.31/0.60 => ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Rats_0 % 0.31/0.60 thf(fact_4970_notin__set__nthsI,axiom, % 0.31/0.60 ! [A: $tType,X3: A,Xs: list @ A,I6: set @ nat] : % 0.31/0.60 ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.60 => ~ ( member @ A @ X3 @ ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % notin_set_nthsI % 0.31/0.60 thf(fact_4971_in__set__nthsD,axiom, % 0.31/0.60 ! [A: $tType,X3: A,Xs: list @ A,I6: set @ nat] : % 0.31/0.60 ( ( member @ A @ X3 @ ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) ) % 0.31/0.60 => ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ). % 0.31/0.60 % 0.31/0.60 % in_set_nthsD % 0.31/0.60 thf(fact_4972_set__nths__subset,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A,I6: set @ nat] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) @ ( set2 @ A @ Xs ) ) ). % 0.31/0.60 % 0.31/0.60 % set_nths_subset % 0.31/0.60 thf(fact_4973_nths__all,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A,I6: set @ nat] : % 0.31/0.60 ( ! [I4: nat] : % 0.31/0.60 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.60 => ( member @ nat @ I4 @ I6 ) ) % 0.31/0.60 => ( ( nths @ A @ Xs @ I6 ) % 0.31/0.60 = Xs ) ) ). % 0.31/0.60 % 0.31/0.60 % nths_all % 0.31/0.60 thf(fact_4974_length__nths,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A,I6: set @ nat] : % 0.31/0.60 ( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I6 ) ) % 0.31/0.60 = ( finite_card @ nat % 0.31/0.60 @ ( collect @ nat % 0.31/0.60 @ ^ [I3: nat] : % 0.31/0.60 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.60 & ( member @ nat @ I3 @ I6 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % length_nths % 0.31/0.60 thf(fact_4975_Nats__altdef1,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ring_1 @ A ) % 0.31/0.60 => ( ( semiring_1_Nats @ A ) % 0.31/0.60 = ( collect @ A % 0.31/0.60 @ ^ [Uu3: A] : % 0.31/0.60 ? [N: int] : % 0.31/0.60 ( ( Uu3 % 0.31/0.60 = ( ring_1_of_int @ A @ N ) ) % 0.31/0.60 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ N ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Nats_altdef1 % 0.31/0.60 thf(fact_4976_rat__floor__lemma,axiom, % 0.31/0.60 ! [A2: int,B2: int] : % 0.31/0.60 ( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ ( divide_divide @ int @ A2 @ B2 ) ) @ ( fract @ A2 @ B2 ) ) % 0.31/0.60 & ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( one_one @ int ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % rat_floor_lemma % 0.31/0.60 thf(fact_4977_image__minus__const__atLeastLessThan__nat,axiom, % 0.31/0.60 ! [C2: nat,Y: nat,X3: nat] : % 0.31/0.60 ( ( ( ord_less @ nat @ C2 @ Y ) % 0.31/0.60 => ( ( image @ nat @ nat % 0.31/0.60 @ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 ) % 0.31/0.60 @ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y ) ) % 0.31/0.60 = ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X3 @ C2 ) @ ( minus_minus @ nat @ Y @ C2 ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less @ nat @ C2 @ Y ) % 0.31/0.60 => ( ( ( ord_less @ nat @ X3 @ Y ) % 0.31/0.60 => ( ( image @ nat @ nat % 0.31/0.60 @ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 ) % 0.31/0.60 @ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y ) ) % 0.31/0.60 = ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less @ nat @ X3 @ Y ) % 0.31/0.60 => ( ( image @ nat @ nat % 0.31/0.60 @ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 ) % 0.31/0.60 @ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y ) ) % 0.31/0.60 = ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_minus_const_atLeastLessThan_nat % 0.31/0.60 thf(fact_4978_image__eqI,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,B2: A,F2: B > A,X3: B,A3: set @ B] : % 0.31/0.60 ( ( B2 % 0.31/0.60 = ( F2 @ X3 ) ) % 0.31/0.60 => ( ( member @ B @ X3 @ A3 ) % 0.31/0.60 => ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_eqI % 0.31/0.60 thf(fact_4979_image__ident,axiom, % 0.31/0.60 ! [A: $tType,Y7: set @ A] : % 0.31/0.60 ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : X % 0.31/0.60 @ Y7 ) % 0.31/0.60 = Y7 ) ). % 0.31/0.60 % 0.31/0.60 % image_ident % 0.31/0.60 thf(fact_4980_image__is__empty,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] : % 0.31/0.60 ( ( ( image @ B @ A @ F2 @ A3 ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 = ( A3 % 0.31/0.60 = ( bot_bot @ ( set @ B ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_is_empty % 0.31/0.60 thf(fact_4981_empty__is__image,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] : % 0.31/0.60 ( ( ( bot_bot @ ( set @ A ) ) % 0.31/0.60 = ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 = ( A3 % 0.31/0.60 = ( bot_bot @ ( set @ B ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % empty_is_image % 0.31/0.60 thf(fact_4982_image__empty,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,F2: B > A] : % 0.31/0.60 ( ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_empty % 0.31/0.60 thf(fact_4983_finite__imageI,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,F5: set @ A,H2: A > B] : % 0.31/0.60 ( ( finite_finite2 @ A @ F5 ) % 0.31/0.60 => ( finite_finite2 @ B @ ( image @ A @ B @ H2 @ F5 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % finite_imageI % 0.31/0.60 thf(fact_4984_image__insert,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F2: B > A,A2: B,B3: set @ B] : % 0.31/0.60 ( ( image @ B @ A @ F2 @ ( insert @ B @ A2 @ B3 ) ) % 0.31/0.60 = ( insert @ A @ ( F2 @ A2 ) @ ( image @ B @ A @ F2 @ B3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_insert % 0.31/0.60 thf(fact_4985_insert__image,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,X3: A,A3: set @ A,F2: A > B] : % 0.31/0.60 ( ( member @ A @ X3 @ A3 ) % 0.31/0.60 => ( ( insert @ B @ ( F2 @ X3 ) @ ( image @ A @ B @ F2 @ A3 ) ) % 0.31/0.60 = ( image @ A @ B @ F2 @ A3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % insert_image % 0.31/0.60 thf(fact_4986_image__add__0,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( comm_monoid_add @ A ) % 0.31/0.60 => ! [S3: set @ A] : % 0.31/0.60 ( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S3 ) % 0.31/0.60 = S3 ) ) ). % 0.31/0.60 % 0.31/0.60 % image_add_0 % 0.31/0.60 thf(fact_4987_image__add__atLeastAtMost,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_semidom @ A ) % 0.31/0.60 => ! [K: A,I: A,J: A] : % 0.31/0.60 ( ( image @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or1337092689740270186AtMost @ A @ I @ J ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_add_atLeastAtMost % 0.31/0.60 thf(fact_4988_image__add__atLeastLessThan,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_semidom @ A ) % 0.31/0.60 => ! [K: A,I: A,J: A] : % 0.31/0.60 ( ( image @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) ) % 0.31/0.60 = ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_add_atLeastLessThan % 0.31/0.60 thf(fact_4989_image__add__atMost,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ordered_ab_group_add @ A ) % 0.31/0.60 => ! [C2: A,A2: A] : % 0.31/0.60 ( ( image @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_ord_atMost @ A @ A2 ) ) % 0.31/0.60 = ( set_ord_atMost @ A @ ( plus_plus @ A @ C2 @ A2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_add_atMost % 0.31/0.60 thf(fact_4990_image__Suc__atLeastAtMost,axiom, % 0.31/0.60 ! [I: nat,J: nat] : % 0.31/0.60 ( ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_Suc_atLeastAtMost % 0.31/0.60 thf(fact_4991_image__Suc__atLeastLessThan,axiom, % 0.31/0.60 ! [I: nat,J: nat] : % 0.31/0.60 ( ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) ) % 0.31/0.60 = ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_Suc_atLeastLessThan % 0.31/0.60 thf(fact_4992_SUP__bot__conv_I2_J,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [B3: B > A,A3: set @ B] : % 0.31/0.60 ( ( ( bot_bot @ A ) % 0.31/0.60 = ( complete_Sup_Sup @ A @ ( image @ B @ A @ B3 @ A3 ) ) ) % 0.31/0.60 = ( ! [X: B] : % 0.31/0.60 ( ( member @ B @ X @ A3 ) % 0.31/0.60 => ( ( B3 @ X ) % 0.31/0.60 = ( bot_bot @ A ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_bot_conv(2) % 0.31/0.60 thf(fact_4993_SUP__bot__conv_I1_J,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [B3: B > A,A3: set @ B] : % 0.31/0.60 ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ B3 @ A3 ) ) % 0.31/0.60 = ( bot_bot @ A ) ) % 0.31/0.60 = ( ! [X: B] : % 0.31/0.60 ( ( member @ B @ X @ A3 ) % 0.31/0.60 => ( ( B3 @ X ) % 0.31/0.60 = ( bot_bot @ A ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_bot_conv(1) % 0.31/0.60 thf(fact_4994_SUP__bot,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [A3: set @ B] : % 0.31/0.60 ( ( complete_Sup_Sup @ A % 0.31/0.60 @ ( image @ B @ A % 0.31/0.60 @ ^ [X: B] : ( bot_bot @ A ) % 0.31/0.60 @ A3 ) ) % 0.31/0.60 = ( bot_bot @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_bot % 0.31/0.60 thf(fact_4995_ccSUP__bot,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( counta3822494911875563373attice @ A ) % 0.31/0.60 => ! [A3: set @ B] : % 0.31/0.60 ( ( complete_Sup_Sup @ A % 0.31/0.60 @ ( image @ B @ A % 0.31/0.60 @ ^ [X: B] : ( bot_bot @ A ) % 0.31/0.60 @ A3 ) ) % 0.31/0.60 = ( bot_bot @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % ccSUP_bot % 0.31/0.60 thf(fact_4996_SUP__const,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [A3: set @ B,F2: A] : % 0.31/0.60 ( ( A3 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( complete_Sup_Sup @ A % 0.31/0.60 @ ( image @ B @ A % 0.31/0.60 @ ^ [I3: B] : F2 % 0.31/0.60 @ A3 ) ) % 0.31/0.60 = F2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_const % 0.31/0.60 thf(fact_4997_ccSUP__const,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( counta3822494911875563373attice @ A ) % 0.31/0.60 => ! [A3: set @ B,F2: A] : % 0.31/0.60 ( ( A3 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( complete_Sup_Sup @ A % 0.31/0.60 @ ( image @ B @ A % 0.31/0.60 @ ^ [I3: B] : F2 % 0.31/0.60 @ A3 ) ) % 0.31/0.60 = F2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % ccSUP_const % 0.31/0.60 thf(fact_4998_cSUP__const,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( condit1219197933456340205attice @ A ) % 0.31/0.60 => ! [A3: set @ B,C2: A] : % 0.31/0.60 ( ( A3 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( complete_Sup_Sup @ A % 0.31/0.60 @ ( image @ B @ A % 0.31/0.60 @ ^ [X: B] : C2 % 0.31/0.60 @ A3 ) ) % 0.31/0.60 = C2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % cSUP_const % 0.31/0.60 thf(fact_4999_image__add__atLeastAtMost_H,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_semidom @ A ) % 0.31/0.60 => ! [K: A,I: A,J: A] : % 0.31/0.60 ( ( image @ A @ A % 0.31/0.60 @ ^ [N: A] : ( plus_plus @ A @ N @ K ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ I @ J ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_add_atLeastAtMost' % 0.31/0.60 thf(fact_5000_INF__const,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [A3: set @ B,F2: A] : % 0.31/0.60 ( ( A3 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( complete_Inf_Inf @ A % 0.31/0.60 @ ( image @ B @ A % 0.31/0.60 @ ^ [I3: B] : F2 % 0.31/0.60 @ A3 ) ) % 0.31/0.60 = F2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_const % 0.31/0.60 thf(fact_5001_ccINF__const,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( counta3822494911875563373attice @ A ) % 0.31/0.60 => ! [A3: set @ B,F2: A] : % 0.31/0.60 ( ( A3 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( complete_Inf_Inf @ A % 0.31/0.60 @ ( image @ B @ A % 0.31/0.60 @ ^ [I3: B] : F2 % 0.31/0.60 @ A3 ) ) % 0.31/0.60 = F2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % ccINF_const % 0.31/0.60 thf(fact_5002_cINF__const,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( condit1219197933456340205attice @ A ) % 0.31/0.60 => ! [A3: set @ B,C2: A] : % 0.31/0.60 ( ( A3 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( complete_Inf_Inf @ A % 0.31/0.60 @ ( image @ B @ A % 0.31/0.60 @ ^ [X: B] : C2 % 0.31/0.60 @ A3 ) ) % 0.31/0.60 = C2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % cINF_const % 0.31/0.60 thf(fact_5003_image__add__atLeastLessThan_H,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_semidom @ A ) % 0.31/0.60 => ! [K: A,I: A,J: A] : % 0.31/0.60 ( ( image @ A @ A % 0.31/0.60 @ ^ [N: A] : ( plus_plus @ A @ N @ K ) % 0.31/0.60 @ ( set_or7035219750837199246ssThan @ A @ I @ J ) ) % 0.31/0.60 = ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_add_atLeastLessThan' % 0.31/0.60 thf(fact_5004_INF__eq__bot__iff,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple5582772986160207858norder @ A ) % 0.31/0.60 => ! [F2: B > A,A3: set @ B] : % 0.31/0.60 ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 = ( bot_bot @ A ) ) % 0.31/0.60 = ( ! [X: A] : % 0.31/0.60 ( ( ord_less @ A @ ( bot_bot @ A ) @ X ) % 0.31/0.60 => ? [Y2: B] : % 0.31/0.60 ( ( member @ B @ Y2 @ A3 ) % 0.31/0.60 & ( ord_less @ A @ ( F2 @ Y2 ) @ X ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_eq_bot_iff % 0.31/0.60 thf(fact_5005_ccSUP__empty,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( counta3822494911875563373attice @ A ) % 0.31/0.60 => ! [F2: B > A] : % 0.31/0.60 ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) ) % 0.31/0.60 = ( bot_bot @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % ccSUP_empty % 0.31/0.60 thf(fact_5006_image__mult__atLeastAtMost,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_field @ A ) % 0.31/0.60 => ! [D2: A,A2: A,B2: A] : % 0.31/0.60 ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 ) % 0.31/0.60 => ( ( image @ A @ A @ ( times_times @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D2 @ A2 ) @ ( times_times @ A @ D2 @ B2 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_mult_atLeastAtMost % 0.31/0.60 thf(fact_5007_image__divide__atLeastAtMost,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_field @ A ) % 0.31/0.60 => ! [D2: A,A2: A,B2: A] : % 0.31/0.60 ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [C4: A] : ( divide_divide @ A @ C4 @ D2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A2 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_divide_atLeastAtMost % 0.31/0.60 thf(fact_5008_less__rat,axiom, % 0.31/0.60 ! [B2: int,D2: int,A2: int,C2: int] : % 0.31/0.60 ( ( B2 % 0.31/0.60 != ( zero_zero @ int ) ) % 0.31/0.60 => ( ( D2 % 0.31/0.60 != ( zero_zero @ int ) ) % 0.31/0.60 => ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( fract @ C2 @ D2 ) ) % 0.31/0.60 = ( ord_less @ int @ ( times_times @ int @ ( times_times @ int @ A2 @ D2 ) @ ( times_times @ int @ B2 @ D2 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % less_rat % 0.31/0.60 thf(fact_5009_setcompr__eq__image,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F2: B > A,P: B > $o] : % 0.31/0.60 ( ( collect @ A % 0.31/0.60 @ ^ [Uu3: A] : % 0.31/0.60 ? [X: B] : % 0.31/0.60 ( ( Uu3 % 0.31/0.60 = ( F2 @ X ) ) % 0.31/0.60 & ( P @ X ) ) ) % 0.31/0.60 = ( image @ B @ A @ F2 @ ( collect @ B @ P ) ) ) ). % 0.31/0.60 % 0.31/0.60 % setcompr_eq_image % 0.31/0.60 thf(fact_5010_Setcompr__eq__image,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] : % 0.31/0.60 ( ( collect @ A % 0.31/0.60 @ ^ [Uu3: A] : % 0.31/0.60 ? [X: B] : % 0.31/0.60 ( ( Uu3 % 0.31/0.60 = ( F2 @ X ) ) % 0.31/0.60 & ( member @ B @ X @ A3 ) ) ) % 0.31/0.60 = ( image @ B @ A @ F2 @ A3 ) ) ). % 0.31/0.60 % 0.31/0.60 % Setcompr_eq_image % 0.31/0.60 thf(fact_5011_of__nat__in__Nats,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_1 @ A ) % 0.31/0.60 => ! [N2: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_1_Nats @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % of_nat_in_Nats % 0.31/0.60 thf(fact_5012_Nats__induct,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_1 @ A ) % 0.31/0.60 => ! [X3: A,P: A > $o] : % 0.31/0.60 ( ( member @ A @ X3 @ ( semiring_1_Nats @ A ) ) % 0.31/0.60 => ( ! [N3: nat] : ( P @ ( semiring_1_of_nat @ A @ N3 ) ) % 0.31/0.60 => ( P @ X3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Nats_induct % 0.31/0.60 thf(fact_5013_Nats__cases,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_1 @ A ) % 0.31/0.60 => ! [X3: A] : % 0.31/0.60 ( ( member @ A @ X3 @ ( semiring_1_Nats @ A ) ) % 0.31/0.60 => ~ ! [N3: nat] : % 0.31/0.60 ( X3 % 0.31/0.60 != ( semiring_1_of_nat @ A @ N3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Nats_cases % 0.31/0.60 thf(fact_5014_image__Fpow__mono,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B3: set @ A] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B3 ) % 0.31/0.60 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F2 ) @ ( finite_Fpow @ B @ A3 ) ) @ ( finite_Fpow @ A @ B3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_Fpow_mono % 0.31/0.60 thf(fact_5015_imageI,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,X3: A,A3: set @ A,F2: A > B] : % 0.31/0.60 ( ( member @ A @ X3 @ A3 ) % 0.31/0.60 => ( member @ B @ ( F2 @ X3 ) @ ( image @ A @ B @ F2 @ A3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % imageI % 0.31/0.60 thf(fact_5016_image__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,Z2: A,F2: B > A,A3: set @ B] : % 0.31/0.60 ( ( member @ A @ Z2 @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 = ( ? [X: B] : % 0.31/0.60 ( ( member @ B @ X @ A3 ) % 0.31/0.60 & ( Z2 % 0.31/0.60 = ( F2 @ X ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_iff % 0.31/0.60 thf(fact_5017_bex__imageD,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] : % 0.31/0.60 ( ? [X2: A] : % 0.31/0.60 ( ( member @ A @ X2 @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 & ( P @ X2 ) ) % 0.31/0.60 => ? [X5: B] : % 0.31/0.60 ( ( member @ B @ X5 @ A3 ) % 0.31/0.60 & ( P @ ( F2 @ X5 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % bex_imageD % 0.31/0.60 thf(fact_5018_image__cong,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,M7: set @ A,N7: set @ A,F2: A > B,G: A > B] : % 0.31/0.60 ( ( M7 = N7 ) % 0.31/0.60 => ( ! [X5: A] : % 0.31/0.60 ( ( member @ A @ X5 @ N7 ) % 0.31/0.60 => ( ( F2 @ X5 ) % 0.31/0.60 = ( G @ X5 ) ) ) % 0.31/0.60 => ( ( image @ A @ B @ F2 @ M7 ) % 0.31/0.60 = ( image @ A @ B @ G @ N7 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_cong % 0.31/0.60 thf(fact_5019_ball__imageD,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] : % 0.31/0.60 ( ! [X5: A] : % 0.31/0.60 ( ( member @ A @ X5 @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 => ( P @ X5 ) ) % 0.31/0.60 => ! [X2: B] : % 0.31/0.60 ( ( member @ B @ X2 @ A3 ) % 0.31/0.60 => ( P @ ( F2 @ X2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % ball_imageD % 0.31/0.60 thf(fact_5020_rev__image__eqI,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,X3: A,A3: set @ A,B2: B,F2: A > B] : % 0.31/0.60 ( ( member @ A @ X3 @ A3 ) % 0.31/0.60 => ( ( B2 % 0.31/0.60 = ( F2 @ X3 ) ) % 0.31/0.60 => ( member @ B @ B2 @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % rev_image_eqI % 0.31/0.60 thf(fact_5021_imageE,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,B2: A,F2: B > A,A3: set @ B] : % 0.31/0.60 ( ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 => ~ ! [X5: B] : % 0.31/0.60 ( ( B2 % 0.31/0.60 = ( F2 @ X5 ) ) % 0.31/0.60 => ~ ( member @ B @ X5 @ A3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % imageE % 0.31/0.60 thf(fact_5022_image__image,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,C: $tType,F2: B > A,G: C > B,A3: set @ C] : % 0.31/0.60 ( ( image @ B @ A @ F2 @ ( image @ C @ B @ G @ A3 ) ) % 0.31/0.60 = ( image @ C @ A % 0.31/0.60 @ ^ [X: C] : ( F2 @ ( G @ X ) ) % 0.31/0.60 @ A3 ) ) ). % 0.31/0.60 % 0.31/0.60 % image_image % 0.31/0.60 thf(fact_5023_Compr__image__eq,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] : % 0.31/0.60 ( ( collect @ A % 0.31/0.60 @ ^ [X: A] : % 0.31/0.60 ( ( member @ A @ X @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 & ( P @ X ) ) ) % 0.31/0.60 = ( image @ B @ A @ F2 % 0.31/0.60 @ ( collect @ B % 0.31/0.60 @ ^ [X: B] : % 0.31/0.60 ( ( member @ B @ X @ A3 ) % 0.31/0.60 & ( P @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Compr_image_eq % 0.31/0.60 thf(fact_5024_all__subset__image,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] : % 0.31/0.60 ( ( ! [B6: set @ A] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 => ( P @ B6 ) ) ) % 0.31/0.60 = ( ! [B6: set @ B] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ B ) @ B6 @ A3 ) % 0.31/0.60 => ( P @ ( image @ B @ A @ F2 @ B6 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % all_subset_image % 0.31/0.60 thf(fact_5025_subset__image__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,B3: set @ A,F2: B > A,A3: set @ B] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 = ( ? [AA: set @ B] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ B ) @ AA @ A3 ) % 0.31/0.60 & ( B3 % 0.31/0.60 = ( image @ B @ A @ F2 @ AA ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % subset_image_iff % 0.31/0.60 thf(fact_5026_image__subset__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B3: set @ A] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B3 ) % 0.31/0.60 = ( ! [X: B] : % 0.31/0.60 ( ( member @ B @ X @ A3 ) % 0.31/0.60 => ( member @ A @ ( F2 @ X ) @ B3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_subset_iff % 0.31/0.60 thf(fact_5027_subset__imageE,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,B3: set @ A,F2: B > A,A3: set @ B] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 => ~ ! [C8: set @ B] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ B ) @ C8 @ A3 ) % 0.31/0.60 => ( B3 % 0.31/0.60 != ( image @ B @ A @ F2 @ C8 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % subset_imageE % 0.31/0.60 thf(fact_5028_image__subsetI,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,A3: set @ A,F2: A > B,B3: set @ B] : % 0.31/0.60 ( ! [X5: A] : % 0.31/0.60 ( ( member @ A @ X5 @ A3 ) % 0.31/0.60 => ( member @ B @ ( F2 @ X5 ) @ B3 ) ) % 0.31/0.60 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B3 ) ) ). % 0.31/0.60 % 0.31/0.60 % image_subsetI % 0.31/0.60 thf(fact_5029_image__mono,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,F2: A > B] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.60 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_mono % 0.31/0.60 thf(fact_5030_image__Pow__surj,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B3: set @ A] : % 0.31/0.60 ( ( ( image @ B @ A @ F2 @ A3 ) % 0.31/0.60 = B3 ) % 0.31/0.60 => ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F2 ) @ ( pow2 @ B @ A3 ) ) % 0.31/0.60 = ( pow2 @ A @ B3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_Pow_surj % 0.31/0.60 thf(fact_5031_image__Collect__subsetI,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,P: A > $o,F2: A > B,B3: set @ B] : % 0.31/0.60 ( ! [X5: A] : % 0.31/0.60 ( ( P @ X5 ) % 0.31/0.60 => ( member @ B @ ( F2 @ X5 ) @ B3 ) ) % 0.31/0.60 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( collect @ A @ P ) ) @ B3 ) ) ). % 0.31/0.60 % 0.31/0.60 % image_Collect_subsetI % 0.31/0.60 thf(fact_5032_image__Pow__mono,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B3: set @ A] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B3 ) % 0.31/0.60 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F2 ) @ ( pow2 @ B @ A3 ) ) @ ( pow2 @ A @ B3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_Pow_mono % 0.31/0.60 thf(fact_5033_Nats__0,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_1 @ A ) % 0.31/0.60 => ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Nats_0 % 0.31/0.60 thf(fact_5034_Nats__add,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_1 @ A ) % 0.31/0.60 => ! [A2: A,B2: A] : % 0.31/0.60 ( ( member @ A @ A2 @ ( semiring_1_Nats @ A ) ) % 0.31/0.60 => ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) ) % 0.31/0.60 => ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Nats_add % 0.31/0.60 thf(fact_5035_Nats__1,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_1 @ A ) % 0.31/0.60 => ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Nats_1 % 0.31/0.60 thf(fact_5036_Nats__mult,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_1 @ A ) % 0.31/0.60 => ! [A2: A,B2: A] : % 0.31/0.60 ( ( member @ A @ A2 @ ( semiring_1_Nats @ A ) ) % 0.31/0.60 => ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) ) % 0.31/0.60 => ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Nats_mult % 0.31/0.60 thf(fact_5037_Nats__numeral,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( semiring_1 @ A ) % 0.31/0.60 => ! [W2: num] : ( member @ A @ ( numeral_numeral @ A @ W2 ) @ ( semiring_1_Nats @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Nats_numeral % 0.31/0.60 thf(fact_5038_UNION__singleton__eq__range,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] : % 0.31/0.60 ( ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.60 @ ( image @ B @ ( set @ A ) % 0.31/0.60 @ ^ [X: B] : ( insert @ A @ ( F2 @ X ) @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 @ A3 ) ) % 0.31/0.60 = ( image @ B @ A @ F2 @ A3 ) ) ). % 0.31/0.60 % 0.31/0.60 % UNION_singleton_eq_range % 0.31/0.60 thf(fact_5039_pigeonhole__infinite,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A3: set @ A,F2: A > B] : % 0.31/0.60 ( ~ ( finite_finite2 @ A @ A3 ) % 0.31/0.60 => ( ( finite_finite2 @ B @ ( image @ A @ B @ F2 @ A3 ) ) % 0.31/0.60 => ? [X5: A] : % 0.31/0.60 ( ( member @ A @ X5 @ A3 ) % 0.31/0.60 & ~ ( finite_finite2 @ A % 0.31/0.60 @ ( collect @ A % 0.31/0.60 @ ^ [A4: A] : % 0.31/0.60 ( ( member @ A @ A4 @ A3 ) % 0.31/0.60 & ( ( F2 @ A4 ) % 0.31/0.60 = ( F2 @ X5 ) ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % pigeonhole_infinite % 0.31/0.60 thf(fact_5040_SUP__eq,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,C: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [A3: set @ B,B3: set @ C,F2: B > A,G: C > A] : % 0.31/0.60 ( ! [I4: B] : % 0.31/0.60 ( ( member @ B @ I4 @ A3 ) % 0.31/0.60 => ? [X2: C] : % 0.31/0.60 ( ( member @ C @ X2 @ B3 ) % 0.31/0.60 & ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ X2 ) ) ) ) % 0.31/0.60 => ( ! [J2: C] : % 0.31/0.60 ( ( member @ C @ J2 @ B3 ) % 0.31/0.60 => ? [X2: B] : % 0.31/0.60 ( ( member @ B @ X2 @ A3 ) % 0.31/0.60 & ( ord_less_eq @ A @ ( G @ J2 ) @ ( F2 @ X2 ) ) ) ) % 0.31/0.60 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 = ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B3 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_eq % 0.31/0.60 thf(fact_5041_INF__eq,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,C: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [A3: set @ B,B3: set @ C,G: C > A,F2: B > A] : % 0.31/0.60 ( ! [I4: B] : % 0.31/0.60 ( ( member @ B @ I4 @ A3 ) % 0.31/0.60 => ? [X2: C] : % 0.31/0.60 ( ( member @ C @ X2 @ B3 ) % 0.31/0.60 & ( ord_less_eq @ A @ ( G @ X2 ) @ ( F2 @ I4 ) ) ) ) % 0.31/0.60 => ( ! [J2: C] : % 0.31/0.60 ( ( member @ C @ J2 @ B3 ) % 0.31/0.60 => ? [X2: B] : % 0.31/0.60 ( ( member @ B @ X2 @ A3 ) % 0.31/0.60 & ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G @ J2 ) ) ) ) % 0.31/0.60 => ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 = ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B3 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_eq % 0.31/0.60 thf(fact_5042_zero__notin__Suc__image,axiom, % 0.31/0.60 ! [A3: set @ nat] : % 0.31/0.60 ~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A3 ) ) ). % 0.31/0.60 % 0.31/0.60 % zero_notin_Suc_image % 0.31/0.60 thf(fact_5043_finite__surj,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,A3: set @ A,B3: set @ B,F2: A > B] : % 0.31/0.60 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.60 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ ( image @ A @ B @ F2 @ A3 ) ) % 0.31/0.60 => ( finite_finite2 @ B @ B3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % finite_surj % 0.31/0.60 thf(fact_5044_finite__subset__image,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,B3: set @ A,F2: B > A,A3: set @ B] : % 0.31/0.60 ( ( finite_finite2 @ A @ B3 ) % 0.31/0.60 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 => ? [C8: set @ B] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ B ) @ C8 @ A3 ) % 0.31/0.60 & ( finite_finite2 @ B @ C8 ) % 0.31/0.60 & ( B3 % 0.31/0.60 = ( image @ B @ A @ F2 @ C8 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % finite_subset_image % 0.31/0.60 thf(fact_5045_ex__finite__subset__image,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] : % 0.31/0.60 ( ( ? [B6: set @ A] : % 0.31/0.60 ( ( finite_finite2 @ A @ B6 ) % 0.31/0.60 & ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 & ( P @ B6 ) ) ) % 0.31/0.60 = ( ? [B6: set @ B] : % 0.31/0.60 ( ( finite_finite2 @ B @ B6 ) % 0.31/0.60 & ( ord_less_eq @ ( set @ B ) @ B6 @ A3 ) % 0.31/0.60 & ( P @ ( image @ B @ A @ F2 @ B6 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % ex_finite_subset_image % 0.31/0.60 thf(fact_5046_all__finite__subset__image,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] : % 0.31/0.60 ( ( ! [B6: set @ A] : % 0.31/0.60 ( ( ( finite_finite2 @ A @ B6 ) % 0.31/0.60 & ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F2 @ A3 ) ) ) % 0.31/0.60 => ( P @ B6 ) ) ) % 0.31/0.60 = ( ! [B6: set @ B] : % 0.31/0.60 ( ( ( finite_finite2 @ B @ B6 ) % 0.31/0.60 & ( ord_less_eq @ ( set @ B ) @ B6 @ A3 ) ) % 0.31/0.60 => ( P @ ( image @ B @ A @ F2 @ B6 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % all_finite_subset_image % 0.31/0.60 thf(fact_5047_translation__Int,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ab_group_add @ A ) % 0.31/0.60 => ! [A2: A,S: set @ A,T2: set @ A] : % 0.31/0.60 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( inf_inf @ ( set @ A ) @ S @ T2 ) ) % 0.31/0.60 = ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % translation_Int % 0.31/0.60 thf(fact_5048_SUP__eq__const,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [I6: set @ B,F2: B > A,X3: A] : % 0.31/0.60 ( ( I6 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ! [I4: B] : % 0.31/0.60 ( ( member @ B @ I4 @ I6 ) % 0.31/0.60 => ( ( F2 @ I4 ) % 0.31/0.60 = X3 ) ) % 0.31/0.60 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ I6 ) ) % 0.31/0.60 = X3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_eq_const % 0.31/0.60 thf(fact_5049_translation__diff,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ab_group_add @ A ) % 0.31/0.60 => ! [A2: A,S: set @ A,T2: set @ A] : % 0.31/0.60 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( minus_minus @ ( set @ A ) @ S @ T2 ) ) % 0.31/0.60 = ( minus_minus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % translation_diff % 0.31/0.60 thf(fact_5050_INF__eq__const,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [I6: set @ B,F2: B > A,X3: A] : % 0.31/0.60 ( ( I6 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ! [I4: B] : % 0.31/0.60 ( ( member @ B @ I4 @ I6 ) % 0.31/0.60 => ( ( F2 @ I4 ) % 0.31/0.60 = X3 ) ) % 0.31/0.60 => ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) ) % 0.31/0.60 = X3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_eq_const % 0.31/0.60 thf(fact_5051_finite__image__absD,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_ring @ A ) % 0.31/0.60 => ! [S3: set @ A] : % 0.31/0.60 ( ( finite_finite2 @ A @ ( image @ A @ A @ ( abs_abs @ A ) @ S3 ) ) % 0.31/0.60 => ( finite_finite2 @ A @ S3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % finite_image_absD % 0.31/0.60 thf(fact_5052_image__Int__subset,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B3: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ ( image @ B @ A @ F2 @ B3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_Int_subset % 0.31/0.60 thf(fact_5053_image__diff__subset,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B3: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ ( image @ B @ A @ F2 @ B3 ) ) @ ( image @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_diff_subset % 0.31/0.60 thf(fact_5054_Rat__induct__pos,axiom, % 0.31/0.60 ! [P: rat > $o,Q4: rat] : % 0.31/0.60 ( ! [A5: int,B5: int] : % 0.31/0.60 ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 ) % 0.31/0.60 => ( P @ ( fract @ A5 @ B5 ) ) ) % 0.31/0.60 => ( P @ Q4 ) ) ). % 0.31/0.60 % 0.31/0.60 % Rat_induct_pos % 0.31/0.60 thf(fact_5055_translation__Compl,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( ab_group_add @ A ) % 0.31/0.60 => ! [A2: A,T2: set @ A] : % 0.31/0.60 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) ) % 0.31/0.60 = ( uminus_uminus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % translation_Compl % 0.31/0.60 thf(fact_5056_quotient__of__eq,axiom, % 0.31/0.60 ! [A2: int,B2: int,P4: int,Q4: int] : % 0.31/0.60 ( ( ( quotient_of @ ( fract @ A2 @ B2 ) ) % 0.31/0.60 = ( product_Pair @ int @ int @ P4 @ Q4 ) ) % 0.31/0.60 => ( ( fract @ P4 @ Q4 ) % 0.31/0.60 = ( fract @ A2 @ B2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % quotient_of_eq % 0.31/0.60 thf(fact_5057_SUP__eqI,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [A3: set @ B,F2: B > A,X3: A] : % 0.31/0.60 ( ! [I4: B] : % 0.31/0.60 ( ( member @ B @ I4 @ A3 ) % 0.31/0.60 => ( ord_less_eq @ A @ ( F2 @ I4 ) @ X3 ) ) % 0.31/0.60 => ( ! [Y4: A] : % 0.31/0.60 ( ! [I5: B] : % 0.31/0.60 ( ( member @ B @ I5 @ A3 ) % 0.31/0.60 => ( ord_less_eq @ A @ ( F2 @ I5 ) @ Y4 ) ) % 0.31/0.60 => ( ord_less_eq @ A @ X3 @ Y4 ) ) % 0.31/0.60 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 = X3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_eqI % 0.31/0.60 thf(fact_5058_SUP__mono,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,C: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [A3: set @ B,B3: set @ C,F2: B > A,G: C > A] : % 0.31/0.60 ( ! [N3: B] : % 0.31/0.60 ( ( member @ B @ N3 @ A3 ) % 0.31/0.60 => ? [X2: C] : % 0.31/0.60 ( ( member @ C @ X2 @ B3 ) % 0.31/0.60 & ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ X2 ) ) ) ) % 0.31/0.60 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_mono % 0.31/0.60 thf(fact_5059_SUP__least,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [A3: set @ B,F2: B > A,U: A] : % 0.31/0.60 ( ! [I4: B] : % 0.31/0.60 ( ( member @ B @ I4 @ A3 ) % 0.31/0.60 => ( ord_less_eq @ A @ ( F2 @ I4 ) @ U ) ) % 0.31/0.60 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_least % 0.31/0.60 thf(fact_5060_SUP__mono_H,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [F2: B > A,G: B > A,A3: set @ B] : % 0.31/0.60 ( ! [X5: B] : ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) % 0.31/0.60 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_mono' % 0.31/0.60 thf(fact_5061_SUP__upper,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [I: B,A3: set @ B,F2: B > A] : % 0.31/0.60 ( ( member @ B @ I @ A3 ) % 0.31/0.60 => ( ord_less_eq @ A @ ( F2 @ I ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_upper % 0.31/0.60 thf(fact_5062_SUP__le__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [F2: B > A,A3: set @ B,U: A] : % 0.31/0.60 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U ) % 0.31/0.60 = ( ! [X: B] : % 0.31/0.60 ( ( member @ B @ X @ A3 ) % 0.31/0.60 => ( ord_less_eq @ A @ ( F2 @ X ) @ U ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_le_iff % 0.31/0.60 thf(fact_5063_SUP__upper2,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [I: B,A3: set @ B,U: A,F2: B > A] : % 0.31/0.60 ( ( member @ B @ I @ A3 ) % 0.31/0.60 => ( ( ord_less_eq @ A @ U @ ( F2 @ I ) ) % 0.31/0.60 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_upper2 % 0.31/0.60 thf(fact_5064_less__SUP__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple5582772986160207858norder @ A ) % 0.31/0.60 => ! [A2: A,F2: B > A,A3: set @ B] : % 0.31/0.60 ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) % 0.31/0.60 = ( ? [X: B] : % 0.31/0.60 ( ( member @ B @ X @ A3 ) % 0.31/0.60 & ( ord_less @ A @ A2 @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % less_SUP_iff % 0.31/0.60 thf(fact_5065_SUP__lessD,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [F2: B > A,A3: set @ B,Y: A,I: B] : % 0.31/0.60 ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ Y ) % 0.31/0.60 => ( ( member @ B @ I @ A3 ) % 0.31/0.60 => ( ord_less @ A @ ( F2 @ I ) @ Y ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_lessD % 0.31/0.60 thf(fact_5066_INF__eqI,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [A3: set @ B,X3: A,F2: B > A] : % 0.31/0.60 ( ! [I4: B] : % 0.31/0.60 ( ( member @ B @ I4 @ A3 ) % 0.31/0.60 => ( ord_less_eq @ A @ X3 @ ( F2 @ I4 ) ) ) % 0.31/0.60 => ( ! [Y4: A] : % 0.31/0.60 ( ! [I5: B] : % 0.31/0.60 ( ( member @ B @ I5 @ A3 ) % 0.31/0.60 => ( ord_less_eq @ A @ Y4 @ ( F2 @ I5 ) ) ) % 0.31/0.60 => ( ord_less_eq @ A @ Y4 @ X3 ) ) % 0.31/0.60 => ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.60 = X3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_eqI % 0.31/0.60 thf(fact_5067_INF__mono,axiom, % 0.31/0.60 ! [C: $tType,A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [B3: set @ B,A3: set @ C,F2: C > A,G: B > A] : % 0.31/0.60 ( ! [M3: B] : % 0.31/0.60 ( ( member @ B @ M3 @ B3 ) % 0.31/0.60 => ? [X2: C] : % 0.31/0.60 ( ( member @ C @ X2 @ A3 ) % 0.31/0.60 & ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G @ M3 ) ) ) ) % 0.31/0.60 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_mono % 0.31/0.60 thf(fact_5068_INF__lower,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [I: B,A3: set @ B,F2: B > A] : % 0.31/0.60 ( ( member @ B @ I @ A3 ) % 0.31/0.60 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( F2 @ I ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_lower % 0.31/0.60 thf(fact_5069_INF__mono_H,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [F2: B > A,G: B > A,A3: set @ B] : % 0.31/0.60 ( ! [X5: B] : ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) % 0.31/0.60 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_mono' % 0.31/0.60 thf(fact_5070_INF__lower2,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [I: B,A3: set @ B,F2: B > A,U: A] : % 0.31/0.60 ( ( member @ B @ I @ A3 ) % 0.31/0.60 => ( ( ord_less_eq @ A @ ( F2 @ I ) @ U ) % 0.31/0.60 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_lower2 % 0.31/0.60 thf(fact_5071_le__INF__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [U: A,F2: B > A,A3: set @ B] : % 0.31/0.60 ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) % 0.31/0.60 = ( ! [X: B] : % 0.31/0.60 ( ( member @ B @ X @ A3 ) % 0.31/0.60 => ( ord_less_eq @ A @ U @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % le_INF_iff % 0.31/0.60 thf(fact_5072_INF__greatest,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [A3: set @ B,U: A,F2: B > A] : % 0.31/0.60 ( ! [I4: B] : % 0.31/0.60 ( ( member @ B @ I4 @ A3 ) % 0.31/0.60 => ( ord_less_eq @ A @ U @ ( F2 @ I4 ) ) ) % 0.31/0.60 => ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_greatest % 0.31/0.60 thf(fact_5073_INF__less__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple5582772986160207858norder @ A ) % 0.31/0.60 => ! [F2: B > A,A3: set @ B,A2: A] : % 0.31/0.60 ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ A2 ) % 0.31/0.60 = ( ? [X: B] : % 0.31/0.60 ( ( member @ B @ X @ A3 ) % 0.31/0.60 & ( ord_less @ A @ ( F2 @ X ) @ A2 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_less_iff % 0.31/0.60 thf(fact_5074_less__INF__D,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [Y: A,F2: B > A,A3: set @ B,I: B] : % 0.31/0.60 ( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) % 0.31/0.60 => ( ( member @ B @ I @ A3 ) % 0.31/0.60 => ( ord_less @ A @ Y @ ( F2 @ I ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % less_INF_D % 0.31/0.60 thf(fact_5075_finite__conv__nat__seg__image,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( finite_finite2 @ A ) % 0.31/0.60 = ( ^ [A6: set @ A] : % 0.31/0.60 ? [N: nat,F3: nat > A] : % 0.31/0.60 ( A6 % 0.31/0.60 = ( image @ nat @ A @ F3 % 0.31/0.60 @ ( collect @ nat % 0.31/0.60 @ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % finite_conv_nat_seg_image % 0.31/0.60 thf(fact_5076_nat__seg__image__imp__finite,axiom, % 0.31/0.60 ! [A: $tType,A3: set @ A,F2: nat > A,N2: nat] : % 0.31/0.60 ( ( A3 % 0.31/0.60 = ( image @ nat @ A @ F2 % 0.31/0.60 @ ( collect @ nat % 0.31/0.60 @ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) ) ) % 0.31/0.60 => ( finite_finite2 @ A @ A3 ) ) ). % 0.31/0.60 % 0.31/0.60 % nat_seg_image_imp_finite % 0.31/0.60 thf(fact_5077_image__constant,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,X3: A,A3: set @ A,C2: B] : % 0.31/0.60 ( ( member @ A @ X3 @ A3 ) % 0.31/0.60 => ( ( image @ A @ B % 0.31/0.60 @ ^ [X: A] : C2 % 0.31/0.60 @ A3 ) % 0.31/0.60 = ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_constant % 0.31/0.60 thf(fact_5078_image__constant__conv,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A3: set @ B,C2: A] : % 0.31/0.60 ( ( ( A3 % 0.31/0.60 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( image @ B @ A % 0.31/0.60 @ ^ [X: B] : C2 % 0.31/0.60 @ A3 ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.60 & ( ( A3 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( image @ B @ A % 0.31/0.60 @ ^ [X: B] : C2 % 0.31/0.60 @ A3 ) % 0.31/0.60 = ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_constant_conv % 0.31/0.60 thf(fact_5079_sum_Oimage__gen,axiom, % 0.31/0.60 ! [A: $tType,C: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_add @ A ) % 0.31/0.60 => ! [S3: set @ B,H2: B > A,G: B > C] : % 0.31/0.60 ( ( finite_finite2 @ B @ S3 ) % 0.31/0.60 => ( ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) % 0.31/0.60 = ( groups7311177749621191930dd_sum @ C @ A % 0.31/0.60 @ ^ [Y2: C] : % 0.31/0.60 ( groups7311177749621191930dd_sum @ B @ A @ H2 % 0.31/0.60 @ ( collect @ B % 0.31/0.60 @ ^ [X: B] : % 0.31/0.60 ( ( member @ B @ X @ S3 ) % 0.31/0.60 & ( ( G @ X ) % 0.31/0.60 = Y2 ) ) ) ) % 0.31/0.60 @ ( image @ B @ C @ G @ S3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sum.image_gen % 0.31/0.60 thf(fact_5080_normalize__eq,axiom, % 0.31/0.60 ! [A2: int,B2: int,P4: int,Q4: int] : % 0.31/0.60 ( ( ( normalize @ ( product_Pair @ int @ int @ A2 @ B2 ) ) % 0.31/0.60 = ( product_Pair @ int @ int @ P4 @ Q4 ) ) % 0.31/0.60 => ( ( fract @ P4 @ Q4 ) % 0.31/0.60 = ( fract @ A2 @ B2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % normalize_eq % 0.31/0.60 thf(fact_5081_prod_Oimage__gen,axiom, % 0.31/0.60 ! [A: $tType,C: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_mult @ A ) % 0.31/0.60 => ! [S3: set @ B,H2: B > A,G: B > C] : % 0.31/0.60 ( ( finite_finite2 @ B @ S3 ) % 0.31/0.60 => ( ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) % 0.31/0.60 = ( groups7121269368397514597t_prod @ C @ A % 0.31/0.60 @ ^ [Y2: C] : % 0.31/0.60 ( groups7121269368397514597t_prod @ B @ A @ H2 % 0.31/0.60 @ ( collect @ B % 0.31/0.60 @ ^ [X: B] : % 0.31/0.60 ( ( member @ B @ X @ S3 ) % 0.31/0.60 & ( ( G @ X ) % 0.31/0.60 = Y2 ) ) ) ) % 0.31/0.60 @ ( image @ B @ C @ G @ S3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % prod.image_gen % 0.31/0.60 thf(fact_5082_Gcd__mono,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( semiring_Gcd @ A ) % 0.31/0.60 => ! [A3: set @ B,F2: B > A,G: B > A] : % 0.31/0.60 ( ! [X5: B] : % 0.31/0.60 ( ( member @ B @ X5 @ A3 ) % 0.31/0.60 => ( dvd_dvd @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 0.31/0.60 => ( dvd_dvd @ A @ ( gcd_Gcd @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( gcd_Gcd @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_mono % 0.31/0.60 thf(fact_5083_the__elem__image__unique,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A3: set @ A,F2: A > B,X3: A] : % 0.31/0.60 ( ( A3 % 0.31/0.60 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 => ( ! [Y4: A] : % 0.31/0.60 ( ( member @ A @ Y4 @ A3 ) % 0.31/0.60 => ( ( F2 @ Y4 ) % 0.31/0.60 = ( F2 @ X3 ) ) ) % 0.31/0.60 => ( ( the_elem @ B @ ( image @ A @ B @ F2 @ A3 ) ) % 0.31/0.60 = ( F2 @ X3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % the_elem_image_unique % 0.31/0.60 thf(fact_5084_le__SUP__iff,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple5582772986160207858norder @ A ) % 0.31/0.60 => ! [X3: A,F2: B > A,A3: set @ B] : % 0.31/0.60 ( ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) % 0.31/0.60 = ( ! [Y2: A] : % 0.31/0.60 ( ( ord_less @ A @ Y2 @ X3 ) % 0.31/0.60 => ? [X: B] : % 0.31/0.60 ( ( member @ B @ X @ A3 ) % 0.31/0.60 & ( ord_less @ A @ Y2 @ ( F2 @ X ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % le_SUP_iff % 0.31/0.60 thf(fact_5085_INF__le__iff,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple5582772986160207858norder @ A ) % 0.31/0.60 => ! [F2: B > A,A3: set @ B,X3: A] : % 0.31/0.60 ( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ X3 ) % 0.31/0.60 = ( ! [Y2: A] : % 0.31/0.60 ( ( ord_less @ A @ X3 @ Y2 ) % 0.31/0.60 => ? [X: B] : % 0.31/0.60 ( ( member @ B @ X @ A3 ) % 0.31/0.60 & ( ord_less @ A @ ( F2 @ X ) @ Y2 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_le_iff % 0.31/0.60 thf(fact_5086_SUP__eq__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [I6: set @ B,C2: A,F2: B > A] : % 0.31/0.60 ( ( I6 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ! [I4: B] : % 0.31/0.60 ( ( member @ B @ I4 @ I6 ) % 0.31/0.60 => ( ord_less_eq @ A @ C2 @ ( F2 @ I4 ) ) ) % 0.31/0.60 => ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ I6 ) ) % 0.31/0.60 = C2 ) % 0.31/0.60 = ( ! [X: B] : % 0.31/0.60 ( ( member @ B @ X @ I6 ) % 0.31/0.60 => ( ( F2 @ X ) % 0.31/0.60 = C2 ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_eq_iff % 0.31/0.60 thf(fact_5087_cSUP__least,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( condit1219197933456340205attice @ A ) % 0.31/0.60 => ! [A3: set @ B,F2: B > A,M7: A] : % 0.31/0.60 ( ( A3 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ! [X5: B] : % 0.31/0.60 ( ( member @ B @ X5 @ A3 ) % 0.31/0.60 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ M7 ) ) % 0.31/0.60 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ M7 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % cSUP_least % 0.31/0.60 thf(fact_5088_INF__eq__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [I6: set @ B,F2: B > A,C2: A] : % 0.31/0.60 ( ( I6 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ! [I4: B] : % 0.31/0.60 ( ( member @ B @ I4 @ I6 ) % 0.31/0.60 => ( ord_less_eq @ A @ ( F2 @ I4 ) @ C2 ) ) % 0.31/0.60 => ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) ) % 0.31/0.60 = C2 ) % 0.31/0.60 = ( ! [X: B] : % 0.31/0.60 ( ( member @ B @ X @ I6 ) % 0.31/0.60 => ( ( F2 @ X ) % 0.31/0.60 = C2 ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_eq_iff % 0.31/0.60 thf(fact_5089_cINF__greatest,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( condit1219197933456340205attice @ A ) % 0.31/0.60 => ! [A3: set @ B,M: A,F2: B > A] : % 0.31/0.60 ( ( A3 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ! [X5: B] : % 0.31/0.60 ( ( member @ B @ X5 @ A3 ) % 0.31/0.60 => ( ord_less_eq @ A @ M @ ( F2 @ X5 ) ) ) % 0.31/0.60 => ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % cINF_greatest % 0.31/0.60 thf(fact_5090_card__image__le,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A3: set @ A,F2: A > B] : % 0.31/0.60 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.60 => ( ord_less_eq @ nat @ ( finite_card @ B @ ( image @ A @ B @ F2 @ A3 ) ) @ ( finite_card @ A @ A3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % card_image_le % 0.31/0.60 thf(fact_5091_Nats__diff,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_idom @ A ) % 0.31/0.60 => ! [A2: A,B2: A] : % 0.31/0.60 ( ( member @ A @ A2 @ ( semiring_1_Nats @ A ) ) % 0.31/0.60 => ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) ) % 0.31/0.60 => ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.60 => ( member @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Nats_diff % 0.31/0.60 thf(fact_5092_SUP__subset__mono,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [A3: set @ B,B3: set @ B,F2: B > A,G: B > A] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 ) % 0.31/0.60 => ( ! [X5: B] : % 0.31/0.60 ( ( member @ B @ X5 @ A3 ) % 0.31/0.60 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 0.31/0.60 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_subset_mono % 0.31/0.60 thf(fact_5093_INF__superset__mono,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [B3: set @ B,A3: set @ B,F2: B > A,G: B > A] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 ) % 0.31/0.60 => ( ! [X5: B] : % 0.31/0.60 ( ( member @ B @ X5 @ B3 ) % 0.31/0.60 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 0.31/0.60 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_superset_mono % 0.31/0.60 thf(fact_5094_SUP__empty,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [F2: B > A] : % 0.31/0.60 ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) ) % 0.31/0.60 = ( bot_bot @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_empty % 0.31/0.60 thf(fact_5095_SUP__constant,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [A3: set @ B,C2: A] : % 0.31/0.60 ( ( ( A3 % 0.31/0.60 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( complete_Sup_Sup @ A % 0.31/0.60 @ ( image @ B @ A % 0.31/0.60 @ ^ [Y2: B] : C2 % 0.31/0.60 @ A3 ) ) % 0.31/0.60 = ( bot_bot @ A ) ) ) % 0.31/0.60 & ( ( A3 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( complete_Sup_Sup @ A % 0.31/0.60 @ ( image @ B @ A % 0.31/0.60 @ ^ [Y2: B] : C2 % 0.31/0.60 @ A3 ) ) % 0.31/0.60 = C2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_constant % 0.31/0.60 thf(fact_5096_sum_Ogroup,axiom, % 0.31/0.60 ! [C: $tType,A: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_add @ A ) % 0.31/0.60 => ! [S3: set @ B,T4: set @ C,G: B > C,H2: B > A] : % 0.31/0.60 ( ( finite_finite2 @ B @ S3 ) % 0.31/0.60 => ( ( finite_finite2 @ C @ T4 ) % 0.31/0.60 => ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S3 ) @ T4 ) % 0.31/0.60 => ( ( groups7311177749621191930dd_sum @ C @ A % 0.31/0.60 @ ^ [Y2: C] : % 0.31/0.60 ( groups7311177749621191930dd_sum @ B @ A @ H2 % 0.31/0.60 @ ( collect @ B % 0.31/0.60 @ ^ [X: B] : % 0.31/0.60 ( ( member @ B @ X @ S3 ) % 0.31/0.60 & ( ( G @ X ) % 0.31/0.60 = Y2 ) ) ) ) % 0.31/0.60 @ T4 ) % 0.31/0.60 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sum.group % 0.31/0.60 thf(fact_5097_INF__inf__const2,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [I6: set @ B,F2: B > A,X3: A] : % 0.31/0.60 ( ( I6 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( complete_Inf_Inf @ A % 0.31/0.60 @ ( image @ B @ A % 0.31/0.60 @ ^ [I3: B] : ( inf_inf @ A @ ( F2 @ I3 ) @ X3 ) % 0.31/0.60 @ I6 ) ) % 0.31/0.60 = ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) ) @ X3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_inf_const2 % 0.31/0.60 thf(fact_5098_INF__inf__const1,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [I6: set @ B,X3: A,F2: B > A] : % 0.31/0.60 ( ( I6 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( complete_Inf_Inf @ A % 0.31/0.60 @ ( image @ B @ A % 0.31/0.60 @ ^ [I3: B] : ( inf_inf @ A @ X3 @ ( F2 @ I3 ) ) % 0.31/0.60 @ I6 ) ) % 0.31/0.60 = ( inf_inf @ A @ X3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_inf_const1 % 0.31/0.60 thf(fact_5099_Nats__subset__Rats,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( field_char_0 @ A ) % 0.31/0.60 => ( ord_less_eq @ ( set @ A ) @ ( semiring_1_Nats @ A ) @ ( field_char_0_Rats @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Nats_subset_Rats % 0.31/0.60 thf(fact_5100_prod_Ogroup,axiom, % 0.31/0.60 ! [C: $tType,A: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_mult @ A ) % 0.31/0.60 => ! [S3: set @ B,T4: set @ C,G: B > C,H2: B > A] : % 0.31/0.60 ( ( finite_finite2 @ B @ S3 ) % 0.31/0.60 => ( ( finite_finite2 @ C @ T4 ) % 0.31/0.60 => ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S3 ) @ T4 ) % 0.31/0.60 => ( ( groups7121269368397514597t_prod @ C @ A % 0.31/0.60 @ ^ [Y2: C] : % 0.31/0.60 ( groups7121269368397514597t_prod @ B @ A @ H2 % 0.31/0.60 @ ( collect @ B % 0.31/0.60 @ ^ [X: B] : % 0.31/0.60 ( ( member @ B @ X @ S3 ) % 0.31/0.60 & ( ( G @ X ) % 0.31/0.60 = Y2 ) ) ) ) % 0.31/0.60 @ T4 ) % 0.31/0.60 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % prod.group % 0.31/0.60 thf(fact_5101_quotient__of__Fract,axiom, % 0.31/0.60 ! [A2: int,B2: int] : % 0.31/0.60 ( ( quotient_of @ ( fract @ A2 @ B2 ) ) % 0.31/0.60 = ( normalize @ ( product_Pair @ int @ int @ A2 @ B2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % quotient_of_Fract % 0.31/0.60 thf(fact_5102_sum_Oreindex__nontrivial,axiom, % 0.31/0.60 ! [C: $tType,A: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_add @ A ) % 0.31/0.60 => ! [A3: set @ B,H2: B > C,G: C > A] : % 0.31/0.60 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.60 => ( ! [X5: B,Y4: B] : % 0.31/0.60 ( ( member @ B @ X5 @ A3 ) % 0.31/0.60 => ( ( member @ B @ Y4 @ A3 ) % 0.31/0.60 => ( ( X5 != Y4 ) % 0.31/0.60 => ( ( ( H2 @ X5 ) % 0.31/0.60 = ( H2 @ Y4 ) ) % 0.31/0.60 => ( ( G @ ( H2 @ X5 ) ) % 0.31/0.60 = ( zero_zero @ A ) ) ) ) ) ) % 0.31/0.60 => ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image @ B @ C @ H2 @ A3 ) ) % 0.31/0.60 = ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sum.reindex_nontrivial % 0.31/0.60 thf(fact_5103_INF__le__SUP,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [A3: set @ B,F2: B > A] : % 0.31/0.60 ( ( A3 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_le_SUP % 0.31/0.60 thf(fact_5104_prod_Oreindex__nontrivial,axiom, % 0.31/0.60 ! [C: $tType,A: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_mult @ A ) % 0.31/0.60 => ! [A3: set @ B,H2: B > C,G: C > A] : % 0.31/0.60 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.60 => ( ! [X5: B,Y4: B] : % 0.31/0.60 ( ( member @ B @ X5 @ A3 ) % 0.31/0.60 => ( ( member @ B @ Y4 @ A3 ) % 0.31/0.60 => ( ( X5 != Y4 ) % 0.31/0.60 => ( ( ( H2 @ X5 ) % 0.31/0.60 = ( H2 @ Y4 ) ) % 0.31/0.60 => ( ( G @ ( H2 @ X5 ) ) % 0.31/0.60 = ( one_one @ A ) ) ) ) ) ) % 0.31/0.60 => ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image @ B @ C @ H2 @ A3 ) ) % 0.31/0.60 = ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % prod.reindex_nontrivial % 0.31/0.60 thf(fact_5105_surj__card__le,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,F2: A > B] : % 0.31/0.60 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.60 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ ( image @ A @ B @ F2 @ A3 ) ) % 0.31/0.60 => ( ord_less_eq @ nat @ ( finite_card @ B @ B3 ) @ ( finite_card @ A @ A3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % surj_card_le % 0.31/0.60 thf(fact_5106_scaleR__image__atLeastAtMost,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( real_V5355595471888546746vector @ A ) % 0.31/0.60 => ! [C2: real,X3: A,Y: A] : % 0.31/0.60 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.60 => ( ( image @ A @ A @ ( real_V8093663219630862766scaleR @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ Y ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % scaleR_image_atLeastAtMost % 0.31/0.60 thf(fact_5107_image__Suc__lessThan,axiom, % 0.31/0.60 ! [N2: nat] : % 0.31/0.60 ( ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ). % 0.31/0.60 % 0.31/0.60 % image_Suc_lessThan % 0.31/0.60 thf(fact_5108_image__Suc__atMost,axiom, % 0.31/0.60 ! [N2: nat] : % 0.31/0.60 ( ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N2 ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_Suc_atMost % 0.31/0.60 thf(fact_5109_atLeast0__atMost__Suc__eq__insert__0,axiom, % 0.31/0.60 ! [N2: nat] : % 0.31/0.60 ( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) % 0.31/0.60 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % atLeast0_atMost_Suc_eq_insert_0 % 0.31/0.60 thf(fact_5110_atLeast0__lessThan__Suc__eq__insert__0,axiom, % 0.31/0.60 ! [N2: nat] : % 0.31/0.60 ( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) % 0.31/0.60 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % atLeast0_lessThan_Suc_eq_insert_0 % 0.31/0.60 thf(fact_5111_lessThan__Suc__eq__insert__0,axiom, % 0.31/0.60 ! [N2: nat] : % 0.31/0.60 ( ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) % 0.31/0.60 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % lessThan_Suc_eq_insert_0 % 0.31/0.60 thf(fact_5112_atMost__Suc__eq__insert__0,axiom, % 0.31/0.60 ! [N2: nat] : % 0.31/0.60 ( ( set_ord_atMost @ nat @ ( suc @ N2 ) ) % 0.31/0.60 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % atMost_Suc_eq_insert_0 % 0.31/0.60 thf(fact_5113_Fract__less__zero__iff,axiom, % 0.31/0.60 ! [B2: int,A2: int] : % 0.31/0.60 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.31/0.60 => ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( zero_zero @ rat ) ) % 0.31/0.60 = ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Fract_less_zero_iff % 0.31/0.60 thf(fact_5114_zero__less__Fract__iff,axiom, % 0.31/0.60 ! [B2: int,A2: int] : % 0.31/0.60 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.31/0.60 => ( ( ord_less @ rat @ ( zero_zero @ rat ) @ ( fract @ A2 @ B2 ) ) % 0.31/0.60 = ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % zero_less_Fract_iff % 0.31/0.60 thf(fact_5115_one__less__Fract__iff,axiom, % 0.31/0.60 ! [B2: int,A2: int] : % 0.31/0.60 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.31/0.60 => ( ( ord_less @ rat @ ( one_one @ rat ) @ ( fract @ A2 @ B2 ) ) % 0.31/0.60 = ( ord_less @ int @ B2 @ A2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % one_less_Fract_iff % 0.31/0.60 thf(fact_5116_Fract__less__one__iff,axiom, % 0.31/0.60 ! [B2: int,A2: int] : % 0.31/0.60 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.31/0.60 => ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( one_one @ rat ) ) % 0.31/0.60 = ( ord_less @ int @ A2 @ B2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Fract_less_one_iff % 0.31/0.60 thf(fact_5117_sum__image__le,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,C: $tType] : % 0.31/0.60 ( ( ordere6911136660526730532id_add @ B ) % 0.31/0.60 => ! [I6: set @ C,G: A > B,F2: C > A] : % 0.31/0.60 ( ( finite_finite2 @ C @ I6 ) % 0.31/0.60 => ( ! [I4: C] : % 0.31/0.60 ( ( member @ C @ I4 @ I6 ) % 0.31/0.60 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F2 @ I4 ) ) ) ) % 0.31/0.60 => ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image @ C @ A @ F2 @ I6 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F2 ) @ I6 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sum_image_le % 0.31/0.60 thf(fact_5118_zero__le__Fract__iff,axiom, % 0.31/0.60 ! [B2: int,A2: int] : % 0.31/0.60 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.31/0.60 => ( ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ ( fract @ A2 @ B2 ) ) % 0.31/0.60 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % zero_le_Fract_iff % 0.31/0.60 thf(fact_5119_Fract__le__zero__iff,axiom, % 0.31/0.60 ! [B2: int,A2: int] : % 0.31/0.60 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.31/0.60 => ( ( ord_less_eq @ rat @ ( fract @ A2 @ B2 ) @ ( zero_zero @ rat ) ) % 0.31/0.60 = ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Fract_le_zero_iff % 0.31/0.60 thf(fact_5120_Fract__le__one__iff,axiom, % 0.31/0.60 ! [B2: int,A2: int] : % 0.31/0.60 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.31/0.60 => ( ( ord_less_eq @ rat @ ( fract @ A2 @ B2 ) @ ( one_one @ rat ) ) % 0.31/0.60 = ( ord_less_eq @ int @ A2 @ B2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Fract_le_one_iff % 0.31/0.60 thf(fact_5121_one__le__Fract__iff,axiom, % 0.31/0.60 ! [B2: int,A2: int] : % 0.31/0.60 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 0.31/0.60 => ( ( ord_less_eq @ rat @ ( one_one @ rat ) @ ( fract @ A2 @ B2 ) ) % 0.31/0.60 = ( ord_less_eq @ int @ B2 @ A2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % one_le_Fract_iff % 0.31/0.60 thf(fact_5122_image__mult__atLeastAtMost__if,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_field @ A ) % 0.31/0.60 => ! [C2: A,X3: A,Y: A] : % 0.31/0.60 ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.31/0.60 => ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X3 ) @ ( times_times @ A @ C2 @ Y ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.31/0.60 => ( ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.60 => ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y ) @ ( times_times @ A @ C2 @ X3 ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.60 => ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_mult_atLeastAtMost_if % 0.31/0.60 thf(fact_5123_image__mult__atLeastAtMost__if_H,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_field @ A ) % 0.31/0.60 => ! [X3: A,Y: A,C2: A] : % 0.31/0.60 ( ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.60 => ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : ( times_times @ A @ X @ C2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X3 @ C2 ) @ ( times_times @ A @ Y @ C2 ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : ( times_times @ A @ X @ C2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C2 ) @ ( times_times @ A @ X3 @ C2 ) ) ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : ( times_times @ A @ X @ C2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_mult_atLeastAtMost_if' % 0.31/0.60 thf(fact_5124_image__affinity__atLeastAtMost,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_field @ A ) % 0.31/0.60 => ! [A2: A,B2: A,M: A,C2: A] : % 0.31/0.60 ( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.60 & ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 0.31/0.60 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_affinity_atLeastAtMost % 0.31/0.60 thf(fact_5125_image__affinity__atLeastAtMost__diff,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_field @ A ) % 0.31/0.60 => ! [A2: A,B2: A,M: A,C2: A] : % 0.31/0.60 ( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.60 & ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 0.31/0.60 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_affinity_atLeastAtMost_diff % 0.31/0.60 thf(fact_5126_image__affinity__atLeastAtMost__div,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_field @ A ) % 0.31/0.60 => ! [A2: A,B2: A,M: A,C2: A] : % 0.31/0.60 ( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.60 & ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 0.31/0.60 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_affinity_atLeastAtMost_div % 0.31/0.60 thf(fact_5127_image__affinity__atLeastAtMost__div__diff,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( linordered_field @ A ) % 0.31/0.60 => ! [A2: A,B2: A,M: A,C2: A] : % 0.31/0.60 ( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.60 & ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 0.31/0.60 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) ) % 0.31/0.60 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 0.31/0.60 => ( ( image @ A @ A % 0.31/0.60 @ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 ) % 0.31/0.60 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.60 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % image_affinity_atLeastAtMost_div_diff % 0.31/0.60 thf(fact_5128_sum__fun__comp,axiom, % 0.31/0.60 ! [A: $tType,C: $tType,B: $tType] : % 0.31/0.60 ( ( semiring_1 @ C ) % 0.31/0.60 => ! [S3: set @ A,R: set @ B,G: A > B,F2: B > C] : % 0.31/0.60 ( ( finite_finite2 @ A @ S3 ) % 0.31/0.60 => ( ( finite_finite2 @ B @ R ) % 0.31/0.60 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ G @ S3 ) @ R ) % 0.31/0.60 => ( ( groups7311177749621191930dd_sum @ A @ C % 0.31/0.60 @ ^ [X: A] : ( F2 @ ( G @ X ) ) % 0.31/0.60 @ S3 ) % 0.31/0.60 = ( groups7311177749621191930dd_sum @ B @ C % 0.31/0.60 @ ^ [Y2: B] : % 0.31/0.60 ( times_times @ C % 0.31/0.60 @ ( semiring_1_of_nat @ C % 0.31/0.60 @ ( finite_card @ A % 0.31/0.60 @ ( collect @ A % 0.31/0.60 @ ^ [X: A] : % 0.31/0.60 ( ( member @ A @ X @ S3 ) % 0.31/0.60 & ( ( G @ X ) % 0.31/0.60 = Y2 ) ) ) ) ) % 0.31/0.60 @ ( F2 @ Y2 ) ) % 0.31/0.60 @ R ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sum_fun_comp % 0.31/0.60 thf(fact_5129_INF__nat__binary,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( counta3822494911875563373attice @ A ) % 0.31/0.60 => ! [A3: A,B3: A] : % 0.31/0.60 ( ( inf_inf @ A @ A3 % 0.31/0.60 @ ( complete_Inf_Inf @ A % 0.31/0.60 @ ( image @ nat @ A % 0.31/0.60 @ ^ [X: nat] : B3 % 0.31/0.60 @ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) ) % 0.31/0.60 = ( inf_inf @ A @ A3 @ B3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_nat_binary % 0.31/0.60 thf(fact_5130_empty__natural,axiom, % 0.31/0.60 ! [C: $tType,B: $tType,D: $tType,A: $tType,F2: A > C,G: D > B] : % 0.31/0.60 ( ( comp @ C @ ( set @ B ) @ A % 0.31/0.60 @ ^ [Uu3: C] : ( bot_bot @ ( set @ B ) ) % 0.31/0.60 @ F2 ) % 0.31/0.60 = ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image @ D @ B @ G ) % 0.31/0.60 @ ^ [Uu3: A] : ( bot_bot @ ( set @ D ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % empty_natural % 0.31/0.60 thf(fact_5131_listrel__def,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( listrel @ A @ B ) % 0.31/0.60 = ( ^ [R5: set @ ( product_prod @ A @ B )] : % 0.31/0.60 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) % 0.31/0.60 @ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o % 0.31/0.60 @ ( listrelp @ A @ B % 0.31/0.60 @ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R5 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % listrel_def % 0.31/0.60 thf(fact_5132_pair__imageI,axiom, % 0.31/0.60 ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set @ ( product_prod @ A @ B ),F2: A > B > C] : % 0.31/0.60 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A3 ) % 0.31/0.60 => ( member @ C @ ( F2 @ A2 @ B2 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F2 ) @ A3 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % pair_imageI % 0.31/0.60 thf(fact_5133_Gcd__abs__eq,axiom, % 0.31/0.60 ! [K5: set @ int] : % 0.31/0.60 ( ( gcd_Gcd @ int @ ( image @ int @ int @ ( abs_abs @ int ) @ K5 ) ) % 0.31/0.60 = ( gcd_Gcd @ int @ K5 ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_abs_eq % 0.31/0.60 thf(fact_5134_finite__UN,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] : % 0.31/0.60 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.60 => ( ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) % 0.31/0.60 = ( ! [X: A] : % 0.31/0.60 ( ( member @ A @ X @ A3 ) % 0.31/0.60 => ( finite_finite2 @ B @ ( B3 @ X ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % finite_UN % 0.31/0.60 thf(fact_5135_UN__constant,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A3: set @ B,C2: set @ A] : % 0.31/0.60 ( ( ( A3 % 0.31/0.60 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.60 @ ( image @ B @ ( set @ A ) % 0.31/0.60 @ ^ [Y2: B] : C2 % 0.31/0.60 @ A3 ) ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.60 & ( ( A3 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.60 @ ( image @ B @ ( set @ A ) % 0.31/0.60 @ ^ [Y2: B] : C2 % 0.31/0.60 @ A3 ) ) % 0.31/0.60 = C2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % UN_constant % 0.31/0.60 thf(fact_5136_finite__UN__I,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] : % 0.31/0.60 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.60 => ( ! [A5: A] : % 0.31/0.60 ( ( member @ A @ A5 @ A3 ) % 0.31/0.60 => ( finite_finite2 @ B @ ( B3 @ A5 ) ) ) % 0.31/0.60 => ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % finite_UN_I % 0.31/0.60 thf(fact_5137_finite__INT,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B )] : % 0.31/0.60 ( ? [X2: A] : % 0.31/0.60 ( ( member @ A @ X2 @ I6 ) % 0.31/0.60 & ( finite_finite2 @ B @ ( A3 @ X2 ) ) ) % 0.31/0.60 => ( finite_finite2 @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % finite_INT % 0.31/0.60 thf(fact_5138_Gcd__int__eq,axiom, % 0.31/0.60 ! [N7: set @ nat] : % 0.31/0.60 ( ( gcd_Gcd @ int @ ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ N7 ) ) % 0.31/0.60 = ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ N7 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_int_eq % 0.31/0.60 thf(fact_5139_UN__simps_I1_J,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,C5: set @ B,A2: A,B3: B > ( set @ A )] : % 0.31/0.60 ( ( ( C5 % 0.31/0.60 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.60 @ ( image @ B @ ( set @ A ) % 0.31/0.60 @ ^ [X: B] : ( insert @ A @ A2 @ ( B3 @ X ) ) % 0.31/0.60 @ C5 ) ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.60 & ( ( C5 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.60 @ ( image @ B @ ( set @ A ) % 0.31/0.60 @ ^ [X: B] : ( insert @ A @ A2 @ ( B3 @ X ) ) % 0.31/0.60 @ C5 ) ) % 0.31/0.60 = ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ C5 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % UN_simps(1) % 0.31/0.60 thf(fact_5140_UN__singleton,axiom, % 0.31/0.60 ! [A: $tType,A3: set @ A] : % 0.31/0.60 ( ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.60 @ ( image @ A @ ( set @ A ) % 0.31/0.60 @ ^ [X: A] : ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 @ A3 ) ) % 0.31/0.60 = A3 ) ). % 0.31/0.60 % 0.31/0.60 % UN_singleton % 0.31/0.60 thf(fact_5141_set__concat,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ ( list @ A )] : % 0.31/0.60 ( ( set2 @ A @ ( concat @ A @ Xs ) ) % 0.31/0.60 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xs ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % set_concat % 0.31/0.60 thf(fact_5142_Gcd__nat__abs__eq,axiom, % 0.31/0.60 ! [K5: set @ int] : % 0.31/0.60 ( ( gcd_Gcd @ nat % 0.31/0.60 @ ( image @ int @ nat % 0.31/0.60 @ ^ [K3: int] : ( nat2 @ ( abs_abs @ int @ K3 ) ) % 0.31/0.60 @ K5 ) ) % 0.31/0.60 = ( nat2 @ ( gcd_Gcd @ int @ K5 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_nat_abs_eq % 0.31/0.60 thf(fact_5143_SUP__UN__eq2,axiom, % 0.31/0.60 ! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] : % 0.31/0.60 ( ( complete_Sup_Sup @ ( A > B > $o ) % 0.31/0.60 @ ( image @ C @ ( A > B > $o ) % 0.31/0.60 @ ^ [I3: C,X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( R2 @ I3 ) ) % 0.31/0.60 @ S3 ) ) % 0.31/0.60 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_UN_eq2 % 0.31/0.60 thf(fact_5144_INF__INT__eq2,axiom, % 0.31/0.60 ! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] : % 0.31/0.60 ( ( complete_Inf_Inf @ ( A > B > $o ) % 0.31/0.60 @ ( image @ C @ ( A > B > $o ) % 0.31/0.60 @ ^ [I3: C,X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( R2 @ I3 ) ) % 0.31/0.60 @ S3 ) ) % 0.31/0.60 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_INT_eq2 % 0.31/0.60 thf(fact_5145_INF__Int__eq2,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,S3: set @ ( set @ ( product_prod @ A @ B ) )] : % 0.31/0.60 ( ( complete_Inf_Inf @ ( A > B > $o ) % 0.31/0.60 @ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o ) % 0.31/0.60 @ ^ [I3: set @ ( product_prod @ A @ B ),X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ I3 ) % 0.31/0.60 @ S3 ) ) % 0.31/0.60 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_Int_eq2 % 0.31/0.60 thf(fact_5146_SUP__Sup__eq2,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,S3: set @ ( set @ ( product_prod @ A @ B ) )] : % 0.31/0.60 ( ( complete_Sup_Sup @ ( A > B > $o ) % 0.31/0.60 @ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o ) % 0.31/0.60 @ ^ [I3: set @ ( product_prod @ A @ B ),X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ I3 ) % 0.31/0.60 @ S3 ) ) % 0.31/0.60 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % SUP_Sup_eq2 % 0.31/0.60 thf(fact_5147_Sup__SUP__eq2,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( complete_Sup_Sup @ ( A > B > $o ) ) % 0.31/0.60 = ( ^ [S7: set @ ( A > B > $o ),X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S7 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Sup_SUP_eq2 % 0.31/0.60 thf(fact_5148_Inf__INT__eq2,axiom, % 0.31/0.60 ! [B: $tType,A: $tType] : % 0.31/0.60 ( ( complete_Inf_Inf @ ( A > B > $o ) ) % 0.31/0.60 = ( ^ [S7: set @ ( A > B > $o ),X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S7 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Inf_INT_eq2 % 0.31/0.60 thf(fact_5149_UN__image__subset,axiom, % 0.31/0.60 ! [C: $tType,A: $tType,B: $tType,F2: B > ( set @ A ),G: C > ( set @ B ),X3: C,X8: set @ A] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ ( G @ X3 ) ) ) @ X8 ) % 0.31/0.60 = ( ord_less_eq @ ( set @ B ) @ ( G @ X3 ) % 0.31/0.60 @ ( collect @ B % 0.31/0.60 @ ^ [X: B] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ X ) @ X8 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % UN_image_subset % 0.31/0.60 thf(fact_5150_listrelp_ONil,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,R2: A > B > $o] : ( listrelp @ A @ B @ R2 @ ( nil @ A ) @ ( nil @ B ) ) ). % 0.31/0.60 % 0.31/0.60 % listrelp.Nil % 0.31/0.60 thf(fact_5151_None__notin__image__Some,axiom, % 0.31/0.60 ! [A: $tType,A3: set @ A] : % 0.31/0.60 ~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ) ). % 0.31/0.60 % 0.31/0.60 % None_notin_image_Some % 0.31/0.60 thf(fact_5152_INF__filter__not__bot,axiom, % 0.31/0.60 ! [I7: $tType,A: $tType,B3: set @ I7,F5: I7 > ( filter @ A )] : % 0.31/0.60 ( ! [X10: set @ I7] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ I7 ) @ X10 @ B3 ) % 0.31/0.60 => ( ( finite_finite2 @ I7 @ X10 ) % 0.31/0.60 => ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ I7 @ ( filter @ A ) @ F5 @ X10 ) ) % 0.31/0.60 != ( bot_bot @ ( filter @ A ) ) ) ) ) % 0.31/0.60 => ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ I7 @ ( filter @ A ) @ F5 @ B3 ) ) % 0.31/0.60 != ( bot_bot @ ( filter @ A ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INF_filter_not_bot % 0.31/0.60 thf(fact_5153_finite__int__iff__bounded,axiom, % 0.31/0.60 ( ( finite_finite2 @ int ) % 0.31/0.60 = ( ^ [S7: set @ int] : % 0.31/0.60 ? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S7 ) @ ( set_ord_lessThan @ int @ K3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % finite_int_iff_bounded % 0.31/0.60 thf(fact_5154_finite__int__iff__bounded__le,axiom, % 0.31/0.60 ( ( finite_finite2 @ int ) % 0.31/0.60 = ( ^ [S7: set @ int] : % 0.31/0.60 ? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S7 ) @ ( set_ord_atMost @ int @ K3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % finite_int_iff_bounded_le % 0.31/0.60 thf(fact_5155_UN__empty2,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A3: set @ B] : % 0.31/0.60 ( ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.60 @ ( image @ B @ ( set @ A ) % 0.31/0.60 @ ^ [X: B] : ( bot_bot @ ( set @ A ) ) % 0.31/0.60 @ A3 ) ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % UN_empty2 % 0.31/0.60 thf(fact_5156_UN__empty,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,B3: B > ( set @ A )] : % 0.31/0.60 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % UN_empty % 0.31/0.60 thf(fact_5157_UNION__empty__conv_I1_J,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B] : % 0.31/0.60 ( ( ( bot_bot @ ( set @ A ) ) % 0.31/0.60 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) ) % 0.31/0.60 = ( ! [X: B] : % 0.31/0.60 ( ( member @ B @ X @ A3 ) % 0.31/0.60 => ( ( B3 @ X ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % UNION_empty_conv(1) % 0.31/0.60 thf(fact_5158_UNION__empty__conv_I2_J,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B] : % 0.31/0.60 ( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 = ( ! [X: B] : % 0.31/0.60 ( ( member @ B @ X @ A3 ) % 0.31/0.60 => ( ( B3 @ X ) % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % UNION_empty_conv(2) % 0.31/0.60 thf(fact_5159_UN__subset__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,A3: B > ( set @ A ),I6: set @ B,B3: set @ A] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) @ B3 ) % 0.31/0.60 = ( ! [X: B] : % 0.31/0.60 ( ( member @ B @ X @ I6 ) % 0.31/0.60 => ( ord_less_eq @ ( set @ A ) @ ( A3 @ X ) @ B3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % UN_subset_iff % 0.31/0.60 thf(fact_5160_UN__upper,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A2: A,A3: set @ A,B3: A > ( set @ B )] : % 0.31/0.60 ( ( member @ A @ A2 @ A3 ) % 0.31/0.60 => ( ord_less_eq @ ( set @ B ) @ ( B3 @ A2 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % UN_upper % 0.31/0.60 thf(fact_5161_UN__least,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,A3: set @ A,B3: A > ( set @ B ),C5: set @ B] : % 0.31/0.60 ( ! [X5: A] : % 0.31/0.60 ( ( member @ A @ X5 @ A3 ) % 0.31/0.60 => ( ord_less_eq @ ( set @ B ) @ ( B3 @ X5 ) @ C5 ) ) % 0.31/0.60 => ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) @ C5 ) ) ). % 0.31/0.60 % 0.31/0.60 % UN_least % 0.31/0.60 thf(fact_5162_UN__mono,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,F2: A > ( set @ B ),G: A > ( set @ B )] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.60 => ( ! [X5: A] : % 0.31/0.60 ( ( member @ A @ X5 @ A3 ) % 0.31/0.60 => ( ord_less_eq @ ( set @ B ) @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 0.31/0.60 => ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ B3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % UN_mono % 0.31/0.60 thf(fact_5163_INT__subset__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,B3: set @ A,A3: B > ( set @ A ),I6: set @ B] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) ) % 0.31/0.60 = ( ! [X: B] : % 0.31/0.60 ( ( member @ B @ X @ I6 ) % 0.31/0.60 => ( ord_less_eq @ ( set @ A ) @ B3 @ ( A3 @ X ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INT_subset_iff % 0.31/0.60 thf(fact_5164_INT__anti__mono,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,F2: A > ( set @ B ),G: A > ( set @ B )] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.60 => ( ! [X5: A] : % 0.31/0.60 ( ( member @ A @ X5 @ A3 ) % 0.31/0.60 => ( ord_less_eq @ ( set @ B ) @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 0.31/0.60 => ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ B3 ) ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ A3 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INT_anti_mono % 0.31/0.60 thf(fact_5165_INT__greatest,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A3: set @ A,C5: set @ B,B3: A > ( set @ B )] : % 0.31/0.60 ( ! [X5: A] : % 0.31/0.60 ( ( member @ A @ X5 @ A3 ) % 0.31/0.60 => ( ord_less_eq @ ( set @ B ) @ C5 @ ( B3 @ X5 ) ) ) % 0.31/0.60 => ( ord_less_eq @ ( set @ B ) @ C5 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INT_greatest % 0.31/0.60 thf(fact_5166_INT__lower,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,A2: A,A3: set @ A,B3: A > ( set @ B )] : % 0.31/0.60 ( ( member @ A @ A2 @ A3 ) % 0.31/0.60 => ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) @ ( B3 @ A2 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INT_lower % 0.31/0.60 thf(fact_5167_in__image__insert__iff,axiom, % 0.31/0.60 ! [A: $tType,B3: set @ ( set @ A ),X3: A,A3: set @ A] : % 0.31/0.60 ( ! [C8: set @ A] : % 0.31/0.60 ( ( member @ ( set @ A ) @ C8 @ B3 ) % 0.31/0.60 => ~ ( member @ A @ X3 @ C8 ) ) % 0.31/0.60 => ( ( member @ ( set @ A ) @ A3 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X3 ) @ B3 ) ) % 0.31/0.60 = ( ( member @ A @ X3 @ A3 ) % 0.31/0.60 & ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % in_image_insert_iff % 0.31/0.60 thf(fact_5168_UN__extend__simps_I1_J,axiom, % 0.31/0.60 ! [A: $tType,B: $tType,C5: set @ B,A2: A,B3: B > ( set @ A )] : % 0.31/0.60 ( ( ( C5 % 0.31/0.60 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ C5 ) ) ) % 0.31/0.60 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.60 & ( ( C5 % 0.31/0.60 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.60 => ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ C5 ) ) ) % 0.31/0.60 = ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.60 @ ( image @ B @ ( set @ A ) % 0.31/0.60 @ ^ [X: B] : ( insert @ A @ A2 @ ( B3 @ X ) ) % 0.31/0.60 @ C5 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % UN_extend_simps(1) % 0.31/0.60 thf(fact_5169_INT__extend__simps_I2_J,axiom, % 0.31/0.60 ! [C: $tType,D: $tType,C5: set @ D,A3: set @ C,B3: D > ( set @ C )] : % 0.31/0.60 ( ( ( C5 % 0.31/0.60 = ( bot_bot @ ( set @ D ) ) ) % 0.31/0.60 => ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D @ ( set @ C ) @ B3 @ C5 ) ) ) % 0.31/0.60 = A3 ) ) % 0.31/0.60 & ( ( C5 % 0.31/0.60 != ( bot_bot @ ( set @ D ) ) ) % 0.31/0.60 => ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D @ ( set @ C ) @ B3 @ C5 ) ) ) % 0.31/0.60 = ( complete_Inf_Inf @ ( set @ C ) % 0.31/0.60 @ ( image @ D @ ( set @ C ) % 0.31/0.60 @ ^ [X: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B3 @ X ) ) % 0.31/0.60 @ C5 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INT_extend_simps(2) % 0.31/0.60 thf(fact_5170_INT__extend__simps_I1_J,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,C5: set @ A,A3: A > ( set @ B ),B3: set @ B] : % 0.31/0.60 ( ( ( C5 % 0.31/0.60 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 => ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ C5 ) ) @ B3 ) % 0.31/0.60 = B3 ) ) % 0.31/0.60 & ( ( C5 % 0.31/0.60 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.60 => ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ C5 ) ) @ B3 ) % 0.31/0.60 = ( complete_Inf_Inf @ ( set @ B ) % 0.31/0.60 @ ( image @ A @ ( set @ B ) % 0.31/0.60 @ ^ [X: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X ) @ B3 ) % 0.31/0.60 @ C5 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INT_extend_simps(1) % 0.31/0.60 thf(fact_5171_Int__Inter__eq_I2_J,axiom, % 0.31/0.60 ! [A: $tType,B12: set @ ( set @ A ),A3: set @ A] : % 0.31/0.60 ( ( ( B12 % 0.31/0.60 = ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 0.31/0.60 => ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) @ A3 ) % 0.31/0.60 = A3 ) ) % 0.31/0.60 & ( ( B12 % 0.31/0.60 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 0.31/0.60 => ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) @ A3 ) % 0.31/0.60 = ( complete_Inf_Inf @ ( set @ A ) % 0.31/0.60 @ ( image @ ( set @ A ) @ ( set @ A ) % 0.31/0.60 @ ^ [B6: set @ A] : ( inf_inf @ ( set @ A ) @ B6 @ A3 ) % 0.31/0.60 @ B12 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Int_Inter_eq(2) % 0.31/0.60 thf(fact_5172_Int__Inter__eq_I1_J,axiom, % 0.31/0.60 ! [A: $tType,B12: set @ ( set @ A ),A3: set @ A] : % 0.31/0.60 ( ( ( B12 % 0.31/0.60 = ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 0.31/0.60 => ( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) ) % 0.31/0.60 = A3 ) ) % 0.31/0.60 & ( ( B12 % 0.31/0.60 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 0.31/0.60 => ( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) ) % 0.31/0.60 = ( complete_Inf_Inf @ ( set @ A ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 ) @ B12 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Int_Inter_eq(1) % 0.31/0.60 thf(fact_5173_Collect__split__mono__strong,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,X8: set @ A,A3: set @ ( product_prod @ A @ B ),Y7: set @ B,P: A > B > $o,Q: A > B > $o] : % 0.31/0.60 ( ( X8 % 0.31/0.60 = ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) ) % 0.31/0.60 => ( ( Y7 % 0.31/0.60 = ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) ) % 0.31/0.60 => ( ! [X5: A] : % 0.31/0.60 ( ( member @ A @ X5 @ X8 ) % 0.31/0.60 => ! [Xa3: B] : % 0.31/0.60 ( ( member @ B @ Xa3 @ Y7 ) % 0.31/0.60 => ( ( P @ X5 @ Xa3 ) % 0.31/0.60 => ( Q @ X5 @ Xa3 ) ) ) ) % 0.31/0.60 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) ) % 0.31/0.60 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Collect_split_mono_strong % 0.31/0.60 thf(fact_5174_INT__extend__simps_I4_J,axiom, % 0.31/0.60 ! [G3: $tType,H4: $tType,C5: set @ H4,A3: set @ G3,B3: H4 > ( set @ G3 )] : % 0.31/0.60 ( ( ( C5 % 0.31/0.60 = ( bot_bot @ ( set @ H4 ) ) ) % 0.31/0.60 => ( ( minus_minus @ ( set @ G3 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image @ H4 @ ( set @ G3 ) @ B3 @ C5 ) ) ) % 0.31/0.60 = A3 ) ) % 0.31/0.60 & ( ( C5 % 0.31/0.60 != ( bot_bot @ ( set @ H4 ) ) ) % 0.31/0.60 => ( ( minus_minus @ ( set @ G3 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image @ H4 @ ( set @ G3 ) @ B3 @ C5 ) ) ) % 0.31/0.60 = ( complete_Inf_Inf @ ( set @ G3 ) % 0.31/0.60 @ ( image @ H4 @ ( set @ G3 ) % 0.31/0.60 @ ^ [X: H4] : ( minus_minus @ ( set @ G3 ) @ A3 @ ( B3 @ X ) ) % 0.31/0.60 @ C5 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INT_extend_simps(4) % 0.31/0.60 thf(fact_5175_infinite__int__iff__infinite__nat__abs,axiom, % 0.31/0.60 ! [S3: set @ int] : % 0.31/0.60 ( ( ~ ( finite_finite2 @ int @ S3 ) ) % 0.31/0.60 = ( ~ ( finite_finite2 @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ S3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % infinite_int_iff_infinite_nat_abs % 0.31/0.60 thf(fact_5176_UN__le__add__shift__strict,axiom, % 0.31/0.60 ! [A: $tType,M7: nat > ( set @ A ),K: nat,N2: nat] : % 0.31/0.60 ( ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.60 @ ( image @ nat @ ( set @ A ) % 0.31/0.60 @ ^ [I3: nat] : ( M7 @ ( plus_plus @ nat @ I3 @ K ) ) % 0.31/0.60 @ ( set_ord_lessThan @ nat @ N2 ) ) ) % 0.31/0.60 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ N2 @ K ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % UN_le_add_shift_strict % 0.31/0.60 thf(fact_5177_UN__le__add__shift,axiom, % 0.31/0.60 ! [A: $tType,M7: nat > ( set @ A ),K: nat,N2: nat] : % 0.31/0.60 ( ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.60 @ ( image @ nat @ ( set @ A ) % 0.31/0.60 @ ^ [I3: nat] : ( M7 @ ( plus_plus @ nat @ I3 @ K ) ) % 0.31/0.60 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 0.31/0.60 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or1337092689740270186AtMost @ nat @ K @ ( plus_plus @ nat @ N2 @ K ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % UN_le_add_shift % 0.31/0.60 thf(fact_5178_subset__subseqs,axiom, % 0.31/0.60 ! [A: $tType,X8: set @ A,Xs: list @ A] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ X8 @ ( set2 @ A @ Xs ) ) % 0.31/0.60 => ( member @ ( set @ A ) @ X8 @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % subset_subseqs % 0.31/0.60 thf(fact_5179_subseqs__powset,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A] : % 0.31/0.60 ( ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) % 0.31/0.60 = ( pow2 @ A @ ( set2 @ A @ Xs ) ) ) ). % 0.31/0.60 % 0.31/0.60 % subseqs_powset % 0.31/0.60 thf(fact_5180_conj__subset__def,axiom, % 0.31/0.60 ! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] : % 0.31/0.60 ( ( ord_less_eq @ ( set @ A ) @ A3 % 0.31/0.60 @ ( collect @ A % 0.31/0.60 @ ^ [X: A] : % 0.31/0.60 ( ( P @ X ) % 0.31/0.60 & ( Q @ X ) ) ) ) % 0.31/0.60 = ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) % 0.31/0.60 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ Q ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % conj_subset_def % 0.31/0.60 thf(fact_5181_finite__Inf__Sup,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( finite8700451911770168679attice @ A ) % 0.31/0.60 => ! [A3: set @ ( set @ A )] : % 0.31/0.60 ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A3 ) ) % 0.31/0.60 @ ( complete_Sup_Sup @ A % 0.31/0.60 @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A ) % 0.31/0.60 @ ( collect @ ( set @ A ) % 0.31/0.60 @ ^ [Uu3: set @ A] : % 0.31/0.60 ? [F3: ( set @ A ) > A] : % 0.31/0.60 ( ( Uu3 % 0.31/0.60 = ( image @ ( set @ A ) @ A @ F3 @ A3 ) ) % 0.31/0.60 & ! [X: set @ A] : % 0.31/0.60 ( ( member @ ( set @ A ) @ X @ A3 ) % 0.31/0.60 => ( member @ A @ ( F3 @ X ) @ X ) ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % finite_Inf_Sup % 0.31/0.60 thf(fact_5182_Inf__Sup__le,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( comple592849572758109894attice @ A ) % 0.31/0.60 => ! [A3: set @ ( set @ A )] : % 0.31/0.60 ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A3 ) ) % 0.31/0.60 @ ( complete_Sup_Sup @ A % 0.31/0.60 @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A ) % 0.31/0.60 @ ( collect @ ( set @ A ) % 0.31/0.60 @ ^ [Uu3: set @ A] : % 0.31/0.60 ? [F3: ( set @ A ) > A] : % 0.31/0.60 ( ( Uu3 % 0.31/0.60 = ( image @ ( set @ A ) @ A @ F3 @ A3 ) ) % 0.31/0.60 & ! [X: set @ A] : % 0.31/0.60 ( ( member @ ( set @ A ) @ X @ A3 ) % 0.31/0.60 => ( member @ A @ ( F3 @ X ) @ X ) ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Inf_Sup_le % 0.31/0.60 thf(fact_5183_Sup__Inf__le,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( comple6319245703460814977attice @ A ) % 0.31/0.60 => ! [A3: set @ ( set @ A )] : % 0.31/0.60 ( ord_less_eq @ A % 0.31/0.60 @ ( complete_Sup_Sup @ A % 0.31/0.60 @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A ) % 0.31/0.60 @ ( collect @ ( set @ A ) % 0.31/0.60 @ ^ [Uu3: set @ A] : % 0.31/0.60 ? [F3: ( set @ A ) > A] : % 0.31/0.60 ( ( Uu3 % 0.31/0.60 = ( image @ ( set @ A ) @ A @ F3 @ A3 ) ) % 0.31/0.60 & ! [X: set @ A] : % 0.31/0.60 ( ( member @ ( set @ A ) @ X @ A3 ) % 0.31/0.60 => ( member @ A @ ( F3 @ X ) @ X ) ) ) ) ) ) % 0.31/0.60 @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A3 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Sup_Inf_le % 0.31/0.60 thf(fact_5184_sum_OUNION__disjoint,axiom, % 0.31/0.60 ! [A: $tType,C: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_add @ A ) % 0.31/0.60 => ! [I6: set @ B,A3: B > ( set @ C ),G: C > A] : % 0.31/0.60 ( ( finite_finite2 @ B @ I6 ) % 0.31/0.60 => ( ! [X5: B] : % 0.31/0.60 ( ( member @ B @ X5 @ I6 ) % 0.31/0.60 => ( finite_finite2 @ C @ ( A3 @ X5 ) ) ) % 0.31/0.60 => ( ! [X5: B] : % 0.31/0.60 ( ( member @ B @ X5 @ I6 ) % 0.31/0.60 => ! [Xa3: B] : % 0.31/0.60 ( ( member @ B @ Xa3 @ I6 ) % 0.31/0.60 => ( ( X5 != Xa3 ) % 0.31/0.60 => ( ( inf_inf @ ( set @ C ) @ ( A3 @ X5 ) @ ( A3 @ Xa3 ) ) % 0.31/0.60 = ( bot_bot @ ( set @ C ) ) ) ) ) ) % 0.31/0.60 => ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ A3 @ I6 ) ) ) % 0.31/0.60 = ( groups7311177749621191930dd_sum @ B @ A % 0.31/0.60 @ ^ [X: B] : ( groups7311177749621191930dd_sum @ C @ A @ G @ ( A3 @ X ) ) % 0.31/0.60 @ I6 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % sum.UNION_disjoint % 0.31/0.60 thf(fact_5185_prod_OUNION__disjoint,axiom, % 0.31/0.60 ! [A: $tType,C: $tType,B: $tType] : % 0.31/0.60 ( ( comm_monoid_mult @ A ) % 0.31/0.60 => ! [I6: set @ B,A3: B > ( set @ C ),G: C > A] : % 0.31/0.60 ( ( finite_finite2 @ B @ I6 ) % 0.31/0.60 => ( ! [X5: B] : % 0.31/0.60 ( ( member @ B @ X5 @ I6 ) % 0.31/0.60 => ( finite_finite2 @ C @ ( A3 @ X5 ) ) ) % 0.31/0.60 => ( ! [X5: B] : % 0.31/0.60 ( ( member @ B @ X5 @ I6 ) % 0.31/0.60 => ! [Xa3: B] : % 0.31/0.60 ( ( member @ B @ Xa3 @ I6 ) % 0.31/0.60 => ( ( X5 != Xa3 ) % 0.31/0.60 => ( ( inf_inf @ ( set @ C ) @ ( A3 @ X5 ) @ ( A3 @ Xa3 ) ) % 0.31/0.60 = ( bot_bot @ ( set @ C ) ) ) ) ) ) % 0.31/0.60 => ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ A3 @ I6 ) ) ) % 0.31/0.60 = ( groups7121269368397514597t_prod @ B @ A % 0.31/0.60 @ ^ [X: B] : ( groups7121269368397514597t_prod @ C @ A @ G @ ( A3 @ X ) ) % 0.31/0.60 @ I6 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % prod.UNION_disjoint % 0.31/0.60 thf(fact_5186_card__UN__le,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B )] : % 0.31/0.60 ( ( finite_finite2 @ A @ I6 ) % 0.31/0.60 => ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) % 0.31/0.60 @ ( groups7311177749621191930dd_sum @ A @ nat % 0.31/0.60 @ ^ [I3: A] : ( finite_card @ B @ ( A3 @ I3 ) ) % 0.31/0.60 @ I6 ) ) ) ). % 0.31/0.60 % 0.31/0.60 % card_UN_le % 0.31/0.60 thf(fact_5187_listrelp__listrel__eq,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] : % 0.31/0.60 ( ( listrelp @ A @ B % 0.31/0.60 @ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R2 ) ) % 0.31/0.60 = ( ^ [X: list @ A,Y2: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X @ Y2 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % listrelp_listrel_eq % 0.31/0.60 thf(fact_5188_Gcd__int__def,axiom, % 0.31/0.60 ( ( gcd_Gcd @ int ) % 0.31/0.60 = ( ^ [K6: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K6 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Gcd_int_def % 0.31/0.60 thf(fact_5189_card__UN__disjoint,axiom, % 0.31/0.60 ! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B )] : % 0.31/0.60 ( ( finite_finite2 @ A @ I6 ) % 0.31/0.60 => ( ! [X5: A] : % 0.31/0.60 ( ( member @ A @ X5 @ I6 ) % 0.31/0.60 => ( finite_finite2 @ B @ ( A3 @ X5 ) ) ) % 0.31/0.60 => ( ! [X5: A] : % 0.31/0.60 ( ( member @ A @ X5 @ I6 ) % 0.31/0.60 => ! [Xa3: A] : % 0.31/0.60 ( ( member @ A @ Xa3 @ I6 ) % 0.31/0.60 => ( ( X5 != Xa3 ) % 0.31/0.60 => ( ( inf_inf @ ( set @ B ) @ ( A3 @ X5 ) @ ( A3 @ Xa3 ) ) % 0.31/0.60 = ( bot_bot @ ( set @ B ) ) ) ) ) ) % 0.31/0.60 => ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) % 0.31/0.60 = ( groups7311177749621191930dd_sum @ A @ nat % 0.31/0.60 @ ^ [I3: A] : ( finite_card @ B @ ( A3 @ I3 ) ) % 0.31/0.60 @ I6 ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % card_UN_disjoint % 0.31/0.60 thf(fact_5190_finite__mono__strict__prefix__implies__finite__fixpoint,axiom, % 0.31/0.60 ! [A: $tType,F2: nat > ( set @ A ),S3: set @ A] : % 0.31/0.60 ( ! [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ I4 ) @ S3 ) % 0.31/0.60 => ( ( finite_finite2 @ A @ S3 ) % 0.31/0.60 => ( ? [N8: nat] : % 0.31/0.60 ( ! [N3: nat] : % 0.31/0.60 ( ( ord_less_eq @ nat @ N3 @ N8 ) % 0.31/0.60 => ! [M3: nat] : % 0.31/0.60 ( ( ord_less_eq @ nat @ M3 @ N8 ) % 0.31/0.60 => ( ( ord_less @ nat @ M3 @ N3 ) % 0.31/0.60 => ( ord_less @ ( set @ A ) @ ( F2 @ M3 ) @ ( F2 @ N3 ) ) ) ) ) % 0.31/0.60 & ! [N3: nat] : % 0.31/0.60 ( ( ord_less_eq @ nat @ N8 @ N3 ) % 0.31/0.60 => ( ( F2 @ N8 ) % 0.31/0.60 = ( F2 @ N3 ) ) ) ) % 0.31/0.60 => ( ( F2 @ ( finite_card @ A @ S3 ) ) % 0.31/0.60 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ F2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % finite_mono_strict_prefix_implies_finite_fixpoint % 0.31/0.60 thf(fact_5191_length__remdups__concat,axiom, % 0.31/0.60 ! [A: $tType,Xss: list @ ( list @ A )] : % 0.31/0.60 ( ( size_size @ ( list @ A ) @ ( remdups @ A @ ( concat @ A @ Xss ) ) ) % 0.31/0.60 = ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xss ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % length_remdups_concat % 0.31/0.60 thf(fact_5192_INT__simps_I4_J,axiom, % 0.31/0.60 ! [G3: $tType,H4: $tType,C5: set @ H4,A3: set @ G3,B3: H4 > ( set @ G3 )] : % 0.31/0.60 ( ( ( C5 % 0.31/0.60 = ( bot_bot @ ( set @ H4 ) ) ) % 0.31/0.60 => ( ( complete_Inf_Inf @ ( set @ G3 ) % 0.31/0.60 @ ( image @ H4 @ ( set @ G3 ) % 0.31/0.60 @ ^ [X: H4] : ( minus_minus @ ( set @ G3 ) @ A3 @ ( B3 @ X ) ) % 0.31/0.60 @ C5 ) ) % 0.31/0.60 = ( top_top @ ( set @ G3 ) ) ) ) % 0.31/0.60 & ( ( C5 % 0.31/0.60 != ( bot_bot @ ( set @ H4 ) ) ) % 0.31/0.60 => ( ( complete_Inf_Inf @ ( set @ G3 ) % 0.31/0.60 @ ( image @ H4 @ ( set @ G3 ) % 0.31/0.60 @ ^ [X: H4] : ( minus_minus @ ( set @ G3 ) @ A3 @ ( B3 @ X ) ) % 0.31/0.60 @ C5 ) ) % 0.31/0.60 = ( minus_minus @ ( set @ G3 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image @ H4 @ ( set @ G3 ) @ B3 @ C5 ) ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % INT_simps(4) % 0.31/0.60 thf(fact_5193_top__apply,axiom, % 0.31/0.60 ! [C: $tType,D: $tType] : % 0.31/0.60 ( ( top @ C ) % 0.31/0.60 => ( ( top_top @ ( D > C ) ) % 0.31/0.60 = ( ^ [X: D] : ( top_top @ C ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % top_apply % 0.31/0.60 thf(fact_5194_UNIV__I,axiom, % 0.31/0.60 ! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % UNIV_I % 0.31/0.60 thf(fact_5195_finite__option__UNIV,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) ) % 0.31/0.60 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % finite_option_UNIV % 0.31/0.60 thf(fact_5196_finite__Plus__UNIV__iff,axiom, % 0.31/0.60 ! [A: $tType,B: $tType] : % 0.31/0.60 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) % 0.31/0.60 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) % 0.31/0.60 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % finite_Plus_UNIV_iff % 0.31/0.60 thf(fact_5197_Int__UNIV,axiom, % 0.31/0.60 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.60 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.60 = ( top_top @ ( set @ A ) ) ) % 0.31/0.60 = ( ( A3 % 0.31/0.60 = ( top_top @ ( set @ A ) ) ) % 0.31/0.60 & ( B3 % 0.31/0.60 = ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Int_UNIV % 0.31/0.60 thf(fact_5198_range__mult,axiom, % 0.31/0.60 ! [A2: real] : % 0.31/0.60 ( ( ( A2 % 0.31/0.60 = ( zero_zero @ real ) ) % 0.31/0.60 => ( ( image @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) ) % 0.31/0.60 = ( insert @ real @ ( zero_zero @ real ) @ ( bot_bot @ ( set @ real ) ) ) ) ) % 0.31/0.60 & ( ( A2 % 0.31/0.60 != ( zero_zero @ real ) ) % 0.31/0.60 => ( ( image @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) ) % 0.31/0.60 = ( top_top @ ( set @ real ) ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % range_mult % 0.31/0.60 thf(fact_5199_max__top,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( order_top @ A ) % 0.31/0.60 => ! [X3: A] : % 0.31/0.60 ( ( ord_max @ A @ ( top_top @ A ) @ X3 ) % 0.31/0.60 = ( top_top @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % max_top % 0.31/0.60 thf(fact_5200_max__top2,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( order_top @ A ) % 0.31/0.60 => ! [X3: A] : % 0.31/0.60 ( ( ord_max @ A @ X3 @ ( top_top @ A ) ) % 0.31/0.60 = ( top_top @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % max_top2 % 0.31/0.60 thf(fact_5201_min__top,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( order_top @ A ) % 0.31/0.60 => ! [X3: A] : % 0.31/0.60 ( ( ord_min @ A @ ( top_top @ A ) @ X3 ) % 0.31/0.60 = X3 ) ) ). % 0.31/0.60 % 0.31/0.60 % min_top % 0.31/0.60 thf(fact_5202_min__top2,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( order_top @ A ) % 0.31/0.60 => ! [X3: A] : % 0.31/0.60 ( ( ord_min @ A @ X3 @ ( top_top @ A ) ) % 0.31/0.60 = X3 ) ) ). % 0.31/0.60 % 0.31/0.60 % min_top2 % 0.31/0.60 thf(fact_5203_Pow__UNIV,axiom, % 0.31/0.60 ! [A: $tType] : % 0.31/0.60 ( ( pow2 @ A @ ( top_top @ ( set @ A ) ) ) % 0.31/0.60 = ( top_top @ ( set @ ( set @ A ) ) ) ) ). % 0.31/0.60 % 0.31/0.60 % Pow_UNIV % 0.31/0.60 thf(fact_5204_remdups__eq__nil__right__iff,axiom, % 0.31/0.60 ! [A: $tType,X3: list @ A] : % 0.31/0.60 ( ( ( nil @ A ) % 0.31/0.60 = ( remdups @ A @ X3 ) ) % 0.31/0.60 = ( X3 % 0.31/0.60 = ( nil @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % remdups_eq_nil_right_iff % 0.31/0.60 thf(fact_5205_remdups__eq__nil__iff,axiom, % 0.31/0.60 ! [A: $tType,X3: list @ A] : % 0.31/0.60 ( ( ( remdups @ A @ X3 ) % 0.31/0.60 = ( nil @ A ) ) % 0.31/0.60 = ( X3 % 0.31/0.60 = ( nil @ A ) ) ) ). % 0.31/0.60 % 0.31/0.60 % remdups_eq_nil_iff % 0.31/0.60 thf(fact_5206_set__remdups,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A] : % 0.31/0.60 ( ( set2 @ A @ ( remdups @ A @ Xs ) ) % 0.31/0.60 = ( set2 @ A @ Xs ) ) ). % 0.31/0.60 % 0.31/0.60 % set_remdups % 0.31/0.60 thf(fact_5207_length__remdups__eq,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A] : % 0.31/0.60 ( ( ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) ) % 0.31/0.60 = ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.60 = ( ( remdups @ A @ Xs ) % 0.31/0.60 = Xs ) ) ). % 0.31/0.60 % 0.31/0.60 % length_remdups_eq % 0.31/0.60 thf(fact_5208_remdups__id__iff__distinct,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A] : % 0.31/0.60 ( ( ( remdups @ A @ Xs ) % 0.31/0.60 = Xs ) % 0.31/0.60 = ( distinct @ A @ Xs ) ) ). % 0.31/0.60 % 0.31/0.60 % remdups_id_iff_distinct % 0.31/0.60 thf(fact_5209_distinct__remdups,axiom, % 0.31/0.60 ! [A: $tType,Xs: list @ A] : ( distinct @ A @ ( remdups @ A @ Xs ) ) ). % 0.31/0.60 % 0.31/0.60 % distinct_remdups % 0.31/0.60 thf(fact_5210_Collect__const,axiom, % 0.31/0.60 ! [A: $tType,P: $o] : % 0.31/0.60 ( ( P % 0.31/0.60 => ( ( collect @ A % 0.31/0.60 @ ^ [S8: A] : P ) % 0.31/0.60 = ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 & ( ~ P % 0.31/0.61 => ( ( collect @ A % 0.31/0.61 @ ^ [S8: A] : P ) % 0.31/0.61 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Collect_const % 0.31/0.61 thf(fact_5211_finite__Collect__not,axiom, % 0.31/0.61 ! [A: $tType,P: A > $o] : % 0.31/0.61 ( ( finite_finite2 @ A @ ( collect @ A @ P ) ) % 0.31/0.61 => ( ( finite_finite2 @ A % 0.31/0.61 @ ( collect @ A % 0.31/0.61 @ ^ [X: A] : % 0.31/0.61 ~ ( P @ X ) ) ) % 0.31/0.61 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % finite_Collect_not % 0.31/0.61 thf(fact_5212_range__abs__Nats,axiom, % 0.31/0.61 ( ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) % 0.31/0.61 = ( semiring_1_Nats @ int ) ) ). % 0.31/0.61 % 0.31/0.61 % range_abs_Nats % 0.31/0.61 thf(fact_5213_Collect__const__case__prod,axiom, % 0.31/0.61 ! [B: $tType,A: $tType,P: $o] : % 0.31/0.61 ( ( P % 0.31/0.61 => ( ( collect @ ( product_prod @ A @ B ) % 0.31/0.61 @ ( product_case_prod @ A @ B @ $o % 0.31/0.61 @ ^ [A4: A,B4: B] : P ) ) % 0.31/0.61 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) % 0.31/0.61 & ( ~ P % 0.31/0.61 => ( ( collect @ ( product_prod @ A @ B ) % 0.31/0.61 @ ( product_case_prod @ A @ B @ $o % 0.31/0.61 @ ^ [A4: A,B4: B] : P ) ) % 0.31/0.61 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Collect_const_case_prod % 0.31/0.61 thf(fact_5214_range__add,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( group_add @ A ) % 0.31/0.61 => ! [A2: A] : % 0.31/0.61 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 = ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % range_add % 0.31/0.61 thf(fact_5215_surj__plus,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ab_group_add @ A ) % 0.31/0.61 => ! [A2: A] : % 0.31/0.61 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 = ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % surj_plus % 0.31/0.61 thf(fact_5216_Sup__eq__top__iff,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( comple5582772986160207858norder @ A ) % 0.31/0.61 => ! [A3: set @ A] : % 0.31/0.61 ( ( ( complete_Sup_Sup @ A @ A3 ) % 0.31/0.61 = ( top_top @ A ) ) % 0.31/0.61 = ( ! [X: A] : % 0.31/0.61 ( ( ord_less @ A @ X @ ( top_top @ A ) ) % 0.31/0.61 => ? [Y2: A] : % 0.31/0.61 ( ( member @ A @ Y2 @ A3 ) % 0.31/0.61 & ( ord_less @ A @ X @ Y2 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Sup_eq_top_iff % 0.31/0.61 thf(fact_5217_boolean__algebra_Ocompl__one,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.61 => ( ( uminus_uminus @ A @ ( top_top @ A ) ) % 0.31/0.61 = ( bot_bot @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % boolean_algebra.compl_one % 0.31/0.61 thf(fact_5218_boolean__algebra_Ocompl__zero,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.61 => ( ( uminus_uminus @ A @ ( bot_bot @ A ) ) % 0.31/0.61 = ( top_top @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % boolean_algebra.compl_zero % 0.31/0.61 thf(fact_5219_ball__UNIV,axiom, % 0.31/0.61 ! [A: $tType,P: A > $o] : % 0.31/0.61 ( ( ! [X: A] : % 0.31/0.61 ( ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 => ( P @ X ) ) ) % 0.31/0.61 = ( ! [X6: A] : ( P @ X6 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % ball_UNIV % 0.31/0.61 thf(fact_5220_Inf__UNIV,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( comple6319245703460814977attice @ A ) % 0.31/0.61 => ( ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 = ( bot_bot @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Inf_UNIV % 0.31/0.61 thf(fact_5221_ccInf__empty,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( counta3822494911875563373attice @ A ) % 0.31/0.61 => ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.61 = ( top_top @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % ccInf_empty % 0.31/0.61 thf(fact_5222_Inf__empty,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( comple6319245703460814977attice @ A ) % 0.31/0.61 => ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.61 = ( top_top @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Inf_empty % 0.31/0.61 thf(fact_5223_Diff__UNIV,axiom, % 0.31/0.61 ! [A: $tType,A3: set @ A] : % 0.31/0.61 ( ( minus_minus @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Diff_UNIV % 0.31/0.61 thf(fact_5224_surj__fn,axiom, % 0.31/0.61 ! [A: $tType,F2: A > A,N2: nat] : % 0.31/0.61 ( ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 = ( top_top @ ( set @ A ) ) ) % 0.31/0.61 => ( ( image @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 = ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % surj_fn % 0.31/0.61 thf(fact_5225_finite__compl,axiom, % 0.31/0.61 ! [A: $tType,A3: set @ A] : % 0.31/0.61 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.61 => ( ( finite_finite2 @ A @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) % 0.31/0.61 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % finite_compl % 0.31/0.61 thf(fact_5226_Gcd__UNIV,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( semiring_Gcd @ A ) % 0.31/0.61 => ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 = ( one_one @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Gcd_UNIV % 0.31/0.61 thf(fact_5227_length__remdups__leq,axiom, % 0.31/0.61 ! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ). % 0.31/0.61 % 0.31/0.61 % length_remdups_leq % 0.31/0.61 thf(fact_5228_SUP__eq__top__iff,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( comple5582772986160207858norder @ A ) % 0.31/0.61 => ! [F2: B > A,A3: set @ B] : % 0.31/0.61 ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.61 = ( top_top @ A ) ) % 0.31/0.61 = ( ! [X: A] : % 0.31/0.61 ( ( ord_less @ A @ X @ ( top_top @ A ) ) % 0.31/0.61 => ? [Y2: B] : % 0.31/0.61 ( ( member @ B @ Y2 @ A3 ) % 0.31/0.61 & ( ord_less @ A @ X @ ( F2 @ Y2 ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % SUP_eq_top_iff % 0.31/0.61 thf(fact_5229_range__constant,axiom, % 0.31/0.61 ! [B: $tType,A: $tType,X3: A] : % 0.31/0.61 ( ( image @ B @ A % 0.31/0.61 @ ^ [Uu3: B] : X3 % 0.31/0.61 @ ( top_top @ ( set @ B ) ) ) % 0.31/0.61 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % range_constant % 0.31/0.61 thf(fact_5230_ccINF__empty,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( counta3822494911875563373attice @ A ) % 0.31/0.61 => ! [F2: B > A] : % 0.31/0.61 ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) ) % 0.31/0.61 = ( top_top @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % ccINF_empty % 0.31/0.61 thf(fact_5231_INT__constant,axiom, % 0.31/0.61 ! [B: $tType,A: $tType,A3: set @ B,C2: set @ A] : % 0.31/0.61 ( ( ( A3 % 0.31/0.61 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.61 => ( ( complete_Inf_Inf @ ( set @ A ) % 0.31/0.61 @ ( image @ B @ ( set @ A ) % 0.31/0.61 @ ^ [Y2: B] : C2 % 0.31/0.61 @ A3 ) ) % 0.31/0.61 = ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 & ( ( A3 % 0.31/0.61 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.61 => ( ( complete_Inf_Inf @ ( set @ A ) % 0.31/0.61 @ ( image @ B @ ( set @ A ) % 0.31/0.61 @ ^ [Y2: B] : C2 % 0.31/0.61 @ A3 ) ) % 0.31/0.61 = C2 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % INT_constant % 0.31/0.61 thf(fact_5232_Inf__atMostLessThan,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( comple6319245703460814977attice @ A ) % 0.31/0.61 => ! [X3: A] : % 0.31/0.61 ( ( ord_less @ A @ ( top_top @ A ) @ X3 ) % 0.31/0.61 => ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X3 ) ) % 0.31/0.61 = ( bot_bot @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Inf_atMostLessThan % 0.31/0.61 thf(fact_5233_INT__simps_I1_J,axiom, % 0.31/0.61 ! [A: $tType,B: $tType,C5: set @ A,A3: A > ( set @ B ),B3: set @ B] : % 0.31/0.61 ( ( ( C5 % 0.31/0.61 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.61 => ( ( complete_Inf_Inf @ ( set @ B ) % 0.31/0.61 @ ( image @ A @ ( set @ B ) % 0.31/0.61 @ ^ [X: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X ) @ B3 ) % 0.31/0.61 @ C5 ) ) % 0.31/0.61 = ( top_top @ ( set @ B ) ) ) ) % 0.31/0.61 & ( ( C5 % 0.31/0.61 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.61 => ( ( complete_Inf_Inf @ ( set @ B ) % 0.31/0.61 @ ( image @ A @ ( set @ B ) % 0.31/0.61 @ ^ [X: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X ) @ B3 ) % 0.31/0.61 @ C5 ) ) % 0.31/0.61 = ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ C5 ) ) @ B3 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % INT_simps(1) % 0.31/0.61 thf(fact_5234_INT__simps_I2_J,axiom, % 0.31/0.61 ! [C: $tType,D: $tType,C5: set @ D,A3: set @ C,B3: D > ( set @ C )] : % 0.31/0.61 ( ( ( C5 % 0.31/0.61 = ( bot_bot @ ( set @ D ) ) ) % 0.31/0.61 => ( ( complete_Inf_Inf @ ( set @ C ) % 0.31/0.61 @ ( image @ D @ ( set @ C ) % 0.31/0.61 @ ^ [X: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B3 @ X ) ) % 0.31/0.61 @ C5 ) ) % 0.31/0.61 = ( top_top @ ( set @ C ) ) ) ) % 0.31/0.61 & ( ( C5 % 0.31/0.61 != ( bot_bot @ ( set @ D ) ) ) % 0.31/0.61 => ( ( complete_Inf_Inf @ ( set @ C ) % 0.31/0.61 @ ( image @ D @ ( set @ C ) % 0.31/0.61 @ ^ [X: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B3 @ X ) ) % 0.31/0.61 @ C5 ) ) % 0.31/0.61 = ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D @ ( set @ C ) @ B3 @ C5 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % INT_simps(2) % 0.31/0.61 thf(fact_5235_INT__simps_I3_J,axiom, % 0.31/0.61 ! [E2: $tType,F: $tType,C5: set @ E2,A3: E2 > ( set @ F ),B3: set @ F] : % 0.31/0.61 ( ( ( C5 % 0.31/0.61 = ( bot_bot @ ( set @ E2 ) ) ) % 0.31/0.61 => ( ( complete_Inf_Inf @ ( set @ F ) % 0.31/0.61 @ ( image @ E2 @ ( set @ F ) % 0.31/0.61 @ ^ [X: E2] : ( minus_minus @ ( set @ F ) @ ( A3 @ X ) @ B3 ) % 0.31/0.61 @ C5 ) ) % 0.31/0.61 = ( top_top @ ( set @ F ) ) ) ) % 0.31/0.61 & ( ( C5 % 0.31/0.61 != ( bot_bot @ ( set @ E2 ) ) ) % 0.31/0.61 => ( ( complete_Inf_Inf @ ( set @ F ) % 0.31/0.61 @ ( image @ E2 @ ( set @ F ) % 0.31/0.61 @ ^ [X: E2] : ( minus_minus @ ( set @ F ) @ ( A3 @ X ) @ B3 ) % 0.31/0.61 @ C5 ) ) % 0.31/0.61 = ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E2 @ ( set @ F ) @ A3 @ C5 ) ) @ B3 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % INT_simps(3) % 0.31/0.61 thf(fact_5236_UNIV__option__conv,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( top_top @ ( set @ ( option @ A ) ) ) % 0.31/0.61 = ( insert @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % UNIV_option_conv % 0.31/0.61 thf(fact_5237_rangeI,axiom, % 0.31/0.61 ! [A: $tType,B: $tType,F2: B > A,X3: B] : ( member @ A @ ( F2 @ X3 ) @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % rangeI % 0.31/0.61 thf(fact_5238_range__eqI,axiom, % 0.31/0.61 ! [A: $tType,B: $tType,B2: A,F2: B > A,X3: B] : % 0.31/0.61 ( ( B2 % 0.31/0.61 = ( F2 @ X3 ) ) % 0.31/0.61 => ( member @ A @ B2 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % range_eqI % 0.31/0.61 thf(fact_5239_rangeE,axiom, % 0.31/0.61 ! [A: $tType,B: $tType,B2: A,F2: B > A] : % 0.31/0.61 ( ( member @ A @ B2 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) % 0.31/0.61 => ~ ! [X5: B] : % 0.31/0.61 ( B2 % 0.31/0.61 != ( F2 @ X5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % rangeE % 0.31/0.61 thf(fact_5240_range__composition,axiom, % 0.31/0.61 ! [A: $tType,C: $tType,B: $tType,F2: C > A,G: B > C] : % 0.31/0.61 ( ( image @ B @ A % 0.31/0.61 @ ^ [X: B] : ( F2 @ ( G @ X ) ) % 0.31/0.61 @ ( top_top @ ( set @ B ) ) ) % 0.31/0.61 = ( image @ C @ A @ F2 @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % range_composition % 0.31/0.61 thf(fact_5241_INF__filter__bot__base,axiom, % 0.31/0.61 ! [B: $tType,A: $tType,I6: set @ A,F5: A > ( filter @ B )] : % 0.31/0.61 ( ! [I4: A] : % 0.31/0.61 ( ( member @ A @ I4 @ I6 ) % 0.31/0.61 => ! [J2: A] : % 0.31/0.61 ( ( member @ A @ J2 @ I6 ) % 0.31/0.61 => ? [X2: A] : % 0.31/0.61 ( ( member @ A @ X2 @ I6 ) % 0.31/0.61 & ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X2 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ I4 ) @ ( F5 @ J2 ) ) ) ) ) ) % 0.31/0.61 => ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F5 @ I6 ) ) % 0.31/0.61 = ( bot_bot @ ( filter @ B ) ) ) % 0.31/0.61 = ( ? [X: A] : % 0.31/0.61 ( ( member @ A @ X @ I6 ) % 0.31/0.61 & ( ( F5 @ X ) % 0.31/0.61 = ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % INF_filter_bot_base % 0.31/0.61 thf(fact_5242_remdups_Osimps_I1_J,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( remdups @ A @ ( nil @ A ) ) % 0.31/0.61 = ( nil @ A ) ) ). % 0.31/0.61 % 0.31/0.61 % remdups.simps(1) % 0.31/0.61 thf(fact_5243_Int__UNIV__right,axiom, % 0.31/0.61 ! [A: $tType,A3: set @ A] : % 0.31/0.61 ( ( inf_inf @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 = A3 ) ). % 0.31/0.61 % 0.31/0.61 % Int_UNIV_right % 0.31/0.61 thf(fact_5244_Int__UNIV__left,axiom, % 0.31/0.61 ! [A: $tType,B3: set @ A] : % 0.31/0.61 ( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 ) % 0.31/0.61 = B3 ) ). % 0.31/0.61 % 0.31/0.61 % Int_UNIV_left % 0.31/0.61 thf(fact_5245_remdups__remdups,axiom, % 0.31/0.61 ! [A: $tType,Xs: list @ A] : % 0.31/0.61 ( ( remdups @ A @ ( remdups @ A @ Xs ) ) % 0.31/0.61 = ( remdups @ A @ Xs ) ) ). % 0.31/0.61 % 0.31/0.61 % remdups_remdups % 0.31/0.61 thf(fact_5246_UNIV__eq__I,axiom, % 0.31/0.61 ! [A: $tType,A3: set @ A] : % 0.31/0.61 ( ! [X5: A] : ( member @ A @ X5 @ A3 ) % 0.31/0.61 => ( ( top_top @ ( set @ A ) ) % 0.31/0.61 = A3 ) ) ). % 0.31/0.61 % 0.31/0.61 % UNIV_eq_I % 0.31/0.61 thf(fact_5247_UNIV__witness,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ? [X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % UNIV_witness % 0.31/0.61 thf(fact_5248_top_Oextremum__strict,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( order_top @ A ) % 0.31/0.61 => ! [A2: A] : % 0.31/0.61 ~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ). % 0.31/0.61 % 0.31/0.61 % top.extremum_strict % 0.31/0.61 thf(fact_5249_top_Onot__eq__extremum,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( order_top @ A ) % 0.31/0.61 => ! [A2: A] : % 0.31/0.61 ( ( A2 % 0.31/0.61 != ( top_top @ A ) ) % 0.31/0.61 = ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % top.not_eq_extremum % 0.31/0.61 thf(fact_5250_UNIV__def,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( top_top @ ( set @ A ) ) % 0.31/0.61 = ( collect @ A % 0.31/0.61 @ ^ [X: A] : $true ) ) ). % 0.31/0.61 % 0.31/0.61 % UNIV_def % 0.31/0.61 thf(fact_5251_subset__UNIV,axiom, % 0.31/0.61 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % subset_UNIV % 0.31/0.61 thf(fact_5252_top_Oextremum__uniqueI,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( order_top @ A ) % 0.31/0.61 => ! [A2: A] : % 0.31/0.61 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 ) % 0.31/0.61 => ( A2 % 0.31/0.61 = ( top_top @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % top.extremum_uniqueI % 0.31/0.61 thf(fact_5253_top_Oextremum__unique,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( order_top @ A ) % 0.31/0.61 => ! [A2: A] : % 0.31/0.61 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 ) % 0.31/0.61 = ( A2 % 0.31/0.61 = ( top_top @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % top.extremum_unique % 0.31/0.61 thf(fact_5254_top__greatest,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( order_top @ A ) % 0.31/0.61 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % top_greatest % 0.31/0.61 thf(fact_5255_distinct__remdups__id,axiom, % 0.31/0.61 ! [A: $tType,Xs: list @ A] : % 0.31/0.61 ( ( distinct @ A @ Xs ) % 0.31/0.61 => ( ( remdups @ A @ Xs ) % 0.31/0.61 = Xs ) ) ). % 0.31/0.61 % 0.31/0.61 % distinct_remdups_id % 0.31/0.61 thf(fact_5256_empty__not__UNIV,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( bot_bot @ ( set @ A ) ) % 0.31/0.61 != ( top_top @ ( set @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % empty_not_UNIV % 0.31/0.61 thf(fact_5257_atLeastAtMost__eq__UNIV__iff,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( bounded_lattice @ A ) % 0.31/0.61 => ! [X3: A,Y: A] : % 0.31/0.61 ( ( ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) % 0.31/0.61 = ( top_top @ ( set @ A ) ) ) % 0.31/0.61 = ( ( X3 % 0.31/0.61 = ( bot_bot @ A ) ) % 0.31/0.61 & ( Y % 0.31/0.61 = ( top_top @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % atLeastAtMost_eq_UNIV_iff % 0.31/0.61 thf(fact_5258_insert__UNIV,axiom, % 0.31/0.61 ! [A: $tType,X3: A] : % 0.31/0.61 ( ( insert @ A @ X3 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 = ( top_top @ ( set @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % insert_UNIV % 0.31/0.61 thf(fact_5259_Inf__filter__not__bot,axiom, % 0.31/0.61 ! [A: $tType,B3: set @ ( filter @ A )] : % 0.31/0.61 ( ! [X10: set @ ( filter @ A )] : % 0.31/0.61 ( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X10 @ B3 ) % 0.31/0.61 => ( ( finite_finite2 @ ( filter @ A ) @ X10 ) % 0.31/0.61 => ( ( complete_Inf_Inf @ ( filter @ A ) @ X10 ) % 0.31/0.61 != ( bot_bot @ ( filter @ A ) ) ) ) ) % 0.31/0.61 => ( ( complete_Inf_Inf @ ( filter @ A ) @ B3 ) % 0.31/0.61 != ( bot_bot @ ( filter @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Inf_filter_not_bot % 0.31/0.61 thf(fact_5260_infinite__UNIV__int,axiom, % 0.31/0.61 ~ ( finite_finite2 @ int @ ( top_top @ ( set @ int ) ) ) ). % 0.31/0.61 % 0.31/0.61 % infinite_UNIV_int % 0.31/0.61 thf(fact_5261_nat__not__finite,axiom, % 0.31/0.61 ~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % nat_not_finite % 0.31/0.61 thf(fact_5262_infinite__UNIV__nat,axiom, % 0.31/0.61 ~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % infinite_UNIV_nat % 0.31/0.61 thf(fact_5263_Finite__Set_Ofinite__set,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( finite_finite2 @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) ) % 0.31/0.61 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Finite_Set.finite_set % 0.31/0.61 thf(fact_5264_finite__Prod__UNIV,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 => ( ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) % 0.31/0.61 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % finite_Prod_UNIV % 0.31/0.61 thf(fact_5265_finite__prod,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) % 0.31/0.61 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % finite_prod % 0.31/0.61 thf(fact_5266_finite__fun__UNIVD2,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) ) % 0.31/0.61 => ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % finite_fun_UNIVD2 % 0.31/0.61 thf(fact_5267_finite__UNIV,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( finite_finite @ A ) % 0.31/0.61 => ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % finite_UNIV % 0.31/0.61 thf(fact_5268_ex__new__if__finite,axiom, % 0.31/0.61 ! [A: $tType,A3: set @ A] : % 0.31/0.61 ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.61 => ? [A5: A] : % 0.31/0.61 ~ ( member @ A @ A5 @ A3 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % ex_new_if_finite % 0.31/0.61 thf(fact_5269_infinite__UNIV__char__0,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( semiring_char_0 @ A ) % 0.31/0.61 => ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % infinite_UNIV_char_0 % 0.31/0.61 thf(fact_5270_infinite__UNIV__listI,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ~ ( finite_finite2 @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % infinite_UNIV_listI % 0.31/0.61 thf(fact_5271_range__subsetD,axiom, % 0.31/0.61 ! [B: $tType,A: $tType,F2: B > A,B3: set @ A,I: B] : % 0.31/0.61 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) @ B3 ) % 0.31/0.61 => ( member @ A @ ( F2 @ I ) @ B3 ) ) ). % 0.31/0.61 % 0.31/0.61 % range_subsetD % 0.31/0.61 thf(fact_5272_perfect__space__class_OUNIV__not__singleton,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo8386298272705272623_space @ A ) % 0.31/0.61 => ! [X3: A] : % 0.31/0.61 ( ( top_top @ ( set @ A ) ) % 0.31/0.61 != ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % perfect_space_class.UNIV_not_singleton % 0.31/0.61 thf(fact_5273_bot__finite__def,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( finite_lattice @ A ) % 0.31/0.61 => ( ( bot_bot @ A ) % 0.31/0.61 = ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % bot_finite_def % 0.31/0.61 thf(fact_5274_finite__range__Some,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( finite_finite2 @ ( option @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % finite_range_Some % 0.31/0.61 thf(fact_5275_not__UNIV__le__Icc,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( no_top @ A ) % 0.31/0.61 => ! [L: A,H2: A] : % 0.31/0.61 ~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % not_UNIV_le_Icc % 0.31/0.61 thf(fact_5276_card__eq__UNIV__imp__eq__UNIV,axiom, % 0.31/0.61 ! [A: $tType,A3: set @ A] : % 0.31/0.61 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 => ( ( ( finite_card @ A @ A3 ) % 0.31/0.61 = ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 => ( A3 % 0.31/0.61 = ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % card_eq_UNIV_imp_eq_UNIV % 0.31/0.61 thf(fact_5277_not__UNIV__le__Iic,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( no_top @ A ) % 0.31/0.61 => ! [H2: A] : % 0.31/0.61 ~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atMost @ A @ H2 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % not_UNIV_le_Iic % 0.31/0.61 thf(fact_5278_notin__range__Some,axiom, % 0.31/0.61 ! [A: $tType,X3: option @ A] : % 0.31/0.61 ( ( ~ ( member @ ( option @ A ) @ X3 @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) % 0.31/0.61 = ( X3 % 0.31/0.61 = ( none @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % notin_range_Some % 0.31/0.61 thf(fact_5279_Inter__UNIV,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( complete_Inf_Inf @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) ) % 0.31/0.61 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Inter_UNIV % 0.31/0.61 thf(fact_5280_Compl__UNIV__eq,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Compl_UNIV_eq % 0.31/0.61 thf(fact_5281_Compl__empty__eq,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.61 = ( top_top @ ( set @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Compl_empty_eq % 0.31/0.61 thf(fact_5282_int__in__range__abs,axiom, % 0.31/0.61 ! [N2: nat] : ( member @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % int_in_range_abs % 0.31/0.61 thf(fact_5283_Compl__eq__Diff__UNIV,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( uminus_uminus @ ( set @ A ) ) % 0.31/0.61 = ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Compl_eq_Diff_UNIV % 0.31/0.61 thf(fact_5284_Inter__empty,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( complete_Inf_Inf @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 0.31/0.61 = ( top_top @ ( set @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Inter_empty % 0.31/0.61 thf(fact_5285_full__SetCompr__eq,axiom, % 0.31/0.61 ! [A: $tType,B: $tType,F2: B > A] : % 0.31/0.61 ( ( collect @ A % 0.31/0.61 @ ^ [U2: A] : % 0.31/0.61 ? [X: B] : % 0.31/0.61 ( U2 % 0.31/0.61 = ( F2 @ X ) ) ) % 0.31/0.61 = ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % full_SetCompr_eq % 0.31/0.61 thf(fact_5286_finite__range__imageI,axiom, % 0.31/0.61 ! [C: $tType,A: $tType,B: $tType,G: B > A,F2: A > C] : % 0.31/0.61 ( ( finite_finite2 @ A @ ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) ) ) % 0.31/0.61 => ( finite_finite2 @ C % 0.31/0.61 @ ( image @ B @ C % 0.31/0.61 @ ^ [X: B] : ( F2 @ ( G @ X ) ) % 0.31/0.61 @ ( top_top @ ( set @ B ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % finite_range_imageI % 0.31/0.61 thf(fact_5287_remove1__remdups,axiom, % 0.31/0.61 ! [A: $tType,Xs: list @ A,X3: A] : % 0.31/0.61 ( ( distinct @ A @ Xs ) % 0.31/0.61 => ( ( remove1 @ A @ X3 @ ( remdups @ A @ Xs ) ) % 0.31/0.61 = ( remdups @ A @ ( remove1 @ A @ X3 @ Xs ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % remove1_remdups % 0.31/0.61 thf(fact_5288_finite__fun__UNIVD1,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) ) % 0.31/0.61 => ( ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) ) % 0.31/0.61 != ( suc @ ( zero_zero @ nat ) ) ) % 0.31/0.61 => ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % finite_fun_UNIVD1 % 0.31/0.61 thf(fact_5289_range__eq__singletonD,axiom, % 0.31/0.61 ! [B: $tType,A: $tType,F2: B > A,A2: A,X3: B] : % 0.31/0.61 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) % 0.31/0.61 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.61 => ( ( F2 @ X3 ) % 0.31/0.61 = A2 ) ) ). % 0.31/0.61 % 0.31/0.61 % range_eq_singletonD % 0.31/0.61 thf(fact_5290_INF__empty,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( comple6319245703460814977attice @ A ) % 0.31/0.61 => ! [F2: B > A] : % 0.31/0.61 ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) ) % 0.31/0.61 = ( top_top @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % INF_empty % 0.31/0.61 thf(fact_5291_INF__constant,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( comple6319245703460814977attice @ A ) % 0.31/0.61 => ! [A3: set @ B,C2: A] : % 0.31/0.61 ( ( ( A3 % 0.31/0.61 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.61 => ( ( complete_Inf_Inf @ A % 0.31/0.61 @ ( image @ B @ A % 0.31/0.61 @ ^ [Y2: B] : C2 % 0.31/0.61 @ A3 ) ) % 0.31/0.61 = ( top_top @ A ) ) ) % 0.31/0.61 & ( ( A3 % 0.31/0.61 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.61 => ( ( complete_Inf_Inf @ A % 0.31/0.61 @ ( image @ B @ A % 0.31/0.61 @ ^ [Y2: B] : C2 % 0.31/0.61 @ A3 ) ) % 0.31/0.61 = C2 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % INF_constant % 0.31/0.61 thf(fact_5292_Sup__finite__empty,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( finite_lattice @ A ) % 0.31/0.61 => ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.61 = ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Sup_finite_empty % 0.31/0.61 thf(fact_5293_Inf__finite__empty,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( finite_lattice @ A ) % 0.31/0.61 => ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.61 = ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Inf_finite_empty % 0.31/0.61 thf(fact_5294_surj__Compl__image__subset,axiom, % 0.31/0.61 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] : % 0.31/0.61 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) % 0.31/0.61 = ( top_top @ ( set @ A ) ) ) % 0.31/0.61 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) ) @ ( image @ B @ A @ F2 @ ( uminus_uminus @ ( set @ B ) @ A3 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % surj_Compl_image_subset % 0.31/0.61 thf(fact_5295_length__remdups__card__conv,axiom, % 0.31/0.61 ! [A: $tType,Xs: list @ A] : % 0.31/0.61 ( ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) ) % 0.31/0.61 = ( finite_card @ A @ ( set2 @ A @ Xs ) ) ) ). % 0.31/0.61 % 0.31/0.61 % length_remdups_card_conv % 0.31/0.61 thf(fact_5296_Nats__def,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( semiring_1 @ A ) % 0.31/0.61 => ( ( semiring_1_Nats @ A ) % 0.31/0.61 = ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Nats_def % 0.31/0.61 thf(fact_5297_Collect__ex__eq,axiom, % 0.31/0.61 ! [A: $tType,B: $tType,P: A > B > $o] : % 0.31/0.61 ( ( collect @ A % 0.31/0.61 @ ^ [X: A] : % 0.31/0.61 ? [X6: B] : ( P @ X @ X6 ) ) % 0.31/0.61 = ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.61 @ ( image @ B @ ( set @ A ) % 0.31/0.61 @ ^ [Y2: B] : % 0.31/0.61 ( collect @ A % 0.31/0.61 @ ^ [X: A] : ( P @ X @ Y2 ) ) % 0.31/0.61 @ ( top_top @ ( set @ B ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Collect_ex_eq % 0.31/0.61 thf(fact_5298_INT__empty,axiom, % 0.31/0.61 ! [B: $tType,A: $tType,B3: B > ( set @ A )] : % 0.31/0.61 ( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) % 0.31/0.61 = ( top_top @ ( set @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % INT_empty % 0.31/0.61 thf(fact_5299_UN__UN__finite__eq,axiom, % 0.31/0.61 ! [A: $tType,A3: nat > ( set @ A )] : % 0.31/0.61 ( ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.61 @ ( image @ nat @ ( set @ A ) % 0.31/0.61 @ ^ [N: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) % 0.31/0.61 @ ( top_top @ ( set @ nat ) ) ) ) % 0.31/0.61 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % UN_UN_finite_eq % 0.31/0.61 thf(fact_5300_inf__top_Osemilattice__neutr__order__axioms,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( bounde4346867609351753570nf_top @ A ) % 0.31/0.61 => ( semila1105856199041335345_order @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % inf_top.semilattice_neutr_order_axioms % 0.31/0.61 thf(fact_5301_finite__UNIV__card__ge__0,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % finite_UNIV_card_ge_0 % 0.31/0.61 thf(fact_5302_UN__finite__subset,axiom, % 0.31/0.61 ! [A: $tType,A3: nat > ( set @ A ),C5: set @ A] : % 0.31/0.61 ( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ C5 ) % 0.31/0.61 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) @ C5 ) ) ). % 0.31/0.61 % 0.31/0.61 % UN_finite_subset % 0.31/0.61 thf(fact_5303_UNIV__nat__eq,axiom, % 0.31/0.61 ( ( top_top @ ( set @ nat ) ) % 0.31/0.61 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % UNIV_nat_eq % 0.31/0.61 thf(fact_5304_UN__finite2__eq,axiom, % 0.31/0.61 ! [A: $tType,A3: nat > ( set @ A ),B3: nat > ( set @ A ),K: nat] : % 0.31/0.61 ( ! [N3: nat] : % 0.31/0.61 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) % 0.31/0.61 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) ) % 0.31/0.61 => ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) % 0.31/0.61 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % UN_finite2_eq % 0.31/0.61 thf(fact_5305_INT__extend__simps_I3_J,axiom, % 0.31/0.61 ! [F: $tType,E2: $tType,C5: set @ E2,A3: E2 > ( set @ F ),B3: set @ F] : % 0.31/0.61 ( ( ( C5 % 0.31/0.61 = ( bot_bot @ ( set @ E2 ) ) ) % 0.31/0.61 => ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E2 @ ( set @ F ) @ A3 @ C5 ) ) @ B3 ) % 0.31/0.61 = ( minus_minus @ ( set @ F ) @ ( top_top @ ( set @ F ) ) @ B3 ) ) ) % 0.31/0.61 & ( ( C5 % 0.31/0.61 != ( bot_bot @ ( set @ E2 ) ) ) % 0.31/0.61 => ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E2 @ ( set @ F ) @ A3 @ C5 ) ) @ B3 ) % 0.31/0.61 = ( complete_Inf_Inf @ ( set @ F ) % 0.31/0.61 @ ( image @ E2 @ ( set @ F ) % 0.31/0.61 @ ^ [X: E2] : ( minus_minus @ ( set @ F ) @ ( A3 @ X ) @ B3 ) % 0.31/0.61 @ C5 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % INT_extend_simps(3) % 0.31/0.61 thf(fact_5306_card__range__greater__zero,axiom, % 0.31/0.61 ! [A: $tType,B: $tType,F2: B > A] : % 0.31/0.61 ( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) % 0.31/0.61 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % card_range_greater_zero % 0.31/0.61 thf(fact_5307_UN__finite2__subset,axiom, % 0.31/0.61 ! [A: $tType,A3: nat > ( set @ A ),B3: nat > ( set @ A ),K: nat] : % 0.31/0.61 ( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) ) % 0.31/0.61 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % UN_finite2_subset % 0.31/0.61 thf(fact_5308_range__mod,axiom, % 0.31/0.61 ! [N2: nat] : % 0.31/0.61 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.61 => ( ( image @ nat @ nat % 0.31/0.61 @ ^ [M2: nat] : ( modulo_modulo @ nat @ M2 @ N2 ) % 0.31/0.61 @ ( top_top @ ( set @ nat ) ) ) % 0.31/0.61 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % range_mod % 0.31/0.61 thf(fact_5309_cclfp__def,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( counta3822494911875563373attice @ A ) % 0.31/0.61 => ( ( order_532582986084564980_cclfp @ A ) % 0.31/0.61 = ( ^ [F3: A > A] : % 0.31/0.61 ( complete_Sup_Sup @ A % 0.31/0.61 @ ( image @ nat @ A % 0.31/0.61 @ ^ [I3: nat] : ( compow @ ( A > A ) @ I3 @ F3 @ ( bot_bot @ A ) ) % 0.31/0.61 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % cclfp_def % 0.31/0.61 thf(fact_5310_root__def,axiom, % 0.31/0.61 ( root % 0.31/0.61 = ( ^ [N: nat,X: real] : % 0.31/0.61 ( if @ real % 0.31/0.61 @ ( N % 0.31/0.61 = ( zero_zero @ nat ) ) % 0.31/0.61 @ ( zero_zero @ real ) % 0.31/0.61 @ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) ) % 0.31/0.61 @ ^ [Y2: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N ) ) % 0.31/0.61 @ X ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % root_def % 0.31/0.61 thf(fact_5311_Id__on__def,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( id_on @ A ) % 0.31/0.61 = ( ^ [A6: set @ A] : % 0.31/0.61 ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) % 0.31/0.61 @ ( image @ A @ ( set @ ( product_prod @ A @ A ) ) % 0.31/0.61 @ ^ [X: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) % 0.31/0.61 @ A6 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Id_on_def % 0.31/0.61 thf(fact_5312_Id__onI,axiom, % 0.31/0.61 ! [A: $tType,A2: A,A3: set @ A] : % 0.31/0.61 ( ( member @ A @ A2 @ A3 ) % 0.31/0.61 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A3 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Id_onI % 0.31/0.61 thf(fact_5313_Id__on__empty,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.61 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Id_on_empty % 0.31/0.61 thf(fact_5314_top__empty__eq2,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( top_top @ ( A > B > $o ) ) % 0.31/0.61 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % top_empty_eq2 % 0.31/0.61 thf(fact_5315_top__set__def,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( top_top @ ( set @ A ) ) % 0.31/0.61 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % top_set_def % 0.31/0.61 thf(fact_5316_less__filter__def,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ord_less @ ( filter @ A ) ) % 0.31/0.61 = ( ^ [F9: filter @ A,F10: filter @ A] : % 0.31/0.61 ( ( ord_less_eq @ ( filter @ A ) @ F9 @ F10 ) % 0.31/0.61 & ~ ( ord_less_eq @ ( filter @ A ) @ F10 @ F9 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % less_filter_def % 0.31/0.61 thf(fact_5317_Id__on__iff,axiom, % 0.31/0.61 ! [A: $tType,X3: A,Y: A,A3: set @ A] : % 0.31/0.61 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( id_on @ A @ A3 ) ) % 0.31/0.61 = ( ( X3 = Y ) % 0.31/0.61 & ( member @ A @ X3 @ A3 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Id_on_iff % 0.31/0.61 thf(fact_5318_Id__on__eqI,axiom, % 0.31/0.61 ! [A: $tType,A2: A,B2: A,A3: set @ A] : % 0.31/0.61 ( ( A2 = B2 ) % 0.31/0.61 => ( ( member @ A @ A2 @ A3 ) % 0.31/0.61 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id_on @ A @ A3 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Id_on_eqI % 0.31/0.61 thf(fact_5319_Id__onE,axiom, % 0.31/0.61 ! [A: $tType,C2: product_prod @ A @ A,A3: set @ A] : % 0.31/0.61 ( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A3 ) ) % 0.31/0.61 => ~ ! [X5: A] : % 0.31/0.61 ( ( member @ A @ X5 @ A3 ) % 0.31/0.61 => ( C2 % 0.31/0.61 != ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Id_onE % 0.31/0.61 thf(fact_5320_these__insert__Some,axiom, % 0.31/0.61 ! [A: $tType,X3: A,A3: set @ ( option @ A )] : % 0.31/0.61 ( ( these @ A @ ( insert @ ( option @ A ) @ ( some @ A @ X3 ) @ A3 ) ) % 0.31/0.61 = ( insert @ A @ X3 @ ( these @ A @ A3 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % these_insert_Some % 0.31/0.61 thf(fact_5321_DERIV__even__real__root,axiom, % 0.31/0.61 ! [N2: nat,X3: real] : % 0.31/0.61 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.61 => ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.31/0.61 => ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) % 0.31/0.61 => ( has_field_derivative @ real @ ( root @ N2 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N2 ) ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_even_real_root % 0.31/0.61 thf(fact_5322_DERIV__real__root__generic,axiom, % 0.31/0.61 ! [N2: nat,X3: real,D6: real] : % 0.31/0.61 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.61 => ( ( X3 % 0.31/0.61 != ( zero_zero @ real ) ) % 0.31/0.61 => ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.61 => ( D6 % 0.31/0.61 = ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) % 0.31/0.61 => ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.31/0.61 => ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) % 0.31/0.61 => ( D6 % 0.31/0.61 = ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ) % 0.31/0.61 => ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.31/0.61 => ( D6 % 0.31/0.61 = ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) % 0.31/0.61 => ( has_field_derivative @ real @ ( root @ N2 ) @ D6 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_real_root_generic % 0.31/0.61 thf(fact_5323_top1I,axiom, % 0.31/0.61 ! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ). % 0.31/0.61 % 0.31/0.61 % top1I % 0.31/0.61 thf(fact_5324_top2I,axiom, % 0.31/0.61 ! [A: $tType,B: $tType,X3: A,Y: B] : ( top_top @ ( A > B > $o ) @ X3 @ Y ) ). % 0.31/0.61 % 0.31/0.61 % top2I % 0.31/0.61 thf(fact_5325_these__empty,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) % 0.31/0.61 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % these_empty % 0.31/0.61 thf(fact_5326_at__within__empty,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.61 => ! [A2: A] : % 0.31/0.61 ( ( topolo174197925503356063within @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.61 = ( bot_bot @ ( filter @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % at_within_empty % 0.31/0.61 thf(fact_5327_these__image__Some__eq,axiom, % 0.31/0.61 ! [A: $tType,A3: set @ A] : % 0.31/0.61 ( ( these @ A @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ) % 0.31/0.61 = A3 ) ). % 0.31/0.61 % 0.31/0.61 % these_image_Some_eq % 0.31/0.61 thf(fact_5328_these__insert__None,axiom, % 0.31/0.61 ! [A: $tType,A3: set @ ( option @ A )] : % 0.31/0.61 ( ( these @ A @ ( insert @ ( option @ A ) @ ( none @ A ) @ A3 ) ) % 0.31/0.61 = ( these @ A @ A3 ) ) ). % 0.31/0.61 % 0.31/0.61 % these_insert_None % 0.31/0.61 thf(fact_5329_DERIV__at__within__shift__lemma,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,Y: A,Z2: A,X3: A,S3: set @ A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z2 @ X3 ) @ ( image @ A @ A @ ( plus_plus @ A @ Z2 ) @ S3 ) ) ) % 0.31/0.61 => ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F2 @ ( plus_plus @ A @ Z2 ) ) @ Y @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_at_within_shift_lemma % 0.31/0.61 thf(fact_5330_DERIV__at__within__shift,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,Y: A,Z2: A,X3: A,S3: set @ A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z2 @ X3 ) @ ( image @ A @ A @ ( plus_plus @ A @ Z2 ) @ S3 ) ) ) % 0.31/0.61 = ( has_field_derivative @ A % 0.31/0.61 @ ^ [X: A] : ( F2 @ ( plus_plus @ A @ Z2 @ X ) ) % 0.31/0.61 @ Y % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_at_within_shift % 0.31/0.61 thf(fact_5331_DERIV__neg__dec__right,axiom, % 0.31/0.61 ! [F2: real > real,L: real,X3: real] : % 0.31/0.61 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ L @ ( zero_zero @ real ) ) % 0.31/0.61 => ? [D3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 0.31/0.61 & ! [H5: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 0.31/0.61 => ( ( ord_less @ real @ H5 @ D3 ) % 0.31/0.61 => ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_neg_dec_right % 0.31/0.61 thf(fact_5332_DERIV__pos__inc__right,axiom, % 0.31/0.61 ! [F2: real > real,L: real,X3: real] : % 0.31/0.61 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ L ) % 0.31/0.61 => ? [D3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 0.31/0.61 & ! [H5: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 0.31/0.61 => ( ( ord_less @ real @ H5 @ D3 ) % 0.31/0.61 => ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_pos_inc_right % 0.31/0.61 thf(fact_5333_DERIV__pos__inc__left,axiom, % 0.31/0.61 ! [F2: real > real,L: real,X3: real] : % 0.31/0.61 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ L ) % 0.31/0.61 => ? [D3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 0.31/0.61 & ! [H5: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 0.31/0.61 => ( ( ord_less @ real @ H5 @ D3 ) % 0.31/0.61 => ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_pos_inc_left % 0.31/0.61 thf(fact_5334_DERIV__neg__dec__left,axiom, % 0.31/0.61 ! [F2: real > real,L: real,X3: real] : % 0.31/0.61 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ L @ ( zero_zero @ real ) ) % 0.31/0.61 => ? [D3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 0.31/0.61 & ! [H5: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 0.31/0.61 => ( ( ord_less @ real @ H5 @ D3 ) % 0.31/0.61 => ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_neg_dec_left % 0.31/0.61 thf(fact_5335_DERIV__neg__imp__decreasing,axiom, % 0.31/0.61 ! [A2: real,B2: real,F2: real > real] : % 0.31/0.61 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.61 => ( ! [X5: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ A2 @ X5 ) % 0.31/0.61 => ( ( ord_less_eq @ real @ X5 @ B2 ) % 0.31/0.61 => ? [Y5: real] : % 0.31/0.61 ( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 & ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) ) ) % 0.31/0.61 => ( ord_less @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_neg_imp_decreasing % 0.31/0.61 thf(fact_5336_DERIV__pos__imp__increasing,axiom, % 0.31/0.61 ! [A2: real,B2: real,F2: real > real] : % 0.31/0.61 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.61 => ( ! [X5: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ A2 @ X5 ) % 0.31/0.61 => ( ( ord_less_eq @ real @ X5 @ B2 ) % 0.31/0.61 => ? [Y5: real] : % 0.31/0.61 ( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 & ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) ) ) % 0.31/0.61 => ( ord_less @ real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_pos_imp_increasing % 0.31/0.61 thf(fact_5337_DERIV__subset,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,F11: A,X3: A,S: set @ A,T2: set @ A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( ( ord_less_eq @ ( set @ A ) @ T2 @ S ) % 0.31/0.61 => ( has_field_derivative @ A @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_subset % 0.31/0.61 thf(fact_5338_has__field__derivative__subset,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,Y: A,X3: A,S: set @ A,T2: set @ A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( ( ord_less_eq @ ( set @ A ) @ T2 @ S ) % 0.31/0.61 => ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_field_derivative_subset % 0.31/0.61 thf(fact_5339_DERIV__shift,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,Y: A,X3: A,Z2: A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ X3 @ Z2 ) @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 = ( has_field_derivative @ A % 0.31/0.61 @ ^ [X: A] : ( F2 @ ( plus_plus @ A @ X @ Z2 ) ) % 0.31/0.61 @ Y % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_shift % 0.31/0.61 thf(fact_5340_at__discrete,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo8865339358273720382pology @ A ) % 0.31/0.61 => ( ( topolo174197925503356063within @ A ) % 0.31/0.61 = ( ^ [X: A,S7: set @ A] : ( bot_bot @ ( filter @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % at_discrete % 0.31/0.61 thf(fact_5341_DERIV__inverse_H,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,D6: A,X3: A,S: set @ A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( ( ( F2 @ X3 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( has_field_derivative @ A % 0.31/0.61 @ ^ [X: A] : ( inverse_inverse @ A @ ( F2 @ X ) ) % 0.31/0.61 @ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) @ D6 ) @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_inverse' % 0.31/0.61 thf(fact_5342_has__real__derivative__pos__inc__right,axiom, % 0.31/0.61 ! [F2: real > real,L: real,X3: real,S3: set @ real] : % 0.31/0.61 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S3 ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ L ) % 0.31/0.61 => ? [D3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 0.31/0.61 & ! [H5: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 0.31/0.61 => ( ( member @ real @ ( plus_plus @ real @ X3 @ H5 ) @ S3 ) % 0.31/0.61 => ( ( ord_less @ real @ H5 @ D3 ) % 0.31/0.61 => ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_real_derivative_pos_inc_right % 0.31/0.61 thf(fact_5343_has__real__derivative__neg__dec__right,axiom, % 0.31/0.61 ! [F2: real > real,L: real,X3: real,S3: set @ real] : % 0.31/0.61 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S3 ) ) % 0.31/0.61 => ( ( ord_less @ real @ L @ ( zero_zero @ real ) ) % 0.31/0.61 => ? [D3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 0.31/0.61 & ! [H5: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 0.31/0.61 => ( ( member @ real @ ( plus_plus @ real @ X3 @ H5 ) @ S3 ) % 0.31/0.61 => ( ( ord_less @ real @ H5 @ D3 ) % 0.31/0.61 => ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_real_derivative_neg_dec_right % 0.31/0.61 thf(fact_5344_has__real__derivative__neg__dec__left,axiom, % 0.31/0.61 ! [F2: real > real,L: real,X3: real,S3: set @ real] : % 0.31/0.61 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S3 ) ) % 0.31/0.61 => ( ( ord_less @ real @ L @ ( zero_zero @ real ) ) % 0.31/0.61 => ? [D3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 0.31/0.61 & ! [H5: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 0.31/0.61 => ( ( member @ real @ ( minus_minus @ real @ X3 @ H5 ) @ S3 ) % 0.31/0.61 => ( ( ord_less @ real @ H5 @ D3 ) % 0.31/0.61 => ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_real_derivative_neg_dec_left % 0.31/0.61 thf(fact_5345_has__real__derivative__pos__inc__left,axiom, % 0.31/0.61 ! [F2: real > real,L: real,X3: real,S3: set @ real] : % 0.31/0.61 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S3 ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ L ) % 0.31/0.61 => ? [D3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 0.31/0.61 & ! [H5: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 0.31/0.61 => ( ( member @ real @ ( minus_minus @ real @ X3 @ H5 ) @ S3 ) % 0.31/0.61 => ( ( ord_less @ real @ H5 @ D3 ) % 0.31/0.61 => ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_real_derivative_pos_inc_left % 0.31/0.61 thf(fact_5346_DERIV__const,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [K: A,F5: filter @ A] : % 0.31/0.61 ( has_field_derivative @ A % 0.31/0.61 @ ^ [X: A] : K % 0.31/0.61 @ ( zero_zero @ A ) % 0.31/0.61 @ F5 ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_const % 0.31/0.61 thf(fact_5347_field__differentiable__add,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,F11: A,F5: filter @ A,G: A > A,G7: A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ F11 @ F5 ) % 0.31/0.61 => ( ( has_field_derivative @ A @ G @ G7 @ F5 ) % 0.31/0.61 => ( has_field_derivative @ A % 0.31/0.61 @ ^ [Z6: A] : ( plus_plus @ A @ ( F2 @ Z6 ) @ ( G @ Z6 ) ) % 0.31/0.61 @ ( plus_plus @ A @ F11 @ G7 ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % field_differentiable_add % 0.31/0.61 thf(fact_5348_DERIV__add,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,D6: A,X3: A,S: set @ A,G: A > A,E5: A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( has_field_derivative @ A % 0.31/0.61 @ ^ [X: A] : ( plus_plus @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ( plus_plus @ A @ D6 @ E5 ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_add % 0.31/0.61 thf(fact_5349_DERIV__mult_H,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,D6: A,X3: A,S: set @ A,G: A > A,E5: A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( has_field_derivative @ A % 0.31/0.61 @ ^ [X: A] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X3 ) @ E5 ) @ ( times_times @ A @ D6 @ ( G @ X3 ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_mult' % 0.31/0.61 thf(fact_5350_DERIV__mult,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,Da: A,X3: A,S: set @ A,G: A > A,Db: A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( has_field_derivative @ A % 0.31/0.61 @ ^ [X: A] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ( plus_plus @ A @ ( times_times @ A @ Da @ ( G @ X3 ) ) @ ( times_times @ A @ Db @ ( F2 @ X3 ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_mult % 0.31/0.61 thf(fact_5351_DERIV__divide,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,D6: A,X3: A,S: set @ A,G: A > A,E5: A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( ( ( G @ X3 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( has_field_derivative @ A % 0.31/0.61 @ ^ [X: A] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D6 @ ( G @ X3 ) ) @ ( times_times @ A @ ( F2 @ X3 ) @ E5 ) ) @ ( times_times @ A @ ( G @ X3 ) @ ( G @ X3 ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_divide % 0.31/0.61 thf(fact_5352_in__these__eq,axiom, % 0.31/0.61 ! [A: $tType,X3: A,A3: set @ ( option @ A )] : % 0.31/0.61 ( ( member @ A @ X3 @ ( these @ A @ A3 ) ) % 0.31/0.61 = ( member @ ( option @ A ) @ ( some @ A @ X3 ) @ A3 ) ) ). % 0.31/0.61 % 0.31/0.61 % in_these_eq % 0.31/0.61 thf(fact_5353_at__neq__bot,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo8386298272705272623_space @ A ) % 0.31/0.61 => ! [A2: A] : % 0.31/0.61 ( ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 != ( bot_bot @ ( filter @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % at_neq_bot % 0.31/0.61 thf(fact_5354_at__le,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.61 => ! [S: set @ A,T2: set @ A,X3: A] : % 0.31/0.61 ( ( ord_less_eq @ ( set @ A ) @ S @ T2 ) % 0.31/0.61 => ( ord_less_eq @ ( filter @ A ) @ ( topolo174197925503356063within @ A @ X3 @ S ) @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % at_le % 0.31/0.61 thf(fact_5355_trivial__limit__at__left__real,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( dense_order @ A ) % 0.31/0.61 & ( no_bot @ A ) % 0.31/0.61 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.61 => ! [X3: A] : % 0.31/0.61 ( ( topolo174197925503356063within @ A @ X3 @ ( set_ord_lessThan @ A @ X3 ) ) % 0.31/0.61 != ( bot_bot @ ( filter @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % trivial_limit_at_left_real % 0.31/0.61 thf(fact_5356_MVT2,axiom, % 0.31/0.61 ! [A2: real,B2: real,F2: real > real,F11: real > real] : % 0.31/0.61 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.61 => ( ! [X5: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ A2 @ X5 ) % 0.31/0.61 => ( ( ord_less_eq @ real @ X5 @ B2 ) % 0.31/0.61 => ( has_field_derivative @ real @ F2 @ ( F11 @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) ) % 0.31/0.61 => ? [Z3: real] : % 0.31/0.61 ( ( ord_less @ real @ A2 @ Z3 ) % 0.31/0.61 & ( ord_less @ real @ Z3 @ B2 ) % 0.31/0.61 & ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) % 0.31/0.61 = ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( F11 @ Z3 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % MVT2 % 0.31/0.61 thf(fact_5357_DERIV__local__const,axiom, % 0.31/0.61 ! [F2: real > real,L: real,X3: real,D2: real] : % 0.31/0.61 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 ) % 0.31/0.61 => ( ! [Y4: real] : % 0.31/0.61 ( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y4 ) ) @ D2 ) % 0.31/0.61 => ( ( F2 @ X3 ) % 0.31/0.61 = ( F2 @ Y4 ) ) ) % 0.31/0.61 => ( L % 0.31/0.61 = ( zero_zero @ real ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_local_const % 0.31/0.61 thf(fact_5358_DERIV__ln,axiom, % 0.31/0.61 ! [X3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.61 => ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( inverse_inverse @ real @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_ln % 0.31/0.61 thf(fact_5359_DERIV__cos__add,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [K: A,Xa2: A] : % 0.31/0.61 ( has_field_derivative @ A % 0.31/0.61 @ ^ [X: A] : ( cos @ A @ ( plus_plus @ A @ X @ K ) ) % 0.31/0.61 @ ( uminus_uminus @ A @ ( sin @ A @ ( plus_plus @ A @ Xa2 @ K ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ Xa2 @ ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_cos_add % 0.31/0.61 thf(fact_5360_DERIV__power__Suc,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,D6: A,X3: A,S: set @ A,N2: nat] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( has_field_derivative @ A % 0.31/0.61 @ ^ [X: A] : ( power_power @ A @ ( F2 @ X ) @ ( suc @ N2 ) ) % 0.31/0.61 @ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( times_times @ A @ D6 @ ( power_power @ A @ ( F2 @ X3 ) @ N2 ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_power_Suc % 0.31/0.61 thf(fact_5361_DERIV__inverse,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [X3: A,S: set @ A] : % 0.31/0.61 ( ( X3 % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_inverse % 0.31/0.61 thf(fact_5362_DERIV__power,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,D6: A,X3: A,S: set @ A,N2: nat] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( has_field_derivative @ A % 0.31/0.61 @ ^ [X: A] : ( power_power @ A @ ( F2 @ X ) @ N2 ) % 0.31/0.61 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( times_times @ A @ D6 @ ( power_power @ A @ ( F2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_power % 0.31/0.61 thf(fact_5363_DERIV__local__max,axiom, % 0.31/0.61 ! [F2: real > real,L: real,X3: real,D2: real] : % 0.31/0.61 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 ) % 0.31/0.61 => ( ! [Y4: real] : % 0.31/0.61 ( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y4 ) ) @ D2 ) % 0.31/0.61 => ( ord_less_eq @ real @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) ) % 0.31/0.61 => ( L % 0.31/0.61 = ( zero_zero @ real ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_local_max % 0.31/0.61 thf(fact_5364_DERIV__local__min,axiom, % 0.31/0.61 ! [F2: real > real,L: real,X3: real,D2: real] : % 0.31/0.61 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 ) % 0.31/0.61 => ( ! [Y4: real] : % 0.31/0.61 ( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y4 ) ) @ D2 ) % 0.31/0.61 => ( ord_less_eq @ real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) ) % 0.31/0.61 => ( L % 0.31/0.61 = ( zero_zero @ real ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_local_min % 0.31/0.61 thf(fact_5365_DERIV__ln__divide,axiom, % 0.31/0.61 ! [X3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.61 => ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_ln_divide % 0.31/0.61 thf(fact_5366_DERIV__pow,axiom, % 0.31/0.61 ! [N2: nat,X3: real,S: set @ real] : % 0.31/0.61 ( has_field_derivative @ real % 0.31/0.61 @ ^ [X: real] : ( power_power @ real @ X @ N2 ) % 0.31/0.61 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ X3 @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ real @ X3 @ S ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_pow % 0.31/0.61 thf(fact_5367_at__within__Icc__at,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo2564578578187576103pology @ A ) % 0.31/0.61 => ! [A2: A,X3: A,B2: A] : % 0.31/0.61 ( ( ord_less @ A @ A2 @ X3 ) % 0.31/0.61 => ( ( ord_less @ A @ X3 @ B2 ) % 0.31/0.61 => ( ( topolo174197925503356063within @ A @ X3 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.61 = ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % at_within_Icc_at % 0.31/0.61 thf(fact_5368_at__within__Icc__at__left,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo2564578578187576103pology @ A ) % 0.31/0.61 => ! [A2: A,B2: A] : % 0.31/0.61 ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.61 => ( ( topolo174197925503356063within @ A @ B2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.61 = ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % at_within_Icc_at_left % 0.31/0.61 thf(fact_5369_trivial__limit__at__left__bot,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( order_bot @ A ) % 0.31/0.61 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.61 => ( ( topolo174197925503356063within @ A @ ( bot_bot @ A ) @ ( set_ord_lessThan @ A @ ( bot_bot @ A ) ) ) % 0.31/0.61 = ( bot_bot @ ( filter @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % trivial_limit_at_left_bot % 0.31/0.61 thf(fact_5370_DERIV__quotient,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,D2: A,X3: A,S: set @ A,G: A > A,E3: A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( ( has_field_derivative @ A @ G @ E3 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( ( ( G @ X3 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( has_field_derivative @ A % 0.31/0.61 @ ^ [Y2: A] : ( divide_divide @ A @ ( F2 @ Y2 ) @ ( G @ Y2 ) ) % 0.31/0.61 @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D2 @ ( G @ X3 ) ) @ ( times_times @ A @ E3 @ ( F2 @ X3 ) ) ) @ ( power_power @ A @ ( G @ X3 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_quotient % 0.31/0.61 thf(fact_5371_DERIV__inverse__fun,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,D2: A,X3: A,S: set @ A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( ( ( F2 @ X3 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( has_field_derivative @ A % 0.31/0.61 @ ^ [X: A] : ( inverse_inverse @ A @ ( F2 @ X ) ) % 0.31/0.61 @ ( uminus_uminus @ A @ ( times_times @ A @ D2 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F2 @ X3 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_inverse_fun % 0.31/0.61 thf(fact_5372_termdiffs__sums__strong,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [K5: real,C2: nat > A,F2: A > A,F11: A,Z2: A] : % 0.31/0.61 ( ! [Z3: A] : % 0.31/0.61 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 ) % 0.31/0.61 => ( sums @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ Z3 @ N ) ) % 0.31/0.61 @ ( F2 @ Z3 ) ) ) % 0.31/0.61 => ( ( has_field_derivative @ A @ F2 @ F11 @ ( topolo174197925503356063within @ A @ Z2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K5 ) % 0.31/0.61 => ( sums @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) % 0.31/0.61 @ F11 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % termdiffs_sums_strong % 0.31/0.61 thf(fact_5373_has__real__derivative__powr,axiom, % 0.31/0.61 ! [Z2: real,R2: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ Z2 ) % 0.31/0.61 => ( has_field_derivative @ real % 0.31/0.61 @ ^ [Z6: real] : ( powr @ real @ Z6 @ R2 ) % 0.31/0.61 @ ( times_times @ real @ R2 @ ( powr @ real @ Z2 @ ( minus_minus @ real @ R2 @ ( one_one @ real ) ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_real_derivative_powr % 0.31/0.61 thf(fact_5374_termdiffs__strong_H,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [K5: real,C2: nat > A,Z2: A] : % 0.31/0.61 ( ! [Z3: A] : % 0.31/0.61 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 ) % 0.31/0.61 => ( summable @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ Z3 @ N ) ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K5 ) % 0.31/0.61 => ( has_field_derivative @ A % 0.31/0.61 @ ^ [Z6: A] : % 0.31/0.61 ( suminf @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ Z6 @ N ) ) ) % 0.31/0.61 @ ( suminf @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ Z2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % termdiffs_strong' % 0.31/0.61 thf(fact_5375_termdiffs__strong,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [C2: nat > A,K5: A,X3: A] : % 0.31/0.61 ( ( summable @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ K5 @ N ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) ) % 0.31/0.61 => ( has_field_derivative @ A % 0.31/0.61 @ ^ [X: A] : % 0.31/0.61 ( suminf @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ X @ N ) ) ) % 0.31/0.61 @ ( suminf @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % termdiffs_strong % 0.31/0.61 thf(fact_5376_termdiffs,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [C2: nat > A,K5: A,X3: A] : % 0.31/0.61 ( ( summable @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ K5 @ N ) ) ) % 0.31/0.61 => ( ( summable @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ K5 @ N ) ) ) % 0.31/0.61 => ( ( summable @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N ) @ ( power_power @ A @ K5 @ N ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) ) % 0.31/0.61 => ( has_field_derivative @ A % 0.31/0.61 @ ^ [X: A] : % 0.31/0.61 ( suminf @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ X @ N ) ) ) % 0.31/0.61 @ ( suminf @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % termdiffs % 0.31/0.61 thf(fact_5377_DERIV__log,axiom, % 0.31/0.61 ! [X3: real,B2: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.61 => ( has_field_derivative @ real @ ( log @ B2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( times_times @ real @ ( ln_ln @ real @ B2 ) @ X3 ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_log % 0.31/0.61 thf(fact_5378_DERIV__fun__powr,axiom, % 0.31/0.61 ! [G: real > real,M: real,X3: real,R2: real] : % 0.31/0.61 ( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) % 0.31/0.61 => ( has_field_derivative @ real % 0.31/0.61 @ ^ [X: real] : ( powr @ real @ ( G @ X ) @ R2 ) % 0.31/0.61 @ ( times_times @ real @ ( times_times @ real @ R2 @ ( powr @ real @ ( G @ X3 ) @ ( minus_minus @ real @ R2 @ ( semiring_1_of_nat @ real @ ( one_one @ nat ) ) ) ) ) @ M ) % 0.31/0.61 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_fun_powr % 0.31/0.61 thf(fact_5379_DERIV__powr,axiom, % 0.31/0.61 ! [G: real > real,M: real,X3: real,F2: real > real,R2: real] : % 0.31/0.61 ( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) % 0.31/0.61 => ( ( has_field_derivative @ real @ F2 @ R2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ( has_field_derivative @ real % 0.31/0.61 @ ^ [X: real] : ( powr @ real @ ( G @ X ) @ ( F2 @ X ) ) % 0.31/0.61 @ ( times_times @ real @ ( powr @ real @ ( G @ X3 ) @ ( F2 @ X3 ) ) @ ( plus_plus @ real @ ( times_times @ real @ R2 @ ( ln_ln @ real @ ( G @ X3 ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ M @ ( F2 @ X3 ) ) @ ( G @ X3 ) ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_powr % 0.31/0.61 thf(fact_5380_DERIV__tan,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [X3: A] : % 0.31/0.61 ( ( ( cos @ A @ X3 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( has_field_derivative @ A @ ( tan @ A ) @ ( inverse_inverse @ A @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_tan % 0.31/0.61 thf(fact_5381_arcosh__real__has__field__derivative,axiom, % 0.31/0.61 ! [X3: real,A3: set @ real] : % 0.31/0.61 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 0.31/0.61 => ( has_field_derivative @ real @ ( arcosh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ A3 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % arcosh_real_has_field_derivative % 0.31/0.61 thf(fact_5382_DERIV__real__sqrt,axiom, % 0.31/0.61 ! [X3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.61 => ( has_field_derivative @ real @ sqrt @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_real_sqrt % 0.31/0.61 thf(fact_5383_DERIV__cot,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [X3: A] : % 0.31/0.61 ( ( ( sin @ A @ X3 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( has_field_derivative @ A @ ( cot @ A ) @ ( uminus_uminus @ A @ ( inverse_inverse @ A @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_cot % 0.31/0.61 thf(fact_5384_Option_Othese__def,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( these @ A ) % 0.31/0.61 = ( ^ [A6: set @ ( option @ A )] : % 0.31/0.61 ( image @ ( option @ A ) @ A @ ( the2 @ A ) % 0.31/0.61 @ ( collect @ ( option @ A ) % 0.31/0.61 @ ^ [X: option @ A] : % 0.31/0.61 ( ( member @ ( option @ A ) @ X @ A6 ) % 0.31/0.61 & ( X % 0.31/0.61 != ( none @ A ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Option.these_def % 0.31/0.61 thf(fact_5385_has__field__derivative__tanh,axiom, % 0.31/0.61 ! [A10: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A10 ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A10 ) ) % 0.31/0.61 => ! [G: A10 > A10,X3: A10,Db: A10,S: set @ A10] : % 0.31/0.61 ( ( ( cosh @ A10 @ ( G @ X3 ) ) % 0.31/0.61 != ( zero_zero @ A10 ) ) % 0.31/0.61 => ( ( has_field_derivative @ A10 @ G @ Db @ ( topolo174197925503356063within @ A10 @ X3 @ S ) ) % 0.31/0.61 => ( has_field_derivative @ A10 % 0.31/0.61 @ ^ [X: A10] : ( tanh @ A10 @ ( G @ X ) ) % 0.31/0.61 @ ( times_times @ A10 @ ( minus_minus @ A10 @ ( one_one @ A10 ) @ ( power_power @ A10 @ ( tanh @ A10 @ ( G @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db ) % 0.31/0.61 @ ( topolo174197925503356063within @ A10 @ X3 @ S ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_field_derivative_tanh % 0.31/0.61 thf(fact_5386_DERIV__real__sqrt__generic,axiom, % 0.31/0.61 ! [X3: real,D6: real] : % 0.31/0.61 ( ( X3 % 0.31/0.61 != ( zero_zero @ real ) ) % 0.31/0.61 => ( ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.61 => ( D6 % 0.31/0.61 = ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.61 => ( ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) % 0.31/0.61 => ( D6 % 0.31/0.61 = ( divide_divide @ real @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) % 0.31/0.61 => ( has_field_derivative @ real @ sqrt @ D6 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_real_sqrt_generic % 0.31/0.61 thf(fact_5387_artanh__real__has__field__derivative,axiom, % 0.31/0.61 ! [X3: real,A3: set @ real] : % 0.31/0.61 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 0.31/0.61 => ( has_field_derivative @ real @ ( artanh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ A3 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % artanh_real_has_field_derivative % 0.31/0.61 thf(fact_5388_DERIV__real__root,axiom, % 0.31/0.61 ! [N2: nat,X3: real] : % 0.31/0.61 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.61 => ( has_field_derivative @ real @ ( root @ N2 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_real_root % 0.31/0.61 thf(fact_5389_DERIV__arcsin,axiom, % 0.31/0.61 ! [X3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 0.31/0.61 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 0.31/0.61 => ( has_field_derivative @ real @ arcsin @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_arcsin % 0.31/0.61 thf(fact_5390_these__not__empty__eq,axiom, % 0.31/0.61 ! [A: $tType,B3: set @ ( option @ A )] : % 0.31/0.61 ( ( ( these @ A @ B3 ) % 0.31/0.61 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.61 = ( ( B3 % 0.31/0.61 != ( bot_bot @ ( set @ ( option @ A ) ) ) ) % 0.31/0.61 & ( B3 % 0.31/0.61 != ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % these_not_empty_eq % 0.31/0.61 thf(fact_5391_these__empty__eq,axiom, % 0.31/0.61 ! [A: $tType,B3: set @ ( option @ A )] : % 0.31/0.61 ( ( ( these @ A @ B3 ) % 0.31/0.61 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.61 = ( ( B3 % 0.31/0.61 = ( bot_bot @ ( set @ ( option @ A ) ) ) ) % 0.31/0.61 | ( B3 % 0.31/0.61 = ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % these_empty_eq % 0.31/0.61 thf(fact_5392_Maclaurin__all__le__objl,axiom, % 0.31/0.61 ! [Diff: nat > real > real,F2: real > real,X3: real,N2: nat] : % 0.31/0.61 ( ( ( ( Diff @ ( zero_zero @ nat ) ) % 0.31/0.61 = F2 ) % 0.31/0.61 & ! [M3: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) % 0.31/0.61 => ? [T7: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) ) % 0.31/0.61 & ( ( F2 @ X3 ) % 0.31/0.61 = ( plus_plus @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X3 @ M2 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.61 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Maclaurin_all_le_objl % 0.31/0.61 thf(fact_5393_Maclaurin__all__le,axiom, % 0.31/0.61 ! [Diff: nat > real > real,F2: real > real,X3: real,N2: nat] : % 0.31/0.61 ( ( ( Diff @ ( zero_zero @ nat ) ) % 0.31/0.61 = F2 ) % 0.31/0.61 => ( ! [M3: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ? [T7: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) ) % 0.31/0.61 & ( ( F2 @ X3 ) % 0.31/0.61 = ( plus_plus @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X3 @ M2 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.61 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Maclaurin_all_le % 0.31/0.61 thf(fact_5394_Some__image__these__eq,axiom, % 0.31/0.61 ! [A: $tType,A3: set @ ( option @ A )] : % 0.31/0.61 ( ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A3 ) ) % 0.31/0.61 = ( collect @ ( option @ A ) % 0.31/0.61 @ ^ [X: option @ A] : % 0.31/0.61 ( ( member @ ( option @ A ) @ X @ A3 ) % 0.31/0.61 & ( X % 0.31/0.61 != ( none @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Some_image_these_eq % 0.31/0.61 thf(fact_5395_DERIV__odd__real__root,axiom, % 0.31/0.61 ! [N2: nat,X3: real] : % 0.31/0.61 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.31/0.61 => ( ( X3 % 0.31/0.61 != ( zero_zero @ real ) ) % 0.31/0.61 => ( has_field_derivative @ real @ ( root @ N2 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_odd_real_root % 0.31/0.61 thf(fact_5396_Maclaurin__minus,axiom, % 0.31/0.61 ! [H2: real,N2: nat,Diff: nat > real > real,F2: real > real] : % 0.31/0.61 ( ( ord_less @ real @ H2 @ ( zero_zero @ real ) ) % 0.31/0.61 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.61 => ( ( ( Diff @ ( zero_zero @ nat ) ) % 0.31/0.61 = F2 ) % 0.31/0.61 => ( ! [M3: nat,T7: real] : % 0.31/0.61 ( ( ( ord_less @ nat @ M3 @ N2 ) % 0.31/0.61 & ( ord_less_eq @ real @ H2 @ T7 ) % 0.31/0.61 & ( ord_less_eq @ real @ T7 @ ( zero_zero @ real ) ) ) % 0.31/0.61 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 0.31/0.61 => ? [T7: real] : % 0.31/0.61 ( ( ord_less @ real @ H2 @ T7 ) % 0.31/0.61 & ( ord_less @ real @ T7 @ ( zero_zero @ real ) ) % 0.31/0.61 & ( ( F2 @ H2 ) % 0.31/0.61 = ( plus_plus @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.61 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Maclaurin_minus % 0.31/0.61 thf(fact_5397_Maclaurin2,axiom, % 0.31/0.61 ! [H2: real,Diff: nat > real > real,F2: real > real,N2: nat] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ H2 ) % 0.31/0.61 => ( ( ( Diff @ ( zero_zero @ nat ) ) % 0.31/0.61 = F2 ) % 0.31/0.61 => ( ! [M3: nat,T7: real] : % 0.31/0.61 ( ( ( ord_less @ nat @ M3 @ N2 ) % 0.31/0.61 & ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 ) % 0.31/0.61 & ( ord_less_eq @ real @ T7 @ H2 ) ) % 0.31/0.61 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 0.31/0.61 => ? [T7: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ T7 ) % 0.31/0.61 & ( ord_less_eq @ real @ T7 @ H2 ) % 0.31/0.61 & ( ( F2 @ H2 ) % 0.31/0.61 = ( plus_plus @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.61 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ H2 @ N2 ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Maclaurin2 % 0.31/0.61 thf(fact_5398_Maclaurin,axiom, % 0.31/0.61 ! [H2: real,N2: nat,Diff: nat > real > real,F2: real > real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ H2 ) % 0.31/0.61 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.61 => ( ( ( Diff @ ( zero_zero @ nat ) ) % 0.31/0.61 = F2 ) % 0.31/0.61 => ( ! [M3: nat,T7: real] : % 0.31/0.61 ( ( ( ord_less @ nat @ M3 @ N2 ) % 0.31/0.61 & ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 ) % 0.31/0.61 & ( ord_less_eq @ real @ T7 @ H2 ) ) % 0.31/0.61 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 0.31/0.61 => ? [T7: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ T7 ) % 0.31/0.61 & ( ord_less @ real @ T7 @ H2 ) % 0.31/0.61 & ( ( F2 @ H2 ) % 0.31/0.61 = ( plus_plus @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.61 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Maclaurin % 0.31/0.61 thf(fact_5399_Maclaurin__all__lt,axiom, % 0.31/0.61 ! [Diff: nat > real > real,F2: real > real,N2: nat,X3: real] : % 0.31/0.61 ( ( ( Diff @ ( zero_zero @ nat ) ) % 0.31/0.61 = F2 ) % 0.31/0.61 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.61 => ( ( X3 % 0.31/0.61 != ( zero_zero @ real ) ) % 0.31/0.61 => ( ! [M3: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ? [T7: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T7 ) ) % 0.31/0.61 & ( ord_less @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) ) % 0.31/0.61 & ( ( F2 @ X3 ) % 0.31/0.61 = ( plus_plus @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X3 @ M2 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.61 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Maclaurin_all_lt % 0.31/0.61 thf(fact_5400_Maclaurin__bi__le,axiom, % 0.31/0.61 ! [Diff: nat > real > real,F2: real > real,N2: nat,X3: real] : % 0.31/0.61 ( ( ( Diff @ ( zero_zero @ nat ) ) % 0.31/0.61 = F2 ) % 0.31/0.61 => ( ! [M3: nat,T7: real] : % 0.31/0.61 ( ( ( ord_less @ nat @ M3 @ N2 ) % 0.31/0.61 & ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) ) ) % 0.31/0.61 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 0.31/0.61 => ? [T7: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) ) % 0.31/0.61 & ( ( F2 @ X3 ) % 0.31/0.61 = ( plus_plus @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X3 @ M2 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.61 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Maclaurin_bi_le % 0.31/0.61 thf(fact_5401_Taylor,axiom, % 0.31/0.61 ! [N2: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real,X3: real] : % 0.31/0.61 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.61 => ( ( ( Diff @ ( zero_zero @ nat ) ) % 0.31/0.61 = F2 ) % 0.31/0.61 => ( ! [M3: nat,T7: real] : % 0.31/0.61 ( ( ( ord_less @ nat @ M3 @ N2 ) % 0.31/0.61 & ( ord_less_eq @ real @ A2 @ T7 ) % 0.31/0.61 & ( ord_less_eq @ real @ T7 @ B2 ) ) % 0.31/0.61 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 0.31/0.61 => ( ( ord_less_eq @ real @ A2 @ C2 ) % 0.31/0.61 => ( ( ord_less_eq @ real @ C2 @ B2 ) % 0.31/0.61 => ( ( ord_less_eq @ real @ A2 @ X3 ) % 0.31/0.61 => ( ( ord_less_eq @ real @ X3 @ B2 ) % 0.31/0.61 => ( ( X3 != C2 ) % 0.31/0.61 => ? [T7: real] : % 0.31/0.61 ( ( ( ord_less @ real @ X3 @ C2 ) % 0.31/0.61 => ( ( ord_less @ real @ X3 @ T7 ) % 0.31/0.61 & ( ord_less @ real @ T7 @ C2 ) ) ) % 0.31/0.61 & ( ~ ( ord_less @ real @ X3 @ C2 ) % 0.31/0.61 => ( ( ord_less @ real @ C2 @ T7 ) % 0.31/0.61 & ( ord_less @ real @ T7 @ X3 ) ) ) % 0.31/0.61 & ( ( F2 @ X3 ) % 0.31/0.61 = ( plus_plus @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ C2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X3 @ C2 ) @ M2 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.61 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X3 @ C2 ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Taylor % 0.31/0.61 thf(fact_5402_Taylor__up,axiom, % 0.31/0.61 ! [N2: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real] : % 0.31/0.61 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.61 => ( ( ( Diff @ ( zero_zero @ nat ) ) % 0.31/0.61 = F2 ) % 0.31/0.61 => ( ! [M3: nat,T7: real] : % 0.31/0.61 ( ( ( ord_less @ nat @ M3 @ N2 ) % 0.31/0.61 & ( ord_less_eq @ real @ A2 @ T7 ) % 0.31/0.61 & ( ord_less_eq @ real @ T7 @ B2 ) ) % 0.31/0.61 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 0.31/0.61 => ( ( ord_less_eq @ real @ A2 @ C2 ) % 0.31/0.61 => ( ( ord_less @ real @ C2 @ B2 ) % 0.31/0.61 => ? [T7: real] : % 0.31/0.61 ( ( ord_less @ real @ C2 @ T7 ) % 0.31/0.61 & ( ord_less @ real @ T7 @ B2 ) % 0.31/0.61 & ( ( F2 @ B2 ) % 0.31/0.61 = ( plus_plus @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ C2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ M2 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.61 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ N2 ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Taylor_up % 0.31/0.61 thf(fact_5403_Taylor__down,axiom, % 0.31/0.61 ! [N2: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real] : % 0.31/0.61 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.61 => ( ( ( Diff @ ( zero_zero @ nat ) ) % 0.31/0.61 = F2 ) % 0.31/0.61 => ( ! [M3: nat,T7: real] : % 0.31/0.61 ( ( ( ord_less @ nat @ M3 @ N2 ) % 0.31/0.61 & ( ord_less_eq @ real @ A2 @ T7 ) % 0.31/0.61 & ( ord_less_eq @ real @ T7 @ B2 ) ) % 0.31/0.61 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ A2 @ C2 ) % 0.31/0.61 => ( ( ord_less_eq @ real @ C2 @ B2 ) % 0.31/0.61 => ? [T7: real] : % 0.31/0.61 ( ( ord_less @ real @ A2 @ T7 ) % 0.31/0.61 & ( ord_less @ real @ T7 @ C2 ) % 0.31/0.61 & ( ( F2 @ A2 ) % 0.31/0.61 = ( plus_plus @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ C2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ M2 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.61 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ N2 ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Taylor_down % 0.31/0.61 thf(fact_5404_Maclaurin__lemma2,axiom, % 0.31/0.61 ! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B3: real] : % 0.31/0.61 ( ! [M3: nat,T7: real] : % 0.31/0.61 ( ( ( ord_less @ nat @ M3 @ N2 ) % 0.31/0.61 & ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 ) % 0.31/0.61 & ( ord_less_eq @ real @ T7 @ H2 ) ) % 0.31/0.61 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 0.31/0.61 => ( ( N2 % 0.31/0.61 = ( suc @ K ) ) % 0.31/0.61 => ! [M6: nat,T9: real] : % 0.31/0.61 ( ( ( ord_less @ nat @ M6 @ N2 ) % 0.31/0.61 & ( ord_less_eq @ real @ ( zero_zero @ real ) @ T9 ) % 0.31/0.61 & ( ord_less_eq @ real @ T9 @ H2 ) ) % 0.31/0.61 => ( has_field_derivative @ real % 0.31/0.61 @ ^ [U2: real] : % 0.31/0.61 ( minus_minus @ real @ ( Diff @ M6 @ U2 ) % 0.31/0.61 @ ( plus_plus @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M6 @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ U2 @ P6 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ M6 ) ) ) % 0.31/0.61 @ ( times_times @ real @ B3 @ ( divide_divide @ real @ ( power_power @ real @ U2 @ ( minus_minus @ nat @ N2 @ M6 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N2 @ M6 ) ) ) ) ) ) % 0.31/0.61 @ ( minus_minus @ real @ ( Diff @ ( suc @ M6 ) @ T9 ) % 0.31/0.61 @ ( plus_plus @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M6 ) @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ T9 @ P6 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ M6 ) ) ) ) % 0.31/0.61 @ ( times_times @ real @ B3 @ ( divide_divide @ real @ ( power_power @ real @ T9 @ ( minus_minus @ nat @ N2 @ ( suc @ M6 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N2 @ ( suc @ M6 ) ) ) ) ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ real @ T9 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Maclaurin_lemma2 % 0.31/0.61 thf(fact_5405_DERIV__arctan__series,axiom, % 0.31/0.61 ! [X3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 0.31/0.61 => ( has_field_derivative @ real % 0.31/0.61 @ ^ [X9: real] : % 0.31/0.61 ( suminf @ real % 0.31/0.61 @ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X9 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) % 0.31/0.61 @ ( suminf @ real % 0.31/0.61 @ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( power_power @ real @ X3 @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_arctan_series % 0.31/0.61 thf(fact_5406_DERIV__arccos,axiom, % 0.31/0.61 ! [X3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 0.31/0.61 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 0.31/0.61 => ( has_field_derivative @ real @ arccos @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_arccos % 0.31/0.61 thf(fact_5407_DERIV__power__series_H,axiom, % 0.31/0.61 ! [R: real,F2: nat > real,X0: real] : % 0.31/0.61 ( ! [X5: real] : % 0.31/0.61 ( ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) ) % 0.31/0.61 => ( summable @ real % 0.31/0.61 @ ^ [N: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N ) @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) @ ( power_power @ real @ X5 @ N ) ) ) ) % 0.31/0.61 => ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ R ) % 0.31/0.61 => ( has_field_derivative @ real % 0.31/0.61 @ ^ [X: real] : % 0.31/0.61 ( suminf @ real % 0.31/0.61 @ ^ [N: nat] : ( times_times @ real @ ( F2 @ N ) @ ( power_power @ real @ X @ ( suc @ N ) ) ) ) % 0.31/0.61 @ ( suminf @ real % 0.31/0.61 @ ^ [N: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N ) @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) @ ( power_power @ real @ X0 @ N ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_power_series' % 0.31/0.61 thf(fact_5408_has__derivative__arcsin,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [G: A > real,X3: A,G7: A > real,S: set @ A] : % 0.31/0.61 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X3 ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( G @ X3 ) @ ( one_one @ real ) ) % 0.31/0.61 => ( ( has_derivative @ A @ real @ G @ G7 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( has_derivative @ A @ real % 0.31/0.61 @ ^ [X: A] : ( arcsin @ ( G @ X ) ) % 0.31/0.61 @ ^ [X: A] : ( times_times @ real @ ( G7 @ X ) @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_arcsin % 0.31/0.61 thf(fact_5409_greaterThanLessThan__iff,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ord @ A ) % 0.31/0.61 => ! [I: A,L: A,U: A] : % 0.31/0.61 ( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U ) ) % 0.31/0.61 = ( ( ord_less @ A @ L @ I ) % 0.31/0.61 & ( ord_less @ A @ I @ U ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % greaterThanLessThan_iff % 0.31/0.61 thf(fact_5410_greaterThanLessThan__empty__iff2,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( dense_linorder @ A ) % 0.31/0.61 => ! [A2: A,B2: A] : % 0.31/0.61 ( ( ( bot_bot @ ( set @ A ) ) % 0.31/0.61 = ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) % 0.31/0.61 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % greaterThanLessThan_empty_iff2 % 0.31/0.61 thf(fact_5411_greaterThanLessThan__empty__iff,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( dense_linorder @ A ) % 0.31/0.61 => ! [A2: A,B2: A] : % 0.31/0.61 ( ( ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) % 0.31/0.61 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.61 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % greaterThanLessThan_empty_iff % 0.31/0.61 thf(fact_5412_greaterThanLessThan__empty,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( order @ A ) % 0.31/0.61 => ! [L: A,K: A] : % 0.31/0.61 ( ( ord_less_eq @ A @ L @ K ) % 0.31/0.61 => ( ( set_or5935395276787703475ssThan @ A @ K @ L ) % 0.31/0.61 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % greaterThanLessThan_empty % 0.31/0.61 thf(fact_5413_infinite__Ioo__iff,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( dense_linorder @ A ) % 0.31/0.61 => ! [A2: A,B2: A] : % 0.31/0.61 ( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) % 0.31/0.61 = ( ord_less @ A @ A2 @ B2 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % infinite_Ioo_iff % 0.31/0.61 thf(fact_5414_cSup__greaterThanLessThan,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( condit6923001295902523014norder @ A ) % 0.31/0.61 & ( dense_linorder @ A ) ) % 0.31/0.61 => ! [Y: A,X3: A] : % 0.31/0.61 ( ( ord_less @ A @ Y @ X3 ) % 0.31/0.61 => ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X3 ) ) % 0.31/0.61 = X3 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % cSup_greaterThanLessThan % 0.31/0.61 thf(fact_5415_Sup__greaterThanLessThan,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( comple6319245703460814977attice @ A ) % 0.31/0.61 & ( dense_linorder @ A ) ) % 0.31/0.61 => ! [X3: A,Y: A] : % 0.31/0.61 ( ( ord_less @ A @ X3 @ Y ) % 0.31/0.61 => ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X3 @ Y ) ) % 0.31/0.61 = Y ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Sup_greaterThanLessThan % 0.31/0.61 thf(fact_5416_cInf__greaterThanLessThan,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( condit6923001295902523014norder @ A ) % 0.31/0.61 & ( dense_linorder @ A ) ) % 0.31/0.61 => ! [Y: A,X3: A] : % 0.31/0.61 ( ( ord_less @ A @ Y @ X3 ) % 0.31/0.61 => ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X3 ) ) % 0.31/0.61 = Y ) ) ) ). % 0.31/0.61 % 0.31/0.61 % cInf_greaterThanLessThan % 0.31/0.61 thf(fact_5417_Inf__greaterThanLessThan,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( comple6319245703460814977attice @ A ) % 0.31/0.61 & ( dense_linorder @ A ) ) % 0.31/0.61 => ! [X3: A,Y: A] : % 0.31/0.61 ( ( ord_less @ A @ X3 @ Y ) % 0.31/0.61 => ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X3 @ Y ) ) % 0.31/0.61 = X3 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Inf_greaterThanLessThan % 0.31/0.61 thf(fact_5418_has__derivative__subset,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [F2: A > B,F11: A > B,X3: A,S: set @ A,T2: set @ A] : % 0.31/0.61 ( ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( ( ord_less_eq @ ( set @ A ) @ T2 @ S ) % 0.31/0.61 => ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_subset % 0.31/0.61 thf(fact_5419_has__derivative__scaleR,axiom, % 0.31/0.61 ! [C: $tType,D: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ D ) % 0.31/0.61 & ( real_V822414075346904944vector @ C ) ) % 0.31/0.61 => ! [F2: D > real,F11: D > real,X3: D,S: set @ D,G: D > C,G7: D > C] : % 0.31/0.61 ( ( has_derivative @ D @ real @ F2 @ F11 @ ( topolo174197925503356063within @ D @ X3 @ S ) ) % 0.31/0.61 => ( ( has_derivative @ D @ C @ G @ G7 @ ( topolo174197925503356063within @ D @ X3 @ S ) ) % 0.31/0.61 => ( has_derivative @ D @ C % 0.31/0.61 @ ^ [X: D] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ^ [H: D] : ( plus_plus @ C @ ( real_V8093663219630862766scaleR @ C @ ( F2 @ X3 ) @ ( G7 @ H ) ) @ ( real_V8093663219630862766scaleR @ C @ ( F11 @ H ) @ ( G @ X3 ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ D @ X3 @ S ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_scaleR % 0.31/0.61 thf(fact_5420_has__derivative__add,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [F2: A > B,F11: A > B,F5: filter @ A,G: A > B,G7: A > B] : % 0.31/0.61 ( ( has_derivative @ A @ B @ F2 @ F11 @ F5 ) % 0.31/0.61 => ( ( has_derivative @ A @ B @ G @ G7 @ F5 ) % 0.31/0.61 => ( has_derivative @ A @ B % 0.31/0.61 @ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ^ [X: A] : ( plus_plus @ B @ ( F11 @ X ) @ ( G7 @ X ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_add % 0.31/0.61 thf(fact_5421_has__derivative__const,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [C2: B,F5: filter @ A] : % 0.31/0.61 ( has_derivative @ A @ B % 0.31/0.61 @ ^ [X: A] : C2 % 0.31/0.61 @ ^ [X: A] : ( zero_zero @ B ) % 0.31/0.61 @ F5 ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_const % 0.31/0.61 thf(fact_5422_infinite__Ioo,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( dense_linorder @ A ) % 0.31/0.61 => ! [A2: A,B2: A] : % 0.31/0.61 ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.61 => ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % infinite_Ioo % 0.31/0.61 thf(fact_5423_has__derivative__zero__unique,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [F5: A > B,X3: A] : % 0.31/0.61 ( ( has_derivative @ A @ B % 0.31/0.61 @ ^ [X: A] : ( zero_zero @ B ) % 0.31/0.61 @ F5 % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 => ( F5 % 0.31/0.61 = ( ^ [H: A] : ( zero_zero @ B ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_zero_unique % 0.31/0.61 thf(fact_5424_has__derivative__mult,axiom, % 0.31/0.61 ! [A: $tType,D: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ D ) % 0.31/0.61 & ( real_V4412858255891104859lgebra @ A ) ) % 0.31/0.61 => ! [F2: D > A,F11: D > A,X3: D,S: set @ D,G: D > A,G7: D > A] : % 0.31/0.61 ( ( has_derivative @ D @ A @ F2 @ F11 @ ( topolo174197925503356063within @ D @ X3 @ S ) ) % 0.31/0.61 => ( ( has_derivative @ D @ A @ G @ G7 @ ( topolo174197925503356063within @ D @ X3 @ S ) ) % 0.31/0.61 => ( has_derivative @ D @ A % 0.31/0.61 @ ^ [X: D] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ^ [H: D] : ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X3 ) @ ( G7 @ H ) ) @ ( times_times @ A @ ( F11 @ H ) @ ( G @ X3 ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ D @ X3 @ S ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_mult % 0.31/0.61 thf(fact_5425_has__derivative__in__compose2,axiom, % 0.31/0.61 ! [B: $tType,A: $tType,C: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ C ) % 0.31/0.61 & ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [T2: set @ A,G: A > B,G7: A > A > B,F2: C > A,S: set @ C,X3: C,F11: C > A] : % 0.31/0.61 ( ! [X5: A] : % 0.31/0.61 ( ( member @ A @ X5 @ T2 ) % 0.31/0.61 => ( has_derivative @ A @ B @ G @ ( G7 @ X5 ) @ ( topolo174197925503356063within @ A @ X5 @ T2 ) ) ) % 0.31/0.61 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ S ) @ T2 ) % 0.31/0.61 => ( ( member @ C @ X3 @ S ) % 0.31/0.61 => ( ( has_derivative @ C @ A @ F2 @ F11 @ ( topolo174197925503356063within @ C @ X3 @ S ) ) % 0.31/0.61 => ( has_derivative @ C @ B % 0.31/0.61 @ ^ [X: C] : ( G @ ( F2 @ X ) ) % 0.31/0.61 @ ^ [Y2: C] : ( G7 @ ( F2 @ X3 ) @ ( F11 @ Y2 ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ C @ X3 @ S ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_in_compose2 % 0.31/0.61 thf(fact_5426_greaterThanLessThan__subseteq__greaterThanLessThan,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( dense_linorder @ A ) % 0.31/0.61 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.61 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D2 ) ) % 0.31/0.61 = ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.61 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.61 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % greaterThanLessThan_subseteq_greaterThanLessThan % 0.31/0.61 thf(fact_5427_ivl__disj__int__two_I4_J,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( order @ A ) % 0.31/0.61 => ! [L: A,M: A,U: A] : % 0.31/0.61 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) ) % 0.31/0.61 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % ivl_disj_int_two(4) % 0.31/0.61 thf(fact_5428_ivl__disj__int__two_I5_J,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( order @ A ) % 0.31/0.61 => ! [L: A,M: A,U: A] : % 0.31/0.61 ( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) ) % 0.31/0.61 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % ivl_disj_int_two(5) % 0.31/0.61 thf(fact_5429_ivl__disj__int__two_I1_J,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( order @ A ) % 0.31/0.61 => ! [L: A,M: A,U: A] : % 0.31/0.61 ( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) ) % 0.31/0.61 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % ivl_disj_int_two(1) % 0.31/0.61 thf(fact_5430_ivl__disj__int__one_I1_J,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( order @ A ) % 0.31/0.61 => ! [L: A,U: A] : % 0.31/0.61 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) ) % 0.31/0.61 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % ivl_disj_int_one(1) % 0.31/0.61 thf(fact_5431_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( dense_linorder @ A ) % 0.31/0.61 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.61 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) ) % 0.31/0.61 = ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.61 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.61 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % greaterThanLessThan_subseteq_atLeastAtMost_iff % 0.31/0.61 thf(fact_5432_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( dense_linorder @ A ) % 0.31/0.61 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.61 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) ) % 0.31/0.61 = ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.61 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.61 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % greaterThanLessThan_subseteq_atLeastLessThan_iff % 0.31/0.61 thf(fact_5433_atLeastAtMost__diff__ends,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( linorder @ A ) % 0.31/0.61 => ! [A2: A,B2: A] : % 0.31/0.61 ( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.61 = ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % atLeastAtMost_diff_ends % 0.31/0.61 thf(fact_5434_has__derivative__divide_H,axiom, % 0.31/0.61 ! [A: $tType,C: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ C ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [F2: C > A,F11: C > A,X3: C,S3: set @ C,G: C > A,G7: C > A] : % 0.31/0.61 ( ( has_derivative @ C @ A @ F2 @ F11 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) % 0.31/0.61 => ( ( has_derivative @ C @ A @ G @ G7 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) % 0.31/0.61 => ( ( ( G @ X3 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( has_derivative @ C @ A % 0.31/0.61 @ ^ [X: C] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ^ [H: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F11 @ H ) @ ( G @ X3 ) ) @ ( times_times @ A @ ( F2 @ X3 ) @ ( G7 @ H ) ) ) @ ( times_times @ A @ ( G @ X3 ) @ ( G @ X3 ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_divide' % 0.31/0.61 thf(fact_5435_arccos__less__arccos,axiom, % 0.31/0.61 ! [X3: real,Y: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 0.31/0.61 => ( ( ord_less @ real @ X3 @ Y ) % 0.31/0.61 => ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) ) % 0.31/0.61 => ( ord_less @ real @ ( arccos @ Y ) @ ( arccos @ X3 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % arccos_less_arccos % 0.31/0.61 thf(fact_5436_arccos__less__mono,axiom, % 0.31/0.61 ! [X3: real,Y: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 0.31/0.61 => ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( arccos @ X3 ) @ ( arccos @ Y ) ) % 0.31/0.61 = ( ord_less @ real @ Y @ X3 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % arccos_less_mono % 0.31/0.61 thf(fact_5437_has__derivative__inverse,axiom, % 0.31/0.61 ! [A: $tType,C: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ C ) % 0.31/0.61 & ( real_V8999393235501362500lgebra @ A ) ) % 0.31/0.61 => ! [F2: C > A,X3: C,F11: C > A,S3: set @ C] : % 0.31/0.61 ( ( ( F2 @ X3 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( ( has_derivative @ C @ A @ F2 @ F11 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) % 0.31/0.61 => ( has_derivative @ C @ A % 0.31/0.61 @ ^ [X: C] : ( inverse_inverse @ A @ ( F2 @ X ) ) % 0.31/0.61 @ ^ [H: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) @ ( F11 @ H ) ) @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_inverse % 0.31/0.61 thf(fact_5438_has__derivative__inverse_H,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.61 => ! [X3: A,S3: set @ A] : % 0.31/0.61 ( ( X3 % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( has_derivative @ A @ A @ ( inverse_inverse @ A ) % 0.31/0.61 @ ^ [H: A] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ X3 ) @ H ) @ ( inverse_inverse @ A @ X3 ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_inverse' % 0.31/0.61 thf(fact_5439_DERIV__isconst3,axiom, % 0.31/0.61 ! [A2: real,B2: real,X3: real,Y: real,F2: real > real] : % 0.31/0.61 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.61 => ( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) ) % 0.31/0.61 => ( ( member @ real @ Y @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) ) % 0.31/0.61 => ( ! [X5: real] : % 0.31/0.61 ( ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) ) % 0.31/0.61 => ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) % 0.31/0.61 => ( ( F2 @ X3 ) % 0.31/0.61 = ( F2 @ Y ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_isconst3 % 0.31/0.61 thf(fact_5440_arccos__lt__bounded,axiom, % 0.31/0.61 ! [Y: real] : % 0.31/0.61 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y ) % 0.31/0.61 => ( ( ord_less @ real @ Y @ ( one_one @ real ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) ) % 0.31/0.61 & ( ord_less @ real @ ( arccos @ Y ) @ pi ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % arccos_lt_bounded % 0.31/0.61 thf(fact_5441_has__derivative__ln,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [G: A > real,X3: A,G7: A > real,S: set @ A] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) % 0.31/0.61 => ( ( has_derivative @ A @ real @ G @ G7 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( has_derivative @ A @ real % 0.31/0.61 @ ^ [X: A] : ( ln_ln @ real @ ( G @ X ) ) % 0.31/0.61 @ ^ [X: A] : ( times_times @ real @ ( G7 @ X ) @ ( inverse_inverse @ real @ ( G @ X3 ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_ln % 0.31/0.61 thf(fact_5442_sin__arccos__nonzero,axiom, % 0.31/0.61 ! [X3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 0.31/0.61 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 0.31/0.61 => ( ( sin @ real @ ( arccos @ X3 ) ) % 0.31/0.61 != ( zero_zero @ real ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % sin_arccos_nonzero % 0.31/0.61 thf(fact_5443_has__derivative__arccos,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [G: A > real,X3: A,G7: A > real,S: set @ A] : % 0.31/0.61 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X3 ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( G @ X3 ) @ ( one_one @ real ) ) % 0.31/0.61 => ( ( has_derivative @ A @ real @ G @ G7 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( has_derivative @ A @ real % 0.31/0.61 @ ^ [X: A] : ( arccos @ ( G @ X ) ) % 0.31/0.61 @ ^ [X: A] : ( times_times @ real @ ( G7 @ X ) @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_arccos % 0.31/0.61 thf(fact_5444_has__derivative__divide,axiom, % 0.31/0.61 ! [A: $tType,C: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ C ) % 0.31/0.61 & ( real_V8999393235501362500lgebra @ A ) ) % 0.31/0.61 => ! [F2: C > A,F11: C > A,X3: C,S3: set @ C,G: C > A,G7: C > A] : % 0.31/0.61 ( ( has_derivative @ C @ A @ F2 @ F11 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) % 0.31/0.61 => ( ( has_derivative @ C @ A @ G @ G7 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) % 0.31/0.61 => ( ( ( G @ X3 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( has_derivative @ C @ A % 0.31/0.61 @ ^ [X: C] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ^ [H: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F2 @ X3 ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G @ X3 ) ) @ ( G7 @ H ) ) @ ( inverse_inverse @ A @ ( G @ X3 ) ) ) ) @ ( divide_divide @ A @ ( F11 @ H ) @ ( G @ X3 ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_divide % 0.31/0.61 thf(fact_5445_has__derivative__prod,axiom, % 0.31/0.61 ! [B: $tType,I7: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ B ) ) % 0.31/0.61 => ! [I6: set @ I7,F2: I7 > A > B,F11: I7 > A > B,X3: A,S3: set @ A] : % 0.31/0.61 ( ! [I4: I7] : % 0.31/0.61 ( ( member @ I7 @ I4 @ I6 ) % 0.31/0.61 => ( has_derivative @ A @ B @ ( F2 @ I4 ) @ ( F11 @ I4 ) @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) % 0.31/0.61 => ( has_derivative @ A @ B % 0.31/0.61 @ ^ [X: A] : % 0.31/0.61 ( groups7121269368397514597t_prod @ I7 @ B % 0.31/0.61 @ ^ [I3: I7] : ( F2 @ I3 @ X ) % 0.31/0.61 @ I6 ) % 0.31/0.61 @ ^ [Y2: A] : % 0.31/0.61 ( groups7311177749621191930dd_sum @ I7 @ B % 0.31/0.61 @ ^ [I3: I7] : % 0.31/0.61 ( times_times @ B @ ( F11 @ I3 @ Y2 ) % 0.31/0.61 @ ( groups7121269368397514597t_prod @ I7 @ B % 0.31/0.61 @ ^ [J3: I7] : ( F2 @ J3 @ X3 ) % 0.31/0.61 @ ( minus_minus @ ( set @ I7 ) @ I6 @ ( insert @ I7 @ I3 @ ( bot_bot @ ( set @ I7 ) ) ) ) ) ) % 0.31/0.61 @ I6 ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_prod % 0.31/0.61 thf(fact_5446_has__derivative__powr,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [G: A > real,G7: A > real,X3: A,X8: set @ A,F2: A > real,F11: A > real] : % 0.31/0.61 ( ( has_derivative @ A @ real @ G @ G7 @ ( topolo174197925503356063within @ A @ X3 @ X8 ) ) % 0.31/0.61 => ( ( has_derivative @ A @ real @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ X8 ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) % 0.31/0.61 => ( ( member @ A @ X3 @ X8 ) % 0.31/0.61 => ( has_derivative @ A @ real % 0.31/0.61 @ ^ [X: A] : ( powr @ real @ ( G @ X ) @ ( F2 @ X ) ) % 0.31/0.61 @ ^ [H: A] : ( times_times @ real @ ( powr @ real @ ( G @ X3 ) @ ( F2 @ X3 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( F11 @ H ) @ ( ln_ln @ real @ ( G @ X3 ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G7 @ H ) @ ( F2 @ X3 ) ) @ ( G @ X3 ) ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ X8 ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_powr % 0.31/0.61 thf(fact_5447_has__derivative__real__sqrt,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [G: A > real,X3: A,G7: A > real,S: set @ A] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) % 0.31/0.61 => ( ( has_derivative @ A @ real @ G @ G7 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 => ( has_derivative @ A @ real % 0.31/0.61 @ ^ [X: A] : ( sqrt @ ( G @ X ) ) % 0.31/0.61 @ ^ [X: A] : ( times_times @ real @ ( G7 @ X ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G @ X3 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_real_sqrt % 0.31/0.61 thf(fact_5448_termdiffs__aux,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [C2: nat > A,K5: A,X3: A] : % 0.31/0.61 ( ( summable @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N ) @ ( power_power @ A @ K5 @ N ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) ) % 0.31/0.61 => ( filterlim @ A @ A % 0.31/0.61 @ ^ [H: A] : % 0.31/0.61 ( suminf @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X3 @ H ) @ N ) @ ( power_power @ A @ X3 @ N ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ X3 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % termdiffs_aux % 0.31/0.61 thf(fact_5449_isCont__powser,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [C2: nat > A,K5: A,X3: A] : % 0.31/0.61 ( ( summable @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ K5 @ N ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 @ ^ [X: A] : % 0.31/0.61 ( suminf @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_powser % 0.31/0.61 thf(fact_5450_isCont__powser_H,axiom, % 0.31/0.61 ! [Aa: $tType,A: $tType] : % 0.31/0.61 ( ( ( topological_t2_space @ A ) % 0.31/0.61 & ( real_Vector_banach @ Aa ) % 0.31/0.61 & ( real_V3459762299906320749_field @ Aa ) ) % 0.31/0.61 => ! [A2: A,F2: A > Aa,C2: nat > Aa,K5: Aa] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 ) % 0.31/0.61 => ( ( summable @ Aa % 0.31/0.61 @ ^ [N: nat] : ( times_times @ Aa @ ( C2 @ N ) @ ( power_power @ Aa @ K5 @ N ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ Aa @ ( F2 @ A2 ) ) @ ( real_V7770717601297561774m_norm @ Aa @ K5 ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 @ ^ [X: A] : % 0.31/0.61 ( suminf @ Aa % 0.31/0.61 @ ^ [N: nat] : ( times_times @ Aa @ ( C2 @ N ) @ ( power_power @ Aa @ ( F2 @ X ) @ N ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_powser' % 0.31/0.61 thf(fact_5451_finite__greaterThanLessThan,axiom, % 0.31/0.61 ! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ). % 0.31/0.61 % 0.31/0.61 % finite_greaterThanLessThan % 0.31/0.61 thf(fact_5452_finite__greaterThanLessThan__int,axiom, % 0.31/0.61 ! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ). % 0.31/0.61 % 0.31/0.61 % finite_greaterThanLessThan_int % 0.31/0.61 thf(fact_5453_card__greaterThanLessThan,axiom, % 0.31/0.61 ! [L: nat,U: nat] : % 0.31/0.61 ( ( finite_card @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) % 0.31/0.61 = ( minus_minus @ nat @ U @ ( suc @ L ) ) ) ). % 0.31/0.61 % 0.31/0.61 % card_greaterThanLessThan % 0.31/0.61 thf(fact_5454_tendsto__mult__left__iff,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( ( field @ A ) % 0.31/0.61 & ( topolo4211221413907600880p_mult @ A ) ) % 0.31/0.61 => ! [C2: A,F2: B > A,L: A,F5: filter @ B] : % 0.31/0.61 ( ( C2 % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( ( filterlim @ B @ A % 0.31/0.61 @ ^ [X: B] : ( times_times @ A @ C2 @ ( F2 @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 = ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_mult_left_iff % 0.31/0.61 thf(fact_5455_tendsto__mult__right__iff,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( ( field @ A ) % 0.31/0.61 & ( topolo4211221413907600880p_mult @ A ) ) % 0.31/0.61 => ! [C2: A,F2: B > A,L: A,F5: filter @ B] : % 0.31/0.61 ( ( C2 % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( ( filterlim @ B @ A % 0.31/0.61 @ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ C2 ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C2 ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 = ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_mult_right_iff % 0.31/0.61 thf(fact_5456_power__tendsto__0__iff,axiom, % 0.31/0.61 ! [A: $tType,N2: nat,F2: A > real,F5: filter @ A] : % 0.31/0.61 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.61 => ( ( filterlim @ A @ real % 0.31/0.61 @ ^ [X: A] : ( power_power @ real @ ( F2 @ X ) @ N2 ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 = ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % power_tendsto_0_iff % 0.31/0.61 thf(fact_5457_isCont__Pair,axiom, % 0.31/0.61 ! [C: $tType,B: $tType,A: $tType] : % 0.31/0.61 ( ( ( topological_t2_space @ A ) % 0.31/0.61 & ( topolo4958980785337419405_space @ B ) % 0.31/0.61 & ( topolo4958980785337419405_space @ C ) ) % 0.31/0.61 => ! [A2: A,F2: A > B,G: A > C] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 ) % 0.31/0.61 => ( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 @ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_Pair % 0.31/0.61 thf(fact_5458_isCont__LIM__compose2,axiom, % 0.31/0.61 ! [C: $tType,B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( topolo4958980785337419405_space @ B ) % 0.31/0.61 & ( topolo4958980785337419405_space @ C ) ) % 0.31/0.61 => ! [A2: A,F2: A > B,G: B > C,L: C] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 ) % 0.31/0.61 => ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ L ) @ ( topolo174197925503356063within @ B @ ( F2 @ A2 ) @ ( top_top @ ( set @ B ) ) ) ) % 0.31/0.61 => ( ? [D4: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 ) % 0.31/0.61 & ! [X5: A] : % 0.31/0.61 ( ( ( X5 != A2 ) % 0.31/0.61 & ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X5 @ A2 ) ) @ D4 ) ) % 0.31/0.61 => ( ( F2 @ X5 ) % 0.31/0.61 != ( F2 @ A2 ) ) ) ) % 0.31/0.61 => ( filterlim @ A @ C % 0.31/0.61 @ ^ [X: A] : ( G @ ( F2 @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ C @ L ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_LIM_compose2 % 0.31/0.61 thf(fact_5459_isCont__iff,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.61 => ! [X3: A,F2: A > B] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F2 ) % 0.31/0.61 = ( filterlim @ A @ B % 0.31/0.61 @ ^ [H: A] : ( F2 @ ( plus_plus @ A @ X3 @ H ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X3 ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_iff % 0.31/0.61 thf(fact_5460_LIM__not__zero,axiom, % 0.31/0.61 ! [Aa: $tType,A: $tType] : % 0.31/0.61 ( ( ( topolo8386298272705272623_space @ A ) % 0.31/0.61 & ( zero @ Aa ) % 0.31/0.61 & ( topological_t2_space @ Aa ) ) % 0.31/0.61 => ! [K: Aa,A2: A] : % 0.31/0.61 ( ( K % 0.31/0.61 != ( zero_zero @ Aa ) ) % 0.31/0.61 => ~ ( filterlim @ A @ Aa % 0.31/0.61 @ ^ [X: A] : K % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_not_zero % 0.31/0.61 thf(fact_5461_tendsto__within__subset,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.61 => ! [F2: A > B,L: filter @ B,X3: A,S3: set @ A,T4: set @ A] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ L @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) % 0.31/0.61 => ( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 ) % 0.31/0.61 => ( filterlim @ A @ B @ F2 @ L @ ( topolo174197925503356063within @ A @ X3 @ T4 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_within_subset % 0.31/0.61 thf(fact_5462_tendsto__arcosh,axiom, % 0.31/0.61 ! [B: $tType,F2: B > real,A2: real,F5: filter @ B] : % 0.31/0.61 ( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 ) % 0.31/0.61 => ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 0.31/0.61 => ( filterlim @ B @ real % 0.31/0.61 @ ^ [X: B] : ( arcosh @ real @ ( F2 @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) ) % 0.31/0.61 @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_arcosh % 0.31/0.61 thf(fact_5463_tendsto__const__iff,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( topological_t2_space @ A ) % 0.31/0.61 => ! [F5: filter @ B,A2: A,B2: A] : % 0.31/0.61 ( ( F5 % 0.31/0.61 != ( bot_bot @ ( filter @ B ) ) ) % 0.31/0.61 => ( ( filterlim @ B @ A % 0.31/0.61 @ ^ [X: B] : A2 % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ B2 ) % 0.31/0.61 @ F5 ) % 0.31/0.61 = ( A2 = B2 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_const_iff % 0.31/0.61 thf(fact_5464_tendsto__unique,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( topological_t2_space @ A ) % 0.31/0.61 => ! [F5: filter @ B,F2: B > A,A2: A,B2: A] : % 0.31/0.61 ( ( F5 % 0.31/0.61 != ( bot_bot @ ( filter @ B ) ) ) % 0.31/0.61 => ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 0.31/0.61 => ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 ) % 0.31/0.61 => ( A2 = B2 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_unique % 0.31/0.61 thf(fact_5465_tendsto__bot,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.61 => ! [F2: B > A,A2: A] : ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( bot_bot @ ( filter @ B ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_bot % 0.31/0.61 thf(fact_5466_nhds__neq__bot,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.61 => ! [A2: A] : % 0.31/0.61 ( ( topolo7230453075368039082e_nhds @ A @ A2 ) % 0.31/0.61 != ( bot_bot @ ( filter @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % nhds_neq_bot % 0.31/0.61 thf(fact_5467_continuous__bot,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( topological_t2_space @ A ) % 0.31/0.61 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.61 => ! [F2: A > B] : ( topolo3448309680560233919inuous @ A @ B @ ( bot_bot @ ( filter @ A ) ) @ F2 ) ) ). % 0.31/0.61 % 0.31/0.61 % continuous_bot % 0.31/0.61 thf(fact_5468_continuous__trivial__limit,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( topological_t2_space @ A ) % 0.31/0.61 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.61 => ! [Net: filter @ A,F2: A > B] : % 0.31/0.61 ( ( Net % 0.31/0.61 = ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ B @ Net @ F2 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % continuous_trivial_limit % 0.31/0.61 thf(fact_5469_LIM__offset__zero__cancel,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.61 => ! [F2: A > B,A2: A,L5: B] : % 0.31/0.61 ( ( filterlim @ A @ B % 0.31/0.61 @ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 => ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_offset_zero_cancel % 0.31/0.61 thf(fact_5470_LIM__offset__zero,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.61 => ! [F2: A > B,L5: B,A2: A] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 => ( filterlim @ A @ B % 0.31/0.61 @ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_offset_zero % 0.31/0.61 thf(fact_5471_LIM__isCont__iff,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.61 => ! [F2: A > B,A2: A] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 = ( filterlim @ A @ B % 0.31/0.61 @ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_isCont_iff % 0.31/0.61 thf(fact_5472_tendsto__null__power,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( real_V2822296259951069270ebra_1 @ B ) % 0.31/0.61 => ! [F2: A > B,F5: filter @ A,N2: nat] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) % 0.31/0.61 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.61 => ( filterlim @ A @ B % 0.31/0.61 @ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N2 ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_null_power % 0.31/0.61 thf(fact_5473_LIM__offset,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.61 => ! [F2: A > B,L5: B,A2: A,K: A] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 => ( filterlim @ A @ B % 0.31/0.61 @ ^ [X: A] : ( F2 @ ( plus_plus @ A @ X @ K ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ K ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_offset % 0.31/0.61 thf(fact_5474_tendsto__Pair,axiom, % 0.31/0.61 ! [C: $tType,B: $tType,A: $tType] : % 0.31/0.61 ( ( ( topolo4958980785337419405_space @ B ) % 0.31/0.61 & ( topolo4958980785337419405_space @ C ) ) % 0.31/0.61 => ! [F2: A > B,A2: B,F5: filter @ A,G: A > C,B2: C] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F5 ) % 0.31/0.61 => ( ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ B2 ) @ F5 ) % 0.31/0.61 => ( filterlim @ A @ ( product_prod @ B @ C ) % 0.31/0.61 @ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ B2 ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_Pair % 0.31/0.61 thf(fact_5475_continuous__Pair,axiom, % 0.31/0.61 ! [C: $tType,B: $tType,A: $tType] : % 0.31/0.61 ( ( ( topological_t2_space @ A ) % 0.31/0.61 & ( topolo4958980785337419405_space @ B ) % 0.31/0.61 & ( topolo4958980785337419405_space @ C ) ) % 0.31/0.61 => ! [F5: filter @ A,F2: A > B,G: A > C] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 ) % 0.31/0.61 => ( ( topolo3448309680560233919inuous @ A @ C @ F5 @ G ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F5 % 0.31/0.61 @ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % continuous_Pair % 0.31/0.61 thf(fact_5476_Lim__transform__eq,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [F2: B > A,G: B > A,F5: filter @ B,A2: A] : % 0.31/0.61 ( ( filterlim @ B @ A % 0.31/0.61 @ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 0.31/0.61 = ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Lim_transform_eq % 0.31/0.61 thf(fact_5477_LIM__zero__cancel,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ B ) % 0.31/0.61 => ! [F2: A > B,L: B,F5: filter @ A] : % 0.31/0.61 ( ( filterlim @ A @ B % 0.31/0.61 @ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ L ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_zero_cancel % 0.31/0.61 thf(fact_5478_Lim__transform2,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [F2: B > A,A2: A,F5: filter @ B,G: B > A] : % 0.31/0.61 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 0.31/0.61 => ( ( filterlim @ B @ A % 0.31/0.61 @ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Lim_transform2 % 0.31/0.61 thf(fact_5479_Lim__transform,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [G: B > A,A2: A,F5: filter @ B,F2: B > A] : % 0.31/0.61 ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 0.31/0.61 => ( ( filterlim @ B @ A % 0.31/0.61 @ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Lim_transform % 0.31/0.61 thf(fact_5480_LIM__zero__iff,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ B ) % 0.31/0.61 => ! [F2: A > B,L: B,F5: filter @ A] : % 0.31/0.61 ( ( filterlim @ A @ B % 0.31/0.61 @ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ L ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 = ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_zero_iff % 0.31/0.61 thf(fact_5481_LIM__zero,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ B ) % 0.31/0.61 => ! [F2: A > B,L: B,F5: filter @ A] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) % 0.31/0.61 => ( filterlim @ A @ B % 0.31/0.61 @ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ L ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 0.31/0.61 @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_zero % 0.31/0.61 thf(fact_5482_tendsto__mult__zero,axiom, % 0.31/0.61 ! [A: $tType,D: $tType] : % 0.31/0.61 ( ( real_V4412858255891104859lgebra @ A ) % 0.31/0.61 => ! [F2: D > A,F5: filter @ D,G: D > A] : % 0.31/0.61 ( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 ) % 0.31/0.61 => ( ( filterlim @ D @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 ) % 0.31/0.61 => ( filterlim @ D @ A % 0.31/0.61 @ ^ [X: D] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_mult_zero % 0.31/0.61 thf(fact_5483_tendsto__mult__left__zero,axiom, % 0.31/0.61 ! [A: $tType,D: $tType] : % 0.31/0.61 ( ( real_V4412858255891104859lgebra @ A ) % 0.31/0.61 => ! [F2: D > A,F5: filter @ D,C2: A] : % 0.31/0.61 ( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 ) % 0.31/0.61 => ( filterlim @ D @ A % 0.31/0.61 @ ^ [X: D] : ( times_times @ A @ ( F2 @ X ) @ C2 ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_mult_left_zero % 0.31/0.61 thf(fact_5484_tendsto__mult__right__zero,axiom, % 0.31/0.61 ! [A: $tType,D: $tType] : % 0.31/0.61 ( ( real_V4412858255891104859lgebra @ A ) % 0.31/0.61 => ! [F2: D > A,F5: filter @ D,C2: A] : % 0.31/0.61 ( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 ) % 0.31/0.61 => ( filterlim @ D @ A % 0.31/0.61 @ ^ [X: D] : ( times_times @ A @ C2 @ ( F2 @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_mult_right_zero % 0.31/0.61 thf(fact_5485_tendsto__add__zero,axiom, % 0.31/0.61 ! [B: $tType,D: $tType] : % 0.31/0.61 ( ( topolo6943815403480290642id_add @ B ) % 0.31/0.61 => ! [F2: D > B,F5: filter @ D,G: D > B] : % 0.31/0.61 ( ( filterlim @ D @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) % 0.31/0.61 => ( ( filterlim @ D @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) % 0.31/0.61 => ( filterlim @ D @ B % 0.31/0.61 @ ^ [X: D] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_add_zero % 0.31/0.61 thf(fact_5486_tendsto__inverse,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.61 => ! [F2: B > A,A2: A,F5: filter @ B] : % 0.31/0.61 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 0.31/0.61 => ( ( A2 % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( filterlim @ B @ A % 0.31/0.61 @ ^ [X: B] : ( inverse_inverse @ A @ ( F2 @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A2 ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_inverse % 0.31/0.61 thf(fact_5487_tendsto__norm__zero,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ B ) % 0.31/0.61 => ! [F2: A > B,F5: filter @ A] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) % 0.31/0.61 => ( filterlim @ A @ real % 0.31/0.61 @ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 0.31/0.61 @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_norm_zero % 0.31/0.61 thf(fact_5488_tendsto__norm__zero__iff,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ B ) % 0.31/0.61 => ! [F2: A > B,F5: filter @ A] : % 0.31/0.61 ( ( filterlim @ A @ real % 0.31/0.61 @ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 = ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_norm_zero_iff % 0.31/0.61 thf(fact_5489_tendsto__norm__zero__cancel,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ B ) % 0.31/0.61 => ! [F2: A > B,F5: filter @ A] : % 0.31/0.61 ( ( filterlim @ A @ real % 0.31/0.61 @ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_norm_zero_cancel % 0.31/0.61 thf(fact_5490_tendsto__sgn,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [F2: B > A,L: A,F5: filter @ B] : % 0.31/0.61 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) % 0.31/0.61 => ( ( L % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( filterlim @ B @ A % 0.31/0.61 @ ^ [X: B] : ( sgn_sgn @ A @ ( F2 @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_sgn % 0.31/0.61 thf(fact_5491_tendsto__divide__zero,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: B > A,F5: filter @ B,C2: A] : % 0.31/0.61 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 ) % 0.31/0.61 => ( filterlim @ B @ A % 0.31/0.61 @ ^ [X: B] : ( divide_divide @ A @ ( F2 @ X ) @ C2 ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_divide_zero % 0.31/0.61 thf(fact_5492_tendsto__divide,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: B > A,A2: A,F5: filter @ B,G: B > A,B2: A] : % 0.31/0.61 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 0.31/0.61 => ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 ) % 0.31/0.61 => ( ( B2 % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( filterlim @ B @ A % 0.31/0.61 @ ^ [X: B] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 0.31/0.61 @ F5 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_divide % 0.31/0.61 thf(fact_5493_continuous__add,axiom, % 0.31/0.61 ! [B: $tType,D: $tType] : % 0.31/0.61 ( ( ( topological_t2_space @ D ) % 0.31/0.61 & ( topolo6943815403480290642id_add @ B ) ) % 0.31/0.61 => ! [F5: filter @ D,F2: D > B,G: D > B] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ D @ B @ F5 @ F2 ) % 0.31/0.61 => ( ( topolo3448309680560233919inuous @ D @ B @ F5 @ G ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ D @ B @ F5 % 0.31/0.61 @ ^ [X: D] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % continuous_add % 0.31/0.61 thf(fact_5494_tendsto__add,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( topolo6943815403480290642id_add @ A ) % 0.31/0.61 => ! [F2: B > A,A2: A,F5: filter @ B,G: B > A,B2: A] : % 0.31/0.61 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 0.31/0.61 => ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 ) % 0.31/0.61 => ( filterlim @ B @ A % 0.31/0.61 @ ^ [X: B] : ( plus_plus @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_add % 0.31/0.61 thf(fact_5495_tendsto__add__const__iff,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( topolo1633459387980952147up_add @ A ) % 0.31/0.61 => ! [C2: A,F2: B > A,D2: A,F5: filter @ B] : % 0.31/0.61 ( ( filterlim @ B @ A % 0.31/0.61 @ ^ [X: B] : ( plus_plus @ A @ C2 @ ( F2 @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ C2 @ D2 ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 = ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ D2 ) @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_add_const_iff % 0.31/0.61 thf(fact_5496_tendsto__tan,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [F2: A > A,A2: A,F5: filter @ A] : % 0.31/0.61 ( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 0.31/0.61 => ( ( ( cos @ A @ A2 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( filterlim @ A @ A % 0.31/0.61 @ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( tan @ A @ A2 ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_tan % 0.31/0.61 thf(fact_5497_tendsto__null__sum,axiom, % 0.31/0.61 ! [C: $tType,B: $tType,A: $tType] : % 0.31/0.61 ( ( topolo5987344860129210374id_add @ C ) % 0.31/0.61 => ! [I6: set @ B,F2: A > B > C,F5: filter @ A] : % 0.31/0.61 ( ! [I4: B] : % 0.31/0.61 ( ( member @ B @ I4 @ I6 ) % 0.31/0.61 => ( filterlim @ A @ C % 0.31/0.61 @ ^ [X: A] : ( F2 @ X @ I4 ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) % 0.31/0.61 @ F5 ) ) % 0.31/0.61 => ( filterlim @ A @ C % 0.31/0.61 @ ^ [I3: A] : ( groups7311177749621191930dd_sum @ B @ C @ ( F2 @ I3 ) @ I6 ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) % 0.31/0.61 @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_null_sum % 0.31/0.61 thf(fact_5498_tendsto__tanh,axiom, % 0.31/0.61 ! [A: $tType,C: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [F2: C > A,A2: A,F5: filter @ C] : % 0.31/0.61 ( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 0.31/0.61 => ( ( ( cosh @ A @ A2 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( filterlim @ C @ A % 0.31/0.61 @ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( tanh @ A @ A2 ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_tanh % 0.31/0.61 thf(fact_5499_tendsto__cot,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [F2: A > A,A2: A,F5: filter @ A] : % 0.31/0.61 ( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 0.31/0.61 => ( ( ( sin @ A @ A2 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( filterlim @ A @ A % 0.31/0.61 @ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( cot @ A @ A2 ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_cot % 0.31/0.61 thf(fact_5500_tendsto__log,axiom, % 0.31/0.61 ! [A: $tType,F2: A > real,A2: real,F5: filter @ A,G: A > real,B2: real] : % 0.31/0.61 ( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 ) % 0.31/0.61 => ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.61 => ( ( A2 % 0.31/0.61 != ( one_one @ real ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 0.31/0.61 => ( filterlim @ A @ real % 0.31/0.61 @ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( log @ A2 @ B2 ) ) % 0.31/0.61 @ F5 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_log % 0.31/0.61 thf(fact_5501_atLeastSucLessThan__greaterThanLessThan,axiom, % 0.31/0.61 ! [L: nat,U: nat] : % 0.31/0.61 ( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U ) % 0.31/0.61 = ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ). % 0.31/0.61 % 0.31/0.61 % atLeastSucLessThan_greaterThanLessThan % 0.31/0.61 thf(fact_5502_tendsto__artanh,axiom, % 0.31/0.61 ! [A: $tType,F2: A > real,A2: real,F5: filter @ A] : % 0.31/0.61 ( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 ) % 0.31/0.61 => ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ A2 ) % 0.31/0.61 => ( ( ord_less @ real @ A2 @ ( one_one @ real ) ) % 0.31/0.61 => ( filterlim @ A @ real % 0.31/0.61 @ ^ [X: A] : ( artanh @ real @ ( F2 @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( artanh @ real @ A2 ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_artanh % 0.31/0.61 thf(fact_5503_LIM__offset__zero__iff,axiom, % 0.31/0.61 ! [C: $tType,D: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( topolo4958980785337419405_space @ D ) % 0.31/0.61 & ( zero @ C ) ) % 0.31/0.61 => ! [A2: A,F2: A > D,L5: D] : % 0.31/0.61 ( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 ) % 0.31/0.61 => ( ( filterlim @ A @ D @ F2 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 = ( filterlim @ A @ D % 0.31/0.61 @ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_offset_zero_iff % 0.31/0.61 thf(fact_5504_IVT,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( ( topolo1944317154257567458pology @ B ) % 0.31/0.61 & ( topolo8458572112393995274pology @ A ) ) % 0.31/0.61 => ! [F2: A > B,A2: A,Y: B,B2: A] : % 0.31/0.61 ( ( ord_less_eq @ B @ ( F2 @ A2 ) @ Y ) % 0.31/0.61 => ( ( ord_less_eq @ B @ Y @ ( F2 @ B2 ) ) % 0.31/0.61 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.61 => ( ! [X5: A] : % 0.31/0.61 ( ( ( ord_less_eq @ A @ A2 @ X5 ) % 0.31/0.61 & ( ord_less_eq @ A @ X5 @ B2 ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) @ F2 ) ) % 0.31/0.61 => ? [X5: A] : % 0.31/0.61 ( ( ord_less_eq @ A @ A2 @ X5 ) % 0.31/0.61 & ( ord_less_eq @ A @ X5 @ B2 ) % 0.31/0.61 & ( ( F2 @ X5 ) % 0.31/0.61 = Y ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % IVT % 0.31/0.61 thf(fact_5505_IVT2,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( ( topolo1944317154257567458pology @ B ) % 0.31/0.61 & ( topolo8458572112393995274pology @ A ) ) % 0.31/0.61 => ! [F2: A > B,B2: A,Y: B,A2: A] : % 0.31/0.61 ( ( ord_less_eq @ B @ ( F2 @ B2 ) @ Y ) % 0.31/0.61 => ( ( ord_less_eq @ B @ Y @ ( F2 @ A2 ) ) % 0.31/0.61 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.61 => ( ! [X5: A] : % 0.31/0.61 ( ( ( ord_less_eq @ A @ A2 @ X5 ) % 0.31/0.61 & ( ord_less_eq @ A @ X5 @ B2 ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) @ F2 ) ) % 0.31/0.61 => ? [X5: A] : % 0.31/0.61 ( ( ord_less_eq @ A @ A2 @ X5 ) % 0.31/0.61 & ( ord_less_eq @ A @ X5 @ B2 ) % 0.31/0.61 & ( ( F2 @ X5 ) % 0.31/0.61 = Y ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % IVT2 % 0.31/0.61 thf(fact_5506_LIM__equal2,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.61 => ! [R: real,A2: A,F2: A > B,G: A > B,L: B] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ R ) % 0.31/0.61 => ( ! [X5: A] : % 0.31/0.61 ( ( X5 != A2 ) % 0.31/0.61 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X5 @ A2 ) ) @ R ) % 0.31/0.61 => ( ( F2 @ X5 ) % 0.31/0.61 = ( G @ X5 ) ) ) ) % 0.31/0.61 => ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 => ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_equal2 % 0.31/0.61 thf(fact_5507_LIM__eq,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [F2: A > B,L5: B,A2: A] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 = ( ! [R5: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ R5 ) % 0.31/0.61 => ? [S8: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ S8 ) % 0.31/0.61 & ! [X: A] : % 0.31/0.61 ( ( ( X != A2 ) % 0.31/0.61 & ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ A2 ) ) @ S8 ) ) % 0.31/0.61 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X ) @ L5 ) ) @ R5 ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_eq % 0.31/0.61 thf(fact_5508_LIM__I,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [A2: A,F2: A > B,L5: B] : % 0.31/0.61 ( ! [R4: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 ) % 0.31/0.61 => ? [S9: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ S9 ) % 0.31/0.61 & ! [X5: A] : % 0.31/0.61 ( ( ( X5 != A2 ) % 0.31/0.61 & ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X5 @ A2 ) ) @ S9 ) ) % 0.31/0.61 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X5 ) @ L5 ) ) @ R4 ) ) ) ) % 0.31/0.61 => ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_I % 0.31/0.61 thf(fact_5509_LIM__D,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [F2: A > B,L5: B,A2: A,R2: real] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 ) % 0.31/0.61 => ? [S2: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ S2 ) % 0.31/0.61 & ! [X2: A] : % 0.31/0.61 ( ( ( X2 != A2 ) % 0.31/0.61 & ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ A2 ) ) @ S2 ) ) % 0.31/0.61 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X2 ) @ L5 ) ) @ R2 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_D % 0.31/0.61 thf(fact_5510_DERIV__LIM__iff,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( inverse @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ A ) ) % 0.31/0.61 => ! [F2: A > A,A2: A,D6: A] : % 0.31/0.61 ( ( filterlim @ A @ A % 0.31/0.61 @ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ A2 @ H ) ) @ ( F2 @ A2 ) ) @ H ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ D6 ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 = ( filterlim @ A @ A % 0.31/0.61 @ ^ [X: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ X ) @ ( F2 @ A2 ) ) @ ( minus_minus @ A @ X @ A2 ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ D6 ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_LIM_iff % 0.31/0.61 thf(fact_5511_LIM__fun__less__zero,axiom, % 0.31/0.61 ! [F2: real > real,L: real,C2: real] : % 0.31/0.61 ( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ L @ ( zero_zero @ real ) ) % 0.31/0.61 => ? [R4: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 ) % 0.31/0.61 & ! [X2: real] : % 0.31/0.61 ( ( ( X2 != C2 ) % 0.31/0.61 & ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X2 ) ) @ R4 ) ) % 0.31/0.61 => ( ord_less @ real @ ( F2 @ X2 ) @ ( zero_zero @ real ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_fun_less_zero % 0.31/0.61 thf(fact_5512_LIM__fun__not__zero,axiom, % 0.31/0.61 ! [F2: real > real,L: real,C2: real] : % 0.31/0.61 ( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ( ( L % 0.31/0.61 != ( zero_zero @ real ) ) % 0.31/0.61 => ? [R4: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 ) % 0.31/0.61 & ! [X2: real] : % 0.31/0.61 ( ( ( X2 != C2 ) % 0.31/0.61 & ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X2 ) ) @ R4 ) ) % 0.31/0.61 => ( ( F2 @ X2 ) % 0.31/0.61 != ( zero_zero @ real ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_fun_not_zero % 0.31/0.61 thf(fact_5513_LIM__fun__gt__zero,axiom, % 0.31/0.61 ! [F2: real > real,L: real,C2: real] : % 0.31/0.61 ( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ L ) % 0.31/0.61 => ? [R4: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 ) % 0.31/0.61 & ! [X2: real] : % 0.31/0.61 ( ( ( X2 != C2 ) % 0.31/0.61 & ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X2 ) ) @ R4 ) ) % 0.31/0.61 => ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_fun_gt_zero % 0.31/0.61 thf(fact_5514_LIM__compose2,axiom, % 0.31/0.61 ! [C: $tType,B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( topolo4958980785337419405_space @ B ) % 0.31/0.61 & ( topolo4958980785337419405_space @ C ) ) % 0.31/0.61 => ! [F2: A > B,B2: B,A2: A,G: B > C,C2: C] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 => ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) ) % 0.31/0.61 => ( ? [D4: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 ) % 0.31/0.61 & ! [X5: A] : % 0.31/0.61 ( ( ( X5 != A2 ) % 0.31/0.61 & ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X5 @ A2 ) ) @ D4 ) ) % 0.31/0.61 => ( ( F2 @ X5 ) % 0.31/0.61 != B2 ) ) ) % 0.31/0.61 => ( filterlim @ A @ C % 0.31/0.61 @ ^ [X: A] : ( G @ ( F2 @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ C @ C2 ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_compose2 % 0.31/0.61 thf(fact_5515_continuous__at__within__divide,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( topological_t2_space @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ B ) ) % 0.31/0.61 => ! [A2: A,S: set @ A,F2: A > B,G: A > B] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 ) % 0.31/0.61 => ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ G ) % 0.31/0.61 => ( ( ( G @ A2 ) % 0.31/0.61 != ( zero_zero @ B ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) % 0.31/0.61 @ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % continuous_at_within_divide % 0.31/0.61 thf(fact_5516_isCont__add,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( topological_t2_space @ A ) % 0.31/0.61 & ( topolo6943815403480290642id_add @ B ) ) % 0.31/0.61 => ! [A2: A,F2: A > B,G: A > B] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 ) % 0.31/0.61 => ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 @ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_add % 0.31/0.61 thf(fact_5517_continuous__at__within__inverse,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( topological_t2_space @ A ) % 0.31/0.61 & ( real_V8999393235501362500lgebra @ B ) ) % 0.31/0.61 => ! [A2: A,S: set @ A,F2: A > B] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 ) % 0.31/0.61 => ( ( ( F2 @ A2 ) % 0.31/0.61 != ( zero_zero @ B ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) % 0.31/0.61 @ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % continuous_at_within_inverse % 0.31/0.61 thf(fact_5518_continuous__at__within__sgn,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( topological_t2_space @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [A2: A,S: set @ A,F2: A > B] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 ) % 0.31/0.61 => ( ( ( F2 @ A2 ) % 0.31/0.61 != ( zero_zero @ B ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) % 0.31/0.61 @ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % continuous_at_within_sgn % 0.31/0.61 thf(fact_5519_DERIV__D,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,D6: A,X3: A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 => ( filterlim @ A @ A % 0.31/0.61 @ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X3 @ H ) ) @ ( F2 @ X3 ) ) @ H ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ D6 ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_D % 0.31/0.61 thf(fact_5520_DERIV__def,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,D6: A,X3: A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 = ( filterlim @ A @ A % 0.31/0.61 @ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X3 @ H ) ) @ ( F2 @ X3 ) ) @ H ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ D6 ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_def % 0.31/0.61 thf(fact_5521_lim__exp__minus__1,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ( filterlim @ A @ A % 0.31/0.61 @ ^ [Z6: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z6 ) @ ( one_one @ A ) ) @ Z6 ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % lim_exp_minus_1 % 0.31/0.61 thf(fact_5522_lemma__termdiff4,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [K: real,F2: A > B,K5: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ K ) % 0.31/0.61 => ( ! [H6: A] : % 0.31/0.61 ( ( H6 % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H6 ) @ K ) % 0.31/0.61 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ H6 ) ) @ ( times_times @ real @ K5 @ ( real_V7770717601297561774m_norm @ A @ H6 ) ) ) ) ) % 0.31/0.61 => ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % lemma_termdiff4 % 0.31/0.61 thf(fact_5523_isCont__eq__Lb,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 => ! [A2: real,B2: real,F2: real > A] : % 0.31/0.61 ( ( ord_less_eq @ real @ A2 @ B2 ) % 0.31/0.61 => ( ! [X5: real] : % 0.31/0.61 ( ( ( ord_less_eq @ real @ A2 @ X5 ) % 0.31/0.61 & ( ord_less_eq @ real @ X5 @ B2 ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) % 0.31/0.61 => ? [M8: A] : % 0.31/0.61 ( ! [X2: real] : % 0.31/0.61 ( ( ( ord_less_eq @ real @ A2 @ X2 ) % 0.31/0.61 & ( ord_less_eq @ real @ X2 @ B2 ) ) % 0.31/0.61 => ( ord_less_eq @ A @ M8 @ ( F2 @ X2 ) ) ) % 0.31/0.61 & ? [X5: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ A2 @ X5 ) % 0.31/0.61 & ( ord_less_eq @ real @ X5 @ B2 ) % 0.31/0.61 & ( ( F2 @ X5 ) % 0.31/0.61 = M8 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_eq_Lb % 0.31/0.61 thf(fact_5524_isCont__eq__Ub,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 => ! [A2: real,B2: real,F2: real > A] : % 0.31/0.61 ( ( ord_less_eq @ real @ A2 @ B2 ) % 0.31/0.61 => ( ! [X5: real] : % 0.31/0.61 ( ( ( ord_less_eq @ real @ A2 @ X5 ) % 0.31/0.61 & ( ord_less_eq @ real @ X5 @ B2 ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) % 0.31/0.61 => ? [M8: A] : % 0.31/0.61 ( ! [X2: real] : % 0.31/0.61 ( ( ( ord_less_eq @ real @ A2 @ X2 ) % 0.31/0.61 & ( ord_less_eq @ real @ X2 @ B2 ) ) % 0.31/0.61 => ( ord_less_eq @ A @ ( F2 @ X2 ) @ M8 ) ) % 0.31/0.61 & ? [X5: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ A2 @ X5 ) % 0.31/0.61 & ( ord_less_eq @ real @ X5 @ B2 ) % 0.31/0.61 & ( ( F2 @ X5 ) % 0.31/0.61 = M8 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_eq_Ub % 0.31/0.61 thf(fact_5525_isCont__bounded,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 => ! [A2: real,B2: real,F2: real > A] : % 0.31/0.61 ( ( ord_less_eq @ real @ A2 @ B2 ) % 0.31/0.61 => ( ! [X5: real] : % 0.31/0.61 ( ( ( ord_less_eq @ real @ A2 @ X5 ) % 0.31/0.61 & ( ord_less_eq @ real @ X5 @ B2 ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) % 0.31/0.61 => ? [M8: A] : % 0.31/0.61 ! [X2: real] : % 0.31/0.61 ( ( ( ord_less_eq @ real @ A2 @ X2 ) % 0.31/0.61 & ( ord_less_eq @ real @ X2 @ B2 ) ) % 0.31/0.61 => ( ord_less_eq @ A @ ( F2 @ X2 ) @ M8 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_bounded % 0.31/0.61 thf(fact_5526_isCont__inverse__function2,axiom, % 0.31/0.61 ! [A2: real,X3: real,B2: real,G: real > real,F2: real > real] : % 0.31/0.61 ( ( ord_less @ real @ A2 @ X3 ) % 0.31/0.61 => ( ( ord_less @ real @ X3 @ B2 ) % 0.31/0.61 => ( ! [Z3: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ A2 @ Z3 ) % 0.31/0.61 => ( ( ord_less_eq @ real @ Z3 @ B2 ) % 0.31/0.61 => ( ( G @ ( F2 @ Z3 ) ) % 0.31/0.61 = Z3 ) ) ) % 0.31/0.61 => ( ! [Z3: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ A2 @ Z3 ) % 0.31/0.61 => ( ( ord_less_eq @ real @ Z3 @ B2 ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X3 ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_inverse_function2 % 0.31/0.61 thf(fact_5527_field__has__derivative__at,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,D6: A,X3: A] : % 0.31/0.61 ( ( has_derivative @ A @ A @ F2 @ ( times_times @ A @ D6 ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 = ( filterlim @ A @ A % 0.31/0.61 @ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X3 @ H ) ) @ ( F2 @ X3 ) ) @ H ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ D6 ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % field_has_derivative_at % 0.31/0.61 thf(fact_5528_isCont__divide,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( topological_t2_space @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ B ) ) % 0.31/0.61 => ! [A2: A,F2: A > B,G: A > B] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 ) % 0.31/0.61 => ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G ) % 0.31/0.61 => ( ( ( G @ A2 ) % 0.31/0.61 != ( zero_zero @ B ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 @ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_divide % 0.31/0.61 thf(fact_5529_isCont__sgn,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( topological_t2_space @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [A2: A,F2: A > B] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 ) % 0.31/0.61 => ( ( ( F2 @ A2 ) % 0.31/0.61 != ( zero_zero @ B ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 @ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_sgn % 0.31/0.61 thf(fact_5530_filterlim__at__to__0,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [F2: A > B,F5: filter @ B,A2: A] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 = ( filterlim @ A @ B % 0.31/0.61 @ ^ [X: A] : ( F2 @ ( plus_plus @ A @ X @ A2 ) ) % 0.31/0.61 @ F5 % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_at_to_0 % 0.31/0.61 thf(fact_5531_continuous__within__tan,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [X3: A,S: set @ A,F2: A > A] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ S ) @ F2 ) % 0.31/0.61 => ( ( ( cos @ A @ ( F2 @ X3 ) ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ S ) % 0.31/0.61 @ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % continuous_within_tan % 0.31/0.61 thf(fact_5532_nth__sorted__list__of__set__greaterThanLessThan,axiom, % 0.31/0.61 ! [N2: nat,J: nat,I: nat] : % 0.31/0.61 ( ( ord_less @ nat @ N2 @ ( minus_minus @ nat @ J @ ( suc @ I ) ) ) % 0.31/0.61 => ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) ) @ N2 ) % 0.31/0.61 = ( suc @ ( plus_plus @ nat @ I @ N2 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % nth_sorted_list_of_set_greaterThanLessThan % 0.31/0.61 thf(fact_5533_continuous__within__cot,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [X3: A,S: set @ A,F2: A > A] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ S ) @ F2 ) % 0.31/0.61 => ( ( ( sin @ A @ ( F2 @ X3 ) ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ S ) % 0.31/0.61 @ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % continuous_within_cot % 0.31/0.61 thf(fact_5534_continuous__at__within__tanh,axiom, % 0.31/0.61 ! [A: $tType,C: $tType] : % 0.31/0.61 ( ( ( topological_t2_space @ C ) % 0.31/0.61 & ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [X3: C,A3: set @ C,F2: C > A] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X3 @ A3 ) @ F2 ) % 0.31/0.61 => ( ( ( cosh @ A @ ( F2 @ X3 ) ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X3 @ A3 ) % 0.31/0.61 @ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % continuous_at_within_tanh % 0.31/0.61 thf(fact_5535_isCont__has__Ub,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 => ! [A2: real,B2: real,F2: real > A] : % 0.31/0.61 ( ( ord_less_eq @ real @ A2 @ B2 ) % 0.31/0.61 => ( ! [X5: real] : % 0.31/0.61 ( ( ( ord_less_eq @ real @ A2 @ X5 ) % 0.31/0.61 & ( ord_less_eq @ real @ X5 @ B2 ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) % 0.31/0.61 => ? [M8: A] : % 0.31/0.61 ( ! [X2: real] : % 0.31/0.61 ( ( ( ord_less_eq @ real @ A2 @ X2 ) % 0.31/0.61 & ( ord_less_eq @ real @ X2 @ B2 ) ) % 0.31/0.61 => ( ord_less_eq @ A @ ( F2 @ X2 ) @ M8 ) ) % 0.31/0.61 & ! [N8: A] : % 0.31/0.61 ( ( ord_less @ A @ N8 @ M8 ) % 0.31/0.61 => ? [X5: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ A2 @ X5 ) % 0.31/0.61 & ( ord_less_eq @ real @ X5 @ B2 ) % 0.31/0.61 & ( ord_less @ A @ N8 @ ( F2 @ X5 ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_has_Ub % 0.31/0.61 thf(fact_5536_isCont__tan,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [X3: A] : % 0.31/0.61 ( ( ( cos @ A @ X3 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ ( tan @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_tan % 0.31/0.61 thf(fact_5537_filterlim__shift__iff,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [F2: A > B,D2: A,F5: filter @ B,A2: A] : % 0.31/0.61 ( ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F2 @ ( plus_plus @ A @ D2 ) ) @ F5 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ D2 ) @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 = ( filterlim @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_shift_iff % 0.31/0.61 thf(fact_5538_filterlim__shift,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [F2: A > B,F5: filter @ B,A2: A,D2: A] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 => ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F2 @ ( plus_plus @ A @ D2 ) ) @ F5 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ D2 ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_shift % 0.31/0.61 thf(fact_5539_isCont__cot,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [X3: A] : % 0.31/0.61 ( ( ( sin @ A @ X3 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ ( cot @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_cot % 0.31/0.61 thf(fact_5540_isCont__tanh,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [X3: A] : % 0.31/0.61 ( ( ( cosh @ A @ X3 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ ( tanh @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_tanh % 0.31/0.61 thf(fact_5541_powser__limit__0,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [S: real,A2: nat > A,F2: A > A] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ S ) % 0.31/0.61 => ( ! [X5: A] : % 0.31/0.61 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ S ) % 0.31/0.61 => ( sums @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( A2 @ N ) @ ( power_power @ A @ X5 @ N ) ) % 0.31/0.61 @ ( F2 @ X5 ) ) ) % 0.31/0.61 => ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % powser_limit_0 % 0.31/0.61 thf(fact_5542_powser__limit__0__strong,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [S: real,A2: nat > A,F2: A > A] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ S ) % 0.31/0.61 => ( ! [X5: A] : % 0.31/0.61 ( ( X5 % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ S ) % 0.31/0.61 => ( sums @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( A2 @ N ) @ ( power_power @ A @ X5 @ N ) ) % 0.31/0.61 @ ( F2 @ X5 ) ) ) ) % 0.31/0.61 => ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % powser_limit_0_strong % 0.31/0.61 thf(fact_5543_lemma__termdiff5,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_Vector_banach @ B ) ) % 0.31/0.61 => ! [K: real,F2: nat > real,G: A > nat > B] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ K ) % 0.31/0.61 => ( ( summable @ real @ F2 ) % 0.31/0.61 => ( ! [H6: A,N3: nat] : % 0.31/0.61 ( ( H6 % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H6 ) @ K ) % 0.31/0.61 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( G @ H6 @ N3 ) ) @ ( times_times @ real @ ( F2 @ N3 ) @ ( real_V7770717601297561774m_norm @ A @ H6 ) ) ) ) ) % 0.31/0.61 => ( filterlim @ A @ B % 0.31/0.61 @ ^ [H: A] : ( suminf @ B @ ( G @ H ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % lemma_termdiff5 % 0.31/0.61 thf(fact_5544_isCont__tan_H,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [A2: A,F2: A > A] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 ) % 0.31/0.61 => ( ( ( cos @ A @ ( F2 @ A2 ) ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 @ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_tan' % 0.31/0.61 thf(fact_5545_isCont__arcosh,axiom, % 0.31/0.61 ! [X3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ ( arcosh @ real ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_arcosh % 0.31/0.61 thf(fact_5546_continuous__at__within__log,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topological_t2_space @ A ) % 0.31/0.61 => ! [A2: A,S: set @ A,F2: A > real,G: A > real] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 ) % 0.31/0.61 => ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S ) @ G ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ A2 ) ) % 0.31/0.61 => ( ( ( F2 @ A2 ) % 0.31/0.61 != ( one_one @ real ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A2 ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S ) % 0.31/0.61 @ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % continuous_at_within_log % 0.31/0.61 thf(fact_5547_isCont__cot_H,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [A2: A,F2: A > A] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 ) % 0.31/0.61 => ( ( ( sin @ A @ ( F2 @ A2 ) ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 @ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_cot' % 0.31/0.61 thf(fact_5548_DERIV__inverse__function,axiom, % 0.31/0.61 ! [F2: real > real,D6: real,G: real > real,X3: real,A2: real,B2: real] : % 0.31/0.61 ( ( has_field_derivative @ real @ F2 @ D6 @ ( topolo174197925503356063within @ real @ ( G @ X3 ) @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.61 => ( ( D6 % 0.31/0.61 != ( zero_zero @ real ) ) % 0.31/0.61 => ( ( ord_less @ real @ A2 @ X3 ) % 0.31/0.61 => ( ( ord_less @ real @ X3 @ B2 ) % 0.31/0.61 => ( ! [Y4: real] : % 0.31/0.61 ( ( ord_less @ real @ A2 @ Y4 ) % 0.31/0.61 => ( ( ord_less @ real @ Y4 @ B2 ) % 0.31/0.61 => ( ( F2 @ ( G @ Y4 ) ) % 0.31/0.61 = Y4 ) ) ) % 0.31/0.61 => ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ G ) % 0.31/0.61 => ( has_field_derivative @ real @ G @ ( inverse_inverse @ real @ D6 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % DERIV_inverse_function % 0.31/0.61 thf(fact_5549_isCont__arccos,axiom, % 0.31/0.61 ! [X3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 0.31/0.61 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ arccos ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_arccos % 0.31/0.61 thf(fact_5550_isCont__arcsin,axiom, % 0.31/0.61 ! [X3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 0.31/0.61 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ arcsin ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_arcsin % 0.31/0.61 thf(fact_5551_LIM__less__bound,axiom, % 0.31/0.61 ! [B2: real,X3: real,F2: real > real] : % 0.31/0.61 ( ( ord_less @ real @ B2 @ X3 ) % 0.31/0.61 => ( ! [X5: real] : % 0.31/0.61 ( ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ B2 @ X3 ) ) % 0.31/0.61 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X5 ) ) ) % 0.31/0.61 => ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) % 0.31/0.61 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_less_bound % 0.31/0.61 thf(fact_5552_isCont__log,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topological_t2_space @ A ) % 0.31/0.61 => ! [A2: A,F2: A > real,G: A > real] : % 0.31/0.61 ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 ) % 0.31/0.61 => ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ A2 ) ) % 0.31/0.61 => ( ( ( F2 @ A2 ) % 0.31/0.61 != ( one_one @ real ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A2 ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.61 @ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_log % 0.31/0.61 thf(fact_5553_isCont__artanh,axiom, % 0.31/0.61 ! [X3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 0.31/0.61 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ ( artanh @ real ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_artanh % 0.31/0.61 thf(fact_5554_isCont__inverse__function,axiom, % 0.31/0.61 ! [D2: real,X3: real,G: real > real,F2: real > real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 ) % 0.31/0.61 => ( ! [Z3: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z3 @ X3 ) ) @ D2 ) % 0.31/0.61 => ( ( G @ ( F2 @ Z3 ) ) % 0.31/0.61 = Z3 ) ) % 0.31/0.61 => ( ! [Z3: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z3 @ X3 ) ) @ D2 ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X3 ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % isCont_inverse_function % 0.31/0.61 thf(fact_5555_GMVT_H,axiom, % 0.31/0.61 ! [A2: real,B2: real,F2: real > real,G: real > real,G7: real > real,F11: real > real] : % 0.31/0.61 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.61 => ( ! [Z3: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ A2 @ Z3 ) % 0.31/0.61 => ( ( ord_less_eq @ real @ Z3 @ B2 ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) ) % 0.31/0.61 => ( ! [Z3: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ A2 @ Z3 ) % 0.31/0.61 => ( ( ord_less_eq @ real @ Z3 @ B2 ) % 0.31/0.61 => ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) % 0.31/0.61 => ( ! [Z3: real] : % 0.31/0.61 ( ( ord_less @ real @ A2 @ Z3 ) % 0.31/0.61 => ( ( ord_less @ real @ Z3 @ B2 ) % 0.31/0.61 => ( has_field_derivative @ real @ G @ ( G7 @ Z3 ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) ) % 0.31/0.61 => ( ! [Z3: real] : % 0.31/0.61 ( ( ord_less @ real @ A2 @ Z3 ) % 0.31/0.61 => ( ( ord_less @ real @ Z3 @ B2 ) % 0.31/0.61 => ( has_field_derivative @ real @ F2 @ ( F11 @ Z3 ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) ) % 0.31/0.61 => ? [C3: real] : % 0.31/0.61 ( ( ord_less @ real @ A2 @ C3 ) % 0.31/0.61 & ( ord_less @ real @ C3 @ B2 ) % 0.31/0.61 & ( ( times_times @ real @ ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) @ ( G7 @ C3 ) ) % 0.31/0.61 = ( times_times @ real @ ( minus_minus @ real @ ( G @ B2 ) @ ( G @ A2 ) ) @ ( F11 @ C3 ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % GMVT' % 0.31/0.61 thf(fact_5556_summable__Leibniz_I2_J,axiom, % 0.31/0.61 ! [A2: nat > real] : % 0.31/0.61 ( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ( topological_monoseq @ real @ A2 ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( A2 @ ( zero_zero @ nat ) ) ) % 0.31/0.61 => ! [N6: nat] : % 0.31/0.61 ( member @ real % 0.31/0.61 @ ( suminf @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) % 0.31/0.61 @ ( set_or1337092689740270186AtMost @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) ) ) % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % summable_Leibniz(2) % 0.31/0.61 thf(fact_5557_summable__Leibniz_I3_J,axiom, % 0.31/0.61 ! [A2: nat > real] : % 0.31/0.61 ( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ( topological_monoseq @ real @ A2 ) % 0.31/0.61 => ( ( ord_less @ real @ ( A2 @ ( zero_zero @ nat ) ) @ ( zero_zero @ real ) ) % 0.31/0.61 => ! [N6: nat] : % 0.31/0.61 ( member @ real % 0.31/0.61 @ ( suminf @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) % 0.31/0.61 @ ( set_or1337092689740270186AtMost @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) @ ( one_one @ nat ) ) ) ) % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % summable_Leibniz(3) % 0.31/0.61 thf(fact_5558_summable__Leibniz_H_I5_J,axiom, % 0.31/0.61 ! [A2: nat > real] : % 0.31/0.61 ( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) ) % 0.31/0.61 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) ) % 0.31/0.61 => ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : % 0.31/0.61 ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real % 0.31/0.61 @ ( suminf @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % summable_Leibniz'(5) % 0.31/0.61 thf(fact_5559_trivial__limit__sequentially,axiom, % 0.31/0.61 ( ( at_top @ nat ) % 0.31/0.61 != ( bot_bot @ ( filter @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % trivial_limit_sequentially % 0.31/0.61 thf(fact_5560_tendsto__zero__mult__left__iff,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( field @ A ) % 0.31/0.61 & ( topolo4211221413907600880p_mult @ A ) ) % 0.31/0.61 => ! [C2: A,A2: nat > A] : % 0.31/0.61 ( ( C2 % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ C2 @ ( A2 @ N ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ ( at_top @ nat ) ) % 0.31/0.61 = ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_zero_mult_left_iff % 0.31/0.61 thf(fact_5561_tendsto__zero__mult__right__iff,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( field @ A ) % 0.31/0.61 & ( topolo4211221413907600880p_mult @ A ) ) % 0.31/0.61 => ! [C2: A,A2: nat > A] : % 0.31/0.61 ( ( C2 % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( A2 @ N ) @ C2 ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ ( at_top @ nat ) ) % 0.31/0.61 = ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_zero_mult_right_iff % 0.31/0.61 thf(fact_5562_tendsto__zero__divide__iff,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( field @ A ) % 0.31/0.61 & ( topolo4211221413907600880p_mult @ A ) ) % 0.31/0.61 => ! [C2: A,A2: nat > A] : % 0.31/0.61 ( ( C2 % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( divide_divide @ A @ ( A2 @ N ) @ C2 ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ ( at_top @ nat ) ) % 0.31/0.61 = ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_zero_divide_iff % 0.31/0.61 thf(fact_5563_filterlim__Suc,axiom, % 0.31/0.61 filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_Suc % 0.31/0.61 thf(fact_5564_filterlim__sequentially__Suc,axiom, % 0.31/0.61 ! [A: $tType,F2: nat > A,F5: filter @ A] : % 0.31/0.61 ( ( filterlim @ nat @ A % 0.31/0.61 @ ^ [X: nat] : ( F2 @ ( suc @ X ) ) % 0.31/0.61 @ F5 % 0.31/0.61 @ ( at_top @ nat ) ) % 0.31/0.61 = ( filterlim @ nat @ A @ F2 @ F5 @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_sequentially_Suc % 0.31/0.61 thf(fact_5565_trivial__limit__at__top__linorder,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( linorder @ A ) % 0.31/0.61 => ( ( at_top @ A ) % 0.31/0.61 != ( bot_bot @ ( filter @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % trivial_limit_at_top_linorder % 0.31/0.61 thf(fact_5566_approx__from__above__dense__linorder,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( dense_linorder @ A ) % 0.31/0.61 & ( topolo3112930676232923870pology @ A ) % 0.31/0.61 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.61 => ! [X3: A,Y: A] : % 0.31/0.61 ( ( ord_less @ A @ X3 @ Y ) % 0.31/0.61 => ? [U4: nat > A] : % 0.31/0.61 ( ! [N6: nat] : ( ord_less @ A @ X3 @ ( U4 @ N6 ) ) % 0.31/0.61 & ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % approx_from_above_dense_linorder % 0.31/0.61 thf(fact_5567_approx__from__below__dense__linorder,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( dense_linorder @ A ) % 0.31/0.61 & ( topolo3112930676232923870pology @ A ) % 0.31/0.61 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.61 => ! [Y: A,X3: A] : % 0.31/0.61 ( ( ord_less @ A @ Y @ X3 ) % 0.31/0.61 => ? [U4: nat > A] : % 0.31/0.61 ( ! [N6: nat] : ( ord_less @ A @ ( U4 @ N6 ) @ X3 ) % 0.31/0.61 & ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % approx_from_below_dense_linorder % 0.31/0.61 thf(fact_5568_LIMSEQ__Suc,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.61 => ! [F2: nat > A,L: A] : % 0.31/0.61 ( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( F2 @ ( suc @ N ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ L ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_Suc % 0.31/0.61 thf(fact_5569_LIMSEQ__imp__Suc,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.61 => ! [F2: nat > A,L: A] : % 0.31/0.61 ( ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( F2 @ ( suc @ N ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ L ) % 0.31/0.61 @ ( at_top @ nat ) ) % 0.31/0.61 => ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_imp_Suc % 0.31/0.61 thf(fact_5570_LIMSEQ__offset,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.61 => ! [F2: nat > A,K: nat,A2: A] : % 0.31/0.61 ( ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) % 0.31/0.61 @ ( at_top @ nat ) ) % 0.31/0.61 => ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_offset % 0.31/0.61 thf(fact_5571_LIMSEQ__ignore__initial__segment,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.61 => ! [F2: nat > A,A2: A,K: nat] : % 0.31/0.61 ( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_ignore_initial_segment % 0.31/0.61 thf(fact_5572_lim__mono,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 => ! [N7: nat,X8: nat > A,Y7: nat > A,X3: A,Y: A] : % 0.31/0.61 ( ! [N3: nat] : % 0.31/0.61 ( ( ord_less_eq @ nat @ N7 @ N3 ) % 0.31/0.61 => ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) % 0.31/0.61 => ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ( filterlim @ nat @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % lim_mono % 0.31/0.61 thf(fact_5573_LIMSEQ__le,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 => ! [X8: nat > A,X3: A,Y7: nat > A,Y: A] : % 0.31/0.61 ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ( filterlim @ nat @ A @ Y7 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ? [N8: nat] : % 0.31/0.61 ! [N3: nat] : % 0.31/0.61 ( ( ord_less_eq @ nat @ N8 @ N3 ) % 0.31/0.61 => ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( Y7 @ N3 ) ) ) % 0.31/0.61 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_le % 0.31/0.61 thf(fact_5574_Lim__bounded,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 => ! [F2: nat > A,L: A,M7: nat,C5: A] : % 0.31/0.61 ( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ! [N3: nat] : % 0.31/0.61 ( ( ord_less_eq @ nat @ M7 @ N3 ) % 0.31/0.61 => ( ord_less_eq @ A @ ( F2 @ N3 ) @ C5 ) ) % 0.31/0.61 => ( ord_less_eq @ A @ L @ C5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Lim_bounded % 0.31/0.61 thf(fact_5575_Lim__bounded2,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 => ! [F2: nat > A,L: A,N7: nat,C5: A] : % 0.31/0.61 ( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ! [N3: nat] : % 0.31/0.61 ( ( ord_less_eq @ nat @ N7 @ N3 ) % 0.31/0.61 => ( ord_less_eq @ A @ C5 @ ( F2 @ N3 ) ) ) % 0.31/0.61 => ( ord_less_eq @ A @ C5 @ L ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Lim_bounded2 % 0.31/0.61 thf(fact_5576_LIMSEQ__le__const,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 => ! [X8: nat > A,X3: A,A2: A] : % 0.31/0.61 ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ? [N8: nat] : % 0.31/0.61 ! [N3: nat] : % 0.31/0.61 ( ( ord_less_eq @ nat @ N8 @ N3 ) % 0.31/0.61 => ( ord_less_eq @ A @ A2 @ ( X8 @ N3 ) ) ) % 0.31/0.61 => ( ord_less_eq @ A @ A2 @ X3 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_le_const % 0.31/0.61 thf(fact_5577_LIMSEQ__le__const2,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 => ! [X8: nat > A,X3: A,A2: A] : % 0.31/0.61 ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ? [N8: nat] : % 0.31/0.61 ! [N3: nat] : % 0.31/0.61 ( ( ord_less_eq @ nat @ N8 @ N3 ) % 0.31/0.61 => ( ord_less_eq @ A @ ( X8 @ N3 ) @ A2 ) ) % 0.31/0.61 => ( ord_less_eq @ A @ X3 @ A2 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_le_const2 % 0.31/0.61 thf(fact_5578_Sup__lim,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( comple5582772986160207858norder @ A ) % 0.31/0.61 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.61 => ! [B2: nat > A,S: set @ A,A2: A] : % 0.31/0.61 ( ! [N3: nat] : ( member @ A @ ( B2 @ N3 ) @ S ) % 0.31/0.61 => ( ( filterlim @ nat @ A @ B2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ord_less_eq @ A @ A2 @ ( complete_Sup_Sup @ A @ S ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Sup_lim % 0.31/0.61 thf(fact_5579_Inf__lim,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( comple5582772986160207858norder @ A ) % 0.31/0.61 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.61 => ! [B2: nat > A,S: set @ A,A2: A] : % 0.31/0.61 ( ! [N3: nat] : ( member @ A @ ( B2 @ N3 ) @ S ) % 0.31/0.61 => ( ( filterlim @ nat @ A @ B2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ S ) @ A2 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Inf_lim % 0.31/0.61 thf(fact_5580_Inf__as__limit,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( comple5582772986160207858norder @ A ) % 0.31/0.61 & ( topolo3112930676232923870pology @ A ) % 0.31/0.61 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.61 => ! [A3: set @ A] : % 0.31/0.61 ( ( A3 % 0.31/0.61 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.61 => ? [U4: nat > A] : % 0.31/0.61 ( ! [N6: nat] : ( member @ A @ ( U4 @ N6 ) @ A3 ) % 0.31/0.61 & ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ ( complete_Inf_Inf @ A @ A3 ) ) @ ( at_top @ nat ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % Inf_as_limit % 0.31/0.61 thf(fact_5581_summable__LIMSEQ__zero,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [F2: nat > A] : % 0.31/0.61 ( ( summable @ A @ F2 ) % 0.31/0.61 => ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % summable_LIMSEQ_zero % 0.31/0.61 thf(fact_5582_mult__nat__left__at__top,axiom, % 0.31/0.61 ! [C2: nat] : % 0.31/0.61 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 ) % 0.31/0.61 => ( filterlim @ nat @ nat @ ( times_times @ nat @ C2 ) @ ( at_top @ nat ) @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % mult_nat_left_at_top % 0.31/0.61 thf(fact_5583_mult__nat__right__at__top,axiom, % 0.31/0.61 ! [C2: nat] : % 0.31/0.61 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 ) % 0.31/0.61 => ( filterlim @ nat @ nat % 0.31/0.61 @ ^ [X: nat] : ( times_times @ nat @ X @ C2 ) % 0.31/0.61 @ ( at_top @ nat ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % mult_nat_right_at_top % 0.31/0.61 thf(fact_5584_monoseq__le,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 => ! [A2: nat > A,X3: A] : % 0.31/0.61 ( ( topological_monoseq @ A @ A2 ) % 0.31/0.61 => ( ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ( ! [N6: nat] : ( ord_less_eq @ A @ ( A2 @ N6 ) @ X3 ) % 0.31/0.61 & ! [M6: nat,N6: nat] : % 0.31/0.61 ( ( ord_less_eq @ nat @ M6 @ N6 ) % 0.31/0.61 => ( ord_less_eq @ A @ ( A2 @ M6 ) @ ( A2 @ N6 ) ) ) ) % 0.31/0.61 | ( ! [N6: nat] : ( ord_less_eq @ A @ X3 @ ( A2 @ N6 ) ) % 0.31/0.61 & ! [M6: nat,N6: nat] : % 0.31/0.61 ( ( ord_less_eq @ nat @ M6 @ N6 ) % 0.31/0.61 => ( ord_less_eq @ A @ ( A2 @ N6 ) @ ( A2 @ M6 ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % monoseq_le % 0.31/0.61 thf(fact_5585_lim__const__over__n,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [A2: A] : % 0.31/0.61 ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ N ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % lim_const_over_n % 0.31/0.61 thf(fact_5586_lim__inverse__n,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % lim_inverse_n % 0.31/0.61 thf(fact_5587_LIMSEQ__linear,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.61 => ! [X8: nat > A,X3: A,L: nat] : % 0.31/0.61 ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L ) % 0.31/0.61 => ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( X8 @ ( times_times @ nat @ N @ L ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_linear % 0.31/0.61 thf(fact_5588_telescope__summable_H,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [F2: nat > A,C2: A] : % 0.31/0.61 ( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( summable @ A % 0.31/0.61 @ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % telescope_summable' % 0.31/0.61 thf(fact_5589_telescope__summable,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [F2: nat > A,C2: A] : % 0.31/0.61 ( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( summable @ A % 0.31/0.61 @ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % telescope_summable % 0.31/0.61 thf(fact_5590_nested__sequence__unique,axiom, % 0.31/0.61 ! [F2: nat > real,G: nat > real] : % 0.31/0.61 ( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) ) % 0.31/0.61 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) ) % 0.31/0.61 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ ( G @ N3 ) ) % 0.31/0.61 => ( ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : ( minus_minus @ real @ ( F2 @ N ) @ ( G @ N ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 0.31/0.61 @ ( at_top @ nat ) ) % 0.31/0.61 => ? [L4: real] : % 0.31/0.61 ( ! [N6: nat] : ( ord_less_eq @ real @ ( F2 @ N6 ) @ L4 ) % 0.31/0.61 & ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) ) % 0.31/0.61 & ! [N6: nat] : ( ord_less_eq @ real @ L4 @ ( G @ N6 ) ) % 0.31/0.61 & ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ L4 ) @ ( at_top @ nat ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % nested_sequence_unique % 0.31/0.61 thf(fact_5591_LIMSEQ__inverse__zero,axiom, % 0.31/0.61 ! [X8: nat > real] : % 0.31/0.61 ( ! [R4: real] : % 0.31/0.61 ? [N8: nat] : % 0.31/0.61 ! [N3: nat] : % 0.31/0.61 ( ( ord_less_eq @ nat @ N8 @ N3 ) % 0.31/0.61 => ( ord_less @ real @ R4 @ ( X8 @ N3 ) ) ) % 0.31/0.61 => ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : ( inverse_inverse @ real @ ( X8 @ N ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_inverse_zero % 0.31/0.61 thf(fact_5592_LIMSEQ__root__const,axiom, % 0.31/0.61 ! [C2: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.61 => ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : ( root @ N @ C2 ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_root_const % 0.31/0.61 thf(fact_5593_LIMSEQ__inverse__real__of__nat,axiom, % 0.31/0.61 ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_inverse_real_of_nat % 0.31/0.61 thf(fact_5594_LIMSEQ__inverse__real__of__nat__add,axiom, % 0.31/0.61 ! [R2: real] : % 0.31/0.61 ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : ( plus_plus @ real @ R2 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ R2 ) % 0.31/0.61 @ ( at_top @ nat ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_inverse_real_of_nat_add % 0.31/0.61 thf(fact_5595_increasing__LIMSEQ,axiom, % 0.31/0.61 ! [F2: nat > real,L: real] : % 0.31/0.61 ( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) ) % 0.31/0.61 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( F2 @ N3 ) @ L ) % 0.31/0.61 => ( ! [E: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ E ) % 0.31/0.61 => ? [N6: nat] : ( ord_less_eq @ real @ L @ ( plus_plus @ real @ ( F2 @ N6 ) @ E ) ) ) % 0.31/0.61 => ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( at_top @ nat ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % increasing_LIMSEQ % 0.31/0.61 thf(fact_5596_lim__1__over__n,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % lim_1_over_n % 0.31/0.61 thf(fact_5597_LIMSEQ__Suc__n__over__n,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( semiring_1_of_nat @ A @ N ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_Suc_n_over_n % 0.31/0.61 thf(fact_5598_LIMSEQ__n__over__Suc__n,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_n_over_Suc_n % 0.31/0.61 thf(fact_5599_LIMSEQ__realpow__zero,axiom, % 0.31/0.61 ! [X3: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.61 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 0.31/0.61 => ( filterlim @ nat @ real @ ( power_power @ real @ X3 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_realpow_zero % 0.31/0.61 thf(fact_5600_telescope__sums_H,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [F2: nat > A,C2: A] : % 0.31/0.61 ( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( sums @ A % 0.31/0.61 @ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) ) % 0.31/0.61 @ ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ C2 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % telescope_sums' % 0.31/0.61 thf(fact_5601_telescope__sums,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [F2: nat > A,C2: A] : % 0.31/0.61 ( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( sums @ A % 0.31/0.61 @ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) ) % 0.31/0.61 @ ( minus_minus @ A @ C2 @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % telescope_sums % 0.31/0.61 thf(fact_5602_LIMSEQ__divide__realpow__zero,axiom, % 0.31/0.61 ! [X3: real,A2: real] : % 0.31/0.61 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 0.31/0.61 => ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : ( divide_divide @ real @ A2 @ ( power_power @ real @ X3 @ N ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_divide_realpow_zero % 0.31/0.61 thf(fact_5603_LIMSEQ__abs__realpow__zero2,axiom, % 0.31/0.61 ! [C2: real] : % 0.31/0.61 ( ( ord_less @ real @ ( abs_abs @ real @ C2 ) @ ( one_one @ real ) ) % 0.31/0.61 => ( filterlim @ nat @ real @ ( power_power @ real @ C2 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_abs_realpow_zero2 % 0.31/0.61 thf(fact_5604_LIMSEQ__abs__realpow__zero,axiom, % 0.31/0.61 ! [C2: real] : % 0.31/0.61 ( ( ord_less @ real @ ( abs_abs @ real @ C2 ) @ ( one_one @ real ) ) % 0.31/0.61 => ( filterlim @ nat @ real @ ( power_power @ real @ ( abs_abs @ real @ C2 ) ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_abs_realpow_zero % 0.31/0.61 thf(fact_5605_LIMSEQ__inverse__realpow__zero,axiom, % 0.31/0.61 ! [X3: real] : % 0.31/0.61 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 0.31/0.61 => ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : ( inverse_inverse @ real @ ( power_power @ real @ X3 @ N ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_inverse_realpow_zero % 0.31/0.61 thf(fact_5606_sums__def_H,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( comm_monoid_add @ A ) % 0.31/0.61 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.61 => ( ( sums @ A ) % 0.31/0.61 = ( ^ [F3: nat > A,S8: A] : % 0.31/0.61 ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ S8 ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % sums_def' % 0.31/0.61 thf(fact_5607_root__test__convergence,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_Vector_banach @ A ) % 0.31/0.61 => ! [F2: nat > A,X3: real] : % 0.31/0.61 ( ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : ( root @ N @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N ) ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ X3 ) % 0.31/0.61 @ ( at_top @ nat ) ) % 0.31/0.61 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 0.31/0.61 => ( summable @ A @ F2 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % root_test_convergence % 0.31/0.61 thf(fact_5608_LIMSEQ__inverse__real__of__nat__add__minus,axiom, % 0.31/0.61 ! [R2: real] : % 0.31/0.61 ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : ( plus_plus @ real @ R2 @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ R2 ) % 0.31/0.61 @ ( at_top @ nat ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_inverse_real_of_nat_add_minus % 0.31/0.61 thf(fact_5609_LIMSEQ__D,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [X8: nat > A,L5: A,R2: real] : % 0.31/0.61 ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 ) % 0.31/0.61 => ? [No: nat] : % 0.31/0.61 ! [N6: nat] : % 0.31/0.61 ( ( ord_less_eq @ nat @ No @ N6 ) % 0.31/0.61 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N6 ) @ L5 ) ) @ R2 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_D % 0.31/0.61 thf(fact_5610_LIMSEQ__I,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [X8: nat > A,L5: A] : % 0.31/0.61 ( ! [R4: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 ) % 0.31/0.61 => ? [No2: nat] : % 0.31/0.61 ! [N3: nat] : % 0.31/0.61 ( ( ord_less_eq @ nat @ No2 @ N3 ) % 0.31/0.61 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N3 ) @ L5 ) ) @ R4 ) ) ) % 0.31/0.61 => ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_I % 0.31/0.61 thf(fact_5611_LIMSEQ__iff,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [X8: nat > A,L5: A] : % 0.31/0.61 ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) % 0.31/0.61 = ( ! [R5: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ R5 ) % 0.31/0.61 => ? [No3: nat] : % 0.31/0.61 ! [N: nat] : % 0.31/0.61 ( ( ord_less_eq @ nat @ No3 @ N ) % 0.31/0.61 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N ) @ L5 ) ) @ R5 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_iff % 0.31/0.61 thf(fact_5612_LIMSEQ__power__zero,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V2822296259951069270ebra_1 @ A ) % 0.31/0.61 => ! [X3: A] : % 0.31/0.61 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) ) % 0.31/0.61 => ( filterlim @ nat @ A @ ( power_power @ A @ X3 ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_power_zero % 0.31/0.61 thf(fact_5613_tendsto__at__iff__sequentially,axiom, % 0.31/0.61 ! [C: $tType,A: $tType] : % 0.31/0.61 ( ( ( topolo3112930676232923870pology @ A ) % 0.31/0.61 & ( topolo4958980785337419405_space @ C ) ) % 0.31/0.61 => ! [F2: A > C,A2: C,X3: A,S: set @ A] : % 0.31/0.61 ( ( filterlim @ A @ C @ F2 @ ( topolo7230453075368039082e_nhds @ C @ A2 ) @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.61 = ( ! [X6: nat > A] : % 0.31/0.61 ( ! [I3: nat] : ( member @ A @ ( X6 @ I3 ) @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.61 => ( ( filterlim @ nat @ A @ X6 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( filterlim @ nat @ C @ ( comp @ A @ C @ nat @ F2 @ X6 ) @ ( topolo7230453075368039082e_nhds @ C @ A2 ) @ ( at_top @ nat ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_at_iff_sequentially % 0.31/0.61 thf(fact_5614_tendsto__power__zero,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( real_V2822296259951069270ebra_1 @ A ) % 0.31/0.61 => ! [F2: B > nat,F5: filter @ B,X3: A] : % 0.31/0.61 ( ( filterlim @ B @ nat @ F2 @ ( at_top @ nat ) @ F5 ) % 0.31/0.61 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) ) % 0.31/0.61 => ( filterlim @ B @ A % 0.31/0.61 @ ^ [Y2: B] : ( power_power @ A @ X3 @ ( F2 @ Y2 ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_power_zero % 0.31/0.61 thf(fact_5615_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom, % 0.31/0.61 ! [R2: real] : % 0.31/0.61 ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : ( times_times @ real @ R2 @ ( plus_plus @ real @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) ) ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ R2 ) % 0.31/0.61 @ ( at_top @ nat ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_inverse_real_of_nat_add_minus_mult % 0.31/0.61 thf(fact_5616_LIMSEQ__norm__0,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [F2: nat > A] : % 0.31/0.61 ( ! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) ) % 0.31/0.61 => ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIMSEQ_norm_0 % 0.31/0.61 thf(fact_5617_field__derivative__lim__unique,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.61 => ! [F2: A > A,Df: A,Z2: A,S: nat > A,A2: A] : % 0.31/0.61 ( ( has_field_derivative @ A @ F2 @ Df @ ( topolo174197925503356063within @ A @ Z2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 => ( ( filterlim @ nat @ A @ S @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ! [N3: nat] : % 0.31/0.61 ( ( S @ N3 ) % 0.31/0.61 != ( zero_zero @ A ) ) % 0.31/0.61 => ( ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ Z2 @ ( S @ N ) ) ) @ ( F2 @ Z2 ) ) @ ( S @ N ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) % 0.31/0.61 @ ( at_top @ nat ) ) % 0.31/0.61 => ( Df = A2 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % field_derivative_lim_unique % 0.31/0.61 thf(fact_5618_powser__times__n__limit__0,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V8999393235501362500lgebra @ A ) ) % 0.31/0.61 => ! [X3: A] : % 0.31/0.61 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) ) % 0.31/0.61 => ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ X3 @ N ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % powser_times_n_limit_0 % 0.31/0.61 thf(fact_5619_lim__n__over__pown,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( real_Vector_banach @ A ) % 0.31/0.61 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.61 => ! [X3: A] : % 0.31/0.61 ( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) % 0.31/0.61 => ( filterlim @ nat @ A % 0.31/0.61 @ ^ [N: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ X3 @ N ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % lim_n_over_pown % 0.31/0.61 thf(fact_5620_summable,axiom, % 0.31/0.61 ! [A2: nat > real] : % 0.31/0.61 ( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) ) % 0.31/0.61 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) ) % 0.31/0.61 => ( summable @ real % 0.31/0.61 @ ^ [N: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) @ ( A2 @ N ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % summable % 0.31/0.61 thf(fact_5621_zeroseq__arctan__series,axiom, % 0.31/0.61 ! [X3: real] : % 0.31/0.61 ( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 0.31/0.61 => ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % zeroseq_arctan_series % 0.31/0.61 thf(fact_5622_summable__Leibniz_H_I3_J,axiom, % 0.31/0.61 ! [A2: nat > real] : % 0.31/0.61 ( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) ) % 0.31/0.61 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) ) % 0.31/0.61 => ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : % 0.31/0.61 ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real % 0.31/0.61 @ ( suminf @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % summable_Leibniz'(3) % 0.31/0.61 thf(fact_5623_summable__Leibniz_H_I2_J,axiom, % 0.31/0.61 ! [A2: nat > real,N2: nat] : % 0.31/0.61 ( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) ) % 0.31/0.61 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) ) % 0.31/0.61 => ( ord_less_eq @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 0.31/0.61 @ ( suminf @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % summable_Leibniz'(2) % 0.31/0.61 thf(fact_5624_sums__alternating__upper__lower,axiom, % 0.31/0.61 ! [A2: nat > real] : % 0.31/0.61 ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) ) % 0.31/0.61 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) ) % 0.31/0.61 => ( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) % 0.31/0.61 => ? [L4: real] : % 0.31/0.61 ( ! [N6: nat] : % 0.31/0.61 ( ord_less_eq @ real % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) ) ) % 0.31/0.61 @ L4 ) % 0.31/0.61 & ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : % 0.31/0.61 ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ L4 ) % 0.31/0.61 @ ( at_top @ nat ) ) % 0.31/0.61 & ! [N6: nat] : % 0.31/0.61 ( ord_less_eq @ real @ L4 % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N6 ) @ ( one_one @ nat ) ) ) ) ) % 0.31/0.61 & ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : % 0.31/0.61 ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ L4 ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % sums_alternating_upper_lower % 0.31/0.61 thf(fact_5625_summable__Leibniz_I5_J,axiom, % 0.31/0.61 ! [A2: nat > real] : % 0.31/0.61 ( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ( topological_monoseq @ real @ A2 ) % 0.31/0.61 => ( filterlim @ nat @ real % 0.31/0.61 @ ^ [N: nat] : % 0.31/0.61 ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real % 0.31/0.61 @ ( suminf @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % summable_Leibniz(5) % 0.31/0.61 thf(fact_5626_summable__Leibniz_H_I4_J,axiom, % 0.31/0.61 ! [A2: nat > real,N2: nat] : % 0.31/0.61 ( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) ) % 0.31/0.61 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) ) % 0.31/0.61 => ( ord_less_eq @ real % 0.31/0.61 @ ( suminf @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) ) % 0.31/0.61 @ ( groups7311177749621191930dd_sum @ nat @ real % 0.31/0.61 @ ^ [I3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I3 ) @ ( A2 @ I3 ) ) % 0.31/0.61 @ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % summable_Leibniz'(4) % 0.31/0.61 thf(fact_5627_has__derivative__at2,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [F2: A > B,F11: A > B,X3: A] : % 0.31/0.61 ( ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 = ( ( real_V3181309239436604168linear @ A @ B @ F11 ) % 0.31/0.61 & ( filterlim @ A @ B % 0.31/0.61 @ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X3 ) ) ) @ ( minus_minus @ B @ ( F2 @ Y2 ) @ ( plus_plus @ B @ ( F2 @ X3 ) @ ( F11 @ ( minus_minus @ A @ Y2 @ X3 ) ) ) ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_at2 % 0.31/0.61 thf(fact_5628_summable__Cauchy_H,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_Vector_banach @ A ) % 0.31/0.61 => ! [F2: nat > A,G: nat > real] : % 0.31/0.61 ( ( eventually @ nat % 0.31/0.61 @ ^ [M2: nat] : % 0.31/0.61 ! [N: nat] : % 0.31/0.61 ( ( ord_less_eq @ nat @ M2 @ N ) % 0.31/0.61 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) @ ( G @ M2 ) ) ) % 0.31/0.61 @ ( at_top @ nat ) ) % 0.31/0.61 => ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) % 0.31/0.61 => ( summable @ A @ F2 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % summable_Cauchy' % 0.31/0.61 thf(fact_5629_has__derivative__at,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [F2: A > B,D6: A > B,X3: A] : % 0.31/0.61 ( ( has_derivative @ A @ B @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 = ( ( real_V3181309239436604168linear @ A @ B @ D6 ) % 0.31/0.61 & ( filterlim @ A @ real % 0.31/0.61 @ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ ( plus_plus @ A @ X3 @ H ) ) @ ( F2 @ X3 ) ) @ ( D6 @ H ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_at % 0.31/0.61 thf(fact_5630_eventually__const,axiom, % 0.31/0.61 ! [A: $tType,F5: filter @ A,P: $o] : % 0.31/0.61 ( ( F5 % 0.31/0.61 != ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.61 => ( ( eventually @ A % 0.31/0.61 @ ^ [X: A] : P % 0.31/0.61 @ F5 ) % 0.31/0.61 = P ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_const % 0.31/0.61 thf(fact_5631_eventually__sequentially__Suc,axiom, % 0.31/0.61 ! [P: nat > $o] : % 0.31/0.61 ( ( eventually @ nat % 0.31/0.61 @ ^ [I3: nat] : ( P @ ( suc @ I3 ) ) % 0.31/0.61 @ ( at_top @ nat ) ) % 0.31/0.61 = ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_sequentially_Suc % 0.31/0.61 thf(fact_5632_eventually__sequentially__seg,axiom, % 0.31/0.61 ! [P: nat > $o,K: nat] : % 0.31/0.61 ( ( eventually @ nat % 0.31/0.61 @ ^ [N: nat] : ( P @ ( plus_plus @ nat @ N @ K ) ) % 0.31/0.61 @ ( at_top @ nat ) ) % 0.31/0.61 = ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_sequentially_seg % 0.31/0.61 thf(fact_5633_eventually__at__top__dense,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( linorder @ A ) % 0.31/0.61 & ( no_top @ A ) ) % 0.31/0.61 => ! [P: A > $o] : % 0.31/0.61 ( ( eventually @ A @ P @ ( at_top @ A ) ) % 0.31/0.61 = ( ? [N5: A] : % 0.31/0.61 ! [N: A] : % 0.31/0.61 ( ( ord_less @ A @ N5 @ N ) % 0.31/0.61 => ( P @ N ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_at_top_dense % 0.31/0.61 thf(fact_5634_eventually__at__top__linorderI,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( linorder @ A ) % 0.31/0.61 => ! [C2: A,P: A > $o] : % 0.31/0.61 ( ! [X5: A] : % 0.31/0.61 ( ( ord_less_eq @ A @ C2 @ X5 ) % 0.31/0.61 => ( P @ X5 ) ) % 0.31/0.61 => ( eventually @ A @ P @ ( at_top @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_at_top_linorderI % 0.31/0.61 thf(fact_5635_eventually__at__top__linorder,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( linorder @ A ) % 0.31/0.61 => ! [P: A > $o] : % 0.31/0.61 ( ( eventually @ A @ P @ ( at_top @ A ) ) % 0.31/0.61 = ( ? [N5: A] : % 0.31/0.61 ! [N: A] : % 0.31/0.61 ( ( ord_less_eq @ A @ N5 @ N ) % 0.31/0.61 => ( P @ N ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_at_top_linorder % 0.31/0.61 thf(fact_5636_eventually__sequentiallyI,axiom, % 0.31/0.61 ! [C2: nat,P: nat > $o] : % 0.31/0.61 ( ! [X5: nat] : % 0.31/0.61 ( ( ord_less_eq @ nat @ C2 @ X5 ) % 0.31/0.61 => ( P @ X5 ) ) % 0.31/0.61 => ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_sequentiallyI % 0.31/0.61 thf(fact_5637_eventually__sequentially,axiom, % 0.31/0.61 ! [P: nat > $o] : % 0.31/0.61 ( ( eventually @ nat @ P @ ( at_top @ nat ) ) % 0.31/0.61 = ( ? [N5: nat] : % 0.31/0.61 ! [N: nat] : % 0.31/0.61 ( ( ord_less_eq @ nat @ N5 @ N ) % 0.31/0.61 => ( P @ N ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_sequentially % 0.31/0.61 thf(fact_5638_eventually__bot,axiom, % 0.31/0.61 ! [A: $tType,P: A > $o] : ( eventually @ A @ P @ ( bot_bot @ ( filter @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_bot % 0.31/0.61 thf(fact_5639_eventually__happens,axiom, % 0.31/0.61 ! [A: $tType,P: A > $o,Net: filter @ A] : % 0.31/0.61 ( ( eventually @ A @ P @ Net ) % 0.31/0.61 => ( ( Net % 0.31/0.61 = ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.61 | ? [X_1: A] : ( P @ X_1 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_happens % 0.31/0.61 thf(fact_5640_eventually__happens_H,axiom, % 0.31/0.61 ! [A: $tType,F5: filter @ A,P: A > $o] : % 0.31/0.61 ( ( F5 % 0.31/0.61 != ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.61 => ( ( eventually @ A @ P @ F5 ) % 0.31/0.61 => ? [X_1: A] : ( P @ X_1 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_happens' % 0.31/0.61 thf(fact_5641_False__imp__not__eventually,axiom, % 0.31/0.61 ! [A: $tType,P: A > $o,Net: filter @ A] : % 0.31/0.61 ( ! [X5: A] : % 0.31/0.61 ~ ( P @ X5 ) % 0.31/0.61 => ( ( Net % 0.31/0.61 != ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.61 => ~ ( eventually @ A @ P @ Net ) ) ) ). % 0.31/0.61 % 0.31/0.61 % False_imp_not_eventually % 0.31/0.61 thf(fact_5642_eventually__const__iff,axiom, % 0.31/0.61 ! [A: $tType,P: $o,F5: filter @ A] : % 0.31/0.61 ( ( eventually @ A % 0.31/0.61 @ ^ [X: A] : P % 0.31/0.61 @ F5 ) % 0.31/0.61 = ( P % 0.31/0.61 | ( F5 % 0.31/0.61 = ( bot_bot @ ( filter @ A ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_const_iff % 0.31/0.61 thf(fact_5643_trivial__limit__def,axiom, % 0.31/0.61 ! [A: $tType,F5: filter @ A] : % 0.31/0.61 ( ( F5 % 0.31/0.61 = ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.61 = ( eventually @ A % 0.31/0.61 @ ^ [X: A] : $false % 0.31/0.61 @ F5 ) ) ). % 0.31/0.61 % 0.31/0.61 % trivial_limit_def % 0.31/0.61 thf(fact_5644_bounded__linear__zero,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ( real_V3181309239436604168linear @ A @ B % 0.31/0.61 @ ^ [X: A] : ( zero_zero @ B ) ) ) ). % 0.31/0.61 % 0.31/0.61 % bounded_linear_zero % 0.31/0.61 thf(fact_5645_bounded__linear__add,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [F2: A > B,G: A > B] : % 0.31/0.61 ( ( real_V3181309239436604168linear @ A @ B @ F2 ) % 0.31/0.61 => ( ( real_V3181309239436604168linear @ A @ B @ G ) % 0.31/0.61 => ( real_V3181309239436604168linear @ A @ B % 0.31/0.61 @ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % bounded_linear_add % 0.31/0.61 thf(fact_5646_eventually__ex,axiom, % 0.31/0.61 ! [B: $tType,A: $tType,P: A > B > $o,F5: filter @ A] : % 0.31/0.61 ( ( eventually @ A % 0.31/0.61 @ ^ [X: A] : % 0.31/0.61 ? [X6: B] : ( P @ X @ X6 ) % 0.31/0.61 @ F5 ) % 0.31/0.61 = ( ? [Y9: A > B] : % 0.31/0.61 ( eventually @ A % 0.31/0.61 @ ^ [X: A] : ( P @ X @ ( Y9 @ X ) ) % 0.31/0.61 @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_ex % 0.31/0.61 thf(fact_5647_eventually__ge__at__top,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( linorder @ A ) % 0.31/0.61 => ! [C2: A] : ( eventually @ A @ ( ord_less_eq @ A @ C2 ) @ ( at_top @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_ge_at_top % 0.31/0.61 thf(fact_5648_eventually__gt__at__top,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( linorder @ A ) % 0.31/0.61 & ( no_top @ A ) ) % 0.31/0.61 => ! [C2: A] : ( eventually @ A @ ( ord_less @ A @ C2 ) @ ( at_top @ A ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_gt_at_top % 0.31/0.61 thf(fact_5649_eventually__ball__finite__distrib,axiom, % 0.31/0.61 ! [B: $tType,A: $tType,A3: set @ A,P: B > A > $o,Net: filter @ B] : % 0.31/0.61 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.61 => ( ( eventually @ B % 0.31/0.61 @ ^ [X: B] : % 0.31/0.61 ! [Y2: A] : % 0.31/0.61 ( ( member @ A @ Y2 @ A3 ) % 0.31/0.61 => ( P @ X @ Y2 ) ) % 0.31/0.61 @ Net ) % 0.31/0.61 = ( ! [X: A] : % 0.31/0.61 ( ( member @ A @ X @ A3 ) % 0.31/0.61 => ( eventually @ B % 0.31/0.61 @ ^ [Y2: B] : ( P @ Y2 @ X ) % 0.31/0.61 @ Net ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_ball_finite_distrib % 0.31/0.61 thf(fact_5650_eventually__ball__finite,axiom, % 0.31/0.61 ! [A: $tType,B: $tType,A3: set @ A,P: B > A > $o,Net: filter @ B] : % 0.31/0.61 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.61 => ( ! [X5: A] : % 0.31/0.61 ( ( member @ A @ X5 @ A3 ) % 0.31/0.61 => ( eventually @ B % 0.31/0.61 @ ^ [Y2: B] : ( P @ Y2 @ X5 ) % 0.31/0.61 @ Net ) ) % 0.31/0.61 => ( eventually @ B % 0.31/0.61 @ ^ [X: B] : % 0.31/0.61 ! [Y2: A] : % 0.31/0.61 ( ( member @ A @ Y2 @ A3 ) % 0.31/0.61 => ( P @ X @ Y2 ) ) % 0.31/0.61 @ Net ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_ball_finite % 0.31/0.61 thf(fact_5651_le__sequentially,axiom, % 0.31/0.61 ! [F5: filter @ nat] : % 0.31/0.61 ( ( ord_less_eq @ ( filter @ nat ) @ F5 @ ( at_top @ nat ) ) % 0.31/0.61 = ( ! [N5: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N5 ) @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % le_sequentially % 0.31/0.61 thf(fact_5652_sequentially__offset,axiom, % 0.31/0.61 ! [P: nat > $o,K: nat] : % 0.31/0.61 ( ( eventually @ nat @ P @ ( at_top @ nat ) ) % 0.31/0.61 => ( eventually @ nat % 0.31/0.61 @ ^ [I3: nat] : ( P @ ( plus_plus @ nat @ I3 @ K ) ) % 0.31/0.61 @ ( at_top @ nat ) ) ) ). % 0.31/0.61 % 0.31/0.61 % sequentially_offset % 0.31/0.61 thf(fact_5653_LIM__at__top__divide,axiom, % 0.31/0.61 ! [A: $tType,F2: A > real,A2: real,F5: filter @ A,G: A > real] : % 0.31/0.61 ( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 0.31/0.61 => ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) % 0.31/0.61 => ( ( eventually @ A % 0.31/0.61 @ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( filterlim @ A @ real % 0.31/0.61 @ ^ [X: A] : ( divide_divide @ real @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ( at_top @ real ) % 0.31/0.61 @ F5 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % LIM_at_top_divide % 0.31/0.61 thf(fact_5654_filterlim__inverse__at__top,axiom, % 0.31/0.61 ! [A: $tType,F2: A > real,F5: filter @ A] : % 0.31/0.61 ( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) % 0.31/0.61 => ( ( eventually @ A % 0.31/0.61 @ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( filterlim @ A @ real % 0.31/0.61 @ ^ [X: A] : ( inverse_inverse @ real @ ( F2 @ X ) ) % 0.31/0.61 @ ( at_top @ real ) % 0.31/0.61 @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_inverse_at_top % 0.31/0.61 thf(fact_5655_filterlim__inverse__at__top__iff,axiom, % 0.31/0.61 ! [A: $tType,F2: A > real,F5: filter @ A] : % 0.31/0.61 ( ( eventually @ A % 0.31/0.61 @ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( ( filterlim @ A @ real % 0.31/0.61 @ ^ [X: A] : ( inverse_inverse @ real @ ( F2 @ X ) ) % 0.31/0.61 @ ( at_top @ real ) % 0.31/0.61 @ F5 ) % 0.31/0.61 = ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_inverse_at_top_iff % 0.31/0.61 thf(fact_5656_filterlim__at__top__iff__inverse__0,axiom, % 0.31/0.61 ! [A: $tType,F2: A > real,F5: filter @ A] : % 0.31/0.61 ( ( eventually @ A % 0.31/0.61 @ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F5 ) % 0.31/0.61 = ( filterlim @ A @ real @ ( comp @ real @ real @ A @ ( inverse_inverse @ real ) @ F2 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_at_top_iff_inverse_0 % 0.31/0.61 thf(fact_5657_eventually__nhds__top,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( order_top @ A ) % 0.31/0.61 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.61 => ! [B2: A,P: A > $o] : % 0.31/0.61 ( ( ord_less @ A @ B2 @ ( top_top @ A ) ) % 0.31/0.61 => ( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ ( top_top @ A ) ) ) % 0.31/0.61 = ( ? [B4: A] : % 0.31/0.61 ( ( ord_less @ A @ B4 @ ( top_top @ A ) ) % 0.31/0.61 & ! [Z6: A] : % 0.31/0.61 ( ( ord_less @ A @ B4 @ Z6 ) % 0.31/0.61 => ( P @ Z6 ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_nhds_top % 0.31/0.61 thf(fact_5658_filterlim__at__top__at__top,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( linorder @ A ) % 0.31/0.61 & ( linorder @ B ) ) % 0.31/0.61 => ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A] : % 0.31/0.61 ( ! [X5: A,Y4: A] : % 0.31/0.61 ( ( Q @ X5 ) % 0.31/0.61 => ( ( Q @ Y4 ) % 0.31/0.61 => ( ( ord_less_eq @ A @ X5 @ Y4 ) % 0.31/0.61 => ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) ) ) % 0.31/0.61 => ( ! [X5: B] : % 0.31/0.61 ( ( P @ X5 ) % 0.31/0.61 => ( ( F2 @ ( G @ X5 ) ) % 0.31/0.61 = X5 ) ) % 0.31/0.61 => ( ! [X5: B] : % 0.31/0.61 ( ( P @ X5 ) % 0.31/0.61 => ( Q @ ( G @ X5 ) ) ) % 0.31/0.61 => ( ( eventually @ A @ Q @ ( at_top @ A ) ) % 0.31/0.61 => ( ( eventually @ B @ P @ ( at_top @ B ) ) % 0.31/0.61 => ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_at_top_at_top % 0.31/0.61 thf(fact_5659_eventually__at__left__field,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( ( linordered_field @ A ) % 0.31/0.61 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.61 => ! [P: A > $o,X3: A] : % 0.31/0.61 ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X3 @ ( set_ord_lessThan @ A @ X3 ) ) ) % 0.31/0.61 = ( ? [B4: A] : % 0.31/0.61 ( ( ord_less @ A @ B4 @ X3 ) % 0.31/0.61 & ! [Y2: A] : % 0.31/0.61 ( ( ord_less @ A @ B4 @ Y2 ) % 0.31/0.61 => ( ( ord_less @ A @ Y2 @ X3 ) % 0.31/0.61 => ( P @ Y2 ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_at_left_field % 0.31/0.61 thf(fact_5660_eventually__at__left,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 => ! [Y: A,X3: A,P: A > $o] : % 0.31/0.61 ( ( ord_less @ A @ Y @ X3 ) % 0.31/0.61 => ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X3 @ ( set_ord_lessThan @ A @ X3 ) ) ) % 0.31/0.61 = ( ? [B4: A] : % 0.31/0.61 ( ( ord_less @ A @ B4 @ X3 ) % 0.31/0.61 & ! [Y2: A] : % 0.31/0.61 ( ( ord_less @ A @ B4 @ Y2 ) % 0.31/0.61 => ( ( ord_less @ A @ Y2 @ X3 ) % 0.31/0.61 => ( P @ Y2 ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_at_left % 0.31/0.61 thf(fact_5661_tendsto__sandwich,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( topolo2564578578187576103pology @ A ) % 0.31/0.61 => ! [F2: B > A,G: B > A,Net: filter @ B,H2: B > A,C2: A] : % 0.31/0.61 ( ( eventually @ B % 0.31/0.61 @ ^ [N: B] : ( ord_less_eq @ A @ ( F2 @ N ) @ ( G @ N ) ) % 0.31/0.61 @ Net ) % 0.31/0.61 => ( ( eventually @ B % 0.31/0.61 @ ^ [N: B] : ( ord_less_eq @ A @ ( G @ N ) @ ( H2 @ N ) ) % 0.31/0.61 @ Net ) % 0.31/0.61 => ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net ) % 0.31/0.61 => ( ( filterlim @ B @ A @ H2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net ) % 0.31/0.61 => ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_sandwich % 0.31/0.61 thf(fact_5662_order__tendsto__iff,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( topolo2564578578187576103pology @ A ) % 0.31/0.61 => ! [F2: B > A,X3: A,F5: filter @ B] : % 0.31/0.61 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ F5 ) % 0.31/0.61 = ( ! [L2: A] : % 0.31/0.61 ( ( ord_less @ A @ L2 @ X3 ) % 0.31/0.61 => ( eventually @ B % 0.31/0.61 @ ^ [X: B] : ( ord_less @ A @ L2 @ ( F2 @ X ) ) % 0.31/0.61 @ F5 ) ) % 0.31/0.61 & ! [U2: A] : % 0.31/0.61 ( ( ord_less @ A @ X3 @ U2 ) % 0.31/0.61 => ( eventually @ B % 0.31/0.61 @ ^ [X: B] : ( ord_less @ A @ ( F2 @ X ) @ U2 ) % 0.31/0.61 @ F5 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % order_tendsto_iff % 0.31/0.61 thf(fact_5663_order__tendstoI,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( topolo2564578578187576103pology @ A ) % 0.31/0.61 => ! [Y: A,F2: B > A,F5: filter @ B] : % 0.31/0.61 ( ! [A5: A] : % 0.31/0.61 ( ( ord_less @ A @ A5 @ Y ) % 0.31/0.61 => ( eventually @ B % 0.31/0.61 @ ^ [X: B] : ( ord_less @ A @ A5 @ ( F2 @ X ) ) % 0.31/0.61 @ F5 ) ) % 0.31/0.61 => ( ! [A5: A] : % 0.31/0.61 ( ( ord_less @ A @ Y @ A5 ) % 0.31/0.61 => ( eventually @ B % 0.31/0.61 @ ^ [X: B] : ( ord_less @ A @ ( F2 @ X ) @ A5 ) % 0.31/0.61 @ F5 ) ) % 0.31/0.61 => ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % order_tendstoI % 0.31/0.61 thf(fact_5664_order__tendstoD_I1_J,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( topolo2564578578187576103pology @ A ) % 0.31/0.61 => ! [F2: B > A,Y: A,F5: filter @ B,A2: A] : % 0.31/0.61 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 ) % 0.31/0.61 => ( ( ord_less @ A @ A2 @ Y ) % 0.31/0.61 => ( eventually @ B % 0.31/0.61 @ ^ [X: B] : ( ord_less @ A @ A2 @ ( F2 @ X ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % order_tendstoD(1) % 0.31/0.61 thf(fact_5665_order__tendstoD_I2_J,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( topolo2564578578187576103pology @ A ) % 0.31/0.61 => ! [F2: B > A,Y: A,F5: filter @ B,A2: A] : % 0.31/0.61 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 ) % 0.31/0.61 => ( ( ord_less @ A @ Y @ A2 ) % 0.31/0.61 => ( eventually @ B % 0.31/0.61 @ ^ [X: B] : ( ord_less @ A @ ( F2 @ X ) @ A2 ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % order_tendstoD(2) % 0.31/0.61 thf(fact_5666_eventually__all__ge__at__top,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( linorder @ A ) % 0.31/0.61 => ! [P: A > $o] : % 0.31/0.61 ( ( eventually @ A @ P @ ( at_top @ A ) ) % 0.31/0.61 => ( eventually @ A % 0.31/0.61 @ ^ [X: A] : % 0.31/0.61 ! [Y2: A] : % 0.31/0.61 ( ( ord_less_eq @ A @ X @ Y2 ) % 0.31/0.61 => ( P @ Y2 ) ) % 0.31/0.61 @ ( at_top @ A ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_all_ge_at_top % 0.31/0.61 thf(fact_5667_filterlim__at__top,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( linorder @ B ) % 0.31/0.61 => ! [F2: A > B,F5: filter @ A] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 ) % 0.31/0.61 = ( ! [Z9: B] : % 0.31/0.61 ( eventually @ A % 0.31/0.61 @ ^ [X: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_at_top % 0.31/0.61 thf(fact_5668_filterlim__at__top__ge,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( linorder @ B ) % 0.31/0.61 => ! [F2: A > B,F5: filter @ A,C2: B] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 ) % 0.31/0.61 = ( ! [Z9: B] : % 0.31/0.61 ( ( ord_less_eq @ B @ C2 @ Z9 ) % 0.31/0.61 => ( eventually @ A % 0.31/0.61 @ ^ [X: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X ) ) % 0.31/0.61 @ F5 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_at_top_ge % 0.31/0.61 thf(fact_5669_filterlim__at__top__mono,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( linorder @ A ) % 0.31/0.61 => ! [F2: B > A,F5: filter @ B,G: B > A] : % 0.31/0.61 ( ( filterlim @ B @ A @ F2 @ ( at_top @ A ) @ F5 ) % 0.31/0.61 => ( ( eventually @ B % 0.31/0.61 @ ^ [X: B] : ( ord_less_eq @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( filterlim @ B @ A @ G @ ( at_top @ A ) @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_at_top_mono % 0.31/0.61 thf(fact_5670_filterlim__at__top__dense,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( unboun7993243217541854897norder @ B ) % 0.31/0.61 => ! [F2: A > B,F5: filter @ A] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 ) % 0.31/0.61 = ( ! [Z9: B] : % 0.31/0.61 ( eventually @ A % 0.31/0.61 @ ^ [X: A] : ( ord_less @ B @ Z9 @ ( F2 @ X ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_at_top_dense % 0.31/0.61 thf(fact_5671_bounded__linear_Otendsto__zero,axiom, % 0.31/0.61 ! [B: $tType,A: $tType,C: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [F2: A > B,G: C > A,F5: filter @ C] : % 0.31/0.61 ( ( real_V3181309239436604168linear @ A @ B @ F2 ) % 0.31/0.61 => ( ( filterlim @ C @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 ) % 0.31/0.61 => ( filterlim @ C @ B % 0.31/0.61 @ ^ [X: C] : ( F2 @ ( G @ X ) ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % bounded_linear.tendsto_zero % 0.31/0.61 thf(fact_5672_eventually__Inf__base,axiom, % 0.31/0.61 ! [A: $tType,B3: set @ ( filter @ A ),P: A > $o] : % 0.31/0.61 ( ( B3 % 0.31/0.61 != ( bot_bot @ ( set @ ( filter @ A ) ) ) ) % 0.31/0.61 => ( ! [F6: filter @ A] : % 0.31/0.61 ( ( member @ ( filter @ A ) @ F6 @ B3 ) % 0.31/0.61 => ! [G5: filter @ A] : % 0.31/0.61 ( ( member @ ( filter @ A ) @ G5 @ B3 ) % 0.31/0.61 => ? [X2: filter @ A] : % 0.31/0.61 ( ( member @ ( filter @ A ) @ X2 @ B3 ) % 0.31/0.61 & ( ord_less_eq @ ( filter @ A ) @ X2 @ ( inf_inf @ ( filter @ A ) @ F6 @ G5 ) ) ) ) ) % 0.31/0.61 => ( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B3 ) ) % 0.31/0.61 = ( ? [X: filter @ A] : % 0.31/0.61 ( ( member @ ( filter @ A ) @ X @ B3 ) % 0.31/0.61 & ( eventually @ A @ P @ X ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_Inf_base % 0.31/0.61 thf(fact_5673_eventually__INF__finite,axiom, % 0.31/0.61 ! [B: $tType,A: $tType,A3: set @ A,P: B > $o,F5: A > ( filter @ B )] : % 0.31/0.61 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.61 => ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F5 @ A3 ) ) ) % 0.31/0.61 = ( ? [Q7: A > B > $o] : % 0.31/0.61 ( ! [X: A] : % 0.31/0.61 ( ( member @ A @ X @ A3 ) % 0.31/0.61 => ( eventually @ B @ ( Q7 @ X ) @ ( F5 @ X ) ) ) % 0.31/0.61 & ! [Y2: B] : % 0.31/0.61 ( ! [X: A] : % 0.31/0.61 ( ( member @ A @ X @ A3 ) % 0.31/0.61 => ( Q7 @ X @ Y2 ) ) % 0.31/0.61 => ( P @ Y2 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_INF_finite % 0.31/0.61 thf(fact_5674_eventually__at__leftI,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( topolo2564578578187576103pology @ A ) % 0.31/0.61 => ! [A2: A,B2: A,P: A > $o] : % 0.31/0.61 ( ! [X5: A] : % 0.31/0.61 ( ( member @ A @ X5 @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) % 0.31/0.61 => ( P @ X5 ) ) % 0.31/0.61 => ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.61 => ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_at_leftI % 0.31/0.61 thf(fact_5675_has__derivative__within__singleton__iff,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [F2: A > B,G: A > B,X3: A] : % 0.31/0.61 ( ( has_derivative @ A @ B @ F2 @ G @ ( topolo174197925503356063within @ A @ X3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.61 = ( real_V3181309239436604168linear @ A @ B @ G ) ) ) ). % 0.31/0.61 % 0.31/0.61 % has_derivative_within_singleton_iff % 0.31/0.61 thf(fact_5676_eventually__at__to__0,axiom, % 0.31/0.61 ! [A: $tType] : % 0.31/0.61 ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 => ! [P: A > $o,A2: A] : % 0.31/0.61 ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.61 = ( eventually @ A % 0.31/0.61 @ ^ [X: A] : ( P @ ( plus_plus @ A @ X @ A2 ) ) % 0.31/0.61 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_at_to_0 % 0.31/0.61 thf(fact_5677_decreasing__tendsto,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( topolo2564578578187576103pology @ A ) % 0.31/0.61 => ! [L: A,F2: B > A,F5: filter @ B] : % 0.31/0.61 ( ( eventually @ B % 0.31/0.61 @ ^ [N: B] : ( ord_less_eq @ A @ L @ ( F2 @ N ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( ! [X5: A] : % 0.31/0.61 ( ( ord_less @ A @ L @ X5 ) % 0.31/0.61 => ( eventually @ B % 0.31/0.61 @ ^ [N: B] : ( ord_less @ A @ ( F2 @ N ) @ X5 ) % 0.31/0.61 @ F5 ) ) % 0.31/0.61 => ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % decreasing_tendsto % 0.31/0.61 thf(fact_5678_increasing__tendsto,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( topolo2564578578187576103pology @ A ) % 0.31/0.61 => ! [F2: B > A,L: A,F5: filter @ B] : % 0.31/0.61 ( ( eventually @ B % 0.31/0.61 @ ^ [N: B] : ( ord_less_eq @ A @ ( F2 @ N ) @ L ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( ! [X5: A] : % 0.31/0.61 ( ( ord_less @ A @ X5 @ L ) % 0.31/0.61 => ( eventually @ B % 0.31/0.61 @ ^ [N: B] : ( ord_less @ A @ X5 @ ( F2 @ N ) ) % 0.31/0.61 @ F5 ) ) % 0.31/0.61 => ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % increasing_tendsto % 0.31/0.61 thf(fact_5679_filterlim__at__top__gt,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( unboun7993243217541854897norder @ B ) % 0.31/0.61 => ! [F2: A > B,F5: filter @ A,C2: B] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 ) % 0.31/0.61 = ( ! [Z9: B] : % 0.31/0.61 ( ( ord_less @ B @ C2 @ Z9 ) % 0.31/0.61 => ( eventually @ A % 0.31/0.61 @ ^ [X: A] : ( ord_less_eq @ B @ Z9 @ ( F2 @ X ) ) % 0.31/0.61 @ F5 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_at_top_gt % 0.31/0.61 thf(fact_5680_tendsto__le,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 => ! [F5: filter @ B,F2: B > A,X3: A,G: B > A,Y: A] : % 0.31/0.61 ( ( F5 % 0.31/0.61 != ( bot_bot @ ( filter @ B ) ) ) % 0.31/0.61 => ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ F5 ) % 0.31/0.61 => ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 ) % 0.31/0.61 => ( ( eventually @ B % 0.31/0.61 @ ^ [X: B] : ( ord_less_eq @ A @ ( G @ X ) @ ( F2 @ X ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( ord_less_eq @ A @ Y @ X3 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_le % 0.31/0.61 thf(fact_5681_tendsto__lowerbound,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 => ! [F2: B > A,X3: A,F5: filter @ B,A2: A] : % 0.31/0.61 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ F5 ) % 0.31/0.61 => ( ( eventually @ B % 0.31/0.61 @ ^ [I3: B] : ( ord_less_eq @ A @ A2 @ ( F2 @ I3 ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( ( F5 % 0.31/0.61 != ( bot_bot @ ( filter @ B ) ) ) % 0.31/0.61 => ( ord_less_eq @ A @ A2 @ X3 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_lowerbound % 0.31/0.61 thf(fact_5682_tendsto__upperbound,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 => ! [F2: B > A,X3: A,F5: filter @ B,A2: A] : % 0.31/0.61 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) @ F5 ) % 0.31/0.61 => ( ( eventually @ B % 0.31/0.61 @ ^ [I3: B] : ( ord_less_eq @ A @ ( F2 @ I3 ) @ A2 ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( ( F5 % 0.31/0.61 != ( bot_bot @ ( filter @ B ) ) ) % 0.31/0.61 => ( ord_less_eq @ A @ X3 @ A2 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_upperbound % 0.31/0.61 thf(fact_5683_eventually__INF,axiom, % 0.31/0.61 ! [A: $tType,B: $tType,P: A > $o,F5: B > ( filter @ A ),B3: set @ B] : % 0.31/0.61 ( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ B @ ( filter @ A ) @ F5 @ B3 ) ) ) % 0.31/0.61 = ( ? [X6: set @ B] : % 0.31/0.61 ( ( ord_less_eq @ ( set @ B ) @ X6 @ B3 ) % 0.31/0.61 & ( finite_finite2 @ B @ X6 ) % 0.31/0.61 & ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ B @ ( filter @ A ) @ F5 @ X6 ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_INF % 0.31/0.61 thf(fact_5684_filterlim__pow__at__top,axiom, % 0.31/0.61 ! [A: $tType,N2: nat,F2: A > real,F5: filter @ A] : % 0.31/0.61 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.61 => ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F5 ) % 0.31/0.61 => ( filterlim @ A @ real % 0.31/0.61 @ ^ [X: A] : ( power_power @ real @ ( F2 @ X ) @ N2 ) % 0.31/0.61 @ ( at_top @ real ) % 0.31/0.61 @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_pow_at_top % 0.31/0.61 thf(fact_5685_bounded__linear_Opos__bounded,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [F2: A > B] : % 0.31/0.61 ( ( real_V3181309239436604168linear @ A @ B @ F2 ) % 0.31/0.61 => ? [K8: real] : % 0.31/0.61 ( ( ord_less @ real @ ( zero_zero @ real ) @ K8 ) % 0.31/0.61 & ! [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ K8 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % bounded_linear.pos_bounded % 0.31/0.61 thf(fact_5686_bounded__linear__intro,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [F2: A > B,K5: real] : % 0.31/0.61 ( ! [X5: A,Y4: A] : % 0.31/0.61 ( ( F2 @ ( plus_plus @ A @ X5 @ Y4 ) ) % 0.31/0.61 = ( plus_plus @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.31/0.61 => ( ! [R4: real,X5: A] : % 0.31/0.61 ( ( F2 @ ( real_V8093663219630862766scaleR @ A @ R4 @ X5 ) ) % 0.31/0.61 = ( real_V8093663219630862766scaleR @ B @ R4 @ ( F2 @ X5 ) ) ) % 0.31/0.61 => ( ! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K5 ) ) % 0.31/0.61 => ( real_V3181309239436604168linear @ A @ B @ F2 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % bounded_linear_intro % 0.31/0.61 thf(fact_5687_tendsto__imp__filterlim__at__left,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( topolo2564578578187576103pology @ B ) % 0.31/0.61 => ! [F2: A > B,L5: B,F5: filter @ A] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ F5 ) % 0.31/0.61 => ( ( eventually @ A % 0.31/0.61 @ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ L5 ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L5 @ ( set_ord_lessThan @ B @ L5 ) ) @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_imp_filterlim_at_left % 0.31/0.61 thf(fact_5688_eventually__Inf,axiom, % 0.31/0.61 ! [A: $tType,P: A > $o,B3: set @ ( filter @ A )] : % 0.31/0.61 ( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B3 ) ) % 0.31/0.61 = ( ? [X6: set @ ( filter @ A )] : % 0.31/0.61 ( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X6 @ B3 ) % 0.31/0.61 & ( finite_finite2 @ ( filter @ A ) @ X6 ) % 0.31/0.61 & ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ X6 ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_Inf % 0.31/0.61 thf(fact_5689_filterlim__at__top__mult__tendsto__pos,axiom, % 0.31/0.61 ! [A: $tType,F2: A > real,C2: real,F5: filter @ A,G: A > real] : % 0.31/0.61 ( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F5 ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.61 => ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F5 ) % 0.31/0.61 => ( filterlim @ A @ real % 0.31/0.61 @ ^ [X: A] : ( times_times @ real @ ( G @ X ) @ ( F2 @ X ) ) % 0.31/0.61 @ ( at_top @ real ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_at_top_mult_tendsto_pos % 0.31/0.61 thf(fact_5690_filterlim__tendsto__pos__mult__at__top,axiom, % 0.31/0.61 ! [A: $tType,F2: A > real,C2: real,F5: filter @ A,G: A > real] : % 0.31/0.61 ( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F5 ) % 0.31/0.61 => ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.61 => ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F5 ) % 0.31/0.61 => ( filterlim @ A @ real % 0.31/0.61 @ ^ [X: A] : ( times_times @ real @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.61 @ ( at_top @ real ) % 0.31/0.61 @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_tendsto_pos_mult_at_top % 0.31/0.61 thf(fact_5691_tendsto__neg__powr,axiom, % 0.31/0.61 ! [A: $tType,S: real,F2: A > real,F5: filter @ A] : % 0.31/0.61 ( ( ord_less @ real @ S @ ( zero_zero @ real ) ) % 0.31/0.61 => ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F5 ) % 0.31/0.61 => ( filterlim @ A @ real % 0.31/0.61 @ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ S ) % 0.31/0.61 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 0.31/0.61 @ F5 ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_neg_powr % 0.31/0.61 thf(fact_5692_filterlim__at__top__at__left,axiom, % 0.31/0.61 ! [B: $tType,A: $tType] : % 0.31/0.61 ( ( ( topolo1944317154257567458pology @ A ) % 0.31/0.61 & ( linorder @ B ) ) % 0.31/0.61 => ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A,A2: A] : % 0.31/0.61 ( ! [X5: A,Y4: A] : % 0.31/0.61 ( ( Q @ X5 ) % 0.31/0.61 => ( ( Q @ Y4 ) % 0.31/0.61 => ( ( ord_less_eq @ A @ X5 @ Y4 ) % 0.31/0.61 => ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) ) ) % 0.31/0.61 => ( ! [X5: B] : % 0.31/0.61 ( ( P @ X5 ) % 0.31/0.61 => ( ( F2 @ ( G @ X5 ) ) % 0.31/0.61 = X5 ) ) % 0.31/0.61 => ( ! [X5: B] : % 0.31/0.61 ( ( P @ X5 ) % 0.31/0.61 => ( Q @ ( G @ X5 ) ) ) % 0.31/0.61 => ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) ) % 0.31/0.61 => ( ! [B5: A] : % 0.31/0.61 ( ( Q @ B5 ) % 0.31/0.61 => ( ord_less @ A @ B5 @ A2 ) ) % 0.31/0.61 => ( ( eventually @ B @ P @ ( at_top @ B ) ) % 0.31/0.61 => ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_at_top_at_left % 0.31/0.61 thf(fact_5693_eventually__INF__base,axiom, % 0.31/0.61 ! [B: $tType,A: $tType,B3: set @ A,F5: A > ( filter @ B ),P: B > $o] : % 0.31/0.61 ( ( B3 % 0.31/0.61 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.61 => ( ! [A5: A] : % 0.31/0.61 ( ( member @ A @ A5 @ B3 ) % 0.31/0.61 => ! [B5: A] : % 0.31/0.61 ( ( member @ A @ B5 @ B3 ) % 0.31/0.61 => ? [X2: A] : % 0.31/0.61 ( ( member @ A @ X2 @ B3 ) % 0.31/0.61 & ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X2 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ A5 ) @ ( F5 @ B5 ) ) ) ) ) ) % 0.31/0.61 => ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F5 @ B3 ) ) ) % 0.31/0.61 = ( ? [X: A] : % 0.31/0.61 ( ( member @ A @ X @ B3 ) % 0.31/0.61 & ( eventually @ B @ P @ ( F5 @ X ) ) ) ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % eventually_INF_base % 0.31/0.61 thf(fact_5694_tendsto__0__le,axiom, % 0.31/0.61 ! [B: $tType,C: $tType,A: $tType] : % 0.31/0.61 ( ( ( real_V822414075346904944vector @ C ) % 0.31/0.61 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.61 => ! [F2: A > B,F5: filter @ A,G: A > C,K5: real] : % 0.31/0.61 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) % 0.31/0.61 => ( ( eventually @ A % 0.31/0.61 @ ^ [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G @ X ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) @ K5 ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % tendsto_0_le % 0.31/0.61 thf(fact_5695_filterlim__at__withinI,axiom, % 0.31/0.61 ! [A: $tType,B: $tType] : % 0.31/0.61 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.61 => ! [F2: B > A,C2: A,F5: filter @ B,A3: set @ A] : % 0.31/0.61 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 ) % 0.31/0.61 => ( ( eventually @ B % 0.31/0.61 @ ^ [X: B] : ( member @ A @ ( F2 @ X ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.61 @ F5 ) % 0.31/0.61 => ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ C2 @ A3 ) @ F5 ) ) ) ) ). % 0.31/0.61 % 0.31/0.61 % filterlim_at_withinI % 0.31/0.61 thf(fact_5696_tendsto__powr_H,axiom, % 0.31/0.61 ! [A: $tType,F2: A > real,A2: real,F5: filter @ A,G: A > real,B2: real] : % 0.31/0.61 ( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 ) % 0.31/0.61 => ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 ) % 0.31/0.61 => ( ( ( A2 % 0.31/0.61 != ( zero_zero @ real ) ) % 0.31/0.61 | ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 0.31/0.61 & ( eventually @ A % 0.31/0.61 @ ^ [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X ) ) % 0.31/0.62 @ F5 ) ) ) % 0.31/0.62 => ( filterlim @ A @ real % 0.31/0.62 @ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) ) % 0.31/0.62 @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_powr' % 0.31/0.62 thf(fact_5697_tendsto__powr2,axiom, % 0.31/0.62 ! [A: $tType,F2: A > real,A2: real,F5: filter @ A,G: A > real,B2: real] : % 0.31/0.62 ( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 ) % 0.31/0.62 => ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 ) % 0.31/0.62 => ( ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X ) ) % 0.31/0.62 @ F5 ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 0.31/0.62 => ( filterlim @ A @ real % 0.31/0.62 @ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) ) % 0.31/0.62 @ F5 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_powr2 % 0.31/0.62 thf(fact_5698_tendsto__zero__powrI,axiom, % 0.31/0.62 ! [A: $tType,F2: A > real,F5: filter @ A,G: A > real,B2: real] : % 0.31/0.62 ( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) % 0.31/0.62 => ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 ) % 0.31/0.62 => ( ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X ) ) % 0.31/0.62 @ F5 ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 0.31/0.62 => ( filterlim @ A @ real % 0.31/0.62 @ ^ [X: A] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 0.31/0.62 @ F5 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_zero_powrI % 0.31/0.62 thf(fact_5699_DERIV__neg__imp__decreasing__at__top,axiom, % 0.31/0.62 ! [B2: real,F2: real > real,Flim: real] : % 0.31/0.62 ( ! [X5: real] : % 0.31/0.62 ( ( ord_less_eq @ real @ B2 @ X5 ) % 0.31/0.62 => ? [Y5: real] : % 0.31/0.62 ( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.62 & ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) ) % 0.31/0.62 => ( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_top @ real ) ) % 0.31/0.62 => ( ord_less @ real @ Flim @ ( F2 @ B2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % DERIV_neg_imp_decreasing_at_top % 0.31/0.62 thf(fact_5700_has__derivative__at__within,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.62 => ! [F2: A > B,F11: A > B,X3: A,S: set @ A] : % 0.31/0.62 ( ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.62 = ( ( real_V3181309239436604168linear @ A @ B @ F11 ) % 0.31/0.62 & ( filterlim @ A @ B % 0.31/0.62 @ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X3 ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y2 ) @ ( F2 @ X3 ) ) @ ( F11 @ ( minus_minus @ A @ Y2 @ X3 ) ) ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 0.31/0.62 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % has_derivative_at_within % 0.31/0.62 thf(fact_5701_has__derivativeI,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.62 => ! [F11: A > B,X3: A,F2: A > B,S: set @ A] : % 0.31/0.62 ( ( real_V3181309239436604168linear @ A @ B @ F11 ) % 0.31/0.62 => ( ( filterlim @ A @ B % 0.31/0.62 @ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X3 ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y2 ) @ ( F2 @ X3 ) ) @ ( F11 @ ( minus_minus @ A @ Y2 @ X3 ) ) ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 0.31/0.62 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.62 => ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % has_derivativeI % 0.31/0.62 thf(fact_5702_has__derivative__iff__Ex,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.62 => ! [F2: A > B,F11: A > B,X3: A] : % 0.31/0.62 ( ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.62 = ( ( real_V3181309239436604168linear @ A @ B @ F11 ) % 0.31/0.62 & ? [E4: A > B] : % 0.31/0.62 ( ! [H: A] : % 0.31/0.62 ( ( F2 @ ( plus_plus @ A @ X3 @ H ) ) % 0.31/0.62 = ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X3 ) @ ( F11 @ H ) ) @ ( E4 @ H ) ) ) % 0.31/0.62 & ( filterlim @ A @ real % 0.31/0.62 @ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E4 @ H ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 0.31/0.62 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % has_derivative_iff_Ex % 0.31/0.62 thf(fact_5703_has__derivative__within,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.62 => ! [F2: A > B,F11: A > B,X3: A,S: set @ A] : % 0.31/0.62 ( ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.62 = ( ( real_V3181309239436604168linear @ A @ B @ F11 ) % 0.31/0.62 & ( filterlim @ A @ B % 0.31/0.62 @ ^ [Y2: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ X3 ) ) ) @ ( minus_minus @ B @ ( F2 @ Y2 ) @ ( plus_plus @ B @ ( F2 @ X3 ) @ ( F11 @ ( minus_minus @ A @ Y2 @ X3 ) ) ) ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 0.31/0.62 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % has_derivative_within % 0.31/0.62 thf(fact_5704_polyfun__extremal,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.62 => ! [C2: nat > A,K: nat,N2: nat,B3: real] : % 0.31/0.62 ( ( ( C2 @ K ) % 0.31/0.62 != ( zero_zero @ A ) ) % 0.31/0.62 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K ) % 0.31/0.62 => ( ( ord_less_eq @ nat @ K @ N2 ) % 0.31/0.62 => ( eventually @ A % 0.31/0.62 @ ^ [Z6: A] : % 0.31/0.62 ( ord_less_eq @ real @ B3 % 0.31/0.62 @ ( real_V7770717601297561774m_norm @ A % 0.31/0.62 @ ( groups7311177749621191930dd_sum @ nat @ A % 0.31/0.62 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) ) % 0.31/0.62 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) % 0.31/0.62 @ ( at_infinity @ A ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % polyfun_extremal % 0.31/0.62 thf(fact_5705_summable__bounded__partials,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( real_V8037385150606011577_space @ A ) % 0.31/0.62 & ( real_V822414075346904944vector @ A ) ) % 0.31/0.62 => ! [F2: nat > A,G: nat > real] : % 0.31/0.62 ( ( eventually @ nat % 0.31/0.62 @ ^ [X02: nat] : % 0.31/0.62 ! [A4: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ X02 @ A4 ) % 0.31/0.62 => ! [B4: nat] : % 0.31/0.62 ( ( ord_less @ nat @ A4 @ B4 ) % 0.31/0.62 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or3652927894154168847AtMost @ nat @ A4 @ B4 ) ) ) @ ( G @ A4 ) ) ) ) % 0.31/0.62 @ ( at_top @ nat ) ) % 0.31/0.62 => ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) % 0.31/0.62 => ( summable @ A @ F2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % summable_bounded_partials % 0.31/0.62 thf(fact_5706_has__derivative__def,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.62 => ( ( has_derivative @ A @ B ) % 0.31/0.62 = ( ^ [F3: A > B,F13: A > B,F9: filter @ A] : % 0.31/0.62 ( ( real_V3181309239436604168linear @ A @ B @ F13 ) % 0.31/0.62 & ( filterlim @ A @ B % 0.31/0.62 @ ^ [Y2: A] : % 0.31/0.62 ( real_V8093663219630862766scaleR @ B % 0.31/0.62 @ ( inverse_inverse @ real % 0.31/0.62 @ ( real_V7770717601297561774m_norm @ A % 0.31/0.62 @ ( minus_minus @ A @ Y2 % 0.31/0.62 @ ( topolo3827282254853284352ce_Lim @ A @ A @ F9 % 0.31/0.62 @ ^ [X: A] : X ) ) ) ) % 0.31/0.62 @ ( minus_minus @ B % 0.31/0.62 @ ( minus_minus @ B @ ( F3 @ Y2 ) % 0.31/0.62 @ ( F3 % 0.31/0.62 @ ( topolo3827282254853284352ce_Lim @ A @ A @ F9 % 0.31/0.62 @ ^ [X: A] : X ) ) ) % 0.31/0.62 @ ( F13 % 0.31/0.62 @ ( minus_minus @ A @ Y2 % 0.31/0.62 @ ( topolo3827282254853284352ce_Lim @ A @ A @ F9 % 0.31/0.62 @ ^ [X: A] : X ) ) ) ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 0.31/0.62 @ F9 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % has_derivative_def % 0.31/0.62 thf(fact_5707_finite__greaterThanAtMost,axiom, % 0.31/0.62 ! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ). % 0.31/0.62 % 0.31/0.62 % finite_greaterThanAtMost % 0.31/0.62 thf(fact_5708_greaterThanAtMost__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ord @ A ) % 0.31/0.62 => ! [I: A,L: A,U: A] : % 0.31/0.62 ( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U ) ) % 0.31/0.62 = ( ( ord_less @ A @ L @ I ) % 0.31/0.62 & ( ord_less_eq @ A @ I @ U ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % greaterThanAtMost_iff % 0.31/0.62 thf(fact_5709_greaterThanAtMost__empty,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [L: A,K: A] : % 0.31/0.62 ( ( ord_less_eq @ A @ L @ K ) % 0.31/0.62 => ( ( set_or3652927894154168847AtMost @ A @ K @ L ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % greaterThanAtMost_empty % 0.31/0.62 thf(fact_5710_greaterThanAtMost__empty__iff2,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [K: A,L: A] : % 0.31/0.62 ( ( ( bot_bot @ ( set @ A ) ) % 0.31/0.62 = ( set_or3652927894154168847AtMost @ A @ K @ L ) ) % 0.31/0.62 = ( ~ ( ord_less @ A @ K @ L ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % greaterThanAtMost_empty_iff2 % 0.31/0.62 thf(fact_5711_greaterThanAtMost__empty__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [K: A,L: A] : % 0.31/0.62 ( ( ( set_or3652927894154168847AtMost @ A @ K @ L ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 = ( ~ ( ord_less @ A @ K @ L ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % greaterThanAtMost_empty_iff % 0.31/0.62 thf(fact_5712_infinite__Ioc__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( dense_linorder @ A ) % 0.31/0.62 => ! [A2: A,B2: A] : % 0.31/0.62 ( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) ) % 0.31/0.62 = ( ord_less @ A @ A2 @ B2 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % infinite_Ioc_iff % 0.31/0.62 thf(fact_5713_image__add__greaterThanAtMost,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linordered_semidom @ A ) % 0.31/0.62 => ! [C2: A,A2: A,B2: A] : % 0.31/0.62 ( ( image @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) % 0.31/0.62 = ( set_or3652927894154168847AtMost @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % image_add_greaterThanAtMost % 0.31/0.62 thf(fact_5714_cSup__greaterThanAtMost,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [Y: A,X3: A] : % 0.31/0.62 ( ( ord_less @ A @ Y @ X3 ) % 0.31/0.62 => ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X3 ) ) % 0.31/0.62 = X3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cSup_greaterThanAtMost % 0.31/0.62 thf(fact_5715_Sup__greaterThanAtMost,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( comple6319245703460814977attice @ A ) % 0.31/0.62 => ! [X3: A,Y: A] : % 0.31/0.62 ( ( ord_less @ A @ X3 @ Y ) % 0.31/0.62 => ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X3 @ Y ) ) % 0.31/0.62 = Y ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Sup_greaterThanAtMost % 0.31/0.62 thf(fact_5716_cInf__greaterThanAtMost,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( condit6923001295902523014norder @ A ) % 0.31/0.62 & ( dense_linorder @ A ) ) % 0.31/0.62 => ! [Y: A,X3: A] : % 0.31/0.62 ( ( ord_less @ A @ Y @ X3 ) % 0.31/0.62 => ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X3 ) ) % 0.31/0.62 = Y ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cInf_greaterThanAtMost % 0.31/0.62 thf(fact_5717_Inf__greaterThanAtMost,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( comple6319245703460814977attice @ A ) % 0.31/0.62 & ( dense_linorder @ A ) ) % 0.31/0.62 => ! [X3: A,Y: A] : % 0.31/0.62 ( ( ord_less @ A @ X3 @ Y ) % 0.31/0.62 => ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X3 @ Y ) ) % 0.31/0.62 = X3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Inf_greaterThanAtMost % 0.31/0.62 thf(fact_5718_Ioc__inj,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.62 ( ( ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) % 0.31/0.62 = ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) ) % 0.31/0.62 = ( ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.62 & ( ord_less_eq @ A @ D2 @ C2 ) ) % 0.31/0.62 | ( ( A2 = C2 ) % 0.31/0.62 & ( B2 = D2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Ioc_inj % 0.31/0.62 thf(fact_5719_atLeastSucAtMost__greaterThanAtMost,axiom, % 0.31/0.62 ! [L: nat,U: nat] : % 0.31/0.62 ( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U ) % 0.31/0.62 = ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ). % 0.31/0.62 % 0.31/0.62 % atLeastSucAtMost_greaterThanAtMost % 0.31/0.62 thf(fact_5720_Ioc__subset__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.62 ( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) ) % 0.31/0.62 = ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.62 | ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.62 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Ioc_subset_iff % 0.31/0.62 thf(fact_5721_infinite__Ioc,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( dense_linorder @ A ) % 0.31/0.62 => ! [A2: A,B2: A] : % 0.31/0.62 ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.62 => ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % infinite_Ioc % 0.31/0.62 thf(fact_5722_ivl__disj__int__two_I6_J,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [L: A,M: A,U: A] : % 0.31/0.62 ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % ivl_disj_int_two(6) % 0.31/0.62 thf(fact_5723_Ioc__disjoint,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.62 ( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 = ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.62 | ( ord_less_eq @ A @ D2 @ C2 ) % 0.31/0.62 | ( ord_less_eq @ A @ B2 @ C2 ) % 0.31/0.62 | ( ord_less_eq @ A @ D2 @ A2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Ioc_disjoint % 0.31/0.62 thf(fact_5724_ivl__disj__int__two_I8_J,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [L: A,M: A,U: A] : % 0.31/0.62 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % ivl_disj_int_two(8) % 0.31/0.62 thf(fact_5725_ivl__disj__int__one_I3_J,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [L: A,U: A] : % 0.31/0.62 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % ivl_disj_int_one(3) % 0.31/0.62 thf(fact_5726_ivl__disj__int__two_I2_J,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [L: A,M: A,U: A] : % 0.31/0.62 ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % ivl_disj_int_two(2) % 0.31/0.62 thf(fact_5727_Lim__ident__at,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topological_t2_space @ A ) % 0.31/0.62 => ! [X3: A,S: set @ A] : % 0.31/0.62 ( ( ( topolo174197925503356063within @ A @ X3 @ S ) % 0.31/0.62 != ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.62 => ( ( topolo3827282254853284352ce_Lim @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ S ) % 0.31/0.62 @ ^ [X: A] : X ) % 0.31/0.62 = X3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Lim_ident_at % 0.31/0.62 thf(fact_5728_eventually__at__left__real,axiom, % 0.31/0.62 ! [B2: real,A2: real] : % 0.31/0.62 ( ( ord_less @ real @ B2 @ A2 ) % 0.31/0.62 => ( eventually @ real % 0.31/0.62 @ ^ [X: real] : ( member @ real @ X @ ( set_or5935395276787703475ssThan @ real @ B2 @ A2 ) ) % 0.31/0.62 @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_at_left_real % 0.31/0.62 thf(fact_5729_sum_Ohead,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( comm_monoid_add @ A ) % 0.31/0.62 => ! [M: nat,N2: nat,G: nat > A] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.62 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.31/0.62 = ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % sum.head % 0.31/0.62 thf(fact_5730_not__tendsto__and__filterlim__at__infinity,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [F5: filter @ B,F2: B > A,C2: A] : % 0.31/0.62 ( ( F5 % 0.31/0.62 != ( bot_bot @ ( filter @ B ) ) ) % 0.31/0.62 => ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 ) % 0.31/0.62 => ~ ( filterlim @ B @ A @ F2 @ ( at_infinity @ A ) @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % not_tendsto_and_filterlim_at_infinity % 0.31/0.62 thf(fact_5731_prod_Ohead,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( comm_monoid_mult @ A ) % 0.31/0.62 => ! [M: nat,N2: nat,G: nat > A] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.62 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 0.31/0.62 = ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % prod.head % 0.31/0.62 thf(fact_5732_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( dense_linorder @ A ) % 0.31/0.62 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.62 ( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) ) % 0.31/0.62 = ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.62 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % greaterThanAtMost_subseteq_atLeastAtMost_iff % 0.31/0.62 thf(fact_5733_tendsto__Lim,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( topological_t2_space @ B ) % 0.31/0.62 => ! [Net: filter @ A,F2: A > B,L: B] : % 0.31/0.62 ( ( Net % 0.31/0.62 != ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.62 => ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ Net ) % 0.31/0.62 => ( ( topolo3827282254853284352ce_Lim @ A @ B @ Net @ F2 ) % 0.31/0.62 = L ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_Lim % 0.31/0.62 thf(fact_5734_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( dense_linorder @ A ) % 0.31/0.62 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.62 ( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) ) % 0.31/0.62 = ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.62 & ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % greaterThanAtMost_subseteq_atLeastLessThan_iff % 0.31/0.62 thf(fact_5735_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( dense_linorder @ A ) % 0.31/0.62 => ! [A2: A,B2: A,C2: A,D2: A] : % 0.31/0.62 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) ) % 0.31/0.62 = ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.62 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % greaterThanLessThan_subseteq_greaterThanAtMost_iff % 0.31/0.62 thf(fact_5736_greaterThanAtMost__eq__atLeastAtMost__diff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ( ( set_or3652927894154168847AtMost @ A ) % 0.31/0.62 = ( ^ [A4: A,B4: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A4 @ B4 ) @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % greaterThanAtMost_eq_atLeastAtMost_diff % 0.31/0.62 thf(fact_5737_tendsto__add__filterlim__at__infinity_H,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ B ) % 0.31/0.62 => ! [F2: A > B,F5: filter @ A,G: A > B,C2: B] : % 0.31/0.62 ( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F5 ) % 0.31/0.62 => ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F5 ) % 0.31/0.62 => ( filterlim @ A @ B % 0.31/0.62 @ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.62 @ ( at_infinity @ B ) % 0.31/0.62 @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_add_filterlim_at_infinity' % 0.31/0.62 thf(fact_5738_tendsto__add__filterlim__at__infinity,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ B ) % 0.31/0.62 => ! [F2: A > B,C2: B,F5: filter @ A,G: A > B] : % 0.31/0.62 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F5 ) % 0.31/0.62 => ( ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F5 ) % 0.31/0.62 => ( filterlim @ A @ B % 0.31/0.62 @ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.62 @ ( at_infinity @ B ) % 0.31/0.62 @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_add_filterlim_at_infinity % 0.31/0.62 thf(fact_5739_continuous__divide,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topological_t2_space @ A ) % 0.31/0.62 & ( real_V3459762299906320749_field @ B ) ) % 0.31/0.62 => ! [F5: filter @ A,F2: A > B,G: A > B] : % 0.31/0.62 ( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 ) % 0.31/0.62 => ( ( topolo3448309680560233919inuous @ A @ B @ F5 @ G ) % 0.31/0.62 => ( ( ( G % 0.31/0.62 @ ( topolo3827282254853284352ce_Lim @ A @ A @ F5 % 0.31/0.62 @ ^ [X: A] : X ) ) % 0.31/0.62 != ( zero_zero @ B ) ) % 0.31/0.62 => ( topolo3448309680560233919inuous @ A @ B @ F5 % 0.31/0.62 @ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_divide % 0.31/0.62 thf(fact_5740_continuous__inverse,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topological_t2_space @ A ) % 0.31/0.62 & ( real_V8999393235501362500lgebra @ B ) ) % 0.31/0.62 => ! [F5: filter @ A,F2: A > B] : % 0.31/0.62 ( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 ) % 0.31/0.62 => ( ( ( F2 % 0.31/0.62 @ ( topolo3827282254853284352ce_Lim @ A @ A @ F5 % 0.31/0.62 @ ^ [X: A] : X ) ) % 0.31/0.62 != ( zero_zero @ B ) ) % 0.31/0.62 => ( topolo3448309680560233919inuous @ A @ B @ F5 % 0.31/0.62 @ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_inverse % 0.31/0.62 thf(fact_5741_continuous__sgn,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topological_t2_space @ A ) % 0.31/0.62 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.62 => ! [F5: filter @ A,F2: A > B] : % 0.31/0.62 ( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 ) % 0.31/0.62 => ( ( ( F2 % 0.31/0.62 @ ( topolo3827282254853284352ce_Lim @ A @ A @ F5 % 0.31/0.62 @ ^ [X: A] : X ) ) % 0.31/0.62 != ( zero_zero @ B ) ) % 0.31/0.62 => ( topolo3448309680560233919inuous @ A @ B @ F5 % 0.31/0.62 @ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_sgn % 0.31/0.62 thf(fact_5742_tendsto__inverse__0,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.62 => ( filterlim @ A @ A @ ( inverse_inverse @ A ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_infinity @ A ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_inverse_0 % 0.31/0.62 thf(fact_5743_filterlim__at__infinity__imp__filterlim__at__top,axiom, % 0.31/0.62 ! [A: $tType,F2: A > real,F5: filter @ A] : % 0.31/0.62 ( ( filterlim @ A @ real @ F2 @ ( at_infinity @ real ) @ F5 ) % 0.31/0.62 => ( ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) ) % 0.31/0.62 @ F5 ) % 0.31/0.62 => ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F5 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_at_infinity_imp_filterlim_at_top % 0.31/0.62 thf(fact_5744_tendsto__mult__filterlim__at__infinity,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.62 => ! [F2: B > A,C2: A,F5: filter @ B,G: B > A] : % 0.31/0.62 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 ) % 0.31/0.62 => ( ( C2 % 0.31/0.62 != ( zero_zero @ A ) ) % 0.31/0.62 => ( ( filterlim @ B @ A @ G @ ( at_infinity @ A ) @ F5 ) % 0.31/0.62 => ( filterlim @ B @ A % 0.31/0.62 @ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.62 @ ( at_infinity @ A ) % 0.31/0.62 @ F5 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_mult_filterlim_at_infinity % 0.31/0.62 thf(fact_5745_tendsto__divide__0,axiom, % 0.31/0.62 ! [A: $tType,C: $tType] : % 0.31/0.62 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.62 => ! [F2: C > A,C2: A,F5: filter @ C,G: C > A] : % 0.31/0.62 ( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 ) % 0.31/0.62 => ( ( filterlim @ C @ A @ G @ ( at_infinity @ A ) @ F5 ) % 0.31/0.62 => ( filterlim @ C @ A % 0.31/0.62 @ ^ [X: C] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 0.31/0.62 @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_divide_0 % 0.31/0.62 thf(fact_5746_filterlim__power__at__infinity,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( real_V8999393235501362500lgebra @ B ) % 0.31/0.62 => ! [F2: A > B,F5: filter @ A,N2: nat] : % 0.31/0.62 ( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F5 ) % 0.31/0.62 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.62 => ( filterlim @ A @ B % 0.31/0.62 @ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N2 ) % 0.31/0.62 @ ( at_infinity @ B ) % 0.31/0.62 @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_power_at_infinity % 0.31/0.62 thf(fact_5747_continuous__tan,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( real_Vector_banach @ A ) % 0.31/0.62 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.62 => ! [F5: filter @ A,F2: A > A] : % 0.31/0.62 ( ( topolo3448309680560233919inuous @ A @ A @ F5 @ F2 ) % 0.31/0.62 => ( ( ( cos @ A % 0.31/0.62 @ ( F2 % 0.31/0.62 @ ( topolo3827282254853284352ce_Lim @ A @ A @ F5 % 0.31/0.62 @ ^ [X: A] : X ) ) ) % 0.31/0.62 != ( zero_zero @ A ) ) % 0.31/0.62 => ( topolo3448309680560233919inuous @ A @ A @ F5 % 0.31/0.62 @ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_tan % 0.31/0.62 thf(fact_5748_continuous__cot,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( real_Vector_banach @ A ) % 0.31/0.62 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.62 => ! [F5: filter @ A,F2: A > A] : % 0.31/0.62 ( ( topolo3448309680560233919inuous @ A @ A @ F5 @ F2 ) % 0.31/0.62 => ( ( ( sin @ A % 0.31/0.62 @ ( F2 % 0.31/0.62 @ ( topolo3827282254853284352ce_Lim @ A @ A @ F5 % 0.31/0.62 @ ^ [X: A] : X ) ) ) % 0.31/0.62 != ( zero_zero @ A ) ) % 0.31/0.62 => ( topolo3448309680560233919inuous @ A @ A @ F5 % 0.31/0.62 @ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_cot % 0.31/0.62 thf(fact_5749_continuous__tanh,axiom, % 0.31/0.62 ! [A: $tType,C: $tType] : % 0.31/0.62 ( ( ( topological_t2_space @ C ) % 0.31/0.62 & ( real_Vector_banach @ A ) % 0.31/0.62 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.62 => ! [F5: filter @ C,F2: C > A] : % 0.31/0.62 ( ( topolo3448309680560233919inuous @ C @ A @ F5 @ F2 ) % 0.31/0.62 => ( ( ( cosh @ A % 0.31/0.62 @ ( F2 % 0.31/0.62 @ ( topolo3827282254853284352ce_Lim @ C @ C @ F5 % 0.31/0.62 @ ^ [X: C] : X ) ) ) % 0.31/0.62 != ( zero_zero @ A ) ) % 0.31/0.62 => ( topolo3448309680560233919inuous @ C @ A @ F5 % 0.31/0.62 @ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_tanh % 0.31/0.62 thf(fact_5750_continuous__arcosh,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topological_t2_space @ A ) % 0.31/0.62 => ! [F5: filter @ A,F2: A > real] : % 0.31/0.62 ( ( topolo3448309680560233919inuous @ A @ real @ F5 @ F2 ) % 0.31/0.62 => ( ( ord_less @ real @ ( one_one @ real ) % 0.31/0.62 @ ( F2 % 0.31/0.62 @ ( topolo3827282254853284352ce_Lim @ A @ A @ F5 % 0.31/0.62 @ ^ [X: A] : X ) ) ) % 0.31/0.62 => ( topolo3448309680560233919inuous @ A @ real @ F5 % 0.31/0.62 @ ^ [X: A] : ( arcosh @ real @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_arcosh % 0.31/0.62 thf(fact_5751_filterlim__inverse__at__infinity,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.62 => ( filterlim @ A @ A @ ( inverse_inverse @ A ) @ ( at_infinity @ A ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_inverse_at_infinity % 0.31/0.62 thf(fact_5752_eventually__at__infinity__pos,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [P4: A > $o] : % 0.31/0.62 ( ( eventually @ A @ P4 @ ( at_infinity @ A ) ) % 0.31/0.62 = ( ? [B4: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ B4 ) % 0.31/0.62 & ! [X: A] : % 0.31/0.62 ( ( ord_less_eq @ real @ B4 @ ( real_V7770717601297561774m_norm @ A @ X ) ) % 0.31/0.62 => ( P4 @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_at_infinity_pos % 0.31/0.62 thf(fact_5753_filterlim__inverse__at__iff,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( real_V8999393235501362500lgebra @ B ) % 0.31/0.62 => ! [G: A > B,F5: filter @ A] : % 0.31/0.62 ( ( filterlim @ A @ B % 0.31/0.62 @ ^ [X: A] : ( inverse_inverse @ B @ ( G @ X ) ) % 0.31/0.62 @ ( topolo174197925503356063within @ B @ ( zero_zero @ B ) @ ( top_top @ ( set @ B ) ) ) % 0.31/0.62 @ F5 ) % 0.31/0.62 = ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F5 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_inverse_at_iff % 0.31/0.62 thf(fact_5754_nth__sorted__list__of__set__greaterThanAtMost,axiom, % 0.31/0.62 ! [N2: nat,J: nat,I: nat] : % 0.31/0.62 ( ( ord_less @ nat @ N2 @ ( minus_minus @ nat @ J @ I ) ) % 0.31/0.62 => ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) ) @ N2 ) % 0.31/0.62 = ( suc @ ( plus_plus @ nat @ I @ N2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % nth_sorted_list_of_set_greaterThanAtMost % 0.31/0.62 thf(fact_5755_continuous__log,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topological_t2_space @ A ) % 0.31/0.62 => ! [F5: filter @ A,F2: A > real,G: A > real] : % 0.31/0.62 ( ( topolo3448309680560233919inuous @ A @ real @ F5 @ F2 ) % 0.31/0.62 => ( ( topolo3448309680560233919inuous @ A @ real @ F5 @ G ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) % 0.31/0.62 @ ( F2 % 0.31/0.62 @ ( topolo3827282254853284352ce_Lim @ A @ A @ F5 % 0.31/0.62 @ ^ [X: A] : X ) ) ) % 0.31/0.62 => ( ( ( F2 % 0.31/0.62 @ ( topolo3827282254853284352ce_Lim @ A @ A @ F5 % 0.31/0.62 @ ^ [X: A] : X ) ) % 0.31/0.62 != ( one_one @ real ) ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) % 0.31/0.62 @ ( G % 0.31/0.62 @ ( topolo3827282254853284352ce_Lim @ A @ A @ F5 % 0.31/0.62 @ ^ [X: A] : X ) ) ) % 0.31/0.62 => ( topolo3448309680560233919inuous @ A @ real @ F5 % 0.31/0.62 @ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_log % 0.31/0.62 thf(fact_5756_filterlim__divide__at__infinity,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.62 => ! [F2: A > A,C2: A,F5: filter @ A,G: A > A] : % 0.31/0.62 ( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 ) % 0.31/0.62 => ( ( filterlim @ A @ A @ G @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) @ F5 ) % 0.31/0.62 => ( ( C2 % 0.31/0.62 != ( zero_zero @ A ) ) % 0.31/0.62 => ( filterlim @ A @ A % 0.31/0.62 @ ^ [X: A] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.62 @ ( at_infinity @ A ) % 0.31/0.62 @ F5 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_divide_at_infinity % 0.31/0.62 thf(fact_5757_filterlim__at__infinity,axiom, % 0.31/0.62 ! [C: $tType,A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [C2: real,F2: C > A,F5: filter @ C] : % 0.31/0.62 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.62 => ( ( filterlim @ C @ A @ F2 @ ( at_infinity @ A ) @ F5 ) % 0.31/0.62 = ( ! [R5: real] : % 0.31/0.62 ( ( ord_less @ real @ C2 @ R5 ) % 0.31/0.62 => ( eventually @ C % 0.31/0.62 @ ^ [X: C] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X ) ) ) % 0.31/0.62 @ F5 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_at_infinity % 0.31/0.62 thf(fact_5758_filterlim__realpow__sequentially__gt1,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.62 => ! [X3: A] : % 0.31/0.62 ( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) % 0.31/0.62 => ( filterlim @ nat @ A @ ( power_power @ A @ X3 ) @ ( at_infinity @ A ) @ ( at_top @ nat ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_realpow_sequentially_gt1 % 0.31/0.62 thf(fact_5759_lim__at__infinity__0,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.62 => ! [F2: A > A,L: A] : % 0.31/0.62 ( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) ) % 0.31/0.62 = ( filterlim @ A @ A @ ( comp @ A @ A @ A @ F2 @ ( inverse_inverse @ A ) ) @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % lim_at_infinity_0 % 0.31/0.62 thf(fact_5760_lim__zero__infinity,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.62 => ! [F2: A > A,L: A] : % 0.31/0.62 ( ( filterlim @ A @ A % 0.31/0.62 @ ^ [X: A] : ( F2 @ ( divide_divide @ A @ ( one_one @ A ) @ X ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ A @ L ) % 0.31/0.62 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.62 => ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % lim_zero_infinity % 0.31/0.62 thf(fact_5761_has__derivative__at__within__iff__Ex,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.62 => ! [X3: A,S3: set @ A,F2: A > B,F11: A > B] : % 0.31/0.62 ( ( member @ A @ X3 @ S3 ) % 0.31/0.62 => ( ( topolo1002775350975398744n_open @ A @ S3 ) % 0.31/0.62 => ( ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) % 0.31/0.62 = ( ( real_V3181309239436604168linear @ A @ B @ F11 ) % 0.31/0.62 & ? [E4: A > B] : % 0.31/0.62 ( ! [H: A] : % 0.31/0.62 ( ( member @ A @ ( plus_plus @ A @ X3 @ H ) @ S3 ) % 0.31/0.62 => ( ( F2 @ ( plus_plus @ A @ X3 @ H ) ) % 0.31/0.62 = ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X3 ) @ ( F11 @ H ) ) @ ( E4 @ H ) ) ) ) % 0.31/0.62 & ( filterlim @ A @ real % 0.31/0.62 @ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E4 @ H ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 0.31/0.62 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % has_derivative_at_within_iff_Ex % 0.31/0.62 thf(fact_5762_has__derivativeI__sandwich,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.62 => ! [E3: real,F11: A > B,S: set @ A,X3: A,F2: A > B,H7: A > real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 ) % 0.31/0.62 => ( ( real_V3181309239436604168linear @ A @ B @ F11 ) % 0.31/0.62 => ( ! [Y4: A] : % 0.31/0.62 ( ( member @ A @ Y4 @ S ) % 0.31/0.62 => ( ( Y4 != X3 ) % 0.31/0.62 => ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y4 @ X3 ) @ E3 ) % 0.31/0.62 => ( ord_less_eq @ real @ ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) @ ( F11 @ ( minus_minus @ A @ Y4 @ X3 ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y4 @ X3 ) ) ) @ ( H7 @ Y4 ) ) ) ) ) % 0.31/0.62 => ( ( filterlim @ A @ real @ H7 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.62 => ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % has_derivativeI_sandwich % 0.31/0.62 thf(fact_5763_filterlim__pow__at__bot__even,axiom, % 0.31/0.62 ! [N2: nat,F2: real > real,F5: filter @ real] : % 0.31/0.62 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.62 => ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F5 ) % 0.31/0.62 => ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.31/0.62 => ( filterlim @ real @ real % 0.31/0.62 @ ^ [X: real] : ( power_power @ real @ ( F2 @ X ) @ N2 ) % 0.31/0.62 @ ( at_top @ real ) % 0.31/0.62 @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_pow_at_bot_even % 0.31/0.62 thf(fact_5764_finite__greaterThanAtMost__int,axiom, % 0.31/0.62 ! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ). % 0.31/0.62 % 0.31/0.62 % finite_greaterThanAtMost_int % 0.31/0.62 thf(fact_5765_open__empty,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ( topolo1002775350975398744n_open @ A @ ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % open_empty % 0.31/0.62 thf(fact_5766_dist__add__cancel2,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [B2: A,A2: A,C2: A] : % 0.31/0.62 ( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ B2 @ A2 ) @ ( plus_plus @ A @ C2 @ A2 ) ) % 0.31/0.62 = ( real_V557655796197034286t_dist @ A @ B2 @ C2 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % dist_add_cancel2 % 0.31/0.62 thf(fact_5767_dist__add__cancel,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [A2: A,B2: A,C2: A] : % 0.31/0.62 ( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ A2 @ C2 ) ) % 0.31/0.62 = ( real_V557655796197034286t_dist @ A @ B2 @ C2 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % dist_add_cancel % 0.31/0.62 thf(fact_5768_dist__0__norm,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [X3: A] : % 0.31/0.62 ( ( real_V557655796197034286t_dist @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.62 = ( real_V7770717601297561774m_norm @ A @ X3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % dist_0_norm % 0.31/0.62 thf(fact_5769_zero__less__dist__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X3: A,Y: A] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X3 @ Y ) ) % 0.31/0.62 = ( X3 != Y ) ) ) ). % 0.31/0.62 % 0.31/0.62 % zero_less_dist_iff % 0.31/0.62 thf(fact_5770_open__Inter,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [S3: set @ ( set @ A )] : % 0.31/0.62 ( ( finite_finite2 @ ( set @ A ) @ S3 ) % 0.31/0.62 => ( ! [X5: set @ A] : % 0.31/0.62 ( ( member @ ( set @ A ) @ X5 @ S3 ) % 0.31/0.62 => ( topolo1002775350975398744n_open @ A @ X5 ) ) % 0.31/0.62 => ( topolo1002775350975398744n_open @ A @ ( complete_Inf_Inf @ ( set @ A ) @ S3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % open_Inter % 0.31/0.62 thf(fact_5771_open__INT,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [A3: set @ B,B3: B > ( set @ A )] : % 0.31/0.62 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.62 => ( ! [X5: B] : % 0.31/0.62 ( ( member @ B @ X5 @ A3 ) % 0.31/0.62 => ( topolo1002775350975398744n_open @ A @ ( B3 @ X5 ) ) ) % 0.31/0.62 => ( topolo1002775350975398744n_open @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % open_INT % 0.31/0.62 thf(fact_5772_not__open__singleton,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo8386298272705272623_space @ A ) % 0.31/0.62 => ! [X3: A] : % 0.31/0.62 ~ ( topolo1002775350975398744n_open @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % not_open_singleton % 0.31/0.62 thf(fact_5773_separation__t2,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topological_t2_space @ A ) % 0.31/0.62 => ! [X3: A,Y: A] : % 0.31/0.62 ( ( X3 != Y ) % 0.31/0.62 = ( ? [U5: set @ A,V6: set @ A] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ A @ U5 ) % 0.31/0.62 & ( topolo1002775350975398744n_open @ A @ V6 ) % 0.31/0.62 & ( member @ A @ X3 @ U5 ) % 0.31/0.62 & ( member @ A @ Y @ V6 ) % 0.31/0.62 & ( ( inf_inf @ ( set @ A ) @ U5 @ V6 ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % separation_t2 % 0.31/0.62 thf(fact_5774_hausdorff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topological_t2_space @ A ) % 0.31/0.62 => ! [X3: A,Y: A] : % 0.31/0.62 ( ( X3 != Y ) % 0.31/0.62 => ? [U6: set @ A,V7: set @ A] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ A @ U6 ) % 0.31/0.62 & ( topolo1002775350975398744n_open @ A @ V7 ) % 0.31/0.62 & ( member @ A @ X3 @ U6 ) % 0.31/0.62 & ( member @ A @ Y @ V7 ) % 0.31/0.62 & ( ( inf_inf @ ( set @ A ) @ U6 @ V7 ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % hausdorff % 0.31/0.62 thf(fact_5775_trivial__limit__at__bot__linorder,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ( ( at_bot @ A ) % 0.31/0.62 != ( bot_bot @ ( filter @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % trivial_limit_at_bot_linorder % 0.31/0.62 thf(fact_5776_first__countable__basis,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo3112930676232923870pology @ A ) % 0.31/0.62 => ! [X3: A] : % 0.31/0.62 ? [A8: nat > ( set @ A )] : % 0.31/0.62 ( ! [I5: nat] : % 0.31/0.62 ( ( member @ A @ X3 @ ( A8 @ I5 ) ) % 0.31/0.62 & ( topolo1002775350975398744n_open @ A @ ( A8 @ I5 ) ) ) % 0.31/0.62 & ! [S10: set @ A] : % 0.31/0.62 ( ( ( topolo1002775350975398744n_open @ A @ S10 ) % 0.31/0.62 & ( member @ A @ X3 @ S10 ) ) % 0.31/0.62 => ? [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( A8 @ I4 ) @ S10 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % first_countable_basis % 0.31/0.62 thf(fact_5777_open__subopen,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ( ( topolo1002775350975398744n_open @ A ) % 0.31/0.62 = ( ^ [S7: set @ A] : % 0.31/0.62 ! [X: A] : % 0.31/0.62 ( ( member @ A @ X @ S7 ) % 0.31/0.62 => ? [T8: set @ A] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ A @ T8 ) % 0.31/0.62 & ( member @ A @ X @ T8 ) % 0.31/0.62 & ( ord_less_eq @ ( set @ A ) @ T8 @ S7 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % open_subopen % 0.31/0.62 thf(fact_5778_openI,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [S3: set @ A] : % 0.31/0.62 ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ S3 ) % 0.31/0.62 => ? [T10: set @ A] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ A @ T10 ) % 0.31/0.62 & ( member @ A @ X5 @ T10 ) % 0.31/0.62 & ( ord_less_eq @ ( set @ A ) @ T10 @ S3 ) ) ) % 0.31/0.62 => ( topolo1002775350975398744n_open @ A @ S3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % openI % 0.31/0.62 thf(fact_5779_norm__conv__dist,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ( ( real_V7770717601297561774m_norm @ A ) % 0.31/0.62 = ( ^ [X: A] : ( real_V557655796197034286t_dist @ A @ X @ ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % norm_conv_dist % 0.31/0.62 thf(fact_5780_open__Collect__ex,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [P: B > A > $o] : % 0.31/0.62 ( ! [I4: B] : ( topolo1002775350975398744n_open @ A @ ( collect @ A @ ( P @ I4 ) ) ) % 0.31/0.62 => ( topolo1002775350975398744n_open @ A % 0.31/0.62 @ ( collect @ A % 0.31/0.62 @ ^ [X: A] : % 0.31/0.62 ? [I3: B] : ( P @ I3 @ X ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % open_Collect_ex % 0.31/0.62 thf(fact_5781_dist__triangle__lt,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X3: A,Z2: A,Y: A,E3: real] : % 0.31/0.62 ( ( ord_less @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Z2 ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z2 ) ) @ E3 ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y ) @ E3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % dist_triangle_lt % 0.31/0.62 thf(fact_5782_dist__triangle__less__add,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X1: A,Y: A,E1: real,X22: A,E22: real] : % 0.31/0.62 ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ Y ) @ E1 ) % 0.31/0.62 => ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ Y ) @ E22 ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X22 ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % dist_triangle_less_add % 0.31/0.62 thf(fact_5783_open__ball,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X3: A,D2: real] : % 0.31/0.62 ( topolo1002775350975398744n_open @ A % 0.31/0.62 @ ( collect @ A % 0.31/0.62 @ ^ [Y2: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y2 ) @ D2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % open_ball % 0.31/0.62 thf(fact_5784_dist__commute__lessI,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [Y: A,X3: A,E3: real] : % 0.31/0.62 ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X3 ) @ E3 ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y ) @ E3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % dist_commute_lessI % 0.31/0.62 thf(fact_5785_open__dist,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ( ( topolo1002775350975398744n_open @ A ) % 0.31/0.62 = ( ^ [S7: set @ A] : % 0.31/0.62 ! [X: A] : % 0.31/0.62 ( ( member @ A @ X @ S7 ) % 0.31/0.62 => ? [E4: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 ) % 0.31/0.62 & ! [Y2: A] : % 0.31/0.62 ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y2 @ X ) @ E4 ) % 0.31/0.62 => ( member @ A @ Y2 @ S7 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % open_dist % 0.31/0.62 thf(fact_5786_Inf__notin__open,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo8458572112393995274pology @ A ) % 0.31/0.62 => ! [A3: set @ A,X3: A] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ A @ A3 ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ A3 ) % 0.31/0.62 => ( ord_less @ A @ X3 @ X5 ) ) % 0.31/0.62 => ~ ( member @ A @ ( complete_Inf_Inf @ A @ A3 ) @ A3 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Inf_notin_open % 0.31/0.62 thf(fact_5787_Sup__notin__open,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo8458572112393995274pology @ A ) % 0.31/0.62 => ! [A3: set @ A,X3: A] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ A @ A3 ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ A3 ) % 0.31/0.62 => ( ord_less @ A @ X5 @ X3 ) ) % 0.31/0.62 => ~ ( member @ A @ ( complete_Sup_Sup @ A @ A3 ) @ A3 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Sup_notin_open % 0.31/0.62 thf(fact_5788_dist__pos__lt,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X3: A,Y: A] : % 0.31/0.62 ( ( X3 != Y ) % 0.31/0.62 => ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X3 @ Y ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % dist_pos_lt % 0.31/0.62 thf(fact_5789_dist__not__less__zero,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X3: A,Y: A] : % 0.31/0.62 ~ ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ Y ) @ ( zero_zero @ real ) ) ) ). % 0.31/0.62 % 0.31/0.62 % dist_not_less_zero % 0.31/0.62 thf(fact_5790_open__diagonal__complement,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topological_t2_space @ A ) % 0.31/0.62 => ( topolo1002775350975398744n_open @ ( product_prod @ A @ A ) % 0.31/0.62 @ ( collect @ ( product_prod @ A @ A ) % 0.31/0.62 @ ^ [Uu3: product_prod @ A @ A] : % 0.31/0.62 ? [X: A,Y2: A] : % 0.31/0.62 ( ( Uu3 % 0.31/0.62 = ( product_Pair @ A @ A @ X @ Y2 ) ) % 0.31/0.62 & ( X != Y2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % open_diagonal_complement % 0.31/0.62 thf(fact_5791_at__within__open__subset,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [A2: A,S3: set @ A,T4: set @ A] : % 0.31/0.62 ( ( member @ A @ A2 @ S3 ) % 0.31/0.62 => ( ( topolo1002775350975398744n_open @ A @ S3 ) % 0.31/0.62 => ( ( ord_less_eq @ ( set @ A ) @ S3 @ T4 ) % 0.31/0.62 => ( ( topolo174197925503356063within @ A @ A2 @ T4 ) % 0.31/0.62 = ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % at_within_open_subset % 0.31/0.62 thf(fact_5792_open__right,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 => ! [S3: set @ A,X3: A,Y: A] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ A @ S3 ) % 0.31/0.62 => ( ( member @ A @ X3 @ S3 ) % 0.31/0.62 => ( ( ord_less @ A @ X3 @ Y ) % 0.31/0.62 => ? [B5: A] : % 0.31/0.62 ( ( ord_less @ A @ X3 @ B5 ) % 0.31/0.62 & ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ X3 @ B5 ) @ S3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % open_right % 0.31/0.62 thf(fact_5793_open__left,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 => ! [S3: set @ A,X3: A,Y: A] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ A @ S3 ) % 0.31/0.62 => ( ( member @ A @ X3 @ S3 ) % 0.31/0.62 => ( ( ord_less @ A @ Y @ X3 ) % 0.31/0.62 => ? [B5: A] : % 0.31/0.62 ( ( ord_less @ A @ B5 @ X3 ) % 0.31/0.62 & ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ B5 @ X3 ) @ S3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % open_left % 0.31/0.62 thf(fact_5794_open__subdiagonal,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 => ( topolo1002775350975398744n_open @ ( product_prod @ A @ A ) % 0.31/0.62 @ ( collect @ ( product_prod @ A @ A ) % 0.31/0.62 @ ^ [Uu3: product_prod @ A @ A] : % 0.31/0.62 ? [X: A,Y2: A] : % 0.31/0.62 ( ( Uu3 % 0.31/0.62 = ( product_Pair @ A @ A @ X @ Y2 ) ) % 0.31/0.62 & ( ord_less @ A @ X @ Y2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % open_subdiagonal % 0.31/0.62 thf(fact_5795_open__superdiagonal,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 => ( topolo1002775350975398744n_open @ ( product_prod @ A @ A ) % 0.31/0.62 @ ( collect @ ( product_prod @ A @ A ) % 0.31/0.62 @ ^ [Uu3: product_prod @ A @ A] : % 0.31/0.62 ? [X: A,Y2: A] : % 0.31/0.62 ( ( Uu3 % 0.31/0.62 = ( product_Pair @ A @ A @ X @ Y2 ) ) % 0.31/0.62 & ( ord_less @ A @ Y2 @ X ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % open_superdiagonal % 0.31/0.62 thf(fact_5796_eventually__at__bot__linorder,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [P: A > $o] : % 0.31/0.62 ( ( eventually @ A @ P @ ( at_bot @ A ) ) % 0.31/0.62 = ( ? [N5: A] : % 0.31/0.62 ! [N: A] : % 0.31/0.62 ( ( ord_less_eq @ A @ N @ N5 ) % 0.31/0.62 => ( P @ N ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_at_bot_linorder % 0.31/0.62 thf(fact_5797_eventually__at__bot__dense,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( linorder @ A ) % 0.31/0.62 & ( no_bot @ A ) ) % 0.31/0.62 => ! [P: A > $o] : % 0.31/0.62 ( ( eventually @ A @ P @ ( at_bot @ A ) ) % 0.31/0.62 = ( ? [N5: A] : % 0.31/0.62 ! [N: A] : % 0.31/0.62 ( ( ord_less @ A @ N @ N5 ) % 0.31/0.62 => ( P @ N ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_at_bot_dense % 0.31/0.62 thf(fact_5798_countable__basis__at__decseq,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo3112930676232923870pology @ A ) % 0.31/0.62 => ! [X3: A] : % 0.31/0.62 ~ ! [A8: nat > ( set @ A )] : % 0.31/0.62 ( ! [I5: nat] : ( topolo1002775350975398744n_open @ A @ ( A8 @ I5 ) ) % 0.31/0.62 => ( ! [I5: nat] : ( member @ A @ X3 @ ( A8 @ I5 ) ) % 0.31/0.62 => ~ ! [S10: set @ A] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ A @ S10 ) % 0.31/0.62 => ( ( member @ A @ X3 @ S10 ) % 0.31/0.62 => ( eventually @ nat % 0.31/0.62 @ ^ [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( A8 @ I3 ) @ S10 ) % 0.31/0.62 @ ( at_top @ nat ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % countable_basis_at_decseq % 0.31/0.62 thf(fact_5799_eventually__at,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [P: A > $o,A2: A,S3: set @ A] : % 0.31/0.62 ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ S3 ) ) % 0.31/0.62 = ( ? [D5: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ D5 ) % 0.31/0.62 & ! [X: A] : % 0.31/0.62 ( ( member @ A @ X @ S3 ) % 0.31/0.62 => ( ( ( X != A2 ) % 0.31/0.62 & ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ D5 ) ) % 0.31/0.62 => ( P @ X ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_at % 0.31/0.62 thf(fact_5800_eventually__nhds__metric,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [P: A > $o,A2: A] : % 0.31/0.62 ( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ A2 ) ) % 0.31/0.62 = ( ? [D5: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ D5 ) % 0.31/0.62 & ! [X: A] : % 0.31/0.62 ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ D5 ) % 0.31/0.62 => ( P @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_nhds_metric % 0.31/0.62 thf(fact_5801_has__field__derivative__transform__within,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.62 => ! [F2: A > A,F11: A,A2: A,S3: set @ A,D2: real,G: A > A] : % 0.31/0.62 ( ( has_field_derivative @ A @ F2 @ F11 @ ( topolo174197925503356063within @ A @ A2 @ S3 ) ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 ) % 0.31/0.62 => ( ( member @ A @ A2 @ S3 ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ S3 ) % 0.31/0.62 => ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ D2 ) % 0.31/0.62 => ( ( F2 @ X5 ) % 0.31/0.62 = ( G @ X5 ) ) ) ) % 0.31/0.62 => ( has_field_derivative @ A @ G @ F11 @ ( topolo174197925503356063within @ A @ A2 @ S3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % has_field_derivative_transform_within % 0.31/0.62 thf(fact_5802_has__derivative__transform__within,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.62 => ! [F2: A > B,F11: A > B,X3: A,S: set @ A,D2: real,G: A > B] : % 0.31/0.62 ( ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 ) % 0.31/0.62 => ( ( member @ A @ X3 @ S ) % 0.31/0.62 => ( ! [X17: A] : % 0.31/0.62 ( ( member @ A @ X17 @ S ) % 0.31/0.62 => ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X17 @ X3 ) @ D2 ) % 0.31/0.62 => ( ( F2 @ X17 ) % 0.31/0.62 = ( G @ X17 ) ) ) ) % 0.31/0.62 => ( has_derivative @ A @ B @ G @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % has_derivative_transform_within % 0.31/0.62 thf(fact_5803_eventually__le__at__bot,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [C2: A] : % 0.31/0.62 ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less_eq @ A @ X @ C2 ) % 0.31/0.62 @ ( at_bot @ A ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_le_at_bot % 0.31/0.62 thf(fact_5804_eventually__gt__at__bot,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( unboun7993243217541854897norder @ A ) % 0.31/0.62 => ! [C2: A] : % 0.31/0.62 ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less @ A @ X @ C2 ) % 0.31/0.62 @ ( at_bot @ A ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_gt_at_bot % 0.31/0.62 thf(fact_5805_Cauchy__def,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ( ( topolo3814608138187158403Cauchy @ A ) % 0.31/0.62 = ( ^ [X6: nat > A] : % 0.31/0.62 ! [E4: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 ) % 0.31/0.62 => ? [M9: nat] : % 0.31/0.62 ! [M2: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M9 @ M2 ) % 0.31/0.62 => ! [N: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M9 @ N ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M2 ) @ ( X6 @ N ) ) @ E4 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Cauchy_def % 0.31/0.62 thf(fact_5806_Cauchy__altdef2,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ( ( topolo3814608138187158403Cauchy @ A ) % 0.31/0.62 = ( ^ [S8: nat > A] : % 0.31/0.62 ! [E4: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 ) % 0.31/0.62 => ? [N5: nat] : % 0.31/0.62 ! [N: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ N5 @ N ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S8 @ N ) @ ( S8 @ N5 ) ) @ E4 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Cauchy_altdef2 % 0.31/0.62 thf(fact_5807_metric__CauchyD,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X8: nat > A,E3: real] : % 0.31/0.62 ( ( topolo3814608138187158403Cauchy @ A @ X8 ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 ) % 0.31/0.62 => ? [M8: nat] : % 0.31/0.62 ! [M6: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M8 @ M6 ) % 0.31/0.62 => ! [N6: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M8 @ N6 ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M6 ) @ ( X8 @ N6 ) ) @ E3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % metric_CauchyD % 0.31/0.62 thf(fact_5808_metric__CauchyI,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X8: nat > A] : % 0.31/0.62 ( ! [E: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ E ) % 0.31/0.62 => ? [M10: nat] : % 0.31/0.62 ! [M3: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M10 @ M3 ) % 0.31/0.62 => ! [N3: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M10 @ N3 ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) @ E ) ) ) ) % 0.31/0.62 => ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % metric_CauchyI % 0.31/0.62 thf(fact_5809_lim__explicit,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [F2: nat > A,F0: A] : % 0.31/0.62 ( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ F0 ) @ ( at_top @ nat ) ) % 0.31/0.62 = ( ! [S7: set @ A] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ A @ S7 ) % 0.31/0.62 => ( ( member @ A @ F0 @ S7 ) % 0.31/0.62 => ? [N5: nat] : % 0.31/0.62 ! [N: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ N5 @ N ) % 0.31/0.62 => ( member @ A @ ( F2 @ N ) @ S7 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % lim_explicit % 0.31/0.62 thf(fact_5810_Lim__transform__within,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.62 => ! [F2: A > B,L: B,X3: A,S3: set @ A,D2: real,G: A > B] : % 0.31/0.62 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 ) % 0.31/0.62 => ( ! [X17: A] : % 0.31/0.62 ( ( member @ A @ X17 @ S3 ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X17 @ X3 ) ) % 0.31/0.62 => ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X17 @ X3 ) @ D2 ) % 0.31/0.62 => ( ( F2 @ X17 ) % 0.31/0.62 = ( G @ X17 ) ) ) ) ) % 0.31/0.62 => ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Lim_transform_within % 0.31/0.62 thf(fact_5811_dist__triangle__half__r,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [Y: A,X1: A,E3: real,X22: A] : % 0.31/0.62 ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X1 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.62 => ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X22 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X22 ) @ E3 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % dist_triangle_half_r % 0.31/0.62 thf(fact_5812_dist__triangle__half__l,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X1: A,Y: A,E3: real,X22: A] : % 0.31/0.62 ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ Y ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.62 => ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ Y ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X22 ) @ E3 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % dist_triangle_half_l % 0.31/0.62 thf(fact_5813_eventually__at__le,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [P: A > $o,A2: A,S3: set @ A] : % 0.31/0.62 ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ S3 ) ) % 0.31/0.62 = ( ? [D5: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ D5 ) % 0.31/0.62 & ! [X: A] : % 0.31/0.62 ( ( member @ A @ X @ S3 ) % 0.31/0.62 => ( ( ( X != A2 ) % 0.31/0.62 & ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ D5 ) ) % 0.31/0.62 => ( P @ X ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_at_le % 0.31/0.62 thf(fact_5814_dist__triangle__third,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X1: A,X22: A,E3: real,X32: A,X42: A] : % 0.31/0.62 ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X22 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) % 0.31/0.62 => ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X22 @ X32 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) % 0.31/0.62 => ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X32 @ X42 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X42 ) @ E3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % dist_triangle_third % 0.31/0.62 thf(fact_5815_at__within__nhd,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [X3: A,S3: set @ A,T4: set @ A,U3: set @ A] : % 0.31/0.62 ( ( member @ A @ X3 @ S3 ) % 0.31/0.62 => ( ( topolo1002775350975398744n_open @ A @ S3 ) % 0.31/0.62 => ( ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ T4 @ S3 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.62 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ U3 @ S3 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.62 => ( ( topolo174197925503356063within @ A @ X3 @ T4 ) % 0.31/0.62 = ( topolo174197925503356063within @ A @ X3 @ U3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % at_within_nhd % 0.31/0.62 thf(fact_5816_filterlim__transform__within,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [G: A > B,G6: filter @ B,X3: A,S3: set @ A,F5: filter @ B,D2: real,F2: A > B] : % 0.31/0.62 ( ( filterlim @ A @ B @ G @ G6 @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) % 0.31/0.62 => ( ( ord_less_eq @ ( filter @ B ) @ G6 @ F5 ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 ) % 0.31/0.62 => ( ! [X17: A] : % 0.31/0.62 ( ( member @ A @ X17 @ S3 ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X17 @ X3 ) ) % 0.31/0.62 => ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X17 @ X3 ) @ D2 ) % 0.31/0.62 => ( ( F2 @ X17 ) % 0.31/0.62 = ( G @ X17 ) ) ) ) ) % 0.31/0.62 => ( filterlim @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_transform_within % 0.31/0.62 thf(fact_5817_Cauchy__altdef,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ( ( topolo3814608138187158403Cauchy @ A ) % 0.31/0.62 = ( ^ [F3: nat > A] : % 0.31/0.62 ! [E4: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 ) % 0.31/0.62 => ? [M9: nat] : % 0.31/0.62 ! [M2: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M9 @ M2 ) % 0.31/0.62 => ! [N: nat] : % 0.31/0.62 ( ( ord_less @ nat @ M2 @ N ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ M2 ) @ ( F3 @ N ) ) @ E4 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Cauchy_altdef % 0.31/0.62 thf(fact_5818_CauchyI_H,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X8: nat > A] : % 0.31/0.62 ( ! [E: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ E ) % 0.31/0.62 => ? [M10: nat] : % 0.31/0.62 ! [M3: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M10 @ M3 ) % 0.31/0.62 => ! [N3: nat] : % 0.31/0.62 ( ( ord_less @ nat @ M3 @ N3 ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) @ E ) ) ) ) % 0.31/0.62 => ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % CauchyI' % 0.31/0.62 thf(fact_5819_at__eq__bot__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [A2: A] : % 0.31/0.62 ( ( ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.62 = ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.62 = ( topolo1002775350975398744n_open @ A @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % at_eq_bot_iff % 0.31/0.62 thf(fact_5820_filterlim__at__infinity__imp__filterlim__at__bot,axiom, % 0.31/0.62 ! [A: $tType,F2: A > real,F5: filter @ A] : % 0.31/0.62 ( ( filterlim @ A @ real @ F2 @ ( at_infinity @ real ) @ F5 ) % 0.31/0.62 => ( ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less @ real @ ( F2 @ X ) @ ( zero_zero @ real ) ) % 0.31/0.62 @ F5 ) % 0.31/0.62 => ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F5 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_at_infinity_imp_filterlim_at_bot % 0.31/0.62 thf(fact_5821_tendstoD,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [F2: B > A,L: A,F5: filter @ B,E3: real] : % 0.31/0.62 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 ) % 0.31/0.62 => ( eventually @ B % 0.31/0.62 @ ^ [X: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ L ) @ E3 ) % 0.31/0.62 @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendstoD % 0.31/0.62 thf(fact_5822_tendstoI,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [F2: B > A,L: A,F5: filter @ B] : % 0.31/0.62 ( ! [E: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ E ) % 0.31/0.62 => ( eventually @ B % 0.31/0.62 @ ^ [X: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ L ) @ E ) % 0.31/0.62 @ F5 ) ) % 0.31/0.62 => ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendstoI % 0.31/0.62 thf(fact_5823_tendsto__iff,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [F2: B > A,L: A,F5: filter @ B] : % 0.31/0.62 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) % 0.31/0.62 = ( ! [E4: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 ) % 0.31/0.62 => ( eventually @ B % 0.31/0.62 @ ^ [X: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F2 @ X ) @ L ) @ E4 ) % 0.31/0.62 @ F5 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_iff % 0.31/0.62 thf(fact_5824_filterlim__at__bot__le,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( linorder @ B ) % 0.31/0.62 => ! [F2: A > B,F5: filter @ A,C2: B] : % 0.31/0.62 ( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 ) % 0.31/0.62 = ( ! [Z9: B] : % 0.31/0.62 ( ( ord_less_eq @ B @ Z9 @ C2 ) % 0.31/0.62 => ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ Z9 ) % 0.31/0.62 @ F5 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_at_bot_le % 0.31/0.62 thf(fact_5825_filterlim__at__bot,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( linorder @ B ) % 0.31/0.62 => ! [F2: A > B,F5: filter @ A] : % 0.31/0.62 ( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 ) % 0.31/0.62 = ( ! [Z9: B] : % 0.31/0.62 ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ Z9 ) % 0.31/0.62 @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_at_bot % 0.31/0.62 thf(fact_5826_filterlim__at__bot__dense,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( ( dense_linorder @ B ) % 0.31/0.62 & ( no_bot @ B ) ) % 0.31/0.62 => ! [F2: A > B,F5: filter @ A] : % 0.31/0.62 ( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 ) % 0.31/0.62 = ( ! [Z9: B] : % 0.31/0.62 ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ Z9 ) % 0.31/0.62 @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_at_bot_dense % 0.31/0.62 thf(fact_5827_metric__LIM__equal2,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.62 => ! [G: A > B,L: B,A2: A,R: real,F2: A > B] : % 0.31/0.62 ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) @ R ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( X5 != A2 ) % 0.31/0.62 => ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ R ) % 0.31/0.62 => ( ( F2 @ X5 ) % 0.31/0.62 = ( G @ X5 ) ) ) ) % 0.31/0.62 => ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % metric_LIM_equal2 % 0.31/0.62 thf(fact_5828_metric__LIM__I,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 & ( real_V7819770556892013058_space @ B ) ) % 0.31/0.62 => ! [A2: A,F2: A > B,L5: B] : % 0.31/0.62 ( ! [R4: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 ) % 0.31/0.62 => ? [S9: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ S9 ) % 0.31/0.62 & ! [X5: A] : % 0.31/0.62 ( ( ( X5 != A2 ) % 0.31/0.62 & ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ S9 ) ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X5 ) @ L5 ) @ R4 ) ) ) ) % 0.31/0.62 => ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % metric_LIM_I % 0.31/0.62 thf(fact_5829_metric__LIM__D,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 & ( real_V7819770556892013058_space @ B ) ) % 0.31/0.62 => ! [F2: A > B,L5: B,A2: A,R2: real] : % 0.31/0.62 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 ) % 0.31/0.62 => ? [S2: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ S2 ) % 0.31/0.62 & ! [X2: A] : % 0.31/0.62 ( ( ( X2 != A2 ) % 0.31/0.62 & ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ A2 ) @ S2 ) ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X2 ) @ L5 ) @ R2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % metric_LIM_D % 0.31/0.62 thf(fact_5830_LIM__def,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 & ( real_V7819770556892013058_space @ B ) ) % 0.31/0.62 => ! [F2: A > B,L5: B,A2: A] : % 0.31/0.62 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.62 = ( ! [R5: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ R5 ) % 0.31/0.62 => ? [S8: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ S8 ) % 0.31/0.62 & ! [X: A] : % 0.31/0.62 ( ( ( X != A2 ) % 0.31/0.62 & ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ A2 ) @ S8 ) ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X ) @ L5 ) @ R5 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % LIM_def % 0.31/0.62 thf(fact_5831_metric__LIMSEQ__D,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X8: nat > A,L5: A,R2: real] : % 0.31/0.62 ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 ) % 0.31/0.62 => ? [No: nat] : % 0.31/0.62 ! [N6: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ No @ N6 ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N6 ) @ L5 ) @ R2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % metric_LIMSEQ_D % 0.31/0.62 thf(fact_5832_metric__LIMSEQ__I,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X8: nat > A,L5: A] : % 0.31/0.62 ( ! [R4: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 ) % 0.31/0.62 => ? [No2: nat] : % 0.31/0.62 ! [N3: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ No2 @ N3 ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N3 ) @ L5 ) @ R4 ) ) ) % 0.31/0.62 => ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % metric_LIMSEQ_I % 0.31/0.62 thf(fact_5833_lim__sequentially,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X8: nat > A,L5: A] : % 0.31/0.62 ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) % 0.31/0.62 = ( ! [R5: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ R5 ) % 0.31/0.62 => ? [No3: nat] : % 0.31/0.62 ! [N: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ No3 @ N ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N ) @ L5 ) @ R5 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % lim_sequentially % 0.31/0.62 thf(fact_5834_metric__Cauchy__iff2,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ( ( topolo3814608138187158403Cauchy @ A ) % 0.31/0.62 = ( ^ [X6: nat > A] : % 0.31/0.62 ! [J3: nat] : % 0.31/0.62 ? [M9: nat] : % 0.31/0.62 ! [M2: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M9 @ M2 ) % 0.31/0.62 => ! [N: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M9 @ N ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M2 ) @ ( X6 @ N ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % metric_Cauchy_iff2 % 0.31/0.62 thf(fact_5835_metric__LIM__compose2,axiom, % 0.31/0.62 ! [C: $tType,B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 & ( topolo4958980785337419405_space @ B ) % 0.31/0.62 & ( topolo4958980785337419405_space @ C ) ) % 0.31/0.62 => ! [F2: A > B,B2: B,A2: A,G: B > C,C2: C] : % 0.31/0.62 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.62 => ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) ) % 0.31/0.62 => ( ? [D4: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 ) % 0.31/0.62 & ! [X5: A] : % 0.31/0.62 ( ( ( X5 != A2 ) % 0.31/0.62 & ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ D4 ) ) % 0.31/0.62 => ( ( F2 @ X5 ) % 0.31/0.62 != B2 ) ) ) % 0.31/0.62 => ( filterlim @ A @ C % 0.31/0.62 @ ^ [X: A] : ( G @ ( F2 @ X ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ C @ C2 ) % 0.31/0.62 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % metric_LIM_compose2 % 0.31/0.62 thf(fact_5836_filterlim__tendsto__pos__mult__at__bot,axiom, % 0.31/0.62 ! [A: $tType,F2: A > real,C2: real,F5: filter @ A,G: A > real] : % 0.31/0.62 ( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F5 ) % 0.31/0.62 => ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 0.31/0.62 => ( ( filterlim @ A @ real @ G @ ( at_bot @ real ) @ F5 ) % 0.31/0.62 => ( filterlim @ A @ real % 0.31/0.62 @ ^ [X: A] : ( times_times @ real @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.62 @ ( at_bot @ real ) % 0.31/0.62 @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_tendsto_pos_mult_at_bot % 0.31/0.62 thf(fact_5837_filterlim__at__bot__lt,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( unboun7993243217541854897norder @ B ) % 0.31/0.62 => ! [F2: A > B,F5: filter @ A,C2: B] : % 0.31/0.62 ( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 ) % 0.31/0.62 = ( ! [Z9: B] : % 0.31/0.62 ( ( ord_less @ B @ Z9 @ C2 ) % 0.31/0.62 => ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ Z9 ) % 0.31/0.62 @ F5 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_at_bot_lt % 0.31/0.62 thf(fact_5838_metric__isCont__LIM__compose2,axiom, % 0.31/0.62 ! [D: $tType,C: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 & ( topolo4958980785337419405_space @ C ) % 0.31/0.62 & ( topolo4958980785337419405_space @ D ) ) % 0.31/0.62 => ! [A2: A,F2: A > C,G: C > D,L: D] : % 0.31/0.62 ( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 ) % 0.31/0.62 => ( ( filterlim @ C @ D @ G @ ( topolo7230453075368039082e_nhds @ D @ L ) @ ( topolo174197925503356063within @ C @ ( F2 @ A2 ) @ ( top_top @ ( set @ C ) ) ) ) % 0.31/0.62 => ( ? [D4: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 ) % 0.31/0.62 & ! [X5: A] : % 0.31/0.62 ( ( ( X5 != A2 ) % 0.31/0.62 & ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ D4 ) ) % 0.31/0.62 => ( ( F2 @ X5 ) % 0.31/0.62 != ( F2 @ A2 ) ) ) ) % 0.31/0.62 => ( filterlim @ A @ D % 0.31/0.62 @ ^ [X: A] : ( G @ ( F2 @ X ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ D @ L ) % 0.31/0.62 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % metric_isCont_LIM_compose2 % 0.31/0.62 thf(fact_5839_tendsto__offset__zero__iff,axiom, % 0.31/0.62 ! [C: $tType,D: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 & ( topolo4958980785337419405_space @ D ) % 0.31/0.62 & ( zero @ C ) ) % 0.31/0.62 => ! [A2: A,S3: set @ A,F2: A > D,L5: D] : % 0.31/0.62 ( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 ) % 0.31/0.62 => ( ( member @ A @ A2 @ S3 ) % 0.31/0.62 => ( ( topolo1002775350975398744n_open @ A @ S3 ) % 0.31/0.62 => ( ( filterlim @ A @ D @ F2 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ S3 ) ) % 0.31/0.62 = ( filterlim @ A @ D % 0.31/0.62 @ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) % 0.31/0.62 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_offset_zero_iff % 0.31/0.62 thf(fact_5840_filterlim__inverse__at__bot,axiom, % 0.31/0.62 ! [A: $tType,F2: A > real,F5: filter @ A] : % 0.31/0.62 ( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) % 0.31/0.62 => ( ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less @ real @ ( F2 @ X ) @ ( zero_zero @ real ) ) % 0.31/0.62 @ F5 ) % 0.31/0.62 => ( filterlim @ A @ real % 0.31/0.62 @ ^ [X: A] : ( inverse_inverse @ real @ ( F2 @ X ) ) % 0.31/0.62 @ ( at_bot @ real ) % 0.31/0.62 @ F5 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_inverse_at_bot % 0.31/0.62 thf(fact_5841_filterlim__tendsto__neg__mult__at__bot,axiom, % 0.31/0.62 ! [A: $tType,F2: A > real,C2: real,F5: filter @ A,G: A > real] : % 0.31/0.62 ( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F5 ) % 0.31/0.62 => ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 0.31/0.62 => ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F5 ) % 0.31/0.62 => ( filterlim @ A @ real % 0.31/0.62 @ ^ [X: A] : ( times_times @ real @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.62 @ ( at_bot @ real ) % 0.31/0.62 @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_tendsto_neg_mult_at_bot % 0.31/0.62 thf(fact_5842_totally__bounded__metric,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ( ( topolo6688025880775521714ounded @ A ) % 0.31/0.62 = ( ^ [S7: set @ A] : % 0.31/0.62 ! [E4: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 ) % 0.31/0.62 => ? [K3: set @ A] : % 0.31/0.62 ( ( finite_finite2 @ A @ K3 ) % 0.31/0.62 & ( ord_less_eq @ ( set @ A ) @ S7 % 0.31/0.62 @ ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.62 @ ( image @ A @ ( set @ A ) % 0.31/0.62 @ ^ [X: A] : % 0.31/0.62 ( collect @ A % 0.31/0.62 @ ^ [Y2: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ E4 ) ) % 0.31/0.62 @ K3 ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % totally_bounded_metric % 0.31/0.62 thf(fact_5843_LIMSEQ__iff__nz,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X8: nat > A,L5: A] : % 0.31/0.62 ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) % 0.31/0.62 = ( ! [R5: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ R5 ) % 0.31/0.62 => ? [No3: nat] : % 0.31/0.62 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ No3 ) % 0.31/0.62 & ! [N: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ No3 @ N ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N ) @ L5 ) @ R5 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % LIMSEQ_iff_nz % 0.31/0.62 thf(fact_5844_DERIV__pos__imp__increasing__at__bot,axiom, % 0.31/0.62 ! [B2: real,F2: real > real,Flim: real] : % 0.31/0.62 ( ! [X5: real] : % 0.31/0.62 ( ( ord_less_eq @ real @ X5 @ B2 ) % 0.31/0.62 => ? [Y5: real] : % 0.31/0.62 ( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.62 & ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) ) % 0.31/0.62 => ( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_bot @ real ) ) % 0.31/0.62 => ( ord_less @ real @ Flim @ ( F2 @ B2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % DERIV_pos_imp_increasing_at_bot % 0.31/0.62 thf(fact_5845_filterlim__pow__at__bot__odd,axiom, % 0.31/0.62 ! [N2: nat,F2: real > real,F5: filter @ real] : % 0.31/0.62 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.62 => ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F5 ) % 0.31/0.62 => ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 0.31/0.62 => ( filterlim @ real @ real % 0.31/0.62 @ ^ [X: real] : ( power_power @ real @ ( F2 @ X ) @ N2 ) % 0.31/0.62 @ ( at_bot @ real ) % 0.31/0.62 @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_pow_at_bot_odd % 0.31/0.62 thf(fact_5846_Bfun__metric__def,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ B ) % 0.31/0.62 => ( ( bfun @ A @ B ) % 0.31/0.62 = ( ^ [F3: A > B,F9: filter @ A] : % 0.31/0.62 ? [Y2: B,K6: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ K6 ) % 0.31/0.62 & ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X ) @ Y2 ) @ K6 ) % 0.31/0.62 @ F9 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Bfun_metric_def % 0.31/0.62 thf(fact_5847_Bfun__def,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ B ) % 0.31/0.62 => ( ( bfun @ A @ B ) % 0.31/0.62 = ( ^ [F3: A > B,F9: filter @ A] : % 0.31/0.62 ? [K6: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ K6 ) % 0.31/0.62 & ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X ) ) @ K6 ) % 0.31/0.62 @ F9 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Bfun_def % 0.31/0.62 thf(fact_5848_BfunE,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ B ) % 0.31/0.62 => ! [F2: A > B,F5: filter @ A] : % 0.31/0.62 ( ( bfun @ A @ B @ F2 @ F5 ) % 0.31/0.62 => ~ ! [B10: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ B10 ) % 0.31/0.62 => ~ ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) @ B10 ) % 0.31/0.62 @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % BfunE % 0.31/0.62 thf(fact_5849_Bseq__add__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [F2: nat > A,C2: A] : % 0.31/0.62 ( ( bfun @ nat @ A % 0.31/0.62 @ ^ [X: nat] : ( plus_plus @ A @ ( F2 @ X ) @ C2 ) % 0.31/0.62 @ ( at_top @ nat ) ) % 0.31/0.62 = ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Bseq_add_iff % 0.31/0.62 thf(fact_5850_Bseq__add,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [F2: nat > A,C2: A] : % 0.31/0.62 ( ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) % 0.31/0.62 => ( bfun @ nat @ A % 0.31/0.62 @ ^ [X: nat] : ( plus_plus @ A @ ( F2 @ X ) @ C2 ) % 0.31/0.62 @ ( at_top @ nat ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Bseq_add % 0.31/0.62 thf(fact_5851_Bseq__Suc__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [F2: nat > A] : % 0.31/0.62 ( ( bfun @ nat @ A % 0.31/0.62 @ ^ [N: nat] : ( F2 @ ( suc @ N ) ) % 0.31/0.62 @ ( at_top @ nat ) ) % 0.31/0.62 = ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Bseq_Suc_iff % 0.31/0.62 thf(fact_5852_Bseq__ignore__initial__segment,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X8: nat > A,K: nat] : % 0.31/0.62 ( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) % 0.31/0.62 => ( bfun @ nat @ A % 0.31/0.62 @ ^ [N: nat] : ( X8 @ ( plus_plus @ nat @ N @ K ) ) % 0.31/0.62 @ ( at_top @ nat ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Bseq_ignore_initial_segment % 0.31/0.62 thf(fact_5853_Bseq__offset,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ! [X8: nat > A,K: nat] : % 0.31/0.62 ( ( bfun @ nat @ A % 0.31/0.62 @ ^ [N: nat] : ( X8 @ ( plus_plus @ nat @ N @ K ) ) % 0.31/0.62 @ ( at_top @ nat ) ) % 0.31/0.62 => ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Bseq_offset % 0.31/0.62 thf(fact_5854_Bseq__cmult__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.62 => ! [C2: A,F2: nat > A] : % 0.31/0.62 ( ( C2 % 0.31/0.62 != ( zero_zero @ A ) ) % 0.31/0.62 => ( ( bfun @ nat @ A % 0.31/0.62 @ ^ [X: nat] : ( times_times @ A @ C2 @ ( F2 @ X ) ) % 0.31/0.62 @ ( at_top @ nat ) ) % 0.31/0.62 = ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Bseq_cmult_iff % 0.31/0.62 thf(fact_5855_BseqD,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [X8: nat > A] : % 0.31/0.62 ( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) % 0.31/0.62 => ? [K8: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ K8 ) % 0.31/0.62 & ! [N6: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N6 ) ) @ K8 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % BseqD % 0.31/0.62 thf(fact_5856_BseqE,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [X8: nat > A] : % 0.31/0.62 ( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) % 0.31/0.62 => ~ ! [K8: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ K8 ) % 0.31/0.62 => ~ ! [N6: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N6 ) ) @ K8 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % BseqE % 0.31/0.62 thf(fact_5857_BseqI,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [K5: real,X8: nat > A] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ K5 ) % 0.31/0.62 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N3 ) ) @ K5 ) % 0.31/0.62 => ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % BseqI % 0.31/0.62 thf(fact_5858_Bseq__def,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [X8: nat > A] : % 0.31/0.62 ( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) % 0.31/0.62 = ( ? [K6: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ K6 ) % 0.31/0.62 & ! [N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N ) ) @ K6 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Bseq_def % 0.31/0.62 thf(fact_5859_Bseq__iff1a,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [X8: nat > A] : % 0.31/0.62 ( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) % 0.31/0.62 = ( ? [N5: nat] : % 0.31/0.62 ! [N: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Bseq_iff1a % 0.31/0.62 thf(fact_5860_Bseq__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [X8: nat > A] : % 0.31/0.62 ( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) % 0.31/0.62 = ( ? [N5: nat] : % 0.31/0.62 ! [N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Bseq_iff % 0.31/0.62 thf(fact_5861_Bseq__iff2,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [X8: nat > A] : % 0.31/0.62 ( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) % 0.31/0.62 = ( ? [K3: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ K3 ) % 0.31/0.62 & ? [X: A] : % 0.31/0.62 ! [N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X8 @ N ) @ ( uminus_uminus @ A @ X ) ) ) @ K3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Bseq_iff2 % 0.31/0.62 thf(fact_5862_Bseq__iff3,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 => ! [X8: nat > A] : % 0.31/0.62 ( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) % 0.31/0.62 = ( ? [K3: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ K3 ) % 0.31/0.62 & ? [N5: nat] : % 0.31/0.62 ! [N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X8 @ N ) @ ( uminus_uminus @ A @ ( X8 @ N5 ) ) ) ) @ K3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Bseq_iff3 % 0.31/0.62 thf(fact_5863_Bfun__inverse,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.62 => ! [F2: B > A,A2: A,F5: filter @ B] : % 0.31/0.62 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 0.31/0.62 => ( ( A2 % 0.31/0.62 != ( zero_zero @ A ) ) % 0.31/0.62 => ( bfun @ B @ A % 0.31/0.62 @ ^ [X: B] : ( inverse_inverse @ A @ ( F2 @ X ) ) % 0.31/0.62 @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Bfun_inverse % 0.31/0.62 thf(fact_5864_map__filter__on__comp,axiom, % 0.31/0.62 ! [A: $tType,C: $tType,B: $tType,G: B > A,Y7: set @ B,X8: set @ A,F5: filter @ B,F2: A > C] : % 0.31/0.62 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ Y7 ) @ X8 ) % 0.31/0.62 => ( ( eventually @ B % 0.31/0.62 @ ^ [X: B] : ( member @ B @ X @ Y7 ) % 0.31/0.62 @ F5 ) % 0.31/0.62 => ( ( map_filter_on @ A @ C @ X8 @ F2 @ ( map_filter_on @ B @ A @ Y7 @ G @ F5 ) ) % 0.31/0.62 = ( map_filter_on @ B @ C @ Y7 @ ( comp @ A @ C @ B @ F2 @ G ) @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % map_filter_on_comp % 0.31/0.62 thf(fact_5865_cauchy__filter__metric,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( real_V768167426530841204y_dist @ A ) % 0.31/0.62 & ( topolo7287701948861334536_space @ A ) ) % 0.31/0.62 => ( ( topolo6773858410816713723filter @ A ) % 0.31/0.62 = ( ^ [F9: filter @ A] : % 0.31/0.62 ! [E4: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 ) % 0.31/0.62 => ? [P3: A > $o] : % 0.31/0.62 ( ( eventually @ A @ P3 @ F9 ) % 0.31/0.62 & ! [X: A,Y2: A] : % 0.31/0.62 ( ( ( P3 @ X ) % 0.31/0.62 & ( P3 @ Y2 ) ) % 0.31/0.62 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ E4 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cauchy_filter_metric % 0.31/0.62 thf(fact_5866_GMVT,axiom, % 0.31/0.62 ! [A2: real,B2: real,F2: real > real,G: real > real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.62 => ( ! [X5: real] : % 0.31/0.62 ( ( ( ord_less_eq @ real @ A2 @ X5 ) % 0.31/0.62 & ( ord_less_eq @ real @ X5 @ B2 ) ) % 0.31/0.62 => ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) % 0.31/0.62 => ( ! [X5: real] : % 0.31/0.62 ( ( ( ord_less @ real @ A2 @ X5 ) % 0.31/0.62 & ( ord_less @ real @ X5 @ B2 ) ) % 0.31/0.62 => ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) % 0.31/0.62 => ( ! [X5: real] : % 0.31/0.62 ( ( ( ord_less_eq @ real @ A2 @ X5 ) % 0.31/0.62 & ( ord_less_eq @ real @ X5 @ B2 ) ) % 0.31/0.62 => ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ G ) ) % 0.31/0.62 => ( ! [X5: real] : % 0.31/0.62 ( ( ( ord_less @ real @ A2 @ X5 ) % 0.31/0.62 & ( ord_less @ real @ X5 @ B2 ) ) % 0.31/0.62 => ( differentiable @ real @ real @ G @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) % 0.31/0.62 => ? [G_c: real,F_c: real,C3: real] : % 0.31/0.62 ( ( has_field_derivative @ real @ G @ G_c @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.62 & ( has_field_derivative @ real @ F2 @ F_c @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.62 & ( ord_less @ real @ A2 @ C3 ) % 0.31/0.62 & ( ord_less @ real @ C3 @ B2 ) % 0.31/0.62 & ( ( times_times @ real @ ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) @ G_c ) % 0.31/0.62 = ( times_times @ real @ ( minus_minus @ real @ ( G @ B2 ) @ ( G @ A2 ) ) @ F_c ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % GMVT % 0.31/0.62 thf(fact_5867_differentiable__cmult__right__iff,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ B ) % 0.31/0.62 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.62 => ! [Q4: B > A,C2: A,T2: B] : % 0.31/0.62 ( ( differentiable @ B @ A % 0.31/0.62 @ ^ [T3: B] : ( times_times @ A @ ( Q4 @ T3 ) @ C2 ) % 0.31/0.62 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) % 0.31/0.62 = ( ( C2 % 0.31/0.62 = ( zero_zero @ A ) ) % 0.31/0.62 | ( differentiable @ B @ A @ Q4 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % differentiable_cmult_right_iff % 0.31/0.62 thf(fact_5868_differentiable__cmult__left__iff,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ B ) % 0.31/0.62 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.62 => ! [C2: A,Q4: B > A,T2: B] : % 0.31/0.62 ( ( differentiable @ B @ A % 0.31/0.62 @ ^ [T3: B] : ( times_times @ A @ C2 @ ( Q4 @ T3 ) ) % 0.31/0.62 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) % 0.31/0.62 = ( ( C2 % 0.31/0.62 = ( zero_zero @ A ) ) % 0.31/0.62 | ( differentiable @ B @ A @ Q4 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % differentiable_cmult_left_iff % 0.31/0.62 thf(fact_5869_differentiable__add,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.62 => ! [F2: A > B,F5: filter @ A,G: A > B] : % 0.31/0.62 ( ( differentiable @ A @ B @ F2 @ F5 ) % 0.31/0.62 => ( ( differentiable @ A @ B @ G @ F5 ) % 0.31/0.62 => ( differentiable @ A @ B % 0.31/0.62 @ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.62 @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % differentiable_add % 0.31/0.62 thf(fact_5870_differentiable__within__subset,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.62 => ! [F2: A > B,X3: A,S: set @ A,T2: set @ A] : % 0.31/0.62 ( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.62 => ( ( ord_less_eq @ ( set @ A ) @ T2 @ S ) % 0.31/0.62 => ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % differentiable_within_subset % 0.31/0.62 thf(fact_5871_differentiable__sum,axiom, % 0.31/0.62 ! [C: $tType,A: $tType,B: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ B ) % 0.31/0.62 & ( real_V822414075346904944vector @ C ) ) % 0.31/0.62 => ! [S: set @ A,F2: A > B > C,Net: filter @ B] : % 0.31/0.62 ( ( finite_finite2 @ A @ S ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ S ) % 0.31/0.62 => ( differentiable @ B @ C @ ( F2 @ X5 ) @ Net ) ) % 0.31/0.62 => ( differentiable @ B @ C % 0.31/0.62 @ ^ [X: B] : % 0.31/0.62 ( groups7311177749621191930dd_sum @ A @ C % 0.31/0.62 @ ^ [A4: A] : ( F2 @ A4 @ X ) % 0.31/0.62 @ S ) % 0.31/0.62 @ Net ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % differentiable_sum % 0.31/0.62 thf(fact_5872_differentiable__divide,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 & ( real_V3459762299906320749_field @ B ) ) % 0.31/0.62 => ! [F2: A > B,X3: A,S: set @ A,G: A > B] : % 0.31/0.62 ( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.62 => ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.62 => ( ( ( G @ X3 ) % 0.31/0.62 != ( zero_zero @ B ) ) % 0.31/0.62 => ( differentiable @ A @ B % 0.31/0.62 @ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.62 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % differentiable_divide % 0.31/0.62 thf(fact_5873_differentiable__inverse,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.62 & ( real_V3459762299906320749_field @ B ) ) % 0.31/0.62 => ! [F2: A > B,X3: A,S: set @ A] : % 0.31/0.62 ( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.62 => ( ( ( F2 @ X3 ) % 0.31/0.62 != ( zero_zero @ B ) ) % 0.31/0.62 => ( differentiable @ A @ B % 0.31/0.62 @ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) ) % 0.31/0.62 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % differentiable_inverse % 0.31/0.62 thf(fact_5874_MVT,axiom, % 0.31/0.62 ! [A2: real,B2: real,F2: real > real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 ) % 0.31/0.62 => ( ! [X5: real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ X5 ) % 0.31/0.62 => ( ( ord_less @ real @ X5 @ B2 ) % 0.31/0.62 => ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) ) % 0.31/0.62 => ? [L4: real,Z3: real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ Z3 ) % 0.31/0.62 & ( ord_less @ real @ Z3 @ B2 ) % 0.31/0.62 & ( has_field_derivative @ real @ F2 @ L4 @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.62 & ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) % 0.31/0.62 = ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ L4 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % MVT % 0.31/0.62 thf(fact_5875_eventually__filtercomap__at__topological,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ B ) % 0.31/0.62 => ! [P: A > $o,F2: A > B,A3: B,B3: set @ B] : % 0.31/0.62 ( ( eventually @ A @ P @ ( filtercomap @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ A3 @ B3 ) ) ) % 0.31/0.62 = ( ? [S7: set @ B] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ B @ S7 ) % 0.31/0.62 & ( member @ B @ A3 @ S7 ) % 0.31/0.62 & ! [X: A] : % 0.31/0.62 ( ( member @ B @ ( F2 @ X ) @ ( minus_minus @ ( set @ B ) @ ( inf_inf @ ( set @ B ) @ S7 @ B3 ) @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) % 0.31/0.62 => ( P @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_filtercomap_at_topological % 0.31/0.62 thf(fact_5876_at__within__eq,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ( ( topolo174197925503356063within @ A ) % 0.31/0.62 = ( ^ [X: A,S8: set @ A] : % 0.31/0.62 ( complete_Inf_Inf @ ( filter @ A ) % 0.31/0.62 @ ( image @ ( set @ A ) @ ( filter @ A ) % 0.31/0.62 @ ^ [S7: set @ A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S7 @ S8 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.62 @ ( collect @ ( set @ A ) % 0.31/0.62 @ ^ [S7: set @ A] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ A @ S7 ) % 0.31/0.62 & ( member @ A @ X @ S7 ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % at_within_eq % 0.31/0.62 thf(fact_5877_filtercomap__bot,axiom, % 0.31/0.62 ! [B: $tType,A: $tType,F2: A > B] : % 0.31/0.62 ( ( filtercomap @ A @ B @ F2 @ ( bot_bot @ ( filter @ B ) ) ) % 0.31/0.62 = ( bot_bot @ ( filter @ A ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filtercomap_bot % 0.31/0.62 thf(fact_5878_principal__le__iff,axiom, % 0.31/0.62 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.62 ( ( ord_less_eq @ ( filter @ A ) @ ( principal @ A @ A3 ) @ ( principal @ A @ B3 ) ) % 0.31/0.62 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ). % 0.31/0.62 % 0.31/0.62 % principal_le_iff % 0.31/0.62 thf(fact_5879_continuous__on__sgn,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 & ( real_V822414075346904944vector @ B ) ) % 0.31/0.62 => ! [S: set @ A,F2: A > B] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ S ) % 0.31/0.62 => ( ( F2 @ X5 ) % 0.31/0.62 != ( zero_zero @ B ) ) ) % 0.31/0.62 => ( topolo81223032696312382ous_on @ A @ B @ S % 0.31/0.62 @ ^ [X: A] : ( sgn_sgn @ B @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_sgn % 0.31/0.62 thf(fact_5880_continuous__on__add,axiom, % 0.31/0.62 ! [B: $tType,D: $tType] : % 0.31/0.62 ( ( ( topolo4958980785337419405_space @ D ) % 0.31/0.62 & ( topolo6943815403480290642id_add @ B ) ) % 0.31/0.62 => ! [S: set @ D,F2: D > B,G: D > B] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ D @ B @ S @ F2 ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ D @ B @ S @ G ) % 0.31/0.62 => ( topolo81223032696312382ous_on @ D @ B @ S % 0.31/0.62 @ ^ [X: D] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_add % 0.31/0.62 thf(fact_5881_continuous__on__Pair,axiom, % 0.31/0.62 ! [C: $tType,B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 & ( topolo4958980785337419405_space @ B ) % 0.31/0.62 & ( topolo4958980785337419405_space @ C ) ) % 0.31/0.62 => ! [S: set @ A,F2: A > B,G: A > C] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ A @ C @ S @ G ) % 0.31/0.62 => ( topolo81223032696312382ous_on @ A @ ( product_prod @ B @ C ) @ S % 0.31/0.62 @ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_Pair % 0.31/0.62 thf(fact_5882_continuous__on__divide,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 & ( real_V3459762299906320749_field @ B ) ) % 0.31/0.62 => ! [S: set @ A,F2: A > B,G: A > B] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ A @ B @ S @ G ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ S ) % 0.31/0.62 => ( ( G @ X5 ) % 0.31/0.62 != ( zero_zero @ B ) ) ) % 0.31/0.62 => ( topolo81223032696312382ous_on @ A @ B @ S % 0.31/0.62 @ ^ [X: A] : ( divide_divide @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_divide % 0.31/0.62 thf(fact_5883_IVT2_H,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( ( topolo1944317154257567458pology @ B ) % 0.31/0.62 & ( topolo8458572112393995274pology @ A ) ) % 0.31/0.62 => ! [F2: A > B,B2: A,Y: B,A2: A] : % 0.31/0.62 ( ( ord_less_eq @ B @ ( F2 @ B2 ) @ Y ) % 0.31/0.62 => ( ( ord_less_eq @ B @ Y @ ( F2 @ A2 ) ) % 0.31/0.62 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 ) % 0.31/0.62 => ? [X5: A] : % 0.31/0.62 ( ( ord_less_eq @ A @ A2 @ X5 ) % 0.31/0.62 & ( ord_less_eq @ A @ X5 @ B2 ) % 0.31/0.62 & ( ( F2 @ X5 ) % 0.31/0.62 = Y ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % IVT2' % 0.31/0.62 thf(fact_5884_IVT_H,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( ( topolo1944317154257567458pology @ B ) % 0.31/0.62 & ( topolo8458572112393995274pology @ A ) ) % 0.31/0.62 => ! [F2: A > B,A2: A,Y: B,B2: A] : % 0.31/0.62 ( ( ord_less_eq @ B @ ( F2 @ A2 ) @ Y ) % 0.31/0.62 => ( ( ord_less_eq @ B @ Y @ ( F2 @ B2 ) ) % 0.31/0.62 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 ) % 0.31/0.62 => ? [X5: A] : % 0.31/0.62 ( ( ord_less_eq @ A @ A2 @ X5 ) % 0.31/0.62 & ( ord_less_eq @ A @ X5 @ B2 ) % 0.31/0.62 & ( ( F2 @ X5 ) % 0.31/0.62 = Y ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % IVT' % 0.31/0.62 thf(fact_5885_continuous__on__subset,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.62 => ! [S: set @ A,F2: A > B,T2: set @ A] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 ) % 0.31/0.62 => ( ( ord_less_eq @ ( set @ A ) @ T2 @ S ) % 0.31/0.62 => ( topolo81223032696312382ous_on @ A @ B @ T2 @ F2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_subset % 0.31/0.62 thf(fact_5886_continuous__on__sing,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.62 => ! [X3: A,F2: A > B] : ( topolo81223032696312382ous_on @ A @ B @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ F2 ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_sing % 0.31/0.62 thf(fact_5887_continuous__on__empty,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.62 => ! [F2: A > B] : ( topolo81223032696312382ous_on @ A @ B @ ( bot_bot @ ( set @ A ) ) @ F2 ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_empty % 0.31/0.62 thf(fact_5888_continuous__on__inverse,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 & ( real_V8999393235501362500lgebra @ B ) ) % 0.31/0.62 => ! [S: set @ A,F2: A > B] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ S ) % 0.31/0.62 => ( ( F2 @ X5 ) % 0.31/0.62 != ( zero_zero @ B ) ) ) % 0.31/0.62 => ( topolo81223032696312382ous_on @ A @ B @ S % 0.31/0.62 @ ^ [X: A] : ( inverse_inverse @ B @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_inverse % 0.31/0.62 thf(fact_5889_continuous__on__compose2,axiom, % 0.31/0.62 ! [A: $tType,B: $tType,C: $tType] : % 0.31/0.62 ( ( ( topolo4958980785337419405_space @ C ) % 0.31/0.62 & ( topolo4958980785337419405_space @ B ) % 0.31/0.62 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.62 => ! [T2: set @ A,G: A > B,S: set @ C,F2: C > A] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ A @ B @ T2 @ G ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ C @ A @ S @ F2 ) % 0.31/0.62 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ S ) @ T2 ) % 0.31/0.62 => ( topolo81223032696312382ous_on @ C @ B @ S % 0.31/0.62 @ ^ [X: C] : ( G @ ( F2 @ X ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_compose2 % 0.31/0.62 thf(fact_5890_bot__eq__principal__empty,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( bot_bot @ ( filter @ A ) ) % 0.31/0.62 = ( principal @ A @ ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bot_eq_principal_empty % 0.31/0.62 thf(fact_5891_principal__eq__bot__iff,axiom, % 0.31/0.62 ! [A: $tType,X8: set @ A] : % 0.31/0.62 ( ( ( principal @ A @ X8 ) % 0.31/0.62 = ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.62 = ( X8 % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % principal_eq_bot_iff % 0.31/0.62 thf(fact_5892_filtercomap__neq__bot,axiom, % 0.31/0.62 ! [A: $tType,B: $tType,F5: filter @ A,F2: B > A] : % 0.31/0.62 ( ! [P8: A > $o] : % 0.31/0.62 ( ( eventually @ A @ P8 @ F5 ) % 0.31/0.62 => ? [X2: B] : ( P8 @ ( F2 @ X2 ) ) ) % 0.31/0.62 => ( ( filtercomap @ B @ A @ F2 @ F5 ) % 0.31/0.62 != ( bot_bot @ ( filter @ B ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filtercomap_neq_bot % 0.31/0.62 thf(fact_5893_continuous__onI__mono,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 & ( dense_order @ B ) % 0.31/0.62 & ( topolo1944317154257567458pology @ B ) ) % 0.31/0.62 => ! [F2: A > B,A3: set @ A] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ B @ ( image @ A @ B @ F2 @ A3 ) ) % 0.31/0.62 => ( ! [X5: A,Y4: A] : % 0.31/0.62 ( ( member @ A @ X5 @ A3 ) % 0.31/0.62 => ( ( member @ A @ Y4 @ A3 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ X5 @ Y4 ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) ) ) % 0.31/0.62 => ( topolo81223032696312382ous_on @ A @ B @ A3 @ F2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_onI_mono % 0.31/0.62 thf(fact_5894_open__Collect__less,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 & ( topolo1944317154257567458pology @ B ) ) % 0.31/0.62 => ! [F2: A > B,G: A > B] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G ) % 0.31/0.62 => ( topolo1002775350975398744n_open @ A % 0.31/0.62 @ ( collect @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % open_Collect_less % 0.31/0.62 thf(fact_5895_nhds__discrete,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo8865339358273720382pology @ A ) % 0.31/0.62 => ( ( topolo7230453075368039082e_nhds @ A ) % 0.31/0.62 = ( ^ [X: A] : ( principal @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % nhds_discrete % 0.31/0.62 thf(fact_5896_continuous__on__tan,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( real_Vector_banach @ A ) % 0.31/0.62 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.62 => ! [S: set @ A,F2: A > A] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ A @ A @ S @ F2 ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ S ) % 0.31/0.62 => ( ( cos @ A @ ( F2 @ X5 ) ) % 0.31/0.62 != ( zero_zero @ A ) ) ) % 0.31/0.62 => ( topolo81223032696312382ous_on @ A @ A @ S % 0.31/0.62 @ ^ [X: A] : ( tan @ A @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_tan % 0.31/0.62 thf(fact_5897_open__Collect__less__Int,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [S: set @ A,F2: A > real,G: A > real] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ A @ real @ S @ G ) % 0.31/0.62 => ? [A8: set @ A] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ A @ A8 ) % 0.31/0.62 & ( ( inf_inf @ ( set @ A ) @ A8 @ S ) % 0.31/0.62 = ( collect @ A % 0.31/0.62 @ ^ [X: A] : % 0.31/0.62 ( ( member @ A @ X @ S ) % 0.31/0.62 & ( ord_less @ real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % open_Collect_less_Int % 0.31/0.62 thf(fact_5898_continuous__on__cot,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( real_Vector_banach @ A ) % 0.31/0.62 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.62 => ! [S: set @ A,F2: A > A] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ A @ A @ S @ F2 ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ S ) % 0.31/0.62 => ( ( sin @ A @ ( F2 @ X5 ) ) % 0.31/0.62 != ( zero_zero @ A ) ) ) % 0.31/0.62 => ( topolo81223032696312382ous_on @ A @ A @ S % 0.31/0.62 @ ^ [X: A] : ( cot @ A @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_cot % 0.31/0.62 thf(fact_5899_continuous__on__tanh,axiom, % 0.31/0.62 ! [A: $tType,C: $tType] : % 0.31/0.62 ( ( ( topolo4958980785337419405_space @ C ) % 0.31/0.62 & ( real_Vector_banach @ A ) % 0.31/0.62 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.62 => ! [A3: set @ C,F2: C > A] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ C @ A @ A3 @ F2 ) % 0.31/0.62 => ( ! [X5: C] : % 0.31/0.62 ( ( member @ C @ X5 @ A3 ) % 0.31/0.62 => ( ( cosh @ A @ ( F2 @ X5 ) ) % 0.31/0.62 != ( zero_zero @ A ) ) ) % 0.31/0.62 => ( topolo81223032696312382ous_on @ C @ A @ A3 % 0.31/0.62 @ ^ [X: C] : ( tanh @ A @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_tanh % 0.31/0.62 thf(fact_5900_eventually__filtercomap__at__top__linorder,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [P: B > $o,F2: B > A] : % 0.31/0.62 ( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_top @ A ) ) ) % 0.31/0.62 = ( ? [N5: A] : % 0.31/0.62 ! [X: B] : % 0.31/0.62 ( ( ord_less_eq @ A @ N5 @ ( F2 @ X ) ) % 0.31/0.62 => ( P @ X ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_filtercomap_at_top_linorder % 0.31/0.62 thf(fact_5901_eventually__filtercomap__at__top__dense,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( linorder @ A ) % 0.31/0.62 & ( no_top @ A ) ) % 0.31/0.62 => ! [P: B > $o,F2: B > A] : % 0.31/0.62 ( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_top @ A ) ) ) % 0.31/0.62 = ( ? [N5: A] : % 0.31/0.62 ! [X: B] : % 0.31/0.62 ( ( ord_less @ A @ N5 @ ( F2 @ X ) ) % 0.31/0.62 => ( P @ X ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_filtercomap_at_top_dense % 0.31/0.62 thf(fact_5902_filtercomap__neq__bot__surj,axiom, % 0.31/0.62 ! [A: $tType,B: $tType,F5: filter @ A,F2: B > A] : % 0.31/0.62 ( ( F5 % 0.31/0.62 != ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.62 => ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) % 0.31/0.62 = ( top_top @ ( set @ A ) ) ) % 0.31/0.62 => ( ( filtercomap @ B @ A @ F2 @ F5 ) % 0.31/0.62 != ( bot_bot @ ( filter @ B ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filtercomap_neq_bot_surj % 0.31/0.62 thf(fact_5903_eventually__filtercomap__at__bot__linorder,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [P: B > $o,F2: B > A] : % 0.31/0.62 ( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_bot @ A ) ) ) % 0.31/0.62 = ( ? [N5: A] : % 0.31/0.62 ! [X: B] : % 0.31/0.62 ( ( ord_less_eq @ A @ ( F2 @ X ) @ N5 ) % 0.31/0.62 => ( P @ X ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_filtercomap_at_bot_linorder % 0.31/0.62 thf(fact_5904_eventually__filtercomap__at__bot__dense,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( linorder @ A ) % 0.31/0.62 & ( no_bot @ A ) ) % 0.31/0.62 => ! [P: B > $o,F2: B > A] : % 0.31/0.62 ( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_bot @ A ) ) ) % 0.31/0.62 = ( ? [N5: A] : % 0.31/0.62 ! [X: B] : % 0.31/0.62 ( ( ord_less @ A @ ( F2 @ X ) @ N5 ) % 0.31/0.62 => ( P @ X ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_filtercomap_at_bot_dense % 0.31/0.62 thf(fact_5905_open__Collect__positive,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [S: set @ A,F2: A > real] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 ) % 0.31/0.62 => ? [A8: set @ A] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ A @ A8 ) % 0.31/0.62 & ( ( inf_inf @ ( set @ A ) @ A8 @ S ) % 0.31/0.62 = ( collect @ A % 0.31/0.62 @ ^ [X: A] : % 0.31/0.62 ( ( member @ A @ X @ S ) % 0.31/0.62 & ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % open_Collect_positive % 0.31/0.62 thf(fact_5906_continuous__on__powr_H,axiom, % 0.31/0.62 ! [C: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ C ) % 0.31/0.62 => ! [S: set @ C,F2: C > real,G: C > real] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ C @ real @ S @ F2 ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ C @ real @ S @ G ) % 0.31/0.62 => ( ! [X5: C] : % 0.31/0.62 ( ( member @ C @ X5 @ S ) % 0.31/0.62 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X5 ) ) % 0.31/0.62 & ( ( ( F2 @ X5 ) % 0.31/0.62 = ( zero_zero @ real ) ) % 0.31/0.62 => ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X5 ) ) ) ) ) % 0.31/0.62 => ( topolo81223032696312382ous_on @ C @ real @ S % 0.31/0.62 @ ^ [X: C] : ( powr @ real @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_powr' % 0.31/0.62 thf(fact_5907_continuous__on__log,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [S: set @ A,F2: A > real,G: A > real] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ A @ real @ S @ G ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ S ) % 0.31/0.62 => ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X5 ) ) ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ S ) % 0.31/0.62 => ( ( F2 @ X5 ) % 0.31/0.62 != ( one_one @ real ) ) ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ S ) % 0.31/0.62 => ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X5 ) ) ) % 0.31/0.62 => ( topolo81223032696312382ous_on @ A @ real @ S % 0.31/0.62 @ ^ [X: A] : ( log @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_log % 0.31/0.62 thf(fact_5908_tendsto__principal__singleton,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [F2: B > A,X3: B] : ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( F2 @ X3 ) ) @ ( principal @ B @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_principal_singleton % 0.31/0.62 thf(fact_5909_nhds__discrete__open,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [X3: A] : % 0.31/0.62 ( ( topolo1002775350975398744n_open @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.62 => ( ( topolo7230453075368039082e_nhds @ A @ X3 ) % 0.31/0.62 = ( principal @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % nhds_discrete_open % 0.31/0.62 thf(fact_5910_DERIV__atLeastAtMost__imp__continuous__on,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( ord @ A ) % 0.31/0.62 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.62 => ! [A2: A,B2: A,F2: A > A] : % 0.31/0.62 ( ! [X5: A] : % 0.31/0.62 ( ( ord_less_eq @ A @ A2 @ X5 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ X5 @ B2 ) % 0.31/0.62 => ? [Y5: A] : ( has_field_derivative @ A @ F2 @ Y5 @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ) ) ) % 0.31/0.62 => ( topolo81223032696312382ous_on @ A @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % DERIV_atLeastAtMost_imp_continuous_on % 0.31/0.62 thf(fact_5911_Rolle__deriv,axiom, % 0.31/0.62 ! [A2: real,B2: real,F2: real > real,F11: real > real > real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.62 => ( ( ( F2 @ A2 ) % 0.31/0.62 = ( F2 @ B2 ) ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 ) % 0.31/0.62 => ( ! [X5: real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ X5 ) % 0.31/0.62 => ( ( ord_less @ real @ X5 @ B2 ) % 0.31/0.62 => ( has_derivative @ real @ real @ F2 @ ( F11 @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) ) % 0.31/0.62 => ? [Z3: real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ Z3 ) % 0.31/0.62 & ( ord_less @ real @ Z3 @ B2 ) % 0.31/0.62 & ( ( F11 @ Z3 ) % 0.31/0.62 = ( ^ [V5: real] : ( zero_zero @ real ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Rolle_deriv % 0.31/0.62 thf(fact_5912_mvt,axiom, % 0.31/0.62 ! [A2: real,B2: real,F2: real > real,F11: real > real > real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 ) % 0.31/0.62 => ( ! [X5: real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ X5 ) % 0.31/0.62 => ( ( ord_less @ real @ X5 @ B2 ) % 0.31/0.62 => ( has_derivative @ real @ real @ F2 @ ( F11 @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) ) % 0.31/0.62 => ~ ! [Xi: real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ Xi ) % 0.31/0.62 => ( ( ord_less @ real @ Xi @ B2 ) % 0.31/0.62 => ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) % 0.31/0.62 != ( F11 @ Xi @ ( minus_minus @ real @ B2 @ A2 ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mvt % 0.31/0.62 thf(fact_5913_continuous__on__Icc__at__leftD,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.62 => ! [A2: A,B2: A,F2: A > B] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 ) % 0.31/0.62 => ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.62 => ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ B2 ) ) @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_Icc_at_leftD % 0.31/0.62 thf(fact_5914_filterlim__base__iff,axiom, % 0.31/0.62 ! [A: $tType,C: $tType,B: $tType,D: $tType,I6: set @ A,F5: A > ( set @ B ),F2: B > C,G6: D > ( set @ C ),J4: set @ D] : % 0.31/0.62 ( ( I6 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ! [I4: A] : % 0.31/0.62 ( ( member @ A @ I4 @ I6 ) % 0.31/0.62 => ! [J2: A] : % 0.31/0.62 ( ( member @ A @ J2 @ I6 ) % 0.31/0.62 => ( ( ord_less_eq @ ( set @ B ) @ ( F5 @ I4 ) @ ( F5 @ J2 ) ) % 0.31/0.62 | ( ord_less_eq @ ( set @ B ) @ ( F5 @ J2 ) @ ( F5 @ I4 ) ) ) ) ) % 0.31/0.62 => ( ( filterlim @ B @ C @ F2 % 0.31/0.62 @ ( complete_Inf_Inf @ ( filter @ C ) % 0.31/0.62 @ ( image @ D @ ( filter @ C ) % 0.31/0.62 @ ^ [J3: D] : ( principal @ C @ ( G6 @ J3 ) ) % 0.31/0.62 @ J4 ) ) % 0.31/0.62 @ ( complete_Inf_Inf @ ( filter @ B ) % 0.31/0.62 @ ( image @ A @ ( filter @ B ) % 0.31/0.62 @ ^ [I3: A] : ( principal @ B @ ( F5 @ I3 ) ) % 0.31/0.62 @ I6 ) ) ) % 0.31/0.62 = ( ! [X: D] : % 0.31/0.62 ( ( member @ D @ X @ J4 ) % 0.31/0.62 => ? [Y2: A] : % 0.31/0.62 ( ( member @ A @ Y2 @ I6 ) % 0.31/0.62 & ! [Z6: B] : % 0.31/0.62 ( ( member @ B @ Z6 @ ( F5 @ Y2 ) ) % 0.31/0.62 => ( member @ C @ ( F2 @ Z6 ) @ ( G6 @ X ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_base_iff % 0.31/0.62 thf(fact_5915_DERIV__pos__imp__increasing__open,axiom, % 0.31/0.62 ! [A2: real,B2: real,F2: real > real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.62 => ( ! [X5: real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ X5 ) % 0.31/0.62 => ( ( ord_less @ real @ X5 @ B2 ) % 0.31/0.62 => ? [Y5: real] : % 0.31/0.62 ( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.62 & ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) ) ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 ) % 0.31/0.62 => ( ord_less @ real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % DERIV_pos_imp_increasing_open % 0.31/0.62 thf(fact_5916_DERIV__neg__imp__decreasing__open,axiom, % 0.31/0.62 ! [A2: real,B2: real,F2: real > real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.62 => ( ! [X5: real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ X5 ) % 0.31/0.62 => ( ( ord_less @ real @ X5 @ B2 ) % 0.31/0.62 => ? [Y5: real] : % 0.31/0.62 ( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.62 & ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) ) ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 ) % 0.31/0.62 => ( ord_less @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % DERIV_neg_imp_decreasing_open % 0.31/0.62 thf(fact_5917_DERIV__isconst__end,axiom, % 0.31/0.62 ! [A2: real,B2: real,F2: real > real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 ) % 0.31/0.62 => ( ! [X5: real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ X5 ) % 0.31/0.62 => ( ( ord_less @ real @ X5 @ B2 ) % 0.31/0.62 => ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) ) % 0.31/0.62 => ( ( F2 @ B2 ) % 0.31/0.62 = ( F2 @ A2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % DERIV_isconst_end % 0.31/0.62 thf(fact_5918_INF__principal__finite,axiom, % 0.31/0.62 ! [B: $tType,A: $tType,X8: set @ A,F2: A > ( set @ B )] : % 0.31/0.62 ( ( finite_finite2 @ A @ X8 ) % 0.31/0.62 => ( ( complete_Inf_Inf @ ( filter @ B ) % 0.31/0.62 @ ( image @ A @ ( filter @ B ) % 0.31/0.62 @ ^ [X: A] : ( principal @ B @ ( F2 @ X ) ) % 0.31/0.62 @ X8 ) ) % 0.31/0.62 = ( principal @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ X8 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % INF_principal_finite % 0.31/0.62 thf(fact_5919_at__within__def,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ( ( topolo174197925503356063within @ A ) % 0.31/0.62 = ( ^ [A4: A,S8: set @ A] : ( inf_inf @ ( filter @ A ) @ ( topolo7230453075368039082e_nhds @ A @ A4 ) @ ( principal @ A @ ( minus_minus @ ( set @ A ) @ S8 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % at_within_def % 0.31/0.62 thf(fact_5920_DERIV__isconst2,axiom, % 0.31/0.62 ! [A2: real,B2: real,F2: real > real,X3: real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 ) % 0.31/0.62 => ( ! [X5: real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ X5 ) % 0.31/0.62 => ( ( ord_less @ real @ X5 @ B2 ) % 0.31/0.62 => ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) ) % 0.31/0.62 => ( ( ord_less_eq @ real @ A2 @ X3 ) % 0.31/0.62 => ( ( ord_less_eq @ real @ X3 @ B2 ) % 0.31/0.62 => ( ( F2 @ X3 ) % 0.31/0.62 = ( F2 @ A2 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % DERIV_isconst2 % 0.31/0.62 thf(fact_5921_at__left__eq,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 => ! [Y: A,X3: A] : % 0.31/0.62 ( ( ord_less @ A @ Y @ X3 ) % 0.31/0.62 => ( ( topolo174197925503356063within @ A @ X3 @ ( set_ord_lessThan @ A @ X3 ) ) % 0.31/0.62 = ( complete_Inf_Inf @ ( filter @ A ) % 0.31/0.62 @ ( image @ A @ ( filter @ A ) % 0.31/0.62 @ ^ [A4: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ A4 @ X3 ) ) % 0.31/0.62 @ ( set_ord_lessThan @ A @ X3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % at_left_eq % 0.31/0.62 thf(fact_5922_Rolle,axiom, % 0.31/0.62 ! [A2: real,B2: real,F2: real > real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.62 => ( ( ( F2 @ A2 ) % 0.31/0.62 = ( F2 @ B2 ) ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F2 ) % 0.31/0.62 => ( ! [X5: real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ X5 ) % 0.31/0.62 => ( ( ord_less @ real @ X5 @ B2 ) % 0.31/0.62 => ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) ) % 0.31/0.62 => ? [Z3: real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ Z3 ) % 0.31/0.62 & ( ord_less @ real @ Z3 @ B2 ) % 0.31/0.62 & ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Rolle % 0.31/0.62 thf(fact_5923_complete__uniform,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo7287701948861334536_space @ A ) % 0.31/0.62 => ( ( topolo2479028161051973599mplete @ A ) % 0.31/0.62 = ( ^ [S7: set @ A] : % 0.31/0.62 ! [F9: filter @ A] : % 0.31/0.62 ( ( ord_less_eq @ ( filter @ A ) @ F9 @ ( principal @ A @ S7 ) ) % 0.31/0.62 => ( ( F9 % 0.31/0.62 != ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.62 => ( ( topolo6773858410816713723filter @ A @ F9 ) % 0.31/0.62 => ? [X: A] : % 0.31/0.62 ( ( member @ A @ X @ S7 ) % 0.31/0.62 & ( ord_less_eq @ ( filter @ A ) @ F9 @ ( topolo7230453075368039082e_nhds @ A @ X ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % complete_uniform % 0.31/0.62 thf(fact_5924_at__within__order,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 => ! [X3: A,S: set @ A] : % 0.31/0.62 ( ( ( top_top @ ( set @ A ) ) % 0.31/0.62 != ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.62 => ( ( topolo174197925503356063within @ A @ X3 @ S ) % 0.31/0.62 = ( inf_inf @ ( filter @ A ) % 0.31/0.62 @ ( complete_Inf_Inf @ ( filter @ A ) % 0.31/0.62 @ ( image @ A @ ( filter @ A ) % 0.31/0.62 @ ^ [A4: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A4 ) @ S ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.62 @ ( set_ord_greaterThan @ A @ X3 ) ) ) % 0.31/0.62 @ ( complete_Inf_Inf @ ( filter @ A ) % 0.31/0.62 @ ( image @ A @ ( filter @ A ) % 0.31/0.62 @ ^ [A4: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A4 ) @ S ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.62 @ ( set_ord_lessThan @ A @ X3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % at_within_order % 0.31/0.62 thf(fact_5925_continuous__on__IccI,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.62 => ! [F2: A > B,A2: A,B2: A] : % 0.31/0.62 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) % 0.31/0.62 => ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ B2 ) ) @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( ord_less @ A @ A2 @ X5 ) % 0.31/0.62 => ( ( ord_less @ A @ X5 @ B2 ) % 0.31/0.62 => ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X5 ) ) @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ) ) ) % 0.31/0.62 => ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.62 => ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_IccI % 0.31/0.62 thf(fact_5926_greaterThan__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ord @ A ) % 0.31/0.62 => ! [I: A,K: A] : % 0.31/0.62 ( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) ) % 0.31/0.62 = ( ord_less @ A @ K @ I ) ) ) ). % 0.31/0.62 % 0.31/0.62 % greaterThan_iff % 0.31/0.62 thf(fact_5927_greaterThan__subset__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [X3: A,Y: A] : % 0.31/0.62 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X3 ) @ ( set_ord_greaterThan @ A @ Y ) ) % 0.31/0.62 = ( ord_less_eq @ A @ Y @ X3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % greaterThan_subset_iff % 0.31/0.62 thf(fact_5928_Sup__greaterThanAtLeast,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( comple6319245703460814977attice @ A ) % 0.31/0.62 => ! [X3: A] : % 0.31/0.62 ( ( ord_less @ A @ X3 @ ( top_top @ A ) ) % 0.31/0.62 => ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X3 ) ) % 0.31/0.62 = ( top_top @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Sup_greaterThanAtLeast % 0.31/0.62 thf(fact_5929_greaterThan__non__empty,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( no_top @ A ) % 0.31/0.62 => ! [X3: A] : % 0.31/0.62 ( ( set_ord_greaterThan @ A @ X3 ) % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % greaterThan_non_empty % 0.31/0.62 thf(fact_5930_infinite__Ioi,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( linorder @ A ) % 0.31/0.62 & ( no_top @ A ) ) % 0.31/0.62 => ! [A2: A] : % 0.31/0.62 ~ ( finite_finite2 @ A @ ( set_ord_greaterThan @ A @ A2 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % infinite_Ioi % 0.31/0.62 thf(fact_5931_greaterThan__def,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ord @ A ) % 0.31/0.62 => ( ( set_ord_greaterThan @ A ) % 0.31/0.62 = ( ^ [L2: A] : ( collect @ A @ ( ord_less @ A @ L2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % greaterThan_def % 0.31/0.62 thf(fact_5932_trivial__limit__at__right__real,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( dense_order @ A ) % 0.31/0.62 & ( no_top @ A ) % 0.31/0.62 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.62 => ! [X3: A] : % 0.31/0.62 ( ( topolo174197925503356063within @ A @ X3 @ ( set_ord_greaterThan @ A @ X3 ) ) % 0.31/0.62 != ( bot_bot @ ( filter @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % trivial_limit_at_right_real % 0.31/0.62 thf(fact_5933_INT__greaterThan__UNIV,axiom, % 0.31/0.62 ( ( complete_Inf_Inf @ ( set @ nat ) @ ( image @ nat @ ( set @ nat ) @ ( set_ord_greaterThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) ) % 0.31/0.62 = ( bot_bot @ ( set @ nat ) ) ) ). % 0.31/0.62 % 0.31/0.62 % INT_greaterThan_UNIV % 0.31/0.62 thf(fact_5934_eventually__at__right,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 => ! [X3: A,Y: A,P: A > $o] : % 0.31/0.62 ( ( ord_less @ A @ X3 @ Y ) % 0.31/0.62 => ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X3 @ ( set_ord_greaterThan @ A @ X3 ) ) ) % 0.31/0.62 = ( ? [B4: A] : % 0.31/0.62 ( ( ord_less @ A @ X3 @ B4 ) % 0.31/0.62 & ! [Y2: A] : % 0.31/0.62 ( ( ord_less @ A @ X3 @ Y2 ) % 0.31/0.62 => ( ( ord_less @ A @ Y2 @ B4 ) % 0.31/0.62 => ( P @ Y2 ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_at_right % 0.31/0.62 thf(fact_5935_eventually__at__right__field,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( linordered_field @ A ) % 0.31/0.62 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.62 => ! [P: A > $o,X3: A] : % 0.31/0.62 ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X3 @ ( set_ord_greaterThan @ A @ X3 ) ) ) % 0.31/0.62 = ( ? [B4: A] : % 0.31/0.62 ( ( ord_less @ A @ X3 @ B4 ) % 0.31/0.62 & ! [Y2: A] : % 0.31/0.62 ( ( ord_less @ A @ X3 @ Y2 ) % 0.31/0.62 => ( ( ord_less @ A @ Y2 @ B4 ) % 0.31/0.62 => ( P @ Y2 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_at_right_field % 0.31/0.62 thf(fact_5936_at__within__Icc__at__right,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo2564578578187576103pology @ A ) % 0.31/0.62 => ! [A2: A,B2: A] : % 0.31/0.62 ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.62 => ( ( topolo174197925503356063within @ A @ A2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.62 = ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % at_within_Icc_at_right % 0.31/0.62 thf(fact_5937_ivl__disj__int__one_I7_J,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [L: A,U: A] : % 0.31/0.62 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % ivl_disj_int_one(7) % 0.31/0.62 thf(fact_5938_trivial__limit__at__right__top,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( order_top @ A ) % 0.31/0.62 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.62 => ( ( topolo174197925503356063within @ A @ ( top_top @ A ) @ ( set_ord_greaterThan @ A @ ( top_top @ A ) ) ) % 0.31/0.62 = ( bot_bot @ ( filter @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % trivial_limit_at_right_top % 0.31/0.62 thf(fact_5939_ivl__disj__int__one_I5_J,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [L: A,U: A] : % 0.31/0.62 ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % ivl_disj_int_one(5) % 0.31/0.62 thf(fact_5940_eventually__at__right__less,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( no_top @ A ) % 0.31/0.62 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.62 => ! [X3: A] : ( eventually @ A @ ( ord_less @ A @ X3 ) @ ( topolo174197925503356063within @ A @ X3 @ ( set_ord_greaterThan @ A @ X3 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_at_right_less % 0.31/0.62 thf(fact_5941_less__separate,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [X3: A,Y: A] : % 0.31/0.62 ( ( ord_less @ A @ X3 @ Y ) % 0.31/0.62 => ? [A5: A,B5: A] : % 0.31/0.62 ( ( member @ A @ X3 @ ( set_ord_lessThan @ A @ A5 ) ) % 0.31/0.62 & ( member @ A @ Y @ ( set_ord_greaterThan @ A @ B5 ) ) % 0.31/0.62 & ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A5 ) @ ( set_ord_greaterThan @ A @ B5 ) ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % less_separate % 0.31/0.62 thf(fact_5942_eventually__at__rightI,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo2564578578187576103pology @ A ) % 0.31/0.62 => ! [A2: A,B2: A,P: A > $o] : % 0.31/0.62 ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) % 0.31/0.62 => ( P @ X5 ) ) % 0.31/0.62 => ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.62 => ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_at_rightI % 0.31/0.62 thf(fact_5943_greaterThan__0,axiom, % 0.31/0.62 ( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) ) % 0.31/0.62 = ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % greaterThan_0 % 0.31/0.62 thf(fact_5944_eventually__at__right__real,axiom, % 0.31/0.62 ! [A2: real,B2: real] : % 0.31/0.62 ( ( ord_less @ real @ A2 @ B2 ) % 0.31/0.62 => ( eventually @ real % 0.31/0.62 @ ^ [X: real] : ( member @ real @ X @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) ) % 0.31/0.62 @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_at_right_real % 0.31/0.62 thf(fact_5945_greaterThan__Suc,axiom, % 0.31/0.62 ! [K: nat] : % 0.31/0.62 ( ( set_ord_greaterThan @ nat @ ( suc @ K ) ) % 0.31/0.62 = ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K ) @ ( insert @ nat @ ( suc @ K ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % greaterThan_Suc % 0.31/0.62 thf(fact_5946_filterlim__times__pos,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( ( linordered_field @ A ) % 0.31/0.62 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.62 => ! [F2: B > A,P4: A,F14: filter @ B,C2: A,L: A] : % 0.31/0.62 ( ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ P4 @ ( set_ord_greaterThan @ A @ P4 ) ) @ F14 ) % 0.31/0.62 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.31/0.62 => ( ( L % 0.31/0.62 = ( times_times @ A @ C2 @ P4 ) ) % 0.31/0.62 => ( filterlim @ B @ A % 0.31/0.62 @ ^ [X: B] : ( times_times @ A @ C2 @ ( F2 @ X ) ) % 0.31/0.62 @ ( topolo174197925503356063within @ A @ L @ ( set_ord_greaterThan @ A @ L ) ) % 0.31/0.62 @ F14 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_times_pos % 0.31/0.62 thf(fact_5947_tendsto__imp__filterlim__at__right,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( topolo2564578578187576103pology @ B ) % 0.31/0.62 => ! [F2: A > B,L5: B,F5: filter @ A] : % 0.31/0.62 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ F5 ) % 0.31/0.62 => ( ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( ord_less @ B @ L5 @ ( F2 @ X ) ) % 0.31/0.62 @ F5 ) % 0.31/0.62 => ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L5 @ ( set_ord_greaterThan @ B @ L5 ) ) @ F5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_imp_filterlim_at_right % 0.31/0.62 thf(fact_5948_continuous__on__Icc__at__rightD,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.62 => ! [A2: A,B2: A,F2: A > B] : % 0.31/0.62 ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 ) % 0.31/0.62 => ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.62 => ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_on_Icc_at_rightD % 0.31/0.62 thf(fact_5949_filterlim__at__bot__at__right,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 & ( linorder @ B ) ) % 0.31/0.62 => ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A,A2: A] : % 0.31/0.62 ( ! [X5: A,Y4: A] : % 0.31/0.62 ( ( Q @ X5 ) % 0.31/0.62 => ( ( Q @ Y4 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ X5 @ Y4 ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) ) ) % 0.31/0.62 => ( ! [X5: B] : % 0.31/0.62 ( ( P @ X5 ) % 0.31/0.62 => ( ( F2 @ ( G @ X5 ) ) % 0.31/0.62 = X5 ) ) % 0.31/0.62 => ( ! [X5: B] : % 0.31/0.62 ( ( P @ X5 ) % 0.31/0.62 => ( Q @ ( G @ X5 ) ) ) % 0.31/0.62 => ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) % 0.31/0.62 => ( ! [B5: A] : % 0.31/0.62 ( ( Q @ B5 ) % 0.31/0.62 => ( ord_less @ A @ A2 @ B5 ) ) % 0.31/0.62 => ( ( eventually @ B @ P @ ( at_bot @ B ) ) % 0.31/0.62 => ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_at_bot_at_right % 0.31/0.62 thf(fact_5950_nhds__metric,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V7819770556892013058_space @ A ) % 0.31/0.62 => ( ( topolo7230453075368039082e_nhds @ A ) % 0.31/0.62 = ( ^ [X: A] : % 0.31/0.62 ( complete_Inf_Inf @ ( filter @ A ) % 0.31/0.62 @ ( image @ real @ ( filter @ A ) % 0.31/0.62 @ ^ [E4: real] : % 0.31/0.62 ( principal @ A % 0.31/0.62 @ ( collect @ A % 0.31/0.62 @ ^ [Y2: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y2 @ X ) @ E4 ) ) ) % 0.31/0.62 @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % nhds_metric % 0.31/0.62 thf(fact_5951_at__right__eq,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 => ! [X3: A,Y: A] : % 0.31/0.62 ( ( ord_less @ A @ X3 @ Y ) % 0.31/0.62 => ( ( topolo174197925503356063within @ A @ X3 @ ( set_ord_greaterThan @ A @ X3 ) ) % 0.31/0.62 = ( complete_Inf_Inf @ ( filter @ A ) % 0.31/0.62 @ ( image @ A @ ( filter @ A ) % 0.31/0.62 @ ^ [A4: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ X3 @ A4 ) ) % 0.31/0.62 @ ( set_ord_greaterThan @ A @ X3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % at_right_eq % 0.31/0.62 thf(fact_5952_isCont__If__ge,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 & ( topolo4958980785337419405_space @ B ) ) % 0.31/0.62 => ! [A2: A,G: A > B,F2: A > B] : % 0.31/0.62 ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) @ G ) % 0.31/0.62 => ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( G @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) % 0.31/0.62 => ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.62 @ ^ [X: A] : ( if @ B @ ( ord_less_eq @ A @ X @ A2 ) @ ( G @ X ) @ ( F2 @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % isCont_If_ge % 0.31/0.62 thf(fact_5953_interval__cases,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit6923001295902523014norder @ A ) % 0.31/0.62 => ! [S3: set @ A] : % 0.31/0.62 ( ! [A5: A,B5: A,X5: A] : % 0.31/0.62 ( ( member @ A @ A5 @ S3 ) % 0.31/0.62 => ( ( member @ A @ B5 @ S3 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ A5 @ X5 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ X5 @ B5 ) % 0.31/0.62 => ( member @ A @ X5 @ S3 ) ) ) ) ) % 0.31/0.62 => ? [A5: A,B5: A] : % 0.31/0.62 ( ( S3 % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 | ( S3 % 0.31/0.62 = ( top_top @ ( set @ A ) ) ) % 0.31/0.62 | ( S3 % 0.31/0.62 = ( set_ord_lessThan @ A @ B5 ) ) % 0.31/0.62 | ( S3 % 0.31/0.62 = ( set_ord_atMost @ A @ B5 ) ) % 0.31/0.62 | ( S3 % 0.31/0.62 = ( set_ord_greaterThan @ A @ A5 ) ) % 0.31/0.62 | ( S3 % 0.31/0.62 = ( set_ord_atLeast @ A @ A5 ) ) % 0.31/0.62 | ( S3 % 0.31/0.62 = ( set_or5935395276787703475ssThan @ A @ A5 @ B5 ) ) % 0.31/0.62 | ( S3 % 0.31/0.62 = ( set_or3652927894154168847AtMost @ A @ A5 @ B5 ) ) % 0.31/0.62 | ( S3 % 0.31/0.62 = ( set_or7035219750837199246ssThan @ A @ A5 @ B5 ) ) % 0.31/0.62 | ( S3 % 0.31/0.62 = ( set_or1337092689740270186AtMost @ A @ A5 @ B5 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % interval_cases % 0.31/0.62 thf(fact_5954_sequentially__imp__eventually__at__right,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( topolo3112930676232923870pology @ A ) % 0.31/0.62 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.62 => ! [A2: A,B2: A,P: A > $o] : % 0.31/0.62 ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.62 => ( ! [F4: nat > A] : % 0.31/0.62 ( ! [N6: nat] : ( ord_less @ A @ A2 @ ( F4 @ N6 ) ) % 0.31/0.62 => ( ! [N6: nat] : ( ord_less @ A @ ( F4 @ N6 ) @ B2 ) % 0.31/0.62 => ( ( order_antimono @ nat @ A @ F4 ) % 0.31/0.62 => ( ( filterlim @ nat @ A @ F4 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) % 0.31/0.62 => ( eventually @ nat % 0.31/0.62 @ ^ [N: nat] : ( P @ ( F4 @ N ) ) % 0.31/0.62 @ ( at_top @ nat ) ) ) ) ) ) % 0.31/0.62 => ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % sequentially_imp_eventually_at_right % 0.31/0.62 thf(fact_5955_tendsto__at__right__sequentially,axiom, % 0.31/0.62 ! [C: $tType,B: $tType] : % 0.31/0.62 ( ( ( topolo3112930676232923870pology @ B ) % 0.31/0.62 & ( topolo1944317154257567458pology @ B ) % 0.31/0.62 & ( topolo4958980785337419405_space @ C ) ) % 0.31/0.62 => ! [A2: B,B2: B,X8: B > C,L5: C] : % 0.31/0.62 ( ( ord_less @ B @ A2 @ B2 ) % 0.31/0.62 => ( ! [S5: nat > B] : % 0.31/0.62 ( ! [N6: nat] : ( ord_less @ B @ A2 @ ( S5 @ N6 ) ) % 0.31/0.62 => ( ! [N6: nat] : ( ord_less @ B @ ( S5 @ N6 ) @ B2 ) % 0.31/0.62 => ( ( order_antimono @ nat @ B @ S5 ) % 0.31/0.62 => ( ( filterlim @ nat @ B @ S5 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ ( at_top @ nat ) ) % 0.31/0.62 => ( filterlim @ nat @ C % 0.31/0.62 @ ^ [N: nat] : ( X8 @ ( S5 @ N ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ C @ L5 ) % 0.31/0.62 @ ( at_top @ nat ) ) ) ) ) ) % 0.31/0.62 => ( filterlim @ B @ C @ X8 @ ( topolo7230453075368039082e_nhds @ C @ L5 ) @ ( topolo174197925503356063within @ B @ A2 @ ( set_ord_greaterThan @ B @ A2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_at_right_sequentially % 0.31/0.62 thf(fact_5956_atLeast__0,axiom, % 0.31/0.62 ( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) ) % 0.31/0.62 = ( top_top @ ( set @ nat ) ) ) ). % 0.31/0.62 % 0.31/0.62 % atLeast_0 % 0.31/0.62 thf(fact_5957_atLeast__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ord @ A ) % 0.31/0.62 => ! [I: A,K: A] : % 0.31/0.62 ( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) ) % 0.31/0.62 = ( ord_less_eq @ A @ K @ I ) ) ) ). % 0.31/0.62 % 0.31/0.62 % atLeast_iff % 0.31/0.62 thf(fact_5958_atLeast__empty__triv,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( set_ord_atLeast @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 = ( top_top @ ( set @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % atLeast_empty_triv % 0.31/0.62 thf(fact_5959_atLeast__subset__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [X3: A,Y: A] : % 0.31/0.62 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X3 ) @ ( set_ord_atLeast @ A @ Y ) ) % 0.31/0.62 = ( ord_less_eq @ A @ Y @ X3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % atLeast_subset_iff % 0.31/0.62 thf(fact_5960_image__add__atLeast,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linordered_semidom @ A ) % 0.31/0.62 => ! [K: A,I: A] : % 0.31/0.62 ( ( image @ A @ A @ ( plus_plus @ A @ K ) @ ( set_ord_atLeast @ A @ I ) ) % 0.31/0.62 = ( set_ord_atLeast @ A @ ( plus_plus @ A @ K @ I ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % image_add_atLeast % 0.31/0.62 thf(fact_5961_Icc__subset__Ici__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [L: A,H2: A,L3: A] : % 0.31/0.62 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atLeast @ A @ L3 ) ) % 0.31/0.62 = ( ~ ( ord_less_eq @ A @ L @ H2 ) % 0.31/0.62 | ( ord_less_eq @ A @ L3 @ L ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Icc_subset_Ici_iff % 0.31/0.62 thf(fact_5962_atLeast__Suc__greaterThan,axiom, % 0.31/0.62 ! [K: nat] : % 0.31/0.62 ( ( set_ord_atLeast @ nat @ ( suc @ K ) ) % 0.31/0.62 = ( set_ord_greaterThan @ nat @ K ) ) ). % 0.31/0.62 % 0.31/0.62 % atLeast_Suc_greaterThan % 0.31/0.62 thf(fact_5963_atLeast__eq__UNIV__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order_bot @ A ) % 0.31/0.62 => ! [X3: A] : % 0.31/0.62 ( ( ( set_ord_atLeast @ A @ X3 ) % 0.31/0.62 = ( top_top @ ( set @ A ) ) ) % 0.31/0.62 = ( X3 % 0.31/0.62 = ( bot_bot @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % atLeast_eq_UNIV_iff % 0.31/0.62 thf(fact_5964_infinite__Ici,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( linorder @ A ) % 0.31/0.62 & ( no_top @ A ) ) % 0.31/0.62 => ! [A2: A] : % 0.31/0.62 ~ ( finite_finite2 @ A @ ( set_ord_atLeast @ A @ A2 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % infinite_Ici % 0.31/0.62 thf(fact_5965_atLeast__def,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ord @ A ) % 0.31/0.62 => ( ( set_ord_atLeast @ A ) % 0.31/0.62 = ( ^ [L2: A] : ( collect @ A @ ( ord_less_eq @ A @ L2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % atLeast_def % 0.31/0.62 thf(fact_5966_antimonoD,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( order @ A ) % 0.31/0.62 & ( order @ B ) ) % 0.31/0.62 => ! [F2: A > B,X3: A,Y: A] : % 0.31/0.62 ( ( order_antimono @ A @ B @ F2 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % antimonoD % 0.31/0.62 thf(fact_5967_antimonoE,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( order @ A ) % 0.31/0.62 & ( order @ B ) ) % 0.31/0.62 => ! [F2: A > B,X3: A,Y: A] : % 0.31/0.62 ( ( order_antimono @ A @ B @ F2 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % antimonoE % 0.31/0.62 thf(fact_5968_antimonoI,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( order @ A ) % 0.31/0.62 & ( order @ B ) ) % 0.31/0.62 => ! [F2: A > B] : % 0.31/0.62 ( ! [X5: A,Y4: A] : % 0.31/0.62 ( ( ord_less_eq @ A @ X5 @ Y4 ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F2 @ Y4 ) @ ( F2 @ X5 ) ) ) % 0.31/0.62 => ( order_antimono @ A @ B @ F2 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % antimonoI % 0.31/0.62 thf(fact_5969_antimono__def,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( order @ A ) % 0.31/0.62 & ( order @ B ) ) % 0.31/0.62 => ( ( order_antimono @ A @ B ) % 0.31/0.62 = ( ^ [F3: A > B] : % 0.31/0.62 ! [X: A,Y2: A] : % 0.31/0.62 ( ( ord_less_eq @ A @ X @ Y2 ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % antimono_def % 0.31/0.62 thf(fact_5970_not__empty__eq__Ici__eq__empty,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [L: A] : % 0.31/0.62 ( ( bot_bot @ ( set @ A ) ) % 0.31/0.62 != ( set_ord_atLeast @ A @ L ) ) ) ). % 0.31/0.62 % 0.31/0.62 % not_empty_eq_Ici_eq_empty % 0.31/0.62 thf(fact_5971_not__UNIV__le__Ici,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( no_bot @ A ) % 0.31/0.62 => ! [L: A] : % 0.31/0.62 ~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atLeast @ A @ L ) ) ) ). % 0.31/0.62 % 0.31/0.62 % not_UNIV_le_Ici % 0.31/0.62 thf(fact_5972_not__Ici__le__Icc,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( no_top @ A ) % 0.31/0.62 => ! [L: A,L3: A,H3: A] : % 0.31/0.62 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % not_Ici_le_Icc % 0.31/0.62 thf(fact_5973_not__Iic__le__Ici,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( no_bot @ A ) % 0.31/0.62 => ! [H2: A,L3: A] : % 0.31/0.62 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_ord_atLeast @ A @ L3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % not_Iic_le_Ici % 0.31/0.62 thf(fact_5974_not__Ici__le__Iic,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( no_top @ A ) % 0.31/0.62 => ! [L: A,H3: A] : % 0.31/0.62 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % not_Ici_le_Iic % 0.31/0.62 thf(fact_5975_Ioi__le__Ico,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [A2: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A2 ) @ ( set_ord_atLeast @ A @ A2 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Ioi_le_Ico % 0.31/0.62 thf(fact_5976_decseq__Suc__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ( ( order_antimono @ nat @ A ) % 0.31/0.62 = ( ^ [F3: nat > A] : % 0.31/0.62 ! [N: nat] : ( ord_less_eq @ A @ ( F3 @ ( suc @ N ) ) @ ( F3 @ N ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % decseq_Suc_iff % 0.31/0.62 thf(fact_5977_decseq__SucI,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [X8: nat > A] : % 0.31/0.62 ( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) ) % 0.31/0.62 => ( order_antimono @ nat @ A @ X8 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % decseq_SucI % 0.31/0.62 thf(fact_5978_decseq__SucD,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [A3: nat > A,I: nat] : % 0.31/0.62 ( ( order_antimono @ nat @ A @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( A3 @ ( suc @ I ) ) @ ( A3 @ I ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % decseq_SucD % 0.31/0.62 thf(fact_5979_decseqD,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [F2: nat > A,I: nat,J: nat] : % 0.31/0.62 ( ( order_antimono @ nat @ A @ F2 ) % 0.31/0.62 => ( ( ord_less_eq @ nat @ I @ J ) % 0.31/0.62 => ( ord_less_eq @ A @ ( F2 @ J ) @ ( F2 @ I ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % decseqD % 0.31/0.62 thf(fact_5980_decseq__def,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ( ( order_antimono @ nat @ A ) % 0.31/0.62 = ( ^ [X6: nat > A] : % 0.31/0.62 ! [M2: nat,N: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M2 @ N ) % 0.31/0.62 => ( ord_less_eq @ A @ ( X6 @ N ) @ ( X6 @ M2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % decseq_def % 0.31/0.62 thf(fact_5981_Ici__subset__Ioi__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A2: A,B2: A] : % 0.31/0.62 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A2 ) @ ( set_ord_greaterThan @ A @ B2 ) ) % 0.31/0.62 = ( ord_less @ A @ B2 @ A2 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Ici_subset_Ioi_iff % 0.31/0.62 thf(fact_5982_ivl__disj__int__one_I8_J,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [L: A,U: A] : % 0.31/0.62 ( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % ivl_disj_int_one(8) % 0.31/0.62 thf(fact_5983_ivl__disj__int__one_I6_J,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [L: A,U: A] : % 0.31/0.62 ( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % ivl_disj_int_one(6) % 0.31/0.62 thf(fact_5984_atMost__Int__atLeast,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [N2: A] : % 0.31/0.62 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ N2 ) @ ( set_ord_atLeast @ A @ N2 ) ) % 0.31/0.62 = ( insert @ A @ N2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % atMost_Int_atLeast % 0.31/0.62 thf(fact_5985_decseq__ge,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 => ! [X8: nat > A,L5: A,N2: nat] : % 0.31/0.62 ( ( order_antimono @ nat @ A @ X8 ) % 0.31/0.62 => ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) % 0.31/0.62 => ( ord_less_eq @ A @ L5 @ ( X8 @ N2 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % decseq_ge % 0.31/0.62 thf(fact_5986_atLeast__Suc,axiom, % 0.31/0.62 ! [K: nat] : % 0.31/0.62 ( ( set_ord_atLeast @ nat @ ( suc @ K ) ) % 0.31/0.62 = ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K ) @ ( insert @ nat @ K @ ( bot_bot @ ( set @ nat ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % atLeast_Suc % 0.31/0.62 thf(fact_5987_Gcd__eq__Max,axiom, % 0.31/0.62 ! [M7: set @ nat] : % 0.31/0.62 ( ( finite_finite2 @ nat @ M7 ) % 0.31/0.62 => ( ( M7 % 0.31/0.62 != ( bot_bot @ ( set @ nat ) ) ) % 0.31/0.62 => ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 ) % 0.31/0.62 => ( ( gcd_Gcd @ nat @ M7 ) % 0.31/0.62 = ( lattic643756798349783984er_Max @ nat % 0.31/0.62 @ ( complete_Inf_Inf @ ( set @ nat ) % 0.31/0.62 @ ( image @ nat @ ( set @ nat ) % 0.31/0.62 @ ^ [M2: nat] : % 0.31/0.62 ( collect @ nat % 0.31/0.62 @ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M2 ) ) % 0.31/0.62 @ M7 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Gcd_eq_Max % 0.31/0.62 thf(fact_5988_continuous__at__Sup__antimono,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( condit6923001295902523014norder @ A ) % 0.31/0.62 & ( topolo1944317154257567458pology @ A ) % 0.31/0.62 & ( condit6923001295902523014norder @ B ) % 0.31/0.62 & ( topolo1944317154257567458pology @ B ) ) % 0.31/0.62 => ! [F2: A > B,S3: set @ A] : % 0.31/0.62 ( ( order_antimono @ A @ B @ F2 ) % 0.31/0.62 => ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Sup_Sup @ A @ S3 ) @ ( set_ord_lessThan @ A @ ( complete_Sup_Sup @ A @ S3 ) ) ) @ F2 ) % 0.31/0.62 => ( ( S3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ S3 ) % 0.31/0.62 => ( ( F2 @ ( complete_Sup_Sup @ A @ S3 ) ) % 0.31/0.62 = ( complete_Inf_Inf @ B @ ( image @ A @ B @ F2 @ S3 ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_at_Sup_antimono % 0.31/0.62 thf(fact_5989_bdd__above_OI,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [A3: set @ A,M7: A] : % 0.31/0.62 ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ X5 @ M7 ) ) % 0.31/0.62 => ( condit941137186595557371_above @ A @ A3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_above.I % 0.31/0.62 thf(fact_5990_bdd__above__empty,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ( condit941137186595557371_above @ A @ ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_above_empty % 0.31/0.62 thf(fact_5991_Max__singleton,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [X3: A] : % 0.31/0.62 ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.62 = X3 ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_singleton % 0.31/0.62 thf(fact_5992_Max__divisors__self__nat,axiom, % 0.31/0.62 ! [N2: nat] : % 0.31/0.62 ( ( N2 % 0.31/0.62 != ( zero_zero @ nat ) ) % 0.31/0.62 => ( ( lattic643756798349783984er_Max @ nat % 0.31/0.62 @ ( collect @ nat % 0.31/0.62 @ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ N2 ) ) ) % 0.31/0.62 = N2 ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_divisors_self_nat % 0.31/0.62 thf(fact_5993_Max_Obounded__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,X3: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X3 ) % 0.31/0.62 = ( ! [X: A] : % 0.31/0.62 ( ( member @ A @ X @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max.bounded_iff % 0.31/0.62 thf(fact_5994_Max__less__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,X3: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X3 ) % 0.31/0.62 = ( ! [X: A] : % 0.31/0.62 ( ( member @ A @ X @ A3 ) % 0.31/0.62 => ( ord_less @ A @ X @ X3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_less_iff % 0.31/0.62 thf(fact_5995_Max__const,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ B,C2: A] : % 0.31/0.62 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.62 => ( ( lattic643756798349783984er_Max @ A % 0.31/0.62 @ ( image @ B @ A % 0.31/0.62 @ ^ [Uu3: B] : C2 % 0.31/0.62 @ A3 ) ) % 0.31/0.62 = C2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_const % 0.31/0.62 thf(fact_5996_bdd__above__UN,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( lattice @ A ) % 0.31/0.62 => ! [I6: set @ B,A3: B > ( set @ A )] : % 0.31/0.62 ( ( finite_finite2 @ B @ I6 ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) ) % 0.31/0.62 = ( ! [X: B] : % 0.31/0.62 ( ( member @ B @ X @ I6 ) % 0.31/0.62 => ( condit941137186595557371_above @ A @ ( A3 @ X ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_above_UN % 0.31/0.62 thf(fact_5997_Max__insert,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,X3: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.62 = ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_insert % 0.31/0.62 thf(fact_5998_bdd__above_OI2,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [A3: set @ B,F2: B > A,M7: A] : % 0.31/0.62 ( ! [X5: B] : % 0.31/0.62 ( ( member @ B @ X5 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ M7 ) ) % 0.31/0.62 => ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_above.I2 % 0.31/0.62 thf(fact_5999_Max_Oin__idem,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,X3: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.62 => ( ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) % 0.31/0.62 = ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max.in_idem % 0.31/0.62 thf(fact_6000_Max__in,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( member @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ A3 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_in % 0.31/0.62 thf(fact_6001_bdd__above__mono,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [B3: set @ A,A3: set @ A] : % 0.31/0.62 ( ( condit941137186595557371_above @ A @ B3 ) % 0.31/0.62 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.62 => ( condit941137186595557371_above @ A @ A3 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_above_mono % 0.31/0.62 thf(fact_6002_bdd__above_OE,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [A3: set @ A] : % 0.31/0.62 ( ( condit941137186595557371_above @ A @ A3 ) % 0.31/0.62 => ~ ! [M8: A] : % 0.31/0.62 ~ ! [X2: A] : % 0.31/0.62 ( ( member @ A @ X2 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ X2 @ M8 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_above.E % 0.31/0.62 thf(fact_6003_bdd__above_Ounfold,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A ) % 0.31/0.62 = ( ^ [A6: set @ A] : % 0.31/0.62 ? [M9: A] : % 0.31/0.62 ! [X: A] : % 0.31/0.62 ( ( member @ A @ X @ A6 ) % 0.31/0.62 => ( ord_less_eq @ A @ X @ M9 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_above.unfold % 0.31/0.62 thf(fact_6004_cSup__upper,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [X3: A,X8: set @ A] : % 0.31/0.62 ( ( member @ A @ X3 @ X8 ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ X8 ) % 0.31/0.62 => ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cSup_upper % 0.31/0.62 thf(fact_6005_cSup__upper2,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [X3: A,X8: set @ A,Y: A] : % 0.31/0.62 ( ( member @ A @ X3 @ X8 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ Y @ X3 ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ X8 ) % 0.31/0.62 => ( ord_less_eq @ A @ Y @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cSup_upper2 % 0.31/0.62 thf(fact_6006_Max__ge,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,X3: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_ge % 0.31/0.62 thf(fact_6007_Max__eqI,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,X3: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ! [Y4: A] : % 0.31/0.62 ( ( member @ A @ Y4 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ Y4 @ X3 ) ) % 0.31/0.62 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.62 => ( ( lattic643756798349783984er_Max @ A @ A3 ) % 0.31/0.62 = X3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_eqI % 0.31/0.62 thf(fact_6008_Max__eq__if,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ A3 ) % 0.31/0.62 => ? [Xa: A] : % 0.31/0.62 ( ( member @ A @ Xa @ B3 ) % 0.31/0.62 & ( ord_less_eq @ A @ X5 @ Xa ) ) ) % 0.31/0.62 => ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ B3 ) % 0.31/0.62 => ? [Xa: A] : % 0.31/0.62 ( ( member @ A @ Xa @ A3 ) % 0.31/0.62 & ( ord_less_eq @ A @ X5 @ Xa ) ) ) % 0.31/0.62 => ( ( lattic643756798349783984er_Max @ A @ A3 ) % 0.31/0.62 = ( lattic643756798349783984er_Max @ A @ B3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_eq_if % 0.31/0.62 thf(fact_6009_Max_OcoboundedI,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,A2: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( member @ A @ A2 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ A2 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max.coboundedI % 0.31/0.62 thf(fact_6010_bdd__above__finite,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( lattice @ A ) % 0.31/0.62 => ! [A3: set @ A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( condit941137186595557371_above @ A @ A3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_above_finite % 0.31/0.62 thf(fact_6011_bdd__above__nat,axiom, % 0.31/0.62 ( ( condit941137186595557371_above @ nat ) % 0.31/0.62 = ( finite_finite2 @ nat ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_above_nat % 0.31/0.62 thf(fact_6012_cSUP__upper2,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [F2: B > A,A3: set @ B,X3: B,U: A] : % 0.31/0.62 ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.62 => ( ( member @ B @ X3 @ A3 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ U @ ( F2 @ X3 ) ) % 0.31/0.62 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cSUP_upper2 % 0.31/0.62 thf(fact_6013_cSUP__upper,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [X3: B,A3: set @ B,F2: B > A] : % 0.31/0.62 ( ( member @ B @ X3 @ A3 ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.62 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cSUP_upper % 0.31/0.62 thf(fact_6014_cSup__mono,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [B3: set @ A,A3: set @ A] : % 0.31/0.62 ( ( B3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ A3 ) % 0.31/0.62 => ( ! [B5: A] : % 0.31/0.62 ( ( member @ A @ B5 @ B3 ) % 0.31/0.62 => ? [X2: A] : % 0.31/0.62 ( ( member @ A @ X2 @ A3 ) % 0.31/0.62 & ( ord_less_eq @ A @ B5 @ X2 ) ) ) % 0.31/0.62 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B3 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cSup_mono % 0.31/0.62 thf(fact_6015_cSup__le__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [S3: set @ A,A2: A] : % 0.31/0.62 ( ( S3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ S3 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S3 ) @ A2 ) % 0.31/0.62 = ( ! [X: A] : % 0.31/0.62 ( ( member @ A @ X @ S3 ) % 0.31/0.62 => ( ord_less_eq @ A @ X @ A2 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cSup_le_iff % 0.31/0.62 thf(fact_6016_less__cSup__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit6923001295902523014norder @ A ) % 0.31/0.62 => ! [X8: set @ A,Y: A] : % 0.31/0.62 ( ( X8 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ X8 ) % 0.31/0.62 => ( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X8 ) ) % 0.31/0.62 = ( ? [X: A] : % 0.31/0.62 ( ( member @ A @ X @ X8 ) % 0.31/0.62 & ( ord_less @ A @ Y @ X ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % less_cSup_iff % 0.31/0.62 thf(fact_6017_Max__eq__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,M: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( ( lattic643756798349783984er_Max @ A @ A3 ) % 0.31/0.62 = M ) % 0.31/0.62 = ( ( member @ A @ M @ A3 ) % 0.31/0.62 & ! [X: A] : % 0.31/0.62 ( ( member @ A @ X @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ X @ M ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_eq_iff % 0.31/0.62 thf(fact_6018_Max__ge__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,X3: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( ord_less_eq @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) % 0.31/0.62 = ( ? [X: A] : % 0.31/0.62 ( ( member @ A @ X @ A3 ) % 0.31/0.62 & ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_ge_iff % 0.31/0.62 thf(fact_6019_eq__Max__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,M: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( M % 0.31/0.62 = ( lattic643756798349783984er_Max @ A @ A3 ) ) % 0.31/0.62 = ( ( member @ A @ M @ A3 ) % 0.31/0.62 & ! [X: A] : % 0.31/0.62 ( ( member @ A @ X @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ X @ M ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eq_Max_iff % 0.31/0.62 thf(fact_6020_Max_OboundedE,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,X3: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X3 ) % 0.31/0.62 => ! [A11: A] : % 0.31/0.62 ( ( member @ A @ A11 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ A11 @ X3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max.boundedE % 0.31/0.62 thf(fact_6021_Max_OboundedI,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,X3: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ! [A5: A] : % 0.31/0.62 ( ( member @ A @ A5 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ A5 @ X3 ) ) % 0.31/0.62 => ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max.boundedI % 0.31/0.62 thf(fact_6022_Max__gr__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,X3: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( ord_less @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) % 0.31/0.62 = ( ? [X: A] : % 0.31/0.62 ( ( member @ A @ X @ A3 ) % 0.31/0.62 & ( ord_less @ A @ X3 @ X ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_gr_iff % 0.31/0.62 thf(fact_6023_Max__insert2,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,A2: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ! [B5: A] : % 0.31/0.62 ( ( member @ A @ B5 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ B5 @ A2 ) ) % 0.31/0.62 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ A2 @ A3 ) ) % 0.31/0.62 = A2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_insert2 % 0.31/0.62 thf(fact_6024_Max__Sup,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( comple5582772986160207858norder @ A ) % 0.31/0.62 => ! [A3: set @ A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( lattic643756798349783984er_Max @ A @ A3 ) % 0.31/0.62 = ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_Sup % 0.31/0.62 thf(fact_6025_cSup__eq__Max,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit6923001295902523014norder @ A ) % 0.31/0.62 => ! [X8: set @ A] : % 0.31/0.62 ( ( finite_finite2 @ A @ X8 ) % 0.31/0.62 => ( ( X8 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( complete_Sup_Sup @ A @ X8 ) % 0.31/0.62 = ( lattic643756798349783984er_Max @ A @ X8 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cSup_eq_Max % 0.31/0.62 thf(fact_6026_cSUP__lessD,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [F2: B > A,A3: set @ B,Y: A,I: B] : % 0.31/0.62 ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.62 => ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ Y ) % 0.31/0.62 => ( ( member @ B @ I @ A3 ) % 0.31/0.62 => ( ord_less @ A @ ( F2 @ I ) @ Y ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cSUP_lessD % 0.31/0.62 thf(fact_6027_Max_Oinfinite,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A] : % 0.31/0.62 ( ~ ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( lattic643756798349783984er_Max @ A @ A3 ) % 0.31/0.62 = ( the2 @ A @ ( none @ A ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max.infinite % 0.31/0.62 thf(fact_6028_cSUP__mono,axiom, % 0.31/0.62 ! [B: $tType,A: $tType,C: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [A3: set @ B,G: C > A,B3: set @ C,F2: B > A] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ ( image @ C @ A @ G @ B3 ) ) % 0.31/0.62 => ( ! [N3: B] : % 0.31/0.62 ( ( member @ B @ N3 @ A3 ) % 0.31/0.62 => ? [X2: C] : % 0.31/0.62 ( ( member @ C @ X2 @ B3 ) % 0.31/0.62 & ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ X2 ) ) ) ) % 0.31/0.62 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cSUP_mono % 0.31/0.62 thf(fact_6029_cSUP__le__iff,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [A3: set @ B,F2: B > A,U: A] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.62 => ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U ) % 0.31/0.62 = ( ! [X: B] : % 0.31/0.62 ( ( member @ B @ X @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( F2 @ X ) @ U ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cSUP_le_iff % 0.31/0.62 thf(fact_6030_cSup__subset__mono,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ B3 ) % 0.31/0.62 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cSup_subset_mono % 0.31/0.62 thf(fact_6031_Max__mono,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [M7: set @ A,N7: set @ A] : % 0.31/0.62 ( ( ord_less_eq @ ( set @ A ) @ M7 @ N7 ) % 0.31/0.62 => ( ( M7 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( finite_finite2 @ A @ N7 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M7 ) @ ( lattic643756798349783984er_Max @ A @ N7 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_mono % 0.31/0.62 thf(fact_6032_Max_Osubset__imp,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.62 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ ( lattic643756798349783984er_Max @ A @ B3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max.subset_imp % 0.31/0.62 thf(fact_6033_hom__Max__commute,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [H2: A > A,N7: set @ A] : % 0.31/0.62 ( ! [X5: A,Y4: A] : % 0.31/0.62 ( ( H2 @ ( ord_max @ A @ X5 @ Y4 ) ) % 0.31/0.62 = ( ord_max @ A @ ( H2 @ X5 ) @ ( H2 @ Y4 ) ) ) % 0.31/0.62 => ( ( finite_finite2 @ A @ N7 ) % 0.31/0.62 => ( ( N7 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( H2 @ ( lattic643756798349783984er_Max @ A @ N7 ) ) % 0.31/0.62 = ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ H2 @ N7 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % hom_Max_commute % 0.31/0.62 thf(fact_6034_Max_Osubset,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( B3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.62 => ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B3 ) @ ( lattic643756798349783984er_Max @ A @ A3 ) ) % 0.31/0.62 = ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max.subset % 0.31/0.62 thf(fact_6035_Max_Oinsert__not__elem,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,X3: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ~ ( member @ A @ X3 @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.62 = ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max.insert_not_elem % 0.31/0.62 thf(fact_6036_Max_Oclosed,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ! [X5: A,Y4: A] : ( member @ A @ ( ord_max @ A @ X5 @ Y4 ) @ ( insert @ A @ X5 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.62 => ( member @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ A3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max.closed % 0.31/0.62 thf(fact_6037_less__cSUP__iff,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( condit6923001295902523014norder @ A ) % 0.31/0.62 => ! [A3: set @ B,F2: B > A,A2: A] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.62 => ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) % 0.31/0.62 = ( ? [X: B] : % 0.31/0.62 ( ( member @ B @ X @ A3 ) % 0.31/0.62 & ( ord_less @ A @ A2 @ ( F2 @ X ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % less_cSUP_iff % 0.31/0.62 thf(fact_6038_card__le__Suc__Max,axiom, % 0.31/0.62 ! [S3: set @ nat] : % 0.31/0.62 ( ( finite_finite2 @ nat @ S3 ) % 0.31/0.62 => ( ord_less_eq @ nat @ ( finite_card @ nat @ S3 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S3 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % card_le_Suc_Max % 0.31/0.62 thf(fact_6039_Sup__nat__def,axiom, % 0.31/0.62 ( ( complete_Sup_Sup @ nat ) % 0.31/0.62 = ( ^ [X6: set @ nat] : % 0.31/0.62 ( if @ nat % 0.31/0.62 @ ( X6 % 0.31/0.62 = ( bot_bot @ ( set @ nat ) ) ) % 0.31/0.62 @ ( zero_zero @ nat ) % 0.31/0.62 @ ( lattic643756798349783984er_Max @ nat @ X6 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Sup_nat_def % 0.31/0.62 thf(fact_6040_divide__nat__def,axiom, % 0.31/0.62 ( ( divide_divide @ nat ) % 0.31/0.62 = ( ^ [M2: nat,N: nat] : % 0.31/0.62 ( if @ nat % 0.31/0.62 @ ( N % 0.31/0.62 = ( zero_zero @ nat ) ) % 0.31/0.62 @ ( zero_zero @ nat ) % 0.31/0.62 @ ( lattic643756798349783984er_Max @ nat % 0.31/0.62 @ ( collect @ nat % 0.31/0.62 @ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N ) @ M2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % divide_nat_def % 0.31/0.62 thf(fact_6041_Max__add__commute,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( linord4140545234300271783up_add @ A ) % 0.31/0.62 => ! [S3: set @ B,F2: B > A,K: A] : % 0.31/0.62 ( ( finite_finite2 @ B @ S3 ) % 0.31/0.62 => ( ( S3 % 0.31/0.62 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.62 => ( ( lattic643756798349783984er_Max @ A % 0.31/0.62 @ ( image @ B @ A % 0.31/0.62 @ ^ [X: B] : ( plus_plus @ A @ ( F2 @ X ) @ K ) % 0.31/0.62 @ S3 ) ) % 0.31/0.62 = ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image @ B @ A @ F2 @ S3 ) ) @ K ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_add_commute % 0.31/0.62 thf(fact_6042_gcd__is__Max__divisors__nat,axiom, % 0.31/0.62 ! [N2: nat,M: nat] : % 0.31/0.62 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.62 => ( ( gcd_gcd @ nat @ M @ N2 ) % 0.31/0.62 = ( lattic643756798349783984er_Max @ nat % 0.31/0.62 @ ( collect @ nat % 0.31/0.62 @ ^ [D5: nat] : % 0.31/0.62 ( ( dvd_dvd @ nat @ D5 @ M ) % 0.31/0.62 & ( dvd_dvd @ nat @ D5 @ N2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % gcd_is_Max_divisors_nat % 0.31/0.62 thf(fact_6043_cSUP__subset__mono,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [A3: set @ B,G: B > A,B3: set @ B,F2: B > A] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ G @ B3 ) ) % 0.31/0.62 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 ) % 0.31/0.62 => ( ! [X5: B] : % 0.31/0.62 ( ( member @ B @ X5 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 0.31/0.62 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cSUP_subset_mono % 0.31/0.62 thf(fact_6044_cSup__cInf,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [S3: set @ A] : % 0.31/0.62 ( ( S3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ S3 ) % 0.31/0.62 => ( ( complete_Sup_Sup @ A @ S3 ) % 0.31/0.62 = ( complete_Inf_Inf @ A % 0.31/0.62 @ ( collect @ A % 0.31/0.62 @ ^ [X: A] : % 0.31/0.62 ! [Y2: A] : % 0.31/0.62 ( ( member @ A @ Y2 @ S3 ) % 0.31/0.62 => ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cSup_cInf % 0.31/0.62 thf(fact_6045_Max_Oremove,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,X3: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.62 => ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( lattic643756798349783984er_Max @ A @ A3 ) % 0.31/0.62 = X3 ) ) % 0.31/0.62 & ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( lattic643756798349783984er_Max @ A @ A3 ) % 0.31/0.62 = ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max.remove % 0.31/0.62 thf(fact_6046_Max_Oinsert__remove,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ! [A3: set @ A,X3: A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.62 = X3 ) ) % 0.31/0.62 & ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.62 = ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max.insert_remove % 0.31/0.62 thf(fact_6047_cSUP__UNION,axiom, % 0.31/0.62 ! [B: $tType,D: $tType,C: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ B ) % 0.31/0.62 => ! [A3: set @ C,B3: C > ( set @ D ),F2: D > B] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ C ) ) ) % 0.31/0.62 => ( ! [X5: C] : % 0.31/0.62 ( ( member @ C @ X5 @ A3 ) % 0.31/0.62 => ( ( B3 @ X5 ) % 0.31/0.62 != ( bot_bot @ ( set @ D ) ) ) ) % 0.31/0.62 => ( ( condit941137186595557371_above @ B % 0.31/0.62 @ ( complete_Sup_Sup @ ( set @ B ) % 0.31/0.62 @ ( image @ C @ ( set @ B ) % 0.31/0.62 @ ^ [X: C] : ( image @ D @ B @ F2 @ ( B3 @ X ) ) % 0.31/0.62 @ A3 ) ) ) % 0.31/0.62 => ( ( complete_Sup_Sup @ B @ ( image @ D @ B @ F2 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ B3 @ A3 ) ) ) ) % 0.31/0.62 = ( complete_Sup_Sup @ B % 0.31/0.62 @ ( image @ C @ B % 0.31/0.62 @ ^ [X: C] : ( complete_Sup_Sup @ B @ ( image @ D @ B @ F2 @ ( B3 @ X ) ) ) % 0.31/0.62 @ A3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cSUP_UNION % 0.31/0.62 thf(fact_6048_sum__le__card__Max,axiom, % 0.31/0.62 ! [A: $tType,A3: set @ A,F2: A > nat] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( times_times @ nat @ ( finite_card @ A @ A3 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ A @ nat @ F2 @ A3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % sum_le_card_Max % 0.31/0.62 thf(fact_6049_continuous__at__Inf__antimono,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( condit6923001295902523014norder @ A ) % 0.31/0.62 & ( topolo1944317154257567458pology @ A ) % 0.31/0.62 & ( condit6923001295902523014norder @ B ) % 0.31/0.62 & ( topolo1944317154257567458pology @ B ) ) % 0.31/0.62 => ! [F2: A > B,S3: set @ A] : % 0.31/0.62 ( ( order_antimono @ A @ B @ F2 ) % 0.31/0.62 => ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Inf_Inf @ A @ S3 ) @ ( set_ord_greaterThan @ A @ ( complete_Inf_Inf @ A @ S3 ) ) ) @ F2 ) % 0.31/0.62 => ( ( S3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ S3 ) % 0.31/0.62 => ( ( F2 @ ( complete_Inf_Inf @ A @ S3 ) ) % 0.31/0.62 = ( complete_Sup_Sup @ B @ ( image @ A @ B @ F2 @ S3 ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_at_Inf_antimono % 0.31/0.62 thf(fact_6050_dual__Min,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( linorder @ A ) % 0.31/0.62 => ( ( lattices_Min @ A % 0.31/0.62 @ ^ [X: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X ) ) % 0.31/0.62 = ( lattic643756798349783984er_Max @ A ) ) ) ). % 0.31/0.62 % 0.31/0.62 % dual_Min % 0.31/0.62 thf(fact_6051_bdd__below_OI,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [A3: set @ A,M7: A] : % 0.31/0.62 ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ M7 @ X5 ) ) % 0.31/0.62 => ( condit1013018076250108175_below @ A @ A3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_below.I % 0.31/0.62 thf(fact_6052_bdd__belowI,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [A3: set @ A,M: A] : % 0.31/0.62 ( ! [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ M @ X5 ) ) % 0.31/0.62 => ( condit1013018076250108175_below @ A @ A3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_belowI % 0.31/0.62 thf(fact_6053_bdd__below__empty,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ( condit1013018076250108175_below @ A @ ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_below_empty % 0.31/0.62 thf(fact_6054_Max__divisors__self__int,axiom, % 0.31/0.62 ! [N2: int] : % 0.31/0.62 ( ( N2 % 0.31/0.62 != ( zero_zero @ int ) ) % 0.31/0.62 => ( ( lattic643756798349783984er_Max @ int % 0.31/0.62 @ ( collect @ int % 0.31/0.62 @ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ N2 ) ) ) % 0.31/0.62 = ( abs_abs @ int @ N2 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Max_divisors_self_int % 0.31/0.62 thf(fact_6055_bdd__below__UN,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( lattice @ A ) % 0.31/0.62 => ! [I6: set @ B,A3: B > ( set @ A )] : % 0.31/0.62 ( ( finite_finite2 @ B @ I6 ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) ) % 0.31/0.62 = ( ! [X: B] : % 0.31/0.62 ( ( member @ B @ X @ I6 ) % 0.31/0.62 => ( condit1013018076250108175_below @ A @ ( A3 @ X ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_below_UN % 0.31/0.62 thf(fact_6056_linorder_OMin_Ocong,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( lattices_Min @ A ) % 0.31/0.62 = ( lattices_Min @ A ) ) ). % 0.31/0.62 % 0.31/0.62 % linorder.Min.cong % 0.31/0.62 thf(fact_6057_bdd__below__finite,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( lattice @ A ) % 0.31/0.62 => ! [A3: set @ A] : % 0.31/0.62 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( condit1013018076250108175_below @ A @ A3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_below_finite % 0.31/0.62 thf(fact_6058_bdd__below_Ounfold,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A ) % 0.31/0.62 = ( ^ [A6: set @ A] : % 0.31/0.62 ? [M9: A] : % 0.31/0.62 ! [X: A] : % 0.31/0.62 ( ( member @ A @ X @ A6 ) % 0.31/0.62 => ( ord_less_eq @ A @ M9 @ X ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_below.unfold % 0.31/0.62 thf(fact_6059_bdd__below_OE,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [A3: set @ A] : % 0.31/0.62 ( ( condit1013018076250108175_below @ A @ A3 ) % 0.31/0.62 => ~ ! [M8: A] : % 0.31/0.62 ~ ! [X2: A] : % 0.31/0.62 ( ( member @ A @ X2 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ M8 @ X2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_below.E % 0.31/0.62 thf(fact_6060_bdd__below__mono,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [B3: set @ A,A3: set @ A] : % 0.31/0.62 ( ( condit1013018076250108175_below @ A @ B3 ) % 0.31/0.62 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.62 => ( condit1013018076250108175_below @ A @ A3 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_below_mono % 0.31/0.62 thf(fact_6061_bdd__below_OI2,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [A3: set @ B,M7: A,F2: B > A] : % 0.31/0.62 ( ! [X5: B] : % 0.31/0.62 ( ( member @ B @ X5 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ M7 @ ( F2 @ X5 ) ) ) % 0.31/0.62 => ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_below.I2 % 0.31/0.62 thf(fact_6062_bdd__belowI2,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( preorder @ A ) % 0.31/0.62 => ! [A3: set @ B,M: A,F2: B > A] : % 0.31/0.62 ( ! [X5: B] : % 0.31/0.62 ( ( member @ B @ X5 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ M @ ( F2 @ X5 ) ) ) % 0.31/0.62 => ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % bdd_belowI2 % 0.31/0.62 thf(fact_6063_cInf__lower,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [X3: A,X8: set @ A] : % 0.31/0.62 ( ( member @ A @ X3 @ X8 ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ X8 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X3 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cInf_lower % 0.31/0.62 thf(fact_6064_cInf__lower2,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [X3: A,X8: set @ A,Y: A] : % 0.31/0.62 ( ( member @ A @ X3 @ X8 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ X8 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cInf_lower2 % 0.31/0.62 thf(fact_6065_cINF__lower,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [F2: B > A,A3: set @ B,X3: B] : % 0.31/0.62 ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.62 => ( ( member @ B @ X3 @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( F2 @ X3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cINF_lower % 0.31/0.62 thf(fact_6066_cINF__lower2,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [F2: B > A,A3: set @ B,X3: B,U: A] : % 0.31/0.62 ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.62 => ( ( member @ B @ X3 @ A3 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ ( F2 @ X3 ) @ U ) % 0.31/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cINF_lower2 % 0.31/0.62 thf(fact_6067_le__cInf__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [S3: set @ A,A2: A] : % 0.31/0.62 ( ( S3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ S3 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ A2 @ ( complete_Inf_Inf @ A @ S3 ) ) % 0.31/0.62 = ( ! [X: A] : % 0.31/0.62 ( ( member @ A @ X @ S3 ) % 0.31/0.62 => ( ord_less_eq @ A @ A2 @ X ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % le_cInf_iff % 0.31/0.62 thf(fact_6068_cInf__mono,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [B3: set @ A,A3: set @ A] : % 0.31/0.62 ( ( B3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ A3 ) % 0.31/0.62 => ( ! [B5: A] : % 0.31/0.62 ( ( member @ A @ B5 @ B3 ) % 0.31/0.62 => ? [X2: A] : % 0.31/0.62 ( ( member @ A @ X2 @ A3 ) % 0.31/0.62 & ( ord_less_eq @ A @ X2 @ B5 ) ) ) % 0.31/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cInf_mono % 0.31/0.62 thf(fact_6069_cInf__less__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit6923001295902523014norder @ A ) % 0.31/0.62 => ! [X8: set @ A,Y: A] : % 0.31/0.62 ( ( X8 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ X8 ) % 0.31/0.62 => ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y ) % 0.31/0.62 = ( ? [X: A] : % 0.31/0.62 ( ( member @ A @ X @ X8 ) % 0.31/0.62 & ( ord_less @ A @ X @ Y ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cInf_less_iff % 0.31/0.62 thf(fact_6070_less__cINF__D,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [F2: B > A,A3: set @ B,Y: A,I: B] : % 0.31/0.62 ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.62 => ( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) % 0.31/0.62 => ( ( member @ B @ I @ A3 ) % 0.31/0.62 => ( ord_less @ A @ Y @ ( F2 @ I ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % less_cINF_D % 0.31/0.62 thf(fact_6071_gcd__is__Max__divisors__int,axiom, % 0.31/0.62 ! [N2: int,M: int] : % 0.31/0.62 ( ( N2 % 0.31/0.62 != ( zero_zero @ int ) ) % 0.31/0.62 => ( ( gcd_gcd @ int @ M @ N2 ) % 0.31/0.62 = ( lattic643756798349783984er_Max @ int % 0.31/0.62 @ ( collect @ int % 0.31/0.62 @ ^ [D5: int] : % 0.31/0.62 ( ( dvd_dvd @ int @ D5 @ M ) % 0.31/0.62 & ( dvd_dvd @ int @ D5 @ N2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % gcd_is_Max_divisors_int % 0.31/0.62 thf(fact_6072_le__cINF__iff,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [A3: set @ B,F2: B > A,U: A] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.62 => ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) % 0.31/0.62 = ( ! [X: B] : % 0.31/0.62 ( ( member @ B @ X @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ U @ ( F2 @ X ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % le_cINF_iff % 0.31/0.62 thf(fact_6073_cINF__mono,axiom, % 0.31/0.62 ! [C: $tType,A: $tType,B: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [B3: set @ B,F2: C > A,A3: set @ C,G: B > A] : % 0.31/0.62 ( ( B3 % 0.31/0.62 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ ( image @ C @ A @ F2 @ A3 ) ) % 0.31/0.62 => ( ! [M3: B] : % 0.31/0.62 ( ( member @ B @ M3 @ B3 ) % 0.31/0.62 => ? [X2: C] : % 0.31/0.62 ( ( member @ C @ X2 @ A3 ) % 0.31/0.62 & ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G @ M3 ) ) ) ) % 0.31/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cINF_mono % 0.31/0.62 thf(fact_6074_cInf__superset__mono,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ B3 ) % 0.31/0.62 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B3 ) @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cInf_superset_mono % 0.31/0.62 thf(fact_6075_cInf__insert,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [X8: set @ A,A2: A] : % 0.31/0.62 ( ( X8 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ X8 ) % 0.31/0.62 => ( ( complete_Inf_Inf @ A @ ( insert @ A @ A2 @ X8 ) ) % 0.31/0.62 = ( inf_inf @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cInf_insert % 0.31/0.62 thf(fact_6076_cInf__insert__If,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [X8: set @ A,A2: A] : % 0.31/0.62 ( ( condit1013018076250108175_below @ A @ X8 ) % 0.31/0.62 => ( ( ( X8 % 0.31/0.62 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( complete_Inf_Inf @ A @ ( insert @ A @ A2 @ X8 ) ) % 0.31/0.62 = A2 ) ) % 0.31/0.62 & ( ( X8 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( complete_Inf_Inf @ A @ ( insert @ A @ A2 @ X8 ) ) % 0.31/0.62 = ( inf_inf @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cInf_insert_If % 0.31/0.62 thf(fact_6077_cINF__less__iff,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( condit6923001295902523014norder @ A ) % 0.31/0.62 => ! [A3: set @ B,F2: B > A,A2: A] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.62 => ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ A2 ) % 0.31/0.62 = ( ? [X: B] : % 0.31/0.62 ( ( member @ B @ X @ A3 ) % 0.31/0.62 & ( ord_less @ A @ ( F2 @ X ) @ A2 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cINF_less_iff % 0.31/0.62 thf(fact_6078_cINF__inf__distrib,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [A3: set @ B,F2: B > A,G: B > A] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ G @ A3 ) ) % 0.31/0.62 => ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A3 ) ) ) % 0.31/0.62 = ( complete_Inf_Inf @ A % 0.31/0.62 @ ( image @ B @ A % 0.31/0.62 @ ^ [A4: B] : ( inf_inf @ A @ ( F2 @ A4 ) @ ( G @ A4 ) ) % 0.31/0.62 @ A3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cINF_inf_distrib % 0.31/0.62 thf(fact_6079_cINF__superset__mono,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [A3: set @ B,G: B > A,B3: set @ B,F2: B > A] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ G @ B3 ) ) % 0.31/0.62 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 ) % 0.31/0.62 => ( ! [X5: B] : % 0.31/0.62 ( ( member @ B @ X5 @ B3 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( G @ X5 ) @ ( F2 @ X5 ) ) ) % 0.31/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cINF_superset_mono % 0.31/0.62 thf(fact_6080_less__eq__cInf__inter,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.62 ( ( condit1013018076250108175_below @ A @ A3 ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ B3 ) % 0.31/0.62 => ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % less_eq_cInf_inter % 0.31/0.62 thf(fact_6081_cINF__insert,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [A3: set @ B,F2: B > A,A2: B] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.62 => ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( insert @ B @ A2 @ A3 ) ) ) % 0.31/0.62 = ( inf_inf @ A @ ( F2 @ A2 ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cINF_insert % 0.31/0.62 thf(fact_6082_cInf__le__cSup,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [A3: set @ A] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ A3 ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cInf_le_cSup % 0.31/0.62 thf(fact_6083_cInf__cSup,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 => ! [S3: set @ A] : % 0.31/0.62 ( ( S3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ S3 ) % 0.31/0.62 => ( ( complete_Inf_Inf @ A @ S3 ) % 0.31/0.62 = ( complete_Sup_Sup @ A % 0.31/0.62 @ ( collect @ A % 0.31/0.62 @ ^ [X: A] : % 0.31/0.62 ! [Y2: A] : % 0.31/0.62 ( ( member @ A @ Y2 @ S3 ) % 0.31/0.62 => ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cInf_cSup % 0.31/0.62 thf(fact_6084_cINF__UNION,axiom, % 0.31/0.62 ! [B: $tType,D: $tType,C: $tType] : % 0.31/0.62 ( ( condit1219197933456340205attice @ B ) % 0.31/0.62 => ! [A3: set @ C,B3: C > ( set @ D ),F2: D > B] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ C ) ) ) % 0.31/0.62 => ( ! [X5: C] : % 0.31/0.62 ( ( member @ C @ X5 @ A3 ) % 0.31/0.62 => ( ( B3 @ X5 ) % 0.31/0.62 != ( bot_bot @ ( set @ D ) ) ) ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ B % 0.31/0.62 @ ( complete_Sup_Sup @ ( set @ B ) % 0.31/0.62 @ ( image @ C @ ( set @ B ) % 0.31/0.62 @ ^ [X: C] : ( image @ D @ B @ F2 @ ( B3 @ X ) ) % 0.31/0.62 @ A3 ) ) ) % 0.31/0.62 => ( ( complete_Inf_Inf @ B @ ( image @ D @ B @ F2 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ B3 @ A3 ) ) ) ) % 0.31/0.62 = ( complete_Inf_Inf @ B % 0.31/0.62 @ ( image @ C @ B % 0.31/0.62 @ ^ [X: C] : ( complete_Inf_Inf @ B @ ( image @ D @ B @ F2 @ ( B3 @ X ) ) ) % 0.31/0.62 @ A3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cINF_UNION % 0.31/0.62 thf(fact_6085_uniformity__dist,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V768167426530841204y_dist @ A ) % 0.31/0.62 => ( ( topolo7806501430040627800ormity @ A ) % 0.31/0.62 = ( complete_Inf_Inf @ ( filter @ ( product_prod @ A @ A ) ) % 0.31/0.62 @ ( image @ real @ ( filter @ ( product_prod @ A @ A ) ) % 0.31/0.62 @ ^ [E4: real] : % 0.31/0.62 ( principal @ ( product_prod @ A @ A ) % 0.31/0.62 @ ( collect @ ( product_prod @ A @ A ) % 0.31/0.62 @ ( product_case_prod @ A @ A @ $o % 0.31/0.62 @ ^ [X: A,Y2: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ E4 ) ) ) ) % 0.31/0.62 @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % uniformity_dist % 0.31/0.62 thf(fact_6086_compactE__image,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [S3: set @ A,C5: set @ B,F2: B > ( set @ A )] : % 0.31/0.62 ( ( topolo2193935891317330818ompact @ A @ S3 ) % 0.31/0.62 => ( ! [T5: B] : % 0.31/0.62 ( ( member @ B @ T5 @ C5 ) % 0.31/0.62 => ( topolo1002775350975398744n_open @ A @ ( F2 @ T5 ) ) ) % 0.31/0.62 => ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ C5 ) ) ) % 0.31/0.62 => ~ ! [C9: set @ B] : % 0.31/0.62 ( ( ord_less_eq @ ( set @ B ) @ C9 @ C5 ) % 0.31/0.62 => ( ( finite_finite2 @ B @ C9 ) % 0.31/0.62 => ~ ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ C9 ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % compactE_image % 0.31/0.62 thf(fact_6087_uniformity__bot,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo7287701948861334536_space @ A ) % 0.31/0.62 => ( ( topolo7806501430040627800ormity @ A ) % 0.31/0.62 != ( bot_bot @ ( filter @ ( product_prod @ A @ A ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % uniformity_bot % 0.31/0.62 thf(fact_6088_compact__empty,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ( topolo2193935891317330818ompact @ A @ ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % compact_empty % 0.31/0.62 thf(fact_6089_uniformity__transE,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo7287701948861334536_space @ A ) % 0.31/0.62 => ! [E5: ( product_prod @ A @ A ) > $o] : % 0.31/0.62 ( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) ) % 0.31/0.62 => ~ ! [D7: ( product_prod @ A @ A ) > $o] : % 0.31/0.62 ( ( eventually @ ( product_prod @ A @ A ) @ D7 @ ( topolo7806501430040627800ormity @ A ) ) % 0.31/0.62 => ~ ! [X2: A,Y5: A] : % 0.31/0.62 ( ( D7 @ ( product_Pair @ A @ A @ X2 @ Y5 ) ) % 0.31/0.62 => ! [Z4: A] : % 0.31/0.62 ( ( D7 @ ( product_Pair @ A @ A @ Y5 @ Z4 ) ) % 0.31/0.62 => ( E5 @ ( product_Pair @ A @ A @ X2 @ Z4 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % uniformity_transE % 0.31/0.62 thf(fact_6090_uniformity__trans,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo7287701948861334536_space @ A ) % 0.31/0.62 => ! [E5: ( product_prod @ A @ A ) > $o] : % 0.31/0.62 ( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) ) % 0.31/0.62 => ? [D7: ( product_prod @ A @ A ) > $o] : % 0.31/0.62 ( ( eventually @ ( product_prod @ A @ A ) @ D7 @ ( topolo7806501430040627800ormity @ A ) ) % 0.31/0.62 & ! [X2: A,Y5: A,Z4: A] : % 0.31/0.62 ( ( D7 @ ( product_Pair @ A @ A @ X2 @ Y5 ) ) % 0.31/0.62 => ( ( D7 @ ( product_Pair @ A @ A @ Y5 @ Z4 ) ) % 0.31/0.62 => ( E5 @ ( product_Pair @ A @ A @ X2 @ Z4 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % uniformity_trans % 0.31/0.62 thf(fact_6091_uniformity__refl,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo7287701948861334536_space @ A ) % 0.31/0.62 => ! [E5: ( product_prod @ A @ A ) > $o,X3: A] : % 0.31/0.62 ( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) ) % 0.31/0.62 => ( E5 @ ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % uniformity_refl % 0.31/0.62 thf(fact_6092_compact__attains__inf,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 => ! [S3: set @ A] : % 0.31/0.62 ( ( topolo2193935891317330818ompact @ A @ S3 ) % 0.31/0.62 => ( ( S3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ? [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ S3 ) % 0.31/0.62 & ! [Xa: A] : % 0.31/0.62 ( ( member @ A @ Xa @ S3 ) % 0.31/0.62 => ( ord_less_eq @ A @ X5 @ Xa ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % compact_attains_inf % 0.31/0.62 thf(fact_6093_compact__attains__sup,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 => ! [S3: set @ A] : % 0.31/0.62 ( ( topolo2193935891317330818ompact @ A @ S3 ) % 0.31/0.62 => ( ( S3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ? [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ S3 ) % 0.31/0.62 & ! [Xa: A] : % 0.31/0.62 ( ( member @ A @ Xa @ S3 ) % 0.31/0.62 => ( ord_less_eq @ A @ Xa @ X5 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % compact_attains_sup % 0.31/0.62 thf(fact_6094_uniformity__sym,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo7287701948861334536_space @ A ) % 0.31/0.62 => ! [E5: ( product_prod @ A @ A ) > $o] : % 0.31/0.62 ( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) ) % 0.31/0.62 => ( eventually @ ( product_prod @ A @ A ) % 0.31/0.62 @ ( product_case_prod @ A @ A @ $o % 0.31/0.62 @ ^ [X: A,Y2: A] : ( E5 @ ( product_Pair @ A @ A @ Y2 @ X ) ) ) % 0.31/0.62 @ ( topolo7806501430040627800ormity @ A ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % uniformity_sym % 0.31/0.62 thf(fact_6095_continuous__attains__sup,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 & ( topolo1944317154257567458pology @ B ) ) % 0.31/0.62 => ! [S: set @ A,F2: A > B] : % 0.31/0.62 ( ( topolo2193935891317330818ompact @ A @ S ) % 0.31/0.62 => ( ( S % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 ) % 0.31/0.62 => ? [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ S ) % 0.31/0.62 & ! [Xa: A] : % 0.31/0.62 ( ( member @ A @ Xa @ S ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F2 @ Xa ) @ ( F2 @ X5 ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_attains_sup % 0.31/0.62 thf(fact_6096_continuous__attains__inf,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 & ( topolo1944317154257567458pology @ B ) ) % 0.31/0.62 => ! [S: set @ A,F2: A > B] : % 0.31/0.62 ( ( topolo2193935891317330818ompact @ A @ S ) % 0.31/0.62 => ( ( S % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 ) % 0.31/0.62 => ? [X5: A] : % 0.31/0.62 ( ( member @ A @ X5 @ S ) % 0.31/0.62 & ! [Xa: A] : % 0.31/0.62 ( ( member @ A @ Xa @ S ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Xa ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % continuous_attains_inf % 0.31/0.62 thf(fact_6097_Cauchy__uniform__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo7287701948861334536_space @ A ) % 0.31/0.62 => ( ( topolo3814608138187158403Cauchy @ A ) % 0.31/0.62 = ( ^ [X6: nat > A] : % 0.31/0.62 ! [P3: ( product_prod @ A @ A ) > $o] : % 0.31/0.62 ( ( eventually @ ( product_prod @ A @ A ) @ P3 @ ( topolo7806501430040627800ormity @ A ) ) % 0.31/0.62 => ? [N5: nat] : % 0.31/0.62 ! [N: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ N5 @ N ) % 0.31/0.62 => ! [M2: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ N5 @ M2 ) % 0.31/0.62 => ( P3 @ ( product_Pair @ A @ A @ ( X6 @ N ) @ ( X6 @ M2 ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Cauchy_uniform_iff % 0.31/0.62 thf(fact_6098_uniformity__complex__def,axiom, % 0.31/0.62 ( ( topolo7806501430040627800ormity @ complex ) % 0.31/0.62 = ( complete_Inf_Inf @ ( filter @ ( product_prod @ complex @ complex ) ) % 0.31/0.62 @ ( image @ real @ ( filter @ ( product_prod @ complex @ complex ) ) % 0.31/0.62 @ ^ [E4: real] : % 0.31/0.62 ( principal @ ( product_prod @ complex @ complex ) % 0.31/0.62 @ ( collect @ ( product_prod @ complex @ complex ) % 0.31/0.62 @ ( product_case_prod @ complex @ complex @ $o % 0.31/0.62 @ ^ [X: complex,Y2: complex] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ complex @ X @ Y2 ) @ E4 ) ) ) ) % 0.31/0.62 @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % uniformity_complex_def % 0.31/0.62 thf(fact_6099_uniformity__real__def,axiom, % 0.31/0.62 ( ( topolo7806501430040627800ormity @ real ) % 0.31/0.62 = ( complete_Inf_Inf @ ( filter @ ( product_prod @ real @ real ) ) % 0.31/0.62 @ ( image @ real @ ( filter @ ( product_prod @ real @ real ) ) % 0.31/0.62 @ ^ [E4: real] : % 0.31/0.62 ( principal @ ( product_prod @ real @ real ) % 0.31/0.62 @ ( collect @ ( product_prod @ real @ real ) % 0.31/0.62 @ ( product_case_prod @ real @ real @ $o % 0.31/0.62 @ ^ [X: real,Y2: real] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ real @ X @ Y2 ) @ E4 ) ) ) ) % 0.31/0.62 @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % uniformity_real_def % 0.31/0.62 thf(fact_6100_totally__bounded__def,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo7287701948861334536_space @ A ) % 0.31/0.62 => ( ( topolo6688025880775521714ounded @ A ) % 0.31/0.62 = ( ^ [S7: set @ A] : % 0.31/0.62 ! [E6: ( product_prod @ A @ A ) > $o] : % 0.31/0.62 ( ( eventually @ ( product_prod @ A @ A ) @ E6 @ ( topolo7806501430040627800ormity @ A ) ) % 0.31/0.62 => ? [X6: set @ A] : % 0.31/0.62 ( ( finite_finite2 @ A @ X6 ) % 0.31/0.62 & ! [X: A] : % 0.31/0.62 ( ( member @ A @ X @ S7 ) % 0.31/0.62 => ? [Y2: A] : % 0.31/0.62 ( ( member @ A @ Y2 @ X6 ) % 0.31/0.62 & ( E6 @ ( product_Pair @ A @ A @ Y2 @ X ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % totally_bounded_def % 0.31/0.62 thf(fact_6101_eventually__uniformity__metric,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( real_V768167426530841204y_dist @ A ) % 0.31/0.62 => ! [P: ( product_prod @ A @ A ) > $o] : % 0.31/0.62 ( ( eventually @ ( product_prod @ A @ A ) @ P @ ( topolo7806501430040627800ormity @ A ) ) % 0.31/0.62 = ( ? [E4: real] : % 0.31/0.62 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 ) % 0.31/0.62 & ! [X: A,Y2: A] : % 0.31/0.62 ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ E4 ) % 0.31/0.62 => ( P @ ( product_Pair @ A @ A @ X @ Y2 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_uniformity_metric % 0.31/0.62 thf(fact_6102_tendsto__iff__uniformity,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( topolo7287701948861334536_space @ B ) % 0.31/0.62 => ! [F2: A > B,L: B,F5: filter @ A] : % 0.31/0.62 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) % 0.31/0.62 = ( ! [E6: ( product_prod @ B @ B ) > $o] : % 0.31/0.62 ( ( eventually @ ( product_prod @ B @ B ) @ E6 @ ( topolo7806501430040627800ormity @ B ) ) % 0.31/0.62 => ( eventually @ A % 0.31/0.62 @ ^ [X: A] : ( E6 @ ( product_Pair @ B @ B @ ( F2 @ X ) @ L ) ) % 0.31/0.62 @ F5 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_iff_uniformity % 0.31/0.62 thf(fact_6103_compactE,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [S3: set @ A,T11: set @ ( set @ A )] : % 0.31/0.62 ( ( topolo2193935891317330818ompact @ A @ S3 ) % 0.31/0.62 => ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ T11 ) ) % 0.31/0.62 => ( ! [B10: set @ A] : % 0.31/0.62 ( ( member @ ( set @ A ) @ B10 @ T11 ) % 0.31/0.62 => ( topolo1002775350975398744n_open @ A @ B10 ) ) % 0.31/0.62 => ~ ! [T12: set @ ( set @ A )] : % 0.31/0.62 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ T12 @ T11 ) % 0.31/0.62 => ( ( finite_finite2 @ ( set @ A ) @ T12 ) % 0.31/0.62 => ~ ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ T12 ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % compactE % 0.31/0.62 thf(fact_6104_compactI,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ! [S: set @ A] : % 0.31/0.62 ( ! [C8: set @ ( set @ A )] : % 0.31/0.62 ( ! [X2: set @ A] : % 0.31/0.62 ( ( member @ ( set @ A ) @ X2 @ C8 ) % 0.31/0.62 => ( topolo1002775350975398744n_open @ A @ X2 ) ) % 0.31/0.62 => ( ( ord_less_eq @ ( set @ A ) @ S @ ( complete_Sup_Sup @ ( set @ A ) @ C8 ) ) % 0.31/0.62 => ? [C10: set @ ( set @ A )] : % 0.31/0.62 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C10 @ C8 ) % 0.31/0.62 & ( finite_finite2 @ ( set @ A ) @ C10 ) % 0.31/0.62 & ( ord_less_eq @ ( set @ A ) @ S @ ( complete_Sup_Sup @ ( set @ A ) @ C10 ) ) ) ) ) % 0.31/0.62 => ( topolo2193935891317330818ompact @ A @ S ) ) ) ). % 0.31/0.62 % 0.31/0.62 % compactI % 0.31/0.62 thf(fact_6105_compact__eq__Heine__Borel,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.62 => ( ( topolo2193935891317330818ompact @ A ) % 0.31/0.62 = ( ^ [S7: set @ A] : % 0.31/0.62 ! [C7: set @ ( set @ A )] : % 0.31/0.62 ( ( ! [X: set @ A] : % 0.31/0.62 ( ( member @ ( set @ A ) @ X @ C7 ) % 0.31/0.62 => ( topolo1002775350975398744n_open @ A @ X ) ) % 0.31/0.62 & ( ord_less_eq @ ( set @ A ) @ S7 @ ( complete_Sup_Sup @ ( set @ A ) @ C7 ) ) ) % 0.31/0.62 => ? [D8: set @ ( set @ A )] : % 0.31/0.62 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ D8 @ C7 ) % 0.31/0.62 & ( finite_finite2 @ ( set @ A ) @ D8 ) % 0.31/0.62 & ( ord_less_eq @ ( set @ A ) @ S7 @ ( complete_Sup_Sup @ ( set @ A ) @ D8 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % compact_eq_Heine_Borel % 0.31/0.62 thf(fact_6106_prod__filter__INF2,axiom, % 0.31/0.62 ! [B: $tType,C: $tType,A: $tType,J4: set @ A,A3: filter @ B,B3: A > ( filter @ C )] : % 0.31/0.62 ( ( J4 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( prod_filter @ B @ C @ A3 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image @ A @ ( filter @ C ) @ B3 @ J4 ) ) ) % 0.31/0.62 = ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) ) % 0.31/0.62 @ ( image @ A @ ( filter @ ( product_prod @ B @ C ) ) % 0.31/0.62 @ ^ [I3: A] : ( prod_filter @ B @ C @ A3 @ ( B3 @ I3 ) ) % 0.31/0.62 @ J4 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % prod_filter_INF2 % 0.31/0.62 thf(fact_6107_prod__filter__INF1,axiom, % 0.31/0.62 ! [B: $tType,C: $tType,A: $tType,I6: set @ A,A3: A > ( filter @ B ),B3: filter @ C] : % 0.31/0.62 ( ( I6 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( prod_filter @ B @ C @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ A3 @ I6 ) ) @ B3 ) % 0.31/0.62 = ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) ) % 0.31/0.62 @ ( image @ A @ ( filter @ ( product_prod @ B @ C ) ) % 0.31/0.62 @ ^ [I3: A] : ( prod_filter @ B @ C @ ( A3 @ I3 ) @ B3 ) % 0.31/0.62 @ I6 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % prod_filter_INF1 % 0.31/0.62 thf(fact_6108_prod__filter__eq__bot,axiom, % 0.31/0.62 ! [A: $tType,B: $tType,A3: filter @ A,B3: filter @ B] : % 0.31/0.62 ( ( ( prod_filter @ A @ B @ A3 @ B3 ) % 0.31/0.62 = ( bot_bot @ ( filter @ ( product_prod @ A @ B ) ) ) ) % 0.31/0.62 = ( ( A3 % 0.31/0.62 = ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.62 | ( B3 % 0.31/0.62 = ( bot_bot @ ( filter @ B ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % prod_filter_eq_bot % 0.31/0.62 thf(fact_6109_eventually__prod__same,axiom, % 0.31/0.62 ! [A: $tType,P: ( product_prod @ A @ A ) > $o,F5: filter @ A] : % 0.31/0.62 ( ( eventually @ ( product_prod @ A @ A ) @ P @ ( prod_filter @ A @ A @ F5 @ F5 ) ) % 0.31/0.62 = ( ? [Q7: A > $o] : % 0.31/0.62 ( ( eventually @ A @ Q7 @ F5 ) % 0.31/0.62 & ! [X: A,Y2: A] : % 0.31/0.62 ( ( Q7 @ X ) % 0.31/0.62 => ( ( Q7 @ Y2 ) % 0.31/0.62 => ( P @ ( product_Pair @ A @ A @ X @ Y2 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_prod_same % 0.31/0.62 thf(fact_6110_eventually__prod__filter,axiom, % 0.31/0.62 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,F5: filter @ A,G6: filter @ B] : % 0.31/0.62 ( ( eventually @ ( product_prod @ A @ B ) @ P @ ( prod_filter @ A @ B @ F5 @ G6 ) ) % 0.31/0.62 = ( ? [Pf: A > $o,Pg: B > $o] : % 0.31/0.62 ( ( eventually @ A @ Pf @ F5 ) % 0.31/0.62 & ( eventually @ B @ Pg @ G6 ) % 0.31/0.62 & ! [X: A,Y2: B] : % 0.31/0.62 ( ( Pf @ X ) % 0.31/0.62 => ( ( Pg @ Y2 ) % 0.31/0.62 => ( P @ ( product_Pair @ A @ B @ X @ Y2 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_prod_filter % 0.31/0.62 thf(fact_6111_prod__filter__mono__iff,axiom, % 0.31/0.62 ! [A: $tType,B: $tType,A3: filter @ A,B3: filter @ B,C5: filter @ A,D6: filter @ B] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.62 => ( ( B3 % 0.31/0.62 != ( bot_bot @ ( filter @ B ) ) ) % 0.31/0.62 => ( ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ A3 @ B3 ) @ ( prod_filter @ A @ B @ C5 @ D6 ) ) % 0.31/0.62 = ( ( ord_less_eq @ ( filter @ A ) @ A3 @ C5 ) % 0.31/0.62 & ( ord_less_eq @ ( filter @ B ) @ B3 @ D6 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % prod_filter_mono_iff % 0.31/0.62 thf(fact_6112_eventually__prod__sequentially,axiom, % 0.31/0.62 ! [P: ( product_prod @ nat @ nat ) > $o] : % 0.31/0.62 ( ( eventually @ ( product_prod @ nat @ nat ) @ P @ ( prod_filter @ nat @ nat @ ( at_top @ nat ) @ ( at_top @ nat ) ) ) % 0.31/0.62 = ( ? [N5: nat] : % 0.31/0.62 ! [M2: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ N5 @ M2 ) % 0.31/0.62 => ! [N: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ N5 @ N ) % 0.31/0.62 => ( P @ ( product_Pair @ nat @ nat @ N @ M2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_prod_sequentially % 0.31/0.62 thf(fact_6113_eventually__prod2,axiom, % 0.31/0.62 ! [A: $tType,B: $tType,A3: filter @ A,P: B > $o,B3: filter @ B] : % 0.31/0.62 ( ( A3 % 0.31/0.62 != ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.62 => ( ( eventually @ ( product_prod @ A @ B ) % 0.31/0.62 @ ( product_case_prod @ A @ B @ $o % 0.31/0.62 @ ^ [X: A] : P ) % 0.31/0.62 @ ( prod_filter @ A @ B @ A3 @ B3 ) ) % 0.31/0.62 = ( eventually @ B @ P @ B3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_prod2 % 0.31/0.62 thf(fact_6114_eventually__prod1,axiom, % 0.31/0.62 ! [A: $tType,B: $tType,B3: filter @ A,P: B > $o,A3: filter @ B] : % 0.31/0.62 ( ( B3 % 0.31/0.62 != ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.62 => ( ( eventually @ ( product_prod @ B @ A ) % 0.31/0.62 @ ( product_case_prod @ B @ A @ $o % 0.31/0.62 @ ^ [X: B,Y2: A] : ( P @ X ) ) % 0.31/0.62 @ ( prod_filter @ B @ A @ A3 @ B3 ) ) % 0.31/0.62 = ( eventually @ B @ P @ A3 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % eventually_prod1 % 0.31/0.62 thf(fact_6115_nhds__prod,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( ( topolo4958980785337419405_space @ B ) % 0.31/0.62 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.62 => ! [A2: A,B2: B] : % 0.31/0.62 ( ( topolo7230453075368039082e_nhds @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) ) % 0.31/0.62 = ( prod_filter @ A @ B @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( topolo7230453075368039082e_nhds @ B @ B2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % nhds_prod % 0.31/0.62 thf(fact_6116_filterlim__Pair,axiom, % 0.31/0.62 ! [C: $tType,B: $tType,A: $tType,F2: A > B,G6: filter @ B,F5: filter @ A,G: A > C,H7: filter @ C] : % 0.31/0.62 ( ( filterlim @ A @ B @ F2 @ G6 @ F5 ) % 0.31/0.62 => ( ( filterlim @ A @ C @ G @ H7 @ F5 ) % 0.31/0.62 => ( filterlim @ A @ ( product_prod @ B @ C ) % 0.31/0.62 @ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.62 @ ( prod_filter @ B @ C @ G6 @ H7 ) % 0.31/0.62 @ F5 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % filterlim_Pair % 0.31/0.62 thf(fact_6117_tendsto__add__Pair,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo6943815403480290642id_add @ A ) % 0.31/0.62 => ! [A2: A,B2: A] : % 0.31/0.62 ( filterlim @ ( product_prod @ A @ A ) @ A % 0.31/0.62 @ ^ [X: product_prod @ A @ A] : ( plus_plus @ A @ ( product_fst @ A @ A @ X ) @ ( product_snd @ A @ A @ X ) ) % 0.31/0.62 @ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.31/0.62 @ ( prod_filter @ A @ A @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( topolo7230453075368039082e_nhds @ A @ B2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % tendsto_add_Pair % 0.31/0.62 thf(fact_6118_uniformity__trans_H,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo7287701948861334536_space @ A ) % 0.31/0.62 => ! [E5: ( product_prod @ A @ A ) > $o] : % 0.31/0.62 ( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) ) % 0.31/0.62 => ( eventually @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) % 0.31/0.62 @ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o % 0.31/0.62 @ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o ) % 0.31/0.62 @ ^ [X: A,Y2: A] : % 0.31/0.62 ( product_case_prod @ A @ A @ $o % 0.31/0.62 @ ^ [Y6: A,Z6: A] : % 0.31/0.62 ( ( Y2 = Y6 ) % 0.31/0.62 => ( E5 @ ( product_Pair @ A @ A @ X @ Z6 ) ) ) ) ) ) % 0.31/0.62 @ ( prod_filter @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ ( topolo7806501430040627800ormity @ A ) @ ( topolo7806501430040627800ormity @ A ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % uniformity_trans' % 0.31/0.62 thf(fact_6119_prod__filter__INF,axiom, % 0.31/0.62 ! [C: $tType,D: $tType,B: $tType,A: $tType,I6: set @ A,J4: set @ B,A3: A > ( filter @ C ),B3: B > ( filter @ D )] : % 0.31/0.62 ( ( I6 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( J4 % 0.31/0.62 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.62 => ( ( prod_filter @ C @ D @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image @ A @ ( filter @ C ) @ A3 @ I6 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image @ B @ ( filter @ D ) @ B3 @ J4 ) ) ) % 0.31/0.62 = ( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) ) % 0.31/0.62 @ ( image @ A @ ( filter @ ( product_prod @ C @ D ) ) % 0.31/0.62 @ ^ [I3: A] : % 0.31/0.62 ( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) ) % 0.31/0.62 @ ( image @ B @ ( filter @ ( product_prod @ C @ D ) ) % 0.31/0.62 @ ^ [J3: B] : ( prod_filter @ C @ D @ ( A3 @ I3 ) @ ( B3 @ J3 ) ) % 0.31/0.62 @ J4 ) ) % 0.31/0.62 @ I6 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % prod_filter_INF % 0.31/0.62 thf(fact_6120_sequentially__imp__eventually__at__left,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( topolo3112930676232923870pology @ A ) % 0.31/0.62 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.62 => ! [B2: A,A2: A,P: A > $o] : % 0.31/0.62 ( ( ord_less @ A @ B2 @ A2 ) % 0.31/0.62 => ( ! [F4: nat > A] : % 0.31/0.62 ( ! [N6: nat] : ( ord_less @ A @ B2 @ ( F4 @ N6 ) ) % 0.31/0.62 => ( ! [N6: nat] : ( ord_less @ A @ ( F4 @ N6 ) @ A2 ) % 0.31/0.62 => ( ( order_mono @ nat @ A @ F4 ) % 0.31/0.62 => ( ( filterlim @ nat @ A @ F4 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) % 0.31/0.62 => ( eventually @ nat % 0.31/0.62 @ ^ [N: nat] : ( P @ ( F4 @ N ) ) % 0.31/0.62 @ ( at_top @ nat ) ) ) ) ) ) % 0.31/0.62 => ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % sequentially_imp_eventually_at_left % 0.31/0.62 thf(fact_6121_relpow__finite__bounded1,axiom, % 0.31/0.62 ! [A: $tType,R: set @ ( product_prod @ A @ A ),K: nat] : % 0.31/0.62 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R ) % 0.31/0.62 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 0.31/0.62 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R ) % 0.31/0.62 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) % 0.31/0.62 @ ( image @ nat @ ( set @ ( product_prod @ A @ A ) ) % 0.31/0.62 @ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) % 0.31/0.62 @ ( collect @ nat % 0.31/0.62 @ ^ [N: nat] : % 0.31/0.62 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) % 0.31/0.62 & ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % relpow_finite_bounded1 % 0.31/0.62 thf(fact_6122_finite__relpow,axiom, % 0.31/0.62 ! [A: $tType,R: set @ ( product_prod @ A @ A ),N2: nat] : % 0.31/0.62 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R ) % 0.31/0.62 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.62 => ( finite_finite2 @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % finite_relpow % 0.31/0.62 thf(fact_6123_mono__funpow,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( lattice @ A ) % 0.31/0.62 & ( order_bot @ A ) ) % 0.31/0.62 => ! [Q: A > A] : % 0.31/0.62 ( ( order_mono @ A @ A @ Q ) % 0.31/0.62 => ( order_mono @ nat @ A % 0.31/0.62 @ ^ [I3: nat] : ( compow @ ( A > A ) @ I3 @ Q @ ( bot_bot @ A ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_funpow % 0.31/0.62 thf(fact_6124_relpow__Suc__I2,axiom, % 0.31/0.62 ! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A ),Z2: A,N2: nat] : % 0.31/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R ) % 0.31/0.62 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) % 0.31/0.62 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % relpow_Suc_I2 % 0.31/0.62 thf(fact_6125_relpow__Suc__E2,axiom, % 0.31/0.62 ! [A: $tType,X3: A,Z2: A,N2: nat,R: set @ ( product_prod @ A @ A )] : % 0.31/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) ) % 0.31/0.62 => ~ ! [Y4: A] : % 0.31/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R ) % 0.31/0.62 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % relpow_Suc_E2 % 0.31/0.62 thf(fact_6126_relpow__Suc__D2,axiom, % 0.31/0.62 ! [A: $tType,X3: A,Z2: A,N2: nat,R: set @ ( product_prod @ A @ A )] : % 0.31/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) ) % 0.31/0.62 => ? [Y4: A] : % 0.31/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R ) % 0.31/0.62 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % relpow_Suc_D2 % 0.31/0.62 thf(fact_6127_relpow__Suc__I,axiom, % 0.31/0.62 ! [A: $tType,X3: A,Y: A,N2: nat,R: set @ ( product_prod @ A @ A ),Z2: A] : % 0.31/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) % 0.31/0.62 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R ) % 0.31/0.62 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % relpow_Suc_I % 0.31/0.62 thf(fact_6128_relpow__Suc__E,axiom, % 0.31/0.62 ! [A: $tType,X3: A,Z2: A,N2: nat,R: set @ ( product_prod @ A @ A )] : % 0.31/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) ) % 0.31/0.62 => ~ ! [Y4: A] : % 0.31/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) % 0.31/0.62 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R ) ) ) ). % 0.31/0.62 % 0.31/0.62 % relpow_Suc_E % 0.31/0.62 thf(fact_6129_relpow__0__E,axiom, % 0.31/0.62 ! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] : % 0.31/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R ) ) % 0.31/0.62 => ( X3 = Y ) ) ). % 0.31/0.62 % 0.31/0.62 % relpow_0_E % 0.31/0.62 thf(fact_6130_relpow__0__I,axiom, % 0.31/0.62 ! [A: $tType,X3: A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R ) ) ). % 0.31/0.62 % 0.31/0.62 % relpow_0_I % 0.31/0.62 thf(fact_6131_mono__invE,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( linorder @ A ) % 0.31/0.62 & ( order @ B ) ) % 0.31/0.62 => ! [F2: A > B,X3: A,Y: A] : % 0.31/0.62 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.62 => ( ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) ) % 0.31/0.62 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_invE % 0.31/0.62 thf(fact_6132_funpow__mono,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [F2: A > A,A3: A,B3: A,N2: nat] : % 0.31/0.62 ( ( order_mono @ A @ A @ F2 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ A3 @ B3 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N2 @ F2 @ A3 ) @ ( compow @ ( A > A ) @ N2 @ F2 @ B3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % funpow_mono % 0.31/0.62 thf(fact_6133_incseq__Suc__iff,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ( ( order_mono @ nat @ A ) % 0.31/0.62 = ( ^ [F3: nat > A] : % 0.31/0.62 ! [N: nat] : ( ord_less_eq @ A @ ( F3 @ N ) @ ( F3 @ ( suc @ N ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % incseq_Suc_iff % 0.31/0.62 thf(fact_6134_incseq__SucI,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [X8: nat > A] : % 0.31/0.62 ( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) ) % 0.31/0.62 => ( order_mono @ nat @ A @ X8 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % incseq_SucI % 0.31/0.62 thf(fact_6135_incseq__SucD,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [A3: nat > A,I: nat] : % 0.31/0.62 ( ( order_mono @ nat @ A @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( A3 @ I ) @ ( A3 @ ( suc @ I ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % incseq_SucD % 0.31/0.62 thf(fact_6136_monoD,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( order @ A ) % 0.31/0.62 & ( order @ B ) ) % 0.31/0.62 => ! [F2: A > B,X3: A,Y: A] : % 0.31/0.62 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % monoD % 0.31/0.62 thf(fact_6137_monoE,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( order @ A ) % 0.31/0.62 & ( order @ B ) ) % 0.31/0.62 => ! [F2: A > B,X3: A,Y: A] : % 0.31/0.62 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % monoE % 0.31/0.62 thf(fact_6138_monoI,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( order @ A ) % 0.31/0.62 & ( order @ B ) ) % 0.31/0.62 => ! [F2: A > B] : % 0.31/0.62 ( ! [X5: A,Y4: A] : % 0.31/0.62 ( ( ord_less_eq @ A @ X5 @ Y4 ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.31/0.62 => ( order_mono @ A @ B @ F2 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % monoI % 0.31/0.62 thf(fact_6139_mono__def,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( order @ A ) % 0.31/0.62 & ( order @ B ) ) % 0.31/0.62 => ( ( order_mono @ A @ B ) % 0.31/0.62 = ( ^ [F3: A > B] : % 0.31/0.62 ! [X: A,Y2: A] : % 0.31/0.62 ( ( ord_less_eq @ A @ X @ Y2 ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_def % 0.31/0.62 thf(fact_6140_incseqD,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [F2: nat > A,I: nat,J: nat] : % 0.31/0.62 ( ( order_mono @ nat @ A @ F2 ) % 0.31/0.62 => ( ( ord_less_eq @ nat @ I @ J ) % 0.31/0.62 => ( ord_less_eq @ A @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % incseqD % 0.31/0.62 thf(fact_6141_incseq__def,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ( ( order_mono @ nat @ A ) % 0.31/0.62 = ( ^ [X6: nat > A] : % 0.31/0.62 ! [M2: nat,N: nat] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M2 @ N ) % 0.31/0.62 => ( ord_less_eq @ A @ ( X6 @ M2 ) @ ( X6 @ N ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % incseq_def % 0.31/0.62 thf(fact_6142_mono__inf,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( semilattice_inf @ A ) % 0.31/0.62 & ( semilattice_inf @ B ) ) % 0.31/0.62 => ! [F2: A > B,A3: A,B3: A] : % 0.31/0.62 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F2 @ ( inf_inf @ A @ A3 @ B3 ) ) @ ( inf_inf @ B @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_inf % 0.31/0.62 thf(fact_6143_relpow__Suc__D2_H,axiom, % 0.31/0.62 ! [A: $tType,N2: nat,R: set @ ( product_prod @ A @ A ),X2: A,Y5: A,Z4: A] : % 0.31/0.62 ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) % 0.31/0.62 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z4 ) @ R ) ) % 0.31/0.62 => ? [W: A] : % 0.31/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ W ) @ R ) % 0.31/0.62 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W @ Z4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % relpow_Suc_D2' % 0.31/0.62 thf(fact_6144_mono__add,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ordere6658533253407199908up_add @ A ) % 0.31/0.62 => ! [A2: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A2 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_add % 0.31/0.62 thf(fact_6145_mono__Suc,axiom, % 0.31/0.62 order_mono @ nat @ nat @ suc ). % 0.31/0.62 % 0.31/0.62 % mono_Suc % 0.31/0.62 thf(fact_6146_mono__pow,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( comple6319245703460814977attice @ A ) % 0.31/0.62 => ! [F2: A > A,N2: nat] : % 0.31/0.62 ( ( order_mono @ A @ A @ F2 ) % 0.31/0.62 => ( order_mono @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_pow % 0.31/0.62 thf(fact_6147_mono__strict__invE,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( linorder @ A ) % 0.31/0.62 & ( order @ B ) ) % 0.31/0.62 => ! [F2: A > B,X3: A,Y: A] : % 0.31/0.62 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.62 => ( ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) ) % 0.31/0.62 => ( ord_less @ A @ X3 @ Y ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_strict_invE % 0.31/0.62 thf(fact_6148_cclfp__lowerbound,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( counta3822494911875563373attice @ A ) % 0.31/0.62 => ! [F2: A > A,A3: A] : % 0.31/0.62 ( ( order_mono @ A @ A @ F2 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ ( F2 @ A3 ) @ A3 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( order_532582986084564980_cclfp @ A @ F2 ) @ A3 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % cclfp_lowerbound % 0.31/0.62 thf(fact_6149_mono__times__nat,axiom, % 0.31/0.62 ! [N2: nat] : % 0.31/0.62 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.62 => ( order_mono @ nat @ nat @ ( times_times @ nat @ N2 ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_times_nat % 0.31/0.62 thf(fact_6150_mono__mult,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ordered_semiring @ A ) % 0.31/0.62 => ! [A2: A] : % 0.31/0.62 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.62 => ( order_mono @ A @ A @ ( times_times @ A @ A2 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_mult % 0.31/0.62 thf(fact_6151_mono__image__least,axiom, % 0.31/0.62 ! [A: $tType,B: $tType] : % 0.31/0.62 ( ( ( order @ B ) % 0.31/0.62 & ( order @ A ) ) % 0.31/0.62 => ! [F2: A > B,M: A,N2: A,M5: B,N4: B] : % 0.31/0.62 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.62 => ( ( ( image @ A @ B @ F2 @ ( set_or7035219750837199246ssThan @ A @ M @ N2 ) ) % 0.31/0.62 = ( set_or7035219750837199246ssThan @ B @ M5 @ N4 ) ) % 0.31/0.62 => ( ( ord_less @ A @ M @ N2 ) % 0.31/0.62 => ( ( F2 @ M ) % 0.31/0.62 = M5 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_image_least % 0.31/0.62 thf(fact_6152_Kleene__iter__gpfp,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order_top @ A ) % 0.31/0.62 => ! [F2: A > A,P4: A,K: nat] : % 0.31/0.62 ( ( order_mono @ A @ A @ F2 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ P4 @ ( F2 @ P4 ) ) % 0.31/0.62 => ( ord_less_eq @ A @ P4 @ ( compow @ ( A > A ) @ K @ F2 @ ( top_top @ A ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Kleene_iter_gpfp % 0.31/0.62 thf(fact_6153_Kleene__iter__lpfp,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order_bot @ A ) % 0.31/0.62 => ! [F2: A > A,P4: A,K: nat] : % 0.31/0.62 ( ( order_mono @ A @ A @ F2 ) % 0.31/0.62 => ( ( ord_less_eq @ A @ ( F2 @ P4 ) @ P4 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) @ P4 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % Kleene_iter_lpfp % 0.31/0.62 thf(fact_6154_funpow__mono2,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( order @ A ) % 0.31/0.62 => ! [F2: A > A,I: nat,J: nat,X3: A,Y: A] : % 0.31/0.62 ( ( order_mono @ A @ A @ F2 ) % 0.31/0.62 => ( ( ord_less_eq @ nat @ I @ J ) % 0.31/0.62 => ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.62 => ( ( ord_less_eq @ A @ X3 @ ( F2 @ X3 ) ) % 0.31/0.62 => ( ord_less_eq @ A @ ( compow @ ( A > A ) @ I @ F2 @ X3 ) @ ( compow @ ( A > A ) @ J @ F2 @ Y ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % funpow_mono2 % 0.31/0.62 thf(fact_6155_relpowp__relpow__eq,axiom, % 0.31/0.62 ! [A: $tType,N2: nat,R: set @ ( product_prod @ A @ A )] : % 0.31/0.62 ( ( compow @ ( A > A > $o ) @ N2 % 0.31/0.62 @ ^ [X: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R ) ) % 0.31/0.62 = ( ^ [X: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % relpowp_relpow_eq % 0.31/0.62 thf(fact_6156_mono__Sup,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( comple6319245703460814977attice @ A ) % 0.31/0.62 & ( comple6319245703460814977attice @ B ) ) % 0.31/0.62 => ! [F2: A > B,A3: set @ A] : % 0.31/0.62 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.62 => ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F2 @ A3 ) ) @ ( F2 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_Sup % 0.31/0.62 thf(fact_6157_mono__SUP,axiom, % 0.31/0.62 ! [B: $tType,A: $tType,C: $tType] : % 0.31/0.62 ( ( ( comple6319245703460814977attice @ A ) % 0.31/0.62 & ( comple6319245703460814977attice @ B ) ) % 0.31/0.62 => ! [F2: A > B,A3: C > A,I6: set @ C] : % 0.31/0.62 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.62 => ( ord_less_eq @ B % 0.31/0.62 @ ( complete_Sup_Sup @ B % 0.31/0.62 @ ( image @ C @ B % 0.31/0.62 @ ^ [X: C] : ( F2 @ ( A3 @ X ) ) % 0.31/0.62 @ I6 ) ) % 0.31/0.62 @ ( F2 @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ A3 @ I6 ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_SUP % 0.31/0.62 thf(fact_6158_mono__Inf,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( comple6319245703460814977attice @ A ) % 0.31/0.62 & ( comple6319245703460814977attice @ B ) ) % 0.31/0.62 => ! [F2: A > B,A3: set @ A] : % 0.31/0.62 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ A3 ) ) @ ( complete_Inf_Inf @ B @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_Inf % 0.31/0.62 thf(fact_6159_mono__INF,axiom, % 0.31/0.62 ! [B: $tType,A: $tType,C: $tType] : % 0.31/0.62 ( ( ( comple6319245703460814977attice @ A ) % 0.31/0.62 & ( comple6319245703460814977attice @ B ) ) % 0.31/0.62 => ! [F2: A > B,A3: C > A,I6: set @ C] : % 0.31/0.62 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ A3 @ I6 ) ) ) % 0.31/0.62 @ ( complete_Inf_Inf @ B % 0.31/0.62 @ ( image @ C @ B % 0.31/0.62 @ ^ [X: C] : ( F2 @ ( A3 @ X ) ) % 0.31/0.62 @ I6 ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_INF % 0.31/0.62 thf(fact_6160_relpow__E,axiom, % 0.31/0.62 ! [A: $tType,X3: A,Z2: A,N2: nat,R: set @ ( product_prod @ A @ A )] : % 0.31/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) % 0.31/0.62 => ( ( ( N2 % 0.31/0.62 = ( zero_zero @ nat ) ) % 0.31/0.62 => ( X3 != Z2 ) ) % 0.31/0.62 => ~ ! [Y4: A,M3: nat] : % 0.31/0.62 ( ( N2 % 0.31/0.62 = ( suc @ M3 ) ) % 0.31/0.62 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R ) ) % 0.31/0.62 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % relpow_E % 0.31/0.62 thf(fact_6161_relpow__E2,axiom, % 0.31/0.62 ! [A: $tType,X3: A,Z2: A,N2: nat,R: set @ ( product_prod @ A @ A )] : % 0.31/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) % 0.31/0.62 => ( ( ( N2 % 0.31/0.62 = ( zero_zero @ nat ) ) % 0.31/0.62 => ( X3 != Z2 ) ) % 0.31/0.62 => ~ ! [Y4: A,M3: nat] : % 0.31/0.62 ( ( N2 % 0.31/0.62 = ( suc @ M3 ) ) % 0.31/0.62 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R ) % 0.31/0.62 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % relpow_E2 % 0.31/0.62 thf(fact_6162_antimono__funpow,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( lattice @ A ) % 0.31/0.62 & ( order_top @ A ) ) % 0.31/0.62 => ! [Q: A > A] : % 0.31/0.62 ( ( order_mono @ A @ A @ Q ) % 0.31/0.62 => ( order_antimono @ nat @ A % 0.31/0.62 @ ^ [I3: nat] : ( compow @ ( A > A ) @ I3 @ Q @ ( top_top @ A ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % antimono_funpow % 0.31/0.62 thf(fact_6163_relpow__empty,axiom, % 0.31/0.62 ! [A: $tType,N2: nat] : % 0.31/0.62 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.62 => ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) % 0.31/0.62 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % relpow_empty % 0.31/0.62 thf(fact_6164_incseq__le,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.62 => ! [X8: nat > A,L5: A,N2: nat] : % 0.31/0.62 ( ( order_mono @ nat @ A @ X8 ) % 0.31/0.62 => ( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) % 0.31/0.62 => ( ord_less_eq @ A @ ( X8 @ N2 ) @ L5 ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % incseq_le % 0.31/0.62 thf(fact_6165_funpow__increasing,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( lattice @ A ) % 0.31/0.62 & ( order_top @ A ) ) % 0.31/0.62 => ! [M: nat,N2: nat,F2: A > A] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.62 => ( ( order_mono @ A @ A @ F2 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N2 @ F2 @ ( top_top @ A ) ) @ ( compow @ ( A > A ) @ M @ F2 @ ( top_top @ A ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % funpow_increasing % 0.31/0.62 thf(fact_6166_funpow__decreasing,axiom, % 0.31/0.62 ! [A: $tType] : % 0.31/0.62 ( ( ( lattice @ A ) % 0.31/0.62 & ( order_bot @ A ) ) % 0.31/0.62 => ! [M: nat,N2: nat,F2: A > A] : % 0.31/0.62 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.62 => ( ( order_mono @ A @ A @ F2 ) % 0.31/0.62 => ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M @ F2 @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N2 @ F2 @ ( bot_bot @ A ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % funpow_decreasing % 0.31/0.62 thf(fact_6167_mono__Max__commute,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( linorder @ A ) % 0.31/0.62 & ( linorder @ B ) ) % 0.31/0.62 => ! [F2: A > B,A3: set @ A] : % 0.31/0.62 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.62 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ( F2 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) % 0.31/0.62 = ( lattic643756798349783984er_Max @ B @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_Max_commute % 0.31/0.62 thf(fact_6168_relpow__fun__conv,axiom, % 0.31/0.62 ! [A: $tType,A2: A,B2: A,N2: nat,R: set @ ( product_prod @ A @ A )] : % 0.31/0.62 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) % 0.31/0.62 = ( ? [F3: nat > A] : % 0.31/0.62 ( ( ( F3 @ ( zero_zero @ nat ) ) % 0.31/0.62 = A2 ) % 0.31/0.62 & ( ( F3 @ N2 ) % 0.31/0.62 = B2 ) % 0.31/0.62 & ! [I3: nat] : % 0.31/0.62 ( ( ord_less @ nat @ I3 @ N2 ) % 0.31/0.62 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) @ R ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % relpow_fun_conv % 0.31/0.62 thf(fact_6169_mono__cSup,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 & ( condit1219197933456340205attice @ B ) ) % 0.31/0.62 => ! [F2: A > B,A3: set @ A] : % 0.31/0.62 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F2 @ A3 ) ) @ ( F2 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_cSup % 0.31/0.62 thf(fact_6170_mono__cSUP,axiom, % 0.31/0.62 ! [B: $tType,A: $tType,C: $tType] : % 0.31/0.62 ( ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 & ( condit1219197933456340205attice @ B ) ) % 0.31/0.62 => ! [F2: A > B,A3: C > A,I6: set @ C] : % 0.31/0.62 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.62 => ( ( condit941137186595557371_above @ A @ ( image @ C @ A @ A3 @ I6 ) ) % 0.31/0.62 => ( ( I6 % 0.31/0.62 != ( bot_bot @ ( set @ C ) ) ) % 0.31/0.62 => ( ord_less_eq @ B % 0.31/0.62 @ ( complete_Sup_Sup @ B % 0.31/0.62 @ ( image @ C @ B % 0.31/0.62 @ ^ [X: C] : ( F2 @ ( A3 @ X ) ) % 0.31/0.62 @ I6 ) ) % 0.31/0.62 @ ( F2 @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ A3 @ I6 ) ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_cSUP % 0.31/0.62 thf(fact_6171_mono__cInf,axiom, % 0.31/0.62 ! [B: $tType,A: $tType] : % 0.31/0.62 ( ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 & ( condit1219197933456340205attice @ B ) ) % 0.31/0.62 => ! [F2: A > B,A3: set @ A] : % 0.31/0.62 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.62 => ( ( condit1013018076250108175_below @ A @ A3 ) % 0.31/0.62 => ( ( A3 % 0.31/0.62 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.62 => ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ A3 ) ) @ ( complete_Inf_Inf @ B @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ) ) ) ). % 0.31/0.62 % 0.31/0.62 % mono_cInf % 0.31/0.62 thf(fact_6172_mono__cINF,axiom, % 0.31/0.62 ! [B: $tType,A: $tType,C: $tType] : % 0.31/0.62 ( ( ( condit1219197933456340205attice @ A ) % 0.31/0.62 & ( condit1219197933456340205attice @ B ) ) % 0.31/0.62 => ! [F2: A > B,A3: C > A,I6: set @ C] : % 0.31/0.62 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.63 => ( ( condit1013018076250108175_below @ A @ ( image @ C @ A @ A3 @ I6 ) ) % 0.31/0.63 => ( ( I6 % 0.31/0.63 != ( bot_bot @ ( set @ C ) ) ) % 0.31/0.63 => ( ord_less_eq @ B @ ( F2 @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ A3 @ I6 ) ) ) % 0.31/0.63 @ ( complete_Inf_Inf @ B % 0.31/0.63 @ ( image @ C @ B % 0.31/0.63 @ ^ [X: C] : ( F2 @ ( A3 @ X ) ) % 0.31/0.63 @ I6 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % mono_cINF % 0.31/0.63 thf(fact_6173_relpow__finite__bounded,axiom, % 0.31/0.63 ! [A: $tType,R: set @ ( product_prod @ A @ A ),K: nat] : % 0.31/0.63 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R ) % 0.31/0.63 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R ) % 0.31/0.63 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) % 0.31/0.63 @ ( image @ nat @ ( set @ ( product_prod @ A @ A ) ) % 0.31/0.63 @ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) % 0.31/0.63 @ ( collect @ nat % 0.31/0.63 @ ^ [N: nat] : ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % relpow_finite_bounded % 0.31/0.63 thf(fact_6174_mono__ge2__power__minus__self,axiom, % 0.31/0.63 ! [K: nat] : % 0.31/0.63 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) % 0.31/0.63 => ( order_mono @ nat @ nat % 0.31/0.63 @ ^ [M2: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K @ M2 ) @ M2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % mono_ge2_power_minus_self % 0.31/0.63 thf(fact_6175_finite__mono__remains__stable__implies__strict__prefix,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( order @ A ) % 0.31/0.63 => ! [F2: nat > A] : % 0.31/0.63 ( ( finite_finite2 @ A @ ( image @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) ) % 0.31/0.63 => ( ( order_mono @ nat @ A @ F2 ) % 0.31/0.63 => ( ! [N3: nat] : % 0.31/0.63 ( ( ( F2 @ N3 ) % 0.31/0.63 = ( F2 @ ( suc @ N3 ) ) ) % 0.31/0.63 => ( ( F2 @ ( suc @ N3 ) ) % 0.31/0.63 = ( F2 @ ( suc @ ( suc @ N3 ) ) ) ) ) % 0.31/0.63 => ? [N9: nat] : % 0.31/0.63 ( ! [N6: nat] : % 0.31/0.63 ( ( ord_less_eq @ nat @ N6 @ N9 ) % 0.31/0.63 => ! [M6: nat] : % 0.31/0.63 ( ( ord_less_eq @ nat @ M6 @ N9 ) % 0.31/0.63 => ( ( ord_less @ nat @ M6 @ N6 ) % 0.31/0.63 => ( ord_less @ A @ ( F2 @ M6 ) @ ( F2 @ N6 ) ) ) ) ) % 0.31/0.63 & ! [N6: nat] : % 0.31/0.63 ( ( ord_less_eq @ nat @ N9 @ N6 ) % 0.31/0.63 => ( ( F2 @ N9 ) % 0.31/0.63 = ( F2 @ N6 ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_mono_remains_stable_implies_strict_prefix % 0.31/0.63 thf(fact_6176_tendsto__at__left__sequentially,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( ( topolo3112930676232923870pology @ B ) % 0.31/0.63 & ( topolo1944317154257567458pology @ B ) % 0.31/0.63 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.63 => ! [B2: B,A2: B,X8: B > A,L5: A] : % 0.31/0.63 ( ( ord_less @ B @ B2 @ A2 ) % 0.31/0.63 => ( ! [S5: nat > B] : % 0.31/0.63 ( ! [N6: nat] : ( ord_less @ B @ ( S5 @ N6 ) @ A2 ) % 0.31/0.63 => ( ! [N6: nat] : ( ord_less @ B @ B2 @ ( S5 @ N6 ) ) % 0.31/0.63 => ( ( order_mono @ nat @ B @ S5 ) % 0.31/0.63 => ( ( filterlim @ nat @ B @ S5 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ ( at_top @ nat ) ) % 0.31/0.63 => ( filterlim @ nat @ A % 0.31/0.63 @ ^ [N: nat] : ( X8 @ ( S5 @ N ) ) % 0.31/0.63 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) % 0.31/0.63 @ ( at_top @ nat ) ) ) ) ) ) % 0.31/0.63 => ( filterlim @ B @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( topolo174197925503356063within @ B @ A2 @ ( set_ord_lessThan @ B @ A2 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % tendsto_at_left_sequentially % 0.31/0.63 thf(fact_6177_continuous__at__Sup__mono,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( ( condit6923001295902523014norder @ A ) % 0.31/0.63 & ( topolo1944317154257567458pology @ A ) % 0.31/0.63 & ( condit6923001295902523014norder @ B ) % 0.31/0.63 & ( topolo1944317154257567458pology @ B ) ) % 0.31/0.63 => ! [F2: A > B,S3: set @ A] : % 0.31/0.63 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.63 => ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Sup_Sup @ A @ S3 ) @ ( set_ord_lessThan @ A @ ( complete_Sup_Sup @ A @ S3 ) ) ) @ F2 ) % 0.31/0.63 => ( ( S3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( condit941137186595557371_above @ A @ S3 ) % 0.31/0.63 => ( ( F2 @ ( complete_Sup_Sup @ A @ S3 ) ) % 0.31/0.63 = ( complete_Sup_Sup @ B @ ( image @ A @ B @ F2 @ S3 ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % continuous_at_Sup_mono % 0.31/0.63 thf(fact_6178_continuous__at__Inf__mono,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( ( condit6923001295902523014norder @ A ) % 0.31/0.63 & ( topolo1944317154257567458pology @ A ) % 0.31/0.63 & ( condit6923001295902523014norder @ B ) % 0.31/0.63 & ( topolo1944317154257567458pology @ B ) ) % 0.31/0.63 => ! [F2: A > B,S3: set @ A] : % 0.31/0.63 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.63 => ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Inf_Inf @ A @ S3 ) @ ( set_ord_greaterThan @ A @ ( complete_Inf_Inf @ A @ S3 ) ) ) @ F2 ) % 0.31/0.63 => ( ( S3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( condit1013018076250108175_below @ A @ S3 ) % 0.31/0.63 => ( ( F2 @ ( complete_Inf_Inf @ A @ S3 ) ) % 0.31/0.63 = ( complete_Inf_Inf @ B @ ( image @ A @ B @ F2 @ S3 ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % continuous_at_Inf_mono % 0.31/0.63 thf(fact_6179_ntrancl__def,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( transitive_ntrancl @ A ) % 0.31/0.63 = ( ^ [N: nat,R6: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) % 0.31/0.63 @ ( image @ nat @ ( set @ ( product_prod @ A @ A ) ) % 0.31/0.63 @ ^ [I3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I3 @ R6 ) % 0.31/0.63 @ ( collect @ nat % 0.31/0.63 @ ^ [I3: nat] : % 0.31/0.63 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I3 ) % 0.31/0.63 & ( ord_less_eq @ nat @ I3 @ ( suc @ N ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ntrancl_def % 0.31/0.63 thf(fact_6180_trancl__finite__eq__relpow,axiom, % 0.31/0.63 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R ) % 0.31/0.63 => ( ( transitive_trancl @ A @ R ) % 0.31/0.63 = ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) % 0.31/0.63 @ ( image @ nat @ ( set @ ( product_prod @ A @ A ) ) % 0.31/0.63 @ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) % 0.31/0.63 @ ( collect @ nat % 0.31/0.63 @ ^ [N: nat] : % 0.31/0.63 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) % 0.31/0.63 & ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl_finite_eq_relpow % 0.31/0.63 thf(fact_6181_trancl__empty,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( transitive_trancl @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) % 0.31/0.63 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl_empty % 0.31/0.63 thf(fact_6182_finite__trancl,axiom, % 0.31/0.63 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 = ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_trancl % 0.31/0.63 thf(fact_6183_ntrancl__Zero,axiom, % 0.31/0.63 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( transitive_ntrancl @ A @ ( zero_zero @ nat ) @ R ) % 0.31/0.63 = R ) ). % 0.31/0.63 % 0.31/0.63 % ntrancl_Zero % 0.31/0.63 thf(fact_6184_mono__Int,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: ( set @ A ) > ( set @ B ),A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F2 ) % 0.31/0.63 => ( ord_less_eq @ ( set @ B ) @ ( F2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) @ ( inf_inf @ ( set @ B ) @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % mono_Int % 0.31/0.63 thf(fact_6185_rtranclp_Omono,axiom, % 0.31/0.63 ! [A: $tType,R2: A > A > $o] : % 0.31/0.63 ( order_mono @ ( A > A > $o ) @ ( A > A > $o ) % 0.31/0.63 @ ^ [P6: A > A > $o,X15: A,X24: A] : % 0.31/0.63 ( ? [A4: A] : % 0.31/0.63 ( ( X15 = A4 ) % 0.31/0.63 & ( X24 = A4 ) ) % 0.31/0.63 | ? [A4: A,B4: A,C4: A] : % 0.31/0.63 ( ( X15 = A4 ) % 0.31/0.63 & ( X24 = C4 ) % 0.31/0.63 & ( P6 @ A4 @ B4 ) % 0.31/0.63 & ( R2 @ B4 @ C4 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtranclp.mono % 0.31/0.63 thf(fact_6186_tranclp_Omono,axiom, % 0.31/0.63 ! [A: $tType,R2: A > A > $o] : % 0.31/0.63 ( order_mono @ ( A > A > $o ) @ ( A > A > $o ) % 0.31/0.63 @ ^ [P6: A > A > $o,X15: A,X24: A] : % 0.31/0.63 ( ? [A4: A,B4: A] : % 0.31/0.63 ( ( X15 = A4 ) % 0.31/0.63 & ( X24 = B4 ) % 0.31/0.63 & ( R2 @ A4 @ B4 ) ) % 0.31/0.63 | ? [A4: A,B4: A,C4: A] : % 0.31/0.63 ( ( X15 = A4 ) % 0.31/0.63 & ( X24 = C4 ) % 0.31/0.63 & ( P6 @ A4 @ B4 ) % 0.31/0.63 & ( R2 @ B4 @ C4 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % tranclp.mono % 0.31/0.63 thf(fact_6187_trancl__induct2,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] : % 0.31/0.63 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R2 ) ) % 0.31/0.63 => ( ! [A5: A,B5: B] : % 0.31/0.63 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B5 ) ) @ R2 ) % 0.31/0.63 => ( P @ A5 @ B5 ) ) % 0.31/0.63 => ( ! [A5: A,B5: B,Aa2: A,Ba: B] : % 0.31/0.63 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B5 ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R2 ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 ) % 0.31/0.63 => ( ( P @ A5 @ B5 ) % 0.31/0.63 => ( P @ Aa2 @ Ba ) ) ) ) % 0.31/0.63 => ( P @ Bx @ By ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl_induct2 % 0.31/0.63 thf(fact_6188_converse__trancl__induct,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( ! [Y4: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ B2 ) @ R2 ) % 0.31/0.63 => ( P @ Y4 ) ) % 0.31/0.63 => ( ! [Y4: A,Z3: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( ( P @ Z3 ) % 0.31/0.63 => ( P @ Y4 ) ) ) ) % 0.31/0.63 => ( P @ A2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % converse_trancl_induct % 0.31/0.63 thf(fact_6189_trancl__trans__induct,axiom, % 0.31/0.63 ! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),P: A > A > $o] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( ! [X5: A,Y4: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ R2 ) % 0.31/0.63 => ( P @ X5 @ Y4 ) ) % 0.31/0.63 => ( ! [X5: A,Y4: A,Z3: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( ( P @ X5 @ Y4 ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( ( P @ Y4 @ Z3 ) % 0.31/0.63 => ( P @ X5 @ Z3 ) ) ) ) ) % 0.31/0.63 => ( P @ X3 @ Y ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl_trans_induct % 0.31/0.63 thf(fact_6190_trancl__into__trancl2,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C2: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl_into_trancl2 % 0.31/0.63 thf(fact_6191_Transitive__Closure_Otrancl__into__trancl,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C2: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R2 ) % 0.31/0.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Transitive_Closure.trancl_into_trancl % 0.31/0.63 thf(fact_6192_irrefl__trancl__rD,axiom, % 0.31/0.63 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] : % 0.31/0.63 ( ! [X5: A] : % 0.31/0.63 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) % 0.31/0.63 => ( X3 != Y ) ) ) ). % 0.31/0.63 % 0.31/0.63 % irrefl_trancl_rD % 0.31/0.63 thf(fact_6193_converse__tranclE,axiom, % 0.31/0.63 ! [A: $tType,X3: A,Z2: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ R2 ) % 0.31/0.63 => ~ ! [Y4: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 ) % 0.31/0.63 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % converse_tranclE % 0.31/0.63 thf(fact_6194_r__r__into__trancl,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R ) % 0.31/0.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % r_r_into_trancl % 0.31/0.63 thf(fact_6195_trancl__induct,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( ! [Y4: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ Y4 ) @ R2 ) % 0.31/0.63 => ( P @ Y4 ) ) % 0.31/0.63 => ( ! [Y4: A,Z3: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ Y4 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 ) % 0.31/0.63 => ( ( P @ Y4 ) % 0.31/0.63 => ( P @ Z3 ) ) ) ) % 0.31/0.63 => ( P @ B2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl_induct % 0.31/0.63 thf(fact_6196_trancl__trans,axiom, % 0.31/0.63 ! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl_trans % 0.31/0.63 thf(fact_6197_tranclE,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) % 0.31/0.63 => ~ ! [C3: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C3 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C3 @ B2 ) @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % tranclE % 0.31/0.63 thf(fact_6198_trancl_Or__into__trancl,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) % 0.31/0.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl.r_into_trancl % 0.31/0.63 thf(fact_6199_trancl_Osimps,axiom, % 0.31/0.63 ! [A: $tType,A12: A,A23: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 = ( ? [A4: A,B4: A] : % 0.31/0.63 ( ( A12 = A4 ) % 0.31/0.63 & ( A23 = B4 ) % 0.31/0.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R2 ) ) % 0.31/0.63 | ? [A4: A,B4: A,C4: A] : % 0.31/0.63 ( ( A12 = A4 ) % 0.31/0.63 & ( A23 = C4 ) % 0.31/0.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C4 ) @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl.simps % 0.31/0.63 thf(fact_6200_trancl_Ocases,axiom, % 0.31/0.63 ! [A: $tType,A12: A,A23: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ R2 ) % 0.31/0.63 => ~ ! [B5: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B5 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A23 ) @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl.cases % 0.31/0.63 thf(fact_6201_finite__trancl__ntranl,axiom, % 0.31/0.63 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R ) % 0.31/0.63 => ( ( transitive_trancl @ A @ R ) % 0.31/0.63 = ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ R ) @ ( one_one @ nat ) ) @ R ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_trancl_ntranl % 0.31/0.63 thf(fact_6202_trancl__set__ntrancl,axiom, % 0.31/0.63 ! [A: $tType,Xs: list @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( transitive_trancl @ A @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) % 0.31/0.63 = ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) @ ( one_one @ nat ) ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl_set_ntrancl % 0.31/0.63 thf(fact_6203_trancl__power,axiom, % 0.31/0.63 ! [A: $tType,P4: product_prod @ A @ A,R: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ P4 @ ( transitive_trancl @ A @ R ) ) % 0.31/0.63 = ( ? [N: nat] : % 0.31/0.63 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) % 0.31/0.63 & ( member @ ( product_prod @ A @ A ) @ P4 @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl_power % 0.31/0.63 thf(fact_6204_less__eq,axiom, % 0.31/0.63 ! [M: nat,N2: nat] : % 0.31/0.63 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N2 ) @ ( transitive_trancl @ nat @ pred_nat ) ) % 0.31/0.63 = ( ord_less @ nat @ M @ N2 ) ) ). % 0.31/0.63 % 0.31/0.63 % less_eq % 0.31/0.63 thf(fact_6205_finite_Omono,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( order_mono @ ( ( set @ A ) > $o ) @ ( ( set @ A ) > $o ) % 0.31/0.63 @ ^ [P6: ( set @ A ) > $o,X: set @ A] : % 0.31/0.63 ( ( X % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 | ? [A6: set @ A,A4: A] : % 0.31/0.63 ( ( X % 0.31/0.63 = ( insert @ A @ A4 @ A6 ) ) % 0.31/0.63 & ( P6 @ A6 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite.mono % 0.31/0.63 thf(fact_6206_remdups__adj__altdef,axiom, % 0.31/0.63 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.63 ( ( ( remdups_adj @ A @ Xs ) % 0.31/0.63 = Ys ) % 0.31/0.63 = ( ? [F3: nat > nat] : % 0.31/0.63 ( ( order_mono @ nat @ nat @ F3 ) % 0.31/0.63 & ( ( image @ nat @ nat @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) % 0.31/0.63 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) % 0.31/0.63 & ! [I3: nat] : % 0.31/0.63 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.63 => ( ( nth @ A @ Xs @ I3 ) % 0.31/0.63 = ( nth @ A @ Ys @ ( F3 @ I3 ) ) ) ) % 0.31/0.63 & ! [I3: nat] : % 0.31/0.63 ( ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.63 => ( ( ( nth @ A @ Xs @ I3 ) % 0.31/0.63 = ( nth @ A @ Xs @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) ) % 0.31/0.63 = ( ( F3 @ I3 ) % 0.31/0.63 = ( F3 @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % remdups_adj_altdef % 0.31/0.63 thf(fact_6207_prod__filter__assoc,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,C: $tType,F5: filter @ A,G6: filter @ B,H7: filter @ C] : % 0.31/0.63 ( ( prod_filter @ ( product_prod @ A @ B ) @ C @ ( prod_filter @ A @ B @ F5 @ G6 ) @ H7 ) % 0.31/0.63 = ( filtermap @ ( product_prod @ A @ ( product_prod @ B @ C ) ) @ ( product_prod @ ( product_prod @ A @ B ) @ C ) % 0.31/0.63 @ ( product_case_prod @ A @ ( product_prod @ B @ C ) @ ( product_prod @ ( product_prod @ A @ B ) @ C ) % 0.31/0.63 @ ^ [X: A] : % 0.31/0.63 ( product_case_prod @ B @ C @ ( product_prod @ ( product_prod @ A @ B ) @ C ) % 0.31/0.63 @ ^ [Y2: B] : ( product_Pair @ ( product_prod @ A @ B ) @ C @ ( product_Pair @ A @ B @ X @ Y2 ) ) ) ) % 0.31/0.63 @ ( prod_filter @ A @ ( product_prod @ B @ C ) @ F5 @ ( prod_filter @ B @ C @ G6 @ H7 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % prod_filter_assoc % 0.31/0.63 thf(fact_6208_remdups__adj__Nil__iff,axiom, % 0.31/0.63 ! [A: $tType,Xs: list @ A] : % 0.31/0.63 ( ( ( remdups_adj @ A @ Xs ) % 0.31/0.63 = ( nil @ A ) ) % 0.31/0.63 = ( Xs % 0.31/0.63 = ( nil @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % remdups_adj_Nil_iff % 0.31/0.63 thf(fact_6209_remdups__adj__set,axiom, % 0.31/0.63 ! [A: $tType,Xs: list @ A] : % 0.31/0.63 ( ( set2 @ A @ ( remdups_adj @ A @ Xs ) ) % 0.31/0.63 = ( set2 @ A @ Xs ) ) ). % 0.31/0.63 % 0.31/0.63 % remdups_adj_set % 0.31/0.63 thf(fact_6210_filtermap__bot,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: B > A] : % 0.31/0.63 ( ( filtermap @ B @ A @ F2 @ ( bot_bot @ ( filter @ B ) ) ) % 0.31/0.63 = ( bot_bot @ ( filter @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % filtermap_bot % 0.31/0.63 thf(fact_6211_mono__compose,axiom, % 0.31/0.63 ! [D: $tType,C: $tType,B: $tType,A: $tType] : % 0.31/0.63 ( ( ( order @ A ) % 0.31/0.63 & ( order @ C ) ) % 0.31/0.63 => ! [Q: A > B > C,F2: D > B] : % 0.31/0.63 ( ( order_mono @ A @ ( B > C ) @ Q ) % 0.31/0.63 => ( order_mono @ A @ ( D > C ) % 0.31/0.63 @ ^ [I3: A,X: D] : ( Q @ I3 @ ( F2 @ X ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % mono_compose % 0.31/0.63 thf(fact_6212_filtermap__bot__iff,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,F2: B > A,F5: filter @ B] : % 0.31/0.63 ( ( ( filtermap @ B @ A @ F2 @ F5 ) % 0.31/0.63 = ( bot_bot @ ( filter @ A ) ) ) % 0.31/0.63 = ( F5 % 0.31/0.63 = ( bot_bot @ ( filter @ B ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % filtermap_bot_iff % 0.31/0.63 thf(fact_6213_remdups__adj_Osimps_I1_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( remdups_adj @ A @ ( nil @ A ) ) % 0.31/0.63 = ( nil @ A ) ) ). % 0.31/0.63 % 0.31/0.63 % remdups_adj.simps(1) % 0.31/0.63 thf(fact_6214_remdups__adj__distinct,axiom, % 0.31/0.63 ! [A: $tType,Xs: list @ A] : % 0.31/0.63 ( ( distinct @ A @ Xs ) % 0.31/0.63 => ( ( remdups_adj @ A @ Xs ) % 0.31/0.63 = Xs ) ) ). % 0.31/0.63 % 0.31/0.63 % remdups_adj_distinct % 0.31/0.63 thf(fact_6215_remdups__adj__length,axiom, % 0.31/0.63 ! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ). % 0.31/0.63 % 0.31/0.63 % remdups_adj_length % 0.31/0.63 thf(fact_6216_filtermap__Pair,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,C: $tType,F2: C > A,G: C > B,F5: filter @ C] : % 0.31/0.63 ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) % 0.31/0.63 @ ( filtermap @ C @ ( product_prod @ A @ B ) % 0.31/0.63 @ ^ [X: C] : ( product_Pair @ A @ B @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.63 @ F5 ) % 0.31/0.63 @ ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F2 @ F5 ) @ ( filtermap @ C @ B @ G @ F5 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % filtermap_Pair % 0.31/0.63 thf(fact_6217_filtermap__nhds__times,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.63 => ! [C2: A,A2: A] : % 0.31/0.63 ( ( C2 % 0.31/0.63 != ( zero_zero @ A ) ) % 0.31/0.63 => ( ( filtermap @ A @ A @ ( times_times @ A @ C2 ) @ ( topolo7230453075368039082e_nhds @ A @ A2 ) ) % 0.31/0.63 = ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ A2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % filtermap_nhds_times % 0.31/0.63 thf(fact_6218_remdups__adj__adjacent,axiom, % 0.31/0.63 ! [A: $tType,I: nat,Xs: list @ A] : % 0.31/0.63 ( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) ) % 0.31/0.63 => ( ( nth @ A @ ( remdups_adj @ A @ Xs ) @ I ) % 0.31/0.63 != ( nth @ A @ ( remdups_adj @ A @ Xs ) @ ( suc @ I ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % remdups_adj_adjacent % 0.31/0.63 thf(fact_6219_at__to__0,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( real_V822414075346904944vector @ A ) % 0.31/0.63 => ! [A2: A] : % 0.31/0.63 ( ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.63 = ( filtermap @ A @ A % 0.31/0.63 @ ^ [X: A] : ( plus_plus @ A @ X @ A2 ) % 0.31/0.63 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % at_to_0 % 0.31/0.63 thf(fact_6220_filtermap__times__pos__at__right,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ( linordered_field @ A ) % 0.31/0.63 & ( topolo1944317154257567458pology @ A ) ) % 0.31/0.63 => ! [C2: A,P4: A] : % 0.31/0.63 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 0.31/0.63 => ( ( filtermap @ A @ A @ ( times_times @ A @ C2 ) @ ( topolo174197925503356063within @ A @ P4 @ ( set_ord_greaterThan @ A @ P4 ) ) ) % 0.31/0.63 = ( topolo174197925503356063within @ A @ ( times_times @ A @ C2 @ P4 ) @ ( set_ord_greaterThan @ A @ ( times_times @ A @ C2 @ P4 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % filtermap_times_pos_at_right % 0.31/0.63 thf(fact_6221_at__to__infinity,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.63 => ( ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) % 0.31/0.63 = ( filtermap @ A @ A @ ( inverse_inverse @ A ) @ ( at_infinity @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % at_to_infinity % 0.31/0.63 thf(fact_6222_prod__filter__principal__singleton,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,X3: A,F5: filter @ B] : % 0.31/0.63 ( ( prod_filter @ A @ B @ ( principal @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ F5 ) % 0.31/0.63 = ( filtermap @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ F5 ) ) ). % 0.31/0.63 % 0.31/0.63 % prod_filter_principal_singleton % 0.31/0.63 thf(fact_6223_remdups__adj__length__ge1,axiom, % 0.31/0.63 ! [A: $tType,Xs: list @ A] : % 0.31/0.63 ( ( Xs % 0.31/0.63 != ( nil @ A ) ) % 0.31/0.63 => ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % remdups_adj_length_ge1 % 0.31/0.63 thf(fact_6224_prod__filter__principal__singleton2,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F5: filter @ A,X3: B] : % 0.31/0.63 ( ( prod_filter @ A @ B @ F5 @ ( principal @ B @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) % 0.31/0.63 = ( filtermap @ A @ ( product_prod @ A @ B ) % 0.31/0.63 @ ^ [A4: A] : ( product_Pair @ A @ B @ A4 @ X3 ) % 0.31/0.63 @ F5 ) ) ). % 0.31/0.63 % 0.31/0.63 % prod_filter_principal_singleton2 % 0.31/0.63 thf(fact_6225_cauchy__filter__metric__filtermap,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( ( real_V768167426530841204y_dist @ B ) % 0.31/0.63 & ( topolo7287701948861334536_space @ B ) ) % 0.31/0.63 => ! [F2: A > B,F5: filter @ A] : % 0.31/0.63 ( ( topolo6773858410816713723filter @ B @ ( filtermap @ A @ B @ F2 @ F5 ) ) % 0.31/0.63 = ( ! [E4: real] : % 0.31/0.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 ) % 0.31/0.63 => ? [P3: A > $o] : % 0.31/0.63 ( ( eventually @ A @ P3 @ F5 ) % 0.31/0.63 & ! [X: A,Y2: A] : % 0.31/0.63 ( ( ( P3 @ X ) % 0.31/0.63 & ( P3 @ Y2 ) ) % 0.31/0.63 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X ) @ ( F2 @ Y2 ) ) @ E4 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % cauchy_filter_metric_filtermap % 0.31/0.63 thf(fact_6226_iteratesp_Omono,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( comple9053668089753744459l_ccpo @ A ) % 0.31/0.63 => ! [F2: A > A] : % 0.31/0.63 ( order_mono @ ( A > $o ) @ ( A > $o ) % 0.31/0.63 @ ^ [P6: A > $o,X: A] : % 0.31/0.63 ( ? [Y2: A] : % 0.31/0.63 ( ( X % 0.31/0.63 = ( F2 @ Y2 ) ) % 0.31/0.63 & ( P6 @ Y2 ) ) % 0.31/0.63 | ? [M9: set @ A] : % 0.31/0.63 ( ( X % 0.31/0.63 = ( complete_Sup_Sup @ A @ M9 ) ) % 0.31/0.63 & ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 ) % 0.31/0.63 & ! [Y2: A] : % 0.31/0.63 ( ( member @ A @ Y2 @ M9 ) % 0.31/0.63 => ( P6 @ Y2 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % iteratesp.mono % 0.31/0.63 thf(fact_6227_compact__imp__fip__image,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.63 => ! [S: set @ A,I6: set @ B,F2: B > ( set @ A )] : % 0.31/0.63 ( ( topolo2193935891317330818ompact @ A @ S ) % 0.31/0.63 => ( ! [I4: B] : % 0.31/0.63 ( ( member @ B @ I4 @ I6 ) % 0.31/0.63 => ( topolo7761053866217962861closed @ A @ ( F2 @ I4 ) ) ) % 0.31/0.63 => ( ! [I9: set @ B] : % 0.31/0.63 ( ( finite_finite2 @ B @ I9 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ B ) @ I9 @ I6 ) % 0.31/0.63 => ( ( inf_inf @ ( set @ A ) @ S @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ I9 ) ) ) % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.63 => ( ( inf_inf @ ( set @ A ) @ S @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ I6 ) ) ) % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % compact_imp_fip_image % 0.31/0.63 thf(fact_6228_closed__empty,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.63 => ( topolo7761053866217962861closed @ A @ ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % closed_empty % 0.31/0.63 thf(fact_6229_closed__singleton,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( topological_t1_space @ A ) % 0.31/0.63 => ! [A2: A] : ( topolo7761053866217962861closed @ A @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % closed_singleton % 0.31/0.63 thf(fact_6230_closed__Union,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.63 => ! [S3: set @ ( set @ A )] : % 0.31/0.63 ( ( finite_finite2 @ ( set @ A ) @ S3 ) % 0.31/0.63 => ( ! [X5: set @ A] : % 0.31/0.63 ( ( member @ ( set @ A ) @ X5 @ S3 ) % 0.31/0.63 => ( topolo7761053866217962861closed @ A @ X5 ) ) % 0.31/0.63 => ( topolo7761053866217962861closed @ A @ ( complete_Sup_Sup @ ( set @ A ) @ S3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % closed_Union % 0.31/0.63 thf(fact_6231_closed__UN,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.63 => ! [A3: set @ B,B3: B > ( set @ A )] : % 0.31/0.63 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.63 => ( ! [X5: B] : % 0.31/0.63 ( ( member @ B @ X5 @ A3 ) % 0.31/0.63 => ( topolo7761053866217962861closed @ A @ ( B3 @ X5 ) ) ) % 0.31/0.63 => ( topolo7761053866217962861closed @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % closed_UN % 0.31/0.63 thf(fact_6232_filtermap__sequentually__ne__bot,axiom, % 0.31/0.63 ! [A: $tType,F2: nat > A] : % 0.31/0.63 ( ( filtermap @ nat @ A @ F2 @ ( at_top @ nat ) ) % 0.31/0.63 != ( bot_bot @ ( filter @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % filtermap_sequentually_ne_bot % 0.31/0.63 thf(fact_6233_closed__diagonal,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( topological_t2_space @ A ) % 0.31/0.63 => ( topolo7761053866217962861closed @ ( product_prod @ A @ A ) % 0.31/0.63 @ ( collect @ ( product_prod @ A @ A ) % 0.31/0.63 @ ^ [Y2: product_prod @ A @ A] : % 0.31/0.63 ? [X: A] : % 0.31/0.63 ( Y2 % 0.31/0.63 = ( product_Pair @ A @ A @ X @ X ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % closed_diagonal % 0.31/0.63 thf(fact_6234_chain__subset,axiom, % 0.31/0.63 ! [A: $tType,Ord: A > A > $o,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( comple1602240252501008431_chain @ A @ Ord @ A3 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.63 => ( comple1602240252501008431_chain @ A @ Ord @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % chain_subset % 0.31/0.63 thf(fact_6235_ccpo__Sup__upper,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( comple9053668089753744459l_ccpo @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A3 ) % 0.31/0.63 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.63 => ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ccpo_Sup_upper % 0.31/0.63 thf(fact_6236_ccpo__Sup__least,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( comple9053668089753744459l_ccpo @ A ) % 0.31/0.63 => ! [A3: set @ A,Z2: A] : % 0.31/0.63 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A3 ) % 0.31/0.63 => ( ! [X5: A] : % 0.31/0.63 ( ( member @ A @ X5 @ A3 ) % 0.31/0.63 => ( ord_less_eq @ A @ X5 @ Z2 ) ) % 0.31/0.63 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ Z2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ccpo_Sup_least % 0.31/0.63 thf(fact_6237_finite__imp__closed,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( topological_t1_space @ A ) % 0.31/0.63 => ! [S3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ S3 ) % 0.31/0.63 => ( topolo7761053866217962861closed @ A @ S3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_imp_closed % 0.31/0.63 thf(fact_6238_chain__empty,axiom, % 0.31/0.63 ! [A: $tType,Ord: A > A > $o] : ( comple1602240252501008431_chain @ A @ Ord @ ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % chain_empty % 0.31/0.63 thf(fact_6239_closed__subdiagonal,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.63 => ( topolo7761053866217962861closed @ ( product_prod @ A @ A ) % 0.31/0.63 @ ( collect @ ( product_prod @ A @ A ) % 0.31/0.63 @ ^ [Uu3: product_prod @ A @ A] : % 0.31/0.63 ? [X: A,Y2: A] : % 0.31/0.63 ( ( Uu3 % 0.31/0.63 = ( product_Pair @ A @ A @ X @ Y2 ) ) % 0.31/0.63 & ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % closed_subdiagonal % 0.31/0.63 thf(fact_6240_closed__superdiagonal,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( topolo1944317154257567458pology @ A ) % 0.31/0.63 => ( topolo7761053866217962861closed @ ( product_prod @ A @ A ) % 0.31/0.63 @ ( collect @ ( product_prod @ A @ A ) % 0.31/0.63 @ ^ [Uu3: product_prod @ A @ A] : % 0.31/0.63 ? [X: A,Y2: A] : % 0.31/0.63 ( ( Uu3 % 0.31/0.63 = ( product_Pair @ A @ A @ X @ Y2 ) ) % 0.31/0.63 & ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % closed_superdiagonal % 0.31/0.63 thf(fact_6241_chain__singleton,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( comple9053668089753744459l_ccpo @ A ) % 0.31/0.63 => ! [X3: A] : ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % chain_singleton % 0.31/0.63 thf(fact_6242_closed__Collect__le,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( ( topolo4958980785337419405_space @ A ) % 0.31/0.63 & ( topolo1944317154257567458pology @ B ) ) % 0.31/0.63 => ! [F2: A > B,G: A > B] : % 0.31/0.63 ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) % 0.31/0.63 => ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G ) % 0.31/0.63 => ( topolo7761053866217962861closed @ A % 0.31/0.63 @ ( collect @ A % 0.31/0.63 @ ^ [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % closed_Collect_le % 0.31/0.63 thf(fact_6243_t4__space,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( topological_t4_space @ A ) % 0.31/0.63 => ! [S3: set @ A,T4: set @ A] : % 0.31/0.63 ( ( topolo7761053866217962861closed @ A @ S3 ) % 0.31/0.63 => ( ( topolo7761053866217962861closed @ A @ T4 ) % 0.31/0.63 => ( ( ( inf_inf @ ( set @ A ) @ S3 @ T4 ) % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ? [U6: set @ A,V7: set @ A] : % 0.31/0.63 ( ( topolo1002775350975398744n_open @ A @ U6 ) % 0.31/0.63 & ( topolo1002775350975398744n_open @ A @ V7 ) % 0.31/0.63 & ( ord_less_eq @ ( set @ A ) @ S3 @ U6 ) % 0.31/0.63 & ( ord_less_eq @ ( set @ A ) @ T4 @ V7 ) % 0.31/0.63 & ( ( inf_inf @ ( set @ A ) @ U6 @ V7 ) % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % t4_space % 0.31/0.63 thf(fact_6244_t3__space,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( topological_t3_space @ A ) % 0.31/0.63 => ! [S3: set @ A,Y: A] : % 0.31/0.63 ( ( topolo7761053866217962861closed @ A @ S3 ) % 0.31/0.63 => ( ~ ( member @ A @ Y @ S3 ) % 0.31/0.63 => ? [U6: set @ A,V7: set @ A] : % 0.31/0.63 ( ( topolo1002775350975398744n_open @ A @ U6 ) % 0.31/0.63 & ( topolo1002775350975398744n_open @ A @ V7 ) % 0.31/0.63 & ( member @ A @ Y @ U6 ) % 0.31/0.63 & ( ord_less_eq @ ( set @ A ) @ S3 @ V7 ) % 0.31/0.63 & ( ( inf_inf @ ( set @ A ) @ U6 @ V7 ) % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % t3_space % 0.31/0.63 thf(fact_6245_nhds__closed,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( topological_t3_space @ A ) % 0.31/0.63 => ! [X3: A,A3: set @ A] : % 0.31/0.63 ( ( member @ A @ X3 @ A3 ) % 0.31/0.63 => ( ( topolo1002775350975398744n_open @ A @ A3 ) % 0.31/0.63 => ? [A15: set @ A] : % 0.31/0.63 ( ( member @ A @ X3 @ A15 ) % 0.31/0.63 & ( topolo7761053866217962861closed @ A @ A15 ) % 0.31/0.63 & ( ord_less_eq @ ( set @ A ) @ A15 @ A3 ) % 0.31/0.63 & ( eventually @ A % 0.31/0.63 @ ^ [Y2: A] : ( member @ A @ Y2 @ A15 ) % 0.31/0.63 @ ( topolo7230453075368039082e_nhds @ A @ X3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % nhds_closed % 0.31/0.63 thf(fact_6246_continuous__on__closed__Union,axiom, % 0.31/0.63 ! [A: $tType,C: $tType,B: $tType] : % 0.31/0.63 ( ( ( topolo4958980785337419405_space @ B ) % 0.31/0.63 & ( topolo4958980785337419405_space @ C ) ) % 0.31/0.63 => ! [I6: set @ A,U3: A > ( set @ B ),F2: B > C] : % 0.31/0.63 ( ( finite_finite2 @ A @ I6 ) % 0.31/0.63 => ( ! [I4: A] : % 0.31/0.63 ( ( member @ A @ I4 @ I6 ) % 0.31/0.63 => ( topolo7761053866217962861closed @ B @ ( U3 @ I4 ) ) ) % 0.31/0.63 => ( ! [I4: A] : % 0.31/0.63 ( ( member @ A @ I4 @ I6 ) % 0.31/0.63 => ( topolo81223032696312382ous_on @ B @ C @ ( U3 @ I4 ) @ F2 ) ) % 0.31/0.63 => ( topolo81223032696312382ous_on @ B @ C @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ U3 @ I6 ) ) @ F2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % continuous_on_closed_Union % 0.31/0.63 thf(fact_6247_Lim__in__closed__set,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.63 => ! [S3: set @ A,F2: B > A,F5: filter @ B,L: A] : % 0.31/0.63 ( ( topolo7761053866217962861closed @ A @ S3 ) % 0.31/0.63 => ( ( eventually @ B % 0.31/0.63 @ ^ [X: B] : ( member @ A @ ( F2 @ X ) @ S3 ) % 0.31/0.63 @ F5 ) % 0.31/0.63 => ( ( F5 % 0.31/0.63 != ( bot_bot @ ( filter @ B ) ) ) % 0.31/0.63 => ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) % 0.31/0.63 => ( member @ A @ L @ S3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Lim_in_closed_set % 0.31/0.63 thf(fact_6248_compact__imp__fip,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.63 => ! [S3: set @ A,F5: set @ ( set @ A )] : % 0.31/0.63 ( ( topolo2193935891317330818ompact @ A @ S3 ) % 0.31/0.63 => ( ! [T5: set @ A] : % 0.31/0.63 ( ( member @ ( set @ A ) @ T5 @ F5 ) % 0.31/0.63 => ( topolo7761053866217962861closed @ A @ T5 ) ) % 0.31/0.63 => ( ! [F15: set @ ( set @ A )] : % 0.31/0.63 ( ( finite_finite2 @ ( set @ A ) @ F15 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F15 @ F5 ) % 0.31/0.63 => ( ( inf_inf @ ( set @ A ) @ S3 @ ( complete_Inf_Inf @ ( set @ A ) @ F15 ) ) % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.63 => ( ( inf_inf @ ( set @ A ) @ S3 @ ( complete_Inf_Inf @ ( set @ A ) @ F5 ) ) % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % compact_imp_fip % 0.31/0.63 thf(fact_6249_compact__fip,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( topolo4958980785337419405_space @ A ) % 0.31/0.63 => ( ( topolo2193935891317330818ompact @ A ) % 0.31/0.63 = ( ^ [U5: set @ A] : % 0.31/0.63 ! [A6: set @ ( set @ A )] : % 0.31/0.63 ( ! [X: set @ A] : % 0.31/0.63 ( ( member @ ( set @ A ) @ X @ A6 ) % 0.31/0.63 => ( topolo7761053866217962861closed @ A @ X ) ) % 0.31/0.63 => ( ! [B6: set @ ( set @ A )] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B6 @ A6 ) % 0.31/0.63 => ( ( finite_finite2 @ ( set @ A ) @ B6 ) % 0.31/0.63 => ( ( inf_inf @ ( set @ A ) @ U5 @ ( complete_Inf_Inf @ ( set @ A ) @ B6 ) ) % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.63 => ( ( inf_inf @ ( set @ A ) @ U5 @ ( complete_Inf_Inf @ ( set @ A ) @ A6 ) ) % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % compact_fip % 0.31/0.63 thf(fact_6250_in__chain__finite,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( comple9053668089753744459l_ccpo @ A ) % 0.31/0.63 => ! [A3: set @ A] : % 0.31/0.63 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( member @ A @ ( complete_Sup_Sup @ A @ A3 ) @ A3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % in_chain_finite % 0.31/0.63 thf(fact_6251_inj__sgn__power,axiom, % 0.31/0.63 ! [N2: nat] : % 0.31/0.63 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.63 => ( inj_on @ real @ real % 0.31/0.63 @ ^ [Y2: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N2 ) ) % 0.31/0.63 @ ( top_top @ ( set @ real ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_sgn_power % 0.31/0.63 thf(fact_6252_inj__on__empty,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B] : ( inj_on @ A @ B @ F2 @ ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_empty % 0.31/0.63 thf(fact_6253_inj__mult__left,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( idom @ A ) % 0.31/0.63 => ! [A2: A] : % 0.31/0.63 ( ( inj_on @ A @ A @ ( times_times @ A @ A2 ) @ ( top_top @ ( set @ A ) ) ) % 0.31/0.63 = ( A2 % 0.31/0.63 != ( zero_zero @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_mult_left % 0.31/0.63 thf(fact_6254_inj__divide__right,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( field @ A ) % 0.31/0.63 => ! [A2: A] : % 0.31/0.63 ( ( inj_on @ A @ A % 0.31/0.63 @ ^ [B4: A] : ( divide_divide @ A @ B4 @ A2 ) % 0.31/0.63 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.63 = ( A2 % 0.31/0.63 != ( zero_zero @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_divide_right % 0.31/0.63 thf(fact_6255_inj__apfst,axiom, % 0.31/0.63 ! [B: $tType,C: $tType,A: $tType,F2: A > C] : % 0.31/0.63 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) % 0.31/0.63 = ( inj_on @ A @ C @ F2 @ ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_apfst % 0.31/0.63 thf(fact_6256_inj__apsnd,axiom, % 0.31/0.63 ! [A: $tType,C: $tType,B: $tType,F2: B > C] : % 0.31/0.63 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) % 0.31/0.63 = ( inj_on @ B @ C @ F2 @ ( top_top @ ( set @ B ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_apsnd % 0.31/0.63 thf(fact_6257_inj__on__insert,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B,A2: A,A3: set @ A] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ ( insert @ A @ A2 @ A3 ) ) % 0.31/0.63 = ( ( inj_on @ A @ B @ F2 @ A3 ) % 0.31/0.63 & ~ ( member @ B @ ( F2 @ A2 ) @ ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_insert % 0.31/0.63 thf(fact_6258_inj__on__Inter,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,S3: set @ ( set @ A ),F2: A > B] : % 0.31/0.63 ( ( S3 % 0.31/0.63 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 0.31/0.63 => ( ! [A8: set @ A] : % 0.31/0.63 ( ( member @ ( set @ A ) @ A8 @ S3 ) % 0.31/0.63 => ( inj_on @ A @ B @ F2 @ A8 ) ) % 0.31/0.63 => ( inj_on @ A @ B @ F2 @ ( complete_Inf_Inf @ ( set @ A ) @ S3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_Inter % 0.31/0.63 thf(fact_6259_finite__inverse__image,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,A3: set @ A,F2: B > A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) % 0.31/0.63 => ( finite_finite2 @ B % 0.31/0.63 @ ( collect @ B % 0.31/0.63 @ ^ [J3: B] : ( member @ A @ ( F2 @ J3 ) @ A3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_inverse_image % 0.31/0.63 thf(fact_6260_finite__inverse__image__gen,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,A3: set @ A,F2: B > A,D6: set @ B] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( inj_on @ B @ A @ F2 @ D6 ) % 0.31/0.63 => ( finite_finite2 @ B % 0.31/0.63 @ ( collect @ B % 0.31/0.63 @ ^ [J3: B] : % 0.31/0.63 ( ( member @ B @ J3 @ D6 ) % 0.31/0.63 & ( member @ A @ ( F2 @ J3 ) @ A3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_inverse_image_gen % 0.31/0.63 thf(fact_6261_inj__on__mult,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semidom_divide @ A ) % 0.31/0.63 => ! [A2: A,A3: set @ A] : % 0.31/0.63 ( ( A2 % 0.31/0.63 != ( zero_zero @ A ) ) % 0.31/0.63 => ( inj_on @ A @ A @ ( times_times @ A @ A2 ) @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_mult % 0.31/0.63 thf(fact_6262_inj__on__add_H,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( cancel_semigroup_add @ A ) % 0.31/0.63 => ! [A2: A,A3: set @ A] : % 0.31/0.63 ( inj_on @ A @ A % 0.31/0.63 @ ^ [B4: A] : ( plus_plus @ A @ B4 @ A2 ) % 0.31/0.63 @ A3 ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_add' % 0.31/0.63 thf(fact_6263_inj__on__add,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( cancel_semigroup_add @ A ) % 0.31/0.63 => ! [A2: A,A3: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_add % 0.31/0.63 thf(fact_6264_inj__add__left,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( cancel_semigroup_add @ A ) % 0.31/0.63 => ! [A2: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_add_left % 0.31/0.63 thf(fact_6265_linorder__injI,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [F2: A > B] : % 0.31/0.63 ( ! [X5: A,Y4: A] : % 0.31/0.63 ( ( ord_less @ A @ X5 @ Y4 ) % 0.31/0.63 => ( ( F2 @ X5 ) % 0.31/0.63 != ( F2 @ Y4 ) ) ) % 0.31/0.63 => ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % linorder_injI % 0.31/0.63 thf(fact_6266_sorted__list__of__set_Oinj__on,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ( inj_on @ A @ A % 0.31/0.63 @ ^ [X: A] : X % 0.31/0.63 @ ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sorted_list_of_set.inj_on % 0.31/0.63 thf(fact_6267_inj__fn,axiom, % 0.31/0.63 ! [A: $tType,F2: A > A,N2: nat] : % 0.31/0.63 ( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.63 => ( inj_on @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_fn % 0.31/0.63 thf(fact_6268_inj__on__subset,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ A3 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.63 => ( inj_on @ A @ B @ F2 @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_subset % 0.31/0.63 thf(fact_6269_subset__inj__on,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B,B3: set @ A,A3: set @ A] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ B3 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 => ( inj_on @ A @ B @ F2 @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % subset_inj_on % 0.31/0.63 thf(fact_6270_linorder__inj__onI,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( order @ A ) % 0.31/0.63 => ! [A3: set @ A,F2: A > B] : % 0.31/0.63 ( ! [X5: A,Y4: A] : % 0.31/0.63 ( ( ord_less @ A @ X5 @ Y4 ) % 0.31/0.63 => ( ( member @ A @ X5 @ A3 ) % 0.31/0.63 => ( ( member @ A @ Y4 @ A3 ) % 0.31/0.63 => ( ( F2 @ X5 ) % 0.31/0.63 != ( F2 @ Y4 ) ) ) ) ) % 0.31/0.63 => ( ! [X5: A,Y4: A] : % 0.31/0.63 ( ( member @ A @ X5 @ A3 ) % 0.31/0.63 => ( ( member @ A @ Y4 @ A3 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ X5 @ Y4 ) % 0.31/0.63 | ( ord_less_eq @ A @ Y4 @ X5 ) ) ) ) % 0.31/0.63 => ( inj_on @ A @ B @ F2 @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % linorder_inj_onI % 0.31/0.63 thf(fact_6271_subset__image__inj,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,S3: set @ A,F2: B > A,T4: set @ B] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( image @ B @ A @ F2 @ T4 ) ) % 0.31/0.63 = ( ? [U5: set @ B] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ B ) @ U5 @ T4 ) % 0.31/0.63 & ( inj_on @ B @ A @ F2 @ U5 ) % 0.31/0.63 & ( S3 % 0.31/0.63 = ( image @ B @ A @ F2 @ U5 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % subset_image_inj % 0.31/0.63 thf(fact_6272_inj__on__image__eq__iff,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B,C5: set @ A,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ C5 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 ) % 0.31/0.63 => ( ( ( image @ A @ B @ F2 @ A3 ) % 0.31/0.63 = ( image @ A @ B @ F2 @ B3 ) ) % 0.31/0.63 = ( A3 = B3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_image_eq_iff % 0.31/0.63 thf(fact_6273_inj__on__image__mem__iff,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B,B3: set @ A,A2: A,A3: set @ A] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ B3 ) % 0.31/0.63 => ( ( member @ A @ A2 @ B3 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 => ( ( member @ B @ ( F2 @ A2 ) @ ( image @ A @ B @ F2 @ A3 ) ) % 0.31/0.63 = ( member @ A @ A2 @ A3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_image_mem_iff % 0.31/0.63 thf(fact_6274_finite__image__iff,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B,A3: set @ A] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ ( image @ A @ B @ F2 @ A3 ) ) % 0.31/0.63 = ( finite_finite2 @ A @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_image_iff % 0.31/0.63 thf(fact_6275_finite__imageD,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] : % 0.31/0.63 ( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.63 => ( ( inj_on @ B @ A @ F2 @ A3 ) % 0.31/0.63 => ( finite_finite2 @ B @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_imageD % 0.31/0.63 thf(fact_6276_inj__on__strict__subset,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B,B3: set @ A,A3: set @ A] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ B3 ) % 0.31/0.63 => ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 => ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_strict_subset % 0.31/0.63 thf(fact_6277_finite__UNIV__inj__surj,axiom, % 0.31/0.63 ! [A: $tType,F2: A > A] : % 0.31/0.63 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) % 0.31/0.63 => ( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.63 => ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.63 = ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_UNIV_inj_surj % 0.31/0.63 thf(fact_6278_finite__UNIV__surj__inj,axiom, % 0.31/0.63 ! [A: $tType,F2: A > A] : % 0.31/0.63 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) % 0.31/0.63 => ( ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.63 = ( top_top @ ( set @ A ) ) ) % 0.31/0.63 => ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_UNIV_surj_inj % 0.31/0.63 thf(fact_6279_inj__image__subset__iff,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B3 ) ) % 0.31/0.63 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_image_subset_iff % 0.31/0.63 thf(fact_6280_inj__on__iff__surj,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,A3: set @ A,A16: set @ B] : % 0.31/0.63 ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( ? [F3: A > B] : % 0.31/0.63 ( ( inj_on @ A @ B @ F3 @ A3 ) % 0.31/0.63 & ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A3 ) @ A16 ) ) ) % 0.31/0.63 = ( ? [G2: B > A] : % 0.31/0.63 ( ( image @ B @ A @ G2 @ A16 ) % 0.31/0.63 = A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_iff_surj % 0.31/0.63 thf(fact_6281_finite__surj__inj,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,F2: A > A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( image @ A @ A @ F2 @ A3 ) ) % 0.31/0.63 => ( inj_on @ A @ A @ F2 @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_surj_inj % 0.31/0.63 thf(fact_6282_inj__on__finite,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B,A3: set @ A,B3: set @ B] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ A3 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.63 => ( finite_finite2 @ A @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_finite % 0.31/0.63 thf(fact_6283_endo__inj__surj,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,F2: A > A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ F2 @ A3 ) @ A3 ) % 0.31/0.63 => ( ( inj_on @ A @ A @ F2 @ A3 ) % 0.31/0.63 => ( ( image @ A @ A @ F2 @ A3 ) % 0.31/0.63 = A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % endo_inj_surj % 0.31/0.63 thf(fact_6284_inj__on__image__Int,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B,C5: set @ A,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ C5 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 ) % 0.31/0.63 => ( ( image @ A @ B @ F2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_image_Int % 0.31/0.63 thf(fact_6285_inj__on__iff__eq__card,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,A3: set @ A,F2: A > B] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( inj_on @ A @ B @ F2 @ A3 ) % 0.31/0.63 = ( ( finite_card @ B @ ( image @ A @ B @ F2 @ A3 ) ) % 0.31/0.63 = ( finite_card @ A @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_iff_eq_card % 0.31/0.63 thf(fact_6286_eq__card__imp__inj__on,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,A3: set @ A,F2: A > B] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( ( finite_card @ B @ ( image @ A @ B @ F2 @ A3 ) ) % 0.31/0.63 = ( finite_card @ A @ A3 ) ) % 0.31/0.63 => ( inj_on @ A @ B @ F2 @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % eq_card_imp_inj_on % 0.31/0.63 thf(fact_6287_inj__on__image__set__diff,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B,C5: set @ A,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ C5 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ C5 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 ) % 0.31/0.63 => ( ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_image_set_diff % 0.31/0.63 thf(fact_6288_pigeonhole,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] : % 0.31/0.63 ( ( ord_less @ nat @ ( finite_card @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( finite_card @ B @ A3 ) ) % 0.31/0.63 => ~ ( inj_on @ B @ A @ F2 @ A3 ) ) ). % 0.31/0.63 % 0.31/0.63 % pigeonhole % 0.31/0.63 thf(fact_6289_continuous__inj__imp__mono,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( ( topolo8458572112393995274pology @ A ) % 0.31/0.63 & ( topolo1944317154257567458pology @ B ) ) % 0.31/0.63 => ! [A2: A,X3: A,B2: A,F2: A > B] : % 0.31/0.63 ( ( ord_less @ A @ A2 @ X3 ) % 0.31/0.63 => ( ( ord_less @ A @ X3 @ B2 ) % 0.31/0.63 => ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F2 ) % 0.31/0.63 => ( ( inj_on @ A @ B @ F2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 0.31/0.63 => ( ( ( ord_less @ B @ ( F2 @ A2 ) @ ( F2 @ X3 ) ) % 0.31/0.63 & ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ B2 ) ) ) % 0.31/0.63 | ( ( ord_less @ B @ ( F2 @ B2 ) @ ( F2 @ X3 ) ) % 0.31/0.63 & ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ A2 ) ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % continuous_inj_imp_mono % 0.31/0.63 thf(fact_6290_the__inv__into__into,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B,A3: set @ A,X3: B,B3: set @ A] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ A3 ) % 0.31/0.63 => ( ( member @ B @ X3 @ ( image @ A @ B @ F2 @ A3 ) ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 => ( member @ A @ ( the_inv_into @ A @ B @ A3 @ F2 @ X3 ) @ B3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % the_inv_into_into % 0.31/0.63 thf(fact_6291_inj__on__UNION__chain,axiom, % 0.31/0.63 ! [C: $tType,B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B ),F2: B > C] : % 0.31/0.63 ( ! [I4: A,J2: A] : % 0.31/0.63 ( ( member @ A @ I4 @ I6 ) % 0.31/0.63 => ( ( member @ A @ J2 @ I6 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ B ) @ ( A3 @ I4 ) @ ( A3 @ J2 ) ) % 0.31/0.63 | ( ord_less_eq @ ( set @ B ) @ ( A3 @ J2 ) @ ( A3 @ I4 ) ) ) ) ) % 0.31/0.63 => ( ! [I4: A] : % 0.31/0.63 ( ( member @ A @ I4 @ I6 ) % 0.31/0.63 => ( inj_on @ B @ C @ F2 @ ( A3 @ I4 ) ) ) % 0.31/0.63 => ( inj_on @ B @ C @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_UNION_chain % 0.31/0.63 thf(fact_6292_inj__on__INTER,axiom, % 0.31/0.63 ! [C: $tType,B: $tType,A: $tType,I6: set @ A,F2: B > C,A3: A > ( set @ B )] : % 0.31/0.63 ( ( I6 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ! [I4: A] : % 0.31/0.63 ( ( member @ A @ I4 @ I6 ) % 0.31/0.63 => ( inj_on @ B @ C @ F2 @ ( A3 @ I4 ) ) ) % 0.31/0.63 => ( inj_on @ B @ C @ F2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_INTER % 0.31/0.63 thf(fact_6293_surjective__iff__injective__gen,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,S3: set @ A,T4: set @ B,F2: A > B] : % 0.31/0.63 ( ( finite_finite2 @ A @ S3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ T4 ) % 0.31/0.63 => ( ( ( finite_card @ A @ S3 ) % 0.31/0.63 = ( finite_card @ B @ T4 ) ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ S3 ) @ T4 ) % 0.31/0.63 => ( ( ! [X: B] : % 0.31/0.63 ( ( member @ B @ X @ T4 ) % 0.31/0.63 => ? [Y2: A] : % 0.31/0.63 ( ( member @ A @ Y2 @ S3 ) % 0.31/0.63 & ( ( F2 @ Y2 ) % 0.31/0.63 = X ) ) ) ) % 0.31/0.63 = ( inj_on @ A @ B @ F2 @ S3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % surjective_iff_injective_gen % 0.31/0.63 thf(fact_6294_card__bij__eq,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B3: set @ B,G: B > A] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ A3 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B3 ) % 0.31/0.63 => ( ( inj_on @ B @ A @ G @ B3 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ B3 ) @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.63 => ( ( finite_card @ A @ A3 ) % 0.31/0.63 = ( finite_card @ B @ B3 ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % card_bij_eq % 0.31/0.63 thf(fact_6295_inj__image__Compl__subset,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B,A3: set @ A] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.63 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_image_Compl_subset % 0.31/0.63 thf(fact_6296_image__INT,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,C: $tType,F2: A > B,C5: set @ A,A3: set @ C,B3: C > ( set @ A ),J: C] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ C5 ) % 0.31/0.63 => ( ! [X5: C] : % 0.31/0.63 ( ( member @ C @ X5 @ A3 ) % 0.31/0.63 => ( ord_less_eq @ ( set @ A ) @ ( B3 @ X5 ) @ C5 ) ) % 0.31/0.63 => ( ( member @ C @ J @ A3 ) % 0.31/0.63 => ( ( image @ A @ B @ F2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ C @ ( set @ A ) @ B3 @ A3 ) ) ) % 0.31/0.63 = ( complete_Inf_Inf @ ( set @ B ) % 0.31/0.63 @ ( image @ C @ ( set @ B ) % 0.31/0.63 @ ^ [X: C] : ( image @ A @ B @ F2 @ ( B3 @ X ) ) % 0.31/0.63 @ A3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % image_INT % 0.31/0.63 thf(fact_6297_card__le__inj,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.63 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ B @ B3 ) ) % 0.31/0.63 => ? [F4: A > B] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A3 ) @ B3 ) % 0.31/0.63 & ( inj_on @ A @ B @ F4 @ A3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % card_le_inj % 0.31/0.63 thf(fact_6298_card__inj__on__le,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B3: set @ B] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ A3 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.63 => ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ B @ B3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % card_inj_on_le % 0.31/0.63 thf(fact_6299_inj__on__iff__card__le,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,A3: set @ A,B3: set @ B] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.63 => ( ( ? [F3: A > B] : % 0.31/0.63 ( ( inj_on @ A @ B @ F3 @ A3 ) % 0.31/0.63 & ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A3 ) @ B3 ) ) ) % 0.31/0.63 = ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ B @ B3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_iff_card_le % 0.31/0.63 thf(fact_6300_log__inj,axiom, % 0.31/0.63 ! [B2: real] : % 0.31/0.63 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 0.31/0.63 => ( inj_on @ real @ real @ ( log @ B2 ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % log_inj % 0.31/0.63 thf(fact_6301_funpow__inj__finite,axiom, % 0.31/0.63 ! [A: $tType,P4: A > A,X3: A] : % 0.31/0.63 ( ( inj_on @ A @ A @ P4 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.63 => ( ( finite_finite2 @ A % 0.31/0.63 @ ( collect @ A % 0.31/0.63 @ ^ [Y2: A] : % 0.31/0.63 ? [N: nat] : % 0.31/0.63 ( Y2 % 0.31/0.63 = ( compow @ ( A > A ) @ N @ P4 @ X3 ) ) ) ) % 0.31/0.63 => ~ ! [N3: nat] : % 0.31/0.63 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 0.31/0.63 => ( ( compow @ ( A > A ) @ N3 @ P4 @ X3 ) % 0.31/0.63 != X3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % funpow_inj_finite % 0.31/0.63 thf(fact_6302_all__subset__image__inj,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,F2: B > A,S3: set @ B,P: ( set @ A ) > $o] : % 0.31/0.63 ( ( ! [T8: set @ A] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) @ T8 @ ( image @ B @ A @ F2 @ S3 ) ) % 0.31/0.63 => ( P @ T8 ) ) ) % 0.31/0.63 = ( ! [T8: set @ B] : % 0.31/0.63 ( ( ( ord_less_eq @ ( set @ B ) @ T8 @ S3 ) % 0.31/0.63 & ( inj_on @ B @ A @ F2 @ T8 ) ) % 0.31/0.63 => ( P @ ( image @ B @ A @ F2 @ T8 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % all_subset_image_inj % 0.31/0.63 thf(fact_6303_ex__subset__image__inj,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,F2: B > A,S3: set @ B,P: ( set @ A ) > $o] : % 0.31/0.63 ( ( ? [T8: set @ A] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) @ T8 @ ( image @ B @ A @ F2 @ S3 ) ) % 0.31/0.63 & ( P @ T8 ) ) ) % 0.31/0.63 = ( ? [T8: set @ B] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ B ) @ T8 @ S3 ) % 0.31/0.63 & ( inj_on @ B @ A @ F2 @ T8 ) % 0.31/0.63 & ( P @ ( image @ B @ A @ F2 @ T8 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ex_subset_image_inj % 0.31/0.63 thf(fact_6304_inj__singleton,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A] : % 0.31/0.63 ( inj_on @ A @ ( set @ A ) % 0.31/0.63 @ ^ [X: A] : ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 @ A3 ) ). % 0.31/0.63 % 0.31/0.63 % inj_singleton % 0.31/0.63 thf(fact_6305_inj__Some,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A] : ( inj_on @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ). % 0.31/0.63 % 0.31/0.63 % inj_Some % 0.31/0.63 thf(fact_6306_inj__Suc,axiom, % 0.31/0.63 ! [N7: set @ nat] : ( inj_on @ nat @ nat @ suc @ N7 ) ). % 0.31/0.63 % 0.31/0.63 % inj_Suc % 0.31/0.63 thf(fact_6307_inj__of__nat,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semiring_char_0 @ A ) % 0.31/0.63 => ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_of_nat % 0.31/0.63 thf(fact_6308_inj__on__diff__nat,axiom, % 0.31/0.63 ! [N7: set @ nat,K: nat] : % 0.31/0.63 ( ! [N3: nat] : % 0.31/0.63 ( ( member @ nat @ N3 @ N7 ) % 0.31/0.63 => ( ord_less_eq @ nat @ K @ N3 ) ) % 0.31/0.63 => ( inj_on @ nat @ nat % 0.31/0.63 @ ^ [N: nat] : ( minus_minus @ nat @ N @ K ) % 0.31/0.63 @ N7 ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_diff_nat % 0.31/0.63 thf(fact_6309_inj__on__set__encode,axiom, % 0.31/0.63 inj_on @ ( set @ nat ) @ nat @ nat_set_encode @ ( collect @ ( set @ nat ) @ ( finite_finite2 @ nat ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_set_encode % 0.31/0.63 thf(fact_6310_inj__on__convol__ident,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B,X8: set @ A] : % 0.31/0.63 ( inj_on @ A @ ( product_prod @ A @ B ) % 0.31/0.63 @ ^ [X: A] : ( product_Pair @ A @ B @ X @ ( F2 @ X ) ) % 0.31/0.63 @ X8 ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_convol_ident % 0.31/0.63 thf(fact_6311_swap__inj__on,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] : % 0.31/0.63 ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) % 0.31/0.63 @ ( product_case_prod @ A @ B @ ( product_prod @ B @ A ) % 0.31/0.63 @ ^ [I3: A,J3: B] : ( product_Pair @ B @ A @ J3 @ I3 ) ) % 0.31/0.63 @ A3 ) ). % 0.31/0.63 % 0.31/0.63 % swap_inj_on % 0.31/0.63 thf(fact_6312_range__inj__infinite,axiom, % 0.31/0.63 ! [A: $tType,F2: nat > A] : % 0.31/0.63 ( ( inj_on @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) % 0.31/0.63 => ~ ( finite_finite2 @ A @ ( image @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % range_inj_infinite % 0.31/0.63 thf(fact_6313_finite__imp__nat__seg__image__inj__on,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ? [N3: nat,F4: nat > A] : % 0.31/0.63 ( ( A3 % 0.31/0.63 = ( image @ nat @ A @ F4 % 0.31/0.63 @ ( collect @ nat % 0.31/0.63 @ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) ) ) % 0.31/0.63 & ( inj_on @ nat @ A @ F4 % 0.31/0.63 @ ( collect @ nat % 0.31/0.63 @ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_imp_nat_seg_image_inj_on % 0.31/0.63 thf(fact_6314_finite__imp__inj__to__nat__seg,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ? [F4: A > nat,N3: nat] : % 0.31/0.63 ( ( ( image @ A @ nat @ F4 @ A3 ) % 0.31/0.63 = ( collect @ nat % 0.31/0.63 @ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) ) % 0.31/0.63 & ( inj_on @ A @ nat @ F4 @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_imp_inj_to_nat_seg % 0.31/0.63 thf(fact_6315_inj__on__nth,axiom, % 0.31/0.63 ! [A: $tType,Xs: list @ A,I6: set @ nat] : % 0.31/0.63 ( ( distinct @ A @ Xs ) % 0.31/0.63 => ( ! [X5: nat] : % 0.31/0.63 ( ( member @ nat @ X5 @ I6 ) % 0.31/0.63 => ( ord_less @ nat @ X5 @ ( size_size @ ( list @ A ) @ Xs ) ) ) % 0.31/0.63 => ( inj_on @ nat @ A @ ( nth @ A @ Xs ) @ I6 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_nth % 0.31/0.63 thf(fact_6316_infinite__iff__countable__subset,axiom, % 0.31/0.63 ! [A: $tType,S3: set @ A] : % 0.31/0.63 ( ( ~ ( finite_finite2 @ A @ S3 ) ) % 0.31/0.63 = ( ? [F3: nat > A] : % 0.31/0.63 ( ( inj_on @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) % 0.31/0.63 & ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % infinite_iff_countable_subset % 0.31/0.63 thf(fact_6317_infinite__countable__subset,axiom, % 0.31/0.63 ! [A: $tType,S3: set @ A] : % 0.31/0.63 ( ~ ( finite_finite2 @ A @ S3 ) % 0.31/0.63 => ? [F4: nat > A] : % 0.31/0.63 ( ( inj_on @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) ) % 0.31/0.63 & ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % infinite_countable_subset % 0.31/0.63 thf(fact_6318_inj__on__funpow__least,axiom, % 0.31/0.63 ! [A: $tType,N2: nat,F2: A > A,S: A] : % 0.31/0.63 ( ( ( compow @ ( A > A ) @ N2 @ F2 @ S ) % 0.31/0.63 = S ) % 0.31/0.63 => ( ! [M3: nat] : % 0.31/0.63 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 ) % 0.31/0.63 => ( ( ord_less @ nat @ M3 @ N2 ) % 0.31/0.63 => ( ( compow @ ( A > A ) @ M3 @ F2 @ S ) % 0.31/0.63 != S ) ) ) % 0.31/0.63 => ( inj_on @ nat @ A % 0.31/0.63 @ ^ [K3: nat] : ( compow @ ( A > A ) @ K3 @ F2 @ S ) % 0.31/0.63 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_funpow_least % 0.31/0.63 thf(fact_6319_flat__lub__def,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( partial_flat_lub @ A ) % 0.31/0.63 = ( ^ [B4: A,A6: set @ A] : % 0.31/0.63 ( if @ A @ ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 % 0.31/0.63 @ ( the @ A % 0.31/0.63 @ ^ [X: A] : ( member @ A @ X @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % flat_lub_def % 0.31/0.63 thf(fact_6320_rtrancl__finite__eq__relpow,axiom, % 0.31/0.63 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R ) % 0.31/0.63 => ( ( transitive_rtrancl @ A @ R ) % 0.31/0.63 = ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) % 0.31/0.63 @ ( image @ nat @ ( set @ ( product_prod @ A @ A ) ) % 0.31/0.63 @ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) % 0.31/0.63 @ ( collect @ nat % 0.31/0.63 @ ^ [N: nat] : ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl_finite_eq_relpow % 0.31/0.63 thf(fact_6321_listrel__rtrancl__refl,axiom, % 0.31/0.63 ! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % listrel_rtrancl_refl % 0.31/0.63 thf(fact_6322_listrel__rtrancl__eq__rtrancl__listrel1,axiom, % 0.31/0.63 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 = ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % listrel_rtrancl_eq_rtrancl_listrel1 % 0.31/0.63 thf(fact_6323_listrel1__rtrancl__subset__rtrancl__listrel1,axiom, % 0.31/0.63 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ ( transitive_rtrancl @ A @ R2 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % listrel1_rtrancl_subset_rtrancl_listrel1 % 0.31/0.63 thf(fact_6324_rtrancl__induct2,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] : % 0.31/0.63 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) ) % 0.31/0.63 => ( ( P @ Ax @ Ay ) % 0.31/0.63 => ( ! [A5: A,B5: B,Aa2: A,Ba: B] : % 0.31/0.63 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B5 ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 ) % 0.31/0.63 => ( ( P @ A5 @ B5 ) % 0.31/0.63 => ( P @ Aa2 @ Ba ) ) ) ) % 0.31/0.63 => ( P @ Bx @ By ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl_induct2 % 0.31/0.63 thf(fact_6325_converse__rtranclE2,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,Xa2: A,Xb: B,Za: A,Zb: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )] : % 0.31/0.63 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa2 @ Xb ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) ) % 0.31/0.63 => ( ( ( product_Pair @ A @ B @ Xa2 @ Xb ) % 0.31/0.63 != ( product_Pair @ A @ B @ Za @ Zb ) ) % 0.31/0.63 => ~ ! [A5: A,B5: B] : % 0.31/0.63 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa2 @ Xb ) @ ( product_Pair @ A @ B @ A5 @ B5 ) ) @ R2 ) % 0.31/0.63 => ~ ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % converse_rtranclE2 % 0.31/0.63 thf(fact_6326_converse__rtrancl__induct2,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] : % 0.31/0.63 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) ) % 0.31/0.63 => ( ( P @ Bx @ By ) % 0.31/0.63 => ( ! [A5: A,B5: B,Aa2: A,Ba: B] : % 0.31/0.63 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 ) % 0.31/0.63 => ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) ) % 0.31/0.63 => ( ( P @ Aa2 @ Ba ) % 0.31/0.63 => ( P @ A5 @ B5 ) ) ) ) % 0.31/0.63 => ( P @ Ax @ Ay ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % converse_rtrancl_induct2 % 0.31/0.63 thf(fact_6327_converse__rtrancl__into__rtrancl,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C2: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % converse_rtrancl_into_rtrancl % 0.31/0.63 thf(fact_6328_converse__rtrancl__induct,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ( ( P @ B2 ) % 0.31/0.63 => ( ! [Y4: A,Z3: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ( ( P @ Z3 ) % 0.31/0.63 => ( P @ Y4 ) ) ) ) % 0.31/0.63 => ( P @ A2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % converse_rtrancl_induct % 0.31/0.63 thf(fact_6329_converse__rtranclE,axiom, % 0.31/0.63 ! [A: $tType,X3: A,Z2: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ( ( X3 != Z2 ) % 0.31/0.63 => ~ ! [Y4: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 ) % 0.31/0.63 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % converse_rtranclE % 0.31/0.63 thf(fact_6330_rtrancl__induct,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ( ( P @ A2 ) % 0.31/0.63 => ( ! [Y4: A,Z3: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ Y4 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 ) % 0.31/0.63 => ( ( P @ Y4 ) % 0.31/0.63 => ( P @ Z3 ) ) ) ) % 0.31/0.63 => ( P @ B2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl_induct % 0.31/0.63 thf(fact_6331_rtrancl__trans,axiom, % 0.31/0.63 ! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl_trans % 0.31/0.63 thf(fact_6332_rtranclE,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ( ( A2 != B2 ) % 0.31/0.63 => ~ ! [Y4: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ Y4 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ B2 ) @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtranclE % 0.31/0.63 thf(fact_6333_rtrancl_Ortrancl__into__rtrancl,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C2: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R2 ) % 0.31/0.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl.rtrancl_into_rtrancl % 0.31/0.63 thf(fact_6334_rtrancl_Ortrancl__refl,axiom, % 0.31/0.63 ! [A: $tType,A2: A,R2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl.rtrancl_refl % 0.31/0.63 thf(fact_6335_rtrancl_Osimps,axiom, % 0.31/0.63 ! [A: $tType,A12: A,A23: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 = ( ? [A4: A] : % 0.31/0.63 ( ( A12 = A4 ) % 0.31/0.63 & ( A23 = A4 ) ) % 0.31/0.63 | ? [A4: A,B4: A,C4: A] : % 0.31/0.63 ( ( A12 = A4 ) % 0.31/0.63 & ( A23 = C4 ) % 0.31/0.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C4 ) @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl.simps % 0.31/0.63 thf(fact_6336_rtrancl_Ocases,axiom, % 0.31/0.63 ! [A: $tType,A12: A,A23: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ A23 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ( ( A23 != A12 ) % 0.31/0.63 => ~ ! [B5: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B5 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A23 ) @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl.cases % 0.31/0.63 thf(fact_6337_listrel__rtrancl__trans,axiom, % 0.31/0.63 ! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Zs2: list @ A] : % 0.31/0.63 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) % 0.31/0.63 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % listrel_rtrancl_trans % 0.31/0.63 thf(fact_6338_tranclD,axiom, % 0.31/0.63 ! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R ) ) % 0.31/0.63 => ? [Z3: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z3 ) @ R ) % 0.31/0.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % tranclD % 0.31/0.63 thf(fact_6339_rtranclD,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R ) ) % 0.31/0.63 => ( ( A2 = B2 ) % 0.31/0.63 | ( ( A2 != B2 ) % 0.31/0.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtranclD % 0.31/0.63 thf(fact_6340_tranclD2,axiom, % 0.31/0.63 ! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R ) ) % 0.31/0.63 => ? [Z3: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z3 ) @ ( transitive_rtrancl @ A @ R ) ) % 0.31/0.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ R ) ) ) ). % 0.31/0.63 % 0.31/0.63 % tranclD2 % 0.31/0.63 thf(fact_6341_trancl__into__rtrancl,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl_into_rtrancl % 0.31/0.63 thf(fact_6342_rtrancl__eq__or__trancl,axiom, % 0.31/0.63 ! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R ) ) % 0.31/0.63 = ( ( X3 = Y ) % 0.31/0.63 | ( ( X3 != Y ) % 0.31/0.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl_eq_or_trancl % 0.31/0.63 thf(fact_6343_rtrancl__into__trancl1,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C2: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R2 ) % 0.31/0.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl_into_trancl1 % 0.31/0.63 thf(fact_6344_rtrancl__into__trancl2,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C2: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl_into_trancl2 % 0.31/0.63 thf(fact_6345_rtrancl__trancl__trancl,axiom, % 0.31/0.63 ! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl_trancl_trancl % 0.31/0.63 thf(fact_6346_trancl__rtrancl__trancl,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C2: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl_rtrancl_trancl % 0.31/0.63 thf(fact_6347_rtrancl__listrel1__eq__len,axiom, % 0.31/0.63 ! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) % 0.31/0.63 => ( ( size_size @ ( list @ A ) @ X3 ) % 0.31/0.63 = ( size_size @ ( list @ A ) @ Y ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl_listrel1_eq_len % 0.31/0.63 thf(fact_6348_listrel__reflcl__if__listrel1,axiom, % 0.31/0.63 ! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) % 0.31/0.63 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % listrel_reflcl_if_listrel1 % 0.31/0.63 thf(fact_6349_rtrancl__listrel1__if__listrel,axiom, % 0.31/0.63 ! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ R2 ) ) % 0.31/0.63 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl_listrel1_if_listrel % 0.31/0.63 thf(fact_6350_pred__nat__trancl__eq__le,axiom, % 0.31/0.63 ! [M: nat,N2: nat] : % 0.31/0.63 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N2 ) @ ( transitive_rtrancl @ nat @ pred_nat ) ) % 0.31/0.63 = ( ord_less_eq @ nat @ M @ N2 ) ) ). % 0.31/0.63 % 0.31/0.63 % pred_nat_trancl_eq_le % 0.31/0.63 thf(fact_6351_listrel__subset__rtrancl__listrel1,axiom, % 0.31/0.63 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel @ A @ A @ R2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % listrel_subset_rtrancl_listrel1 % 0.31/0.63 thf(fact_6352_uniformly__continuous__onD,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( ( topolo7287701948861334536_space @ A ) % 0.31/0.63 & ( topolo7287701948861334536_space @ B ) ) % 0.31/0.63 => ! [S: set @ A,F2: A > B,E5: ( product_prod @ B @ B ) > $o] : % 0.31/0.63 ( ( topolo6026614971017936543ous_on @ A @ B @ S @ F2 ) % 0.31/0.63 => ( ( eventually @ ( product_prod @ B @ B ) @ E5 @ ( topolo7806501430040627800ormity @ B ) ) % 0.31/0.63 => ( eventually @ ( product_prod @ A @ A ) % 0.31/0.63 @ ( product_case_prod @ A @ A @ $o % 0.31/0.63 @ ^ [X: A,Y2: A] : % 0.31/0.63 ( ( member @ A @ X @ S ) % 0.31/0.63 => ( ( member @ A @ Y2 @ S ) % 0.31/0.63 => ( E5 @ ( product_Pair @ B @ B @ ( F2 @ X ) @ ( F2 @ Y2 ) ) ) ) ) ) % 0.31/0.63 @ ( topolo7806501430040627800ormity @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % uniformly_continuous_onD % 0.31/0.63 thf(fact_6353_measure__function__int,axiom, % 0.31/0.63 fun_is_measure @ int @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ). % 0.31/0.63 % 0.31/0.63 % measure_function_int % 0.31/0.63 thf(fact_6354_measure__size,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( size @ A ) % 0.31/0.63 => ( fun_is_measure @ A @ ( size_size @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % measure_size % 0.31/0.63 thf(fact_6355_uniformly__continuous__on__def,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( ( real_V7819770556892013058_space @ A ) % 0.31/0.63 & ( real_V7819770556892013058_space @ B ) ) % 0.31/0.63 => ( ( topolo6026614971017936543ous_on @ A @ B ) % 0.31/0.63 = ( ^ [S8: set @ A,F3: A > B] : % 0.31/0.63 ! [E4: real] : % 0.31/0.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 ) % 0.31/0.63 => ? [D5: real] : % 0.31/0.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ D5 ) % 0.31/0.63 & ! [X: A] : % 0.31/0.63 ( ( member @ A @ X @ S8 ) % 0.31/0.63 => ! [Y2: A] : % 0.31/0.63 ( ( member @ A @ Y2 @ S8 ) % 0.31/0.63 => ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y2 @ X ) @ D5 ) % 0.31/0.63 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) ) @ E4 ) ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % uniformly_continuous_on_def % 0.31/0.63 thf(fact_6356_isUCont__def,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( ( real_V7819770556892013058_space @ A ) % 0.31/0.63 & ( real_V7819770556892013058_space @ B ) ) % 0.31/0.63 => ! [F2: A > B] : % 0.31/0.63 ( ( topolo6026614971017936543ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) % 0.31/0.63 = ( ! [R5: real] : % 0.31/0.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ R5 ) % 0.31/0.63 => ? [S8: real] : % 0.31/0.63 ( ( ord_less @ real @ ( zero_zero @ real ) @ S8 ) % 0.31/0.63 & ! [X: A,Y2: A] : % 0.31/0.63 ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ S8 ) % 0.31/0.63 => ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F2 @ X ) @ ( F2 @ Y2 ) ) @ R5 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % isUCont_def % 0.31/0.63 thf(fact_6357_set__list__bind,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,Xs: list @ B,F2: B > ( list @ A )] : % 0.31/0.63 ( ( set2 @ A @ ( bind @ B @ A @ Xs @ F2 ) ) % 0.31/0.63 = ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.63 @ ( image @ B @ ( set @ A ) % 0.31/0.63 @ ^ [X: B] : ( set2 @ A @ ( F2 @ X ) ) % 0.31/0.63 @ ( set2 @ B @ Xs ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % set_list_bind % 0.31/0.63 thf(fact_6358_has__derivative__power__int,axiom, % 0.31/0.63 ! [A: $tType,C: $tType] : % 0.31/0.63 ( ( ( real_V822414075346904944vector @ C ) % 0.31/0.63 & ( real_V3459762299906320749_field @ A ) ) % 0.31/0.63 => ! [F2: C > A,X3: C,F11: C > A,S3: set @ C,N2: int] : % 0.31/0.63 ( ( ( F2 @ X3 ) % 0.31/0.63 != ( zero_zero @ A ) ) % 0.31/0.63 => ( ( has_derivative @ C @ A @ F2 @ F11 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) % 0.31/0.63 => ( has_derivative @ C @ A % 0.31/0.63 @ ^ [X: C] : ( power_int @ A @ ( F2 @ X ) @ N2 ) % 0.31/0.63 @ ^ [H: C] : ( times_times @ A @ ( F11 @ H ) @ ( times_times @ A @ ( ring_1_of_int @ A @ N2 ) @ ( power_int @ A @ ( F2 @ X3 ) @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) ) ) ) % 0.31/0.63 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % has_derivative_power_int % 0.31/0.63 thf(fact_6359_power__int__1__left,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [N2: int] : % 0.31/0.63 ( ( power_int @ A @ ( one_one @ A ) @ N2 ) % 0.31/0.63 = ( one_one @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_1_left % 0.31/0.63 thf(fact_6360_power__int__mult__numeral,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [X3: A,M: num,N2: num] : % 0.31/0.63 ( ( power_int @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) ) % 0.31/0.63 = ( power_int @ A @ X3 @ ( numeral_numeral @ int @ ( times_times @ num @ M @ N2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_mult_numeral % 0.31/0.63 thf(fact_6361_power__int__1__right,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ( inverse @ A ) % 0.31/0.63 & ( monoid_mult @ A ) ) % 0.31/0.63 => ! [Y: A] : % 0.31/0.63 ( ( power_int @ A @ Y @ ( one_one @ int ) ) % 0.31/0.63 = Y ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_1_right % 0.31/0.63 thf(fact_6362_power__int__sgn,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [A2: A,N2: int] : % 0.31/0.63 ( ( sgn_sgn @ A @ ( power_int @ A @ A2 @ N2 ) ) % 0.31/0.63 = ( power_int @ A @ ( sgn_sgn @ A @ A2 ) @ N2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_sgn % 0.31/0.63 thf(fact_6363_bind__simps_I1_J,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: B > ( list @ A )] : % 0.31/0.63 ( ( bind @ B @ A @ ( nil @ B ) @ F2 ) % 0.31/0.63 = ( nil @ A ) ) ). % 0.31/0.63 % 0.31/0.63 % bind_simps(1) % 0.31/0.63 thf(fact_6364_power__int__mult__distrib__numeral1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( field @ A ) % 0.31/0.63 => ! [W2: num,Y: A,M: int] : % 0.31/0.63 ( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) @ M ) % 0.31/0.63 = ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M ) @ ( power_int @ A @ Y @ M ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_mult_distrib_numeral1 % 0.31/0.63 thf(fact_6365_power__int__mult__distrib__numeral2,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( field @ A ) % 0.31/0.63 => ! [X3: A,W2: num,M: int] : % 0.31/0.63 ( ( power_int @ A @ ( times_times @ A @ X3 @ ( numeral_numeral @ A @ W2 ) ) @ M ) % 0.31/0.63 = ( times_times @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_mult_distrib_numeral2 % 0.31/0.63 thf(fact_6366_power__int__0__left,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [M: int] : % 0.31/0.63 ( ( M % 0.31/0.63 != ( zero_zero @ int ) ) % 0.31/0.63 => ( ( power_int @ A @ ( zero_zero @ A ) @ M ) % 0.31/0.63 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_0_left % 0.31/0.63 thf(fact_6367_power__int__eq__0__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [X3: A,N2: int] : % 0.31/0.63 ( ( ( power_int @ A @ X3 @ N2 ) % 0.31/0.63 = ( zero_zero @ A ) ) % 0.31/0.63 = ( ( X3 % 0.31/0.63 = ( zero_zero @ A ) ) % 0.31/0.63 & ( N2 % 0.31/0.63 != ( zero_zero @ int ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_eq_0_iff % 0.31/0.63 thf(fact_6368_power__int__0__right,axiom, % 0.31/0.63 ! [B: $tType] : % 0.31/0.63 ( ( ( inverse @ B ) % 0.31/0.63 & ( power @ B ) ) % 0.31/0.63 => ! [X3: B] : % 0.31/0.63 ( ( power_int @ B @ X3 @ ( zero_zero @ int ) ) % 0.31/0.63 = ( one_one @ B ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_0_right % 0.31/0.63 thf(fact_6369_abs__power__int__minus,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [A2: A,N2: int] : % 0.31/0.63 ( ( abs_abs @ A @ ( power_int @ A @ ( uminus_uminus @ A @ A2 ) @ N2 ) ) % 0.31/0.63 = ( abs_abs @ A @ ( power_int @ A @ A2 @ N2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % abs_power_int_minus % 0.31/0.63 thf(fact_6370_power__int__of__nat,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ( inverse @ A ) % 0.31/0.63 & ( power @ A ) ) % 0.31/0.63 => ! [X3: A,N2: nat] : % 0.31/0.63 ( ( power_int @ A @ X3 @ ( semiring_1_of_nat @ int @ N2 ) ) % 0.31/0.63 = ( power_power @ A @ X3 @ N2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_of_nat % 0.31/0.63 thf(fact_6371_power__int__numeral,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ( inverse @ A ) % 0.31/0.63 & ( power @ A ) ) % 0.31/0.63 => ! [X3: A,N2: num] : % 0.31/0.63 ( ( power_int @ A @ X3 @ ( numeral_numeral @ int @ N2 ) ) % 0.31/0.63 = ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ N2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_numeral % 0.31/0.63 thf(fact_6372_power__int__minus1__right,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ( inverse @ A ) % 0.31/0.63 & ( monoid_mult @ A ) ) % 0.31/0.63 => ! [Y: A] : % 0.31/0.63 ( ( power_int @ A @ Y @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) % 0.31/0.63 = ( inverse_inverse @ A @ Y ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_minus1_right % 0.31/0.63 thf(fact_6373_power__int__add__numeral,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [X3: A,M: num,N2: num] : % 0.31/0.63 ( ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ M ) ) @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ N2 ) ) ) % 0.31/0.63 = ( power_int @ A @ X3 @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_add_numeral % 0.31/0.63 thf(fact_6374_power__int__add__numeral2,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [X3: A,M: num,N2: num,B2: A] : % 0.31/0.63 ( ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ M ) ) @ ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ N2 ) ) @ B2 ) ) % 0.31/0.63 = ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N2 ) ) ) @ B2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_add_numeral2 % 0.31/0.63 thf(fact_6375_power__int__mono__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [A2: A,B2: A,N2: int] : % 0.31/0.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 0.31/0.63 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ ( power_int @ A @ A2 @ N2 ) @ ( power_int @ A @ B2 @ N2 ) ) % 0.31/0.63 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_mono_iff % 0.31/0.63 thf(fact_6376_list__bind__cong,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F2: A > ( list @ B ),G: A > ( list @ B )] : % 0.31/0.63 ( ( Xs = Ys ) % 0.31/0.63 => ( ! [X5: A] : % 0.31/0.63 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.63 => ( ( F2 @ X5 ) % 0.31/0.63 = ( G @ X5 ) ) ) % 0.31/0.63 => ( ( bind @ A @ B @ Xs @ F2 ) % 0.31/0.63 = ( bind @ A @ B @ Ys @ G ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % list_bind_cong % 0.31/0.63 thf(fact_6377_continuous__on__power__int,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( ( topolo4958980785337419405_space @ A ) % 0.31/0.63 & ( real_V8999393235501362500lgebra @ B ) ) % 0.31/0.63 => ! [S: set @ A,F2: A > B,N2: int] : % 0.31/0.63 ( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 ) % 0.31/0.63 => ( ! [X5: A] : % 0.31/0.63 ( ( member @ A @ X5 @ S ) % 0.31/0.63 => ( ( F2 @ X5 ) % 0.31/0.63 != ( zero_zero @ B ) ) ) % 0.31/0.63 => ( topolo81223032696312382ous_on @ A @ B @ S % 0.31/0.63 @ ^ [X: A] : ( power_int @ B @ ( F2 @ X ) @ N2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % continuous_on_power_int % 0.31/0.63 thf(fact_6378_power__int__abs,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [A2: A,N2: int] : % 0.31/0.63 ( ( abs_abs @ A @ ( power_int @ A @ A2 @ N2 ) ) % 0.31/0.63 = ( power_int @ A @ ( abs_abs @ A @ A2 ) @ N2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_abs % 0.31/0.63 thf(fact_6379_power__int__mult__distrib,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( field @ A ) % 0.31/0.63 => ! [X3: A,Y: A,M: int] : % 0.31/0.63 ( ( power_int @ A @ ( times_times @ A @ X3 @ Y ) @ M ) % 0.31/0.63 = ( times_times @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ Y @ M ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_mult_distrib % 0.31/0.63 thf(fact_6380_power__int__commutes,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [X3: A,N2: int] : % 0.31/0.63 ( ( times_times @ A @ ( power_int @ A @ X3 @ N2 ) @ X3 ) % 0.31/0.63 = ( times_times @ A @ X3 @ ( power_int @ A @ X3 @ N2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_commutes % 0.31/0.63 thf(fact_6381_power__int__mult,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [X3: A,M: int,N2: int] : % 0.31/0.63 ( ( power_int @ A @ X3 @ ( times_times @ int @ M @ N2 ) ) % 0.31/0.63 = ( power_int @ A @ ( power_int @ A @ X3 @ M ) @ N2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_mult % 0.31/0.63 thf(fact_6382_power__int__inverse,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [X3: A,N2: int] : % 0.31/0.63 ( ( power_int @ A @ ( inverse_inverse @ A @ X3 ) @ N2 ) % 0.31/0.63 = ( inverse_inverse @ A @ ( power_int @ A @ X3 @ N2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_inverse % 0.31/0.63 thf(fact_6383_power__int__divide__distrib,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( field @ A ) % 0.31/0.63 => ! [X3: A,Y: A,M: int] : % 0.31/0.63 ( ( power_int @ A @ ( divide_divide @ A @ X3 @ Y ) @ M ) % 0.31/0.63 = ( divide_divide @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ Y @ M ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_divide_distrib % 0.31/0.63 thf(fact_6384_power__int__not__zero,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [X3: A,N2: int] : % 0.31/0.63 ( ( ( X3 % 0.31/0.63 != ( zero_zero @ A ) ) % 0.31/0.63 | ( N2 % 0.31/0.63 = ( zero_zero @ int ) ) ) % 0.31/0.63 => ( ( power_int @ A @ X3 @ N2 ) % 0.31/0.63 != ( zero_zero @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_not_zero % 0.31/0.63 thf(fact_6385_power__int__one__over,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [X3: A,N2: int] : % 0.31/0.63 ( ( power_int @ A @ ( divide_divide @ A @ ( one_one @ A ) @ X3 ) @ N2 ) % 0.31/0.63 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_int @ A @ X3 @ N2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_one_over % 0.31/0.63 thf(fact_6386_zero__less__power__int,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [X3: A,N2: int] : % 0.31/0.63 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.63 => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X3 @ N2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % zero_less_power_int % 0.31/0.63 thf(fact_6387_power__int__minus,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [X3: A,N2: int] : % 0.31/0.63 ( ( power_int @ A @ X3 @ ( uminus_uminus @ int @ N2 ) ) % 0.31/0.63 = ( inverse_inverse @ A @ ( power_int @ A @ X3 @ N2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_minus % 0.31/0.63 thf(fact_6388_zero__le__power__int,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [X3: A,N2: int] : % 0.31/0.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.63 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X3 @ N2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % zero_le_power_int % 0.31/0.63 thf(fact_6389_power__int__0__left__If,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [M: int] : % 0.31/0.63 ( ( ( M % 0.31/0.63 = ( zero_zero @ int ) ) % 0.31/0.63 => ( ( power_int @ A @ ( zero_zero @ A ) @ M ) % 0.31/0.63 = ( one_one @ A ) ) ) % 0.31/0.63 & ( ( M % 0.31/0.63 != ( zero_zero @ int ) ) % 0.31/0.63 => ( ( power_int @ A @ ( zero_zero @ A ) @ M ) % 0.31/0.63 = ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_0_left_If % 0.31/0.63 thf(fact_6390_power__int__increasing,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [N2: int,N7: int,A2: A] : % 0.31/0.63 ( ( ord_less_eq @ int @ N2 @ N7 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) % 0.31/0.63 => ( ord_less_eq @ A @ ( power_int @ A @ A2 @ N2 ) @ ( power_int @ A @ A2 @ N7 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_increasing % 0.31/0.63 thf(fact_6391_power__int__strict__increasing,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [N2: int,N7: int,A2: A] : % 0.31/0.63 ( ( ord_less @ int @ N2 @ N7 ) % 0.31/0.63 => ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 0.31/0.63 => ( ord_less @ A @ ( power_int @ A @ A2 @ N2 ) @ ( power_int @ A @ A2 @ N7 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_strict_increasing % 0.31/0.63 thf(fact_6392_power__int__diff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( field @ A ) % 0.31/0.63 => ! [X3: A,M: int,N2: int] : % 0.31/0.63 ( ( ( X3 % 0.31/0.63 != ( zero_zero @ A ) ) % 0.31/0.63 | ( M != N2 ) ) % 0.31/0.63 => ( ( power_int @ A @ X3 @ ( minus_minus @ int @ M @ N2 ) ) % 0.31/0.63 = ( divide_divide @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ X3 @ N2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_diff % 0.31/0.63 thf(fact_6393_tendsto__power__int,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( real_V8999393235501362500lgebra @ A ) % 0.31/0.63 => ! [F2: B > A,A2: A,F5: filter @ B,N2: int] : % 0.31/0.63 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 0.31/0.63 => ( ( A2 % 0.31/0.63 != ( zero_zero @ A ) ) % 0.31/0.63 => ( filterlim @ B @ A % 0.31/0.63 @ ^ [X: B] : ( power_int @ A @ ( F2 @ X ) @ N2 ) % 0.31/0.63 @ ( topolo7230453075368039082e_nhds @ A @ ( power_int @ A @ A2 @ N2 ) ) % 0.31/0.63 @ F5 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % tendsto_power_int % 0.31/0.63 thf(fact_6394_continuous__at__within__power__int,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( ( topological_t2_space @ A ) % 0.31/0.63 & ( real_V8999393235501362500lgebra @ B ) ) % 0.31/0.63 => ! [A2: A,S: set @ A,F2: A > B,N2: int] : % 0.31/0.63 ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F2 ) % 0.31/0.63 => ( ( ( F2 @ A2 ) % 0.31/0.63 != ( zero_zero @ B ) ) % 0.31/0.63 => ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) % 0.31/0.63 @ ^ [X: A] : ( power_int @ B @ ( F2 @ X ) @ N2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % continuous_at_within_power_int % 0.31/0.63 thf(fact_6395_differentiable__power__int,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( ( real_V822414075346904944vector @ A ) % 0.31/0.63 & ( real_V3459762299906320749_field @ B ) ) % 0.31/0.63 => ! [F2: A > B,X3: A,S: set @ A,N2: int] : % 0.31/0.63 ( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.63 => ( ( ( F2 @ X3 ) % 0.31/0.63 != ( zero_zero @ B ) ) % 0.31/0.63 => ( differentiable @ A @ B % 0.31/0.63 @ ^ [X: A] : ( power_int @ B @ ( F2 @ X ) @ N2 ) % 0.31/0.63 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % differentiable_power_int % 0.31/0.63 thf(fact_6396_continuous__power__int,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( ( topological_t2_space @ A ) % 0.31/0.63 & ( real_V8999393235501362500lgebra @ B ) ) % 0.31/0.63 => ! [F5: filter @ A,F2: A > B,N2: int] : % 0.31/0.63 ( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 ) % 0.31/0.63 => ( ( ( F2 % 0.31/0.63 @ ( topolo3827282254853284352ce_Lim @ A @ A @ F5 % 0.31/0.63 @ ^ [X: A] : X ) ) % 0.31/0.63 != ( zero_zero @ B ) ) % 0.31/0.63 => ( topolo3448309680560233919inuous @ A @ B @ F5 % 0.31/0.63 @ ^ [X: A] : ( power_int @ B @ ( F2 @ X ) @ N2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % continuous_power_int % 0.31/0.63 thf(fact_6397_power__int__strict__decreasing,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [N2: int,N7: int,A2: A] : % 0.31/0.63 ( ( ord_less @ int @ N2 @ N7 ) % 0.31/0.63 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.63 => ( ( ord_less @ A @ A2 @ ( one_one @ A ) ) % 0.31/0.63 => ( ord_less @ A @ ( power_int @ A @ A2 @ N7 ) @ ( power_int @ A @ A2 @ N2 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_strict_decreasing % 0.31/0.63 thf(fact_6398_power__int__mono,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [X3: A,Y: A,N2: int] : % 0.31/0.63 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.63 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.63 => ( ord_less_eq @ A @ ( power_int @ A @ X3 @ N2 ) @ ( power_int @ A @ Y @ N2 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_mono % 0.31/0.63 thf(fact_6399_power__int__strict__antimono,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [A2: A,B2: A,N2: int] : % 0.31/0.63 ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.63 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.63 => ( ( ord_less @ int @ N2 @ ( zero_zero @ int ) ) % 0.31/0.63 => ( ord_less @ A @ ( power_int @ A @ B2 @ N2 ) @ ( power_int @ A @ A2 @ N2 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_strict_antimono % 0.31/0.63 thf(fact_6400_one__le__power__int,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [X3: A,N2: int] : % 0.31/0.63 ( ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) % 0.31/0.63 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) % 0.31/0.63 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X3 @ N2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % one_le_power_int % 0.31/0.63 thf(fact_6401_one__less__power__int,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [A2: A,N2: int] : % 0.31/0.63 ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 0.31/0.63 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 ) % 0.31/0.63 => ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A2 @ N2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % one_less_power_int % 0.31/0.63 thf(fact_6402_power__int__add,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [X3: A,M: int,N2: int] : % 0.31/0.63 ( ( ( X3 % 0.31/0.63 != ( zero_zero @ A ) ) % 0.31/0.63 | ( ( plus_plus @ int @ M @ N2 ) % 0.31/0.63 != ( zero_zero @ int ) ) ) % 0.31/0.63 => ( ( power_int @ A @ X3 @ ( plus_plus @ int @ M @ N2 ) ) % 0.31/0.63 = ( times_times @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ X3 @ N2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_add % 0.31/0.63 thf(fact_6403_power__int__antimono,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [A2: A,B2: A,N2: int] : % 0.31/0.63 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.63 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.63 => ( ( ord_less @ int @ N2 @ ( zero_zero @ int ) ) % 0.31/0.63 => ( ord_less_eq @ A @ ( power_int @ A @ B2 @ N2 ) @ ( power_int @ A @ A2 @ N2 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_antimono % 0.31/0.63 thf(fact_6404_power__int__strict__mono,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [A2: A,B2: A,N2: int] : % 0.31/0.63 ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.63 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 ) % 0.31/0.63 => ( ord_less @ A @ ( power_int @ A @ A2 @ N2 ) @ ( power_int @ A @ B2 @ N2 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_strict_mono % 0.31/0.63 thf(fact_6405_power__int__decreasing,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [N2: int,N7: int,A2: A] : % 0.31/0.63 ( ( ord_less_eq @ int @ N2 @ N7 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) % 0.31/0.63 => ( ( ( A2 % 0.31/0.63 != ( zero_zero @ A ) ) % 0.31/0.63 | ( N7 % 0.31/0.63 != ( zero_zero @ int ) ) % 0.31/0.63 | ( N2 % 0.31/0.63 = ( zero_zero @ int ) ) ) % 0.31/0.63 => ( ord_less_eq @ A @ ( power_int @ A @ A2 @ N7 ) @ ( power_int @ A @ A2 @ N2 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_decreasing % 0.31/0.63 thf(fact_6406_power__int__le__one,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [X3: A,N2: int] : % 0.31/0.63 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 0.31/0.63 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ X3 @ ( one_one @ A ) ) % 0.31/0.63 => ( ord_less_eq @ A @ ( power_int @ A @ X3 @ N2 ) @ ( one_one @ A ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_le_one % 0.31/0.63 thf(fact_6407_power__int__le__imp__le__exp,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [X3: A,M: int,N2: int] : % 0.31/0.63 ( ( ord_less @ A @ ( one_one @ A ) @ X3 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ X3 @ N2 ) ) % 0.31/0.63 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) % 0.31/0.63 => ( ord_less_eq @ int @ M @ N2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_le_imp_le_exp % 0.31/0.63 thf(fact_6408_power__int__le__imp__less__exp,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [X3: A,M: int,N2: int] : % 0.31/0.63 ( ( ord_less @ A @ ( one_one @ A ) @ X3 ) % 0.31/0.63 => ( ( ord_less @ A @ ( power_int @ A @ X3 @ M ) @ ( power_int @ A @ X3 @ N2 ) ) % 0.31/0.63 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) % 0.31/0.63 => ( ord_less @ int @ M @ N2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_le_imp_less_exp % 0.31/0.63 thf(fact_6409_power__int__minus__mult,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( field @ A ) % 0.31/0.63 => ! [X3: A,N2: int] : % 0.31/0.63 ( ( ( X3 % 0.31/0.63 != ( zero_zero @ A ) ) % 0.31/0.63 | ( N2 % 0.31/0.63 != ( zero_zero @ int ) ) ) % 0.31/0.63 => ( ( times_times @ A @ ( power_int @ A @ X3 @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) ) @ X3 ) % 0.31/0.63 = ( power_int @ A @ X3 @ N2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_minus_mult % 0.31/0.63 thf(fact_6410_power__int__add__1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [X3: A,M: int] : % 0.31/0.63 ( ( ( X3 % 0.31/0.63 != ( zero_zero @ A ) ) % 0.31/0.63 | ( M % 0.31/0.63 != ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) % 0.31/0.63 => ( ( power_int @ A @ X3 @ ( plus_plus @ int @ M @ ( one_one @ int ) ) ) % 0.31/0.63 = ( times_times @ A @ ( power_int @ A @ X3 @ M ) @ X3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_add_1 % 0.31/0.63 thf(fact_6411_power__int__add__1_H,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( division_ring @ A ) % 0.31/0.63 => ! [X3: A,M: int] : % 0.31/0.63 ( ( ( X3 % 0.31/0.63 != ( zero_zero @ A ) ) % 0.31/0.63 | ( M % 0.31/0.63 != ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) % 0.31/0.63 => ( ( power_int @ A @ X3 @ ( plus_plus @ int @ M @ ( one_one @ int ) ) ) % 0.31/0.63 = ( times_times @ A @ X3 @ ( power_int @ A @ X3 @ M ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_add_1' % 0.31/0.63 thf(fact_6412_power__int__def,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ( inverse @ A ) % 0.31/0.63 & ( power @ A ) ) % 0.31/0.63 => ( ( power_int @ A ) % 0.31/0.63 = ( ^ [X: A,N: int] : ( if @ A @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N ) @ ( power_power @ A @ X @ ( nat2 @ N ) ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ ( nat2 @ ( uminus_uminus @ int @ N ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_def % 0.31/0.63 thf(fact_6413_powr__real__of__int_H,axiom, % 0.31/0.63 ! [X3: real,N2: int] : % 0.31/0.63 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 0.31/0.63 => ( ( ( X3 % 0.31/0.63 != ( zero_zero @ real ) ) % 0.31/0.63 | ( ord_less @ int @ ( zero_zero @ int ) @ N2 ) ) % 0.31/0.63 => ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ N2 ) ) % 0.31/0.63 = ( power_int @ real @ X3 @ N2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % powr_real_of_int' % 0.31/0.63 thf(fact_6414_DERIV__power__int,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.63 => ! [F2: A > A,D2: A,X3: A,S: set @ A,N2: int] : % 0.31/0.63 ( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 0.31/0.63 => ( ( ( F2 @ X3 ) % 0.31/0.63 != ( zero_zero @ A ) ) % 0.31/0.63 => ( has_field_derivative @ A % 0.31/0.63 @ ^ [X: A] : ( power_int @ A @ ( F2 @ X ) @ N2 ) % 0.31/0.63 @ ( times_times @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ N2 ) @ ( power_int @ A @ ( F2 @ X3 ) @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) ) ) @ D2 ) % 0.31/0.63 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % DERIV_power_int % 0.31/0.63 thf(fact_6415_has__derivative__power__int_H,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( real_V3459762299906320749_field @ A ) % 0.31/0.63 => ! [X3: A,N2: int,S3: set @ A] : % 0.31/0.63 ( ( X3 % 0.31/0.63 != ( zero_zero @ A ) ) % 0.31/0.63 => ( has_derivative @ A @ A % 0.31/0.63 @ ^ [X: A] : ( power_int @ A @ X @ N2 ) % 0.31/0.63 @ ^ [Y2: A] : ( times_times @ A @ Y2 @ ( times_times @ A @ ( ring_1_of_int @ A @ N2 ) @ ( power_int @ A @ X3 @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) ) ) ) % 0.31/0.63 @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % has_derivative_power_int' % 0.31/0.63 thf(fact_6416_power__int__numeral__neg__numeral,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_field @ A ) % 0.31/0.63 => ! [M: num,N2: num] : % 0.31/0.63 ( ( power_int @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) % 0.31/0.63 = ( inverse_inverse @ A @ ( numeral_numeral @ A @ ( pow @ M @ N2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % power_int_numeral_neg_numeral % 0.31/0.63 thf(fact_6417_eventually__less__ceiling,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( ( archim2362893244070406136eiling @ B ) % 0.31/0.63 & ( topolo2564578578187576103pology @ B ) ) % 0.31/0.63 => ! [F2: A > B,L: B,F5: filter @ A] : % 0.31/0.63 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) % 0.31/0.63 => ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) ) % 0.31/0.63 => ( eventually @ A % 0.31/0.63 @ ^ [X: A] : ( ord_less @ B @ ( F2 @ X ) @ ( ring_1_of_int @ B @ ( archimedean_ceiling @ B @ L ) ) ) % 0.31/0.63 @ F5 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % eventually_less_ceiling % 0.31/0.63 thf(fact_6418_Ints__sum,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( ring_1 @ B ) % 0.31/0.63 => ! [A3: set @ A,F2: A > B] : % 0.31/0.63 ( ! [X5: A] : % 0.31/0.63 ( ( member @ A @ X5 @ A3 ) % 0.31/0.63 => ( member @ B @ ( F2 @ X5 ) @ ( ring_1_Ints @ B ) ) ) % 0.31/0.63 => ( member @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( ring_1_Ints @ B ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_sum % 0.31/0.63 thf(fact_6419_Ints__prod,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( ( comm_monoid_mult @ B ) % 0.31/0.63 & ( ring_1 @ B ) ) % 0.31/0.63 => ! [A3: set @ A,F2: A > B] : % 0.31/0.63 ( ! [X5: A] : % 0.31/0.63 ( ( member @ A @ X5 @ A3 ) % 0.31/0.63 => ( member @ B @ ( F2 @ X5 ) @ ( ring_1_Ints @ B ) ) ) % 0.31/0.63 => ( member @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) @ ( ring_1_Ints @ B ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_prod % 0.31/0.63 thf(fact_6420_frac__eq__0__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.63 => ! [X3: A] : % 0.31/0.63 ( ( ( archimedean_frac @ A @ X3 ) % 0.31/0.63 = ( zero_zero @ A ) ) % 0.31/0.63 = ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % frac_eq_0_iff % 0.31/0.63 thf(fact_6421_floor__add2,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.63 => ! [X3: A,Y: A] : % 0.31/0.63 ( ( ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 | ( member @ A @ Y @ ( ring_1_Ints @ A ) ) ) % 0.31/0.63 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 0.31/0.63 = ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % floor_add2 % 0.31/0.63 thf(fact_6422_frac__gt__0__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.63 => ! [X3: A] : % 0.31/0.63 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X3 ) ) % 0.31/0.63 = ( ~ ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % frac_gt_0_iff % 0.31/0.63 thf(fact_6423_Ints__double__eq__0__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_char_0 @ A ) % 0.31/0.63 => ! [A2: A] : % 0.31/0.63 ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( ( ( plus_plus @ A @ A2 @ A2 ) % 0.31/0.63 = ( zero_zero @ A ) ) % 0.31/0.63 = ( A2 % 0.31/0.63 = ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_double_eq_0_iff % 0.31/0.63 thf(fact_6424_Ints__0,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_1 @ A ) % 0.31/0.63 => ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_0 % 0.31/0.63 thf(fact_6425_Ints__minus,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_1 @ A ) % 0.31/0.63 => ! [A2: A] : % 0.31/0.63 ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( member @ A @ ( uminus_uminus @ A @ A2 ) @ ( ring_1_Ints @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_minus % 0.31/0.63 thf(fact_6426_minus__in__Ints__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_1 @ A ) % 0.31/0.63 => ! [X3: A] : % 0.31/0.63 ( ( member @ A @ ( uminus_uminus @ A @ X3 ) @ ( ring_1_Ints @ A ) ) % 0.31/0.63 = ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % minus_in_Ints_iff % 0.31/0.63 thf(fact_6427_Ints__of__nat,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_1 @ A ) % 0.31/0.63 => ! [N2: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( ring_1_Ints @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_of_nat % 0.31/0.63 thf(fact_6428_Ints__power,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_1 @ A ) % 0.31/0.63 => ! [A2: A,N2: nat] : % 0.31/0.63 ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( member @ A @ ( power_power @ A @ A2 @ N2 ) @ ( ring_1_Ints @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_power % 0.31/0.63 thf(fact_6429_Ints__of__int,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_1 @ A ) % 0.31/0.63 => ! [Z2: int] : ( member @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( ring_1_Ints @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_of_int % 0.31/0.63 thf(fact_6430_Ints__induct,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_1 @ A ) % 0.31/0.63 => ! [Q4: A,P: A > $o] : % 0.31/0.63 ( ( member @ A @ Q4 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( ! [Z3: int] : ( P @ ( ring_1_of_int @ A @ Z3 ) ) % 0.31/0.63 => ( P @ Q4 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_induct % 0.31/0.63 thf(fact_6431_Ints__cases,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_1 @ A ) % 0.31/0.63 => ! [Q4: A] : % 0.31/0.63 ( ( member @ A @ Q4 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ~ ! [Z3: int] : % 0.31/0.63 ( Q4 % 0.31/0.63 != ( ring_1_of_int @ A @ Z3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_cases % 0.31/0.63 thf(fact_6432_Ints__add,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_1 @ A ) % 0.31/0.63 => ! [A2: A,B2: A] : % 0.31/0.63 ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_add % 0.31/0.63 thf(fact_6433_Ints__1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_1 @ A ) % 0.31/0.63 => ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_1 % 0.31/0.63 thf(fact_6434_Ints__diff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_1 @ A ) % 0.31/0.63 => ! [A2: A,B2: A] : % 0.31/0.63 ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( member @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_diff % 0.31/0.63 thf(fact_6435_Ints__mult,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_1 @ A ) % 0.31/0.63 => ! [A2: A,B2: A] : % 0.31/0.63 ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_mult % 0.31/0.63 thf(fact_6436_Ints__numeral,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_1 @ A ) % 0.31/0.63 => ! [N2: num] : ( member @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_Ints @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_numeral % 0.31/0.63 thf(fact_6437_Ints__abs,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_idom @ A ) % 0.31/0.63 => ! [A2: A] : % 0.31/0.63 ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( member @ A @ ( abs_abs @ A @ A2 ) @ ( ring_1_Ints @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_abs % 0.31/0.63 thf(fact_6438_Ints__subset__Rats,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( field_char_0 @ A ) % 0.31/0.63 => ( ord_less_eq @ ( set @ A ) @ ( ring_1_Ints @ A ) @ ( field_char_0_Rats @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_subset_Rats % 0.31/0.63 thf(fact_6439_Nats__subset__Ints,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_1 @ A ) % 0.31/0.63 => ( ord_less_eq @ ( set @ A ) @ ( semiring_1_Nats @ A ) @ ( ring_1_Ints @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Nats_subset_Ints % 0.31/0.63 thf(fact_6440_finite__int__segment,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.63 => ! [A2: A,B2: A] : % 0.31/0.63 ( finite_finite2 @ A % 0.31/0.63 @ ( collect @ A % 0.31/0.63 @ ^ [X: A] : % 0.31/0.63 ( ( member @ A @ X @ ( ring_1_Ints @ A ) ) % 0.31/0.63 & ( ord_less_eq @ A @ A2 @ X ) % 0.31/0.63 & ( ord_less_eq @ A @ X @ B2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_int_segment % 0.31/0.63 thf(fact_6441_Ints__odd__nonzero,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_char_0 @ A ) % 0.31/0.63 => ! [A2: A] : % 0.31/0.63 ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ A2 ) % 0.31/0.63 != ( zero_zero @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_odd_nonzero % 0.31/0.63 thf(fact_6442_of__int__divide__in__Ints,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( idom_divide @ A ) % 0.31/0.63 => ! [B2: int,A2: int] : % 0.31/0.63 ( ( dvd_dvd @ int @ B2 @ A2 ) % 0.31/0.63 => ( member @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A2 ) @ ( ring_1_of_int @ A @ B2 ) ) @ ( ring_1_Ints @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % of_int_divide_in_Ints % 0.31/0.63 thf(fact_6443_finite__abs__int__segment,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.63 => ! [A2: A] : % 0.31/0.63 ( finite_finite2 @ A % 0.31/0.63 @ ( collect @ A % 0.31/0.63 @ ^ [K3: A] : % 0.31/0.63 ( ( member @ A @ K3 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 & ( ord_less_eq @ A @ ( abs_abs @ A @ K3 ) @ A2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_abs_int_segment % 0.31/0.63 thf(fact_6444_Nats__altdef2,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_idom @ A ) % 0.31/0.63 => ( ( semiring_1_Nats @ A ) % 0.31/0.63 = ( collect @ A % 0.31/0.63 @ ^ [N: A] : % 0.31/0.63 ( ( member @ A @ N @ ( ring_1_Ints @ A ) ) % 0.31/0.63 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ N ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Nats_altdef2 % 0.31/0.63 thf(fact_6445_Ints__def,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ring_1 @ A ) % 0.31/0.63 => ( ( ring_1_Ints @ A ) % 0.31/0.63 = ( image @ int @ A @ ( ring_1_of_int @ A ) @ ( top_top @ ( set @ int ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_def % 0.31/0.63 thf(fact_6446_Ints__odd__less__0,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_idom @ A ) % 0.31/0.63 => ! [A2: A] : % 0.31/0.63 ( ( member @ A @ A2 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ A2 ) @ ( zero_zero @ A ) ) % 0.31/0.63 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_odd_less_0 % 0.31/0.63 thf(fact_6447_Ints__nonzero__abs__ge1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_idom @ A ) % 0.31/0.63 => ! [X3: A] : % 0.31/0.63 ( ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( ( X3 % 0.31/0.63 != ( zero_zero @ A ) ) % 0.31/0.63 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_nonzero_abs_ge1 % 0.31/0.63 thf(fact_6448_Ints__nonzero__abs__less1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_idom @ A ) % 0.31/0.63 => ! [X3: A] : % 0.31/0.63 ( ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( ( ord_less @ A @ ( abs_abs @ A @ X3 ) @ ( one_one @ A ) ) % 0.31/0.63 => ( X3 % 0.31/0.63 = ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_nonzero_abs_less1 % 0.31/0.63 thf(fact_6449_Ints__eq__abs__less1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linordered_idom @ A ) % 0.31/0.63 => ! [X3: A,Y: A] : % 0.31/0.63 ( ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( ( member @ A @ Y @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( ( X3 = Y ) % 0.31/0.63 = ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ints_eq_abs_less1 % 0.31/0.63 thf(fact_6450_frac__neg,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.63 => ! [X3: A] : % 0.31/0.63 ( ( ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X3 ) ) % 0.31/0.63 = ( zero_zero @ A ) ) ) % 0.31/0.63 & ( ~ ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) % 0.31/0.63 => ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X3 ) ) % 0.31/0.63 = ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % frac_neg % 0.31/0.63 thf(fact_6451_le__mult__floor__Ints,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( ( archim2362893244070406136eiling @ B ) % 0.31/0.63 & ( linordered_idom @ A ) ) % 0.31/0.63 => ! [A2: B,B2: B] : % 0.31/0.63 ( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A2 ) % 0.31/0.63 => ( ( member @ B @ A2 @ ( ring_1_Ints @ B ) ) % 0.31/0.63 => ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A2 ) @ ( archim6421214686448440834_floor @ B @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A2 @ B2 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % le_mult_floor_Ints % 0.31/0.63 thf(fact_6452_frac__unique__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( archim2362893244070406136eiling @ A ) % 0.31/0.63 => ! [X3: A,A2: A] : % 0.31/0.63 ( ( ( archimedean_frac @ A @ X3 ) % 0.31/0.63 = A2 ) % 0.31/0.63 = ( ( member @ A @ ( minus_minus @ A @ X3 @ A2 ) @ ( ring_1_Ints @ A ) ) % 0.31/0.63 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 0.31/0.63 & ( ord_less @ A @ A2 @ ( one_one @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % frac_unique_iff % 0.31/0.63 thf(fact_6453_mult__ceiling__le__Ints,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( ( archim2362893244070406136eiling @ B ) % 0.31/0.63 & ( linordered_idom @ A ) ) % 0.31/0.63 => ! [A2: B,B2: B] : % 0.31/0.63 ( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A2 ) % 0.31/0.63 => ( ( member @ B @ A2 @ ( ring_1_Ints @ B ) ) % 0.31/0.63 => ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A2 @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A2 ) @ ( archimedean_ceiling @ B @ B2 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % mult_ceiling_le_Ints % 0.31/0.63 thf(fact_6454_eventually__floor__less,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( ( archim2362893244070406136eiling @ B ) % 0.31/0.63 & ( topolo2564578578187576103pology @ B ) ) % 0.31/0.63 => ! [F2: A > B,L: B,F5: filter @ A] : % 0.31/0.63 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) % 0.31/0.63 => ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) ) % 0.31/0.63 => ( eventually @ A % 0.31/0.63 @ ^ [X: A] : ( ord_less @ B @ ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ B @ L ) ) @ ( F2 @ X ) ) % 0.31/0.63 @ F5 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % eventually_floor_less % 0.31/0.63 thf(fact_6455_Max_Oeq__fold_H,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ( ( lattic643756798349783984er_Max @ A ) % 0.31/0.63 = ( ^ [A6: set @ A] : % 0.31/0.63 ( the2 @ A % 0.31/0.63 @ ( finite_fold @ A @ ( option @ A ) % 0.31/0.63 @ ^ [X: A,Y2: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( ord_max @ A @ X ) @ Y2 ) ) % 0.31/0.63 @ ( none @ A ) % 0.31/0.63 @ A6 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Max.eq_fold' % 0.31/0.63 thf(fact_6456_UN__le__eq__Un0,axiom, % 0.31/0.63 ! [A: $tType,M7: nat > ( set @ A ),N2: nat] : % 0.31/0.63 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ) @ ( M7 @ ( zero_zero @ nat ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % UN_le_eq_Un0 % 0.31/0.63 thf(fact_6457_UnCI,axiom, % 0.31/0.63 ! [A: $tType,C2: A,B3: set @ A,A3: set @ A] : % 0.31/0.63 ( ( ~ ( member @ A @ C2 @ B3 ) % 0.31/0.63 => ( member @ A @ C2 @ A3 ) ) % 0.31/0.63 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % UnCI % 0.31/0.63 thf(fact_6458_Un__iff,axiom, % 0.31/0.63 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( ( member @ A @ C2 @ A3 ) % 0.31/0.63 | ( member @ A @ C2 @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_iff % 0.31/0.63 thf(fact_6459_sup_Obounded__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [B2: A,C2: A,A2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 ) % 0.31/0.63 = ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.63 & ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.bounded_iff % 0.31/0.63 thf(fact_6460_le__sup__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [X3: A,Y: A,Z2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y ) @ Z2 ) % 0.31/0.63 = ( ( ord_less_eq @ A @ X3 @ Z2 ) % 0.31/0.63 & ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % le_sup_iff % 0.31/0.63 thf(fact_6461_sup__bot_Oright__neutral,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( bounde4967611905675639751up_bot @ A ) % 0.31/0.63 => ! [A2: A] : % 0.31/0.63 ( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) ) % 0.31/0.63 = A2 ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_bot.right_neutral % 0.31/0.63 thf(fact_6462_sup__bot_Oneutr__eq__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( bounde4967611905675639751up_bot @ A ) % 0.31/0.63 => ! [A2: A,B2: A] : % 0.31/0.63 ( ( ( bot_bot @ A ) % 0.31/0.63 = ( sup_sup @ A @ A2 @ B2 ) ) % 0.31/0.63 = ( ( A2 % 0.31/0.63 = ( bot_bot @ A ) ) % 0.31/0.63 & ( B2 % 0.31/0.63 = ( bot_bot @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_bot.neutr_eq_iff % 0.31/0.63 thf(fact_6463_sup__bot_Oleft__neutral,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( bounde4967611905675639751up_bot @ A ) % 0.31/0.63 => ! [A2: A] : % 0.31/0.63 ( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 ) % 0.31/0.63 = A2 ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_bot.left_neutral % 0.31/0.63 thf(fact_6464_sup__bot_Oeq__neutr__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( bounde4967611905675639751up_bot @ A ) % 0.31/0.63 => ! [A2: A,B2: A] : % 0.31/0.63 ( ( ( sup_sup @ A @ A2 @ B2 ) % 0.31/0.63 = ( bot_bot @ A ) ) % 0.31/0.63 = ( ( A2 % 0.31/0.63 = ( bot_bot @ A ) ) % 0.31/0.63 & ( B2 % 0.31/0.63 = ( bot_bot @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_bot.eq_neutr_iff % 0.31/0.63 thf(fact_6465_sup__eq__bot__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( bounde4967611905675639751up_bot @ A ) % 0.31/0.63 => ! [X3: A,Y: A] : % 0.31/0.63 ( ( ( sup_sup @ A @ X3 @ Y ) % 0.31/0.63 = ( bot_bot @ A ) ) % 0.31/0.63 = ( ( X3 % 0.31/0.63 = ( bot_bot @ A ) ) % 0.31/0.63 & ( Y % 0.31/0.63 = ( bot_bot @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_eq_bot_iff % 0.31/0.63 thf(fact_6466_bot__eq__sup__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( bounde4967611905675639751up_bot @ A ) % 0.31/0.63 => ! [X3: A,Y: A] : % 0.31/0.63 ( ( ( bot_bot @ A ) % 0.31/0.63 = ( sup_sup @ A @ X3 @ Y ) ) % 0.31/0.63 = ( ( X3 % 0.31/0.63 = ( bot_bot @ A ) ) % 0.31/0.63 & ( Y % 0.31/0.63 = ( bot_bot @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % bot_eq_sup_iff % 0.31/0.63 thf(fact_6467_sup__bot__right,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( bounde4967611905675639751up_bot @ A ) % 0.31/0.63 => ! [X3: A] : % 0.31/0.63 ( ( sup_sup @ A @ X3 @ ( bot_bot @ A ) ) % 0.31/0.63 = X3 ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_bot_right % 0.31/0.63 thf(fact_6468_sup__bot__left,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( bounde4967611905675639751up_bot @ A ) % 0.31/0.63 => ! [X3: A] : % 0.31/0.63 ( ( sup_sup @ A @ ( bot_bot @ A ) @ X3 ) % 0.31/0.63 = X3 ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_bot_left % 0.31/0.63 thf(fact_6469_Un__empty,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( ( sup_sup @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 = ( ( A3 % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 & ( B3 % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_empty % 0.31/0.63 thf(fact_6470_finite__Un,axiom, % 0.31/0.63 ! [A: $tType,F5: set @ A,G6: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G6 ) ) % 0.31/0.63 = ( ( finite_finite2 @ A @ F5 ) % 0.31/0.63 & ( finite_finite2 @ A @ G6 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_Un % 0.31/0.63 thf(fact_6471_Un__subset__iff,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C5 ) % 0.31/0.63 = ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) % 0.31/0.63 & ( ord_less_eq @ ( set @ A ) @ B3 @ C5 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_subset_iff % 0.31/0.63 thf(fact_6472_Un__insert__right,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) % 0.31/0.63 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_insert_right % 0.31/0.63 thf(fact_6473_Un__insert__left,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B3: set @ A,C5: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C5 ) % 0.31/0.63 = ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_insert_left % 0.31/0.63 thf(fact_6474_Int__Un__eq_I4_J,axiom, % 0.31/0.63 ! [A: $tType,T4: set @ A,S3: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ T4 @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) ) % 0.31/0.63 = T4 ) ). % 0.31/0.63 % 0.31/0.63 % Int_Un_eq(4) % 0.31/0.63 thf(fact_6475_Int__Un__eq_I3_J,axiom, % 0.31/0.63 ! [A: $tType,S3: set @ A,T4: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ S3 @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) ) % 0.31/0.63 = S3 ) ). % 0.31/0.63 % 0.31/0.63 % Int_Un_eq(3) % 0.31/0.63 thf(fact_6476_Int__Un__eq_I2_J,axiom, % 0.31/0.63 ! [A: $tType,S3: set @ A,T4: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) @ T4 ) % 0.31/0.63 = T4 ) ). % 0.31/0.63 % 0.31/0.63 % Int_Un_eq(2) % 0.31/0.63 thf(fact_6477_Int__Un__eq_I1_J,axiom, % 0.31/0.63 ! [A: $tType,S3: set @ A,T4: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S3 @ T4 ) @ S3 ) % 0.31/0.63 = S3 ) ). % 0.31/0.63 % 0.31/0.63 % Int_Un_eq(1) % 0.31/0.63 thf(fact_6478_Un__Int__eq_I4_J,axiom, % 0.31/0.63 ! [A: $tType,T4: set @ A,S3: set @ A] : % 0.31/0.63 ( ( inf_inf @ ( set @ A ) @ T4 @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) ) % 0.31/0.63 = T4 ) ). % 0.31/0.63 % 0.31/0.63 % Un_Int_eq(4) % 0.31/0.63 thf(fact_6479_Un__Int__eq_I3_J,axiom, % 0.31/0.63 ! [A: $tType,S3: set @ A,T4: set @ A] : % 0.31/0.63 ( ( inf_inf @ ( set @ A ) @ S3 @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) ) % 0.31/0.63 = S3 ) ). % 0.31/0.63 % 0.31/0.63 % Un_Int_eq(3) % 0.31/0.63 thf(fact_6480_Un__Int__eq_I2_J,axiom, % 0.31/0.63 ! [A: $tType,S3: set @ A,T4: set @ A] : % 0.31/0.63 ( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) @ T4 ) % 0.31/0.63 = T4 ) ). % 0.31/0.63 % 0.31/0.63 % Un_Int_eq(2) % 0.31/0.63 thf(fact_6481_Un__Int__eq_I1_J,axiom, % 0.31/0.63 ! [A: $tType,S3: set @ A,T4: set @ A] : % 0.31/0.63 ( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) @ S3 ) % 0.31/0.63 = S3 ) ). % 0.31/0.63 % 0.31/0.63 % Un_Int_eq(1) % 0.31/0.63 thf(fact_6482_Un__Diff__cancel2,axiom, % 0.31/0.63 ! [A: $tType,B3: set @ A,A3: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) @ A3 ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ B3 @ A3 ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_Diff_cancel2 % 0.31/0.63 thf(fact_6483_Un__Diff__cancel,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_Diff_cancel % 0.31/0.63 thf(fact_6484_fold__empty,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: B > A > A,Z2: A] : % 0.31/0.63 ( ( finite_fold @ B @ A @ F2 @ Z2 @ ( bot_bot @ ( set @ B ) ) ) % 0.31/0.63 = Z2 ) ). % 0.31/0.63 % 0.31/0.63 % fold_empty % 0.31/0.63 thf(fact_6485_fold__infinite,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,A3: set @ A,F2: A > B > B,Z2: B] : % 0.31/0.63 ( ~ ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( finite_fold @ A @ B @ F2 @ Z2 @ A3 ) % 0.31/0.63 = Z2 ) ) ). % 0.31/0.63 % 0.31/0.63 % fold_infinite % 0.31/0.63 thf(fact_6486_Compl__Diff__eq,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ B3 ) ) ). % 0.31/0.63 % 0.31/0.63 % Compl_Diff_eq % 0.31/0.63 thf(fact_6487_if__image__distrib,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,P: B > $o,F2: B > A,G: B > A,S3: set @ B] : % 0.31/0.63 ( ( image @ B @ A % 0.31/0.63 @ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.63 @ S3 ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ S3 @ ( collect @ B @ P ) ) ) % 0.31/0.63 @ ( image @ B @ A @ G % 0.31/0.63 @ ( inf_inf @ ( set @ B ) @ S3 % 0.31/0.63 @ ( collect @ B % 0.31/0.63 @ ^ [X: B] : % 0.31/0.63 ~ ( P @ X ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % if_image_distrib % 0.31/0.63 thf(fact_6488_set__union,axiom, % 0.31/0.63 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.63 ( ( set2 @ A @ ( union @ A @ Xs @ Ys ) ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ). % 0.31/0.63 % 0.31/0.63 % set_union % 0.31/0.63 thf(fact_6489_UN__simps_I2_J,axiom, % 0.31/0.63 ! [C: $tType,D: $tType,C5: set @ C,A3: C > ( set @ D ),B3: set @ D] : % 0.31/0.63 ( ( ( C5 % 0.31/0.63 = ( bot_bot @ ( set @ C ) ) ) % 0.31/0.63 => ( ( complete_Sup_Sup @ ( set @ D ) % 0.31/0.63 @ ( image @ C @ ( set @ D ) % 0.31/0.63 @ ^ [X: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X ) @ B3 ) % 0.31/0.63 @ C5 ) ) % 0.31/0.63 = ( bot_bot @ ( set @ D ) ) ) ) % 0.31/0.63 & ( ( C5 % 0.31/0.63 != ( bot_bot @ ( set @ C ) ) ) % 0.31/0.63 => ( ( complete_Sup_Sup @ ( set @ D ) % 0.31/0.63 @ ( image @ C @ ( set @ D ) % 0.31/0.63 @ ^ [X: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X ) @ B3 ) % 0.31/0.63 @ C5 ) ) % 0.31/0.63 = ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ A3 @ C5 ) ) @ B3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % UN_simps(2) % 0.31/0.63 thf(fact_6490_UN__simps_I3_J,axiom, % 0.31/0.63 ! [E2: $tType,F: $tType,C5: set @ F,A3: set @ E2,B3: F > ( set @ E2 )] : % 0.31/0.63 ( ( ( C5 % 0.31/0.63 = ( bot_bot @ ( set @ F ) ) ) % 0.31/0.63 => ( ( complete_Sup_Sup @ ( set @ E2 ) % 0.31/0.63 @ ( image @ F @ ( set @ E2 ) % 0.31/0.63 @ ^ [X: F] : ( sup_sup @ ( set @ E2 ) @ A3 @ ( B3 @ X ) ) % 0.31/0.63 @ C5 ) ) % 0.31/0.63 = ( bot_bot @ ( set @ E2 ) ) ) ) % 0.31/0.63 & ( ( C5 % 0.31/0.63 != ( bot_bot @ ( set @ F ) ) ) % 0.31/0.63 => ( ( complete_Sup_Sup @ ( set @ E2 ) % 0.31/0.63 @ ( image @ F @ ( set @ E2 ) % 0.31/0.63 @ ^ [X: F] : ( sup_sup @ ( set @ E2 ) @ A3 @ ( B3 @ X ) ) % 0.31/0.63 @ C5 ) ) % 0.31/0.63 = ( sup_sup @ ( set @ E2 ) @ A3 @ ( complete_Sup_Sup @ ( set @ E2 ) @ ( image @ F @ ( set @ E2 ) @ B3 @ C5 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % UN_simps(3) % 0.31/0.63 thf(fact_6491_rtrancl__Un__separator__converseE,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) ) % 0.31/0.63 => ( ! [X5: A,Y4: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X5 ) @ Q ) % 0.31/0.63 => ( Y4 = X5 ) ) ) % 0.31/0.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl_Un_separator_converseE % 0.31/0.63 thf(fact_6492_rtrancl__Un__separatorE,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) ) % 0.31/0.63 => ( ! [X5: A,Y4: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X5 ) @ ( transitive_rtrancl @ A @ P ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ Q ) % 0.31/0.63 => ( X5 = Y4 ) ) ) % 0.31/0.63 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl_Un_separatorE % 0.31/0.63 thf(fact_6493_mono__Un,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: ( set @ A ) > ( set @ B ),A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F2 ) % 0.31/0.63 => ( ord_less_eq @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) @ ( F2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % mono_Un % 0.31/0.63 thf(fact_6494_mono__sup,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( ( semilattice_sup @ A ) % 0.31/0.63 & ( semilattice_sup @ B ) ) % 0.31/0.63 => ! [F2: A > B,A3: A,B3: A] : % 0.31/0.63 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.63 => ( ord_less_eq @ B @ ( sup_sup @ B @ ( F2 @ A3 ) @ ( F2 @ B3 ) ) @ ( F2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % mono_sup % 0.31/0.63 thf(fact_6495_insert__is__Un,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( insert @ A ) % 0.31/0.63 = ( ^ [A4: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % insert_is_Un % 0.31/0.63 thf(fact_6496_Un__singleton__iff,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,X3: A] : % 0.31/0.63 ( ( ( sup_sup @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 = ( ( ( A3 % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 & ( B3 % 0.31/0.63 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.63 | ( ( A3 % 0.31/0.63 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 & ( B3 % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 | ( ( A3 % 0.31/0.63 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 & ( B3 % 0.31/0.63 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_singleton_iff % 0.31/0.63 thf(fact_6497_singleton__Un__iff,axiom, % 0.31/0.63 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( ( ( A3 % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 & ( B3 % 0.31/0.63 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.63 | ( ( A3 % 0.31/0.63 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 & ( B3 % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 | ( ( A3 % 0.31/0.63 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 & ( B3 % 0.31/0.63 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % singleton_Un_iff % 0.31/0.63 thf(fact_6498_boolean__algebra_Odisj__zero__right,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.63 => ! [X3: A] : % 0.31/0.63 ( ( sup_sup @ A @ X3 @ ( bot_bot @ A ) ) % 0.31/0.63 = X3 ) ) ). % 0.31/0.63 % 0.31/0.63 % boolean_algebra.disj_zero_right % 0.31/0.63 thf(fact_6499_Un__empty__right,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 = A3 ) ). % 0.31/0.63 % 0.31/0.63 % Un_empty_right % 0.31/0.63 thf(fact_6500_Un__empty__left,axiom, % 0.31/0.63 ! [A: $tType,B3: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 ) % 0.31/0.63 = B3 ) ). % 0.31/0.63 % 0.31/0.63 % Un_empty_left % 0.31/0.63 thf(fact_6501_Sup__fold__sup,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( comple6319245703460814977attice @ A ) % 0.31/0.63 => ! [A3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( complete_Sup_Sup @ A @ A3 ) % 0.31/0.63 = ( finite_fold @ A @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fold_sup % 0.31/0.63 thf(fact_6502_cSup__union__distrib,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( condit1219197933456340205attice @ A ) % 0.31/0.63 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( condit941137186595557371_above @ A @ A3 ) % 0.31/0.63 => ( ( B3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( condit941137186595557371_above @ A @ B3 ) % 0.31/0.63 => ( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % cSup_union_distrib % 0.31/0.63 thf(fact_6503_Compl__partition2,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ A3 ) % 0.31/0.63 = ( top_top @ ( set @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Compl_partition2 % 0.31/0.63 thf(fact_6504_Compl__partition,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) % 0.31/0.63 = ( top_top @ ( set @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Compl_partition % 0.31/0.63 thf(fact_6505_sup__Sup__fold__sup,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( comple6319245703460814977attice @ A ) % 0.31/0.63 => ! [A3: set @ A,B3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ B3 ) % 0.31/0.63 = ( finite_fold @ A @ A @ ( sup_sup @ A ) @ B3 @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_Sup_fold_sup % 0.31/0.63 thf(fact_6506_infinite__Un,axiom, % 0.31/0.63 ! [A: $tType,S3: set @ A,T4: set @ A] : % 0.31/0.63 ( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) ) ) % 0.31/0.63 = ( ~ ( finite_finite2 @ A @ S3 ) % 0.31/0.63 | ~ ( finite_finite2 @ A @ T4 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % infinite_Un % 0.31/0.63 thf(fact_6507_Un__infinite,axiom, % 0.31/0.63 ! [A: $tType,S3: set @ A,T4: set @ A] : % 0.31/0.63 ( ~ ( finite_finite2 @ A @ S3 ) % 0.31/0.63 => ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S3 @ T4 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_infinite % 0.31/0.63 thf(fact_6508_finite__UnI,axiom, % 0.31/0.63 ! [A: $tType,F5: set @ A,G6: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ F5 ) % 0.31/0.63 => ( ( finite_finite2 @ A @ G6 ) % 0.31/0.63 => ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G6 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_UnI % 0.31/0.63 thf(fact_6509_insert__def,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( insert @ A ) % 0.31/0.63 = ( ^ [A4: A] : % 0.31/0.63 ( sup_sup @ ( set @ A ) % 0.31/0.63 @ ( collect @ A % 0.31/0.63 @ ^ [X: A] : ( X = A4 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % insert_def % 0.31/0.63 thf(fact_6510_Un__Int__crazy,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) @ ( inf_inf @ ( set @ A ) @ C5 @ A3 ) ) % 0.31/0.63 = ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) ) @ ( sup_sup @ ( set @ A ) @ C5 @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_Int_crazy % 0.31/0.63 thf(fact_6511_Int__Un__distrib,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.31/0.63 ( ( inf_inf @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( inf_inf @ ( set @ A ) @ A3 @ C5 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Int_Un_distrib % 0.31/0.63 thf(fact_6512_Un__Int__distrib,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) % 0.31/0.63 = ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ ( sup_sup @ ( set @ A ) @ A3 @ C5 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_Int_distrib % 0.31/0.63 thf(fact_6513_Int__Un__distrib2,axiom, % 0.31/0.63 ! [A: $tType,B3: set @ A,C5: set @ A,A3: set @ A] : % 0.31/0.63 ( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) @ A3 ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B3 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C5 @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Int_Un_distrib2 % 0.31/0.63 thf(fact_6514_Un__Int__distrib2,axiom, % 0.31/0.63 ! [A: $tType,B3: set @ A,C5: set @ A,A3: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) @ A3 ) % 0.31/0.63 = ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B3 @ A3 ) @ ( sup_sup @ ( set @ A ) @ C5 @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_Int_distrib2 % 0.31/0.63 thf(fact_6515_Un__Diff,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.31/0.63 ( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C5 ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ C5 ) @ ( minus_minus @ ( set @ A ) @ B3 @ C5 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_Diff % 0.31/0.63 thf(fact_6516_UnE,axiom, % 0.31/0.63 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 => ( ~ ( member @ A @ C2 @ A3 ) % 0.31/0.63 => ( member @ A @ C2 @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % UnE % 0.31/0.63 thf(fact_6517_UnI1,axiom, % 0.31/0.63 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( member @ A @ C2 @ A3 ) % 0.31/0.63 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % UnI1 % 0.31/0.63 thf(fact_6518_UnI2,axiom, % 0.31/0.63 ! [A: $tType,C2: A,B3: set @ A,A3: set @ A] : % 0.31/0.63 ( ( member @ A @ C2 @ B3 ) % 0.31/0.63 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % UnI2 % 0.31/0.63 thf(fact_6519_bex__Un,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] : % 0.31/0.63 ( ( ? [X: A] : % 0.31/0.63 ( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 & ( P @ X ) ) ) % 0.31/0.63 = ( ? [X: A] : % 0.31/0.63 ( ( member @ A @ X @ A3 ) % 0.31/0.63 & ( P @ X ) ) % 0.31/0.63 | ? [X: A] : % 0.31/0.63 ( ( member @ A @ X @ B3 ) % 0.31/0.63 & ( P @ X ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % bex_Un % 0.31/0.63 thf(fact_6520_ball__Un,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] : % 0.31/0.63 ( ( ! [X: A] : % 0.31/0.63 ( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 => ( P @ X ) ) ) % 0.31/0.63 = ( ! [X: A] : % 0.31/0.63 ( ( member @ A @ X @ A3 ) % 0.31/0.63 => ( P @ X ) ) % 0.31/0.63 & ! [X: A] : % 0.31/0.63 ( ( member @ A @ X @ B3 ) % 0.31/0.63 => ( P @ X ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ball_Un % 0.31/0.63 thf(fact_6521_Un__assoc,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C5 ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_assoc % 0.31/0.63 thf(fact_6522_Un__absorb,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ A3 @ A3 ) % 0.31/0.63 = A3 ) ). % 0.31/0.63 % 0.31/0.63 % Un_absorb % 0.31/0.63 thf(fact_6523_Un__commute,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) ) % 0.31/0.63 = ( ^ [A6: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A6 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_commute % 0.31/0.63 thf(fact_6524_Un__left__absorb,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_left_absorb % 0.31/0.63 thf(fact_6525_Un__left__commute,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ C5 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_left_commute % 0.31/0.63 thf(fact_6526_Un__def,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) ) % 0.31/0.63 = ( ^ [A6: set @ A,B6: set @ A] : % 0.31/0.63 ( collect @ A % 0.31/0.63 @ ^ [X: A] : % 0.31/0.63 ( ( member @ A @ X @ A6 ) % 0.31/0.63 | ( member @ A @ X @ B6 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_def % 0.31/0.63 thf(fact_6527_Collect__disj__eq,axiom, % 0.31/0.63 ! [A: $tType,P: A > $o,Q: A > $o] : % 0.31/0.63 ( ( collect @ A % 0.31/0.63 @ ^ [X: A] : % 0.31/0.63 ( ( P @ X ) % 0.31/0.63 | ( Q @ X ) ) ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Collect_disj_eq % 0.31/0.63 thf(fact_6528_Collect__imp__eq,axiom, % 0.31/0.63 ! [A: $tType,P: A > $o,Q: A > $o] : % 0.31/0.63 ( ( collect @ A % 0.31/0.63 @ ^ [X: A] : % 0.31/0.63 ( ( P @ X ) % 0.31/0.63 => ( Q @ X ) ) ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Collect_imp_eq % 0.31/0.63 thf(fact_6529_less__supI1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [X3: A,A2: A,B2: A] : % 0.31/0.63 ( ( ord_less @ A @ X3 @ A2 ) % 0.31/0.63 => ( ord_less @ A @ X3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % less_supI1 % 0.31/0.63 thf(fact_6530_less__supI2,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [X3: A,B2: A,A2: A] : % 0.31/0.63 ( ( ord_less @ A @ X3 @ B2 ) % 0.31/0.63 => ( ord_less @ A @ X3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % less_supI2 % 0.31/0.63 thf(fact_6531_sup_Oabsorb3,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [B2: A,A2: A] : % 0.31/0.63 ( ( ord_less @ A @ B2 @ A2 ) % 0.31/0.63 => ( ( sup_sup @ A @ A2 @ B2 ) % 0.31/0.63 = A2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.absorb3 % 0.31/0.63 thf(fact_6532_sup_Oabsorb4,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A2: A,B2: A] : % 0.31/0.63 ( ( ord_less @ A @ A2 @ B2 ) % 0.31/0.63 => ( ( sup_sup @ A @ A2 @ B2 ) % 0.31/0.63 = B2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.absorb4 % 0.31/0.63 thf(fact_6533_sup_Ostrict__boundedE,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [B2: A,C2: A,A2: A] : % 0.31/0.63 ( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 ) % 0.31/0.63 => ~ ( ( ord_less @ A @ B2 @ A2 ) % 0.31/0.63 => ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.strict_boundedE % 0.31/0.63 thf(fact_6534_sup_Ostrict__order__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ( ( ord_less @ A ) % 0.31/0.63 = ( ^ [B4: A,A4: A] : % 0.31/0.63 ( ( A4 % 0.31/0.63 = ( sup_sup @ A @ A4 @ B4 ) ) % 0.31/0.63 & ( A4 != B4 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.strict_order_iff % 0.31/0.63 thf(fact_6535_sup_Ostrict__coboundedI1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [C2: A,A2: A,B2: A] : % 0.31/0.63 ( ( ord_less @ A @ C2 @ A2 ) % 0.31/0.63 => ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.strict_coboundedI1 % 0.31/0.63 thf(fact_6536_sup_Ostrict__coboundedI2,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [C2: A,B2: A,A2: A] : % 0.31/0.63 ( ( ord_less @ A @ C2 @ B2 ) % 0.31/0.63 => ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.strict_coboundedI2 % 0.31/0.63 thf(fact_6537_Un__UNIV__right,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.63 = ( top_top @ ( set @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_UNIV_right % 0.31/0.63 thf(fact_6538_Un__UNIV__left,axiom, % 0.31/0.63 ! [A: $tType,B3: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 ) % 0.31/0.63 = ( top_top @ ( set @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_UNIV_left % 0.31/0.63 thf(fact_6539_Diff__Un,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.31/0.63 ( ( minus_minus @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) ) % 0.31/0.63 = ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A3 @ C5 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Diff_Un % 0.31/0.63 thf(fact_6540_Diff__Int,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.31/0.63 ( ( minus_minus @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A3 @ C5 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Diff_Int % 0.31/0.63 thf(fact_6541_Int__Diff__Un,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = A3 ) ). % 0.31/0.63 % 0.31/0.63 % Int_Diff_Un % 0.31/0.63 thf(fact_6542_Un__Diff__Int,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = A3 ) ). % 0.31/0.63 % 0.31/0.63 % Un_Diff_Int % 0.31/0.63 thf(fact_6543_Compl__Int,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Compl_Int % 0.31/0.63 thf(fact_6544_Compl__Un,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Compl_Un % 0.31/0.63 thf(fact_6545_set__shuffles,axiom, % 0.31/0.63 ! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys: list @ A] : % 0.31/0.63 ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) ) % 0.31/0.63 => ( ( set2 @ A @ Zs2 ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % set_shuffles % 0.31/0.63 thf(fact_6546_Diff__subset__conv,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ C5 ) % 0.31/0.63 = ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Diff_subset_conv % 0.31/0.63 thf(fact_6547_Diff__partition,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) % 0.31/0.63 = B3 ) ) ). % 0.31/0.63 % 0.31/0.63 % Diff_partition % 0.31/0.63 thf(fact_6548_Un__Int__assoc__eq,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 0.31/0.63 ( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C5 ) % 0.31/0.63 = ( inf_inf @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C5 ) ) ) % 0.31/0.63 = ( ord_less_eq @ ( set @ A ) @ C5 @ A3 ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_Int_assoc_eq % 0.31/0.63 thf(fact_6549_distrib__sup__le,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( lattice @ A ) % 0.31/0.63 => ! [X3: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y ) @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % distrib_sup_le % 0.31/0.63 thf(fact_6550_distrib__inf__le,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( lattice @ A ) % 0.31/0.63 => ! [X3: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y ) @ ( inf_inf @ A @ X3 @ Z2 ) ) @ ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % distrib_inf_le % 0.31/0.63 thf(fact_6551_subset__Un__eq,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) ) % 0.31/0.63 = ( ^ [A6: set @ A,B6: set @ A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ A6 @ B6 ) % 0.31/0.63 = B6 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % subset_Un_eq % 0.31/0.63 thf(fact_6552_subset__UnE,axiom, % 0.31/0.63 ! [A: $tType,C5: set @ A,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) @ C5 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 => ~ ! [A15: set @ A] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) @ A15 @ A3 ) % 0.31/0.63 => ! [B14: set @ A] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) @ B14 @ B3 ) % 0.31/0.63 => ( C5 % 0.31/0.63 != ( sup_sup @ ( set @ A ) @ A15 @ B14 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % subset_UnE % 0.31/0.63 thf(fact_6553_Un__absorb2,axiom, % 0.31/0.63 ! [A: $tType,B3: set @ A,A3: set @ A] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 = A3 ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_absorb2 % 0.31/0.63 thf(fact_6554_Un__absorb1,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 = B3 ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_absorb1 % 0.31/0.63 thf(fact_6555_Un__upper2,axiom, % 0.31/0.63 ! [A: $tType,B3: set @ A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_upper2 % 0.31/0.63 thf(fact_6556_Un__upper1,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_upper1 % 0.31/0.63 thf(fact_6557_Un__least,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,C5: set @ A,B3: set @ A] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 ) % 0.31/0.63 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C5 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_least % 0.31/0.63 thf(fact_6558_Un__mono,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,C5: set @ A,B3: set @ A,D6: set @ A] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ D6 ) % 0.31/0.63 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ ( sup_sup @ ( set @ A ) @ C5 @ D6 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_mono % 0.31/0.63 thf(fact_6559_Un__Pow__subset,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B3 ) ) @ ( pow2 @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Un_Pow_subset % 0.31/0.63 thf(fact_6560_inf__sup__ord_I4_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( lattice @ A ) % 0.31/0.63 => ! [Y: A,X3: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X3 @ Y ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inf_sup_ord(4) % 0.31/0.63 thf(fact_6561_inf__sup__ord_I3_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( lattice @ A ) % 0.31/0.63 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inf_sup_ord(3) % 0.31/0.63 thf(fact_6562_le__supE,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A2: A,B2: A,X3: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X3 ) % 0.31/0.63 => ~ ( ( ord_less_eq @ A @ A2 @ X3 ) % 0.31/0.63 => ~ ( ord_less_eq @ A @ B2 @ X3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % le_supE % 0.31/0.63 thf(fact_6563_le__supI,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A2: A,X3: A,B2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ A2 @ X3 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ B2 @ X3 ) % 0.31/0.63 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % le_supI % 0.31/0.63 thf(fact_6564_sup__ge1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_ge1 % 0.31/0.63 thf(fact_6565_sup__ge2,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [Y: A,X3: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X3 @ Y ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_ge2 % 0.31/0.63 thf(fact_6566_le__supI1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [X3: A,A2: A,B2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ X3 @ A2 ) % 0.31/0.63 => ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % le_supI1 % 0.31/0.63 thf(fact_6567_le__supI2,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [X3: A,B2: A,A2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ X3 @ B2 ) % 0.31/0.63 => ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % le_supI2 % 0.31/0.63 thf(fact_6568_sup_Omono,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [C2: A,A2: A,D2: A,B2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ D2 @ B2 ) % 0.31/0.63 => ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D2 ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.mono % 0.31/0.63 thf(fact_6569_sup__mono,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A2: A,C2: A,B2: A,D2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ A2 @ C2 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ B2 @ D2 ) % 0.31/0.63 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C2 @ D2 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_mono % 0.31/0.63 thf(fact_6570_sup__least,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [Y: A,X3: A,Z2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ Y @ X3 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ Z2 @ X3 ) % 0.31/0.63 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_least % 0.31/0.63 thf(fact_6571_le__iff__sup,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ( ( ord_less_eq @ A ) % 0.31/0.63 = ( ^ [X: A,Y2: A] : % 0.31/0.63 ( ( sup_sup @ A @ X @ Y2 ) % 0.31/0.63 = Y2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % le_iff_sup % 0.31/0.63 thf(fact_6572_sup_OorderE,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [B2: A,A2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.63 => ( A2 % 0.31/0.63 = ( sup_sup @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.orderE % 0.31/0.63 thf(fact_6573_sup_OorderI,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A2: A,B2: A] : % 0.31/0.63 ( ( A2 % 0.31/0.63 = ( sup_sup @ A @ A2 @ B2 ) ) % 0.31/0.63 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.orderI % 0.31/0.63 thf(fact_6574_sup__unique,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [F2: A > A > A,X3: A,Y: A] : % 0.31/0.63 ( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ X5 @ ( F2 @ X5 @ Y4 ) ) % 0.31/0.63 => ( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ ( F2 @ X5 @ Y4 ) ) % 0.31/0.63 => ( ! [X5: A,Y4: A,Z3: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ Y4 @ X5 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ Z3 @ X5 ) % 0.31/0.63 => ( ord_less_eq @ A @ ( F2 @ Y4 @ Z3 ) @ X5 ) ) ) % 0.31/0.63 => ( ( sup_sup @ A @ X3 @ Y ) % 0.31/0.63 = ( F2 @ X3 @ Y ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_unique % 0.31/0.63 thf(fact_6575_sup_Oabsorb1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [B2: A,A2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.63 => ( ( sup_sup @ A @ A2 @ B2 ) % 0.31/0.63 = A2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.absorb1 % 0.31/0.63 thf(fact_6576_sup_Oabsorb2,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A2: A,B2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ A2 @ B2 ) % 0.31/0.63 => ( ( sup_sup @ A @ A2 @ B2 ) % 0.31/0.63 = B2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.absorb2 % 0.31/0.63 thf(fact_6577_sup__absorb1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [Y: A,X3: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ Y @ X3 ) % 0.31/0.63 => ( ( sup_sup @ A @ X3 @ Y ) % 0.31/0.63 = X3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_absorb1 % 0.31/0.63 thf(fact_6578_sup__absorb2,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [X3: A,Y: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.63 => ( ( sup_sup @ A @ X3 @ Y ) % 0.31/0.63 = Y ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_absorb2 % 0.31/0.63 thf(fact_6579_sup_OboundedE,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [B2: A,C2: A,A2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 ) % 0.31/0.63 => ~ ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.63 => ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.boundedE % 0.31/0.63 thf(fact_6580_sup_OboundedI,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [B2: A,A2: A,C2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ B2 @ A2 ) % 0.31/0.63 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.63 => ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.boundedI % 0.31/0.63 thf(fact_6581_sup_Oorder__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ( ( ord_less_eq @ A ) % 0.31/0.63 = ( ^ [B4: A,A4: A] : % 0.31/0.63 ( A4 % 0.31/0.63 = ( sup_sup @ A @ A4 @ B4 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.order_iff % 0.31/0.63 thf(fact_6582_sup_Ocobounded1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.cobounded1 % 0.31/0.63 thf(fact_6583_sup_Ocobounded2,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.cobounded2 % 0.31/0.63 thf(fact_6584_sup_Oabsorb__iff1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ( ( ord_less_eq @ A ) % 0.31/0.63 = ( ^ [B4: A,A4: A] : % 0.31/0.63 ( ( sup_sup @ A @ A4 @ B4 ) % 0.31/0.63 = A4 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.absorb_iff1 % 0.31/0.63 thf(fact_6585_sup_Oabsorb__iff2,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ( ( ord_less_eq @ A ) % 0.31/0.63 = ( ^ [A4: A,B4: A] : % 0.31/0.63 ( ( sup_sup @ A @ A4 @ B4 ) % 0.31/0.63 = B4 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.absorb_iff2 % 0.31/0.63 thf(fact_6586_sup_OcoboundedI1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [C2: A,A2: A,B2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ C2 @ A2 ) % 0.31/0.63 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.coboundedI1 % 0.31/0.63 thf(fact_6587_sup_OcoboundedI2,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [C2: A,B2: A,A2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ C2 @ B2 ) % 0.31/0.63 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup.coboundedI2 % 0.31/0.63 thf(fact_6588_ivl__disj__un__two_I3_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,M: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ M ) % 0.31/0.63 => ( ( ord_less_eq @ A @ M @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) ) % 0.31/0.63 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_two(3) % 0.31/0.63 thf(fact_6589_ivl__disj__un__two__touch_I4_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,M: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ M ) % 0.31/0.63 => ( ( ord_less_eq @ A @ M @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) ) % 0.31/0.63 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_two_touch(4) % 0.31/0.63 thf(fact_6590_ivl__disj__un__two_I6_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,M: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ M ) % 0.31/0.63 => ( ( ord_less_eq @ A @ M @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) ) % 0.31/0.63 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_two(6) % 0.31/0.63 thf(fact_6591_sup__SUP__fold__sup,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( comple6319245703460814977attice @ A ) % 0.31/0.63 => ! [A3: set @ B,B3: A,F2: B > A] : % 0.31/0.63 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.63 => ( ( sup_sup @ A @ B3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) % 0.31/0.63 = ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F2 ) @ B3 @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_SUP_fold_sup % 0.31/0.63 thf(fact_6592_image__Un,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B3: set @ B] : % 0.31/0.63 ( ( image @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.63 = ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ ( image @ B @ A @ F2 @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % image_Un % 0.31/0.63 thf(fact_6593_cSUP__union,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( condit1219197933456340205attice @ A ) % 0.31/0.63 => ! [A3: set @ B,F2: B > A,B3: set @ B] : % 0.31/0.63 ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.63 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.63 => ( ( B3 % 0.31/0.63 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.63 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ B3 ) ) % 0.31/0.63 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) ) % 0.31/0.63 = ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ B3 ) ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % cSUP_union % 0.31/0.63 thf(fact_6594_SUP__fold__sup,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( comple6319245703460814977attice @ A ) % 0.31/0.63 => ! [A3: set @ B,F2: B > A] : % 0.31/0.63 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.63 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.63 = ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F2 ) @ ( bot_bot @ A ) @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % SUP_fold_sup % 0.31/0.63 thf(fact_6595_sup__shunt,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.63 => ! [X3: A,Y: A] : % 0.31/0.63 ( ( ( sup_sup @ A @ X3 @ Y ) % 0.31/0.63 = ( top_top @ A ) ) % 0.31/0.63 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_shunt % 0.31/0.63 thf(fact_6596_sup__neg__inf,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.63 => ! [P4: A,Q4: A,R2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ P4 @ ( sup_sup @ A @ Q4 @ R2 ) ) % 0.31/0.63 = ( ord_less_eq @ A @ ( inf_inf @ A @ P4 @ ( uminus_uminus @ A @ Q4 ) ) @ R2 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_neg_inf % 0.31/0.63 thf(fact_6597_shunt2,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.63 => ! [X3: A,Y: A,Z2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) @ Z2 ) % 0.31/0.63 = ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % shunt2 % 0.31/0.63 thf(fact_6598_shunt1,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.63 => ! [X3: A,Y: A,Z2: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ Z2 ) % 0.31/0.63 = ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y ) @ Z2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % shunt1 % 0.31/0.63 thf(fact_6599_boolean__algebra_Ocomplement__unique,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.63 => ! [A2: A,X3: A,Y: A] : % 0.31/0.63 ( ( ( inf_inf @ A @ A2 @ X3 ) % 0.31/0.63 = ( bot_bot @ A ) ) % 0.31/0.63 => ( ( ( sup_sup @ A @ A2 @ X3 ) % 0.31/0.63 = ( top_top @ A ) ) % 0.31/0.63 => ( ( ( inf_inf @ A @ A2 @ Y ) % 0.31/0.63 = ( bot_bot @ A ) ) % 0.31/0.63 => ( ( ( sup_sup @ A @ A2 @ Y ) % 0.31/0.63 = ( top_top @ A ) ) % 0.31/0.63 => ( X3 = Y ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % boolean_algebra.complement_unique % 0.31/0.63 thf(fact_6600_finite__Sup__in,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( comple6319245703460814977attice @ A ) % 0.31/0.63 => ! [A3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ! [X5: A,Y4: A] : % 0.31/0.63 ( ( member @ A @ X5 @ A3 ) % 0.31/0.63 => ( ( member @ A @ Y4 @ A3 ) % 0.31/0.63 => ( member @ A @ ( sup_sup @ A @ X5 @ Y4 ) @ A3 ) ) ) % 0.31/0.63 => ( member @ A @ ( complete_Sup_Sup @ A @ A3 ) @ A3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_Sup_in % 0.31/0.63 thf(fact_6601_less__eq__Inf__inter,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( comple6319245703460814977attice @ A ) % 0.31/0.63 => ! [A3: set @ A,B3: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % less_eq_Inf_inter % 0.31/0.63 thf(fact_6602_ivl__disj__un__two_I7_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,M: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ M ) % 0.31/0.63 => ( ( ord_less_eq @ A @ M @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) ) % 0.31/0.63 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_two(7) % 0.31/0.63 thf(fact_6603_inf__Inf__fold__inf,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( comple6319245703460814977attice @ A ) % 0.31/0.63 => ! [A3: set @ A,B3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A3 ) @ B3 ) % 0.31/0.63 = ( finite_fold @ A @ A @ ( inf_inf @ A ) @ B3 @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inf_Inf_fold_inf % 0.31/0.63 thf(fact_6604_ivl__disj__un__one_I2_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) ) % 0.31/0.63 = ( set_ord_lessThan @ A @ U ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_one(2) % 0.31/0.63 thf(fact_6605_Union__image__empty,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,A3: set @ A,F2: B > ( set @ A )] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ ( bot_bot @ ( set @ B ) ) ) ) ) % 0.31/0.63 = A3 ) ). % 0.31/0.63 % 0.31/0.63 % Union_image_empty % 0.31/0.63 thf(fact_6606_card__Un__le,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % card_Un_le % 0.31/0.63 thf(fact_6607_atLeastLessThan__add__Un,axiom, % 0.31/0.63 ! [I: nat,J: nat,K: nat] : % 0.31/0.63 ( ( ord_less_eq @ nat @ I @ J ) % 0.31/0.63 => ( ( set_or7035219750837199246ssThan @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) % 0.31/0.63 = ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) @ ( set_or7035219750837199246ssThan @ nat @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % atLeastLessThan_add_Un % 0.31/0.63 thf(fact_6608_ivl__disj__un__two_I8_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,M: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ M ) % 0.31/0.63 => ( ( ord_less_eq @ A @ M @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) ) % 0.31/0.63 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_two(8) % 0.31/0.63 thf(fact_6609_ivl__disj__un__one_I8_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) ) % 0.31/0.63 = ( set_ord_atLeast @ A @ L ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_one(8) % 0.31/0.63 thf(fact_6610_ivl__disj__un__one_I3_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) ) % 0.31/0.63 = ( set_ord_atMost @ A @ U ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_one(3) % 0.31/0.63 thf(fact_6611_ivl__disj__un__one_I5_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) ) % 0.31/0.63 = ( set_ord_greaterThan @ A @ L ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_one(5) % 0.31/0.63 thf(fact_6612_Pow__insert,axiom, % 0.31/0.63 ! [A: $tType,A2: A,A3: set @ A] : % 0.31/0.63 ( ( pow2 @ A @ ( insert @ A @ A2 @ A3 ) ) % 0.31/0.63 = ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ A2 ) @ ( pow2 @ A @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Pow_insert % 0.31/0.63 thf(fact_6613_UN__extend__simps_I2_J,axiom, % 0.31/0.63 ! [D: $tType,C: $tType,C5: set @ C,A3: C > ( set @ D ),B3: set @ D] : % 0.31/0.63 ( ( ( C5 % 0.31/0.63 = ( bot_bot @ ( set @ C ) ) ) % 0.31/0.63 => ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ A3 @ C5 ) ) @ B3 ) % 0.31/0.63 = B3 ) ) % 0.31/0.63 & ( ( C5 % 0.31/0.63 != ( bot_bot @ ( set @ C ) ) ) % 0.31/0.63 => ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ A3 @ C5 ) ) @ B3 ) % 0.31/0.63 = ( complete_Sup_Sup @ ( set @ D ) % 0.31/0.63 @ ( image @ C @ ( set @ D ) % 0.31/0.63 @ ^ [X: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X ) @ B3 ) % 0.31/0.63 @ C5 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % UN_extend_simps(2) % 0.31/0.63 thf(fact_6614_UN__extend__simps_I3_J,axiom, % 0.31/0.63 ! [E2: $tType,F: $tType,C5: set @ F,A3: set @ E2,B3: F > ( set @ E2 )] : % 0.31/0.63 ( ( ( C5 % 0.31/0.63 = ( bot_bot @ ( set @ F ) ) ) % 0.31/0.63 => ( ( sup_sup @ ( set @ E2 ) @ A3 @ ( complete_Sup_Sup @ ( set @ E2 ) @ ( image @ F @ ( set @ E2 ) @ B3 @ C5 ) ) ) % 0.31/0.63 = A3 ) ) % 0.31/0.63 & ( ( C5 % 0.31/0.63 != ( bot_bot @ ( set @ F ) ) ) % 0.31/0.63 => ( ( sup_sup @ ( set @ E2 ) @ A3 @ ( complete_Sup_Sup @ ( set @ E2 ) @ ( image @ F @ ( set @ E2 ) @ B3 @ C5 ) ) ) % 0.31/0.63 = ( complete_Sup_Sup @ ( set @ E2 ) % 0.31/0.63 @ ( image @ F @ ( set @ E2 ) % 0.31/0.63 @ ^ [X: F] : ( sup_sup @ ( set @ E2 ) @ A3 @ ( B3 @ X ) ) % 0.31/0.63 @ C5 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % UN_extend_simps(3) % 0.31/0.63 thf(fact_6615_Inter__Un__subset,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B3 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A3 @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inter_Un_subset % 0.31/0.63 thf(fact_6616_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.63 => ! [X3: A,Y: A] : % 0.31/0.63 ( ( ( inf_inf @ A @ X3 @ Y ) % 0.31/0.63 = ( bot_bot @ A ) ) % 0.31/0.63 => ( ( ( sup_sup @ A @ X3 @ Y ) % 0.31/0.63 = ( top_top @ A ) ) % 0.31/0.63 => ( ( uminus_uminus @ A @ X3 ) % 0.31/0.63 = Y ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % boolean_algebra_class.boolean_algebra.compl_unique % 0.31/0.63 thf(fact_6617_cSup__insert__If,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( condit1219197933456340205attice @ A ) % 0.31/0.63 => ! [X8: set @ A,A2: A] : % 0.31/0.63 ( ( condit941137186595557371_above @ A @ X8 ) % 0.31/0.63 => ( ( ( X8 % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ X8 ) ) % 0.31/0.63 = A2 ) ) % 0.31/0.63 & ( ( X8 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ X8 ) ) % 0.31/0.63 = ( sup_sup @ A @ A2 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % cSup_insert_If % 0.31/0.63 thf(fact_6618_cSup__insert,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( condit1219197933456340205attice @ A ) % 0.31/0.63 => ! [X8: set @ A,A2: A] : % 0.31/0.63 ( ( X8 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( condit941137186595557371_above @ A @ X8 ) % 0.31/0.63 => ( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ X8 ) ) % 0.31/0.63 = ( sup_sup @ A @ A2 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % cSup_insert % 0.31/0.63 thf(fact_6619_ivl__disj__un__two__touch_I2_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,M: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ M ) % 0.31/0.63 => ( ( ord_less @ A @ M @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) ) % 0.31/0.63 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_two_touch(2) % 0.31/0.63 thf(fact_6620_sum_Ounion__inter,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( comm_monoid_add @ A ) % 0.31/0.63 => ! [A3: set @ B,B3: set @ B,G: B > A] : % 0.31/0.63 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.63 => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) ) % 0.31/0.63 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sum.union_inter % 0.31/0.63 thf(fact_6621_prod_Ounion__inter,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( comm_monoid_mult @ A ) % 0.31/0.63 => ! [A3: set @ B,B3: set @ B,G: B > A] : % 0.31/0.63 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.63 => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) ) % 0.31/0.63 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % prod.union_inter % 0.31/0.63 thf(fact_6622_Inf__fold__inf,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( comple6319245703460814977attice @ A ) % 0.31/0.63 => ! [A3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( complete_Inf_Inf @ A @ A3 ) % 0.31/0.63 = ( finite_fold @ A @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fold_inf % 0.31/0.63 thf(fact_6623_card__Un__Int,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.63 => ( ( plus_plus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) % 0.31/0.63 = ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % card_Un_Int % 0.31/0.63 thf(fact_6624_ivl__disj__un__two__touch_I3_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,M: A,U: A] : % 0.31/0.63 ( ( ord_less @ A @ L @ M ) % 0.31/0.63 => ( ( ord_less_eq @ A @ M @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) ) % 0.31/0.63 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_two_touch(3) % 0.31/0.63 thf(fact_6625_ivl__disj__un__two_I1_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,M: A,U: A] : % 0.31/0.63 ( ( ord_less @ A @ L @ M ) % 0.31/0.63 => ( ( ord_less_eq @ A @ M @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) ) % 0.31/0.63 = ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_two(1) % 0.31/0.63 thf(fact_6626_ivl__disj__un__one_I4_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) ) % 0.31/0.63 = ( set_ord_atMost @ A @ U ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_one(4) % 0.31/0.63 thf(fact_6627_ivl__disj__un__singleton_I2_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [U: A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 = ( set_ord_atMost @ A @ U ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_singleton(2) % 0.31/0.63 thf(fact_6628_sum_Oeq__fold,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( comm_monoid_add @ A ) % 0.31/0.63 => ( ( groups7311177749621191930dd_sum @ B @ A ) % 0.31/0.63 = ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sum.eq_fold % 0.31/0.63 thf(fact_6629_cInf__union__distrib,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( condit1219197933456340205attice @ A ) % 0.31/0.63 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( condit1013018076250108175_below @ A @ A3 ) % 0.31/0.63 => ( ( B3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( condit1013018076250108175_below @ A @ B3 ) % 0.31/0.63 => ( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % cInf_union_distrib % 0.31/0.63 thf(fact_6630_ivl__disj__un__two_I2_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,M: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ M ) % 0.31/0.63 => ( ( ord_less @ A @ M @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) ) % 0.31/0.63 = ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_two(2) % 0.31/0.63 thf(fact_6631_Max_Ounion,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.63 => ( ( B3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ ( lattic643756798349783984er_Max @ A @ B3 ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Max.union % 0.31/0.63 thf(fact_6632_ivl__disj__un__one_I1_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,U: A] : % 0.31/0.63 ( ( ord_less @ A @ L @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) ) % 0.31/0.63 = ( set_ord_lessThan @ A @ U ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_one(1) % 0.31/0.63 thf(fact_6633_ivl__disj__un__one_I7_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) ) % 0.31/0.63 = ( set_ord_atLeast @ A @ L ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_one(7) % 0.31/0.63 thf(fact_6634_ivl__disj__un__singleton_I1_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_greaterThan @ A @ L ) ) % 0.31/0.63 = ( set_ord_atLeast @ A @ L ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_singleton(1) % 0.31/0.63 thf(fact_6635_Max_Oeq__fold,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.63 = ( finite_fold @ A @ A @ ( ord_max @ A ) @ X3 @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Max.eq_fold % 0.31/0.63 thf(fact_6636_ivl__disj__un__two__touch_I1_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,M: A,U: A] : % 0.31/0.63 ( ( ord_less @ A @ L @ M ) % 0.31/0.63 => ( ( ord_less @ A @ M @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) ) % 0.31/0.63 = ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_two_touch(1) % 0.31/0.63 thf(fact_6637_SUP__nat__binary,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( counta3822494911875563373attice @ A ) % 0.31/0.63 => ! [A3: A,B3: A] : % 0.31/0.63 ( ( sup_sup @ A @ A3 % 0.31/0.63 @ ( complete_Sup_Sup @ A % 0.31/0.63 @ ( image @ nat @ A % 0.31/0.63 @ ^ [X: nat] : B3 % 0.31/0.63 @ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) ) % 0.31/0.63 = ( sup_sup @ A @ A3 @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % SUP_nat_binary % 0.31/0.63 thf(fact_6638_ivl__disj__un__one_I6_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,U: A] : % 0.31/0.63 ( ( ord_less @ A @ L @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) ) % 0.31/0.63 = ( set_ord_greaterThan @ A @ L ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_one(6) % 0.31/0.63 thf(fact_6639_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( condit1219197933456340205attice @ A ) % 0.31/0.63 => ! [A3: set @ B,F2: B > A,G: B > A] : % 0.31/0.63 ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.63 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.63 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ G @ A3 ) ) % 0.31/0.63 => ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A3 ) ) ) % 0.31/0.63 = ( complete_Sup_Sup @ A % 0.31/0.63 @ ( image @ B @ A % 0.31/0.63 @ ^ [A4: B] : ( sup_sup @ A @ ( F2 @ A4 ) @ ( G @ A4 ) ) % 0.31/0.63 @ A3 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % conditionally_complete_lattice_class.SUP_sup_distrib % 0.31/0.63 thf(fact_6640_inj__on__disjoint__Un,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,F2: A > B,A3: set @ A,G: A > B,B3: set @ A] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ A3 ) % 0.31/0.63 => ( ( inj_on @ A @ B @ G @ B3 ) % 0.31/0.63 => ( ( ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ G @ B3 ) ) % 0.31/0.63 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.63 => ( inj_on @ A @ B % 0.31/0.63 @ ^ [X: A] : ( if @ B @ ( member @ A @ X @ A3 ) @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.63 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_disjoint_Un % 0.31/0.63 thf(fact_6641_sup__bot_Osemilattice__neutr__order__axioms,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( bounde4967611905675639751up_bot @ A ) % 0.31/0.63 => ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) % 0.31/0.63 @ ^ [X: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X ) % 0.31/0.63 @ ^ [X: A,Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_bot.semilattice_neutr_order_axioms % 0.31/0.63 thf(fact_6642_rtrancl__insert,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( transitive_rtrancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) % 0.31/0.63 = ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R2 ) % 0.31/0.63 @ ( collect @ ( product_prod @ A @ A ) % 0.31/0.63 @ ( product_case_prod @ A @ A @ $o % 0.31/0.63 @ ^ [X: A,Y2: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % rtrancl_insert % 0.31/0.63 thf(fact_6643_trancl__insert2,axiom, % 0.31/0.63 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) % 0.31/0.63 = ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 ) % 0.31/0.63 @ ( collect @ ( product_prod @ A @ A ) % 0.31/0.63 @ ( product_case_prod @ A @ A @ $o % 0.31/0.63 @ ^ [X: A,Y2: A] : % 0.31/0.63 ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 | ( X = A2 ) ) % 0.31/0.63 & ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y2 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.63 | ( Y2 = B2 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl_insert2 % 0.31/0.63 thf(fact_6644_cSup__inter__less__eq,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( condit1219197933456340205attice @ A ) % 0.31/0.63 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( condit941137186595557371_above @ A @ A3 ) % 0.31/0.63 => ( ( condit941137186595557371_above @ A @ B3 ) % 0.31/0.63 => ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % cSup_inter_less_eq % 0.31/0.63 thf(fact_6645_cSUP__insert,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( condit1219197933456340205attice @ A ) % 0.31/0.63 => ! [A3: set @ B,F2: B > A,A2: B] : % 0.31/0.63 ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.63 => ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.63 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( insert @ B @ A2 @ A3 ) ) ) % 0.31/0.63 = ( sup_sup @ A @ ( F2 @ A2 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % cSUP_insert % 0.31/0.63 thf(fact_6646_sum_Ounion__inter__neutral,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( comm_monoid_add @ A ) % 0.31/0.63 => ! [A3: set @ B,B3: set @ B,G: B > A] : % 0.31/0.63 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.63 => ( ! [X5: B] : % 0.31/0.63 ( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.63 => ( ( G @ X5 ) % 0.31/0.63 = ( zero_zero @ A ) ) ) % 0.31/0.63 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.63 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B3 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sum.union_inter_neutral % 0.31/0.63 thf(fact_6647_sum__Un,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( ab_group_add @ A ) % 0.31/0.63 => ! [A3: set @ B,B3: set @ B,F2: B > A] : % 0.31/0.63 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.63 => ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.63 = ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sum_Un % 0.31/0.63 thf(fact_6648_sum_Ounion__disjoint,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( comm_monoid_add @ A ) % 0.31/0.63 => ! [A3: set @ B,B3: set @ B,G: B > A] : % 0.31/0.63 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.63 => ( ( ( inf_inf @ ( set @ B ) @ A3 @ B3 ) % 0.31/0.63 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.63 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.63 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B3 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sum.union_disjoint % 0.31/0.63 thf(fact_6649_prod_Ounion__inter__neutral,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( comm_monoid_mult @ A ) % 0.31/0.63 => ! [A3: set @ B,B3: set @ B,G: B > A] : % 0.31/0.63 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.63 => ( ! [X5: B] : % 0.31/0.63 ( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.63 => ( ( G @ X5 ) % 0.31/0.63 = ( one_one @ A ) ) ) % 0.31/0.63 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.63 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B3 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % prod.union_inter_neutral % 0.31/0.63 thf(fact_6650_prod_Ounion__disjoint,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( comm_monoid_mult @ A ) % 0.31/0.63 => ! [A3: set @ B,B3: set @ B,G: B > A] : % 0.31/0.63 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.63 => ( ( ( inf_inf @ ( set @ B ) @ A3 @ B3 ) % 0.31/0.63 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.63 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.63 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B3 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % prod.union_disjoint % 0.31/0.63 thf(fact_6651_ivl__disj__un__singleton_I6_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_singleton(6) % 0.31/0.63 thf(fact_6652_sum__Un2,axiom, % 0.31/0.63 ! [B: $tType,A: $tType] : % 0.31/0.63 ( ( comm_monoid_add @ B ) % 0.31/0.63 => ! [A3: set @ A,B3: set @ A,F2: A > B] : % 0.31/0.63 ( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 => ( ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sum_Un2 % 0.31/0.63 thf(fact_6653_sum_Ounion__diff2,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( comm_monoid_add @ A ) % 0.31/0.63 => ! [A3: set @ B,B3: set @ B,G: B > A] : % 0.31/0.63 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.63 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.63 = ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B3 @ A3 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sum.union_diff2 % 0.31/0.63 thf(fact_6654_prod_Ounion__diff2,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( comm_monoid_mult @ A ) % 0.31/0.63 => ! [A3: set @ B,B3: set @ B,G: B > A] : % 0.31/0.63 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.63 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.63 = ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B3 @ A3 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % prod.union_diff2 % 0.31/0.63 thf(fact_6655_card__Un__disjoint,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.63 => ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( plus_plus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % card_Un_disjoint % 0.31/0.63 thf(fact_6656_inj__on__Un,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( inj_on @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( ( inj_on @ A @ B @ F2 @ A3 ) % 0.31/0.63 & ( inj_on @ A @ B @ F2 @ B3 ) % 0.31/0.63 & ( ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) @ ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) % 0.31/0.63 = ( bot_bot @ ( set @ B ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inj_on_Un % 0.31/0.63 thf(fact_6657_ivl__disj__un__singleton_I5_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) ) % 0.31/0.63 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_singleton(5) % 0.31/0.63 thf(fact_6658_ivl__disj__un__two_I4_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,M: A,U: A] : % 0.31/0.63 ( ( ord_less_eq @ A @ L @ M ) % 0.31/0.63 => ( ( ord_less @ A @ M @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) ) % 0.31/0.63 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_two(4) % 0.31/0.63 thf(fact_6659_cINF__union,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( condit1219197933456340205attice @ A ) % 0.31/0.63 => ! [A3: set @ B,F2: B > A,B3: set @ B] : % 0.31/0.63 ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.63 => ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.63 => ( ( B3 % 0.31/0.63 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.63 => ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ B3 ) ) % 0.31/0.63 => ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) ) % 0.31/0.63 = ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ B3 ) ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % cINF_union % 0.31/0.63 thf(fact_6660_ivl__disj__un__singleton_I3_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,U: A] : % 0.31/0.63 ( ( ord_less @ A @ L @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) ) % 0.31/0.63 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_singleton(3) % 0.31/0.63 thf(fact_6661_inf__INF__fold__inf,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( comple6319245703460814977attice @ A ) % 0.31/0.63 => ! [A3: set @ B,B3: A,F2: B > A] : % 0.31/0.63 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.63 => ( ( inf_inf @ A @ B3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) % 0.31/0.63 = ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F2 ) @ B3 @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inf_INF_fold_inf % 0.31/0.63 thf(fact_6662_sum__Un__nat,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,F2: A > nat] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.63 => ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sum_Un_nat % 0.31/0.63 thf(fact_6663_ivl__disj__un__two_I5_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,M: A,U: A] : % 0.31/0.63 ( ( ord_less @ A @ L @ M ) % 0.31/0.63 => ( ( ord_less_eq @ A @ M @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) ) % 0.31/0.63 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_two(5) % 0.31/0.63 thf(fact_6664_ivl__disj__un__singleton_I4_J,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ! [L: A,U: A] : % 0.31/0.63 ( ( ord_less @ A @ L @ U ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % ivl_disj_un_singleton(4) % 0.31/0.63 thf(fact_6665_trancl__insert,axiom, % 0.31/0.63 ! [A: $tType,Y: A,X3: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ R2 ) ) % 0.31/0.63 = ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 ) % 0.31/0.63 @ ( collect @ ( product_prod @ A @ A ) % 0.31/0.63 @ ( product_case_prod @ A @ A @ $o % 0.31/0.63 @ ^ [A4: A,B4: A] : % 0.31/0.63 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B4 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % trancl_insert % 0.31/0.63 thf(fact_6666_prod__Un,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( field @ A ) % 0.31/0.63 => ! [A3: set @ B,B3: set @ B,F2: B > A] : % 0.31/0.63 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.63 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.63 => ( ! [X5: B] : % 0.31/0.63 ( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.63 => ( ( F2 @ X5 ) % 0.31/0.63 != ( zero_zero @ A ) ) ) % 0.31/0.63 => ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.63 = ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % prod_Un % 0.31/0.63 thf(fact_6667_INF__fold__inf,axiom, % 0.31/0.63 ! [A: $tType,B: $tType] : % 0.31/0.63 ( ( comple6319245703460814977attice @ A ) % 0.31/0.63 => ! [A3: set @ B,F2: B > A] : % 0.31/0.63 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.63 => ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.63 = ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F2 ) @ ( top_top @ A ) @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % INF_fold_inf % 0.31/0.63 thf(fact_6668_If__the__inv__into__in__Func,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,G: A > B,C5: set @ A,B3: set @ A,X3: A] : % 0.31/0.63 ( ( inj_on @ A @ B @ G @ C5 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ C5 @ ( sup_sup @ ( set @ A ) @ B3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.63 => ( member @ ( B > A ) % 0.31/0.63 @ ^ [I3: B] : ( if @ A @ ( member @ B @ I3 @ ( image @ A @ B @ G @ C5 ) ) @ ( the_inv_into @ A @ B @ C5 @ G @ I3 ) @ X3 ) % 0.31/0.63 @ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % If_the_inv_into_in_Func % 0.31/0.63 thf(fact_6669_min__weak__def,axiom, % 0.31/0.63 ( fun_min_weak % 0.31/0.63 = ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % min_weak_def % 0.31/0.63 thf(fact_6670_Ball__fold,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,P: A > $o] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( ! [X: A] : % 0.31/0.63 ( ( member @ A @ X @ A3 ) % 0.31/0.63 => ( P @ X ) ) ) % 0.31/0.63 = ( finite_fold @ A @ $o % 0.31/0.63 @ ^ [K3: A,S8: $o] : % 0.31/0.63 ( S8 % 0.31/0.63 & ( P @ K3 ) ) % 0.31/0.63 @ $true % 0.31/0.63 @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Ball_fold % 0.31/0.63 thf(fact_6671_sup__int__def,axiom, % 0.31/0.63 ( ( sup_sup @ int ) % 0.31/0.63 = ( ord_max @ int ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_int_def % 0.31/0.63 thf(fact_6672_sup__set__def,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( sup_sup @ ( set @ A ) ) % 0.31/0.63 = ( ^ [A6: set @ A,B6: set @ A] : % 0.31/0.63 ( collect @ A % 0.31/0.63 @ ( sup_sup @ ( A > $o ) % 0.31/0.63 @ ^ [X: A] : ( member @ A @ X @ A6 ) % 0.31/0.63 @ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_set_def % 0.31/0.63 thf(fact_6673_sup__nat__def,axiom, % 0.31/0.63 ( ( sup_sup @ nat ) % 0.31/0.63 = ( ord_max @ nat ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_nat_def % 0.31/0.63 thf(fact_6674_sup__Un__eq2,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] : % 0.31/0.63 ( ( sup_sup @ ( A > B > $o ) % 0.31/0.63 @ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R ) % 0.31/0.63 @ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S3 ) ) % 0.31/0.63 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_Un_eq2 % 0.31/0.63 thf(fact_6675_union__fold__insert,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( sup_sup @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 = ( finite_fold @ A @ ( set @ A ) @ ( insert @ A ) @ B3 @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % union_fold_insert % 0.31/0.63 thf(fact_6676_card_Oeq__fold,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( finite_card @ A ) % 0.31/0.63 = ( finite_fold @ A @ nat % 0.31/0.63 @ ^ [Uu3: A] : suc % 0.31/0.63 @ ( zero_zero @ nat ) ) ) ). % 0.31/0.63 % 0.31/0.63 % card.eq_fold % 0.31/0.63 thf(fact_6677_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( linorder @ A ) % 0.31/0.63 => ( ( linord4507533701916653071of_set @ A ) % 0.31/0.63 = ( finite_fold @ A @ ( list @ A ) % 0.31/0.63 @ ( linorder_insort_key @ A @ A % 0.31/0.63 @ ^ [X: A] : X ) % 0.31/0.63 @ ( nil @ A ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sorted_list_of_set.fold_insort_key.eq_fold % 0.31/0.63 thf(fact_6678_image__fold__insert,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,A3: set @ A,F2: A > B] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( image @ A @ B @ F2 @ A3 ) % 0.31/0.63 = ( finite_fold @ A @ ( set @ B ) % 0.31/0.63 @ ^ [K3: A] : ( insert @ B @ ( F2 @ K3 ) ) % 0.31/0.63 @ ( bot_bot @ ( set @ B ) ) % 0.31/0.63 @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % image_fold_insert % 0.31/0.63 thf(fact_6679_Id__on__fold,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( id_on @ A @ A3 ) % 0.31/0.63 = ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) ) % 0.31/0.63 @ ^ [X: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) ) % 0.31/0.63 @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) % 0.31/0.63 @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Id_on_fold % 0.31/0.63 thf(fact_6680_fold__union__pair,axiom, % 0.31/0.63 ! [B: $tType,A: $tType,B3: set @ A,X3: B,A3: set @ ( product_prod @ B @ A )] : % 0.31/0.63 ( ( finite_finite2 @ A @ B3 ) % 0.31/0.63 => ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) % 0.31/0.63 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) % 0.31/0.63 @ ( image @ A @ ( set @ ( product_prod @ B @ A ) ) % 0.31/0.63 @ ^ [Y2: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y2 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) % 0.31/0.63 @ B3 ) ) % 0.31/0.63 @ A3 ) % 0.31/0.63 = ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) ) % 0.31/0.63 @ ^ [Y2: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y2 ) ) % 0.31/0.63 @ A3 % 0.31/0.63 @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % fold_union_pair % 0.31/0.63 thf(fact_6681_Pow__fold,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( pow2 @ A @ A3 ) % 0.31/0.63 = ( finite_fold @ A @ ( set @ ( set @ A ) ) % 0.31/0.63 @ ^ [X: A,A6: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A6 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X ) @ A6 ) ) % 0.31/0.63 @ ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 0.31/0.63 @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Pow_fold % 0.31/0.63 thf(fact_6682_max__weak__def,axiom, % 0.31/0.63 ( fun_max_weak % 0.31/0.63 = ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % max_weak_def % 0.31/0.63 thf(fact_6683_Func__map__surj,axiom, % 0.31/0.63 ! [C: $tType,A: $tType,D: $tType,B: $tType,F1: B > A,A17: set @ B,B1: set @ A,F22: C > D,B22: set @ C,A26: set @ D] : % 0.31/0.63 ( ( ( image @ B @ A @ F1 @ A17 ) % 0.31/0.63 = B1 ) % 0.31/0.63 => ( ( inj_on @ C @ D @ F22 @ B22 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ D ) @ ( image @ C @ D @ F22 @ B22 ) @ A26 ) % 0.31/0.63 => ( ( ( B22 % 0.31/0.63 = ( bot_bot @ ( set @ C ) ) ) % 0.31/0.63 => ( A26 % 0.31/0.63 = ( bot_bot @ ( set @ D ) ) ) ) % 0.31/0.63 => ( ( bNF_Wellorder_Func @ C @ A @ B22 @ B1 ) % 0.31/0.63 = ( image @ ( D > B ) @ ( C > A ) @ ( bNF_We4925052301507509544nc_map @ C @ B @ A @ D @ B22 @ F1 @ F22 ) @ ( bNF_Wellorder_Func @ D @ B @ A26 @ A17 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Func_map_surj % 0.31/0.63 thf(fact_6684_max__ext__additive,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,B3: set @ A,R: set @ ( product_prod @ A @ A ),C5: set @ A,D6: set @ A] : % 0.31/0.63 ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A3 @ B3 ) @ ( max_ext @ A @ R ) ) % 0.31/0.63 => ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ C5 @ D6 ) @ ( max_ext @ A @ R ) ) % 0.31/0.63 => ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ C5 ) @ ( sup_sup @ ( set @ A ) @ B3 @ D6 ) ) @ ( max_ext @ A @ R ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % max_ext_additive % 0.31/0.63 thf(fact_6685_Func__map,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,D: $tType,C: $tType,G: A > B,A26: set @ A,A17: set @ B,F1: B > C,B1: set @ C,F22: D > A,B22: set @ D] : % 0.31/0.63 ( ( member @ ( A > B ) @ G @ ( bNF_Wellorder_Func @ A @ B @ A26 @ A17 ) ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ F1 @ A17 ) @ B1 ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ D @ A @ F22 @ B22 ) @ A26 ) % 0.31/0.63 => ( member @ ( D > C ) @ ( bNF_We4925052301507509544nc_map @ D @ B @ C @ A @ B22 @ F1 @ F22 @ G ) @ ( bNF_Wellorder_Func @ D @ C @ B22 @ B1 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Func_map % 0.31/0.63 thf(fact_6686_max__ext_Ocases,axiom, % 0.31/0.63 ! [A: $tType,A12: set @ A,A23: set @ A,R: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A12 @ A23 ) @ ( max_ext @ A @ R ) ) % 0.31/0.63 => ~ ( ( finite_finite2 @ A @ A12 ) % 0.31/0.63 => ( ( finite_finite2 @ A @ A23 ) % 0.31/0.63 => ( ( A23 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ~ ! [X2: A] : % 0.31/0.63 ( ( member @ A @ X2 @ A12 ) % 0.31/0.63 => ? [Xa3: A] : % 0.31/0.63 ( ( member @ A @ Xa3 @ A23 ) % 0.31/0.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa3 ) @ R ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % max_ext.cases % 0.31/0.63 thf(fact_6687_max__ext_Osimps,axiom, % 0.31/0.63 ! [A: $tType,A12: set @ A,A23: set @ A,R: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A12 @ A23 ) @ ( max_ext @ A @ R ) ) % 0.31/0.63 = ( ( finite_finite2 @ A @ A12 ) % 0.31/0.63 & ( finite_finite2 @ A @ A23 ) % 0.31/0.63 & ( A23 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 & ! [X: A] : % 0.31/0.63 ( ( member @ A @ X @ A12 ) % 0.31/0.63 => ? [Y2: A] : % 0.31/0.63 ( ( member @ A @ Y2 @ A23 ) % 0.31/0.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % max_ext.simps % 0.31/0.63 thf(fact_6688_max__ext_Omax__extI,axiom, % 0.31/0.63 ! [A: $tType,X8: set @ A,Y7: set @ A,R: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( ( finite_finite2 @ A @ X8 ) % 0.31/0.63 => ( ( finite_finite2 @ A @ Y7 ) % 0.31/0.63 => ( ( Y7 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ! [X5: A] : % 0.31/0.63 ( ( member @ A @ X5 @ X8 ) % 0.31/0.63 => ? [Xa: A] : % 0.31/0.63 ( ( member @ A @ Xa @ Y7 ) % 0.31/0.63 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa ) @ R ) ) ) % 0.31/0.63 => ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X8 @ Y7 ) @ ( max_ext @ A @ R ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % max_ext.max_extI % 0.31/0.63 thf(fact_6689_Func__non__emp,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,B3: set @ A,A3: set @ B] : % 0.31/0.63 ( ( B3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( bNF_Wellorder_Func @ B @ A @ A3 @ B3 ) % 0.31/0.63 != ( bot_bot @ ( set @ ( B > A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Func_non_emp % 0.31/0.63 thf(fact_6690_Func__is__emp,axiom, % 0.31/0.63 ! [A: $tType,B: $tType,A3: set @ A,B3: set @ B] : % 0.31/0.63 ( ( ( bNF_Wellorder_Func @ A @ B @ A3 @ B3 ) % 0.31/0.63 = ( bot_bot @ ( set @ ( A > B ) ) ) ) % 0.31/0.63 = ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 & ( B3 % 0.31/0.63 = ( bot_bot @ ( set @ B ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Func_is_emp % 0.31/0.63 thf(fact_6691_Set__filter__fold,axiom, % 0.31/0.63 ! [A: $tType,A3: set @ A,P: A > $o] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( filter3 @ A @ P @ A3 ) % 0.31/0.63 = ( finite_fold @ A @ ( set @ A ) % 0.31/0.63 @ ^ [X: A,A18: set @ A] : ( if @ ( set @ A ) @ ( P @ X ) @ ( insert @ A @ X @ A18 ) @ A18 ) % 0.31/0.63 @ ( bot_bot @ ( set @ A ) ) % 0.31/0.63 @ A3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Set_filter_fold % 0.31/0.63 thf(fact_6692_Sup__fin_Oeq__fold_H,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ( ( lattic5882676163264333800up_fin @ A ) % 0.31/0.63 = ( ^ [A6: set @ A] : % 0.31/0.63 ( the2 @ A % 0.31/0.63 @ ( finite_fold @ A @ ( option @ A ) % 0.31/0.63 @ ^ [X: A,Y2: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( sup_sup @ A @ X ) @ Y2 ) ) % 0.31/0.63 @ ( none @ A ) % 0.31/0.63 @ A6 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.eq_fold' % 0.31/0.63 thf(fact_6693_member__filter,axiom, % 0.31/0.63 ! [A: $tType,X3: A,P: A > $o,A3: set @ A] : % 0.31/0.63 ( ( member @ A @ X3 @ ( filter3 @ A @ P @ A3 ) ) % 0.31/0.63 = ( ( member @ A @ X3 @ A3 ) % 0.31/0.63 & ( P @ X3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % member_filter % 0.31/0.63 thf(fact_6694_inf__Sup__absorb,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( lattice @ A ) % 0.31/0.63 => ! [A3: set @ A,A2: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( member @ A @ A2 @ A3 ) % 0.31/0.63 => ( ( inf_inf @ A @ A2 @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) % 0.31/0.63 = A2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inf_Sup_absorb % 0.31/0.63 thf(fact_6695_Sup__fin_Osingleton,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [X3: A] : % 0.31/0.63 ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 = X3 ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.singleton % 0.31/0.63 thf(fact_6696_Sup__fin_Oinsert,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.63 = ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.insert % 0.31/0.63 thf(fact_6697_Sup__fin_Oin__idem,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.63 => ( ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) % 0.31/0.63 = ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.in_idem % 0.31/0.63 thf(fact_6698_Sup__fin_OcoboundedI,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A3: set @ A,A2: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( member @ A @ A2 @ A3 ) % 0.31/0.63 => ( ord_less_eq @ A @ A2 @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.coboundedI % 0.31/0.63 thf(fact_6699_Sup__fin__Max,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( ( semilattice_sup @ A ) % 0.31/0.63 & ( linorder @ A ) ) % 0.31/0.63 => ( ( lattic5882676163264333800up_fin @ A ) % 0.31/0.63 = ( lattic643756798349783984er_Max @ A ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin_Max % 0.31/0.63 thf(fact_6700_Set_Ofilter__def,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( filter3 @ A ) % 0.31/0.63 = ( ^ [P3: A > $o,A6: set @ A] : % 0.31/0.63 ( collect @ A % 0.31/0.63 @ ^ [A4: A] : % 0.31/0.63 ( ( member @ A @ A4 @ A6 ) % 0.31/0.63 & ( P3 @ A4 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Set.filter_def % 0.31/0.63 thf(fact_6701_finite__filter,axiom, % 0.31/0.63 ! [A: $tType,S3: set @ A,P: A > $o] : % 0.31/0.63 ( ( finite_finite2 @ A @ S3 ) % 0.31/0.63 => ( finite_finite2 @ A @ ( filter3 @ A @ P @ S3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % finite_filter % 0.31/0.63 thf(fact_6702_Sup__fin_Obounded__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X3 ) % 0.31/0.63 = ( ! [X: A] : % 0.31/0.63 ( ( member @ A @ X @ A3 ) % 0.31/0.63 => ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.bounded_iff % 0.31/0.63 thf(fact_6703_Sup__fin_OboundedI,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ! [A5: A] : % 0.31/0.63 ( ( member @ A @ A5 @ A3 ) % 0.31/0.63 => ( ord_less_eq @ A @ A5 @ X3 ) ) % 0.31/0.63 => ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.boundedI % 0.31/0.63 thf(fact_6704_Sup__fin_OboundedE,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X3 ) % 0.31/0.63 => ! [A11: A] : % 0.31/0.63 ( ( member @ A @ A11 @ A3 ) % 0.31/0.63 => ( ord_less_eq @ A @ A11 @ X3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.boundedE % 0.31/0.63 thf(fact_6705_cSup__eq__Sup__fin,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( condit1219197933456340205attice @ A ) % 0.31/0.63 => ! [X8: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ X8 ) % 0.31/0.63 => ( ( X8 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( complete_Sup_Sup @ A @ X8 ) % 0.31/0.63 = ( lattic5882676163264333800up_fin @ A @ X8 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % cSup_eq_Sup_fin % 0.31/0.63 thf(fact_6706_Sup__fin__Sup,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( comple6319245703460814977attice @ A ) % 0.31/0.63 => ! [A3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic5882676163264333800up_fin @ A @ A3 ) % 0.31/0.63 = ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin_Sup % 0.31/0.63 thf(fact_6707_Sup__fin_Oinfinite,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A3: set @ A] : % 0.31/0.63 ( ~ ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( lattic5882676163264333800up_fin @ A @ A3 ) % 0.31/0.63 = ( the2 @ A @ ( none @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.infinite % 0.31/0.63 thf(fact_6708_Sup__fin_Osubset__imp,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.63 => ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.subset_imp % 0.31/0.63 thf(fact_6709_Sup__fin_Ohom__commute,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [H2: A > A,N7: set @ A] : % 0.31/0.63 ( ! [X5: A,Y4: A] : % 0.31/0.63 ( ( H2 @ ( sup_sup @ A @ X5 @ Y4 ) ) % 0.31/0.63 = ( sup_sup @ A @ ( H2 @ X5 ) @ ( H2 @ Y4 ) ) ) % 0.31/0.63 => ( ( finite_finite2 @ A @ N7 ) % 0.31/0.63 => ( ( N7 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( H2 @ ( lattic5882676163264333800up_fin @ A @ N7 ) ) % 0.31/0.63 = ( lattic5882676163264333800up_fin @ A @ ( image @ A @ A @ H2 @ N7 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.hom_commute % 0.31/0.63 thf(fact_6710_Sup__fin_Osubset,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( B3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.63 => ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B3 ) @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) % 0.31/0.63 = ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.subset % 0.31/0.63 thf(fact_6711_Sup__fin_Oinsert__not__elem,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ~ ( member @ A @ X3 @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.63 = ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.insert_not_elem % 0.31/0.63 thf(fact_6712_Sup__fin_Oclosed,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ! [X5: A,Y4: A] : ( member @ A @ ( sup_sup @ A @ X5 @ Y4 ) @ ( insert @ A @ X5 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.63 => ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ A3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.closed % 0.31/0.63 thf(fact_6713_Sup__fin_Ounion,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.63 => ( ( B3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B3 ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.union % 0.31/0.63 thf(fact_6714_Sup__fin_Oeq__fold,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.63 = ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X3 @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.eq_fold % 0.31/0.63 thf(fact_6715_inter__Set__filter,axiom, % 0.31/0.63 ! [A: $tType,B3: set @ A,A3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ B3 ) % 0.31/0.63 => ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 = ( filter3 @ A % 0.31/0.63 @ ^ [X: A] : ( member @ A @ X @ A3 ) % 0.31/0.63 @ B3 ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inter_Set_filter % 0.31/0.63 thf(fact_6716_inf__Sup1__distrib,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( distrib_lattice @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( inf_inf @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) % 0.31/0.63 = ( lattic5882676163264333800up_fin @ A % 0.31/0.63 @ ( collect @ A % 0.31/0.63 @ ^ [Uu3: A] : % 0.31/0.63 ? [A4: A] : % 0.31/0.63 ( ( Uu3 % 0.31/0.63 = ( inf_inf @ A @ X3 @ A4 ) ) % 0.31/0.63 & ( member @ A @ A4 @ A3 ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inf_Sup1_distrib % 0.31/0.63 thf(fact_6717_inf__Sup2__distrib,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( distrib_lattice @ A ) % 0.31/0.63 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.63 => ( ( B3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B3 ) ) % 0.31/0.63 = ( lattic5882676163264333800up_fin @ A % 0.31/0.63 @ ( collect @ A % 0.31/0.63 @ ^ [Uu3: A] : % 0.31/0.63 ? [A4: A,B4: A] : % 0.31/0.63 ( ( Uu3 % 0.31/0.63 = ( inf_inf @ A @ A4 @ B4 ) ) % 0.31/0.63 & ( member @ A @ A4 @ A3 ) % 0.31/0.63 & ( member @ A @ B4 @ B3 ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % inf_Sup2_distrib % 0.31/0.63 thf(fact_6718_Sup__fin_Oinsert__remove,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.63 = X3 ) ) % 0.31/0.63 & ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.63 = ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.insert_remove % 0.31/0.63 thf(fact_6719_Sup__fin_Oremove,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_sup @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.63 => ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic5882676163264333800up_fin @ A @ A3 ) % 0.31/0.63 = X3 ) ) % 0.31/0.63 & ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic5882676163264333800up_fin @ A @ A3 ) % 0.31/0.63 = ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Sup_fin.remove % 0.31/0.63 thf(fact_6720_Inf__fin_Oeq__fold_H,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ( ( lattic7752659483105999362nf_fin @ A ) % 0.31/0.63 = ( ^ [A6: set @ A] : % 0.31/0.63 ( the2 @ A % 0.31/0.63 @ ( finite_fold @ A @ ( option @ A ) % 0.31/0.63 @ ^ [X: A,Y2: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( inf_inf @ A @ X ) @ Y2 ) ) % 0.31/0.63 @ ( none @ A ) % 0.31/0.63 @ A6 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.eq_fold' % 0.31/0.63 thf(fact_6721_max__ext__def,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( max_ext @ A ) % 0.31/0.63 = ( ^ [R6: set @ ( product_prod @ A @ A )] : % 0.31/0.63 ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) % 0.31/0.63 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o % 0.31/0.63 @ ( max_extp @ A % 0.31/0.63 @ ^ [X: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R6 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % max_ext_def % 0.31/0.63 thf(fact_6722_Inf__fin_Osingleton,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [X3: A] : % 0.31/0.63 ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 = X3 ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.singleton % 0.31/0.63 thf(fact_6723_sup__Inf__absorb,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( lattice @ A ) % 0.31/0.63 => ! [A3: set @ A,A2: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( member @ A @ A2 @ A3 ) % 0.31/0.63 => ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ A2 ) % 0.31/0.63 = A2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_Inf_absorb % 0.31/0.63 thf(fact_6724_Inf__fin_Oinsert,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.63 = ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.insert % 0.31/0.63 thf(fact_6725_Inf__fin_OcoboundedI,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [A3: set @ A,A2: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( member @ A @ A2 @ A3 ) % 0.31/0.63 => ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ A2 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.coboundedI % 0.31/0.63 thf(fact_6726_Inf__fin_Oin__idem,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.63 => ( ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) % 0.31/0.63 = ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.in_idem % 0.31/0.63 thf(fact_6727_Inf__fin_OboundedE,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( ord_less_eq @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) % 0.31/0.63 => ! [A11: A] : % 0.31/0.63 ( ( member @ A @ A11 @ A3 ) % 0.31/0.63 => ( ord_less_eq @ A @ X3 @ A11 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.boundedE % 0.31/0.63 thf(fact_6728_Inf__fin_OboundedI,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ! [A5: A] : % 0.31/0.63 ( ( member @ A @ A5 @ A3 ) % 0.31/0.63 => ( ord_less_eq @ A @ X3 @ A5 ) ) % 0.31/0.63 => ( ord_less_eq @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.boundedI % 0.31/0.63 thf(fact_6729_Inf__fin_Obounded__iff,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( ord_less_eq @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) % 0.31/0.63 = ( ! [X: A] : % 0.31/0.63 ( ( member @ A @ X @ A3 ) % 0.31/0.63 => ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.bounded_iff % 0.31/0.63 thf(fact_6730_Inf__fin__Inf,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( comple6319245703460814977attice @ A ) % 0.31/0.63 => ! [A3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic7752659483105999362nf_fin @ A @ A3 ) % 0.31/0.63 = ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin_Inf % 0.31/0.63 thf(fact_6731_cInf__eq__Inf__fin,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( condit1219197933456340205attice @ A ) % 0.31/0.63 => ! [X8: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ X8 ) % 0.31/0.63 => ( ( X8 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( complete_Inf_Inf @ A @ X8 ) % 0.31/0.63 = ( lattic7752659483105999362nf_fin @ A @ X8 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % cInf_eq_Inf_fin % 0.31/0.63 thf(fact_6732_Inf__fin_Oinfinite,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [A3: set @ A] : % 0.31/0.63 ( ~ ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( lattic7752659483105999362nf_fin @ A @ A3 ) % 0.31/0.63 = ( the2 @ A @ ( none @ A ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.infinite % 0.31/0.63 thf(fact_6733_Inf__fin_Osubset__imp,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.63 => ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B3 ) @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.subset_imp % 0.31/0.63 thf(fact_6734_Inf__fin_Ohom__commute,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [H2: A > A,N7: set @ A] : % 0.31/0.63 ( ! [X5: A,Y4: A] : % 0.31/0.63 ( ( H2 @ ( inf_inf @ A @ X5 @ Y4 ) ) % 0.31/0.63 = ( inf_inf @ A @ ( H2 @ X5 ) @ ( H2 @ Y4 ) ) ) % 0.31/0.63 => ( ( finite_finite2 @ A @ N7 ) % 0.31/0.63 => ( ( N7 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( H2 @ ( lattic7752659483105999362nf_fin @ A @ N7 ) ) % 0.31/0.63 = ( lattic7752659483105999362nf_fin @ A @ ( image @ A @ A @ H2 @ N7 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.hom_commute % 0.31/0.63 thf(fact_6735_Inf__fin_Osubset,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( B3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.63 => ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B3 ) @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) % 0.31/0.63 = ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.subset % 0.31/0.63 thf(fact_6736_Inf__fin_Oclosed,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [A3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ! [X5: A,Y4: A] : ( member @ A @ ( inf_inf @ A @ X5 @ Y4 ) @ ( insert @ A @ X5 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.63 => ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ A3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.closed % 0.31/0.63 thf(fact_6737_Inf__fin_Oinsert__not__elem,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ~ ( member @ A @ X3 @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.63 = ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.insert_not_elem % 0.31/0.63 thf(fact_6738_Inf__fin_Ounion,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.63 => ( ( B3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.63 = ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic7752659483105999362nf_fin @ A @ B3 ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.union % 0.31/0.63 thf(fact_6739_Inf__fin__le__Sup__fin,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( lattice @ A ) % 0.31/0.63 => ! [A3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin_le_Sup_fin % 0.31/0.63 thf(fact_6740_Inf__fin_Oeq__fold,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.63 = ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X3 @ A3 ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.eq_fold % 0.31/0.63 thf(fact_6741_sup__Inf2__distrib,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( distrib_lattice @ A ) % 0.31/0.63 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.63 => ( ( B3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic7752659483105999362nf_fin @ A @ B3 ) ) % 0.31/0.63 = ( lattic7752659483105999362nf_fin @ A % 0.31/0.63 @ ( collect @ A % 0.31/0.63 @ ^ [Uu3: A] : % 0.31/0.63 ? [A4: A,B4: A] : % 0.31/0.63 ( ( Uu3 % 0.31/0.63 = ( sup_sup @ A @ A4 @ B4 ) ) % 0.31/0.63 & ( member @ A @ A4 @ A3 ) % 0.31/0.63 & ( member @ A @ B4 @ B3 ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_Inf2_distrib % 0.31/0.63 thf(fact_6742_sup__Inf1__distrib,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( distrib_lattice @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( A3 % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( sup_sup @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) % 0.31/0.63 = ( lattic7752659483105999362nf_fin @ A % 0.31/0.63 @ ( collect @ A % 0.31/0.63 @ ^ [Uu3: A] : % 0.31/0.63 ? [A4: A] : % 0.31/0.63 ( ( Uu3 % 0.31/0.63 = ( sup_sup @ A @ X3 @ A4 ) ) % 0.31/0.63 & ( member @ A @ A4 @ A3 ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % sup_Inf1_distrib % 0.31/0.63 thf(fact_6743_Inf__fin_Oremove,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.63 => ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic7752659483105999362nf_fin @ A @ A3 ) % 0.31/0.63 = X3 ) ) % 0.31/0.63 & ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic7752659483105999362nf_fin @ A @ A3 ) % 0.31/0.63 = ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.63 % 0.31/0.63 % Inf_fin.remove % 0.31/0.63 thf(fact_6744_Inf__fin_Oinsert__remove,axiom, % 0.31/0.63 ! [A: $tType] : % 0.31/0.63 ( ( semilattice_inf @ A ) % 0.31/0.63 => ! [A3: set @ A,X3: A] : % 0.31/0.63 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.63 => ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.63 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.63 => ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.64 = X3 ) ) % 0.31/0.64 & ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.64 = ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Inf_fin.insert_remove % 0.31/0.64 thf(fact_6745_max__extp__max__ext__eq,axiom, % 0.31/0.64 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( max_extp @ A % 0.31/0.64 @ ^ [X: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R ) ) % 0.31/0.64 = ( ^ [X: set @ A,Y2: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X @ Y2 ) @ ( max_ext @ A @ R ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % max_extp_max_ext_eq % 0.31/0.64 thf(fact_6746_max__extp__eq,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( max_extp @ A ) % 0.31/0.64 = ( ^ [R5: A > A > $o,X: set @ A,Y2: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X @ Y2 ) @ ( max_ext @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % max_extp_eq % 0.31/0.64 thf(fact_6747_Min_Oeq__fold_H,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ( ( lattic643756798350308766er_Min @ A ) % 0.31/0.64 = ( ^ [A6: set @ A] : % 0.31/0.64 ( the2 @ A % 0.31/0.64 @ ( finite_fold @ A @ ( option @ A ) % 0.31/0.64 @ ^ [X: A,Y2: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( ord_min @ A @ X ) @ Y2 ) ) % 0.31/0.64 @ ( none @ A ) % 0.31/0.64 @ A6 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min.eq_fold' % 0.31/0.64 thf(fact_6748_insert__relcomp__union__fold,axiom, % 0.31/0.64 ! [C: $tType,B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B ),X3: product_prod @ C @ A,X8: set @ ( product_prod @ C @ B )] : % 0.31/0.64 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 ) % 0.31/0.64 => ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X3 @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S3 ) @ X8 ) % 0.31/0.64 = ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) ) % 0.31/0.64 @ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) ) % 0.31/0.64 @ ^ [W3: A,Z6: B,A18: set @ ( product_prod @ C @ B )] : % 0.31/0.64 ( if @ ( set @ ( product_prod @ C @ B ) ) % 0.31/0.64 @ ( ( product_snd @ C @ A @ X3 ) % 0.31/0.64 = W3 ) % 0.31/0.64 @ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X3 ) @ Z6 ) @ A18 ) % 0.31/0.64 @ A18 ) ) % 0.31/0.64 @ X8 % 0.31/0.64 @ S3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % insert_relcomp_union_fold % 0.31/0.64 thf(fact_6749_relcomp__empty1,axiom, % 0.31/0.64 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ C @ B )] : % 0.31/0.64 ( ( relcomp @ A @ C @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) ) @ R ) % 0.31/0.64 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % relcomp_empty1 % 0.31/0.64 thf(fact_6750_relcomp__empty2,axiom, % 0.31/0.64 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ C )] : % 0.31/0.64 ( ( relcomp @ A @ C @ B @ R @ ( bot_bot @ ( set @ ( product_prod @ C @ B ) ) ) ) % 0.31/0.64 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % relcomp_empty2 % 0.31/0.64 thf(fact_6751_finite__relcomp,axiom, % 0.31/0.64 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] : % 0.31/0.64 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R ) % 0.31/0.64 => ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S3 ) % 0.31/0.64 => ( finite_finite2 @ ( product_prod @ A @ C ) @ ( relcomp @ A @ B @ C @ R @ S3 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % finite_relcomp % 0.31/0.64 thf(fact_6752_Min__singleton,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [X3: A] : % 0.31/0.64 ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.64 = X3 ) ) ). % 0.31/0.64 % 0.31/0.64 % Min_singleton % 0.31/0.64 thf(fact_6753_Min_Obounded__iff,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,X3: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( ord_less_eq @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) ) % 0.31/0.64 = ( ! [X: A] : % 0.31/0.64 ( ( member @ A @ X @ A3 ) % 0.31/0.64 => ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min.bounded_iff % 0.31/0.64 thf(fact_6754_Min__gr__iff,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,X3: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( ord_less @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) ) % 0.31/0.64 = ( ! [X: A] : % 0.31/0.64 ( ( member @ A @ X @ A3 ) % 0.31/0.64 => ( ord_less @ A @ X3 @ X ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min_gr_iff % 0.31/0.64 thf(fact_6755_Min__const,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ B,C2: A] : % 0.31/0.64 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.64 => ( ( lattic643756798350308766er_Min @ A % 0.31/0.64 @ ( image @ B @ A % 0.31/0.64 @ ^ [Uu3: B] : C2 % 0.31/0.64 @ A3 ) ) % 0.31/0.64 = C2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min_const % 0.31/0.64 thf(fact_6756_Min__insert,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,X3: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.64 = ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min_insert % 0.31/0.64 thf(fact_6757_minus__Max__eq__Min,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linord5086331880401160121up_add @ A ) % 0.31/0.64 => ! [S3: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ S3 ) % 0.31/0.64 => ( ( S3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S3 ) ) % 0.31/0.64 = ( lattic643756798350308766er_Min @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % minus_Max_eq_Min % 0.31/0.64 thf(fact_6758_minus__Min__eq__Max,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linord5086331880401160121up_add @ A ) % 0.31/0.64 => ! [S3: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ S3 ) % 0.31/0.64 => ( ( S3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S3 ) ) % 0.31/0.64 = ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % minus_Min_eq_Max % 0.31/0.64 thf(fact_6759_Inf__fin__Min,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( ( semilattice_inf @ A ) % 0.31/0.64 & ( linorder @ A ) ) % 0.31/0.64 => ( ( lattic7752659483105999362nf_fin @ A ) % 0.31/0.64 = ( lattic643756798350308766er_Min @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Inf_fin_Min % 0.31/0.64 thf(fact_6760_relcomp_Ocases,axiom, % 0.31/0.64 ! [A: $tType,C: $tType,B: $tType,A12: A,A23: C,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A12 @ A23 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) ) % 0.31/0.64 => ~ ! [B5: B] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A12 @ B5 ) @ R2 ) % 0.31/0.64 => ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ A23 ) @ S ) ) ) ). % 0.31/0.64 % 0.31/0.64 % relcomp.cases % 0.31/0.64 thf(fact_6761_relcomp_Osimps,axiom, % 0.31/0.64 ! [B: $tType,C: $tType,A: $tType,A12: A,A23: C,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A12 @ A23 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) ) % 0.31/0.64 = ( ? [A4: A,B4: B,C4: C] : % 0.31/0.64 ( ( A12 = A4 ) % 0.31/0.64 & ( A23 = C4 ) % 0.31/0.64 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R2 ) % 0.31/0.64 & ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ C4 ) @ S ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % relcomp.simps % 0.31/0.64 thf(fact_6762_relcomp_OrelcompI,axiom, % 0.31/0.64 ! [A: $tType,C: $tType,B: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),C2: C,S: set @ ( product_prod @ B @ C )] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 ) % 0.31/0.64 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ C2 ) @ S ) % 0.31/0.64 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A2 @ C2 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % relcomp.relcompI % 0.31/0.64 thf(fact_6763_relcompE,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R2: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R2 @ S ) ) % 0.31/0.64 => ~ ! [X5: A,Y4: C,Z3: B] : % 0.31/0.64 ( ( Xz % 0.31/0.64 = ( product_Pair @ A @ B @ X5 @ Z3 ) ) % 0.31/0.64 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X5 @ Y4 ) @ R2 ) % 0.31/0.64 => ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y4 @ Z3 ) @ S ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % relcompE % 0.31/0.64 thf(fact_6764_relcompEpair,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,C: $tType,A2: A,C2: B,R2: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ C2 ) @ ( relcomp @ A @ C @ B @ R2 @ S ) ) % 0.31/0.64 => ~ ! [B5: C] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A2 @ B5 ) @ R2 ) % 0.31/0.64 => ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B5 @ C2 ) @ S ) ) ) ). % 0.31/0.64 % 0.31/0.64 % relcompEpair % 0.31/0.64 thf(fact_6765_Min_OcoboundedI,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,A2: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( member @ A @ A2 @ A3 ) % 0.31/0.64 => ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ A2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min.coboundedI % 0.31/0.64 thf(fact_6766_Min__eqI,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,X3: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ! [Y4: A] : % 0.31/0.64 ( ( member @ A @ Y4 @ A3 ) % 0.31/0.64 => ( ord_less_eq @ A @ X3 @ Y4 ) ) % 0.31/0.64 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.64 => ( ( lattic643756798350308766er_Min @ A @ A3 ) % 0.31/0.64 = X3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min_eqI % 0.31/0.64 thf(fact_6767_Min__le,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,X3: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.64 => ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ X3 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min_le % 0.31/0.64 thf(fact_6768_relpow__add,axiom, % 0.31/0.64 ! [A: $tType,M: nat,N2: nat,R: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( plus_plus @ nat @ M @ N2 ) @ R ) % 0.31/0.64 = ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M @ R ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ). % 0.31/0.64 % 0.31/0.64 % relpow_add % 0.31/0.64 thf(fact_6769_relpow_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,R: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R ) % 0.31/0.64 = ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) @ R ) ) ). % 0.31/0.64 % 0.31/0.64 % relpow.simps(2) % 0.31/0.64 thf(fact_6770_Min_Oin__idem,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,X3: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.64 => ( ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) ) % 0.31/0.64 = ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min.in_idem % 0.31/0.64 thf(fact_6771_Min__in,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( member @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ A3 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min_in % 0.31/0.64 thf(fact_6772_union__comp__emptyL,axiom, % 0.31/0.64 ! [A: $tType,A3: set @ ( product_prod @ A @ A ),C5: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( ( relcomp @ A @ A @ A @ A3 @ C5 ) % 0.31/0.64 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) % 0.31/0.64 => ( ( ( relcomp @ A @ A @ A @ B3 @ C5 ) % 0.31/0.64 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) % 0.31/0.64 => ( ( relcomp @ A @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B3 ) @ C5 ) % 0.31/0.64 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % union_comp_emptyL % 0.31/0.64 thf(fact_6773_union__comp__emptyR,axiom, % 0.31/0.64 ! [A: $tType,A3: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A ),C5: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( ( relcomp @ A @ A @ A @ A3 @ B3 ) % 0.31/0.64 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) % 0.31/0.64 => ( ( ( relcomp @ A @ A @ A @ A3 @ C5 ) % 0.31/0.64 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) % 0.31/0.64 => ( ( relcomp @ A @ A @ A @ A3 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ B3 @ C5 ) ) % 0.31/0.64 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % union_comp_emptyR % 0.31/0.64 thf(fact_6774_Min_OboundedI,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,X3: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ! [A5: A] : % 0.31/0.64 ( ( member @ A @ A5 @ A3 ) % 0.31/0.64 => ( ord_less_eq @ A @ X3 @ A5 ) ) % 0.31/0.64 => ( ord_less_eq @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min.boundedI % 0.31/0.64 thf(fact_6775_Min_OboundedE,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,X3: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( ord_less_eq @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) ) % 0.31/0.64 => ! [A11: A] : % 0.31/0.64 ( ( member @ A @ A11 @ A3 ) % 0.31/0.64 => ( ord_less_eq @ A @ X3 @ A11 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min.boundedE % 0.31/0.64 thf(fact_6776_eq__Min__iff,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,M: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( M % 0.31/0.64 = ( lattic643756798350308766er_Min @ A @ A3 ) ) % 0.31/0.64 = ( ( member @ A @ M @ A3 ) % 0.31/0.64 & ! [X: A] : % 0.31/0.64 ( ( member @ A @ X @ A3 ) % 0.31/0.64 => ( ord_less_eq @ A @ M @ X ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % eq_Min_iff % 0.31/0.64 thf(fact_6777_Min__le__iff,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,X3: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ X3 ) % 0.31/0.64 = ( ? [X: A] : % 0.31/0.64 ( ( member @ A @ X @ A3 ) % 0.31/0.64 & ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min_le_iff % 0.31/0.64 thf(fact_6778_Min__eq__iff,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,M: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( ( lattic643756798350308766er_Min @ A @ A3 ) % 0.31/0.64 = M ) % 0.31/0.64 = ( ( member @ A @ M @ A3 ) % 0.31/0.64 & ! [X: A] : % 0.31/0.64 ( ( member @ A @ X @ A3 ) % 0.31/0.64 => ( ord_less_eq @ A @ M @ X ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min_eq_iff % 0.31/0.64 thf(fact_6779_Min__less__iff,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,X3: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ X3 ) % 0.31/0.64 = ( ? [X: A] : % 0.31/0.64 ( ( member @ A @ X @ A3 ) % 0.31/0.64 & ( ord_less @ A @ X @ X3 ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min_less_iff % 0.31/0.64 thf(fact_6780_Min__insert2,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,A2: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ! [B5: A] : % 0.31/0.64 ( ( member @ A @ B5 @ A3 ) % 0.31/0.64 => ( ord_less_eq @ A @ A2 @ B5 ) ) % 0.31/0.64 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ A2 @ A3 ) ) % 0.31/0.64 = A2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min_insert2 % 0.31/0.64 thf(fact_6781_cInf__eq__Min,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( condit6923001295902523014norder @ A ) % 0.31/0.64 => ! [X8: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ X8 ) % 0.31/0.64 => ( ( X8 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( complete_Inf_Inf @ A @ X8 ) % 0.31/0.64 = ( lattic643756798350308766er_Min @ A @ X8 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % cInf_eq_Min % 0.31/0.64 thf(fact_6782_Min__Inf,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( comple5582772986160207858norder @ A ) % 0.31/0.64 => ! [A3: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( lattic643756798350308766er_Min @ A @ A3 ) % 0.31/0.64 = ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min_Inf % 0.31/0.64 thf(fact_6783_relcomp__unfold,axiom, % 0.31/0.64 ! [B: $tType,C: $tType,A: $tType] : % 0.31/0.64 ( ( relcomp @ A @ C @ B ) % 0.31/0.64 = ( ^ [R5: set @ ( product_prod @ A @ C ),S8: set @ ( product_prod @ C @ B )] : % 0.31/0.64 ( collect @ ( product_prod @ A @ B ) % 0.31/0.64 @ ( product_case_prod @ A @ B @ $o % 0.31/0.64 @ ^ [X: A,Z6: B] : % 0.31/0.64 ? [Y2: C] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ Y2 ) @ R5 ) % 0.31/0.64 & ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y2 @ Z6 ) @ S8 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % relcomp_unfold % 0.31/0.64 thf(fact_6784_Min_Oinfinite,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A] : % 0.31/0.64 ( ~ ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( lattic643756798350308766er_Min @ A @ A3 ) % 0.31/0.64 = ( the2 @ A @ ( none @ A ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min.infinite % 0.31/0.64 thf(fact_6785_Min_Osubset__imp,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.64 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.64 => ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B3 ) @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min.subset_imp % 0.31/0.64 thf(fact_6786_Min__antimono,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [M7: set @ A,N7: set @ A] : % 0.31/0.64 ( ( ord_less_eq @ ( set @ A ) @ M7 @ N7 ) % 0.31/0.64 => ( ( M7 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( finite_finite2 @ A @ N7 ) % 0.31/0.64 => ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N7 ) @ ( lattic643756798350308766er_Min @ A @ M7 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min_antimono % 0.31/0.64 thf(fact_6787_max__ext__compat,axiom, % 0.31/0.64 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S3 ) @ R ) % 0.31/0.64 => ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S3 ) @ ( insert @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( max_ext @ A @ R ) ) ) ). % 0.31/0.64 % 0.31/0.64 % max_ext_compat % 0.31/0.64 thf(fact_6788_hom__Min__commute,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [H2: A > A,N7: set @ A] : % 0.31/0.64 ( ! [X5: A,Y4: A] : % 0.31/0.64 ( ( H2 @ ( ord_min @ A @ X5 @ Y4 ) ) % 0.31/0.64 = ( ord_min @ A @ ( H2 @ X5 ) @ ( H2 @ Y4 ) ) ) % 0.31/0.64 => ( ( finite_finite2 @ A @ N7 ) % 0.31/0.64 => ( ( N7 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( H2 @ ( lattic643756798350308766er_Min @ A @ N7 ) ) % 0.31/0.64 = ( lattic643756798350308766er_Min @ A @ ( image @ A @ A @ H2 @ N7 ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % hom_Min_commute % 0.31/0.64 thf(fact_6789_min__ext__compat,axiom, % 0.31/0.64 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S3 ) @ R ) % 0.31/0.64 => ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S3 ) @ ( insert @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( min_ext @ A @ R ) ) ) ). % 0.31/0.64 % 0.31/0.64 % min_ext_compat % 0.31/0.64 thf(fact_6790_Min_Osubset,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( B3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.64 => ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B3 ) @ ( lattic643756798350308766er_Min @ A @ A3 ) ) % 0.31/0.64 = ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min.subset % 0.31/0.64 thf(fact_6791_Min_Oclosed,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ! [X5: A,Y4: A] : ( member @ A @ ( ord_min @ A @ X5 @ Y4 ) @ ( insert @ A @ X5 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.64 => ( member @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ A3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min.closed % 0.31/0.64 thf(fact_6792_Min_Oinsert__not__elem,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,X3: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ~ ( member @ A @ X3 @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.64 = ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min.insert_not_elem % 0.31/0.64 thf(fact_6793_mono__Min__commute,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( ( linorder @ A ) % 0.31/0.64 & ( linorder @ B ) ) % 0.31/0.64 => ! [F2: A > B,A3: set @ A] : % 0.31/0.64 ( ( order_mono @ A @ B @ F2 ) % 0.31/0.64 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( F2 @ ( lattic643756798350308766er_Min @ A @ A3 ) ) % 0.31/0.64 = ( lattic643756798350308766er_Min @ B @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % mono_Min_commute % 0.31/0.64 thf(fact_6794_Min_Ounion,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,B3: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.64 => ( ( B3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.64 = ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ ( lattic643756798350308766er_Min @ A @ B3 ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min.union % 0.31/0.64 thf(fact_6795_Min_Oeq__fold,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,X3: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.64 = ( finite_fold @ A @ A @ ( ord_min @ A ) @ X3 @ A3 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min.eq_fold % 0.31/0.64 thf(fact_6796_Min__add__commute,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( linord4140545234300271783up_add @ A ) % 0.31/0.64 => ! [S3: set @ B,F2: B > A,K: A] : % 0.31/0.64 ( ( finite_finite2 @ B @ S3 ) % 0.31/0.64 => ( ( S3 % 0.31/0.64 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.64 => ( ( lattic643756798350308766er_Min @ A % 0.31/0.64 @ ( image @ B @ A % 0.31/0.64 @ ^ [X: B] : ( plus_plus @ A @ ( F2 @ X ) @ K ) % 0.31/0.64 @ S3 ) ) % 0.31/0.64 = ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image @ B @ A @ F2 @ S3 ) ) @ K ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min_add_commute % 0.31/0.64 thf(fact_6797_relcomp__fold,axiom, % 0.31/0.64 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] : % 0.31/0.64 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R ) % 0.31/0.64 => ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S3 ) % 0.31/0.64 => ( ( relcomp @ A @ B @ C @ R @ S3 ) % 0.31/0.64 = ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ A @ C ) ) % 0.31/0.64 @ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) % 0.31/0.64 @ ^ [X: A,Y2: B,A6: set @ ( product_prod @ A @ C )] : % 0.31/0.64 ( finite_fold @ ( product_prod @ B @ C ) @ ( set @ ( product_prod @ A @ C ) ) % 0.31/0.64 @ ( product_case_prod @ B @ C @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) % 0.31/0.64 @ ^ [W3: B,Z6: C,A18: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y2 = W3 ) @ ( insert @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ Z6 ) @ A18 ) @ A18 ) ) % 0.31/0.64 @ A6 % 0.31/0.64 @ S3 ) ) % 0.31/0.64 @ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) ) % 0.31/0.64 @ R ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % relcomp_fold % 0.31/0.64 thf(fact_6798_Min_Oinsert__remove,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,X3: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.64 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.64 = X3 ) ) % 0.31/0.64 & ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.64 = ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min.insert_remove % 0.31/0.64 thf(fact_6799_Min_Oremove,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A,X3: A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.64 => ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.64 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( lattic643756798350308766er_Min @ A @ A3 ) % 0.31/0.64 = X3 ) ) % 0.31/0.64 & ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( lattic643756798350308766er_Min @ A @ A3 ) % 0.31/0.64 = ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Min.remove % 0.31/0.64 thf(fact_6800_insert__relcomp__fold,axiom, % 0.31/0.64 ! [C: $tType,B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B ),X3: product_prod @ C @ A,R: set @ ( product_prod @ C @ A )] : % 0.31/0.64 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 ) % 0.31/0.64 => ( ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X3 @ R ) @ S3 ) % 0.31/0.64 = ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) ) % 0.31/0.64 @ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) ) % 0.31/0.64 @ ^ [W3: A,Z6: B,A18: set @ ( product_prod @ C @ B )] : % 0.31/0.64 ( if @ ( set @ ( product_prod @ C @ B ) ) % 0.31/0.64 @ ( ( product_snd @ C @ A @ X3 ) % 0.31/0.64 = W3 ) % 0.31/0.64 @ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X3 ) @ Z6 ) @ A18 ) % 0.31/0.64 @ A18 ) ) % 0.31/0.64 @ ( relcomp @ C @ A @ B @ R @ S3 ) % 0.31/0.64 @ S3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % insert_relcomp_fold % 0.31/0.64 thf(fact_6801_card__Min__le__sum,axiom, % 0.31/0.64 ! [A: $tType,A3: set @ A,F2: A > nat] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A3 ) @ ( lattic643756798350308766er_Min @ nat @ ( image @ A @ nat @ F2 @ A3 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % card_Min_le_sum % 0.31/0.64 thf(fact_6802_comp__fun__commute__relcomp__fold,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,C: $tType,S3: set @ ( product_prod @ A @ B )] : % 0.31/0.64 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 ) % 0.31/0.64 => ( finite6289374366891150609ommute @ ( product_prod @ C @ A ) @ ( set @ ( product_prod @ C @ B ) ) % 0.31/0.64 @ ( product_case_prod @ C @ A @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) ) % 0.31/0.64 @ ^ [X: C,Y2: A,A6: set @ ( product_prod @ C @ B )] : % 0.31/0.64 ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) ) % 0.31/0.64 @ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) ) % 0.31/0.64 @ ^ [W3: A,Z6: B,A18: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y2 = W3 ) @ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X @ Z6 ) @ A18 ) @ A18 ) ) % 0.31/0.64 @ A6 % 0.31/0.64 @ S3 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % comp_fun_commute_relcomp_fold % 0.31/0.64 thf(fact_6803_dual__Max,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ( ( lattices_Max @ A % 0.31/0.64 @ ^ [X: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X ) ) % 0.31/0.64 = ( lattic643756798350308766er_Min @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % dual_Max % 0.31/0.64 thf(fact_6804_linorder_OMax_Ocong,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( lattices_Max @ A ) % 0.31/0.64 = ( lattices_Max @ A ) ) ). % 0.31/0.64 % 0.31/0.64 % linorder.Max.cong % 0.31/0.64 thf(fact_6805_comp__fun__commute__product__fold,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,B3: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ B3 ) % 0.31/0.64 => ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) ) % 0.31/0.64 @ ^ [X: B,Z6: set @ ( product_prod @ B @ A )] : % 0.31/0.64 ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) ) % 0.31/0.64 @ ^ [Y2: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y2 ) ) % 0.31/0.64 @ Z6 % 0.31/0.64 @ B3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % comp_fun_commute_product_fold % 0.31/0.64 thf(fact_6806_f__arg__min__list__f,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( linorder @ B ) % 0.31/0.64 => ! [Xs: list @ A,F2: A > B] : % 0.31/0.64 ( ( Xs % 0.31/0.64 != ( nil @ A ) ) % 0.31/0.64 => ( ( F2 @ ( arg_min_list @ A @ B @ F2 @ Xs ) ) % 0.31/0.64 = ( lattic643756798350308766er_Min @ B @ ( image @ A @ B @ F2 @ ( set2 @ A @ Xs ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % f_arg_min_list_f % 0.31/0.64 thf(fact_6807_UNION__fun__upd,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,A3: B > ( set @ A ),I: B,B3: set @ A,J4: set @ B] : % 0.31/0.64 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ ( fun_upd @ B @ ( set @ A ) @ A3 @ I @ B3 ) @ J4 ) ) % 0.31/0.64 = ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ B ) @ J4 @ ( insert @ B @ I @ ( bot_bot @ ( set @ B ) ) ) ) ) ) @ ( if @ ( set @ A ) @ ( member @ B @ I @ J4 ) @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % UNION_fun_upd % 0.31/0.64 thf(fact_6808_finite__update__induct,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > B,C2: B,P: ( A > B ) > $o] : % 0.31/0.64 ( ( finite_finite2 @ A % 0.31/0.64 @ ( collect @ A % 0.31/0.64 @ ^ [A4: A] : % 0.31/0.64 ( ( F2 @ A4 ) % 0.31/0.64 != C2 ) ) ) % 0.31/0.64 => ( ( P % 0.31/0.64 @ ^ [A4: A] : C2 ) % 0.31/0.64 => ( ! [A5: A,B5: B,F4: A > B] : % 0.31/0.64 ( ( finite_finite2 @ A % 0.31/0.64 @ ( collect @ A % 0.31/0.64 @ ^ [C4: A] : % 0.31/0.64 ( ( F4 @ C4 ) % 0.31/0.64 != C2 ) ) ) % 0.31/0.64 => ( ( ( F4 @ A5 ) % 0.31/0.64 = C2 ) % 0.31/0.64 => ( ( B5 != C2 ) % 0.31/0.64 => ( ( P @ F4 ) % 0.31/0.64 => ( P @ ( fun_upd @ A @ B @ F4 @ A5 @ B5 ) ) ) ) ) ) % 0.31/0.64 => ( P @ F2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % finite_update_induct % 0.31/0.64 thf(fact_6809_arg__min__list__in,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( linorder @ B ) % 0.31/0.64 => ! [Xs: list @ A,F2: A > B] : % 0.31/0.64 ( ( Xs % 0.31/0.64 != ( nil @ A ) ) % 0.31/0.64 => ( member @ A @ ( arg_min_list @ A @ B @ F2 @ Xs ) @ ( set2 @ A @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % arg_min_list_in % 0.31/0.64 thf(fact_6810_fun__upd__image,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,X3: B,A3: set @ B,F2: B > A,Y: A] : % 0.31/0.64 ( ( ( member @ B @ X3 @ A3 ) % 0.31/0.64 => ( ( image @ B @ A @ ( fun_upd @ B @ A @ F2 @ X3 @ Y ) @ A3 ) % 0.31/0.64 = ( insert @ A @ Y @ ( image @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) % 0.31/0.64 & ( ~ ( member @ B @ X3 @ A3 ) % 0.31/0.64 => ( ( image @ B @ A @ ( fun_upd @ B @ A @ F2 @ X3 @ Y ) @ A3 ) % 0.31/0.64 = ( image @ B @ A @ F2 @ A3 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % fun_upd_image % 0.31/0.64 thf(fact_6811_comp__fun__commute__on_Ofold__set__union__disj,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,A3: set @ A,B3: set @ A,Z2: B] : % 0.31/0.64 ( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ S3 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ S3 ) % 0.31/0.64 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( finite_finite2 @ A @ B3 ) % 0.31/0.64 => ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.64 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) % 0.31/0.64 = ( finite_fold @ A @ B @ F2 @ ( finite_fold @ A @ B @ F2 @ Z2 @ A3 ) @ B3 ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % comp_fun_commute_on.fold_set_union_disj % 0.31/0.64 thf(fact_6812_comp__fun__commute__on_Ofold__insert__remove,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] : % 0.31/0.64 ( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 ) % 0.31/0.64 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.64 = ( F2 @ X3 @ ( finite_fold @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % comp_fun_commute_on.fold_insert_remove % 0.31/0.64 thf(fact_6813_Finite__Set_Ofold__cong,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,G: A > B > B,A3: set @ A,S: B,T2: B,B3: set @ A] : % 0.31/0.64 ( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( finite4664212375090638736ute_on @ A @ B @ S3 @ G ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ S3 ) % 0.31/0.64 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ! [X5: A] : % 0.31/0.64 ( ( member @ A @ X5 @ A3 ) % 0.31/0.64 => ( ( F2 @ X5 ) % 0.31/0.64 = ( G @ X5 ) ) ) % 0.31/0.64 => ( ( S = T2 ) % 0.31/0.64 => ( ( A3 = B3 ) % 0.31/0.64 => ( ( finite_fold @ A @ B @ F2 @ S @ A3 ) % 0.31/0.64 = ( finite_fold @ A @ B @ G @ T2 @ B3 ) ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Finite_Set.fold_cong % 0.31/0.64 thf(fact_6814_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,C: $tType,S3: set @ A,F2: A > B > B,G: C > A,R: set @ C] : % 0.31/0.64 ( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ G @ ( top_top @ ( set @ C ) ) ) @ S3 ) % 0.31/0.64 => ( finite4664212375090638736ute_on @ C @ B @ R @ ( comp @ A @ ( B > B ) @ C @ F2 @ G ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % comp_fun_commute_on.comp_comp_fun_commute_on % 0.31/0.64 thf(fact_6815_comp__fun__commute__on_Ofold__insert,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] : % 0.31/0.64 ( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 ) % 0.31/0.64 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ~ ( member @ A @ X3 @ A3 ) % 0.31/0.64 => ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.64 = ( F2 @ X3 @ ( finite_fold @ A @ B @ F2 @ Z2 @ A3 ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % comp_fun_commute_on.fold_insert % 0.31/0.64 thf(fact_6816_comp__fun__commute__on_Ofold__insert2,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] : % 0.31/0.64 ( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 ) % 0.31/0.64 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ~ ( member @ A @ X3 @ A3 ) % 0.31/0.64 => ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.64 = ( finite_fold @ A @ B @ F2 @ ( F2 @ X3 @ Z2 ) @ A3 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % comp_fun_commute_on.fold_insert2 % 0.31/0.64 thf(fact_6817_comp__fun__commute__on_Ofold__fun__left__comm,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] : % 0.31/0.64 ( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 ) % 0.31/0.64 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( F2 @ X3 @ ( finite_fold @ A @ B @ F2 @ Z2 @ A3 ) ) % 0.31/0.64 = ( finite_fold @ A @ B @ F2 @ ( F2 @ X3 @ Z2 ) @ A3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % comp_fun_commute_on.fold_fun_left_comm % 0.31/0.64 thf(fact_6818_comp__fun__commute__on_Ofold__rec,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,A3: set @ A,X3: A,Z2: B] : % 0.31/0.64 ( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ S3 ) % 0.31/0.64 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.64 => ( ( finite_fold @ A @ B @ F2 @ Z2 @ A3 ) % 0.31/0.64 = ( F2 @ X3 @ ( finite_fold @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % comp_fun_commute_on.fold_rec % 0.31/0.64 thf(fact_6819_image__map__upd,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,X3: A,A3: set @ A,M: A > ( option @ B ),Y: B] : % 0.31/0.64 ( ~ ( member @ A @ X3 @ A3 ) % 0.31/0.64 => ( ( image @ A @ ( option @ B ) @ ( fun_upd @ A @ ( option @ B ) @ M @ X3 @ ( some @ B @ Y ) ) @ A3 ) % 0.31/0.64 = ( image @ A @ ( option @ B ) @ M @ A3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % image_map_upd % 0.31/0.64 thf(fact_6820_finite__range__updI,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > ( option @ A ),A2: B,B2: A] : % 0.31/0.64 ( ( finite_finite2 @ ( option @ A ) @ ( image @ B @ ( option @ A ) @ F2 @ ( top_top @ ( set @ B ) ) ) ) % 0.31/0.64 => ( finite_finite2 @ ( option @ A ) @ ( image @ B @ ( option @ A ) @ ( fun_upd @ B @ ( option @ A ) @ F2 @ A2 @ ( some @ A @ B2 ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % finite_range_updI % 0.31/0.64 thf(fact_6821_empty__upd__none,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,X3: A] : % 0.31/0.64 ( ( fun_upd @ A @ ( option @ B ) % 0.31/0.64 @ ^ [X: A] : ( none @ B ) % 0.31/0.64 @ X3 % 0.31/0.64 @ ( none @ B ) ) % 0.31/0.64 = ( ^ [X: A] : ( none @ B ) ) ) ). % 0.31/0.64 % 0.31/0.64 % empty_upd_none % 0.31/0.64 thf(fact_6822_map__upd__Some__unfold,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,M: B > ( option @ A ),A2: B,B2: A,X3: B,Y: A] : % 0.31/0.64 ( ( ( fun_upd @ B @ ( option @ A ) @ M @ A2 @ ( some @ A @ B2 ) @ X3 ) % 0.31/0.64 = ( some @ A @ Y ) ) % 0.31/0.64 = ( ( ( X3 = A2 ) % 0.31/0.64 & ( B2 = Y ) ) % 0.31/0.64 | ( ( X3 != A2 ) % 0.31/0.64 & ( ( M @ X3 ) % 0.31/0.64 = ( some @ A @ Y ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_upd_Some_unfold % 0.31/0.64 thf(fact_6823_map__upd__triv,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,T2: B > ( option @ A ),K: B,X3: A] : % 0.31/0.64 ( ( ( T2 @ K ) % 0.31/0.64 = ( some @ A @ X3 ) ) % 0.31/0.64 => ( ( fun_upd @ B @ ( option @ A ) @ T2 @ K @ ( some @ A @ X3 ) ) % 0.31/0.64 = T2 ) ) ). % 0.31/0.64 % 0.31/0.64 % map_upd_triv % 0.31/0.64 thf(fact_6824_map__upd__eqD1,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,M: A > ( option @ B ),A2: A,X3: B,N2: A > ( option @ B ),Y: B] : % 0.31/0.64 ( ( ( fun_upd @ A @ ( option @ B ) @ M @ A2 @ ( some @ B @ X3 ) ) % 0.31/0.64 = ( fun_upd @ A @ ( option @ B ) @ N2 @ A2 @ ( some @ B @ Y ) ) ) % 0.31/0.64 => ( X3 = Y ) ) ). % 0.31/0.64 % 0.31/0.64 % map_upd_eqD1 % 0.31/0.64 thf(fact_6825_map__upd__nonempty,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,T2: A > ( option @ B ),K: A,X3: B] : % 0.31/0.64 ( ( fun_upd @ A @ ( option @ B ) @ T2 @ K @ ( some @ B @ X3 ) ) % 0.31/0.64 != ( ^ [X: A] : ( none @ B ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_upd_nonempty % 0.31/0.64 thf(fact_6826_graph__map__upd,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,M: A > ( option @ B ),K: A,V2: B] : % 0.31/0.64 ( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( some @ B @ V2 ) ) ) % 0.31/0.64 = ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( none @ B ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % graph_map_upd % 0.31/0.64 thf(fact_6827_restrict__upd__same,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,M: A > ( option @ B ),X3: A,Y: B] : % 0.31/0.64 ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X3 @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.64 = ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % restrict_upd_same % 0.31/0.64 thf(fact_6828_restrict__out,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,X3: A,A3: set @ A,M: A > ( option @ B )] : % 0.31/0.64 ( ~ ( member @ A @ X3 @ A3 ) % 0.31/0.64 => ( ( restrict_map @ A @ B @ M @ A3 @ X3 ) % 0.31/0.64 = ( none @ B ) ) ) ). % 0.31/0.64 % 0.31/0.64 % restrict_out % 0.31/0.64 thf(fact_6829_restrict__map__empty,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,D6: set @ A] : % 0.31/0.64 ( ( restrict_map @ A @ B % 0.31/0.64 @ ^ [X: A] : ( none @ B ) % 0.31/0.64 @ D6 ) % 0.31/0.64 = ( ^ [X: A] : ( none @ B ) ) ) ). % 0.31/0.64 % 0.31/0.64 % restrict_map_empty % 0.31/0.64 thf(fact_6830_restrict__map__to__empty,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,M: A > ( option @ B )] : % 0.31/0.64 ( ( restrict_map @ A @ B @ M @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 = ( ^ [X: A] : ( none @ B ) ) ) ). % 0.31/0.64 % 0.31/0.64 % restrict_map_to_empty % 0.31/0.64 thf(fact_6831_graph__empty,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( graph @ A @ B % 0.31/0.64 @ ^ [X: A] : ( none @ B ) ) % 0.31/0.64 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % graph_empty % 0.31/0.64 thf(fact_6832_fun__upd__restrict__conv,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,X3: A,D6: set @ A,M: A > ( option @ B ),Y: option @ B] : % 0.31/0.64 ( ( member @ A @ X3 @ D6 ) % 0.31/0.64 => ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D6 ) @ X3 @ Y ) % 0.31/0.64 = ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X3 @ Y ) ) ) ). % 0.31/0.64 % 0.31/0.64 % fun_upd_restrict_conv % 0.31/0.64 thf(fact_6833_restrict__fun__upd,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,X3: A,D6: set @ A,M: A > ( option @ B ),Y: option @ B] : % 0.31/0.64 ( ( ( member @ A @ X3 @ D6 ) % 0.31/0.64 => ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X3 @ Y ) @ D6 ) % 0.31/0.64 = ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X3 @ Y ) ) ) % 0.31/0.64 & ( ~ ( member @ A @ X3 @ D6 ) % 0.31/0.64 => ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X3 @ Y ) @ D6 ) % 0.31/0.64 = ( restrict_map @ A @ B @ M @ D6 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % restrict_fun_upd % 0.31/0.64 thf(fact_6834_fun__upd__None__restrict,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,X3: A,D6: set @ A,M: A > ( option @ B )] : % 0.31/0.64 ( ( ( member @ A @ X3 @ D6 ) % 0.31/0.64 => ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D6 ) @ X3 @ ( none @ B ) ) % 0.31/0.64 = ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) % 0.31/0.64 & ( ~ ( member @ A @ X3 @ D6 ) % 0.31/0.64 => ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D6 ) @ X3 @ ( none @ B ) ) % 0.31/0.64 = ( restrict_map @ A @ B @ M @ D6 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % fun_upd_None_restrict % 0.31/0.64 thf(fact_6835_graph__restrictD_I1_J,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,K: A,V2: B,M: A > ( option @ B ),A3: set @ A] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M @ A3 ) ) ) % 0.31/0.64 => ( member @ A @ K @ A3 ) ) ). % 0.31/0.64 % 0.31/0.64 % graph_restrictD(1) % 0.31/0.64 thf(fact_6836_graph__restrictD_I2_J,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,K: A,V2: B,M: A > ( option @ B ),A3: set @ A] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M @ A3 ) ) ) % 0.31/0.64 => ( ( M @ K ) % 0.31/0.64 = ( some @ B @ V2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % graph_restrictD(2) % 0.31/0.64 thf(fact_6837_in__graphD,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,K: A,V2: B,M: A > ( option @ B )] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ M ) ) % 0.31/0.64 => ( ( M @ K ) % 0.31/0.64 = ( some @ B @ V2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % in_graphD % 0.31/0.64 thf(fact_6838_in__graphI,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,M: B > ( option @ A ),K: B,V2: A] : % 0.31/0.64 ( ( ( M @ K ) % 0.31/0.64 = ( some @ A @ V2 ) ) % 0.31/0.64 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V2 ) @ ( graph @ B @ A @ M ) ) ) ). % 0.31/0.64 % 0.31/0.64 % in_graphI % 0.31/0.64 thf(fact_6839_restrict__map__def,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( restrict_map @ A @ B ) % 0.31/0.64 = ( ^ [M2: A > ( option @ B ),A6: set @ A,X: A] : ( if @ ( option @ B ) @ ( member @ A @ X @ A6 ) @ ( M2 @ X ) @ ( none @ B ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % restrict_map_def % 0.31/0.64 thf(fact_6840_graph__def,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( graph @ A @ B ) % 0.31/0.64 = ( ^ [M2: A > ( option @ B )] : % 0.31/0.64 ( collect @ ( product_prod @ A @ B ) % 0.31/0.64 @ ^ [Uu3: product_prod @ A @ B] : % 0.31/0.64 ? [A4: A,B4: B] : % 0.31/0.64 ( ( Uu3 % 0.31/0.64 = ( product_Pair @ A @ B @ A4 @ B4 ) ) % 0.31/0.64 & ( ( M2 @ A4 ) % 0.31/0.64 = ( some @ B @ B4 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % graph_def % 0.31/0.64 thf(fact_6841_graph__fun__upd__None,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,M: A > ( option @ B ),K: A] : % 0.31/0.64 ( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( none @ B ) ) ) % 0.31/0.64 = ( collect @ ( product_prod @ A @ B ) % 0.31/0.64 @ ^ [E4: product_prod @ A @ B] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ B ) @ E4 @ ( graph @ A @ B @ M ) ) % 0.31/0.64 & ( ( product_fst @ A @ B @ E4 ) % 0.31/0.64 != K ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % graph_fun_upd_None % 0.31/0.64 thf(fact_6842_fun__upd__restrict,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,M: A > ( option @ B ),D6: set @ A,X3: A,Y: option @ B] : % 0.31/0.64 ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D6 ) @ X3 @ Y ) % 0.31/0.64 = ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X3 @ Y ) ) ). % 0.31/0.64 % 0.31/0.64 % fun_upd_restrict % 0.31/0.64 thf(fact_6843_restrict__complement__singleton__eq,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: A > ( option @ B ),X3: A] : % 0.31/0.64 ( ( restrict_map @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 0.31/0.64 = ( fun_upd @ A @ ( option @ B ) @ F2 @ X3 @ ( none @ B ) ) ) ). % 0.31/0.64 % 0.31/0.64 % restrict_complement_singleton_eq % 0.31/0.64 thf(fact_6844_restrict__map__upds,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,D6: set @ A,M: A > ( option @ B )] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ D6 ) % 0.31/0.64 => ( ( restrict_map @ A @ B @ ( map_upds @ A @ B @ M @ Xs @ Ys ) @ D6 ) % 0.31/0.64 = ( map_upds @ A @ B @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D6 @ ( set2 @ A @ Xs ) ) ) @ Xs @ Ys ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % restrict_map_upds % 0.31/0.64 thf(fact_6845_ran__map__upd,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,M: B > ( option @ A ),A2: B,B2: A] : % 0.31/0.64 ( ( ( M @ A2 ) % 0.31/0.64 = ( none @ A ) ) % 0.31/0.64 => ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M @ A2 @ ( some @ A @ B2 ) ) ) % 0.31/0.64 = ( insert @ A @ B2 @ ( ran @ B @ A @ M ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % ran_map_upd % 0.31/0.64 thf(fact_6846_map__upds__apply__nontin,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,X3: A,Xs: list @ A,F2: A > ( option @ B ),Ys: list @ B] : % 0.31/0.64 ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ( ( map_upds @ A @ B @ F2 @ Xs @ Ys @ X3 ) % 0.31/0.64 = ( F2 @ X3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_upds_apply_nontin % 0.31/0.64 thf(fact_6847_ran__empty,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( ran @ B @ A % 0.31/0.64 @ ^ [X: B] : ( none @ A ) ) % 0.31/0.64 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % ran_empty % 0.31/0.64 thf(fact_6848_map__upds__list__update2__drop,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xs: list @ A,I: nat,M: A > ( option @ B ),Ys: list @ B,Y: B] : % 0.31/0.64 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I ) % 0.31/0.64 => ( ( map_upds @ A @ B @ M @ Xs @ ( list_update @ B @ Ys @ I @ Y ) ) % 0.31/0.64 = ( map_upds @ A @ B @ M @ Xs @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_upds_list_update2_drop % 0.31/0.64 thf(fact_6849_map__upds__twist,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,A2: A,As: list @ A,M: A > ( option @ B ),B2: B,Bs: list @ B] : % 0.31/0.64 ( ~ ( member @ A @ A2 @ ( set2 @ A @ As ) ) % 0.31/0.64 => ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ A2 @ ( some @ B @ B2 ) ) @ As @ Bs ) % 0.31/0.64 = ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ As @ Bs ) @ A2 @ ( some @ B @ B2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_upds_twist % 0.31/0.64 thf(fact_6850_ranI,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,M: B > ( option @ A ),A2: B,B2: A] : % 0.31/0.64 ( ( ( M @ A2 ) % 0.31/0.64 = ( some @ A @ B2 ) ) % 0.31/0.64 => ( member @ A @ B2 @ ( ran @ B @ A @ M ) ) ) ). % 0.31/0.64 % 0.31/0.64 % ranI % 0.31/0.64 thf(fact_6851_ran__restrictD,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Y: A,M: B > ( option @ A ),A3: set @ B] : % 0.31/0.64 ( ( member @ A @ Y @ ( ran @ B @ A @ ( restrict_map @ B @ A @ M @ A3 ) ) ) % 0.31/0.64 => ? [X5: B] : % 0.31/0.64 ( ( member @ B @ X5 @ A3 ) % 0.31/0.64 & ( ( M @ X5 ) % 0.31/0.64 = ( some @ A @ Y ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % ran_restrictD % 0.31/0.64 thf(fact_6852_ran__def,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( ran @ A @ B ) % 0.31/0.64 = ( ^ [M2: A > ( option @ B )] : % 0.31/0.64 ( collect @ B % 0.31/0.64 @ ^ [B4: B] : % 0.31/0.64 ? [A4: A] : % 0.31/0.64 ( ( M2 @ A4 ) % 0.31/0.64 = ( some @ B @ B4 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % ran_def % 0.31/0.64 thf(fact_6853_ran__map__upd__Some,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,M: B > ( option @ A ),X3: B,Y: A,Z2: A] : % 0.31/0.64 ( ( ( M @ X3 ) % 0.31/0.64 = ( some @ A @ Y ) ) % 0.31/0.64 => ( ( inj_on @ B @ ( option @ A ) @ M @ ( dom @ B @ A @ M ) ) % 0.31/0.64 => ( ~ ( member @ A @ Z2 @ ( ran @ B @ A @ M ) ) % 0.31/0.64 => ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M @ X3 @ ( some @ A @ Z2 ) ) ) % 0.31/0.64 = ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % ran_map_upd_Some % 0.31/0.64 thf(fact_6854_map__upds__fold__map__upd,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( map_upds @ A @ B ) % 0.31/0.64 = ( ^ [M2: A > ( option @ B ),Ks: list @ A,Vs: list @ B] : % 0.31/0.64 ( foldl @ ( A > ( option @ B ) ) @ ( product_prod @ A @ B ) % 0.31/0.64 @ ^ [N: A > ( option @ B )] : % 0.31/0.64 ( product_case_prod @ A @ B @ ( A > ( option @ B ) ) % 0.31/0.64 @ ^ [K3: A,V5: B] : ( fun_upd @ A @ ( option @ B ) @ N @ K3 @ ( some @ B @ V5 ) ) ) % 0.31/0.64 @ M2 % 0.31/0.64 @ ( zip @ A @ B @ Ks @ Vs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_upds_fold_map_upd % 0.31/0.64 thf(fact_6855_dom__eq__empty__conv,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > ( option @ B )] : % 0.31/0.64 ( ( ( dom @ A @ B @ F2 ) % 0.31/0.64 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 = ( F2 % 0.31/0.64 = ( ^ [X: A] : ( none @ B ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % dom_eq_empty_conv % 0.31/0.64 thf(fact_6856_fun__upd__None__if__notin__dom,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,K: A,M: A > ( option @ B )] : % 0.31/0.64 ( ~ ( member @ A @ K @ ( dom @ A @ B @ M ) ) % 0.31/0.64 => ( ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( none @ B ) ) % 0.31/0.64 = M ) ) ). % 0.31/0.64 % 0.31/0.64 % fun_upd_None_if_notin_dom % 0.31/0.64 thf(fact_6857_dom__const,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > B] : % 0.31/0.64 ( ( dom @ A @ B % 0.31/0.64 @ ^ [X: A] : ( some @ B @ ( F2 @ X ) ) ) % 0.31/0.64 = ( top_top @ ( set @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % dom_const % 0.31/0.64 thf(fact_6858_dom__empty,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( dom @ A @ B % 0.31/0.64 @ ^ [X: A] : ( none @ B ) ) % 0.31/0.64 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % dom_empty % 0.31/0.64 thf(fact_6859_finite__graph__iff__finite__dom,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,M: A > ( option @ B )] : % 0.31/0.64 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( graph @ A @ B @ M ) ) % 0.31/0.64 = ( finite_finite2 @ A @ ( dom @ A @ B @ M ) ) ) ). % 0.31/0.64 % 0.31/0.64 % finite_graph_iff_finite_dom % 0.31/0.64 thf(fact_6860_dom__fun__upd,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Y: option @ B,F2: A > ( option @ B ),X3: A] : % 0.31/0.64 ( ( ( Y % 0.31/0.64 = ( none @ B ) ) % 0.31/0.64 => ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X3 @ Y ) ) % 0.31/0.64 = ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F2 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) % 0.31/0.64 & ( ( Y % 0.31/0.64 != ( none @ B ) ) % 0.31/0.64 => ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X3 @ Y ) ) % 0.31/0.64 = ( insert @ A @ X3 @ ( dom @ A @ B @ F2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % dom_fun_upd % 0.31/0.64 thf(fact_6861_finite__ran,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,P4: A > ( option @ B )] : % 0.31/0.64 ( ( finite_finite2 @ A @ ( dom @ A @ B @ P4 ) ) % 0.31/0.64 => ( finite_finite2 @ B @ ( ran @ A @ B @ P4 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % finite_ran % 0.31/0.64 thf(fact_6862_insert__dom,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > ( option @ A ),X3: B,Y: A] : % 0.31/0.64 ( ( ( F2 @ X3 ) % 0.31/0.64 = ( some @ A @ Y ) ) % 0.31/0.64 => ( ( insert @ B @ X3 @ ( dom @ B @ A @ F2 ) ) % 0.31/0.64 = ( dom @ B @ A @ F2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % insert_dom % 0.31/0.64 thf(fact_6863_domD,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,A2: A,M: A > ( option @ B )] : % 0.31/0.64 ( ( member @ A @ A2 @ ( dom @ A @ B @ M ) ) % 0.31/0.64 => ? [B5: B] : % 0.31/0.64 ( ( M @ A2 ) % 0.31/0.64 = ( some @ B @ B5 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % domD % 0.31/0.64 thf(fact_6864_domI,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,M: B > ( option @ A ),A2: B,B2: A] : % 0.31/0.64 ( ( ( M @ A2 ) % 0.31/0.64 = ( some @ A @ B2 ) ) % 0.31/0.64 => ( member @ B @ A2 @ ( dom @ B @ A @ M ) ) ) ). % 0.31/0.64 % 0.31/0.64 % domI % 0.31/0.64 thf(fact_6865_domIff,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,A2: A,M: A > ( option @ B )] : % 0.31/0.64 ( ( member @ A @ A2 @ ( dom @ A @ B @ M ) ) % 0.31/0.64 = ( ( M @ A2 ) % 0.31/0.64 != ( none @ B ) ) ) ). % 0.31/0.64 % 0.31/0.64 % domIff % 0.31/0.64 thf(fact_6866_dom__def,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( dom @ A @ B ) % 0.31/0.64 = ( ^ [M2: A > ( option @ B )] : % 0.31/0.64 ( collect @ A % 0.31/0.64 @ ^ [A4: A] : % 0.31/0.64 ( ( M2 @ A4 ) % 0.31/0.64 != ( none @ B ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % dom_def % 0.31/0.64 thf(fact_6867_foldl__cong,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,A2: A,B2: A,L: list @ B,K: list @ B,F2: A > B > A,G: A > B > A] : % 0.31/0.64 ( ( A2 = B2 ) % 0.31/0.64 => ( ( L = K ) % 0.31/0.64 => ( ! [A5: A,X5: B] : % 0.31/0.64 ( ( member @ B @ X5 @ ( set2 @ B @ L ) ) % 0.31/0.64 => ( ( F2 @ A5 @ X5 ) % 0.31/0.64 = ( G @ A5 @ X5 ) ) ) % 0.31/0.64 => ( ( foldl @ A @ B @ F2 @ A2 @ L ) % 0.31/0.64 = ( foldl @ A @ B @ G @ B2 @ K ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % foldl_cong % 0.31/0.64 thf(fact_6868_foldl__Nil,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A > B,A2: B] : % 0.31/0.64 ( ( foldl @ B @ A @ F2 @ A2 @ ( nil @ A ) ) % 0.31/0.64 = A2 ) ). % 0.31/0.64 % 0.31/0.64 % foldl_Nil % 0.31/0.64 thf(fact_6869_finite__map__freshness,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: A > ( option @ B )] : % 0.31/0.64 ( ( finite_finite2 @ A @ ( dom @ A @ B @ F2 ) ) % 0.31/0.64 => ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) % 0.31/0.64 => ? [X5: A] : % 0.31/0.64 ( ( F2 @ X5 ) % 0.31/0.64 = ( none @ B ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % finite_map_freshness % 0.31/0.64 thf(fact_6870_dom__minus,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > ( option @ A ),X3: B,A3: set @ B] : % 0.31/0.64 ( ( ( F2 @ X3 ) % 0.31/0.64 = ( none @ A ) ) % 0.31/0.64 => ( ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F2 ) @ ( insert @ B @ X3 @ A3 ) ) % 0.31/0.64 = ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F2 ) @ A3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % dom_minus % 0.31/0.64 thf(fact_6871_finite__set__of__finite__maps,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.64 => ( finite_finite2 @ ( A > ( option @ B ) ) % 0.31/0.64 @ ( collect @ ( A > ( option @ B ) ) % 0.31/0.64 @ ^ [M2: A > ( option @ B )] : % 0.31/0.64 ( ( ( dom @ A @ B @ M2 ) % 0.31/0.64 = A3 ) % 0.31/0.64 & ( ord_less_eq @ ( set @ B ) @ ( ran @ A @ B @ M2 ) @ B3 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % finite_set_of_finite_maps % 0.31/0.64 thf(fact_6872_graph__eq__to__snd__dom,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( graph @ A @ B ) % 0.31/0.64 = ( ^ [M2: A > ( option @ B )] : % 0.31/0.64 ( image @ A @ ( product_prod @ A @ B ) % 0.31/0.64 @ ^ [X: A] : ( product_Pair @ A @ B @ X @ ( the2 @ B @ ( M2 @ X ) ) ) % 0.31/0.64 @ ( dom @ A @ B @ M2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % graph_eq_to_snd_dom % 0.31/0.64 thf(fact_6873_finite__Map__induct,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,M: A > ( option @ B ),P: ( A > ( option @ B ) ) > $o] : % 0.31/0.64 ( ( finite_finite2 @ A @ ( dom @ A @ B @ M ) ) % 0.31/0.64 => ( ( P % 0.31/0.64 @ ^ [X: A] : ( none @ B ) ) % 0.31/0.64 => ( ! [K2: A,V3: B,M3: A > ( option @ B )] : % 0.31/0.64 ( ( finite_finite2 @ A @ ( dom @ A @ B @ M3 ) ) % 0.31/0.64 => ( ~ ( member @ A @ K2 @ ( dom @ A @ B @ M3 ) ) % 0.31/0.64 => ( ( P @ M3 ) % 0.31/0.64 => ( P @ ( fun_upd @ A @ ( option @ B ) @ M3 @ K2 @ ( some @ B @ V3 ) ) ) ) ) ) % 0.31/0.64 => ( P @ M ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % finite_Map_induct % 0.31/0.64 thf(fact_6874_dom__eq__singleton__conv,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: A > ( option @ B ),X3: A] : % 0.31/0.64 ( ( ( dom @ A @ B @ F2 ) % 0.31/0.64 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.64 = ( ? [V5: B] : % 0.31/0.64 ( F2 % 0.31/0.64 = ( fun_upd @ A @ ( option @ B ) % 0.31/0.64 @ ^ [X: A] : ( none @ B ) % 0.31/0.64 @ X3 % 0.31/0.64 @ ( some @ B @ V5 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % dom_eq_singleton_conv % 0.31/0.64 thf(fact_6875_min__list__Min,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [Xs: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 != ( nil @ A ) ) % 0.31/0.64 => ( ( min_list @ A @ Xs ) % 0.31/0.64 = ( lattic643756798350308766er_Min @ A @ ( set2 @ A @ Xs ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % min_list_Min % 0.31/0.64 thf(fact_6876_minus__fold__remove,axiom, % 0.31/0.64 ! [A: $tType,A3: set @ A,B3: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( minus_minus @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.64 = ( finite_fold @ A @ ( set @ A ) @ ( remove @ A ) @ B3 @ A3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % minus_fold_remove % 0.31/0.64 thf(fact_6877_member__remove,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Y: A,A3: set @ A] : % 0.31/0.64 ( ( member @ A @ X3 @ ( remove @ A @ Y @ A3 ) ) % 0.31/0.64 = ( ( member @ A @ X3 @ A3 ) % 0.31/0.64 & ( X3 != Y ) ) ) ). % 0.31/0.64 % 0.31/0.64 % member_remove % 0.31/0.64 thf(fact_6878_remove__code_I1_J,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( remove @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 = ( set2 @ A @ ( removeAll @ A @ X3 @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remove_code(1) % 0.31/0.64 thf(fact_6879_remove__def,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( remove @ A ) % 0.31/0.64 = ( ^ [X: A,A6: set @ A] : ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remove_def % 0.31/0.64 thf(fact_6880_finite__subsets__at__top__finite,axiom, % 0.31/0.64 ! [A: $tType,A3: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( finite5375528669736107172at_top @ A @ A3 ) % 0.31/0.64 = ( principal @ ( set @ A ) @ ( insert @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % finite_subsets_at_top_finite % 0.31/0.64 thf(fact_6881_pos__deriv__imp__strict__mono,axiom, % 0.31/0.64 ! [F2: real > real,F11: real > real] : % 0.31/0.64 ( ! [X5: real] : ( has_field_derivative @ real @ F2 @ ( F11 @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) % 0.31/0.64 => ( ! [X5: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F11 @ X5 ) ) % 0.31/0.64 => ( order_strict_mono @ real @ real @ F2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % pos_deriv_imp_strict_mono % 0.31/0.64 thf(fact_6882_eventually__finite__subsets__at__top__weakI,axiom, % 0.31/0.64 ! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] : % 0.31/0.64 ( ! [X10: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ X10 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ X10 @ A3 ) % 0.31/0.64 => ( P @ X10 ) ) ) % 0.31/0.64 => ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % eventually_finite_subsets_at_top_weakI % 0.31/0.64 thf(fact_6883_eventually__finite__subsets__at__top__finite,axiom, % 0.31/0.64 ! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A3 ) ) % 0.31/0.64 = ( P @ A3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % eventually_finite_subsets_at_top_finite % 0.31/0.64 thf(fact_6884_strict__mono__mono,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( ( order @ A ) % 0.31/0.64 & ( order @ B ) ) % 0.31/0.64 => ! [F2: A > B] : % 0.31/0.64 ( ( order_strict_mono @ A @ B @ F2 ) % 0.31/0.64 => ( order_mono @ A @ B @ F2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % strict_mono_mono % 0.31/0.64 thf(fact_6885_strict__mono__eq,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( ( linorder @ A ) % 0.31/0.64 & ( order @ B ) ) % 0.31/0.64 => ! [F2: A > B,X3: A,Y: A] : % 0.31/0.64 ( ( order_strict_mono @ A @ B @ F2 ) % 0.31/0.64 => ( ( ( F2 @ X3 ) % 0.31/0.64 = ( F2 @ Y ) ) % 0.31/0.64 = ( X3 = Y ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % strict_mono_eq % 0.31/0.64 thf(fact_6886_strict__mono__less,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( ( linorder @ A ) % 0.31/0.64 & ( order @ B ) ) % 0.31/0.64 => ! [F2: A > B,X3: A,Y: A] : % 0.31/0.64 ( ( order_strict_mono @ A @ B @ F2 ) % 0.31/0.64 => ( ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) ) % 0.31/0.64 = ( ord_less @ A @ X3 @ Y ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % strict_mono_less % 0.31/0.64 thf(fact_6887_strict__mono__def,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( ( order @ A ) % 0.31/0.64 & ( order @ B ) ) % 0.31/0.64 => ( ( order_strict_mono @ A @ B ) % 0.31/0.64 = ( ^ [F3: A > B] : % 0.31/0.64 ! [X: A,Y2: A] : % 0.31/0.64 ( ( ord_less @ A @ X @ Y2 ) % 0.31/0.64 => ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % strict_mono_def % 0.31/0.64 thf(fact_6888_strict__monoI,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( ( order @ A ) % 0.31/0.64 & ( order @ B ) ) % 0.31/0.64 => ! [F2: A > B] : % 0.31/0.64 ( ! [X5: A,Y4: A] : % 0.31/0.64 ( ( ord_less @ A @ X5 @ Y4 ) % 0.31/0.64 => ( ord_less @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.31/0.64 => ( order_strict_mono @ A @ B @ F2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % strict_monoI % 0.31/0.64 thf(fact_6889_strict__monoD,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( ( order @ A ) % 0.31/0.64 & ( order @ B ) ) % 0.31/0.64 => ! [F2: A > B,X3: A,Y: A] : % 0.31/0.64 ( ( order_strict_mono @ A @ B @ F2 ) % 0.31/0.64 => ( ( ord_less @ A @ X3 @ Y ) % 0.31/0.64 => ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % strict_monoD % 0.31/0.64 thf(fact_6890_strict__mono__add,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linordered_semidom @ A ) % 0.31/0.64 => ! [K: A] : % 0.31/0.64 ( order_strict_mono @ A @ A % 0.31/0.64 @ ^ [N: A] : ( plus_plus @ A @ N @ K ) ) ) ). % 0.31/0.64 % 0.31/0.64 % strict_mono_add % 0.31/0.64 thf(fact_6891_strict__mono__less__eq,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( ( linorder @ A ) % 0.31/0.64 & ( order @ B ) ) % 0.31/0.64 => ! [F2: A > B,X3: A,Y: A] : % 0.31/0.64 ( ( order_strict_mono @ A @ B @ F2 ) % 0.31/0.64 => ( ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) ) % 0.31/0.64 = ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % strict_mono_less_eq % 0.31/0.64 thf(fact_6892_strict__mono__leD,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( ( order @ A ) % 0.31/0.64 & ( order @ B ) ) % 0.31/0.64 => ! [R2: A > B,M: A,N2: A] : % 0.31/0.64 ( ( order_strict_mono @ A @ B @ R2 ) % 0.31/0.64 => ( ( ord_less_eq @ A @ M @ N2 ) % 0.31/0.64 => ( ord_less_eq @ B @ ( R2 @ M ) @ ( R2 @ N2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % strict_mono_leD % 0.31/0.64 thf(fact_6893_finite__subsets__at__top__neq__bot,axiom, % 0.31/0.64 ! [A: $tType,A3: set @ A] : % 0.31/0.64 ( ( finite5375528669736107172at_top @ A @ A3 ) % 0.31/0.64 != ( bot_bot @ ( filter @ ( set @ A ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % finite_subsets_at_top_neq_bot % 0.31/0.64 thf(fact_6894_eventually__finite__subsets__at__top,axiom, % 0.31/0.64 ! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] : % 0.31/0.64 ( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A3 ) ) % 0.31/0.64 = ( ? [X6: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ X6 ) % 0.31/0.64 & ( ord_less_eq @ ( set @ A ) @ X6 @ A3 ) % 0.31/0.64 & ! [Y9: set @ A] : % 0.31/0.64 ( ( ( finite_finite2 @ A @ Y9 ) % 0.31/0.64 & ( ord_less_eq @ ( set @ A ) @ X6 @ Y9 ) % 0.31/0.64 & ( ord_less_eq @ ( set @ A ) @ Y9 @ A3 ) ) % 0.31/0.64 => ( P @ Y9 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % eventually_finite_subsets_at_top % 0.31/0.64 thf(fact_6895_finite__subsets__at__top__def,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( finite5375528669736107172at_top @ A ) % 0.31/0.64 = ( ^ [A6: set @ A] : % 0.31/0.64 ( complete_Inf_Inf @ ( filter @ ( set @ A ) ) % 0.31/0.64 @ ( image @ ( set @ A ) @ ( filter @ ( set @ A ) ) % 0.31/0.64 @ ^ [X6: set @ A] : % 0.31/0.64 ( principal @ ( set @ A ) % 0.31/0.64 @ ( collect @ ( set @ A ) % 0.31/0.64 @ ^ [Y9: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ Y9 ) % 0.31/0.64 & ( ord_less_eq @ ( set @ A ) @ X6 @ Y9 ) % 0.31/0.64 & ( ord_less_eq @ ( set @ A ) @ Y9 @ A6 ) ) ) ) % 0.31/0.64 @ ( collect @ ( set @ A ) % 0.31/0.64 @ ^ [X6: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ X6 ) % 0.31/0.64 & ( ord_less_eq @ ( set @ A ) @ X6 @ A6 ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % finite_subsets_at_top_def % 0.31/0.64 thf(fact_6896_filterlim__finite__subsets__at__top,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: A > ( set @ B ),A3: set @ B,F5: filter @ A] : % 0.31/0.64 ( ( filterlim @ A @ ( set @ B ) @ F2 @ ( finite5375528669736107172at_top @ B @ A3 ) @ F5 ) % 0.31/0.64 = ( ! [X6: set @ B] : % 0.31/0.64 ( ( ( finite_finite2 @ B @ X6 ) % 0.31/0.64 & ( ord_less_eq @ ( set @ B ) @ X6 @ A3 ) ) % 0.31/0.64 => ( eventually @ A % 0.31/0.64 @ ^ [Y2: A] : % 0.31/0.64 ( ( finite_finite2 @ B @ ( F2 @ Y2 ) ) % 0.31/0.64 & ( ord_less_eq @ ( set @ B ) @ X6 @ ( F2 @ Y2 ) ) % 0.31/0.64 & ( ord_less_eq @ ( set @ B ) @ ( F2 @ Y2 ) @ A3 ) ) % 0.31/0.64 @ F5 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % filterlim_finite_subsets_at_top % 0.31/0.64 thf(fact_6897_ring__1__class_Oof__int__def,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( ring_1 @ A ) % 0.31/0.64 => ( ( ring_1_of_int @ A ) % 0.31/0.64 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ A @ A @ rep_Integ @ ( id @ A ) % 0.31/0.64 @ ( product_case_prod @ nat @ nat @ A % 0.31/0.64 @ ^ [I3: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % ring_1_class.of_int_def % 0.31/0.64 thf(fact_6898_sorted__key__list__of__set__def,axiom, % 0.31/0.64 ! [A: $tType,B: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ( ( linord144544945434240204of_set @ B @ A ) % 0.31/0.64 = ( ^ [F3: B > A] : ( finite_folding_F @ B @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F3 ) @ ( nil @ B ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % sorted_key_list_of_set_def % 0.31/0.64 thf(fact_6899_of__nat__eq__id,axiom, % 0.31/0.64 ( ( semiring_1_of_nat @ nat ) % 0.31/0.64 = ( id @ nat ) ) ). % 0.31/0.64 % 0.31/0.64 % of_nat_eq_id % 0.31/0.64 thf(fact_6900_case__prod__Pair,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) ) % 0.31/0.64 = ( id @ ( product_prod @ A @ B ) ) ) ). % 0.31/0.64 % 0.31/0.64 % case_prod_Pair % 0.31/0.64 thf(fact_6901_id__funpow,axiom, % 0.31/0.64 ! [A: $tType,N2: nat] : % 0.31/0.64 ( ( compow @ ( A > A ) @ N2 @ ( id @ A ) ) % 0.31/0.64 = ( id @ A ) ) ). % 0.31/0.64 % 0.31/0.64 % id_funpow % 0.31/0.64 thf(fact_6902_apfst__id,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( product_apfst @ A @ A @ B @ ( id @ A ) ) % 0.31/0.64 = ( id @ ( product_prod @ A @ B ) ) ) ). % 0.31/0.64 % 0.31/0.64 % apfst_id % 0.31/0.64 thf(fact_6903_apsnd__id,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( product_apsnd @ B @ B @ A @ ( id @ B ) ) % 0.31/0.64 = ( id @ ( product_prod @ A @ B ) ) ) ). % 0.31/0.64 % 0.31/0.64 % apsnd_id % 0.31/0.64 thf(fact_6904_push__bit__0__id,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.64 => ( ( bit_se4730199178511100633sh_bit @ A @ ( zero_zero @ nat ) ) % 0.31/0.64 = ( id @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % push_bit_0_id % 0.31/0.64 thf(fact_6905_drop__bit__0,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.64 => ( ( bit_se4197421643247451524op_bit @ A @ ( zero_zero @ nat ) ) % 0.31/0.64 = ( id @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % drop_bit_0 % 0.31/0.64 thf(fact_6906_comp__the__Some,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( comp @ ( option @ A ) @ A @ A @ ( the2 @ A ) @ ( some @ A ) ) % 0.31/0.64 = ( id @ A ) ) ). % 0.31/0.64 % 0.31/0.64 % comp_the_Some % 0.31/0.64 thf(fact_6907_infinite__enumerate,axiom, % 0.31/0.64 ! [S3: set @ nat] : % 0.31/0.64 ( ~ ( finite_finite2 @ nat @ S3 ) % 0.31/0.64 => ? [R4: nat > nat] : % 0.31/0.64 ( ( order_strict_mono @ nat @ nat @ R4 ) % 0.31/0.64 & ! [N6: nat] : ( member @ nat @ ( R4 @ N6 ) @ S3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % infinite_enumerate % 0.31/0.64 thf(fact_6908_strict__mono__imp__increasing,axiom, % 0.31/0.64 ! [F2: nat > nat,N2: nat] : % 0.31/0.64 ( ( order_strict_mono @ nat @ nat @ F2 ) % 0.31/0.64 => ( ord_less_eq @ nat @ N2 @ ( F2 @ N2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % strict_mono_imp_increasing % 0.31/0.64 thf(fact_6909_strict__mono__Suc__iff,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( order @ A ) % 0.31/0.64 => ( ( order_strict_mono @ nat @ A ) % 0.31/0.64 = ( ^ [F3: nat > A] : % 0.31/0.64 ! [N: nat] : ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ ( suc @ N ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % strict_mono_Suc_iff % 0.31/0.64 thf(fact_6910_less__int__def,axiom, % 0.31/0.64 ( ( ord_less @ int ) % 0.31/0.64 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) ) % 0.31/0.64 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o ) % 0.31/0.64 @ ^ [X: nat,Y2: nat] : % 0.31/0.64 ( product_case_prod @ nat @ nat @ $o % 0.31/0.64 @ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y2 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % less_int_def % 0.31/0.64 thf(fact_6911_nat__def,axiom, % 0.31/0.64 ( nat2 % 0.31/0.64 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ nat @ nat @ rep_Integ @ ( id @ nat ) @ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nat_def % 0.31/0.64 thf(fact_6912_less__eq__int__def,axiom, % 0.31/0.64 ( ( ord_less_eq @ int ) % 0.31/0.64 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) ) % 0.31/0.64 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o ) % 0.31/0.64 @ ^ [X: nat,Y2: nat] : % 0.31/0.64 ( product_case_prod @ nat @ nat @ $o % 0.31/0.64 @ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y2 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % less_eq_int_def % 0.31/0.64 thf(fact_6913_funpow__simps__right_I1_J,axiom, % 0.31/0.64 ! [A: $tType,F2: A > A] : % 0.31/0.64 ( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F2 ) % 0.31/0.64 = ( id @ A ) ) ). % 0.31/0.64 % 0.31/0.64 % funpow_simps_right(1) % 0.31/0.64 thf(fact_6914_fst__diag__id,axiom, % 0.31/0.64 ! [A: $tType,Z2: A] : % 0.31/0.64 ( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A ) % 0.31/0.64 @ ^ [X: A] : ( product_Pair @ A @ A @ X @ X ) % 0.31/0.64 @ Z2 ) % 0.31/0.64 = ( id @ A @ Z2 ) ) ). % 0.31/0.64 % 0.31/0.64 % fst_diag_id % 0.31/0.64 thf(fact_6915_snd__diag__id,axiom, % 0.31/0.64 ! [A: $tType,Z2: A] : % 0.31/0.64 ( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A ) % 0.31/0.64 @ ^ [X: A] : ( product_Pair @ A @ A @ X @ X ) % 0.31/0.64 @ Z2 ) % 0.31/0.64 = ( id @ A @ Z2 ) ) ). % 0.31/0.64 % 0.31/0.64 % snd_diag_id % 0.31/0.64 thf(fact_6916_summable__mono__reindex,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( ( comm_monoid_add @ A ) % 0.31/0.64 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.64 => ! [G: nat > nat,F2: nat > A] : % 0.31/0.64 ( ( order_strict_mono @ nat @ nat @ G ) % 0.31/0.64 => ( ! [N3: nat] : % 0.31/0.64 ( ~ ( member @ nat @ N3 @ ( image @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) ) % 0.31/0.64 => ( ( F2 @ N3 ) % 0.31/0.64 = ( zero_zero @ A ) ) ) % 0.31/0.64 => ( ( summable @ A % 0.31/0.64 @ ^ [N: nat] : ( F2 @ ( G @ N ) ) ) % 0.31/0.64 = ( summable @ A @ F2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % summable_mono_reindex % 0.31/0.64 thf(fact_6917_sums__mono__reindex,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( ( comm_monoid_add @ A ) % 0.31/0.64 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.64 => ! [G: nat > nat,F2: nat > A,C2: A] : % 0.31/0.64 ( ( order_strict_mono @ nat @ nat @ G ) % 0.31/0.64 => ( ! [N3: nat] : % 0.31/0.64 ( ~ ( member @ nat @ N3 @ ( image @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) ) % 0.31/0.64 => ( ( F2 @ N3 ) % 0.31/0.64 = ( zero_zero @ A ) ) ) % 0.31/0.64 => ( ( sums @ A % 0.31/0.64 @ ^ [N: nat] : ( F2 @ ( G @ N ) ) % 0.31/0.64 @ C2 ) % 0.31/0.64 = ( sums @ A @ F2 @ C2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % sums_mono_reindex % 0.31/0.64 thf(fact_6918_suminf__mono__reindex,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( ( comm_monoid_add @ A ) % 0.31/0.64 & ( topological_t2_space @ A ) ) % 0.31/0.64 => ! [G: nat > nat,F2: nat > A] : % 0.31/0.64 ( ( order_strict_mono @ nat @ nat @ G ) % 0.31/0.64 => ( ! [N3: nat] : % 0.31/0.64 ( ~ ( member @ nat @ N3 @ ( image @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) ) % 0.31/0.64 => ( ( F2 @ N3 ) % 0.31/0.64 = ( zero_zero @ A ) ) ) % 0.31/0.64 => ( ( suminf @ A % 0.31/0.64 @ ^ [N: nat] : ( F2 @ ( G @ N ) ) ) % 0.31/0.64 = ( suminf @ A @ F2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % suminf_mono_reindex % 0.31/0.64 thf(fact_6919_increasing__Bseq__subseq__iff,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( real_V822414075346904944vector @ A ) % 0.31/0.64 => ! [F2: nat > A,G: nat > nat] : % 0.31/0.64 ( ! [X5: nat,Y4: nat] : % 0.31/0.64 ( ( ord_less_eq @ nat @ X5 @ Y4 ) % 0.31/0.64 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ X5 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ Y4 ) ) ) ) % 0.31/0.64 => ( ( order_strict_mono @ nat @ nat @ G ) % 0.31/0.64 => ( ( bfun @ nat @ A % 0.31/0.64 @ ^ [X: nat] : ( F2 @ ( G @ X ) ) % 0.31/0.64 @ ( at_top @ nat ) ) % 0.31/0.64 = ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % increasing_Bseq_subseq_iff % 0.31/0.64 thf(fact_6920_folding__on_Oinsert__remove,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] : % 0.31/0.64 ( ( finite_folding_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 ) % 0.31/0.64 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.64 = ( F2 @ X3 @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % folding_on.insert_remove % 0.31/0.64 thf(fact_6921_folding__on_Oremove,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,A3: set @ A,X3: A,Z2: B] : % 0.31/0.64 ( ( finite_folding_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ S3 ) % 0.31/0.64 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.64 => ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ A3 ) % 0.31/0.64 = ( F2 @ X3 @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % folding_on.remove % 0.31/0.64 thf(fact_6922_card_Ofolding__on__axioms,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( finite_folding_on @ A @ nat @ ( top_top @ ( set @ A ) ) % 0.31/0.64 @ ^ [Uu3: A] : suc ) ). % 0.31/0.64 % 0.31/0.64 % card.folding_on_axioms % 0.31/0.64 thf(fact_6923_folding__on_Oempty,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,S3: set @ A,F2: A > B > B,Z2: B] : % 0.31/0.64 ( ( finite_folding_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 = Z2 ) ) ). % 0.31/0.64 % 0.31/0.64 % folding_on.empty % 0.31/0.64 thf(fact_6924_folding__on_Oinfinite,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,S3: set @ A,F2: A > B > B,A3: set @ A,Z2: B] : % 0.31/0.64 ( ( finite_folding_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ~ ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ A3 ) % 0.31/0.64 = Z2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % folding_on.infinite % 0.31/0.64 thf(fact_6925_sorted__list__of__set_Ofold__insort__key_Ofolding__on__axioms,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ( finite_folding_on @ A @ ( list @ A ) @ ( top_top @ ( set @ A ) ) % 0.31/0.64 @ ( linorder_insort_key @ A @ A % 0.31/0.64 @ ^ [X: A] : X ) ) ) ). % 0.31/0.64 % 0.31/0.64 % sorted_list_of_set.fold_insort_key.folding_on_axioms % 0.31/0.64 thf(fact_6926_card__def,axiom, % 0.31/0.64 ! [B: $tType] : % 0.31/0.64 ( ( finite_card @ B ) % 0.31/0.64 = ( finite_folding_F @ B @ nat % 0.31/0.64 @ ^ [Uu3: B] : suc % 0.31/0.64 @ ( zero_zero @ nat ) ) ) ). % 0.31/0.64 % 0.31/0.64 % card_def % 0.31/0.64 thf(fact_6927_folding__on_Oinsert,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] : % 0.31/0.64 ( ( finite_folding_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 ) % 0.31/0.64 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ~ ( member @ A @ X3 @ A3 ) % 0.31/0.64 => ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.64 = ( F2 @ X3 @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ A3 ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % folding_on.insert % 0.31/0.64 thf(fact_6928_folding__idem__on_Oinsert__idem,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] : % 0.31/0.64 ( ( finite1890593828518410140dem_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 ) % 0.31/0.64 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.64 = ( F2 @ X3 @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ A3 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % folding_idem_on.insert_idem % 0.31/0.64 thf(fact_6929_comp__fun__idem__on_Ofold__insert__idem,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] : % 0.31/0.64 ( ( finite673082921795544331dem_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 ) % 0.31/0.64 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.64 = ( F2 @ X3 @ ( finite_fold @ A @ B @ F2 @ Z2 @ A3 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % comp_fun_idem_on.fold_insert_idem % 0.31/0.64 thf(fact_6930_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,C: $tType,S3: set @ A,F2: A > B > B,G: C > A,R: set @ C] : % 0.31/0.64 ( ( finite673082921795544331dem_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ G @ ( top_top @ ( set @ C ) ) ) @ S3 ) % 0.31/0.64 => ( finite673082921795544331dem_on @ C @ B @ R @ ( comp @ A @ ( B > B ) @ C @ F2 @ G ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % comp_fun_idem_on.comp_comp_fun_idem_on % 0.31/0.64 thf(fact_6931_comp__fun__idem__on_Ofold__insert__idem2,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X3: A,A3: set @ A,Z2: B] : % 0.31/0.64 ( ( finite673082921795544331dem_on @ A @ B @ S3 @ F2 ) % 0.31/0.64 => ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ S3 ) % 0.31/0.64 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert @ A @ X3 @ A3 ) ) % 0.31/0.64 = ( finite_fold @ A @ B @ F2 @ ( F2 @ X3 @ Z2 ) @ A3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % comp_fun_idem_on.fold_insert_idem2 % 0.31/0.64 thf(fact_6932_positive__rat,axiom, % 0.31/0.64 ! [A2: int,B2: int] : % 0.31/0.64 ( ( positive @ ( fract @ A2 @ B2 ) ) % 0.31/0.64 = ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ A2 @ B2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % positive_rat % 0.31/0.64 thf(fact_6933_nth__image,axiom, % 0.31/0.64 ! [A: $tType,L: nat,Xs: list @ A] : % 0.31/0.64 ( ( ord_less_eq @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( image @ nat @ A @ ( nth @ A @ Xs ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ L ) ) % 0.31/0.64 = ( set2 @ A @ ( take @ A @ L @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_image % 0.31/0.64 thf(fact_6934_take__take,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,M: nat,Xs: list @ A] : % 0.31/0.64 ( ( take @ A @ N2 @ ( take @ A @ M @ Xs ) ) % 0.31/0.64 = ( take @ A @ ( ord_min @ nat @ N2 @ M ) @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % take_take % 0.31/0.64 thf(fact_6935_take__eq__Nil2,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( ( nil @ A ) % 0.31/0.64 = ( take @ A @ N2 @ Xs ) ) % 0.31/0.64 = ( ( N2 % 0.31/0.64 = ( zero_zero @ nat ) ) % 0.31/0.64 | ( Xs % 0.31/0.64 = ( nil @ A ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % take_eq_Nil2 % 0.31/0.64 thf(fact_6936_take__eq__Nil,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( ( take @ A @ N2 @ Xs ) % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 = ( ( N2 % 0.31/0.64 = ( zero_zero @ nat ) ) % 0.31/0.64 | ( Xs % 0.31/0.64 = ( nil @ A ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % take_eq_Nil % 0.31/0.64 thf(fact_6937_take0,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( take @ A @ ( zero_zero @ nat ) ) % 0.31/0.64 = ( ^ [Xs2: list @ A] : ( nil @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % take0 % 0.31/0.64 thf(fact_6938_take__all__iff,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( ( take @ A @ N2 @ Xs ) % 0.31/0.64 = Xs ) % 0.31/0.64 = ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ). % 0.31/0.64 % 0.31/0.64 % take_all_iff % 0.31/0.64 thf(fact_6939_take__all,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,N2: nat] : % 0.31/0.64 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) % 0.31/0.64 => ( ( take @ A @ N2 @ Xs ) % 0.31/0.64 = Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % take_all % 0.31/0.64 thf(fact_6940_nth__take,axiom, % 0.31/0.64 ! [A: $tType,I: nat,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( ord_less @ nat @ I @ N2 ) % 0.31/0.64 => ( ( nth @ A @ ( take @ A @ N2 @ Xs ) @ I ) % 0.31/0.64 = ( nth @ A @ Xs @ I ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_take % 0.31/0.64 thf(fact_6941_take__update__cancel,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,M: nat,Xs: list @ A,Y: A] : % 0.31/0.64 ( ( ord_less_eq @ nat @ N2 @ M ) % 0.31/0.64 => ( ( take @ A @ N2 @ ( list_update @ A @ Xs @ M @ Y ) ) % 0.31/0.64 = ( take @ A @ N2 @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % take_update_cancel % 0.31/0.64 thf(fact_6942_length__take,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( size_size @ ( list @ A ) @ ( take @ A @ N2 @ Xs ) ) % 0.31/0.64 = ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ). % 0.31/0.64 % 0.31/0.64 % length_take % 0.31/0.64 thf(fact_6943_nths__upt__eq__take,axiom, % 0.31/0.64 ! [A: $tType,L: list @ A,N2: nat] : % 0.31/0.64 ( ( nths @ A @ L @ ( set_ord_lessThan @ nat @ N2 ) ) % 0.31/0.64 = ( take @ A @ N2 @ L ) ) ). % 0.31/0.64 % 0.31/0.64 % nths_upt_eq_take % 0.31/0.64 thf(fact_6944_take__replicate,axiom, % 0.31/0.64 ! [A: $tType,I: nat,K: nat,X3: A] : % 0.31/0.64 ( ( take @ A @ I @ ( replicate @ A @ K @ X3 ) ) % 0.31/0.64 = ( replicate @ A @ ( ord_min @ nat @ I @ K ) @ X3 ) ) ). % 0.31/0.64 % 0.31/0.64 % take_replicate % 0.31/0.64 thf(fact_6945_dom__map__upds,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,M: A > ( option @ B ),Xs: list @ A,Ys: list @ B] : % 0.31/0.64 ( ( dom @ A @ B @ ( map_upds @ A @ B @ M @ Xs @ Ys ) ) % 0.31/0.64 = ( sup_sup @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) @ ( dom @ A @ B @ M ) ) ) ). % 0.31/0.64 % 0.31/0.64 % dom_map_upds % 0.31/0.64 thf(fact_6946_in__set__takeD,axiom, % 0.31/0.64 ! [A: $tType,X3: A,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( member @ A @ X3 @ ( set2 @ A @ ( take @ A @ N2 @ Xs ) ) ) % 0.31/0.64 => ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % in_set_takeD % 0.31/0.64 thf(fact_6947_take__zip,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,N2: nat,Xs: list @ A,Ys: list @ B] : % 0.31/0.64 ( ( take @ ( product_prod @ A @ B ) @ N2 @ ( zip @ A @ B @ Xs @ Ys ) ) % 0.31/0.64 = ( zip @ A @ B @ ( take @ A @ N2 @ Xs ) @ ( take @ B @ N2 @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % take_zip % 0.31/0.64 thf(fact_6948_take__Nil,axiom, % 0.31/0.64 ! [A: $tType,N2: nat] : % 0.31/0.64 ( ( take @ A @ N2 @ ( nil @ A ) ) % 0.31/0.64 = ( nil @ A ) ) ). % 0.31/0.64 % 0.31/0.64 % take_Nil % 0.31/0.64 thf(fact_6949_take__update__swap,axiom, % 0.31/0.64 ! [A: $tType,M: nat,Xs: list @ A,N2: nat,X3: A] : % 0.31/0.64 ( ( take @ A @ M @ ( list_update @ A @ Xs @ N2 @ X3 ) ) % 0.31/0.64 = ( list_update @ A @ ( take @ A @ M @ Xs ) @ N2 @ X3 ) ) ). % 0.31/0.64 % 0.31/0.64 % take_update_swap % 0.31/0.64 thf(fact_6950_take__equalityI,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ! [I4: nat] : % 0.31/0.64 ( ( take @ A @ I4 @ Xs ) % 0.31/0.64 = ( take @ A @ I4 @ Ys ) ) % 0.31/0.64 => ( Xs = Ys ) ) ). % 0.31/0.64 % 0.31/0.64 % take_equalityI % 0.31/0.64 thf(fact_6951_distinct__take,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,I: nat] : % 0.31/0.64 ( ( distinct @ A @ Xs ) % 0.31/0.64 => ( distinct @ A @ ( take @ A @ I @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % distinct_take % 0.31/0.64 thf(fact_6952_take__0,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A] : % 0.31/0.64 ( ( take @ A @ ( zero_zero @ nat ) @ Xs ) % 0.31/0.64 = ( nil @ A ) ) ). % 0.31/0.64 % 0.31/0.64 % take_0 % 0.31/0.64 thf(fact_6953_set__take__subset,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ N2 @ Xs ) ) @ ( set2 @ A @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % set_take_subset % 0.31/0.64 thf(fact_6954_set__take__subset__set__take,axiom, % 0.31/0.64 ! [A: $tType,M: nat,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.64 => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ M @ Xs ) ) @ ( set2 @ A @ ( take @ A @ N2 @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % set_take_subset_set_take % 0.31/0.64 thf(fact_6955_zip__obtain__same__length,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ ( product_prod @ A @ B ) ) > $o] : % 0.31/0.64 ( ! [Zs: list @ A,Ws: list @ B,N3: nat] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Zs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ws ) ) % 0.31/0.64 => ( ( N3 % 0.31/0.64 = ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) % 0.31/0.64 => ( ( Zs % 0.31/0.64 = ( take @ A @ N3 @ Xs ) ) % 0.31/0.64 => ( ( Ws % 0.31/0.64 = ( take @ B @ N3 @ Ys ) ) % 0.31/0.64 => ( P @ ( zip @ A @ B @ Zs @ Ws ) ) ) ) ) ) % 0.31/0.64 => ( P @ ( zip @ A @ B @ Xs @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % zip_obtain_same_length % 0.31/0.64 thf(fact_6956_nth__take__lemma,axiom, % 0.31/0.64 ! [A: $tType,K: nat,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) ) % 0.31/0.64 => ( ! [I4: nat] : % 0.31/0.64 ( ( ord_less @ nat @ I4 @ K ) % 0.31/0.64 => ( ( nth @ A @ Xs @ I4 ) % 0.31/0.64 = ( nth @ A @ Ys @ I4 ) ) ) % 0.31/0.64 => ( ( take @ A @ K @ Xs ) % 0.31/0.64 = ( take @ A @ K @ Ys ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_take_lemma % 0.31/0.64 thf(fact_6957_less__rat__def,axiom, % 0.31/0.64 ( ( ord_less @ rat ) % 0.31/0.64 = ( ^ [X: rat,Y2: rat] : ( positive @ ( minus_minus @ rat @ Y2 @ X ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % less_rat_def % 0.31/0.64 thf(fact_6958_map__upd__upds__conv__if,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,X3: A,Ys: list @ B,Xs: list @ A,F2: A > ( option @ B ),Y: B] : % 0.31/0.64 ( ( ( member @ A @ X3 @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) ) % 0.31/0.64 => ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X3 @ ( some @ B @ Y ) ) @ Xs @ Ys ) % 0.31/0.64 = ( map_upds @ A @ B @ F2 @ Xs @ Ys ) ) ) % 0.31/0.64 & ( ~ ( member @ A @ X3 @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) ) % 0.31/0.64 => ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X3 @ ( some @ B @ Y ) ) @ Xs @ Ys ) % 0.31/0.64 = ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ F2 @ Xs @ Ys ) @ X3 @ ( some @ B @ Y ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_upd_upds_conv_if % 0.31/0.64 thf(fact_6959_Rat_Opositive_Orep__eq,axiom, % 0.31/0.64 ( positive % 0.31/0.64 = ( ^ [X: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ ( rep_Rat @ X ) ) @ ( product_snd @ int @ int @ ( rep_Rat @ X ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Rat.positive.rep_eq % 0.31/0.64 thf(fact_6960_lexord__take__index__conv,axiom, % 0.31/0.64 ! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) ) % 0.31/0.64 = ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X3 ) @ ( size_size @ ( list @ A ) @ Y ) ) % 0.31/0.64 & ( ( take @ A @ ( size_size @ ( list @ A ) @ X3 ) @ Y ) % 0.31/0.64 = X3 ) ) % 0.31/0.64 | ? [I3: nat] : % 0.31/0.64 ( ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X3 ) @ ( size_size @ ( list @ A ) @ Y ) ) ) % 0.31/0.64 & ( ( take @ A @ I3 @ X3 ) % 0.31/0.64 = ( take @ A @ I3 @ Y ) ) % 0.31/0.64 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X3 @ I3 ) @ ( nth @ A @ Y @ I3 ) ) @ R2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_take_index_conv % 0.31/0.64 thf(fact_6961_lexord__linear,axiom, % 0.31/0.64 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: list @ A,Y: list @ A] : % 0.31/0.64 ( ! [A5: A,B5: A] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ R2 ) % 0.31/0.64 | ( A5 = B5 ) % 0.31/0.64 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A5 ) @ R2 ) ) % 0.31/0.64 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) ) % 0.31/0.64 | ( X3 = Y ) % 0.31/0.64 | ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X3 ) @ ( lexord @ A @ R2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_linear % 0.31/0.64 thf(fact_6962_lexord__irreflexive,axiom, % 0.31/0.64 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] : % 0.31/0.64 ( ! [X5: A] : % 0.31/0.64 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 ) % 0.31/0.64 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_irreflexive % 0.31/0.64 thf(fact_6963_lexord__Nil__right,axiom, % 0.31/0.64 ! [A: $tType,X3: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ ( nil @ A ) ) @ ( lexord @ A @ R2 ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_Nil_right % 0.31/0.64 thf(fact_6964_lexord__partial__trans,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A ),Ys: list @ A,Zs2: list @ A] : % 0.31/0.64 ( ! [X5: A,Y4: A,Z3: A] : % 0.31/0.64 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ R2 ) % 0.31/0.64 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R2 ) % 0.31/0.64 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Z3 ) @ R2 ) ) ) ) % 0.31/0.64 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R2 ) ) % 0.31/0.64 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lexord @ A @ R2 ) ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( lexord @ A @ R2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_partial_trans % 0.31/0.64 thf(fact_6965_List_Olexordp__def,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( lexordp @ A ) % 0.31/0.64 = ( ^ [R5: A > A > $o,Xs2: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % List.lexordp_def % 0.31/0.64 thf(fact_6966_Rat_Opositive__def,axiom, % 0.31/0.64 ( positive % 0.31/0.64 = ( map_fun @ rat @ ( product_prod @ int @ int ) @ $o @ $o @ rep_Rat @ ( id @ $o ) % 0.31/0.64 @ ^ [X: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Rat.positive_def % 0.31/0.64 thf(fact_6967_lex__take__index,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) ) % 0.31/0.64 => ~ ! [I4: nat] : % 0.31/0.64 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Ys ) ) % 0.31/0.64 => ( ( ( take @ A @ I4 @ Xs ) % 0.31/0.64 = ( take @ A @ I4 @ Ys ) ) % 0.31/0.64 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Ys @ I4 ) ) @ R2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lex_take_index % 0.31/0.64 thf(fact_6968_ntrancl__Suc,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,R: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( transitive_ntrancl @ A @ ( suc @ N2 ) @ R ) % 0.31/0.64 = ( relcomp @ A @ A @ A @ ( transitive_ntrancl @ A @ N2 @ R ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ R ) ) ) ). % 0.31/0.64 % 0.31/0.64 % ntrancl_Suc % 0.31/0.64 thf(fact_6969_IdI,axiom, % 0.31/0.64 ! [A: $tType,A2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id2 @ A ) ) ). % 0.31/0.64 % 0.31/0.64 % IdI % 0.31/0.64 thf(fact_6970_pair__in__Id__conv,axiom, % 0.31/0.64 ! [A: $tType,A2: A,B2: A] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id2 @ A ) ) % 0.31/0.64 = ( A2 = B2 ) ) ). % 0.31/0.64 % 0.31/0.64 % pair_in_Id_conv % 0.31/0.64 thf(fact_6971_rtrancl__empty,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( transitive_rtrancl @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) % 0.31/0.64 = ( id2 @ A ) ) ). % 0.31/0.64 % 0.31/0.64 % rtrancl_empty % 0.31/0.64 thf(fact_6972_Id__def,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( id2 @ A ) % 0.31/0.64 = ( collect @ ( product_prod @ A @ A ) % 0.31/0.64 @ ^ [P6: product_prod @ A @ A] : % 0.31/0.64 ? [X: A] : % 0.31/0.64 ( P6 % 0.31/0.64 = ( product_Pair @ A @ A @ X @ X ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Id_def % 0.31/0.64 thf(fact_6973_relpow_Osimps_I1_J,axiom, % 0.31/0.64 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R ) % 0.31/0.64 = ( id2 @ A ) ) ). % 0.31/0.64 % 0.31/0.64 % relpow.simps(1) % 0.31/0.64 thf(fact_6974_IdD,axiom, % 0.31/0.64 ! [A: $tType,A2: A,B2: A] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id2 @ A ) ) % 0.31/0.64 => ( A2 = B2 ) ) ). % 0.31/0.64 % 0.31/0.64 % IdD % 0.31/0.64 thf(fact_6975_IdE,axiom, % 0.31/0.64 ! [A: $tType,P4: product_prod @ A @ A] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ A ) @ P4 @ ( id2 @ A ) ) % 0.31/0.64 => ~ ! [X5: A] : % 0.31/0.64 ( P4 % 0.31/0.64 != ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % IdE % 0.31/0.64 thf(fact_6976_Nil__notin__lex,axiom, % 0.31/0.64 ! [A: $tType,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) @ ( lex @ A @ R2 ) ) ). % 0.31/0.64 % 0.31/0.64 % Nil_notin_lex % 0.31/0.64 thf(fact_6977_Nil2__notin__lex,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( lex @ A @ R2 ) ) ). % 0.31/0.64 % 0.31/0.64 % Nil2_notin_lex % 0.31/0.64 thf(fact_6978_reflcl__set__eq,axiom, % 0.31/0.64 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( sup_sup @ ( A > A > $o ) % 0.31/0.64 @ ^ [X: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 ) % 0.31/0.64 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) ) % 0.31/0.64 = ( ^ [X: A,Y2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % reflcl_set_eq % 0.31/0.64 thf(fact_6979_lexord__lex,axiom, % 0.31/0.64 ! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lex @ A @ R2 ) ) % 0.31/0.64 = ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) ) % 0.31/0.64 & ( ( size_size @ ( list @ A ) @ X3 ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ Y ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_lex % 0.31/0.64 thf(fact_6980_lenlex__conv,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( lenlex @ A ) % 0.31/0.64 = ( ^ [R5: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) % 0.31/0.64 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o % 0.31/0.64 @ ^ [Xs2: list @ A,Ys3: list @ A] : % 0.31/0.64 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys3 ) ) % 0.31/0.64 | ( ( ( size_size @ ( list @ A ) @ Xs2 ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ Ys3 ) ) % 0.31/0.64 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys3 ) @ ( lex @ A @ R5 ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lenlex_conv % 0.31/0.64 thf(fact_6981_map__of__zip__nth,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,I: nat] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( distinct @ A @ Xs ) % 0.31/0.64 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ ( nth @ A @ Xs @ I ) ) % 0.31/0.64 = ( some @ B @ ( nth @ B @ Ys @ I ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_zip_nth % 0.31/0.64 thf(fact_6982_Nil__lenlex__iff1,axiom, % 0.31/0.64 ! [A: $tType,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R2 ) ) % 0.31/0.64 = ( Ns % 0.31/0.64 != ( nil @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Nil_lenlex_iff1 % 0.31/0.64 thf(fact_6983_map__of__eq__empty__iff,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] : % 0.31/0.64 ( ( ( map_of @ A @ B @ Xys ) % 0.31/0.64 = ( ^ [X: A] : ( none @ B ) ) ) % 0.31/0.64 = ( Xys % 0.31/0.64 = ( nil @ ( product_prod @ A @ B ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_eq_empty_iff % 0.31/0.64 thf(fact_6984_empty__eq__map__of__iff,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] : % 0.31/0.64 ( ( ( ^ [X: A] : ( none @ B ) ) % 0.31/0.64 = ( map_of @ A @ B @ Xys ) ) % 0.31/0.64 = ( Xys % 0.31/0.64 = ( nil @ ( product_prod @ A @ B ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % empty_eq_map_of_iff % 0.31/0.64 thf(fact_6985_map__of__zip__is__None,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,X3: A] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ X3 ) % 0.31/0.64 = ( none @ B ) ) % 0.31/0.64 = ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_zip_is_None % 0.31/0.64 thf(fact_6986_dom__map__of__zip,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( dom @ A @ B @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) ) % 0.31/0.64 = ( set2 @ A @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % dom_map_of_zip % 0.31/0.64 thf(fact_6987_finite__dom__map__of,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,L: list @ ( product_prod @ A @ B )] : ( finite_finite2 @ A @ ( dom @ A @ B @ ( map_of @ A @ B @ L ) ) ) ). % 0.31/0.64 % 0.31/0.64 % finite_dom_map_of % 0.31/0.64 thf(fact_6988_finite__graph__map__of,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Al: list @ ( product_prod @ A @ B )] : ( finite_finite2 @ ( product_prod @ A @ B ) @ ( graph @ A @ B @ ( map_of @ A @ B @ Al ) ) ) ). % 0.31/0.64 % 0.31/0.64 % finite_graph_map_of % 0.31/0.64 thf(fact_6989_map__of__Cons__code_I1_J,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,K: B] : % 0.31/0.64 ( ( map_of @ B @ A @ ( nil @ ( product_prod @ B @ A ) ) @ K ) % 0.31/0.64 = ( none @ A ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_Cons_code(1) % 0.31/0.64 thf(fact_6990_map__of_Osimps_I1_J,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( map_of @ A @ B @ ( nil @ ( product_prod @ A @ B ) ) ) % 0.31/0.64 = ( ^ [X: A] : ( none @ B ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of.simps(1) % 0.31/0.64 thf(fact_6991_lenlex__irreflexive,axiom, % 0.31/0.64 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] : % 0.31/0.64 ( ! [X5: A] : % 0.31/0.64 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 ) % 0.31/0.64 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lenlex_irreflexive % 0.31/0.64 thf(fact_6992_Nil__lenlex__iff2,axiom, % 0.31/0.64 ! [A: $tType,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R2 ) ) ). % 0.31/0.64 % 0.31/0.64 % Nil_lenlex_iff2 % 0.31/0.64 thf(fact_6993_weak__map__of__SomeI,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,K: A,X3: B,L: list @ ( product_prod @ A @ B )] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ X3 ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) ) % 0.31/0.64 => ? [X5: B] : % 0.31/0.64 ( ( map_of @ A @ B @ L @ K ) % 0.31/0.64 = ( some @ B @ X5 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % weak_map_of_SomeI % 0.31/0.64 thf(fact_6994_map__of__SomeD,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Y: A] : % 0.31/0.64 ( ( ( map_of @ B @ A @ Xs @ K ) % 0.31/0.64 = ( some @ A @ Y ) ) % 0.31/0.64 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ Y ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_SomeD % 0.31/0.64 thf(fact_6995_map__of__zip__inject,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Ys: list @ A,Xs: list @ B,Zs2: list @ A] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Ys ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Xs ) ) % 0.31/0.64 => ( ( ( size_size @ ( list @ A ) @ Zs2 ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Xs ) ) % 0.31/0.64 => ( ( distinct @ B @ Xs ) % 0.31/0.64 => ( ( ( map_of @ B @ A @ ( zip @ B @ A @ Xs @ Ys ) ) % 0.31/0.64 = ( map_of @ B @ A @ ( zip @ B @ A @ Xs @ Zs2 ) ) ) % 0.31/0.64 => ( Ys = Zs2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_zip_inject % 0.31/0.64 thf(fact_6996_finite__range__map__of,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A )] : ( finite_finite2 @ ( option @ A ) @ ( image @ B @ ( option @ A ) @ ( map_of @ B @ A @ Xys ) @ ( top_top @ ( set @ B ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % finite_range_map_of % 0.31/0.64 thf(fact_6997_map__of__zip__is__Some,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X3: A] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 = ( ? [Y2: B] : % 0.31/0.64 ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ X3 ) % 0.31/0.64 = ( some @ B @ Y2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_zip_is_Some % 0.31/0.64 thf(fact_6998_lenlex__length,axiom, % 0.31/0.64 ! [A: $tType,Ms: list @ A,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R2 ) ) % 0.31/0.64 => ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lenlex_length % 0.31/0.64 thf(fact_6999_map__of__eq__None__iff,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A ),X3: B] : % 0.31/0.64 ( ( ( map_of @ B @ A @ Xys @ X3 ) % 0.31/0.64 = ( none @ A ) ) % 0.31/0.64 = ( ~ ( member @ B @ X3 @ ( image @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ ( set2 @ ( product_prod @ B @ A ) @ Xys ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_eq_None_iff % 0.31/0.64 thf(fact_7000_map__of__zip__upd,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Ys: list @ B,Xs: list @ A,Zs2: list @ B,X3: A,Y: B,Z2: B] : % 0.31/0.64 ( ( ( size_size @ ( list @ B ) @ Ys ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( ( size_size @ ( list @ B ) @ Zs2 ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ( ( ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) @ X3 @ ( some @ B @ Y ) ) % 0.31/0.64 = ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs2 ) ) @ X3 @ ( some @ B @ Z2 ) ) ) % 0.31/0.64 => ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) % 0.31/0.64 = ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs2 ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_zip_upd % 0.31/0.64 thf(fact_7001_ran__map__of__zip,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( distinct @ A @ Xs ) % 0.31/0.64 => ( ( ran @ A @ B @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) ) % 0.31/0.64 = ( set2 @ B @ Ys ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % ran_map_of_zip % 0.31/0.64 thf(fact_7002_map__of__is__SomeI,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xys: list @ ( product_prod @ A @ B ),X3: A,Y: B] : % 0.31/0.64 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) ) % 0.31/0.64 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) % 0.31/0.64 => ( ( map_of @ A @ B @ Xys @ X3 ) % 0.31/0.64 = ( some @ B @ Y ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_is_SomeI % 0.31/0.64 thf(fact_7003_Some__eq__map__of__iff,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),Y: B,X3: A] : % 0.31/0.64 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) ) % 0.31/0.64 => ( ( ( some @ B @ Y ) % 0.31/0.64 = ( map_of @ A @ B @ Xys @ X3 ) ) % 0.31/0.64 = ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Some_eq_map_of_iff % 0.31/0.64 thf(fact_7004_map__ident,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( map @ A @ A % 0.31/0.64 @ ^ [X: A] : X ) % 0.31/0.64 = ( ^ [Xs2: list @ A] : Xs2 ) ) ). % 0.31/0.64 % 0.31/0.64 % map_ident % 0.31/0.64 thf(fact_7005_list_Omap__disc__iff,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > B,A2: list @ A] : % 0.31/0.64 ( ( ( map @ A @ B @ F2 @ A2 ) % 0.31/0.64 = ( nil @ B ) ) % 0.31/0.64 = ( A2 % 0.31/0.64 = ( nil @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.map_disc_iff % 0.31/0.64 thf(fact_7006_Nil__is__map__conv,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] : % 0.31/0.64 ( ( ( nil @ A ) % 0.31/0.64 = ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.64 = ( Xs % 0.31/0.64 = ( nil @ B ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Nil_is_map_conv % 0.31/0.64 thf(fact_7007_map__is__Nil__conv,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] : % 0.31/0.64 ( ( ( map @ B @ A @ F2 @ Xs ) % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 = ( Xs % 0.31/0.64 = ( nil @ B ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_is_Nil_conv % 0.31/0.64 thf(fact_7008_map__map,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,C: $tType,F2: B > A,G: C > B,Xs: list @ C] : % 0.31/0.64 ( ( map @ B @ A @ F2 @ ( map @ C @ B @ G @ Xs ) ) % 0.31/0.64 = ( map @ C @ A @ ( comp @ B @ A @ C @ F2 @ G ) @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % map_map % 0.31/0.64 thf(fact_7009_List_Omap_Ocompositionality,axiom, % 0.31/0.64 ! [B: $tType,C: $tType,A: $tType,F2: B > C,G: A > B,List: list @ A] : % 0.31/0.64 ( ( map @ B @ C @ F2 @ ( map @ A @ B @ G @ List ) ) % 0.31/0.64 = ( map @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) @ List ) ) ). % 0.31/0.64 % 0.31/0.64 % List.map.compositionality % 0.31/0.64 thf(fact_7010_list_Omap__comp,axiom, % 0.31/0.64 ! [B: $tType,C: $tType,A: $tType,G: B > C,F2: A > B,V2: list @ A] : % 0.31/0.64 ( ( map @ B @ C @ G @ ( map @ A @ B @ F2 @ V2 ) ) % 0.31/0.64 = ( map @ A @ C @ ( comp @ B @ C @ A @ G @ F2 ) @ V2 ) ) ). % 0.31/0.64 % 0.31/0.64 % list.map_comp % 0.31/0.64 thf(fact_7011_map__eq__conv,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,G: B > A] : % 0.31/0.64 ( ( ( map @ B @ A @ F2 @ Xs ) % 0.31/0.64 = ( map @ B @ A @ G @ Xs ) ) % 0.31/0.64 = ( ! [X: B] : % 0.31/0.64 ( ( member @ B @ X @ ( set2 @ B @ Xs ) ) % 0.31/0.64 => ( ( F2 @ X ) % 0.31/0.64 = ( G @ X ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_eq_conv % 0.31/0.64 thf(fact_7012_length__map,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] : % 0.31/0.64 ( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % length_map % 0.31/0.64 thf(fact_7013_list_Omap__id0,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( map @ A @ A @ ( id @ A ) ) % 0.31/0.64 = ( id @ ( list @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.map_id0 % 0.31/0.64 thf(fact_7014_map__replicate,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,N2: nat,X3: B] : % 0.31/0.64 ( ( map @ B @ A @ F2 @ ( replicate @ B @ N2 @ X3 ) ) % 0.31/0.64 = ( replicate @ A @ N2 @ ( F2 @ X3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_replicate % 0.31/0.64 thf(fact_7015_list_Oset__map,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > B,V2: list @ A] : % 0.31/0.64 ( ( set2 @ B @ ( map @ A @ B @ F2 @ V2 ) ) % 0.31/0.64 = ( image @ A @ B @ F2 @ ( set2 @ A @ V2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.set_map % 0.31/0.64 thf(fact_7016_map__snd__enumerate,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( map @ ( product_prod @ nat @ A ) @ A @ ( product_snd @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) ) % 0.31/0.64 = Xs ) ). % 0.31/0.64 % 0.31/0.64 % map_snd_enumerate % 0.31/0.64 thf(fact_7017_inj__map__eq__map,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > B,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.64 => ( ( ( map @ A @ B @ F2 @ Xs ) % 0.31/0.64 = ( map @ A @ B @ F2 @ Ys ) ) % 0.31/0.64 = ( Xs = Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % inj_map_eq_map % 0.31/0.64 thf(fact_7018_map__fun__upd,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Y: A,Xs: list @ A,F2: A > B,V2: B] : % 0.31/0.64 ( ~ ( member @ A @ Y @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ( ( map @ A @ B @ ( fun_upd @ A @ B @ F2 @ Y @ V2 ) @ Xs ) % 0.31/0.64 = ( map @ A @ B @ F2 @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_fun_upd % 0.31/0.64 thf(fact_7019_map__comp__map,axiom, % 0.31/0.64 ! [B: $tType,C: $tType,A: $tType,F2: C > B,G: A > C] : % 0.31/0.64 ( ( comp @ ( list @ C ) @ ( list @ B ) @ ( list @ A ) @ ( map @ C @ B @ F2 ) @ ( map @ A @ C @ G ) ) % 0.31/0.64 = ( map @ A @ B @ ( comp @ C @ B @ A @ F2 @ G ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_comp_map % 0.31/0.64 thf(fact_7020_List_Omap_Ocomp,axiom, % 0.31/0.64 ! [C: $tType,B: $tType,A: $tType,F2: B > C,G: A > B] : % 0.31/0.64 ( ( comp @ ( list @ B ) @ ( list @ C ) @ ( list @ A ) @ ( map @ B @ C @ F2 ) @ ( map @ A @ B @ G ) ) % 0.31/0.64 = ( map @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) ) ) ). % 0.31/0.64 % 0.31/0.64 % List.map.comp % 0.31/0.64 thf(fact_7021_size__list__map,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: A > nat,G: B > A,Xs: list @ B] : % 0.31/0.64 ( ( size_list @ A @ F2 @ ( map @ B @ A @ G @ Xs ) ) % 0.31/0.64 = ( size_list @ B @ ( comp @ A @ nat @ B @ F2 @ G ) @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % size_list_map % 0.31/0.64 thf(fact_7022_nth__map,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,N2: nat,Xs: list @ A,F2: A > B] : % 0.31/0.64 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( nth @ B @ ( map @ A @ B @ F2 @ Xs ) @ N2 ) % 0.31/0.64 = ( F2 @ ( nth @ A @ Xs @ N2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_map % 0.31/0.64 thf(fact_7023_map__fst__zip,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) % 0.31/0.64 = Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % map_fst_zip % 0.31/0.64 thf(fact_7024_map__snd__zip,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) % 0.31/0.64 = Ys ) ) ). % 0.31/0.64 % 0.31/0.64 % map_snd_zip % 0.31/0.64 thf(fact_7025_inj__map,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > B] : % 0.31/0.64 ( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) % 0.31/0.64 = ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % inj_map % 0.31/0.64 thf(fact_7026_inj__mapI,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > B] : % 0.31/0.64 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.64 => ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % inj_mapI % 0.31/0.64 thf(fact_7027_map__of__eq__Some__iff,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),X3: A,Y: B] : % 0.31/0.64 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) ) % 0.31/0.64 => ( ( ( map_of @ A @ B @ Xys @ X3 ) % 0.31/0.64 = ( some @ B @ Y ) ) % 0.31/0.64 = ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_eq_Some_iff % 0.31/0.64 thf(fact_7028_List_Omap_Oidentity,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( map @ A @ A % 0.31/0.64 @ ^ [X: A] : X ) % 0.31/0.64 = ( id @ ( list @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % List.map.identity % 0.31/0.64 thf(fact_7029_list_Omap__id,axiom, % 0.31/0.64 ! [A: $tType,T2: list @ A] : % 0.31/0.64 ( ( map @ A @ A @ ( id @ A ) @ T2 ) % 0.31/0.64 = T2 ) ). % 0.31/0.64 % 0.31/0.64 % list.map_id % 0.31/0.64 thf(fact_7030_list_Osize__gen__o__map,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: B > nat,G: A > B] : % 0.31/0.64 ( ( comp @ ( list @ B ) @ nat @ ( list @ A ) @ ( size_list @ B @ F2 ) @ ( map @ A @ B @ G ) ) % 0.31/0.64 = ( size_list @ A @ ( comp @ B @ nat @ A @ F2 @ G ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.size_gen_o_map % 0.31/0.64 thf(fact_7031_map2__map__map,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,C: $tType,D: $tType,H2: B > C > A,F2: D > B,Xs: list @ D,G: D > C] : % 0.31/0.64 ( ( map @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ H2 ) @ ( zip @ B @ C @ ( map @ D @ B @ F2 @ Xs ) @ ( map @ D @ C @ G @ Xs ) ) ) % 0.31/0.64 = ( map @ D @ A % 0.31/0.64 @ ^ [X: D] : ( H2 @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.64 @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % map2_map_map % 0.31/0.64 thf(fact_7032_zip__map__map,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,C: $tType,D: $tType,F2: C > A,Xs: list @ C,G: D > B,Ys: list @ D] : % 0.31/0.64 ( ( zip @ A @ B @ ( map @ C @ A @ F2 @ Xs ) @ ( map @ D @ B @ G @ Ys ) ) % 0.31/0.64 = ( map @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) % 0.31/0.64 @ ( product_case_prod @ C @ D @ ( product_prod @ A @ B ) % 0.31/0.64 @ ^ [X: C,Y2: D] : ( product_Pair @ A @ B @ ( F2 @ X ) @ ( G @ Y2 ) ) ) % 0.31/0.64 @ ( zip @ C @ D @ Xs @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % zip_map_map % 0.31/0.64 thf(fact_7033_zip__map2,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,F2: C > B,Ys: list @ C] : % 0.31/0.64 ( ( zip @ A @ B @ Xs @ ( map @ C @ B @ F2 @ Ys ) ) % 0.31/0.64 = ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B ) % 0.31/0.64 @ ( product_case_prod @ A @ C @ ( product_prod @ A @ B ) % 0.31/0.64 @ ^ [X: A,Y2: C] : ( product_Pair @ A @ B @ X @ ( F2 @ Y2 ) ) ) % 0.31/0.64 @ ( zip @ A @ C @ Xs @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % zip_map2 % 0.31/0.64 thf(fact_7034_zip__map1,axiom, % 0.31/0.64 ! [A: $tType,C: $tType,B: $tType,F2: C > A,Xs: list @ C,Ys: list @ B] : % 0.31/0.64 ( ( zip @ A @ B @ ( map @ C @ A @ F2 @ Xs ) @ Ys ) % 0.31/0.64 = ( map @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B ) % 0.31/0.64 @ ( product_case_prod @ C @ B @ ( product_prod @ A @ B ) % 0.31/0.64 @ ^ [X: C] : ( product_Pair @ A @ B @ ( F2 @ X ) ) ) % 0.31/0.64 @ ( zip @ C @ B @ Xs @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % zip_map1 % 0.31/0.64 thf(fact_7035_map__zip__map2,axiom, % 0.31/0.64 ! [C: $tType,A: $tType,B: $tType,D: $tType,F2: ( product_prod @ B @ C ) > A,Xs: list @ B,G: D > C,Ys: list @ D] : % 0.31/0.64 ( ( map @ ( product_prod @ B @ C ) @ A @ F2 @ ( zip @ B @ C @ Xs @ ( map @ D @ C @ G @ Ys ) ) ) % 0.31/0.64 = ( map @ ( product_prod @ B @ D ) @ A % 0.31/0.64 @ ( product_case_prod @ B @ D @ A % 0.31/0.64 @ ^ [X: B,Y2: D] : ( F2 @ ( product_Pair @ B @ C @ X @ ( G @ Y2 ) ) ) ) % 0.31/0.64 @ ( zip @ B @ D @ Xs @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_zip_map2 % 0.31/0.64 thf(fact_7036_map__zip__map,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,D: $tType,C: $tType,F2: ( product_prod @ B @ C ) > A,G: D > B,Xs: list @ D,Ys: list @ C] : % 0.31/0.64 ( ( map @ ( product_prod @ B @ C ) @ A @ F2 @ ( zip @ B @ C @ ( map @ D @ B @ G @ Xs ) @ Ys ) ) % 0.31/0.64 = ( map @ ( product_prod @ D @ C ) @ A % 0.31/0.64 @ ( product_case_prod @ D @ C @ A % 0.31/0.64 @ ^ [X: D,Y2: C] : ( F2 @ ( product_Pair @ B @ C @ ( G @ X ) @ Y2 ) ) ) % 0.31/0.64 @ ( zip @ D @ C @ Xs @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_zip_map % 0.31/0.64 thf(fact_7037_map__injective,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ B] : % 0.31/0.64 ( ( ( map @ B @ A @ F2 @ Xs ) % 0.31/0.64 = ( map @ B @ A @ F2 @ Ys ) ) % 0.31/0.64 => ( ( inj_on @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) % 0.31/0.64 => ( Xs = Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_injective % 0.31/0.64 thf(fact_7038_remdups__map__remdups,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] : % 0.31/0.64 ( ( remdups @ A @ ( map @ B @ A @ F2 @ ( remdups @ B @ Xs ) ) ) % 0.31/0.64 = ( remdups @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remdups_map_remdups % 0.31/0.64 thf(fact_7039_map__eq__imp__length__eq,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,C: $tType,F2: B > A,Xs: list @ B,G: C > A,Ys: list @ C] : % 0.31/0.64 ( ( ( map @ B @ A @ F2 @ Xs ) % 0.31/0.64 = ( map @ C @ A @ G @ Ys ) ) % 0.31/0.64 => ( ( size_size @ ( list @ B ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ C ) @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_eq_imp_length_eq % 0.31/0.64 thf(fact_7040_rotate1__map,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] : % 0.31/0.64 ( ( rotate1 @ A @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.64 = ( map @ B @ A @ F2 @ ( rotate1 @ B @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % rotate1_map % 0.31/0.64 thf(fact_7041_list_Omap__ident,axiom, % 0.31/0.64 ! [A: $tType,T2: list @ A] : % 0.31/0.64 ( ( map @ A @ A % 0.31/0.64 @ ^ [X: A] : X % 0.31/0.64 @ T2 ) % 0.31/0.64 = T2 ) ). % 0.31/0.64 % 0.31/0.64 % list.map_ident % 0.31/0.64 thf(fact_7042_map__concat,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ ( list @ B )] : % 0.31/0.64 ( ( map @ B @ A @ F2 @ ( concat @ B @ Xs ) ) % 0.31/0.64 = ( concat @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_concat % 0.31/0.64 thf(fact_7043_map__update,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,K: nat,Y: B] : % 0.31/0.64 ( ( map @ B @ A @ F2 @ ( list_update @ B @ Xs @ K @ Y ) ) % 0.31/0.64 = ( list_update @ A @ ( map @ B @ A @ F2 @ Xs ) @ K @ ( F2 @ Y ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_update % 0.31/0.64 thf(fact_7044_list_Osimps_I8_J,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: A > B] : % 0.31/0.64 ( ( map @ A @ B @ F2 @ ( nil @ A ) ) % 0.31/0.64 = ( nil @ B ) ) ). % 0.31/0.64 % 0.31/0.64 % list.simps(8) % 0.31/0.64 thf(fact_7045_list_Omap__cong,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,X3: list @ A,Ya: list @ A,F2: A > B,G: A > B] : % 0.31/0.64 ( ( X3 = Ya ) % 0.31/0.64 => ( ! [Z3: A] : % 0.31/0.64 ( ( member @ A @ Z3 @ ( set2 @ A @ Ya ) ) % 0.31/0.64 => ( ( F2 @ Z3 ) % 0.31/0.64 = ( G @ Z3 ) ) ) % 0.31/0.64 => ( ( map @ A @ B @ F2 @ X3 ) % 0.31/0.64 = ( map @ A @ B @ G @ Ya ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.map_cong % 0.31/0.64 thf(fact_7046_list_Omap__cong0,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,X3: list @ A,F2: A > B,G: A > B] : % 0.31/0.64 ( ! [Z3: A] : % 0.31/0.64 ( ( member @ A @ Z3 @ ( set2 @ A @ X3 ) ) % 0.31/0.64 => ( ( F2 @ Z3 ) % 0.31/0.64 = ( G @ Z3 ) ) ) % 0.31/0.64 => ( ( map @ A @ B @ F2 @ X3 ) % 0.31/0.64 = ( map @ A @ B @ G @ X3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.map_cong0 % 0.31/0.64 thf(fact_7047_list_Oinj__map__strong,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,X3: list @ A,Xa2: list @ A,F2: A > B,Fa: A > B] : % 0.31/0.64 ( ! [Z3: A,Za2: A] : % 0.31/0.64 ( ( member @ A @ Z3 @ ( set2 @ A @ X3 ) ) % 0.31/0.64 => ( ( member @ A @ Za2 @ ( set2 @ A @ Xa2 ) ) % 0.31/0.64 => ( ( ( F2 @ Z3 ) % 0.31/0.64 = ( Fa @ Za2 ) ) % 0.31/0.64 => ( Z3 = Za2 ) ) ) ) % 0.31/0.64 => ( ( ( map @ A @ B @ F2 @ X3 ) % 0.31/0.64 = ( map @ A @ B @ Fa @ Xa2 ) ) % 0.31/0.64 => ( X3 = Xa2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.inj_map_strong % 0.31/0.64 thf(fact_7048_map__ext,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ A,F2: A > B,G: A > B] : % 0.31/0.64 ( ! [X5: A] : % 0.31/0.64 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ( ( F2 @ X5 ) % 0.31/0.64 = ( G @ X5 ) ) ) % 0.31/0.64 => ( ( map @ A @ B @ F2 @ Xs ) % 0.31/0.64 = ( map @ A @ B @ G @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_ext % 0.31/0.64 thf(fact_7049_map__idI,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,F2: A > A] : % 0.31/0.64 ( ! [X5: A] : % 0.31/0.64 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ( ( F2 @ X5 ) % 0.31/0.64 = X5 ) ) % 0.31/0.64 => ( ( map @ A @ A @ F2 @ Xs ) % 0.31/0.64 = Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % map_idI % 0.31/0.64 thf(fact_7050_map__cong,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F2: A > B,G: A > B] : % 0.31/0.64 ( ( Xs = Ys ) % 0.31/0.64 => ( ! [X5: A] : % 0.31/0.64 ( ( member @ A @ X5 @ ( set2 @ A @ Ys ) ) % 0.31/0.64 => ( ( F2 @ X5 ) % 0.31/0.64 = ( G @ X5 ) ) ) % 0.31/0.64 => ( ( map @ A @ B @ F2 @ Xs ) % 0.31/0.64 = ( map @ A @ B @ G @ Ys ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_cong % 0.31/0.64 thf(fact_7051_ex__map__conv,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Ys: list @ B,F2: A > B] : % 0.31/0.64 ( ( ? [Xs2: list @ A] : % 0.31/0.64 ( Ys % 0.31/0.64 = ( map @ A @ B @ F2 @ Xs2 ) ) ) % 0.31/0.64 = ( ! [X: B] : % 0.31/0.64 ( ( member @ B @ X @ ( set2 @ B @ Ys ) ) % 0.31/0.64 => ? [Y2: A] : % 0.31/0.64 ( X % 0.31/0.64 = ( F2 @ Y2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % ex_map_conv % 0.31/0.64 thf(fact_7052_nths__map,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,I6: set @ nat] : % 0.31/0.64 ( ( nths @ A @ ( map @ B @ A @ F2 @ Xs ) @ I6 ) % 0.31/0.64 = ( map @ B @ A @ F2 @ ( nths @ B @ Xs @ I6 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nths_map % 0.31/0.64 thf(fact_7053_pair__list__eqI,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] : % 0.31/0.64 ( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) % 0.31/0.64 = ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) % 0.31/0.64 => ( ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Xs ) % 0.31/0.64 = ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Ys ) ) % 0.31/0.64 => ( Xs = Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % pair_list_eqI % 0.31/0.64 thf(fact_7054_enumerate__Suc__eq,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( enumerate @ A @ ( suc @ N2 ) @ Xs ) % 0.31/0.64 = ( map @ ( product_prod @ nat @ A ) @ ( product_prod @ nat @ A ) @ ( product_apfst @ nat @ nat @ A @ suc ) @ ( enumerate @ A @ N2 @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % enumerate_Suc_eq % 0.31/0.64 thf(fact_7055_map__replicate__const,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,K: A,Lst: list @ B] : % 0.31/0.64 ( ( map @ B @ A % 0.31/0.64 @ ^ [X: B] : K % 0.31/0.64 @ Lst ) % 0.31/0.64 = ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K ) ) ). % 0.31/0.64 % 0.31/0.64 % map_replicate_const % 0.31/0.64 thf(fact_7056_foldl__map,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,C: $tType,G: A > B > A,A2: A,F2: C > B,Xs: list @ C] : % 0.31/0.64 ( ( foldl @ A @ B @ G @ A2 @ ( map @ C @ B @ F2 @ Xs ) ) % 0.31/0.64 = ( foldl @ A @ C % 0.31/0.64 @ ^ [A4: A,X: C] : ( G @ A4 @ ( F2 @ X ) ) % 0.31/0.64 @ A2 % 0.31/0.64 @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % foldl_map % 0.31/0.64 thf(fact_7057_image__set,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] : % 0.31/0.64 ( ( image @ B @ A @ F2 @ ( set2 @ B @ Xs ) ) % 0.31/0.64 = ( set2 @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % image_set % 0.31/0.64 thf(fact_7058_take__map,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,N2: nat,F2: B > A,Xs: list @ B] : % 0.31/0.64 ( ( take @ A @ N2 @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.64 = ( map @ B @ A @ F2 @ ( take @ B @ N2 @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % take_map % 0.31/0.64 thf(fact_7059_inj__on__map__eq__map,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > B,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( inj_on @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) % 0.31/0.64 => ( ( ( map @ A @ B @ F2 @ Xs ) % 0.31/0.64 = ( map @ A @ B @ F2 @ Ys ) ) % 0.31/0.64 = ( Xs = Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % inj_on_map_eq_map % 0.31/0.64 thf(fact_7060_map__inj__on,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ B] : % 0.31/0.64 ( ( ( map @ B @ A @ F2 @ Xs ) % 0.31/0.64 = ( map @ B @ A @ F2 @ Ys ) ) % 0.31/0.64 => ( ( inj_on @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) ) % 0.31/0.64 => ( Xs = Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_inj_on % 0.31/0.64 thf(fact_7061_distinct__map,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] : % 0.31/0.64 ( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.64 = ( ( distinct @ B @ Xs ) % 0.31/0.64 & ( inj_on @ B @ A @ F2 @ ( set2 @ B @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % distinct_map % 0.31/0.64 thf(fact_7062_remdups__adj__map__injective,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > B,Xs: list @ A] : % 0.31/0.64 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.64 => ( ( remdups_adj @ B @ ( map @ A @ B @ F2 @ Xs ) ) % 0.31/0.64 = ( map @ A @ B @ F2 @ ( remdups_adj @ A @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remdups_adj_map_injective % 0.31/0.64 thf(fact_7063_zip__map__fst__snd,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Zs2: list @ ( product_prod @ A @ B )] : % 0.31/0.64 ( ( zip @ A @ B @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs2 ) @ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs2 ) ) % 0.31/0.64 = Zs2 ) ). % 0.31/0.64 % 0.31/0.64 % zip_map_fst_snd % 0.31/0.64 thf(fact_7064_map__of__eqI,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] : % 0.31/0.64 ( ( ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) % 0.31/0.64 = ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) ) % 0.31/0.64 => ( ! [X5: A] : % 0.31/0.64 ( ( member @ A @ X5 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) ) % 0.31/0.64 => ( ( map_of @ A @ B @ Xs @ X5 ) % 0.31/0.64 = ( map_of @ A @ B @ Ys @ X5 ) ) ) % 0.31/0.64 => ( ( map_of @ A @ B @ Xs ) % 0.31/0.64 = ( map_of @ A @ B @ Ys ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_eqI % 0.31/0.64 thf(fact_7065_map__removeAll__inj,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > B,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.64 => ( ( map @ A @ B @ F2 @ ( removeAll @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( removeAll @ B @ ( F2 @ X3 ) @ ( map @ A @ B @ F2 @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_removeAll_inj % 0.31/0.64 thf(fact_7066_inj__mapD,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > B] : % 0.31/0.64 ( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) % 0.31/0.64 => ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % inj_mapD % 0.31/0.64 thf(fact_7067_zip__assoc,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ C] : % 0.31/0.64 ( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs2 ) ) % 0.31/0.64 = ( map @ ( product_prod @ ( product_prod @ A @ B ) @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) ) % 0.31/0.64 @ ( product_case_prod @ ( product_prod @ A @ B ) @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) ) % 0.31/0.64 @ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ ( product_prod @ B @ C ) ) ) % 0.31/0.64 @ ^ [X: A,Y2: B,Z6: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C @ Y2 @ Z6 ) ) ) ) % 0.31/0.64 @ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys ) @ Zs2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % zip_assoc % 0.31/0.64 thf(fact_7068_zip__left__commute,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ C] : % 0.31/0.64 ( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs2 ) ) % 0.31/0.64 = ( map @ ( product_prod @ B @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( product_prod @ B @ C ) ) % 0.31/0.64 @ ( product_case_prod @ B @ ( product_prod @ A @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) ) % 0.31/0.64 @ ^ [Y2: B] : % 0.31/0.64 ( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) ) % 0.31/0.64 @ ^ [X: A,Z6: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C @ Y2 @ Z6 ) ) ) ) % 0.31/0.64 @ ( zip @ B @ ( product_prod @ A @ C ) @ Ys @ ( zip @ A @ C @ Xs @ Zs2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % zip_left_commute % 0.31/0.64 thf(fact_7069_zip__commute,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( zip @ A @ B ) % 0.31/0.64 = ( ^ [Xs2: list @ A,Ys3: list @ B] : % 0.31/0.64 ( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) % 0.31/0.64 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B ) % 0.31/0.64 @ ^ [X: B,Y2: A] : ( product_Pair @ A @ B @ Y2 @ X ) ) % 0.31/0.64 @ ( zip @ B @ A @ Ys3 @ Xs2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % zip_commute % 0.31/0.64 thf(fact_7070_zip__eq__conv,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ ( product_prod @ A @ B )] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( ( zip @ A @ B @ Xs @ Ys ) % 0.31/0.64 = Zs2 ) % 0.31/0.64 = ( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs2 ) % 0.31/0.64 = Xs ) % 0.31/0.64 & ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs2 ) % 0.31/0.64 = Ys ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % zip_eq_conv % 0.31/0.64 thf(fact_7071_distinct__insort__key,axiom, % 0.31/0.64 ! [A: $tType,B: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [F2: B > A,X3: B,Xs: list @ B] : % 0.31/0.64 ( ( distinct @ A @ ( map @ B @ A @ F2 @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) ) % 0.31/0.64 = ( ~ ( member @ A @ ( F2 @ X3 ) @ ( image @ B @ A @ F2 @ ( set2 @ B @ Xs ) ) ) % 0.31/0.64 & ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % distinct_insort_key % 0.31/0.64 thf(fact_7072_map__removeAll__inj__on,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > B,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( inj_on @ A @ B @ F2 @ ( insert @ A @ X3 @ ( set2 @ A @ Xs ) ) ) % 0.31/0.64 => ( ( map @ A @ B @ F2 @ ( removeAll @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( removeAll @ B @ ( F2 @ X3 ) @ ( map @ A @ B @ F2 @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_removeAll_inj_on % 0.31/0.64 thf(fact_7073_eq__key__imp__eq__value,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),K: A,V1: B,V22: B] : % 0.31/0.64 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) % 0.31/0.64 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V1 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) ) % 0.31/0.64 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V22 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) ) % 0.31/0.64 => ( V1 = V22 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % eq_key_imp_eq_value % 0.31/0.64 thf(fact_7074_inj__on__mapI,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > B,A3: set @ ( list @ A )] : % 0.31/0.64 ( ( inj_on @ A @ B @ F2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A3 ) ) ) % 0.31/0.64 => ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F2 ) @ A3 ) ) ). % 0.31/0.64 % 0.31/0.64 % inj_on_mapI % 0.31/0.64 thf(fact_7075_map__of__zip__map,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ A,F2: A > B] : % 0.31/0.64 ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ ( map @ A @ B @ F2 @ Xs ) ) ) % 0.31/0.64 = ( ^ [X: A] : ( if @ ( option @ B ) @ ( member @ A @ X @ ( set2 @ A @ Xs ) ) @ ( some @ B @ ( F2 @ X ) ) @ ( none @ B ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_zip_map % 0.31/0.64 thf(fact_7076_map__fst__zip__take,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] : % 0.31/0.64 ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) % 0.31/0.64 = ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % map_fst_zip_take % 0.31/0.64 thf(fact_7077_map__snd__zip__take,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ B,Ys: list @ A] : % 0.31/0.64 ( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( zip @ B @ A @ Xs @ Ys ) ) % 0.31/0.64 = ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ Ys ) ) ). % 0.31/0.64 % 0.31/0.64 % map_snd_zip_take % 0.31/0.64 thf(fact_7078_map__of__mapk__SomeI,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,C: $tType,F2: A > B,T2: list @ ( product_prod @ A @ C ),K: A,X3: C] : % 0.31/0.64 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.64 => ( ( ( map_of @ A @ C @ T2 @ K ) % 0.31/0.64 = ( some @ C @ X3 ) ) % 0.31/0.64 => ( ( map_of @ B @ C % 0.31/0.64 @ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) % 0.31/0.64 @ ( product_case_prod @ A @ C @ ( product_prod @ B @ C ) % 0.31/0.64 @ ^ [K3: A] : ( product_Pair @ B @ C @ ( F2 @ K3 ) ) ) % 0.31/0.64 @ T2 ) % 0.31/0.64 @ ( F2 @ K ) ) % 0.31/0.64 = ( some @ C @ X3 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_mapk_SomeI % 0.31/0.64 thf(fact_7079_set__map__of__compr,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B )] : % 0.31/0.64 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) % 0.31/0.64 => ( ( set2 @ ( product_prod @ A @ B ) @ Xs ) % 0.31/0.64 = ( collect @ ( product_prod @ A @ B ) % 0.31/0.64 @ ( product_case_prod @ A @ B @ $o % 0.31/0.64 @ ^ [K3: A,V5: B] : % 0.31/0.64 ( ( map_of @ A @ B @ Xs @ K3 ) % 0.31/0.64 = ( some @ B @ V5 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % set_map_of_compr % 0.31/0.64 thf(fact_7080_set__relcomp,axiom, % 0.31/0.64 ! [B: $tType,C: $tType,A: $tType,Xys: list @ ( product_prod @ A @ C ),Yzs: list @ ( product_prod @ C @ B )] : % 0.31/0.64 ( ( relcomp @ A @ C @ B @ ( set2 @ ( product_prod @ A @ C ) @ Xys ) @ ( set2 @ ( product_prod @ C @ B ) @ Yzs ) ) % 0.31/0.64 = ( set2 @ ( product_prod @ A @ B ) % 0.31/0.64 @ ( concat @ ( product_prod @ A @ B ) % 0.31/0.64 @ ( map @ ( product_prod @ A @ C ) @ ( list @ ( product_prod @ A @ B ) ) % 0.31/0.64 @ ^ [Xy: product_prod @ A @ C] : % 0.31/0.64 ( concat @ ( product_prod @ A @ B ) % 0.31/0.64 @ ( map @ ( product_prod @ C @ B ) @ ( list @ ( product_prod @ A @ B ) ) % 0.31/0.64 @ ^ [Yz: product_prod @ C @ B] : % 0.31/0.64 ( if @ ( list @ ( product_prod @ A @ B ) ) % 0.31/0.64 @ ( ( product_snd @ A @ C @ Xy ) % 0.31/0.64 = ( product_fst @ C @ B @ Yz ) ) % 0.31/0.64 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Xy ) @ ( product_snd @ C @ B @ Yz ) ) @ ( nil @ ( product_prod @ A @ B ) ) ) % 0.31/0.64 @ ( nil @ ( product_prod @ A @ B ) ) ) % 0.31/0.64 @ Yzs ) ) % 0.31/0.64 @ Xys ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % set_relcomp % 0.31/0.64 thf(fact_7081_dual__max,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ( ( max @ A % 0.31/0.64 @ ^ [X: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X ) ) % 0.31/0.64 = ( ord_min @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % dual_max % 0.31/0.64 thf(fact_7082_list_Oinject,axiom, % 0.31/0.64 ! [A: $tType,X21: A,X222: list @ A,Y21: A,Y222: list @ A] : % 0.31/0.64 ( ( ( cons @ A @ X21 @ X222 ) % 0.31/0.64 = ( cons @ A @ Y21 @ Y222 ) ) % 0.31/0.64 = ( ( X21 = Y21 ) % 0.31/0.64 & ( X222 = Y222 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.inject % 0.31/0.64 thf(fact_7083_list_Osimps_I15_J,axiom, % 0.31/0.64 ! [A: $tType,X21: A,X222: list @ A] : % 0.31/0.64 ( ( set2 @ A @ ( cons @ A @ X21 @ X222 ) ) % 0.31/0.64 = ( insert @ A @ X21 @ ( set2 @ A @ X222 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.simps(15) % 0.31/0.64 thf(fact_7084_nth__Cons__Suc,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,N2: nat] : % 0.31/0.64 ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ ( suc @ N2 ) ) % 0.31/0.64 = ( nth @ A @ Xs @ N2 ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_Cons_Suc % 0.31/0.64 thf(fact_7085_nth__Cons__0,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ ( zero_zero @ nat ) ) % 0.31/0.64 = X3 ) ). % 0.31/0.64 % 0.31/0.64 % nth_Cons_0 % 0.31/0.64 thf(fact_7086_take__Suc__Cons,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( take @ A @ ( suc @ N2 ) @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( cons @ A @ X3 @ ( take @ A @ N2 @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % take_Suc_Cons % 0.31/0.64 thf(fact_7087_nths__singleton,axiom, % 0.31/0.64 ! [A: $tType,A3: set @ nat,X3: A] : % 0.31/0.64 ( ( ( member @ nat @ ( zero_zero @ nat ) @ A3 ) % 0.31/0.64 => ( ( nths @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ A3 ) % 0.31/0.64 = ( cons @ A @ X3 @ ( nil @ A ) ) ) ) % 0.31/0.64 & ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 ) % 0.31/0.64 => ( ( nths @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ A3 ) % 0.31/0.64 = ( nil @ A ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nths_singleton % 0.31/0.64 thf(fact_7088_Cons__listrel1__Cons,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R2 ) ) % 0.31/0.64 = ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) % 0.31/0.64 & ( Xs = Ys ) ) % 0.31/0.64 | ( ( X3 = Y ) % 0.31/0.64 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Cons_listrel1_Cons % 0.31/0.64 thf(fact_7089_horner__sum__simps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,B: $tType] : % 0.31/0.64 ( ( comm_semiring_0 @ A ) % 0.31/0.64 => ! [F2: B > A,A2: A,X3: B,Xs: list @ B] : % 0.31/0.64 ( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ ( cons @ B @ X3 @ Xs ) ) % 0.31/0.64 = ( plus_plus @ A @ ( F2 @ X3 ) @ ( times_times @ A @ A2 @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ Xs ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % horner_sum_simps(2) % 0.31/0.64 thf(fact_7090_lexord__cons__cons,axiom, % 0.31/0.64 ! [A: $tType,A2: A,X3: list @ A,B2: A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A2 @ X3 ) @ ( cons @ A @ B2 @ Y ) ) @ ( lexord @ A @ R2 ) ) % 0.31/0.64 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) % 0.31/0.64 | ( ( A2 = B2 ) % 0.31/0.64 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_cons_cons % 0.31/0.64 thf(fact_7091_lexord__Nil__left,axiom, % 0.31/0.64 ! [A: $tType,Y: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y ) @ ( lexord @ A @ R2 ) ) % 0.31/0.64 = ( ? [A4: A,X: list @ A] : % 0.31/0.64 ( Y % 0.31/0.64 = ( cons @ A @ A4 @ X ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_Nil_left % 0.31/0.64 thf(fact_7092_enumerate__simps_I2_J,axiom, % 0.31/0.64 ! [B: $tType,N2: nat,X3: B,Xs: list @ B] : % 0.31/0.64 ( ( enumerate @ B @ N2 @ ( cons @ B @ X3 @ Xs ) ) % 0.31/0.64 = ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N2 @ X3 ) @ ( enumerate @ B @ ( suc @ N2 ) @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % enumerate_simps(2) % 0.31/0.64 thf(fact_7093_map__upds__Cons,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,M: A > ( option @ B ),A2: A,As: list @ A,B2: B,Bs: list @ B] : % 0.31/0.64 ( ( map_upds @ A @ B @ M @ ( cons @ A @ A2 @ As ) @ ( cons @ B @ B2 @ Bs ) ) % 0.31/0.64 = ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ A2 @ ( some @ B @ B2 ) ) @ As @ Bs ) ) ). % 0.31/0.64 % 0.31/0.64 % map_upds_Cons % 0.31/0.64 thf(fact_7094_zip__Cons__Cons,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,X3: A,Xs: list @ A,Y: B,Ys: list @ B] : % 0.31/0.64 ( ( zip @ A @ B @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys ) ) % 0.31/0.64 = ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % zip_Cons_Cons % 0.31/0.64 thf(fact_7095_nth__Cons__numeral,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,V2: num] : % 0.31/0.64 ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ ( numeral_numeral @ nat @ V2 ) ) % 0.31/0.64 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_Cons_numeral % 0.31/0.64 thf(fact_7096_take__Cons__numeral,axiom, % 0.31/0.64 ! [A: $tType,V2: num,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( take @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( cons @ A @ X3 @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % take_Cons_numeral % 0.31/0.64 thf(fact_7097_Cons__in__lex,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( lex @ A @ R2 ) ) % 0.31/0.64 = ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) % 0.31/0.64 & ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ Ys ) ) ) % 0.31/0.64 | ( ( X3 = Y ) % 0.31/0.64 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Cons_in_lex % 0.31/0.64 thf(fact_7098_concat__map__singleton,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] : % 0.31/0.64 ( ( concat @ A % 0.31/0.64 @ ( map @ B @ ( list @ A ) % 0.31/0.64 @ ^ [X: B] : ( cons @ A @ ( F2 @ X ) @ ( nil @ A ) ) % 0.31/0.64 @ Xs ) ) % 0.31/0.64 = ( map @ B @ A @ F2 @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % concat_map_singleton % 0.31/0.64 thf(fact_7099_nth__Cons__pos,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.64 => ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N2 ) % 0.31/0.64 = ( nth @ A @ Xs @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_Cons_pos % 0.31/0.64 thf(fact_7100_zip__same__conv__map,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A] : % 0.31/0.64 ( ( zip @ A @ A @ Xs @ Xs ) % 0.31/0.64 = ( map @ A @ ( product_prod @ A @ A ) % 0.31/0.64 @ ^ [X: A] : ( product_Pair @ A @ A @ X @ X ) % 0.31/0.64 @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % zip_same_conv_map % 0.31/0.64 thf(fact_7101_product__concat__map,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( product @ A @ B ) % 0.31/0.64 = ( ^ [Xs2: list @ A,Ys3: list @ B] : % 0.31/0.64 ( concat @ ( product_prod @ A @ B ) % 0.31/0.64 @ ( map @ A @ ( list @ ( product_prod @ A @ B ) ) % 0.31/0.64 @ ^ [X: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys3 ) % 0.31/0.64 @ Xs2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % product_concat_map % 0.31/0.64 thf(fact_7102_n__lists_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( n_lists @ A @ ( suc @ N2 ) @ Xs ) % 0.31/0.64 = ( concat @ ( list @ A ) % 0.31/0.64 @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) ) % 0.31/0.64 @ ^ [Ys3: list @ A] : % 0.31/0.64 ( map @ A @ ( list @ A ) % 0.31/0.64 @ ^ [Y2: A] : ( cons @ A @ Y2 @ Ys3 ) % 0.31/0.64 @ Xs ) % 0.31/0.64 @ ( n_lists @ A @ N2 @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % n_lists.simps(2) % 0.31/0.64 thf(fact_7103_map__eq__Cons__conv,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Y: A,Ys: list @ A] : % 0.31/0.64 ( ( ( map @ B @ A @ F2 @ Xs ) % 0.31/0.64 = ( cons @ A @ Y @ Ys ) ) % 0.31/0.64 = ( ? [Z6: B,Zs3: list @ B] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( cons @ B @ Z6 @ Zs3 ) ) % 0.31/0.64 & ( ( F2 @ Z6 ) % 0.31/0.64 = Y ) % 0.31/0.64 & ( ( map @ B @ A @ F2 @ Zs3 ) % 0.31/0.64 = Ys ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_eq_Cons_conv % 0.31/0.64 thf(fact_7104_Cons__eq__map__conv,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,X3: A,Xs: list @ A,F2: B > A,Ys: list @ B] : % 0.31/0.64 ( ( ( cons @ A @ X3 @ Xs ) % 0.31/0.64 = ( map @ B @ A @ F2 @ Ys ) ) % 0.31/0.64 = ( ? [Z6: B,Zs3: list @ B] : % 0.31/0.64 ( ( Ys % 0.31/0.64 = ( cons @ B @ Z6 @ Zs3 ) ) % 0.31/0.64 & ( X3 % 0.31/0.64 = ( F2 @ Z6 ) ) % 0.31/0.64 & ( Xs % 0.31/0.64 = ( map @ B @ A @ F2 @ Zs3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Cons_eq_map_conv % 0.31/0.64 thf(fact_7105_map__eq__Cons__D,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: B > A,Xs: list @ B,Y: A,Ys: list @ A] : % 0.31/0.64 ( ( ( map @ B @ A @ F2 @ Xs ) % 0.31/0.64 = ( cons @ A @ Y @ Ys ) ) % 0.31/0.64 => ? [Z3: B,Zs: list @ B] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( cons @ B @ Z3 @ Zs ) ) % 0.31/0.64 & ( ( F2 @ Z3 ) % 0.31/0.64 = Y ) % 0.31/0.64 & ( ( map @ B @ A @ F2 @ Zs ) % 0.31/0.64 = Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_eq_Cons_D % 0.31/0.64 thf(fact_7106_Cons__eq__map__D,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,X3: A,Xs: list @ A,F2: B > A,Ys: list @ B] : % 0.31/0.64 ( ( ( cons @ A @ X3 @ Xs ) % 0.31/0.64 = ( map @ B @ A @ F2 @ Ys ) ) % 0.31/0.64 => ? [Z3: B,Zs: list @ B] : % 0.31/0.64 ( ( Ys % 0.31/0.64 = ( cons @ B @ Z3 @ Zs ) ) % 0.31/0.64 & ( X3 % 0.31/0.64 = ( F2 @ Z3 ) ) % 0.31/0.64 & ( Xs % 0.31/0.64 = ( map @ B @ A @ F2 @ Zs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Cons_eq_map_D % 0.31/0.64 thf(fact_7107_list_Osimps_I9_J,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > B,X21: A,X222: list @ A] : % 0.31/0.64 ( ( map @ A @ B @ F2 @ ( cons @ A @ X21 @ X222 ) ) % 0.31/0.64 = ( cons @ B @ ( F2 @ X21 ) @ ( map @ A @ B @ F2 @ X222 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.simps(9) % 0.31/0.64 thf(fact_7108_product__lists_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Xss: list @ ( list @ A )] : % 0.31/0.64 ( ( product_lists @ A @ ( cons @ ( list @ A ) @ Xs @ Xss ) ) % 0.31/0.64 = ( concat @ ( list @ A ) % 0.31/0.64 @ ( map @ A @ ( list @ ( list @ A ) ) % 0.31/0.64 @ ^ [X: A] : ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( product_lists @ A @ Xss ) ) % 0.31/0.64 @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % product_lists.simps(2) % 0.31/0.64 thf(fact_7109_List_Obind__def,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( bind @ A @ B ) % 0.31/0.64 = ( ^ [Xs2: list @ A,F3: A > ( list @ B )] : ( concat @ B @ ( map @ A @ ( list @ B ) @ F3 @ Xs2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % List.bind_def % 0.31/0.64 thf(fact_7110_take__Cons,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( take @ A @ N2 @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( case_nat @ ( list @ A ) @ ( nil @ A ) % 0.31/0.64 @ ^ [M2: nat] : ( cons @ A @ X3 @ ( take @ A @ M2 @ Xs ) ) % 0.31/0.64 @ N2 ) ) ). % 0.31/0.64 % 0.31/0.64 % take_Cons % 0.31/0.64 thf(fact_7111_shuffles_Osimps_I3_J,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,Y: A,Ys: list @ A] : % 0.31/0.64 ( ( shuffles @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) % 0.31/0.64 = ( sup_sup @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X3 @ Xs ) @ Ys ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % shuffles.simps(3) % 0.31/0.64 thf(fact_7112_Cons__shuffles__subset1,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ ( cons @ A @ X3 @ Xs ) @ Ys ) ) ). % 0.31/0.64 % 0.31/0.64 % Cons_shuffles_subset1 % 0.31/0.64 thf(fact_7113_Cons__shuffles__subset2,axiom, % 0.31/0.64 ! [A: $tType,Y: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Cons_shuffles_subset2 % 0.31/0.64 thf(fact_7114_nth__Cons,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,N2: nat] : % 0.31/0.64 ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N2 ) % 0.31/0.64 = ( case_nat @ A @ X3 @ ( nth @ A @ Xs ) @ N2 ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_Cons % 0.31/0.64 thf(fact_7115_foldl__Cons,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: B > A > B,A2: B,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( foldl @ B @ A @ F2 @ A2 @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( foldl @ B @ A @ F2 @ ( F2 @ A2 @ X3 ) @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % foldl_Cons % 0.31/0.64 thf(fact_7116_gen__length__code_I2_J,axiom, % 0.31/0.64 ! [B: $tType,N2: nat,X3: B,Xs: list @ B] : % 0.31/0.64 ( ( gen_length @ B @ N2 @ ( cons @ B @ X3 @ Xs ) ) % 0.31/0.64 = ( gen_length @ B @ ( suc @ N2 ) @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % gen_length_code(2) % 0.31/0.64 thf(fact_7117_arg__min__list_Osimps_I1_J,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( linorder @ B ) % 0.31/0.64 => ! [F2: A > B,X3: A] : % 0.31/0.64 ( ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) % 0.31/0.64 = X3 ) ) ). % 0.31/0.64 % 0.31/0.64 % arg_min_list.simps(1) % 0.31/0.64 thf(fact_7118_find_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,P: A > $o,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( ( P @ X3 ) % 0.31/0.64 => ( ( find @ A @ P @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( some @ A @ X3 ) ) ) % 0.31/0.64 & ( ~ ( P @ X3 ) % 0.31/0.64 => ( ( find @ A @ P @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( find @ A @ P @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % find.simps(2) % 0.31/0.64 thf(fact_7119_shufflesE,axiom, % 0.31/0.64 ! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) ) % 0.31/0.64 => ( ( ( Zs2 = Xs ) % 0.31/0.64 => ( Ys % 0.31/0.64 != ( nil @ A ) ) ) % 0.31/0.64 => ( ( ( Zs2 = Ys ) % 0.31/0.64 => ( Xs % 0.31/0.64 != ( nil @ A ) ) ) % 0.31/0.64 => ( ! [X5: A,Xs4: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( cons @ A @ X5 @ Xs4 ) ) % 0.31/0.64 => ! [Z3: A,Zs4: list @ A] : % 0.31/0.64 ( ( Zs2 % 0.31/0.64 = ( cons @ A @ Z3 @ Zs4 ) ) % 0.31/0.64 => ( ( X5 = Z3 ) % 0.31/0.64 => ~ ( member @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs4 @ Ys ) ) ) ) ) % 0.31/0.64 => ~ ! [Y4: A,Ys5: list @ A] : % 0.31/0.64 ( ( Ys % 0.31/0.64 = ( cons @ A @ Y4 @ Ys5 ) ) % 0.31/0.64 => ! [Z3: A,Zs4: list @ A] : % 0.31/0.64 ( ( Zs2 % 0.31/0.64 = ( cons @ A @ Z3 @ Zs4 ) ) % 0.31/0.64 => ( ( Y4 = Z3 ) % 0.31/0.64 => ~ ( member @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs @ Ys5 ) ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % shufflesE % 0.31/0.64 thf(fact_7120_insort__key_Osimps_I1_J,axiom, % 0.31/0.64 ! [A: $tType,B: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [F2: B > A,X3: B] : % 0.31/0.64 ( ( linorder_insort_key @ B @ A @ F2 @ X3 @ ( nil @ B ) ) % 0.31/0.64 = ( cons @ B @ X3 @ ( nil @ B ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % insort_key.simps(1) % 0.31/0.64 thf(fact_7121_list__update__code_I2_J,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,Y: A] : % 0.31/0.64 ( ( list_update @ A @ ( cons @ A @ X3 @ Xs ) @ ( zero_zero @ nat ) @ Y ) % 0.31/0.64 = ( cons @ A @ Y @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % list_update_code(2) % 0.31/0.64 thf(fact_7122_distinct__singleton,axiom, % 0.31/0.64 ! [A: $tType,X3: A] : ( distinct @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % distinct_singleton % 0.31/0.64 thf(fact_7123_distinct_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( distinct @ A @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 & ( distinct @ A @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % distinct.simps(2) % 0.31/0.64 thf(fact_7124_length__Suc__conv,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,N2: nat] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( suc @ N2 ) ) % 0.31/0.64 = ( ? [Y2: A,Ys3: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( cons @ A @ Y2 @ Ys3 ) ) % 0.31/0.64 & ( ( size_size @ ( list @ A ) @ Ys3 ) % 0.31/0.64 = N2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % length_Suc_conv % 0.31/0.64 thf(fact_7125_Suc__length__conv,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( ( suc @ N2 ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 = ( ? [Y2: A,Ys3: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( cons @ A @ Y2 @ Ys3 ) ) % 0.31/0.64 & ( ( size_size @ ( list @ A ) @ Ys3 ) % 0.31/0.64 = N2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Suc_length_conv % 0.31/0.64 thf(fact_7126_length__Cons,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( size_size @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % length_Cons % 0.31/0.64 thf(fact_7127_list__induct4,axiom, % 0.31/0.64 ! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ C,Ws2: list @ D,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D ) > $o] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( ( size_size @ ( list @ B ) @ Ys ) % 0.31/0.64 = ( size_size @ ( list @ C ) @ Zs2 ) ) % 0.31/0.64 => ( ( ( size_size @ ( list @ C ) @ Zs2 ) % 0.31/0.64 = ( size_size @ ( list @ D ) @ Ws2 ) ) % 0.31/0.64 => ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) ) % 0.31/0.64 => ( ! [X5: A,Xs3: list @ A,Y4: B,Ys4: list @ B,Z3: C,Zs: list @ C,W: D,Ws: list @ D] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs3 ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys4 ) ) % 0.31/0.64 => ( ( ( size_size @ ( list @ B ) @ Ys4 ) % 0.31/0.64 = ( size_size @ ( list @ C ) @ Zs ) ) % 0.31/0.64 => ( ( ( size_size @ ( list @ C ) @ Zs ) % 0.31/0.64 = ( size_size @ ( list @ D ) @ Ws ) ) % 0.31/0.64 => ( ( P @ Xs3 @ Ys4 @ Zs @ Ws ) % 0.31/0.64 => ( P @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) @ ( cons @ C @ Z3 @ Zs ) @ ( cons @ D @ W @ Ws ) ) ) ) ) ) % 0.31/0.64 => ( P @ Xs @ Ys @ Zs2 @ Ws2 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list_induct4 % 0.31/0.64 thf(fact_7128_list__induct3,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( ( size_size @ ( list @ B ) @ Ys ) % 0.31/0.64 = ( size_size @ ( list @ C ) @ Zs2 ) ) % 0.31/0.64 => ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) ) % 0.31/0.64 => ( ! [X5: A,Xs3: list @ A,Y4: B,Ys4: list @ B,Z3: C,Zs: list @ C] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs3 ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys4 ) ) % 0.31/0.64 => ( ( ( size_size @ ( list @ B ) @ Ys4 ) % 0.31/0.64 = ( size_size @ ( list @ C ) @ Zs ) ) % 0.31/0.64 => ( ( P @ Xs3 @ Ys4 @ Zs ) % 0.31/0.64 => ( P @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) @ ( cons @ C @ Z3 @ Zs ) ) ) ) ) % 0.31/0.64 => ( P @ Xs @ Ys @ Zs2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list_induct3 % 0.31/0.64 thf(fact_7129_list__induct2,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( P @ ( nil @ A ) @ ( nil @ B ) ) % 0.31/0.64 => ( ! [X5: A,Xs3: list @ A,Y4: B,Ys4: list @ B] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs3 ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys4 ) ) % 0.31/0.64 => ( ( P @ Xs3 @ Ys4 ) % 0.31/0.64 => ( P @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) ) ) ) % 0.31/0.64 => ( P @ Xs @ Ys ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list_induct2 % 0.31/0.64 thf(fact_7130_listrelp_OCons,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,R2: A > B > $o,X3: A,Y: B,Xs: list @ A,Ys: list @ B] : % 0.31/0.64 ( ( R2 @ X3 @ Y ) % 0.31/0.64 => ( ( listrelp @ A @ B @ R2 @ Xs @ Ys ) % 0.31/0.64 => ( listrelp @ A @ B @ R2 @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % listrelp.Cons % 0.31/0.64 thf(fact_7131_list__update_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,I: nat,V2: A] : % 0.31/0.64 ( ( list_update @ A @ ( cons @ A @ X3 @ Xs ) @ I @ V2 ) % 0.31/0.64 = ( case_nat @ ( list @ A ) @ ( cons @ A @ V2 @ Xs ) % 0.31/0.64 @ ^ [J3: nat] : ( cons @ A @ X3 @ ( list_update @ A @ Xs @ J3 @ V2 ) ) % 0.31/0.64 @ I ) ) ). % 0.31/0.64 % 0.31/0.64 % list_update.simps(2) % 0.31/0.64 thf(fact_7132_set__ConsD,axiom, % 0.31/0.64 ! [A: $tType,Y: A,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X3 @ Xs ) ) ) % 0.31/0.64 => ( ( Y = X3 ) % 0.31/0.64 | ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % set_ConsD % 0.31/0.64 thf(fact_7133_list_Oset__cases,axiom, % 0.31/0.64 ! [A: $tType,E3: A,A2: list @ A] : % 0.31/0.64 ( ( member @ A @ E3 @ ( set2 @ A @ A2 ) ) % 0.31/0.64 => ( ! [Z23: list @ A] : % 0.31/0.64 ( A2 % 0.31/0.64 != ( cons @ A @ E3 @ Z23 ) ) % 0.31/0.64 => ~ ! [Z12: A,Z23: list @ A] : % 0.31/0.64 ( ( A2 % 0.31/0.64 = ( cons @ A @ Z12 @ Z23 ) ) % 0.31/0.64 => ~ ( member @ A @ E3 @ ( set2 @ A @ Z23 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.set_cases % 0.31/0.64 thf(fact_7134_list_Oset__intros_I1_J,axiom, % 0.31/0.64 ! [A: $tType,X21: A,X222: list @ A] : ( member @ A @ X21 @ ( set2 @ A @ ( cons @ A @ X21 @ X222 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.set_intros(1) % 0.31/0.64 thf(fact_7135_list_Oset__intros_I2_J,axiom, % 0.31/0.64 ! [A: $tType,Y: A,X222: list @ A,X21: A] : % 0.31/0.64 ( ( member @ A @ Y @ ( set2 @ A @ X222 ) ) % 0.31/0.64 => ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X21 @ X222 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.set_intros(2) % 0.31/0.64 thf(fact_7136_list__nonempty__induct,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] : % 0.31/0.64 ( ( Xs % 0.31/0.64 != ( nil @ A ) ) % 0.31/0.64 => ( ! [X5: A] : ( P @ ( cons @ A @ X5 @ ( nil @ A ) ) ) % 0.31/0.64 => ( ! [X5: A,Xs3: list @ A] : % 0.31/0.64 ( ( Xs3 % 0.31/0.64 != ( nil @ A ) ) % 0.31/0.64 => ( ( P @ Xs3 ) % 0.31/0.64 => ( P @ ( cons @ A @ X5 @ Xs3 ) ) ) ) % 0.31/0.64 => ( P @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list_nonempty_induct % 0.31/0.64 thf(fact_7137_list__induct2_H,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] : % 0.31/0.64 ( ( P @ ( nil @ A ) @ ( nil @ B ) ) % 0.31/0.64 => ( ! [X5: A,Xs3: list @ A] : ( P @ ( cons @ A @ X5 @ Xs3 ) @ ( nil @ B ) ) % 0.31/0.64 => ( ! [Y4: B,Ys4: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y4 @ Ys4 ) ) % 0.31/0.64 => ( ! [X5: A,Xs3: list @ A,Y4: B,Ys4: list @ B] : % 0.31/0.64 ( ( P @ Xs3 @ Ys4 ) % 0.31/0.64 => ( P @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ B @ Y4 @ Ys4 ) ) ) % 0.31/0.64 => ( P @ Xs @ Ys ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list_induct2' % 0.31/0.64 thf(fact_7138_neq__Nil__conv,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 != ( nil @ A ) ) % 0.31/0.64 = ( ? [Y2: A,Ys3: list @ A] : % 0.31/0.64 ( Xs % 0.31/0.64 = ( cons @ A @ Y2 @ Ys3 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % neq_Nil_conv % 0.31/0.64 thf(fact_7139_remdups__adj_Ocases,axiom, % 0.31/0.64 ! [A: $tType,X3: list @ A] : % 0.31/0.64 ( ( X3 % 0.31/0.64 != ( nil @ A ) ) % 0.31/0.64 => ( ! [X5: A] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( cons @ A @ X5 @ ( nil @ A ) ) ) % 0.31/0.64 => ~ ! [X5: A,Y4: A,Xs3: list @ A] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remdups_adj.cases % 0.31/0.64 thf(fact_7140_transpose_Ocases,axiom, % 0.31/0.64 ! [A: $tType,X3: list @ ( list @ A )] : % 0.31/0.64 ( ( X3 % 0.31/0.64 != ( nil @ ( list @ A ) ) ) % 0.31/0.64 => ( ! [Xss2: list @ ( list @ A )] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) % 0.31/0.64 => ~ ! [X5: A,Xs3: list @ A,Xss2: list @ ( list @ A )] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % transpose.cases % 0.31/0.64 thf(fact_7141_min__list_Ocases,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( ord @ A ) % 0.31/0.64 => ! [X3: list @ A] : % 0.31/0.64 ( ! [X5: A,Xs3: list @ A] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( cons @ A @ X5 @ Xs3 ) ) % 0.31/0.64 => ( X3 % 0.31/0.64 = ( nil @ A ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % min_list.cases % 0.31/0.64 thf(fact_7142_list_Oexhaust,axiom, % 0.31/0.64 ! [A: $tType,Y: list @ A] : % 0.31/0.64 ( ( Y % 0.31/0.64 != ( nil @ A ) ) % 0.31/0.64 => ~ ! [X212: A,X223: list @ A] : % 0.31/0.64 ( Y % 0.31/0.64 != ( cons @ A @ X212 @ X223 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.exhaust % 0.31/0.64 thf(fact_7143_list_OdiscI,axiom, % 0.31/0.64 ! [A: $tType,List: list @ A,X21: A,X222: list @ A] : % 0.31/0.64 ( ( List % 0.31/0.64 = ( cons @ A @ X21 @ X222 ) ) % 0.31/0.64 => ( List % 0.31/0.64 != ( nil @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.discI % 0.31/0.64 thf(fact_7144_list_Odistinct_I1_J,axiom, % 0.31/0.64 ! [A: $tType,X21: A,X222: list @ A] : % 0.31/0.64 ( ( nil @ A ) % 0.31/0.64 != ( cons @ A @ X21 @ X222 ) ) ). % 0.31/0.64 % 0.31/0.64 % list.distinct(1) % 0.31/0.64 thf(fact_7145_removeAll_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Y: A,Xs: list @ A] : % 0.31/0.64 ( ( ( X3 = Y ) % 0.31/0.64 => ( ( removeAll @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) % 0.31/0.64 = ( removeAll @ A @ X3 @ Xs ) ) ) % 0.31/0.64 & ( ( X3 != Y ) % 0.31/0.64 => ( ( removeAll @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) % 0.31/0.64 = ( cons @ A @ Y @ ( removeAll @ A @ X3 @ Xs ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % removeAll.simps(2) % 0.31/0.64 thf(fact_7146_remove1_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Y: A,Xs: list @ A] : % 0.31/0.64 ( ( ( X3 = Y ) % 0.31/0.64 => ( ( remove1 @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) % 0.31/0.64 = Xs ) ) % 0.31/0.64 & ( ( X3 != Y ) % 0.31/0.64 => ( ( remove1 @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) % 0.31/0.64 = ( cons @ A @ Y @ ( remove1 @ A @ X3 @ Xs ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remove1.simps(2) % 0.31/0.64 thf(fact_7147_Cons__in__shuffles__rightI,axiom, % 0.31/0.64 ! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys: list @ A,Z2: A] : % 0.31/0.64 ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) ) % 0.31/0.64 => ( member @ ( list @ A ) @ ( cons @ A @ Z2 @ Zs2 ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Z2 @ Ys ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Cons_in_shuffles_rightI % 0.31/0.64 thf(fact_7148_Cons__in__shuffles__leftI,axiom, % 0.31/0.64 ! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys: list @ A,Z2: A] : % 0.31/0.64 ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) ) % 0.31/0.64 => ( member @ ( list @ A ) @ ( cons @ A @ Z2 @ Zs2 ) @ ( shuffles @ A @ ( cons @ A @ Z2 @ Xs ) @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Cons_in_shuffles_leftI % 0.31/0.64 thf(fact_7149_not__Cons__self2,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( cons @ A @ X3 @ Xs ) % 0.31/0.64 != Xs ) ). % 0.31/0.64 % 0.31/0.64 % not_Cons_self2 % 0.31/0.64 thf(fact_7150_ord_Omax__def,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( max @ A ) % 0.31/0.64 = ( ^ [Less_eq: A > A > $o,A4: A,B4: A] : ( if @ A @ ( Less_eq @ A4 @ B4 ) @ B4 @ A4 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % ord.max_def % 0.31/0.64 thf(fact_7151_ord_Omax_Ocong,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( max @ A ) % 0.31/0.64 = ( max @ A ) ) ). % 0.31/0.64 % 0.31/0.64 % ord.max.cong % 0.31/0.64 thf(fact_7152_distinct__length__2__or__more,axiom, % 0.31/0.64 ! [A: $tType,A2: A,B2: A,Xs: list @ A] : % 0.31/0.64 ( ( distinct @ A @ ( cons @ A @ A2 @ ( cons @ A @ B2 @ Xs ) ) ) % 0.31/0.64 = ( ( A2 != B2 ) % 0.31/0.64 & ( distinct @ A @ ( cons @ A @ A2 @ Xs ) ) % 0.31/0.64 & ( distinct @ A @ ( cons @ A @ B2 @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % distinct_length_2_or_more % 0.31/0.64 thf(fact_7153_list__update__code_I3_J,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,I: nat,Y: A] : % 0.31/0.64 ( ( list_update @ A @ ( cons @ A @ X3 @ Xs ) @ ( suc @ I ) @ Y ) % 0.31/0.64 = ( cons @ A @ X3 @ ( list_update @ A @ Xs @ I @ Y ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list_update_code(3) % 0.31/0.64 thf(fact_7154_replicate__Suc,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,X3: A] : % 0.31/0.64 ( ( replicate @ A @ ( suc @ N2 ) @ X3 ) % 0.31/0.64 = ( cons @ A @ X3 @ ( replicate @ A @ N2 @ X3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % replicate_Suc % 0.31/0.64 thf(fact_7155_remdups_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ( ( remdups @ A @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( remdups @ A @ Xs ) ) ) % 0.31/0.64 & ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ( ( remdups @ A @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( cons @ A @ X3 @ ( remdups @ A @ Xs ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remdups.simps(2) % 0.31/0.64 thf(fact_7156_remdups__adj_Osimps_I3_J,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Y: A,Xs: list @ A] : % 0.31/0.64 ( ( ( X3 = Y ) % 0.31/0.64 => ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) ) % 0.31/0.64 = ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) ) % 0.31/0.64 & ( ( X3 != Y ) % 0.31/0.64 => ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) ) % 0.31/0.64 = ( cons @ A @ X3 @ ( remdups_adj @ A @ ( cons @ A @ Y @ Xs ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remdups_adj.simps(3) % 0.31/0.64 thf(fact_7157_remdups__adj_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,X3: A] : % 0.31/0.64 ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) ) % 0.31/0.64 = ( cons @ A @ X3 @ ( nil @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remdups_adj.simps(2) % 0.31/0.64 thf(fact_7158_remdups__adj_Oelims,axiom, % 0.31/0.64 ! [A: $tType,X3: list @ A,Y: list @ A] : % 0.31/0.64 ( ( ( remdups_adj @ A @ X3 ) % 0.31/0.64 = Y ) % 0.31/0.64 => ( ( ( X3 % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 => ( Y % 0.31/0.64 != ( nil @ A ) ) ) % 0.31/0.64 => ( ! [X5: A] : % 0.31/0.64 ( ( X3 % 0.31/0.64 = ( cons @ A @ X5 @ ( nil @ A ) ) ) % 0.31/0.64 => ( Y % 0.31/0.64 != ( cons @ A @ X5 @ ( nil @ A ) ) ) ) % 0.31/0.64 => ~ ! [X5: A,Y4: A,Xs3: list @ A] : % 0.31/0.64 ( ( X3 % 0.31/0.64 = ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Xs3 ) ) ) % 0.31/0.64 => ~ ( ( ( X5 = Y4 ) % 0.31/0.64 => ( Y % 0.31/0.64 = ( remdups_adj @ A @ ( cons @ A @ X5 @ Xs3 ) ) ) ) % 0.31/0.64 & ( ( X5 != Y4 ) % 0.31/0.64 => ( Y % 0.31/0.64 = ( cons @ A @ X5 @ ( remdups_adj @ A @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remdups_adj.elims % 0.31/0.64 thf(fact_7159_zip__eq__ConsE,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Xy2: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] : % 0.31/0.64 ( ( ( zip @ A @ B @ Xs @ Ys ) % 0.31/0.64 = ( cons @ ( product_prod @ A @ B ) @ Xy2 @ Xys ) ) % 0.31/0.64 => ~ ! [X5: A,Xs4: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( cons @ A @ X5 @ Xs4 ) ) % 0.31/0.64 => ! [Y4: B,Ys5: list @ B] : % 0.31/0.64 ( ( Ys % 0.31/0.64 = ( cons @ B @ Y4 @ Ys5 ) ) % 0.31/0.64 => ( ( Xy2 % 0.31/0.64 = ( product_Pair @ A @ B @ X5 @ Y4 ) ) % 0.31/0.64 => ( Xys % 0.31/0.64 != ( zip @ A @ B @ Xs4 @ Ys5 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % zip_eq_ConsE % 0.31/0.64 thf(fact_7160_successively_Ocases,axiom, % 0.31/0.64 ! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( list @ A )] : % 0.31/0.64 ( ! [P8: A > A > $o] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( nil @ A ) ) ) % 0.31/0.64 => ( ! [P8: A > A > $o,X5: A] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) % 0.31/0.64 => ~ ! [P8: A > A > $o,X5: A,Y4: A,Xs3: list @ A] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % successively.cases % 0.31/0.64 thf(fact_7161_arg__min__list_Ocases,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( linorder @ B ) % 0.31/0.64 => ! [X3: product_prod @ ( A > B ) @ ( list @ A )] : % 0.31/0.64 ( ! [F4: A > B,X5: A] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) % 0.31/0.64 => ( ! [F4: A > B,X5: A,Y4: A,Zs: list @ A] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Zs ) ) ) ) % 0.31/0.64 => ~ ! [A5: A > B] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( product_Pair @ ( A > B ) @ ( list @ A ) @ A5 @ ( nil @ A ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % arg_min_list.cases % 0.31/0.64 thf(fact_7162_sorted__wrt_Ocases,axiom, % 0.31/0.64 ! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( list @ A )] : % 0.31/0.64 ( ! [P8: A > A > $o] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( nil @ A ) ) ) % 0.31/0.64 => ~ ! [P8: A > A > $o,X5: A,Ys4: list @ A] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X5 @ Ys4 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % sorted_wrt.cases % 0.31/0.64 thf(fact_7163_splice_Ocases,axiom, % 0.31/0.64 ! [A: $tType,X3: product_prod @ ( list @ A ) @ ( list @ A )] : % 0.31/0.64 ( ! [Ys4: list @ A] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) ) % 0.31/0.64 => ~ ! [X5: A,Xs3: list @ A,Ys4: list @ A] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Ys4 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % splice.cases % 0.31/0.64 thf(fact_7164_shuffles_Ocases,axiom, % 0.31/0.64 ! [A: $tType,X3: product_prod @ ( list @ A ) @ ( list @ A )] : % 0.31/0.64 ( ! [Ys4: list @ A] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) ) % 0.31/0.64 => ( ! [Xs3: list @ A] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ ( nil @ A ) ) ) % 0.31/0.64 => ~ ! [X5: A,Xs3: list @ A,Y4: A,Ys4: list @ A] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ A @ Y4 @ Ys4 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % shuffles.cases % 0.31/0.64 thf(fact_7165_map__tailrec__rev_Ocases,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,X3: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] : % 0.31/0.64 ( ! [F4: A > B,Bs2: list @ B] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) ) % 0.31/0.64 => ~ ! [F4: A > B,A5: A,As2: list @ A,Bs2: list @ B] : % 0.31/0.64 ( X3 % 0.31/0.64 != ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As2 ) @ Bs2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_tailrec_rev.cases % 0.31/0.64 thf(fact_7166_listrel1I2,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),X3: A] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ X3 @ Ys ) ) @ ( listrel1 @ A @ R2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % listrel1I2 % 0.31/0.64 thf(fact_7167_inj__on__Cons1,axiom, % 0.31/0.64 ! [A: $tType,X3: A,A3: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ A3 ) ). % 0.31/0.64 % 0.31/0.64 % inj_on_Cons1 % 0.31/0.64 thf(fact_7168_listrelp_Ocases,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,R2: A > B > $o,A12: list @ A,A23: list @ B] : % 0.31/0.64 ( ( listrelp @ A @ B @ R2 @ A12 @ A23 ) % 0.31/0.64 => ( ( ( A12 % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 => ( A23 % 0.31/0.64 != ( nil @ B ) ) ) % 0.31/0.64 => ~ ! [X5: A,Y4: B,Xs3: list @ A] : % 0.31/0.64 ( ( A12 % 0.31/0.64 = ( cons @ A @ X5 @ Xs3 ) ) % 0.31/0.64 => ! [Ys4: list @ B] : % 0.31/0.64 ( ( A23 % 0.31/0.64 = ( cons @ B @ Y4 @ Ys4 ) ) % 0.31/0.64 => ( ( R2 @ X5 @ Y4 ) % 0.31/0.64 => ~ ( listrelp @ A @ B @ R2 @ Xs3 @ Ys4 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % listrelp.cases % 0.31/0.64 thf(fact_7169_listrelp_Osimps,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( listrelp @ A @ B ) % 0.31/0.64 = ( ^ [R5: A > B > $o,A1: list @ A,A22: list @ B] : % 0.31/0.64 ( ( ( A1 % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 & ( A22 % 0.31/0.64 = ( nil @ B ) ) ) % 0.31/0.64 | ? [X: A,Y2: B,Xs2: list @ A,Ys3: list @ B] : % 0.31/0.64 ( ( A1 % 0.31/0.64 = ( cons @ A @ X @ Xs2 ) ) % 0.31/0.64 & ( A22 % 0.31/0.64 = ( cons @ B @ Y2 @ Ys3 ) ) % 0.31/0.64 & ( R5 @ X @ Y2 ) % 0.31/0.64 & ( listrelp @ A @ B @ R5 @ Xs2 @ Ys3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % listrelp.simps % 0.31/0.64 thf(fact_7170_set__subset__Cons,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,X3: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X3 @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % set_subset_Cons % 0.31/0.64 thf(fact_7171_impossible__Cons,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A,X3: A] : % 0.31/0.64 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) % 0.31/0.64 => ( Xs % 0.31/0.64 != ( cons @ A @ X3 @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % impossible_Cons % 0.31/0.64 thf(fact_7172_insort__key_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,B: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [F2: B > A,X3: B,Y: B,Ys: list @ B] : % 0.31/0.64 ( ( ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y ) ) % 0.31/0.64 => ( ( linorder_insort_key @ B @ A @ F2 @ X3 @ ( cons @ B @ Y @ Ys ) ) % 0.31/0.64 = ( cons @ B @ X3 @ ( cons @ B @ Y @ Ys ) ) ) ) % 0.31/0.64 & ( ~ ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y ) ) % 0.31/0.64 => ( ( linorder_insort_key @ B @ A @ F2 @ X3 @ ( cons @ B @ Y @ Ys ) ) % 0.31/0.64 = ( cons @ B @ Y @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Ys ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % insort_key.simps(2) % 0.31/0.64 thf(fact_7173_Cons__in__subseqsD,axiom, % 0.31/0.64 ! [A: $tType,Y: A,Ys: list @ A,Xs: list @ A] : % 0.31/0.64 ( ( member @ ( list @ A ) @ ( cons @ A @ Y @ Ys ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) % 0.31/0.64 => ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Cons_in_subseqsD % 0.31/0.64 thf(fact_7174_inj__split__Cons,axiom, % 0.31/0.64 ! [A: $tType,X8: set @ ( product_prod @ ( list @ A ) @ A )] : % 0.31/0.64 ( inj_on @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A ) % 0.31/0.64 @ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A ) % 0.31/0.64 @ ^ [Xs2: list @ A,N: A] : ( cons @ A @ N @ Xs2 ) ) % 0.31/0.64 @ X8 ) ). % 0.31/0.64 % 0.31/0.64 % inj_split_Cons % 0.31/0.64 thf(fact_7175_ord_Olexordp_Omono,axiom, % 0.31/0.64 ! [A: $tType,Less: A > A > $o] : % 0.31/0.64 ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o ) % 0.31/0.64 @ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X15: list @ A,X24: list @ A] : % 0.31/0.64 ( ? [Y2: A,Ys3: list @ A] : % 0.31/0.64 ( ( X15 % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 & ( X24 % 0.31/0.64 = ( cons @ A @ Y2 @ Ys3 ) ) ) % 0.31/0.64 | ? [X: A,Y2: A,Xs2: list @ A,Ys3: list @ A] : % 0.31/0.64 ( ( X15 % 0.31/0.64 = ( cons @ A @ X @ Xs2 ) ) % 0.31/0.64 & ( X24 % 0.31/0.64 = ( cons @ A @ Y2 @ Ys3 ) ) % 0.31/0.64 & ( Less @ X @ Y2 ) ) % 0.31/0.64 | ? [X: A,Y2: A,Xs2: list @ A,Ys3: list @ A] : % 0.31/0.64 ( ( X15 % 0.31/0.64 = ( cons @ A @ X @ Xs2 ) ) % 0.31/0.64 & ( X24 % 0.31/0.64 = ( cons @ A @ Y2 @ Ys3 ) ) % 0.31/0.64 & ~ ( Less @ X @ Y2 ) % 0.31/0.64 & ~ ( Less @ Y2 @ X ) % 0.31/0.64 & ( P6 @ Xs2 @ Ys3 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % ord.lexordp.mono % 0.31/0.64 thf(fact_7176_arg__min__list_Osimps_I2_J,axiom, % 0.31/0.64 ! [B: $tType,A: $tType] : % 0.31/0.64 ( ( linorder @ B ) % 0.31/0.64 => ! [F2: A > B,X3: A,Y: A,Zs2: list @ A] : % 0.31/0.64 ( ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Zs2 ) ) ) % 0.31/0.64 = ( if @ A @ ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ Y @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ Y @ Zs2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % arg_min_list.simps(2) % 0.31/0.64 thf(fact_7177_Suc__le__length__iff,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 = ( ? [X: A,Ys3: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( cons @ A @ X @ Ys3 ) ) % 0.31/0.64 & ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Suc_le_length_iff % 0.31/0.64 thf(fact_7178_distinct__set__subseqs,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A] : % 0.31/0.64 ( ( distinct @ A @ Xs ) % 0.31/0.64 => ( distinct @ ( set @ A ) @ ( map @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % distinct_set_subseqs % 0.31/0.64 thf(fact_7179_insort__is__Cons,axiom, % 0.31/0.64 ! [A: $tType,B: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [Xs: list @ B,F2: B > A,A2: B] : % 0.31/0.64 ( ! [X5: B] : % 0.31/0.64 ( ( member @ B @ X5 @ ( set2 @ B @ Xs ) ) % 0.31/0.64 => ( ord_less_eq @ A @ ( F2 @ A2 ) @ ( F2 @ X5 ) ) ) % 0.31/0.64 => ( ( linorder_insort_key @ B @ A @ F2 @ A2 @ Xs ) % 0.31/0.64 = ( cons @ B @ A2 @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % insort_is_Cons % 0.31/0.64 thf(fact_7180_map__of__Cons__code_I2_J,axiom, % 0.31/0.64 ! [C: $tType,B: $tType,L: B,K: B,V2: C,Ps: list @ ( product_prod @ B @ C )] : % 0.31/0.64 ( ( ( L = K ) % 0.31/0.64 => ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V2 ) @ Ps ) @ K ) % 0.31/0.64 = ( some @ C @ V2 ) ) ) % 0.31/0.64 & ( ( L != K ) % 0.31/0.64 => ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V2 ) @ Ps ) @ K ) % 0.31/0.64 = ( map_of @ B @ C @ Ps @ K ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_Cons_code(2) % 0.31/0.64 thf(fact_7181_listrel1I1,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % listrel1I1 % 0.31/0.64 thf(fact_7182_Cons__listrel1E1,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ Ys ) @ ( listrel1 @ A @ R2 ) ) % 0.31/0.64 => ( ! [Y4: A] : % 0.31/0.64 ( ( Ys % 0.31/0.64 = ( cons @ A @ Y4 @ Xs ) ) % 0.31/0.64 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R2 ) ) % 0.31/0.64 => ~ ! [Zs: list @ A] : % 0.31/0.64 ( ( Ys % 0.31/0.64 = ( cons @ A @ X3 @ Zs ) ) % 0.31/0.64 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Cons_listrel1E1 % 0.31/0.64 thf(fact_7183_Cons__listrel1E2,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R2 ) ) % 0.31/0.64 => ( ! [X5: A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( cons @ A @ X5 @ Ys ) ) % 0.31/0.64 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y ) @ R2 ) ) % 0.31/0.64 => ~ ! [Zs: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( cons @ A @ Y @ Zs ) ) % 0.31/0.64 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Cons_listrel1E2 % 0.31/0.64 thf(fact_7184_listrel_OCons,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,X3: A,Y: B,R2: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys: list @ B] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R2 ) % 0.31/0.64 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % listrel.Cons % 0.31/0.64 thf(fact_7185_listrel__Cons1,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Y: A,Ys: list @ A,Xs: list @ B,R2: set @ ( product_prod @ A @ B )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys ) @ Xs ) @ ( listrel @ A @ B @ R2 ) ) % 0.31/0.64 => ~ ! [Y4: B,Ys4: list @ B] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( cons @ B @ Y4 @ Ys4 ) ) % 0.31/0.64 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y4 ) @ R2 ) % 0.31/0.64 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys @ Ys4 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % listrel_Cons1 % 0.31/0.64 thf(fact_7186_listrel__Cons2,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R2 ) ) % 0.31/0.64 => ~ ! [X5: A,Xs3: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( cons @ A @ X5 @ Xs3 ) ) % 0.31/0.64 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y ) @ R2 ) % 0.31/0.64 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % listrel_Cons2 % 0.31/0.64 thf(fact_7187_count__list_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Y: A,Xs: list @ A] : % 0.31/0.64 ( ( ( X3 = Y ) % 0.31/0.64 => ( ( count_list @ A @ ( cons @ A @ X3 @ Xs ) @ Y ) % 0.31/0.64 = ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y ) @ ( one_one @ nat ) ) ) ) % 0.31/0.64 & ( ( X3 != Y ) % 0.31/0.64 => ( ( count_list @ A @ ( cons @ A @ X3 @ Xs ) @ Y ) % 0.31/0.64 = ( count_list @ A @ Xs @ Y ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % count_list.simps(2) % 0.31/0.64 thf(fact_7188_the__elem__set,axiom, % 0.31/0.64 ! [A: $tType,X3: A] : % 0.31/0.64 ( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) % 0.31/0.64 = X3 ) ). % 0.31/0.64 % 0.31/0.64 % the_elem_set % 0.31/0.64 thf(fact_7189_rtrancl__listrel1__ConsI1,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),X3: A] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ X3 @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % rtrancl_listrel1_ConsI1 % 0.31/0.64 thf(fact_7190_lexordp_Omono,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( ord @ A ) % 0.31/0.64 => ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o ) % 0.31/0.64 @ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X15: list @ A,X24: list @ A] : % 0.31/0.64 ( ? [Y2: A,Ys3: list @ A] : % 0.31/0.64 ( ( X15 % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 & ( X24 % 0.31/0.64 = ( cons @ A @ Y2 @ Ys3 ) ) ) % 0.31/0.64 | ? [X: A,Y2: A,Xs2: list @ A,Ys3: list @ A] : % 0.31/0.64 ( ( X15 % 0.31/0.64 = ( cons @ A @ X @ Xs2 ) ) % 0.31/0.64 & ( X24 % 0.31/0.64 = ( cons @ A @ Y2 @ Ys3 ) ) % 0.31/0.64 & ( ord_less @ A @ X @ Y2 ) ) % 0.31/0.64 | ? [X: A,Y2: A,Xs2: list @ A,Ys3: list @ A] : % 0.31/0.64 ( ( X15 % 0.31/0.64 = ( cons @ A @ X @ Xs2 ) ) % 0.31/0.64 & ( X24 % 0.31/0.64 = ( cons @ A @ Y2 @ Ys3 ) ) % 0.31/0.64 & ~ ( ord_less @ A @ X @ Y2 ) % 0.31/0.64 & ~ ( ord_less @ A @ Y2 @ X ) % 0.31/0.64 & ( P6 @ Xs2 @ Ys3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexordp.mono % 0.31/0.64 thf(fact_7191_zip__replicate1,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,N2: nat,X3: A,Ys: list @ B] : % 0.31/0.64 ( ( zip @ A @ B @ ( replicate @ A @ N2 @ X3 ) @ Ys ) % 0.31/0.64 = ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ ( take @ B @ N2 @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % zip_replicate1 % 0.31/0.64 thf(fact_7192_list_Osize_I4_J,axiom, % 0.31/0.64 ! [A: $tType,X21: A,X222: list @ A] : % 0.31/0.64 ( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X222 ) ) % 0.31/0.64 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X222 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.size(4) % 0.31/0.64 thf(fact_7193_nth__Cons_H,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( ( N2 % 0.31/0.64 = ( zero_zero @ nat ) ) % 0.31/0.64 => ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N2 ) % 0.31/0.64 = X3 ) ) % 0.31/0.64 & ( ( N2 % 0.31/0.64 != ( zero_zero @ nat ) ) % 0.31/0.64 => ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N2 ) % 0.31/0.64 = ( nth @ A @ Xs @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_Cons' % 0.31/0.64 thf(fact_7194_zip__replicate2,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ A,N2: nat,Y: B] : % 0.31/0.64 ( ( zip @ A @ B @ Xs @ ( replicate @ B @ N2 @ Y ) ) % 0.31/0.64 = ( map @ A @ ( product_prod @ A @ B ) % 0.31/0.64 @ ^ [X: A] : ( product_Pair @ A @ B @ X @ Y ) % 0.31/0.64 @ ( take @ A @ N2 @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % zip_replicate2 % 0.31/0.64 thf(fact_7195_listrel_Ocases,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,A12: list @ A,A23: list @ B,R2: set @ ( product_prod @ A @ B )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A12 @ A23 ) @ ( listrel @ A @ B @ R2 ) ) % 0.31/0.64 => ( ( ( A12 % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 => ( A23 % 0.31/0.64 != ( nil @ B ) ) ) % 0.31/0.64 => ~ ! [X5: A,Y4: B,Xs3: list @ A] : % 0.31/0.64 ( ( A12 % 0.31/0.64 = ( cons @ A @ X5 @ Xs3 ) ) % 0.31/0.64 => ! [Ys4: list @ B] : % 0.31/0.64 ( ( A23 % 0.31/0.64 = ( cons @ B @ Y4 @ Ys4 ) ) % 0.31/0.64 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ R2 ) % 0.31/0.64 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys4 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % listrel.cases % 0.31/0.64 thf(fact_7196_listrel_Osimps,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,A12: list @ A,A23: list @ B,R2: set @ ( product_prod @ A @ B )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A12 @ A23 ) @ ( listrel @ A @ B @ R2 ) ) % 0.31/0.64 = ( ( ( A12 % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 & ( A23 % 0.31/0.64 = ( nil @ B ) ) ) % 0.31/0.64 | ? [X: A,Y2: B,Xs2: list @ A,Ys3: list @ B] : % 0.31/0.64 ( ( A12 % 0.31/0.64 = ( cons @ A @ X @ Xs2 ) ) % 0.31/0.64 & ( A23 % 0.31/0.64 = ( cons @ B @ Y2 @ Ys3 ) ) % 0.31/0.64 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R2 ) % 0.31/0.64 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys3 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % listrel.simps % 0.31/0.64 thf(fact_7197_remdups__adj__replicate,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,X3: A] : % 0.31/0.64 ( ( ( N2 % 0.31/0.64 = ( zero_zero @ nat ) ) % 0.31/0.64 => ( ( remdups_adj @ A @ ( replicate @ A @ N2 @ X3 ) ) % 0.31/0.64 = ( nil @ A ) ) ) % 0.31/0.64 & ( ( N2 % 0.31/0.64 != ( zero_zero @ nat ) ) % 0.31/0.64 => ( ( remdups_adj @ A @ ( replicate @ A @ N2 @ X3 ) ) % 0.31/0.64 = ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remdups_adj_replicate % 0.31/0.64 thf(fact_7198_remdups__adj__singleton,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,X3: A] : % 0.31/0.64 ( ( ( remdups_adj @ A @ Xs ) % 0.31/0.64 = ( cons @ A @ X3 @ ( nil @ A ) ) ) % 0.31/0.64 => ( Xs % 0.31/0.64 = ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remdups_adj_singleton % 0.31/0.64 thf(fact_7199_Id__on__set,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A] : % 0.31/0.64 ( ( id_on @ A @ ( set2 @ A @ Xs ) ) % 0.31/0.64 = ( set2 @ ( product_prod @ A @ A ) % 0.31/0.64 @ ( map @ A @ ( product_prod @ A @ A ) % 0.31/0.64 @ ^ [X: A] : ( product_Pair @ A @ A @ X @ X ) % 0.31/0.64 @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Id_on_set % 0.31/0.64 thf(fact_7200_list_Osize__gen_I2_J,axiom, % 0.31/0.64 ! [A: $tType,X3: A > nat,X21: A,X222: list @ A] : % 0.31/0.64 ( ( size_list @ A @ X3 @ ( cons @ A @ X21 @ X222 ) ) % 0.31/0.64 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( X3 @ X21 ) @ ( size_list @ A @ X3 @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list.size_gen(2) % 0.31/0.64 thf(fact_7201_shuffles_Oelims,axiom, % 0.31/0.64 ! [A: $tType,X3: list @ A,Xa2: list @ A,Y: set @ ( list @ A )] : % 0.31/0.64 ( ( ( shuffles @ A @ X3 @ Xa2 ) % 0.31/0.64 = Y ) % 0.31/0.64 => ( ( ( X3 % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 => ( Y % 0.31/0.64 != ( insert @ ( list @ A ) @ Xa2 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) % 0.31/0.64 => ( ( ( Xa2 % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 => ( Y % 0.31/0.64 != ( insert @ ( list @ A ) @ X3 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) % 0.31/0.64 => ~ ! [X5: A,Xs3: list @ A] : % 0.31/0.64 ( ( X3 % 0.31/0.64 = ( cons @ A @ X5 @ Xs3 ) ) % 0.31/0.64 => ! [Y4: A,Ys4: list @ A] : % 0.31/0.64 ( ( Xa2 % 0.31/0.64 = ( cons @ A @ Y4 @ Ys4 ) ) % 0.31/0.64 => ( Y % 0.31/0.64 != ( sup_sup @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 ) @ ( shuffles @ A @ Xs3 @ ( cons @ A @ Y4 @ Ys4 ) ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y4 ) @ ( shuffles @ A @ ( cons @ A @ X5 @ Xs3 ) @ Ys4 ) ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % shuffles.elims % 0.31/0.64 thf(fact_7202_nth__equal__first__eq,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,N2: nat] : % 0.31/0.64 ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ( ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N2 ) % 0.31/0.64 = X3 ) % 0.31/0.64 = ( N2 % 0.31/0.64 = ( zero_zero @ nat ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_equal_first_eq % 0.31/0.64 thf(fact_7203_nth__non__equal__first__eq,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Y: A,Xs: list @ A,N2: nat] : % 0.31/0.64 ( ( X3 != Y ) % 0.31/0.64 => ( ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N2 ) % 0.31/0.64 = Y ) % 0.31/0.64 = ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) % 0.31/0.64 = Y ) % 0.31/0.64 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_non_equal_first_eq % 0.31/0.64 thf(fact_7204_take__Cons_H,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( ( N2 % 0.31/0.64 = ( zero_zero @ nat ) ) % 0.31/0.64 => ( ( take @ A @ N2 @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( nil @ A ) ) ) % 0.31/0.64 & ( ( N2 % 0.31/0.64 != ( zero_zero @ nat ) ) % 0.31/0.64 => ( ( take @ A @ N2 @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( cons @ A @ X3 @ ( take @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ Xs ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % take_Cons' % 0.31/0.64 thf(fact_7205_Cons__replicate__eq,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,N2: nat,Y: A] : % 0.31/0.64 ( ( ( cons @ A @ X3 @ Xs ) % 0.31/0.64 = ( replicate @ A @ N2 @ Y ) ) % 0.31/0.64 = ( ( X3 = Y ) % 0.31/0.64 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.64 & ( Xs % 0.31/0.64 = ( replicate @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ X3 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Cons_replicate_eq % 0.31/0.64 thf(fact_7206_map__of__map__restrict,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,F2: A > B,Ks2: list @ A] : % 0.31/0.64 ( ( map_of @ A @ B % 0.31/0.64 @ ( map @ A @ ( product_prod @ A @ B ) % 0.31/0.64 @ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( F2 @ K3 ) ) % 0.31/0.64 @ Ks2 ) ) % 0.31/0.64 = ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F2 ) @ ( set2 @ A @ Ks2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_map_restrict % 0.31/0.64 thf(fact_7207_map__of__map__keys,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ A,M: A > ( option @ B )] : % 0.31/0.64 ( ( ( set2 @ A @ Xs ) % 0.31/0.64 = ( dom @ A @ B @ M ) ) % 0.31/0.64 => ( ( map_of @ A @ B % 0.31/0.64 @ ( map @ A @ ( product_prod @ A @ B ) % 0.31/0.64 @ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( the2 @ B @ ( M @ K3 ) ) ) % 0.31/0.64 @ Xs ) ) % 0.31/0.64 = M ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of_map_keys % 0.31/0.64 thf(fact_7208_Cons__lenlex__iff,axiom, % 0.31/0.64 ! [A: $tType,M: A,Ms: list @ A,N2: A,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M @ Ms ) @ ( cons @ A @ N2 @ Ns ) ) @ ( lenlex @ A @ R2 ) ) % 0.31/0.64 = ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) % 0.31/0.64 | ( ( ( size_size @ ( list @ A ) @ Ms ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ Ns ) ) % 0.31/0.64 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ N2 ) @ R2 ) ) % 0.31/0.64 | ( ( M = N2 ) % 0.31/0.64 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Cons_lenlex_iff % 0.31/0.64 thf(fact_7209_map__of_Osimps_I2_J,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,P4: product_prod @ A @ B,Ps: list @ ( product_prod @ A @ B )] : % 0.31/0.64 ( ( map_of @ A @ B @ ( cons @ ( product_prod @ A @ B ) @ P4 @ Ps ) ) % 0.31/0.64 = ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Ps ) @ ( product_fst @ A @ B @ P4 ) @ ( some @ B @ ( product_snd @ A @ B @ P4 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_of.simps(2) % 0.31/0.64 thf(fact_7210_rtrancl__listrel1__ConsI2,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.64 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % rtrancl_listrel1_ConsI2 % 0.31/0.64 thf(fact_7211_Pow__set_I2_J,axiom, % 0.31/0.64 ! [B: $tType,X3: B,Xs: list @ B] : % 0.31/0.64 ( ( pow2 @ B @ ( set2 @ B @ ( cons @ B @ X3 @ Xs ) ) ) % 0.31/0.64 = ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) @ ( image @ ( set @ B ) @ ( set @ B ) @ ( insert @ B @ X3 ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Pow_set(2) % 0.31/0.64 thf(fact_7212_sorted__list__of__set__nonempty,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( linorder @ A ) % 0.31/0.64 => ! [A3: set @ A] : % 0.31/0.64 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.64 => ( ( A3 % 0.31/0.64 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.64 => ( ( linord4507533701916653071of_set @ A @ A3 ) % 0.31/0.64 = ( cons @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % sorted_list_of_set_nonempty % 0.31/0.64 thf(fact_7213_product__code,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] : % 0.31/0.64 ( ( product_product @ A @ B @ ( set2 @ A @ Xs ) @ ( set2 @ B @ Ys ) ) % 0.31/0.64 = ( set2 @ ( product_prod @ A @ B ) % 0.31/0.64 @ ( concat @ ( product_prod @ A @ B ) % 0.31/0.64 @ ( map @ A @ ( list @ ( product_prod @ A @ B ) ) % 0.31/0.64 @ ^ [X: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys ) % 0.31/0.64 @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % product_code % 0.31/0.64 thf(fact_7214_set__Cons__sing__Nil,axiom, % 0.31/0.64 ! [A: $tType,A3: set @ A] : % 0.31/0.64 ( ( set_Cons @ A @ A3 @ ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) % 0.31/0.64 = ( image @ A @ ( list @ A ) % 0.31/0.64 @ ^ [X: A] : ( cons @ A @ X @ ( nil @ A ) ) % 0.31/0.64 @ A3 ) ) ). % 0.31/0.64 % 0.31/0.64 % set_Cons_sing_Nil % 0.31/0.64 thf(fact_7215_n__lists__Nil,axiom, % 0.31/0.64 ! [A: $tType,N2: nat] : % 0.31/0.64 ( ( ( N2 % 0.31/0.64 = ( zero_zero @ nat ) ) % 0.31/0.64 => ( ( n_lists @ A @ N2 @ ( nil @ A ) ) % 0.31/0.64 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ) % 0.31/0.64 & ( ( N2 % 0.31/0.64 != ( zero_zero @ nat ) ) % 0.31/0.64 => ( ( n_lists @ A @ N2 @ ( nil @ A ) ) % 0.31/0.64 = ( nil @ ( list @ A ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % n_lists_Nil % 0.31/0.64 thf(fact_7216_listset_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,A3: set @ A,As3: list @ ( set @ A )] : % 0.31/0.64 ( ( listset @ A @ ( cons @ ( set @ A ) @ A3 @ As3 ) ) % 0.31/0.64 = ( set_Cons @ A @ A3 @ ( listset @ A @ As3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % listset.simps(2) % 0.31/0.64 thf(fact_7217_subseqs_Osimps_I1_J,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( subseqs @ A @ ( nil @ A ) ) % 0.31/0.64 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % subseqs.simps(1) % 0.31/0.64 thf(fact_7218_product__lists_Osimps_I1_J,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( product_lists @ A @ ( nil @ ( list @ A ) ) ) % 0.31/0.64 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % product_lists.simps(1) % 0.31/0.64 thf(fact_7219_n__lists_Osimps_I1_J,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A] : % 0.31/0.64 ( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs ) % 0.31/0.64 = ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % n_lists.simps(1) % 0.31/0.64 thf(fact_7220_sorted__list__of__set__greaterThanAtMost,axiom, % 0.31/0.64 ! [I: nat,J: nat] : % 0.31/0.64 ( ( ord_less_eq @ nat @ ( suc @ I ) @ J ) % 0.31/0.64 => ( ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) ) % 0.31/0.64 = ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ ( suc @ I ) @ J ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % sorted_list_of_set_greaterThanAtMost % 0.31/0.64 thf(fact_7221_sorted__list__of__set__greaterThanLessThan,axiom, % 0.31/0.64 ! [I: nat,J: nat] : % 0.31/0.64 ( ( ord_less @ nat @ ( suc @ I ) @ J ) % 0.31/0.64 => ( ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) ) % 0.31/0.64 = ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ ( suc @ I ) @ J ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % sorted_list_of_set_greaterThanLessThan % 0.31/0.64 thf(fact_7222_set__Cons__def,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( set_Cons @ A ) % 0.31/0.64 = ( ^ [A6: set @ A,XS2: set @ ( list @ A )] : % 0.31/0.64 ( collect @ ( list @ A ) % 0.31/0.64 @ ^ [Z6: list @ A] : % 0.31/0.64 ? [X: A,Xs2: list @ A] : % 0.31/0.64 ( ( Z6 % 0.31/0.64 = ( cons @ A @ X @ Xs2 ) ) % 0.31/0.64 & ( member @ A @ X @ A6 ) % 0.31/0.64 & ( member @ ( list @ A ) @ Xs2 @ XS2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % set_Cons_def % 0.31/0.64 thf(fact_7223_concat__inth,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A] : % 0.31/0.64 ( ( nth @ A @ ( append @ A @ Xs @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 = X3 ) ). % 0.31/0.64 % 0.31/0.64 % concat_inth % 0.31/0.64 thf(fact_7224_map__upds__append1,axiom, % 0.31/0.64 ! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,M: A > ( option @ B ),X3: A] : % 0.31/0.64 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( map_upds @ A @ B @ M @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ Ys ) % 0.31/0.64 = ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ Xs @ Ys ) @ X3 @ ( some @ B @ ( nth @ B @ Ys @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_upds_append1 % 0.31/0.64 thf(fact_7225_append_Oassoc,axiom, % 0.31/0.64 ! [A: $tType,A2: list @ A,B2: list @ A,C2: list @ A] : % 0.31/0.64 ( ( append @ A @ ( append @ A @ A2 @ B2 ) @ C2 ) % 0.31/0.64 = ( append @ A @ A2 @ ( append @ A @ B2 @ C2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % append.assoc % 0.31/0.64 thf(fact_7226_append__assoc,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] : % 0.31/0.64 ( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs2 ) % 0.31/0.64 = ( append @ A @ Xs @ ( append @ A @ Ys @ Zs2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % append_assoc % 0.31/0.64 thf(fact_7227_append__same__eq,axiom, % 0.31/0.64 ! [A: $tType,Ys: list @ A,Xs: list @ A,Zs2: list @ A] : % 0.31/0.64 ( ( ( append @ A @ Ys @ Xs ) % 0.31/0.64 = ( append @ A @ Zs2 @ Xs ) ) % 0.31/0.64 = ( Ys = Zs2 ) ) ). % 0.31/0.64 % 0.31/0.64 % append_same_eq % 0.31/0.64 thf(fact_7228_same__append__eq,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] : % 0.31/0.64 ( ( ( append @ A @ Xs @ Ys ) % 0.31/0.64 = ( append @ A @ Xs @ Zs2 ) ) % 0.31/0.64 = ( Ys = Zs2 ) ) ). % 0.31/0.64 % 0.31/0.64 % same_append_eq % 0.31/0.64 thf(fact_7229_append__is__Nil__conv,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( ( append @ A @ Xs @ Ys ) % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 = ( ( Xs % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 & ( Ys % 0.31/0.64 = ( nil @ A ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % append_is_Nil_conv % 0.31/0.64 thf(fact_7230_Nil__is__append__conv,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( ( nil @ A ) % 0.31/0.64 = ( append @ A @ Xs @ Ys ) ) % 0.31/0.64 = ( ( Xs % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 & ( Ys % 0.31/0.64 = ( nil @ A ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Nil_is_append_conv % 0.31/0.64 thf(fact_7231_self__append__conv2,axiom, % 0.31/0.64 ! [A: $tType,Y: list @ A,Xs: list @ A] : % 0.31/0.64 ( ( Y % 0.31/0.64 = ( append @ A @ Xs @ Y ) ) % 0.31/0.64 = ( Xs % 0.31/0.64 = ( nil @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % self_append_conv2 % 0.31/0.64 thf(fact_7232_append__self__conv2,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( ( append @ A @ Xs @ Ys ) % 0.31/0.64 = Ys ) % 0.31/0.64 = ( Xs % 0.31/0.64 = ( nil @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % append_self_conv2 % 0.31/0.64 thf(fact_7233_self__append__conv,axiom, % 0.31/0.64 ! [A: $tType,Y: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( Y % 0.31/0.64 = ( append @ A @ Y @ Ys ) ) % 0.31/0.64 = ( Ys % 0.31/0.64 = ( nil @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % self_append_conv % 0.31/0.64 thf(fact_7234_append__self__conv,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( ( append @ A @ Xs @ Ys ) % 0.31/0.64 = Xs ) % 0.31/0.64 = ( Ys % 0.31/0.64 = ( nil @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % append_self_conv % 0.31/0.64 thf(fact_7235_append__Nil2,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A] : % 0.31/0.64 ( ( append @ A @ Xs @ ( nil @ A ) ) % 0.31/0.64 = Xs ) ). % 0.31/0.64 % 0.31/0.64 % append_Nil2 % 0.31/0.64 thf(fact_7236_append_Oright__neutral,axiom, % 0.31/0.64 ! [A: $tType,A2: list @ A] : % 0.31/0.64 ( ( append @ A @ A2 @ ( nil @ A ) ) % 0.31/0.64 = A2 ) ). % 0.31/0.64 % 0.31/0.64 % append.right_neutral % 0.31/0.64 thf(fact_7237_append__eq__append__conv,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A,Us: list @ A,Vs2: list @ A] : % 0.31/0.64 ( ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ Ys ) ) % 0.31/0.64 | ( ( size_size @ ( list @ A ) @ Us ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ Vs2 ) ) ) % 0.31/0.64 => ( ( ( append @ A @ Xs @ Us ) % 0.31/0.64 = ( append @ A @ Ys @ Vs2 ) ) % 0.31/0.64 = ( ( Xs = Ys ) % 0.31/0.64 & ( Us = Vs2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % append_eq_append_conv % 0.31/0.64 thf(fact_7238_map__append,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ B] : % 0.31/0.64 ( ( map @ B @ A @ F2 @ ( append @ B @ Xs @ Ys ) ) % 0.31/0.64 = ( append @ A @ ( map @ B @ A @ F2 @ Xs ) @ ( map @ B @ A @ F2 @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_append % 0.31/0.64 thf(fact_7239_concat__append,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] : % 0.31/0.64 ( ( concat @ A @ ( append @ ( list @ A ) @ Xs @ Ys ) ) % 0.31/0.64 = ( append @ A @ ( concat @ A @ Xs ) @ ( concat @ A @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % concat_append % 0.31/0.64 thf(fact_7240_removeAll__append,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( removeAll @ A @ X3 @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.64 = ( append @ A @ ( removeAll @ A @ X3 @ Xs ) @ ( removeAll @ A @ X3 @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % removeAll_append % 0.31/0.64 thf(fact_7241_foldl__append,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: A > B > A,A2: A,Xs: list @ B,Ys: list @ B] : % 0.31/0.64 ( ( foldl @ A @ B @ F2 @ A2 @ ( append @ B @ Xs @ Ys ) ) % 0.31/0.64 = ( foldl @ A @ B @ F2 @ ( foldl @ A @ B @ F2 @ A2 @ Xs ) @ Ys ) ) ). % 0.31/0.64 % 0.31/0.64 % foldl_append % 0.31/0.64 thf(fact_7242_append1__eq__conv,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A,Y: A] : % 0.31/0.64 ( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) % 0.31/0.64 = ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) % 0.31/0.64 = ( ( Xs = Ys ) % 0.31/0.64 & ( X3 = Y ) ) ) ). % 0.31/0.64 % 0.31/0.64 % append1_eq_conv % 0.31/0.64 thf(fact_7243_length__append,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.64 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % length_append % 0.31/0.64 thf(fact_7244_set__append,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( set2 @ A @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.64 = ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % set_append % 0.31/0.64 thf(fact_7245_zip__append,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xs: list @ A,Us: list @ B,Ys: list @ A,Vs2: list @ B] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Us ) ) % 0.31/0.64 => ( ( zip @ A @ B @ ( append @ A @ Xs @ Ys ) @ ( append @ B @ Us @ Vs2 ) ) % 0.31/0.64 = ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Us ) @ ( zip @ A @ B @ Ys @ Vs2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % zip_append % 0.31/0.64 thf(fact_7246_fun__upds__append__drop,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,M: A > ( option @ B ),Zs2: list @ A] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( map_upds @ A @ B @ M @ ( append @ A @ Xs @ Zs2 ) @ Ys ) % 0.31/0.64 = ( map_upds @ A @ B @ M @ Xs @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % fun_upds_append_drop % 0.31/0.64 thf(fact_7247_fun__upds__append2__drop,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,M: A > ( option @ B ),Zs2: list @ B] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.64 => ( ( map_upds @ A @ B @ M @ Xs @ ( append @ B @ Ys @ Zs2 ) ) % 0.31/0.64 = ( map_upds @ A @ B @ M @ Xs @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % fun_upds_append2_drop % 0.31/0.64 thf(fact_7248_size__list__append,axiom, % 0.31/0.64 ! [A: $tType,F2: A > nat,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( size_list @ A @ F2 @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.64 = ( plus_plus @ nat @ ( size_list @ A @ F2 @ Xs ) @ ( size_list @ A @ F2 @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % size_list_append % 0.31/0.64 thf(fact_7249_bind__simps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,X3: B,Xs: list @ B,F2: B > ( list @ A )] : % 0.31/0.64 ( ( bind @ B @ A @ ( cons @ B @ X3 @ Xs ) @ F2 ) % 0.31/0.64 = ( append @ A @ ( F2 @ X3 ) @ ( bind @ B @ A @ Xs @ F2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % bind_simps(2) % 0.31/0.64 thf(fact_7250_nth__append__length,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A] : % 0.31/0.64 ( ( nth @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 = X3 ) ). % 0.31/0.64 % 0.31/0.64 % nth_append_length % 0.31/0.64 thf(fact_7251_nth__append__length__plus,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A,N2: nat] : % 0.31/0.64 ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) % 0.31/0.64 = ( nth @ A @ Ys @ N2 ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_append_length_plus % 0.31/0.64 thf(fact_7252_take__append,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( take @ A @ N2 @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.64 = ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( take @ A @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % take_append % 0.31/0.64 thf(fact_7253_list__update__length,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A,Y: A] : % 0.31/0.64 ( ( list_update @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) @ Y ) % 0.31/0.64 = ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list_update_length % 0.31/0.64 thf(fact_7254_sorted__list__of__set__lessThan__Suc,axiom, % 0.31/0.64 ! [K: nat] : % 0.31/0.64 ( ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ ( suc @ K ) ) ) % 0.31/0.64 = ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ K ) ) @ ( cons @ nat @ K @ ( nil @ nat ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % sorted_list_of_set_lessThan_Suc % 0.31/0.64 thf(fact_7255_sorted__list__of__set__atMost__Suc,axiom, % 0.31/0.64 ! [K: nat] : % 0.31/0.64 ( ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ ( suc @ K ) ) ) % 0.31/0.64 = ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ K ) ) @ ( cons @ nat @ ( suc @ K ) @ ( nil @ nat ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % sorted_list_of_set_atMost_Suc % 0.31/0.64 thf(fact_7256_distinct__append,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( distinct @ A @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.64 = ( ( distinct @ A @ Xs ) % 0.31/0.64 & ( distinct @ A @ Ys ) % 0.31/0.64 & ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) % 0.31/0.64 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % distinct_append % 0.31/0.64 thf(fact_7257_map__eq__append__conv,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ A,Zs2: list @ A] : % 0.31/0.64 ( ( ( map @ B @ A @ F2 @ Xs ) % 0.31/0.64 = ( append @ A @ Ys @ Zs2 ) ) % 0.31/0.64 = ( ? [Us2: list @ B,Vs: list @ B] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( append @ B @ Us2 @ Vs ) ) % 0.31/0.64 & ( Ys % 0.31/0.64 = ( map @ B @ A @ F2 @ Us2 ) ) % 0.31/0.64 & ( Zs2 % 0.31/0.64 = ( map @ B @ A @ F2 @ Vs ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % map_eq_append_conv % 0.31/0.64 thf(fact_7258_append__eq__map__conv,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,Ys: list @ A,Zs2: list @ A,F2: B > A,Xs: list @ B] : % 0.31/0.64 ( ( ( append @ A @ Ys @ Zs2 ) % 0.31/0.64 = ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.64 = ( ? [Us2: list @ B,Vs: list @ B] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( append @ B @ Us2 @ Vs ) ) % 0.31/0.64 & ( Ys % 0.31/0.64 = ( map @ B @ A @ F2 @ Us2 ) ) % 0.31/0.64 & ( Zs2 % 0.31/0.64 = ( map @ B @ A @ F2 @ Vs ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % append_eq_map_conv % 0.31/0.64 thf(fact_7259_split__list,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ? [Ys4: list @ A,Zs: list @ A] : % 0.31/0.64 ( Xs % 0.31/0.64 = ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % split_list % 0.31/0.64 thf(fact_7260_split__list__last,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ? [Ys4: list @ A,Zs: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs ) ) ) % 0.31/0.64 & ~ ( member @ A @ X3 @ ( set2 @ A @ Zs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % split_list_last % 0.31/0.64 thf(fact_7261_split__list__prop,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.31/0.64 ( ? [X2: A] : % 0.31/0.64 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 & ( P @ X2 ) ) % 0.31/0.64 => ? [Ys4: list @ A,X5: A] : % 0.31/0.64 ( ? [Zs: list @ A] : % 0.31/0.64 ( Xs % 0.31/0.64 = ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) ) % 0.31/0.64 & ( P @ X5 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % split_list_prop % 0.31/0.64 thf(fact_7262_split__list__first,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ? [Ys4: list @ A,Zs: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs ) ) ) % 0.31/0.64 & ~ ( member @ A @ X3 @ ( set2 @ A @ Ys4 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % split_list_first % 0.31/0.64 thf(fact_7263_split__list__propE,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.31/0.64 ( ? [X2: A] : % 0.31/0.64 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 & ( P @ X2 ) ) % 0.31/0.64 => ~ ! [Ys4: list @ A,X5: A] : % 0.31/0.64 ( ? [Zs: list @ A] : % 0.31/0.64 ( Xs % 0.31/0.64 = ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) ) % 0.31/0.64 => ~ ( P @ X5 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % split_list_propE % 0.31/0.64 thf(fact_7264_append__Cons__eq__iff,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A,Xs5: list @ A,Ys6: list @ A] : % 0.31/0.64 ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ( ~ ( member @ A @ X3 @ ( set2 @ A @ Ys ) ) % 0.31/0.64 => ( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ Ys ) ) % 0.31/0.64 = ( append @ A @ Xs5 @ ( cons @ A @ X3 @ Ys6 ) ) ) % 0.31/0.64 = ( ( Xs = Xs5 ) % 0.31/0.64 & ( Ys = Ys6 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % append_Cons_eq_iff % 0.31/0.64 thf(fact_7265_in__set__conv__decomp,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 = ( ? [Ys3: list @ A,Zs3: list @ A] : % 0.31/0.64 ( Xs % 0.31/0.64 = ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % in_set_conv_decomp % 0.31/0.64 thf(fact_7266_split__list__last__prop,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.31/0.64 ( ? [X2: A] : % 0.31/0.64 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 & ( P @ X2 ) ) % 0.31/0.64 => ? [Ys4: list @ A,X5: A,Zs: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) ) % 0.31/0.64 & ( P @ X5 ) % 0.31/0.64 & ! [Xa: A] : % 0.31/0.64 ( ( member @ A @ Xa @ ( set2 @ A @ Zs ) ) % 0.31/0.64 => ~ ( P @ Xa ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % split_list_last_prop % 0.31/0.64 thf(fact_7267_split__list__first__prop,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.31/0.64 ( ? [X2: A] : % 0.31/0.64 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 & ( P @ X2 ) ) % 0.31/0.64 => ? [Ys4: list @ A,X5: A] : % 0.31/0.64 ( ? [Zs: list @ A] : % 0.31/0.64 ( Xs % 0.31/0.64 = ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) ) % 0.31/0.64 & ( P @ X5 ) % 0.31/0.64 & ! [Xa: A] : % 0.31/0.64 ( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) ) % 0.31/0.64 => ~ ( P @ Xa ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % split_list_first_prop % 0.31/0.64 thf(fact_7268_split__list__last__propE,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.31/0.64 ( ? [X2: A] : % 0.31/0.64 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 & ( P @ X2 ) ) % 0.31/0.64 => ~ ! [Ys4: list @ A,X5: A,Zs: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) ) % 0.31/0.64 => ( ( P @ X5 ) % 0.31/0.64 => ~ ! [Xa: A] : % 0.31/0.64 ( ( member @ A @ Xa @ ( set2 @ A @ Zs ) ) % 0.31/0.64 => ~ ( P @ Xa ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % split_list_last_propE % 0.31/0.64 thf(fact_7269_split__list__first__propE,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.31/0.64 ( ? [X2: A] : % 0.31/0.64 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 & ( P @ X2 ) ) % 0.31/0.64 => ~ ! [Ys4: list @ A,X5: A] : % 0.31/0.64 ( ? [Zs: list @ A] : % 0.31/0.64 ( Xs % 0.31/0.64 = ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs ) ) ) % 0.31/0.64 => ( ( P @ X5 ) % 0.31/0.64 => ~ ! [Xa: A] : % 0.31/0.64 ( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) ) % 0.31/0.64 => ~ ( P @ Xa ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % split_list_first_propE % 0.31/0.64 thf(fact_7270_in__set__conv__decomp__last,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 = ( ? [Ys3: list @ A,Zs3: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs3 ) ) ) % 0.31/0.64 & ~ ( member @ A @ X3 @ ( set2 @ A @ Zs3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % in_set_conv_decomp_last % 0.31/0.64 thf(fact_7271_in__set__conv__decomp__first,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 = ( ? [Ys3: list @ A,Zs3: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs3 ) ) ) % 0.31/0.64 & ~ ( member @ A @ X3 @ ( set2 @ A @ Ys3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % in_set_conv_decomp_first % 0.31/0.64 thf(fact_7272_split__list__last__prop__iff,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.31/0.64 ( ( ? [X: A] : % 0.31/0.64 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.64 & ( P @ X ) ) ) % 0.31/0.64 = ( ? [Ys3: list @ A,X: A,Zs3: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) ) % 0.31/0.64 & ( P @ X ) % 0.31/0.64 & ! [Y2: A] : % 0.31/0.64 ( ( member @ A @ Y2 @ ( set2 @ A @ Zs3 ) ) % 0.31/0.64 => ~ ( P @ Y2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % split_list_last_prop_iff % 0.31/0.64 thf(fact_7273_split__list__first__prop__iff,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.31/0.64 ( ( ? [X: A] : % 0.31/0.64 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.64 & ( P @ X ) ) ) % 0.31/0.64 = ( ? [Ys3: list @ A,X: A] : % 0.31/0.64 ( ? [Zs3: list @ A] : % 0.31/0.64 ( Xs % 0.31/0.64 = ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) ) % 0.31/0.64 & ( P @ X ) % 0.31/0.64 & ! [Y2: A] : % 0.31/0.64 ( ( member @ A @ Y2 @ ( set2 @ A @ Ys3 ) ) % 0.31/0.64 => ~ ( P @ Y2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % split_list_first_prop_iff % 0.31/0.64 thf(fact_7274_concat__eq__appendD,axiom, % 0.31/0.64 ! [A: $tType,Xss: list @ ( list @ A ),Ys: list @ A,Zs2: list @ A] : % 0.31/0.64 ( ( ( concat @ A @ Xss ) % 0.31/0.64 = ( append @ A @ Ys @ Zs2 ) ) % 0.31/0.64 => ( ( Xss % 0.31/0.64 != ( nil @ ( list @ A ) ) ) % 0.31/0.64 => ? [Xss1: list @ ( list @ A ),Xs3: list @ A,Xs4: list @ A,Xss22: list @ ( list @ A )] : % 0.31/0.64 ( ( Xss % 0.31/0.64 = ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs4 ) @ Xss22 ) ) ) % 0.31/0.64 & ( Ys % 0.31/0.64 = ( append @ A @ ( concat @ A @ Xss1 ) @ Xs3 ) ) % 0.31/0.64 & ( Zs2 % 0.31/0.64 = ( append @ A @ Xs4 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % concat_eq_appendD % 0.31/0.64 thf(fact_7275_concat__eq__append__conv,axiom, % 0.31/0.64 ! [A: $tType,Xss: list @ ( list @ A ),Ys: list @ A,Zs2: list @ A] : % 0.31/0.64 ( ( ( concat @ A @ Xss ) % 0.31/0.64 = ( append @ A @ Ys @ Zs2 ) ) % 0.31/0.64 = ( ( ( Xss % 0.31/0.64 = ( nil @ ( list @ A ) ) ) % 0.31/0.64 => ( ( Ys % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 & ( Zs2 % 0.31/0.64 = ( nil @ A ) ) ) ) % 0.31/0.64 & ( ( Xss % 0.31/0.64 != ( nil @ ( list @ A ) ) ) % 0.31/0.64 => ? [Xss12: list @ ( list @ A ),Xs2: list @ A,Xs6: list @ A,Xss23: list @ ( list @ A )] : % 0.31/0.64 ( ( Xss % 0.31/0.64 = ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs2 @ Xs6 ) @ Xss23 ) ) ) % 0.31/0.64 & ( Ys % 0.31/0.64 = ( append @ A @ ( concat @ A @ Xss12 ) @ Xs2 ) ) % 0.31/0.64 & ( Zs2 % 0.31/0.64 = ( append @ A @ Xs6 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % concat_eq_append_conv % 0.31/0.64 thf(fact_7276_rev__nonempty__induct,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] : % 0.31/0.64 ( ( Xs % 0.31/0.64 != ( nil @ A ) ) % 0.31/0.64 => ( ! [X5: A] : ( P @ ( cons @ A @ X5 @ ( nil @ A ) ) ) % 0.31/0.64 => ( ! [X5: A,Xs3: list @ A] : % 0.31/0.64 ( ( Xs3 % 0.31/0.64 != ( nil @ A ) ) % 0.31/0.64 => ( ( P @ Xs3 ) % 0.31/0.64 => ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) ) ) % 0.31/0.64 => ( P @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % rev_nonempty_induct % 0.31/0.64 thf(fact_7277_append__eq__Cons__conv,axiom, % 0.31/0.64 ! [A: $tType,Ys: list @ A,Zs2: list @ A,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( ( append @ A @ Ys @ Zs2 ) % 0.31/0.64 = ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( ( ( Ys % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 & ( Zs2 % 0.31/0.64 = ( cons @ A @ X3 @ Xs ) ) ) % 0.31/0.64 | ? [Ys7: list @ A] : % 0.31/0.64 ( ( Ys % 0.31/0.64 = ( cons @ A @ X3 @ Ys7 ) ) % 0.31/0.64 & ( ( append @ A @ Ys7 @ Zs2 ) % 0.31/0.64 = Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % append_eq_Cons_conv % 0.31/0.64 thf(fact_7278_Cons__eq__append__conv,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A,Zs2: list @ A] : % 0.31/0.64 ( ( ( cons @ A @ X3 @ Xs ) % 0.31/0.64 = ( append @ A @ Ys @ Zs2 ) ) % 0.31/0.64 = ( ( ( Ys % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 & ( ( cons @ A @ X3 @ Xs ) % 0.31/0.64 = Zs2 ) ) % 0.31/0.64 | ? [Ys7: list @ A] : % 0.31/0.64 ( ( ( cons @ A @ X3 @ Ys7 ) % 0.31/0.64 = Ys ) % 0.31/0.64 & ( Xs % 0.31/0.64 = ( append @ A @ Ys7 @ Zs2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Cons_eq_append_conv % 0.31/0.64 thf(fact_7279_rev__exhaust,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 != ( nil @ A ) ) % 0.31/0.64 => ~ ! [Ys4: list @ A,Y4: A] : % 0.31/0.64 ( Xs % 0.31/0.64 != ( append @ A @ Ys4 @ ( cons @ A @ Y4 @ ( nil @ A ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % rev_exhaust % 0.31/0.64 thf(fact_7280_rev__induct,axiom, % 0.31/0.64 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] : % 0.31/0.64 ( ( P @ ( nil @ A ) ) % 0.31/0.64 => ( ! [X5: A,Xs3: list @ A] : % 0.31/0.64 ( ( P @ Xs3 ) % 0.31/0.64 => ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) ) % 0.31/0.64 => ( P @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % rev_induct % 0.31/0.64 thf(fact_7281_append__Cons,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( append @ A @ ( cons @ A @ X3 @ Xs ) @ Ys ) % 0.31/0.64 = ( cons @ A @ X3 @ ( append @ A @ Xs @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % append_Cons % 0.31/0.64 thf(fact_7282_Cons__eq__appendI,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs2: list @ A] : % 0.31/0.64 ( ( ( cons @ A @ X3 @ Xs1 ) % 0.31/0.64 = Ys ) % 0.31/0.64 => ( ( Xs % 0.31/0.64 = ( append @ A @ Xs1 @ Zs2 ) ) % 0.31/0.64 => ( ( cons @ A @ X3 @ Xs ) % 0.31/0.64 = ( append @ A @ Ys @ Zs2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % Cons_eq_appendI % 0.31/0.64 thf(fact_7283_concat_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,X3: list @ A,Xs: list @ ( list @ A )] : % 0.31/0.64 ( ( concat @ A @ ( cons @ ( list @ A ) @ X3 @ Xs ) ) % 0.31/0.64 = ( append @ A @ X3 @ ( concat @ A @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % concat.simps(2) % 0.31/0.64 thf(fact_7284_replicate__app__Cons__same,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( append @ A @ ( replicate @ A @ N2 @ X3 ) @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( cons @ A @ X3 @ ( append @ A @ ( replicate @ A @ N2 @ X3 ) @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % replicate_app_Cons_same % 0.31/0.64 thf(fact_7285_append__listrel1I,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Us: list @ A,Vs2: list @ A] : % 0.31/0.64 ( ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) % 0.31/0.64 & ( Us = Vs2 ) ) % 0.31/0.64 | ( ( Xs = Ys ) % 0.31/0.64 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs2 ) @ ( listrel1 @ A @ R2 ) ) ) ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs2 ) ) @ ( listrel1 @ A @ R2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % append_listrel1I % 0.31/0.64 thf(fact_7286_lexord__append__leftI,axiom, % 0.31/0.64 ! [A: $tType,U: list @ A,V2: list @ A,R2: set @ ( product_prod @ A @ A ),X3: list @ A] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R2 ) ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X3 @ U ) @ ( append @ A @ X3 @ V2 ) ) @ ( lexord @ A @ R2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_append_leftI % 0.31/0.64 thf(fact_7287_lex__append__leftI,axiom, % 0.31/0.64 ! [A: $tType,Ys: list @ A,Zs2: list @ A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R2 ) ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lex_append_leftI % 0.31/0.64 thf(fact_7288_comm__append__are__replicate,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( ( append @ A @ Xs @ Ys ) % 0.31/0.64 = ( append @ A @ Ys @ Xs ) ) % 0.31/0.64 => ? [M3: nat,N3: nat,Zs: list @ A] : % 0.31/0.64 ( ( ( concat @ A @ ( replicate @ ( list @ A ) @ M3 @ Zs ) ) % 0.31/0.64 = Xs ) % 0.31/0.64 & ( ( concat @ A @ ( replicate @ ( list @ A ) @ N3 @ Zs ) ) % 0.31/0.64 = Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % comm_append_are_replicate % 0.31/0.64 thf(fact_7289_replicate__add,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,M: nat,X3: A] : % 0.31/0.64 ( ( replicate @ A @ ( plus_plus @ nat @ N2 @ M ) @ X3 ) % 0.31/0.64 = ( append @ A @ ( replicate @ A @ N2 @ X3 ) @ ( replicate @ A @ M @ X3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % replicate_add % 0.31/0.64 thf(fact_7290_eq__Nil__appendI,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( Xs = Ys ) % 0.31/0.64 => ( Xs % 0.31/0.64 = ( append @ A @ ( nil @ A ) @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % eq_Nil_appendI % 0.31/0.64 thf(fact_7291_append_Oleft__neutral,axiom, % 0.31/0.64 ! [A: $tType,A2: list @ A] : % 0.31/0.64 ( ( append @ A @ ( nil @ A ) @ A2 ) % 0.31/0.64 = A2 ) ). % 0.31/0.64 % 0.31/0.64 % append.left_neutral % 0.31/0.64 thf(fact_7292_append__Nil,axiom, % 0.31/0.64 ! [A: $tType,Ys: list @ A] : % 0.31/0.64 ( ( append @ A @ ( nil @ A ) @ Ys ) % 0.31/0.64 = Ys ) ). % 0.31/0.64 % 0.31/0.64 % append_Nil % 0.31/0.64 thf(fact_7293_append__eq__appendI,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs2: list @ A,Ys: list @ A,Us: list @ A] : % 0.31/0.64 ( ( ( append @ A @ Xs @ Xs1 ) % 0.31/0.64 = Zs2 ) % 0.31/0.64 => ( ( Ys % 0.31/0.64 = ( append @ A @ Xs1 @ Us ) ) % 0.31/0.64 => ( ( append @ A @ Xs @ Ys ) % 0.31/0.64 = ( append @ A @ Zs2 @ Us ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % append_eq_appendI % 0.31/0.64 thf(fact_7294_append__eq__append__conv2,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A,Ts: list @ A] : % 0.31/0.64 ( ( ( append @ A @ Xs @ Ys ) % 0.31/0.64 = ( append @ A @ Zs2 @ Ts ) ) % 0.31/0.64 = ( ? [Us2: list @ A] : % 0.31/0.64 ( ( ( Xs % 0.31/0.64 = ( append @ A @ Zs2 @ Us2 ) ) % 0.31/0.64 & ( ( append @ A @ Us2 @ Ys ) % 0.31/0.64 = Ts ) ) % 0.31/0.64 | ( ( ( append @ A @ Xs @ Us2 ) % 0.31/0.64 = Zs2 ) % 0.31/0.64 & ( Ys % 0.31/0.64 = ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % append_eq_append_conv2 % 0.31/0.64 thf(fact_7295_append__replicate__commute,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,X3: A,K: nat] : % 0.31/0.64 ( ( append @ A @ ( replicate @ A @ N2 @ X3 ) @ ( replicate @ A @ K @ X3 ) ) % 0.31/0.64 = ( append @ A @ ( replicate @ A @ K @ X3 ) @ ( replicate @ A @ N2 @ X3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % append_replicate_commute % 0.31/0.64 thf(fact_7296_enumerate__append__eq,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( enumerate @ A @ N2 @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.64 = ( append @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) @ ( enumerate @ A @ ( plus_plus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % enumerate_append_eq % 0.31/0.64 thf(fact_7297_remdups__append2,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( remdups @ A @ ( append @ A @ Xs @ ( remdups @ A @ Ys ) ) ) % 0.31/0.64 = ( remdups @ A @ ( append @ A @ Xs @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remdups_append2 % 0.31/0.64 thf(fact_7298_remove1__append,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ( ( remove1 @ A @ X3 @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.64 = ( append @ A @ ( remove1 @ A @ X3 @ Xs ) @ Ys ) ) ) % 0.31/0.64 & ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ( ( remove1 @ A @ X3 @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.64 = ( append @ A @ Xs @ ( remove1 @ A @ X3 @ Ys ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remove1_append % 0.31/0.64 thf(fact_7299_same__length__different,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( Xs != Ys ) % 0.31/0.64 => ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ Ys ) ) % 0.31/0.64 => ? [Pre: list @ A,X5: A,Xs4: list @ A,Y4: A,Ys5: list @ A] : % 0.31/0.64 ( ( X5 != Y4 ) % 0.31/0.64 & ( Xs % 0.31/0.64 = ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X5 @ ( nil @ A ) ) @ Xs4 ) ) ) % 0.31/0.64 & ( Ys % 0.31/0.64 = ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Ys5 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % same_length_different % 0.31/0.64 thf(fact_7300_not__distinct__decomp,axiom, % 0.31/0.64 ! [A: $tType,Ws2: list @ A] : % 0.31/0.64 ( ~ ( distinct @ A @ Ws2 ) % 0.31/0.64 => ? [Xs3: list @ A,Ys4: list @ A,Zs: list @ A,Y4: A] : % 0.31/0.64 ( Ws2 % 0.31/0.64 = ( append @ A @ Xs3 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ ( append @ A @ Ys4 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Zs ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % not_distinct_decomp % 0.31/0.64 thf(fact_7301_not__distinct__conv__prefix,axiom, % 0.31/0.64 ! [A: $tType,As: list @ A] : % 0.31/0.64 ( ( ~ ( distinct @ A @ As ) ) % 0.31/0.64 = ( ? [Xs2: list @ A,Y2: A,Ys3: list @ A] : % 0.31/0.64 ( ( member @ A @ Y2 @ ( set2 @ A @ Xs2 ) ) % 0.31/0.64 & ( distinct @ A @ Xs2 ) % 0.31/0.64 & ( As % 0.31/0.64 = ( append @ A @ Xs2 @ ( cons @ A @ Y2 @ Ys3 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % not_distinct_conv_prefix % 0.31/0.64 thf(fact_7302_replicate__append__same,axiom, % 0.31/0.64 ! [A: $tType,I: nat,X3: A] : % 0.31/0.64 ( ( append @ A @ ( replicate @ A @ I @ X3 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) % 0.31/0.64 = ( cons @ A @ X3 @ ( replicate @ A @ I @ X3 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % replicate_append_same % 0.31/0.64 thf(fact_7303_list__update__append1,axiom, % 0.31/0.64 ! [A: $tType,I: nat,Xs: list @ A,Ys: list @ A,X3: A] : % 0.31/0.64 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ I @ X3 ) % 0.31/0.64 = ( append @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list_update_append1 % 0.31/0.64 thf(fact_7304_remdups__adj__append__two,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,X3: A,Y: A] : % 0.31/0.64 ( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) % 0.31/0.64 = ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) @ ( if @ ( list @ A ) @ ( X3 = Y ) @ ( nil @ A ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remdups_adj_append_two % 0.31/0.64 thf(fact_7305_remove1__split,axiom, % 0.31/0.64 ! [A: $tType,A2: A,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( member @ A @ A2 @ ( set2 @ A @ Xs ) ) % 0.31/0.64 => ( ( ( remove1 @ A @ A2 @ Xs ) % 0.31/0.64 = Ys ) % 0.31/0.64 = ( ? [Ls: list @ A,Rs: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( append @ A @ Ls @ ( cons @ A @ A2 @ Rs ) ) ) % 0.31/0.64 & ~ ( member @ A @ A2 @ ( set2 @ A @ Ls ) ) % 0.31/0.64 & ( Ys % 0.31/0.64 = ( append @ A @ Ls @ Rs ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % remove1_split % 0.31/0.64 thf(fact_7306_lexord__append__leftD,axiom, % 0.31/0.64 ! [A: $tType,X3: list @ A,U: list @ A,V2: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X3 @ U ) @ ( append @ A @ X3 @ V2 ) ) @ ( lexord @ A @ R2 ) ) % 0.31/0.64 => ( ! [A5: A] : % 0.31/0.64 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R2 ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_append_leftD % 0.31/0.64 thf(fact_7307_lexord__append__rightI,axiom, % 0.31/0.64 ! [A: $tType,Y: list @ A,X3: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ? [B11: A,Z4: list @ A] : % 0.31/0.64 ( Y % 0.31/0.64 = ( cons @ A @ B11 @ Z4 ) ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ ( append @ A @ X3 @ Y ) ) @ ( lexord @ A @ R2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_append_rightI % 0.31/0.64 thf(fact_7308_lexord__sufE,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Zs2: list @ A,Ys: list @ A,Qs: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Zs2 ) @ ( append @ A @ Ys @ Qs ) ) @ ( lexord @ A @ R2 ) ) % 0.31/0.64 => ( ( Xs != Ys ) % 0.31/0.64 => ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ Ys ) ) % 0.31/0.64 => ( ( ( size_size @ ( list @ A ) @ Zs2 ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ Qs ) ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R2 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_sufE % 0.31/0.64 thf(fact_7309_lex__append__leftD,axiom, % 0.31/0.64 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: list @ A] : % 0.31/0.64 ( ! [X5: A] : % 0.31/0.64 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 ) % 0.31/0.64 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R2 ) ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lex_append_leftD % 0.31/0.64 thf(fact_7310_lex__append__left__iff,axiom, % 0.31/0.64 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: list @ A] : % 0.31/0.64 ( ! [X5: A] : % 0.31/0.64 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 ) % 0.31/0.64 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R2 ) ) % 0.31/0.64 = ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lex_append_left_iff % 0.31/0.64 thf(fact_7311_rotate1_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( rotate1 @ A @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % rotate1.simps(2) % 0.31/0.64 thf(fact_7312_lex__append__rightI,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Vs2: list @ A,Us: list @ A] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) ) % 0.31/0.64 => ( ( ( size_size @ ( list @ A ) @ Vs2 ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ Us ) ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs2 ) ) @ ( lex @ A @ R2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lex_append_rightI % 0.31/0.64 thf(fact_7313_lenlex__append1,axiom, % 0.31/0.64 ! [A: $tType,Us: list @ A,Xs: list @ A,R: set @ ( product_prod @ A @ A ),Vs2: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs ) @ ( lenlex @ A @ R ) ) % 0.31/0.64 => ( ( ( size_size @ ( list @ A ) @ Vs2 ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ Ys ) ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs2 ) @ ( append @ A @ Xs @ Ys ) ) @ ( lenlex @ A @ R ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lenlex_append1 % 0.31/0.64 thf(fact_7314_nths__append,axiom, % 0.31/0.64 ! [A: $tType,L: list @ A,L3: list @ A,A3: set @ nat] : % 0.31/0.64 ( ( nths @ A @ ( append @ A @ L @ L3 ) @ A3 ) % 0.31/0.64 = ( append @ A @ ( nths @ A @ L @ A3 ) % 0.31/0.64 @ ( nths @ A @ L3 % 0.31/0.64 @ ( collect @ nat % 0.31/0.64 @ ^ [J3: nat] : ( member @ nat @ ( plus_plus @ nat @ J3 @ ( size_size @ ( list @ A ) @ L ) ) @ A3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nths_append % 0.31/0.64 thf(fact_7315_length__Suc__conv__rev,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,N2: nat] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( suc @ N2 ) ) % 0.31/0.64 = ( ? [Y2: A,Ys3: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( append @ A @ Ys3 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) % 0.31/0.64 & ( ( size_size @ ( list @ A ) @ Ys3 ) % 0.31/0.64 = N2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % length_Suc_conv_rev % 0.31/0.64 thf(fact_7316_length__append__singleton,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,X3: A] : % 0.31/0.64 ( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) % 0.31/0.64 = ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % length_append_singleton % 0.31/0.64 thf(fact_7317_subseqs_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( subseqs @ A @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( append @ ( list @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( subseqs @ A @ Xs ) ) @ ( subseqs @ A @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % subseqs.simps(2) % 0.31/0.64 thf(fact_7318_nth__append,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N2 ) % 0.31/0.64 = ( nth @ A @ Xs @ N2 ) ) ) % 0.31/0.64 & ( ~ ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N2 ) % 0.31/0.64 = ( nth @ A @ Ys @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_append % 0.31/0.64 thf(fact_7319_list__update__append,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A,Ys: list @ A,X3: A] : % 0.31/0.64 ( ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N2 @ X3 ) % 0.31/0.64 = ( append @ A @ ( list_update @ A @ Xs @ N2 @ X3 ) @ Ys ) ) ) % 0.31/0.64 & ( ~ ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N2 @ X3 ) % 0.31/0.64 = ( append @ A @ Xs @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) @ X3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % list_update_append % 0.31/0.64 thf(fact_7320_listrel1E,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) % 0.31/0.64 => ~ ! [X5: A,Y4: A] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ R2 ) % 0.31/0.64 => ! [Us3: list @ A,Vs3: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 = ( append @ A @ Us3 @ ( cons @ A @ X5 @ Vs3 ) ) ) % 0.31/0.64 => ( Ys % 0.31/0.64 != ( append @ A @ Us3 @ ( cons @ A @ Y4 @ Vs3 ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % listrel1E % 0.31/0.64 thf(fact_7321_listrel1I,axiom, % 0.31/0.64 ! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Us: list @ A,Vs2: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) % 0.31/0.64 => ( ( Xs % 0.31/0.64 = ( append @ A @ Us @ ( cons @ A @ X3 @ Vs2 ) ) ) % 0.31/0.64 => ( ( Ys % 0.31/0.64 = ( append @ A @ Us @ ( cons @ A @ Y @ Vs2 ) ) ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % listrel1I % 0.31/0.64 thf(fact_7322_lexord__append__left__rightI,axiom, % 0.31/0.64 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),U: list @ A,X3: list @ A,Y: list @ A] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A2 @ X3 ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y ) ) ) @ ( lexord @ A @ R2 ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_append_left_rightI % 0.31/0.64 thf(fact_7323_lexord__same__pref__iff,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lexord @ A @ R2 ) ) % 0.31/0.64 = ( ? [X: A] : % 0.31/0.64 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.64 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R2 ) ) % 0.31/0.64 | ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lexord @ A @ R2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_same_pref_iff % 0.31/0.64 thf(fact_7324_lexord__sufI,axiom, % 0.31/0.64 ! [A: $tType,U: list @ A,W2: list @ A,R2: set @ ( product_prod @ A @ A ),V2: list @ A,Z2: list @ A] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W2 ) @ ( lexord @ A @ R2 ) ) % 0.31/0.64 => ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W2 ) @ ( size_size @ ( list @ A ) @ U ) ) % 0.31/0.64 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V2 ) @ ( append @ A @ W2 @ Z2 ) ) @ ( lexord @ A @ R2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_sufI % 0.31/0.64 thf(fact_7325_product_Osimps_I2_J,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,X3: A,Xs: list @ A,Ys: list @ B] : % 0.31/0.64 ( ( product @ A @ B @ ( cons @ A @ X3 @ Xs ) @ Ys ) % 0.31/0.64 = ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % product.simps(2) % 0.31/0.64 thf(fact_7326_snoc__listrel1__snoc__iff,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A,Y: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R2 ) ) % 0.31/0.64 = ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) % 0.31/0.64 & ( X3 = Y ) ) % 0.31/0.64 | ( ( Xs = Ys ) % 0.31/0.64 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % snoc_listrel1_snoc_iff % 0.31/0.64 thf(fact_7327_horner__sum__append,axiom, % 0.31/0.64 ! [A: $tType,B: $tType] : % 0.31/0.64 ( ( comm_semiring_1 @ A ) % 0.31/0.64 => ! [F2: B > A,A2: A,Xs: list @ B,Ys: list @ B] : % 0.31/0.64 ( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ ( append @ B @ Xs @ Ys ) ) % 0.31/0.64 = ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A2 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ Ys ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % horner_sum_append % 0.31/0.64 thf(fact_7328_nths__Cons,axiom, % 0.31/0.64 ! [A: $tType,X3: A,L: list @ A,A3: set @ nat] : % 0.31/0.64 ( ( nths @ A @ ( cons @ A @ X3 @ L ) @ A3 ) % 0.31/0.64 = ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A3 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( nil @ A ) ) % 0.31/0.64 @ ( nths @ A @ L % 0.31/0.64 @ ( collect @ nat % 0.31/0.64 @ ^ [J3: nat] : ( member @ nat @ ( suc @ J3 ) @ A3 ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nths_Cons % 0.31/0.64 thf(fact_7329_comm__append__is__replicate,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( Xs % 0.31/0.64 != ( nil @ A ) ) % 0.31/0.64 => ( ( Ys % 0.31/0.64 != ( nil @ A ) ) % 0.31/0.64 => ( ( ( append @ A @ Xs @ Ys ) % 0.31/0.64 = ( append @ A @ Ys @ Xs ) ) % 0.31/0.64 => ? [N3: nat,Zs: list @ A] : % 0.31/0.64 ( ( ord_less @ nat @ ( one_one @ nat ) @ N3 ) % 0.31/0.64 & ( ( concat @ A @ ( replicate @ ( list @ A ) @ N3 @ Zs ) ) % 0.31/0.64 = ( append @ A @ Xs @ Ys ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % comm_append_is_replicate % 0.31/0.64 thf(fact_7330_listrel1__def,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( listrel1 @ A ) % 0.31/0.64 = ( ^ [R5: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) % 0.31/0.64 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o % 0.31/0.64 @ ^ [Xs2: list @ A,Ys3: list @ A] : % 0.31/0.64 ? [Us2: list @ A,Z6: A,Z7: A,Vs: list @ A] : % 0.31/0.64 ( ( Xs2 % 0.31/0.64 = ( append @ A @ Us2 @ ( cons @ A @ Z6 @ Vs ) ) ) % 0.31/0.64 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ Z7 ) @ R5 ) % 0.31/0.64 & ( Ys3 % 0.31/0.64 = ( append @ A @ Us2 @ ( cons @ A @ Z7 @ Vs ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % listrel1_def % 0.31/0.64 thf(fact_7331_lexord__def,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( lexord @ A ) % 0.31/0.64 = ( ^ [R5: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) % 0.31/0.64 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o % 0.31/0.64 @ ^ [X: list @ A,Y2: list @ A] : % 0.31/0.64 ? [A4: A,V5: list @ A] : % 0.31/0.64 ( ( Y2 % 0.31/0.64 = ( append @ A @ X @ ( cons @ A @ A4 @ V5 ) ) ) % 0.31/0.64 | ? [U2: list @ A,B4: A,C4: A,W3: list @ A,Z6: list @ A] : % 0.31/0.64 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C4 ) @ R5 ) % 0.31/0.64 & ( X % 0.31/0.64 = ( append @ A @ U2 @ ( cons @ A @ B4 @ W3 ) ) ) % 0.31/0.64 & ( Y2 % 0.31/0.64 = ( append @ A @ U2 @ ( cons @ A @ C4 @ Z6 ) ) ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lexord_def % 0.31/0.64 thf(fact_7332_take__Suc__conv__app__nth,axiom, % 0.31/0.64 ! [A: $tType,I: nat,Xs: list @ A] : % 0.31/0.64 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( take @ A @ ( suc @ I ) @ Xs ) % 0.31/0.64 = ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( nil @ A ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % take_Suc_conv_app_nth % 0.31/0.64 thf(fact_7333_lex__conv,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( lex @ A ) % 0.31/0.64 = ( ^ [R5: set @ ( product_prod @ A @ A )] : % 0.31/0.64 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) % 0.31/0.64 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o % 0.31/0.64 @ ^ [Xs2: list @ A,Ys3: list @ A] : % 0.31/0.64 ( ( ( size_size @ ( list @ A ) @ Xs2 ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ Ys3 ) ) % 0.31/0.64 & ? [Xys2: list @ A,X: A,Y2: A,Xs6: list @ A,Ys7: list @ A] : % 0.31/0.64 ( ( Xs2 % 0.31/0.64 = ( append @ A @ Xys2 @ ( cons @ A @ X @ Xs6 ) ) ) % 0.31/0.64 & ( Ys3 % 0.31/0.64 = ( append @ A @ Xys2 @ ( cons @ A @ Y2 @ Ys7 ) ) ) % 0.31/0.64 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R5 ) ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % lex_conv % 0.31/0.64 thf(fact_7334_nth__repl,axiom, % 0.31/0.64 ! [A: $tType,M: nat,Xs: list @ A,N2: nat,X3: A] : % 0.31/0.64 ( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( M != N2 ) % 0.31/0.64 => ( ( nth @ A @ ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( drop @ A @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ Xs ) ) ) @ M ) % 0.31/0.64 = ( nth @ A @ Xs @ M ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_repl % 0.31/0.64 thf(fact_7335_pos__n__replace,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A,Y: A] : % 0.31/0.64 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.64 = ( size_size @ ( list @ A ) @ ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( append @ A @ ( cons @ A @ Y @ ( nil @ A ) ) @ ( drop @ A @ ( suc @ N2 ) @ Xs ) ) ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % pos_n_replace % 0.31/0.64 thf(fact_7336_drop0,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( drop @ A @ ( zero_zero @ nat ) ) % 0.31/0.64 = ( ^ [X: list @ A] : X ) ) ). % 0.31/0.64 % 0.31/0.64 % drop0 % 0.31/0.64 thf(fact_7337_drop__drop,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,M: nat,Xs: list @ A] : % 0.31/0.64 ( ( drop @ A @ N2 @ ( drop @ A @ M @ Xs ) ) % 0.31/0.64 = ( drop @ A @ ( plus_plus @ nat @ N2 @ M ) @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % drop_drop % 0.31/0.64 thf(fact_7338_drop__Suc__Cons,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( drop @ A @ ( suc @ N2 ) @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( drop @ A @ N2 @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % drop_Suc_Cons % 0.31/0.64 thf(fact_7339_length__drop,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( size_size @ ( list @ A ) @ ( drop @ A @ N2 @ Xs ) ) % 0.31/0.64 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ). % 0.31/0.64 % 0.31/0.64 % length_drop % 0.31/0.64 thf(fact_7340_append__take__drop__id,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( drop @ A @ N2 @ Xs ) ) % 0.31/0.64 = Xs ) ). % 0.31/0.64 % 0.31/0.64 % append_take_drop_id % 0.31/0.64 thf(fact_7341_drop__update__cancel,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,M: nat,Xs: list @ A,X3: A] : % 0.31/0.64 ( ( ord_less @ nat @ N2 @ M ) % 0.31/0.64 => ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N2 @ X3 ) ) % 0.31/0.64 = ( drop @ A @ M @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % drop_update_cancel % 0.31/0.64 thf(fact_7342_drop__replicate,axiom, % 0.31/0.64 ! [A: $tType,I: nat,K: nat,X3: A] : % 0.31/0.64 ( ( drop @ A @ I @ ( replicate @ A @ K @ X3 ) ) % 0.31/0.64 = ( replicate @ A @ ( minus_minus @ nat @ K @ I ) @ X3 ) ) ). % 0.31/0.64 % 0.31/0.64 % drop_replicate % 0.31/0.64 thf(fact_7343_drop__eq__Nil2,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( ( nil @ A ) % 0.31/0.64 = ( drop @ A @ N2 @ Xs ) ) % 0.31/0.64 = ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ). % 0.31/0.64 % 0.31/0.64 % drop_eq_Nil2 % 0.31/0.64 thf(fact_7344_drop__eq__Nil,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( ( drop @ A @ N2 @ Xs ) % 0.31/0.64 = ( nil @ A ) ) % 0.31/0.64 = ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ). % 0.31/0.64 % 0.31/0.64 % drop_eq_Nil % 0.31/0.64 thf(fact_7345_drop__all,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,N2: nat] : % 0.31/0.64 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) % 0.31/0.64 => ( ( drop @ A @ N2 @ Xs ) % 0.31/0.64 = ( nil @ A ) ) ) ). % 0.31/0.64 % 0.31/0.64 % drop_all % 0.31/0.64 thf(fact_7346_drop__append,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A,Ys: list @ A] : % 0.31/0.64 ( ( drop @ A @ N2 @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.64 = ( append @ A @ ( drop @ A @ N2 @ Xs ) @ ( drop @ A @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % drop_append % 0.31/0.64 thf(fact_7347_drop__Cons__numeral,axiom, % 0.31/0.64 ! [A: $tType,V2: num,X3: A,Xs: list @ A] : % 0.31/0.64 ( ( drop @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.64 = ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % drop_Cons_numeral % 0.31/0.64 thf(fact_7348_nth__drop,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A,I: nat] : % 0.31/0.64 ( ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.64 => ( ( nth @ A @ ( drop @ A @ N2 @ Xs ) @ I ) % 0.31/0.64 = ( nth @ A @ Xs @ ( plus_plus @ nat @ N2 @ I ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_drop % 0.31/0.64 thf(fact_7349_nth__via__drop,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A,Y: A,Ys: list @ A] : % 0.31/0.64 ( ( ( drop @ A @ N2 @ Xs ) % 0.31/0.64 = ( cons @ A @ Y @ Ys ) ) % 0.31/0.64 => ( ( nth @ A @ Xs @ N2 ) % 0.31/0.64 = Y ) ) ). % 0.31/0.64 % 0.31/0.64 % nth_via_drop % 0.31/0.64 thf(fact_7350_set__drop__subset__set__drop,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,M: nat,Xs: list @ A] : % 0.31/0.64 ( ( ord_less_eq @ nat @ N2 @ M ) % 0.31/0.64 => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ M @ Xs ) ) @ ( set2 @ A @ ( drop @ A @ N2 @ Xs ) ) ) ) ). % 0.31/0.64 % 0.31/0.64 % set_drop_subset_set_drop % 0.31/0.64 thf(fact_7351_drop__zip,axiom, % 0.31/0.64 ! [A: $tType,B: $tType,N2: nat,Xs: list @ A,Ys: list @ B] : % 0.31/0.64 ( ( drop @ ( product_prod @ A @ B ) @ N2 @ ( zip @ A @ B @ Xs @ Ys ) ) % 0.31/0.64 = ( zip @ A @ B @ ( drop @ A @ N2 @ Xs ) @ ( drop @ B @ N2 @ Ys ) ) ) ). % 0.31/0.64 % 0.31/0.64 % drop_zip % 0.31/0.64 thf(fact_7352_drop__0,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A] : % 0.31/0.64 ( ( drop @ A @ ( zero_zero @ nat ) @ Xs ) % 0.31/0.64 = Xs ) ). % 0.31/0.64 % 0.31/0.64 % drop_0 % 0.31/0.64 thf(fact_7353_drop__Nil,axiom, % 0.31/0.64 ! [A: $tType,N2: nat] : % 0.31/0.64 ( ( drop @ A @ N2 @ ( nil @ A ) ) % 0.31/0.64 = ( nil @ A ) ) ). % 0.31/0.64 % 0.31/0.64 % drop_Nil % 0.31/0.64 thf(fact_7354_in__set__dropD,axiom, % 0.31/0.64 ! [A: $tType,X3: A,N2: nat,Xs: list @ A] : % 0.31/0.64 ( ( member @ A @ X3 @ ( set2 @ A @ ( drop @ A @ N2 @ Xs ) ) ) % 0.31/0.64 => ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % in_set_dropD % 0.31/0.64 thf(fact_7355_take__drop,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,M: nat,Xs: list @ A] : % 0.31/0.64 ( ( take @ A @ N2 @ ( drop @ A @ M @ Xs ) ) % 0.31/0.64 = ( drop @ A @ M @ ( take @ A @ ( plus_plus @ nat @ N2 @ M ) @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % take_drop % 0.31/0.64 thf(fact_7356_drop__take,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,M: nat,Xs: list @ A] : % 0.31/0.64 ( ( drop @ A @ N2 @ ( take @ A @ M @ Xs ) ) % 0.31/0.64 = ( take @ A @ ( minus_minus @ nat @ M @ N2 ) @ ( drop @ A @ N2 @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % drop_take % 0.31/0.64 thf(fact_7357_distinct__drop,axiom, % 0.31/0.64 ! [A: $tType,Xs: list @ A,I: nat] : % 0.31/0.64 ( ( distinct @ A @ Xs ) % 0.31/0.64 => ( distinct @ A @ ( drop @ A @ I @ Xs ) ) ) ). % 0.31/0.64 % 0.31/0.64 % distinct_drop % 0.31/0.64 thf(fact_7358_set__drop__subset,axiom, % 0.31/0.64 ! [A: $tType,N2: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ N2 @ Xs ) ) @ ( set2 @ A @ Xs ) ) ). % 0.31/0.64 % 0.31/0.64 % set_drop_subset % 0.31/0.64 thf(fact_7359_drop__eq__nths,axiom, % 0.31/0.64 ! [A: $tType] : % 0.31/0.64 ( ( drop @ A ) % 0.31/0.65 = ( ^ [N: nat,Xs2: list @ A] : ( nths @ A @ Xs2 @ ( collect @ nat @ ( ord_less_eq @ nat @ N ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % drop_eq_nths % 0.31/0.65 thf(fact_7360_drop__map,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,N2: nat,F2: B > A,Xs: list @ B] : % 0.31/0.65 ( ( drop @ A @ N2 @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 = ( map @ B @ A @ F2 @ ( drop @ B @ N2 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % drop_map % 0.31/0.65 thf(fact_7361_take__add,axiom, % 0.31/0.65 ! [A: $tType,I: nat,J: nat,Xs: list @ A] : % 0.31/0.65 ( ( take @ A @ ( plus_plus @ nat @ I @ J ) @ Xs ) % 0.31/0.65 = ( append @ A @ ( take @ A @ I @ Xs ) @ ( take @ A @ J @ ( drop @ A @ I @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % take_add % 0.31/0.65 thf(fact_7362_append__eq__conv__conj,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] : % 0.31/0.65 ( ( ( append @ A @ Xs @ Ys ) % 0.31/0.65 = Zs2 ) % 0.31/0.65 = ( ( Xs % 0.31/0.65 = ( take @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs2 ) ) % 0.31/0.65 & ( Ys % 0.31/0.65 = ( drop @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % append_eq_conv_conj % 0.31/0.65 thf(fact_7363_drop__update__swap,axiom, % 0.31/0.65 ! [A: $tType,M: nat,N2: nat,Xs: list @ A,X3: A] : % 0.31/0.65 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.65 => ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N2 @ X3 ) ) % 0.31/0.65 = ( list_update @ A @ ( drop @ A @ M @ Xs ) @ ( minus_minus @ nat @ N2 @ M ) @ X3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % drop_update_swap % 0.31/0.65 thf(fact_7364_drop__Cons,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( drop @ A @ N2 @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( case_nat @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) % 0.31/0.65 @ ^ [M2: nat] : ( drop @ A @ M2 @ Xs ) % 0.31/0.65 @ N2 ) ) ). % 0.31/0.65 % 0.31/0.65 % drop_Cons % 0.31/0.65 thf(fact_7365_nths__drop,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A,I6: set @ nat] : % 0.31/0.65 ( ( nths @ A @ ( drop @ A @ N2 @ Xs ) @ I6 ) % 0.31/0.65 = ( nths @ A @ Xs @ ( image @ nat @ nat @ ( plus_plus @ nat @ N2 ) @ I6 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % nths_drop % 0.31/0.65 thf(fact_7366_drop__Cons_H,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( ( N2 % 0.31/0.65 = ( zero_zero @ nat ) ) % 0.31/0.65 => ( ( drop @ A @ N2 @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( cons @ A @ X3 @ Xs ) ) ) % 0.31/0.65 & ( ( N2 % 0.31/0.65 != ( zero_zero @ nat ) ) % 0.31/0.65 => ( ( drop @ A @ N2 @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( drop @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % drop_Cons' % 0.31/0.65 thf(fact_7367_append__eq__append__conv__if,axiom, % 0.31/0.65 ! [A: $tType,Xs_1: list @ A,Xs_2: list @ A,Ys_1: list @ A,Ys_2: list @ A] : % 0.31/0.65 ( ( ( append @ A @ Xs_1 @ Xs_2 ) % 0.31/0.65 = ( append @ A @ Ys_1 @ Ys_2 ) ) % 0.31/0.65 = ( ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs_1 ) @ ( size_size @ ( list @ A ) @ Ys_1 ) ) % 0.31/0.65 => ( ( Xs_1 % 0.31/0.65 = ( take @ A @ ( size_size @ ( list @ A ) @ Xs_1 ) @ Ys_1 ) ) % 0.31/0.65 & ( Xs_2 % 0.31/0.65 = ( append @ A @ ( drop @ A @ ( size_size @ ( list @ A ) @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) ) % 0.31/0.65 & ( ~ ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs_1 ) @ ( size_size @ ( list @ A ) @ Ys_1 ) ) % 0.31/0.65 => ( ( ( take @ A @ ( size_size @ ( list @ A ) @ Ys_1 ) @ Xs_1 ) % 0.31/0.65 = Ys_1 ) % 0.31/0.65 & ( ( append @ A @ ( drop @ A @ ( size_size @ ( list @ A ) @ Ys_1 ) @ Xs_1 ) @ Xs_2 ) % 0.31/0.65 = Ys_2 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % append_eq_append_conv_if % 0.31/0.65 thf(fact_7368_zip__append2,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ B] : % 0.31/0.65 ( ( zip @ A @ B @ Xs @ ( append @ B @ Ys @ Zs2 ) ) % 0.31/0.65 = ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) @ Ys ) @ ( zip @ A @ B @ ( drop @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) @ Zs2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % zip_append2 % 0.31/0.65 thf(fact_7369_zip__append1,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ B] : % 0.31/0.65 ( ( zip @ A @ B @ ( append @ A @ Xs @ Ys ) @ Zs2 ) % 0.31/0.65 = ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ ( take @ B @ ( size_size @ ( list @ A ) @ Xs ) @ Zs2 ) ) @ ( zip @ A @ B @ Ys @ ( drop @ B @ ( size_size @ ( list @ A ) @ Xs ) @ Zs2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % zip_append1 % 0.31/0.65 thf(fact_7370_Cons__nth__drop__Suc,axiom, % 0.31/0.65 ! [A: $tType,I: nat,Xs: list @ A] : % 0.31/0.65 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) ) % 0.31/0.65 = ( drop @ A @ I @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Cons_nth_drop_Suc % 0.31/0.65 thf(fact_7371_set__take__disj__set__drop__if__distinct,axiom, % 0.31/0.65 ! [A: $tType,Vs2: list @ A,I: nat,J: nat] : % 0.31/0.65 ( ( distinct @ A @ Vs2 ) % 0.31/0.65 => ( ( ord_less_eq @ nat @ I @ J ) % 0.31/0.65 => ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I @ Vs2 ) ) @ ( set2 @ A @ ( drop @ A @ J @ Vs2 ) ) ) % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % set_take_disj_set_drop_if_distinct % 0.31/0.65 thf(fact_7372_id__take__nth__drop,axiom, % 0.31/0.65 ! [A: $tType,I: nat,Xs: list @ A] : % 0.31/0.65 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( Xs % 0.31/0.65 = ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % id_take_nth_drop % 0.31/0.65 thf(fact_7373_upd__conv__take__nth__drop,axiom, % 0.31/0.65 ! [A: $tType,I: nat,Xs: list @ A,A2: A] : % 0.31/0.65 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ( list_update @ A @ Xs @ I @ A2 ) % 0.31/0.65 = ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ A2 @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upd_conv_take_nth_drop % 0.31/0.65 thf(fact_7374_lexn__conv,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( lexn @ A ) % 0.31/0.65 = ( ^ [R5: set @ ( product_prod @ A @ A ),N: nat] : % 0.31/0.65 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) % 0.31/0.65 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o % 0.31/0.65 @ ^ [Xs2: list @ A,Ys3: list @ A] : % 0.31/0.65 ( ( ( size_size @ ( list @ A ) @ Xs2 ) % 0.31/0.65 = N ) % 0.31/0.65 & ( ( size_size @ ( list @ A ) @ Ys3 ) % 0.31/0.65 = N ) % 0.31/0.65 & ? [Xys2: list @ A,X: A,Y2: A,Xs6: list @ A,Ys7: list @ A] : % 0.31/0.65 ( ( Xs2 % 0.31/0.65 = ( append @ A @ Xys2 @ ( cons @ A @ X @ Xs6 ) ) ) % 0.31/0.65 & ( Ys3 % 0.31/0.65 = ( append @ A @ Xys2 @ ( cons @ A @ Y2 @ Ys7 ) ) ) % 0.31/0.65 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R5 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % lexn_conv % 0.31/0.65 thf(fact_7375_list__encode_Oelims,axiom, % 0.31/0.65 ! [X3: list @ nat,Y: nat] : % 0.31/0.65 ( ( ( nat_list_encode @ X3 ) % 0.31/0.65 = Y ) % 0.31/0.65 => ( ( ( X3 % 0.31/0.65 = ( nil @ nat ) ) % 0.31/0.65 => ( Y % 0.31/0.65 != ( zero_zero @ nat ) ) ) % 0.31/0.65 => ~ ! [X5: nat,Xs3: list @ nat] : % 0.31/0.65 ( ( X3 % 0.31/0.65 = ( cons @ nat @ X5 @ Xs3 ) ) % 0.31/0.65 => ( Y % 0.31/0.65 != ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % list_encode.elims % 0.31/0.65 thf(fact_7376_lexn_Osimps_I1_J,axiom, % 0.31/0.65 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.65 ( ( lexn @ A @ R2 @ ( zero_zero @ nat ) ) % 0.31/0.65 = ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % lexn.simps(1) % 0.31/0.65 thf(fact_7377_list__encode_Osimps_I1_J,axiom, % 0.31/0.65 ( ( nat_list_encode @ ( nil @ nat ) ) % 0.31/0.65 = ( zero_zero @ nat ) ) ). % 0.31/0.65 % 0.31/0.65 % list_encode.simps(1) % 0.31/0.65 thf(fact_7378_lexn__length,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),N2: nat] : % 0.31/0.65 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexn @ A @ R2 @ N2 ) ) % 0.31/0.65 => ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.65 = N2 ) % 0.31/0.65 & ( ( size_size @ ( list @ A ) @ Ys ) % 0.31/0.65 = N2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % lexn_length % 0.31/0.65 thf(fact_7379_lex__def,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( lex @ A ) % 0.31/0.65 = ( ^ [R5: set @ ( product_prod @ A @ A )] : ( complete_Sup_Sup @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image @ nat @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( lexn @ A @ R5 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % lex_def % 0.31/0.65 thf(fact_7380_list__encode_Osimps_I2_J,axiom, % 0.31/0.65 ! [X3: nat,Xs: list @ nat] : % 0.31/0.65 ( ( nat_list_encode @ ( cons @ nat @ X3 @ Xs ) ) % 0.31/0.65 = ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % list_encode.simps(2) % 0.31/0.65 thf(fact_7381_upto__aux__rec,axiom, % 0.31/0.65 ( upto_aux % 0.31/0.65 = ( ^ [I3: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_aux_rec % 0.31/0.65 thf(fact_7382_list_Osimps_I5_J,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F1: B,F22: A > ( list @ A ) > B,X21: A,X222: list @ A] : % 0.31/0.65 ( ( case_list @ B @ A @ F1 @ F22 @ ( cons @ A @ X21 @ X222 ) ) % 0.31/0.65 = ( F22 @ X21 @ X222 ) ) ). % 0.31/0.65 % 0.31/0.65 % list.simps(5) % 0.31/0.65 thf(fact_7383_list_Ocase__distrib,axiom, % 0.31/0.65 ! [B: $tType,C: $tType,A: $tType,H2: B > C,F1: B,F22: A > ( list @ A ) > B,List: list @ A] : % 0.31/0.65 ( ( H2 @ ( case_list @ B @ A @ F1 @ F22 @ List ) ) % 0.31/0.65 = ( case_list @ C @ A @ ( H2 @ F1 ) % 0.31/0.65 @ ^ [X15: A,X24: list @ A] : ( H2 @ ( F22 @ X15 @ X24 ) ) % 0.31/0.65 @ List ) ) ). % 0.31/0.65 % 0.31/0.65 % list.case_distrib % 0.31/0.65 thf(fact_7384_list_Osimps_I4_J,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F1: B,F22: A > ( list @ A ) > B] : % 0.31/0.65 ( ( case_list @ B @ A @ F1 @ F22 @ ( nil @ A ) ) % 0.31/0.65 = F1 ) ). % 0.31/0.65 % 0.31/0.65 % list.simps(4) % 0.31/0.65 thf(fact_7385_remdups__adj__Cons,axiom, % 0.31/0.65 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( case_list @ ( list @ A ) @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) % 0.31/0.65 @ ^ [Y2: A,Xs2: list @ A] : ( if @ ( list @ A ) @ ( X3 = Y2 ) @ ( cons @ A @ Y2 @ Xs2 ) @ ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Xs2 ) ) ) % 0.31/0.65 @ ( remdups_adj @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % remdups_adj_Cons % 0.31/0.65 thf(fact_7386_min__list_Osimps,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( ord @ A ) % 0.31/0.65 => ! [X3: A,Xs: list @ A] : % 0.31/0.65 ( ( min_list @ A @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( case_list @ A @ A @ X3 % 0.31/0.65 @ ^ [A4: A,List2: list @ A] : ( ord_min @ A @ X3 @ ( min_list @ A @ Xs ) ) % 0.31/0.65 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % min_list.simps % 0.31/0.65 thf(fact_7387_transpose_Oelims,axiom, % 0.31/0.65 ! [A: $tType,X3: list @ ( list @ A ),Y: list @ ( list @ A )] : % 0.31/0.65 ( ( ( transpose @ A @ X3 ) % 0.31/0.65 = Y ) % 0.31/0.65 => ( ( ( X3 % 0.31/0.65 = ( nil @ ( list @ A ) ) ) % 0.31/0.65 => ( Y % 0.31/0.65 != ( nil @ ( list @ A ) ) ) ) % 0.31/0.65 => ( ! [Xss2: list @ ( list @ A )] : % 0.31/0.65 ( ( X3 % 0.31/0.65 = ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) % 0.31/0.65 => ( Y % 0.31/0.65 != ( transpose @ A @ Xss2 ) ) ) % 0.31/0.65 => ~ ! [X5: A,Xs3: list @ A,Xss2: list @ ( list @ A )] : % 0.31/0.65 ( ( X3 % 0.31/0.65 = ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) ) % 0.31/0.65 => ( Y % 0.31/0.65 != ( cons @ ( list @ A ) % 0.31/0.65 @ ( cons @ A @ X5 % 0.31/0.65 @ ( concat @ A % 0.31/0.65 @ ( map @ ( list @ A ) @ ( list @ A ) % 0.31/0.65 @ ( case_list @ ( list @ A ) @ A @ ( nil @ A ) % 0.31/0.65 @ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) ) % 0.31/0.65 @ Xss2 ) ) ) % 0.31/0.65 @ ( transpose @ A % 0.31/0.65 @ ( cons @ ( list @ A ) @ Xs3 % 0.31/0.65 @ ( concat @ ( list @ A ) % 0.31/0.65 @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) ) % 0.31/0.65 @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) ) % 0.31/0.65 @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) ) % 0.31/0.65 @ Xss2 ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose.elims % 0.31/0.65 thf(fact_7388_transpose_Osimps_I3_J,axiom, % 0.31/0.65 ! [A: $tType,X3: A,Xs: list @ A,Xss: list @ ( list @ A )] : % 0.31/0.65 ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ Xss ) ) % 0.31/0.65 = ( cons @ ( list @ A ) % 0.31/0.65 @ ( cons @ A @ X3 % 0.31/0.65 @ ( concat @ A % 0.31/0.65 @ ( map @ ( list @ A ) @ ( list @ A ) % 0.31/0.65 @ ( case_list @ ( list @ A ) @ A @ ( nil @ A ) % 0.31/0.65 @ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) ) % 0.31/0.65 @ Xss ) ) ) % 0.31/0.65 @ ( transpose @ A % 0.31/0.65 @ ( cons @ ( list @ A ) @ Xs % 0.31/0.65 @ ( concat @ ( list @ A ) % 0.31/0.65 @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) ) % 0.31/0.65 @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) ) % 0.31/0.65 @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) ) % 0.31/0.65 @ Xss ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose.simps(3) % 0.31/0.65 thf(fact_7389_list_Odisc__eq__case_I1_J,axiom, % 0.31/0.65 ! [A: $tType,List: list @ A] : % 0.31/0.65 ( ( List % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 = ( case_list @ $o @ A @ $true % 0.31/0.65 @ ^ [Uu3: A,Uv3: list @ A] : $false % 0.31/0.65 @ List ) ) ). % 0.31/0.65 % 0.31/0.65 % list.disc_eq_case(1) % 0.31/0.65 thf(fact_7390_list_Odisc__eq__case_I2_J,axiom, % 0.31/0.65 ! [A: $tType,List: list @ A] : % 0.31/0.65 ( ( List % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 = ( case_list @ $o @ A @ $false % 0.31/0.65 @ ^ [Uu3: A,Uv3: list @ A] : $true % 0.31/0.65 @ List ) ) ). % 0.31/0.65 % 0.31/0.65 % list.disc_eq_case(2) % 0.31/0.65 thf(fact_7391_transpose_Osimps_I2_J,axiom, % 0.31/0.65 ! [A: $tType,Xss: list @ ( list @ A )] : % 0.31/0.65 ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) ) % 0.31/0.65 = ( transpose @ A @ Xss ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose.simps(2) % 0.31/0.65 thf(fact_7392_transpose_Osimps_I1_J,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( transpose @ A @ ( nil @ ( list @ A ) ) ) % 0.31/0.65 = ( nil @ ( list @ A ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose.simps(1) % 0.31/0.65 thf(fact_7393_transpose__map__map,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ ( list @ B )] : % 0.31/0.65 ( ( transpose @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ Xs ) ) % 0.31/0.65 = ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F2 ) @ ( transpose @ B @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose_map_map % 0.31/0.65 thf(fact_7394_transpose__empty,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ ( list @ A )] : % 0.31/0.65 ( ( ( transpose @ A @ Xs ) % 0.31/0.65 = ( nil @ ( list @ A ) ) ) % 0.31/0.65 = ( ! [X: list @ A] : % 0.31/0.65 ( ( member @ ( list @ A ) @ X @ ( set2 @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( X % 0.31/0.65 = ( nil @ A ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose_empty % 0.31/0.65 thf(fact_7395_zip__Cons,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B] : % 0.31/0.65 ( ( zip @ A @ B @ Xs @ ( cons @ B @ Y @ Ys ) ) % 0.31/0.65 = ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ A @ ( nil @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ^ [Z6: A,Zs3: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z6 @ Y ) @ ( zip @ A @ B @ Zs3 @ Ys ) ) % 0.31/0.65 @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % zip_Cons % 0.31/0.65 thf(fact_7396_zip__Cons1,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,X3: A,Xs: list @ A,Ys: list @ B] : % 0.31/0.65 ( ( zip @ A @ B @ ( cons @ A @ X3 @ Xs ) @ Ys ) % 0.31/0.65 = ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ^ [Y2: B,Ys3: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ ( zip @ A @ B @ Xs @ Ys3 ) ) % 0.31/0.65 @ Ys ) ) ). % 0.31/0.65 % 0.31/0.65 % zip_Cons1 % 0.31/0.65 thf(fact_7397_upto_Opsimps,axiom, % 0.31/0.65 ! [I: int,J: int] : % 0.31/0.65 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I @ J ) ) % 0.31/0.65 => ( ( ( ord_less_eq @ int @ I @ J ) % 0.31/0.65 => ( ( upto @ I @ J ) % 0.31/0.65 = ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) % 0.31/0.65 & ( ~ ( ord_less_eq @ int @ I @ J ) % 0.31/0.65 => ( ( upto @ I @ J ) % 0.31/0.65 = ( nil @ int ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto.psimps % 0.31/0.65 thf(fact_7398_upto_Opelims,axiom, % 0.31/0.65 ! [X3: int,Xa2: int,Y: list @ int] : % 0.31/0.65 ( ( ( upto @ X3 @ Xa2 ) % 0.31/0.65 = Y ) % 0.31/0.65 => ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X3 @ Xa2 ) ) % 0.31/0.65 => ~ ( ( ( ( ord_less_eq @ int @ X3 @ Xa2 ) % 0.31/0.65 => ( Y % 0.31/0.65 = ( cons @ int @ X3 @ ( upto @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) @ Xa2 ) ) ) ) % 0.31/0.65 & ( ~ ( ord_less_eq @ int @ X3 @ Xa2 ) % 0.31/0.65 => ( Y % 0.31/0.65 = ( nil @ int ) ) ) ) % 0.31/0.65 => ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X3 @ Xa2 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto.pelims % 0.31/0.65 thf(fact_7399_upto__empty,axiom, % 0.31/0.65 ! [J: int,I: int] : % 0.31/0.65 ( ( ord_less @ int @ J @ I ) % 0.31/0.65 => ( ( upto @ I @ J ) % 0.31/0.65 = ( nil @ int ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_empty % 0.31/0.65 thf(fact_7400_upto__Nil2,axiom, % 0.31/0.65 ! [I: int,J: int] : % 0.31/0.65 ( ( ( nil @ int ) % 0.31/0.65 = ( upto @ I @ J ) ) % 0.31/0.65 = ( ord_less @ int @ J @ I ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_Nil2 % 0.31/0.65 thf(fact_7401_upto__Nil,axiom, % 0.31/0.65 ! [I: int,J: int] : % 0.31/0.65 ( ( ( upto @ I @ J ) % 0.31/0.65 = ( nil @ int ) ) % 0.31/0.65 = ( ord_less @ int @ J @ I ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_Nil % 0.31/0.65 thf(fact_7402_upto__single,axiom, % 0.31/0.65 ! [I: int] : % 0.31/0.65 ( ( upto @ I @ I ) % 0.31/0.65 = ( cons @ int @ I @ ( nil @ int ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_single % 0.31/0.65 thf(fact_7403_nth__upto,axiom, % 0.31/0.65 ! [I: int,K: nat,J: int] : % 0.31/0.65 ( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) @ J ) % 0.31/0.65 => ( ( nth @ int @ ( upto @ I @ J ) @ K ) % 0.31/0.65 = ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % nth_upto % 0.31/0.65 thf(fact_7404_length__upto,axiom, % 0.31/0.65 ! [I: int,J: int] : % 0.31/0.65 ( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) ) % 0.31/0.65 = ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % length_upto % 0.31/0.65 thf(fact_7405_upto__rec__numeral_I1_J,axiom, % 0.31/0.65 ! [M: num,N2: num] : % 0.31/0.65 ( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) ) % 0.31/0.65 => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) ) % 0.31/0.65 = ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N2 ) ) ) ) ) % 0.31/0.65 & ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) ) % 0.31/0.65 => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) ) % 0.31/0.65 = ( nil @ int ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_rec_numeral(1) % 0.31/0.65 thf(fact_7406_upto__rec__numeral_I2_J,axiom, % 0.31/0.65 ! [M: num,N2: num] : % 0.31/0.65 ( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) % 0.31/0.65 => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) % 0.31/0.65 = ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ) % 0.31/0.65 & ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) % 0.31/0.65 => ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) % 0.31/0.65 = ( nil @ int ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_rec_numeral(2) % 0.31/0.65 thf(fact_7407_upto__rec__numeral_I3_J,axiom, % 0.31/0.65 ! [M: num,N2: num] : % 0.31/0.65 ( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) ) % 0.31/0.65 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) ) % 0.31/0.65 = ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N2 ) ) ) ) ) % 0.31/0.65 & ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) ) % 0.31/0.65 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) ) % 0.31/0.65 = ( nil @ int ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_rec_numeral(3) % 0.31/0.65 thf(fact_7408_upto__rec__numeral_I4_J,axiom, % 0.31/0.65 ! [M: num,N2: num] : % 0.31/0.65 ( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) % 0.31/0.65 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) % 0.31/0.65 = ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ) % 0.31/0.65 & ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) % 0.31/0.65 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) % 0.31/0.65 = ( nil @ int ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_rec_numeral(4) % 0.31/0.65 thf(fact_7409_atLeastAtMost__upto,axiom, % 0.31/0.65 ( ( set_or1337092689740270186AtMost @ int ) % 0.31/0.65 = ( ^ [I3: int,J3: int] : ( set2 @ int @ ( upto @ I3 @ J3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % atLeastAtMost_upto % 0.31/0.65 thf(fact_7410_distinct__upto,axiom, % 0.31/0.65 ! [I: int,J: int] : ( distinct @ int @ ( upto @ I @ J ) ) ). % 0.31/0.65 % 0.31/0.65 % distinct_upto % 0.31/0.65 thf(fact_7411_upto__aux__def,axiom, % 0.31/0.65 ( upto_aux % 0.31/0.65 = ( ^ [I3: int,J3: int] : ( append @ int @ ( upto @ I3 @ J3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_aux_def % 0.31/0.65 thf(fact_7412_upto__code,axiom, % 0.31/0.65 ( upto % 0.31/0.65 = ( ^ [I3: int,J3: int] : ( upto_aux @ I3 @ J3 @ ( nil @ int ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_code % 0.31/0.65 thf(fact_7413_upto__split2,axiom, % 0.31/0.65 ! [I: int,J: int,K: int] : % 0.31/0.65 ( ( ord_less_eq @ int @ I @ J ) % 0.31/0.65 => ( ( ord_less_eq @ int @ J @ K ) % 0.31/0.65 => ( ( upto @ I @ K ) % 0.31/0.65 = ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_split2 % 0.31/0.65 thf(fact_7414_upto__split1,axiom, % 0.31/0.65 ! [I: int,J: int,K: int] : % 0.31/0.65 ( ( ord_less_eq @ int @ I @ J ) % 0.31/0.65 => ( ( ord_less_eq @ int @ J @ K ) % 0.31/0.65 => ( ( upto @ I @ K ) % 0.31/0.65 = ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( upto @ J @ K ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_split1 % 0.31/0.65 thf(fact_7415_atLeastLessThan__upto,axiom, % 0.31/0.65 ( ( set_or7035219750837199246ssThan @ int ) % 0.31/0.65 = ( ^ [I3: int,J3: int] : ( set2 @ int @ ( upto @ I3 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % atLeastLessThan_upto % 0.31/0.65 thf(fact_7416_greaterThanAtMost__upto,axiom, % 0.31/0.65 ( ( set_or3652927894154168847AtMost @ int ) % 0.31/0.65 = ( ^ [I3: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % greaterThanAtMost_upto % 0.31/0.65 thf(fact_7417_upto__rec1,axiom, % 0.31/0.65 ! [I: int,J: int] : % 0.31/0.65 ( ( ord_less_eq @ int @ I @ J ) % 0.31/0.65 => ( ( upto @ I @ J ) % 0.31/0.65 = ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_rec1 % 0.31/0.65 thf(fact_7418_upto_Osimps,axiom, % 0.31/0.65 ( upto % 0.31/0.65 = ( ^ [I3: int,J3: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I3 @ J3 ) @ ( cons @ int @ I3 @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) @ ( nil @ int ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto.simps % 0.31/0.65 thf(fact_7419_upto_Oelims,axiom, % 0.31/0.65 ! [X3: int,Xa2: int,Y: list @ int] : % 0.31/0.65 ( ( ( upto @ X3 @ Xa2 ) % 0.31/0.65 = Y ) % 0.31/0.65 => ( ( ( ord_less_eq @ int @ X3 @ Xa2 ) % 0.31/0.65 => ( Y % 0.31/0.65 = ( cons @ int @ X3 @ ( upto @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) @ Xa2 ) ) ) ) % 0.31/0.65 & ( ~ ( ord_less_eq @ int @ X3 @ Xa2 ) % 0.31/0.65 => ( Y % 0.31/0.65 = ( nil @ int ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto.elims % 0.31/0.65 thf(fact_7420_upto__rec2,axiom, % 0.31/0.65 ! [I: int,J: int] : % 0.31/0.65 ( ( ord_less_eq @ int @ I @ J ) % 0.31/0.65 => ( ( upto @ I @ J ) % 0.31/0.65 = ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( nil @ int ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_rec2 % 0.31/0.65 thf(fact_7421_greaterThanLessThan__upto,axiom, % 0.31/0.65 ( ( set_or5935395276787703475ssThan @ int ) % 0.31/0.65 = ( ^ [I3: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % greaterThanLessThan_upto % 0.31/0.65 thf(fact_7422_upto__split3,axiom, % 0.31/0.65 ! [I: int,J: int,K: int] : % 0.31/0.65 ( ( ord_less_eq @ int @ I @ J ) % 0.31/0.65 => ( ( ord_less_eq @ int @ J @ K ) % 0.31/0.65 => ( ( upto @ I @ K ) % 0.31/0.65 = ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upto_split3 % 0.31/0.65 thf(fact_7423_transpose_Opelims,axiom, % 0.31/0.65 ! [A: $tType,X3: list @ ( list @ A ),Y: list @ ( list @ A )] : % 0.31/0.65 ( ( ( transpose @ A @ X3 ) % 0.31/0.65 = Y ) % 0.31/0.65 => ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X3 ) % 0.31/0.65 => ( ( ( X3 % 0.31/0.65 = ( nil @ ( list @ A ) ) ) % 0.31/0.65 => ( ( Y % 0.31/0.65 = ( nil @ ( list @ A ) ) ) % 0.31/0.65 => ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) ) % 0.31/0.65 => ( ! [Xss2: list @ ( list @ A )] : % 0.31/0.65 ( ( X3 % 0.31/0.65 = ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) % 0.31/0.65 => ( ( Y % 0.31/0.65 = ( transpose @ A @ Xss2 ) ) % 0.31/0.65 => ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) ) % 0.31/0.65 => ~ ! [X5: A,Xs3: list @ A,Xss2: list @ ( list @ A )] : % 0.31/0.65 ( ( X3 % 0.31/0.65 = ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) ) % 0.31/0.65 => ( ( Y % 0.31/0.65 = ( cons @ ( list @ A ) % 0.31/0.65 @ ( cons @ A @ X5 % 0.31/0.65 @ ( concat @ A % 0.31/0.65 @ ( map @ ( list @ A ) @ ( list @ A ) % 0.31/0.65 @ ( case_list @ ( list @ A ) @ A @ ( nil @ A ) % 0.31/0.65 @ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) ) % 0.31/0.65 @ Xss2 ) ) ) % 0.31/0.65 @ ( transpose @ A % 0.31/0.65 @ ( cons @ ( list @ A ) @ Xs3 % 0.31/0.65 @ ( concat @ ( list @ A ) % 0.31/0.65 @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) ) % 0.31/0.65 @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) ) % 0.31/0.65 @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) ) % 0.31/0.65 @ Xss2 ) ) ) ) ) ) % 0.31/0.65 => ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose.pelims % 0.31/0.65 thf(fact_7424_transpose_Opsimps_I3_J,axiom, % 0.31/0.65 ! [A: $tType,X3: A,Xs: list @ A,Xss: list @ ( list @ A )] : % 0.31/0.65 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ Xss ) ) % 0.31/0.65 => ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ Xss ) ) % 0.31/0.65 = ( cons @ ( list @ A ) % 0.31/0.65 @ ( cons @ A @ X3 % 0.31/0.65 @ ( concat @ A % 0.31/0.65 @ ( map @ ( list @ A ) @ ( list @ A ) % 0.31/0.65 @ ( case_list @ ( list @ A ) @ A @ ( nil @ A ) % 0.31/0.65 @ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) ) % 0.31/0.65 @ Xss ) ) ) % 0.31/0.65 @ ( transpose @ A % 0.31/0.65 @ ( cons @ ( list @ A ) @ Xs % 0.31/0.65 @ ( concat @ ( list @ A ) % 0.31/0.65 @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) ) % 0.31/0.65 @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) ) % 0.31/0.65 @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) ) % 0.31/0.65 @ Xss ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose.psimps(3) % 0.31/0.65 thf(fact_7425_transpose_Opsimps_I2_J,axiom, % 0.31/0.65 ! [A: $tType,Xss: list @ ( list @ A )] : % 0.31/0.65 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) ) % 0.31/0.65 => ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) ) % 0.31/0.65 = ( transpose @ A @ Xss ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose.psimps(2) % 0.31/0.65 thf(fact_7426_transpose_Opsimps_I1_J,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) % 0.31/0.65 => ( ( transpose @ A @ ( nil @ ( list @ A ) ) ) % 0.31/0.65 = ( nil @ ( list @ A ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose.psimps(1) % 0.31/0.65 thf(fact_7427_transpose_Opinduct,axiom, % 0.31/0.65 ! [A: $tType,A0: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > $o] : % 0.31/0.65 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ A0 ) % 0.31/0.65 => ( ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) % 0.31/0.65 => ( P @ ( nil @ ( list @ A ) ) ) ) % 0.31/0.65 => ( ! [Xss2: list @ ( list @ A )] : % 0.31/0.65 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) % 0.31/0.65 => ( ( P @ Xss2 ) % 0.31/0.65 => ( P @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) ) % 0.31/0.65 => ( ! [X5: A,Xs3: list @ A,Xss2: list @ ( list @ A )] : % 0.31/0.65 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) ) % 0.31/0.65 => ( ( P % 0.31/0.65 @ ( cons @ ( list @ A ) @ Xs3 % 0.31/0.65 @ ( concat @ ( list @ A ) % 0.31/0.65 @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) ) % 0.31/0.65 @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) ) % 0.31/0.65 @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) ) % 0.31/0.65 @ Xss2 ) ) ) ) % 0.31/0.65 => ( P @ ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) ) ) ) % 0.31/0.65 => ( P @ A0 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose.pinduct % 0.31/0.65 thf(fact_7428_remdups__adj_Opelims,axiom, % 0.31/0.65 ! [A: $tType,X3: list @ A,Y: list @ A] : % 0.31/0.65 ( ( ( remdups_adj @ A @ X3 ) % 0.31/0.65 = Y ) % 0.31/0.65 => ( ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ X3 ) % 0.31/0.65 => ( ( ( X3 % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 => ( ( Y % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 => ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( nil @ A ) ) ) ) % 0.31/0.65 => ( ! [X5: A] : % 0.31/0.65 ( ( X3 % 0.31/0.65 = ( cons @ A @ X5 @ ( nil @ A ) ) ) % 0.31/0.65 => ( ( Y % 0.31/0.65 = ( cons @ A @ X5 @ ( nil @ A ) ) ) % 0.31/0.65 => ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) ) % 0.31/0.65 => ~ ! [X5: A,Y4: A,Xs3: list @ A] : % 0.31/0.65 ( ( X3 % 0.31/0.65 = ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Xs3 ) ) ) % 0.31/0.65 => ( ( ( ( X5 = Y4 ) % 0.31/0.65 => ( Y % 0.31/0.65 = ( remdups_adj @ A @ ( cons @ A @ X5 @ Xs3 ) ) ) ) % 0.31/0.65 & ( ( X5 != Y4 ) % 0.31/0.65 => ( Y % 0.31/0.65 = ( cons @ A @ X5 @ ( remdups_adj @ A @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) ) ) % 0.31/0.65 => ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % remdups_adj.pelims % 0.31/0.65 thf(fact_7429_take__hd__drop,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( cons @ A @ ( hd @ A @ ( drop @ A @ N2 @ Xs ) ) @ ( nil @ A ) ) ) % 0.31/0.65 = ( take @ A @ ( suc @ N2 ) @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % take_hd_drop % 0.31/0.65 thf(fact_7430_hd__remdups__adj,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( hd @ A @ ( remdups_adj @ A @ Xs ) ) % 0.31/0.65 = ( hd @ A @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % hd_remdups_adj % 0.31/0.65 thf(fact_7431_hd__append2,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( Xs % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( hd @ A @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( hd @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % hd_append2 % 0.31/0.65 thf(fact_7432_hd__replicate,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,X3: A] : % 0.31/0.65 ( ( N2 % 0.31/0.65 != ( zero_zero @ nat ) ) % 0.31/0.65 => ( ( hd @ A @ ( replicate @ A @ N2 @ X3 ) ) % 0.31/0.65 = X3 ) ) ). % 0.31/0.65 % 0.31/0.65 % hd_replicate % 0.31/0.65 thf(fact_7433_hd__take,axiom, % 0.31/0.65 ! [A: $tType,J: nat,Xs: list @ A] : % 0.31/0.65 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ J ) % 0.31/0.65 => ( ( hd @ A @ ( take @ A @ J @ Xs ) ) % 0.31/0.65 = ( hd @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % hd_take % 0.31/0.65 thf(fact_7434_hd__in__set,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( Xs % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( member @ A @ ( hd @ A @ Xs ) @ ( set2 @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % hd_in_set % 0.31/0.65 thf(fact_7435_list_Oset__sel_I1_J,axiom, % 0.31/0.65 ! [A: $tType,A2: list @ A] : % 0.31/0.65 ( ( A2 % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( member @ A @ ( hd @ A @ A2 ) @ ( set2 @ A @ A2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % list.set_sel(1) % 0.31/0.65 thf(fact_7436_hd__append,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( ( Xs % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 => ( ( hd @ A @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( hd @ A @ Ys ) ) ) % 0.31/0.65 & ( ( Xs % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( hd @ A @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( hd @ A @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % hd_append % 0.31/0.65 thf(fact_7437_longest__common__prefix,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ? [Ps2: list @ A,Xs4: list @ A,Ys5: list @ A] : % 0.31/0.65 ( ( Xs % 0.31/0.65 = ( append @ A @ Ps2 @ Xs4 ) ) % 0.31/0.65 & ( Ys % 0.31/0.65 = ( append @ A @ Ps2 @ Ys5 ) ) % 0.31/0.65 & ( ( Xs4 % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 | ( Ys5 % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 | ( ( hd @ A @ Xs4 ) % 0.31/0.65 != ( hd @ A @ Ys5 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % longest_common_prefix % 0.31/0.65 thf(fact_7438_list_Osel_I1_J,axiom, % 0.31/0.65 ! [A: $tType,X21: A,X222: list @ A] : % 0.31/0.65 ( ( hd @ A @ ( cons @ A @ X21 @ X222 ) ) % 0.31/0.65 = X21 ) ). % 0.31/0.65 % 0.31/0.65 % list.sel(1) % 0.31/0.65 thf(fact_7439_hd__zip,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 0.31/0.65 ( ( Xs % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( Ys % 0.31/0.65 != ( nil @ B ) ) % 0.31/0.65 => ( ( hd @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) % 0.31/0.65 = ( product_Pair @ A @ B @ ( hd @ A @ Xs ) @ ( hd @ B @ Ys ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % hd_zip % 0.31/0.65 thf(fact_7440_hd__concat,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ ( list @ A )] : % 0.31/0.65 ( ( Xs % 0.31/0.65 != ( nil @ ( list @ A ) ) ) % 0.31/0.65 => ( ( ( hd @ ( list @ A ) @ Xs ) % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( hd @ A @ ( concat @ A @ Xs ) ) % 0.31/0.65 = ( hd @ A @ ( hd @ ( list @ A ) @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % hd_concat % 0.31/0.65 thf(fact_7441_hd__map,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,Xs: list @ A,F2: A > B] : % 0.31/0.65 ( ( Xs % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( hd @ B @ ( map @ A @ B @ F2 @ Xs ) ) % 0.31/0.65 = ( F2 @ ( hd @ A @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % hd_map % 0.31/0.65 thf(fact_7442_list_Omap__sel_I1_J,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A2: list @ A,F2: A > B] : % 0.31/0.65 ( ( A2 % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( hd @ B @ ( map @ A @ B @ F2 @ A2 ) ) % 0.31/0.65 = ( F2 @ ( hd @ A @ A2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % list.map_sel(1) % 0.31/0.65 thf(fact_7443_hd__conv__nth,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( Xs % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( hd @ A @ Xs ) % 0.31/0.65 = ( nth @ A @ Xs @ ( zero_zero @ nat ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % hd_conv_nth % 0.31/0.65 thf(fact_7444_hd__drop__conv__nth,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ( hd @ A @ ( drop @ A @ N2 @ Xs ) ) % 0.31/0.65 = ( nth @ A @ Xs @ N2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % hd_drop_conv_nth % 0.31/0.65 thf(fact_7445_remdups__adj__singleton__iff,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) % 0.31/0.65 = ( suc @ ( zero_zero @ nat ) ) ) % 0.31/0.65 = ( ( Xs % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 & ( Xs % 0.31/0.65 = ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ ( hd @ A @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % remdups_adj_singleton_iff % 0.31/0.65 thf(fact_7446_list__encode_Opelims,axiom, % 0.31/0.65 ! [X3: list @ nat,Y: nat] : % 0.31/0.65 ( ( ( nat_list_encode @ X3 ) % 0.31/0.65 = Y ) % 0.31/0.65 => ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X3 ) % 0.31/0.65 => ( ( ( X3 % 0.31/0.65 = ( nil @ nat ) ) % 0.31/0.65 => ( ( Y % 0.31/0.65 = ( zero_zero @ nat ) ) % 0.31/0.65 => ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) ) % 0.31/0.65 => ~ ! [X5: nat,Xs3: list @ nat] : % 0.31/0.65 ( ( X3 % 0.31/0.65 = ( cons @ nat @ X5 @ Xs3 ) ) % 0.31/0.65 => ( ( Y % 0.31/0.65 = ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) ) % 0.31/0.65 => ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X5 @ Xs3 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % list_encode.pelims % 0.31/0.65 thf(fact_7447_horner__sum__bit__eq__take__bit,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( bit_se359711467146920520ations @ A ) % 0.31/0.65 => ! [A2: A,N2: nat] : % 0.31/0.65 ( ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) % 0.31/0.65 = ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % horner_sum_bit_eq_take_bit % 0.31/0.65 thf(fact_7448_remdups__upt,axiom, % 0.31/0.65 ! [M: nat,N2: nat] : % 0.31/0.65 ( ( remdups @ nat @ ( upt @ M @ N2 ) ) % 0.31/0.65 = ( upt @ M @ N2 ) ) ). % 0.31/0.65 % 0.31/0.65 % remdups_upt % 0.31/0.65 thf(fact_7449_hd__upt,axiom, % 0.31/0.65 ! [I: nat,J: nat] : % 0.31/0.65 ( ( ord_less @ nat @ I @ J ) % 0.31/0.65 => ( ( hd @ nat @ ( upt @ I @ J ) ) % 0.31/0.65 = I ) ) ). % 0.31/0.65 % 0.31/0.65 % hd_upt % 0.31/0.65 thf(fact_7450_drop__upt,axiom, % 0.31/0.65 ! [M: nat,I: nat,J: nat] : % 0.31/0.65 ( ( drop @ nat @ M @ ( upt @ I @ J ) ) % 0.31/0.65 = ( upt @ ( plus_plus @ nat @ I @ M ) @ J ) ) ). % 0.31/0.65 % 0.31/0.65 % drop_upt % 0.31/0.65 thf(fact_7451_length__upt,axiom, % 0.31/0.65 ! [I: nat,J: nat] : % 0.31/0.65 ( ( size_size @ ( list @ nat ) @ ( upt @ I @ J ) ) % 0.31/0.65 = ( minus_minus @ nat @ J @ I ) ) ). % 0.31/0.65 % 0.31/0.65 % length_upt % 0.31/0.65 thf(fact_7452_take__upt,axiom, % 0.31/0.65 ! [I: nat,M: nat,N2: nat] : % 0.31/0.65 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ M ) @ N2 ) % 0.31/0.65 => ( ( take @ nat @ M @ ( upt @ I @ N2 ) ) % 0.31/0.65 = ( upt @ I @ ( plus_plus @ nat @ I @ M ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % take_upt % 0.31/0.65 thf(fact_7453_upt__conv__Nil,axiom, % 0.31/0.65 ! [J: nat,I: nat] : % 0.31/0.65 ( ( ord_less_eq @ nat @ J @ I ) % 0.31/0.65 => ( ( upt @ I @ J ) % 0.31/0.65 = ( nil @ nat ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upt_conv_Nil % 0.31/0.65 thf(fact_7454_sorted__list__of__set__range,axiom, % 0.31/0.65 ! [M: nat,N2: nat] : % 0.31/0.65 ( ( linord4507533701916653071of_set @ nat @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) % 0.31/0.65 = ( upt @ M @ N2 ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_list_of_set_range % 0.31/0.65 thf(fact_7455_upt__eq__Nil__conv,axiom, % 0.31/0.65 ! [I: nat,J: nat] : % 0.31/0.65 ( ( ( upt @ I @ J ) % 0.31/0.65 = ( nil @ nat ) ) % 0.31/0.65 = ( ( J % 0.31/0.65 = ( zero_zero @ nat ) ) % 0.31/0.65 | ( ord_less_eq @ nat @ J @ I ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upt_eq_Nil_conv % 0.31/0.65 thf(fact_7456_nth__upt,axiom, % 0.31/0.65 ! [I: nat,K: nat,J: nat] : % 0.31/0.65 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) % 0.31/0.65 => ( ( nth @ nat @ ( upt @ I @ J ) @ K ) % 0.31/0.65 = ( plus_plus @ nat @ I @ K ) ) ) ). % 0.31/0.65 % 0.31/0.65 % nth_upt % 0.31/0.65 thf(fact_7457_map__fst__enumerate,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) ) % 0.31/0.65 = ( upt @ N2 @ ( plus_plus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % map_fst_enumerate % 0.31/0.65 thf(fact_7458_upt__rec__numeral,axiom, % 0.31/0.65 ! [M: num,N2: num] : % 0.31/0.65 ( ( ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) % 0.31/0.65 => ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) % 0.31/0.65 = ( cons @ nat @ ( numeral_numeral @ nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N2 ) ) ) ) ) % 0.31/0.65 & ( ~ ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) % 0.31/0.65 => ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) % 0.31/0.65 = ( nil @ nat ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upt_rec_numeral % 0.31/0.65 thf(fact_7459_upt__0,axiom, % 0.31/0.65 ! [I: nat] : % 0.31/0.65 ( ( upt @ I @ ( zero_zero @ nat ) ) % 0.31/0.65 = ( nil @ nat ) ) ). % 0.31/0.65 % 0.31/0.65 % upt_0 % 0.31/0.65 thf(fact_7460_upt__conv__Cons__Cons,axiom, % 0.31/0.65 ! [M: nat,N2: nat,Ns: list @ nat,Q4: nat] : % 0.31/0.65 ( ( ( cons @ nat @ M @ ( cons @ nat @ N2 @ Ns ) ) % 0.31/0.65 = ( upt @ M @ Q4 ) ) % 0.31/0.65 = ( ( cons @ nat @ N2 @ Ns ) % 0.31/0.65 = ( upt @ ( suc @ M ) @ Q4 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upt_conv_Cons_Cons % 0.31/0.65 thf(fact_7461_enumerate__map__upt,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,F2: nat > A,M: nat] : % 0.31/0.65 ( ( enumerate @ A @ N2 @ ( map @ nat @ A @ F2 @ ( upt @ N2 @ M ) ) ) % 0.31/0.65 = ( map @ nat @ ( product_prod @ nat @ A ) % 0.31/0.65 @ ^ [K3: nat] : ( product_Pair @ nat @ A @ K3 @ ( F2 @ K3 ) ) % 0.31/0.65 @ ( upt @ N2 @ M ) ) ) ). % 0.31/0.65 % 0.31/0.65 % enumerate_map_upt % 0.31/0.65 thf(fact_7462_map__replicate__trivial,axiom, % 0.31/0.65 ! [A: $tType,X3: A,I: nat] : % 0.31/0.65 ( ( map @ nat @ A % 0.31/0.65 @ ^ [I3: nat] : X3 % 0.31/0.65 @ ( upt @ ( zero_zero @ nat ) @ I ) ) % 0.31/0.65 = ( replicate @ A @ I @ X3 ) ) ). % 0.31/0.65 % 0.31/0.65 % map_replicate_trivial % 0.31/0.65 thf(fact_7463_map__add__upt,axiom, % 0.31/0.65 ! [N2: nat,M: nat] : % 0.31/0.65 ( ( map @ nat @ nat % 0.31/0.65 @ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ N2 ) % 0.31/0.65 @ ( upt @ ( zero_zero @ nat ) @ M ) ) % 0.31/0.65 = ( upt @ N2 @ ( plus_plus @ nat @ M @ N2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % map_add_upt % 0.31/0.65 thf(fact_7464_map__Suc__upt,axiom, % 0.31/0.65 ! [M: nat,N2: nat] : % 0.31/0.65 ( ( map @ nat @ nat @ suc @ ( upt @ M @ N2 ) ) % 0.31/0.65 = ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % map_Suc_upt % 0.31/0.65 thf(fact_7465_atMost__upto,axiom, % 0.31/0.65 ( ( set_ord_atMost @ nat ) % 0.31/0.65 = ( ^ [N: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % atMost_upto % 0.31/0.65 thf(fact_7466_greaterThanAtMost__upt,axiom, % 0.31/0.65 ( ( set_or3652927894154168847AtMost @ nat ) % 0.31/0.65 = ( ^ [N: nat,M2: nat] : ( set2 @ nat @ ( upt @ ( suc @ N ) @ ( suc @ M2 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % greaterThanAtMost_upt % 0.31/0.65 thf(fact_7467_atLeast__upt,axiom, % 0.31/0.65 ( ( set_ord_lessThan @ nat ) % 0.31/0.65 = ( ^ [N: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % atLeast_upt % 0.31/0.65 thf(fact_7468_atLeastAtMost__upt,axiom, % 0.31/0.65 ( ( set_or1337092689740270186AtMost @ nat ) % 0.31/0.65 = ( ^ [N: nat,M2: nat] : ( set2 @ nat @ ( upt @ N @ ( suc @ M2 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % atLeastAtMost_upt % 0.31/0.65 thf(fact_7469_atLeastLessThan__upt,axiom, % 0.31/0.65 ( ( set_or7035219750837199246ssThan @ nat ) % 0.31/0.65 = ( ^ [I3: nat,J3: nat] : ( set2 @ nat @ ( upt @ I3 @ J3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % atLeastLessThan_upt % 0.31/0.65 thf(fact_7470_greaterThanLessThan__upt,axiom, % 0.31/0.65 ( ( set_or5935395276787703475ssThan @ nat ) % 0.31/0.65 = ( ^ [N: nat,M2: nat] : ( set2 @ nat @ ( upt @ ( suc @ N ) @ M2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % greaterThanLessThan_upt % 0.31/0.65 thf(fact_7471_distinct__upt,axiom, % 0.31/0.65 ! [I: nat,J: nat] : ( distinct @ nat @ ( upt @ I @ J ) ) ). % 0.31/0.65 % 0.31/0.65 % distinct_upt % 0.31/0.65 thf(fact_7472_upt__conv__Cons,axiom, % 0.31/0.65 ! [I: nat,J: nat] : % 0.31/0.65 ( ( ord_less @ nat @ I @ J ) % 0.31/0.65 => ( ( upt @ I @ J ) % 0.31/0.65 = ( cons @ nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upt_conv_Cons % 0.31/0.65 thf(fact_7473_enumerate__eq__zip,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( enumerate @ A ) % 0.31/0.65 = ( ^ [N: nat,Xs2: list @ A] : ( zip @ nat @ A @ ( upt @ N @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) @ Xs2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % enumerate_eq_zip % 0.31/0.65 thf(fact_7474_map__upt__Suc,axiom, % 0.31/0.65 ! [A: $tType,F2: nat > A,N2: nat] : % 0.31/0.65 ( ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 0.31/0.65 = ( cons @ A @ ( F2 @ ( zero_zero @ nat ) ) % 0.31/0.65 @ ( map @ nat @ A % 0.31/0.65 @ ^ [I3: nat] : ( F2 @ ( suc @ I3 ) ) % 0.31/0.65 @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % map_upt_Suc % 0.31/0.65 thf(fact_7475_map__decr__upt,axiom, % 0.31/0.65 ! [M: nat,N2: nat] : % 0.31/0.65 ( ( map @ nat @ nat % 0.31/0.65 @ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) % 0.31/0.65 @ ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 0.31/0.65 = ( upt @ M @ N2 ) ) ). % 0.31/0.65 % 0.31/0.65 % map_decr_upt % 0.31/0.65 thf(fact_7476_map__nth,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( map @ nat @ A @ ( nth @ A @ Xs ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) % 0.31/0.65 = Xs ) ). % 0.31/0.65 % 0.31/0.65 % map_nth % 0.31/0.65 thf(fact_7477_upt__add__eq__append,axiom, % 0.31/0.65 ! [I: nat,J: nat,K: nat] : % 0.31/0.65 ( ( ord_less_eq @ nat @ I @ J ) % 0.31/0.65 => ( ( upt @ I @ ( plus_plus @ nat @ J @ K ) ) % 0.31/0.65 = ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upt_add_eq_append % 0.31/0.65 thf(fact_7478_nth__map__upt,axiom, % 0.31/0.65 ! [A: $tType,I: nat,N2: nat,M: nat,F2: nat > A] : % 0.31/0.65 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N2 @ M ) ) % 0.31/0.65 => ( ( nth @ A @ ( map @ nat @ A @ F2 @ ( upt @ M @ N2 ) ) @ I ) % 0.31/0.65 = ( F2 @ ( plus_plus @ nat @ M @ I ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % nth_map_upt % 0.31/0.65 thf(fact_7479_upt__eq__Cons__conv,axiom, % 0.31/0.65 ! [I: nat,J: nat,X3: nat,Xs: list @ nat] : % 0.31/0.65 ( ( ( upt @ I @ J ) % 0.31/0.65 = ( cons @ nat @ X3 @ Xs ) ) % 0.31/0.65 = ( ( ord_less @ nat @ I @ J ) % 0.31/0.65 & ( I = X3 ) % 0.31/0.65 & ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J ) % 0.31/0.65 = Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upt_eq_Cons_conv % 0.31/0.65 thf(fact_7480_upt__rec,axiom, % 0.31/0.65 ( upt % 0.31/0.65 = ( ^ [I3: nat,J3: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I3 @ J3 ) @ ( cons @ nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ ( nil @ nat ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upt_rec % 0.31/0.65 thf(fact_7481_enumerate__replicate__eq,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,M: nat,A2: A] : % 0.31/0.65 ( ( enumerate @ A @ N2 @ ( replicate @ A @ M @ A2 ) ) % 0.31/0.65 = ( map @ nat @ ( product_prod @ nat @ A ) % 0.31/0.65 @ ^ [Q5: nat] : ( product_Pair @ nat @ A @ Q5 @ A2 ) % 0.31/0.65 @ ( upt @ N2 @ ( plus_plus @ nat @ N2 @ M ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % enumerate_replicate_eq % 0.31/0.65 thf(fact_7482_map__upt__eqI,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,N2: nat,M: nat,F2: nat > A] : % 0.31/0.65 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.65 = ( minus_minus @ nat @ N2 @ M ) ) % 0.31/0.65 => ( ! [I4: nat] : % 0.31/0.65 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ( nth @ A @ Xs @ I4 ) % 0.31/0.65 = ( F2 @ ( plus_plus @ nat @ M @ I4 ) ) ) ) % 0.31/0.65 => ( ( map @ nat @ A @ F2 @ ( upt @ M @ N2 ) ) % 0.31/0.65 = Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % map_upt_eqI % 0.31/0.65 thf(fact_7483_upt__Suc,axiom, % 0.31/0.65 ! [I: nat,J: nat] : % 0.31/0.65 ( ( ( ord_less_eq @ nat @ I @ J ) % 0.31/0.65 => ( ( upt @ I @ ( suc @ J ) ) % 0.31/0.65 = ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) ) % 0.31/0.65 & ( ~ ( ord_less_eq @ nat @ I @ J ) % 0.31/0.65 => ( ( upt @ I @ ( suc @ J ) ) % 0.31/0.65 = ( nil @ nat ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upt_Suc % 0.31/0.65 thf(fact_7484_upt__Suc__append,axiom, % 0.31/0.65 ! [I: nat,J: nat] : % 0.31/0.65 ( ( ord_less_eq @ nat @ I @ J ) % 0.31/0.65 => ( ( upt @ I @ ( suc @ J ) ) % 0.31/0.65 = ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % upt_Suc_append % 0.31/0.65 thf(fact_7485_transpose__rectangle,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ ( list @ A ),N2: nat] : % 0.31/0.65 ( ( ( Xs % 0.31/0.65 = ( nil @ ( list @ A ) ) ) % 0.31/0.65 => ( N2 % 0.31/0.65 = ( zero_zero @ nat ) ) ) % 0.31/0.65 => ( ! [I4: nat] : % 0.31/0.65 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) % 0.31/0.65 => ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I4 ) ) % 0.31/0.65 = N2 ) ) % 0.31/0.65 => ( ( transpose @ A @ Xs ) % 0.31/0.65 = ( map @ nat @ ( list @ A ) % 0.31/0.65 @ ^ [I3: nat] : % 0.31/0.65 ( map @ nat @ A % 0.31/0.65 @ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J3 ) @ I3 ) % 0.31/0.65 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) ) % 0.31/0.65 @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose_rectangle % 0.31/0.65 thf(fact_7486_extract__SomeE,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A,Y: A,Zs2: list @ A] : % 0.31/0.65 ( ( ( extract @ A @ P @ Xs ) % 0.31/0.65 = ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs2 ) ) ) ) % 0.31/0.65 => ( ( Xs % 0.31/0.65 = ( append @ A @ Ys @ ( cons @ A @ Y @ Zs2 ) ) ) % 0.31/0.65 & ( P @ Y ) % 0.31/0.65 & ~ ? [X2: A] : % 0.31/0.65 ( ( member @ A @ X2 @ ( set2 @ A @ Ys ) ) % 0.31/0.65 & ( P @ X2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % extract_SomeE % 0.31/0.65 thf(fact_7487_extract__Some__iff,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A,Y: A,Zs2: list @ A] : % 0.31/0.65 ( ( ( extract @ A @ P @ Xs ) % 0.31/0.65 = ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs2 ) ) ) ) % 0.31/0.65 = ( ( Xs % 0.31/0.65 = ( append @ A @ Ys @ ( cons @ A @ Y @ Zs2 ) ) ) % 0.31/0.65 & ( P @ Y ) % 0.31/0.65 & ~ ? [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Ys ) ) % 0.31/0.65 & ( P @ X ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % extract_Some_iff % 0.31/0.65 thf(fact_7488_extract__None__iff,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( ( extract @ A @ P @ Xs ) % 0.31/0.65 = ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) % 0.31/0.65 = ( ~ ? [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.65 & ( P @ X ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % extract_None_iff % 0.31/0.65 thf(fact_7489_extract__Nil__code,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o] : % 0.31/0.65 ( ( extract @ A @ P @ ( nil @ A ) ) % 0.31/0.65 = ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % extract_Nil_code % 0.31/0.65 thf(fact_7490_extract__Cons__code,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( ( P @ X3 ) % 0.31/0.65 => ( ( extract @ A @ P @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( nil @ A ) @ ( product_Pair @ A @ ( list @ A ) @ X3 @ Xs ) ) ) ) ) % 0.31/0.65 & ( ~ ( P @ X3 ) % 0.31/0.65 => ( ( extract @ A @ P @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( case_option @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) % 0.31/0.65 @ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) % 0.31/0.65 @ ^ [Ys3: list @ A] : % 0.31/0.65 ( product_case_prod @ A @ ( list @ A ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) % 0.31/0.65 @ ^ [Y2: A,Zs3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( cons @ A @ X3 @ Ys3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y2 @ Zs3 ) ) ) ) ) % 0.31/0.65 @ ( extract @ A @ P @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % extract_Cons_code % 0.31/0.65 thf(fact_7491_sum__list__map__eq__sum__count2,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,X8: set @ A,F2: A > nat] : % 0.31/0.65 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X8 ) % 0.31/0.65 => ( ( finite_finite2 @ A @ X8 ) % 0.31/0.65 => ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) ) % 0.31/0.65 = ( groups7311177749621191930dd_sum @ A @ nat % 0.31/0.65 @ ^ [X: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X ) @ ( F2 @ X ) ) % 0.31/0.65 @ X8 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_map_eq_sum_count2 % 0.31/0.65 thf(fact_7492_nth__transpose,axiom, % 0.31/0.65 ! [A: $tType,I: nat,Xs: list @ ( list @ A )] : % 0.31/0.65 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) ) % 0.31/0.65 => ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I ) % 0.31/0.65 = ( map @ ( list @ A ) @ A % 0.31/0.65 @ ^ [Xs2: list @ A] : ( nth @ A @ Xs2 @ I ) % 0.31/0.65 @ ( filter2 @ ( list @ A ) % 0.31/0.65 @ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) ) % 0.31/0.65 @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % nth_transpose % 0.31/0.65 thf(fact_7493_filter__filter,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Q: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( filter2 @ A @ P @ ( filter2 @ A @ Q @ Xs ) ) % 0.31/0.65 = ( filter2 @ A % 0.31/0.65 @ ^ [X: A] : % 0.31/0.65 ( ( Q @ X ) % 0.31/0.65 & ( P @ X ) ) % 0.31/0.65 @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_filter % 0.31/0.65 thf(fact_7494_filter__True,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.31/0.65 ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( P @ X5 ) ) % 0.31/0.65 => ( ( filter2 @ A @ P @ Xs ) % 0.31/0.65 = Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_True % 0.31/0.65 thf(fact_7495_filter__append,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( filter2 @ A @ P @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( append @ A @ ( filter2 @ A @ P @ Xs ) @ ( filter2 @ A @ P @ Ys ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_append % 0.31/0.65 thf(fact_7496_remove1__filter__not,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,X3: A,Xs: list @ A] : % 0.31/0.65 ( ~ ( P @ X3 ) % 0.31/0.65 => ( ( remove1 @ A @ X3 @ ( filter2 @ A @ P @ Xs ) ) % 0.31/0.65 = ( filter2 @ A @ P @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % remove1_filter_not % 0.31/0.65 thf(fact_7497_removeAll__filter__not,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,X3: A,Xs: list @ A] : % 0.31/0.65 ( ~ ( P @ X3 ) % 0.31/0.65 => ( ( removeAll @ A @ X3 @ ( filter2 @ A @ P @ Xs ) ) % 0.31/0.65 = ( filter2 @ A @ P @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % removeAll_filter_not % 0.31/0.65 thf(fact_7498_set__filter,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( set2 @ A @ ( filter2 @ A @ P @ Xs ) ) % 0.31/0.65 = ( collect @ A % 0.31/0.65 @ ^ [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.65 & ( P @ X ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % set_filter % 0.31/0.65 thf(fact_7499_sum__list_ONil,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( monoid_add @ A ) % 0.31/0.65 => ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) ) % 0.31/0.65 = ( zero_zero @ A ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list.Nil % 0.31/0.65 thf(fact_7500_sum__list__eq__0__iff,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( canoni5634975068530333245id_add @ A ) % 0.31/0.65 => ! [Ns: list @ A] : % 0.31/0.65 ( ( ( groups8242544230860333062m_list @ A @ Ns ) % 0.31/0.65 = ( zero_zero @ A ) ) % 0.31/0.65 = ( ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Ns ) ) % 0.31/0.65 => ( X % 0.31/0.65 = ( zero_zero @ A ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_eq_0_iff % 0.31/0.65 thf(fact_7501_sum__list_OCons,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( monoid_add @ A ) % 0.31/0.65 => ! [X3: A,Xs: list @ A] : % 0.31/0.65 ( ( groups8242544230860333062m_list @ A @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( plus_plus @ A @ X3 @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list.Cons % 0.31/0.65 thf(fact_7502_filter__False,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.31/0.65 ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ~ ( P @ X5 ) ) % 0.31/0.65 => ( ( filter2 @ A @ P @ Xs ) % 0.31/0.65 = ( nil @ A ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_False % 0.31/0.65 thf(fact_7503_sum__list__append,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( monoid_add @ A ) % 0.31/0.65 => ! [Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( groups8242544230860333062m_list @ A @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( groups8242544230860333062m_list @ A @ Ys ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_append % 0.31/0.65 thf(fact_7504_sum__list__0,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( monoid_add @ A ) % 0.31/0.65 => ! [Xs: list @ B] : % 0.31/0.65 ( ( groups8242544230860333062m_list @ A % 0.31/0.65 @ ( map @ B @ A % 0.31/0.65 @ ^ [X: B] : ( zero_zero @ A ) % 0.31/0.65 @ Xs ) ) % 0.31/0.65 = ( zero_zero @ A ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_0 % 0.31/0.65 thf(fact_7505_length__filter__map,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] : % 0.31/0.65 ( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ ( map @ B @ A @ F2 @ Xs ) ) ) % 0.31/0.65 = ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % length_filter_map % 0.31/0.65 thf(fact_7506_sum__list__upt,axiom, % 0.31/0.65 ! [M: nat,N2: nat] : % 0.31/0.65 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.65 => ( ( groups8242544230860333062m_list @ nat @ ( upt @ M @ N2 ) ) % 0.31/0.65 = ( groups7311177749621191930dd_sum @ nat @ nat % 0.31/0.65 @ ^ [X: nat] : X % 0.31/0.65 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_upt % 0.31/0.65 thf(fact_7507_filter__map,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] : % 0.31/0.65 ( ( filter2 @ A @ P @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 = ( map @ B @ A @ F2 @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_map % 0.31/0.65 thf(fact_7508_sum__list__filter__le__nat,axiom, % 0.31/0.65 ! [A: $tType,F2: A > nat,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ ( filter2 @ A @ P @ Xs ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_filter_le_nat % 0.31/0.65 thf(fact_7509_sum__list__addf,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( comm_monoid_add @ A ) % 0.31/0.65 => ! [F2: B > A,G: B > A,Xs: list @ B] : % 0.31/0.65 ( ( groups8242544230860333062m_list @ A % 0.31/0.65 @ ( map @ B @ A % 0.31/0.65 @ ^ [X: B] : ( plus_plus @ A @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.65 @ Xs ) ) % 0.31/0.65 = ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_addf % 0.31/0.65 thf(fact_7510_sum__list__map__filter_H,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( monoid_add @ A ) % 0.31/0.65 => ! [F2: B > A,P: B > $o,Xs: list @ B] : % 0.31/0.65 ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) ) % 0.31/0.65 = ( groups8242544230860333062m_list @ A % 0.31/0.65 @ ( map @ B @ A % 0.31/0.65 @ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( F2 @ X ) @ ( zero_zero @ A ) ) % 0.31/0.65 @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_map_filter' % 0.31/0.65 thf(fact_7511_filter__concat,axiom, % 0.31/0.65 ! [A: $tType,P4: A > $o,Xs: list @ ( list @ A )] : % 0.31/0.65 ( ( filter2 @ A @ P4 @ ( concat @ A @ Xs ) ) % 0.31/0.65 = ( concat @ A @ ( map @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ P4 ) @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_concat % 0.31/0.65 thf(fact_7512_sum__list__map__filter,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( monoid_add @ A ) % 0.31/0.65 => ! [Xs: list @ B,P: B > $o,F2: B > A] : % 0.31/0.65 ( ! [X5: B] : % 0.31/0.65 ( ( member @ B @ X5 @ ( set2 @ B @ Xs ) ) % 0.31/0.65 => ( ~ ( P @ X5 ) % 0.31/0.65 => ( ( F2 @ X5 ) % 0.31/0.65 = ( zero_zero @ A ) ) ) ) % 0.31/0.65 => ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) ) % 0.31/0.65 = ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_map_filter % 0.31/0.65 thf(fact_7513_distinct__map__filter,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,P: B > $o] : % 0.31/0.65 ( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 => ( distinct @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % distinct_map_filter % 0.31/0.65 thf(fact_7514_length__concat,axiom, % 0.31/0.65 ! [B: $tType,Xss: list @ ( list @ B )] : % 0.31/0.65 ( ( size_size @ ( list @ B ) @ ( concat @ B @ Xss ) ) % 0.31/0.65 = ( groups8242544230860333062m_list @ nat @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) ) ) ). % 0.31/0.65 % 0.31/0.65 % length_concat % 0.31/0.65 thf(fact_7515_sum__list__nonpos,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( ordere6911136660526730532id_add @ A ) % 0.31/0.65 => ! [Xs: list @ A] : % 0.31/0.65 ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( ord_less_eq @ A @ X5 @ ( zero_zero @ A ) ) ) % 0.31/0.65 => ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( zero_zero @ A ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_nonpos % 0.31/0.65 thf(fact_7516_sum__list__nonneg__eq__0__iff,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( ordere6911136660526730532id_add @ A ) % 0.31/0.65 => ! [Xs: list @ A] : % 0.31/0.65 ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 ) ) % 0.31/0.65 => ( ( ( groups8242544230860333062m_list @ A @ Xs ) % 0.31/0.65 = ( zero_zero @ A ) ) % 0.31/0.65 = ( ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( X % 0.31/0.65 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_nonneg_eq_0_iff % 0.31/0.65 thf(fact_7517_Groups__List_Osum__list__nonneg,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( ordere6911136660526730532id_add @ A ) % 0.31/0.65 => ! [Xs: list @ A] : % 0.31/0.65 ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 ) ) % 0.31/0.65 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Groups_List.sum_list_nonneg % 0.31/0.65 thf(fact_7518_empty__filter__conv,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( ( nil @ A ) % 0.31/0.65 = ( filter2 @ A @ P @ Xs ) ) % 0.31/0.65 = ( ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ~ ( P @ X ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % empty_filter_conv % 0.31/0.65 thf(fact_7519_filter__empty__conv,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( ( filter2 @ A @ P @ Xs ) % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 = ( ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ~ ( P @ X ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_empty_conv % 0.31/0.65 thf(fact_7520_Cons__eq__filterD,axiom, % 0.31/0.65 ! [A: $tType,X3: A,Xs: list @ A,P: A > $o,Ys: list @ A] : % 0.31/0.65 ( ( ( cons @ A @ X3 @ Xs ) % 0.31/0.65 = ( filter2 @ A @ P @ Ys ) ) % 0.31/0.65 => ? [Us3: list @ A,Vs3: list @ A] : % 0.31/0.65 ( ( Ys % 0.31/0.65 = ( append @ A @ Us3 @ ( cons @ A @ X3 @ Vs3 ) ) ) % 0.31/0.65 & ! [X2: A] : % 0.31/0.65 ( ( member @ A @ X2 @ ( set2 @ A @ Us3 ) ) % 0.31/0.65 => ~ ( P @ X2 ) ) % 0.31/0.65 & ( P @ X3 ) % 0.31/0.65 & ( Xs % 0.31/0.65 = ( filter2 @ A @ P @ Vs3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Cons_eq_filterD % 0.31/0.65 thf(fact_7521_filter__eq__ConsD,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Ys: list @ A,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( ( filter2 @ A @ P @ Ys ) % 0.31/0.65 = ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 => ? [Us3: list @ A,Vs3: list @ A] : % 0.31/0.65 ( ( Ys % 0.31/0.65 = ( append @ A @ Us3 @ ( cons @ A @ X3 @ Vs3 ) ) ) % 0.31/0.65 & ! [X2: A] : % 0.31/0.65 ( ( member @ A @ X2 @ ( set2 @ A @ Us3 ) ) % 0.31/0.65 => ~ ( P @ X2 ) ) % 0.31/0.65 & ( P @ X3 ) % 0.31/0.65 & ( Xs % 0.31/0.65 = ( filter2 @ A @ P @ Vs3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_eq_ConsD % 0.31/0.65 thf(fact_7522_Cons__eq__filter__iff,axiom, % 0.31/0.65 ! [A: $tType,X3: A,Xs: list @ A,P: A > $o,Ys: list @ A] : % 0.31/0.65 ( ( ( cons @ A @ X3 @ Xs ) % 0.31/0.65 = ( filter2 @ A @ P @ Ys ) ) % 0.31/0.65 = ( ? [Us2: list @ A,Vs: list @ A] : % 0.31/0.65 ( ( Ys % 0.31/0.65 = ( append @ A @ Us2 @ ( cons @ A @ X3 @ Vs ) ) ) % 0.31/0.65 & ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Us2 ) ) % 0.31/0.65 => ~ ( P @ X ) ) % 0.31/0.65 & ( P @ X3 ) % 0.31/0.65 & ( Xs % 0.31/0.65 = ( filter2 @ A @ P @ Vs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Cons_eq_filter_iff % 0.31/0.65 thf(fact_7523_filter__eq__Cons__iff,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Ys: list @ A,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( ( filter2 @ A @ P @ Ys ) % 0.31/0.65 = ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( ? [Us2: list @ A,Vs: list @ A] : % 0.31/0.65 ( ( Ys % 0.31/0.65 = ( append @ A @ Us2 @ ( cons @ A @ X3 @ Vs ) ) ) % 0.31/0.65 & ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Us2 ) ) % 0.31/0.65 => ~ ( P @ X ) ) % 0.31/0.65 & ( P @ X3 ) % 0.31/0.65 & ( Xs % 0.31/0.65 = ( filter2 @ A @ P @ Vs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_eq_Cons_iff % 0.31/0.65 thf(fact_7524_inj__on__filter__key__eq,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F2: A > B,Y: A,Xs: list @ A] : % 0.31/0.65 ( ( inj_on @ A @ B @ F2 @ ( insert @ A @ Y @ ( set2 @ A @ Xs ) ) ) % 0.31/0.65 => ( ( filter2 @ A % 0.31/0.65 @ ^ [X: A] : % 0.31/0.65 ( ( F2 @ Y ) % 0.31/0.65 = ( F2 @ X ) ) % 0.31/0.65 @ Xs ) % 0.31/0.65 = ( filter2 @ A % 0.31/0.65 @ ( ^ [Y3: A,Z: A] : ( Y3 = Z ) % 0.31/0.65 @ Y ) % 0.31/0.65 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % inj_on_filter_key_eq % 0.31/0.65 thf(fact_7525_member__le__sum__list,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( canoni5634975068530333245id_add @ A ) % 0.31/0.65 => ! [X3: A,Xs: list @ A] : % 0.31/0.65 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( ord_less_eq @ A @ X3 @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % member_le_sum_list % 0.31/0.65 thf(fact_7526_length__filter__le,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % length_filter_le % 0.31/0.65 thf(fact_7527_filter__is__subset,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( filter2 @ A @ P @ Xs ) ) @ ( set2 @ A @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_is_subset % 0.31/0.65 thf(fact_7528_distinct__filter,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.31/0.65 ( ( distinct @ A @ Xs ) % 0.31/0.65 => ( distinct @ A @ ( filter2 @ A @ P @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % distinct_filter % 0.31/0.65 thf(fact_7529_length__filter__less,axiom, % 0.31/0.65 ! [A: $tType,X3: A,Xs: list @ A,P: A > $o] : % 0.31/0.65 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( ~ ( P @ X3 ) % 0.31/0.65 => ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % length_filter_less % 0.31/0.65 thf(fact_7530_filter__set,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( filter3 @ A @ P @ ( set2 @ A @ Xs ) ) % 0.31/0.65 = ( set2 @ A @ ( filter2 @ A @ P @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_set % 0.31/0.65 thf(fact_7531_replicate__length__filter,axiom, % 0.31/0.65 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( replicate @ A % 0.31/0.65 @ ( size_size @ ( list @ A ) % 0.31/0.65 @ ( filter2 @ A % 0.31/0.65 @ ( ^ [Y3: A,Z: A] : ( Y3 = Z ) % 0.31/0.65 @ X3 ) % 0.31/0.65 @ Xs ) ) % 0.31/0.65 @ X3 ) % 0.31/0.65 = ( filter2 @ A % 0.31/0.65 @ ( ^ [Y3: A,Z: A] : ( Y3 = Z ) % 0.31/0.65 @ X3 ) % 0.31/0.65 @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % replicate_length_filter % 0.31/0.65 thf(fact_7532_filter__id__conv,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( ( filter2 @ A @ P @ Xs ) % 0.31/0.65 = Xs ) % 0.31/0.65 = ( ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( P @ X ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_id_conv % 0.31/0.65 thf(fact_7533_filter__cong,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] : % 0.31/0.65 ( ( Xs = Ys ) % 0.31/0.65 => ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ Ys ) ) % 0.31/0.65 => ( ( P @ X5 ) % 0.31/0.65 = ( Q @ X5 ) ) ) % 0.31/0.65 => ( ( filter2 @ A @ P @ Xs ) % 0.31/0.65 = ( filter2 @ A @ Q @ Ys ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_cong % 0.31/0.65 thf(fact_7534_inter__set__filter,axiom, % 0.31/0.65 ! [A: $tType,A3: set @ A,Xs: list @ A] : % 0.31/0.65 ( ( inf_inf @ ( set @ A ) @ A3 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 = ( set2 @ A % 0.31/0.65 @ ( filter2 @ A % 0.31/0.65 @ ^ [X: A] : ( member @ A @ X @ A3 ) % 0.31/0.65 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % inter_set_filter % 0.31/0.65 thf(fact_7535_sum__length__filter__compl,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) % 0.31/0.65 @ ( size_size @ ( list @ A ) % 0.31/0.65 @ ( filter2 @ A % 0.31/0.65 @ ^ [X: A] : % 0.31/0.65 ~ ( P @ X ) % 0.31/0.65 @ Xs ) ) ) % 0.31/0.65 = ( size_size @ ( list @ A ) @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_length_filter_compl % 0.31/0.65 thf(fact_7536_remdups__filter,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( remdups @ A @ ( filter2 @ A @ P @ Xs ) ) % 0.31/0.65 = ( filter2 @ A @ P @ ( remdups @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % remdups_filter % 0.31/0.65 thf(fact_7537_filter__replicate,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,X3: A,N2: nat] : % 0.31/0.65 ( ( ( P @ X3 ) % 0.31/0.65 => ( ( filter2 @ A @ P @ ( replicate @ A @ N2 @ X3 ) ) % 0.31/0.65 = ( replicate @ A @ N2 @ X3 ) ) ) % 0.31/0.65 & ( ~ ( P @ X3 ) % 0.31/0.65 => ( ( filter2 @ A @ P @ ( replicate @ A @ N2 @ X3 ) ) % 0.31/0.65 = ( nil @ A ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_replicate % 0.31/0.65 thf(fact_7538_filter_Osimps_I1_J,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o] : % 0.31/0.65 ( ( filter2 @ A @ P @ ( nil @ A ) ) % 0.31/0.65 = ( nil @ A ) ) ). % 0.31/0.65 % 0.31/0.65 % filter.simps(1) % 0.31/0.65 thf(fact_7539_filter_Osimps_I2_J,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( ( P @ X3 ) % 0.31/0.65 => ( ( filter2 @ A @ P @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( cons @ A @ X3 @ ( filter2 @ A @ P @ Xs ) ) ) ) % 0.31/0.65 & ( ~ ( P @ X3 ) % 0.31/0.65 => ( ( filter2 @ A @ P @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( filter2 @ A @ P @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter.simps(2) % 0.31/0.65 thf(fact_7540_filter__shuffles,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( image @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ P ) @ ( shuffles @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( shuffles @ A @ ( filter2 @ A @ P @ Xs ) @ ( filter2 @ A @ P @ Ys ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_shuffles % 0.31/0.65 thf(fact_7541_removeAll__filter__not__eq,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( removeAll @ A ) % 0.31/0.65 = ( ^ [X: A] : % 0.31/0.65 ( filter2 @ A % 0.31/0.65 @ ^ [Y2: A] : ( X != Y2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % removeAll_filter_not_eq % 0.31/0.65 thf(fact_7542_filter__remove1,axiom, % 0.31/0.65 ! [A: $tType,Q: A > $o,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( filter2 @ A @ Q @ ( remove1 @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( remove1 @ A @ X3 @ ( filter2 @ A @ Q @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_remove1 % 0.31/0.65 thf(fact_7543_partition__in__shuffles,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.31/0.65 ( member @ ( list @ A ) @ Xs % 0.31/0.65 @ ( shuffles @ A @ ( filter2 @ A @ P @ Xs ) % 0.31/0.65 @ ( filter2 @ A % 0.31/0.65 @ ^ [X: A] : % 0.31/0.65 ~ ( P @ X ) % 0.31/0.65 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % partition_in_shuffles % 0.31/0.65 thf(fact_7544_filter__insort__triv,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [P: B > $o,X3: B,F2: B > A,Xs: list @ B] : % 0.31/0.65 ( ~ ( P @ X3 ) % 0.31/0.65 => ( ( filter2 @ B @ P @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) % 0.31/0.65 = ( filter2 @ B @ P @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_insort_triv % 0.31/0.65 thf(fact_7545_sum__list__abs,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( ordere166539214618696060dd_abs @ A ) % 0.31/0.65 => ! [Xs: list @ A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( groups8242544230860333062m_list @ A @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ A @ A @ ( abs_abs @ A ) @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_abs % 0.31/0.65 thf(fact_7546_filter__in__nths,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,S: set @ nat] : % 0.31/0.65 ( ( distinct @ A @ Xs ) % 0.31/0.65 => ( ( filter2 @ A % 0.31/0.65 @ ^ [X: A] : ( member @ A @ X @ ( set2 @ A @ ( nths @ A @ Xs @ S ) ) ) % 0.31/0.65 @ Xs ) % 0.31/0.65 = ( nths @ A @ Xs @ S ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_in_nths % 0.31/0.65 thf(fact_7547_sum__list__replicate,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( semiring_1 @ A ) % 0.31/0.65 => ! [N2: nat,C2: A] : % 0.31/0.65 ( ( groups8242544230860333062m_list @ A @ ( replicate @ A @ N2 @ C2 ) ) % 0.31/0.65 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ C2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_replicate % 0.31/0.65 thf(fact_7548_sum__list__mono,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( ( monoid_add @ B ) % 0.31/0.65 & ( ordere6658533253407199908up_add @ B ) ) % 0.31/0.65 => ! [Xs: list @ A,F2: A > B,G: A > B] : % 0.31/0.65 ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 0.31/0.65 => ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_mono % 0.31/0.65 thf(fact_7549_distinct__sum__list__conv__Sum,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( comm_monoid_add @ A ) % 0.31/0.65 => ! [Xs: list @ A] : % 0.31/0.65 ( ( distinct @ A @ Xs ) % 0.31/0.65 => ( ( groups8242544230860333062m_list @ A @ Xs ) % 0.31/0.65 = ( groups7311177749621191930dd_sum @ A @ A % 0.31/0.65 @ ^ [X: A] : X % 0.31/0.65 @ ( set2 @ A @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % distinct_sum_list_conv_Sum % 0.31/0.65 thf(fact_7550_set__minus__filter__out,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Y: A] : % 0.31/0.65 ( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.65 = ( set2 @ A % 0.31/0.65 @ ( filter2 @ A % 0.31/0.65 @ ^ [X: A] : ( X != Y ) % 0.31/0.65 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % set_minus_filter_out % 0.31/0.65 thf(fact_7551_filter__shuffles__disjoint1_I2_J,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] : % 0.31/0.65 ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.65 => ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) ) % 0.31/0.65 => ( ( filter2 @ A % 0.31/0.65 @ ^ [X: A] : % 0.31/0.65 ~ ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.65 @ Zs2 ) % 0.31/0.65 = Ys ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_shuffles_disjoint1(2) % 0.31/0.65 thf(fact_7552_filter__shuffles__disjoint1_I1_J,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] : % 0.31/0.65 ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.65 => ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) ) % 0.31/0.65 => ( ( filter2 @ A % 0.31/0.65 @ ^ [X: A] : ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.65 @ Zs2 ) % 0.31/0.65 = Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_shuffles_disjoint1(1) % 0.31/0.65 thf(fact_7553_filter__shuffles__disjoint2_I2_J,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] : % 0.31/0.65 ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.65 => ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) ) % 0.31/0.65 => ( ( filter2 @ A % 0.31/0.65 @ ^ [X: A] : % 0.31/0.65 ~ ( member @ A @ X @ ( set2 @ A @ Ys ) ) % 0.31/0.65 @ Zs2 ) % 0.31/0.65 = Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_shuffles_disjoint2(2) % 0.31/0.65 thf(fact_7554_filter__shuffles__disjoint2_I1_J,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] : % 0.31/0.65 ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.65 => ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) ) % 0.31/0.65 => ( ( filter2 @ A % 0.31/0.65 @ ^ [X: A] : ( member @ A @ X @ ( set2 @ A @ Ys ) ) % 0.31/0.65 @ Zs2 ) % 0.31/0.65 = Ys ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_shuffles_disjoint2(1) % 0.31/0.65 thf(fact_7555_filter__eq__nths,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( filter2 @ A ) % 0.31/0.65 = ( ^ [P3: A > $o,Xs2: list @ A] : % 0.31/0.65 ( nths @ A @ Xs2 % 0.31/0.65 @ ( collect @ nat % 0.31/0.65 @ ^ [I3: nat] : % 0.31/0.65 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) % 0.31/0.65 & ( P3 @ ( nth @ A @ Xs2 @ I3 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_eq_nths % 0.31/0.65 thf(fact_7556_length__filter__conv__card,axiom, % 0.31/0.65 ! [A: $tType,P4: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P4 @ Xs ) ) % 0.31/0.65 = ( finite_card @ nat % 0.31/0.65 @ ( collect @ nat % 0.31/0.65 @ ^ [I3: nat] : % 0.31/0.65 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 & ( P4 @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % length_filter_conv_card % 0.31/0.65 thf(fact_7557_sum__list__strict__mono,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( ( monoid_add @ B ) % 0.31/0.65 & ( strict9044650504122735259up_add @ B ) ) % 0.31/0.65 => ! [Xs: list @ A,F2: A > B,G: A > B] : % 0.31/0.65 ( ( Xs % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( ord_less @ B @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 0.31/0.65 => ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_strict_mono % 0.31/0.65 thf(fact_7558_elem__le__sum__list,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( canoni5634975068530333245id_add @ A ) % 0.31/0.65 => ! [K: nat,Ns: list @ A] : % 0.31/0.65 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Ns ) ) % 0.31/0.65 => ( ord_less_eq @ A @ ( nth @ A @ Ns @ K ) @ ( groups8242544230860333062m_list @ A @ Ns ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % elem_le_sum_list % 0.31/0.65 thf(fact_7559_sum__list__distinct__conv__sum__set,axiom, % 0.31/0.65 ! [C: $tType,B: $tType] : % 0.31/0.65 ( ( comm_monoid_add @ C ) % 0.31/0.65 => ! [Xs: list @ B,F2: B > C] : % 0.31/0.65 ( ( distinct @ B @ Xs ) % 0.31/0.65 => ( ( groups8242544230860333062m_list @ C @ ( map @ B @ C @ F2 @ Xs ) ) % 0.31/0.65 = ( groups7311177749621191930dd_sum @ B @ C @ F2 @ ( set2 @ B @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_distinct_conv_sum_set % 0.31/0.65 thf(fact_7560_sum_Odistinct__set__conv__list,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( comm_monoid_add @ A ) % 0.31/0.65 => ! [Xs: list @ B,G: B > A] : % 0.31/0.65 ( ( distinct @ B @ Xs ) % 0.31/0.65 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set2 @ B @ Xs ) ) % 0.31/0.65 = ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum.distinct_set_conv_list % 0.31/0.65 thf(fact_7561_sum__list__map__remove1,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( comm_monoid_add @ A ) % 0.31/0.65 => ! [X3: B,Xs: list @ B,F2: B > A] : % 0.31/0.65 ( ( member @ B @ X3 @ ( set2 @ B @ Xs ) ) % 0.31/0.65 => ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 = ( plus_plus @ A @ ( F2 @ X3 ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( remove1 @ B @ X3 @ Xs ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_map_remove1 % 0.31/0.65 thf(fact_7562_sum__code,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( comm_monoid_add @ A ) % 0.31/0.65 => ! [G: B > A,Xs: list @ B] : % 0.31/0.65 ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set2 @ B @ Xs ) ) % 0.31/0.65 = ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ ( remdups @ B @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_code % 0.31/0.65 thf(fact_7563_size__list__conv__sum__list,axiom, % 0.31/0.65 ! [B: $tType] : % 0.31/0.65 ( ( size_list @ B ) % 0.31/0.65 = ( ^ [F3: B > nat,Xs2: list @ B] : ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ B @ nat @ F3 @ Xs2 ) ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % size_list_conv_sum_list % 0.31/0.65 thf(fact_7564_sum__list__triv,axiom, % 0.31/0.65 ! [C: $tType,B: $tType] : % 0.31/0.65 ( ( semiring_1 @ B ) % 0.31/0.65 => ! [R2: B,Xs: list @ C] : % 0.31/0.65 ( ( groups8242544230860333062m_list @ B % 0.31/0.65 @ ( map @ C @ B % 0.31/0.65 @ ^ [X: C] : R2 % 0.31/0.65 @ Xs ) ) % 0.31/0.65 = ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_triv % 0.31/0.65 thf(fact_7565_sum__list__Suc,axiom, % 0.31/0.65 ! [A: $tType,F2: A > nat,Xs: list @ A] : % 0.31/0.65 ( ( groups8242544230860333062m_list @ nat % 0.31/0.65 @ ( map @ A @ nat % 0.31/0.65 @ ^ [X: A] : ( suc @ ( F2 @ X ) ) % 0.31/0.65 @ Xs ) ) % 0.31/0.65 = ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_Suc % 0.31/0.65 thf(fact_7566_distinct__length__filter,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.31/0.65 ( ( distinct @ A @ Xs ) % 0.31/0.65 => ( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) % 0.31/0.65 = ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( set2 @ A @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % distinct_length_filter % 0.31/0.65 thf(fact_7567_sum__list__sum__nth,axiom, % 0.31/0.65 ! [B: $tType] : % 0.31/0.65 ( ( comm_monoid_add @ B ) % 0.31/0.65 => ( ( groups8242544230860333062m_list @ B ) % 0.31/0.65 = ( ^ [Xs2: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_sum_nth % 0.31/0.65 thf(fact_7568_card__length__sum__list__rec,axiom, % 0.31/0.65 ! [M: nat,N7: nat] : % 0.31/0.65 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M ) % 0.31/0.65 => ( ( finite_card @ ( list @ nat ) % 0.31/0.65 @ ( collect @ ( list @ nat ) % 0.31/0.65 @ ^ [L2: list @ nat] : % 0.31/0.65 ( ( ( size_size @ ( list @ nat ) @ L2 ) % 0.31/0.65 = M ) % 0.31/0.65 & ( ( groups8242544230860333062m_list @ nat @ L2 ) % 0.31/0.65 = N7 ) ) ) ) % 0.31/0.65 = ( plus_plus @ nat % 0.31/0.65 @ ( finite_card @ ( list @ nat ) % 0.31/0.65 @ ( collect @ ( list @ nat ) % 0.31/0.65 @ ^ [L2: list @ nat] : % 0.31/0.65 ( ( ( size_size @ ( list @ nat ) @ L2 ) % 0.31/0.65 = ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) % 0.31/0.65 & ( ( groups8242544230860333062m_list @ nat @ L2 ) % 0.31/0.65 = N7 ) ) ) ) % 0.31/0.65 @ ( finite_card @ ( list @ nat ) % 0.31/0.65 @ ( collect @ ( list @ nat ) % 0.31/0.65 @ ^ [L2: list @ nat] : % 0.31/0.65 ( ( ( size_size @ ( list @ nat ) @ L2 ) % 0.31/0.65 = M ) % 0.31/0.65 & ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L2 ) @ ( one_one @ nat ) ) % 0.31/0.65 = N7 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % card_length_sum_list_rec % 0.31/0.65 thf(fact_7569_card__length__sum__list,axiom, % 0.31/0.65 ! [M: nat,N7: nat] : % 0.31/0.65 ( ( finite_card @ ( list @ nat ) % 0.31/0.65 @ ( collect @ ( list @ nat ) % 0.31/0.65 @ ^ [L2: list @ nat] : % 0.31/0.65 ( ( ( size_size @ ( list @ nat ) @ L2 ) % 0.31/0.65 = M ) % 0.31/0.65 & ( ( groups8242544230860333062m_list @ nat @ L2 ) % 0.31/0.65 = N7 ) ) ) ) % 0.31/0.65 = ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N7 @ M ) @ ( one_one @ nat ) ) @ N7 ) ) ). % 0.31/0.65 % 0.31/0.65 % card_length_sum_list % 0.31/0.65 thf(fact_7570_sum__list__map__eq__sum__count,axiom, % 0.31/0.65 ! [A: $tType,F2: A > nat,Xs: list @ A] : % 0.31/0.65 ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) ) % 0.31/0.65 = ( groups7311177749621191930dd_sum @ A @ nat % 0.31/0.65 @ ^ [X: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X ) @ ( F2 @ X ) ) % 0.31/0.65 @ ( set2 @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_map_eq_sum_count % 0.31/0.65 thf(fact_7571_sum__list__update,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( ordere1170586879665033532d_diff @ A ) % 0.31/0.65 => ! [K: nat,Xs: list @ A,X3: A] : % 0.31/0.65 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X3 ) ) % 0.31/0.65 = ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X3 ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list_update % 0.31/0.65 thf(fact_7572_transpose__aux__filter__head,axiom, % 0.31/0.65 ! [A: $tType,Xss: list @ ( list @ A )] : % 0.31/0.65 ( ( concat @ A % 0.31/0.65 @ ( map @ ( list @ A ) @ ( list @ A ) % 0.31/0.65 @ ( case_list @ ( list @ A ) @ A @ ( nil @ A ) % 0.31/0.65 @ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) ) % 0.31/0.65 @ Xss ) ) % 0.31/0.65 = ( map @ ( list @ A ) @ A @ ( hd @ A ) % 0.31/0.65 @ ( filter2 @ ( list @ A ) % 0.31/0.65 @ ^ [Ys3: list @ A] : % 0.31/0.65 ( Ys3 % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 @ Xss ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose_aux_filter_head % 0.31/0.65 thf(fact_7573_map__filter__def,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( map_filter @ A @ B ) % 0.31/0.65 = ( ^ [F3: A > ( option @ B ),Xs2: list @ A] : % 0.31/0.65 ( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F3 ) % 0.31/0.65 @ ( filter2 @ A % 0.31/0.65 @ ^ [X: A] : % 0.31/0.65 ( ( F3 @ X ) % 0.31/0.65 != ( none @ B ) ) % 0.31/0.65 @ Xs2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % map_filter_def % 0.31/0.65 thf(fact_7574_transpose__aux__filter__tail,axiom, % 0.31/0.65 ! [A: $tType,Xss: list @ ( list @ A )] : % 0.31/0.65 ( ( concat @ ( list @ A ) % 0.31/0.65 @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) ) % 0.31/0.65 @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) ) % 0.31/0.65 @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) ) % 0.31/0.65 @ Xss ) ) % 0.31/0.65 = ( map @ ( list @ A ) @ ( list @ A ) @ ( tl @ A ) % 0.31/0.65 @ ( filter2 @ ( list @ A ) % 0.31/0.65 @ ^ [Ys3: list @ A] : % 0.31/0.65 ( Ys3 % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 @ Xss ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose_aux_filter_tail % 0.31/0.65 thf(fact_7575_tl__upt,axiom, % 0.31/0.65 ! [M: nat,N2: nat] : % 0.31/0.65 ( ( tl @ nat @ ( upt @ M @ N2 ) ) % 0.31/0.65 = ( upt @ ( suc @ M ) @ N2 ) ) ). % 0.31/0.65 % 0.31/0.65 % tl_upt % 0.31/0.65 thf(fact_7576_tl__append2,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( Xs % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( tl @ A @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( append @ A @ ( tl @ A @ Xs ) @ Ys ) ) ) ). % 0.31/0.65 % 0.31/0.65 % tl_append2 % 0.31/0.65 thf(fact_7577_remdups__adj__Cons__alt,axiom, % 0.31/0.65 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( cons @ A @ X3 @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) ) % 0.31/0.65 = ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % remdups_adj_Cons_alt % 0.31/0.65 thf(fact_7578_length__tl,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) % 0.31/0.65 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ). % 0.31/0.65 % 0.31/0.65 % length_tl % 0.31/0.65 thf(fact_7579_list_Ocollapse,axiom, % 0.31/0.65 ! [A: $tType,List: list @ A] : % 0.31/0.65 ( ( List % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) % 0.31/0.65 = List ) ) ). % 0.31/0.65 % 0.31/0.65 % list.collapse % 0.31/0.65 thf(fact_7580_hd__Cons__tl,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( Xs % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( cons @ A @ ( hd @ A @ Xs ) @ ( tl @ A @ Xs ) ) % 0.31/0.65 = Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % hd_Cons_tl % 0.31/0.65 thf(fact_7581_tl__replicate,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,X3: A] : % 0.31/0.65 ( ( tl @ A @ ( replicate @ A @ N2 @ X3 ) ) % 0.31/0.65 = ( replicate @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ X3 ) ) ). % 0.31/0.65 % 0.31/0.65 % tl_replicate % 0.31/0.65 thf(fact_7582_list_Omap__sel_I2_J,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A2: list @ A,F2: A > B] : % 0.31/0.65 ( ( A2 % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( tl @ B @ ( map @ A @ B @ F2 @ A2 ) ) % 0.31/0.65 = ( map @ A @ B @ F2 @ ( tl @ A @ A2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % list.map_sel(2) % 0.31/0.65 thf(fact_7583_map__tl,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] : % 0.31/0.65 ( ( map @ B @ A @ F2 @ ( tl @ B @ Xs ) ) % 0.31/0.65 = ( tl @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % map_tl % 0.31/0.65 thf(fact_7584_map__of__filter__in,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Z2: A,P: B > A > $o] : % 0.31/0.65 ( ( ( map_of @ B @ A @ Xs @ K ) % 0.31/0.65 = ( some @ A @ Z2 ) ) % 0.31/0.65 => ( ( P @ K @ Z2 ) % 0.31/0.65 => ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) @ Xs ) @ K ) % 0.31/0.65 = ( some @ A @ Z2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % map_of_filter_in % 0.31/0.65 thf(fact_7585_tl__append,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( tl @ A @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( case_list @ ( list @ A ) @ A @ ( tl @ A @ Ys ) % 0.31/0.65 @ ^ [Z6: A,Zs3: list @ A] : ( append @ A @ Zs3 @ Ys ) % 0.31/0.65 @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % tl_append % 0.31/0.65 thf(fact_7586_list_Osel_I3_J,axiom, % 0.31/0.65 ! [A: $tType,X21: A,X222: list @ A] : % 0.31/0.65 ( ( tl @ A @ ( cons @ A @ X21 @ X222 ) ) % 0.31/0.65 = X222 ) ). % 0.31/0.65 % 0.31/0.65 % list.sel(3) % 0.31/0.65 thf(fact_7587_Nil__tl,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( ( nil @ A ) % 0.31/0.65 = ( tl @ A @ Xs ) ) % 0.31/0.65 = ( ( Xs % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 | ? [X: A] : % 0.31/0.65 ( Xs % 0.31/0.65 = ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Nil_tl % 0.31/0.65 thf(fact_7588_tl__Nil,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( ( tl @ A @ Xs ) % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 = ( ( Xs % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 | ? [X: A] : % 0.31/0.65 ( Xs % 0.31/0.65 = ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % tl_Nil % 0.31/0.65 thf(fact_7589_list_Osel_I2_J,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( tl @ A @ ( nil @ A ) ) % 0.31/0.65 = ( nil @ A ) ) ). % 0.31/0.65 % 0.31/0.65 % list.sel(2) % 0.31/0.65 thf(fact_7590_map__filter__simps_I2_J,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F2: B > ( option @ A )] : % 0.31/0.65 ( ( map_filter @ B @ A @ F2 @ ( nil @ B ) ) % 0.31/0.65 = ( nil @ A ) ) ). % 0.31/0.65 % 0.31/0.65 % map_filter_simps(2) % 0.31/0.65 thf(fact_7591_tl__def,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( tl @ A ) % 0.31/0.65 = ( case_list @ ( list @ A ) @ A @ ( nil @ A ) % 0.31/0.65 @ ^ [X213: A,X224: list @ A] : X224 ) ) ). % 0.31/0.65 % 0.31/0.65 % tl_def % 0.31/0.65 thf(fact_7592_take__tl,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( take @ A @ N2 @ ( tl @ A @ Xs ) ) % 0.31/0.65 = ( tl @ A @ ( take @ A @ ( suc @ N2 ) @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % take_tl % 0.31/0.65 thf(fact_7593_drop__Suc,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( drop @ A @ ( suc @ N2 ) @ Xs ) % 0.31/0.65 = ( drop @ A @ N2 @ ( tl @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % drop_Suc % 0.31/0.65 thf(fact_7594_tl__drop,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( tl @ A @ ( drop @ A @ N2 @ Xs ) ) % 0.31/0.65 = ( drop @ A @ N2 @ ( tl @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % tl_drop % 0.31/0.65 thf(fact_7595_distinct__tl,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( distinct @ A @ Xs ) % 0.31/0.65 => ( distinct @ A @ ( tl @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % distinct_tl % 0.31/0.65 thf(fact_7596_list_Oset__sel_I2_J,axiom, % 0.31/0.65 ! [A: $tType,A2: list @ A,X3: A] : % 0.31/0.65 ( ( A2 % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( member @ A @ X3 @ ( set2 @ A @ ( tl @ A @ A2 ) ) ) % 0.31/0.65 => ( member @ A @ X3 @ ( set2 @ A @ A2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % list.set_sel(2) % 0.31/0.65 thf(fact_7597_list_Oexpand,axiom, % 0.31/0.65 ! [A: $tType,List: list @ A,List3: list @ A] : % 0.31/0.65 ( ( ( List % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 = ( List3 % 0.31/0.65 = ( nil @ A ) ) ) % 0.31/0.65 => ( ( ( List % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( List3 % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( ( hd @ A @ List ) % 0.31/0.65 = ( hd @ A @ List3 ) ) % 0.31/0.65 & ( ( tl @ A @ List ) % 0.31/0.65 = ( tl @ A @ List3 ) ) ) ) ) % 0.31/0.65 => ( List = List3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % list.expand % 0.31/0.65 thf(fact_7598_list_Oexhaust__sel,axiom, % 0.31/0.65 ! [A: $tType,List: list @ A] : % 0.31/0.65 ( ( List % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( List % 0.31/0.65 = ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % list.exhaust_sel % 0.31/0.65 thf(fact_7599_map__filter__simps_I1_J,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: B > ( option @ A ),X3: B,Xs: list @ B] : % 0.31/0.65 ( ( map_filter @ B @ A @ F2 @ ( cons @ B @ X3 @ Xs ) ) % 0.31/0.65 = ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F2 @ Xs ) % 0.31/0.65 @ ^ [Y2: A] : ( cons @ A @ Y2 @ ( map_filter @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 @ ( F2 @ X3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % map_filter_simps(1) % 0.31/0.65 thf(fact_7600_tl__take,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( tl @ A @ ( take @ A @ N2 @ Xs ) ) % 0.31/0.65 = ( take @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( tl @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % tl_take % 0.31/0.65 thf(fact_7601_list_Ocase__eq__if,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( case_list @ B @ A ) % 0.31/0.65 = ( ^ [F12: B,F23: A > ( list @ A ) > B,List2: list @ A] : % 0.31/0.65 ( if @ B % 0.31/0.65 @ ( List2 % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 @ F12 % 0.31/0.65 @ ( F23 @ ( hd @ A @ List2 ) @ ( tl @ A @ List2 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % list.case_eq_if % 0.31/0.65 thf(fact_7602_Nitpick_Osize__list__simp_I2_J,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( size_size @ ( list @ A ) ) % 0.31/0.65 = ( ^ [Xs2: list @ A] : % 0.31/0.65 ( if @ nat % 0.31/0.65 @ ( Xs2 % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 @ ( zero_zero @ nat ) % 0.31/0.65 @ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs2 ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Nitpick.size_list_simp(2) % 0.31/0.65 thf(fact_7603_nth__tl,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) ) % 0.31/0.65 => ( ( nth @ A @ ( tl @ A @ Xs ) @ N2 ) % 0.31/0.65 = ( nth @ A @ Xs @ ( suc @ N2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % nth_tl % 0.31/0.65 thf(fact_7604_remdups__adj__append,axiom, % 0.31/0.65 ! [A: $tType,Xs_1: list @ A,X3: A,Xs_2: list @ A] : % 0.31/0.65 ( ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X3 @ Xs_2 ) ) ) % 0.31/0.65 = ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs_2 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % remdups_adj_append % 0.31/0.65 thf(fact_7605_Cons__in__shuffles__iff,axiom, % 0.31/0.65 ! [A: $tType,Z2: A,Zs2: list @ A,Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( member @ ( list @ A ) @ ( cons @ A @ Z2 @ Zs2 ) @ ( shuffles @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( ( ( Xs % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 & ( ( hd @ A @ Xs ) % 0.31/0.65 = Z2 ) % 0.31/0.65 & ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ ( tl @ A @ Xs ) @ Ys ) ) ) % 0.31/0.65 | ( ( Ys % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 & ( ( hd @ A @ Ys ) % 0.31/0.65 = Z2 ) % 0.31/0.65 & ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ ( tl @ A @ Ys ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Cons_in_shuffles_iff % 0.31/0.65 thf(fact_7606_list_Osplit__sel__asm,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] : % 0.31/0.65 ( ( P @ ( case_list @ B @ A @ F1 @ F22 @ List ) ) % 0.31/0.65 = ( ~ ( ( ( List % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 & ~ ( P @ F1 ) ) % 0.31/0.65 | ( ( List % 0.31/0.65 = ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) % 0.31/0.65 & ~ ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % list.split_sel_asm % 0.31/0.65 thf(fact_7607_list_Osplit__sel,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] : % 0.31/0.65 ( ( P @ ( case_list @ B @ A @ F1 @ F22 @ List ) ) % 0.31/0.65 = ( ( ( List % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 => ( P @ F1 ) ) % 0.31/0.65 & ( ( List % 0.31/0.65 = ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) % 0.31/0.65 => ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % list.split_sel % 0.31/0.65 thf(fact_7608_nths__shift__lemma__Suc,axiom, % 0.31/0.65 ! [A: $tType,P: nat > $o,Xs: list @ A,Is: list @ nat] : % 0.31/0.65 ( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat ) % 0.31/0.65 @ ( filter2 @ ( product_prod @ A @ nat ) % 0.31/0.65 @ ^ [P6: product_prod @ A @ nat] : ( P @ ( suc @ ( product_snd @ A @ nat @ P6 ) ) ) % 0.31/0.65 @ ( zip @ A @ nat @ Xs @ Is ) ) ) % 0.31/0.65 = ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat ) % 0.31/0.65 @ ( filter2 @ ( product_prod @ A @ nat ) % 0.31/0.65 @ ^ [P6: product_prod @ A @ nat] : ( P @ ( product_snd @ A @ nat @ P6 ) ) % 0.31/0.65 @ ( zip @ A @ nat @ Xs @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % nths_shift_lemma_Suc % 0.31/0.65 thf(fact_7609_take__Suc,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,N2: nat] : % 0.31/0.65 ( ( Xs % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( take @ A @ ( suc @ N2 ) @ Xs ) % 0.31/0.65 = ( cons @ A @ ( hd @ A @ Xs ) @ ( take @ A @ N2 @ ( tl @ A @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % take_Suc % 0.31/0.65 thf(fact_7610_rotate1__hd__tl,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( Xs % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( rotate1 @ A @ Xs ) % 0.31/0.65 = ( append @ A @ ( tl @ A @ Xs ) @ ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rotate1_hd_tl % 0.31/0.65 thf(fact_7611_nths__shift__lemma,axiom, % 0.31/0.65 ! [A: $tType,A3: set @ nat,Xs: list @ A,I: nat] : % 0.31/0.65 ( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat ) % 0.31/0.65 @ ( filter2 @ ( product_prod @ A @ nat ) % 0.31/0.65 @ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A3 ) % 0.31/0.65 @ ( zip @ A @ nat @ Xs @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) % 0.31/0.65 = ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat ) % 0.31/0.65 @ ( filter2 @ ( product_prod @ A @ nat ) % 0.31/0.65 @ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P6 ) @ I ) @ A3 ) % 0.31/0.65 @ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % nths_shift_lemma % 0.31/0.65 thf(fact_7612_Nitpick_Osize__list__simp_I1_J,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( size_list @ A ) % 0.31/0.65 = ( ^ [F3: A > nat,Xs2: list @ A] : % 0.31/0.65 ( if @ nat % 0.31/0.65 @ ( Xs2 % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 @ ( zero_zero @ nat ) % 0.31/0.65 @ ( suc @ ( plus_plus @ nat @ ( F3 @ ( hd @ A @ Xs2 ) ) @ ( size_list @ A @ F3 @ ( tl @ A @ Xs2 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Nitpick.size_list_simp(1) % 0.31/0.65 thf(fact_7613_nths__def,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( nths @ A ) % 0.31/0.65 = ( ^ [Xs2: list @ A,A6: set @ nat] : % 0.31/0.65 ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat ) % 0.31/0.65 @ ( filter2 @ ( product_prod @ A @ nat ) % 0.31/0.65 @ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A6 ) % 0.31/0.65 @ ( zip @ A @ nat @ Xs2 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % nths_def % 0.31/0.65 thf(fact_7614_map__filter__map__filter,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: B > A,P: B > $o,Xs: list @ B] : % 0.31/0.65 ( ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) % 0.31/0.65 = ( map_filter @ B @ A % 0.31/0.65 @ ^ [X: B] : ( if @ ( option @ A ) @ ( P @ X ) @ ( some @ A @ ( F2 @ X ) ) @ ( none @ A ) ) % 0.31/0.65 @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % map_filter_map_filter % 0.31/0.65 thf(fact_7615_transpose__max__length,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ ( list @ A )] : % 0.31/0.65 ( ( foldr @ ( list @ A ) @ nat % 0.31/0.65 @ ^ [Xs2: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) ) % 0.31/0.65 @ ( transpose @ A @ Xs ) % 0.31/0.65 @ ( zero_zero @ nat ) ) % 0.31/0.65 = ( size_size @ ( list @ ( list @ A ) ) % 0.31/0.65 @ ( filter2 @ ( list @ A ) % 0.31/0.65 @ ^ [X: list @ A] : % 0.31/0.65 ( X % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose_max_length % 0.31/0.65 thf(fact_7616_transpose__aux__max,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Xs: list @ A,Xss: list @ ( list @ B )] : % 0.31/0.65 ( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 @ ( foldr @ ( list @ B ) @ nat % 0.31/0.65 @ ^ [Xs2: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs2 ) ) % 0.31/0.65 @ Xss % 0.31/0.65 @ ( zero_zero @ nat ) ) ) % 0.31/0.65 = ( suc % 0.31/0.65 @ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.65 @ ( foldr @ ( list @ B ) @ nat % 0.31/0.65 @ ^ [X: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X ) @ ( suc @ ( zero_zero @ nat ) ) ) ) % 0.31/0.65 @ ( filter2 @ ( list @ B ) % 0.31/0.65 @ ^ [Ys3: list @ B] : % 0.31/0.65 ( Ys3 % 0.31/0.65 != ( nil @ B ) ) % 0.31/0.65 @ Xss ) % 0.31/0.65 @ ( zero_zero @ nat ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose_aux_max % 0.31/0.65 thf(fact_7617_foldr__append,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F2: B > A > A,Xs: list @ B,Ys: list @ B,A2: A] : % 0.31/0.65 ( ( foldr @ B @ A @ F2 @ ( append @ B @ Xs @ Ys ) @ A2 ) % 0.31/0.65 = ( foldr @ B @ A @ F2 @ Xs @ ( foldr @ B @ A @ F2 @ Ys @ A2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % foldr_append % 0.31/0.65 thf(fact_7618_foldr__replicate,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: B > A > A,N2: nat,X3: B] : % 0.31/0.65 ( ( foldr @ B @ A @ F2 @ ( replicate @ B @ N2 @ X3 ) ) % 0.31/0.65 = ( compow @ ( A > A ) @ N2 @ ( F2 @ X3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % foldr_replicate % 0.31/0.65 thf(fact_7619_foldr__map,axiom, % 0.31/0.65 ! [C: $tType,B: $tType,A: $tType,G: B > A > A,F2: C > B,Xs: list @ C,A2: A] : % 0.31/0.65 ( ( foldr @ B @ A @ G @ ( map @ C @ B @ F2 @ Xs ) @ A2 ) % 0.31/0.65 = ( foldr @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G @ F2 ) @ Xs @ A2 ) ) ). % 0.31/0.65 % 0.31/0.65 % foldr_map % 0.31/0.65 thf(fact_7620_foldr__Cons,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F2: A > B > B,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( foldr @ A @ B @ F2 @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( foldr @ A @ B @ F2 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % foldr_Cons % 0.31/0.65 thf(fact_7621_foldr__cong,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A2: A,B2: A,L: list @ B,K: list @ B,F2: B > A > A,G: B > A > A] : % 0.31/0.65 ( ( A2 = B2 ) % 0.31/0.65 => ( ( L = K ) % 0.31/0.65 => ( ! [A5: A,X5: B] : % 0.31/0.65 ( ( member @ B @ X5 @ ( set2 @ B @ L ) ) % 0.31/0.65 => ( ( F2 @ X5 @ A5 ) % 0.31/0.65 = ( G @ X5 @ A5 ) ) ) % 0.31/0.65 => ( ( foldr @ B @ A @ F2 @ L @ A2 ) % 0.31/0.65 = ( foldr @ B @ A @ G @ K @ B2 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % foldr_cong % 0.31/0.65 thf(fact_7622_foldr__Nil,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: A > B > B] : % 0.31/0.65 ( ( foldr @ A @ B @ F2 @ ( nil @ A ) ) % 0.31/0.65 = ( id @ B ) ) ). % 0.31/0.65 % 0.31/0.65 % foldr_Nil % 0.31/0.65 thf(fact_7623_foldr__filter,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: B > A > A,P: B > $o,Xs: list @ B] : % 0.31/0.65 ( ( foldr @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) % 0.31/0.65 = ( foldr @ B @ A % 0.31/0.65 @ ^ [X: B] : ( if @ ( A > A ) @ ( P @ X ) @ ( F2 @ X ) @ ( id @ A ) ) % 0.31/0.65 @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % foldr_filter % 0.31/0.65 thf(fact_7624_sum__list_Oeq__foldr,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( monoid_add @ A ) % 0.31/0.65 => ( ( groups8242544230860333062m_list @ A ) % 0.31/0.65 = ( ^ [Xs2: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs2 @ ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_list.eq_foldr % 0.31/0.65 thf(fact_7625_horner__sum__foldr,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( comm_semiring_0 @ A ) % 0.31/0.65 => ( ( groups4207007520872428315er_sum @ B @ A ) % 0.31/0.65 = ( ^ [F3: B > A,A4: A,Xs2: list @ B] : % 0.31/0.65 ( foldr @ B @ A % 0.31/0.65 @ ^ [X: B,B4: A] : ( plus_plus @ A @ ( F3 @ X ) @ ( times_times @ A @ A4 @ B4 ) ) % 0.31/0.65 @ Xs2 % 0.31/0.65 @ ( zero_zero @ A ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % horner_sum_foldr % 0.31/0.65 thf(fact_7626_length__transpose,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ ( list @ A )] : % 0.31/0.65 ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) % 0.31/0.65 = ( foldr @ ( list @ A ) @ nat % 0.31/0.65 @ ^ [Xs2: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) ) % 0.31/0.65 @ Xs % 0.31/0.65 @ ( zero_zero @ nat ) ) ) ). % 0.31/0.65 % 0.31/0.65 % length_transpose % 0.31/0.65 thf(fact_7627_sorted__wrt__less__sum__mono__lowerbound,axiom, % 0.31/0.65 ! [B: $tType] : % 0.31/0.65 ( ( ordere6911136660526730532id_add @ B ) % 0.31/0.65 => ! [F2: nat > B,Ns: list @ nat] : % 0.31/0.65 ( ! [X5: nat,Y4: nat] : % 0.31/0.65 ( ( ord_less_eq @ nat @ X5 @ Y4 ) % 0.31/0.65 => ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 0.31/0.65 => ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns ) % 0.31/0.65 => ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F2 @ Ns ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt_less_sum_mono_lowerbound % 0.31/0.65 thf(fact_7628_length__product__lists,axiom, % 0.31/0.65 ! [B: $tType,Xss: list @ ( list @ B )] : % 0.31/0.65 ( ( size_size @ ( list @ ( list @ B ) ) @ ( product_lists @ B @ Xss ) ) % 0.31/0.65 = ( foldr @ nat @ nat @ ( times_times @ nat ) @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) @ ( one_one @ nat ) ) ) ). % 0.31/0.65 % 0.31/0.65 % length_product_lists % 0.31/0.65 thf(fact_7629_sorted__map__same,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [F2: B > A,G: ( list @ B ) > A,Xs: list @ B] : % 0.31/0.65 ( sorted_wrt @ A @ ( ord_less_eq @ A ) % 0.31/0.65 @ ( map @ B @ A @ F2 % 0.31/0.65 @ ( filter2 @ B % 0.31/0.65 @ ^ [X: B] : % 0.31/0.65 ( ( F2 @ X ) % 0.31/0.65 = ( G @ Xs ) ) % 0.31/0.65 @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_map_same % 0.31/0.65 thf(fact_7630_sorted__map,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [F2: B > A,Xs: list @ B] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 = ( sorted_wrt @ B % 0.31/0.65 @ ^ [X: B,Y2: B] : ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y2 ) ) % 0.31/0.65 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_map % 0.31/0.65 thf(fact_7631_sorted__filter,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [F2: B > A,Xs: list @ B,P: B > $o] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_filter % 0.31/0.65 thf(fact_7632_sorted__wrt__map,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,R: A > A > $o,F2: B > A,Xs: list @ B] : % 0.31/0.65 ( ( sorted_wrt @ A @ R @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 = ( sorted_wrt @ B % 0.31/0.65 @ ^ [X: B,Y2: B] : ( R @ ( F2 @ X ) @ ( F2 @ Y2 ) ) % 0.31/0.65 @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt_map % 0.31/0.65 thf(fact_7633_sorted__insort__key,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [F2: B > A,X3: B,Xs: list @ B] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) ) % 0.31/0.65 = ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_insort_key % 0.31/0.65 thf(fact_7634_sorted__map__remove1,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [F2: B > A,Xs: list @ B,X3: B] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ ( remove1 @ B @ X3 @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_map_remove1 % 0.31/0.65 thf(fact_7635_sorted__same,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [G: ( list @ A ) > A,Xs: list @ A] : % 0.31/0.65 ( sorted_wrt @ A @ ( ord_less_eq @ A ) % 0.31/0.65 @ ( filter2 @ A % 0.31/0.65 @ ^ [X: A] : % 0.31/0.65 ( X % 0.31/0.65 = ( G @ Xs ) ) % 0.31/0.65 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_same % 0.31/0.65 thf(fact_7636_sorted__wrt__filter,axiom, % 0.31/0.65 ! [A: $tType,F2: A > A > $o,Xs: list @ A,P: A > $o] : % 0.31/0.65 ( ( sorted_wrt @ A @ F2 @ Xs ) % 0.31/0.65 => ( sorted_wrt @ A @ F2 @ ( filter2 @ A @ P @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt_filter % 0.31/0.65 thf(fact_7637_sorted__tl,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( tl @ A @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_tl % 0.31/0.65 thf(fact_7638_sorted__upt,axiom, % 0.31/0.65 ! [M: nat,N2: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M @ N2 ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_upt % 0.31/0.65 thf(fact_7639_sorted__wrt__upt,axiom, % 0.31/0.65 ! [M: nat,N2: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M @ N2 ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt_upt % 0.31/0.65 thf(fact_7640_sorted__wrt__true,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( sorted_wrt @ A % 0.31/0.65 @ ^ [Uu3: A,Uv3: A] : $true % 0.31/0.65 @ Xs ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt_true % 0.31/0.65 thf(fact_7641_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [A3: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_list_of_set.strict_sorted_key_list_of_set % 0.31/0.65 thf(fact_7642_sorted__remdups__adj,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups_adj @ A @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_remdups_adj % 0.31/0.65 thf(fact_7643_sorted__insort,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [X3: A,Xs: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) % 0.31/0.65 @ ( linorder_insort_key @ A @ A % 0.31/0.65 @ ^ [X: A] : X % 0.31/0.65 @ X3 % 0.31/0.65 @ Xs ) ) % 0.31/0.65 = ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_insort % 0.31/0.65 thf(fact_7644_strict__sorted__imp__sorted,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs ) % 0.31/0.65 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % strict_sorted_imp_sorted % 0.31/0.65 thf(fact_7645_sorted__replicate,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [N2: nat,X3: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N2 @ X3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_replicate % 0.31/0.65 thf(fact_7646_sorted__nths,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A,I6: set @ nat] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nths @ A @ Xs @ I6 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_nths % 0.31/0.65 thf(fact_7647_sorted__remove1,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A,A2: A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A2 @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_remove1 % 0.31/0.65 thf(fact_7648_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [A3: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_list_of_set.sorted_sorted_key_list_of_set % 0.31/0.65 thf(fact_7649_sorted2,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [X3: A,Y: A,Zs2: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Zs2 ) ) ) % 0.31/0.65 = ( ( ord_less_eq @ A @ X3 @ Y ) % 0.31/0.65 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y @ Zs2 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted2 % 0.31/0.65 thf(fact_7650_sorted1,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [X3: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted1 % 0.31/0.65 thf(fact_7651_sorted__wrt1,axiom, % 0.31/0.65 ! [A: $tType,P: A > A > $o,X3: A] : ( sorted_wrt @ A @ P @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt1 % 0.31/0.65 thf(fact_7652_strict__sorted__simps_I1_J,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ). % 0.31/0.65 % 0.31/0.65 % strict_sorted_simps(1) % 0.31/0.65 thf(fact_7653_sorted__wrt_Osimps_I1_J,axiom, % 0.31/0.65 ! [A: $tType,P: A > A > $o] : ( sorted_wrt @ A @ P @ ( nil @ A ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt.simps(1) % 0.31/0.65 thf(fact_7654_sorted0,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted0 % 0.31/0.65 thf(fact_7655_sorted__remdups,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups @ A @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_remdups % 0.31/0.65 thf(fact_7656_sorted__take,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A,N2: nat] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( take @ A @ N2 @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_take % 0.31/0.65 thf(fact_7657_sorted__wrt__take,axiom, % 0.31/0.65 ! [A: $tType,F2: A > A > $o,Xs: list @ A,N2: nat] : % 0.31/0.65 ( ( sorted_wrt @ A @ F2 @ Xs ) % 0.31/0.65 => ( sorted_wrt @ A @ F2 @ ( take @ A @ N2 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt_take % 0.31/0.65 thf(fact_7658_sorted__drop,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A,N2: nat] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( drop @ A @ N2 @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_drop % 0.31/0.65 thf(fact_7659_sorted__wrt__drop,axiom, % 0.31/0.65 ! [A: $tType,F2: A > A > $o,Xs: list @ A,N2: nat] : % 0.31/0.65 ( ( sorted_wrt @ A @ F2 @ Xs ) % 0.31/0.65 => ( sorted_wrt @ A @ F2 @ ( drop @ A @ N2 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt_drop % 0.31/0.65 thf(fact_7660_strict__sorted__equal,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs ) % 0.31/0.65 => ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys ) % 0.31/0.65 => ( ( ( set2 @ A @ Ys ) % 0.31/0.65 = ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( Ys = Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % strict_sorted_equal % 0.31/0.65 thf(fact_7661_sorted__wrt__mono__rel,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,P: A > A > $o,Q: A > A > $o] : % 0.31/0.65 ( ! [X5: A,Y4: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( ( P @ X5 @ Y4 ) % 0.31/0.65 => ( Q @ X5 @ Y4 ) ) ) ) % 0.31/0.65 => ( ( sorted_wrt @ A @ P @ Xs ) % 0.31/0.65 => ( sorted_wrt @ A @ Q @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt_mono_rel % 0.31/0.65 thf(fact_7662_sorted__wrt__iff__nth__less,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( sorted_wrt @ A ) % 0.31/0.65 = ( ^ [P3: A > A > $o,Xs2: list @ A] : % 0.31/0.65 ! [I3: nat,J3: nat] : % 0.31/0.65 ( ( ord_less @ nat @ I3 @ J3 ) % 0.31/0.65 => ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) % 0.31/0.65 => ( P3 @ ( nth @ A @ Xs2 @ I3 ) @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt_iff_nth_less % 0.31/0.65 thf(fact_7663_sorted__wrt__nth__less,axiom, % 0.31/0.65 ! [A: $tType,P: A > A > $o,Xs: list @ A,I: nat,J: nat] : % 0.31/0.65 ( ( sorted_wrt @ A @ P @ Xs ) % 0.31/0.65 => ( ( ord_less @ nat @ I @ J ) % 0.31/0.65 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( P @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt_nth_less % 0.31/0.65 thf(fact_7664_sorted__wrt01,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,P: A > A > $o] : % 0.31/0.65 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) % 0.31/0.65 => ( sorted_wrt @ A @ P @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt01 % 0.31/0.65 thf(fact_7665_sorted__distinct__set__unique,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 => ( ( distinct @ A @ Xs ) % 0.31/0.65 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys ) % 0.31/0.65 => ( ( distinct @ A @ Ys ) % 0.31/0.65 => ( ( ( set2 @ A @ Xs ) % 0.31/0.65 = ( set2 @ A @ Ys ) ) % 0.31/0.65 => ( Xs = Ys ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_distinct_set_unique % 0.31/0.65 thf(fact_7666_strict__sorted__iff,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [L: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L ) % 0.31/0.65 = ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L ) % 0.31/0.65 & ( distinct @ A @ L ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % strict_sorted_iff % 0.31/0.65 thf(fact_7667_sorted__wrt__append,axiom, % 0.31/0.65 ! [A: $tType,P: A > A > $o,Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ P @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( ( sorted_wrt @ A @ P @ Xs ) % 0.31/0.65 & ( sorted_wrt @ A @ P @ Ys ) % 0.31/0.65 & ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ! [Y2: A] : % 0.31/0.65 ( ( member @ A @ Y2 @ ( set2 @ A @ Ys ) ) % 0.31/0.65 => ( P @ X @ Y2 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt_append % 0.31/0.65 thf(fact_7668_sorted__append,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys ) % 0.31/0.65 & ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ! [Y2: A] : % 0.31/0.65 ( ( member @ A @ Y2 @ ( set2 @ A @ Ys ) ) % 0.31/0.65 => ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_append % 0.31/0.65 thf(fact_7669_sorted__simps_I2_J,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [X3: A,Ys: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X3 @ Ys ) ) % 0.31/0.65 = ( ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Ys ) ) % 0.31/0.65 => ( ord_less_eq @ A @ X3 @ X ) ) % 0.31/0.65 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_simps(2) % 0.31/0.65 thf(fact_7670_strict__sorted__simps_I2_J,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [X3: A,Ys: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X3 @ Ys ) ) % 0.31/0.65 = ( ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Ys ) ) % 0.31/0.65 => ( ord_less @ A @ X3 @ X ) ) % 0.31/0.65 & ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % strict_sorted_simps(2) % 0.31/0.65 thf(fact_7671_sorted__wrt_Oelims_I3_J,axiom, % 0.31/0.65 ! [A: $tType,X3: A > A > $o,Xa2: list @ A] : % 0.31/0.65 ( ~ ( sorted_wrt @ A @ X3 @ Xa2 ) % 0.31/0.65 => ~ ! [X5: A,Ys4: list @ A] : % 0.31/0.65 ( ( Xa2 % 0.31/0.65 = ( cons @ A @ X5 @ Ys4 ) ) % 0.31/0.65 => ( ! [Xa3: A] : % 0.31/0.65 ( ( member @ A @ Xa3 @ ( set2 @ A @ Ys4 ) ) % 0.31/0.65 => ( X3 @ X5 @ Xa3 ) ) % 0.31/0.65 & ( sorted_wrt @ A @ X3 @ Ys4 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt.elims(3) % 0.31/0.65 thf(fact_7672_sorted__wrt_Osimps_I2_J,axiom, % 0.31/0.65 ! [A: $tType,P: A > A > $o,X3: A,Ys: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ P @ ( cons @ A @ X3 @ Ys ) ) % 0.31/0.65 = ( ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Ys ) ) % 0.31/0.65 => ( P @ X3 @ X ) ) % 0.31/0.65 & ( sorted_wrt @ A @ P @ Ys ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt.simps(2) % 0.31/0.65 thf(fact_7673_sorted01,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A] : % 0.31/0.65 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) % 0.31/0.65 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted01 % 0.31/0.65 thf(fact_7674_sorted__iff__nth__mono__less,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 = ( ! [I3: nat,J3: nat] : % 0.31/0.65 ( ( ord_less @ nat @ I3 @ J3 ) % 0.31/0.65 => ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_iff_nth_mono_less % 0.31/0.65 thf(fact_7675_sorted__wrt_Oelims_I1_J,axiom, % 0.31/0.65 ! [A: $tType,X3: A > A > $o,Xa2: list @ A,Y: $o] : % 0.31/0.65 ( ( ( sorted_wrt @ A @ X3 @ Xa2 ) % 0.31/0.65 = Y ) % 0.31/0.65 => ( ( ( Xa2 % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 => ~ Y ) % 0.31/0.65 => ~ ! [X5: A,Ys4: list @ A] : % 0.31/0.65 ( ( Xa2 % 0.31/0.65 = ( cons @ A @ X5 @ Ys4 ) ) % 0.31/0.65 => ( Y % 0.31/0.65 = ( ~ ( ! [Y2: A] : % 0.31/0.65 ( ( member @ A @ Y2 @ ( set2 @ A @ Ys4 ) ) % 0.31/0.65 => ( X3 @ X5 @ Y2 ) ) % 0.31/0.65 & ( sorted_wrt @ A @ X3 @ Ys4 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt.elims(1) % 0.31/0.65 thf(fact_7676_sorted__wrt_Oelims_I2_J,axiom, % 0.31/0.65 ! [A: $tType,X3: A > A > $o,Xa2: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ X3 @ Xa2 ) % 0.31/0.65 => ( ( Xa2 % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ~ ! [X5: A,Ys4: list @ A] : % 0.31/0.65 ( ( Xa2 % 0.31/0.65 = ( cons @ A @ X5 @ Ys4 ) ) % 0.31/0.65 => ~ ( ! [Xa: A] : % 0.31/0.65 ( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) ) % 0.31/0.65 => ( X3 @ X5 @ Xa ) ) % 0.31/0.65 & ( sorted_wrt @ A @ X3 @ Ys4 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt.elims(2) % 0.31/0.65 thf(fact_7677_finite__sorted__distinct__unique,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [A3: set @ A] : % 0.31/0.65 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.65 => ? [X5: list @ A] : % 0.31/0.65 ( ( ( set2 @ A @ X5 ) % 0.31/0.65 = A3 ) % 0.31/0.65 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X5 ) % 0.31/0.65 & ( distinct @ A @ X5 ) % 0.31/0.65 & ! [Y5: list @ A] : % 0.31/0.65 ( ( ( ( set2 @ A @ Y5 ) % 0.31/0.65 = A3 ) % 0.31/0.65 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y5 ) % 0.31/0.65 & ( distinct @ A @ Y5 ) ) % 0.31/0.65 => ( Y5 = X5 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % finite_sorted_distinct_unique % 0.31/0.65 thf(fact_7678_sorted__list__of__set_Oidem__if__sorted__distinct,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 => ( ( distinct @ A @ Xs ) % 0.31/0.65 => ( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs ) ) % 0.31/0.65 = Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_list_of_set.idem_if_sorted_distinct % 0.31/0.65 thf(fact_7679_filter__insort,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [F2: B > A,Xs: list @ B,P: B > $o,X3: B] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 => ( ( P @ X3 ) % 0.31/0.65 => ( ( filter2 @ B @ P @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) % 0.31/0.65 = ( linorder_insort_key @ B @ A @ F2 @ X3 @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_insort % 0.31/0.65 thf(fact_7680_concat__conv__foldr,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( concat @ A ) % 0.31/0.65 = ( ^ [Xss3: list @ ( list @ A )] : ( foldr @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ Xss3 @ ( nil @ A ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % concat_conv_foldr % 0.31/0.65 thf(fact_7681_insort__remove1,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [A2: A,Xs: list @ A] : % 0.31/0.65 ( ( member @ A @ A2 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 => ( ( linorder_insort_key @ A @ A % 0.31/0.65 @ ^ [X: A] : X % 0.31/0.65 @ A2 % 0.31/0.65 @ ( remove1 @ A @ A2 @ Xs ) ) % 0.31/0.65 = Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % insort_remove1 % 0.31/0.65 thf(fact_7682_sorted__iff__nth__Suc,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 = ( ! [I3: nat] : % 0.31/0.65 ( ( ord_less @ nat @ ( suc @ I3 ) @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ ( suc @ I3 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_iff_nth_Suc % 0.31/0.65 thf(fact_7683_sorted__list__of__set_Ofinite__set__strict__sorted,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [A3: set @ A] : % 0.31/0.65 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.65 => ~ ! [L4: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L4 ) % 0.31/0.65 => ( ( ( set2 @ A @ L4 ) % 0.31/0.65 = A3 ) % 0.31/0.65 => ( ( size_size @ ( list @ A ) @ L4 ) % 0.31/0.65 != ( finite_card @ A @ A3 ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_list_of_set.finite_set_strict_sorted % 0.31/0.65 thf(fact_7684_sorted__nth__mono,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A,I: nat,J: nat] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 => ( ( ord_less_eq @ nat @ I @ J ) % 0.31/0.65 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_nth_mono % 0.31/0.65 thf(fact_7685_sorted__iff__nth__mono,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 = ( ! [I3: nat,J3: nat] : % 0.31/0.65 ( ( ord_less_eq @ nat @ I3 @ J3 ) % 0.31/0.65 => ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_iff_nth_mono % 0.31/0.65 thf(fact_7686_sorted__wrt__less__idx,axiom, % 0.31/0.65 ! [Ns: list @ nat,I: nat] : % 0.31/0.65 ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns ) % 0.31/0.65 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ nat ) @ Ns ) ) % 0.31/0.65 => ( ord_less_eq @ nat @ I @ ( nth @ nat @ Ns @ I ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt_less_idx % 0.31/0.65 thf(fact_7687_sorted__find__Min,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A,P: A > $o] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 => ( ? [X2: A] : % 0.31/0.65 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 & ( P @ X2 ) ) % 0.31/0.65 => ( ( find @ A @ P @ Xs ) % 0.31/0.65 = ( some @ A % 0.31/0.65 @ ( lattic643756798350308766er_Min @ A % 0.31/0.65 @ ( collect @ A % 0.31/0.65 @ ^ [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.65 & ( P @ X ) ) ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_find_Min % 0.31/0.65 thf(fact_7688_sorted__enumerate,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_enumerate % 0.31/0.65 thf(fact_7689_map__sorted__distinct__set__unique,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [F2: B > A,Xs: list @ B,Ys: list @ B] : % 0.31/0.65 ( ( inj_on @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) ) % 0.31/0.65 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 => ( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Ys ) ) % 0.31/0.65 => ( ( distinct @ A @ ( map @ B @ A @ F2 @ Ys ) ) % 0.31/0.65 => ( ( ( set2 @ B @ Xs ) % 0.31/0.65 = ( set2 @ B @ Ys ) ) % 0.31/0.65 => ( Xs = Ys ) ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % map_sorted_distinct_set_unique % 0.31/0.65 thf(fact_7690_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [A3: set @ A,L: list @ A] : % 0.31/0.65 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.65 => ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L ) % 0.31/0.65 & ( ( set2 @ A @ L ) % 0.31/0.65 = A3 ) % 0.31/0.65 & ( ( size_size @ ( list @ A ) @ L ) % 0.31/0.65 = ( finite_card @ A @ A3 ) ) ) % 0.31/0.65 = ( ( linord4507533701916653071of_set @ A @ A3 ) % 0.31/0.65 = L ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_list_of_set.sorted_key_list_of_set_unique % 0.31/0.65 thf(fact_7691_sorted__insort__is__snoc,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A,A2: A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 => ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( ord_less_eq @ A @ X5 @ A2 ) ) % 0.31/0.65 => ( ( linorder_insort_key @ A @ A % 0.31/0.65 @ ^ [X: A] : X % 0.31/0.65 @ A2 % 0.31/0.65 @ Xs ) % 0.31/0.65 = ( append @ A @ Xs @ ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_insort_is_snoc % 0.31/0.65 thf(fact_7692_insort__key__remove1,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [A2: B,Xs: list @ B,F2: B > A] : % 0.31/0.65 ( ( member @ B @ A2 @ ( set2 @ B @ Xs ) ) % 0.31/0.65 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 => ( ( ( hd @ B % 0.31/0.65 @ ( filter2 @ B % 0.31/0.65 @ ^ [X: B] : % 0.31/0.65 ( ( F2 @ A2 ) % 0.31/0.65 = ( F2 @ X ) ) % 0.31/0.65 @ Xs ) ) % 0.31/0.65 = A2 ) % 0.31/0.65 => ( ( linorder_insort_key @ B @ A @ F2 @ A2 @ ( remove1 @ B @ A2 @ Xs ) ) % 0.31/0.65 = Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % insort_key_remove1 % 0.31/0.65 thf(fact_7693_nth__nth__transpose__sorted,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ ( list @ A ),I: nat,J: nat] : % 0.31/0.65 ( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) ) % 0.31/0.65 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) ) % 0.31/0.65 => ( ( ord_less @ nat @ J % 0.31/0.65 @ ( size_size @ ( list @ ( list @ A ) ) % 0.31/0.65 @ ( filter2 @ ( list @ A ) % 0.31/0.65 @ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) ) % 0.31/0.65 @ Xs ) ) ) % 0.31/0.65 => ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I ) @ J ) % 0.31/0.65 = ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J ) @ I ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % nth_nth_transpose_sorted % 0.31/0.65 thf(fact_7694_transpose__column,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ ( list @ A ),I: nat] : % 0.31/0.65 ( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) ) % 0.31/0.65 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) % 0.31/0.65 => ( ( map @ ( list @ A ) @ A % 0.31/0.65 @ ^ [Ys3: list @ A] : ( nth @ A @ Ys3 @ I ) % 0.31/0.65 @ ( filter2 @ ( list @ A ) % 0.31/0.65 @ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) ) % 0.31/0.65 @ ( transpose @ A @ Xs ) ) ) % 0.31/0.65 = ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose_column % 0.31/0.65 thf(fact_7695_rev__rev__ident,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( rev @ A @ ( rev @ A @ Xs ) ) % 0.31/0.65 = Xs ) ). % 0.31/0.65 % 0.31/0.65 % rev_rev_ident % 0.31/0.65 thf(fact_7696_rev__is__rev__conv,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( ( rev @ A @ Xs ) % 0.31/0.65 = ( rev @ A @ Ys ) ) % 0.31/0.65 = ( Xs = Ys ) ) ). % 0.31/0.65 % 0.31/0.65 % rev_is_rev_conv % 0.31/0.65 thf(fact_7697_rev__is__Nil__conv,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( ( rev @ A @ Xs ) % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 = ( Xs % 0.31/0.65 = ( nil @ A ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rev_is_Nil_conv % 0.31/0.65 thf(fact_7698_Nil__is__rev__conv,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( ( nil @ A ) % 0.31/0.65 = ( rev @ A @ Xs ) ) % 0.31/0.65 = ( Xs % 0.31/0.65 = ( nil @ A ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Nil_is_rev_conv % 0.31/0.65 thf(fact_7699_set__rev,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( set2 @ A @ ( rev @ A @ Xs ) ) % 0.31/0.65 = ( set2 @ A @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % set_rev % 0.31/0.65 thf(fact_7700_length__rev,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) ) % 0.31/0.65 = ( size_size @ ( list @ A ) @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % length_rev % 0.31/0.65 thf(fact_7701_rev__append,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( rev @ A @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( append @ A @ ( rev @ A @ Ys ) @ ( rev @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rev_append % 0.31/0.65 thf(fact_7702_distinct__rev,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( distinct @ A @ ( rev @ A @ Xs ) ) % 0.31/0.65 = ( distinct @ A @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % distinct_rev % 0.31/0.65 thf(fact_7703_rev__replicate,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,X3: A] : % 0.31/0.65 ( ( rev @ A @ ( replicate @ A @ N2 @ X3 ) ) % 0.31/0.65 = ( replicate @ A @ N2 @ X3 ) ) ). % 0.31/0.65 % 0.31/0.65 % rev_replicate % 0.31/0.65 thf(fact_7704_remdups__adj__rev,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A] : % 0.31/0.65 ( ( remdups_adj @ A @ ( rev @ A @ Xs ) ) % 0.31/0.65 = ( rev @ A @ ( remdups_adj @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % remdups_adj_rev % 0.31/0.65 thf(fact_7705_inj__on__rev,axiom, % 0.31/0.65 ! [A: $tType,A3: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( rev @ A ) @ A3 ) ). % 0.31/0.65 % 0.31/0.65 % inj_on_rev % 0.31/0.65 thf(fact_7706_singleton__rev__conv,axiom, % 0.31/0.65 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( ( cons @ A @ X3 @ ( nil @ A ) ) % 0.31/0.65 = ( rev @ A @ Xs ) ) % 0.31/0.65 = ( ( cons @ A @ X3 @ ( nil @ A ) ) % 0.31/0.65 = Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % singleton_rev_conv % 0.31/0.65 thf(fact_7707_rev__singleton__conv,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,X3: A] : % 0.31/0.65 ( ( ( rev @ A @ Xs ) % 0.31/0.65 = ( cons @ A @ X3 @ ( nil @ A ) ) ) % 0.31/0.65 = ( Xs % 0.31/0.65 = ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rev_singleton_conv % 0.31/0.65 thf(fact_7708_rev__eq__Cons__iff,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A] : % 0.31/0.65 ( ( ( rev @ A @ Xs ) % 0.31/0.65 = ( cons @ A @ Y @ Ys ) ) % 0.31/0.65 = ( Xs % 0.31/0.65 = ( append @ A @ ( rev @ A @ Ys ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rev_eq_Cons_iff % 0.31/0.65 thf(fact_7709_foldl__conv__foldr,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( foldl @ A @ B ) % 0.31/0.65 = ( ^ [F3: A > B > A,A4: A,Xs2: list @ B] : % 0.31/0.65 ( foldr @ B @ A % 0.31/0.65 @ ^ [X: B,Y2: A] : ( F3 @ Y2 @ X ) % 0.31/0.65 @ ( rev @ B @ Xs2 ) % 0.31/0.65 @ A4 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % foldl_conv_foldr % 0.31/0.65 thf(fact_7710_foldr__conv__foldl,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( foldr @ B @ A ) % 0.31/0.65 = ( ^ [F3: B > A > A,Xs2: list @ B,A4: A] : % 0.31/0.65 ( foldl @ A @ B % 0.31/0.65 @ ^ [X: A,Y2: B] : ( F3 @ Y2 @ X ) % 0.31/0.65 @ A4 % 0.31/0.65 @ ( rev @ B @ Xs2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % foldr_conv_foldl % 0.31/0.65 thf(fact_7711_sorted__wrt__rev,axiom, % 0.31/0.65 ! [A: $tType,P: A > A > $o,Xs: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ P @ ( rev @ A @ Xs ) ) % 0.31/0.65 = ( sorted_wrt @ A % 0.31/0.65 @ ^ [X: A,Y2: A] : ( P @ Y2 @ X ) % 0.31/0.65 @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt_rev % 0.31/0.65 thf(fact_7712_sorted__upto,axiom, % 0.31/0.65 ! [M: int,N2: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M @ N2 ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_upto % 0.31/0.65 thf(fact_7713_sorted__wrt__upto,axiom, % 0.31/0.65 ! [I: int,J: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I @ J ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_wrt_upto % 0.31/0.65 thf(fact_7714_rev_Osimps_I1_J,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( rev @ A @ ( nil @ A ) ) % 0.31/0.65 = ( nil @ A ) ) ). % 0.31/0.65 % 0.31/0.65 % rev.simps(1) % 0.31/0.65 thf(fact_7715_rev__swap,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( ( rev @ A @ Xs ) % 0.31/0.65 = Ys ) % 0.31/0.65 = ( Xs % 0.31/0.65 = ( rev @ A @ Ys ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rev_swap % 0.31/0.65 thf(fact_7716_rev_Osimps_I2_J,axiom, % 0.31/0.65 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( rev @ A @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( append @ A @ ( rev @ A @ Xs ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rev.simps(2) % 0.31/0.65 thf(fact_7717_zip__rev,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 0.31/0.65 ( ( ( size_size @ ( list @ A ) @ Xs ) % 0.31/0.65 = ( size_size @ ( list @ B ) @ Ys ) ) % 0.31/0.65 => ( ( zip @ A @ B @ ( rev @ A @ Xs ) @ ( rev @ B @ Ys ) ) % 0.31/0.65 = ( rev @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % zip_rev % 0.31/0.65 thf(fact_7718_rev__filter,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( rev @ A @ ( filter2 @ A @ P @ Xs ) ) % 0.31/0.65 = ( filter2 @ A @ P @ ( rev @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rev_filter % 0.31/0.65 thf(fact_7719_rev__map,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] : % 0.31/0.65 ( ( rev @ A @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 = ( map @ B @ A @ F2 @ ( rev @ B @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rev_map % 0.31/0.65 thf(fact_7720_rev__concat,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ ( list @ A )] : % 0.31/0.65 ( ( rev @ A @ ( concat @ A @ Xs ) ) % 0.31/0.65 = ( concat @ A @ ( map @ ( list @ A ) @ ( list @ A ) @ ( rev @ A ) @ ( rev @ ( list @ A ) @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rev_concat % 0.31/0.65 thf(fact_7721_take__rev,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( take @ A @ N2 @ ( rev @ A @ Xs ) ) % 0.31/0.65 = ( rev @ A @ ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % take_rev % 0.31/0.65 thf(fact_7722_rev__take,axiom, % 0.31/0.65 ! [A: $tType,I: nat,Xs: list @ A] : % 0.31/0.65 ( ( rev @ A @ ( take @ A @ I @ Xs ) ) % 0.31/0.65 = ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I ) @ ( rev @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rev_take % 0.31/0.65 thf(fact_7723_rev__drop,axiom, % 0.31/0.65 ! [A: $tType,I: nat,Xs: list @ A] : % 0.31/0.65 ( ( rev @ A @ ( drop @ A @ I @ Xs ) ) % 0.31/0.65 = ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I ) @ ( rev @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rev_drop % 0.31/0.65 thf(fact_7724_drop__rev,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( drop @ A @ N2 @ ( rev @ A @ Xs ) ) % 0.31/0.65 = ( rev @ A @ ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % drop_rev % 0.31/0.65 thf(fact_7725_rev__nth,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ( nth @ A @ ( rev @ A @ Xs ) @ N2 ) % 0.31/0.65 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N2 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rev_nth % 0.31/0.65 thf(fact_7726_rev__update,axiom, % 0.31/0.65 ! [A: $tType,K: nat,Xs: list @ A,Y: A] : % 0.31/0.65 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ( rev @ A @ ( list_update @ A @ Xs @ K @ Y ) ) % 0.31/0.65 = ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) @ ( one_one @ nat ) ) @ Y ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rev_update % 0.31/0.65 thf(fact_7727_sorted__transpose,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ ( list @ A )] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_transpose % 0.31/0.65 thf(fact_7728_sorted__rev__iff__nth__Suc,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) ) % 0.31/0.65 = ( ! [I3: nat] : % 0.31/0.65 ( ( ord_less @ nat @ ( suc @ I3 ) @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I3 ) ) @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_rev_iff_nth_Suc % 0.31/0.65 thf(fact_7729_sorted__rev__nth__mono,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A,I: nat,J: nat] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) ) % 0.31/0.65 => ( ( ord_less_eq @ nat @ I @ J ) % 0.31/0.65 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_rev_nth_mono % 0.31/0.65 thf(fact_7730_sorted__rev__iff__nth__mono,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) ) % 0.31/0.65 = ( ! [I3: nat,J3: nat] : % 0.31/0.65 ( ( ord_less_eq @ nat @ I3 @ J3 ) % 0.31/0.65 => ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ord_less_eq @ A @ ( nth @ A @ Xs @ J3 ) @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_rev_iff_nth_mono % 0.31/0.65 thf(fact_7731_foldr__max__sorted,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A,Y: A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) ) % 0.31/0.65 => ( ( ( Xs % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 => ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y ) % 0.31/0.65 = Y ) ) % 0.31/0.65 & ( ( Xs % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y ) % 0.31/0.65 = ( ord_max @ A @ ( nth @ A @ Xs @ ( zero_zero @ nat ) ) @ Y ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % foldr_max_sorted % 0.31/0.65 thf(fact_7732_length__transpose__sorted,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ ( list @ A )] : % 0.31/0.65 ( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) ) % 0.31/0.65 => ( ( ( Xs % 0.31/0.65 = ( nil @ ( list @ A ) ) ) % 0.31/0.65 => ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) % 0.31/0.65 = ( zero_zero @ nat ) ) ) % 0.31/0.65 & ( ( Xs % 0.31/0.65 != ( nil @ ( list @ A ) ) ) % 0.31/0.65 => ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) % 0.31/0.65 = ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ ( zero_zero @ nat ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % length_transpose_sorted % 0.31/0.65 thf(fact_7733_transpose__column__length,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ ( list @ A ),I: nat] : % 0.31/0.65 ( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) ) % 0.31/0.65 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) % 0.31/0.65 => ( ( size_size @ ( list @ ( list @ A ) ) % 0.31/0.65 @ ( filter2 @ ( list @ A ) % 0.31/0.65 @ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) ) % 0.31/0.65 @ ( transpose @ A @ Xs ) ) ) % 0.31/0.65 = ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose_column_length % 0.31/0.65 thf(fact_7734_folding__insort__key_Ofinite__set__strict__sorted,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 ) % 0.31/0.65 => ( ( finite_finite2 @ B @ A3 ) % 0.31/0.65 => ~ ! [L4: list @ B] : % 0.31/0.65 ( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F2 @ L4 ) ) % 0.31/0.65 => ( ( ( set2 @ B @ L4 ) % 0.31/0.65 = A3 ) % 0.31/0.65 => ( ( size_size @ ( list @ B ) @ L4 ) % 0.31/0.65 != ( finite_card @ B @ A3 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.finite_set_strict_sorted % 0.31/0.65 thf(fact_7735_transpose__transpose,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ ( list @ A )] : % 0.31/0.65 ( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) ) % 0.31/0.65 => ( ( transpose @ A @ ( transpose @ A @ Xs ) ) % 0.31/0.65 = ( takeWhile @ ( list @ A ) % 0.31/0.65 @ ^ [X: list @ A] : % 0.31/0.65 ( X % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % transpose_transpose % 0.31/0.65 thf(fact_7736_takeWhile__idem,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( takeWhile @ A @ P @ ( takeWhile @ A @ P @ Xs ) ) % 0.31/0.65 = ( takeWhile @ A @ P @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_idem % 0.31/0.65 thf(fact_7737_takeWhile__eq__all__conv,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( ( takeWhile @ A @ P @ Xs ) % 0.31/0.65 = Xs ) % 0.31/0.65 = ( ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( P @ X ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_eq_all_conv % 0.31/0.65 thf(fact_7738_takeWhile__append1,axiom, % 0.31/0.65 ! [A: $tType,X3: A,Xs: list @ A,P: A > $o,Ys: list @ A] : % 0.31/0.65 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( ~ ( P @ X3 ) % 0.31/0.65 => ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( takeWhile @ A @ P @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_append1 % 0.31/0.65 thf(fact_7739_takeWhile__append2,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] : % 0.31/0.65 ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( P @ X5 ) ) % 0.31/0.65 => ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( append @ A @ Xs @ ( takeWhile @ A @ P @ Ys ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_append2 % 0.31/0.65 thf(fact_7740_takeWhile__replicate,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,X3: A,N2: nat] : % 0.31/0.65 ( ( ( P @ X3 ) % 0.31/0.65 => ( ( takeWhile @ A @ P @ ( replicate @ A @ N2 @ X3 ) ) % 0.31/0.65 = ( replicate @ A @ N2 @ X3 ) ) ) % 0.31/0.65 & ( ~ ( P @ X3 ) % 0.31/0.65 => ( ( takeWhile @ A @ P @ ( replicate @ A @ N2 @ X3 ) ) % 0.31/0.65 = ( nil @ A ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_replicate % 0.31/0.65 thf(fact_7741_length__concat__rev,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ ( list @ A )] : % 0.31/0.65 ( ( size_size @ ( list @ A ) @ ( concat @ A @ ( rev @ ( list @ A ) @ Xs ) ) ) % 0.31/0.65 = ( size_size @ ( list @ A ) @ ( concat @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % length_concat_rev % 0.31/0.65 thf(fact_7742_sorted__takeWhile,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A,P: A > $o] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_takeWhile % 0.31/0.65 thf(fact_7743_nth__length__takeWhile,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ~ ( P @ ( nth @ A @ Xs @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % nth_length_takeWhile % 0.31/0.65 thf(fact_7744_takeWhile__nth,axiom, % 0.31/0.65 ! [A: $tType,J: nat,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) % 0.31/0.65 => ( ( nth @ A @ ( takeWhile @ A @ P @ Xs ) @ J ) % 0.31/0.65 = ( nth @ A @ Xs @ J ) ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_nth % 0.31/0.65 thf(fact_7745_takeWhile__map,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] : % 0.31/0.65 ( ( takeWhile @ A @ P @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 = ( map @ B @ A @ F2 @ ( takeWhile @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_map % 0.31/0.65 thf(fact_7746_folding__insort__key_Odistinct__if__distinct__map,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,Xs: list @ B] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( distinct @ A @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 => ( distinct @ B @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.distinct_if_distinct_map % 0.31/0.65 thf(fact_7747_zip__takeWhile__snd,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Xs: list @ A,P: B > $o,Ys: list @ B] : % 0.31/0.65 ( ( zip @ A @ B @ Xs @ ( takeWhile @ B @ P @ Ys ) ) % 0.31/0.65 = ( takeWhile @ ( product_prod @ A @ B ) @ ( comp @ B @ $o @ ( product_prod @ A @ B ) @ P @ ( product_snd @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ). % 0.31/0.65 % 0.31/0.65 % zip_takeWhile_snd % 0.31/0.65 thf(fact_7748_zip__takeWhile__fst,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,P: A > $o,Xs: list @ A,Ys: list @ B] : % 0.31/0.65 ( ( zip @ A @ B @ ( takeWhile @ A @ P @ Xs ) @ Ys ) % 0.31/0.65 = ( takeWhile @ ( product_prod @ A @ B ) @ ( comp @ A @ $o @ ( product_prod @ A @ B ) @ P @ ( product_fst @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ). % 0.31/0.65 % 0.31/0.65 % zip_takeWhile_fst % 0.31/0.65 thf(fact_7749_takeWhile__tail,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,X3: A,Xs: list @ A,L: list @ A] : % 0.31/0.65 ( ~ ( P @ X3 ) % 0.31/0.65 => ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ ( cons @ A @ X3 @ L ) ) ) % 0.31/0.65 = ( takeWhile @ A @ P @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_tail % 0.31/0.65 thf(fact_7750_folding__insort__key_Oinj__on,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( inj_on @ B @ A @ F2 @ S3 ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.inj_on % 0.31/0.65 thf(fact_7751_takeWhile_Osimps_I2_J,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( ( P @ X3 ) % 0.31/0.65 => ( ( takeWhile @ A @ P @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( cons @ A @ X3 @ ( takeWhile @ A @ P @ Xs ) ) ) ) % 0.31/0.65 & ( ~ ( P @ X3 ) % 0.31/0.65 => ( ( takeWhile @ A @ P @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( nil @ A ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile.simps(2) % 0.31/0.65 thf(fact_7752_takeWhile_Osimps_I1_J,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o] : % 0.31/0.65 ( ( takeWhile @ A @ P @ ( nil @ A ) ) % 0.31/0.65 = ( nil @ A ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile.simps(1) % 0.31/0.65 thf(fact_7753_set__takeWhileD,axiom, % 0.31/0.65 ! [A: $tType,X3: A,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( member @ A @ X3 @ ( set2 @ A @ ( takeWhile @ A @ P @ Xs ) ) ) % 0.31/0.65 => ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 & ( P @ X3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % set_takeWhileD % 0.31/0.65 thf(fact_7754_takeWhile__cong,axiom, % 0.31/0.65 ! [A: $tType,L: list @ A,K: list @ A,P: A > $o,Q: A > $o] : % 0.31/0.65 ( ( L = K ) % 0.31/0.65 => ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ L ) ) % 0.31/0.65 => ( ( P @ X5 ) % 0.31/0.65 = ( Q @ X5 ) ) ) % 0.31/0.65 => ( ( takeWhile @ A @ P @ L ) % 0.31/0.65 = ( takeWhile @ A @ Q @ K ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_cong % 0.31/0.65 thf(fact_7755_takeWhile__eq__take,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( takeWhile @ A ) % 0.31/0.65 = ( ^ [P3: A > $o,Xs2: list @ A] : ( take @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs2 ) ) @ Xs2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_eq_take % 0.31/0.65 thf(fact_7756_distinct__takeWhile,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.31/0.65 ( ( distinct @ A @ Xs ) % 0.31/0.65 => ( distinct @ A @ ( takeWhile @ A @ P @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % distinct_takeWhile % 0.31/0.65 thf(fact_7757_length__takeWhile__le,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % length_takeWhile_le % 0.31/0.65 thf(fact_7758_takeWhile__eq__Nil__iff,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( ( takeWhile @ A @ P @ Xs ) % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 = ( ( Xs % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 | ~ ( P @ ( hd @ A @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_eq_Nil_iff % 0.31/0.65 thf(fact_7759_takeWhile__append,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] : % 0.31/0.65 ( ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( P @ X5 ) ) % 0.31/0.65 => ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( append @ A @ Xs @ ( takeWhile @ A @ P @ Ys ) ) ) ) % 0.31/0.65 & ( ~ ! [X2: A] : % 0.31/0.65 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( P @ X2 ) ) % 0.31/0.65 => ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( takeWhile @ A @ P @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_append % 0.31/0.65 thf(fact_7760_length__takeWhile__less__P__nth,axiom, % 0.31/0.65 ! [A: $tType,J: nat,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ! [I4: nat] : % 0.31/0.65 ( ( ord_less @ nat @ I4 @ J ) % 0.31/0.65 => ( P @ ( nth @ A @ Xs @ I4 ) ) ) % 0.31/0.65 => ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % length_takeWhile_less_P_nth % 0.31/0.65 thf(fact_7761_takeWhile__eq__take__P__nth,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A,P: A > $o] : % 0.31/0.65 ( ! [I4: nat] : % 0.31/0.65 ( ( ord_less @ nat @ I4 @ N2 ) % 0.31/0.65 => ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ( P @ ( nth @ A @ Xs @ I4 ) ) ) ) % 0.31/0.65 => ( ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 => ~ ( P @ ( nth @ A @ Xs @ N2 ) ) ) % 0.31/0.65 => ( ( takeWhile @ A @ P @ Xs ) % 0.31/0.65 = ( take @ A @ N2 @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_eq_take_P_nth % 0.31/0.65 thf(fact_7762_sorted__list__of__set_Ofolding__insort__key__axioms,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ( folding_insort_key @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ ( set @ A ) ) % 0.31/0.65 @ ^ [X: A] : X ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_list_of_set.folding_insort_key_axioms % 0.31/0.65 thf(fact_7763_filter__equals__takeWhile__sorted__rev,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [F2: B > A,Xs: list @ B,T2: A] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) % 0.31/0.65 => ( ( filter2 @ B % 0.31/0.65 @ ^ [X: B] : ( ord_less @ A @ T2 @ ( F2 @ X ) ) % 0.31/0.65 @ Xs ) % 0.31/0.65 = ( takeWhile @ B % 0.31/0.65 @ ^ [X: B] : ( ord_less @ A @ T2 @ ( F2 @ X ) ) % 0.31/0.65 @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % filter_equals_takeWhile_sorted_rev % 0.31/0.65 thf(fact_7764_folding__insort__key_Osorted__key__list__of__set__unique,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B,L: list @ B] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 ) % 0.31/0.65 => ( ( finite_finite2 @ B @ A3 ) % 0.31/0.65 => ( ( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F2 @ L ) ) % 0.31/0.65 & ( ( set2 @ B @ L ) % 0.31/0.65 = A3 ) % 0.31/0.65 & ( ( size_size @ ( list @ B ) @ L ) % 0.31/0.65 = ( finite_card @ B @ A3 ) ) ) % 0.31/0.65 = ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) % 0.31/0.65 = L ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.sorted_key_list_of_set_unique % 0.31/0.65 thf(fact_7765_folding__insort__key_Osorted__key__list__of__set__remove,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,X3: B,A3: set @ B] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X3 @ A3 ) @ S3 ) % 0.31/0.65 => ( ( finite_finite2 @ B @ A3 ) % 0.31/0.65 => ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) % 0.31/0.65 = ( remove1 @ B @ X3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.sorted_key_list_of_set_remove % 0.31/0.65 thf(fact_7766_linorder_Osorted__key__list__of__set_Ocong,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( sorted8670434370408473282of_set @ A @ B ) % 0.31/0.65 = ( sorted8670434370408473282of_set @ A @ B ) ) ). % 0.31/0.65 % 0.31/0.65 % linorder.sorted_key_list_of_set.cong % 0.31/0.65 thf(fact_7767_folding__insort__key_Osorted__key__list__of__set__inject,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B,B3: set @ B] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ S3 ) % 0.31/0.65 => ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) % 0.31/0.65 = ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ B3 ) ) % 0.31/0.65 => ( ( finite_finite2 @ B @ A3 ) % 0.31/0.65 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.65 => ( A3 = B3 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.sorted_key_list_of_set_inject % 0.31/0.65 thf(fact_7768_folding__insort__key_Osorted__key__list__of__set__empty,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( bot_bot @ ( set @ B ) ) ) % 0.31/0.65 = ( nil @ B ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.sorted_key_list_of_set_empty % 0.31/0.65 thf(fact_7769_folding__insort__key_Oset__sorted__key__list__of__set,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 ) % 0.31/0.65 => ( ( finite_finite2 @ B @ A3 ) % 0.31/0.65 => ( ( set2 @ B @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) ) % 0.31/0.65 = A3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.set_sorted_key_list_of_set % 0.31/0.65 thf(fact_7770_folding__insort__key_Olength__sorted__key__list__of__set,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 ) % 0.31/0.65 => ( ( size_size @ ( list @ B ) @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) ) % 0.31/0.65 = ( finite_card @ B @ A3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.length_sorted_key_list_of_set % 0.31/0.65 thf(fact_7771_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 ) % 0.31/0.65 => ( distinct @ A @ ( map @ B @ A @ F2 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.distinct_sorted_key_list_of_set % 0.31/0.65 thf(fact_7772_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 ) % 0.31/0.65 => ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F2 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.strict_sorted_key_list_of_set % 0.31/0.65 thf(fact_7773_folding__insort__key_Osorted__sorted__key__list__of__set,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 ) % 0.31/0.65 => ( sorted_wrt @ A @ Less_eq2 @ ( map @ B @ A @ F2 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.sorted_sorted_key_list_of_set % 0.31/0.65 thf(fact_7774_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A3: set @ B] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 ) % 0.31/0.65 => ( ( finite_finite2 @ B @ A3 ) % 0.31/0.65 => ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) % 0.31/0.65 = ( nil @ B ) ) % 0.31/0.65 = ( A3 % 0.31/0.65 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.sorted_key_list_of_set_eq_Nil_iff % 0.31/0.65 thf(fact_7775_folding__insort__key_Oidem__if__sorted__distinct,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,Xs: list @ B] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ ( set2 @ B @ Xs ) @ S3 ) % 0.31/0.65 => ( ( sorted_wrt @ A @ Less_eq2 @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 => ( ( distinct @ B @ Xs ) % 0.31/0.65 => ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( set2 @ B @ Xs ) ) % 0.31/0.65 = Xs ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.idem_if_sorted_distinct % 0.31/0.65 thf(fact_7776_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,X3: B,A3: set @ B] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X3 @ A3 ) @ S3 ) % 0.31/0.65 => ( ( finite_finite2 @ B @ A3 ) % 0.31/0.65 => ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( insert @ B @ X3 @ A3 ) ) % 0.31/0.65 = ( insort_key @ A @ B @ Less_eq2 @ F2 @ X3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.sorted_key_list_of_set_insert_remove % 0.31/0.65 thf(fact_7777_folding__insort__key_Osorted__key__list__of__set__insert,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,X3: B,A3: set @ B] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X3 @ A3 ) @ S3 ) % 0.31/0.65 => ( ( finite_finite2 @ B @ A3 ) % 0.31/0.65 => ( ~ ( member @ B @ X3 @ A3 ) % 0.31/0.65 => ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ ( insert @ B @ X3 @ A3 ) ) % 0.31/0.65 = ( insort_key @ A @ B @ Less_eq2 @ F2 @ X3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq2 @ F2 @ A3 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.sorted_key_list_of_set_insert % 0.31/0.65 thf(fact_7778_linorder_Oinsort__key_Ocong,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( insort_key @ A @ B ) % 0.31/0.65 = ( insort_key @ A @ B ) ) ). % 0.31/0.65 % 0.31/0.65 % linorder.insort_key.cong % 0.31/0.65 thf(fact_7779_folding__insort__key_Oinsort__key__commute,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,X3: B,Y: B] : % 0.31/0.65 ( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F2 ) % 0.31/0.65 => ( ( member @ B @ X3 @ S3 ) % 0.31/0.65 => ( ( member @ B @ Y @ S3 ) % 0.31/0.65 => ( ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq2 @ F2 @ Y ) @ ( insort_key @ A @ B @ Less_eq2 @ F2 @ X3 ) ) % 0.31/0.65 = ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq2 @ F2 @ X3 ) @ ( insort_key @ A @ B @ Less_eq2 @ F2 @ Y ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % folding_insort_key.insort_key_commute % 0.31/0.65 thf(fact_7780_extract__def,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( extract @ A ) % 0.31/0.65 = ( ^ [P3: A > $o,Xs2: list @ A] : % 0.31/0.65 ( case_list @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ A @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) % 0.31/0.65 @ ^ [Y2: A,Ys3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( takeWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs2 ) @ ( product_Pair @ A @ ( list @ A ) @ Y2 @ Ys3 ) ) ) % 0.31/0.65 @ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % extract_def % 0.31/0.65 thf(fact_7781_lenlex__append2,axiom, % 0.31/0.65 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Us: list @ A,Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ( irrefl @ A @ R ) % 0.31/0.65 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Xs ) @ ( append @ A @ Us @ Ys ) ) @ ( lenlex @ A @ R ) ) % 0.31/0.65 = ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lenlex @ A @ R ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % lenlex_append2 % 0.31/0.65 thf(fact_7782_dropWhile__idem,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( dropWhile @ A @ P @ ( dropWhile @ A @ P @ Xs ) ) % 0.31/0.65 = ( dropWhile @ A @ P @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile_idem % 0.31/0.65 thf(fact_7783_dropWhile__eq__Nil__conv,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( ( dropWhile @ A @ P @ Xs ) % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 = ( ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( P @ X ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile_eq_Nil_conv % 0.31/0.65 thf(fact_7784_dropWhile__append2,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] : % 0.31/0.65 ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( P @ X5 ) ) % 0.31/0.65 => ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( dropWhile @ A @ P @ Ys ) ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile_append2 % 0.31/0.65 thf(fact_7785_dropWhile__append1,axiom, % 0.31/0.65 ! [A: $tType,X3: A,Xs: list @ A,P: A > $o,Ys: list @ A] : % 0.31/0.65 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( ~ ( P @ X3 ) % 0.31/0.65 => ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ Ys ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile_append1 % 0.31/0.65 thf(fact_7786_dropWhile__replicate,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,X3: A,N2: nat] : % 0.31/0.65 ( ( ( P @ X3 ) % 0.31/0.65 => ( ( dropWhile @ A @ P @ ( replicate @ A @ N2 @ X3 ) ) % 0.31/0.65 = ( nil @ A ) ) ) % 0.31/0.65 & ( ~ ( P @ X3 ) % 0.31/0.65 => ( ( dropWhile @ A @ P @ ( replicate @ A @ N2 @ X3 ) ) % 0.31/0.65 = ( replicate @ A @ N2 @ X3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile_replicate % 0.31/0.65 thf(fact_7787_takeWhile__dropWhile__id,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( append @ A @ ( takeWhile @ A @ P @ Xs ) @ ( dropWhile @ A @ P @ Xs ) ) % 0.31/0.65 = Xs ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_dropWhile_id % 0.31/0.65 thf(fact_7788_lexord__same__pref__if__irrefl,axiom, % 0.31/0.65 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: list @ A] : % 0.31/0.65 ( ( irrefl @ A @ R2 ) % 0.31/0.65 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lexord @ A @ R2 ) ) % 0.31/0.65 = ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lexord @ A @ R2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % lexord_same_pref_if_irrefl % 0.31/0.65 thf(fact_7789_sorted__dropWhile,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( linorder @ A ) % 0.31/0.65 => ! [Xs: list @ A,P: A > $o] : % 0.31/0.65 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) % 0.31/0.65 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( dropWhile @ A @ P @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sorted_dropWhile % 0.31/0.65 thf(fact_7790_hd__dropWhile,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( ( dropWhile @ A @ P @ Xs ) % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ~ ( P @ ( hd @ A @ ( dropWhile @ A @ P @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % hd_dropWhile % 0.31/0.65 thf(fact_7791_dropWhile__eq__self__iff,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( ( dropWhile @ A @ P @ Xs ) % 0.31/0.65 = Xs ) % 0.31/0.65 = ( ( Xs % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 | ~ ( P @ ( hd @ A @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile_eq_self_iff % 0.31/0.65 thf(fact_7792_length__dropWhile__le,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % length_dropWhile_le % 0.31/0.65 thf(fact_7793_distinct__dropWhile,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,P: A > $o] : % 0.31/0.65 ( ( distinct @ A @ Xs ) % 0.31/0.65 => ( distinct @ A @ ( dropWhile @ A @ P @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % distinct_dropWhile % 0.31/0.65 thf(fact_7794_set__dropWhileD,axiom, % 0.31/0.65 ! [A: $tType,X3: A,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( member @ A @ X3 @ ( set2 @ A @ ( dropWhile @ A @ P @ Xs ) ) ) % 0.31/0.65 => ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % set_dropWhileD % 0.31/0.65 thf(fact_7795_dropWhile__cong,axiom, % 0.31/0.65 ! [A: $tType,L: list @ A,K: list @ A,P: A > $o,Q: A > $o] : % 0.31/0.65 ( ( L = K ) % 0.31/0.65 => ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ L ) ) % 0.31/0.65 => ( ( P @ X5 ) % 0.31/0.65 = ( Q @ X5 ) ) ) % 0.31/0.65 => ( ( dropWhile @ A @ P @ L ) % 0.31/0.65 = ( dropWhile @ A @ Q @ K ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile_cong % 0.31/0.65 thf(fact_7796_irreflI,axiom, % 0.31/0.65 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : % 0.31/0.65 ( ! [A5: A] : % 0.31/0.65 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R ) % 0.31/0.65 => ( irrefl @ A @ R ) ) ). % 0.31/0.65 % 0.31/0.65 % irreflI % 0.31/0.65 thf(fact_7797_irrefl__def,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( irrefl @ A ) % 0.31/0.65 = ( ^ [R5: set @ ( product_prod @ A @ A )] : % 0.31/0.65 ! [A4: A] : % 0.31/0.65 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R5 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % irrefl_def % 0.31/0.65 thf(fact_7798_dropWhile_Osimps_I1_J,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o] : % 0.31/0.65 ( ( dropWhile @ A @ P @ ( nil @ A ) ) % 0.31/0.65 = ( nil @ A ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile.simps(1) % 0.31/0.65 thf(fact_7799_dropWhile_Osimps_I2_J,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( ( P @ X3 ) % 0.31/0.65 => ( ( dropWhile @ A @ P @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( dropWhile @ A @ P @ Xs ) ) ) % 0.31/0.65 & ( ~ ( P @ X3 ) % 0.31/0.65 => ( ( dropWhile @ A @ P @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( cons @ A @ X3 @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile.simps(2) % 0.31/0.65 thf(fact_7800_remdups__adj__Cons_H,axiom, % 0.31/0.65 ! [A: $tType,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( cons @ A @ X3 % 0.31/0.65 @ ( remdups_adj @ A % 0.31/0.65 @ ( dropWhile @ A % 0.31/0.65 @ ^ [Y2: A] : ( Y2 = X3 ) % 0.31/0.65 @ Xs ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % remdups_adj_Cons' % 0.31/0.65 thf(fact_7801_irrefl__lex,axiom, % 0.31/0.65 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.65 ( ( irrefl @ A @ R2 ) % 0.31/0.65 => ( irrefl @ ( list @ A ) @ ( lex @ A @ R2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % irrefl_lex % 0.31/0.65 thf(fact_7802_lexord__irrefl,axiom, % 0.31/0.65 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : % 0.31/0.65 ( ( irrefl @ A @ R ) % 0.31/0.65 => ( irrefl @ ( list @ A ) @ ( lexord @ A @ R ) ) ) ). % 0.31/0.65 % 0.31/0.65 % lexord_irrefl % 0.31/0.65 thf(fact_7803_dropWhile__append3,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Y: A,Xs: list @ A,Ys: list @ A] : % 0.31/0.65 ( ~ ( P @ Y ) % 0.31/0.65 => ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) % 0.31/0.65 = ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile_append3 % 0.31/0.65 thf(fact_7804_dropWhile__map,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] : % 0.31/0.65 ( ( dropWhile @ A @ P @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 = ( map @ B @ A @ F2 @ ( dropWhile @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile_map % 0.31/0.65 thf(fact_7805_dropWhile__eq__Cons__conv,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A,Y: A,Ys: list @ A] : % 0.31/0.65 ( ( ( dropWhile @ A @ P @ Xs ) % 0.31/0.65 = ( cons @ A @ Y @ Ys ) ) % 0.31/0.65 = ( ( Xs % 0.31/0.65 = ( append @ A @ ( takeWhile @ A @ P @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) % 0.31/0.65 & ~ ( P @ Y ) ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile_eq_Cons_conv % 0.31/0.65 thf(fact_7806_takeWhile__eq__filter,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ ( dropWhile @ A @ P @ Xs ) ) ) % 0.31/0.65 => ~ ( P @ X5 ) ) % 0.31/0.65 => ( ( takeWhile @ A @ P @ Xs ) % 0.31/0.65 = ( filter2 @ A @ P @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_eq_filter % 0.31/0.65 thf(fact_7807_dropWhile__eq__drop,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( dropWhile @ A ) % 0.31/0.65 = ( ^ [P3: A > $o,Xs2: list @ A] : ( drop @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs2 ) ) @ Xs2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile_eq_drop % 0.31/0.65 thf(fact_7808_dropWhile__append,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] : % 0.31/0.65 ( ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( P @ X5 ) ) % 0.31/0.65 => ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( dropWhile @ A @ P @ Ys ) ) ) % 0.31/0.65 & ( ~ ! [X2: A] : % 0.31/0.65 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( P @ X2 ) ) % 0.31/0.65 => ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) ) % 0.31/0.65 = ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ Ys ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile_append % 0.31/0.65 thf(fact_7809_lexl__not__refl,axiom, % 0.31/0.65 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: list @ A] : % 0.31/0.65 ( ( irrefl @ A @ R2 ) % 0.31/0.65 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ X3 ) @ ( lex @ A @ R2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % lexl_not_refl % 0.31/0.65 thf(fact_7810_find__dropWhile,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( find @ A ) % 0.31/0.65 = ( ^ [P3: A > $o,Xs2: list @ A] : % 0.31/0.65 ( case_list @ ( option @ A ) @ A @ ( none @ A ) % 0.31/0.65 @ ^ [X: A,Xa4: list @ A] : ( some @ A @ X ) % 0.31/0.65 @ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % find_dropWhile % 0.31/0.65 thf(fact_7811_remdups__adj__append__dropWhile,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A] : % 0.31/0.65 ( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) % 0.31/0.65 = ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) % 0.31/0.65 @ ( remdups_adj @ A % 0.31/0.65 @ ( dropWhile @ A % 0.31/0.65 @ ^ [X: A] : ( X = Y ) % 0.31/0.65 @ Ys ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % remdups_adj_append_dropWhile % 0.31/0.65 thf(fact_7812_tl__remdups__adj,axiom, % 0.31/0.65 ! [A: $tType,Ys: list @ A] : % 0.31/0.65 ( ( Ys % 0.31/0.65 != ( nil @ A ) ) % 0.31/0.65 => ( ( tl @ A @ ( remdups_adj @ A @ Ys ) ) % 0.31/0.65 = ( remdups_adj @ A % 0.31/0.65 @ ( dropWhile @ A % 0.31/0.65 @ ^ [X: A] : % 0.31/0.65 ( X % 0.31/0.65 = ( hd @ A @ Ys ) ) % 0.31/0.65 @ ( tl @ A @ Ys ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % tl_remdups_adj % 0.31/0.65 thf(fact_7813_dropWhile__nth,axiom, % 0.31/0.65 ! [A: $tType,J: nat,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) ) % 0.31/0.65 => ( ( nth @ A @ ( dropWhile @ A @ P @ Xs ) @ J ) % 0.31/0.65 = ( nth @ A @ Xs @ ( plus_plus @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile_nth % 0.31/0.65 thf(fact_7814_dropWhile__neq__rev,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,X3: A] : % 0.31/0.65 ( ( distinct @ A @ Xs ) % 0.31/0.65 => ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( ( dropWhile @ A % 0.31/0.65 @ ^ [Y2: A] : ( Y2 != X3 ) % 0.31/0.65 @ ( rev @ A @ Xs ) ) % 0.31/0.65 = ( cons @ A @ X3 % 0.31/0.65 @ ( rev @ A % 0.31/0.65 @ ( takeWhile @ A % 0.31/0.65 @ ^ [Y2: A] : ( Y2 != X3 ) % 0.31/0.65 @ Xs ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % dropWhile_neq_rev % 0.31/0.65 thf(fact_7815_takeWhile__neq__rev,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,X3: A] : % 0.31/0.65 ( ( distinct @ A @ Xs ) % 0.31/0.65 => ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 0.31/0.65 => ( ( takeWhile @ A % 0.31/0.65 @ ^ [Y2: A] : ( Y2 != X3 ) % 0.31/0.65 @ ( rev @ A @ Xs ) ) % 0.31/0.65 = ( rev @ A % 0.31/0.65 @ ( tl @ A % 0.31/0.65 @ ( dropWhile @ A % 0.31/0.65 @ ^ [Y2: A] : ( Y2 != X3 ) % 0.31/0.65 @ Xs ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % takeWhile_neq_rev % 0.31/0.65 thf(fact_7816_partition__filter__conv,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( partition @ A ) % 0.31/0.65 = ( ^ [F3: A > $o,Xs2: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ F3 @ Xs2 ) @ ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ F3 ) @ Xs2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % partition_filter_conv % 0.31/0.65 thf(fact_7817_min__list_Oelims,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( ord @ A ) % 0.31/0.65 => ! [X3: list @ A,Y: A] : % 0.31/0.65 ( ( ( min_list @ A @ X3 ) % 0.31/0.65 = Y ) % 0.31/0.65 => ( ! [X5: A,Xs3: list @ A] : % 0.31/0.65 ( ( X3 % 0.31/0.65 = ( cons @ A @ X5 @ Xs3 ) ) % 0.31/0.65 => ( Y % 0.31/0.65 != ( case_list @ A @ A @ X5 % 0.31/0.65 @ ^ [A4: A,List2: list @ A] : ( ord_min @ A @ X5 @ ( min_list @ A @ Xs3 ) ) % 0.31/0.65 @ Xs3 ) ) ) % 0.31/0.65 => ~ ( ( X3 % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 => ( Y % 0.31/0.65 != ( undefined @ A ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % min_list.elims % 0.31/0.65 thf(fact_7818_hd__def,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( hd @ A ) % 0.31/0.65 = ( case_list @ A @ A @ ( undefined @ A ) % 0.31/0.65 @ ^ [X213: A,X224: list @ A] : X213 ) ) ). % 0.31/0.65 % 0.31/0.65 % hd_def % 0.31/0.65 thf(fact_7819_option_Othe__def,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( the2 @ A ) % 0.31/0.65 = ( case_option @ A @ A @ ( undefined @ A ) % 0.31/0.65 @ ^ [X24: A] : X24 ) ) ). % 0.31/0.65 % 0.31/0.65 % option.the_def % 0.31/0.65 thf(fact_7820_partition__filter1,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( partition @ A @ P @ Xs ) ) % 0.31/0.65 = ( filter2 @ A @ P @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % partition_filter1 % 0.31/0.65 thf(fact_7821_partition_Osimps_I1_J,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o] : % 0.31/0.65 ( ( partition @ A @ P @ ( nil @ A ) ) % 0.31/0.65 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) ) ). % 0.31/0.65 % 0.31/0.65 % partition.simps(1) % 0.31/0.65 thf(fact_7822_partition__P,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A,Yes: list @ A,No4: list @ A] : % 0.31/0.65 ( ( ( partition @ A @ P @ Xs ) % 0.31/0.65 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No4 ) ) % 0.31/0.65 => ( ! [X2: A] : % 0.31/0.65 ( ( member @ A @ X2 @ ( set2 @ A @ Yes ) ) % 0.31/0.65 => ( P @ X2 ) ) % 0.31/0.65 & ! [X2: A] : % 0.31/0.65 ( ( member @ A @ X2 @ ( set2 @ A @ No4 ) ) % 0.31/0.65 => ~ ( P @ X2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % partition_P % 0.31/0.65 thf(fact_7823_partition_Osimps_I2_J,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,X3: A,Xs: list @ A] : % 0.31/0.65 ( ( partition @ A @ P @ ( cons @ A @ X3 @ Xs ) ) % 0.31/0.65 = ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) % 0.31/0.65 @ ^ [Yes2: list @ A,No3: list @ A] : ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Yes2 ) @ No3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ ( cons @ A @ X3 @ No3 ) ) ) % 0.31/0.65 @ ( partition @ A @ P @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % partition.simps(2) % 0.31/0.65 thf(fact_7824_partition__filter2,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A] : % 0.31/0.65 ( ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( partition @ A @ P @ Xs ) ) % 0.31/0.65 = ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % partition_filter2 % 0.31/0.65 thf(fact_7825_Func__empty,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,B3: set @ B] : % 0.31/0.65 ( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B3 ) % 0.31/0.65 = ( insert @ ( A > B ) % 0.31/0.65 @ ^ [X: A] : ( undefined @ B ) % 0.31/0.65 @ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Func_empty % 0.31/0.65 thf(fact_7826_partition__set,axiom, % 0.31/0.65 ! [A: $tType,P: A > $o,Xs: list @ A,Yes: list @ A,No4: list @ A] : % 0.31/0.65 ( ( ( partition @ A @ P @ Xs ) % 0.31/0.65 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No4 ) ) % 0.31/0.65 => ( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Yes ) @ ( set2 @ A @ No4 ) ) % 0.31/0.65 = ( set2 @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % partition_set % 0.31/0.65 thf(fact_7827_arg__min__list_Oelims,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( linorder @ B ) % 0.31/0.65 => ! [X3: A > B,Xa2: list @ A,Y: A] : % 0.31/0.65 ( ( ( arg_min_list @ A @ B @ X3 @ Xa2 ) % 0.31/0.65 = Y ) % 0.31/0.65 => ( ! [X5: A] : % 0.31/0.65 ( ( Xa2 % 0.31/0.65 = ( cons @ A @ X5 @ ( nil @ A ) ) ) % 0.31/0.65 => ( Y != X5 ) ) % 0.31/0.65 => ( ! [X5: A,Y4: A,Zs: list @ A] : % 0.31/0.65 ( ( Xa2 % 0.31/0.65 = ( cons @ A @ X5 @ ( cons @ A @ Y4 @ Zs ) ) ) % 0.31/0.65 => ( Y % 0.31/0.65 != ( if @ A @ ( ord_less_eq @ B @ ( X3 @ X5 ) @ ( X3 @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y4 @ Zs ) ) ) ) @ X5 @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y4 @ Zs ) ) ) ) ) % 0.31/0.65 => ~ ( ( Xa2 % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 => ( Y % 0.31/0.65 != ( undefined @ A ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % arg_min_list.elims % 0.31/0.65 thf(fact_7828_min__list_Opelims,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( ord @ A ) % 0.31/0.65 => ! [X3: list @ A,Y: A] : % 0.31/0.65 ( ( ( min_list @ A @ X3 ) % 0.31/0.65 = Y ) % 0.31/0.65 => ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X3 ) % 0.31/0.65 => ( ! [X5: A,Xs3: list @ A] : % 0.31/0.65 ( ( X3 % 0.31/0.65 = ( cons @ A @ X5 @ Xs3 ) ) % 0.31/0.65 => ( ( Y % 0.31/0.65 = ( case_list @ A @ A @ X5 % 0.31/0.65 @ ^ [A4: A,List2: list @ A] : ( ord_min @ A @ X5 @ ( min_list @ A @ Xs3 ) ) % 0.31/0.65 @ Xs3 ) ) % 0.31/0.65 => ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( cons @ A @ X5 @ Xs3 ) ) ) ) % 0.31/0.65 => ~ ( ( X3 % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 => ( ( Y % 0.31/0.65 = ( undefined @ A ) ) % 0.31/0.65 => ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % min_list.pelims % 0.31/0.65 thf(fact_7829_lists__length__Suc__eq,axiom, % 0.31/0.65 ! [A: $tType,A3: set @ A,N2: nat] : % 0.31/0.65 ( ( collect @ ( list @ A ) % 0.31/0.65 @ ^ [Xs2: list @ A] : % 0.31/0.65 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) % 0.31/0.65 & ( ( size_size @ ( list @ A ) @ Xs2 ) % 0.31/0.65 = ( suc @ N2 ) ) ) ) % 0.31/0.65 = ( image @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A ) % 0.31/0.65 @ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A ) % 0.31/0.65 @ ^ [Xs2: list @ A,N: A] : ( cons @ A @ N @ Xs2 ) ) % 0.31/0.65 @ ( product_Sigma @ ( list @ A ) @ A % 0.31/0.65 @ ( collect @ ( list @ A ) % 0.31/0.65 @ ^ [Xs2: list @ A] : % 0.31/0.65 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) % 0.31/0.65 & ( ( size_size @ ( list @ A ) @ Xs2 ) % 0.31/0.65 = N2 ) ) ) % 0.31/0.65 @ ^ [Uu3: list @ A] : A3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % lists_length_Suc_eq % 0.31/0.65 thf(fact_7830_mem__Sigma__iff,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] : % 0.31/0.65 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) % 0.31/0.65 = ( ( member @ A @ A2 @ A3 ) % 0.31/0.65 & ( member @ B @ B2 @ ( B3 @ A2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % mem_Sigma_iff % 0.31/0.65 thf(fact_7831_SigmaI,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A2: A,A3: set @ A,B2: B,B3: A > ( set @ B )] : % 0.31/0.65 ( ( member @ A @ A2 @ A3 ) % 0.31/0.65 => ( ( member @ B @ B2 @ ( B3 @ A2 ) ) % 0.31/0.65 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % SigmaI % 0.31/0.65 thf(fact_7832_Collect__case__prod,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,P: A > $o,Q: B > $o] : % 0.31/0.65 ( ( collect @ ( product_prod @ A @ B ) % 0.31/0.65 @ ( product_case_prod @ A @ B @ $o % 0.31/0.65 @ ^ [A4: A,B4: B] : % 0.31/0.65 ( ( P @ A4 ) % 0.31/0.65 & ( Q @ B4 ) ) ) ) % 0.31/0.65 = ( product_Sigma @ A @ B @ ( collect @ A @ P ) % 0.31/0.65 @ ^ [Uu3: A] : ( collect @ B @ Q ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Collect_case_prod % 0.31/0.65 thf(fact_7833_Sigma__empty1,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,B3: A > ( set @ B )] : % 0.31/0.65 ( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B3 ) % 0.31/0.65 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Sigma_empty1 % 0.31/0.65 thf(fact_7834_Compl__Times__UNIV2,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A] : % 0.31/0.65 ( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) ) % 0.31/0.65 = ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A3 ) % 0.31/0.65 @ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Compl_Times_UNIV2 % 0.31/0.65 thf(fact_7835_Compl__Times__UNIV1,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ B] : % 0.31/0.65 ( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) ) % 0.31/0.65 @ ^ [Uu3: A] : A3 ) ) % 0.31/0.65 = ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) ) % 0.31/0.65 @ ^ [Uu3: A] : ( uminus_uminus @ ( set @ B ) @ A3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Compl_Times_UNIV1 % 0.31/0.65 thf(fact_7836_Times__empty,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,A3: set @ A,B3: set @ B] : % 0.31/0.65 ( ( ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) % 0.31/0.65 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) % 0.31/0.65 = ( ( A3 % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.65 | ( B3 % 0.31/0.65 = ( bot_bot @ ( set @ B ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Times_empty % 0.31/0.65 thf(fact_7837_Sigma__empty2,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A] : % 0.31/0.65 ( ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : ( bot_bot @ ( set @ B ) ) ) % 0.31/0.65 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Sigma_empty2 % 0.31/0.65 thf(fact_7838_finite__SigmaI,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] : % 0.31/0.65 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.65 => ( ! [A5: A] : % 0.31/0.65 ( ( member @ A @ A5 @ A3 ) % 0.31/0.65 => ( finite_finite2 @ B @ ( B3 @ A5 ) ) ) % 0.31/0.65 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % finite_SigmaI % 0.31/0.65 thf(fact_7839_UNIV__Times__UNIV,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) ) % 0.31/0.65 @ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) % 0.31/0.65 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % UNIV_Times_UNIV % 0.31/0.65 thf(fact_7840_fst__image__times,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,B3: set @ B,A3: set @ A] : % 0.31/0.65 ( ( ( B3 % 0.31/0.65 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.65 => ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) ) % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.65 & ( ( B3 % 0.31/0.65 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.65 => ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) ) % 0.31/0.65 = A3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % fst_image_times % 0.31/0.65 thf(fact_7841_snd__image__times,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ B,B3: set @ A] : % 0.31/0.65 ( ( ( A3 % 0.31/0.65 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.65 => ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) % 0.31/0.65 @ ( product_Sigma @ B @ A @ A3 % 0.31/0.65 @ ^ [Uu3: B] : B3 ) ) % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.65 & ( ( A3 % 0.31/0.65 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.65 => ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) % 0.31/0.65 @ ( product_Sigma @ B @ A @ A3 % 0.31/0.65 @ ^ [Uu3: B] : B3 ) ) % 0.31/0.65 = B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % snd_image_times % 0.31/0.65 thf(fact_7842_set__product,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] : % 0.31/0.65 ( ( set2 @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) ) % 0.31/0.65 = ( product_Sigma @ A @ B @ ( set2 @ A @ Xs ) % 0.31/0.65 @ ^ [Uu3: A] : ( set2 @ B @ Ys ) ) ) ). % 0.31/0.65 % 0.31/0.65 % set_product % 0.31/0.65 thf(fact_7843_insert__Times__insert,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A2: A,A3: set @ A,B2: B,B3: set @ B] : % 0.31/0.65 ( ( product_Sigma @ A @ B @ ( insert @ A @ A2 @ A3 ) % 0.31/0.65 @ ^ [Uu3: A] : ( insert @ B @ B2 @ B3 ) ) % 0.31/0.65 = ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) % 0.31/0.65 @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : ( insert @ B @ B2 @ B3 ) ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ ( insert @ A @ A2 @ A3 ) % 0.31/0.65 @ ^ [Uu3: A] : B3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % insert_Times_insert % 0.31/0.65 thf(fact_7844_card__SigmaI,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] : % 0.31/0.65 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.65 => ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ A3 ) % 0.31/0.65 => ( finite_finite2 @ B @ ( B3 @ X5 ) ) ) % 0.31/0.65 => ( ( finite_card @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) % 0.31/0.65 = ( groups7311177749621191930dd_sum @ A @ nat % 0.31/0.65 @ ^ [A4: A] : ( finite_card @ B @ ( B3 @ A4 ) ) % 0.31/0.65 @ A3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % card_SigmaI % 0.31/0.65 thf(fact_7845_inj__on__apfst,axiom, % 0.31/0.65 ! [B: $tType,C: $tType,A: $tType,F2: A > C,A3: set @ A] : % 0.31/0.65 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) ) % 0.31/0.65 = ( inj_on @ A @ C @ F2 @ A3 ) ) ). % 0.31/0.65 % 0.31/0.65 % inj_on_apfst % 0.31/0.65 thf(fact_7846_inj__on__apsnd,axiom, % 0.31/0.65 ! [A: $tType,C: $tType,B: $tType,F2: B > C,A3: set @ B] : % 0.31/0.65 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) ) % 0.31/0.65 @ ^ [Uu3: A] : A3 ) ) % 0.31/0.65 = ( inj_on @ B @ C @ F2 @ A3 ) ) ). % 0.31/0.65 % 0.31/0.65 % inj_on_apsnd % 0.31/0.65 thf(fact_7847_fst__image__Sigma,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] : % 0.31/0.65 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) % 0.31/0.65 = ( collect @ A % 0.31/0.65 @ ^ [X: A] : % 0.31/0.65 ( ( member @ A @ X @ A3 ) % 0.31/0.65 & ( ( B3 @ X ) % 0.31/0.65 != ( bot_bot @ ( set @ B ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % fst_image_Sigma % 0.31/0.65 thf(fact_7848_trancl__subset__Sigma__aux,axiom, % 0.31/0.65 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),A3: set @ A] : % 0.31/0.65 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 % 0.31/0.65 @ ( product_Sigma @ A @ A @ A3 % 0.31/0.65 @ ^ [Uu3: A] : A3 ) ) % 0.31/0.65 => ( ( A2 = B2 ) % 0.31/0.65 | ( member @ A @ A2 @ A3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % trancl_subset_Sigma_aux % 0.31/0.65 thf(fact_7849_SigmaE2,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] : % 0.31/0.65 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) % 0.31/0.65 => ~ ( ( member @ A @ A2 @ A3 ) % 0.31/0.65 => ~ ( member @ B @ B2 @ ( B3 @ A2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % SigmaE2 % 0.31/0.65 thf(fact_7850_SigmaD2,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] : % 0.31/0.65 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) % 0.31/0.65 => ( member @ B @ B2 @ ( B3 @ A2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % SigmaD2 % 0.31/0.65 thf(fact_7851_SigmaD1,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A2: A,B2: B,A3: set @ A,B3: A > ( set @ B )] : % 0.31/0.65 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) % 0.31/0.65 => ( member @ A @ A2 @ A3 ) ) ). % 0.31/0.65 % 0.31/0.65 % SigmaD1 % 0.31/0.65 thf(fact_7852_SigmaE,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,C2: product_prod @ A @ B,A3: set @ A,B3: A > ( set @ B )] : % 0.31/0.65 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) % 0.31/0.65 => ~ ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ A3 ) % 0.31/0.65 => ! [Y4: B] : % 0.31/0.65 ( ( member @ B @ Y4 @ ( B3 @ X5 ) ) % 0.31/0.65 => ( C2 % 0.31/0.65 != ( product_Pair @ A @ B @ X5 @ Y4 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % SigmaE % 0.31/0.65 thf(fact_7853_times__eq__iff,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,A3: set @ A,B3: set @ B,C5: set @ A,D6: set @ B] : % 0.31/0.65 ( ( ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) % 0.31/0.65 = ( product_Sigma @ A @ B @ C5 % 0.31/0.65 @ ^ [Uu3: A] : D6 ) ) % 0.31/0.65 = ( ( ( A3 = C5 ) % 0.31/0.65 & ( B3 = D6 ) ) % 0.31/0.65 | ( ( ( A3 % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.65 | ( B3 % 0.31/0.65 = ( bot_bot @ ( set @ B ) ) ) ) % 0.31/0.65 & ( ( C5 % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.65 | ( D6 % 0.31/0.65 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % times_eq_iff % 0.31/0.65 thf(fact_7854_Sigma__empty__iff,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,I6: set @ A,X8: A > ( set @ B )] : % 0.31/0.65 ( ( ( product_Sigma @ A @ B @ I6 @ X8 ) % 0.31/0.65 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) % 0.31/0.65 = ( ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ I6 ) % 0.31/0.65 => ( ( X8 @ X ) % 0.31/0.65 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Sigma_empty_iff % 0.31/0.65 thf(fact_7855_Sigma__Union,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,X8: set @ ( set @ A ),B3: A > ( set @ B )] : % 0.31/0.65 ( ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ X8 ) @ B3 ) % 0.31/0.65 = ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ( image @ ( set @ A ) @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ^ [A6: set @ A] : ( product_Sigma @ A @ B @ A6 @ B3 ) % 0.31/0.65 @ X8 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Sigma_Union % 0.31/0.65 thf(fact_7856_Times__subset__cancel2,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,X3: A,C5: set @ A,A3: set @ B,B3: set @ B] : % 0.31/0.65 ( ( member @ A @ X3 @ C5 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) % 0.31/0.65 @ ( product_Sigma @ B @ A @ A3 % 0.31/0.65 @ ^ [Uu3: B] : C5 ) % 0.31/0.65 @ ( product_Sigma @ B @ A @ B3 % 0.31/0.65 @ ^ [Uu3: B] : C5 ) ) % 0.31/0.65 = ( ord_less_eq @ ( set @ B ) @ A3 @ B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Times_subset_cancel2 % 0.31/0.65 thf(fact_7857_Sigma__mono,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,C5: set @ A,B3: A > ( set @ B ),D6: A > ( set @ B )] : % 0.31/0.65 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) % 0.31/0.65 => ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ A3 ) % 0.31/0.65 => ( ord_less_eq @ ( set @ B ) @ ( B3 @ X5 ) @ ( D6 @ X5 ) ) ) % 0.31/0.65 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ C5 @ D6 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Sigma_mono % 0.31/0.65 thf(fact_7858_Restr__subset,axiom, % 0.31/0.65 ! [A: $tType,A3: set @ A,B3: set @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.65 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.65 => ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) % 0.31/0.65 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 % 0.31/0.65 @ ( product_Sigma @ A @ A @ B3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) ) % 0.31/0.65 @ ( product_Sigma @ A @ A @ A3 % 0.31/0.65 @ ^ [Uu3: A] : A3 ) ) % 0.31/0.65 = ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 % 0.31/0.65 @ ( product_Sigma @ A @ A @ A3 % 0.31/0.65 @ ^ [Uu3: A] : A3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Restr_subset % 0.31/0.65 thf(fact_7859_Times__Un__distrib1,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C5: set @ B] : % 0.31/0.65 ( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.65 @ ^ [Uu3: A] : C5 ) % 0.31/0.65 = ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : C5 ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ B3 % 0.31/0.65 @ ^ [Uu3: A] : C5 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Times_Un_distrib1 % 0.31/0.65 thf(fact_7860_Sigma__Un__distrib2,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B ),B3: A > ( set @ B )] : % 0.31/0.65 ( ( product_Sigma @ A @ B @ I6 % 0.31/0.65 @ ^ [I3: A] : ( sup_sup @ ( set @ B ) @ ( A3 @ I3 ) @ ( B3 @ I3 ) ) ) % 0.31/0.65 = ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I6 @ A3 ) @ ( product_Sigma @ A @ B @ I6 @ B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Sigma_Un_distrib2 % 0.31/0.65 thf(fact_7861_Sigma__Un__distrib1,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,I6: set @ A,J4: set @ A,C5: A > ( set @ B )] : % 0.31/0.65 ( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I6 @ J4 ) @ C5 ) % 0.31/0.65 = ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I6 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Sigma_Un_distrib1 % 0.31/0.65 thf(fact_7862_member__product,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,X3: product_prod @ A @ B,A3: set @ A,B3: set @ B] : % 0.31/0.65 ( ( member @ ( product_prod @ A @ B ) @ X3 @ ( product_product @ A @ B @ A3 @ B3 ) ) % 0.31/0.65 = ( member @ ( product_prod @ A @ B ) @ X3 % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % member_product % 0.31/0.65 thf(fact_7863_Product__Type_Oproduct__def,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( product_product @ A @ B ) % 0.31/0.65 = ( ^ [A6: set @ A,B6: set @ B] : % 0.31/0.65 ( product_Sigma @ A @ B @ A6 % 0.31/0.65 @ ^ [Uu3: A] : B6 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Product_Type.product_def % 0.31/0.65 thf(fact_7864_finite__cartesian__product,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] : % 0.31/0.65 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.65 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.65 => ( finite_finite2 @ ( product_prod @ A @ B ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % finite_cartesian_product % 0.31/0.65 thf(fact_7865_Times__Int__distrib1,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C5: set @ B] : % 0.31/0.65 ( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.65 @ ^ [Uu3: A] : C5 ) % 0.31/0.65 = ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : C5 ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ B3 % 0.31/0.65 @ ^ [Uu3: A] : C5 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Times_Int_distrib1 % 0.31/0.65 thf(fact_7866_Sigma__Int__distrib2,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B ),B3: A > ( set @ B )] : % 0.31/0.65 ( ( product_Sigma @ A @ B @ I6 % 0.31/0.65 @ ^ [I3: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ I3 ) @ ( B3 @ I3 ) ) ) % 0.31/0.65 = ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I6 @ A3 ) @ ( product_Sigma @ A @ B @ I6 @ B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Sigma_Int_distrib2 % 0.31/0.65 thf(fact_7867_Times__Int__Times,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,C5: set @ A,D6: set @ B] : % 0.31/0.65 ( ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ C5 % 0.31/0.65 @ ^ [Uu3: A] : D6 ) ) % 0.31/0.65 = ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A3 @ C5 ) % 0.31/0.65 @ ^ [Uu3: A] : ( inf_inf @ ( set @ B ) @ B3 @ D6 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Times_Int_Times % 0.31/0.65 thf(fact_7868_Sigma__Int__distrib1,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,I6: set @ A,J4: set @ A,C5: A > ( set @ B )] : % 0.31/0.65 ( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ I6 @ J4 ) @ C5 ) % 0.31/0.65 = ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I6 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Sigma_Int_distrib1 % 0.31/0.65 thf(fact_7869_Sigma__Diff__distrib1,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,I6: set @ A,J4: set @ A,C5: A > ( set @ B )] : % 0.31/0.65 ( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ I6 @ J4 ) @ C5 ) % 0.31/0.65 = ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I6 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Sigma_Diff_distrib1 % 0.31/0.65 thf(fact_7870_Sigma__cong,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C5: A > ( set @ B ),D6: A > ( set @ B )] : % 0.31/0.65 ( ( A3 = B3 ) % 0.31/0.65 => ( ! [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ B3 ) % 0.31/0.65 => ( ( C5 @ X5 ) % 0.31/0.65 = ( D6 @ X5 ) ) ) % 0.31/0.65 => ( ( product_Sigma @ A @ B @ A3 @ C5 ) % 0.31/0.65 = ( product_Sigma @ A @ B @ B3 @ D6 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Sigma_cong % 0.31/0.65 thf(fact_7871_Times__eq__cancel2,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,X3: A,C5: set @ A,A3: set @ B,B3: set @ B] : % 0.31/0.65 ( ( member @ A @ X3 @ C5 ) % 0.31/0.65 => ( ( ( product_Sigma @ B @ A @ A3 % 0.31/0.65 @ ^ [Uu3: B] : C5 ) % 0.31/0.65 = ( product_Sigma @ B @ A @ B3 % 0.31/0.65 @ ^ [Uu3: B] : C5 ) ) % 0.31/0.65 = ( A3 = B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Times_eq_cancel2 % 0.31/0.65 thf(fact_7872_Times__Diff__distrib1,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C5: set @ B] : % 0.31/0.65 ( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.65 @ ^ [Uu3: A] : C5 ) % 0.31/0.65 = ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : C5 ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ B3 % 0.31/0.65 @ ^ [Uu3: A] : C5 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Times_Diff_distrib1 % 0.31/0.65 thf(fact_7873_Sigma__Diff__distrib2,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B ),B3: A > ( set @ B )] : % 0.31/0.65 ( ( product_Sigma @ A @ B @ I6 % 0.31/0.65 @ ^ [I3: A] : ( minus_minus @ ( set @ B ) @ ( A3 @ I3 ) @ ( B3 @ I3 ) ) ) % 0.31/0.65 = ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I6 @ A3 ) @ ( product_Sigma @ A @ B @ I6 @ B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Sigma_Diff_distrib2 % 0.31/0.65 thf(fact_7874_Collect__case__prod__Sigma,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] : % 0.31/0.65 ( ( collect @ ( product_prod @ A @ B ) % 0.31/0.65 @ ( product_case_prod @ A @ B @ $o % 0.31/0.65 @ ^ [X: A,Y2: B] : % 0.31/0.65 ( ( P @ X ) % 0.31/0.65 & ( Q @ X @ Y2 ) ) ) ) % 0.31/0.65 = ( product_Sigma @ A @ B @ ( collect @ A @ P ) % 0.31/0.65 @ ^ [X: A] : ( collect @ B @ ( Q @ X ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Collect_case_prod_Sigma % 0.31/0.65 thf(fact_7875_swap__product,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ B,B3: set @ A] : % 0.31/0.65 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) % 0.31/0.65 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B ) % 0.31/0.65 @ ^ [I3: B,J3: A] : ( product_Pair @ A @ B @ J3 @ I3 ) ) % 0.31/0.65 @ ( product_Sigma @ B @ A @ A3 % 0.31/0.65 @ ^ [Uu3: B] : B3 ) ) % 0.31/0.65 = ( product_Sigma @ A @ B @ B3 % 0.31/0.65 @ ^ [Uu3: A] : A3 ) ) ). % 0.31/0.65 % 0.31/0.65 % swap_product % 0.31/0.65 thf(fact_7876_times__subset__iff,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,A3: set @ A,C5: set @ B,B3: set @ A,D6: set @ B] : % 0.31/0.65 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : C5 ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ B3 % 0.31/0.65 @ ^ [Uu3: A] : D6 ) ) % 0.31/0.65 = ( ( A3 % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.65 | ( C5 % 0.31/0.65 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.65 | ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.65 & ( ord_less_eq @ ( set @ B ) @ C5 @ D6 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % times_subset_iff % 0.31/0.65 thf(fact_7877_mem__Times__iff,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,X3: product_prod @ A @ B,A3: set @ A,B3: set @ B] : % 0.31/0.65 ( ( member @ ( product_prod @ A @ B ) @ X3 % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) ) % 0.31/0.65 = ( ( member @ A @ ( product_fst @ A @ B @ X3 ) @ A3 ) % 0.31/0.65 & ( member @ B @ ( product_snd @ A @ B @ X3 ) @ B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % mem_Times_iff % 0.31/0.65 thf(fact_7878_image__paired__Times,axiom, % 0.31/0.65 ! [C: $tType,D: $tType,B: $tType,A: $tType,F2: C > A,G: D > B,A3: set @ C,B3: set @ D] : % 0.31/0.65 ( ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) % 0.31/0.65 @ ( product_case_prod @ C @ D @ ( product_prod @ A @ B ) % 0.31/0.65 @ ^ [X: C,Y2: D] : ( product_Pair @ A @ B @ ( F2 @ X ) @ ( G @ Y2 ) ) ) % 0.31/0.65 @ ( product_Sigma @ C @ D @ A3 % 0.31/0.65 @ ^ [Uu3: C] : B3 ) ) % 0.31/0.65 = ( product_Sigma @ A @ B @ ( image @ C @ A @ F2 @ A3 ) % 0.31/0.65 @ ^ [Uu3: A] : ( image @ D @ B @ G @ B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % image_paired_Times % 0.31/0.65 thf(fact_7879_finite__SigmaI2,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] : % 0.31/0.65 ( ( finite_finite2 @ A % 0.31/0.65 @ ( collect @ A % 0.31/0.65 @ ^ [X: A] : % 0.31/0.65 ( ( member @ A @ X @ A3 ) % 0.31/0.65 & ( ( B3 @ X ) % 0.31/0.65 != ( bot_bot @ ( set @ B ) ) ) ) ) ) % 0.31/0.65 => ( ! [A5: A] : % 0.31/0.65 ( ( member @ A @ A5 @ A3 ) % 0.31/0.65 => ( finite_finite2 @ B @ ( B3 @ A5 ) ) ) % 0.31/0.65 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % finite_SigmaI2 % 0.31/0.65 thf(fact_7880_finite__cartesian__productD1,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] : % 0.31/0.65 ( ( finite_finite2 @ ( product_prod @ A @ B ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) ) % 0.31/0.65 => ( ( B3 % 0.31/0.65 != ( bot_bot @ ( set @ B ) ) ) % 0.31/0.65 => ( finite_finite2 @ A @ A3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % finite_cartesian_productD1 % 0.31/0.65 thf(fact_7881_finite__cartesian__productD2,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,A3: set @ A,B3: set @ B] : % 0.31/0.65 ( ( finite_finite2 @ ( product_prod @ A @ B ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) ) % 0.31/0.65 => ( ( A3 % 0.31/0.65 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.65 => ( finite_finite2 @ B @ B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % finite_cartesian_productD2 % 0.31/0.65 thf(fact_7882_finite__cartesian__product__iff,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,A3: set @ A,B3: set @ B] : % 0.31/0.65 ( ( finite_finite2 @ ( product_prod @ A @ B ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) ) % 0.31/0.65 = ( ( A3 % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.65 | ( B3 % 0.31/0.65 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.65 | ( ( finite_finite2 @ A @ A3 ) % 0.31/0.65 & ( finite_finite2 @ B @ B3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % finite_cartesian_product_iff % 0.31/0.65 thf(fact_7883_UN__Times__distrib,axiom, % 0.31/0.65 ! [C: $tType,D: $tType,B: $tType,A: $tType,E5: C > ( set @ A ),F5: D > ( set @ B ),A3: set @ C,B3: set @ D] : % 0.31/0.65 ( ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ( image @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ( product_case_prod @ C @ D @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ^ [A4: C,B4: D] : % 0.31/0.65 ( product_Sigma @ A @ B @ ( E5 @ A4 ) % 0.31/0.65 @ ^ [Uu3: A] : ( F5 @ B4 ) ) ) % 0.31/0.65 @ ( product_Sigma @ C @ D @ A3 % 0.31/0.65 @ ^ [Uu3: C] : B3 ) ) ) % 0.31/0.65 = ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ C @ ( set @ A ) @ E5 @ A3 ) ) % 0.31/0.65 @ ^ [Uu3: A] : ( complete_Sup_Sup @ ( set @ B ) @ ( image @ D @ ( set @ B ) @ F5 @ B3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % UN_Times_distrib % 0.31/0.65 thf(fact_7884_snd__image__Sigma,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,A3: set @ B,B3: B > ( set @ A )] : % 0.31/0.65 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( product_Sigma @ B @ A @ A3 @ B3 ) ) % 0.31/0.65 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % snd_image_Sigma % 0.31/0.65 thf(fact_7885_subset__fst__imageI,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,S3: set @ ( product_prod @ A @ B ),Y: B] : % 0.31/0.65 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) % 0.31/0.65 @ S3 ) % 0.31/0.65 => ( ( member @ B @ Y @ B3 ) % 0.31/0.65 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % subset_fst_imageI % 0.31/0.65 thf(fact_7886_subset__snd__imageI,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,S3: set @ ( product_prod @ A @ B ),X3: A] : % 0.31/0.65 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) % 0.31/0.65 @ S3 ) % 0.31/0.65 => ( ( member @ A @ X3 @ A3 ) % 0.31/0.65 => ( ord_less_eq @ ( set @ B ) @ B3 @ ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % subset_snd_imageI % 0.31/0.65 thf(fact_7887_subset__fst__snd,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] : % 0.31/0.65 ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 % 0.31/0.65 @ ( product_Sigma @ A @ B @ ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) % 0.31/0.65 @ ^ [Uu3: A] : ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % subset_fst_snd % 0.31/0.65 thf(fact_7888_card__cartesian__product__singleton,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,X3: A,A3: set @ B] : % 0.31/0.65 ( ( finite_card @ ( product_prod @ A @ B ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.65 @ ^ [Uu3: A] : A3 ) ) % 0.31/0.65 = ( finite_card @ B @ A3 ) ) ). % 0.31/0.65 % 0.31/0.65 % card_cartesian_product_singleton % 0.31/0.65 thf(fact_7889_Sigma__interval__disjoint,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( order @ A ) % 0.31/0.65 => ! [A3: set @ B,V2: B > A,W2: A] : % 0.31/0.65 ( ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) % 0.31/0.65 @ ( product_Sigma @ B @ A @ A3 % 0.31/0.65 @ ^ [I3: B] : ( set_ord_atMost @ A @ ( V2 @ I3 ) ) ) % 0.31/0.65 @ ( product_Sigma @ B @ A @ A3 % 0.31/0.65 @ ^ [I3: B] : ( set_or3652927894154168847AtMost @ A @ ( V2 @ I3 ) @ W2 ) ) ) % 0.31/0.65 = ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Sigma_interval_disjoint % 0.31/0.65 thf(fact_7890_sum_OSigma,axiom, % 0.31/0.65 ! [A: $tType,C: $tType,B: $tType] : % 0.31/0.65 ( ( comm_monoid_add @ A ) % 0.31/0.65 => ! [A3: set @ B,B3: B > ( set @ C ),G: B > C > A] : % 0.31/0.65 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.65 => ( ! [X5: B] : % 0.31/0.65 ( ( member @ B @ X5 @ A3 ) % 0.31/0.65 => ( finite_finite2 @ C @ ( B3 @ X5 ) ) ) % 0.31/0.65 => ( ( groups7311177749621191930dd_sum @ B @ A % 0.31/0.65 @ ^ [X: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ X ) @ ( B3 @ X ) ) % 0.31/0.65 @ A3 ) % 0.31/0.65 = ( groups7311177749621191930dd_sum @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G ) @ ( product_Sigma @ B @ C @ A3 @ B3 ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum.Sigma % 0.31/0.65 thf(fact_7891_prod_OSigma,axiom, % 0.31/0.65 ! [A: $tType,C: $tType,B: $tType] : % 0.31/0.65 ( ( comm_monoid_mult @ A ) % 0.31/0.65 => ! [A3: set @ B,B3: B > ( set @ C ),G: B > C > A] : % 0.31/0.65 ( ( finite_finite2 @ B @ A3 ) % 0.31/0.65 => ( ! [X5: B] : % 0.31/0.65 ( ( member @ B @ X5 @ A3 ) % 0.31/0.65 => ( finite_finite2 @ C @ ( B3 @ X5 ) ) ) % 0.31/0.65 => ( ( groups7121269368397514597t_prod @ B @ A % 0.31/0.65 @ ^ [X: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ X ) @ ( B3 @ X ) ) % 0.31/0.65 @ A3 ) % 0.31/0.65 = ( groups7121269368397514597t_prod @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G ) @ ( product_Sigma @ B @ C @ A3 @ B3 ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % prod.Sigma % 0.31/0.65 thf(fact_7892_Sigma__def,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( product_Sigma @ A @ B ) % 0.31/0.65 = ( ^ [A6: set @ A,B6: A > ( set @ B )] : % 0.31/0.65 ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ( image @ A @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ^ [X: A] : % 0.31/0.65 ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ( image @ B @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ^ [Y2: B] : ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) % 0.31/0.65 @ ( B6 @ X ) ) ) % 0.31/0.65 @ A6 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Sigma_def % 0.31/0.65 thf(fact_7893_product__fold,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] : % 0.31/0.65 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.65 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.65 => ( ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) % 0.31/0.65 = ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ^ [X: A,Z6: set @ ( product_prod @ A @ B )] : % 0.31/0.65 ( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) ) % 0.31/0.65 @ ^ [Y2: B] : ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) ) % 0.31/0.65 @ Z6 % 0.31/0.65 @ B3 ) % 0.31/0.65 @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) % 0.31/0.65 @ A3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % product_fold % 0.31/0.65 thf(fact_7894_uniformly__continuous__on__uniformity,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( ( topolo7287701948861334536_space @ A ) % 0.31/0.65 & ( topolo7287701948861334536_space @ B ) ) % 0.31/0.65 => ( ( topolo6026614971017936543ous_on @ A @ B ) % 0.31/0.65 = ( ^ [S8: set @ A,F3: A > B] : % 0.31/0.65 ( filterlim @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) % 0.31/0.65 @ ( product_case_prod @ A @ A @ ( product_prod @ B @ B ) % 0.31/0.65 @ ^ [X: A,Y2: A] : ( product_Pair @ B @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) ) % 0.31/0.65 @ ( topolo7806501430040627800ormity @ B ) % 0.31/0.65 @ ( inf_inf @ ( filter @ ( product_prod @ A @ A ) ) @ ( topolo7806501430040627800ormity @ A ) % 0.31/0.65 @ ( principal @ ( product_prod @ A @ A ) % 0.31/0.65 @ ( product_Sigma @ A @ A @ S8 % 0.31/0.65 @ ^ [Uu3: A] : S8 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % uniformly_continuous_on_uniformity % 0.31/0.65 thf(fact_7895_sum__mult__sum__if__inj,axiom, % 0.31/0.65 ! [A: $tType,C: $tType,B: $tType] : % 0.31/0.65 ( ( semiring_0 @ B ) % 0.31/0.65 => ! [F2: A > B,G: C > B,A3: set @ A,B3: set @ C] : % 0.31/0.65 ( ( inj_on @ ( product_prod @ A @ C ) @ B % 0.31/0.65 @ ( product_case_prod @ A @ C @ B % 0.31/0.65 @ ^ [A4: A,B4: C] : ( times_times @ B @ ( F2 @ A4 ) @ ( G @ B4 ) ) ) % 0.31/0.65 @ ( product_Sigma @ A @ C @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) ) % 0.31/0.65 => ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B3 ) ) % 0.31/0.65 = ( groups7311177749621191930dd_sum @ B @ B @ ( id @ B ) % 0.31/0.65 @ ( collect @ B % 0.31/0.65 @ ^ [Uu3: B] : % 0.31/0.65 ? [A4: A,B4: C] : % 0.31/0.65 ( ( Uu3 % 0.31/0.65 = ( times_times @ B @ ( F2 @ A4 ) @ ( G @ B4 ) ) ) % 0.31/0.65 & ( member @ A @ A4 @ A3 ) % 0.31/0.65 & ( member @ C @ B4 @ B3 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % sum_mult_sum_if_inj % 0.31/0.65 thf(fact_7896_infinite__cartesian__product,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] : % 0.31/0.65 ( ~ ( finite_finite2 @ A @ A3 ) % 0.31/0.65 => ( ~ ( finite_finite2 @ B @ B3 ) % 0.31/0.65 => ~ ( finite_finite2 @ ( product_prod @ A @ B ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % infinite_cartesian_product % 0.31/0.65 thf(fact_7897_Gr__incl,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,A3: set @ A,F2: A > B,B3: set @ B] : % 0.31/0.65 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A3 @ F2 ) % 0.31/0.65 @ ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : B3 ) ) % 0.31/0.65 = ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B3 ) ) ). % 0.31/0.65 % 0.31/0.65 % Gr_incl % 0.31/0.65 thf(fact_7898_GrD2,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,X3: A,Fx: B,A3: set @ A,F2: A > B] : % 0.31/0.65 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Fx ) @ ( bNF_Gr @ A @ B @ A3 @ F2 ) ) % 0.31/0.65 => ( ( F2 @ X3 ) % 0.31/0.65 = Fx ) ) ). % 0.31/0.65 % 0.31/0.65 % GrD2 % 0.31/0.65 thf(fact_7899_GrD1,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,X3: A,Fx: B,A3: set @ A,F2: A > B] : % 0.31/0.65 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Fx ) @ ( bNF_Gr @ A @ B @ A3 @ F2 ) ) % 0.31/0.65 => ( member @ A @ X3 @ A3 ) ) ). % 0.31/0.65 % 0.31/0.65 % GrD1 % 0.31/0.65 thf(fact_7900_Gr__def,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( bNF_Gr @ A @ B ) % 0.31/0.65 = ( ^ [A6: set @ A,F3: A > B] : % 0.31/0.65 ( collect @ ( product_prod @ A @ B ) % 0.31/0.65 @ ^ [Uu3: product_prod @ A @ B] : % 0.31/0.65 ? [A4: A] : % 0.31/0.65 ( ( Uu3 % 0.31/0.65 = ( product_Pair @ A @ B @ A4 @ ( F3 @ A4 ) ) ) % 0.31/0.65 & ( member @ A @ A4 @ A6 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Gr_def % 0.31/0.65 thf(fact_7901_pairs__le__eq__Sigma,axiom, % 0.31/0.65 ! [M: nat] : % 0.31/0.65 ( ( collect @ ( product_prod @ nat @ nat ) % 0.31/0.65 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.65 @ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ M ) ) ) % 0.31/0.65 = ( product_Sigma @ nat @ nat @ ( set_ord_atMost @ nat @ M ) % 0.31/0.65 @ ^ [R5: nat] : ( set_ord_atMost @ nat @ ( minus_minus @ nat @ M @ R5 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % pairs_le_eq_Sigma % 0.31/0.65 thf(fact_7902_relChain__def,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( ord @ B ) % 0.31/0.65 => ( ( bNF_Ca3754400796208372196lChain @ A @ B ) % 0.31/0.65 = ( ^ [R5: set @ ( product_prod @ A @ A ),As4: A > B] : % 0.31/0.65 ! [I3: A,J3: A] : % 0.31/0.65 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I3 @ J3 ) @ R5 ) % 0.31/0.65 => ( ord_less_eq @ B @ ( As4 @ I3 ) @ ( As4 @ J3 ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % relChain_def % 0.31/0.65 thf(fact_7903_natLess__def,axiom, % 0.31/0.65 ( bNF_Ca8459412986667044542atLess % 0.31/0.65 = ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % natLess_def % 0.31/0.65 thf(fact_7904_rotate__drop__take,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( rotate @ A ) % 0.31/0.65 = ( ^ [N: nat,Xs2: list @ A] : ( append @ A @ ( drop @ A @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ Xs2 ) @ ( take @ A @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ Xs2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rotate_drop_take % 0.31/0.65 thf(fact_7905_image__split__eq__Sigma,axiom, % 0.31/0.65 ! [C: $tType,B: $tType,A: $tType,F2: C > A,G: C > B,A3: set @ C] : % 0.31/0.65 ( ( image @ C @ ( product_prod @ A @ B ) % 0.31/0.65 @ ^ [X: C] : ( product_Pair @ A @ B @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.65 @ A3 ) % 0.31/0.65 = ( product_Sigma @ A @ B @ ( image @ C @ A @ F2 @ A3 ) % 0.31/0.65 @ ^ [X: A] : ( image @ C @ B @ G @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % image_split_eq_Sigma % 0.31/0.65 thf(fact_7906_vimage__eq,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,A2: A,F2: A > B,B3: set @ B] : % 0.31/0.65 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F2 @ B3 ) ) % 0.31/0.65 = ( member @ B @ ( F2 @ A2 ) @ B3 ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_eq % 0.31/0.65 thf(fact_7907_vimageI,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F2: B > A,A2: B,B2: A,B3: set @ A] : % 0.31/0.65 ( ( ( F2 @ A2 ) % 0.31/0.65 = B2 ) % 0.31/0.65 => ( ( member @ A @ B2 @ B3 ) % 0.31/0.65 => ( member @ B @ A2 @ ( vimage @ B @ A @ F2 @ B3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimageI % 0.31/0.65 thf(fact_7908_vimage__Collect__eq,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F2: A > B,P: B > $o] : % 0.31/0.65 ( ( vimage @ A @ B @ F2 @ ( collect @ B @ P ) ) % 0.31/0.65 = ( collect @ A % 0.31/0.65 @ ^ [Y2: A] : ( P @ ( F2 @ Y2 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_Collect_eq % 0.31/0.65 thf(fact_7909_vimage__ident,axiom, % 0.31/0.65 ! [A: $tType,Y7: set @ A] : % 0.31/0.65 ( ( vimage @ A @ A % 0.31/0.65 @ ^ [X: A] : X % 0.31/0.65 @ Y7 ) % 0.31/0.65 = Y7 ) ). % 0.31/0.65 % 0.31/0.65 % vimage_ident % 0.31/0.65 thf(fact_7910_vimage__UNIV,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F2: A > B] : % 0.31/0.65 ( ( vimage @ A @ B @ F2 @ ( top_top @ ( set @ B ) ) ) % 0.31/0.65 = ( top_top @ ( set @ A ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_UNIV % 0.31/0.65 thf(fact_7911_vimage__empty,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F2: A > B] : % 0.31/0.65 ( ( vimage @ A @ B @ F2 @ ( bot_bot @ ( set @ B ) ) ) % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_empty % 0.31/0.65 thf(fact_7912_vimage__Int,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: A > B,A3: set @ B,B3: set @ B] : % 0.31/0.65 ( ( vimage @ A @ B @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.65 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A3 ) @ ( vimage @ A @ B @ F2 @ B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_Int % 0.31/0.65 thf(fact_7913_vimage__Un,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: A > B,A3: set @ B,B3: set @ B] : % 0.31/0.65 ( ( vimage @ A @ B @ F2 @ ( sup_sup @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.65 = ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A3 ) @ ( vimage @ A @ B @ F2 @ B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_Un % 0.31/0.65 thf(fact_7914_rotate__is__Nil__conv,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( ( rotate @ A @ N2 @ Xs ) % 0.31/0.65 = ( nil @ A ) ) % 0.31/0.65 = ( Xs % 0.31/0.65 = ( nil @ A ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rotate_is_Nil_conv % 0.31/0.65 thf(fact_7915_set__rotate,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( set2 @ A @ ( rotate @ A @ N2 @ Xs ) ) % 0.31/0.65 = ( set2 @ A @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % set_rotate % 0.31/0.65 thf(fact_7916_length__rotate,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( size_size @ ( list @ A ) @ ( rotate @ A @ N2 @ Xs ) ) % 0.31/0.65 = ( size_size @ ( list @ A ) @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % length_rotate % 0.31/0.65 thf(fact_7917_distinct__rotate,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( distinct @ A @ ( rotate @ A @ N2 @ Xs ) ) % 0.31/0.65 = ( distinct @ A @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % distinct_rotate % 0.31/0.65 thf(fact_7918_vimage__const,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,C2: B,A3: set @ B] : % 0.31/0.65 ( ( ( member @ B @ C2 @ A3 ) % 0.31/0.65 => ( ( vimage @ A @ B % 0.31/0.65 @ ^ [X: A] : C2 % 0.31/0.65 @ A3 ) % 0.31/0.65 = ( top_top @ ( set @ A ) ) ) ) % 0.31/0.65 & ( ~ ( member @ B @ C2 @ A3 ) % 0.31/0.65 => ( ( vimage @ A @ B % 0.31/0.65 @ ^ [X: A] : C2 % 0.31/0.65 @ A3 ) % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_const % 0.31/0.65 thf(fact_7919_rotate__Suc,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( rotate @ A @ ( suc @ N2 ) @ Xs ) % 0.31/0.65 = ( rotate1 @ A @ ( rotate @ A @ N2 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rotate_Suc % 0.31/0.65 thf(fact_7920_rotate0,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( rotate @ A @ ( zero_zero @ nat ) ) % 0.31/0.65 = ( id @ ( list @ A ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rotate0 % 0.31/0.65 thf(fact_7921_image__vimage__eq,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: B > A,A3: set @ A] : % 0.31/0.65 ( ( image @ B @ A @ F2 @ ( vimage @ B @ A @ F2 @ A3 ) ) % 0.31/0.65 = ( inf_inf @ ( set @ A ) @ A3 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % image_vimage_eq % 0.31/0.65 thf(fact_7922_rotate__length01,axiom, % 0.31/0.65 ! [A: $tType,Xs: list @ A,N2: nat] : % 0.31/0.65 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) % 0.31/0.65 => ( ( rotate @ A @ N2 @ Xs ) % 0.31/0.65 = Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % rotate_length01 % 0.31/0.65 thf(fact_7923_rotate__id,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.65 = ( zero_zero @ nat ) ) % 0.31/0.65 => ( ( rotate @ A @ N2 @ Xs ) % 0.31/0.65 = Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % rotate_id % 0.31/0.65 thf(fact_7924_vimage__if,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,C2: B,A3: set @ B,D2: B,B3: set @ A] : % 0.31/0.65 ( ( ( member @ B @ C2 @ A3 ) % 0.31/0.65 => ( ( ( member @ B @ D2 @ A3 ) % 0.31/0.65 => ( ( vimage @ A @ B % 0.31/0.65 @ ^ [X: A] : ( if @ B @ ( member @ A @ X @ B3 ) @ C2 @ D2 ) % 0.31/0.65 @ A3 ) % 0.31/0.65 = ( top_top @ ( set @ A ) ) ) ) % 0.31/0.65 & ( ~ ( member @ B @ D2 @ A3 ) % 0.31/0.65 => ( ( vimage @ A @ B % 0.31/0.65 @ ^ [X: A] : ( if @ B @ ( member @ A @ X @ B3 ) @ C2 @ D2 ) % 0.31/0.65 @ A3 ) % 0.31/0.65 = B3 ) ) ) ) % 0.31/0.65 & ( ~ ( member @ B @ C2 @ A3 ) % 0.31/0.65 => ( ( ( member @ B @ D2 @ A3 ) % 0.31/0.65 => ( ( vimage @ A @ B % 0.31/0.65 @ ^ [X: A] : ( if @ B @ ( member @ A @ X @ B3 ) @ C2 @ D2 ) % 0.31/0.65 @ A3 ) % 0.31/0.65 = ( uminus_uminus @ ( set @ A ) @ B3 ) ) ) % 0.31/0.65 & ( ~ ( member @ B @ D2 @ A3 ) % 0.31/0.65 => ( ( vimage @ A @ B % 0.31/0.65 @ ^ [X: A] : ( if @ B @ ( member @ A @ X @ B3 ) @ C2 @ D2 ) % 0.31/0.65 @ A3 ) % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_if % 0.31/0.65 thf(fact_7925_Pair__vimage__Sigma,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,X3: B,A3: set @ B,F2: B > ( set @ A )] : % 0.31/0.65 ( ( ( member @ B @ X3 @ A3 ) % 0.31/0.65 => ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 ) @ ( product_Sigma @ B @ A @ A3 @ F2 ) ) % 0.31/0.65 = ( F2 @ X3 ) ) ) % 0.31/0.65 & ( ~ ( member @ B @ X3 @ A3 ) % 0.31/0.65 => ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 ) @ ( product_Sigma @ B @ A @ A3 @ F2 ) ) % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % Pair_vimage_Sigma % 0.31/0.65 thf(fact_7926_vimage__snd,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ B] : % 0.31/0.65 ( ( vimage @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) % 0.31/0.65 = ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) ) % 0.31/0.65 @ ^ [Uu3: A] : A3 ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_snd % 0.31/0.65 thf(fact_7927_vimage__fst,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A] : % 0.31/0.65 ( ( vimage @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) % 0.31/0.65 = ( product_Sigma @ A @ B @ A3 % 0.31/0.65 @ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_fst % 0.31/0.65 thf(fact_7928_vimage__Times,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,C: $tType,F2: A > ( product_prod @ B @ C ),A3: set @ B,B3: set @ C] : % 0.31/0.65 ( ( vimage @ A @ ( product_prod @ B @ C ) @ F2 % 0.31/0.65 @ ( product_Sigma @ B @ C @ A3 % 0.31/0.65 @ ^ [Uu3: B] : B3 ) ) % 0.31/0.65 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F2 ) @ A3 ) @ ( vimage @ A @ C @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F2 ) @ B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_Times % 0.31/0.65 thf(fact_7929_vimage__singleton__eq,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,A2: A,F2: A > B,B2: B] : % 0.31/0.65 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F2 @ ( insert @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) ) % 0.31/0.65 = ( ( F2 @ A2 ) % 0.31/0.65 = B2 ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_singleton_eq % 0.31/0.65 thf(fact_7930_vimage__insert,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: A > B,A2: B,B3: set @ B] : % 0.31/0.65 ( ( vimage @ A @ B @ F2 @ ( insert @ B @ A2 @ B3 ) ) % 0.31/0.65 = ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( vimage @ A @ B @ F2 @ B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_insert % 0.31/0.65 thf(fact_7931_vimage__Suc__insert__Suc,axiom, % 0.31/0.65 ! [N2: nat,A3: set @ nat] : % 0.31/0.65 ( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( suc @ N2 ) @ A3 ) ) % 0.31/0.65 = ( insert @ nat @ N2 @ ( vimage @ nat @ nat @ suc @ A3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_Suc_insert_Suc % 0.31/0.65 thf(fact_7932_vimage__Suc__insert__0,axiom, % 0.31/0.65 ! [A3: set @ nat] : % 0.31/0.65 ( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( zero_zero @ nat ) @ A3 ) ) % 0.31/0.65 = ( vimage @ nat @ nat @ suc @ A3 ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_Suc_insert_0 % 0.31/0.65 thf(fact_7933_rotate__add,axiom, % 0.31/0.65 ! [A: $tType,M: nat,N2: nat] : % 0.31/0.65 ( ( rotate @ A @ ( plus_plus @ nat @ M @ N2 ) ) % 0.31/0.65 = ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( rotate @ A @ M ) @ ( rotate @ A @ N2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rotate_add % 0.31/0.65 thf(fact_7934_rotate__map,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,N2: nat,F2: B > A,Xs: list @ B] : % 0.31/0.65 ( ( rotate @ A @ N2 @ ( map @ B @ A @ F2 @ Xs ) ) % 0.31/0.65 = ( map @ B @ A @ F2 @ ( rotate @ B @ N2 @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rotate_map % 0.31/0.65 thf(fact_7935_rotate__def,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( rotate @ A ) % 0.31/0.65 = ( ^ [N: nat] : ( compow @ ( ( list @ A ) > ( list @ A ) ) @ N @ ( rotate1 @ A ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rotate_def % 0.31/0.65 thf(fact_7936_rotate__rotate,axiom, % 0.31/0.65 ! [A: $tType,M: nat,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( rotate @ A @ M @ ( rotate @ A @ N2 @ Xs ) ) % 0.31/0.65 = ( rotate @ A @ ( plus_plus @ nat @ M @ N2 ) @ Xs ) ) ). % 0.31/0.65 % 0.31/0.65 % rotate_rotate % 0.31/0.65 thf(fact_7937_vimage__def,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( vimage @ A @ B ) % 0.31/0.65 = ( ^ [F3: A > B,B6: set @ B] : % 0.31/0.65 ( collect @ A % 0.31/0.65 @ ^ [X: A] : ( member @ B @ ( F3 @ X ) @ B6 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_def % 0.31/0.65 thf(fact_7938_vimage__Collect,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,P: B > $o,F2: A > B,Q: A > $o] : % 0.31/0.65 ( ! [X5: A] : % 0.31/0.65 ( ( P @ ( F2 @ X5 ) ) % 0.31/0.65 = ( Q @ X5 ) ) % 0.31/0.65 => ( ( vimage @ A @ B @ F2 @ ( collect @ B @ P ) ) % 0.31/0.65 = ( collect @ A @ Q ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_Collect % 0.31/0.65 thf(fact_7939_vimageI2,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F2: B > A,A2: B,A3: set @ A] : % 0.31/0.65 ( ( member @ A @ ( F2 @ A2 ) @ A3 ) % 0.31/0.65 => ( member @ B @ A2 @ ( vimage @ B @ A @ F2 @ A3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimageI2 % 0.31/0.65 thf(fact_7940_vimageE,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,A2: A,F2: A > B,B3: set @ B] : % 0.31/0.65 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F2 @ B3 ) ) % 0.31/0.65 => ( member @ B @ ( F2 @ A2 ) @ B3 ) ) ). % 0.31/0.65 % 0.31/0.65 % vimageE % 0.31/0.65 thf(fact_7941_vimageD,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,A2: A,F2: A > B,A3: set @ B] : % 0.31/0.65 ( ( member @ A @ A2 @ ( vimage @ A @ B @ F2 @ A3 ) ) % 0.31/0.65 => ( member @ B @ ( F2 @ A2 ) @ A3 ) ) ). % 0.31/0.65 % 0.31/0.65 % vimageD % 0.31/0.65 thf(fact_7942_rotate1__rotate__swap,axiom, % 0.31/0.65 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.65 ( ( rotate1 @ A @ ( rotate @ A @ N2 @ Xs ) ) % 0.31/0.65 = ( rotate @ A @ N2 @ ( rotate1 @ A @ Xs ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rotate1_rotate_swap % 0.31/0.65 thf(fact_7943_finite__vimage__Suc__iff,axiom, % 0.31/0.65 ! [F5: set @ nat] : % 0.31/0.65 ( ( finite_finite2 @ nat @ ( vimage @ nat @ nat @ suc @ F5 ) ) % 0.31/0.65 = ( finite_finite2 @ nat @ F5 ) ) ). % 0.31/0.65 % 0.31/0.65 % finite_vimage_Suc_iff % 0.31/0.65 thf(fact_7944_continuous__imp__open__vimage,axiom, % 0.31/0.65 ! [A: $tType,B: $tType] : % 0.31/0.65 ( ( ( topolo4958980785337419405_space @ B ) % 0.31/0.65 & ( topolo4958980785337419405_space @ A ) ) % 0.31/0.65 => ! [S: set @ A,F2: A > B,B3: set @ B] : % 0.31/0.65 ( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 ) % 0.31/0.65 => ( ( topolo1002775350975398744n_open @ A @ S ) % 0.31/0.65 => ( ( topolo1002775350975398744n_open @ B @ B3 ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ B3 ) @ S ) % 0.31/0.65 => ( topolo1002775350975398744n_open @ A @ ( vimage @ A @ B @ F2 @ B3 ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % continuous_imp_open_vimage % 0.31/0.65 thf(fact_7945_vimage__mono,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,F2: B > A] : % 0.31/0.65 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.65 => ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ A3 ) @ ( vimage @ B @ A @ F2 @ B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_mono % 0.31/0.65 thf(fact_7946_subset__vimage__iff,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,A3: set @ A,F2: A > B,B3: set @ B] : % 0.31/0.65 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( vimage @ A @ B @ F2 @ B3 ) ) % 0.31/0.65 = ( ! [X: A] : % 0.31/0.65 ( ( member @ A @ X @ A3 ) % 0.31/0.65 => ( member @ B @ ( F2 @ X ) @ B3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % subset_vimage_iff % 0.31/0.65 thf(fact_7947_rotate__conv__mod,axiom, % 0.31/0.65 ! [A: $tType] : % 0.31/0.65 ( ( rotate @ A ) % 0.31/0.65 = ( ^ [N: nat,Xs2: list @ A] : ( rotate @ A @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ Xs2 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % rotate_conv_mod % 0.31/0.65 thf(fact_7948_rotate__append,axiom, % 0.31/0.65 ! [A: $tType,L: list @ A,Q4: list @ A] : % 0.31/0.65 ( ( rotate @ A @ ( size_size @ ( list @ A ) @ L ) @ ( append @ A @ L @ Q4 ) ) % 0.31/0.65 = ( append @ A @ Q4 @ L ) ) ). % 0.31/0.65 % 0.31/0.65 % rotate_append % 0.31/0.65 thf(fact_7949_finite__vimage__IntI,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F5: set @ A,H2: B > A,A3: set @ B] : % 0.31/0.65 ( ( finite_finite2 @ A @ F5 ) % 0.31/0.65 => ( ( inj_on @ B @ A @ H2 @ A3 ) % 0.31/0.65 => ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ F5 ) @ A3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % finite_vimage_IntI % 0.31/0.65 thf(fact_7950_finite__vimageI,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F5: set @ A,H2: B > A] : % 0.31/0.65 ( ( finite_finite2 @ A @ F5 ) % 0.31/0.65 => ( ( inj_on @ B @ A @ H2 @ ( top_top @ ( set @ B ) ) ) % 0.31/0.65 => ( finite_finite2 @ B @ ( vimage @ B @ A @ H2 @ F5 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % finite_vimageI % 0.31/0.65 thf(fact_7951_vimage__inter__cong,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,S3: set @ A,F2: A > B,G: A > B,Y: set @ B] : % 0.31/0.65 ( ! [W: A] : % 0.31/0.65 ( ( member @ A @ W @ S3 ) % 0.31/0.65 => ( ( F2 @ W ) % 0.31/0.65 = ( G @ W ) ) ) % 0.31/0.65 => ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ Y ) @ S3 ) % 0.31/0.65 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G @ Y ) @ S3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_inter_cong % 0.31/0.65 thf(fact_7952_vimage__Diff,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: A > B,A3: set @ B,B3: set @ B] : % 0.31/0.65 ( ( vimage @ A @ B @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) % 0.31/0.65 = ( minus_minus @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A3 ) @ ( vimage @ A @ B @ F2 @ B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_Diff % 0.31/0.65 thf(fact_7953_vimage__Compl,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: A > B,A3: set @ B] : % 0.31/0.65 ( ( vimage @ A @ B @ F2 @ ( uminus_uminus @ ( set @ B ) @ A3 ) ) % 0.31/0.65 = ( uminus_uminus @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_Compl % 0.31/0.65 thf(fact_7954_surj__vimage__empty,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F2: B > A,A3: set @ A] : % 0.31/0.65 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) % 0.31/0.65 = ( top_top @ ( set @ A ) ) ) % 0.31/0.65 => ( ( ( vimage @ B @ A @ F2 @ A3 ) % 0.31/0.65 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.65 = ( A3 % 0.31/0.65 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % surj_vimage_empty % 0.31/0.65 thf(fact_7955_image__subset__iff__subset__vimage,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B3: set @ A] : % 0.31/0.65 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B3 ) % 0.31/0.65 = ( ord_less_eq @ ( set @ B ) @ A3 @ ( vimage @ B @ A @ F2 @ B3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % image_subset_iff_subset_vimage % 0.31/0.65 thf(fact_7956_image__vimage__subset,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F2: B > A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( vimage @ B @ A @ F2 @ A3 ) ) @ A3 ) ). % 0.31/0.65 % 0.31/0.65 % image_vimage_subset % 0.31/0.65 thf(fact_7957_vimage__subsetD,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: B > A,B3: set @ A,A3: set @ B] : % 0.31/0.65 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) % 0.31/0.65 = ( top_top @ ( set @ A ) ) ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ B3 ) @ A3 ) % 0.31/0.65 => ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_subsetD % 0.31/0.65 thf(fact_7958_finite__vimageD,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,H2: A > B,F5: set @ B] : % 0.31/0.65 ( ( finite_finite2 @ A @ ( vimage @ A @ B @ H2 @ F5 ) ) % 0.31/0.65 => ( ( ( image @ A @ B @ H2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.65 = ( top_top @ ( set @ B ) ) ) % 0.31/0.65 => ( finite_finite2 @ B @ F5 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % finite_vimageD % 0.31/0.65 thf(fact_7959_finite__vimageD_H,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: A > B,A3: set @ B] : % 0.31/0.65 ( ( finite_finite2 @ A @ ( vimage @ A @ B @ F2 @ A3 ) ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.65 => ( finite_finite2 @ B @ A3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % finite_vimageD' % 0.31/0.65 thf(fact_7960_inf__img__fin__domE,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F2: B > A,A3: set @ B] : % 0.31/0.65 ( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.65 => ( ~ ( finite_finite2 @ B @ A3 ) % 0.31/0.65 => ~ ! [Y4: A] : % 0.31/0.65 ( ( member @ A @ Y4 @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.65 => ( finite_finite2 @ B @ ( vimage @ B @ A @ F2 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % inf_img_fin_domE % 0.31/0.65 thf(fact_7961_inf__img__fin__dom,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F2: B > A,A3: set @ B] : % 0.31/0.65 ( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.65 => ( ~ ( finite_finite2 @ B @ A3 ) % 0.31/0.65 => ? [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.65 & ~ ( finite_finite2 @ B @ ( vimage @ B @ A @ F2 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % inf_img_fin_dom % 0.31/0.65 thf(fact_7962_vimage__subsetI,axiom, % 0.31/0.65 ! [B: $tType,A: $tType,F2: A > B,B3: set @ B,A3: set @ A] : % 0.31/0.65 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.65 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ ( image @ A @ B @ F2 @ A3 ) ) % 0.31/0.65 => ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ B3 ) @ A3 ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_subsetI % 0.31/0.65 thf(fact_7963_finite__finite__vimage__IntI,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F5: set @ A,H2: B > A,A3: set @ B] : % 0.31/0.65 ( ( finite_finite2 @ A @ F5 ) % 0.31/0.65 => ( ! [Y4: A] : % 0.31/0.65 ( ( member @ A @ Y4 @ F5 ) % 0.31/0.65 => ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) % 0.31/0.65 => ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ F5 ) @ A3 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % finite_finite_vimage_IntI % 0.31/0.65 thf(fact_7964_vimage__eq__UN,axiom, % 0.31/0.65 ! [B: $tType,A: $tType] : % 0.31/0.65 ( ( vimage @ A @ B ) % 0.31/0.65 = ( ^ [F3: A > B,B6: set @ B] : % 0.31/0.65 ( complete_Sup_Sup @ ( set @ A ) % 0.31/0.65 @ ( image @ B @ ( set @ A ) % 0.31/0.65 @ ^ [Y2: B] : ( vimage @ A @ B @ F3 @ ( insert @ B @ Y2 @ ( bot_bot @ ( set @ B ) ) ) ) % 0.31/0.65 @ B6 ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % vimage_eq_UN % 0.31/0.65 thf(fact_7965_inf__img__fin__dom_H,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] : % 0.31/0.65 ( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.65 => ( ~ ( finite_finite2 @ B @ A3 ) % 0.31/0.65 => ? [X5: A] : % 0.31/0.65 ( ( member @ A @ X5 @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.65 & ~ ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) ) ) ). % 0.31/0.65 % 0.31/0.65 % inf_img_fin_dom' % 0.31/0.65 thf(fact_7966_inf__img__fin__domE_H,axiom, % 0.31/0.65 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] : % 0.31/0.65 ( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.65 => ( ~ ( finite_finite2 @ B @ A3 ) % 0.31/0.65 => ~ ! [Y4: A] : % 0.31/0.65 ( ( member @ A @ Y4 @ ( image @ B @ A @ F2 @ A3 ) ) % 0.31/0.65 => ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F2 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % inf_img_fin_domE' % 0.31/0.66 thf(fact_7967_card__vimage__inj,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,F2: A > B,A3: set @ B] : % 0.31/0.66 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) % 0.31/0.66 => ( ( finite_card @ A @ ( vimage @ A @ B @ F2 @ A3 ) ) % 0.31/0.66 = ( finite_card @ B @ A3 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % card_vimage_inj % 0.31/0.66 thf(fact_7968_card__vimage__inj__on__le,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,F2: A > B,D6: set @ A,A3: set @ B] : % 0.31/0.66 ( ( inj_on @ A @ B @ F2 @ D6 ) % 0.31/0.66 => ( ( finite_finite2 @ B @ A3 ) % 0.31/0.66 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A3 ) @ D6 ) ) @ ( finite_card @ B @ A3 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % card_vimage_inj_on_le % 0.31/0.66 thf(fact_7969_rotate__rev,axiom, % 0.31/0.66 ! [A: $tType,N2: nat,Xs: list @ A] : % 0.31/0.66 ( ( rotate @ A @ N2 @ ( rev @ A @ Xs ) ) % 0.31/0.66 = ( rev @ A @ ( rotate @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) @ Xs ) ) ) ). % 0.31/0.66 % 0.31/0.66 % rotate_rev % 0.31/0.66 thf(fact_7970_set__decode__div__2,axiom, % 0.31/0.66 ! [X3: nat] : % 0.31/0.66 ( ( nat_set_decode @ ( divide_divide @ nat @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 0.31/0.66 = ( vimage @ nat @ nat @ suc @ ( nat_set_decode @ X3 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % set_decode_div_2 % 0.31/0.66 thf(fact_7971_set__encode__vimage__Suc,axiom, % 0.31/0.66 ! [A3: set @ nat] : % 0.31/0.66 ( ( nat_set_encode @ ( vimage @ nat @ nat @ suc @ A3 ) ) % 0.31/0.66 = ( divide_divide @ nat @ ( nat_set_encode @ A3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % set_encode_vimage_Suc % 0.31/0.66 thf(fact_7972_inj__vimage__singleton,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,F2: A > B,A2: B] : % 0.31/0.66 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.66 => ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) % 0.31/0.66 @ ( insert @ A % 0.31/0.66 @ ( the @ A % 0.31/0.66 @ ^ [X: A] : % 0.31/0.66 ( ( F2 @ X ) % 0.31/0.66 = A2 ) ) % 0.31/0.66 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % inj_vimage_singleton % 0.31/0.66 thf(fact_7973_inj__on__vimage__singleton,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,F2: A > B,A3: set @ A,A2: B] : % 0.31/0.66 ( ( inj_on @ A @ B @ F2 @ A3 ) % 0.31/0.66 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) @ A3 ) % 0.31/0.66 @ ( insert @ A % 0.31/0.66 @ ( the @ A % 0.31/0.66 @ ^ [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A3 ) % 0.31/0.66 & ( ( F2 @ X ) % 0.31/0.66 = A2 ) ) ) % 0.31/0.66 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % inj_on_vimage_singleton % 0.31/0.66 thf(fact_7974_inv__image__partition,axiom, % 0.31/0.66 ! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] : % 0.31/0.66 ( ! [X5: A] : % 0.31/0.66 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 0.31/0.66 => ( P @ X5 ) ) % 0.31/0.66 => ( ! [Y4: A] : % 0.31/0.66 ( ( member @ A @ Y4 @ ( set2 @ A @ Ys ) ) % 0.31/0.66 => ~ ( P @ Y4 ) ) % 0.31/0.66 => ( ( vimage @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( partition @ A @ P ) @ ( insert @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) ) % 0.31/0.66 = ( shuffles @ A @ Xs @ Ys ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % inv_image_partition % 0.31/0.66 thf(fact_7975_nth__rotate,axiom, % 0.31/0.66 ! [A: $tType,N2: nat,Xs: list @ A,M: nat] : % 0.31/0.66 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 0.31/0.66 => ( ( nth @ A @ ( rotate @ A @ M @ Xs ) @ N2 ) % 0.31/0.66 = ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % nth_rotate % 0.31/0.66 thf(fact_7976_hd__rotate__conv__nth,axiom, % 0.31/0.66 ! [A: $tType,Xs: list @ A,N2: nat] : % 0.31/0.66 ( ( Xs % 0.31/0.66 != ( nil @ A ) ) % 0.31/0.66 => ( ( hd @ A @ ( rotate @ A @ N2 @ Xs ) ) % 0.31/0.66 = ( nth @ A @ Xs @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % hd_rotate_conv_nth % 0.31/0.66 thf(fact_7977_dual__min,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( linorder @ A ) % 0.31/0.66 => ( ( min @ A % 0.31/0.66 @ ^ [X: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ X ) ) % 0.31/0.66 = ( ord_max @ A ) ) ) ). % 0.31/0.66 % 0.31/0.66 % dual_min % 0.31/0.66 thf(fact_7978_listrel1__subset__listrel,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R3 ) % 0.31/0.66 => ( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R3 ) % 0.31/0.66 => ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R2 ) @ ( listrel @ A @ A @ R3 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % listrel1_subset_listrel % 0.31/0.66 thf(fact_7979_refl__onD2,axiom, % 0.31/0.66 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] : % 0.31/0.66 ( ( refl_on @ A @ A3 @ R2 ) % 0.31/0.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) % 0.31/0.66 => ( member @ A @ Y @ A3 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % refl_onD2 % 0.31/0.66 thf(fact_7980_refl__onD1,axiom, % 0.31/0.66 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] : % 0.31/0.66 ( ( refl_on @ A @ A3 @ R2 ) % 0.31/0.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) % 0.31/0.66 => ( member @ A @ X3 @ A3 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % refl_onD1 % 0.31/0.66 thf(fact_7981_refl__onD,axiom, % 0.31/0.66 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A] : % 0.31/0.66 ( ( refl_on @ A @ A3 @ R2 ) % 0.31/0.66 => ( ( member @ A @ A2 @ A3 ) % 0.31/0.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % refl_onD % 0.31/0.66 thf(fact_7982_refl__on__empty,axiom, % 0.31/0.66 ! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % refl_on_empty % 0.31/0.66 thf(fact_7983_ord_Omin__def,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( min @ A ) % 0.31/0.66 = ( ^ [Less_eq: A > A > $o,A4: A,B4: A] : ( if @ A @ ( Less_eq @ A4 @ B4 ) @ A4 @ B4 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % ord.min_def % 0.31/0.66 thf(fact_7984_ord_Omin_Ocong,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( min @ A ) % 0.31/0.66 = ( min @ A ) ) ). % 0.31/0.66 % 0.31/0.66 % ord.min.cong % 0.31/0.66 thf(fact_7985_refl__on__def,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( refl_on @ A ) % 0.31/0.66 = ( ^ [A6: set @ A,R5: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 % 0.31/0.66 @ ( product_Sigma @ A @ A @ A6 % 0.31/0.66 @ ^ [Uu3: A] : A6 ) ) % 0.31/0.66 & ! [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A6 ) % 0.31/0.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R5 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % refl_on_def % 0.31/0.66 thf(fact_7986_refl__onI,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A] : % 0.31/0.66 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 % 0.31/0.66 @ ( product_Sigma @ A @ A @ A3 % 0.31/0.66 @ ^ [Uu3: A] : A3 ) ) % 0.31/0.66 => ( ! [X5: A] : % 0.31/0.66 ( ( member @ A @ X5 @ A3 ) % 0.31/0.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 ) ) % 0.31/0.66 => ( refl_on @ A @ A3 @ R2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % refl_onI % 0.31/0.66 thf(fact_7987_refl__on__def_H,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( refl_on @ A ) % 0.31/0.66 = ( ^ [A6: set @ A,R5: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ! [X: product_prod @ A @ A] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ X @ R5 ) % 0.31/0.66 => ( product_case_prod @ A @ A @ $o % 0.31/0.66 @ ^ [Y2: A,Z6: A] : % 0.31/0.66 ( ( member @ A @ Y2 @ A6 ) % 0.31/0.66 & ( member @ A @ Z6 @ A6 ) ) % 0.31/0.66 @ X ) ) % 0.31/0.66 & ! [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A6 ) % 0.31/0.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R5 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % refl_on_def' % 0.31/0.66 thf(fact_7988_refl__on__singleton,axiom, % 0.31/0.66 ! [A: $tType,X3: A] : ( refl_on @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % refl_on_singleton % 0.31/0.66 thf(fact_7989_refl__on__domain,axiom, % 0.31/0.66 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] : % 0.31/0.66 ( ( refl_on @ A @ A3 @ R2 ) % 0.31/0.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) % 0.31/0.66 => ( ( member @ A @ A2 @ A3 ) % 0.31/0.66 & ( member @ A @ B2 @ A3 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % refl_on_domain % 0.31/0.66 thf(fact_7990_Restr__natLeq,axiom, % 0.31/0.66 ! [N2: nat] : % 0.31/0.66 ( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq % 0.31/0.66 @ ( product_Sigma @ nat @ nat % 0.31/0.66 @ ( collect @ nat % 0.31/0.66 @ ^ [X: nat] : ( ord_less @ nat @ X @ N2 ) ) % 0.31/0.66 @ ^ [Uu3: nat] : % 0.31/0.66 ( collect @ nat % 0.31/0.66 @ ^ [X: nat] : ( ord_less @ nat @ X @ N2 ) ) ) ) % 0.31/0.66 = ( collect @ ( product_prod @ nat @ nat ) % 0.31/0.66 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.66 @ ^ [X: nat,Y2: nat] : % 0.31/0.66 ( ( ord_less @ nat @ X @ N2 ) % 0.31/0.66 & ( ord_less @ nat @ Y2 @ N2 ) % 0.31/0.66 & ( ord_less_eq @ nat @ X @ Y2 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Restr_natLeq % 0.31/0.66 thf(fact_7991_natLeq__def,axiom, % 0.31/0.66 ( bNF_Ca8665028551170535155natLeq % 0.31/0.66 = ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less_eq @ nat ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % natLeq_def % 0.31/0.66 thf(fact_7992_Restr__natLeq2,axiom, % 0.31/0.66 ! [N2: nat] : % 0.31/0.66 ( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq % 0.31/0.66 @ ( product_Sigma @ nat @ nat @ ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N2 ) % 0.31/0.66 @ ^ [Uu3: nat] : ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N2 ) ) ) % 0.31/0.66 = ( collect @ ( product_prod @ nat @ nat ) % 0.31/0.66 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.66 @ ^ [X: nat,Y2: nat] : % 0.31/0.66 ( ( ord_less @ nat @ X @ N2 ) % 0.31/0.66 & ( ord_less @ nat @ Y2 @ N2 ) % 0.31/0.66 & ( ord_less_eq @ nat @ X @ Y2 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Restr_natLeq2 % 0.31/0.66 thf(fact_7993_linear__order__on__singleton,axiom, % 0.31/0.66 ! [A: $tType,X3: A] : ( order_679001287576687338der_on @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % linear_order_on_singleton % 0.31/0.66 thf(fact_7994_lnear__order__on__empty,axiom, % 0.31/0.66 ! [A: $tType] : ( order_679001287576687338der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % lnear_order_on_empty % 0.31/0.66 thf(fact_7995_underS__def,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( order_underS @ A ) % 0.31/0.66 = ( ^ [R5: set @ ( product_prod @ A @ A ),A4: A] : % 0.31/0.66 ( collect @ A % 0.31/0.66 @ ^ [B4: A] : % 0.31/0.66 ( ( B4 != A4 ) % 0.31/0.66 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ R5 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % underS_def % 0.31/0.66 thf(fact_7996_natLeq__underS__less,axiom, % 0.31/0.66 ! [N2: nat] : % 0.31/0.66 ( ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N2 ) % 0.31/0.66 = ( collect @ nat % 0.31/0.66 @ ^ [X: nat] : ( ord_less @ nat @ X @ N2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % natLeq_underS_less % 0.31/0.66 thf(fact_7997_underS__I,axiom, % 0.31/0.66 ! [A: $tType,I: A,J: A,R: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( I != J ) % 0.31/0.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R ) % 0.31/0.66 => ( member @ A @ I @ ( order_underS @ A @ R @ J ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % underS_I % 0.31/0.66 thf(fact_7998_underS__E,axiom, % 0.31/0.66 ! [A: $tType,I: A,R: set @ ( product_prod @ A @ A ),J: A] : % 0.31/0.66 ( ( member @ A @ I @ ( order_underS @ A @ R @ J ) ) % 0.31/0.66 => ( ( I != J ) % 0.31/0.66 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R ) ) ) ). % 0.31/0.66 % 0.31/0.66 % underS_E % 0.31/0.66 thf(fact_7999_total__on__singleton,axiom, % 0.31/0.66 ! [A: $tType,X3: A] : ( total_on @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % total_on_singleton % 0.31/0.66 thf(fact_8000_Arg__bounded,axiom, % 0.31/0.66 ! [Z2: complex] : % 0.31/0.66 ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z2 ) ) % 0.31/0.66 & ( ord_less_eq @ real @ ( arg @ Z2 ) @ pi ) ) ). % 0.31/0.66 % 0.31/0.66 % Arg_bounded % 0.31/0.66 thf(fact_8001_total__lenlex,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R2 ) % 0.31/0.66 => ( total_on @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) @ ( lenlex @ A @ R2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % total_lenlex % 0.31/0.66 thf(fact_8002_total__lexord,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R2 ) % 0.31/0.66 => ( total_on @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) @ ( lexord @ A @ R2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % total_lexord % 0.31/0.66 thf(fact_8003_total__on__empty,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R2 ) ). % 0.31/0.66 % 0.31/0.66 % total_on_empty % 0.31/0.66 thf(fact_8004_total__onI,axiom, % 0.31/0.66 ! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ! [X5: A,Y4: A] : % 0.31/0.66 ( ( member @ A @ X5 @ A3 ) % 0.31/0.66 => ( ( member @ A @ Y4 @ A3 ) % 0.31/0.66 => ( ( X5 != Y4 ) % 0.31/0.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ R2 ) % 0.31/0.66 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X5 ) @ R2 ) ) ) ) ) % 0.31/0.66 => ( total_on @ A @ A3 @ R2 ) ) ). % 0.31/0.66 % 0.31/0.66 % total_onI % 0.31/0.66 thf(fact_8005_total__on__def,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( total_on @ A ) % 0.31/0.66 = ( ^ [A6: set @ A,R5: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ! [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A6 ) % 0.31/0.66 => ! [Y2: A] : % 0.31/0.66 ( ( member @ A @ Y2 @ A6 ) % 0.31/0.66 => ( ( X != Y2 ) % 0.31/0.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R5 ) % 0.31/0.66 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ R5 ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % total_on_def % 0.31/0.66 thf(fact_8006_Arg__correct,axiom, % 0.31/0.66 ! [Z2: complex] : % 0.31/0.66 ( ( Z2 % 0.31/0.66 != ( zero_zero @ complex ) ) % 0.31/0.66 => ( ( ( sgn_sgn @ complex @ Z2 ) % 0.31/0.66 = ( cis @ ( arg @ Z2 ) ) ) % 0.31/0.66 & ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z2 ) ) % 0.31/0.66 & ( ord_less_eq @ real @ ( arg @ Z2 ) @ pi ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Arg_correct % 0.31/0.66 thf(fact_8007_Total__subset__Id,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( total_on @ A @ ( field2 @ A @ R2 ) @ R2 ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) % 0.31/0.66 => ( ( R2 % 0.31/0.66 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) % 0.31/0.66 | ? [A5: A] : % 0.31/0.66 ( R2 % 0.31/0.66 = ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Total_subset_Id % 0.31/0.66 thf(fact_8008_Field__empty,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( field2 @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) % 0.31/0.66 = ( bot_bot @ ( set @ A ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Field_empty % 0.31/0.66 thf(fact_8009_Field__insert,axiom, % 0.31/0.66 ! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( field2 @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) % 0.31/0.66 = ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Field_insert % 0.31/0.66 thf(fact_8010_underS__Field2,axiom, % 0.31/0.66 ! [A: $tType,A2: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( member @ A @ A2 @ ( field2 @ A @ R2 ) ) % 0.31/0.66 => ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R2 @ A2 ) @ ( field2 @ A @ R2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % underS_Field2 % 0.31/0.66 thf(fact_8011_Order__Relation_OunderS__Field,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A] : ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A2 ) @ ( field2 @ A @ R2 ) ) ). % 0.31/0.66 % 0.31/0.66 % Order_Relation.underS_Field % 0.31/0.66 thf(fact_8012_underS__empty,axiom, % 0.31/0.66 ! [A: $tType,A2: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ~ ( member @ A @ A2 @ ( field2 @ A @ R2 ) ) % 0.31/0.66 => ( ( order_underS @ A @ R2 @ A2 ) % 0.31/0.66 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % underS_empty % 0.31/0.66 thf(fact_8013_underS__Field3,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A] : % 0.31/0.66 ( ( ( field2 @ A @ R2 ) % 0.31/0.66 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.66 => ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R2 @ A2 ) @ ( field2 @ A @ R2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % underS_Field3 % 0.31/0.66 thf(fact_8014_FieldI2,axiom, % 0.31/0.66 ! [A: $tType,I: A,J: A,R: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R ) % 0.31/0.66 => ( member @ A @ J @ ( field2 @ A @ R ) ) ) ). % 0.31/0.66 % 0.31/0.66 % FieldI2 % 0.31/0.66 thf(fact_8015_FieldI1,axiom, % 0.31/0.66 ! [A: $tType,I: A,J: A,R: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R ) % 0.31/0.66 => ( member @ A @ I @ ( field2 @ A @ R ) ) ) ). % 0.31/0.66 % 0.31/0.66 % FieldI1 % 0.31/0.66 thf(fact_8016_mono__Field,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) % 0.31/0.66 => ( ord_less_eq @ ( set @ A ) @ ( field2 @ A @ R2 ) @ ( field2 @ A @ S ) ) ) ). % 0.31/0.66 % 0.31/0.66 % mono_Field % 0.31/0.66 thf(fact_8017_finite__Field,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 ) % 0.31/0.66 => ( finite_finite2 @ A @ ( field2 @ A @ R2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % finite_Field % 0.31/0.66 thf(fact_8018_Field__Restr__subset,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A] : % 0.31/0.66 ( ord_less_eq @ ( set @ A ) % 0.31/0.66 @ ( field2 @ A % 0.31/0.66 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 % 0.31/0.66 @ ( product_Sigma @ A @ A @ A3 % 0.31/0.66 @ ^ [Uu3: A] : A3 ) ) ) % 0.31/0.66 @ A3 ) ). % 0.31/0.66 % 0.31/0.66 % Field_Restr_subset % 0.31/0.66 thf(fact_8019_Field__natLeq__on,axiom, % 0.31/0.66 ! [N2: nat] : % 0.31/0.66 ( ( field2 @ nat % 0.31/0.66 @ ( collect @ ( product_prod @ nat @ nat ) % 0.31/0.66 @ ( product_case_prod @ nat @ nat @ $o % 0.31/0.66 @ ^ [X: nat,Y2: nat] : % 0.31/0.66 ( ( ord_less @ nat @ X @ N2 ) % 0.31/0.66 & ( ord_less @ nat @ Y2 @ N2 ) % 0.31/0.66 & ( ord_less_eq @ nat @ X @ Y2 ) ) ) ) ) % 0.31/0.66 = ( collect @ nat % 0.31/0.66 @ ^ [X: nat] : ( ord_less @ nat @ X @ N2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Field_natLeq_on % 0.31/0.66 thf(fact_8020_underS__incl__iff,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] : % 0.31/0.66 ( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) % 0.31/0.66 => ( ( member @ A @ A2 @ ( field2 @ A @ R2 ) ) % 0.31/0.66 => ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A2 ) @ ( order_underS @ A @ R2 @ B2 ) ) % 0.31/0.66 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % underS_incl_iff % 0.31/0.66 thf(fact_8021_Linear__order__in__diff__Id,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] : % 0.31/0.66 ( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) % 0.31/0.66 => ( ( member @ A @ A2 @ ( field2 @ A @ R2 ) ) % 0.31/0.66 => ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) ) % 0.31/0.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) % 0.31/0.66 = ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A2 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Linear_order_in_diff_Id % 0.31/0.66 thf(fact_8022_Refl__Field__Restr2,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A] : % 0.31/0.66 ( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( field2 @ A @ R2 ) ) % 0.31/0.66 => ( ( field2 @ A % 0.31/0.66 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 % 0.31/0.66 @ ( product_Sigma @ A @ A @ A3 % 0.31/0.66 @ ^ [Uu3: A] : A3 ) ) ) % 0.31/0.66 = A3 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Refl_Field_Restr2 % 0.31/0.66 thf(fact_8023_cis__Arg__unique,axiom, % 0.31/0.66 ! [Z2: complex,X3: real] : % 0.31/0.66 ( ( ( sgn_sgn @ complex @ Z2 ) % 0.31/0.66 = ( cis @ X3 ) ) % 0.31/0.66 => ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X3 ) % 0.31/0.66 => ( ( ord_less_eq @ real @ X3 @ pi ) % 0.31/0.66 => ( ( arg @ Z2 ) % 0.31/0.66 = X3 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % cis_Arg_unique % 0.31/0.66 thf(fact_8024_Under__def,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( order_Under @ A ) % 0.31/0.66 = ( ^ [R5: set @ ( product_prod @ A @ A ),A6: set @ A] : % 0.31/0.66 ( collect @ A % 0.31/0.66 @ ^ [B4: A] : % 0.31/0.66 ( ( member @ A @ B4 @ ( field2 @ A @ R5 ) ) % 0.31/0.66 & ! [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A6 ) % 0.31/0.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ X ) @ R5 ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Under_def % 0.31/0.66 thf(fact_8025_UnderS__def,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( order_UnderS @ A ) % 0.31/0.66 = ( ^ [R5: set @ ( product_prod @ A @ A ),A6: set @ A] : % 0.31/0.66 ( collect @ A % 0.31/0.66 @ ^ [B4: A] : % 0.31/0.66 ( ( member @ A @ B4 @ ( field2 @ A @ R5 ) ) % 0.31/0.66 & ! [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A6 ) % 0.31/0.66 => ( ( B4 != X ) % 0.31/0.66 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ X ) @ R5 ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % UnderS_def % 0.31/0.66 thf(fact_8026_Above__def,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( order_Above @ A ) % 0.31/0.66 = ( ^ [R5: set @ ( product_prod @ A @ A ),A6: set @ A] : % 0.31/0.66 ( collect @ A % 0.31/0.66 @ ^ [B4: A] : % 0.31/0.66 ( ( member @ A @ B4 @ ( field2 @ A @ R5 ) ) % 0.31/0.66 & ! [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A6 ) % 0.31/0.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ B4 ) @ R5 ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Above_def % 0.31/0.66 thf(fact_8027_bij__betw__roots__unity,axiom, % 0.31/0.66 ! [N2: nat] : % 0.31/0.66 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.66 => ( bij_betw @ nat @ complex % 0.31/0.66 @ ^ [K3: nat] : ( cis @ ( divide_divide @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( semiring_1_of_nat @ real @ K3 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) % 0.31/0.66 @ ( set_ord_lessThan @ nat @ N2 ) % 0.31/0.66 @ ( collect @ complex % 0.31/0.66 @ ^ [Z6: complex] : % 0.31/0.66 ( ( power_power @ complex @ Z6 @ N2 ) % 0.31/0.66 = ( one_one @ complex ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_roots_unity % 0.31/0.66 thf(fact_8028_bij__betw__add,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( cancel_semigroup_add @ A ) % 0.31/0.66 => ! [A2: A,A3: set @ A,B3: set @ A] : % 0.31/0.66 ( ( bij_betw @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 @ B3 ) % 0.31/0.66 = ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 ) % 0.31/0.66 = B3 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_add % 0.31/0.66 thf(fact_8029_bij__betw__combine,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B3: set @ B,C5: set @ A,D6: set @ B] : % 0.31/0.66 ( ( bij_betw @ A @ B @ F2 @ A3 @ B3 ) % 0.31/0.66 => ( ( bij_betw @ A @ B @ F2 @ C5 @ D6 ) % 0.31/0.66 => ( ( ( inf_inf @ ( set @ B ) @ B3 @ D6 ) % 0.31/0.66 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.66 => ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ C5 ) @ ( sup_sup @ ( set @ B ) @ B3 @ D6 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_combine % 0.31/0.66 thf(fact_8030_bij__betw__partition,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,F2: A > B,A3: set @ A,C5: set @ A,B3: set @ B,D6: set @ B] : % 0.31/0.66 ( ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ C5 ) @ ( sup_sup @ ( set @ B ) @ B3 @ D6 ) ) % 0.31/0.66 => ( ( bij_betw @ A @ B @ F2 @ C5 @ D6 ) % 0.31/0.66 => ( ( ( inf_inf @ ( set @ A ) @ A3 @ C5 ) % 0.31/0.66 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.66 => ( ( ( inf_inf @ ( set @ B ) @ B3 @ D6 ) % 0.31/0.66 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.66 => ( bij_betw @ A @ B @ F2 @ A3 @ B3 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_partition % 0.31/0.66 thf(fact_8031_bij__betw__disjoint__Un,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,F2: A > B,A3: set @ A,C5: set @ B,G: A > B,B3: set @ A,D6: set @ B] : % 0.31/0.66 ( ( bij_betw @ A @ B @ F2 @ A3 @ C5 ) % 0.31/0.66 => ( ( bij_betw @ A @ B @ G @ B3 @ D6 ) % 0.31/0.66 => ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.66 = ( bot_bot @ ( set @ A ) ) ) % 0.31/0.66 => ( ( ( inf_inf @ ( set @ B ) @ C5 @ D6 ) % 0.31/0.66 = ( bot_bot @ ( set @ B ) ) ) % 0.31/0.66 => ( bij_betw @ A @ B % 0.31/0.66 @ ^ [X: A] : ( if @ B @ ( member @ A @ X @ A3 ) @ ( F2 @ X ) @ ( G @ X ) ) % 0.31/0.66 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) % 0.31/0.66 @ ( sup_sup @ ( set @ B ) @ C5 @ D6 ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_disjoint_Un % 0.31/0.66 thf(fact_8032_bij__betw__subset,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,F2: A > B,A3: set @ A,A16: set @ B,B3: set @ A,B15: set @ B] : % 0.31/0.66 ( ( bij_betw @ A @ B @ F2 @ A3 @ A16 ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 0.31/0.66 => ( ( ( image @ A @ B @ F2 @ B3 ) % 0.31/0.66 = B15 ) % 0.31/0.66 => ( bij_betw @ A @ B @ F2 @ B3 @ B15 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_subset % 0.31/0.66 thf(fact_8033_bij__betw__byWitness,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,A3: set @ A,F11: B > A,F2: A > B,A16: set @ B] : % 0.31/0.66 ( ! [X5: A] : % 0.31/0.66 ( ( member @ A @ X5 @ A3 ) % 0.31/0.66 => ( ( F11 @ ( F2 @ X5 ) ) % 0.31/0.66 = X5 ) ) % 0.31/0.66 => ( ! [X5: B] : % 0.31/0.66 ( ( member @ B @ X5 @ A16 ) % 0.31/0.66 => ( ( F2 @ ( F11 @ X5 ) ) % 0.31/0.66 = X5 ) ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ A16 ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F11 @ A16 ) @ A3 ) % 0.31/0.66 => ( bij_betw @ A @ B @ F2 @ A3 @ A16 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_byWitness % 0.31/0.66 thf(fact_8034_bij__betw__UNION__chain,axiom, % 0.31/0.66 ! [B: $tType,C: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B ),F2: B > C,A16: A > ( set @ C )] : % 0.31/0.66 ( ! [I4: A,J2: A] : % 0.31/0.66 ( ( member @ A @ I4 @ I6 ) % 0.31/0.66 => ( ( member @ A @ J2 @ I6 ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ B ) @ ( A3 @ I4 ) @ ( A3 @ J2 ) ) % 0.31/0.66 | ( ord_less_eq @ ( set @ B ) @ ( A3 @ J2 ) @ ( A3 @ I4 ) ) ) ) ) % 0.31/0.66 => ( ! [I4: A] : % 0.31/0.66 ( ( member @ A @ I4 @ I6 ) % 0.31/0.66 => ( bij_betw @ B @ C @ F2 @ ( A3 @ I4 ) @ ( A16 @ I4 ) ) ) % 0.31/0.66 => ( bij_betw @ B @ C @ F2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ A @ ( set @ C ) @ A16 @ I6 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_UNION_chain % 0.31/0.66 thf(fact_8035_bij__betw__comp__iff2,axiom, % 0.31/0.66 ! [C: $tType,A: $tType,B: $tType,F11: A > B,A16: set @ A,A19: set @ B,F2: C > A,A3: set @ C] : % 0.31/0.66 ( ( bij_betw @ A @ B @ F11 @ A16 @ A19 ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ A3 ) @ A16 ) % 0.31/0.66 => ( ( bij_betw @ C @ A @ F2 @ A3 @ A16 ) % 0.31/0.66 = ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F11 @ F2 ) @ A3 @ A19 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_comp_iff2 % 0.31/0.66 thf(fact_8036_finite__same__card__bij,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,A3: set @ A,B3: set @ B] : % 0.31/0.66 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.66 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.66 => ( ( ( finite_card @ A @ A3 ) % 0.31/0.66 = ( finite_card @ B @ B3 ) ) % 0.31/0.66 => ? [H6: A > B] : ( bij_betw @ A @ B @ H6 @ A3 @ B3 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % finite_same_card_bij % 0.31/0.66 thf(fact_8037_bij__betw__iff__card,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,A3: set @ A,B3: set @ B] : % 0.31/0.66 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.66 => ( ( finite_finite2 @ B @ B3 ) % 0.31/0.66 => ( ( ? [F3: A > B] : ( bij_betw @ A @ B @ F3 @ A3 @ B3 ) ) % 0.31/0.66 = ( ( finite_card @ A @ A3 ) % 0.31/0.66 = ( finite_card @ B @ B3 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_iff_card % 0.31/0.66 thf(fact_8038_bij__betw__empty2,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,F2: A > B,A3: set @ A] : % 0.31/0.66 ( ( bij_betw @ A @ B @ F2 @ A3 @ ( bot_bot @ ( set @ B ) ) ) % 0.31/0.66 => ( A3 % 0.31/0.66 = ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_empty2 % 0.31/0.66 thf(fact_8039_bij__betw__empty1,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,F2: A > B,A3: set @ B] : % 0.31/0.66 ( ( bij_betw @ A @ B @ F2 @ ( bot_bot @ ( set @ A ) ) @ A3 ) % 0.31/0.66 => ( A3 % 0.31/0.66 = ( bot_bot @ ( set @ B ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_empty1 % 0.31/0.66 thf(fact_8040_bij__betw__finite,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B3: set @ B] : % 0.31/0.66 ( ( bij_betw @ A @ B @ F2 @ A3 @ B3 ) % 0.31/0.66 => ( ( finite_finite2 @ A @ A3 ) % 0.31/0.66 = ( finite_finite2 @ B @ B3 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_finite % 0.31/0.66 thf(fact_8041_notIn__Un__bij__betw,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,B2: A,A3: set @ A,F2: A > B,A16: set @ B] : % 0.31/0.66 ( ~ ( member @ A @ B2 @ A3 ) % 0.31/0.66 => ( ~ ( member @ B @ ( F2 @ B2 ) @ A16 ) % 0.31/0.66 => ( ( bij_betw @ A @ B @ F2 @ A3 @ A16 ) % 0.31/0.66 => ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A16 @ ( insert @ B @ ( F2 @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % notIn_Un_bij_betw % 0.31/0.66 thf(fact_8042_notIn__Un__bij__betw3,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,B2: A,A3: set @ A,F2: A > B,A16: set @ B] : % 0.31/0.66 ( ~ ( member @ A @ B2 @ A3 ) % 0.31/0.66 => ( ~ ( member @ B @ ( F2 @ B2 ) @ A16 ) % 0.31/0.66 => ( ( bij_betw @ A @ B @ F2 @ A3 @ A16 ) % 0.31/0.66 = ( bij_betw @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A16 @ ( insert @ B @ ( F2 @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % notIn_Un_bij_betw3 % 0.31/0.66 thf(fact_8043_bij__betw__funpow,axiom, % 0.31/0.66 ! [A: $tType,F2: A > A,S3: set @ A,N2: nat] : % 0.31/0.66 ( ( bij_betw @ A @ A @ F2 @ S3 @ S3 ) % 0.31/0.66 => ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ S3 @ S3 ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_funpow % 0.31/0.66 thf(fact_8044_bij__fn,axiom, % 0.31/0.66 ! [A: $tType,F2: A > A,N2: nat] : % 0.31/0.66 ( ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) % 0.31/0.66 => ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_fn % 0.31/0.66 thf(fact_8045_finite__vimage__iff,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,H2: A > B,F5: set @ B] : % 0.31/0.66 ( ( bij_betw @ A @ B @ H2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) % 0.31/0.66 => ( ( finite_finite2 @ A @ ( vimage @ A @ B @ H2 @ F5 ) ) % 0.31/0.66 = ( finite_finite2 @ B @ F5 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % finite_vimage_iff % 0.31/0.66 thf(fact_8046_infinite__imp__bij__betw2,axiom, % 0.31/0.66 ! [A: $tType,A3: set @ A,A2: A] : % 0.31/0.66 ( ~ ( finite_finite2 @ A @ A3 ) % 0.31/0.66 => ? [H6: A > A] : ( bij_betw @ A @ A @ H6 @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % infinite_imp_bij_betw2 % 0.31/0.66 thf(fact_8047_infinite__imp__bij__betw,axiom, % 0.31/0.66 ! [A: $tType,A3: set @ A,A2: A] : % 0.31/0.66 ( ~ ( finite_finite2 @ A @ A3 ) % 0.31/0.66 => ? [H6: A > A] : ( bij_betw @ A @ A @ H6 @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % infinite_imp_bij_betw % 0.31/0.66 thf(fact_8048_vimage__subset__eq,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,F2: A > B,B3: set @ B,A3: set @ A] : % 0.31/0.66 ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ B3 ) @ A3 ) % 0.31/0.66 = ( ord_less_eq @ ( set @ B ) @ B3 @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % vimage_subset_eq % 0.31/0.66 thf(fact_8049_ex__bij__betw__nat__finite,axiom, % 0.31/0.66 ! [A: $tType,M7: set @ A] : % 0.31/0.66 ( ( finite_finite2 @ A @ M7 ) % 0.31/0.66 => ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ). % 0.31/0.66 % 0.31/0.66 % ex_bij_betw_nat_finite % 0.31/0.66 thf(fact_8050_ex__bij__betw__nat__finite__1,axiom, % 0.31/0.66 ! [A: $tType,M7: set @ A] : % 0.31/0.66 ( ( finite_finite2 @ A @ M7 ) % 0.31/0.66 => ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ). % 0.31/0.66 % 0.31/0.66 % ex_bij_betw_nat_finite_1 % 0.31/0.66 thf(fact_8051_sum_Oreindex__bij__betw__not__neutral,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,C: $tType] : % 0.31/0.66 ( ( comm_monoid_add @ A ) % 0.31/0.66 => ! [S6: set @ B,T6: set @ C,H2: B > C,S3: set @ B,T4: set @ C,G: C > A] : % 0.31/0.66 ( ( finite_finite2 @ B @ S6 ) % 0.31/0.66 => ( ( finite_finite2 @ C @ T6 ) % 0.31/0.66 => ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) % 0.31/0.66 => ( ! [A5: B] : % 0.31/0.66 ( ( member @ B @ A5 @ S6 ) % 0.31/0.66 => ( ( G @ ( H2 @ A5 ) ) % 0.31/0.66 = ( zero_zero @ A ) ) ) % 0.31/0.66 => ( ! [B5: C] : % 0.31/0.66 ( ( member @ C @ B5 @ T6 ) % 0.31/0.66 => ( ( G @ B5 ) % 0.31/0.66 = ( zero_zero @ A ) ) ) % 0.31/0.66 => ( ( groups7311177749621191930dd_sum @ B @ A % 0.31/0.66 @ ^ [X: B] : ( G @ ( H2 @ X ) ) % 0.31/0.66 @ S3 ) % 0.31/0.66 = ( groups7311177749621191930dd_sum @ C @ A @ G @ T4 ) ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % sum.reindex_bij_betw_not_neutral % 0.31/0.66 thf(fact_8052_prod_Oreindex__bij__betw__not__neutral,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,C: $tType] : % 0.31/0.66 ( ( comm_monoid_mult @ A ) % 0.31/0.66 => ! [S6: set @ B,T6: set @ C,H2: B > C,S3: set @ B,T4: set @ C,G: C > A] : % 0.31/0.66 ( ( finite_finite2 @ B @ S6 ) % 0.31/0.66 => ( ( finite_finite2 @ C @ T6 ) % 0.31/0.66 => ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) % 0.31/0.66 => ( ! [A5: B] : % 0.31/0.66 ( ( member @ B @ A5 @ S6 ) % 0.31/0.66 => ( ( G @ ( H2 @ A5 ) ) % 0.31/0.66 = ( one_one @ A ) ) ) % 0.31/0.66 => ( ! [B5: C] : % 0.31/0.66 ( ( member @ C @ B5 @ T6 ) % 0.31/0.66 => ( ( G @ B5 ) % 0.31/0.66 = ( one_one @ A ) ) ) % 0.31/0.66 => ( ( groups7121269368397514597t_prod @ B @ A % 0.31/0.66 @ ^ [X: B] : ( G @ ( H2 @ X ) ) % 0.31/0.66 @ S3 ) % 0.31/0.66 = ( groups7121269368397514597t_prod @ C @ A @ G @ T4 ) ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % prod.reindex_bij_betw_not_neutral % 0.31/0.66 thf(fact_8053_bij__betw__nth,axiom, % 0.31/0.66 ! [A: $tType,Xs: list @ A,A3: set @ nat,B3: set @ A] : % 0.31/0.66 ( ( distinct @ A @ Xs ) % 0.31/0.66 => ( ( A3 % 0.31/0.66 = ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) ) % 0.31/0.66 => ( ( B3 % 0.31/0.66 = ( set2 @ A @ Xs ) ) % 0.31/0.66 => ( bij_betw @ nat @ A @ ( nth @ A @ Xs ) @ A3 @ B3 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_nth % 0.31/0.66 thf(fact_8054_ex__bij__betw__strict__mono__card,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( wellorder @ A ) % 0.31/0.66 => ! [M7: set @ A] : % 0.31/0.66 ( ( finite_finite2 @ A @ M7 ) % 0.31/0.66 => ~ ! [H6: nat > A] : % 0.31/0.66 ( ( bij_betw @ nat @ A @ H6 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ M7 ) ) @ M7 ) % 0.31/0.66 => ~ ( strict_mono_on @ nat @ A @ H6 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ M7 ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % ex_bij_betw_strict_mono_card % 0.31/0.66 thf(fact_8055_Schroeder__Bernstein,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,F2: A > B,A3: set @ A,B3: set @ B,G: B > A] : % 0.31/0.66 ( ( inj_on @ A @ B @ F2 @ A3 ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B3 ) % 0.31/0.66 => ( ( inj_on @ B @ A @ G @ B3 ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ B3 ) @ A3 ) % 0.31/0.66 => ? [H6: A > B] : ( bij_betw @ A @ B @ H6 @ A3 @ B3 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Schroeder_Bernstein % 0.31/0.66 thf(fact_8056_bij__betw__nth__root__unity,axiom, % 0.31/0.66 ! [C2: complex,N2: nat] : % 0.31/0.66 ( ( C2 % 0.31/0.66 != ( zero_zero @ complex ) ) % 0.31/0.66 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 0.31/0.66 => ( bij_betw @ complex @ complex @ ( times_times @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( root @ N2 @ ( real_V7770717601297561774m_norm @ complex @ C2 ) ) ) @ ( cis @ ( divide_divide @ real @ ( arg @ C2 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) % 0.31/0.66 @ ( collect @ complex % 0.31/0.66 @ ^ [Z6: complex] : % 0.31/0.66 ( ( power_power @ complex @ Z6 @ N2 ) % 0.31/0.66 = ( one_one @ complex ) ) ) % 0.31/0.66 @ ( collect @ complex % 0.31/0.66 @ ^ [Z6: complex] : % 0.31/0.66 ( ( power_power @ complex @ Z6 @ N2 ) % 0.31/0.66 = C2 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_nth_root_unity % 0.31/0.66 thf(fact_8057_bij__betw__Suc,axiom, % 0.31/0.66 ! [M7: set @ nat,N7: set @ nat] : % 0.31/0.66 ( ( bij_betw @ nat @ nat @ suc @ M7 @ N7 ) % 0.31/0.66 = ( ( image @ nat @ nat @ suc @ M7 ) % 0.31/0.66 = N7 ) ) ). % 0.31/0.66 % 0.31/0.66 % bij_betw_Suc % 0.31/0.66 thf(fact_8058_le__rel__bool__arg__iff,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( ord @ A ) % 0.31/0.66 => ( ( ord_less_eq @ ( $o > A ) ) % 0.31/0.66 = ( ^ [X6: $o > A,Y9: $o > A] : % 0.31/0.66 ( ( ord_less_eq @ A @ ( X6 @ $false ) @ ( Y9 @ $false ) ) % 0.31/0.66 & ( ord_less_eq @ A @ ( X6 @ $true ) @ ( Y9 @ $true ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % le_rel_bool_arg_iff % 0.31/0.66 thf(fact_8059_ex__bij__betw__finite__nat,axiom, % 0.31/0.66 ! [A: $tType,M7: set @ A] : % 0.31/0.66 ( ( finite_finite2 @ A @ M7 ) % 0.31/0.66 => ? [H6: A > nat] : ( bij_betw @ A @ nat @ H6 @ M7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % ex_bij_betw_finite_nat % 0.31/0.66 thf(fact_8060_finite__Linear__order__induct,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,P: A > $o] : % 0.31/0.66 ( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) % 0.31/0.66 => ( ( member @ A @ X3 @ ( field2 @ A @ R2 ) ) % 0.31/0.66 => ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 ) % 0.31/0.66 => ( ! [X5: A] : % 0.31/0.66 ( ( member @ A @ X5 @ ( field2 @ A @ R2 ) ) % 0.31/0.66 => ( ! [Y5: A] : % 0.31/0.66 ( ( member @ A @ Y5 @ ( order_aboveS @ A @ R2 @ X5 ) ) % 0.31/0.66 => ( P @ Y5 ) ) % 0.31/0.66 => ( P @ X5 ) ) ) % 0.31/0.66 => ( P @ X3 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % finite_Linear_order_induct % 0.31/0.66 thf(fact_8061_Arg__def,axiom, % 0.31/0.66 ( arg % 0.31/0.66 = ( ^ [Z6: complex] : % 0.31/0.66 ( if @ real % 0.31/0.66 @ ( Z6 % 0.31/0.66 = ( zero_zero @ complex ) ) % 0.31/0.66 @ ( zero_zero @ real ) % 0.31/0.66 @ ( fChoice @ real % 0.31/0.66 @ ^ [A4: real] : % 0.31/0.66 ( ( ( sgn_sgn @ complex @ Z6 ) % 0.31/0.66 = ( cis @ A4 ) ) % 0.31/0.66 & ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ A4 ) % 0.31/0.66 & ( ord_less_eq @ real @ A4 @ pi ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Arg_def % 0.31/0.66 thf(fact_8062_some__in__eq,axiom, % 0.31/0.66 ! [A: $tType,A3: set @ A] : % 0.31/0.66 ( ( member @ A % 0.31/0.66 @ ( fChoice @ A % 0.31/0.66 @ ^ [X: A] : ( member @ A @ X @ A3 ) ) % 0.31/0.66 @ A3 ) % 0.31/0.66 = ( A3 % 0.31/0.66 != ( bot_bot @ ( set @ A ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % some_in_eq % 0.31/0.66 thf(fact_8063_exE__some,axiom, % 0.31/0.66 ! [A: $tType,P: A > $o,C2: A] : % 0.31/0.66 ( ? [X_12: A] : ( P @ X_12 ) % 0.31/0.66 => ( ( C2 % 0.31/0.66 = ( fChoice @ A @ P ) ) % 0.31/0.66 => ( P @ C2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % exE_some % 0.31/0.66 thf(fact_8064_aboveS__def,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( order_aboveS @ A ) % 0.31/0.66 = ( ^ [R5: set @ ( product_prod @ A @ A ),A4: A] : % 0.31/0.66 ( collect @ A % 0.31/0.66 @ ^ [B4: A] : % 0.31/0.66 ( ( B4 != A4 ) % 0.31/0.66 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R5 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % aboveS_def % 0.31/0.66 thf(fact_8065_arg__min__SOME__Min,axiom, % 0.31/0.66 ! [B: $tType,A: $tType] : % 0.31/0.66 ( ( linorder @ B ) % 0.31/0.66 => ! [S3: set @ A,F2: A > B] : % 0.31/0.66 ( ( finite_finite2 @ A @ S3 ) % 0.31/0.66 => ( ( lattic7623131987881927897min_on @ A @ B @ F2 @ S3 ) % 0.31/0.66 = ( fChoice @ A % 0.31/0.66 @ ^ [Y2: A] : % 0.31/0.66 ( ( member @ A @ Y2 @ S3 ) % 0.31/0.66 & ( ( F2 @ Y2 ) % 0.31/0.66 = ( lattic643756798350308766er_Min @ B @ ( image @ A @ B @ F2 @ S3 ) ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % arg_min_SOME_Min % 0.31/0.66 thf(fact_8066_cofinal__def,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bNF_Ca7293521722713021262ofinal @ A ) % 0.31/0.66 = ( ^ [A6: set @ A,R5: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ! [X: A] : % 0.31/0.66 ( ( member @ A @ X @ ( field2 @ A @ R5 ) ) % 0.31/0.66 => ? [Y2: A] : % 0.31/0.66 ( ( member @ A @ Y2 @ A6 ) % 0.31/0.66 & ( X != Y2 ) % 0.31/0.66 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R5 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % cofinal_def % 0.31/0.66 thf(fact_8067_bsqr__def,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bNF_Wellorder_bsqr @ A ) % 0.31/0.66 = ( ^ [R5: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( collect @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) % 0.31/0.66 @ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o % 0.31/0.66 @ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o ) % 0.31/0.66 @ ^ [A1: A,A22: A] : % 0.31/0.66 ( product_case_prod @ A @ A @ $o % 0.31/0.66 @ ^ [B16: A,B23: A] : % 0.31/0.66 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A1 @ ( insert @ A @ A22 @ ( insert @ A @ B16 @ ( insert @ A @ B23 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R5 ) ) % 0.31/0.66 & ( ( ( A1 = B16 ) % 0.31/0.66 & ( A22 = B23 ) ) % 0.31/0.66 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A1 @ A22 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B16 @ B23 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) % 0.31/0.66 | ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A1 @ A22 ) % 0.31/0.66 = ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B16 @ B23 ) ) % 0.31/0.66 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B16 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) ) % 0.31/0.66 | ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A1 @ A22 ) % 0.31/0.66 = ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B16 @ B23 ) ) % 0.31/0.66 & ( A1 = B16 ) % 0.31/0.66 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A22 @ B23 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bsqr_def % 0.31/0.66 thf(fact_8068_Eps__case__prod__eq,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,X3: A,Y: B] : % 0.31/0.66 ( ( fChoice @ ( product_prod @ A @ B ) % 0.31/0.66 @ ( product_case_prod @ A @ B @ $o % 0.31/0.66 @ ^ [X9: A,Y6: B] : % 0.31/0.66 ( ( X3 = X9 ) % 0.31/0.66 & ( Y = Y6 ) ) ) ) % 0.31/0.66 = ( product_Pair @ A @ B @ X3 @ Y ) ) ). % 0.31/0.66 % 0.31/0.66 % Eps_case_prod_eq % 0.31/0.66 thf(fact_8069_split__paired__Eps,axiom, % 0.31/0.66 ! [B: $tType,A: $tType] : % 0.31/0.66 ( ( fChoice @ ( product_prod @ A @ B ) ) % 0.31/0.66 = ( ^ [P3: ( product_prod @ A @ B ) > $o] : % 0.31/0.66 ( fChoice @ ( product_prod @ A @ B ) % 0.31/0.66 @ ( product_case_prod @ A @ B @ $o % 0.31/0.66 @ ^ [A4: A,B4: B] : ( P3 @ ( product_Pair @ A @ B @ A4 @ B4 ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % split_paired_Eps % 0.31/0.66 thf(fact_8070_to__nat__on__def,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( countable_to_nat_on @ A ) % 0.31/0.66 = ( ^ [S7: set @ A] : % 0.31/0.66 ( fChoice @ ( A > nat ) % 0.31/0.66 @ ^ [F3: A > nat] : % 0.31/0.66 ( ( ( finite_finite2 @ A @ S7 ) % 0.31/0.66 => ( bij_betw @ A @ nat @ F3 @ S7 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S7 ) ) ) ) % 0.31/0.66 & ( ~ ( finite_finite2 @ A @ S7 ) % 0.31/0.66 => ( bij_betw @ A @ nat @ F3 @ S7 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % to_nat_on_def % 0.31/0.66 thf(fact_8071_Refl__under__underS,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A] : % 0.31/0.66 ( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 ) % 0.31/0.66 => ( ( member @ A @ A2 @ ( field2 @ A @ R2 ) ) % 0.31/0.66 => ( ( order_under @ A @ R2 @ A2 ) % 0.31/0.66 = ( sup_sup @ ( set @ A ) @ ( order_underS @ A @ R2 @ A2 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Refl_under_underS % 0.31/0.66 thf(fact_8072_under__def,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( order_under @ A ) % 0.31/0.66 = ( ^ [R5: set @ ( product_prod @ A @ A ),A4: A] : % 0.31/0.66 ( collect @ A % 0.31/0.66 @ ^ [B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ R5 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % under_def % 0.31/0.66 thf(fact_8073_underS__subset__under,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A] : ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A2 ) @ ( order_under @ A @ R2 @ A2 ) ) ). % 0.31/0.66 % 0.31/0.66 % underS_subset_under % 0.31/0.66 thf(fact_8074_under__Field,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A] : ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R2 @ A2 ) @ ( field2 @ A @ R2 ) ) ). % 0.31/0.66 % 0.31/0.66 % under_Field % 0.31/0.66 thf(fact_8075_to__nat__on__finite,axiom, % 0.31/0.66 ! [A: $tType,S3: set @ A] : % 0.31/0.66 ( ( finite_finite2 @ A @ S3 ) % 0.31/0.66 => ( bij_betw @ A @ nat @ ( countable_to_nat_on @ A @ S3 ) @ S3 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S3 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % to_nat_on_finite % 0.31/0.66 thf(fact_8076_Linear__order__wf__diff__Id,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) % 0.31/0.66 => ( ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) % 0.31/0.66 = ( ! [A6: set @ A] : % 0.31/0.66 ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R2 ) ) % 0.31/0.66 => ( ( A6 % 0.31/0.66 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.66 => ? [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A6 ) % 0.31/0.66 & ! [Y2: A] : % 0.31/0.66 ( ( member @ A @ Y2 @ A6 ) % 0.31/0.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 ) ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Linear_order_wf_diff_Id % 0.31/0.66 thf(fact_8077_slice__eq__mask,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.66 => ! [N2: nat,M: nat,A2: A] : % 0.31/0.66 ( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) ) ) % 0.31/0.66 = ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ M @ N2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % slice_eq_mask % 0.31/0.66 thf(fact_8078_wf__empty,axiom, % 0.31/0.66 ! [A: $tType] : ( wf @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_empty % 0.31/0.66 thf(fact_8079_wf__listrel1__iff,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( wf @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) % 0.31/0.66 = ( wf @ A @ R2 ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_listrel1_iff % 0.31/0.66 thf(fact_8080_wf__lex,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( wf @ A @ R2 ) % 0.31/0.66 => ( wf @ ( list @ A ) @ ( lex @ A @ R2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_lex % 0.31/0.66 thf(fact_8081_wf__lenlex,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( wf @ A @ R2 ) % 0.31/0.66 => ( wf @ ( list @ A ) @ ( lenlex @ A @ R2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_lenlex % 0.31/0.66 thf(fact_8082_not__negative__int__iff,axiom, % 0.31/0.66 ! [K: int] : % 0.31/0.66 ( ( ord_less @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ ( zero_zero @ int ) ) % 0.31/0.66 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ). % 0.31/0.66 % 0.31/0.66 % not_negative_int_iff % 0.31/0.66 thf(fact_8083_not__nonnegative__int__iff,axiom, % 0.31/0.66 ! [K: int] : % 0.31/0.66 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ K ) ) % 0.31/0.66 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ). % 0.31/0.66 % 0.31/0.66 % not_nonnegative_int_iff % 0.31/0.66 thf(fact_8084_bit_Oconj__cancel__left,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.66 => ! [X3: A] : % 0.31/0.66 ( ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X3 ) @ X3 ) % 0.31/0.66 = ( zero_zero @ A ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bit.conj_cancel_left % 0.31/0.66 thf(fact_8085_bit_Oconj__cancel__right,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.66 => ! [X3: A] : % 0.31/0.66 ( ( bit_se5824344872417868541ns_and @ A @ X3 @ ( bit_ri4277139882892585799ns_not @ A @ X3 ) ) % 0.31/0.66 = ( zero_zero @ A ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bit.conj_cancel_right % 0.31/0.66 thf(fact_8086_bit_Ocompl__one,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.66 => ( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.31/0.66 = ( zero_zero @ A ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bit.compl_one % 0.31/0.66 thf(fact_8087_bit_Ocompl__zero,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.66 => ( ( bit_ri4277139882892585799ns_not @ A @ ( zero_zero @ A ) ) % 0.31/0.66 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bit.compl_zero % 0.31/0.66 thf(fact_8088_wf__insert,axiom, % 0.31/0.66 ! [A: $tType,Y: A,X3: A,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( wf @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ R2 ) ) % 0.31/0.66 = ( ( wf @ A @ R2 ) % 0.31/0.66 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_insert % 0.31/0.66 thf(fact_8089_reduction__pairI,axiom, % 0.31/0.66 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ( wf @ A @ R ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S3 ) @ R ) % 0.31/0.66 => ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ S3 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % reduction_pairI % 0.31/0.66 thf(fact_8090_wf__bounded__measure,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > nat,F2: A > nat] : % 0.31/0.66 ( ! [A5: A,B5: A] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A5 ) @ R2 ) % 0.31/0.66 => ( ( ord_less_eq @ nat @ ( Ub @ B5 ) @ ( Ub @ A5 ) ) % 0.31/0.66 & ( ord_less_eq @ nat @ ( F2 @ B5 ) @ ( Ub @ A5 ) ) % 0.31/0.66 & ( ord_less @ nat @ ( F2 @ A5 ) @ ( F2 @ B5 ) ) ) ) % 0.31/0.66 => ( wf @ A @ R2 ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_bounded_measure % 0.31/0.66 thf(fact_8091_wfI,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,B3: set @ A] : % 0.31/0.66 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 % 0.31/0.66 @ ( product_Sigma @ A @ A @ A3 % 0.31/0.66 @ ^ [Uu3: A] : B3 ) ) % 0.31/0.66 => ( ! [X5: A,P8: A > $o] : % 0.31/0.66 ( ! [Xa: A] : % 0.31/0.66 ( ! [Y4: A] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa ) @ R2 ) % 0.31/0.66 => ( P8 @ Y4 ) ) % 0.31/0.66 => ( P8 @ Xa ) ) % 0.31/0.66 => ( ( member @ A @ X5 @ A3 ) % 0.31/0.66 => ( ( member @ A @ X5 @ B3 ) % 0.31/0.66 => ( P8 @ X5 ) ) ) ) % 0.31/0.66 => ( wf @ A @ R2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wfI % 0.31/0.66 thf(fact_8092_wf__lexn,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),N2: nat] : % 0.31/0.66 ( ( wf @ A @ R2 ) % 0.31/0.66 => ( wf @ ( list @ A ) @ ( lexn @ A @ R2 @ N2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_lexn % 0.31/0.66 thf(fact_8093_wf__int__ge__less__than2,axiom, % 0.31/0.66 ! [D2: int] : ( wf @ int @ ( int_ge_less_than2 @ D2 ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_int_ge_less_than2 % 0.31/0.66 thf(fact_8094_wf__int__ge__less__than,axiom, % 0.31/0.66 ! [D2: int] : ( wf @ int @ ( int_ge_less_than @ D2 ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_int_ge_less_than % 0.31/0.66 thf(fact_8095_wf__no__loop,axiom, % 0.31/0.66 ! [B: $tType,R: set @ ( product_prod @ B @ B )] : % 0.31/0.66 ( ( ( relcomp @ B @ B @ B @ R @ R ) % 0.31/0.66 = ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) ) % 0.31/0.66 => ( wf @ B @ R ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_no_loop % 0.31/0.66 thf(fact_8096_wfE__min_H,axiom, % 0.31/0.66 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Q: set @ A] : % 0.31/0.66 ( ( wf @ A @ R ) % 0.31/0.66 => ( ( Q % 0.31/0.66 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.66 => ~ ! [Z3: A] : % 0.31/0.66 ( ( member @ A @ Z3 @ Q ) % 0.31/0.66 => ~ ! [Y5: A] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ R ) % 0.31/0.66 => ~ ( member @ A @ Y5 @ Q ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wfE_min' % 0.31/0.66 thf(fact_8097_wf__linord__ex__has__least,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P: B > $o,K: B,M: B > A] : % 0.31/0.66 ( ( wf @ A @ R2 ) % 0.31/0.66 => ( ! [X5: A,Y4: A] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y4 ) @ ( transitive_trancl @ A @ R2 ) ) % 0.31/0.66 = ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X5 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) % 0.31/0.66 => ( ( P @ K ) % 0.31/0.66 => ? [X5: B] : % 0.31/0.66 ( ( P @ X5 ) % 0.31/0.66 & ! [Y5: B] : % 0.31/0.66 ( ( P @ Y5 ) % 0.31/0.66 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M @ X5 ) @ ( M @ Y5 ) ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_linord_ex_has_least % 0.31/0.66 thf(fact_8098_wf__def,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( wf @ A ) % 0.31/0.66 = ( ^ [R5: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ! [P3: A > $o] : % 0.31/0.66 ( ! [X: A] : % 0.31/0.66 ( ! [Y2: A] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ R5 ) % 0.31/0.66 => ( P3 @ Y2 ) ) % 0.31/0.66 => ( P3 @ X ) ) % 0.31/0.66 => ! [X6: A] : ( P3 @ X6 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_def % 0.31/0.66 thf(fact_8099_wfE__min,axiom, % 0.31/0.66 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X3: A,Q: set @ A] : % 0.31/0.66 ( ( wf @ A @ R ) % 0.31/0.66 => ( ( member @ A @ X3 @ Q ) % 0.31/0.66 => ~ ! [Z3: A] : % 0.31/0.66 ( ( member @ A @ Z3 @ Q ) % 0.31/0.66 => ~ ! [Y5: A] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ R ) % 0.31/0.66 => ~ ( member @ A @ Y5 @ Q ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wfE_min % 0.31/0.66 thf(fact_8100_wfI__min,axiom, % 0.31/0.66 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ! [X5: A,Q8: set @ A] : % 0.31/0.66 ( ( member @ A @ X5 @ Q8 ) % 0.31/0.66 => ? [Xa: A] : % 0.31/0.66 ( ( member @ A @ Xa @ Q8 ) % 0.31/0.66 & ! [Y4: A] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa ) @ R ) % 0.31/0.66 => ~ ( member @ A @ Y4 @ Q8 ) ) ) ) % 0.31/0.66 => ( wf @ A @ R ) ) ). % 0.31/0.66 % 0.31/0.66 % wfI_min % 0.31/0.66 thf(fact_8101_wfUNIVI,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( ! [P8: A > $o,X5: A] : % 0.31/0.66 ( ! [Xa: A] : % 0.31/0.66 ( ! [Y4: A] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa ) @ R2 ) % 0.31/0.66 => ( P8 @ Y4 ) ) % 0.31/0.66 => ( P8 @ Xa ) ) % 0.31/0.66 => ( P8 @ X5 ) ) % 0.31/0.66 => ( wf @ A @ R2 ) ) ). % 0.31/0.66 % 0.31/0.66 % wfUNIVI % 0.31/0.66 thf(fact_8102_wf__asym,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,X3: A] : % 0.31/0.66 ( ( wf @ A @ R2 ) % 0.31/0.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X3 ) @ R2 ) % 0.31/0.66 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A2 ) @ R2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_asym % 0.31/0.66 thf(fact_8103_wf__induct,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A2: A] : % 0.31/0.66 ( ( wf @ A @ R2 ) % 0.31/0.66 => ( ! [X5: A] : % 0.31/0.66 ( ! [Y5: A] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X5 ) @ R2 ) % 0.31/0.66 => ( P @ Y5 ) ) % 0.31/0.66 => ( P @ X5 ) ) % 0.31/0.66 => ( P @ A2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_induct % 0.31/0.66 thf(fact_8104_wf__irrefl,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A] : % 0.31/0.66 ( ( wf @ A @ R2 ) % 0.31/0.66 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R2 ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_irrefl % 0.31/0.66 thf(fact_8105_wf__not__sym,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,X3: A] : % 0.31/0.66 ( ( wf @ A @ R2 ) % 0.31/0.66 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X3 ) @ R2 ) % 0.31/0.66 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A2 ) @ R2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_not_sym % 0.31/0.66 thf(fact_8106_wf__not__refl,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A] : % 0.31/0.66 ( ( wf @ A @ R2 ) % 0.31/0.66 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R2 ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_not_refl % 0.31/0.66 thf(fact_8107_wf__eq__minimal,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( wf @ A ) % 0.31/0.66 = ( ^ [R5: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ! [Q7: set @ A] : % 0.31/0.66 ( ? [X: A] : ( member @ A @ X @ Q7 ) % 0.31/0.66 => ? [X: A] : % 0.31/0.66 ( ( member @ A @ X @ Q7 ) % 0.31/0.66 & ! [Y2: A] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ R5 ) % 0.31/0.66 => ~ ( member @ A @ Y2 @ Q7 ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_eq_minimal % 0.31/0.66 thf(fact_8108_wf__induct__rule,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A2: A] : % 0.31/0.66 ( ( wf @ A @ R2 ) % 0.31/0.66 => ( ! [X5: A] : % 0.31/0.66 ( ! [Y5: A] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X5 ) @ R2 ) % 0.31/0.66 => ( P @ Y5 ) ) % 0.31/0.66 => ( P @ X5 ) ) % 0.31/0.66 => ( P @ A2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_induct_rule % 0.31/0.66 thf(fact_8109_wf__iff__no__infinite__down__chain,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( wf @ A ) % 0.31/0.66 = ( ^ [R5: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ~ ? [F3: nat > A] : % 0.31/0.66 ! [I3: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F3 @ ( suc @ I3 ) ) @ ( F3 @ I3 ) ) @ R5 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_iff_no_infinite_down_chain % 0.31/0.66 thf(fact_8110_wf__no__infinite__down__chainE,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),F2: nat > A] : % 0.31/0.66 ( ( wf @ A @ R2 ) % 0.31/0.66 => ~ ! [K2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ ( suc @ K2 ) ) @ ( F2 @ K2 ) ) @ R2 ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_no_infinite_down_chainE % 0.31/0.66 thf(fact_8111_minus__eq__not__plus__1,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.66 => ( ( uminus_uminus @ A ) % 0.31/0.66 = ( ^ [A4: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A4 ) @ ( one_one @ A ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % minus_eq_not_plus_1 % 0.31/0.66 thf(fact_8112_not__diff__distrib,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.66 => ! [A2: A,B2: A] : % 0.31/0.66 ( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A2 @ B2 ) ) % 0.31/0.66 = ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ B2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % not_diff_distrib % 0.31/0.66 thf(fact_8113_not__add__distrib,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.66 => ! [A2: A,B2: A] : % 0.31/0.66 ( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A2 @ B2 ) ) % 0.31/0.66 = ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ B2 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % not_add_distrib % 0.31/0.66 thf(fact_8114_wf,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( wellorder @ A ) % 0.31/0.66 => ( wf @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf % 0.31/0.66 thf(fact_8115_wf__if__measure,axiom, % 0.31/0.66 ! [A: $tType,P: A > $o,F2: A > nat,G: A > A] : % 0.31/0.66 ( ! [X5: A] : % 0.31/0.66 ( ( P @ X5 ) % 0.31/0.66 => ( ord_less @ nat @ ( F2 @ ( G @ X5 ) ) @ ( F2 @ X5 ) ) ) % 0.31/0.66 => ( wf @ A % 0.31/0.66 @ ( collect @ ( product_prod @ A @ A ) % 0.31/0.66 @ ( product_case_prod @ A @ A @ $o % 0.31/0.66 @ ^ [Y2: A,X: A] : % 0.31/0.66 ( ( P @ X ) % 0.31/0.66 & ( Y2 % 0.31/0.66 = ( G @ X ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_if_measure % 0.31/0.66 thf(fact_8116_wf__less,axiom, % 0.31/0.66 wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_less % 0.31/0.66 thf(fact_8117_wf__eq__minimal2,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( wf @ A ) % 0.31/0.66 = ( ^ [R5: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ! [A6: set @ A] : % 0.31/0.66 ( ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R5 ) ) % 0.31/0.66 & ( A6 % 0.31/0.66 != ( bot_bot @ ( set @ A ) ) ) ) % 0.31/0.66 => ? [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A6 ) % 0.31/0.66 & ! [Y2: A] : % 0.31/0.66 ( ( member @ A @ Y2 @ A6 ) % 0.31/0.66 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ R5 ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_eq_minimal2 % 0.31/0.66 thf(fact_8118_wf__bounded__set,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F2: A > ( set @ B )] : % 0.31/0.66 ( ! [A5: A,B5: A] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A5 ) @ R2 ) % 0.31/0.66 => ( ( finite_finite2 @ B @ ( Ub @ A5 ) ) % 0.31/0.66 & ( ord_less_eq @ ( set @ B ) @ ( Ub @ B5 ) @ ( Ub @ A5 ) ) % 0.31/0.66 & ( ord_less_eq @ ( set @ B ) @ ( F2 @ B5 ) @ ( Ub @ A5 ) ) % 0.31/0.66 & ( ord_less @ ( set @ B ) @ ( F2 @ A5 ) @ ( F2 @ B5 ) ) ) ) % 0.31/0.66 => ( wf @ A @ R2 ) ) ). % 0.31/0.66 % 0.31/0.66 % wf_bounded_set % 0.31/0.66 thf(fact_8119_take__bit__not__mask__eq__0,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.66 => ! [M: nat,N2: nat] : % 0.31/0.66 ( ( ord_less_eq @ nat @ M @ N2 ) % 0.31/0.66 => ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) % 0.31/0.66 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % take_bit_not_mask_eq_0 % 0.31/0.66 thf(fact_8120_push__bit__mask__eq,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.66 => ! [M: nat,N2: nat] : % 0.31/0.66 ( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) % 0.31/0.66 = ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ N2 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % push_bit_mask_eq % 0.31/0.66 thf(fact_8121_finite__subset__wf,axiom, % 0.31/0.66 ! [A: $tType,A3: set @ A] : % 0.31/0.66 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.66 => ( wf @ ( set @ A ) % 0.31/0.66 @ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) % 0.31/0.66 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o % 0.31/0.66 @ ^ [X6: set @ A,Y9: set @ A] : % 0.31/0.66 ( ( ord_less @ ( set @ A ) @ X6 @ Y9 ) % 0.31/0.66 & ( ord_less_eq @ ( set @ A ) @ Y9 @ A3 ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % finite_subset_wf % 0.31/0.66 thf(fact_8122_bit_Ocompl__unique,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.66 => ! [X3: A,Y: A] : % 0.31/0.66 ( ( ( bit_se5824344872417868541ns_and @ A @ X3 @ Y ) % 0.31/0.66 = ( zero_zero @ A ) ) % 0.31/0.66 => ( ( ( bit_se1065995026697491101ons_or @ A @ X3 @ Y ) % 0.31/0.66 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 0.31/0.66 => ( ( bit_ri4277139882892585799ns_not @ A @ X3 ) % 0.31/0.66 = Y ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bit.compl_unique % 0.31/0.66 thf(fact_8123_bit__not__iff__eq,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.66 => ! [A2: A,N2: nat] : % 0.31/0.66 ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ N2 ) % 0.31/0.66 = ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) % 0.31/0.66 != ( zero_zero @ A ) ) % 0.31/0.66 & ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bit_not_iff_eq % 0.31/0.66 thf(fact_8124_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.66 => ( boolea3799213064322606851m_diff @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bit_se5824344971392196577ns_xor @ A ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bit.abstract_boolean_algebra_sym_diff_axioms % 0.31/0.66 thf(fact_8125_dependent__wf__choice,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P: ( A > B ) > A > B > $o] : % 0.31/0.66 ( ( wf @ A @ R ) % 0.31/0.66 => ( ! [F4: A > B,G4: A > B,X5: A,R4: B] : % 0.31/0.66 ( ! [Z4: A] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X5 ) @ R ) % 0.31/0.66 => ( ( F4 @ Z4 ) % 0.31/0.66 = ( G4 @ Z4 ) ) ) % 0.31/0.66 => ( ( P @ F4 @ X5 @ R4 ) % 0.31/0.66 = ( P @ G4 @ X5 @ R4 ) ) ) % 0.31/0.66 => ( ! [X5: A,F4: A > B] : % 0.31/0.66 ( ! [Y5: A] : % 0.31/0.66 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X5 ) @ R ) % 0.31/0.66 => ( P @ F4 @ Y5 @ ( F4 @ Y5 ) ) ) % 0.31/0.66 => ? [X_12: B] : ( P @ F4 @ X5 @ X_12 ) ) % 0.31/0.66 => ? [F4: A > B] : % 0.31/0.66 ! [X2: A] : ( P @ F4 @ X2 @ ( F4 @ X2 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % dependent_wf_choice % 0.31/0.66 thf(fact_8126_dependent__wellorder__choice,axiom, % 0.31/0.66 ! [B: $tType,A: $tType] : % 0.31/0.66 ( ( wellorder @ A ) % 0.31/0.66 => ! [P: ( A > B ) > A > B > $o] : % 0.31/0.66 ( ! [R4: B,F4: A > B,G4: A > B,X5: A] : % 0.31/0.66 ( ! [Y5: A] : % 0.31/0.66 ( ( ord_less @ A @ Y5 @ X5 ) % 0.31/0.66 => ( ( F4 @ Y5 ) % 0.31/0.66 = ( G4 @ Y5 ) ) ) % 0.31/0.66 => ( ( P @ F4 @ X5 @ R4 ) % 0.31/0.66 = ( P @ G4 @ X5 @ R4 ) ) ) % 0.31/0.66 => ( ! [X5: A,F4: A > B] : % 0.31/0.66 ( ! [Y5: A] : % 0.31/0.66 ( ( ord_less @ A @ Y5 @ X5 ) % 0.31/0.66 => ( P @ F4 @ Y5 @ ( F4 @ Y5 ) ) ) % 0.31/0.66 => ? [X_12: B] : ( P @ F4 @ X5 @ X_12 ) ) % 0.31/0.66 => ? [F4: A > B] : % 0.31/0.66 ! [X2: A] : ( P @ F4 @ X2 @ ( F4 @ X2 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % dependent_wellorder_choice % 0.31/0.66 thf(fact_8127_chains__extend,axiom, % 0.31/0.66 ! [A: $tType,C2: set @ ( set @ A ),S3: set @ ( set @ A ),Z2: set @ A] : % 0.31/0.66 ( ( member @ ( set @ ( set @ A ) ) @ C2 @ ( chains @ A @ S3 ) ) % 0.31/0.66 => ( ( member @ ( set @ A ) @ Z2 @ S3 ) % 0.31/0.66 => ( ! [X5: set @ A] : % 0.31/0.66 ( ( member @ ( set @ A ) @ X5 @ C2 ) % 0.31/0.66 => ( ord_less_eq @ ( set @ A ) @ X5 @ Z2 ) ) % 0.31/0.66 => ( member @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ Z2 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C2 ) @ ( chains @ A @ S3 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % chains_extend % 0.31/0.66 thf(fact_8128_bit_Oabstract__boolean__algebra__axioms,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_ri3973907225187159222ations @ A ) % 0.31/0.66 => ( boolea2506097494486148201lgebra @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bit.abstract_boolean_algebra_axioms % 0.31/0.66 thf(fact_8129_chainsD,axiom, % 0.31/0.66 ! [A: $tType,C2: set @ ( set @ A ),S3: set @ ( set @ A ),X3: set @ A,Y: set @ A] : % 0.31/0.66 ( ( member @ ( set @ ( set @ A ) ) @ C2 @ ( chains @ A @ S3 ) ) % 0.31/0.66 => ( ( member @ ( set @ A ) @ X3 @ C2 ) % 0.31/0.66 => ( ( member @ ( set @ A ) @ Y @ C2 ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ A ) @ X3 @ Y ) % 0.31/0.66 | ( ord_less_eq @ ( set @ A ) @ Y @ X3 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % chainsD % 0.31/0.66 thf(fact_8130_Zorn__Lemma2,axiom, % 0.31/0.66 ! [A: $tType,A3: set @ ( set @ A )] : % 0.31/0.66 ( ! [X5: set @ ( set @ A )] : % 0.31/0.66 ( ( member @ ( set @ ( set @ A ) ) @ X5 @ ( chains @ A @ A3 ) ) % 0.31/0.66 => ? [Xa: set @ A] : % 0.31/0.66 ( ( member @ ( set @ A ) @ Xa @ A3 ) % 0.31/0.66 & ! [Xb2: set @ A] : % 0.31/0.66 ( ( member @ ( set @ A ) @ Xb2 @ X5 ) % 0.31/0.66 => ( ord_less_eq @ ( set @ A ) @ Xb2 @ Xa ) ) ) ) % 0.31/0.66 => ? [X5: set @ A] : % 0.31/0.66 ( ( member @ ( set @ A ) @ X5 @ A3 ) % 0.31/0.66 & ! [Xa: set @ A] : % 0.31/0.66 ( ( member @ ( set @ A ) @ Xa @ A3 ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ A ) @ X5 @ Xa ) % 0.31/0.66 => ( Xa = X5 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Zorn_Lemma2 % 0.31/0.66 thf(fact_8131_Zorn__Lemma,axiom, % 0.31/0.66 ! [A: $tType,A3: set @ ( set @ A )] : % 0.31/0.66 ( ! [X5: set @ ( set @ A )] : % 0.31/0.66 ( ( member @ ( set @ ( set @ A ) ) @ X5 @ ( chains @ A @ A3 ) ) % 0.31/0.66 => ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ X5 ) @ A3 ) ) % 0.31/0.66 => ? [X5: set @ A] : % 0.31/0.66 ( ( member @ ( set @ A ) @ X5 @ A3 ) % 0.31/0.66 & ! [Xa: set @ A] : % 0.31/0.66 ( ( member @ ( set @ A ) @ Xa @ A3 ) % 0.31/0.66 => ( ( ord_less_eq @ ( set @ A ) @ X5 @ Xa ) % 0.31/0.66 => ( Xa = X5 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Zorn_Lemma % 0.31/0.66 thf(fact_8132_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( boolea8198339166811842893lgebra @ A ) % 0.31/0.66 => ( boolea2506097494486148201lgebra @ A @ ( inf_inf @ A ) @ ( sup_sup @ A ) @ ( uminus_uminus @ A ) @ ( bot_bot @ A ) @ ( top_top @ A ) ) ) ). % 0.31/0.66 % 0.31/0.66 % boolean_algebra.abstract_boolean_algebra_axioms % 0.31/0.66 thf(fact_8133_and__not__num__eq__None__iff,axiom, % 0.31/0.66 ! [M: num,N2: num] : % 0.31/0.66 ( ( ( bit_and_not_num @ M @ N2 ) % 0.31/0.66 = ( none @ num ) ) % 0.31/0.66 = ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) ) % 0.31/0.66 = ( zero_zero @ int ) ) ) ). % 0.31/0.66 % 0.31/0.66 % and_not_num_eq_None_iff % 0.31/0.66 thf(fact_8134_partial__order__on__well__order__on,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A] : % 0.31/0.66 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 ) % 0.31/0.66 => ( ( order_7125193373082350890der_on @ A @ A3 @ R2 ) % 0.31/0.66 => ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % partial_order_on_well_order_on % 0.31/0.66 thf(fact_8135_partial__order__on__empty,axiom, % 0.31/0.66 ! [A: $tType] : ( order_7125193373082350890der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % partial_order_on_empty % 0.31/0.66 thf(fact_8136_and__not__num_Osimps_I1_J,axiom, % 0.31/0.66 ( ( bit_and_not_num @ one2 @ one2 ) % 0.31/0.66 = ( none @ num ) ) ). % 0.31/0.66 % 0.31/0.66 % and_not_num.simps(1) % 0.31/0.66 thf(fact_8137_and__not__num_Osimps_I3_J,axiom, % 0.31/0.66 ! [N2: num] : % 0.31/0.66 ( ( bit_and_not_num @ one2 @ ( bit1 @ N2 ) ) % 0.31/0.66 = ( none @ num ) ) ). % 0.31/0.66 % 0.31/0.66 % and_not_num.simps(3) % 0.31/0.66 thf(fact_8138_finite__Partial__order__induct,axiom, % 0.31/0.66 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,P: A > $o] : % 0.31/0.66 ( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) % 0.31/0.66 => ( ( member @ A @ X3 @ ( field2 @ A @ R2 ) ) % 0.31/0.66 => ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 ) % 0.31/0.66 => ( ! [X5: A] : % 0.31/0.66 ( ( member @ A @ X5 @ ( field2 @ A @ R2 ) ) % 0.31/0.66 => ( ! [Y5: A] : % 0.31/0.66 ( ( member @ A @ Y5 @ ( order_aboveS @ A @ R2 @ X5 ) ) % 0.31/0.66 => ( P @ Y5 ) ) % 0.31/0.66 => ( P @ X5 ) ) ) % 0.31/0.66 => ( P @ X3 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % finite_Partial_order_induct % 0.31/0.66 thf(fact_8139_and__not__num_Oelims,axiom, % 0.31/0.66 ! [X3: num,Xa2: num,Y: option @ num] : % 0.31/0.66 ( ( ( bit_and_not_num @ X3 @ Xa2 ) % 0.31/0.66 = Y ) % 0.31/0.66 => ( ( ( X3 = one2 ) % 0.31/0.66 => ( ( Xa2 = one2 ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( none @ num ) ) ) ) % 0.31/0.66 => ( ( ( X3 = one2 ) % 0.31/0.66 => ( ? [N3: num] : % 0.31/0.66 ( Xa2 % 0.31/0.66 = ( bit0 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( some @ num @ one2 ) ) ) ) % 0.31/0.66 => ( ( ( X3 = one2 ) % 0.31/0.66 => ( ? [N3: num] : % 0.31/0.66 ( Xa2 % 0.31/0.66 = ( bit1 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( none @ num ) ) ) ) % 0.31/0.66 => ( ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit0 @ M3 ) ) % 0.31/0.66 => ( ( Xa2 = one2 ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( some @ num @ ( bit0 @ M3 ) ) ) ) ) % 0.31/0.66 => ( ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit0 @ M3 ) ) % 0.31/0.66 => ! [N3: num] : % 0.31/0.66 ( ( Xa2 % 0.31/0.66 = ( bit0 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) ) % 0.31/0.66 => ( ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit0 @ M3 ) ) % 0.31/0.66 => ! [N3: num] : % 0.31/0.66 ( ( Xa2 % 0.31/0.66 = ( bit1 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) ) % 0.31/0.66 => ( ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit1 @ M3 ) ) % 0.31/0.66 => ( ( Xa2 = one2 ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( some @ num @ ( bit0 @ M3 ) ) ) ) ) % 0.31/0.66 => ( ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit1 @ M3 ) ) % 0.31/0.66 => ! [N3: num] : % 0.31/0.66 ( ( Xa2 % 0.31/0.66 = ( bit0 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 ) % 0.31/0.66 @ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) ) % 0.31/0.66 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) ) % 0.31/0.66 => ~ ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit1 @ M3 ) ) % 0.31/0.66 => ! [N3: num] : % 0.31/0.66 ( ( Xa2 % 0.31/0.66 = ( bit1 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % and_not_num.elims % 0.31/0.66 thf(fact_8140_max__ext__eq,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( max_ext @ A ) % 0.31/0.66 = ( ^ [R6: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) % 0.31/0.66 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o % 0.31/0.66 @ ^ [X6: set @ A,Y9: set @ A] : % 0.31/0.66 ( ( finite_finite2 @ A @ X6 ) % 0.31/0.66 & ( finite_finite2 @ A @ Y9 ) % 0.31/0.66 & ( Y9 % 0.31/0.66 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.66 & ! [X: A] : % 0.31/0.66 ( ( member @ A @ X @ X6 ) % 0.31/0.66 => ? [Y2: A] : % 0.31/0.66 ( ( member @ A @ Y2 @ Y9 ) % 0.31/0.66 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R6 ) ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % max_ext_eq % 0.31/0.66 thf(fact_8141_bex__empty,axiom, % 0.31/0.66 ! [A: $tType,P: A > $o] : % 0.31/0.66 ~ ? [X2: A] : % 0.31/0.66 ( ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) % 0.31/0.66 & ( P @ X2 ) ) ). % 0.31/0.66 % 0.31/0.66 % bex_empty % 0.31/0.66 thf(fact_8142_map__option__eq__Some,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,F2: B > A,Xo: option @ B,Y: A] : % 0.31/0.66 ( ( ( map_option @ B @ A @ F2 @ Xo ) % 0.31/0.66 = ( some @ A @ Y ) ) % 0.31/0.66 = ( ? [Z6: B] : % 0.31/0.66 ( ( Xo % 0.31/0.66 = ( some @ B @ Z6 ) ) % 0.31/0.66 & ( ( F2 @ Z6 ) % 0.31/0.66 = Y ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % map_option_eq_Some % 0.31/0.66 thf(fact_8143_option_Omap__disc__iff,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,F2: A > B,A2: option @ A] : % 0.31/0.66 ( ( ( map_option @ A @ B @ F2 @ A2 ) % 0.31/0.66 = ( none @ B ) ) % 0.31/0.66 = ( A2 % 0.31/0.66 = ( none @ A ) ) ) ). % 0.31/0.66 % 0.31/0.66 % option.map_disc_iff % 0.31/0.66 thf(fact_8144_map__option__is__None,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,F2: B > A,Opt: option @ B] : % 0.31/0.66 ( ( ( map_option @ B @ A @ F2 @ Opt ) % 0.31/0.66 = ( none @ A ) ) % 0.31/0.66 = ( Opt % 0.31/0.66 = ( none @ B ) ) ) ). % 0.31/0.66 % 0.31/0.66 % map_option_is_None % 0.31/0.66 thf(fact_8145_None__eq__map__option__iff,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,F2: B > A,X3: option @ B] : % 0.31/0.66 ( ( ( none @ A ) % 0.31/0.66 = ( map_option @ B @ A @ F2 @ X3 ) ) % 0.31/0.66 = ( X3 % 0.31/0.66 = ( none @ B ) ) ) ). % 0.31/0.66 % 0.31/0.66 % None_eq_map_option_iff % 0.31/0.66 thf(fact_8146_finite__Collect__bex,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,A3: set @ A,Q: B > A > $o] : % 0.31/0.66 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.66 => ( ( finite_finite2 @ B % 0.31/0.66 @ ( collect @ B % 0.31/0.66 @ ^ [X: B] : % 0.31/0.66 ? [Y2: A] : % 0.31/0.66 ( ( member @ A @ Y2 @ A3 ) % 0.31/0.66 & ( Q @ X @ Y2 ) ) ) ) % 0.31/0.66 = ( ! [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A3 ) % 0.31/0.66 => ( finite_finite2 @ B % 0.31/0.66 @ ( collect @ B % 0.31/0.66 @ ^ [Y2: B] : ( Q @ Y2 @ X ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % finite_Collect_bex % 0.31/0.66 thf(fact_8147_map__option__o__empty,axiom, % 0.31/0.66 ! [C: $tType,B: $tType,A: $tType,F2: C > B] : % 0.31/0.66 ( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F2 ) % 0.31/0.66 @ ^ [X: A] : ( none @ C ) ) % 0.31/0.66 = ( ^ [X: A] : ( none @ B ) ) ) ). % 0.31/0.66 % 0.31/0.66 % map_option_o_empty % 0.31/0.66 thf(fact_8148_bex__UNIV,axiom, % 0.31/0.66 ! [A: $tType,P: A > $o] : % 0.31/0.66 ( ( ? [X: A] : % 0.31/0.66 ( ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) % 0.31/0.66 & ( P @ X ) ) ) % 0.31/0.66 = ( ? [X6: A] : ( P @ X6 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % bex_UNIV % 0.31/0.66 thf(fact_8149_case__map__option,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,C: $tType,G: A,H2: B > A,F2: C > B,X3: option @ C] : % 0.31/0.66 ( ( case_option @ A @ B @ G @ H2 @ ( map_option @ C @ B @ F2 @ X3 ) ) % 0.31/0.66 = ( case_option @ A @ C @ G @ ( comp @ B @ A @ C @ H2 @ F2 ) @ X3 ) ) ). % 0.31/0.66 % 0.31/0.66 % case_map_option % 0.31/0.66 thf(fact_8150_map__option__o__map__upd,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,C: $tType,F2: C > B,M: A > ( option @ C ),A2: A,B2: C] : % 0.31/0.66 ( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F2 ) @ ( fun_upd @ A @ ( option @ C ) @ M @ A2 @ ( some @ C @ B2 ) ) ) % 0.31/0.66 = ( fun_upd @ A @ ( option @ B ) @ ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F2 ) @ M ) @ A2 @ ( some @ B @ ( F2 @ B2 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % map_option_o_map_upd % 0.31/0.66 thf(fact_8151_nths__nths,axiom, % 0.31/0.66 ! [A: $tType,Xs: list @ A,A3: set @ nat,B3: set @ nat] : % 0.31/0.66 ( ( nths @ A @ ( nths @ A @ Xs @ A3 ) @ B3 ) % 0.31/0.66 = ( nths @ A @ Xs % 0.31/0.66 @ ( collect @ nat % 0.31/0.66 @ ^ [I3: nat] : % 0.31/0.66 ( ( member @ nat @ I3 @ A3 ) % 0.31/0.66 & ( member @ nat % 0.31/0.66 @ ( finite_card @ nat % 0.31/0.66 @ ( collect @ nat % 0.31/0.66 @ ^ [I10: nat] : % 0.31/0.66 ( ( member @ nat @ I10 @ A3 ) % 0.31/0.66 & ( ord_less @ nat @ I10 @ I3 ) ) ) ) % 0.31/0.66 @ B3 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % nths_nths % 0.31/0.66 thf(fact_8152_vimage__image__eq,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,F2: A > B,A3: set @ A] : % 0.31/0.66 ( ( vimage @ A @ B @ F2 @ ( image @ A @ B @ F2 @ A3 ) ) % 0.31/0.66 = ( collect @ A % 0.31/0.66 @ ^ [Y2: A] : % 0.31/0.66 ? [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A3 ) % 0.31/0.66 & ( ( F2 @ X ) % 0.31/0.66 = ( F2 @ Y2 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % vimage_image_eq % 0.31/0.66 thf(fact_8153_option_Osize__gen__o__map,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,F2: B > nat,G: A > B] : % 0.31/0.66 ( ( comp @ ( option @ B ) @ nat @ ( option @ A ) @ ( size_option @ B @ F2 ) @ ( map_option @ A @ B @ G ) ) % 0.31/0.66 = ( size_option @ A @ ( comp @ B @ nat @ A @ F2 @ G ) ) ) ). % 0.31/0.66 % 0.31/0.66 % option.size_gen_o_map % 0.31/0.66 thf(fact_8154_map__option_Ocomp,axiom, % 0.31/0.66 ! [C: $tType,B: $tType,A: $tType,F2: B > C,G: A > B] : % 0.31/0.66 ( ( comp @ ( option @ B ) @ ( option @ C ) @ ( option @ A ) @ ( map_option @ B @ C @ F2 ) @ ( map_option @ A @ B @ G ) ) % 0.31/0.66 = ( map_option @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) ) ) ). % 0.31/0.66 % 0.31/0.66 % map_option.comp % 0.31/0.66 thf(fact_8155_option_Omap__comp,axiom, % 0.31/0.66 ! [B: $tType,C: $tType,A: $tType,G: B > C,F2: A > B,V2: option @ A] : % 0.31/0.66 ( ( map_option @ B @ C @ G @ ( map_option @ A @ B @ F2 @ V2 ) ) % 0.31/0.66 = ( map_option @ A @ C @ ( comp @ B @ C @ A @ G @ F2 ) @ V2 ) ) ). % 0.31/0.66 % 0.31/0.66 % option.map_comp % 0.31/0.66 thf(fact_8156_map__option_Ocompositionality,axiom, % 0.31/0.66 ! [B: $tType,C: $tType,A: $tType,F2: B > C,G: A > B,Option: option @ A] : % 0.31/0.66 ( ( map_option @ B @ C @ F2 @ ( map_option @ A @ B @ G @ Option ) ) % 0.31/0.66 = ( map_option @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) @ Option ) ) ). % 0.31/0.66 % 0.31/0.66 % map_option.compositionality % 0.31/0.66 thf(fact_8157_option_Osimps_I8_J,axiom, % 0.31/0.66 ! [A: $tType,B: $tType,F2: A > B] : % 0.31/0.66 ( ( map_option @ A @ B @ F2 @ ( none @ A ) ) % 0.31/0.66 = ( none @ B ) ) ). % 0.31/0.66 % 0.31/0.66 % option.simps(8) % 0.31/0.66 thf(fact_8158_option_Omap__sel,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,A2: option @ A,F2: A > B] : % 0.31/0.66 ( ( A2 % 0.31/0.66 != ( none @ A ) ) % 0.31/0.66 => ( ( the2 @ B @ ( map_option @ A @ B @ F2 @ A2 ) ) % 0.31/0.66 = ( F2 @ ( the2 @ A @ A2 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % option.map_sel % 0.31/0.66 thf(fact_8159_option_Omap__id,axiom, % 0.31/0.66 ! [A: $tType,T2: option @ A] : % 0.31/0.66 ( ( map_option @ A @ A @ ( id @ A ) @ T2 ) % 0.31/0.66 = T2 ) ). % 0.31/0.66 % 0.31/0.66 % option.map_id % 0.31/0.66 thf(fact_8160_option_Omap__id0,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( map_option @ A @ A @ ( id @ A ) ) % 0.31/0.66 = ( id @ ( option @ A ) ) ) ). % 0.31/0.66 % 0.31/0.66 % option.map_id0 % 0.31/0.66 thf(fact_8161_map__option_Oidentity,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( map_option @ A @ A % 0.31/0.66 @ ^ [X: A] : X ) % 0.31/0.66 = ( id @ ( option @ A ) ) ) ). % 0.31/0.66 % 0.31/0.66 % map_option.identity % 0.31/0.66 thf(fact_8162_option_Omap__ident,axiom, % 0.31/0.66 ! [A: $tType,T2: option @ A] : % 0.31/0.66 ( ( map_option @ A @ A % 0.31/0.66 @ ^ [X: A] : X % 0.31/0.66 @ T2 ) % 0.31/0.66 = T2 ) ). % 0.31/0.66 % 0.31/0.66 % option.map_ident % 0.31/0.66 thf(fact_8163_Bex__def,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bex @ A ) % 0.31/0.66 = ( ^ [A6: set @ A,P3: A > $o] : % 0.31/0.66 ? [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A6 ) % 0.31/0.66 & ( P3 @ X ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Bex_def % 0.31/0.66 thf(fact_8164_Bex__def__raw,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bex @ A ) % 0.31/0.66 = ( ^ [A6: set @ A,P3: A > $o] : % 0.31/0.66 ? [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A6 ) % 0.31/0.66 & ( P3 @ X ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Bex_def_raw % 0.31/0.66 thf(fact_8165_map__option__cong,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,X3: option @ A,Y: option @ A,F2: A > B,G: A > B] : % 0.31/0.66 ( ( X3 = Y ) % 0.31/0.66 => ( ! [A5: A] : % 0.31/0.66 ( ( Y % 0.31/0.66 = ( some @ A @ A5 ) ) % 0.31/0.66 => ( ( F2 @ A5 ) % 0.31/0.66 = ( G @ A5 ) ) ) % 0.31/0.66 => ( ( map_option @ A @ B @ F2 @ X3 ) % 0.31/0.66 = ( map_option @ A @ B @ G @ Y ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % map_option_cong % 0.31/0.66 thf(fact_8166_option_Osimps_I9_J,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,F2: A > B,X22: A] : % 0.31/0.66 ( ( map_option @ A @ B @ F2 @ ( some @ A @ X22 ) ) % 0.31/0.66 = ( some @ B @ ( F2 @ X22 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % option.simps(9) % 0.31/0.66 thf(fact_8167_Bex__fold,axiom, % 0.31/0.66 ! [A: $tType,A3: set @ A,P: A > $o] : % 0.31/0.66 ( ( finite_finite2 @ A @ A3 ) % 0.31/0.66 => ( ( ? [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A3 ) % 0.31/0.66 & ( P @ X ) ) ) % 0.31/0.66 = ( finite_fold @ A @ $o % 0.31/0.66 @ ^ [K3: A,S8: $o] : % 0.31/0.66 ( S8 % 0.31/0.66 | ( P @ K3 ) ) % 0.31/0.66 @ $false % 0.31/0.66 @ A3 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Bex_fold % 0.31/0.66 thf(fact_8168_option_Oinj__map,axiom, % 0.31/0.66 ! [B: $tType,A: $tType,F2: A > B] : % 0.31/0.66 ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) % 0.31/0.66 => ( inj_on @ ( option @ A ) @ ( option @ B ) @ ( map_option @ A @ B @ F2 ) @ ( top_top @ ( set @ ( option @ A ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % option.inj_map % 0.31/0.66 thf(fact_8169_image__def,axiom, % 0.31/0.66 ! [B: $tType,A: $tType] : % 0.31/0.66 ( ( image @ A @ B ) % 0.31/0.66 = ( ^ [F3: A > B,A6: set @ A] : % 0.31/0.66 ( collect @ B % 0.31/0.66 @ ^ [Y2: B] : % 0.31/0.66 ? [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A6 ) % 0.31/0.66 & ( Y2 % 0.31/0.66 = ( F3 @ X ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % image_def % 0.31/0.66 thf(fact_8170_map__option__case,axiom, % 0.31/0.66 ! [A: $tType,B: $tType] : % 0.31/0.66 ( ( map_option @ B @ A ) % 0.31/0.66 = ( ^ [F3: B > A] : % 0.31/0.66 ( case_option @ ( option @ A ) @ B @ ( none @ A ) % 0.31/0.66 @ ^ [X: B] : ( some @ A @ ( F3 @ X ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % map_option_case % 0.31/0.66 thf(fact_8171_max__extp_Ocases,axiom, % 0.31/0.66 ! [A: $tType,R: A > A > $o,A12: set @ A,A23: set @ A] : % 0.31/0.66 ( ( max_extp @ A @ R @ A12 @ A23 ) % 0.31/0.66 => ~ ( ( finite_finite2 @ A @ A12 ) % 0.31/0.66 => ( ( finite_finite2 @ A @ A23 ) % 0.31/0.66 => ( ( A23 % 0.31/0.66 != ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) % 0.31/0.66 => ~ ! [X2: A] : % 0.31/0.66 ( ( member @ A @ X2 @ A12 ) % 0.31/0.66 => ? [Xa3: A] : % 0.31/0.66 ( ( member @ A @ Xa3 @ A23 ) % 0.31/0.66 & ( R @ X2 @ Xa3 ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % max_extp.cases % 0.31/0.66 thf(fact_8172_max__extp_Osimps,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( max_extp @ A ) % 0.31/0.66 = ( ^ [R6: A > A > $o,A1: set @ A,A22: set @ A] : % 0.31/0.66 ( ( finite_finite2 @ A @ A1 ) % 0.31/0.66 & ( finite_finite2 @ A @ A22 ) % 0.31/0.66 & ( A22 % 0.31/0.66 != ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) % 0.31/0.66 & ! [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A1 ) % 0.31/0.66 => ? [Y2: A] : % 0.31/0.66 ( ( member @ A @ Y2 @ A22 ) % 0.31/0.66 & ( R6 @ X @ Y2 ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % max_extp.simps % 0.31/0.66 thf(fact_8173_max__extp_Omax__extI,axiom, % 0.31/0.66 ! [A: $tType,X8: set @ A,Y7: set @ A,R: A > A > $o] : % 0.31/0.66 ( ( finite_finite2 @ A @ X8 ) % 0.31/0.66 => ( ( finite_finite2 @ A @ Y7 ) % 0.31/0.66 => ( ( Y7 % 0.31/0.66 != ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) % 0.31/0.66 => ( ! [X5: A] : % 0.31/0.66 ( ( member @ A @ X5 @ X8 ) % 0.31/0.66 => ? [Xa: A] : % 0.31/0.66 ( ( member @ A @ Xa @ Y7 ) % 0.31/0.66 & ( R @ X5 @ Xa ) ) ) % 0.31/0.66 => ( max_extp @ A @ R @ X8 @ Y7 ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % max_extp.max_extI % 0.31/0.66 thf(fact_8174_map__of__map,axiom, % 0.31/0.66 ! [B: $tType,C: $tType,A: $tType,F2: C > B,Xs: list @ ( product_prod @ A @ C )] : % 0.31/0.66 ( ( map_of @ A @ B % 0.31/0.66 @ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B ) % 0.31/0.66 @ ( product_case_prod @ A @ C @ ( product_prod @ A @ B ) % 0.31/0.66 @ ^ [K3: A,V5: C] : ( product_Pair @ A @ B @ K3 @ ( F2 @ V5 ) ) ) % 0.31/0.66 @ Xs ) ) % 0.31/0.66 = ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F2 ) @ ( map_of @ A @ C @ Xs ) ) ) ). % 0.31/0.66 % 0.31/0.66 % map_of_map % 0.31/0.66 thf(fact_8175_min__ext__def,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( min_ext @ A ) % 0.31/0.66 = ( ^ [R5: set @ ( product_prod @ A @ A )] : % 0.31/0.66 ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) % 0.31/0.66 @ ^ [Uu3: product_prod @ ( set @ A ) @ ( set @ A )] : % 0.31/0.66 ? [X6: set @ A,Y9: set @ A] : % 0.31/0.66 ( ( Uu3 % 0.31/0.66 = ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X6 @ Y9 ) ) % 0.31/0.66 & ( X6 % 0.31/0.66 != ( bot_bot @ ( set @ A ) ) ) % 0.31/0.66 & ! [X: A] : % 0.31/0.66 ( ( member @ A @ X @ Y9 ) % 0.31/0.66 => ? [Y2: A] : % 0.31/0.66 ( ( member @ A @ Y2 @ X6 ) % 0.31/0.66 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ R5 ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % min_ext_def % 0.31/0.66 thf(fact_8176_map__project__def,axiom, % 0.31/0.66 ! [B: $tType,A: $tType] : % 0.31/0.66 ( ( map_project @ A @ B ) % 0.31/0.66 = ( ^ [F3: A > ( option @ B ),A6: set @ A] : % 0.31/0.66 ( collect @ B % 0.31/0.66 @ ^ [B4: B] : % 0.31/0.66 ? [X: A] : % 0.31/0.66 ( ( member @ A @ X @ A6 ) % 0.31/0.66 & ( ( F3 @ X ) % 0.31/0.66 = ( some @ B @ B4 ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % map_project_def % 0.31/0.66 thf(fact_8177_and__num_Oelims,axiom, % 0.31/0.66 ! [X3: num,Xa2: num,Y: option @ num] : % 0.31/0.66 ( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 ) % 0.31/0.66 = Y ) % 0.31/0.66 => ( ( ( X3 = one2 ) % 0.31/0.66 => ( ( Xa2 = one2 ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( some @ num @ one2 ) ) ) ) % 0.31/0.66 => ( ( ( X3 = one2 ) % 0.31/0.66 => ( ? [N3: num] : % 0.31/0.66 ( Xa2 % 0.31/0.66 = ( bit0 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( none @ num ) ) ) ) % 0.31/0.66 => ( ( ( X3 = one2 ) % 0.31/0.66 => ( ? [N3: num] : % 0.31/0.66 ( Xa2 % 0.31/0.66 = ( bit1 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( some @ num @ one2 ) ) ) ) % 0.31/0.66 => ( ( ? [M3: num] : % 0.31/0.66 ( X3 % 0.31/0.66 = ( bit0 @ M3 ) ) % 0.31/0.66 => ( ( Xa2 = one2 ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( none @ num ) ) ) ) % 0.31/0.66 => ( ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit0 @ M3 ) ) % 0.31/0.66 => ! [N3: num] : % 0.31/0.66 ( ( Xa2 % 0.31/0.66 = ( bit0 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) ) % 0.31/0.66 => ( ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit0 @ M3 ) ) % 0.31/0.66 => ! [N3: num] : % 0.31/0.66 ( ( Xa2 % 0.31/0.66 = ( bit1 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) ) % 0.31/0.66 => ( ( ? [M3: num] : % 0.31/0.66 ( X3 % 0.31/0.66 = ( bit1 @ M3 ) ) % 0.31/0.66 => ( ( Xa2 = one2 ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( some @ num @ one2 ) ) ) ) % 0.31/0.66 => ( ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit1 @ M3 ) ) % 0.31/0.66 => ! [N3: num] : % 0.31/0.66 ( ( Xa2 % 0.31/0.66 = ( bit0 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) ) % 0.31/0.66 => ~ ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit1 @ M3 ) ) % 0.31/0.66 => ! [N3: num] : % 0.31/0.66 ( ( Xa2 % 0.31/0.66 = ( bit1 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 ) % 0.31/0.66 @ ^ [N10: num] : ( some @ num @ ( bit1 @ N10 ) ) % 0.31/0.66 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % and_num.elims % 0.31/0.66 thf(fact_8178_and__num_Osimps_I4_J,axiom, % 0.31/0.66 ! [M: num] : % 0.31/0.66 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one2 ) % 0.31/0.66 = ( none @ num ) ) ). % 0.31/0.66 % 0.31/0.66 % and_num.simps(4) % 0.31/0.66 thf(fact_8179_and__num_Osimps_I2_J,axiom, % 0.31/0.66 ! [N2: num] : % 0.31/0.66 ( ( bit_un7362597486090784418nd_num @ one2 @ ( bit0 @ N2 ) ) % 0.31/0.66 = ( none @ num ) ) ). % 0.31/0.66 % 0.31/0.66 % and_num.simps(2) % 0.31/0.66 thf(fact_8180_and__num__eq__None__iff,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_un5681908812861735899ations @ A ) % 0.31/0.66 => ! [M: num,N2: num] : % 0.31/0.66 ( ( ( bit_un7362597486090784418nd_num @ M @ N2 ) % 0.31/0.66 = ( none @ num ) ) % 0.31/0.66 = ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) % 0.31/0.66 = ( zero_zero @ A ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % and_num_eq_None_iff % 0.31/0.66 thf(fact_8181_numeral__and__num,axiom, % 0.31/0.66 ! [A: $tType] : % 0.31/0.66 ( ( bit_un5681908812861735899ations @ A ) % 0.31/0.66 => ! [M: num,N2: num] : % 0.31/0.66 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) % 0.31/0.66 = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % numeral_and_num % 0.31/0.66 thf(fact_8182_xor__num_Oelims,axiom, % 0.31/0.66 ! [X3: num,Xa2: num,Y: option @ num] : % 0.31/0.66 ( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 ) % 0.31/0.66 = Y ) % 0.31/0.66 => ( ( ( X3 = one2 ) % 0.31/0.66 => ( ( Xa2 = one2 ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( none @ num ) ) ) ) % 0.31/0.66 => ( ( ( X3 = one2 ) % 0.31/0.66 => ! [N3: num] : % 0.31/0.66 ( ( Xa2 % 0.31/0.66 = ( bit0 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( some @ num @ ( bit1 @ N3 ) ) ) ) ) % 0.31/0.66 => ( ( ( X3 = one2 ) % 0.31/0.66 => ! [N3: num] : % 0.31/0.66 ( ( Xa2 % 0.31/0.66 = ( bit1 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( some @ num @ ( bit0 @ N3 ) ) ) ) ) % 0.31/0.66 => ( ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit0 @ M3 ) ) % 0.31/0.66 => ( ( Xa2 = one2 ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( some @ num @ ( bit1 @ M3 ) ) ) ) ) % 0.31/0.66 => ( ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit0 @ M3 ) ) % 0.31/0.66 => ! [N3: num] : % 0.31/0.66 ( ( Xa2 % 0.31/0.66 = ( bit0 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) % 0.31/0.66 => ( ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit0 @ M3 ) ) % 0.31/0.66 => ! [N3: num] : % 0.31/0.66 ( ( Xa2 % 0.31/0.66 = ( bit1 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) ) % 0.31/0.66 => ( ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit1 @ M3 ) ) % 0.31/0.66 => ( ( Xa2 = one2 ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( some @ num @ ( bit0 @ M3 ) ) ) ) ) % 0.31/0.66 => ( ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit1 @ M3 ) ) % 0.31/0.66 => ! [N3: num] : % 0.31/0.66 ( ( Xa2 % 0.31/0.66 = ( bit0 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) ) % 0.31/0.66 => ~ ! [M3: num] : % 0.31/0.66 ( ( X3 % 0.31/0.66 = ( bit1 @ M3 ) ) % 0.31/0.66 => ! [N3: num] : % 0.31/0.66 ( ( Xa2 % 0.31/0.66 = ( bit1 @ N3 ) ) % 0.31/0.66 => ( Y % 0.31/0.66 != ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % xor_num.elims % 0.31/0.66 thf(fact_8183_option_Orec__o__map,axiom, % 0.31/0.66 ! [B: $tType,C: $tType,A: $tType,G: C,Ga: B > C,F2: A > B] : % 0.31/0.66 ( ( comp @ ( option @ B ) @ C @ ( option @ A ) @ ( rec_option @ C @ B @ G @ Ga ) @ ( map_option @ A @ B @ F2 ) ) % 0.31/0.66 = ( rec_option @ C @ A @ G % 0.31/0.66 @ ^ [X: A] : ( Ga @ ( F2 @ X ) ) ) ) ). % 0.31/0.66 % 0.31/0.66 % option.rec_o_map % 0.31/0.66 thf(fact_8184_option_Osimps_I6_J,axiom, % 0.31/0.66 ! [A: $tType,C: $tType,F1: C,F22: A > C] : % 0.31/0.66 ( ( rec_option @ C @ A @ F1 @ F22 @ ( none @ A ) ) % 0.31/0.66 = F1 ) ). % 0.31/0.66 % 0.31/0.66 % option.simps(6) % 0.31/0.66 thf(fact_8185_xor__num_Osimps_I1_J,axiom, % 0.31/0.66 ( ( bit_un2480387367778600638or_num @ one2 @ one2 ) % 0.31/0.66 = ( none @ num ) ) ). % 0.31/0.66 % 0.31/0.66 % xor_num.simps(1) % 0.31/0.66 % 0.31/0.66 % Type constructors (813) % 0.31/0.66 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( comple6319245703460814977attice @ A20 ) % 0.31/0.66 => ( condit1219197933456340205attice @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( counta3822494911875563373attice @ A20 ) % 0.31/0.66 => ( counta3822494911875563373attice @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( comple592849572758109894attice @ A20 ) % 0.31/0.66 => ( comple592849572758109894attice @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( bounded_lattice @ A20 ) % 0.31/0.66 => ( bounde4967611905675639751up_bot @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( bounded_lattice @ A20 ) % 0.31/0.66 => ( bounde4346867609351753570nf_top @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( comple6319245703460814977attice @ A20 ) % 0.31/0.66 => ( comple6319245703460814977attice @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( boolea8198339166811842893lgebra @ A20 ) % 0.31/0.66 => ( boolea8198339166811842893lgebra @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( bounded_lattice @ A20 ) % 0.31/0.66 => ( bounded_lattice_bot @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Complete__Partial__Order_Occpo,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( comple6319245703460814977attice @ A20 ) % 0.31/0.66 => ( comple9053668089753744459l_ccpo @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Lattices_Osemilattice__sup,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( semilattice_sup @ A20 ) % 0.31/0.66 => ( semilattice_sup @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Lattices_Osemilattice__inf,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( semilattice_inf @ A20 ) % 0.31/0.66 => ( semilattice_inf @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Lattices_Odistrib__lattice,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( distrib_lattice @ A20 ) % 0.31/0.66 => ( distrib_lattice @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Lattices_Obounded__lattice,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( bounded_lattice @ A20 ) % 0.31/0.66 => ( bounded_lattice @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Orderings_Oorder__top,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( order_top @ A20 ) % 0.31/0.66 => ( order_top @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Orderings_Oorder__bot,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( order_bot @ A20 ) % 0.31/0.66 => ( order_bot @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Orderings_Opreorder,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( preorder @ A20 ) % 0.31/0.66 => ( preorder @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Finite__Set_Ofinite,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( ( finite_finite @ A10 ) % 0.31/0.66 & ( finite_finite @ A20 ) ) % 0.31/0.66 => ( finite_finite @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Lattices_Olattice,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( lattice @ A20 ) % 0.31/0.66 => ( lattice @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Orderings_Oorder,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( order @ A20 ) % 0.31/0.66 => ( order @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Orderings_Otop,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( top @ A20 ) % 0.31/0.66 => ( top @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Orderings_Oord,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( ord @ A20 ) % 0.31/0.66 => ( ord @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Orderings_Obot,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( bot @ A20 ) % 0.31/0.66 => ( bot @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_fun___Groups_Ouminus,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( uminus @ A20 ) % 0.31/0.66 => ( uminus @ ( A10 > A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom, % 0.31/0.66 condit6923001295902523014norder @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,axiom, % 0.31/0.66 condit1219197933456340205attice @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom, % 0.31/0.66 bit_un5681908812861735899ations @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom, % 0.31/0.66 semiri1453513574482234551roduct @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom, % 0.31/0.66 euclid5411537665997757685th_nat @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom, % 0.31/0.66 ordere1937475149494474687imp_le @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom, % 0.31/0.66 euclid3128863361964157862miring @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom, % 0.31/0.66 euclid4440199948858584721cancel @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom, % 0.31/0.66 unique1627219031080169319umeral @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom, % 0.31/0.66 euclid8851590272496341667cancel @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom, % 0.31/0.66 semiri6575147826004484403cancel @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom, % 0.31/0.66 strict9044650504122735259up_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom, % 0.31/0.66 ordere580206878836729694up_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom, % 0.31/0.66 ordere2412721322843649153imp_le @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom, % 0.31/0.66 bit_se359711467146920520ations @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom, % 0.31/0.66 linord2810124833399127020strict @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom, % 0.31/0.66 strict7427464778891057005id_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom, % 0.31/0.66 ordere8940638589300402666id_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom, % 0.31/0.66 euclid3725896446679973847miring @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom, % 0.31/0.66 topolo4958980785337419405_space @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom, % 0.31/0.66 topolo1944317154257567458pology @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Topological__Spaces_Odiscrete__topology,axiom, % 0.31/0.66 topolo8865339358273720382pology @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom, % 0.31/0.66 linord715952674999750819strict @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom, % 0.31/0.66 topolo5987344860129210374id_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom, % 0.31/0.66 linord4140545234300271783up_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom, % 0.31/0.66 bit_ri3973907225187159222ations @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom, % 0.31/0.66 topolo2564578578187576103pology @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom, % 0.31/0.66 semiri2026040879449505780visors @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom, % 0.31/0.66 linord181362715937106298miring @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom, % 0.31/0.66 topolo4211221413907600880p_mult @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom, % 0.31/0.66 linord8928482502909563296strict @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom, % 0.31/0.66 semiri3467727345109120633visors @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom, % 0.31/0.66 ordere6658533253407199908up_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom, % 0.31/0.66 ordere166539214618696060dd_abs @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom, % 0.31/0.66 semiri6843258321239162965malize @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom, % 0.31/0.66 ordere6911136660526730532id_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom, % 0.31/0.66 linord5086331880401160121up_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom, % 0.31/0.66 cancel2418104881723323429up_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom, % 0.31/0.66 ring_15535105094025558882visors @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom, % 0.31/0.66 topolo6943815403480290642id_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom, % 0.31/0.66 cancel1802427076303600483id_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom, % 0.31/0.66 linord4710134922213307826strict @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom, % 0.31/0.66 comm_s4317794764714335236cancel @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom, % 0.31/0.66 bit_semiring_bits @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom, % 0.31/0.66 topological_t2_space @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Topological__Spaces_Ot1__space,axiom, % 0.31/0.66 topological_t1_space @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom, % 0.31/0.66 ordere2520102378445227354miring @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom, % 0.31/0.66 linord6961819062388156250ring_1 @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom, % 0.31/0.66 ordered_ab_group_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom, % 0.31/0.66 cancel_semigroup_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom, % 0.31/0.66 linordered_semiring @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom, % 0.31/0.66 ordered_semiring_0 @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom, % 0.31/0.66 linordered_semidom @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_2,axiom, % 0.31/0.66 semilattice_sup @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Lattices_Osemilattice__inf_3,axiom, % 0.31/0.66 semilattice_inf @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Lattices_Odistrib__lattice_4,axiom, % 0.31/0.66 distrib_lattice @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom, % 0.31/0.66 ab_semigroup_mult @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom, % 0.31/0.66 semiring_1_cancel @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom, % 0.31/0.66 algebraic_semidom @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom, % 0.31/0.66 comm_monoid_mult @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom, % 0.31/0.66 ab_semigroup_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom, % 0.31/0.66 ordered_semiring @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom, % 0.31/0.66 ordered_ring_abs @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom, % 0.31/0.66 semiring_parity @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom, % 0.31/0.66 comm_monoid_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom, % 0.31/0.66 semiring_modulo @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom, % 0.31/0.66 linordered_ring @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom, % 0.31/0.66 linordered_idom @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom, % 0.31/0.66 comm_semiring_1 @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom, % 0.31/0.66 comm_semiring_0 @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom, % 0.31/0.66 semigroup_mult @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom, % 0.31/0.66 semidom_modulo @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom, % 0.31/0.66 semidom_divide @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom, % 0.31/0.66 semiring_numeral @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom, % 0.31/0.66 semigroup_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom, % 0.31/0.66 zero_less_one @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom, % 0.31/0.66 comm_semiring @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom, % 0.31/0.66 semiring_char_0 @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Oab__group__add,axiom, % 0.31/0.66 ab_group_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom, % 0.31/0.66 zero_neq_one @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Oordered__ring,axiom, % 0.31/0.66 ordered_ring @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom, % 0.31/0.66 idom_abs_sgn @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Parity_Oring__parity,axiom, % 0.31/0.66 ring_parity @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Orderings_Opreorder_5,axiom, % 0.31/0.66 preorder @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Orderings_Olinorder,axiom, % 0.31/0.66 linorder @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom, % 0.31/0.66 monoid_mult @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Oidom__modulo,axiom, % 0.31/0.66 idom_modulo @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Oidom__divide,axiom, % 0.31/0.66 idom_divide @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom, % 0.31/0.66 comm_ring_1 @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Omonoid__add,axiom, % 0.31/0.66 monoid_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom, % 0.31/0.66 semiring_1 @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Osemiring__0,axiom, % 0.31/0.66 semiring_0 @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Orderings_Ono__top,axiom, % 0.31/0.66 no_top @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom, % 0.31/0.66 no_bot @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Lattices_Olattice_6,axiom, % 0.31/0.66 lattice @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom, % 0.31/0.66 group_add @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___GCD_Osemiring__gcd,axiom, % 0.31/0.66 semiring_gcd @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom, % 0.31/0.66 semiring_Gcd @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Omult__zero,axiom, % 0.31/0.66 mult_zero @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom, % 0.31/0.66 comm_ring @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Orderings_Oorder_7,axiom, % 0.31/0.66 order @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Num_Oneg__numeral,axiom, % 0.31/0.66 neg_numeral @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom, % 0.31/0.66 ring_char_0 @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Osemiring,axiom, % 0.31/0.66 semiring @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Osemidom,axiom, % 0.31/0.66 semidom @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Orderings_Oord_8,axiom, % 0.31/0.66 ord @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Ouminus_9,axiom, % 0.31/0.66 uminus @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Oring__1,axiom, % 0.31/0.66 ring_1 @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Oabs__if,axiom, % 0.31/0.66 abs_if @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___GCD_Oring__gcd,axiom, % 0.31/0.66 ring_gcd @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Power_Opower,axiom, % 0.31/0.66 power @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Num_Onumeral,axiom, % 0.31/0.66 numeral @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Ozero,axiom, % 0.31/0.66 zero @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Oplus,axiom, % 0.31/0.66 plus @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Oring,axiom, % 0.31/0.66 ring @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Oidom,axiom, % 0.31/0.66 idom @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Groups_Oone,axiom, % 0.31/0.66 one @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Int_Oint___Rings_Odvd,axiom, % 0.31/0.66 dvd @ int ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_10,axiom, % 0.31/0.66 condit6923001295902523014norder @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11,axiom, % 0.31/0.66 condit1219197933456340205attice @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_12,axiom, % 0.31/0.66 bit_un5681908812861735899ations @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_13,axiom, % 0.31/0.66 semiri1453513574482234551roduct @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_14,axiom, % 0.31/0.66 euclid5411537665997757685th_nat @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_15,axiom, % 0.31/0.66 ordere1937475149494474687imp_le @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_16,axiom, % 0.31/0.66 euclid3128863361964157862miring @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_17,axiom, % 0.31/0.66 euclid4440199948858584721cancel @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_18,axiom, % 0.31/0.66 unique1627219031080169319umeral @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_19,axiom, % 0.31/0.66 semiri6575147826004484403cancel @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_20,axiom, % 0.31/0.66 strict9044650504122735259up_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom, % 0.31/0.66 ordere1170586879665033532d_diff @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_21,axiom, % 0.31/0.66 ordere580206878836729694up_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_22,axiom, % 0.31/0.66 ordere2412721322843649153imp_le @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_23,axiom, % 0.31/0.66 bit_se359711467146920520ations @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_24,axiom, % 0.31/0.66 linord2810124833399127020strict @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_25,axiom, % 0.31/0.66 strict7427464778891057005id_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_26,axiom, % 0.31/0.66 ordere8940638589300402666id_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom, % 0.31/0.66 canoni5634975068530333245id_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_27,axiom, % 0.31/0.66 euclid3725896446679973847miring @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_28,axiom, % 0.31/0.66 topolo4958980785337419405_space @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_29,axiom, % 0.31/0.66 topolo1944317154257567458pology @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_30,axiom, % 0.31/0.66 topolo8865339358273720382pology @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_31,axiom, % 0.31/0.66 topolo5987344860129210374id_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_32,axiom, % 0.31/0.66 linord4140545234300271783up_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_33,axiom, % 0.31/0.66 topolo2564578578187576103pology @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_34,axiom, % 0.31/0.66 semiri2026040879449505780visors @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_35,axiom, % 0.31/0.66 linord181362715937106298miring @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_36,axiom, % 0.31/0.66 topolo4211221413907600880p_mult @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_37,axiom, % 0.31/0.66 linord8928482502909563296strict @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_38,axiom, % 0.31/0.66 semiri3467727345109120633visors @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_39,axiom, % 0.31/0.66 ordere6658533253407199908up_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_40,axiom, % 0.31/0.66 semiri6843258321239162965malize @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_41,axiom, % 0.31/0.66 ordere6911136660526730532id_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_42,axiom, % 0.31/0.66 cancel2418104881723323429up_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_43,axiom, % 0.31/0.66 topolo6943815403480290642id_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_44,axiom, % 0.31/0.66 cancel1802427076303600483id_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_45,axiom, % 0.31/0.66 comm_s4317794764714335236cancel @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_46,axiom, % 0.31/0.66 bit_semiring_bits @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_47,axiom, % 0.31/0.66 topological_t2_space @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space_48,axiom, % 0.31/0.66 topological_t1_space @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_49,axiom, % 0.31/0.66 ordere2520102378445227354miring @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_50,axiom, % 0.31/0.66 cancel_semigroup_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Olinordered__semiring_51,axiom, % 0.31/0.66 linordered_semiring @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_52,axiom, % 0.31/0.66 ordered_semiring_0 @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_53,axiom, % 0.31/0.66 linordered_semidom @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_54,axiom, % 0.31/0.66 semilattice_sup @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_55,axiom, % 0.31/0.66 semilattice_inf @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_56,axiom, % 0.31/0.66 distrib_lattice @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_57,axiom, % 0.31/0.66 ab_semigroup_mult @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_58,axiom, % 0.31/0.66 semiring_1_cancel @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_59,axiom, % 0.31/0.66 algebraic_semidom @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_60,axiom, % 0.31/0.66 comm_monoid_mult @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom, % 0.31/0.66 comm_monoid_diff @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_61,axiom, % 0.31/0.66 ab_semigroup_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Oordered__semiring_62,axiom, % 0.31/0.66 ordered_semiring @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Parity_Osemiring__parity_63,axiom, % 0.31/0.66 semiring_parity @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_64,axiom, % 0.31/0.66 comm_monoid_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Osemiring__modulo_65,axiom, % 0.31/0.66 semiring_modulo @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_66,axiom, % 0.31/0.66 comm_semiring_1 @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_67,axiom, % 0.31/0.66 comm_semiring_0 @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Osemigroup__mult_68,axiom, % 0.31/0.66 semigroup_mult @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Osemidom__modulo_69,axiom, % 0.31/0.66 semidom_modulo @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Osemidom__divide_70,axiom, % 0.31/0.66 semidom_divide @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Num_Osemiring__numeral_71,axiom, % 0.31/0.66 semiring_numeral @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Osemigroup__add_72,axiom, % 0.31/0.66 semigroup_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Ozero__less__one_73,axiom, % 0.31/0.66 zero_less_one @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Ocomm__semiring_74,axiom, % 0.31/0.66 comm_semiring @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom, % 0.31/0.66 wellorder @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Orderings_Oorder__bot_75,axiom, % 0.31/0.66 order_bot @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_76,axiom, % 0.31/0.66 semiring_char_0 @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Ozero__neq__one_77,axiom, % 0.31/0.66 zero_neq_one @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Orderings_Opreorder_78,axiom, % 0.31/0.66 preorder @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Orderings_Olinorder_79,axiom, % 0.31/0.66 linorder @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Omonoid__mult_80,axiom, % 0.31/0.66 monoid_mult @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Omonoid__add_81,axiom, % 0.31/0.66 monoid_add @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Osemiring__1_82,axiom, % 0.31/0.66 semiring_1 @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Osemiring__0_83,axiom, % 0.31/0.66 semiring_0 @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Orderings_Ono__top_84,axiom, % 0.31/0.66 no_top @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Lattices_Olattice_85,axiom, % 0.31/0.66 lattice @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___GCD_Osemiring__gcd_86,axiom, % 0.31/0.66 semiring_gcd @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_87,axiom, % 0.31/0.66 semiring_Gcd @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Omult__zero_88,axiom, % 0.31/0.66 mult_zero @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Orderings_Oorder_89,axiom, % 0.31/0.66 order @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Osemiring_90,axiom, % 0.31/0.66 semiring @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Osemidom_91,axiom, % 0.31/0.66 semidom @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Orderings_Oord_92,axiom, % 0.31/0.66 ord @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Orderings_Obot_93,axiom, % 0.31/0.66 bot @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Power_Opower_94,axiom, % 0.31/0.66 power @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Num_Onumeral_95,axiom, % 0.31/0.66 numeral @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Ozero_96,axiom, % 0.31/0.66 zero @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Oplus_97,axiom, % 0.31/0.66 plus @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Groups_Oone_98,axiom, % 0.31/0.66 one @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Rings_Odvd_99,axiom, % 0.31/0.66 dvd @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Nat_Onat___Nat_Osize,axiom, % 0.31/0.66 size @ nat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Num_Onum___Orderings_Opreorder_100,axiom, % 0.31/0.66 preorder @ num ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Num_Onum___Orderings_Olinorder_101,axiom, % 0.31/0.66 linorder @ num ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Num_Onum___Orderings_Oorder_102,axiom, % 0.31/0.66 order @ num ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Num_Onum___Orderings_Oord_103,axiom, % 0.31/0.66 ord @ num ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Num_Onum___Groups_Oplus_104,axiom, % 0.31/0.66 plus @ num ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Num_Onum___Nat_Osize_105,axiom, % 0.31/0.66 size @ num ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_106,axiom, % 0.31/0.66 semiri1453513574482234551roduct @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_107,axiom, % 0.31/0.66 ordere1937475149494474687imp_le @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_108,axiom, % 0.31/0.66 semiri6575147826004484403cancel @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_109,axiom, % 0.31/0.66 strict9044650504122735259up_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_110,axiom, % 0.31/0.66 ordere580206878836729694up_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_111,axiom, % 0.31/0.66 ordere2412721322843649153imp_le @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_112,axiom, % 0.31/0.66 linord2810124833399127020strict @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_113,axiom, % 0.31/0.66 strict7427464778891057005id_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_114,axiom, % 0.31/0.66 ordere8940638589300402666id_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom, % 0.31/0.66 archim462609752435547400_field @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_115,axiom, % 0.31/0.66 linord715952674999750819strict @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom, % 0.31/0.66 unboun7993243217541854897norder @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_116,axiom, % 0.31/0.66 linord4140545234300271783up_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_117,axiom, % 0.31/0.66 semiri2026040879449505780visors @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_118,axiom, % 0.31/0.66 linord181362715937106298miring @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_119,axiom, % 0.31/0.66 linord8928482502909563296strict @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_120,axiom, % 0.31/0.66 semiri3467727345109120633visors @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_121,axiom, % 0.31/0.66 ordere6658533253407199908up_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_122,axiom, % 0.31/0.66 ordere166539214618696060dd_abs @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom, % 0.31/0.66 archim2362893244070406136eiling @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_123,axiom, % 0.31/0.66 ordere6911136660526730532id_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_124,axiom, % 0.31/0.66 linord5086331880401160121up_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_125,axiom, % 0.31/0.66 cancel2418104881723323429up_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_126,axiom, % 0.31/0.66 ring_15535105094025558882visors @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_127,axiom, % 0.31/0.66 cancel1802427076303600483id_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_128,axiom, % 0.31/0.66 linord4710134922213307826strict @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_129,axiom, % 0.31/0.66 comm_s4317794764714335236cancel @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_130,axiom, % 0.31/0.66 ordere2520102378445227354miring @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_131,axiom, % 0.31/0.66 linord6961819062388156250ring_1 @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_132,axiom, % 0.31/0.66 ordered_ab_group_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_133,axiom, % 0.31/0.66 cancel_semigroup_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Olinordered__semiring_134,axiom, % 0.31/0.66 linordered_semiring @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_135,axiom, % 0.31/0.66 ordered_semiring_0 @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Olinordered__semidom_136,axiom, % 0.31/0.66 linordered_semidom @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom, % 0.31/0.66 dense_linorder @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_137,axiom, % 0.31/0.66 semilattice_sup @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_138,axiom, % 0.31/0.66 semilattice_inf @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_139,axiom, % 0.31/0.66 distrib_lattice @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_140,axiom, % 0.31/0.66 ab_semigroup_mult @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_141,axiom, % 0.31/0.66 semiring_1_cancel @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_142,axiom, % 0.31/0.66 comm_monoid_mult @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_143,axiom, % 0.31/0.66 ab_semigroup_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom, % 0.31/0.66 linordered_field @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Oordered__semiring_144,axiom, % 0.31/0.66 ordered_semiring @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_145,axiom, % 0.31/0.66 ordered_ring_abs @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_146,axiom, % 0.31/0.66 comm_monoid_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Olinordered__ring_147,axiom, % 0.31/0.66 linordered_ring @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Olinordered__idom_148,axiom, % 0.31/0.66 linordered_idom @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_149,axiom, % 0.31/0.66 comm_semiring_1 @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_150,axiom, % 0.31/0.66 comm_semiring_0 @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Orderings_Odense__order,axiom, % 0.31/0.66 dense_order @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Osemigroup__mult_151,axiom, % 0.31/0.66 semigroup_mult @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Osemidom__divide_152,axiom, % 0.31/0.66 semidom_divide @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Num_Osemiring__numeral_153,axiom, % 0.31/0.66 semiring_numeral @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Osemigroup__add_154,axiom, % 0.31/0.66 semigroup_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Fields_Ofield__abs__sgn,axiom, % 0.31/0.66 field_abs_sgn @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom, % 0.31/0.66 division_ring @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Ozero__less__one_155,axiom, % 0.31/0.66 zero_less_one @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Ocomm__semiring_156,axiom, % 0.31/0.66 comm_semiring @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Nat_Osemiring__char__0_157,axiom, % 0.31/0.66 semiring_char_0 @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Oab__group__add_158,axiom, % 0.31/0.66 ab_group_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom, % 0.31/0.66 field_char_0 @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Ozero__neq__one_159,axiom, % 0.31/0.66 zero_neq_one @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Oordered__ring_160,axiom, % 0.31/0.66 ordered_ring @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_161,axiom, % 0.31/0.66 idom_abs_sgn @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Orderings_Opreorder_162,axiom, % 0.31/0.66 preorder @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Orderings_Olinorder_163,axiom, % 0.31/0.66 linorder @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Omonoid__mult_164,axiom, % 0.31/0.66 monoid_mult @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Oidom__divide_165,axiom, % 0.31/0.66 idom_divide @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_166,axiom, % 0.31/0.66 comm_ring_1 @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Omonoid__add_167,axiom, % 0.31/0.66 monoid_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Osemiring__1_168,axiom, % 0.31/0.66 semiring_1 @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Osemiring__0_169,axiom, % 0.31/0.66 semiring_0 @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Orderings_Ono__top_170,axiom, % 0.31/0.66 no_top @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Orderings_Ono__bot_171,axiom, % 0.31/0.66 no_bot @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Lattices_Olattice_172,axiom, % 0.31/0.66 lattice @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Ogroup__add_173,axiom, % 0.31/0.66 group_add @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Omult__zero_174,axiom, % 0.31/0.66 mult_zero @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Ocomm__ring_175,axiom, % 0.31/0.66 comm_ring @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Orderings_Oorder_176,axiom, % 0.31/0.66 order @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Num_Oneg__numeral_177,axiom, % 0.31/0.66 neg_numeral @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Nat_Oring__char__0_178,axiom, % 0.31/0.66 ring_char_0 @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Osemiring_179,axiom, % 0.31/0.66 semiring @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Fields_Oinverse,axiom, % 0.31/0.66 inverse @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Osemidom_180,axiom, % 0.31/0.66 semidom @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Orderings_Oord_181,axiom, % 0.31/0.66 ord @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Ouminus_182,axiom, % 0.31/0.66 uminus @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Oring__1_183,axiom, % 0.31/0.66 ring_1 @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Oabs__if_184,axiom, % 0.31/0.66 abs_if @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Fields_Ofield,axiom, % 0.31/0.66 field @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Power_Opower_185,axiom, % 0.31/0.66 power @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Num_Onumeral_186,axiom, % 0.31/0.66 numeral @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Ozero_187,axiom, % 0.31/0.66 zero @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Oplus_188,axiom, % 0.31/0.66 plus @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Oring_189,axiom, % 0.31/0.66 ring @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Oidom_190,axiom, % 0.31/0.66 idom @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Groups_Oone_191,axiom, % 0.31/0.66 one @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Rat_Orat___Rings_Odvd_192,axiom, % 0.31/0.66 dvd @ rat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_193,axiom, % 0.31/0.66 ! [A10: $tType] : ( condit1219197933456340205attice @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_194,axiom, % 0.31/0.66 ! [A10: $tType] : ( counta3822494911875563373attice @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_195,axiom, % 0.31/0.66 ! [A10: $tType] : ( comple592849572758109894attice @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_196,axiom, % 0.31/0.66 ! [A10: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_197,axiom, % 0.31/0.66 ! [A10: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_198,axiom, % 0.31/0.66 ! [A10: $tType] : ( comple6319245703460814977attice @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_199,axiom, % 0.31/0.66 ! [A10: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_200,axiom, % 0.31/0.66 ! [A10: $tType] : ( bounded_lattice_bot @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_201,axiom, % 0.31/0.66 ! [A10: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_202,axiom, % 0.31/0.66 ! [A10: $tType] : ( semilattice_sup @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_203,axiom, % 0.31/0.66 ! [A10: $tType] : ( semilattice_inf @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_204,axiom, % 0.31/0.66 ! [A10: $tType] : ( distrib_lattice @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Lattices_Obounded__lattice_205,axiom, % 0.31/0.66 ! [A10: $tType] : ( bounded_lattice @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Orderings_Oorder__top_206,axiom, % 0.31/0.66 ! [A10: $tType] : ( order_top @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Orderings_Oorder__bot_207,axiom, % 0.31/0.66 ! [A10: $tType] : ( order_bot @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Orderings_Opreorder_208,axiom, % 0.31/0.66 ! [A10: $tType] : ( preorder @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Finite__Set_Ofinite_209,axiom, % 0.31/0.66 ! [A10: $tType] : % 0.31/0.66 ( ( finite_finite @ A10 ) % 0.31/0.66 => ( finite_finite @ ( set @ A10 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Lattices_Olattice_210,axiom, % 0.31/0.66 ! [A10: $tType] : ( lattice @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Orderings_Oorder_211,axiom, % 0.31/0.66 ! [A10: $tType] : ( order @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Orderings_Otop_212,axiom, % 0.31/0.66 ! [A10: $tType] : ( top @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Orderings_Oord_213,axiom, % 0.31/0.66 ! [A10: $tType] : ( ord @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Orderings_Obot_214,axiom, % 0.31/0.66 ! [A10: $tType] : ( bot @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Set_Oset___Groups_Ouminus_215,axiom, % 0.31/0.66 ! [A10: $tType] : ( uminus @ ( set @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_216,axiom, % 0.31/0.66 condit1219197933456340205attice @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_217,axiom, % 0.31/0.66 counta3822494911875563373attice @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_218,axiom, % 0.31/0.66 comple592849572758109894attice @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_219,axiom, % 0.31/0.66 topolo4958980785337419405_space @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_220,axiom, % 0.31/0.66 topolo1944317154257567458pology @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_221,axiom, % 0.31/0.66 topolo8865339358273720382pology @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_222,axiom, % 0.31/0.66 bounde4967611905675639751up_bot @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_223,axiom, % 0.31/0.66 bounde4346867609351753570nf_top @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_224,axiom, % 0.31/0.66 comple6319245703460814977attice @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_225,axiom, % 0.31/0.66 topolo2564578578187576103pology @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_226,axiom, % 0.31/0.66 boolea8198339166811842893lgebra @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_227,axiom, % 0.31/0.66 bounded_lattice_bot @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_228,axiom, % 0.31/0.66 topological_t2_space @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_229,axiom, % 0.31/0.66 topological_t1_space @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_230,axiom, % 0.31/0.66 comple9053668089753744459l_ccpo @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_231,axiom, % 0.31/0.66 semilattice_sup @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_232,axiom, % 0.31/0.66 semilattice_inf @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_233,axiom, % 0.31/0.66 distrib_lattice @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_234,axiom, % 0.31/0.66 bounded_lattice @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Orderings_Oorder__top_235,axiom, % 0.31/0.66 order_top @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Orderings_Oorder__bot_236,axiom, % 0.31/0.66 order_bot @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Orderings_Opreorder_237,axiom, % 0.31/0.66 preorder @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Orderings_Olinorder_238,axiom, % 0.31/0.66 linorder @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Finite__Set_Ofinite_239,axiom, % 0.31/0.66 finite_finite @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Lattices_Olattice_240,axiom, % 0.31/0.66 lattice @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Orderings_Oorder_241,axiom, % 0.31/0.66 order @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Orderings_Otop_242,axiom, % 0.31/0.66 top @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Orderings_Oord_243,axiom, % 0.31/0.66 ord @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Orderings_Obot_244,axiom, % 0.31/0.66 bot @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_HOL_Obool___Groups_Ouminus_245,axiom, % 0.31/0.66 uminus @ $o ). % 0.31/0.66 % 0.31/0.66 thf(tcon_List_Olist___Nat_Osize_246,axiom, % 0.31/0.66 ! [A10: $tType] : ( size @ ( list @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_247,axiom, % 0.31/0.66 condit6923001295902523014norder @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_248,axiom, % 0.31/0.66 condit1219197933456340205attice @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_249,axiom, % 0.31/0.66 semiri1453513574482234551roduct @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom, % 0.31/0.66 condit5016429287641298734tinuum @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_250,axiom, % 0.31/0.66 ordere1937475149494474687imp_le @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Topological__Spaces_Olinear__continuum__topology,axiom, % 0.31/0.66 topolo8458572112393995274pology @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Topological__Spaces_Ofirst__countable__topology,axiom, % 0.31/0.66 topolo3112930676232923870pology @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__div__algebra,axiom, % 0.31/0.66 real_V8999393235501362500lgebra @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra__1,axiom, % 0.31/0.66 real_V2822296259951069270ebra_1 @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_251,axiom, % 0.31/0.66 semiri6575147826004484403cancel @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra,axiom, % 0.31/0.66 real_V4412858255891104859lgebra @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oordered__real__vector,axiom, % 0.31/0.66 real_V5355595471888546746vector @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add_252,axiom, % 0.31/0.66 strict9044650504122735259up_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_253,axiom, % 0.31/0.66 ordere580206878836729694up_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_254,axiom, % 0.31/0.66 ordere2412721322843649153imp_le @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_255,axiom, % 0.31/0.66 linord2810124833399127020strict @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom, % 0.31/0.66 real_V822414075346904944vector @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Ostrict__ordered__comm__monoid__add_256,axiom, % 0.31/0.66 strict7427464778891057005id_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_257,axiom, % 0.31/0.66 ordere8940638589300402666id_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_258,axiom, % 0.31/0.66 topolo4958980785337419405_space @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_259,axiom, % 0.31/0.66 topolo1944317154257567458pology @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom, % 0.31/0.66 real_V3459762299906320749_field @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom, % 0.31/0.66 real_V5047593784448816457lgebra @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_260,axiom, % 0.31/0.66 archim462609752435547400_field @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_261,axiom, % 0.31/0.66 linord715952674999750819strict @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ouniformity__dist,axiom, % 0.31/0.66 real_V768167426530841204y_dist @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_262,axiom, % 0.31/0.66 unboun7993243217541854897norder @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_263,axiom, % 0.31/0.66 topolo5987344860129210374id_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_264,axiom, % 0.31/0.66 linord4140545234300271783up_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_265,axiom, % 0.31/0.66 topolo2564578578187576103pology @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_266,axiom, % 0.31/0.66 semiri2026040879449505780visors @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_267,axiom, % 0.31/0.66 linord181362715937106298miring @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1,axiom, % 0.31/0.66 real_V2191834092415804123ebra_1 @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space,axiom, % 0.31/0.66 real_V8037385150606011577_space @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Limits_Otopological__semigroup__mult_268,axiom, % 0.31/0.66 topolo4211221413907600880p_mult @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom, % 0.31/0.66 topolo7287701948861334536_space @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom, % 0.31/0.66 topolo8386298272705272623_space @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_269,axiom, % 0.31/0.66 linord8928482502909563296strict @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_270,axiom, % 0.31/0.66 semiri3467727345109120633visors @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom, % 0.31/0.66 real_V7819770556892013058_space @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom, % 0.31/0.66 topolo1287966508704411220up_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_271,axiom, % 0.31/0.66 ordere6658533253407199908up_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_272,axiom, % 0.31/0.66 ordere166539214618696060dd_abs @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_273,axiom, % 0.31/0.66 archim2362893244070406136eiling @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom, % 0.31/0.66 real_V4867850818363320053vector @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add_274,axiom, % 0.31/0.66 ordere6911136660526730532id_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_275,axiom, % 0.31/0.66 linord5086331880401160121up_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_276,axiom, % 0.31/0.66 cancel2418104881723323429up_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_277,axiom, % 0.31/0.66 ring_15535105094025558882visors @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__field,axiom, % 0.31/0.66 real_V7773925162809079976_field @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Limits_Otopological__monoid__add_278,axiom, % 0.31/0.66 topolo6943815403480290642id_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_279,axiom, % 0.31/0.66 cancel1802427076303600483id_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_280,axiom, % 0.31/0.66 linord4710134922213307826strict @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_281,axiom, % 0.31/0.66 comm_s4317794764714335236cancel @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Limits_Otopological__group__add,axiom, % 0.31/0.66 topolo1633459387980952147up_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_282,axiom, % 0.31/0.66 topological_t2_space @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Topological__Spaces_Ot1__space_283,axiom, % 0.31/0.66 topological_t1_space @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_284,axiom, % 0.31/0.66 ordere2520102378445227354miring @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_285,axiom, % 0.31/0.66 linord6961819062388156250ring_1 @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_286,axiom, % 0.31/0.66 ordered_ab_group_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_287,axiom, % 0.31/0.66 cancel_semigroup_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Olinordered__semiring_288,axiom, % 0.31/0.66 linordered_semiring @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Real__Vector__Spaces_Obanach,axiom, % 0.31/0.66 real_Vector_banach @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Oordered__semiring__0_289,axiom, % 0.31/0.66 ordered_semiring_0 @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_290,axiom, % 0.31/0.66 linordered_semidom @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Orderings_Odense__linorder_291,axiom, % 0.31/0.66 dense_linorder @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_292,axiom, % 0.31/0.66 semilattice_sup @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Lattices_Osemilattice__inf_293,axiom, % 0.31/0.66 semilattice_inf @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Lattices_Odistrib__lattice_294,axiom, % 0.31/0.66 distrib_lattice @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_295,axiom, % 0.31/0.66 ab_semigroup_mult @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_296,axiom, % 0.31/0.66 semiring_1_cancel @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_297,axiom, % 0.31/0.66 comm_monoid_mult @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_298,axiom, % 0.31/0.66 ab_semigroup_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Fields_Olinordered__field_299,axiom, % 0.31/0.66 linordered_field @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Oordered__semiring_300,axiom, % 0.31/0.66 ordered_semiring @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_301,axiom, % 0.31/0.66 ordered_ring_abs @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_302,axiom, % 0.31/0.66 comm_monoid_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Olinordered__ring_303,axiom, % 0.31/0.66 linordered_ring @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Olinordered__idom_304,axiom, % 0.31/0.66 linordered_idom @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_305,axiom, % 0.31/0.66 comm_semiring_1 @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_306,axiom, % 0.31/0.66 comm_semiring_0 @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Orderings_Odense__order_307,axiom, % 0.31/0.66 dense_order @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Osemigroup__mult_308,axiom, % 0.31/0.66 semigroup_mult @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Osemidom__divide_309,axiom, % 0.31/0.66 semidom_divide @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Num_Osemiring__numeral_310,axiom, % 0.31/0.66 semiring_numeral @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Osemigroup__add_311,axiom, % 0.31/0.66 semigroup_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Fields_Ofield__abs__sgn_312,axiom, % 0.31/0.66 field_abs_sgn @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Fields_Odivision__ring_313,axiom, % 0.31/0.66 division_ring @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Ozero__less__one_314,axiom, % 0.31/0.66 zero_less_one @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Ocomm__semiring_315,axiom, % 0.31/0.66 comm_semiring @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_316,axiom, % 0.31/0.66 semiring_char_0 @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Oab__group__add_317,axiom, % 0.31/0.66 ab_group_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Fields_Ofield__char__0_318,axiom, % 0.31/0.66 field_char_0 @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Ozero__neq__one_319,axiom, % 0.31/0.66 zero_neq_one @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Oordered__ring_320,axiom, % 0.31/0.66 ordered_ring @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_321,axiom, % 0.31/0.66 idom_abs_sgn @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Orderings_Opreorder_322,axiom, % 0.31/0.66 preorder @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Orderings_Olinorder_323,axiom, % 0.31/0.66 linorder @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Omonoid__mult_324,axiom, % 0.31/0.66 monoid_mult @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Transcendental_Oln,axiom, % 0.31/0.66 ln @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Oidom__divide_325,axiom, % 0.31/0.66 idom_divide @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_326,axiom, % 0.31/0.66 comm_ring_1 @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Omonoid__add_327,axiom, % 0.31/0.66 monoid_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Osemiring__1_328,axiom, % 0.31/0.66 semiring_1 @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Osemiring__0_329,axiom, % 0.31/0.66 semiring_0 @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Orderings_Ono__top_330,axiom, % 0.31/0.66 no_top @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Orderings_Ono__bot_331,axiom, % 0.31/0.66 no_bot @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Lattices_Olattice_332,axiom, % 0.31/0.66 lattice @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Ogroup__add_333,axiom, % 0.31/0.66 group_add @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Omult__zero_334,axiom, % 0.31/0.66 mult_zero @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Ocomm__ring_335,axiom, % 0.31/0.66 comm_ring @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Orderings_Oorder_336,axiom, % 0.31/0.66 order @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Num_Oneg__numeral_337,axiom, % 0.31/0.66 neg_numeral @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Nat_Oring__char__0_338,axiom, % 0.31/0.66 ring_char_0 @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Osemiring_339,axiom, % 0.31/0.66 semiring @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Fields_Oinverse_340,axiom, % 0.31/0.66 inverse @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Osemidom_341,axiom, % 0.31/0.66 semidom @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Orderings_Oord_342,axiom, % 0.31/0.66 ord @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Ouminus_343,axiom, % 0.31/0.66 uminus @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Oring__1_344,axiom, % 0.31/0.66 ring_1 @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Oabs__if_345,axiom, % 0.31/0.66 abs_if @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Fields_Ofield_346,axiom, % 0.31/0.66 field @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Power_Opower_347,axiom, % 0.31/0.66 power @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Num_Onumeral_348,axiom, % 0.31/0.66 numeral @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Ozero_349,axiom, % 0.31/0.66 zero @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Oplus_350,axiom, % 0.31/0.66 plus @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Oring_351,axiom, % 0.31/0.66 ring @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Oidom_352,axiom, % 0.31/0.66 idom @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Groups_Oone_353,axiom, % 0.31/0.66 one @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Real_Oreal___Rings_Odvd_354,axiom, % 0.31/0.66 dvd @ real ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_355,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( ( finite_finite @ A10 ) % 0.31/0.66 & ( finite_finite @ A20 ) ) % 0.31/0.66 => ( finite_finite @ ( sum_sum @ A10 @ A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Sum__Type_Osum___Nat_Osize_356,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : ( size @ ( sum_sum @ A10 @ A20 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_357,axiom, % 0.31/0.66 ! [A10: $tType] : ( condit1219197933456340205attice @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_358,axiom, % 0.31/0.66 ! [A10: $tType] : ( counta3822494911875563373attice @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_359,axiom, % 0.31/0.66 ! [A10: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_360,axiom, % 0.31/0.66 ! [A10: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_361,axiom, % 0.31/0.66 ! [A10: $tType] : ( comple6319245703460814977attice @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_362,axiom, % 0.31/0.66 ! [A10: $tType] : ( bounded_lattice_bot @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_363,axiom, % 0.31/0.66 ! [A10: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_364,axiom, % 0.31/0.66 ! [A10: $tType] : ( semilattice_sup @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_365,axiom, % 0.31/0.66 ! [A10: $tType] : ( semilattice_inf @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_366,axiom, % 0.31/0.66 ! [A10: $tType] : ( distrib_lattice @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_367,axiom, % 0.31/0.66 ! [A10: $tType] : ( bounded_lattice @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Orderings_Oorder__top_368,axiom, % 0.31/0.66 ! [A10: $tType] : ( order_top @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_369,axiom, % 0.31/0.66 ! [A10: $tType] : ( order_bot @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Orderings_Opreorder_370,axiom, % 0.31/0.66 ! [A10: $tType] : ( preorder @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Lattices_Olattice_371,axiom, % 0.31/0.66 ! [A10: $tType] : ( lattice @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Orderings_Oorder_372,axiom, % 0.31/0.66 ! [A10: $tType] : ( order @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Orderings_Otop_373,axiom, % 0.31/0.66 ! [A10: $tType] : ( top @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Orderings_Oord_374,axiom, % 0.31/0.66 ! [A10: $tType] : ( ord @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Filter_Ofilter___Orderings_Obot_375,axiom, % 0.31/0.66 ! [A10: $tType] : ( bot @ ( filter @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Option_Ooption___Finite__Set_Ofinite_376,axiom, % 0.31/0.66 ! [A10: $tType] : % 0.31/0.66 ( ( finite_finite @ A10 ) % 0.31/0.66 => ( finite_finite @ ( option @ A10 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Option_Ooption___Nat_Osize_377,axiom, % 0.31/0.66 ! [A10: $tType] : ( size @ ( option @ A10 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_378,axiom, % 0.31/0.66 semiri1453513574482234551roduct @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_379,axiom, % 0.31/0.66 topolo3112930676232923870pology @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_380,axiom, % 0.31/0.66 real_V8999393235501362500lgebra @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_381,axiom, % 0.31/0.66 real_V2822296259951069270ebra_1 @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_382,axiom, % 0.31/0.66 semiri6575147826004484403cancel @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_383,axiom, % 0.31/0.66 real_V4412858255891104859lgebra @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_384,axiom, % 0.31/0.66 real_V822414075346904944vector @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_385,axiom, % 0.31/0.66 topolo4958980785337419405_space @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_386,axiom, % 0.31/0.66 real_V3459762299906320749_field @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_387,axiom, % 0.31/0.66 real_V5047593784448816457lgebra @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_388,axiom, % 0.31/0.66 real_V768167426530841204y_dist @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_389,axiom, % 0.31/0.66 topolo5987344860129210374id_add @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_390,axiom, % 0.31/0.66 semiri2026040879449505780visors @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_391,axiom, % 0.31/0.66 real_V2191834092415804123ebra_1 @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_392,axiom, % 0.31/0.66 real_V8037385150606011577_space @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_393,axiom, % 0.31/0.66 topolo4211221413907600880p_mult @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_394,axiom, % 0.31/0.66 topolo7287701948861334536_space @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_395,axiom, % 0.31/0.66 topolo8386298272705272623_space @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_396,axiom, % 0.31/0.66 semiri3467727345109120633visors @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_397,axiom, % 0.31/0.66 real_V7819770556892013058_space @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_398,axiom, % 0.31/0.66 topolo1287966508704411220up_add @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_399,axiom, % 0.31/0.66 real_V4867850818363320053vector @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_400,axiom, % 0.31/0.66 cancel2418104881723323429up_add @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_401,axiom, % 0.31/0.66 ring_15535105094025558882visors @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_402,axiom, % 0.31/0.66 real_V7773925162809079976_field @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_403,axiom, % 0.31/0.66 topolo6943815403480290642id_add @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_404,axiom, % 0.31/0.66 cancel1802427076303600483id_add @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_405,axiom, % 0.31/0.66 comm_s4317794764714335236cancel @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Limits_Otopological__group__add_406,axiom, % 0.31/0.66 topolo1633459387980952147up_add @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_407,axiom, % 0.31/0.66 topological_t2_space @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_408,axiom, % 0.31/0.66 topological_t1_space @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_409,axiom, % 0.31/0.66 cancel_semigroup_add @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_410,axiom, % 0.31/0.66 real_Vector_banach @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_411,axiom, % 0.31/0.66 ab_semigroup_mult @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_412,axiom, % 0.31/0.66 semiring_1_cancel @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_413,axiom, % 0.31/0.66 comm_monoid_mult @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_414,axiom, % 0.31/0.66 ab_semigroup_add @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_415,axiom, % 0.31/0.66 comm_monoid_add @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_416,axiom, % 0.31/0.66 comm_semiring_1 @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_417,axiom, % 0.31/0.66 comm_semiring_0 @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_418,axiom, % 0.31/0.66 semigroup_mult @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_419,axiom, % 0.31/0.66 semidom_divide @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_420,axiom, % 0.31/0.66 semiring_numeral @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_421,axiom, % 0.31/0.66 semigroup_add @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_422,axiom, % 0.31/0.66 field_abs_sgn @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_423,axiom, % 0.31/0.66 division_ring @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_424,axiom, % 0.31/0.66 comm_semiring @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_425,axiom, % 0.31/0.66 semiring_char_0 @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_426,axiom, % 0.31/0.66 ab_group_add @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_427,axiom, % 0.31/0.66 field_char_0 @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_428,axiom, % 0.31/0.66 zero_neq_one @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_429,axiom, % 0.31/0.66 idom_abs_sgn @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_430,axiom, % 0.31/0.66 monoid_mult @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_431,axiom, % 0.31/0.66 idom_divide @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_432,axiom, % 0.31/0.66 comm_ring_1 @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_433,axiom, % 0.31/0.66 monoid_add @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_434,axiom, % 0.31/0.66 semiring_1 @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_435,axiom, % 0.31/0.66 semiring_0 @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_436,axiom, % 0.31/0.66 group_add @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Omult__zero_437,axiom, % 0.31/0.66 mult_zero @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_438,axiom, % 0.31/0.66 comm_ring @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_439,axiom, % 0.31/0.66 neg_numeral @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_440,axiom, % 0.31/0.66 ring_char_0 @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Osemiring_441,axiom, % 0.31/0.66 semiring @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Fields_Oinverse_442,axiom, % 0.31/0.66 inverse @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Osemidom_443,axiom, % 0.31/0.66 semidom @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Ouminus_444,axiom, % 0.31/0.66 uminus @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Oring__1_445,axiom, % 0.31/0.66 ring_1 @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Fields_Ofield_446,axiom, % 0.31/0.66 field @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Power_Opower_447,axiom, % 0.31/0.66 power @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Num_Onumeral_448,axiom, % 0.31/0.66 numeral @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Ozero_449,axiom, % 0.31/0.66 zero @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Oplus_450,axiom, % 0.31/0.66 plus @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Oring_451,axiom, % 0.31/0.66 ring @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Oidom_452,axiom, % 0.31/0.66 idom @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Groups_Oone_453,axiom, % 0.31/0.66 one @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Complex_Ocomplex___Rings_Odvd_454,axiom, % 0.31/0.66 dvd @ complex ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_455,axiom, % 0.31/0.66 condit6923001295902523014norder @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_456,axiom, % 0.31/0.66 condit1219197933456340205attice @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_457,axiom, % 0.31/0.66 counta3822494911875563373attice @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_458,axiom, % 0.31/0.66 comple592849572758109894attice @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_459,axiom, % 0.31/0.66 strict9044650504122735259up_add @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_460,axiom, % 0.31/0.66 strict7427464778891057005id_add @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_461,axiom, % 0.31/0.66 canoni5634975068530333245id_add @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_462,axiom, % 0.31/0.66 bounde4967611905675639751up_bot @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_463,axiom, % 0.31/0.66 bounde4346867609351753570nf_top @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder,axiom, % 0.31/0.66 comple5582772986160207858norder @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_464,axiom, % 0.31/0.66 linord4140545234300271783up_add @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_465,axiom, % 0.31/0.66 comple6319245703460814977attice @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_466,axiom, % 0.31/0.66 linord181362715937106298miring @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_467,axiom, % 0.31/0.66 semiri3467727345109120633visors @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_468,axiom, % 0.31/0.66 ordere6658533253407199908up_add @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_469,axiom, % 0.31/0.66 ordere6911136660526730532id_add @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_470,axiom, % 0.31/0.66 bounded_lattice_bot @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_471,axiom, % 0.31/0.66 ordere2520102378445227354miring @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_472,axiom, % 0.31/0.66 comple9053668089753744459l_ccpo @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_473,axiom, % 0.31/0.66 semilattice_sup @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_474,axiom, % 0.31/0.66 semilattice_inf @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_475,axiom, % 0.31/0.66 distrib_lattice @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_476,axiom, % 0.31/0.66 bounded_lattice @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_477,axiom, % 0.31/0.66 ab_semigroup_mult @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_478,axiom, % 0.31/0.66 comm_monoid_mult @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_479,axiom, % 0.31/0.66 ab_semigroup_add @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_480,axiom, % 0.31/0.66 ordered_semiring @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_481,axiom, % 0.31/0.66 comm_monoid_add @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_482,axiom, % 0.31/0.66 comm_semiring_1 @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_483,axiom, % 0.31/0.66 comm_semiring_0 @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_484,axiom, % 0.31/0.66 semigroup_mult @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_485,axiom, % 0.31/0.66 semiring_numeral @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_486,axiom, % 0.31/0.66 semigroup_add @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_487,axiom, % 0.31/0.66 zero_less_one @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_488,axiom, % 0.31/0.66 comm_semiring @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_489,axiom, % 0.31/0.66 wellorder @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_490,axiom, % 0.31/0.66 order_top @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_491,axiom, % 0.31/0.66 order_bot @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_492,axiom, % 0.31/0.66 semiring_char_0 @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_493,axiom, % 0.31/0.66 zero_neq_one @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_494,axiom, % 0.31/0.66 preorder @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_495,axiom, % 0.31/0.66 linorder @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_496,axiom, % 0.31/0.66 monoid_mult @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_497,axiom, % 0.31/0.66 monoid_add @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_498,axiom, % 0.31/0.66 semiring_1 @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_499,axiom, % 0.31/0.66 semiring_0 @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_500,axiom, % 0.31/0.66 lattice @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_501,axiom, % 0.31/0.66 mult_zero @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_502,axiom, % 0.31/0.66 order @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_503,axiom, % 0.31/0.66 semiring @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Orderings_Otop_504,axiom, % 0.31/0.66 top @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_505,axiom, % 0.31/0.66 ord @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Orderings_Obot_506,axiom, % 0.31/0.66 bot @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Power_Opower_507,axiom, % 0.31/0.66 power @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Num_Onumeral_508,axiom, % 0.31/0.66 numeral @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_509,axiom, % 0.31/0.66 zero @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Oplus_510,axiom, % 0.31/0.66 plus @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Groups_Oone_511,axiom, % 0.31/0.66 one @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Extended__Nat_Oenat___Rings_Odvd_512,axiom, % 0.31/0.66 dvd @ extended_enat ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_513,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( ( topolo4958980785337419405_space @ A10 ) % 0.31/0.66 & ( topolo4958980785337419405_space @ A20 ) ) % 0.31/0.66 => ( topolo4958980785337419405_space @ ( product_prod @ A10 @ A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_514,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( ( topological_t2_space @ A10 ) % 0.31/0.66 & ( topological_t2_space @ A20 ) ) % 0.31/0.66 => ( topological_t2_space @ ( product_prod @ A10 @ A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_515,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( ( topological_t1_space @ A10 ) % 0.31/0.66 & ( topological_t1_space @ A20 ) ) % 0.31/0.66 => ( topological_t1_space @ ( product_prod @ A10 @ A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_516,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : % 0.31/0.66 ( ( ( finite_finite @ A10 ) % 0.31/0.66 & ( finite_finite @ A20 ) ) % 0.31/0.66 => ( finite_finite @ ( product_prod @ A10 @ A20 ) ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Oprod___Nat_Osize_517,axiom, % 0.31/0.66 ! [A10: $tType,A20: $tType] : ( size @ ( product_prod @ A10 @ A20 ) ) ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_518,axiom, % 0.31/0.66 condit6923001295902523014norder @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_519,axiom, % 0.31/0.66 condit1219197933456340205attice @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_520,axiom, % 0.31/0.66 counta3822494911875563373attice @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_521,axiom, % 0.31/0.66 comple592849572758109894attice @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_522,axiom, % 0.31/0.66 bounde4967611905675639751up_bot @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_523,axiom, % 0.31/0.66 bounde4346867609351753570nf_top @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_524,axiom, % 0.31/0.66 comple5582772986160207858norder @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_525,axiom, % 0.31/0.66 comple6319245703460814977attice @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_526,axiom, % 0.31/0.66 boolea8198339166811842893lgebra @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_527,axiom, % 0.31/0.66 bounded_lattice_bot @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_528,axiom, % 0.31/0.66 comple9053668089753744459l_ccpo @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_529,axiom, % 0.31/0.66 semilattice_sup @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_530,axiom, % 0.31/0.66 semilattice_inf @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_531,axiom, % 0.31/0.66 distrib_lattice @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_532,axiom, % 0.31/0.66 bounded_lattice @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Orderings_Owellorder_533,axiom, % 0.31/0.66 wellorder @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_534,axiom, % 0.31/0.66 order_top @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_535,axiom, % 0.31/0.66 order_bot @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_536,axiom, % 0.31/0.66 preorder @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_537,axiom, % 0.31/0.66 linorder @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_538,axiom, % 0.31/0.66 finite_finite @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Lattices_Olattice_539,axiom, % 0.31/0.66 lattice @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Orderings_Oorder_540,axiom, % 0.31/0.66 order @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Orderings_Otop_541,axiom, % 0.31/0.66 top @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Orderings_Oord_542,axiom, % 0.31/0.66 ord @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Orderings_Obot_543,axiom, % 0.31/0.66 bot @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Product__Type_Ounit___Groups_Ouminus_544,axiom, % 0.31/0.66 uminus @ product_unit ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_545,axiom, % 0.31/0.66 bit_un5681908812861735899ations @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_546,axiom, % 0.31/0.66 semiri1453513574482234551roduct @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_547,axiom, % 0.31/0.66 euclid5411537665997757685th_nat @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_548,axiom, % 0.31/0.66 ordere1937475149494474687imp_le @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_549,axiom, % 0.31/0.66 euclid3128863361964157862miring @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_550,axiom, % 0.31/0.66 euclid4440199948858584721cancel @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_551,axiom, % 0.31/0.66 unique1627219031080169319umeral @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_552,axiom, % 0.31/0.66 euclid8851590272496341667cancel @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_553,axiom, % 0.31/0.66 semiri6575147826004484403cancel @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_554,axiom, % 0.31/0.66 strict9044650504122735259up_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_555,axiom, % 0.31/0.66 ordere580206878836729694up_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_556,axiom, % 0.31/0.66 ordere2412721322843649153imp_le @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_557,axiom, % 0.31/0.66 bit_se359711467146920520ations @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_558,axiom, % 0.31/0.66 linord2810124833399127020strict @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_559,axiom, % 0.31/0.66 strict7427464778891057005id_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_560,axiom, % 0.31/0.66 ordere8940638589300402666id_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_561,axiom, % 0.31/0.66 euclid3725896446679973847miring @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_562,axiom, % 0.31/0.66 linord715952674999750819strict @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_563,axiom, % 0.31/0.66 linord4140545234300271783up_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_564,axiom, % 0.31/0.66 bit_ri3973907225187159222ations @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_565,axiom, % 0.31/0.66 semiri2026040879449505780visors @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_566,axiom, % 0.31/0.66 linord181362715937106298miring @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_567,axiom, % 0.31/0.66 linord8928482502909563296strict @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_568,axiom, % 0.31/0.66 semiri3467727345109120633visors @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_569,axiom, % 0.31/0.66 ordere6658533253407199908up_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_570,axiom, % 0.31/0.66 ordere166539214618696060dd_abs @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_571,axiom, % 0.31/0.66 ordere6911136660526730532id_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_572,axiom, % 0.31/0.66 linord5086331880401160121up_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_573,axiom, % 0.31/0.66 cancel2418104881723323429up_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_574,axiom, % 0.31/0.66 ring_15535105094025558882visors @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_575,axiom, % 0.31/0.66 cancel1802427076303600483id_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_576,axiom, % 0.31/0.66 linord4710134922213307826strict @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_577,axiom, % 0.31/0.66 comm_s4317794764714335236cancel @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_578,axiom, % 0.31/0.66 bit_semiring_bits @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_579,axiom, % 0.31/0.66 ordere2520102378445227354miring @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_580,axiom, % 0.31/0.66 linord6961819062388156250ring_1 @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_581,axiom, % 0.31/0.66 ordered_ab_group_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_582,axiom, % 0.31/0.66 cancel_semigroup_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_583,axiom, % 0.31/0.66 linordered_semiring @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_584,axiom, % 0.31/0.66 ordered_semiring_0 @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_585,axiom, % 0.31/0.66 linordered_semidom @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_586,axiom, % 0.31/0.66 ab_semigroup_mult @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_587,axiom, % 0.31/0.66 semiring_1_cancel @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_588,axiom, % 0.31/0.66 algebraic_semidom @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_589,axiom, % 0.31/0.66 comm_monoid_mult @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_590,axiom, % 0.31/0.66 ab_semigroup_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_591,axiom, % 0.31/0.66 ordered_semiring @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_592,axiom, % 0.31/0.66 ordered_ring_abs @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_593,axiom, % 0.31/0.66 semiring_parity @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_594,axiom, % 0.31/0.66 comm_monoid_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_595,axiom, % 0.31/0.66 semiring_modulo @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_596,axiom, % 0.31/0.66 linordered_ring @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_597,axiom, % 0.31/0.66 linordered_idom @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_598,axiom, % 0.31/0.66 comm_semiring_1 @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_599,axiom, % 0.31/0.66 comm_semiring_0 @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_600,axiom, % 0.31/0.66 semigroup_mult @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_601,axiom, % 0.31/0.66 semidom_modulo @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_602,axiom, % 0.31/0.66 semidom_divide @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_603,axiom, % 0.31/0.66 semiring_numeral @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_604,axiom, % 0.31/0.66 semigroup_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_605,axiom, % 0.31/0.66 zero_less_one @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_606,axiom, % 0.31/0.66 comm_semiring @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_607,axiom, % 0.31/0.66 semiring_char_0 @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_608,axiom, % 0.31/0.66 ab_group_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_609,axiom, % 0.31/0.66 zero_neq_one @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_610,axiom, % 0.31/0.66 ordered_ring @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_611,axiom, % 0.31/0.66 idom_abs_sgn @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_612,axiom, % 0.31/0.66 ring_parity @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_613,axiom, % 0.31/0.66 preorder @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_614,axiom, % 0.31/0.66 linorder @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_615,axiom, % 0.31/0.66 monoid_mult @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_616,axiom, % 0.31/0.66 idom_modulo @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_617,axiom, % 0.31/0.66 idom_divide @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_618,axiom, % 0.31/0.66 comm_ring_1 @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_619,axiom, % 0.31/0.66 monoid_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_620,axiom, % 0.31/0.66 semiring_1 @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_621,axiom, % 0.31/0.66 semiring_0 @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_622,axiom, % 0.31/0.66 group_add @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_623,axiom, % 0.31/0.66 mult_zero @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_624,axiom, % 0.31/0.66 comm_ring @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_625,axiom, % 0.31/0.66 order @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_626,axiom, % 0.31/0.66 neg_numeral @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_627,axiom, % 0.31/0.66 ring_char_0 @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_628,axiom, % 0.31/0.66 semiring @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_629,axiom, % 0.31/0.66 semidom @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_630,axiom, % 0.31/0.66 ord @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_631,axiom, % 0.31/0.66 uminus @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_632,axiom, % 0.31/0.66 ring_1 @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_633,axiom, % 0.31/0.66 abs_if @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Power_Opower_634,axiom, % 0.31/0.66 power @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_635,axiom, % 0.31/0.66 numeral @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_636,axiom, % 0.31/0.66 zero @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_637,axiom, % 0.31/0.66 plus @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Oring_638,axiom, % 0.31/0.66 ring @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_639,axiom, % 0.31/0.66 idom @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Groups_Oone_640,axiom, % 0.31/0.66 one @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_641,axiom, % 0.31/0.66 dvd @ code_integer ). % 0.31/0.66 % 0.31/0.66 thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_642,axiom, % 0.31/0.66 size @ vEBT_VEBT ). % 0.31/0.66 % 0.31/0.66 % Helper facts (4) % 0.31/0.66 thf(help_If_3_1_T,axiom, % 0.31/0.66 ! [P: $o] : % 0.31/0.66 ( ( P = $true ) % 0.31/0.66 | ( P = $false ) ) ). % 0.31/0.66 % 0.31/0.66 thf(help_If_2_1_T,axiom, % 0.31/0.66 ! [A: $tType,X3: A,Y: A] : % 0.31/0.66 ( ( if @ A @ $false @ X3 @ Y ) % 0.31/0.66 = Y ) ). % 0.31/0.66 % 0.31/0.66 thf(help_If_1_1_T,axiom, % 0.31/0.66 ! [A: $tType,X3: A,Y: A] : % 0.31/0.66 ( ( if @ A @ $true @ X3 @ Y ) % 0.31/0.66 = X3 ) ). % 0.31/0.66 % 0.31/0.66 thf(help_fChoice_1_1_T,axiom, % 0.31/0.66 ! [A: $tType,P: A > $o] : % 0.31/0.66 ( ( P @ ( fChoice @ A @ P ) ) % 0.31/0.66 = ( ? [X6: A] : ( P @ X6 ) ) ) ). % 0.31/0.66 % 0.31/0.66 % Conjectures (2) % 0.31/0.66 thf(conj_0,hypothesis, % 0.31/0.66 ! [X2: nat] : % 0.31/0.66 ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ maxs ) @ X2 ) % 0.31/0.66 => thesis ) ). % 0.31/0.66 % 0.31/0.66 thf(conj_1,conjecture, % 0.31/0.66 thesis ). % 0.31/0.66 % 0.31/0.66 %------------------------------------------------------------------------------ % 0.31/0.66 ------------------------ % 6.41/2.68 ---- Embedded in THF --- % 6.41/2.68 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9). % 6.41/2.68 %%% Generated on Thu Feb 26 15:54:29 EST 2026 % 6.41/2.68 %%% using '$$dhol' embedding, version 1.3.0. % 6.41/2.68 %%% Logic specification used: % 6.41/2.68 %%% thf(spec, logic, $$dhol). % 6.41/2.68 % 6.41/2.68 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.0zb77TbOdX/DTF2DTF3756.p % 6.41/2.68 thf(ty_t_VEBT__Definitions_OVEBT, type, vEBT_VEBT: $tType). % 6.41/2.68 thf(ty_t_Code__Numeral_Ointeger, type, code_integer: $tType). % 6.41/2.68 thf(ty_t_Code__Evaluation_Oterm, type, code_term: $tType). % 6.41/2.68 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType). % 6.41/2.68 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))). % 6.41/2.68 thf(ty_t_Extended__Nat_Oenat, type, extended_enat: $tType). % 6.41/2.68 thf(ty_t_Complex_Ocomplex, type, complex: $tType). % 6.41/2.68 thf(ty_t_Sum__Type_Osum, type, sum_sum: ($tType > ($tType > $tType))). % 6.41/2.68 thf(ty_t_Option_Ooption, type, option: ($tType > $tType)). % 6.41/2.68 thf(ty_t_Filter_Ofilter, type, filter: ($tType > $tType)). % 6.41/2.68 thf(ty_t_Real_Oreal, type, real: $tType). % 6.41/2.68 thf(ty_t_List_Olist, type, list: ($tType > $tType)). % 6.41/2.68 thf(ty_t_Set_Oset, type, set: ($tType > $tType)). % 6.41/2.68 thf(ty_t_Rat_Orat, type, rat: $tType). % 6.41/2.68 thf(ty_t_Num_Onum, type, num: $tType). % 6.41/2.68 thf(ty_t_Nat_Onat, type, nat: $tType). % 6.41/2.68 thf(ty_t_Int_Oint, type, int: $tType). % 6.41/2.68 thf(ty_t_itself, type, itself: ($tType > $tType)). % 6.41/2.68 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Nat_Osize, type, size: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Odvd, type, dvd: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Oidom, type, idom: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Oring, type, ring: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Oplus, type, plus: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Num_Onumeral, type, numeral: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Power_Opower, type, power: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Fields_Ofield, type, field: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_GCD_Oring__gcd, type, ring_gcd: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Oabs__if, type, abs_if: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Oring__1, type, ring_1: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Ouminus, type, uminus: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Osemidom, type, semidom: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Fields_Oinverse, type, inverse: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Osemiring, type, semiring: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Nat_Oring__char__0, type, ring_char_0: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Num_Oneg__numeral, type, neg_numeral: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Ocomm__ring, type, comm_ring: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Omult__zero, type, mult_zero: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_GCD_Osemiring__Gcd, type, semiring_Gcd: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_GCD_Osemiring__gcd, type, semiring_gcd: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Osemiring__0, type, semiring_0: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Omonoid__add, type, monoid_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Ocomm__ring__1, type, comm_ring_1: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Oidom__divide, type, idom_divide: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Oidom__modulo, type, idom_modulo: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Transcendental_Oln, type, ln: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Omonoid__mult, type, monoid_mult: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Parity_Oring__parity, type, ring_parity: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Oidom__abs__sgn, type, idom_abs_sgn: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Oordered__ring, type, ordered_ring: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Ozero__neq__one, type, zero_neq_one: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Enum_Ofinite__lattice, type, finite_lattice: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Fields_Ofield__char__0, type, field_char_0: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Ocomm__semiring, type, comm_semiring: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Ozero__less__one, type, zero_less_one: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Fields_Odivision__ring, type, division_ring: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Fields_Ofield__abs__sgn, type, field_abs_sgn: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Num_Osemiring__numeral, type, semiring_numeral: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Osemidom__divide, type, semidom_divide: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Osemidom__modulo, type, semidom_modulo: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Osemigroup__mult, type, semigroup_mult: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Ocomm__semiring__0, type, comm_semiring_0: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Ocomm__semiring__1, type, comm_semiring_1: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Olinordered__ring, type, linordered_ring: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Osemiring__modulo, type, semiring_modulo: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Parity_Osemiring__parity, type, semiring_parity: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Oordered__ring__abs, type, ordered_ring_abs: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Oordered__semiring, type, ordered_semiring: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Ocomm__monoid__diff, type, comm_monoid_diff: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Ocomm__monoid__mult, type, comm_monoid_mult: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Oalgebraic__semidom, type, algebraic_semidom: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Osemiring__1__cancel, type, semiring_1_cancel: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Oab__semigroup__mult, type, ab_semigroup_mult: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Lattices_Odistrib__lattice, type, distrib_lattice: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Lattices_Osemilattice__inf, type, semilattice_inf: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Oordered__semiring__0, type, ordered_semiring_0: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Real__Vector__Spaces_Obanach, type, real_Vector_banach: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Olinordered__semiring, type, linordered_semiring: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Complete__Partial__Order_Occpo, type, comple9053668089753744459l_ccpo: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Enum_Ofinite__distrib__lattice, type, finite8700451911770168679attice: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Olinordered__semiring__1, type, linord6961819062388156250ring_1: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Oordered__comm__semiring, type, ordere2520102378445227354miring: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Topological__Spaces_Ot1__space, type, topological_t1_space: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Topological__Spaces_Ot2__space, type, topological_t2_space: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Topological__Spaces_Ot3__space, type, topological_t3_space: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Topological__Spaces_Ot4__space, type, topological_t4_space: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Bit__Operations_Osemiring__bits, type, bit_semiring_bits: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Lattices_Obounded__lattice__bot, type, bounded_lattice_bot: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Limits_Otopological__group__add, type, topolo1633459387980952147up_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Ocomm__semiring__1__cancel, type, comm_s4317794764714335236cancel: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Olinordered__ring__strict, type, linord4710134922213307826strict: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1802427076303600483id_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Limits_Otopological__monoid__add, type, topolo6943815403480290642id_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Real__Vector__Spaces_Oreal__field, type, real_V7773925162809079976_field: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Oring__1__no__zero__divisors, type, ring_15535105094025558882visors: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel2418104881723323429up_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord5086331880401160121up_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Oordered__comm__monoid__add, type, ordere6911136660526730532id_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Real__Vector__Spaces_Oreal__vector, type, real_V4867850818363320053vector: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Archimedean__Field_Ofloor__ceiling, type, archim2362893244070406136eiling: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_GCD_Osemiring__gcd__mult__normalize, type, semiri6843258321239162965malize: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Oordered__ab__group__add__abs, type, ordere166539214618696060dd_abs: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere6658533253407199908up_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Limits_Otopological__ab__group__add, type, topolo1287966508704411220up_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Real__Vector__Spaces_Ometric__space, type, real_V7819770556892013058_space: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Osemiring__no__zero__divisors, type, semiri3467727345109120633visors: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Boolean__Algebras_Oboolean__algebra, type, boolea8198339166811842893lgebra: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Olinordered__semiring__strict, type, linord8928482502909563296strict: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Topological__Spaces_Operfect__space, type, topolo8386298272705272623_space: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Topological__Spaces_Ouniform__space, type, topolo7287701948861334536_space: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Limits_Otopological__semigroup__mult, type, topolo4211221413907600880p_mult: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Real__Vector__Spaces_Ocomplete__space, type, real_V8037385150606011577_space: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Real__Vector__Spaces_Oreal__algebra__1, type, real_V2191834092415804123ebra_1: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord181362715937106298miring: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors, type, semiri2026040879449505780visors: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Topological__Spaces_Oorder__topology, type, topolo2564578578187576103pology: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Bit__Operations_Oring__bit__operations, type, bit_ri3973907225187159222ations: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple6319245703460814977attice: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Olinordered__ab__semigroup__add, type, linord4140545234300271783up_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Limits_Otopological__comm__monoid__add, type, topolo5987344860129210374id_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Orderings_Ounbounded__dense__linorder, type, unboun7993243217541854897norder: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Real__Vector__Spaces_Ouniformity__dist, type, real_V768167426530841204y_dist: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Olinordered__semiring__1__strict, type, linord715952674999750819strict: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Archimedean__Field_Oarchimedean__field, type, archim462609752435547400_field: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Complete__Lattices_Ocomplete__linorder, type, comple5582772986160207858norder: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Real__Vector__Spaces_Oreal__div__algebra, type, real_V5047593784448816457lgebra: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Lattices_Obounded__semilattice__inf__top, type, bounde4346867609351753570nf_top: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot, type, bounde4967611905675639751up_bot: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__field, type, real_V3459762299906320749_field: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Topological__Spaces_Odiscrete__topology, type, topolo8865339358273720382pology: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Topological__Spaces_Olinorder__topology, type, topolo1944317154257567458pology: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Topological__Spaces_Otopological__space, type, topolo4958980785337419405_space: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring, type, euclid3725896446679973847miring: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni5634975068530333245id_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add, type, ordere8940638589300402666id_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add, type, strict7427464778891057005id_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector, type, real_V822414075346904944vector: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Olinordered__comm__semiring__strict, type, linord2810124833399127020strict: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Bit__Operations_Osemiring__bit__operations, type, bit_se359711467146920520ations: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere2412721322843649153imp_le: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add, type, ordere580206878836729694up_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff, type, ordere1170586879665033532d_diff: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add, type, strict9044650504122735259up_add: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Real__Vector__Spaces_Oordered__real__vector, type, real_V5355595471888546746vector: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra, type, real_V4412858255891104859lgebra: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel, type, semiri6575147826004484403cancel: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel, type, euclid8851590272496341667cancel: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra__1, type, real_V2822296259951069270ebra_1: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral, type, unique1627219031080169319umeral: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice, type, comple592849572758109894attice: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__div__algebra, type, real_V8999393235501362500lgebra: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Topological__Spaces_Ofirst__countable__topology, type, topolo3112930676232923870pology: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel, type, euclid4440199948858584721cancel: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring, type, euclid3128863361964157862miring: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Topological__Spaces_Olinear__continuum__topology, type, topolo8458572112393995274pology: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le, type, ordere1937475149494474687imp_le: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit5016429287641298734tinuum: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat, type, euclid5411537665997757685th_nat: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Countable__Complete__Lattices_Ocountable__complete__lattice, type, counta3822494911875563373attice: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, type, semiri1453513574482234551roduct: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations, type, bit_un5681908812861735899ations: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice, type, condit1219197933456340205attice: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit6923001295902523014norder: (!> [A:$tType]: ($o))). % 6.41/2.68 thf(sy_c_Archimedean__Field_Oceiling, type, archimedean_ceiling: (!> [A:$tType]: ((A > int)))). % 6.41/2.68 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor, type, archim6421214686448440834_floor: (!> [A:$tType]: ((A > int)))). % 6.41/2.68 thf(sy_c_Archimedean__Field_Ofrac, type, archimedean_frac: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Archimedean__Field_Oround, type, archimedean_round: (!> [A:$tType]: ((A > int)))). % 6.41/2.68 thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal, type, bNF_Ca7293521722713021262ofinal: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))). % 6.41/2.68 thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq, type, bNF_Ca8665028551170535155natLeq: (set @ ((product_prod @ nat) @ nat))). % 6.41/2.68 thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess, type, bNF_Ca8459412986667044542atLess: (set @ ((product_prod @ nat) @ nat))). % 6.41/2.68 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca3754400796208372196lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))). % 6.41/2.68 thf(sy_c_BNF__Def_OGr, type, bNF_Gr: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (set @ ((product_prod @ A) @ B))))))). % 6.41/2.68 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2, type, bNF_Greatest_image2: (!> [C:$tType,A:$tType,B:$tType]: (((set @ C) > ((C > A) > ((C > B) > (set @ ((product_prod @ A) @ B)))))))). % 6.41/2.68 thf(sy_c_BNF__Wellorder__Constructions_OFunc, type, bNF_Wellorder_Func: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ (A > B))))))). % 6.41/2.68 thf(sy_c_BNF__Wellorder__Constructions_OFunc__map, type, bNF_We4925052301507509544nc_map: (!> [B:$tType,C:$tType,A:$tType,D:$tType]: (((set @ B) > ((C > A) > ((B > D) > ((D > C) > (B > A)))))))). % 6.41/2.68 thf(sy_c_BNF__Wellorder__Constructions_Obsqr, type, bNF_Wellorder_bsqr: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A))))))). % 6.41/2.68 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2, type, bNF_We1388413361240627857o_max2: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (A > A)))))). % 6.41/2.68 thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod, type, basic_BNF_size_prod: (!> [A:$tType,B:$tType]: (((A > nat) > ((B > nat) > (((product_prod @ A) @ B) > nat)))))). % 6.41/2.68 thf(sy_c_Binomial_Obinomial, type, binomial: (nat > (nat > nat))). % 6.41/2.68 thf(sy_c_Binomial_Ogbinomial, type, gbinomial: (!> [A:$tType]: ((A > (nat > A))))). % 6.41/2.68 thf(sy_c_Bit__Operations_Oand__int__rel, type, bit_and_int_rel: (((product_prod @ int) @ int) > (((product_prod @ int) @ int) > $o))). % 6.41/2.68 thf(sy_c_Bit__Operations_Oand__not__num, type, bit_and_not_num: (num > (num > (option @ num)))). % 6.41/2.68 thf(sy_c_Bit__Operations_Oconcat__bit, type, bit_concat_bit: (nat > (int > (int > int)))). % 6.41/2.68 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot, type, bit_ri4277139882892585799ns_not: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit, type, bit_ri4674362597316999326ke_bit: (!> [A:$tType]: ((nat > (A > A))))). % 6.41/2.68 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand, type, bit_se5824344872417868541ns_and: (!> [A:$tType]: ((A > (A > A))))). % 6.41/2.68 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit, type, bit_se4197421643247451524op_bit: (!> [A:$tType]: ((nat > (A > A))))). % 6.41/2.68 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit, type, bit_se8732182000553998342ip_bit: (!> [A:$tType]: ((nat > (A > A))))). % 6.41/2.68 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask, type, bit_se2239418461657761734s_mask: (!> [A:$tType]: ((nat > A)))). % 6.41/2.68 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor, type, bit_se1065995026697491101ons_or: (!> [A:$tType]: ((A > (A > A))))). % 6.41/2.68 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit, type, bit_se4730199178511100633sh_bit: (!> [A:$tType]: ((nat > (A > A))))). % 6.41/2.68 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit, type, bit_se5668285175392031749et_bit: (!> [A:$tType]: ((nat > (A > A))))). % 6.41/2.68 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit, type, bit_se2584673776208193580ke_bit: (!> [A:$tType]: ((nat > (A > A))))). % 6.41/2.68 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit, type, bit_se2638667681897837118et_bit: (!> [A:$tType]: ((nat > (A > A))))). % 6.41/2.68 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor, type, bit_se5824344971392196577ns_xor: (!> [A:$tType]: ((A > (A > A))))). % 6.41/2.68 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit, type, bit_se5641148757651400278ts_bit: (!> [A:$tType]: ((A > (nat > $o))))). % 6.41/2.68 thf(sy_c_Bit__Operations_Otake__bit__num, type, bit_take_bit_num: (nat > (num > (option @ num)))). % 6.41/2.68 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num, type, bit_un7362597486090784418nd_num: (num > (num > (option @ num)))). % 6.41/2.68 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num, type, bit_un2480387367778600638or_num: (num > (num > (option @ num)))). % 6.41/2.68 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra, type, boolea2506097494486148201lgebra: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > ((A > A) > (A > (A > $o)))))))). % 6.41/2.68 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff, type, boolea3799213064322606851m_diff: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > ((A > A) > (A > (A > ((A > (A > A)) > $o))))))))). % 6.41/2.68 thf(sy_c_Code__Numeral_Obit__cut__integer, type, code_bit_cut_integer: (code_integer > ((product_prod @ code_integer) @ $o))). % 6.41/2.68 thf(sy_c_Code__Numeral_Odivmod__abs, type, code_divmod_abs: (code_integer > (code_integer > ((product_prod @ code_integer) @ code_integer)))). % 6.41/2.68 thf(sy_c_Code__Numeral_Odivmod__integer, type, code_divmod_integer: (code_integer > (code_integer > ((product_prod @ code_integer) @ code_integer)))). % 6.41/2.68 thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer, type, code_int_of_integer: (code_integer > int)). % 6.41/2.68 thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int, type, code_integer_of_int: (int > code_integer)). % 6.41/2.68 thf(sy_c_Code__Numeral_Onat__of__integer, type, code_nat_of_integer: (code_integer > nat)). % 6.41/2.68 thf(sy_c_Complete__Lattices_OInf__class_OInf, type, complete_Inf_Inf: (!> [A:$tType]: (((set @ A) > A)))). % 6.41/2.68 thf(sy_c_Complete__Lattices_OSup__class_OSup, type, complete_Sup_Sup: (!> [A:$tType]: (((set @ A) > A)))). % 6.41/2.68 thf(sy_c_Complete__Partial__Order_Ochain, type, comple1602240252501008431_chain: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))). % 6.41/2.68 thf(sy_c_Complex_OArg, type, arg: (complex > real)). % 6.41/2.68 thf(sy_c_Complex_Ocis, type, cis: (real > complex)). % 6.41/2.68 thf(sy_c_Complex_Ocomplex_OComplex, type, complex2: (real > (real > complex))). % 6.41/2.68 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above, type, condit941137186595557371_above: (!> [A:$tType]: (((set @ A) > $o)))). % 6.41/2.68 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below, type, condit1013018076250108175_below: (!> [A:$tType]: (((set @ A) > $o)))). % 6.41/2.68 thf(sy_c_Countable__Set_Oto__nat__on, type, countable_to_nat_on: (!> [A:$tType]: (((set @ A) > (A > nat))))). % 6.41/2.68 thf(sy_c_Deriv_Odifferentiable, type, differentiable: (!> [A:$tType,B:$tType]: (((A > B) > ((filter @ A) > $o))))). % 6.41/2.68 thf(sy_c_Deriv_Ohas__derivative, type, has_derivative: (!> [A:$tType,B:$tType]: (((A > B) > ((A > B) > ((filter @ A) > $o)))))). % 6.41/2.68 thf(sy_c_Deriv_Ohas__field__derivative, type, has_field_derivative: (!> [A:$tType]: (((A > A) > (A > ((filter @ A) > $o)))))). % 6.41/2.68 thf(sy_c_Divides_Oadjust__div, type, adjust_div: (((product_prod @ int) @ int) > int)). % 6.41/2.68 thf(sy_c_Divides_Odivmod__nat, type, divmod_nat: (nat > (nat > ((product_prod @ nat) @ nat)))). % 6.41/2.68 thf(sy_c_Divides_Oeucl__rel__int, type, eucl_rel_int: (int > (int > (((product_prod @ int) @ int) > $o)))). % 6.41/2.68 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux, type, unique5940410009612947441es_aux: (!> [A:$tType]: ((((product_prod @ A) @ A) > $o)))). % 6.41/2.68 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod, type, unique8689654367752047608divmod: (!> [A:$tType]: ((num > (num > ((product_prod @ A) @ A)))))). % 6.41/2.68 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step, type, unique1321980374590559556d_step: (!> [A:$tType]: ((num > (((product_prod @ A) @ A) > ((product_prod @ A) @ A)))))). % 6.41/2.68 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer, type, comm_s3205402744901411588hammer: (!> [A:$tType]: ((A > (nat > A))))). % 6.41/2.68 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact, type, semiring_char_0_fact: (!> [A:$tType]: ((nat > A)))). % 6.41/2.68 thf(sy_c_Fields_Oinverse__class_Oinverse, type, inverse_inverse: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Filter_Oat__bot, type, at_bot: (!> [A:$tType]: ((filter @ A)))). % 6.41/2.68 thf(sy_c_Filter_Oat__top, type, at_top: (!> [A:$tType]: ((filter @ A)))). % 6.41/2.68 thf(sy_c_Filter_Oeventually, type, eventually: (!> [A:$tType]: (((A > $o) > ((filter @ A) > $o))))). % 6.41/2.68 thf(sy_c_Filter_Ofiltercomap, type, filtercomap: (!> [A:$tType,B:$tType]: (((A > B) > ((filter @ B) > (filter @ A)))))). % 6.41/2.68 thf(sy_c_Filter_Ofilterlim, type, filterlim: (!> [A:$tType,B:$tType]: (((A > B) > ((filter @ B) > ((filter @ A) > $o)))))). % 6.41/2.68 thf(sy_c_Filter_Ofiltermap, type, filtermap: (!> [A:$tType,B:$tType]: (((A > B) > ((filter @ A) > (filter @ B)))))). % 6.41/2.68 thf(sy_c_Filter_Ofinite__subsets__at__top, type, finite5375528669736107172at_top: (!> [A:$tType]: (((set @ A) > (filter @ (set @ A)))))). % 6.41/2.68 thf(sy_c_Filter_Omap__filter__on, type, map_filter_on: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > ((filter @ A) > (filter @ B))))))). % 6.41/2.68 thf(sy_c_Filter_Oprincipal, type, principal: (!> [A:$tType]: (((set @ A) > (filter @ A))))). % 6.41/2.68 thf(sy_c_Filter_Oprod__filter, type, prod_filter: (!> [A:$tType,B:$tType]: (((filter @ A) > ((filter @ B) > (filter @ ((product_prod @ A) @ B))))))). % 6.41/2.68 thf(sy_c_Finite__Set_OFpow, type, finite_Fpow: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))). % 6.41/2.68 thf(sy_c_Finite__Set_Ocard, type, finite_card: (!> [B:$tType]: (((set @ B) > nat)))). % 6.41/2.68 thf(sy_c_Finite__Set_Ocomp__fun__commute, type, finite6289374366891150609ommute: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))). % 6.41/2.68 thf(sy_c_Finite__Set_Ocomp__fun__commute__on, type, finite4664212375090638736ute_on: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (B > B)) > $o))))). % 6.41/2.68 thf(sy_c_Finite__Set_Ocomp__fun__idem__on, type, finite673082921795544331dem_on: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (B > B)) > $o))))). % 6.41/2.68 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))). % 6.41/2.68 thf(sy_c_Finite__Set_Ofold, type, finite_fold: (!> [A:$tType,B:$tType]: (((A > (B > B)) > (B > ((set @ A) > B)))))). % 6.41/2.68 thf(sy_c_Finite__Set_Ofolding__idem__on, type, finite1890593828518410140dem_on: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (B > B)) > $o))))). % 6.41/2.68 thf(sy_c_Finite__Set_Ofolding__on, type, finite_folding_on: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (B > B)) > $o))))). % 6.41/2.68 thf(sy_c_Finite__Set_Ofolding__on_OF, type, finite_folding_F: (!> [A:$tType,B:$tType]: (((A > (B > B)) > (B > ((set @ A) > B)))))). % 6.41/2.68 thf(sy_c_Fun_Obij__betw, type, bij_betw: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > ((set @ B) > $o)))))). % 6.41/2.68 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))). % 6.41/2.68 thf(sy_c_Fun_Ofun__upd, type, fun_upd: (!> [A:$tType,B:$tType]: (((A > B) > (A > (B > (A > B))))))). % 6.41/2.68 thf(sy_c_Fun_Oid, type, id: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Fun_Oinj__on, type, inj_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))). % 6.41/2.68 thf(sy_c_Fun_Omap__fun, type, map_fun: (!> [C:$tType,A:$tType,B:$tType,D:$tType]: (((C > A) > ((B > D) > ((A > B) > (C > D))))))). % 6.41/2.68 thf(sy_c_Fun_Ostrict__mono__on, type, strict_mono_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))). % 6.41/2.68 thf(sy_c_Fun_Othe__inv__into, type, the_inv_into: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (B > A)))))). % 6.41/2.68 thf(sy_c_Fun__Def_Ois__measure, type, fun_is_measure: (!> [A:$tType]: (((A > nat) > $o)))). % 6.41/2.68 thf(sy_c_Fun__Def_Omax__strict, type, fun_max_strict: (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))). % 6.41/2.68 thf(sy_c_Fun__Def_Omax__weak, type, fun_max_weak: (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))). % 6.41/2.68 thf(sy_c_Fun__Def_Omin__strict, type, fun_min_strict: (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))). % 6.41/2.68 thf(sy_c_Fun__Def_Omin__weak, type, fun_min_weak: (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))). % 6.41/2.68 thf(sy_c_Fun__Def_Opair__leq, type, fun_pair_leq: (set @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)))). % 6.41/2.68 thf(sy_c_Fun__Def_Opair__less, type, fun_pair_less: (set @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)))). % 6.41/2.68 thf(sy_c_Fun__Def_Oreduction__pair, type, fun_reduction_pair: (!> [A:$tType]: ((((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) > $o)))). % 6.41/2.68 thf(sy_c_GCD_OGcd__class_OGcd, type, gcd_Gcd: (!> [A:$tType]: (((set @ A) > A)))). % 6.41/2.68 thf(sy_c_GCD_Obezw, type, bezw: (nat > (nat > ((product_prod @ int) @ int)))). % 6.41/2.68 thf(sy_c_GCD_Obezw__rel, type, bezw_rel: (((product_prod @ nat) @ nat) > (((product_prod @ nat) @ nat) > $o))). % 6.41/2.68 thf(sy_c_GCD_Ogcd__class_Ogcd, type, gcd_gcd: (!> [A:$tType]: ((A > (A > A))))). % 6.41/2.68 thf(sy_c_GCD_Ogcd__nat__rel, type, gcd_nat_rel: (((product_prod @ nat) @ nat) > (((product_prod @ nat) @ nat) > $o))). % 6.41/2.68 thf(sy_c_GCD_Osemiring__1__class_Osemiring__char, type, semiri4206861660011772517g_char: (!> [A:$tType]: (((itself @ A) > nat)))). % 6.41/2.68 thf(sy_c_Groups_Oabs__class_Oabs, type, abs_abs: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))). % 6.41/2.68 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))). % 6.41/2.68 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))). % 6.41/2.68 thf(sy_c_Groups_Osgn__class_Osgn, type, sgn_sgn: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Groups_Otimes__class_Otimes, type, times_times: (!> [A:$tType]: ((A > (A > A))))). % 6.41/2.68 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))). % 6.41/2.68 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum, type, groups7311177749621191930dd_sum: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > A))))). % 6.41/2.68 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H, type, groups1027152243600224163dd_sum: (!> [C:$tType,A:$tType]: (((C > A) > ((set @ C) > A))))). % 6.41/2.68 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod, type, groups7121269368397514597t_prod: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > A))))). % 6.41/2.68 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H, type, groups1962203154675924110t_prod: (!> [C:$tType,A:$tType]: (((C > A) > ((set @ C) > A))))). % 6.41/2.68 thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum, type, groups4207007520872428315er_sum: (!> [B:$tType,A:$tType]: (((B > A) > (A > ((list @ B) > A)))))). % 6.41/2.68 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list, type, groups8242544230860333062m_list: (!> [A:$tType]: (((list @ A) > A)))). % 6.41/2.68 thf(sy_c_HOL_ONO__MATCH, type, nO_MATCH: (!> [A:$tType,B:$tType]: ((A > (B > $o))))). % 6.41/2.68 thf(sy_c_HOL_OThe, type, the: (!> [A:$tType]: (((A > $o) > A)))). % 6.41/2.68 thf(sy_c_HOL_Oundefined, type, undefined: (!> [A:$tType]: (A))). % 6.41/2.68 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))). % 6.41/2.68 thf(sy_c_Int_OAbs__Integ, type, abs_Integ: (((product_prod @ nat) @ nat) > int)). % 6.41/2.68 thf(sy_c_Int_ORep__Integ, type, rep_Integ: (int > ((product_prod @ nat) @ nat))). % 6.41/2.68 thf(sy_c_Int_Oint__ge__less__than, type, int_ge_less_than: (int > (set @ ((product_prod @ int) @ int)))). % 6.41/2.68 thf(sy_c_Int_Oint__ge__less__than2, type, int_ge_less_than2: (int > (set @ ((product_prod @ int) @ int)))). % 6.41/2.68 thf(sy_c_Int_Onat, type, nat2: (int > nat)). % 6.41/2.68 thf(sy_c_Int_Opower__int, type, power_int: (!> [A:$tType]: ((A > (int > A))))). % 6.41/2.68 thf(sy_c_Int_Oring__1__class_OInts, type, ring_1_Ints: (!> [A:$tType]: ((set @ A)))). % 6.41/2.68 thf(sy_c_Int_Oring__1__class_Oof__int, type, ring_1_of_int: (!> [A:$tType]: ((int > A)))). % 6.41/2.68 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))). % 6.41/2.68 thf(sy_c_Lattices_Osemilattice__neutr__order, type, semila1105856199041335345_order: (!> [A:$tType]: (((A > (A > A)) > (A > ((A > (A > $o)) > ((A > (A > $o)) > $o))))))). % 6.41/2.68 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))). % 6.41/2.68 thf(sy_c_Lattices__Big_Olinorder_OMax, type, lattices_Max: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > A))))). % 6.41/2.68 thf(sy_c_Lattices__Big_Olinorder_OMin, type, lattices_Min: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > A))))). % 6.41/2.68 thf(sy_c_Lattices__Big_Olinorder__class_OMax, type, lattic643756798349783984er_Max: (!> [A:$tType]: (((set @ A) > A)))). % 6.41/2.68 thf(sy_c_Lattices__Big_Olinorder__class_OMin, type, lattic643756798350308766er_Min: (!> [A:$tType]: (((set @ A) > A)))). % 6.41/2.68 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on, type, lattic7623131987881927897min_on: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > B))))). % 6.41/2.68 thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin, type, lattic7752659483105999362nf_fin: (!> [A:$tType]: (((set @ A) > A)))). % 6.41/2.68 thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin, type, lattic5882676163264333800up_fin: (!> [A:$tType]: (((set @ A) > A)))). % 6.41/2.68 thf(sy_c_Limits_OBfun, type, bfun: (!> [A:$tType,B:$tType]: (((A > B) > ((filter @ A) > $o))))). % 6.41/2.68 thf(sy_c_Limits_Oat__infinity, type, at_infinity: (!> [A:$tType]: ((filter @ A)))). % 6.41/2.68 thf(sy_c_List_Oappend, type, append: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))). % 6.41/2.68 thf(sy_c_List_Oarg__min__list, type, arg_min_list: (!> [A:$tType,B:$tType]: (((A > B) > ((list @ A) > A))))). % 6.41/2.68 thf(sy_c_List_Obind, type, bind: (!> [A:$tType,B:$tType]: (((list @ A) > ((A > (list @ B)) > (list @ B)))))). % 6.41/2.68 thf(sy_c_List_Oconcat, type, concat: (!> [A:$tType]: (((list @ (list @ A)) > (list @ A))))). % 6.41/2.68 thf(sy_c_List_Ocount__list, type, count_list: (!> [A:$tType]: (((list @ A) > (A > nat))))). % 6.41/2.68 thf(sy_c_List_Odistinct, type, distinct: (!> [A:$tType]: (((list @ A) > $o)))). % 6.41/2.68 thf(sy_c_List_Odrop, type, drop: (!> [A:$tType]: ((nat > ((list @ A) > (list @ A)))))). % 6.41/2.68 thf(sy_c_List_OdropWhile, type, dropWhile: (!> [A:$tType]: (((A > $o) > ((list @ A) > (list @ A)))))). % 6.41/2.68 thf(sy_c_List_Oenumerate, type, enumerate: (!> [A:$tType]: ((nat > ((list @ A) > (list @ ((product_prod @ nat) @ A))))))). % 6.41/2.68 thf(sy_c_List_Oextract, type, extract: (!> [A:$tType]: (((A > $o) > ((list @ A) > (option @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))))))))). % 6.41/2.68 thf(sy_c_List_Ofilter, type, filter2: (!> [A:$tType]: (((A > $o) > ((list @ A) > (list @ A)))))). % 6.41/2.68 thf(sy_c_List_Ofind, type, find: (!> [A:$tType]: (((A > $o) > ((list @ A) > (option @ A)))))). % 6.41/2.68 thf(sy_c_List_Ofolding__insort__key, type, folding_insort_key: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > ((set @ B) > ((B > A) > $o))))))). % 6.41/2.68 thf(sy_c_List_Ofoldl, type, foldl: (!> [B:$tType,A:$tType]: (((B > (A > B)) > (B > ((list @ A) > B)))))). % 6.41/2.68 thf(sy_c_List_Ofoldr, type, foldr: (!> [A:$tType,B:$tType]: (((A > (B > B)) > ((list @ A) > (B > B)))))). % 6.41/2.68 thf(sy_c_List_Ogen__length, type, gen_length: (!> [A:$tType]: ((nat > ((list @ A) > nat))))). % 6.41/2.68 thf(sy_c_List_Olenlex, type, lenlex: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (list @ A)) @ (list @ A))))))). % 6.41/2.68 thf(sy_c_List_Olex, type, lex: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (list @ A)) @ (list @ A))))))). % 6.41/2.68 thf(sy_c_List_Olexn, type, lexn: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (nat > (set @ ((product_prod @ (list @ A)) @ (list @ A)))))))). % 6.41/2.68 thf(sy_c_List_Olexord, type, lexord: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (list @ A)) @ (list @ A))))))). % 6.41/2.68 thf(sy_c_List_Olexordp, type, lexordp: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > ((list @ A) > $o)))))). % 6.41/2.68 thf(sy_c_List_Olinorder_Oinsort__key, type, insort_key: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((B > A) > (B > ((list @ B) > (list @ B)))))))). % 6.41/2.68 thf(sy_c_List_Olinorder_Osorted__key__list__of__set, type, sorted8670434370408473282of_set: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((B > A) > ((set @ B) > (list @ B))))))). % 6.41/2.68 thf(sy_c_List_Olinorder__class_Oinsort__key, type, linorder_insort_key: (!> [B:$tType,A:$tType]: (((B > A) > (B > ((list @ B) > (list @ B))))))). % 6.41/2.68 thf(sy_c_List_Olinorder__class_Osorted__key__list__of__set, type, linord144544945434240204of_set: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > (list @ B)))))). % 6.41/2.68 thf(sy_c_List_Olinorder__class_Osorted__list__of__set, type, linord4507533701916653071of_set: (!> [A:$tType]: (((set @ A) > (list @ A))))). % 6.41/2.68 thf(sy_c_List_Olist_OCons, type, cons: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))). % 6.41/2.68 thf(sy_c_List_Olist_ONil, type, nil: (!> [A:$tType]: ((list @ A)))). % 6.41/2.68 thf(sy_c_List_Olist_Ocase__list, type, case_list: (!> [B:$tType,A:$tType]: ((B > ((A > ((list @ A) > B)) > ((list @ A) > B)))))). % 6.41/2.68 thf(sy_c_List_Olist_Ohd, type, hd: (!> [A:$tType]: (((list @ A) > A)))). % 6.41/2.68 thf(sy_c_List_Olist_Omap, type, map: (!> [A:$tType,Aa:$tType]: (((A > Aa) > ((list @ A) > (list @ Aa)))))). % 6.41/2.68 thf(sy_c_List_Olist_Oset, type, set2: (!> [A:$tType]: (((list @ A) > (set @ A))))). % 6.41/2.68 thf(sy_c_List_Olist_Osize__list, type, size_list: (!> [A:$tType]: (((A > nat) > ((list @ A) > nat))))). % 6.41/2.68 thf(sy_c_List_Olist_Otl, type, tl: (!> [A:$tType]: (((list @ A) > (list @ A))))). % 6.41/2.68 thf(sy_c_List_Olist__update, type, list_update: (!> [A:$tType]: (((list @ A) > (nat > (A > (list @ A))))))). % 6.41/2.68 thf(sy_c_List_Olistrel, type, listrel: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ (list @ A)) @ (list @ B))))))). % 6.41/2.68 thf(sy_c_List_Olistrel1, type, listrel1: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (list @ A)) @ (list @ A))))))). % 6.41/2.68 thf(sy_c_List_Olistrel1p, type, listrel1p: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > ((list @ A) > $o)))))). % 6.41/2.68 thf(sy_c_List_Olistrelp, type, listrelp: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((list @ A) > ((list @ B) > $o)))))). % 6.41/2.68 thf(sy_c_List_Olistset, type, listset: (!> [A:$tType]: (((list @ (set @ A)) > (set @ (list @ A)))))). % 6.41/2.68 thf(sy_c_List_Omap__filter, type, map_filter: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > ((list @ A) > (list @ B)))))). % 6.41/2.68 thf(sy_c_List_Omap__project, type, map_project: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > ((set @ A) > (set @ B)))))). % 6.41/2.68 thf(sy_c_List_Omin__list, type, min_list: (!> [A:$tType]: (((list @ A) > A)))). % 6.41/2.68 thf(sy_c_List_Omin__list__rel, type, min_list_rel: (!> [A:$tType]: (((list @ A) > ((list @ A) > $o))))). % 6.41/2.68 thf(sy_c_List_On__lists, type, n_lists: (!> [A:$tType]: ((nat > ((list @ A) > (list @ (list @ A))))))). % 6.41/2.68 thf(sy_c_List_Onth, type, nth: (!> [A:$tType]: (((list @ A) > (nat > A))))). % 6.41/2.68 thf(sy_c_List_Onths, type, nths: (!> [A:$tType]: (((list @ A) > ((set @ nat) > (list @ A)))))). % 6.41/2.68 thf(sy_c_List_Opartition, type, partition: (!> [A:$tType]: (((A > $o) > ((list @ A) > ((product_prod @ (list @ A)) @ (list @ A))))))). % 6.41/2.68 thf(sy_c_List_Oproduct, type, product: (!> [A:$tType,B:$tType]: (((list @ A) > ((list @ B) > (list @ ((product_prod @ A) @ B))))))). % 6.41/2.68 thf(sy_c_List_Oproduct__lists, type, product_lists: (!> [A:$tType]: (((list @ (list @ A)) > (list @ (list @ A)))))). % 6.41/2.68 thf(sy_c_List_Oremdups, type, remdups: (!> [A:$tType]: (((list @ A) > (list @ A))))). % 6.41/2.68 thf(sy_c_List_Oremdups__adj, type, remdups_adj: (!> [A:$tType]: (((list @ A) > (list @ A))))). % 6.41/2.68 thf(sy_c_List_Oremdups__adj__rel, type, remdups_adj_rel: (!> [A:$tType]: (((list @ A) > ((list @ A) > $o))))). % 6.41/2.68 thf(sy_c_List_Oremove1, type, remove1: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))). % 6.41/2.68 thf(sy_c_List_OremoveAll, type, removeAll: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))). % 6.41/2.68 thf(sy_c_List_Oreplicate, type, replicate: (!> [A:$tType]: ((nat > (A > (list @ A)))))). % 6.41/2.68 thf(sy_c_List_Orev, type, rev: (!> [A:$tType]: (((list @ A) > (list @ A))))). % 6.41/2.68 thf(sy_c_List_Orotate, type, rotate: (!> [A:$tType]: ((nat > ((list @ A) > (list @ A)))))). % 6.41/2.68 thf(sy_c_List_Orotate1, type, rotate1: (!> [A:$tType]: (((list @ A) > (list @ A))))). % 6.41/2.68 thf(sy_c_List_Oset__Cons, type, set_Cons: (!> [A:$tType]: (((set @ A) > ((set @ (list @ A)) > (set @ (list @ A))))))). % 6.41/2.68 thf(sy_c_List_Oshuffles, type, shuffles: (!> [A:$tType]: (((list @ A) > ((list @ A) > (set @ (list @ A))))))). % 6.41/2.68 thf(sy_c_List_Osorted__wrt, type, sorted_wrt: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > $o))))). % 6.41/2.68 thf(sy_c_List_Osubseqs, type, subseqs: (!> [A:$tType]: (((list @ A) > (list @ (list @ A)))))). % 6.41/2.68 thf(sy_c_List_Otake, type, take: (!> [A:$tType]: ((nat > ((list @ A) > (list @ A)))))). % 6.41/2.68 thf(sy_c_List_OtakeWhile, type, takeWhile: (!> [A:$tType]: (((A > $o) > ((list @ A) > (list @ A)))))). % 6.41/2.68 thf(sy_c_List_Otranspose, type, transpose: (!> [A:$tType]: (((list @ (list @ A)) > (list @ (list @ A)))))). % 6.41/2.68 thf(sy_c_List_Otranspose__rel, type, transpose_rel: (!> [A:$tType]: (((list @ (list @ A)) > ((list @ (list @ A)) > $o))))). % 6.41/2.68 thf(sy_c_List_Ounion, type, union: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))). % 6.41/2.68 thf(sy_c_List_Oupt, type, upt: (nat > (nat > (list @ nat)))). % 6.41/2.68 thf(sy_c_List_Oupto, type, upto: (int > (int > (list @ int)))). % 6.41/2.68 thf(sy_c_List_Oupto__aux, type, upto_aux: (int > (int > ((list @ int) > (list @ int))))). % 6.41/2.68 thf(sy_c_List_Oupto__rel, type, upto_rel: (((product_prod @ int) @ int) > (((product_prod @ int) @ int) > $o))). % 6.41/2.68 thf(sy_c_List_Ozip, type, zip: (!> [A:$tType,B:$tType]: (((list @ A) > ((list @ B) > (list @ ((product_prod @ A) @ B))))))). % 6.41/2.68 thf(sy_c_Map_Odom, type, dom: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > (set @ A))))). % 6.41/2.68 thf(sy_c_Map_Ograph, type, graph: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > (set @ ((product_prod @ A) @ B)))))). % 6.41/2.68 thf(sy_c_Map_Omap__of, type, map_of: (!> [A:$tType,B:$tType]: (((list @ ((product_prod @ A) @ B)) > (A > (option @ B)))))). % 6.41/2.68 thf(sy_c_Map_Omap__upds, type, map_upds: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > ((list @ A) > ((list @ B) > (A > (option @ B)))))))). % 6.41/2.68 thf(sy_c_Map_Oran, type, ran: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > (set @ B))))). % 6.41/2.68 thf(sy_c_Map_Orestrict__map, type, restrict_map: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > ((set @ A) > (A > (option @ B))))))). % 6.41/2.68 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)). % 6.41/2.68 thf(sy_c_Nat_Ocompow, type, compow: (!> [A:$tType]: ((nat > (A > A))))). % 6.41/2.68 thf(sy_c_Nat_Ofunpow, type, funpow: (!> [A:$tType]: ((nat > ((A > A) > (A > A)))))). % 6.41/2.68 thf(sy_c_Nat_Onat_Ocase__nat, type, case_nat: (!> [A:$tType]: ((A > ((nat > A) > (nat > A)))))). % 6.41/2.68 thf(sy_c_Nat_Onat_Opred, type, pred: (nat > nat)). % 6.41/2.68 thf(sy_c_Nat_Oold_Onat_Orec__nat, type, rec_nat: (!> [T:$tType]: ((T > ((nat > (T > T)) > (nat > T)))))). % 6.41/2.68 thf(sy_c_Nat_Oold_Onat_Orec__set__nat, type, rec_set_nat: (!> [T:$tType]: ((T > ((nat > (T > T)) > (nat > (T > $o))))))). % 6.41/2.68 thf(sy_c_Nat_Osemiring__1__class_ONats, type, semiring_1_Nats: (!> [A:$tType]: ((set @ A)))). % 6.41/2.68 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))). % 6.41/2.68 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux, type, semiri8178284476397505188at_aux: (!> [A:$tType]: (((A > A) > (nat > (A > A)))))). % 6.41/2.68 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))). % 6.41/2.68 thf(sy_c_Nat__Bijection_Olist__encode, type, nat_list_encode: ((list @ nat) > nat)). % 6.41/2.68 thf(sy_c_Nat__Bijection_Olist__encode__rel, type, nat_list_encode_rel: ((list @ nat) > ((list @ nat) > $o))). % 6.41/2.68 thf(sy_c_Nat__Bijection_Oprod__decode__aux, type, nat_prod_decode_aux: (nat > (nat > ((product_prod @ nat) @ nat)))). % 6.41/2.68 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel, type, nat_pr5047031295181774490ux_rel: (((product_prod @ nat) @ nat) > (((product_prod @ nat) @ nat) > $o))). % 6.41/2.68 thf(sy_c_Nat__Bijection_Oprod__encode, type, nat_prod_encode: (((product_prod @ nat) @ nat) > nat)). % 6.41/2.68 thf(sy_c_Nat__Bijection_Oset__decode, type, nat_set_decode: (nat > (set @ nat))). % 6.41/2.68 thf(sy_c_Nat__Bijection_Oset__encode, type, nat_set_encode: ((set @ nat) > nat)). % 6.41/2.68 thf(sy_c_Nat__Bijection_Otriangle, type, nat_triangle: (nat > nat)). % 6.41/2.68 thf(sy_c_NthRoot_Oroot, type, root: (nat > (real > real))). % 6.41/2.68 thf(sy_c_NthRoot_Osqrt, type, sqrt: (real > real)). % 6.41/2.68 thf(sy_c_Num_OBitM, type, bitM: (num > num)). % 6.41/2.68 thf(sy_c_Num_Oinc, type, inc: (num > num)). % 6.41/2.68 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec, type, neg_numeral_dbl_dec: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Num_Oneg__numeral__class_Osub, type, neg_numeral_sub: (!> [A:$tType]: ((num > (num > A))))). % 6.41/2.68 thf(sy_c_Num_Onum_OBit0, type, bit0: (num > num)). % 6.41/2.68 thf(sy_c_Num_Onum_OBit1, type, bit1: (num > num)). % 6.41/2.68 thf(sy_c_Num_Onum_OOne, type, one2: num). % 6.41/2.68 thf(sy_c_Num_Onum_Ocase__num, type, case_num: (!> [A:$tType]: ((A > ((num > A) > ((num > A) > (num > A))))))). % 6.41/2.68 thf(sy_c_Num_Onum_Osize__num, type, size_num: (num > nat)). % 6.41/2.68 thf(sy_c_Num_Onum__of__nat, type, num_of_nat: (nat > num)). % 6.41/2.68 thf(sy_c_Num_Onumeral__class_Onumeral, type, numeral_numeral: (!> [A:$tType]: ((num > A)))). % 6.41/2.68 thf(sy_c_Num_Opow, type, pow: (num > (num > num))). % 6.41/2.68 thf(sy_c_Num_Opred__numeral, type, pred_numeral: (num > nat)). % 6.41/2.68 thf(sy_c_Option_Ooption_ONone, type, none: (!> [A:$tType]: ((option @ A)))). % 6.41/2.68 thf(sy_c_Option_Ooption_OSome, type, some: (!> [A:$tType]: ((A > (option @ A))))). % 6.41/2.68 thf(sy_c_Option_Ooption_Ocase__option, type, case_option: (!> [B:$tType,A:$tType]: ((B > ((A > B) > ((option @ A) > B)))))). % 6.41/2.68 thf(sy_c_Option_Ooption_Omap__option, type, map_option: (!> [A:$tType,Aa:$tType]: (((A > Aa) > ((option @ A) > (option @ Aa)))))). % 6.41/2.68 thf(sy_c_Option_Ooption_Orec__option, type, rec_option: (!> [C:$tType,A:$tType]: ((C > ((A > C) > ((option @ A) > C)))))). % 6.41/2.68 thf(sy_c_Option_Ooption_Osize__option, type, size_option: (!> [A:$tType]: (((A > nat) > ((option @ A) > nat))))). % 6.41/2.68 thf(sy_c_Option_Ooption_Othe, type, the2: (!> [A:$tType]: (((option @ A) > A)))). % 6.41/2.68 thf(sy_c_Option_Othese, type, these: (!> [A:$tType]: (((set @ (option @ A)) > (set @ A))))). % 6.41/2.68 thf(sy_c_Order__Continuity_Ocountable__complete__lattice__class_Occlfp, type, order_532582986084564980_cclfp: (!> [A:$tType]: (((A > A) > A)))). % 6.41/2.68 thf(sy_c_Order__Relation_OAbove, type, order_Above: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ A) > (set @ A)))))). % 6.41/2.68 thf(sy_c_Order__Relation_OUnder, type, order_Under: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ A) > (set @ A)))))). % 6.41/2.68 thf(sy_c_Order__Relation_OUnderS, type, order_UnderS: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ A) > (set @ A)))))). % 6.41/2.68 thf(sy_c_Order__Relation_OaboveS, type, order_aboveS: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))). % 6.41/2.68 thf(sy_c_Order__Relation_Olinear__order__on, type, order_679001287576687338der_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))). % 6.41/2.68 thf(sy_c_Order__Relation_Opartial__order__on, type, order_7125193373082350890der_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))). % 6.41/2.68 thf(sy_c_Order__Relation_Ounder, type, order_under: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))). % 6.41/2.68 thf(sy_c_Order__Relation_OunderS, type, order_underS: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))). % 6.41/2.68 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))). % 6.41/2.68 thf(sy_c_Orderings_Oord_OLeast, type, least: (!> [A:$tType]: (((A > (A > $o)) > ((A > $o) > A))))). % 6.41/2.68 thf(sy_c_Orderings_Oord_Omax, type, max: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > A)))))). % 6.41/2.68 thf(sy_c_Orderings_Oord_Omin, type, min: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > A)))))). % 6.41/2.68 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))). % 6.41/2.68 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))). % 6.41/2.68 thf(sy_c_Orderings_Oord__class_Omax, type, ord_max: (!> [A:$tType]: ((A > (A > A))))). % 6.41/2.68 thf(sy_c_Orderings_Oord__class_Omin, type, ord_min: (!> [A:$tType]: ((A > (A > A))))). % 6.41/2.68 thf(sy_c_Orderings_Oorder__class_OGreatest, type, order_Greatest: (!> [A:$tType]: (((A > $o) > A)))). % 6.41/2.68 thf(sy_c_Orderings_Oorder__class_Oantimono, type, order_antimono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))). % 6.41/2.68 thf(sy_c_Orderings_Oorder__class_Omono, type, order_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))). % 6.41/2.68 thf(sy_c_Orderings_Oorder__class_Ostrict__mono, type, order_strict_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))). % 6.41/2.68 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))). % 6.41/2.68 thf(sy_c_Partial__Function_Oflat__lub, type, partial_flat_lub: (!> [A:$tType]: ((A > ((set @ A) > A))))). % 6.41/2.68 thf(sy_c_Power_Opower__class_Opower, type, power_power: (!> [A:$tType]: ((A > (nat > A))))). % 6.41/2.68 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))). % 6.41/2.68 thf(sy_c_Product__Type_OSigma, type, product_Sigma: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ ((product_prod @ A) @ B))))))). % 6.41/2.68 thf(sy_c_Product__Type_Oapfst, type, product_apfst: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > (((product_prod @ A) @ B) > ((product_prod @ C) @ B)))))). % 6.41/2.68 thf(sy_c_Product__Type_Oapsnd, type, product_apsnd: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > (((product_prod @ A) @ B) > ((product_prod @ A) @ C)))))). % 6.41/2.68 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))). % 6.41/2.68 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod, type, product_rec_set_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > (T > $o)))))). % 6.41/2.68 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))). % 6.41/2.68 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))). % 6.41/2.68 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))). % 6.41/2.68 thf(sy_c_Product__Type_Oproduct, type, product_product: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ ((product_prod @ A) @ B))))))). % 6.41/2.68 thf(sy_c_Rat_OFract, type, fract: (int > (int > rat))). % 6.41/2.68 thf(sy_c_Rat_OFrct, type, frct: (((product_prod @ int) @ int) > rat)). % 6.41/2.68 thf(sy_c_Rat_ORep__Rat, type, rep_Rat: (rat > ((product_prod @ int) @ int))). % 6.41/2.68 thf(sy_c_Rat_Ofield__char__0__class_ORats, type, field_char_0_Rats: (!> [A:$tType]: ((set @ A)))). % 6.41/2.68 thf(sy_c_Rat_Onormalize, type, normalize: (((product_prod @ int) @ int) > ((product_prod @ int) @ int))). % 6.41/2.68 thf(sy_c_Rat_Oof__int, type, of_int: (int > rat)). % 6.41/2.68 thf(sy_c_Rat_Opositive, type, positive: (rat > $o)). % 6.41/2.68 thf(sy_c_Rat_Oquotient__of, type, quotient_of: (rat > ((product_prod @ int) @ int))). % 6.41/2.68 thf(sy_c_Real__Vector__Spaces_Obounded__linear, type, real_V3181309239436604168linear: (!> [A:$tType,B:$tType]: (((A > B) > $o)))). % 6.41/2.68 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist, type, real_V557655796197034286t_dist: (!> [A:$tType]: ((A > (A > real))))). % 6.41/2.68 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm, type, real_V7770717601297561774m_norm: (!> [A:$tType]: ((A > real)))). % 6.41/2.68 thf(sy_c_Real__Vector__Spaces_Oof__real, type, real_Vector_of_real: (!> [A:$tType]: ((real > A)))). % 6.41/2.68 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR, type, real_V8093663219630862766scaleR: (!> [A:$tType]: ((real > (A > A))))). % 6.41/2.68 thf(sy_c_Relation_OField, type, field2: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ A))))). % 6.41/2.68 thf(sy_c_Relation_OId, type, id2: (!> [A:$tType]: ((set @ ((product_prod @ A) @ A))))). % 6.41/2.68 thf(sy_c_Relation_OId__on, type, id_on: (!> [A:$tType]: (((set @ A) > (set @ ((product_prod @ A) @ A)))))). % 6.41/2.68 thf(sy_c_Relation_Oirrefl, type, irrefl: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))). % 6.41/2.68 thf(sy_c_Relation_Orefl__on, type, refl_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))). % 6.41/2.68 thf(sy_c_Relation_Orelcomp, type, relcomp: (!> [A:$tType,B:$tType,C:$tType]: (((set @ ((product_prod @ A) @ B)) > ((set @ ((product_prod @ B) @ C)) > (set @ ((product_prod @ A) @ C))))))). % 6.41/2.68 thf(sy_c_Relation_Ototal__on, type, total_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))). % 6.41/2.68 thf(sy_c_Rings_Odivide__class_Odivide, type, divide_divide: (!> [A:$tType]: ((A > (A > A))))). % 6.41/2.68 thf(sy_c_Rings_Odvd__class_Odvd, type, dvd_dvd: (!> [A:$tType]: ((A > (A > $o))))). % 6.41/2.68 thf(sy_c_Rings_Omodulo__class_Omodulo, type, modulo_modulo: (!> [A:$tType]: ((A > (A > A))))). % 6.41/2.68 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool, type, zero_neq_one_of_bool: (!> [A:$tType]: (($o > A)))). % 6.41/2.68 thf(sy_c_Series_Osuminf, type, suminf: (!> [A:$tType]: (((nat > A) > A)))). % 6.41/2.68 thf(sy_c_Series_Osummable, type, summable: (!> [A:$tType]: (((nat > A) > $o)))). % 6.41/2.68 thf(sy_c_Series_Osums, type, sums: (!> [A:$tType]: (((nat > A) > (A > $o))))). % 6.41/2.68 thf(sy_c_Set_OBall, type, ball: (!> [A:$tType]: (((set @ A) > ((A > $o) > $o))))). % 6.41/2.68 thf(sy_c_Set_OBex, type, bex: (!> [A:$tType]: (((set @ A) > ((A > $o) > $o))))). % 6.41/2.68 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))). % 6.41/2.68 thf(sy_c_Set_OPow, type, pow2: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))). % 6.41/2.68 thf(sy_c_Set_Ofilter, type, filter3: (!> [A:$tType]: (((A > $o) > ((set @ A) > (set @ A)))))). % 6.41/2.68 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))). % 6.41/2.68 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))). % 6.41/2.68 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))). % 6.41/2.68 thf(sy_c_Set_Oremove, type, remove: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))). % 6.41/2.68 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))). % 6.41/2.68 thf(sy_c_Set_Ovimage, type, vimage: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ B) > (set @ A)))))). % 6.41/2.68 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat, type, set_fo6178422350223883121st_nat: (!> [A:$tType]: (((nat > (A > A)) > (nat > (nat > (A > A))))))). % 6.41/2.68 thf(sy_c_Set__Interval_Oord__class_OatLeast, type, set_ord_atLeast: (!> [A:$tType]: ((A > (set @ A))))). % 6.41/2.68 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost, type, set_or1337092689740270186AtMost: (!> [A:$tType]: ((A > (A > (set @ A)))))). % 6.41/2.68 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan, type, set_or7035219750837199246ssThan: (!> [A:$tType]: ((A > (A > (set @ A)))))). % 6.41/2.68 thf(sy_c_Set__Interval_Oord__class_OatMost, type, set_ord_atMost: (!> [A:$tType]: ((A > (set @ A))))). % 6.41/2.68 thf(sy_c_Set__Interval_Oord__class_OgreaterThan, type, set_ord_greaterThan: (!> [A:$tType]: ((A > (set @ A))))). % 6.41/2.68 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost, type, set_or3652927894154168847AtMost: (!> [A:$tType]: ((A > (A > (set @ A)))))). % 6.41/2.68 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan, type, set_or5935395276787703475ssThan: (!> [A:$tType]: ((A > (A > (set @ A)))))). % 6.41/2.68 thf(sy_c_Set__Interval_Oord__class_OlessThan, type, set_ord_lessThan: (!> [A:$tType]: ((A > (set @ A))))). % 6.41/2.68 thf(sy_c_Topological__Spaces_Ocontinuous, type, topolo3448309680560233919inuous: (!> [A:$tType,B:$tType]: (((filter @ A) > ((A > B) > $o))))). % 6.41/2.68 thf(sy_c_Topological__Spaces_Ocontinuous__on, type, topolo81223032696312382ous_on: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > $o))))). % 6.41/2.68 thf(sy_c_Topological__Spaces_Omonoseq, type, topological_monoseq: (!> [A:$tType]: (((nat > A) > $o)))). % 6.41/2.68 thf(sy_c_Topological__Spaces_Oopen__class_Oopen, type, topolo1002775350975398744n_open: (!> [A:$tType]: (((set @ A) > $o)))). % 6.41/2.68 thf(sy_c_Topological__Spaces_Ot2__space__class_OLim, type, topolo3827282254853284352ce_Lim: (!> [F:$tType,A:$tType]: (((filter @ F) > ((F > A) > A))))). % 6.41/2.68 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within, type, topolo174197925503356063within: (!> [A:$tType]: ((A > ((set @ A) > (filter @ A)))))). % 6.41/2.68 thf(sy_c_Topological__Spaces_Otopological__space__class_Oclosed, type, topolo7761053866217962861closed: (!> [A:$tType]: (((set @ A) > $o)))). % 6.41/2.68 thf(sy_c_Topological__Spaces_Otopological__space__class_Ocompact, type, topolo2193935891317330818ompact: (!> [A:$tType]: (((set @ A) > $o)))). % 6.41/2.68 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds, type, topolo7230453075368039082e_nhds: (!> [A:$tType]: ((A > (filter @ A))))). % 6.41/2.68 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy, type, topolo3814608138187158403Cauchy: (!> [A:$tType]: (((nat > A) > $o)))). % 6.41/2.68 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter, type, topolo6773858410816713723filter: (!> [A:$tType]: (((filter @ A) > $o)))). % 6.41/2.68 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocomplete, type, topolo2479028161051973599mplete: (!> [A:$tType]: (((set @ A) > $o)))). % 6.41/2.68 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded, type, topolo6688025880775521714ounded: (!> [A:$tType]: (((set @ A) > $o)))). % 6.41/2.68 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity, type, topolo7806501430040627800ormity: (!> [A:$tType]: ((filter @ ((product_prod @ A) @ A))))). % 6.41/2.68 thf(sy_c_Topological__Spaces_Ouniformly__continuous__on, type, topolo6026614971017936543ous_on: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > $o))))). % 6.41/2.68 thf(sy_c_Transcendental_Oarccos, type, arccos: (real > real)). % 6.41/2.68 thf(sy_c_Transcendental_Oarcosh, type, arcosh: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Transcendental_Oarcsin, type, arcsin: (real > real)). % 6.41/2.68 thf(sy_c_Transcendental_Oarctan, type, arctan: (real > real)). % 6.41/2.68 thf(sy_c_Transcendental_Oarsinh, type, arsinh: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Transcendental_Oartanh, type, artanh: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Transcendental_Ocos, type, cos: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Transcendental_Ocos__coeff, type, cos_coeff: (nat > real)). % 6.41/2.68 thf(sy_c_Transcendental_Ocosh, type, cosh: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Transcendental_Ocot, type, cot: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Transcendental_Odiffs, type, diffs: (!> [A:$tType]: (((nat > A) > (nat > A))))). % 6.41/2.68 thf(sy_c_Transcendental_Oexp, type, exp: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Transcendental_Oln__class_Oln, type, ln_ln: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Transcendental_Olog, type, log: (real > (real > real))). % 6.41/2.68 thf(sy_c_Transcendental_Opi, type, pi: real). % 6.41/2.68 thf(sy_c_Transcendental_Opowr, type, powr: (!> [A:$tType]: ((A > (A > A))))). % 6.41/2.68 thf(sy_c_Transcendental_Osin, type, sin: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Transcendental_Osin__coeff, type, sin_coeff: (nat > real)). % 6.41/2.68 thf(sy_c_Transcendental_Osinh, type, sinh: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Transcendental_Otan, type, tan: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Transcendental_Otanh, type, tanh: (!> [A:$tType]: ((A > A)))). % 6.41/2.68 thf(sy_c_Transitive__Closure_Ontrancl, type, transitive_ntrancl: (!> [A:$tType]: ((nat > ((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A))))))). % 6.41/2.68 thf(sy_c_Transitive__Closure_Ortrancl, type, transitive_rtrancl: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A)))))). % 6.41/2.68 thf(sy_c_Transitive__Closure_Otrancl, type, transitive_trancl: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A)))))). % 6.41/2.68 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf, type, vEBT_Leaf: ($o > ($o > vEBT_VEBT))). % 6.41/2.68 thf(sy_c_VEBT__Definitions_OVEBT_ONode, type, vEBT_Node: ((option @ ((product_prod @ nat) @ nat)) > (nat > ((list @ vEBT_VEBT) > (vEBT_VEBT > vEBT_VEBT))))). % 6.41/2.68 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT, type, vEBT_size_VEBT: (vEBT_VEBT > nat)). % 6.41/2.68 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options, type, vEBT_V8194947554948674370ptions: (vEBT_VEBT > (nat > $o))). % 6.41/2.68 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh, type, vEBT_VEBT_high: (nat > (nat > nat))). % 6.41/2.68 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children, type, vEBT_V5917875025757280293ildren: (nat > ((list @ vEBT_VEBT) > (nat > $o)))). % 6.41/2.68 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow, type, vEBT_VEBT_low: (nat > (nat > nat))). % 6.41/2.68 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima, type, vEBT_VEBT_membermima: (vEBT_VEBT > (nat > $o))). % 6.41/2.68 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel, type, vEBT_V4351362008482014158ma_rel: (((product_prod @ vEBT_VEBT) @ nat) > (((product_prod @ vEBT_VEBT) @ nat) > $o))). % 6.41/2.68 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member, type, vEBT_V5719532721284313246member: (vEBT_VEBT > (nat > $o))). % 6.41/2.68 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel, type, vEBT_V5765760719290551771er_rel: (((product_prod @ vEBT_VEBT) @ nat) > (((product_prod @ vEBT_VEBT) @ nat) > $o))). % 6.41/2.68 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H, type, vEBT_VEBT_valid: (vEBT_VEBT > (nat > $o))). % 6.41/2.68 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel, type, vEBT_VEBT_valid_rel: (((product_prod @ vEBT_VEBT) @ nat) > (((product_prod @ vEBT_VEBT) @ nat) > $o))). % 6.41/2.68 thf(sy_c_VEBT__Definitions_Oinvar__vebt, type, vEBT_invar_vebt: (vEBT_VEBT > (nat > $o))). % 6.41/2.68 thf(sy_c_VEBT__Definitions_Oset__vebt, type, vEBT_set_vebt: (vEBT_VEBT > (set @ nat))). % 6.41/2.68 thf(sy_c_VEBT__Definitions_Ovebt__buildup, type, vEBT_vebt_buildup: (nat > vEBT_VEBT)). % 6.41/2.68 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel, type, vEBT_v4011308405150292612up_rel: (nat > (nat > $o))). % 6.41/2.68 thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat, type, vEBT_VEBT_bit_concat: (nat > (nat > (nat > nat)))). % 6.41/2.68 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull, type, vEBT_VEBT_minNull: (vEBT_VEBT > $o)). % 6.41/2.68 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel, type, vEBT_V6963167321098673237ll_rel: (vEBT_VEBT > (vEBT_VEBT > $o))). % 6.41/2.68 thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H, type, vEBT_VEBT_set_vebt: (vEBT_VEBT > (set @ nat))). % 6.41/2.68 thf(sy_c_VEBT__Member_Ovebt__member, type, vEBT_vebt_member: (vEBT_VEBT > (nat > $o))). % 6.41/2.68 thf(sy_c_VEBT__Member_Ovebt__member__rel, type, vEBT_vebt_member_rel: (((product_prod @ vEBT_VEBT) @ nat) > (((product_prod @ vEBT_VEBT) @ nat) > $o))). % 6.41/2.68 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd, type, vEBT_VEBT_add: ((option @ nat) > ((option @ nat) > (option @ nat)))). % 6.41/2.68 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater, type, vEBT_VEBT_greater: ((option @ nat) > ((option @ nat) > $o))). % 6.41/2.68 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless, type, vEBT_VEBT_less: ((option @ nat) > ((option @ nat) > $o))). % 6.41/2.68 thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq, type, vEBT_VEBT_lesseq: ((option @ nat) > ((option @ nat) > $o))). % 6.41/2.68 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set, type, vEBT_VEBT_max_in_set: ((set @ nat) > (nat > $o))). % 6.41/2.68 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set, type, vEBT_VEBT_min_in_set: ((set @ nat) > (nat > $o))). % 6.41/2.68 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul, type, vEBT_VEBT_mul: ((option @ nat) > ((option @ nat) > (option @ nat)))). % 6.41/2.68 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift, type, vEBT_V2048590022279873568_shift: (!> [A:$tType]: (((A > (A > A)) > ((option @ A) > ((option @ A) > (option @ A))))))). % 6.41/2.68 thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower, type, vEBT_VEBT_power: ((option @ nat) > ((option @ nat) > (option @ nat)))). % 6.41/2.68 thf(sy_c_VEBT__MinMax_Ovebt__maxt, type, vEBT_vebt_maxt: (vEBT_VEBT > (option @ nat))). % 6.41/2.68 thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel, type, vEBT_vebt_maxt_rel: (vEBT_VEBT > (vEBT_VEBT > $o))). % 6.41/2.68 thf(sy_c_VEBT__MinMax_Ovebt__mint, type, vEBT_vebt_mint: (vEBT_VEBT > (option @ nat))). % 6.41/2.68 thf(sy_c_VEBT__MinMax_Ovebt__mint__rel, type, vEBT_vebt_mint_rel: (vEBT_VEBT > (vEBT_VEBT > $o))). % 6.41/2.68 thf(sy_c_Wellfounded_Oaccp, type, accp: (!> [A:$tType]: (((A > (A > $o)) > (A > $o))))). % 6.41/2.68 thf(sy_c_Wellfounded_Ofinite__psubset, type, finite_psubset: (!> [A:$tType]: ((set @ ((product_prod @ (set @ A)) @ (set @ A)))))). % 6.41/2.68 thf(sy_c_Wellfounded_Olex__prod, type, lex_prod: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))). % 6.41/2.68 thf(sy_c_Wellfounded_Omax__ext, type, max_ext: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (set @ A)) @ (set @ A))))))). % 6.41/2.68 thf(sy_c_Wellfounded_Omax__extp, type, max_extp: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > ((set @ A) > $o)))))). % 6.41/2.68 thf(sy_c_Wellfounded_Omeasure, type, measure: (!> [A:$tType]: (((A > nat) > (set @ ((product_prod @ A) @ A)))))). % 6.41/2.68 thf(sy_c_Wellfounded_Omin__ext, type, min_ext: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (set @ A)) @ (set @ A))))))). % 6.41/2.68 thf(sy_c_Wellfounded_Omlex__prod, type, mlex_prod: (!> [A:$tType]: (((A > nat) > ((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A))))))). % 6.41/2.68 thf(sy_c_Wellfounded_Opred__nat, type, pred_nat: (set @ ((product_prod @ nat) @ nat))). % 6.41/2.68 thf(sy_c_Wellfounded_Owf, type, wf: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))). % 6.41/2.68 thf(sy_c_Wfrec_Osame__fst, type, same_fst: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))). % 6.41/2.68 thf(sy_c_Zorn_Ochains, type, chains: (!> [A:$tType]: (((set @ (set @ A)) > (set @ (set @ (set @ A))))))). % 6.41/2.68 thf(sy_c_fChoice, type, fChoice: (!> [A:$tType]: (((A > $o) > A)))). % 6.41/2.68 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))). % 6.41/2.68 thf(sy_v_deg, type, deg: nat). % 6.41/2.68 thf(sy_v_m____, type, m: nat). % 6.41/2.68 thf(sy_v_ma, type, ma: nat). % 6.41/2.68 thf(sy_v_maxs____, type, maxs: nat). % 6.41/2.68 thf(sy_v_mi, type, mi: nat). % 6.41/2.68 thf(sy_v_n____, type, n: nat). % 6.41/2.68 thf(sy_v_summary, type, summary: vEBT_VEBT). % 6.41/2.68 thf(sy_v_thesis____, type, thesis: $o). % 6.41/2.68 thf(sy_v_treeList, type, treeList: (list @ vEBT_VEBT)). % 6.41/2.68 thf(fact_0__092_060open_062_092_060exists_062x_O_Aboth__member__options_A_ItreeList_A_B_Amaxs_J_Ax_092_060close_062, axiom, (? [X_1:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ treeList) @ maxs)) @ X_1)))). % 6.41/2.68 thf(fact_1_not__min__Null__member, axiom, (! [T2:vEBT_VEBT]: (((~ ((vEBT_VEBT_minNull @ T2))) => (? [X_1:nat]: (((vEBT_V8194947554948674370ptions @ T2) @ X_1))))))). % 6.41/2.68 thf(fact_2__092_060open_062Some_Amaxs_A_061_Avebt__maxt_Asummary_092_060close_062, axiom, (((some @ nat) @ maxs) = (vEBT_vebt_maxt @ summary))). % 6.41/2.68 thf(fact_3_both__member__options__def, axiom, (vEBT_V8194947554948674370ptions = (^ [T3:vEBT_VEBT,X:nat]: ((((vEBT_V5719532721284313246member @ T3) @ X) | ((vEBT_VEBT_membermima @ T3) @ X)))))). % 6.41/2.68 thf(fact_4__C5_C_I7_J, axiom, ((mi = ma) => (! [X2:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X2) @ ((set2 @ vEBT_VEBT) @ treeList)) => (~ ((? [X_12:nat]: (((vEBT_V8194947554948674370ptions @ X2) @ X_12)))))))))). % 6.41/2.68 thf(fact_5__C5_C_I1_J, axiom, (! [X2:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X2) @ ((set2 @ vEBT_VEBT) @ treeList)) => ((vEBT_invar_vebt @ X2) @ n))))). % 6.41/2.68 thf(fact_6_in__children__def, axiom, (vEBT_V5917875025757280293ildren = (^ [N:nat,TreeList:(list @ vEBT_VEBT),X:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList) @ ((vEBT_VEBT_high @ X) @ N))) @ ((vEBT_VEBT_low @ X) @ N)))))). % 6.41/2.68 thf(fact_7_both__member__options__equiv__member, axiom, (! [T2:vEBT_VEBT,N2:nat,X3:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => (((vEBT_V8194947554948674370ptions @ T2) @ X3) = ((vEBT_vebt_member @ T2) @ X3)))))). % 6.41/2.68 thf(fact_8_valid__member__both__member__options, axiom, (! [T2:vEBT_VEBT,N2:nat,X3:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => (((vEBT_V8194947554948674370ptions @ T2) @ X3) => ((vEBT_vebt_member @ T2) @ X3)))))). % 6.41/2.68 thf(fact_9_intind, axiom, (! [A:$tType,I:nat,N2:nat,P:(A > $o),X3:A]: (((((ord_less @ nat) @ I) @ N2) => ((P @ X3) => (P @ (((nth @ A) @ (((replicate @ A) @ N2) @ X3)) @ I))))))). % 6.41/2.68 thf(fact_10_assms_I2_J, axiom, (mi != ma)). % 6.41/2.68 thf(fact_11_list__update__id, axiom, (! [A:$tType,Xs:(list @ A),I:nat]: ((((((list_update @ A) @ Xs) @ I) @ (((nth @ A) @ Xs) @ I)) = Xs)))). % 6.41/2.68 thf(fact_12_nth__list__update__neq, axiom, (! [A:$tType,I:nat,J:nat,Xs:(list @ A),X3:A]: (((I != J) => ((((nth @ A) @ ((((list_update @ A) @ Xs) @ I) @ X3)) @ J) = (((nth @ A) @ Xs) @ J)))))). % 6.41/2.68 thf(fact_13_min__Null__member, axiom, (! [T2:vEBT_VEBT,X3:nat]: (((vEBT_VEBT_minNull @ T2) => (~ (((vEBT_vebt_member @ T2) @ X3))))))). % 6.41/2.68 thf(fact_14__C5_C_I2_J, axiom, ((vEBT_invar_vebt @ summary) @ m)). % 6.41/2.68 thf(fact_15_bit__split__inv, axiom, (! [X3:nat,D2:nat]: (((((vEBT_VEBT_bit_concat @ ((vEBT_VEBT_high @ X3) @ D2)) @ ((vEBT_VEBT_low @ X3) @ D2)) @ D2) = X3)))). % 6.41/2.68 thf(fact_16__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxs_O_ASome_Amaxs_A_061_Avebt__maxt_Asummary_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [Maxs:nat]: ((((some @ nat) @ Maxs) != (vEBT_vebt_maxt @ summary))))))). % 6.41/2.68 thf(fact_17_member__valid__both__member__options, axiom, (! [Tree:vEBT_VEBT,N2:nat,X3:nat]: ((((vEBT_invar_vebt @ Tree) @ N2) => (((vEBT_vebt_member @ Tree) @ X3) => (((vEBT_V5719532721284313246member @ Tree) @ X3) | ((vEBT_VEBT_membermima @ Tree) @ X3))))))). % 6.41/2.68 thf(fact_18_maxt__member, axiom, (! [T2:vEBT_VEBT,N2:nat,Maxi:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => (((vEBT_vebt_maxt @ T2) = ((some @ nat) @ Maxi)) => ((vEBT_vebt_member @ T2) @ Maxi)))))). % 6.41/2.68 thf(fact_19_list__update__overwrite, axiom, (! [A:$tType,Xs:(list @ A),I:nat,X3:A,Y:A]: ((((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I) @ X3)) @ I) @ Y) = ((((list_update @ A) @ Xs) @ I) @ Y))))). % 6.41/2.68 thf(fact_20_member__correct, axiom, (! [T2:vEBT_VEBT,N2:nat,X3:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => (((vEBT_vebt_member @ T2) @ X3) = (((member @ nat) @ X3) @ (vEBT_set_vebt @ T2))))))). % 6.41/2.68 thf(fact_21__092_060open_062both__member__options_Asummary_A_Ihigh_Ama_An_J_092_060close_062, axiom, ((vEBT_V8194947554948674370ptions @ summary) @ ((vEBT_VEBT_high @ ma) @ n))). % 6.41/2.68 thf(fact_22_nth__replicate, axiom, (! [A:$tType,I:nat,N2:nat,X3:A]: (((((ord_less @ nat) @ I) @ N2) => ((((nth @ A) @ (((replicate @ A) @ N2) @ X3)) @ I) = X3))))). % 6.41/2.68 thf(fact_23__C5_C_I8_J, axiom, (((ord_less_eq @ nat) @ mi) @ ma)). % 6.41/2.68 thf(fact_24_greater__shift, axiom, ((ord_less @ nat) = (^ [Y2:nat,X:nat]: (((vEBT_VEBT_greater @ ((some @ nat) @ X)) @ ((some @ nat) @ Y2)))))). % 6.41/2.68 thf(fact_25_less__shift, axiom, ((ord_less @ nat) = (^ [X:nat,Y2:nat]: (((vEBT_VEBT_less @ ((some @ nat) @ X)) @ ((some @ nat) @ Y2)))))). % 6.41/2.68 thf(fact_26__092_060open_062high_Ama_An_A_060_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062, axiom, (((ord_less @ nat) @ ((vEBT_VEBT_high @ ma) @ n)) @ ((the2 @ nat) @ (vEBT_vebt_maxt @ summary)))). % 6.41/2.68 thf(fact_27_mint__member, axiom, (! [T2:vEBT_VEBT,N2:nat,Maxi:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => (((vEBT_vebt_mint @ T2) = ((some @ nat) @ Maxi)) => ((vEBT_vebt_member @ T2) @ Maxi)))))). % 6.41/2.68 thf(fact_28_list__update__swap, axiom, (! [A:$tType,I:nat,I2:nat,Xs:(list @ A),X3:A,X4:A]: (((I != I2) => (((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I) @ X3)) @ I2) @ X4) = ((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I2) @ X4)) @ I) @ X3)))))). % 6.41/2.68 thf(fact_29_maxt__corr__help, axiom, (! [T2:vEBT_VEBT,N2:nat,Maxi:nat,X3:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => (((vEBT_vebt_maxt @ T2) = ((some @ nat) @ Maxi)) => (((vEBT_vebt_member @ T2) @ X3) => (((ord_less_eq @ nat) @ X3) @ Maxi))))))). % 6.41/2.68 thf(fact_30_maxt__corr, axiom, (! [T2:vEBT_VEBT,N2:nat,X3:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => (((vEBT_vebt_maxt @ T2) = ((some @ nat) @ X3)) => ((vEBT_VEBT_max_in_set @ (vEBT_VEBT_set_vebt @ T2)) @ X3)))))). % 6.41/2.68 thf(fact_31_maxt__sound, axiom, (! [T2:vEBT_VEBT,N2:nat,X3:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => (((vEBT_VEBT_max_in_set @ (vEBT_VEBT_set_vebt @ T2)) @ X3) => ((vEBT_vebt_maxt @ T2) = ((some @ nat) @ X3))))))). % 6.41/2.68 thf(fact_32_buildup__nothing__in__leaf, axiom, (! [N2:nat,X3:nat]: ((~ (((vEBT_V5719532721284313246member @ (vEBT_vebt_buildup @ N2)) @ X3)))))). % 6.41/2.68 thf(fact_33_buildup__nothing__in__min__max, axiom, (! [N2:nat,X3:nat]: ((~ (((vEBT_VEBT_membermima @ (vEBT_vebt_buildup @ N2)) @ X3)))))). % 6.41/2.68 thf(fact_34_option_Oinject, axiom, (! [A:$tType,X22:A,Y22:A]: (((((some @ A) @ X22) = ((some @ A) @ Y22)) = (X22 = Y22))))). % 6.41/2.68 thf(fact_35_valid__eq, axiom, (vEBT_VEBT_valid = vEBT_invar_vebt)). % 6.41/2.68 thf(fact_36_valid__eq1, axiom, (! [T2:vEBT_VEBT,D2:nat]: ((((vEBT_invar_vebt @ T2) @ D2) => ((vEBT_VEBT_valid @ T2) @ D2))))). % 6.41/2.68 thf(fact_37_valid__eq2, axiom, (! [T2:vEBT_VEBT,D2:nat]: ((((vEBT_VEBT_valid @ T2) @ D2) => ((vEBT_invar_vebt @ T2) @ D2))))). % 6.41/2.68 thf(fact_38_inthall, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o),N2:nat]: (((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (P @ X5)))) => ((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => (P @ (((nth @ A) @ Xs) @ N2))))))). % 6.41/2.68 thf(fact_39_deg__not__0, axiom, (! [T2:vEBT_VEBT,N2:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))). % 6.41/2.68 thf(fact_40__092_060open_062high_Ama_An_A_092_060le_062_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062, axiom, (((ord_less_eq @ nat) @ ((vEBT_VEBT_high @ ma) @ n)) @ ((the2 @ nat) @ (vEBT_vebt_maxt @ summary)))). % 6.41/2.68 thf(fact_41_valid__tree__deg__neq__0, axiom, (! [T2:vEBT_VEBT]: ((~ (((vEBT_invar_vebt @ T2) @ (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_42_valid__0__not, axiom, (! [T2:vEBT_VEBT]: ((~ (((vEBT_invar_vebt @ T2) @ (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_43_min__in__set__def, axiom, (vEBT_VEBT_min_in_set = (^ [Xs2:(set @ nat),X:nat]: (((((member @ nat) @ X) @ Xs2) & (! [Y2:nat]: (((((member @ nat) @ Y2) @ Xs2) => (((ord_less_eq @ nat) @ X) @ Y2))))))))). % 6.41/2.68 thf(fact_44_max__in__set__def, axiom, (vEBT_VEBT_max_in_set = (^ [Xs2:(set @ nat),X:nat]: (((((member @ nat) @ X) @ Xs2) & (! [Y2:nat]: (((((member @ nat) @ Y2) @ Xs2) => (((ord_less_eq @ nat) @ Y2) @ X))))))))). % 6.41/2.68 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))). % 6.41/2.68 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A3)))) = A3)))). % 6.41/2.68 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) = (Q @ X5)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))). % 6.41/2.68 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F2:(A > B),G:(A > B)]: (((! [X5:A]: (((F2 @ X5) = (G @ X5)))) => (F2 = G))))). % 6.41/2.68 thf(fact_49_buildup__gives__valid, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((vEBT_invar_vebt @ (vEBT_vebt_buildup @ N2)) @ N2))))). % 6.41/2.68 thf(fact_50_set__vebt__set__vebt_H__valid, axiom, (! [T2:vEBT_VEBT,N2:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => ((vEBT_set_vebt @ T2) = (vEBT_VEBT_set_vebt @ T2)))))). % 6.41/2.68 thf(fact_51_mint__corr__help, axiom, (! [T2:vEBT_VEBT,N2:nat,Mini:nat,X3:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => (((vEBT_vebt_mint @ T2) = ((some @ nat) @ Mini)) => (((vEBT_vebt_member @ T2) @ X3) => (((ord_less_eq @ nat) @ Mini) @ X3))))))). % 6.41/2.68 thf(fact_52_replicate__eq__replicate, axiom, (! [A:$tType,M:nat,X3:A,N2:nat,Y:A]: ((((((replicate @ A) @ M) @ X3) = (((replicate @ A) @ N2) @ Y)) = ((M = N2) & ((M != (zero_zero @ nat)) => (X3 = Y))))))). % 6.41/2.68 thf(fact_53_length__replicate, axiom, (! [A:$tType,N2:nat,X3:A]: ((((size_size @ (list @ A)) @ (((replicate @ A) @ N2) @ X3)) = N2)))). % 6.41/2.68 thf(fact_54_length__list__update, axiom, (! [A:$tType,Xs:(list @ A),I:nat,X3:A]: ((((size_size @ (list @ A)) @ ((((list_update @ A) @ Xs) @ I) @ X3)) = ((size_size @ (list @ A)) @ Xs))))). % 6.41/2.68 thf(fact_55_mint__sound, axiom, (! [T2:vEBT_VEBT,N2:nat,X3:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => (((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ T2)) @ X3) => ((vEBT_vebt_mint @ T2) = ((some @ nat) @ X3))))))). % 6.41/2.68 thf(fact_56_mint__corr, axiom, (! [T2:vEBT_VEBT,N2:nat,X3:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => (((vEBT_vebt_mint @ T2) = ((some @ nat) @ X3)) => ((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ T2)) @ X3)))))). % 6.41/2.68 thf(fact_57_Ball__set__replicate, axiom, (! [A:$tType,N2:nat,A2:A,P:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ (((replicate @ A) @ N2) @ A2))) => (P @ X)))) = ((P @ A2) | (N2 = (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_58_Bex__set__replicate, axiom, (! [A:$tType,N2:nat,A2:A,P:(A > $o)]: (((? [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ (((replicate @ A) @ N2) @ A2))) & (P @ X)))) = ((P @ A2) & (N2 != (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_59_in__set__replicate, axiom, (! [A:$tType,X3:A,N2:nat,Y:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ (((replicate @ A) @ N2) @ Y))) = ((X3 = Y) & (N2 != (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_60_list__update__beyond, axiom, (! [A:$tType,Xs:(list @ A),I:nat,X3:A]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ I) => (((((list_update @ A) @ Xs) @ I) @ X3) = Xs))))). % 6.41/2.68 thf(fact_61__C5_C_I4_J, axiom, (m = (suc @ n))). % 6.41/2.68 thf(fact_62_nth__list__update__eq, axiom, (! [A:$tType,I:nat,Xs:(list @ A),X3:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ ((((list_update @ A) @ Xs) @ I) @ X3)) @ I) = X3))))). % 6.41/2.68 thf(fact_63_lesseq__shift, axiom, ((ord_less_eq @ nat) = (^ [X:nat,Y2:nat]: (((vEBT_VEBT_lesseq @ ((some @ nat) @ X)) @ ((some @ nat) @ Y2)))))). % 6.41/2.68 thf(fact_64_set__swap, axiom, (! [A:$tType,I:nat,Xs:(list @ A),J:nat]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((set2 @ A) @ ((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I) @ (((nth @ A) @ Xs) @ J))) @ J) @ (((nth @ A) @ Xs) @ I))) = ((set2 @ A) @ Xs))))))). % 6.41/2.68 thf(fact_65_Ex__list__of__length, axiom, (! [A:$tType,N2:nat]: ((? [Xs3:(list @ A)]: ((((size_size @ (list @ A)) @ Xs3) = N2)))))). % 6.41/2.68 thf(fact_66_neq__if__length__neq, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) != ((size_size @ (list @ A)) @ Ys)) => (Xs != Ys))))). % 6.41/2.68 thf(fact_67_option_Osel, axiom, (! [A:$tType,X22:A]: ((((the2 @ A) @ ((some @ A) @ X22)) = X22)))). % 6.41/2.68 thf(fact_68_length__pos__if__in__set, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs)))))). % 6.41/2.68 thf(fact_69_length__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((! [Xs3:(list @ A)]: (((! [Ys2:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ys2)) @ ((size_size @ (list @ A)) @ Xs3)) => (P @ Ys2)))) => (P @ Xs3)))) => (P @ Xs))))). % 6.41/2.68 thf(fact_70_list__eq__iff__nth__eq, axiom, (! [A:$tType]: (((^ [Y3:(list @ A),Z:(list @ A)]: ((Y3 = Z))) = (^ [Xs2:(list @ A),Ys3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) = ((size_size @ (list @ A)) @ Ys3)) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs2)) => ((((nth @ A) @ Xs2) @ I3) = (((nth @ A) @ Ys3) @ I3)))))))))))). % 6.41/2.68 thf(fact_71_Skolem__list__nth, axiom, (! [A:$tType,K:nat,P:(nat > (A > $o))]: (((! [I3:nat]: (((((ord_less @ nat) @ I3) @ K) => (? [X6:A]: (((P @ I3) @ X6)))))) = (? [Xs2:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) = K) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ K) => ((P @ I3) @ (((nth @ A) @ Xs2) @ I3)))))))))))). % 6.41/2.68 thf(fact_72_nth__equalityI, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys)) => ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ Xs) @ I4) = (((nth @ A) @ Ys) @ I4))))) => (Xs = Ys)))))). % 6.41/2.68 thf(fact_73_replicate__length__same, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: (((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (X5 = X3)))) => ((((replicate @ A) @ ((size_size @ (list @ A)) @ Xs)) @ X3) = Xs))))). % 6.41/2.68 thf(fact_74_replicate__eqI, axiom, (! [A:$tType,Xs:(list @ A),N2:nat,X3:A]: (((((size_size @ (list @ A)) @ Xs) = N2) => ((! [Y4:A]: (((((member @ A) @ Y4) @ ((set2 @ A) @ Xs)) => (Y4 = X3)))) => (Xs = (((replicate @ A) @ N2) @ X3))))))). % 6.41/2.68 thf(fact_75_all__set__conv__all__nth, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (P @ X)))) = (! [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) => (P @ (((nth @ A) @ Xs) @ I3))))))))). % 6.41/2.68 thf(fact_76_all__nth__imp__all__set, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o),X3:A]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Xs)) => (P @ (((nth @ A) @ Xs) @ I4))))) => ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (P @ X3)))))). % 6.41/2.68 thf(fact_77_in__set__conv__nth, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) = (? [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) & ((((nth @ A) @ Xs) @ I3) = X3)))))))). % 6.41/2.68 thf(fact_78_list__ball__nth, axiom, (! [A:$tType,N2:nat,Xs:(list @ A),P:(A > $o)]: (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => ((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (P @ X5)))) => (P @ (((nth @ A) @ Xs) @ N2))))))). % 6.41/2.68 thf(fact_79_nth__mem, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ A) @ (((nth @ A) @ Xs) @ N2)) @ ((set2 @ A) @ Xs)))))). % 6.41/2.68 thf(fact_80_set__update__memI, axiom, (! [A:$tType,N2:nat,Xs:(list @ A),X3:A]: (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ A) @ X3) @ ((set2 @ A) @ ((((list_update @ A) @ Xs) @ N2) @ X3))))))). % 6.41/2.68 thf(fact_81_list__update__same__conv, axiom, (! [A:$tType,I:nat,Xs:(list @ A),X3:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((((list_update @ A) @ Xs) @ I) @ X3) = Xs) = ((((nth @ A) @ Xs) @ I) = X3)))))). % 6.41/2.68 thf(fact_82_nth__list__update, axiom, (! [A:$tType,I:nat,Xs:(list @ A),J:nat,X3:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => (((I = J) => ((((nth @ A) @ ((((list_update @ A) @ Xs) @ I) @ X3)) @ J) = X3)) & ((I != J) => ((((nth @ A) @ ((((list_update @ A) @ Xs) @ I) @ X3)) @ J) = (((nth @ A) @ Xs) @ J)))))))). % 6.41/2.68 thf(fact_83_le0, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N2)))). % 6.41/2.68 thf(fact_84_bot__nat__0_Oextremum, axiom, (! [A2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A2)))). % 6.41/2.68 thf(fact_85_less__nat__zero__code, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_86_neq0__conv, axiom, (! [N2:nat]: (((N2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))). % 6.41/2.68 thf(fact_87_bot__nat__0_Onot__eq__extremum, axiom, (! [A2:nat]: (((A2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A2))))). % 6.41/2.68 thf(fact_88_not__gr__zero, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [N2:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N2))) = (N2 = (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_89_le__zero__eq, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [N2:A]: (((((ord_less_eq @ A) @ N2) @ (zero_zero @ A)) = (N2 = (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_90__C5_C_I5_J, axiom, (deg = (((plus_plus @ nat) @ n) @ m))). % 6.41/2.68 thf(fact_91_ex__least__nat__le, axiom, (! [P:(nat > $o),N2:nat]: (((P @ N2) => ((~ ((P @ (zero_zero @ nat)))) => (? [K2:nat]: (((((ord_less_eq @ nat) @ K2) @ N2) & ((! [I5:nat]: (((((ord_less @ nat) @ I5) @ K2) => (~ ((P @ I5)))))) & (P @ K2)))))))))). % 6.41/2.68 thf(fact_92_buildup__gives__empty, axiom, (! [N2:nat]: (((vEBT_VEBT_set_vebt @ (vEBT_vebt_buildup @ N2)) = (bot_bot @ (set @ nat)))))). % 6.41/2.68 thf(fact_93_dual__order_Orefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))). % 6.41/2.68 thf(fact_94_even__odd__cases, axiom, (! [X3:nat]: (((! [N3:nat]: ((X3 != (((plus_plus @ nat) @ N3) @ N3)))) => (~ ((! [N3:nat]: ((X3 != (((plus_plus @ nat) @ N3) @ (suc @ N3))))))))))). % 6.41/2.68 thf(fact_95_add__shift, axiom, (! [X3:nat,Y:nat,Z2:nat]: ((((((plus_plus @ nat) @ X3) @ Y) = Z2) = (((vEBT_VEBT_add @ ((some @ nat) @ X3)) @ ((some @ nat) @ Y)) = ((some @ nat) @ Z2)))))). % 6.41/2.68 thf(fact_96_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ X3) @ X3))))))). % 6.41/2.68 thf(fact_97_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C2)) = (B2 = C2)))))))). % 6.41/2.68 thf(fact_98_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C2) @ A2)) = (B2 = C2)))))))). % 6.41/2.68 thf(fact_99_bot__apply, axiom, (! [C:$tType,D:$tType]: (((bot @ C) => ((bot_bot @ (D > C)) = (^ [X:D]: ((bot_bot @ C)))))))). % 6.41/2.68 thf(fact_100_nat_Oinject, axiom, (! [X22:nat,Y22:nat]: ((((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))). % 6.41/2.68 thf(fact_101_old_Onat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % 6.41/2.68 thf(fact_102_add__le__cancel__left, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))). % 6.41/2.68 thf(fact_103_add__le__cancel__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))). % 6.41/2.68 thf(fact_104_add_Oright__neutral, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))). % 6.41/2.68 thf(fact_105_double__zero__sym, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A2:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ A2) @ A2)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_106_add__cancel__left__left, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [B2:A,A2:A]: ((((((plus_plus @ A) @ B2) @ A2) = A2) = (B2 = (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_107_add__cancel__left__right, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A2:A,B2:A]: ((((((plus_plus @ A) @ A2) @ B2) = A2) = (B2 = (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_108_add__cancel__right__left, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A2:A,B2:A]: (((A2 = (((plus_plus @ A) @ B2) @ A2)) = (B2 = (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_109_add__cancel__right__right, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A2:A,B2:A]: (((A2 = (((plus_plus @ A) @ A2) @ B2)) = (B2 = (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_110_add__eq__0__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [X3:A,Y:A]: ((((((plus_plus @ A) @ X3) @ Y) = (zero_zero @ A)) = ((X3 = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_111_zero__eq__add__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [X3:A,Y:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ X3) @ Y)) = ((X3 = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_112_add__0, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))). % 6.41/2.68 thf(fact_113_add__less__cancel__left, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2)) = (((ord_less @ A) @ A2) @ B2)))))))). % 6.41/2.68 thf(fact_114_add__less__cancel__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((ord_less @ A) @ A2) @ B2)))))))). % 6.41/2.68 thf(fact_115_lessI, axiom, (! [N2:nat]: ((((ord_less @ nat) @ N2) @ (suc @ N2))))). % 6.41/2.68 thf(fact_116_Suc__mono, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ N2) => (((ord_less @ nat) @ (suc @ M)) @ (suc @ N2)))))). % 6.41/2.68 thf(fact_117_Suc__less__eq, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (suc @ M)) @ (suc @ N2)) = (((ord_less @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_118_Suc__le__mono, axiom, (! [N2:nat,M:nat]: (((((ord_less_eq @ nat) @ (suc @ N2)) @ (suc @ M)) = (((ord_less_eq @ nat) @ N2) @ M))))). % 6.41/2.68 thf(fact_119_add__Suc__right, axiom, (! [M:nat,N2:nat]: (((((plus_plus @ nat) @ M) @ (suc @ N2)) = (suc @ (((plus_plus @ nat) @ M) @ N2)))))). % 6.41/2.68 thf(fact_120_add__is__0, axiom, (! [M:nat,N2:nat]: ((((((plus_plus @ nat) @ M) @ N2) = (zero_zero @ nat)) = ((M = (zero_zero @ nat)) & (N2 = (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_121_Nat_Oadd__0__right, axiom, (! [M:nat]: (((((plus_plus @ nat) @ M) @ (zero_zero @ nat)) = M)))). % 6.41/2.68 thf(fact_122_nat__add__left__cancel__less, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N2)) = (((ord_less @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_123_nat__add__left__cancel__le, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N2)) = (((ord_less_eq @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_124_add__le__same__cancel1, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ B2) @ A2)) @ B2) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_125_add__le__same__cancel2, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_126_le__add__same__cancel1, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))). % 6.41/2.68 thf(fact_127_le__add__same__cancel2, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ B2) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))). % 6.41/2.68 thf(fact_128_double__add__le__zero__iff__single__add__le__zero, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_129_zero__le__double__add__iff__zero__le__single__add, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))). % 6.41/2.68 thf(fact_130_zero__less__double__add__iff__zero__less__single__add, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))). % 6.41/2.68 thf(fact_131_double__add__less__zero__iff__single__add__less__zero, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_132_less__add__same__cancel2, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ B2) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))). % 6.41/2.68 thf(fact_133_less__add__same__cancel1, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B2)) = (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))). % 6.41/2.68 thf(fact_134_add__less__same__cancel2, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_135_add__less__same__cancel1, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ B2) @ A2)) @ B2) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_136_zero__less__Suc, axiom, (! [N2:nat]: ((((ord_less @ nat) @ (zero_zero @ nat)) @ (suc @ N2))))). % 6.41/2.68 thf(fact_137_less__Suc0, axiom, (! [N2:nat]: (((((ord_less @ nat) @ N2) @ (suc @ (zero_zero @ nat))) = (N2 = (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_138_add__gr__0, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ M) @ N2)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) | (((ord_less @ nat) @ (zero_zero @ nat)) @ N2)))))). % 6.41/2.68 thf(fact_139_one__is__add, axiom, (! [M:nat,N2:nat]: ((((suc @ (zero_zero @ nat)) = (((plus_plus @ nat) @ M) @ N2)) = (((M = (suc @ (zero_zero @ nat))) & (N2 = (zero_zero @ nat))) | ((M = (zero_zero @ nat)) & (N2 = (suc @ (zero_zero @ nat))))))))). % 6.41/2.68 thf(fact_140_add__is__1, axiom, (! [M:nat,N2:nat]: ((((((plus_plus @ nat) @ M) @ N2) = (suc @ (zero_zero @ nat))) = (((M = (suc @ (zero_zero @ nat))) & (N2 = (zero_zero @ nat))) | ((M = (zero_zero @ nat)) & (N2 = (suc @ (zero_zero @ nat))))))))). % 6.41/2.68 thf(fact_141_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))). % 6.41/2.68 thf(fact_142_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))). % 6.41/2.68 thf(fact_143_nat__arith_Osuc1, axiom, (! [A3:nat,K:nat,A2:nat]: (((A3 = (((plus_plus @ nat) @ K) @ A2)) => ((suc @ A3) = (((plus_plus @ nat) @ K) @ (suc @ A2))))))). % 6.41/2.68 thf(fact_144_add__Suc, axiom, (! [M:nat,N2:nat]: (((((plus_plus @ nat) @ (suc @ M)) @ N2) = (suc @ (((plus_plus @ nat) @ M) @ N2)))))). % 6.41/2.68 thf(fact_145_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A3:A,K:A,A2:A,B2:A]: (((A3 = (((plus_plus @ A) @ K) @ A2)) => ((((plus_plus @ A) @ A3) @ B2) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_146_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B3:A,K:A,B2:A,A2:A]: (((B3 = (((plus_plus @ A) @ K) @ B2)) => ((((plus_plus @ A) @ A2) @ B3) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_147_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))). % 6.41/2.68 thf(fact_148_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C2)) = (B2 = C2)))))))). % 6.41/2.68 thf(fact_149_add_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B2:A,A2:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C2) @ A2)) = (B2 = C2)))))))). % 6.41/2.68 thf(fact_150_add_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A4:A,B4:A]: ((((plus_plus @ A) @ B4) @ A4)))))))). % 6.41/2.68 thf(fact_151_add_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [B2:A,A2:A,C2:A]: (((((plus_plus @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C2)) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))). % 6.41/2.68 thf(fact_152_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C2)) => (B2 = C2)))))))). % 6.41/2.68 thf(fact_153_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C2) @ A2)) => (B2 = C2)))))))). % 6.41/2.68 thf(fact_154_Suc__inject, axiom, (! [X3:nat,Y:nat]: ((((suc @ X3) = (suc @ Y)) => (X3 = Y))))). % 6.41/2.68 thf(fact_155_n__not__Suc__n, axiom, (! [N2:nat]: ((N2 != (suc @ N2))))). % 6.41/2.68 thf(fact_156_add__Suc__shift, axiom, (! [M:nat,N2:nat]: (((((plus_plus @ nat) @ (suc @ M)) @ N2) = (((plus_plus @ nat) @ M) @ (suc @ N2)))))). % 6.41/2.68 thf(fact_157_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X:A]: ((bot_bot @ B)))))))). % 6.41/2.68 thf(fact_158_less__imp__Suc__add, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ N2) => (? [K2:nat]: ((N2 = (suc @ (((plus_plus @ nat) @ M) @ K2))))))))). % 6.41/2.68 thf(fact_159_less__iff__Suc__add, axiom, ((ord_less @ nat) = (^ [M2:nat,N:nat]: ((? [K3:nat]: ((N = (suc @ (((plus_plus @ nat) @ M2) @ K3))))))))). % 6.41/2.68 thf(fact_160_less__add__Suc2, axiom, (! [I:nat,M:nat]: ((((ord_less @ nat) @ I) @ (suc @ (((plus_plus @ nat) @ M) @ I)))))). % 6.41/2.68 thf(fact_161_less__add__Suc1, axiom, (! [I:nat,M:nat]: ((((ord_less @ nat) @ I) @ (suc @ (((plus_plus @ nat) @ I) @ M)))))). % 6.41/2.68 thf(fact_162_less__natE, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ N2) => (~ ((! [Q2:nat]: ((N2 != (suc @ (((plus_plus @ nat) @ M) @ Q2))))))))))). % 6.41/2.68 thf(fact_163_bot_Oextremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A2))))))). % 6.41/2.68 thf(fact_164_bot_Oextremum__unique, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) = (A2 = (bot_bot @ A))))))))). % 6.41/2.68 thf(fact_165_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) => (A2 = (bot_bot @ A))))))))). % 6.41/2.68 thf(fact_166_bot_Oextremum__strict, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ (bot_bot @ A)))))))))). % 6.41/2.68 thf(fact_167_bot_Onot__eq__extremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((A2 != (bot_bot @ A)) = (((ord_less @ A) @ (bot_bot @ A)) @ A2)))))))). % 6.41/2.68 thf(fact_168_add__mono__thms__linordered__semiring_I3_J, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (K = L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))). % 6.41/2.68 thf(fact_169_add__mono__thms__linordered__semiring_I2_J, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))). % 6.41/2.68 thf(fact_170_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))). % 6.41/2.68 thf(fact_171_add__mono, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ D2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ D2)))))))))). % 6.41/2.68 thf(fact_172_add__left__mono, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2))))))))). % 6.41/2.68 thf(fact_173_less__eqE, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (~ ((! [C3:A]: ((B2 != (((plus_plus @ A) @ A2) @ C3))))))))))))). % 6.41/2.68 thf(fact_174_add__right__mono, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2))))))))). % 6.41/2.68 thf(fact_175_le__iff__add, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: ((? [C4:A]: ((B4 = (((plus_plus @ A) @ A4) @ C4))))))))))). % 6.41/2.68 thf(fact_176_add__le__imp__le__left, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2)) => (((ord_less_eq @ A) @ A2) @ B2)))))))). % 6.41/2.68 thf(fact_177_add__le__imp__le__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) => (((ord_less_eq @ A) @ A2) @ B2)))))))). % 6.41/2.68 thf(fact_178_comm__monoid__add__class_Oadd__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))). % 6.41/2.68 thf(fact_179_add_Ocomm__neutral, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))). % 6.41/2.68 thf(fact_180_add_Ogroup__left__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A2) = A2))))))). % 6.41/2.68 thf(fact_181_add__mono__thms__linordered__field_I5_J, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))). % 6.41/2.68 thf(fact_182_add__mono__thms__linordered__field_I2_J, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))). % 6.41/2.68 thf(fact_183_add__mono__thms__linordered__field_I1_J, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (K = L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))). % 6.41/2.68 thf(fact_184_add__strict__mono, axiom, (! [A:$tType]: (((strict9044650504122735259up_add @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ C2) @ D2) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ D2)))))))))). % 6.41/2.68 thf(fact_185_add__strict__left__mono, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2))))))))). % 6.41/2.68 thf(fact_186_add__strict__right__mono, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2))))))))). % 6.41/2.68 thf(fact_187_add__less__imp__less__left, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2)) => (((ord_less @ A) @ A2) @ B2)))))))). % 6.41/2.68 thf(fact_188_add__less__imp__less__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) => (((ord_less @ A) @ A2) @ B2)))))))). % 6.41/2.68 thf(fact_189_nat_Odistinct_I1_J, axiom, (! [X22:nat]: (((zero_zero @ nat) != (suc @ X22))))). % 6.41/2.68 thf(fact_190_old_Onat_Odistinct_I2_J, axiom, (! [Nat3:nat]: (((suc @ Nat3) != (zero_zero @ nat))))). % 6.41/2.68 thf(fact_191_old_Onat_Odistinct_I1_J, axiom, (! [Nat2:nat]: (((zero_zero @ nat) != (suc @ Nat2))))). % 6.41/2.68 thf(fact_192_nat_OdiscI, axiom, (! [Nat:nat,X22:nat]: (((Nat = (suc @ X22)) => (Nat != (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_193_old_Onat_Oexhaust, axiom, (! [Y:nat]: (((Y != (zero_zero @ nat)) => (~ ((! [Nat4:nat]: ((Y != (suc @ Nat4)))))))))). % 6.41/2.68 thf(fact_194_nat__induct, axiom, (! [P:(nat > $o),N2:nat]: (((P @ (zero_zero @ nat)) => ((! [N3:nat]: (((P @ N3) => (P @ (suc @ N3))))) => (P @ N2)))))). % 6.41/2.68 thf(fact_195_diff__induct, axiom, (! [P:(nat > (nat > $o)),M:nat,N2:nat]: (((! [X5:nat]: (((P @ X5) @ (zero_zero @ nat)))) => ((! [Y4:nat]: (((P @ (zero_zero @ nat)) @ (suc @ Y4)))) => ((! [X5:nat,Y4:nat]: ((((P @ X5) @ Y4) => ((P @ (suc @ X5)) @ (suc @ Y4))))) => ((P @ M) @ N2))))))). % 6.41/2.68 thf(fact_196_zero__induct, axiom, (! [P:(nat > $o),K:nat]: (((P @ K) => ((! [N3:nat]: (((P @ (suc @ N3)) => (P @ N3)))) => (P @ (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_197_Suc__neq__Zero, axiom, (! [M:nat]: (((suc @ M) != (zero_zero @ nat))))). % 6.41/2.68 thf(fact_198_Zero__neq__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))). % 6.41/2.68 thf(fact_199_Zero__not__Suc, axiom, (! [M:nat]: (((zero_zero @ nat) != (suc @ M))))). % 6.41/2.68 thf(fact_200_not0__implies__Suc, axiom, (! [N2:nat]: (((N2 != (zero_zero @ nat)) => (? [M3:nat]: ((N2 = (suc @ M3)))))))). % 6.41/2.68 thf(fact_201_plus__nat_Oadd__0, axiom, (! [N2:nat]: (((((plus_plus @ nat) @ (zero_zero @ nat)) @ N2) = N2)))). % 6.41/2.68 thf(fact_202_add__eq__self__zero, axiom, (! [M:nat,N2:nat]: ((((((plus_plus @ nat) @ M) @ N2) = M) => (N2 = (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_203_Nat_OlessE, axiom, (! [I:nat,K:nat]: (((((ord_less @ nat) @ I) @ K) => ((K != (suc @ I)) => (~ ((! [J2:nat]: (((((ord_less @ nat) @ I) @ J2) => (K != (suc @ J2)))))))))))). % 6.41/2.68 thf(fact_204_Suc__lessD, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (suc @ M)) @ N2) => (((ord_less @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_205_Suc__lessE, axiom, (! [I:nat,K:nat]: (((((ord_less @ nat) @ (suc @ I)) @ K) => (~ ((! [J2:nat]: (((((ord_less @ nat) @ I) @ J2) => (K != (suc @ J2))))))))))). % 6.41/2.68 thf(fact_206_Suc__lessI, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ N2) => (((suc @ M) != N2) => (((ord_less @ nat) @ (suc @ M)) @ N2)))))). % 6.41/2.68 thf(fact_207_less__SucE, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ (suc @ N2)) => ((~ ((((ord_less @ nat) @ M) @ N2))) => (M = N2)))))). % 6.41/2.68 thf(fact_208_less__SucI, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ N2) => (((ord_less @ nat) @ M) @ (suc @ N2)))))). % 6.41/2.68 thf(fact_209_Ex__less__Suc, axiom, (! [N2:nat,P:(nat > $o)]: (((? [I3:nat]: (((((ord_less @ nat) @ I3) @ (suc @ N2)) & (P @ I3)))) = ((P @ N2) | (? [I3:nat]: (((((ord_less @ nat) @ I3) @ N2) & (P @ I3))))))))). % 6.41/2.68 thf(fact_210_less__Suc__eq, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ (suc @ N2)) = ((((ord_less @ nat) @ M) @ N2) | (M = N2)))))). % 6.41/2.68 thf(fact_211_not__less__eq, axiom, (! [M:nat,N2:nat]: (((~ ((((ord_less @ nat) @ M) @ N2))) = (((ord_less @ nat) @ N2) @ (suc @ M)))))). % 6.41/2.68 thf(fact_212_All__less__Suc, axiom, (! [N2:nat,P:(nat > $o)]: (((! [I3:nat]: (((((ord_less @ nat) @ I3) @ (suc @ N2)) => (P @ I3)))) = ((P @ N2) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ N2) => (P @ I3))))))))). % 6.41/2.68 thf(fact_213_Suc__less__eq2, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (suc @ N2)) @ M) = (? [M4:nat]: (((M = (suc @ M4)) & (((ord_less @ nat) @ N2) @ M4)))))))). % 6.41/2.68 thf(fact_214_less__antisym, axiom, (! [N2:nat,M:nat]: (((~ ((((ord_less @ nat) @ N2) @ M))) => ((((ord_less @ nat) @ N2) @ (suc @ M)) => (M = N2)))))). % 6.41/2.68 thf(fact_215_Suc__less__SucD, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (suc @ M)) @ (suc @ N2)) => (((ord_less @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_216_less__trans__Suc, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ J) @ K) => (((ord_less @ nat) @ (suc @ I)) @ K)))))). % 6.41/2.68 thf(fact_217_less__Suc__induct, axiom, (! [I:nat,J:nat,P:(nat > (nat > $o))]: (((((ord_less @ nat) @ I) @ J) => ((! [I4:nat]: (((P @ I4) @ (suc @ I4)))) => ((! [I4:nat,J2:nat,K2:nat]: (((((ord_less @ nat) @ I4) @ J2) => ((((ord_less @ nat) @ J2) @ K2) => (((P @ I4) @ J2) => (((P @ J2) @ K2) => ((P @ I4) @ K2))))))) => ((P @ I) @ J))))))). % 6.41/2.68 thf(fact_218_strict__inc__induct, axiom, (! [I:nat,J:nat,P:(nat > $o)]: (((((ord_less @ nat) @ I) @ J) => ((! [I4:nat]: (((J = (suc @ I4)) => (P @ I4)))) => ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ J) => ((P @ (suc @ I4)) => (P @ I4))))) => (P @ I))))))). % 6.41/2.68 thf(fact_219_not__less__less__Suc__eq, axiom, (! [N2:nat,M:nat]: (((~ ((((ord_less @ nat) @ N2) @ M))) => ((((ord_less @ nat) @ N2) @ (suc @ M)) = (N2 = M)))))). % 6.41/2.68 thf(fact_220_add__lessD1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ K) => (((ord_less @ nat) @ I) @ K))))). % 6.41/2.68 thf(fact_221_add__less__mono, axiom, (! [I:nat,J:nat,K:nat,L:nat]: (((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ K) @ L) => (((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ L))))))). % 6.41/2.68 thf(fact_222_not__add__less1, axiom, (! [I:nat,J:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ I)))))). % 6.41/2.68 thf(fact_223_not__add__less2, axiom, (! [J:nat,I:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ J) @ I)) @ I)))))). % 6.41/2.68 thf(fact_224_add__less__mono1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ K)))))). % 6.41/2.68 thf(fact_225_trans__less__add1, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ I) @ (((plus_plus @ nat) @ J) @ M)))))). % 6.41/2.68 thf(fact_226_trans__less__add2, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ I) @ (((plus_plus @ nat) @ M) @ J)))))). % 6.41/2.68 thf(fact_227_less__add__eq__less, axiom, (! [K:nat,L:nat,M:nat,N2:nat]: (((((ord_less @ nat) @ K) @ L) => (((((plus_plus @ nat) @ M) @ L) = (((plus_plus @ nat) @ K) @ N2)) => (((ord_less @ nat) @ M) @ N2)))))). % 6.41/2.68 thf(fact_228_Suc__leD, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N2) => (((ord_less_eq @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_229_le__SucE, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N2)) => ((~ ((((ord_less_eq @ nat) @ M) @ N2))) => (M = (suc @ N2))))))). % 6.41/2.68 thf(fact_230_le__SucI, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => (((ord_less_eq @ nat) @ M) @ (suc @ N2)))))). % 6.41/2.68 thf(fact_231_Suc__le__D, axiom, (! [N2:nat,M5:nat]: (((((ord_less_eq @ nat) @ (suc @ N2)) @ M5) => (? [M3:nat]: ((M5 = (suc @ M3)))))))). % 6.41/2.68 thf(fact_232_le__Suc__eq, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N2)) = ((((ord_less_eq @ nat) @ M) @ N2) | (M = (suc @ N2))))))). % 6.41/2.68 thf(fact_233_Suc__n__not__le__n, axiom, (! [N2:nat]: ((~ ((((ord_less_eq @ nat) @ (suc @ N2)) @ N2)))))). % 6.41/2.68 thf(fact_234_not__less__eq__eq, axiom, (! [M:nat,N2:nat]: (((~ ((((ord_less_eq @ nat) @ M) @ N2))) = (((ord_less_eq @ nat) @ (suc @ N2)) @ M))))). % 6.41/2.68 thf(fact_235_full__nat__induct, axiom, (! [P:(nat > $o),N2:nat]: (((! [N3:nat]: (((! [M6:nat]: (((((ord_less_eq @ nat) @ (suc @ M6)) @ N3) => (P @ M6)))) => (P @ N3)))) => (P @ N2))))). % 6.41/2.68 thf(fact_236_nat__induct__at__least, axiom, (! [M:nat,N2:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ M) @ N2) => ((P @ M) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ M) @ N3) => ((P @ N3) => (P @ (suc @ N3)))))) => (P @ N2))))))). % 6.41/2.68 thf(fact_237_transitive__stepwise__le, axiom, (! [M:nat,N2:nat,R:(nat > (nat > $o))]: (((((ord_less_eq @ nat) @ M) @ N2) => ((! [X5:nat]: (((R @ X5) @ X5))) => ((! [X5:nat,Y4:nat,Z3:nat]: ((((R @ X5) @ Y4) => (((R @ Y4) @ Z3) => ((R @ X5) @ Z3))))) => ((! [N3:nat]: (((R @ N3) @ (suc @ N3)))) => ((R @ M) @ N2)))))))). % 6.41/2.68 thf(fact_238_add__leE, axiom, (! [M:nat,K:nat,N2:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ N2) => (~ (((((ord_less_eq @ nat) @ M) @ N2) => (~ ((((ord_less_eq @ nat) @ K) @ N2)))))))))). % 6.41/2.68 thf(fact_239_le__add1, axiom, (! [N2:nat,M:nat]: ((((ord_less_eq @ nat) @ N2) @ (((plus_plus @ nat) @ N2) @ M))))). % 6.41/2.68 thf(fact_240_le__add2, axiom, (! [N2:nat,M:nat]: ((((ord_less_eq @ nat) @ N2) @ (((plus_plus @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_241_add__leD1, axiom, (! [M:nat,K:nat,N2:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ N2) => (((ord_less_eq @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_242_add__leD2, axiom, (! [M:nat,K:nat,N2:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ N2) => (((ord_less_eq @ nat) @ K) @ N2))))). % 6.41/2.68 thf(fact_243_le__Suc__ex, axiom, (! [K:nat,L:nat]: (((((ord_less_eq @ nat) @ K) @ L) => (? [N3:nat]: ((L = (((plus_plus @ nat) @ K) @ N3)))))))). % 6.41/2.68 thf(fact_244_add__le__mono, axiom, (! [I:nat,J:nat,K:nat,L:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ K) @ L) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ L))))))). % 6.41/2.68 thf(fact_245_add__le__mono1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ K)))))). % 6.41/2.68 thf(fact_246_trans__le__add1, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ I) @ (((plus_plus @ nat) @ J) @ M)))))). % 6.41/2.68 thf(fact_247_trans__le__add2, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ I) @ (((plus_plus @ nat) @ M) @ J)))))). % 6.41/2.68 thf(fact_248_nat__le__iff__add, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N:nat]: ((? [K3:nat]: ((N = (((plus_plus @ nat) @ M2) @ K3)))))))). % 6.41/2.68 thf(fact_249_subset__code_I1_J, axiom, (! [A:$tType,Xs:(list @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ B3) = (! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (((member @ A) @ X) @ B3)))))))). % 6.41/2.68 thf(fact_250_add__decreasing, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ B2))))))))). % 6.41/2.68 thf(fact_251_add__increasing, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C2)))))))))). % 6.41/2.68 thf(fact_252_add__decreasing2, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ B2))))))))). % 6.41/2.68 thf(fact_253_add__increasing2, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [C2:A,B2:A,A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less_eq @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C2)))))))))). % 6.41/2.68 thf(fact_254_add__nonneg__nonneg, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_255_add__nonpos__nonpos, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_256_add__nonneg__eq__0__iff, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((((plus_plus @ A) @ X3) @ Y) = (zero_zero @ A)) = ((X3 = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))))). % 6.41/2.68 thf(fact_257_add__nonpos__eq__0__iff, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ Y) @ (zero_zero @ A)) => (((((plus_plus @ A) @ X3) @ Y) = (zero_zero @ A)) = ((X3 = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))))). % 6.41/2.68 thf(fact_258_add__less__le__mono, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ D2) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ D2)))))))))). % 6.41/2.68 thf(fact_259_add__le__less__mono, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ C2) @ D2) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ D2)))))))))). % 6.41/2.68 thf(fact_260_add__mono__thms__linordered__field_I3_J, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))). % 6.41/2.68 thf(fact_261_add__mono__thms__linordered__field_I4_J, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))). % 6.41/2.68 thf(fact_262_pos__add__strict, axiom, (! [A:$tType]: (((strict7427464778891057005id_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C2)))))))))). % 6.41/2.68 thf(fact_263_canonically__ordered__monoid__add__class_OlessE, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((! [C3:A]: (((B2 = (((plus_plus @ A) @ A2) @ C3)) => (C3 = (zero_zero @ A)))))))))))))). % 6.41/2.68 thf(fact_264_add__pos__pos, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_265_add__neg__neg, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_266_lift__Suc__mono__less, axiom, (! [A:$tType]: (((order @ A) => (! [F2:(nat > A),N2:nat,N4:nat]: (((! [N3:nat]: ((((ord_less @ A) @ (F2 @ N3)) @ (F2 @ (suc @ N3))))) => ((((ord_less @ nat) @ N2) @ N4) => (((ord_less @ A) @ (F2 @ N2)) @ (F2 @ N4)))))))))). % 6.41/2.68 thf(fact_267_lift__Suc__mono__less__iff, axiom, (! [A:$tType]: (((order @ A) => (! [F2:(nat > A),N2:nat,M:nat]: (((! [N3:nat]: ((((ord_less @ A) @ (F2 @ N3)) @ (F2 @ (suc @ N3))))) => ((((ord_less @ A) @ (F2 @ N2)) @ (F2 @ M)) = (((ord_less @ nat) @ N2) @ M))))))))). % 6.41/2.68 thf(fact_268_lift__Suc__mono__le, axiom, (! [A:$tType]: (((order @ A) => (! [F2:(nat > A),N2:nat,N4:nat]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (F2 @ N3)) @ (F2 @ (suc @ N3))))) => ((((ord_less_eq @ nat) @ N2) @ N4) => (((ord_less_eq @ A) @ (F2 @ N2)) @ (F2 @ N4)))))))))). % 6.41/2.68 thf(fact_269_lift__Suc__antimono__le, axiom, (! [A:$tType]: (((order @ A) => (! [F2:(nat > A),N2:nat,N4:nat]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (F2 @ (suc @ N3))) @ (F2 @ N3)))) => ((((ord_less_eq @ nat) @ N2) @ N4) => (((ord_less_eq @ A) @ (F2 @ N4)) @ (F2 @ N2)))))))))). % 6.41/2.68 thf(fact_270_less__Suc__eq__0__disj, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ (suc @ N2)) = ((M = (zero_zero @ nat)) | (? [J3:nat]: (((M = (suc @ J3)) & (((ord_less @ nat) @ J3) @ N2))))))))). % 6.41/2.68 thf(fact_271_gr0__implies__Suc, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (? [M3:nat]: ((N2 = (suc @ M3)))))))). % 6.41/2.68 thf(fact_272_All__less__Suc2, axiom, (! [N2:nat,P:(nat > $o)]: (((! [I3:nat]: (((((ord_less @ nat) @ I3) @ (suc @ N2)) => (P @ I3)))) = ((P @ (zero_zero @ nat)) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ N2) => (P @ (suc @ I3)))))))))). % 6.41/2.68 thf(fact_273_gr0__conv__Suc, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) = (? [M2:nat]: ((N2 = (suc @ M2)))))))). % 6.41/2.68 thf(fact_274_Ex__less__Suc2, axiom, (! [N2:nat,P:(nat > $o)]: (((? [I3:nat]: (((((ord_less @ nat) @ I3) @ (suc @ N2)) & (P @ I3)))) = ((P @ (zero_zero @ nat)) | (? [I3:nat]: (((((ord_less @ nat) @ I3) @ N2) & (P @ (suc @ I3)))))))))). % 6.41/2.68 thf(fact_275_less__imp__add__positive, axiom, (! [I:nat,J:nat]: (((((ord_less @ nat) @ I) @ J) => (? [K2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K2) & ((((plus_plus @ nat) @ I) @ K2) = J)))))))). % 6.41/2.68 thf(fact_276_le__imp__less__Suc, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => (((ord_less @ nat) @ M) @ (suc @ N2)))))). % 6.41/2.68 thf(fact_277_less__eq__Suc__le, axiom, ((ord_less @ nat) = (^ [N:nat]: (((ord_less_eq @ nat) @ (suc @ N)))))). % 6.41/2.68 thf(fact_278_less__Suc__eq__le, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ (suc @ N2)) = (((ord_less_eq @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_279_le__less__Suc__eq, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((ord_less @ nat) @ N2) @ (suc @ M)) = (N2 = M)))))). % 6.41/2.68 thf(fact_280_Suc__le__lessD, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N2) => (((ord_less @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_281_inc__induct, axiom, (! [I:nat,J:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ I) @ J) => ((P @ J) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ I) @ N3) => ((((ord_less @ nat) @ N3) @ J) => ((P @ (suc @ N3)) => (P @ N3)))))) => (P @ I))))))). % 6.41/2.68 thf(fact_282_dec__induct, axiom, (! [I:nat,J:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ I) @ J) => ((P @ I) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ I) @ N3) => ((((ord_less @ nat) @ N3) @ J) => ((P @ N3) => (P @ (suc @ N3))))))) => (P @ J))))))). % 6.41/2.68 thf(fact_283_Suc__le__eq, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N2) = (((ord_less @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_284_Suc__leI, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ N2) => (((ord_less_eq @ nat) @ (suc @ M)) @ N2))))). % 6.41/2.68 thf(fact_285_mono__nat__linear__lb, axiom, (! [F2:(nat > nat),M:nat,K:nat]: (((! [M3:nat,N3:nat]: (((((ord_less @ nat) @ M3) @ N3) => (((ord_less @ nat) @ (F2 @ M3)) @ (F2 @ N3))))) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (F2 @ M)) @ K)) @ (F2 @ (((plus_plus @ nat) @ M) @ K))))))). % 6.41/2.68 thf(fact_286_add__strict__increasing2, axiom, (! [A:$tType]: (((ordere8940638589300402666id_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C2)))))))))). % 6.41/2.68 thf(fact_287_add__strict__increasing, axiom, (! [A:$tType]: (((ordere8940638589300402666id_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C2)))))))))). % 6.41/2.68 thf(fact_288_add__pos__nonneg, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_289_add__nonpos__neg, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_290_add__nonneg__pos, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_291_add__neg__nonpos, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_292_ex__least__nat__less, axiom, (! [P:(nat > $o),N2:nat]: (((P @ N2) => ((~ ((P @ (zero_zero @ nat)))) => (? [K2:nat]: (((((ord_less @ nat) @ K2) @ N2) & ((! [I5:nat]: (((((ord_less_eq @ nat) @ I5) @ K2) => (~ ((P @ I5)))))) & (P @ (suc @ K2))))))))))). % 6.41/2.68 thf(fact_293_set__update__subsetI, axiom, (! [A:$tType,Xs:(list @ A),A3:(set @ A),X3:A,I:nat]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ A3) => ((((member @ A) @ X3) @ A3) => (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ ((((list_update @ A) @ Xs) @ I) @ X3))) @ A3)))))). % 6.41/2.68 thf(fact_294_nle__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((~ ((((ord_less_eq @ A) @ A2) @ B2))) = ((((ord_less_eq @ A) @ B2) @ A2) & (B2 != A2))))))))). % 6.41/2.68 thf(fact_295_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A,Z2:A]: ((((((ord_less_eq @ A) @ X3) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z2)))) => (((((ord_less_eq @ A) @ Y) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Z2)))) => (((((ord_less_eq @ A) @ X3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y)))) => (((((ord_less_eq @ A) @ Z2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X3)))) => (((((ord_less_eq @ A) @ Y) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X3)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Y))))))))))))))))). % 6.41/2.68 thf(fact_296_order__class_Oorder__eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) & (((ord_less_eq @ A) @ Y2) @ X))))))))). % 6.41/2.68 thf(fact_297_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))). % 6.41/2.68 thf(fact_298_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))). % 6.41/2.68 thf(fact_299_order__antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_less_eq @ A) @ Y) @ X3) => (X3 = Y))))))))). % 6.41/2.68 thf(fact_300_order_Otrans, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))). % 6.41/2.68 thf(fact_301_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less_eq @ A) @ X3) @ Z2))))))))). % 6.41/2.68 thf(fact_302_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((P @ A5) @ B5)))) => ((! [A5:A,B5:A]: ((((P @ B5) @ A5) => ((P @ A5) @ B5)))) => ((P @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_303_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (((ord_less_eq @ A) @ A4) @ B4))))))))). % 6.41/2.68 thf(fact_304_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))). % 6.41/2.68 thf(fact_305_dual__order_Otrans, axiom, (! [A:$tType]: (((preorder @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ A2))))))))). % 6.41/2.68 thf(fact_306_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))). % 6.41/2.68 thf(fact_307_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G) => (((ord_less_eq @ B) @ (F2 @ X3)) @ (G @ X3))))))))). % 6.41/2.68 thf(fact_308_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G) => (((ord_less_eq @ B) @ (F2 @ X3)) @ (G @ X3))))))))). % 6.41/2.68 thf(fact_309_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G:(A > B)]: (((! [X5:A]: ((((ord_less_eq @ B) @ (F2 @ X5)) @ (G @ X5)))) => (((ord_less_eq @ (A > B)) @ F2) @ G)))))))). % 6.41/2.68 thf(fact_310_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F3:(A > B),G2:(A > B)]: ((! [X:A]: ((((ord_less_eq @ B) @ (F3 @ X)) @ (G2 @ X))))))))))). % 6.41/2.68 thf(fact_311_Orderings_Oorder__eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (((ord_less_eq @ A) @ B4) @ A4))))))))). % 6.41/2.68 thf(fact_312_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F2:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F2 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X5:B,Y4:B]: (((((ord_less_eq @ B) @ X5) @ Y4) => (((ord_less_eq @ A) @ (F2 @ X5)) @ (F2 @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F2 @ C2))))))))))). % 6.41/2.68 thf(fact_313_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F2:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F2 @ B2)) @ C2) => ((! [X5:A,Y4:A]: (((((ord_less_eq @ A) @ X5) @ Y4) => (((ord_less_eq @ C) @ (F2 @ X5)) @ (F2 @ Y4))))) => (((ord_less_eq @ C) @ (F2 @ A2)) @ C2)))))))))). % 6.41/2.68 thf(fact_314_order__eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y:A]: (((X3 = Y) => (((ord_less_eq @ A) @ X3) @ Y)))))))). % 6.41/2.68 thf(fact_315_linorder__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) | (((ord_less_eq @ A) @ Y) @ X3)))))))). % 6.41/2.68 thf(fact_316_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F2:(B > A),B2:B,C2:B]: (((A2 = (F2 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X5:B,Y4:B]: (((((ord_less_eq @ B) @ X5) @ Y4) => (((ord_less_eq @ A) @ (F2 @ X5)) @ (F2 @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F2 @ C2))))))))))). % 6.41/2.68 thf(fact_317_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F2:(A > B),C2:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F2 @ B2) = C2) => ((! [X5:A,Y4:A]: (((((ord_less_eq @ A) @ X5) @ Y4) => (((ord_less_eq @ B) @ (F2 @ X5)) @ (F2 @ Y4))))) => (((ord_less_eq @ B) @ (F2 @ A2)) @ C2)))))))))). % 6.41/2.68 thf(fact_318_linorder__le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y))) => (((ord_less_eq @ A) @ Y) @ X3)))))))). % 6.41/2.68 thf(fact_319_order__antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X3:A]: (((((ord_less_eq @ A) @ Y) @ X3) => ((((ord_less_eq @ A) @ X3) @ Y) = (X3 = Y))))))))). % 6.41/2.68 thf(fact_320_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X3:A]: ((((zero_zero @ A) = X3) = (X3 = (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_321_lt__ex, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X3:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X3))))))))). % 6.41/2.68 thf(fact_322_gt__ex, axiom, (! [A:$tType]: (((no_top @ A) => (! [X3:A]: ((? [X_1:A]: ((((ord_less @ A) @ X3) @ X_1))))))))). % 6.41/2.68 thf(fact_323_dense, axiom, (! [A:$tType]: (((dense_order @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (? [Z3:A]: (((((ord_less @ A) @ X3) @ Z3) & (((ord_less @ A) @ Z3) @ Y))))))))))). % 6.41/2.68 thf(fact_324_less__imp__neq, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (X3 != Y)))))))). % 6.41/2.68 thf(fact_325_order_Oasym, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))). % 6.41/2.68 thf(fact_326_ord__eq__less__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((A2 = B2) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))). % 6.41/2.68 thf(fact_327_ord__less__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((B2 = C2) => (((ord_less @ A) @ A2) @ C2))))))))). % 6.41/2.68 thf(fact_328_less__induct, axiom, (! [A:$tType]: (((wellorder @ A) => (! [P:(A > $o),A2:A]: (((! [X5:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X5) => (P @ Y5)))) => (P @ X5)))) => (P @ A2)))))))). % 6.41/2.68 thf(fact_329_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X3:A]: (((~ ((((ord_less @ A) @ Y) @ X3))) => ((~ ((((ord_less @ A) @ X3) @ Y))) = (X3 = Y))))))))). % 6.41/2.68 thf(fact_330_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((~ ((((ord_less @ A) @ X3) @ Y))) => ((X3 != Y) => (((ord_less @ A) @ Y) @ X3))))))))). % 6.41/2.68 thf(fact_331_dual__order_Oasym, axiom, (! [A:$tType]: (((preorder @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (~ ((((ord_less @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_332_dual__order_Oirrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))). % 6.41/2.68 thf(fact_333_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P2:(A > $o)]: ((? [X7:A]: ((P2 @ X7))))) = (^ [P3:(A > $o)]: ((? [N:A]: (((P3 @ N) & (! [M2:A]: (((((ord_less @ A) @ M2) @ N) => (~ ((P3 @ M2)))))))))))))))). % 6.41/2.68 thf(fact_334_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A5:A,B5:A]: (((((ord_less @ A) @ A5) @ B5) => ((P @ A5) @ B5)))) => ((! [A5:A]: (((P @ A5) @ A5))) => ((! [A5:A,B5:A]: ((((P @ B5) @ A5) => ((P @ A5) @ B5)))) => ((P @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_335_order_Ostrict__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))). % 6.41/2.68 thf(fact_336_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((~ ((((ord_less @ A) @ X3) @ Y))) = ((((ord_less @ A) @ Y) @ X3) | (X3 = Y))))))))). % 6.41/2.68 thf(fact_337_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ A2))))))))). % 6.41/2.68 thf(fact_338_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (A2 != B2)))))))). % 6.41/2.68 thf(fact_339_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (A2 != B2)))))))). % 6.41/2.68 thf(fact_340_measure__induct, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F2:(A > B),P:(A > $o),A2:A]: (((! [X5:A]: (((! [Y5:A]: (((((ord_less @ B) @ (F2 @ Y5)) @ (F2 @ X5)) => (P @ Y5)))) => (P @ X5)))) => (P @ A2)))))))). % 6.41/2.68 thf(fact_341_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F2:(A > B),P:(A > $o),A2:A]: (((! [X5:A]: (((! [Y5:A]: (((((ord_less @ B) @ (F2 @ Y5)) @ (F2 @ X5)) => (P @ Y5)))) => (P @ X5)))) => (P @ A2)))))))). % 6.41/2.68 thf(fact_342_linorder__neqE, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((X3 != Y) => ((~ ((((ord_less @ A) @ X3) @ Y))) => (((ord_less @ A) @ Y) @ X3))))))))). % 6.41/2.68 thf(fact_343_order__less__asym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (~ ((((ord_less @ A) @ Y) @ X3)))))))))). % 6.41/2.68 thf(fact_344_linorder__neq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((X3 != Y) = ((((ord_less @ A) @ X3) @ Y) | (((ord_less @ A) @ Y) @ X3))))))))). % 6.41/2.68 thf(fact_345_order__less__asym_H, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))). % 6.41/2.68 thf(fact_346_order__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y:A,Z2:A]: (((((ord_less @ A) @ X3) @ Y) => ((((ord_less @ A) @ Y) @ Z2) => (((ord_less @ A) @ X3) @ Z2))))))))). % 6.41/2.68 thf(fact_347_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F2:(B > A),B2:B,C2:B]: (((A2 = (F2 @ B2)) => ((((ord_less @ B) @ B2) @ C2) => ((! [X5:B,Y4:B]: (((((ord_less @ B) @ X5) @ Y4) => (((ord_less @ A) @ (F2 @ X5)) @ (F2 @ Y4))))) => (((ord_less @ A) @ A2) @ (F2 @ C2))))))))))). % 6.41/2.68 thf(fact_348_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F2:(A > B),C2:B]: (((((ord_less @ A) @ A2) @ B2) => (((F2 @ B2) = C2) => ((! [X5:A,Y4:A]: (((((ord_less @ A) @ X5) @ Y4) => (((ord_less @ B) @ (F2 @ X5)) @ (F2 @ Y4))))) => (((ord_less @ B) @ (F2 @ A2)) @ C2)))))))))). % 6.41/2.68 thf(fact_349_order__less__irrefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A]: ((~ ((((ord_less @ A) @ X3) @ X3))))))))). % 6.41/2.68 thf(fact_350_order__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F2:(B > A),B2:B,C2:B]: (((((ord_less @ A) @ A2) @ (F2 @ B2)) => ((((ord_less @ B) @ B2) @ C2) => ((! [X5:B,Y4:B]: (((((ord_less @ B) @ X5) @ Y4) => (((ord_less @ A) @ (F2 @ X5)) @ (F2 @ Y4))))) => (((ord_less @ A) @ A2) @ (F2 @ C2))))))))))). % 6.41/2.68 thf(fact_351_order__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F2:(A > C),C2:C]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ C) @ (F2 @ B2)) @ C2) => ((! [X5:A,Y4:A]: (((((ord_less @ A) @ X5) @ Y4) => (((ord_less @ C) @ (F2 @ X5)) @ (F2 @ Y4))))) => (((ord_less @ C) @ (F2 @ A2)) @ C2)))))))))). % 6.41/2.68 thf(fact_352_order__less__not__sym, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (~ ((((ord_less @ A) @ Y) @ X3)))))))))). % 6.41/2.68 thf(fact_353_order__less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y:A,P:$o]: (((((ord_less @ A) @ X3) @ Y) => ((((ord_less @ A) @ Y) @ X3) => P)))))))). % 6.41/2.68 thf(fact_354_linorder__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) | ((X3 = Y) | (((ord_less @ A) @ Y) @ X3))))))))). % 6.41/2.68 thf(fact_355_order__less__imp__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (X3 != Y)))))))). % 6.41/2.68 thf(fact_356_order__less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (Y != X3)))))))). % 6.41/2.68 thf(fact_357_order__less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (~ ((((ord_less @ A) @ Y) @ X3)))))))))). % 6.41/2.68 thf(fact_358_nat__neq__iff, axiom, (! [M:nat,N2:nat]: (((M != N2) = ((((ord_less @ nat) @ M) @ N2) | (((ord_less @ nat) @ N2) @ M)))))). % 6.41/2.68 thf(fact_359_less__not__refl, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ N2)))))). % 6.41/2.68 thf(fact_360_less__not__refl2, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ N2) @ M) => (M != N2))))). % 6.41/2.68 thf(fact_361_less__not__refl3, axiom, (! [S:nat,T2:nat]: (((((ord_less @ nat) @ S) @ T2) => (S != T2))))). % 6.41/2.68 thf(fact_362_less__irrefl__nat, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ N2)))))). % 6.41/2.68 thf(fact_363_nat__less__induct, axiom, (! [P:(nat > $o),N2:nat]: (((! [N3:nat]: (((! [M6:nat]: (((((ord_less @ nat) @ M6) @ N3) => (P @ M6)))) => (P @ N3)))) => (P @ N2))))). % 6.41/2.68 thf(fact_364_infinite__descent, axiom, (! [P:(nat > $o),N2:nat]: (((! [N3:nat]: (((~ ((P @ N3))) => (? [M6:nat]: (((((ord_less @ nat) @ M6) @ N3) & (~ ((P @ M6))))))))) => (P @ N2))))). % 6.41/2.68 thf(fact_365_linorder__neqE__nat, axiom, (! [X3:nat,Y:nat]: (((X3 != Y) => ((~ ((((ord_less @ nat) @ X3) @ Y))) => (((ord_less @ nat) @ Y) @ X3)))))). % 6.41/2.68 thf(fact_366_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X3:A]: (((! [X5:A]: (((~ ((P @ X5))) => (? [Y5:A]: (((((ord_less @ nat) @ (V @ Y5)) @ (V @ X5)) & (~ ((P @ Y5))))))))) => (P @ X3))))). % 6.41/2.68 thf(fact_367_le__refl, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ N2) @ N2)))). % 6.41/2.68 thf(fact_368_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))). % 6.41/2.68 thf(fact_369_eq__imp__le, axiom, (! [M:nat,N2:nat]: (((M = N2) => (((ord_less_eq @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_370_le__antisym, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((ord_less_eq @ nat) @ N2) @ M) => (M = N2)))))). % 6.41/2.68 thf(fact_371_nat__le__linear, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) | (((ord_less_eq @ nat) @ N2) @ M))))). % 6.41/2.68 thf(fact_372_Nat_Oex__has__greatest__nat, axiom, (! [P:(nat > $o),K:nat,B2:nat]: (((P @ K) => ((! [Y4:nat]: (((P @ Y4) => (((ord_less_eq @ nat) @ Y4) @ B2)))) => (? [X5:nat]: (((P @ X5) & (! [Y5:nat]: (((P @ Y5) => (((ord_less_eq @ nat) @ Y5) @ X5)))))))))))). % 6.41/2.68 thf(fact_373_size__neq__size__imp__neq, axiom, (! [A:$tType]: (((size @ A) => (! [X3:A,Y:A]: (((((size_size @ A) @ X3) != ((size_size @ A) @ Y)) => (X3 != Y)))))))). % 6.41/2.68 thf(fact_374_zero__le, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3))))))). % 6.41/2.68 thf(fact_375_leD, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X3:A]: (((((ord_less_eq @ A) @ Y) @ X3) => (~ ((((ord_less @ A) @ X3) @ Y)))))))))). % 6.41/2.68 thf(fact_376_leI, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((~ ((((ord_less @ A) @ X3) @ Y))) => (((ord_less_eq @ A) @ Y) @ X3)))))))). % 6.41/2.68 thf(fact_377_nless__le, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((~ ((((ord_less @ A) @ A2) @ B2))) = ((~ ((((ord_less_eq @ A) @ A2) @ B2))) | (A2 = B2))))))))). % 6.41/2.68 thf(fact_378_antisym__conv1, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y:A]: (((~ ((((ord_less @ A) @ X3) @ Y))) => ((((ord_less_eq @ A) @ X3) @ Y) = (X3 = Y))))))))). % 6.41/2.68 thf(fact_379_antisym__conv2, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => ((~ ((((ord_less @ A) @ X3) @ Y))) = (X3 = Y))))))))). % 6.41/2.68 thf(fact_380_dense__ge, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,Y:A]: (((! [X5:A]: (((((ord_less @ A) @ Z2) @ X5) => (((ord_less_eq @ A) @ Y) @ X5)))) => (((ord_less_eq @ A) @ Y) @ Z2)))))))). % 6.41/2.68 thf(fact_381_dense__le, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Y:A,Z2:A]: (((! [X5:A]: (((((ord_less @ A) @ X5) @ Y) => (((ord_less_eq @ A) @ X5) @ Z2)))) => (((ord_less_eq @ A) @ Y) @ Z2)))))))). % 6.41/2.68 thf(fact_382_less__le__not__le, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) & (~ ((((ord_less_eq @ A) @ Y2) @ X))))))))))). % 6.41/2.68 thf(fact_383_not__le__imp__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X3:A]: (((~ ((((ord_less_eq @ A) @ Y) @ X3))) => (((ord_less @ A) @ X3) @ Y)))))))). % 6.41/2.68 thf(fact_384_order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) | (A4 = B4))))))))). % 6.41/2.68 thf(fact_385_order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (A4 != B4))))))))). % 6.41/2.68 thf(fact_386_order_Ostrict__trans1, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))). % 6.41/2.68 thf(fact_387_order_Ostrict__trans2, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less @ A) @ A2) @ C2))))))))). % 6.41/2.68 thf(fact_388_order_Ostrict__iff__not, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (~ ((((ord_less_eq @ A) @ B4) @ A4))))))))))). % 6.41/2.68 thf(fact_389_dense__ge__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [Z2:A,X3:A,Y:A]: (((((ord_less @ A) @ Z2) @ X3) => ((! [W:A]: (((((ord_less @ A) @ Z2) @ W) => ((((ord_less @ A) @ W) @ X3) => (((ord_less_eq @ A) @ Y) @ W))))) => (((ord_less_eq @ A) @ Y) @ Z2))))))))). % 6.41/2.68 thf(fact_390_dense__le__bounded, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [X3:A,Y:A,Z2:A]: (((((ord_less @ A) @ X3) @ Y) => ((! [W:A]: (((((ord_less @ A) @ X3) @ W) => ((((ord_less @ A) @ W) @ Y) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y) @ Z2))))))))). % 6.41/2.68 thf(fact_391_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: (((((ord_less @ A) @ B4) @ A4) | (A4 = B4))))))))). % 6.41/2.68 thf(fact_392_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [B4:A,A4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (A4 != B4))))))))). % 6.41/2.68 thf(fact_393_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: (((preorder @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ A2))))))))). % 6.41/2.68 thf(fact_394_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: (((preorder @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ A2))))))))). % 6.41/2.68 thf(fact_395_dual__order_Ostrict__iff__not, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ A) = (^ [B4:A,A4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (~ ((((ord_less_eq @ A) @ A4) @ B4))))))))))). % 6.41/2.68 thf(fact_396_order_Ostrict__implies__order, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ A2) @ B2)))))))). % 6.41/2.68 thf(fact_397_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: (((preorder @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ B2) @ A2)))))))). % 6.41/2.68 thf(fact_398_order__le__less, axiom, (! [A:$tType]: (((order @ A) => ((ord_less_eq @ A) = (^ [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) | (X = Y2))))))))). % 6.41/2.68 thf(fact_399_order__less__le, axiom, (! [A:$tType]: (((order @ A) => ((ord_less @ A) = (^ [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) & (X != Y2))))))))). % 6.41/2.68 thf(fact_400_linorder__not__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y))) = (((ord_less @ A) @ Y) @ X3)))))))). % 6.41/2.68 thf(fact_401_linorder__not__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((~ ((((ord_less @ A) @ X3) @ Y))) = (((ord_less_eq @ A) @ Y) @ X3)))))))). % 6.41/2.68 thf(fact_402_order__less__imp__le, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (((ord_less_eq @ A) @ X3) @ Y)))))))). % 6.41/2.68 thf(fact_403_order__le__neq__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((A2 != B2) => (((ord_less @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_404_order__neq__le__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((A2 != B2) => ((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_405_order__le__less__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_less @ A) @ Y) @ Z2) => (((ord_less @ A) @ X3) @ Z2))))))))). % 6.41/2.68 thf(fact_406_order__less__le__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y:A,Z2:A]: (((((ord_less @ A) @ X3) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less @ A) @ X3) @ Z2))))))))). % 6.41/2.68 thf(fact_407_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F2:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F2 @ B2)) => ((((ord_less @ B) @ B2) @ C2) => ((! [X5:B,Y4:B]: (((((ord_less @ B) @ X5) @ Y4) => (((ord_less @ A) @ (F2 @ X5)) @ (F2 @ Y4))))) => (((ord_less @ A) @ A2) @ (F2 @ C2))))))))))). % 6.41/2.68 thf(fact_408_order__le__less__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F2:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ C) @ (F2 @ B2)) @ C2) => ((! [X5:A,Y4:A]: (((((ord_less_eq @ A) @ X5) @ Y4) => (((ord_less_eq @ C) @ (F2 @ X5)) @ (F2 @ Y4))))) => (((ord_less @ C) @ (F2 @ A2)) @ C2)))))))))). % 6.41/2.68 thf(fact_409_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F2:(B > A),B2:B,C2:B]: (((((ord_less @ A) @ A2) @ (F2 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X5:B,Y4:B]: (((((ord_less_eq @ B) @ X5) @ Y4) => (((ord_less_eq @ A) @ (F2 @ X5)) @ (F2 @ Y4))))) => (((ord_less @ A) @ A2) @ (F2 @ C2))))))))))). % 6.41/2.68 thf(fact_410_order__less__le__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F2:(A > C),C2:C]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F2 @ B2)) @ C2) => ((! [X5:A,Y4:A]: (((((ord_less @ A) @ X5) @ Y4) => (((ord_less @ C) @ (F2 @ X5)) @ (F2 @ Y4))))) => (((ord_less @ C) @ (F2 @ A2)) @ C2)))))))))). % 6.41/2.68 thf(fact_411_linorder__le__less__linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) | (((ord_less @ A) @ Y) @ X3)))))))). % 6.41/2.68 thf(fact_412_order__le__imp__less__or__eq, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_less @ A) @ X3) @ Y) | (X3 = Y))))))))). % 6.41/2.68 thf(fact_413_gr__zeroI, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [N2:A]: (((N2 != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N2)))))))). % 6.41/2.68 thf(fact_414_not__less__zero, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [N2:A]: ((~ ((((ord_less @ A) @ N2) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_415_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [M:A,N2:A]: (((((ord_less @ A) @ M) @ N2) => (N2 != (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_416_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [N2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N2) = (N2 != (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_417_bot__nat__0_Oextremum__strict, axiom, (! [A2:nat]: ((~ ((((ord_less @ nat) @ A2) @ (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_418_gr0I, axiom, (! [N2:nat]: (((N2 != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))). % 6.41/2.68 thf(fact_419_not__gr0, axiom, (! [N2:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2))) = (N2 = (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_420_not__less0, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_421_less__zeroE, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_422_gr__implies__not0, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ N2) => (N2 != (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_423_infinite__descent0, axiom, (! [P:(nat > $o),N2:nat]: (((P @ (zero_zero @ nat)) => ((! [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) => ((~ ((P @ N3))) => (? [M6:nat]: (((((ord_less @ nat) @ M6) @ N3) & (~ ((P @ M6)))))))))) => (P @ N2)))))). % 6.41/2.68 thf(fact_424_infinite__descent0__measure, axiom, (! [A:$tType,V:(A > nat),P:(A > $o),X3:A]: (((! [X5:A]: ((((V @ X5) = (zero_zero @ nat)) => (P @ X5)))) => ((! [X5:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V @ X5)) => ((~ ((P @ X5))) => (? [Y5:A]: (((((ord_less @ nat) @ (V @ Y5)) @ (V @ X5)) & (~ ((P @ Y5)))))))))) => (P @ X3)))))). % 6.41/2.68 thf(fact_425_less__eq__nat_Osimps_I1_J, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N2)))). % 6.41/2.68 thf(fact_426_bot__nat__0_Oextremum__unique, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) = (A2 = (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_427_bot__nat__0_Oextremum__uniqueI, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) => (A2 = (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_428_le__0__eq, axiom, (! [N2:nat]: (((((ord_less_eq @ nat) @ N2) @ (zero_zero @ nat)) = (N2 = (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_429_nat__less__le, axiom, ((ord_less @ nat) = (^ [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) & (M2 != N)))))). % 6.41/2.68 thf(fact_430_less__imp__le__nat, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ N2) => (((ord_less_eq @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_431_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N:nat]: (((((ord_less @ nat) @ M2) @ N) | (M2 = N)))))). % 6.41/2.68 thf(fact_432_less__or__eq__imp__le, axiom, (! [M:nat,N2:nat]: ((((((ord_less @ nat) @ M) @ N2) | (M = N2)) => (((ord_less_eq @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_433_le__neq__implies__less, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => ((M != N2) => (((ord_less @ nat) @ M) @ N2)))))). % 6.41/2.68 thf(fact_434_less__mono__imp__le__mono, axiom, (! [F2:(nat > nat),I:nat,J:nat]: (((! [I4:nat,J2:nat]: (((((ord_less @ nat) @ I4) @ J2) => (((ord_less @ nat) @ (F2 @ I4)) @ (F2 @ J2))))) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (F2 @ I)) @ (F2 @ J))))))). % 6.41/2.68 thf(fact_435_double__eq__0__iff, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A2:A]: ((((((plus_plus @ A) @ A2) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_436_assms_I1_J, axiom, ((vEBT_invar_vebt @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ mi) @ ma))) @ deg) @ treeList) @ summary)) @ deg)). % 6.41/2.68 thf(fact_437_mint__corr__help__empty, axiom, (! [T2:vEBT_VEBT,N2:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => (((vEBT_vebt_mint @ T2) = (none @ nat)) => ((vEBT_VEBT_set_vebt @ T2) = (bot_bot @ (set @ nat)))))))). % 6.41/2.68 thf(fact_438_maxt__corr__help__empty, axiom, (! [T2:vEBT_VEBT,N2:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => (((vEBT_vebt_maxt @ T2) = (none @ nat)) => ((vEBT_VEBT_set_vebt @ T2) = (bot_bot @ (set @ nat)))))))). % 6.41/2.68 thf(fact_439_field__le__epsilon, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: (((! [E:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E) => (((ord_less_eq @ A) @ X3) @ (((plus_plus @ A) @ Y) @ E))))) => (((ord_less_eq @ A) @ X3) @ Y)))))))). % 6.41/2.68 thf(fact_440_ex__has__greatest__nat__lemma, axiom, (! [A:$tType,P:(A > $o),K:A,F2:(A > nat),N2:nat]: (((P @ K) => ((! [X5:A]: (((P @ X5) => (? [Y5:A]: (((P @ Y5) & (~ ((((ord_less_eq @ nat) @ (F2 @ Y5)) @ (F2 @ X5)))))))))) => (? [Y4:A]: (((P @ Y4) & (~ ((((ord_less @ nat) @ (F2 @ Y4)) @ (((plus_plus @ nat) @ (F2 @ K)) @ N2)))))))))))). % 6.41/2.68 thf(fact_441_option_Osize__gen_I2_J, axiom, (! [A:$tType,X3:(A > nat),X22:A]: (((((size_option @ A) @ X3) @ ((some @ A) @ X22)) = (((plus_plus @ nat) @ (X3 @ X22)) @ (suc @ (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_442_add__less__zeroD, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ X3) @ Y)) @ (zero_zero @ A)) => ((((ord_less @ A) @ X3) @ (zero_zero @ A)) | (((ord_less @ A) @ Y) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_443_empty__subsetI, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3)))). % 6.41/2.68 thf(fact_444_subset__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = (A3 = (bot_bot @ (set @ A))))))). % 6.41/2.68 thf(fact_445_empty__iff, axiom, (! [A:$tType,C2:A]: ((~ ((((member @ A) @ C2) @ (bot_bot @ (set @ A)))))))). % 6.41/2.68 thf(fact_446_all__not__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X:A]: ((~ ((((member @ A) @ X) @ A3))))) = (A3 = (bot_bot @ (set @ A))))))). % 6.41/2.68 thf(fact_447_deg__deg__n, axiom, (! [Info:(option @ ((product_prod @ nat) @ nat)),Deg:nat,TreeList2:(list @ vEBT_VEBT),Summary:vEBT_VEBT,N2:nat]: ((((vEBT_invar_vebt @ ((((vEBT_Node @ Info) @ Deg) @ TreeList2) @ Summary)) @ N2) => (Deg = N2))))). % 6.41/2.68 thf(fact_448_deg__SUcn__Node, axiom, (! [Tree:vEBT_VEBT,N2:nat]: ((((vEBT_invar_vebt @ Tree) @ (suc @ (suc @ N2))) => (? [Info2:(option @ ((product_prod @ nat) @ nat)),TreeList3:(list @ vEBT_VEBT),S2:vEBT_VEBT]: ((Tree = ((((vEBT_Node @ Info2) @ (suc @ (suc @ N2))) @ TreeList3) @ S2)))))))). % 6.41/2.68 thf(fact_449_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X:A]: ((~ ((P @ X))))))))). % 6.41/2.68 thf(fact_450_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X:A]: ((~ ((P @ X))))))))). % 6.41/2.68 thf(fact_451_psubsetI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((A3 != B3) => (((ord_less @ (set @ A)) @ A3) @ B3)))))). % 6.41/2.68 thf(fact_452_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => (A3 = B3)))))). % 6.41/2.68 thf(fact_453_subsetI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((member @ A) @ X5) @ B3)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))). % 6.41/2.68 thf(fact_454_mi__eq__ma__no__ch, axiom, (! [Mi:nat,Ma:nat,Deg:nat,TreeList2:(list @ vEBT_VEBT),Summary:vEBT_VEBT]: ((((vEBT_invar_vebt @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi) @ Ma))) @ Deg) @ TreeList2) @ Summary)) @ Deg) => ((Mi = Ma) => ((! [X2:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X2) @ ((set2 @ vEBT_VEBT) @ TreeList2)) => (~ ((? [X_12:nat]: (((vEBT_V8194947554948674370ptions @ X2) @ X_12)))))))) & (~ ((? [X_12:nat]: (((vEBT_V8194947554948674370ptions @ Summary) @ X_12))))))))))). % 6.41/2.68 thf(fact_455_not__Some__eq, axiom, (! [A:$tType,X3:(option @ A)]: (((! [Y2:A]: ((X3 != ((some @ A) @ Y2)))) = (X3 = (none @ A)))))). % 6.41/2.68 thf(fact_456_not__None__eq, axiom, (! [A:$tType,X3:(option @ A)]: (((X3 != (none @ A)) = (? [Y2:A]: ((X3 = ((some @ A) @ Y2)))))))). % 6.41/2.68 thf(fact_457_option_Ocollapse, axiom, (! [A:$tType,Option:(option @ A)]: (((Option != (none @ A)) => (((some @ A) @ ((the2 @ A) @ Option)) = Option))))). % 6.41/2.68 thf(fact_458_add__def, axiom, (vEBT_VEBT_add = ((vEBT_V2048590022279873568_shift @ nat) @ (plus_plus @ nat)))). % 6.41/2.68 thf(fact_459_psubsetE, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B3) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ (set @ A)) @ B3) @ A3)))))))). % 6.41/2.68 thf(fact_460_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) & (A6 != B6)))))))). % 6.41/2.68 thf(fact_461_less__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less @ (A > B)) = (^ [F3:(A > B),G2:(A > B)]: (((((ord_less_eq @ (A > B)) @ F3) @ G2) & (~ ((((ord_less_eq @ (A > B)) @ G2) @ F3))))))))))). % 6.41/2.68 thf(fact_462_psubset__imp__subset, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))). % 6.41/2.68 thf(fact_463_psubset__subset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C5) => (((ord_less @ (set @ A)) @ A3) @ C5)))))). % 6.41/2.68 thf(fact_464_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) & (~ ((((ord_less_eq @ (set @ A)) @ B6) @ A6)))))))))). % 6.41/2.68 thf(fact_465_subset__psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less @ (set @ A)) @ B3) @ C5) => (((ord_less @ (set @ A)) @ A3) @ C5)))))). % 6.41/2.68 thf(fact_466_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A6) @ B6) | (A6 = B6)))))))). % 6.41/2.68 thf(fact_467_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X:A]: (((P @ X) => (Q @ X)))))))). % 6.41/2.68 thf(fact_468_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y3:(set @ A),Z:(set @ A)]: ((Y3 = Z))) = (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A6)))))))). % 6.41/2.68 thf(fact_469_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C5) => (((ord_less_eq @ (set @ A)) @ A3) @ C5)))))). % 6.41/2.68 thf(fact_470_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) => (Q @ X5)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))). % 6.41/2.68 thf(fact_471_subset__refl, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ A3)))). % 6.41/2.68 thf(fact_472_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A6) => (((member @ A) @ T3) @ B6)))))))))). % 6.41/2.68 thf(fact_473_equalityD2, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (((ord_less_eq @ (set @ A)) @ B3) @ A3))))). % 6.41/2.68 thf(fact_474_equalityD1, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))). % 6.41/2.68 thf(fact_475_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ A6) => (((member @ A) @ X) @ B6)))))))))). % 6.41/2.68 thf(fact_476_equalityE, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (~ ((((ord_less_eq @ (set @ A)) @ B3) @ A3)))))))))). % 6.41/2.68 thf(fact_477_subsetD, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B3)))))). % 6.41/2.68 thf(fact_478_in__mono, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ X3) @ A3) => (((member @ A) @ X3) @ B3)))))). % 6.41/2.68 thf(fact_479_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))). % 6.41/2.68 thf(fact_480_bot__nat__def, axiom, ((bot_bot @ nat) = (zero_zero @ nat))). % 6.41/2.68 thf(fact_481_combine__options__cases, axiom, (! [A:$tType,B:$tType,X3:(option @ A),P:((option @ A) > ((option @ B) > $o)),Y:(option @ B)]: ((((X3 = (none @ A)) => ((P @ X3) @ Y)) => (((Y = (none @ B)) => ((P @ X3) @ Y)) => ((! [A5:A,B5:B]: (((X3 = ((some @ A) @ A5)) => ((Y = ((some @ B) @ B5)) => ((P @ X3) @ Y))))) => ((P @ X3) @ Y))))))). % 6.41/2.68 thf(fact_482_split__option__all, axiom, (! [A:$tType]: (((^ [P2:((option @ A) > $o)]: ((! [X7:(option @ A)]: ((P2 @ X7))))) = (^ [P3:((option @ A) > $o)]: (((P3 @ (none @ A)) & (! [X:A]: ((P3 @ ((some @ A) @ X))))))))))). % 6.41/2.68 thf(fact_483_split__option__ex, axiom, (! [A:$tType]: (((^ [P2:((option @ A) > $o)]: ((? [X7:(option @ A)]: ((P2 @ X7))))) = (^ [P3:((option @ A) > $o)]: (((P3 @ (none @ A)) | (? [X:A]: ((P3 @ ((some @ A) @ X))))))))))). % 6.41/2.68 thf(fact_484_option_Oexhaust, axiom, (! [A:$tType,Y:(option @ A)]: (((Y != (none @ A)) => (~ ((! [X23:A]: ((Y != ((some @ A) @ X23)))))))))). % 6.41/2.68 thf(fact_485_option_OdiscI, axiom, (! [A:$tType,Option:(option @ A),X22:A]: (((Option = ((some @ A) @ X22)) => (Option != (none @ A)))))). % 6.41/2.68 thf(fact_486_option_Odistinct_I1_J, axiom, (! [A:$tType,X22:A]: (((none @ A) != ((some @ A) @ X22))))). % 6.41/2.68 thf(fact_487_vebt__maxt_Osimps_I3_J, axiom, (! [Mi:nat,Ma:nat,Ux:nat,Uy:(list @ vEBT_VEBT),Uz:vEBT_VEBT]: (((vEBT_vebt_maxt @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi) @ Ma))) @ Ux) @ Uy) @ Uz)) = ((some @ nat) @ Ma))))). % 6.41/2.68 thf(fact_488_vebt__mint_Osimps_I3_J, axiom, (! [Mi:nat,Ma:nat,Ux:nat,Uy:(list @ vEBT_VEBT),Uz:vEBT_VEBT]: (((vEBT_vebt_mint @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi) @ Ma))) @ Ux) @ Uy) @ Uz)) = ((some @ nat) @ Mi))))). % 6.41/2.68 thf(fact_489_option_Oexpand, axiom, (! [A:$tType,Option:(option @ A),Option2:(option @ A)]: ((((Option = (none @ A)) = (Option2 = (none @ A))) => (((Option != (none @ A)) => ((Option2 != (none @ A)) => (((the2 @ A) @ Option) = ((the2 @ A) @ Option2)))) => (Option = Option2)))))). % 6.41/2.68 thf(fact_490_option_Osize_I3_J, axiom, (! [A:$tType]: ((((size_size @ (option @ A)) @ (none @ A)) = (suc @ (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_491_option_Osize__gen_I1_J, axiom, (! [A:$tType,X3:(A > nat)]: (((((size_option @ A) @ X3) @ (none @ A)) = (suc @ (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_492_option_Oexhaust__sel, axiom, (! [A:$tType,Option:(option @ A)]: (((Option != (none @ A)) => (Option = ((some @ A) @ ((the2 @ A) @ Option))))))). % 6.41/2.68 thf(fact_493_option_Osize__neq, axiom, (! [A:$tType,X3:(option @ A)]: ((((size_size @ (option @ A)) @ X3) != (zero_zero @ nat))))). % 6.41/2.68 thf(fact_494_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A,Y:A]: (((X3 != Y) => ((~ ((((ord_less @ A) @ X3) @ Y))) => (((ord_less @ A) @ Y) @ X3))))))))). % 6.41/2.68 thf(fact_495_linordered__field__no__ub, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X2:A]: ((? [X_1:A]: ((((ord_less @ A) @ X2) @ X_1))))))))). % 6.41/2.68 thf(fact_496_linordered__field__no__lb, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X2:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X2))))))))). % 6.41/2.68 thf(fact_497_not__psubset__empty, axiom, (! [A:$tType,A3:(set @ A)]: ((~ ((((ord_less @ (set @ A)) @ A3) @ (bot_bot @ (set @ A)))))))). % 6.41/2.68 thf(fact_498_ex__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((? [X:A]: ((((member @ A) @ X) @ A3))) = (A3 != (bot_bot @ (set @ A))))))). % 6.41/2.68 thf(fact_499_equals0I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [Y4:A]: ((~ ((((member @ A) @ Y4) @ A3))))) => (A3 = (bot_bot @ (set @ A))))))). % 6.41/2.68 thf(fact_500_equals0D, axiom, (! [A:$tType,A3:(set @ A),A2:A]: (((A3 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A3))))))). % 6.41/2.68 thf(fact_501_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))). % 6.41/2.68 thf(fact_502_ex__has__least__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M:(A > nat)]: (((P @ K) => (? [X5:A]: (((P @ X5) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M @ X5)) @ (M @ Y5)))))))))))). % 6.41/2.68 thf(fact_503_Lattices__Big_Oex__has__greatest__nat, axiom, (! [A:$tType,P:(A > $o),K:A,F2:(A > nat),B2:nat]: (((P @ K) => ((! [Y4:A]: (((P @ Y4) => (((ord_less @ nat) @ (F2 @ Y4)) @ B2)))) => (? [X5:A]: (((P @ X5) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (F2 @ Y5)) @ (F2 @ X5))))))))))))). % 6.41/2.68 thf(fact_504_option_Osize_I4_J, axiom, (! [A:$tType,X22:A]: ((((size_size @ (option @ A)) @ ((some @ A) @ X22)) = (suc @ (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_505_VEBT__internal_Omembermima_Osimps_I3_J, axiom, (! [Mi:nat,Ma:nat,Va:(list @ vEBT_VEBT),Vb:vEBT_VEBT,X3:nat]: ((((vEBT_VEBT_membermima @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi) @ Ma))) @ (zero_zero @ nat)) @ Va) @ Vb)) @ X3) = ((X3 = Mi) | (X3 = Ma)))))). % 6.41/2.68 thf(fact_506_vebt__member_Osimps_I4_J, axiom, (! [V2:((product_prod @ nat) @ nat),Vb:(list @ vEBT_VEBT),Vc:vEBT_VEBT,X3:nat]: ((~ (((vEBT_vebt_member @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V2)) @ (suc @ (zero_zero @ nat))) @ Vb) @ Vc)) @ X3)))))). % 6.41/2.68 thf(fact_507_vebt__member_Osimps_I3_J, axiom, (! [V2:((product_prod @ nat) @ nat),Uy:(list @ vEBT_VEBT),Uz:vEBT_VEBT,X3:nat]: ((~ (((vEBT_vebt_member @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V2)) @ (zero_zero @ nat)) @ Uy) @ Uz)) @ X3)))))). % 6.41/2.68 thf(fact_508_VEBT__internal_OminNull_Osimps_I5_J, axiom, (! [Uz:((product_prod @ nat) @ nat),Va:nat,Vb:(list @ vEBT_VEBT),Vc:vEBT_VEBT]: ((~ ((vEBT_VEBT_minNull @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ Uz)) @ Va) @ Vb) @ Vc))))))). % 6.41/2.68 thf(fact_509_VEBT__internal_Onaive__member_Osimps_I2_J, axiom, (! [Uu:(option @ ((product_prod @ nat) @ nat)),Uv:(list @ vEBT_VEBT),Uw:vEBT_VEBT,Ux:nat]: ((~ (((vEBT_V5719532721284313246member @ ((((vEBT_Node @ Uu) @ (zero_zero @ nat)) @ Uv) @ Uw)) @ Ux)))))). % 6.41/2.68 thf(fact_510_VEBT_Oinject_I1_J, axiom, (! [X11:(option @ ((product_prod @ nat) @ nat)),X12:nat,X13:(list @ vEBT_VEBT),X14:vEBT_VEBT,Y11:(option @ ((product_prod @ nat) @ nat)),Y12:nat,Y13:(list @ vEBT_VEBT),Y14:vEBT_VEBT]: (((((((vEBT_Node @ X11) @ X12) @ X13) @ X14) = ((((vEBT_Node @ Y11) @ Y12) @ Y13) @ Y14)) = ((X11 = Y11) & ((X12 = Y12) & ((X13 = Y13) & (X14 = Y14)))))))). % 6.41/2.68 thf(fact_511_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A7:A,B7:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A7) @ B7)) = ((A2 = A7) & (B2 = B7)))))). % 6.41/2.68 thf(fact_512_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X1 = Y1) & (X22 = Y22)))))). % 6.41/2.68 thf(fact_513_nth__enumerate__eq, axiom, (! [A:$tType,M:nat,Xs:(list @ A),N2:nat]: (((((ord_less @ nat) @ M) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ ((product_prod @ nat) @ A)) @ (((enumerate @ A) @ N2) @ Xs)) @ M) = ((((product_Pair @ nat) @ A) @ (((plus_plus @ nat) @ N2) @ M)) @ (((nth @ A) @ Xs) @ M))))))). % 6.41/2.68 thf(fact_514_set__vebt__finite, axiom, (! [T2:vEBT_VEBT,N2:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => ((finite_finite2 @ nat) @ (vEBT_VEBT_set_vebt @ T2)))))). % 6.41/2.68 thf(fact_515_Euclid__induct, axiom, (! [P:(nat > (nat > $o)),A2:nat,B2:nat]: (((! [A5:nat,B5:nat]: ((((P @ A5) @ B5) = ((P @ B5) @ A5)))) => ((! [A5:nat]: (((P @ A5) @ (zero_zero @ nat)))) => ((! [A5:nat,B5:nat]: ((((P @ A5) @ B5) => ((P @ A5) @ (((plus_plus @ nat) @ A5) @ B5))))) => ((P @ A2) @ B2))))))). % 6.41/2.68 thf(fact_516_List_Ofinite__set, axiom, (! [A:$tType,Xs:(list @ A)]: (((finite_finite2 @ A) @ ((set2 @ A) @ Xs))))). % 6.41/2.68 thf(fact_517_length__enumerate, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((size_size @ (list @ ((product_prod @ nat) @ A))) @ (((enumerate @ A) @ N2) @ Xs)) = ((size_size @ (list @ A)) @ Xs))))). % 6.41/2.68 thf(fact_518_psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B3) => ((((ord_less @ (set @ A)) @ B3) @ C5) => (((ord_less @ (set @ A)) @ A3) @ C5)))))). % 6.41/2.68 thf(fact_519_psubsetD, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:A]: (((((ord_less @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B3)))))). % 6.41/2.68 thf(fact_520_VEBT__internal_Ooption__shift_Ocases, axiom, (! [A:$tType,X3:((product_prod @ (A > (A > A))) @ ((product_prod @ (option @ A)) @ (option @ A)))]: (((! [Uu2:(A > (A > A)),Uv2:(option @ A)]: ((X3 != ((((product_Pair @ (A > (A > A))) @ ((product_prod @ (option @ A)) @ (option @ A))) @ Uu2) @ ((((product_Pair @ (option @ A)) @ (option @ A)) @ (none @ A)) @ Uv2))))) => ((! [Uw2:(A > (A > A)),V3:A]: ((X3 != ((((product_Pair @ (A > (A > A))) @ ((product_prod @ (option @ A)) @ (option @ A))) @ Uw2) @ ((((product_Pair @ (option @ A)) @ (option @ A)) @ ((some @ A) @ V3)) @ (none @ A)))))) => (~ ((! [F4:(A > (A > A)),A5:A,B5:A]: ((X3 != ((((product_Pair @ (A > (A > A))) @ ((product_prod @ (option @ A)) @ (option @ A))) @ F4) @ ((((product_Pair @ (option @ A)) @ (option @ A)) @ ((some @ A) @ A5)) @ ((some @ A) @ B5))))))))))))). % 6.41/2.68 thf(fact_521_VEBT__internal_Ooption__comp__shift_Ocases, axiom, (! [A:$tType,X3:((product_prod @ (A > (A > $o))) @ ((product_prod @ (option @ A)) @ (option @ A)))]: (((! [Uu2:(A > (A > $o)),Uv2:(option @ A)]: ((X3 != ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (option @ A)) @ (option @ A))) @ Uu2) @ ((((product_Pair @ (option @ A)) @ (option @ A)) @ (none @ A)) @ Uv2))))) => ((! [Uw2:(A > (A > $o)),V3:A]: ((X3 != ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (option @ A)) @ (option @ A))) @ Uw2) @ ((((product_Pair @ (option @ A)) @ (option @ A)) @ ((some @ A) @ V3)) @ (none @ A)))))) => (~ ((! [F4:(A > (A > $o)),X5:A,Y4:A]: ((X3 != ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (option @ A)) @ (option @ A))) @ F4) @ ((((product_Pair @ (option @ A)) @ (option @ A)) @ ((some @ A) @ X5)) @ ((some @ A) @ Y4))))))))))))). % 6.41/2.68 thf(fact_522_finite__list, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (? [Xs3:(list @ A)]: ((((set2 @ A) @ Xs3) = A3))))))). % 6.41/2.68 thf(fact_523_vebt__member_Osimps_I2_J, axiom, (! [Uu:nat,Uv:(list @ vEBT_VEBT),Uw:vEBT_VEBT,X3:nat]: ((~ (((vEBT_vebt_member @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu) @ Uv) @ Uw)) @ X3)))))). % 6.41/2.68 thf(fact_524_VEBT__internal_OminNull_Osimps_I4_J, axiom, (! [Uw:nat,Ux:(list @ vEBT_VEBT),Uy:vEBT_VEBT]: ((vEBT_VEBT_minNull @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uw) @ Ux) @ Uy))))). % 6.41/2.68 thf(fact_525_ex__min__if__finite, axiom, (! [A:$tType]: (((order @ A) => (! [S3:(set @ A)]: ((((finite_finite2 @ A) @ S3) => ((S3 != (bot_bot @ (set @ A))) => (? [X5:A]: (((((member @ A) @ X5) @ S3) & (~ ((? [Xa:A]: (((((member @ A) @ Xa) @ S3) & (((ord_less @ A) @ Xa) @ X5))))))))))))))))). % 6.41/2.68 thf(fact_526_infinite__growing, axiom, (! [A:$tType]: (((linorder @ A) => (! [X8:(set @ A)]: (((X8 != (bot_bot @ (set @ A))) => ((! [X5:A]: (((((member @ A) @ X5) @ X8) => (? [Xa:A]: (((((member @ A) @ Xa) @ X8) & (((ord_less @ A) @ X5) @ Xa))))))) => (~ (((finite_finite2 @ A) @ X8))))))))))). % 6.41/2.68 thf(fact_527_VEBT__internal_Omembermima_Osimps_I2_J, axiom, (! [Ux:(list @ vEBT_VEBT),Uy:vEBT_VEBT,Uz:nat]: ((~ (((vEBT_VEBT_membermima @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (zero_zero @ nat)) @ Ux) @ Uy)) @ Uz)))))). % 6.41/2.68 thf(fact_528_VEBT__internal_Ooption__shift_Osimps_I3_J, axiom, (! [A:$tType,F2:(A > (A > A)),A2:A,B2:A]: ((((((vEBT_V2048590022279873568_shift @ A) @ F2) @ ((some @ A) @ A2)) @ ((some @ A) @ B2)) = ((some @ A) @ ((F2 @ A2) @ B2)))))). % 6.41/2.68 thf(fact_529_VEBT__internal_Ooption__shift_Osimps_I1_J, axiom, (! [A:$tType,Uu:(A > (A > A)),Uv:(option @ A)]: ((((((vEBT_V2048590022279873568_shift @ A) @ Uu) @ (none @ A)) @ Uv) = (none @ A))))). % 6.41/2.68 thf(fact_530_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B5:B]: ((Y != ((((product_Pair @ A) @ B) @ A5) @ B5))))))))). % 6.41/2.68 thf(fact_531_surj__pair, axiom, (! [A:$tType,B:$tType,P4:((product_prod @ A) @ B)]: ((? [X5:A,Y4:B]: ((P4 = ((((product_Pair @ A) @ B) @ X5) @ Y4))))))). % 6.41/2.68 thf(fact_532_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P4:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P @ P4))))). % 6.41/2.68 thf(fact_533_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A7:A,B7:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A7) @ B7)) => (~ (((A2 = A7) => (B2 != B7)))))))). % 6.41/2.68 thf(fact_534_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B5:B,C3:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C3)))))))))). % 6.41/2.68 thf(fact_535_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,C3:C,D3: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) @ C3) @ D3))))))))))). % 6.41/2.68 thf(fact_536_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2))))]: ((~ ((! [A5:A,B5:B,C3:C,D3:D,E:E2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E2)) @ C3) @ ((((product_Pair @ D) @ E2) @ D3) @ E)))))))))))). % 6.41/2.68 thf(fact_537_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E2:$tType,F:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F)))))]: ((~ ((! [A5:A,B5:B,C3:C,D3:D,E:E2,F4:F]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E2) @ F)) @ D3) @ ((((product_Pair @ E2) @ F) @ E) @ F4))))))))))))). % 6.41/2.68 thf(fact_538_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E2:$tType,F:$tType,G3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F) @ G3))))))]: ((~ ((! [A5:A,B5:B,C3:C,D3:D,E:E2,F4:F,G4:G3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F) @ G3))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F) @ G3)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E2) @ ((product_prod @ F) @ G3))) @ D3) @ ((((product_Pair @ E2) @ ((product_prod @ F) @ G3)) @ E) @ ((((product_Pair @ F) @ G3) @ F4) @ G4)))))))))))))). % 6.41/2.68 thf(fact_539_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B5:B,C3:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C3))))) => (P @ X3))))). % 6.41/2.68 thf(fact_540_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A5:A,B5:B,C3:C,D3:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C3) @ D3)))))) => (P @ X3))))). % 6.41/2.68 thf(fact_541_prod__induct5, axiom, (! [E2:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2)))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2))))]: (((! [A5:A,B5:B,C3:C,D3:D,E:E2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E2))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E2)) @ C3) @ ((((product_Pair @ D) @ E2) @ D3) @ E))))))) => (P @ X3))))). % 6.41/2.68 thf(fact_542_prod__induct6, axiom, (! [F:$tType,E2:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F)))))]: (((! [A5:A,B5:B,C3:C,D3:D,E:E2,F4:F]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ F))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E2) @ F)) @ D3) @ ((((product_Pair @ E2) @ F) @ E) @ F4)))))))) => (P @ X3))))). % 6.41/2.68 thf(fact_543_prod__induct7, axiom, (! [G3:$tType,F:$tType,E2:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F) @ G3)))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F) @ G3))))))]: (((! [A5:A,B5:B,C3:C,D3:D,E:E2,F4:F,G4:G3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F) @ G3))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E2) @ ((product_prod @ F) @ G3)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E2) @ ((product_prod @ F) @ G3))) @ D3) @ ((((product_Pair @ E2) @ ((product_prod @ F) @ G3)) @ E) @ ((((product_Pair @ F) @ G3) @ F4) @ G4))))))))) => (P @ X3))))). % 6.41/2.68 thf(fact_544_vebt__maxt_Osimps_I2_J, axiom, (! [Uu:nat,Uv:(list @ vEBT_VEBT),Uw:vEBT_VEBT]: (((vEBT_vebt_maxt @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu) @ Uv) @ Uw)) = (none @ nat))))). % 6.41/2.68 thf(fact_545_vebt__mint_Osimps_I2_J, axiom, (! [Uu:nat,Uv:(list @ vEBT_VEBT),Uw:vEBT_VEBT]: (((vEBT_vebt_mint @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu) @ Uv) @ Uw)) = (none @ nat))))). % 6.41/2.68 thf(fact_546_VEBT__internal_Ooption__shift_Osimps_I2_J, axiom, (! [A:$tType,Uw:(A > (A > A)),V2:A]: ((((((vEBT_V2048590022279873568_shift @ A) @ Uw) @ ((some @ A) @ V2)) @ (none @ A)) = (none @ A))))). % 6.41/2.68 thf(fact_547_VEBT__internal_Ooption__shift_Oelims, axiom, (! [A:$tType,X3:(A > (A > A)),Xa2:(option @ A),Xb:(option @ A),Y:(option @ A)]: (((((((vEBT_V2048590022279873568_shift @ A) @ X3) @ Xa2) @ Xb) = Y) => (((Xa2 = (none @ A)) => (Y != (none @ A))) => (((? [V3:A]: ((Xa2 = ((some @ A) @ V3)))) => ((Xb = (none @ A)) => (Y != (none @ A)))) => (~ ((! [A5:A]: (((Xa2 = ((some @ A) @ A5)) => (! [B5:A]: (((Xb = ((some @ A) @ B5)) => (Y != ((some @ A) @ ((X3 @ A5) @ B5))))))))))))))))). % 6.41/2.68 thf(fact_548_vebt__buildup_Ocases, axiom, (! [X3:nat]: (((X3 != (zero_zero @ nat)) => ((X3 != (suc @ (zero_zero @ nat))) => (~ ((! [Va2:nat]: ((X3 != (suc @ (suc @ Va2)))))))))))). % 6.41/2.68 thf(fact_549_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))). % 6.41/2.68 thf(fact_550_finite__has__minimal, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (? [X5:A]: (((((member @ A) @ X5) @ A3) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ Xa) @ X5) => (X5 = Xa)))))))))))))))). % 6.41/2.68 thf(fact_551_finite__has__maximal, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (? [X5:A]: (((((member @ A) @ X5) @ A3) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ X5) @ Xa) => (X5 = Xa)))))))))))))))). % 6.41/2.68 thf(fact_552_finite__code, axiom, (! [A:$tType]: (((finite_finite @ A) => ((finite_finite2 @ A) = (^ [A6:(set @ A)]: ($true))))))). % 6.41/2.68 thf(fact_553_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X:A]: ((((member @ A) @ X) @ (bot_bot @ (set @ A))))))))). % 6.41/2.68 thf(fact_554_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))). % 6.41/2.68 thf(fact_555_infinite__nat__iff__unbounded__le, axiom, (! [S3:(set @ nat)]: (((~ (((finite_finite2 @ nat) @ S3))) = (! [M2:nat]: ((? [N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) & (((member @ nat) @ N) @ S3)))))))))). % 6.41/2.68 thf(fact_556_finite__nat__set__iff__bounded__le, axiom, ((finite_finite2 @ nat) = (^ [N5:(set @ nat)]: ((? [M2:nat]: ((! [X:nat]: (((((member @ nat) @ X) @ N5) => (((ord_less_eq @ nat) @ X) @ M2)))))))))). % 6.41/2.68 thf(fact_557_unbounded__k__infinite, axiom, (! [K:nat,S3:(set @ nat)]: (((! [M3:nat]: (((((ord_less @ nat) @ K) @ M3) => (? [N6:nat]: (((((ord_less @ nat) @ M3) @ N6) & (((member @ nat) @ N6) @ S3))))))) => (~ (((finite_finite2 @ nat) @ S3))))))). % 6.41/2.68 thf(fact_558_bounded__nat__set__is__finite, axiom, (! [N7:(set @ nat),N2:nat]: (((! [X5:nat]: (((((member @ nat) @ X5) @ N7) => (((ord_less @ nat) @ X5) @ N2)))) => ((finite_finite2 @ nat) @ N7))))). % 6.41/2.68 thf(fact_559_finite__maxlen, axiom, (! [A:$tType,M7:(set @ (list @ A))]: ((((finite_finite2 @ (list @ A)) @ M7) => (? [N3:nat]: ((! [X2:(list @ A)]: (((((member @ (list @ A)) @ X2) @ M7) => (((ord_less @ nat) @ ((size_size @ (list @ A)) @ X2)) @ N3)))))))))). % 6.41/2.68 thf(fact_560_subrelI, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((! [X5:A,Y4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y4)) @ R2) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y4)) @ S)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R2) @ S))))). % 6.41/2.68 thf(fact_561_finite__set__choice, axiom, (! [B:$tType,A:$tType,A3:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A3) => ((! [X5:A]: (((((member @ A) @ X5) @ A3) => (? [X_12:B]: (((P @ X5) @ X_12)))))) => (? [F4:(A > B)]: ((! [X2:A]: (((((member @ A) @ X2) @ A3) => ((P @ X2) @ (F4 @ X2)))))))))))). % 6.41/2.68 thf(fact_562_finite, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A3:(set @ A)]: (((finite_finite2 @ A) @ A3))))))). % 6.41/2.68 thf(fact_563_bounded__Max__nat, axiom, (! [P:(nat > $o),X3:nat,M7:nat]: (((P @ X3) => ((! [X5:nat]: (((P @ X5) => (((ord_less_eq @ nat) @ X5) @ M7)))) => (~ ((! [M3:nat]: (((P @ M3) => (~ ((! [X2:nat]: (((P @ X2) => (((ord_less_eq @ nat) @ X2) @ M3)))))))))))))))). % 6.41/2.68 thf(fact_564_fold__atLeastAtMost__nat_Ocases, axiom, (! [A:$tType,X3:((product_prod @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A)))]: ((~ ((! [F4:(nat > (A > A)),A5:nat,B5:nat,Acc:A]: ((X3 != ((((product_Pair @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A))) @ F4) @ ((((product_Pair @ nat) @ ((product_prod @ nat) @ A)) @ A5) @ ((((product_Pair @ nat) @ A) @ B5) @ Acc))))))))))). % 6.41/2.68 thf(fact_565_finite__has__minimal2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A2) @ A3) => (? [X5:A]: (((((member @ A) @ X5) @ A3) & ((((ord_less_eq @ A) @ X5) @ A2) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ Xa) @ X5) => (X5 = Xa))))))))))))))))). % 6.41/2.68 thf(fact_566_finite__has__maximal2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A2) @ A3) => (? [X5:A]: (((((member @ A) @ X5) @ A3) & ((((ord_less_eq @ A) @ A2) @ X5) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ X5) @ Xa) => (X5 = Xa))))))))))))))))). % 6.41/2.68 thf(fact_567_finite__transitivity__chain, axiom, (! [A:$tType,A3:(set @ A),R:(A > (A > $o))]: ((((finite_finite2 @ A) @ A3) => ((! [X5:A]: ((~ (((R @ X5) @ X5))))) => ((! [X5:A,Y4:A,Z3:A]: ((((R @ X5) @ Y4) => (((R @ Y4) @ Z3) => ((R @ X5) @ Z3))))) => ((! [X5:A]: (((((member @ A) @ X5) @ A3) => (? [Y5:A]: (((((member @ A) @ Y5) @ A3) & ((R @ X5) @ Y5))))))) => (A3 = (bot_bot @ (set @ A)))))))))). % 6.41/2.68 thf(fact_568_infinite__imp__nonempty, axiom, (! [A:$tType,S3:(set @ A)]: (((~ (((finite_finite2 @ A) @ S3))) => (S3 != (bot_bot @ (set @ A))))))). % 6.41/2.68 thf(fact_569_finite_OemptyI, axiom, (! [A:$tType]: (((finite_finite2 @ A) @ (bot_bot @ (set @ A)))))). % 6.41/2.68 thf(fact_570_rev__finite__subset, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((finite_finite2 @ A) @ A3)))))). % 6.41/2.68 thf(fact_571_infinite__super, axiom, (! [A:$tType,S3:(set @ A),T4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S3) @ T4) => ((~ (((finite_finite2 @ A) @ S3))) => (~ (((finite_finite2 @ A) @ T4)))))))). % 6.41/2.68 thf(fact_572_finite__subset, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((finite_finite2 @ A) @ B3) => ((finite_finite2 @ A) @ A3)))))). % 6.41/2.68 thf(fact_573_finite__psubset__induct, axiom, (! [A:$tType,A3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A3) => ((! [A8:(set @ A)]: ((((finite_finite2 @ A) @ A8) => ((! [B8:(set @ A)]: (((((ord_less @ (set @ A)) @ B8) @ A8) => (P @ B8)))) => (P @ A8))))) => (P @ A3)))))). % 6.41/2.68 thf(fact_574_finite__nat__set__iff__bounded, axiom, ((finite_finite2 @ nat) = (^ [N5:(set @ nat)]: ((? [M2:nat]: ((! [X:nat]: (((((member @ nat) @ X) @ N5) => (((ord_less @ nat) @ X) @ M2)))))))))). % 6.41/2.68 thf(fact_575_infinite__nat__iff__unbounded, axiom, (! [S3:(set @ nat)]: (((~ (((finite_finite2 @ nat) @ S3))) = (! [M2:nat]: ((? [N:nat]: (((((ord_less @ nat) @ M2) @ N) & (((member @ nat) @ N) @ S3)))))))))). % 6.41/2.68 thf(fact_576_arg__min__if__finite_I2_J, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [S3:(set @ A),F2:(A > B)]: ((((finite_finite2 @ A) @ S3) => ((S3 != (bot_bot @ (set @ A))) => (~ ((? [X2:A]: (((((member @ A) @ X2) @ S3) & (((ord_less @ B) @ (F2 @ X2)) @ (F2 @ ((((lattic7623131987881927897min_on @ A) @ B) @ F2) @ S3)))))))))))))))). % 6.41/2.68 thf(fact_577_arg__min__least, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [S3:(set @ A),Y:A,F2:(A > B)]: ((((finite_finite2 @ A) @ S3) => ((S3 != (bot_bot @ (set @ A))) => ((((member @ A) @ Y) @ S3) => (((ord_less_eq @ B) @ (F2 @ ((((lattic7623131987881927897min_on @ A) @ B) @ F2) @ S3))) @ (F2 @ Y))))))))))). % 6.41/2.68 thf(fact_578_nat__descend__induct, axiom, (! [N2:nat,P:(nat > $o),M:nat]: (((! [K2:nat]: (((((ord_less @ nat) @ N2) @ K2) => (P @ K2)))) => ((! [K2:nat]: (((((ord_less_eq @ nat) @ K2) @ N2) => ((! [I5:nat]: (((((ord_less @ nat) @ K2) @ I5) => (P @ I5)))) => (P @ K2))))) => (P @ M)))))). % 6.41/2.68 thf(fact_579_subset__emptyI, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X5:A]: ((~ ((((member @ A) @ X5) @ A3))))) => (((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))))))). % 6.41/2.68 thf(fact_580_exists__least__lemma, axiom, (! [P:(nat > $o)]: (((~ ((P @ (zero_zero @ nat)))) => ((? [X_12:nat]: ((P @ X_12))) => (? [N3:nat]: (((~ ((P @ N3))) & (P @ (suc @ N3)))))))))). % 6.41/2.68 thf(fact_581_dependent__nat__choice, axiom, (! [A:$tType,P:(nat > (A > $o)),Q:(nat > (A > (A > $o)))]: (((? [X_12:A]: (((P @ (zero_zero @ nat)) @ X_12))) => ((! [X5:A,N3:nat]: ((((P @ N3) @ X5) => (? [Y5:A]: ((((P @ (suc @ N3)) @ Y5) & (((Q @ N3) @ X5) @ Y5))))))) => (? [F4:(nat > A)]: ((! [N6:nat]: ((((P @ N6) @ (F4 @ N6)) & (((Q @ N6) @ (F4 @ N6)) @ (F4 @ (suc @ N6))))))))))))). % 6.41/2.68 thf(fact_582_list__decode_Ocases, axiom, (! [X3:nat]: (((X3 != (zero_zero @ nat)) => (~ ((! [N3:nat]: ((X3 != (suc @ N3)))))))))). % 6.41/2.68 thf(fact_583_verit__sum__simplify, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ (zero_zero @ A)) = A2))))))). % 6.41/2.68 thf(fact_584_add__0__iff, axiom, (! [A:$tType]: (((semiri1453513574482234551roduct @ A) => (! [B2:A,A2:A]: (((B2 = (((plus_plus @ A) @ B2) @ A2)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_585_field__lbound__gt__zero, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [D1:A,D22:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ D1) => ((((ord_less @ A) @ (zero_zero @ A)) @ D22) => (? [E:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E) & ((((ord_less @ A) @ E) @ D1) & (((ord_less @ A) @ E) @ D22))))))))))))). % 6.41/2.68 thf(fact_586_less__by__empty, axiom, (! [A:$tType,A3:(set @ ((product_prod @ A) @ A)),B3:(set @ ((product_prod @ A) @ A))]: (((A3 = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ A3) @ B3))))). % 6.41/2.68 thf(fact_587_verit__comp__simplify1_I2_J, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))). % 6.41/2.68 thf(fact_588_verit__la__disequality, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((A2 = B2) | ((~ ((((ord_less_eq @ A) @ A2) @ B2))) | (~ ((((ord_less_eq @ A) @ B2) @ A2))))))))))). % 6.41/2.68 thf(fact_589_verit__comp__simplify1_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))). % 6.41/2.68 thf(fact_590_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R2:A,S:B,R:(set @ ((product_prod @ A) @ B)),S4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S)) @ R) => ((S4 = S) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R2) @ S4)) @ R)))))). % 6.41/2.68 thf(fact_591_prod__decode__aux_Ocases, axiom, (! [X3:((product_prod @ nat) @ nat)]: ((~ ((! [K2:nat,M3:nat]: ((X3 != ((((product_Pair @ nat) @ nat) @ K2) @ M3))))))))). % 6.41/2.68 thf(fact_592_arg__min__if__finite_I1_J, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [S3:(set @ A),F2:(A > B)]: ((((finite_finite2 @ A) @ S3) => ((S3 != (bot_bot @ (set @ A))) => (((member @ A) @ ((((lattic7623131987881927897min_on @ A) @ B) @ F2) @ S3)) @ S3))))))))). % 6.41/2.68 thf(fact_593_verit__comp__simplify1_I3_J, axiom, (! [B:$tType]: (((linorder @ B) => (! [B7:B,A7:B]: (((~ ((((ord_less_eq @ B) @ B7) @ A7))) = (((ord_less @ B) @ A7) @ B7)))))))). % 6.41/2.68 thf(fact_594_triangle__Suc, axiom, (! [N2:nat]: (((nat_triangle @ (suc @ N2)) = (((plus_plus @ nat) @ (nat_triangle @ N2)) @ (suc @ N2)))))). % 6.41/2.68 thf(fact_595_less__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))). % 6.41/2.68 thf(fact_596_minf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z3) => (~ ((((ord_less_eq @ A) @ T2) @ X2)))))))))))))). % 6.41/2.68 thf(fact_597_minf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z3) => (((ord_less_eq @ A) @ X2) @ T2)))))))))))). % 6.41/2.68 thf(fact_598_pinf_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ Z3) @ X2) => (((ord_less_eq @ A) @ T2) @ X2)))))))))))). % 6.41/2.68 thf(fact_599_pinf_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ Z3) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ T2)))))))))))))). % 6.41/2.68 thf(fact_600_complete__interval, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [A2:A,B2:A,P:(A > $o)]: (((((ord_less @ A) @ A2) @ B2) => ((P @ A2) => ((~ ((P @ B2))) => (? [C3:A]: (((((ord_less_eq @ A) @ A2) @ C3) & ((((ord_less_eq @ A) @ C3) @ B2) & ((! [X2:A]: ((((((ord_less_eq @ A) @ A2) @ X2) & (((ord_less @ A) @ X2) @ C3)) => (P @ X2)))) & (! [D4:A]: (((! [X5:A]: ((((((ord_less_eq @ A) @ A2) @ X5) & (((ord_less @ A) @ X5) @ D4)) => (P @ X5)))) => (((ord_less_eq @ A) @ D4) @ C3)))))))))))))))))). % 6.41/2.68 thf(fact_601_le__numeral__extra_I3_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))). % 6.41/2.68 thf(fact_602_find__Some__iff, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),X3:A]: ((((((find @ A) @ P) @ Xs) = ((some @ A) @ X3)) = (? [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) & ((P @ (((nth @ A) @ Xs) @ I3)) & ((X3 = (((nth @ A) @ Xs) @ I3)) & (! [J3:nat]: (((((ord_less @ nat) @ J3) @ I3) => (~ ((P @ (((nth @ A) @ Xs) @ J3)))))))))))))))). % 6.41/2.68 thf(fact_603_find__Some__iff2, axiom, (! [A:$tType,X3:A,P:(A > $o),Xs:(list @ A)]: (((((some @ A) @ X3) = (((find @ A) @ P) @ Xs)) = (? [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) & ((P @ (((nth @ A) @ Xs) @ I3)) & ((X3 = (((nth @ A) @ Xs) @ I3)) & (! [J3:nat]: (((((ord_less @ nat) @ J3) @ I3) => (~ ((P @ (((nth @ A) @ Xs) @ J3)))))))))))))))). % 6.41/2.68 thf(fact_604_triangle__0, axiom, ((nat_triangle @ (zero_zero @ nat)) = (zero_zero @ nat))). % 6.41/2.68 thf(fact_605_find__cong, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),P:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Ys)) => ((P @ X5) = (Q @ X5))))) => ((((find @ A) @ P) @ Xs) = (((find @ A) @ Q) @ Ys))))))). % 6.41/2.68 thf(fact_606_ex__gt__or__lt, axiom, (! [A:$tType]: (((condit5016429287641298734tinuum @ A) => (! [A2:A]: ((? [B5:A]: (((((ord_less @ A) @ A2) @ B5) | (((ord_less @ A) @ B5) @ A2)))))))))). % 6.41/2.68 thf(fact_607_pinf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P5:(A > $o),Q:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((P @ X5) = (P5 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((Q @ X5) = (Q3 @ X5))))))) => (? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ Z3) @ X2) => (((P @ X2) & (Q @ X2)) = ((P5 @ X2) & (Q3 @ X2)))))))))))))))). % 6.41/2.68 thf(fact_608_pinf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P5:(A > $o),Q:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((P @ X5) = (P5 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((Q @ X5) = (Q3 @ X5))))))) => (? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ Z3) @ X2) => (((P @ X2) | (Q @ X2)) = ((P5 @ X2) | (Q3 @ X2)))))))))))))))). % 6.41/2.68 thf(fact_609_pinf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ Z3) @ X2) => (X2 != T2)))))))))))). % 6.41/2.68 thf(fact_610_pinf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ Z3) @ X2) => (X2 != T2)))))))))))). % 6.41/2.68 thf(fact_611_pinf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ Z3) @ X2) => (~ ((((ord_less @ A) @ X2) @ T2)))))))))))))). % 6.41/2.68 thf(fact_612_pinf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ Z3) @ X2) => (((ord_less @ A) @ T2) @ X2)))))))))))). % 6.41/2.68 thf(fact_613_pinf_I11_J, axiom, (! [C:$tType,D:$tType]: (((ord @ C) => (! [F5:D]: ((? [Z3:C]: ((! [X2:C]: (((((ord_less @ C) @ Z3) @ X2) => (F5 = F5)))))))))))). % 6.41/2.68 thf(fact_614_minf_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P5:(A > $o),Q:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((P @ X5) = (P5 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((Q @ X5) = (Q3 @ X5))))))) => (? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z3) => (((P @ X2) & (Q @ X2)) = ((P5 @ X2) & (Q3 @ X2)))))))))))))))). % 6.41/2.68 thf(fact_615_minf_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o),P5:(A > $o),Q:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((P @ X5) = (P5 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((Q @ X5) = (Q3 @ X5))))))) => (? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z3) => (((P @ X2) | (Q @ X2)) = ((P5 @ X2) | (Q3 @ X2)))))))))))))))). % 6.41/2.68 thf(fact_616_minf_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z3) => (X2 != T2)))))))))))). % 6.41/2.68 thf(fact_617_minf_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z3) => (X2 != T2)))))))))))). % 6.41/2.68 thf(fact_618_minf_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z3) => (((ord_less @ A) @ X2) @ T2)))))))))))). % 6.41/2.68 thf(fact_619_minf_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [T2:A]: ((? [Z3:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z3) => (~ ((((ord_less @ A) @ T2) @ X2)))))))))))))). % 6.41/2.68 thf(fact_620_minf_I11_J, axiom, (! [C:$tType,D:$tType]: (((ord @ C) => (! [F5:D]: ((? [Z3:C]: ((! [X2:C]: (((((ord_less @ C) @ X2) @ Z3) => (F5 = F5)))))))))))). % 6.41/2.68 thf(fact_621_is__num__normalize_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))). % 6.41/2.68 thf(fact_622_find__None__iff2, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((none @ A) = (((find @ A) @ P) @ Xs)) = (~ ((? [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) & (P @ X)))))))))). % 6.41/2.68 thf(fact_623_find__None__iff, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((((find @ A) @ P) @ Xs) = (none @ A)) = (~ ((? [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) & (P @ X)))))))))). % 6.41/2.68 thf(fact_624_nth__zip, axiom, (! [A:$tType,B:$tType,I:nat,Xs:(list @ A),Ys:(list @ B)]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ B)) @ Ys)) => ((((nth @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys)) @ I) = ((((product_Pair @ A) @ B) @ (((nth @ A) @ Xs) @ I)) @ (((nth @ B) @ Ys) @ I)))))))). % 6.41/2.68 thf(fact_625__C5_C_I10_J, axiom, ((mi != ma) => (! [I5:nat]: (((((ord_less @ nat) @ I5) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ m)) => (((((vEBT_VEBT_high @ ma) @ n) = I5) => ((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ treeList) @ I5)) @ ((vEBT_VEBT_low @ ma) @ n))) & (! [X2:nat]: ((((((vEBT_VEBT_high @ X2) @ n) = I5) & ((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ treeList) @ I5)) @ ((vEBT_VEBT_low @ X2) @ n))) => ((((ord_less @ nat) @ mi) @ X2) & (((ord_less_eq @ nat) @ X2) @ ma))))))))))). % 6.41/2.68 thf(fact_626_vebt__member_Ocases, axiom, (! [X3:((product_prod @ vEBT_VEBT) @ nat)]: (((! [A5:$o,B5:$o,X5:nat]: ((X3 != ((((product_Pair @ vEBT_VEBT) @ nat) @ ((vEBT_Leaf @ A5) @ B5)) @ X5)))) => ((! [Uu2:nat,Uv2:(list @ vEBT_VEBT),Uw2:vEBT_VEBT,X5:nat]: ((X3 != ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu2) @ Uv2) @ Uw2)) @ X5)))) => ((! [V3:((product_prod @ nat) @ nat),Uy2:(list @ vEBT_VEBT),Uz2:vEBT_VEBT,X5:nat]: ((X3 != ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V3)) @ (zero_zero @ nat)) @ Uy2) @ Uz2)) @ X5)))) => ((! [V3:((product_prod @ nat) @ nat),Vb2:(list @ vEBT_VEBT),Vc2:vEBT_VEBT,X5:nat]: ((X3 != ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V3)) @ (suc @ (zero_zero @ nat))) @ Vb2) @ Vc2)) @ X5)))) => (~ ((! [Mi2:nat,Ma2:nat,Va2:nat,TreeList3:(list @ vEBT_VEBT),Summary2:vEBT_VEBT,X5:nat]: ((X3 != ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ (suc @ Va2))) @ TreeList3) @ Summary2)) @ X5))))))))))))). % 6.41/2.68 thf(fact_627_VEBT__internal_Omembermima_Ocases, axiom, (! [X3:((product_prod @ vEBT_VEBT) @ nat)]: (((! [Uu2:$o,Uv2:$o,Uw2:nat]: ((X3 != ((((product_Pair @ vEBT_VEBT) @ nat) @ ((vEBT_Leaf @ Uu2) @ Uv2)) @ Uw2)))) => ((! [Ux2:(list @ vEBT_VEBT),Uy2:vEBT_VEBT,Uz2:nat]: ((X3 != ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (zero_zero @ nat)) @ Ux2) @ Uy2)) @ Uz2)))) => ((! [Mi2:nat,Ma2:nat,Va3:(list @ vEBT_VEBT),Vb2:vEBT_VEBT,X5:nat]: ((X3 != ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (zero_zero @ nat)) @ Va3) @ Vb2)) @ X5)))) => ((! [Mi2:nat,Ma2:nat,V3:nat,TreeList3:(list @ vEBT_VEBT),Vc2:vEBT_VEBT,X5:nat]: ((X3 != ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ V3)) @ TreeList3) @ Vc2)) @ X5)))) => (~ ((! [V3:nat,TreeList3:(list @ vEBT_VEBT),Vd:vEBT_VEBT,X5:nat]: ((X3 != ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ V3)) @ TreeList3) @ Vd)) @ X5))))))))))))). % 6.41/2.68 thf(fact_628_count__notin, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)))) => ((((count_list @ A) @ Xs) @ X3) = (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_629_listrel1__iff__update, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((listrel1 @ A) @ R2)) = (? [Y2:A,N:nat]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (((nth @ A) @ Xs) @ N)) @ Y2)) @ R2) & ((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) & (Ys = ((((list_update @ A) @ Xs) @ N) @ Y2)))))))))). % 6.41/2.68 thf(fact_630_pair__lessI2, axiom, (! [A2:nat,B2:nat,S:nat,T2:nat]: (((((ord_less_eq @ nat) @ A2) @ B2) => ((((ord_less @ nat) @ S) @ T2) => (((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ A2) @ S)) @ ((((product_Pair @ nat) @ nat) @ B2) @ T2))) @ fun_pair_less)))))). % 6.41/2.68 thf(fact_631_pair__less__iff1, axiom, (! [X3:nat,Y:nat,Z2:nat]: (((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ X3) @ Y)) @ ((((product_Pair @ nat) @ nat) @ X3) @ Z2))) @ fun_pair_less) = (((ord_less @ nat) @ Y) @ Z2))))). % 6.41/2.68 thf(fact_632_gen__length__def, axiom, (! [A:$tType]: (((gen_length @ A) = (^ [N:nat,Xs2:(list @ A)]: ((((plus_plus @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs2)))))))). % 6.41/2.68 thf(fact_633_set__encode__empty, axiom, ((nat_set_encode @ (bot_bot @ (set @ nat))) = (zero_zero @ nat))). % 6.41/2.68 thf(fact_634_Leaf__0__not, axiom, (! [A2:$o,B2:$o]: ((~ (((vEBT_invar_vebt @ ((vEBT_Leaf @ A2) @ B2)) @ (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_635_power__shift, axiom, (! [X3:nat,Y:nat,Z2:nat]: ((((((power_power @ nat) @ X3) @ Y) = Z2) = (((vEBT_VEBT_power @ ((some @ nat) @ X3)) @ ((some @ nat) @ Y)) = ((some @ nat) @ Z2)))))). % 6.41/2.68 thf(fact_636__C5_C_I3_J, axiom, (((size_size @ (list @ vEBT_VEBT)) @ treeList) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ m))). % 6.41/2.68 thf(fact_637__C5_C_I9_J, axiom, (((ord_less @ nat) @ ma) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ deg))). % 6.41/2.68 thf(fact_638_VEBT_Oinject_I2_J, axiom, (! [X21:$o,X222:$o,Y21:$o,Y222:$o]: (((((vEBT_Leaf @ X21) @ X222) = ((vEBT_Leaf @ Y21) @ Y222)) = ((X21 = Y21) & (X222 = Y222)))))). % 6.41/2.68 thf(fact_639_local_Opower__def, axiom, (vEBT_VEBT_power = ((vEBT_V2048590022279873568_shift @ nat) @ (power_power @ nat)))). % 6.41/2.68 thf(fact_640_high__bound__aux, axiom, (! [Ma:nat,N2:nat,M:nat]: (((((ord_less @ nat) @ Ma) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ N2) @ M))) => (((ord_less @ nat) @ ((vEBT_VEBT_high @ Ma) @ N2)) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)))))). % 6.41/2.68 thf(fact_641_member__bound, axiom, (! [Tree:vEBT_VEBT,X3:nat,N2:nat]: ((((vEBT_vebt_member @ Tree) @ X3) => (((vEBT_invar_vebt @ Tree) @ N2) => (((ord_less @ nat) @ X3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))))))). % 6.41/2.68 thf(fact_642__C5_C_I6_J, axiom, (! [I5:nat]: (((((ord_less @ nat) @ I5) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ m)) => ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ treeList) @ I5)) @ X6))) = ((vEBT_V8194947554948674370ptions @ summary) @ I5)))))). % 6.41/2.68 thf(fact_643_numeral__le__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:num,N2:num]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N2)) = (((ord_less_eq @ num) @ M) @ N2)))))))). % 6.41/2.68 thf(fact_644_numeral__less__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:num,N2:num]: (((((ord_less @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N2)) = (((ord_less @ num) @ M) @ N2)))))))). % 6.41/2.68 thf(fact_645_numeral__plus__numeral, axiom, (! [A:$tType]: (((numeral @ A) => (! [M:num,N2:num]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N2)) = ((numeral_numeral @ A) @ (((plus_plus @ num) @ M) @ N2))))))))). % 6.41/2.68 thf(fact_646_add__numeral__left, axiom, (! [A:$tType]: (((numeral @ A) => (! [V2:num,W2:num,Z2:A]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ V2)) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ W2)) @ Z2)) = (((plus_plus @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ V2) @ W2))) @ Z2)))))))). % 6.41/2.68 thf(fact_647_mi__ma__2__deg, axiom, (! [Mi:nat,Ma:nat,Deg:nat,TreeList2:(list @ vEBT_VEBT),Summary:vEBT_VEBT,N2:nat]: ((((vEBT_invar_vebt @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi) @ Ma))) @ Deg) @ TreeList2) @ Summary)) @ N2) => ((((ord_less_eq @ nat) @ Mi) @ Ma) & (((ord_less @ nat) @ Ma) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg))))))). % 6.41/2.68 thf(fact_648_Suc__numeral, axiom, (! [N2:num]: (((suc @ ((numeral_numeral @ nat) @ N2)) = ((numeral_numeral @ nat) @ (((plus_plus @ num) @ N2) @ one2)))))). % 6.41/2.68 thf(fact_649_add__2__eq__Suc_H, axiom, (! [N2:nat]: (((((plus_plus @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (suc @ (suc @ N2)))))). % 6.41/2.68 thf(fact_650_add__2__eq__Suc, axiom, (! [N2:nat]: (((((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2) = (suc @ (suc @ N2)))))). % 6.41/2.68 thf(fact_651_numeral__Bit0, axiom, (! [A:$tType]: (((numeral @ A) => (! [N2:num]: ((((numeral_numeral @ A) @ (bit0 @ N2)) = (((plus_plus @ A) @ ((numeral_numeral @ A) @ N2)) @ ((numeral_numeral @ A) @ N2))))))))). % 6.41/2.68 thf(fact_652_numeral__2__eq__2, axiom, (((numeral_numeral @ nat) @ (bit0 @ one2)) = (suc @ (suc @ (zero_zero @ nat))))). % 6.41/2.68 thf(fact_653_less__2__cases, axiom, (! [N2:nat]: (((((ord_less @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) => ((N2 = (zero_zero @ nat)) | (N2 = (suc @ (zero_zero @ nat)))))))). % 6.41/2.68 thf(fact_654_less__2__cases__iff, axiom, (! [N2:nat]: (((((ord_less @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = ((N2 = (zero_zero @ nat)) | (N2 = (suc @ (zero_zero @ nat)))))))). % 6.41/2.68 thf(fact_655_VEBT__internal_Oexp__split__high__low_I1_J, axiom, (! [X3:nat,N2:nat,M:nat]: (((((ord_less @ nat) @ X3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ N2) @ M))) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((ord_less @ nat) @ ((vEBT_VEBT_high @ X3) @ N2)) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)))))))). % 6.41/2.68 thf(fact_656_VEBT__internal_Oexp__split__high__low_I2_J, axiom, (! [X3:nat,N2:nat,M:nat]: (((((ord_less @ nat) @ X3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ N2) @ M))) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((ord_less @ nat) @ ((vEBT_VEBT_low @ X3) @ N2)) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)))))))). % 6.41/2.68 thf(fact_657_numeral__1__eq__Suc__0, axiom, (((numeral_numeral @ nat) @ one2) = (suc @ (zero_zero @ nat)))). % 6.41/2.68 thf(fact_658_Suc__nat__number__of__add, axiom, (! [V2:num,N2:nat]: (((suc @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ V2)) @ N2)) = (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (((plus_plus @ num) @ V2) @ one2))) @ N2))))). % 6.41/2.68 thf(fact_659_VEBT_Osize_I4_J, axiom, (! [X21:$o,X222:$o]: ((((size_size @ vEBT_VEBT) @ ((vEBT_Leaf @ X21) @ X222)) = (zero_zero @ nat))))). % 6.41/2.68 thf(fact_660_VEBT__internal_Ovalid_H_Ocases, axiom, (! [X3:((product_prod @ vEBT_VEBT) @ nat)]: (((! [Uu2:$o,Uv2:$o,D3:nat]: ((X3 != ((((product_Pair @ vEBT_VEBT) @ nat) @ ((vEBT_Leaf @ Uu2) @ Uv2)) @ D3)))) => (~ ((! [Mima:(option @ ((product_prod @ nat) @ nat)),Deg2:nat,TreeList3:(list @ vEBT_VEBT),Summary2:vEBT_VEBT,Deg3:nat]: ((X3 != ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ Mima) @ Deg2) @ TreeList3) @ Summary2)) @ Deg3)))))))))). % 6.41/2.68 thf(fact_661_VEBT_Odistinct_I1_J, axiom, (! [X11:(option @ ((product_prod @ nat) @ nat)),X12:nat,X13:(list @ vEBT_VEBT),X14:vEBT_VEBT,X21:$o,X222:$o]: ((((((vEBT_Node @ X11) @ X12) @ X13) @ X14) != ((vEBT_Leaf @ X21) @ X222))))). % 6.41/2.68 thf(fact_662_VEBT_Oexhaust, axiom, (! [Y:vEBT_VEBT]: (((! [X112:(option @ ((product_prod @ nat) @ nat)),X122:nat,X132:(list @ vEBT_VEBT),X142:vEBT_VEBT]: ((Y != ((((vEBT_Node @ X112) @ X122) @ X132) @ X142)))) => (~ ((! [X212:$o,X223:$o]: ((Y != ((vEBT_Leaf @ X212) @ X223)))))))))). % 6.41/2.68 thf(fact_663_zero__neq__numeral, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N2:num]: (((zero_zero @ A) != ((numeral_numeral @ A) @ N2)))))))). % 6.41/2.68 thf(fact_664_VEBT__internal_Omembermima_Osimps_I1_J, axiom, (! [Uu:$o,Uv:$o,Uw:nat]: ((~ (((vEBT_VEBT_membermima @ ((vEBT_Leaf @ Uu) @ Uv)) @ Uw)))))). % 6.41/2.68 thf(fact_665_num_Osize_I4_J, axiom, (((size_size @ num) @ one2) = (zero_zero @ nat))). % 6.41/2.68 thf(fact_666_listrel1__eq__len, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((listrel1 @ A) @ R2)) => (((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys)))))). % 6.41/2.68 thf(fact_667_listrel1__mono, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R2) @ S) => (((ord_less_eq @ (set @ ((product_prod @ (list @ A)) @ (list @ A)))) @ ((listrel1 @ A) @ R2)) @ ((listrel1 @ A) @ S)))))). % 6.41/2.68 thf(fact_668_VEBT__internal_OminNull_Osimps_I3_J, axiom, (! [Uu:$o]: ((~ ((vEBT_VEBT_minNull @ ((vEBT_Leaf @ Uu) @ $true))))))). % 6.41/2.68 thf(fact_669_VEBT__internal_OminNull_Osimps_I2_J, axiom, (! [Uv:$o]: ((~ ((vEBT_VEBT_minNull @ ((vEBT_Leaf @ $true) @ Uv))))))). % 6.41/2.68 thf(fact_670_VEBT__internal_OminNull_Osimps_I1_J, axiom, (vEBT_VEBT_minNull @ ((vEBT_Leaf @ $false) @ $false))). % 6.41/2.68 thf(fact_671_invar__vebt_Ointros_I2_J, axiom, (! [TreeList2:(list @ vEBT_VEBT),N2:nat,Summary:vEBT_VEBT,M:nat,Deg:nat]: (((! [X5:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X5) @ ((set2 @ vEBT_VEBT) @ TreeList2)) => ((vEBT_invar_vebt @ X5) @ N2)))) => (((vEBT_invar_vebt @ Summary) @ M) => ((((size_size @ (list @ vEBT_VEBT)) @ TreeList2) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) => ((M = N2) => ((Deg = (((plus_plus @ nat) @ N2) @ M)) => ((~ ((? [X_1:nat]: (((vEBT_V8194947554948674370ptions @ Summary) @ X_1))))) => ((! [X5:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X5) @ ((set2 @ vEBT_VEBT) @ TreeList2)) => (~ ((? [X_1:nat]: (((vEBT_V8194947554948674370ptions @ X5) @ X_1)))))))) => ((vEBT_invar_vebt @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Deg) @ TreeList2) @ Summary)) @ Deg))))))))))). % 6.41/2.68 thf(fact_672_invar__vebt_Ointros_I3_J, axiom, (! [TreeList2:(list @ vEBT_VEBT),N2:nat,Summary:vEBT_VEBT,M:nat,Deg:nat]: (((! [X5:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X5) @ ((set2 @ vEBT_VEBT) @ TreeList2)) => ((vEBT_invar_vebt @ X5) @ N2)))) => (((vEBT_invar_vebt @ Summary) @ M) => ((((size_size @ (list @ vEBT_VEBT)) @ TreeList2) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) => ((M = (suc @ N2)) => ((Deg = (((plus_plus @ nat) @ N2) @ M)) => ((~ ((? [X_1:nat]: (((vEBT_V8194947554948674370ptions @ Summary) @ X_1))))) => ((! [X5:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X5) @ ((set2 @ vEBT_VEBT) @ TreeList2)) => (~ ((? [X_1:nat]: (((vEBT_V8194947554948674370ptions @ X5) @ X_1)))))))) => ((vEBT_invar_vebt @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Deg) @ TreeList2) @ Summary)) @ Deg))))))))))). % 6.41/2.68 thf(fact_673_zip__same, axiom, (! [A:$tType,A2:A,B2:A,Xs:(list @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((set2 @ ((product_prod @ A) @ A)) @ ((((zip @ A) @ A) @ Xs) @ Xs))) = ((((member @ A) @ A2) @ ((set2 @ A) @ Xs)) & (A2 = B2)))))). % 6.41/2.68 thf(fact_674_in__set__zipE, axiom, (! [A:$tType,B:$tType,X3:A,Y:B,Xs:(list @ A),Ys:(list @ B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) @ ((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys))) => (~ (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (~ ((((member @ B) @ Y) @ ((set2 @ B) @ Ys))))))))))). % 6.41/2.68 thf(fact_675_set__zip__leftD, axiom, (! [B:$tType,A:$tType,X3:A,Y:B,Xs:(list @ A),Ys:(list @ B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) @ ((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys))) => (((member @ A) @ X3) @ ((set2 @ A) @ Xs)))))). % 6.41/2.68 thf(fact_676_set__zip__rightD, axiom, (! [A:$tType,B:$tType,X3:A,Y:B,Xs:(list @ A),Ys:(list @ B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) @ ((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys))) => (((member @ B) @ Y) @ ((set2 @ B) @ Ys)))))). % 6.41/2.68 thf(fact_677_zip__update, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),I:nat,X3:A,Ys:(list @ B),Y:B]: ((((((zip @ A) @ B) @ ((((list_update @ A) @ Xs) @ I) @ X3)) @ ((((list_update @ B) @ Ys) @ I) @ Y)) = ((((list_update @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys)) @ I) @ ((((product_Pair @ A) @ B) @ X3) @ Y)))))). % 6.41/2.68 thf(fact_678_zero__le__numeral, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N2:num]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ N2)))))))). % 6.41/2.68 thf(fact_679_not__numeral__le__zero, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N2:num]: ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ N2)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_680_not__numeral__less__zero, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N2:num]: ((~ ((((ord_less @ A) @ ((numeral_numeral @ A) @ N2)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_681_zero__less__numeral, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N2:num]: ((((ord_less @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ N2)))))))). % 6.41/2.68 thf(fact_682_set__encode__eq, axiom, (! [A3:(set @ nat),B3:(set @ nat)]: ((((finite_finite2 @ nat) @ A3) => (((finite_finite2 @ nat) @ B3) => (((nat_set_encode @ A3) = (nat_set_encode @ B3)) = (A3 = B3))))))). % 6.41/2.68 thf(fact_683_vebt__buildup_Osimps_I1_J, axiom, ((vEBT_vebt_buildup @ (zero_zero @ nat)) = ((vEBT_Leaf @ $false) @ $false))). % 6.41/2.68 thf(fact_684_in__set__impl__in__set__zip2, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B),Y:B]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => ((((member @ B) @ Y) @ ((set2 @ B) @ Ys)) => (~ ((! [X5:A]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y)) @ ((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys)))))))))))))). % 6.41/2.68 thf(fact_685_in__set__impl__in__set__zip1, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B),X3:A]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (~ ((! [Y4:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y4)) @ ((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys)))))))))))))). % 6.41/2.68 thf(fact_686_VEBT__internal_Onaive__member_Ocases, axiom, (! [X3:((product_prod @ vEBT_VEBT) @ nat)]: (((! [A5:$o,B5:$o,X5:nat]: ((X3 != ((((product_Pair @ vEBT_VEBT) @ nat) @ ((vEBT_Leaf @ A5) @ B5)) @ X5)))) => ((! [Uu2:(option @ ((product_prod @ nat) @ nat)),Uv2:(list @ vEBT_VEBT),Uw2:vEBT_VEBT,Ux2:nat]: ((X3 != ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ Uu2) @ (zero_zero @ nat)) @ Uv2) @ Uw2)) @ Ux2)))) => (~ ((! [Uy2:(option @ ((product_prod @ nat) @ nat)),V3:nat,TreeList3:(list @ vEBT_VEBT),S2:vEBT_VEBT,X5:nat]: ((X3 != ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ Uy2) @ (suc @ V3)) @ TreeList3) @ S2)) @ X5))))))))))). % 6.41/2.68 thf(fact_687_invar__vebt_Ointros_I1_J, axiom, (! [A2:$o,B2:$o]: (((vEBT_invar_vebt @ ((vEBT_Leaf @ A2) @ B2)) @ (suc @ (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_688_vebt__buildup_Osimps_I2_J, axiom, ((vEBT_vebt_buildup @ (suc @ (zero_zero @ nat))) = ((vEBT_Leaf @ $false) @ $false))). % 6.41/2.68 thf(fact_689_invar__vebt_Osimps, axiom, (vEBT_invar_vebt = (^ [A1:vEBT_VEBT,A22:nat]: ((((? [A4:$o,B4:$o]: ((A1 = ((vEBT_Leaf @ A4) @ B4)))) & (A22 = (suc @ (zero_zero @ nat)))) | ((? [TreeList:(list @ vEBT_VEBT),N:nat,Summary3:vEBT_VEBT]: (((A1 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ A22) @ TreeList) @ Summary3)) & ((! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList)) => ((vEBT_invar_vebt @ X) @ N)))) & (((vEBT_invar_vebt @ Summary3) @ N) & ((((size_size @ (list @ vEBT_VEBT)) @ TreeList) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) & ((A22 = (((plus_plus @ nat) @ N) @ N)) & ((~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ Summary3) @ X6))))) & (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6)))))))))))))))) | ((? [TreeList:(list @ vEBT_VEBT),N:nat,Summary3:vEBT_VEBT]: (((A1 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ A22) @ TreeList) @ Summary3)) & ((! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList)) => ((vEBT_invar_vebt @ X) @ N)))) & (((vEBT_invar_vebt @ Summary3) @ (suc @ N)) & ((((size_size @ (list @ vEBT_VEBT)) @ TreeList) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ N))) & ((A22 = (((plus_plus @ nat) @ N) @ (suc @ N))) & ((~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ Summary3) @ X6))))) & (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6)))))))))))))))) | ((? [TreeList:(list @ vEBT_VEBT),N:nat,Summary3:vEBT_VEBT,Mi3:nat,Ma3:nat]: (((A1 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi3) @ Ma3))) @ A22) @ TreeList) @ Summary3)) & ((! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList)) => ((vEBT_invar_vebt @ X) @ N)))) & (((vEBT_invar_vebt @ Summary3) @ N) & ((((size_size @ (list @ vEBT_VEBT)) @ TreeList) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) & ((A22 = (((plus_plus @ nat) @ N) @ N)) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) => ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList) @ I3)) @ X6))) = ((vEBT_V8194947554948674370ptions @ Summary3) @ I3))))) & (((Mi3 = Ma3) => (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6))))))))) & ((((ord_less_eq @ nat) @ Mi3) @ Ma3) & ((((ord_less @ nat) @ Ma3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ A22)) & ((Mi3 != Ma3) => (! [I3:nat]: (((((ord_less @ nat) @ I3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) => (((((vEBT_VEBT_high @ Ma3) @ N) = I3) => ((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList) @ I3)) @ ((vEBT_VEBT_low @ Ma3) @ N))) & (! [X:nat]: ((((((vEBT_VEBT_high @ X) @ N) = I3) & ((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList) @ I3)) @ ((vEBT_VEBT_low @ X) @ N))) => ((((ord_less @ nat) @ Mi3) @ X) & (((ord_less_eq @ nat) @ X) @ Ma3))))))))))))))))))))) | (? [TreeList:(list @ vEBT_VEBT),N:nat,Summary3:vEBT_VEBT,Mi3:nat,Ma3:nat]: (((A1 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi3) @ Ma3))) @ A22) @ TreeList) @ Summary3)) & ((! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList)) => ((vEBT_invar_vebt @ X) @ N)))) & (((vEBT_invar_vebt @ Summary3) @ (suc @ N)) & ((((size_size @ (list @ vEBT_VEBT)) @ TreeList) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ N))) & ((A22 = (((plus_plus @ nat) @ N) @ (suc @ N))) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ N))) => ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList) @ I3)) @ X6))) = ((vEBT_V8194947554948674370ptions @ Summary3) @ I3))))) & (((Mi3 = Ma3) => (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6))))))))) & ((((ord_less_eq @ nat) @ Mi3) @ Ma3) & ((((ord_less @ nat) @ Ma3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ A22)) & ((Mi3 != Ma3) => (! [I3:nat]: (((((ord_less @ nat) @ I3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ N))) => (((((vEBT_VEBT_high @ Ma3) @ N) = I3) => ((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList) @ I3)) @ ((vEBT_VEBT_low @ Ma3) @ N))) & (! [X:nat]: ((((((vEBT_VEBT_high @ X) @ N) = I3) & ((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList) @ I3)) @ ((vEBT_VEBT_low @ X) @ N))) => ((((ord_less @ nat) @ Mi3) @ X) & (((ord_less_eq @ nat) @ X) @ Ma3))))))))))))))))))))))))))))). % 6.41/2.68 thf(fact_690_invar__vebt_Ocases, axiom, (! [A12:vEBT_VEBT,A23:nat]: ((((vEBT_invar_vebt @ A12) @ A23) => (((? [A5:$o,B5:$o]: ((A12 = ((vEBT_Leaf @ A5) @ B5)))) => (A23 != (suc @ (zero_zero @ nat)))) => ((! [TreeList3:(list @ vEBT_VEBT),N3:nat,Summary2:vEBT_VEBT,M3:nat,Deg2:nat]: (((A12 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Deg2) @ TreeList3) @ Summary2)) => ((A23 = Deg2) => ((! [X2:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X2) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => ((vEBT_invar_vebt @ X2) @ N3)))) => (((vEBT_invar_vebt @ Summary2) @ M3) => ((((size_size @ (list @ vEBT_VEBT)) @ TreeList3) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M3)) => ((M3 = N3) => ((Deg2 = (((plus_plus @ nat) @ N3) @ M3)) => ((~ ((? [X_12:nat]: (((vEBT_V8194947554948674370ptions @ Summary2) @ X_12))))) => (~ ((! [X2:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X2) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X_12:nat]: (((vEBT_V8194947554948674370ptions @ X2) @ X_12)))))))))))))))))))) => ((! [TreeList3:(list @ vEBT_VEBT),N3:nat,Summary2:vEBT_VEBT,M3:nat,Deg2:nat]: (((A12 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Deg2) @ TreeList3) @ Summary2)) => ((A23 = Deg2) => ((! [X2:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X2) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => ((vEBT_invar_vebt @ X2) @ N3)))) => (((vEBT_invar_vebt @ Summary2) @ M3) => ((((size_size @ (list @ vEBT_VEBT)) @ TreeList3) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M3)) => ((M3 = (suc @ N3)) => ((Deg2 = (((plus_plus @ nat) @ N3) @ M3)) => ((~ ((? [X_12:nat]: (((vEBT_V8194947554948674370ptions @ Summary2) @ X_12))))) => (~ ((! [X2:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X2) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X_12:nat]: (((vEBT_V8194947554948674370ptions @ X2) @ X_12)))))))))))))))))))) => ((! [TreeList3:(list @ vEBT_VEBT),N3:nat,Summary2:vEBT_VEBT,M3:nat,Deg2:nat,Mi2:nat,Ma2:nat]: (((A12 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ Deg2) @ TreeList3) @ Summary2)) => ((A23 = Deg2) => ((! [X2:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X2) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => ((vEBT_invar_vebt @ X2) @ N3)))) => (((vEBT_invar_vebt @ Summary2) @ M3) => ((((size_size @ (list @ vEBT_VEBT)) @ TreeList3) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M3)) => ((M3 = N3) => ((Deg2 = (((plus_plus @ nat) @ N3) @ M3)) => ((! [I5:nat]: (((((ord_less @ nat) @ I5) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M3)) => ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList3) @ I5)) @ X6))) = ((vEBT_V8194947554948674370ptions @ Summary2) @ I5))))) => (((Mi2 = Ma2) => (! [X2:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X2) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X_12:nat]: (((vEBT_V8194947554948674370ptions @ X2) @ X_12))))))))) => ((((ord_less_eq @ nat) @ Mi2) @ Ma2) => ((((ord_less @ nat) @ Ma2) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg2)) => (~ (((Mi2 != Ma2) => (! [I5:nat]: (((((ord_less @ nat) @ I5) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M3)) => (((((vEBT_VEBT_high @ Ma2) @ N3) = I5) => ((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList3) @ I5)) @ ((vEBT_VEBT_low @ Ma2) @ N3))) & (! [X2:nat]: ((((((vEBT_VEBT_high @ X2) @ N3) = I5) & ((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList3) @ I5)) @ ((vEBT_VEBT_low @ X2) @ N3))) => ((((ord_less @ nat) @ Mi2) @ X2) & (((ord_less_eq @ nat) @ X2) @ Ma2))))))))))))))))))))))))) => (~ ((! [TreeList3:(list @ vEBT_VEBT),N3:nat,Summary2:vEBT_VEBT,M3:nat,Deg2:nat,Mi2:nat,Ma2:nat]: (((A12 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ Deg2) @ TreeList3) @ Summary2)) => ((A23 = Deg2) => ((! [X2:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X2) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => ((vEBT_invar_vebt @ X2) @ N3)))) => (((vEBT_invar_vebt @ Summary2) @ M3) => ((((size_size @ (list @ vEBT_VEBT)) @ TreeList3) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M3)) => ((M3 = (suc @ N3)) => ((Deg2 = (((plus_plus @ nat) @ N3) @ M3)) => ((! [I5:nat]: (((((ord_less @ nat) @ I5) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M3)) => ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList3) @ I5)) @ X6))) = ((vEBT_V8194947554948674370ptions @ Summary2) @ I5))))) => (((Mi2 = Ma2) => (! [X2:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X2) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X_12:nat]: (((vEBT_V8194947554948674370ptions @ X2) @ X_12))))))))) => ((((ord_less_eq @ nat) @ Mi2) @ Ma2) => ((((ord_less @ nat) @ Ma2) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg2)) => (~ (((Mi2 != Ma2) => (! [I5:nat]: (((((ord_less @ nat) @ I5) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M3)) => (((((vEBT_VEBT_high @ Ma2) @ N3) = I5) => ((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList3) @ I5)) @ ((vEBT_VEBT_low @ Ma2) @ N3))) & (! [X2:nat]: ((((((vEBT_VEBT_high @ X2) @ N3) = I5) & ((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList3) @ I5)) @ ((vEBT_VEBT_low @ X2) @ N3))) => ((((ord_less @ nat) @ Mi2) @ X2) & (((ord_less_eq @ nat) @ X2) @ Ma2))))))))))))))))))))))))))))))))))). % 6.41/2.68 thf(fact_691_VEBT__internal_OminNull_Ocases, axiom, (! [X3:vEBT_VEBT]: (((X3 != ((vEBT_Leaf @ $false) @ $false)) => ((! [Uv2:$o]: ((X3 != ((vEBT_Leaf @ $true) @ Uv2)))) => ((! [Uu2:$o]: ((X3 != ((vEBT_Leaf @ Uu2) @ $true)))) => ((! [Uw2:nat,Ux2:(list @ vEBT_VEBT),Uy2:vEBT_VEBT]: ((X3 != ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uw2) @ Ux2) @ Uy2)))) => (~ ((! [Uz2:((product_prod @ nat) @ nat),Va3:nat,Vb2:(list @ vEBT_VEBT),Vc2:vEBT_VEBT]: ((X3 != ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ Uz2)) @ Va3) @ Vb2) @ Vc2))))))))))))). % 6.41/2.68 thf(fact_692_num_Osize_I5_J, axiom, (! [X22:num]: ((((size_size @ num) @ (bit0 @ X22)) = (((plus_plus @ nat) @ ((size_size @ num) @ X22)) @ (suc @ (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_693_invar__vebt_Ointros_I4_J, axiom, (! [TreeList2:(list @ vEBT_VEBT),N2:nat,Summary:vEBT_VEBT,M:nat,Deg:nat,Mi:nat,Ma:nat]: (((! [X5:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X5) @ ((set2 @ vEBT_VEBT) @ TreeList2)) => ((vEBT_invar_vebt @ X5) @ N2)))) => (((vEBT_invar_vebt @ Summary) @ M) => ((((size_size @ (list @ vEBT_VEBT)) @ TreeList2) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) => ((M = N2) => ((Deg = (((plus_plus @ nat) @ N2) @ M)) => ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) => ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList2) @ I4)) @ X6))) = ((vEBT_V8194947554948674370ptions @ Summary) @ I4))))) => (((Mi = Ma) => (! [X5:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X5) @ ((set2 @ vEBT_VEBT) @ TreeList2)) => (~ ((? [X_1:nat]: (((vEBT_V8194947554948674370ptions @ X5) @ X_1))))))))) => ((((ord_less_eq @ nat) @ Mi) @ Ma) => ((((ord_less @ nat) @ Ma) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg)) => (((Mi != Ma) => (! [I4:nat]: (((((ord_less @ nat) @ I4) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) => (((((vEBT_VEBT_high @ Ma) @ N2) = I4) => ((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList2) @ I4)) @ ((vEBT_VEBT_low @ Ma) @ N2))) & (! [X5:nat]: ((((((vEBT_VEBT_high @ X5) @ N2) = I4) & ((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList2) @ I4)) @ ((vEBT_VEBT_low @ X5) @ N2))) => ((((ord_less @ nat) @ Mi) @ X5) & (((ord_less_eq @ nat) @ X5) @ Ma)))))))))) => ((vEBT_invar_vebt @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi) @ Ma))) @ Deg) @ TreeList2) @ Summary)) @ Deg)))))))))))))). % 6.41/2.68 thf(fact_694_VEBT__internal_OminNull_Oelims_I3_J, axiom, (! [X3:vEBT_VEBT]: (((~ ((vEBT_VEBT_minNull @ X3))) => ((! [Uv2:$o]: ((X3 != ((vEBT_Leaf @ $true) @ Uv2)))) => ((! [Uu2:$o]: ((X3 != ((vEBT_Leaf @ Uu2) @ $true)))) => (~ ((! [Uz2:((product_prod @ nat) @ nat),Va3:nat,Vb2:(list @ vEBT_VEBT),Vc2:vEBT_VEBT]: ((X3 != ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ Uz2)) @ Va3) @ Vb2) @ Vc2)))))))))))). % 6.41/2.68 thf(fact_695_VEBT__internal_OminNull_Oelims_I2_J, axiom, (! [X3:vEBT_VEBT]: (((vEBT_VEBT_minNull @ X3) => ((X3 != ((vEBT_Leaf @ $false) @ $false)) => (~ ((! [Uw2:nat,Ux2:(list @ vEBT_VEBT),Uy2:vEBT_VEBT]: ((X3 != ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uw2) @ Ux2) @ Uy2))))))))))). % 6.41/2.68 thf(fact_696_invar__vebt_Ointros_I5_J, axiom, (! [TreeList2:(list @ vEBT_VEBT),N2:nat,Summary:vEBT_VEBT,M:nat,Deg:nat,Mi:nat,Ma:nat]: (((! [X5:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X5) @ ((set2 @ vEBT_VEBT) @ TreeList2)) => ((vEBT_invar_vebt @ X5) @ N2)))) => (((vEBT_invar_vebt @ Summary) @ M) => ((((size_size @ (list @ vEBT_VEBT)) @ TreeList2) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) => ((M = (suc @ N2)) => ((Deg = (((plus_plus @ nat) @ N2) @ M)) => ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) => ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList2) @ I4)) @ X6))) = ((vEBT_V8194947554948674370ptions @ Summary) @ I4))))) => (((Mi = Ma) => (! [X5:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X5) @ ((set2 @ vEBT_VEBT) @ TreeList2)) => (~ ((? [X_1:nat]: (((vEBT_V8194947554948674370ptions @ X5) @ X_1))))))))) => ((((ord_less_eq @ nat) @ Mi) @ Ma) => ((((ord_less @ nat) @ Ma) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg)) => (((Mi != Ma) => (! [I4:nat]: (((((ord_less @ nat) @ I4) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) => (((((vEBT_VEBT_high @ Ma) @ N2) = I4) => ((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList2) @ I4)) @ ((vEBT_VEBT_low @ Ma) @ N2))) & (! [X5:nat]: ((((((vEBT_VEBT_high @ X5) @ N2) = I4) & ((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList2) @ I4)) @ ((vEBT_VEBT_low @ X5) @ N2))) => ((((ord_less @ nat) @ Mi) @ X5) & (((ord_less_eq @ nat) @ X5) @ Ma)))))))))) => ((vEBT_invar_vebt @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi) @ Ma))) @ Deg) @ TreeList2) @ Summary)) @ Deg)))))))))))))). % 6.41/2.68 thf(fact_697_set__encode__inf, axiom, (! [A3:(set @ nat)]: (((~ (((finite_finite2 @ nat) @ A3))) => ((nat_set_encode @ A3) = (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_698_pair__lessI1, axiom, (! [A2:nat,B2:nat,S:nat,T2:nat]: (((((ord_less @ nat) @ A2) @ B2) => (((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ A2) @ S)) @ ((((product_Pair @ nat) @ nat) @ B2) @ T2))) @ fun_pair_less))))). % 6.41/2.68 thf(fact_699_count__le__length, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((ord_less_eq @ nat) @ (((count_list @ A) @ Xs) @ X3)) @ ((size_size @ (list @ A)) @ Xs))))). % 6.41/2.68 thf(fact_700_vebt__mint_Ocases, axiom, (! [X3:vEBT_VEBT]: (((! [A5:$o,B5:$o]: ((X3 != ((vEBT_Leaf @ A5) @ B5)))) => ((! [Uu2:nat,Uv2:(list @ vEBT_VEBT),Uw2:vEBT_VEBT]: ((X3 != ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu2) @ Uv2) @ Uw2)))) => (~ ((! [Mi2:nat,Ma2:nat,Ux2:nat,Uy2:(list @ vEBT_VEBT),Uz2:vEBT_VEBT]: ((X3 != ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ Ux2) @ Uy2) @ Uz2))))))))))). % 6.41/2.68 thf(fact_701_VEBT__internal_OminNull_Oelims_I1_J, axiom, (! [X3:vEBT_VEBT,Y:$o]: ((((vEBT_VEBT_minNull @ X3) = Y) => (((X3 = ((vEBT_Leaf @ $false) @ $false)) => (~ (Y))) => (((? [Uv2:$o]: ((X3 = ((vEBT_Leaf @ $true) @ Uv2)))) => Y) => (((? [Uu2:$o]: ((X3 = ((vEBT_Leaf @ Uu2) @ $true)))) => Y) => (((? [Uw2:nat,Ux2:(list @ vEBT_VEBT),Uy2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uw2) @ Ux2) @ Uy2)))) => (~ (Y))) => (~ (((? [Uz2:((product_prod @ nat) @ nat),Va3:nat,Vb2:(list @ vEBT_VEBT),Vc2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ Uz2)) @ Va3) @ Vb2) @ Vc2)))) => Y))))))))))). % 6.41/2.68 thf(fact_702_length__code, axiom, (! [A:$tType]: (((size_size @ (list @ A)) = ((gen_length @ A) @ (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_703_sum__power2__eq__zero__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A,Y:A]: ((((((plus_plus @ A) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (zero_zero @ A)) = ((X3 = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_704_zero__less__power2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (A2 != (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_705_power2__less__eq__zero__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_706_power2__eq__iff__nonneg, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (X3 = Y)))))))))). % 6.41/2.68 thf(fact_707_zero__eq__power2, axiom, (! [A:$tType]: (((semiri2026040879449505780visors @ A) => (! [A2:A]: ((((((power_power @ A) @ A2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_708_power__mono__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B2:A,N2:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N2)) @ (((power_power @ A) @ B2) @ N2)) = (((ord_less_eq @ A) @ A2) @ B2))))))))))). % 6.41/2.68 thf(fact_709_power__eq__0__iff, axiom, (! [A:$tType]: (((semiri2026040879449505780visors @ A) => (! [A2:A,N2:nat]: ((((((power_power @ A) @ A2) @ N2) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))))))). % 6.41/2.68 thf(fact_710_member__inv, axiom, (! [Mi:nat,Ma:nat,Deg:nat,TreeList2:(list @ vEBT_VEBT),Summary:vEBT_VEBT,X3:nat]: ((((vEBT_vebt_member @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi) @ Ma))) @ Deg) @ TreeList2) @ Summary)) @ X3) => ((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg) & ((X3 = Mi) | ((X3 = Ma) | ((((ord_less @ nat) @ X3) @ Ma) & ((((ord_less @ nat) @ Mi) @ X3) & ((((ord_less @ nat) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList2)) & ((vEBT_vebt_member @ (((nth @ vEBT_VEBT) @ TreeList2) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ X3) @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))))))). % 6.41/2.68 thf(fact_711_both__member__options__ding, axiom, (! [Info:(option @ ((product_prod @ nat) @ nat)),Deg:nat,TreeList2:(list @ vEBT_VEBT),Summary:vEBT_VEBT,N2:nat,X3:nat]: ((((vEBT_invar_vebt @ ((((vEBT_Node @ Info) @ Deg) @ TreeList2) @ Summary)) @ N2) => ((((ord_less @ nat) @ X3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg)) => (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList2) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ X3) @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) => ((vEBT_V8194947554948674370ptions @ ((((vEBT_Node @ Info) @ Deg) @ TreeList2) @ Summary)) @ X3))))))). % 6.41/2.68 thf(fact_712_nat__zero__less__power__iff, axiom, (! [X3:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((power_power @ nat) @ X3) @ N2)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ X3) | (N2 = (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_713_pow__sum, axiom, (! [A2:nat,B2:nat]: (((((divide_divide @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ A2) @ B2))) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ A2)) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ B2))))). % 6.41/2.68 thf(fact_714_high__def, axiom, (vEBT_VEBT_high = (^ [X:nat,N:nat]: ((((divide_divide @ nat) @ X) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))))). % 6.41/2.68 thf(fact_715_division__ring__divide__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))). % 6.41/2.68 thf(fact_716_divide__cancel__right, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,C2:A,B2:A]: ((((((divide_divide @ A) @ A2) @ C2) = (((divide_divide @ A) @ B2) @ C2)) = ((C2 = (zero_zero @ A)) | (A2 = B2))))))))). % 6.41/2.68 thf(fact_717_divide__cancel__left, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A2:A,B2:A]: ((((((divide_divide @ A) @ C2) @ A2) = (((divide_divide @ A) @ C2) @ B2)) = ((C2 = (zero_zero @ A)) | (A2 = B2))))))))). % 6.41/2.68 thf(fact_718_div__by__0, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))). % 6.41/2.68 thf(fact_719_divide__eq__0__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A]: ((((((divide_divide @ A) @ A2) @ B2) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_720_div__0, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A2:A]: (((((divide_divide @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))). % 6.41/2.68 thf(fact_721_power__0__Suc, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N2:nat]: (((((power_power @ A) @ (zero_zero @ A)) @ (suc @ N2)) = (zero_zero @ A)))))))). % 6.41/2.68 thf(fact_722_power__zero__numeral, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [K:num]: (((((power_power @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ nat) @ K)) = (zero_zero @ A)))))))). % 6.41/2.68 thf(fact_723_power__Suc0__right, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A2:A]: (((((power_power @ A) @ A2) @ (suc @ (zero_zero @ nat))) = A2))))))). % 6.41/2.68 thf(fact_724_power__Suc__0, axiom, (! [N2:nat]: (((((power_power @ nat) @ (suc @ (zero_zero @ nat))) @ N2) = (suc @ (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_725_nat__power__eq__Suc__0__iff, axiom, (! [X3:nat,M:nat]: ((((((power_power @ nat) @ X3) @ M) = (suc @ (zero_zero @ nat))) = ((M = (zero_zero @ nat)) | (X3 = (suc @ (zero_zero @ nat)))))))). % 6.41/2.68 thf(fact_726_add__divide__distrib, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ C2)) @ (((divide_divide @ A) @ B2) @ C2))))))))). % 6.41/2.68 thf(fact_727_divide__right__mono__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ (((divide_divide @ A) @ A2) @ C2)))))))))). % 6.41/2.68 thf(fact_728_divide__nonpos__nonpos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ Y) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X3) @ Y)))))))))). % 6.41/2.68 thf(fact_729_divide__nonpos__nonneg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_730_divide__nonneg__nonpos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less_eq @ A) @ Y) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_731_divide__nonneg__nonneg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X3) @ Y)))))))))). % 6.41/2.68 thf(fact_732_zero__le__divide__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ B2)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))))))))))). % 6.41/2.68 thf(fact_733_divide__right__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ C2)) @ (((divide_divide @ A) @ B2) @ C2)))))))))). % 6.41/2.68 thf(fact_734_divide__le__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ (zero_zero @ A)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))))). % 6.41/2.68 thf(fact_735_divide__neg__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ (zero_zero @ A)) => ((((ord_less @ A) @ Y) @ (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X3) @ Y)))))))))). % 6.41/2.68 thf(fact_736_divide__neg__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y) => (((ord_less @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_737_divide__pos__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less @ A) @ Y) @ (zero_zero @ A)) => (((ord_less @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_738_divide__pos__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X3) @ Y)))))))))). % 6.41/2.68 thf(fact_739_divide__less__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ (zero_zero @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))))). % 6.41/2.68 thf(fact_740_divide__less__cancel, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ A2) @ C2)) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A2) @ B2)) & (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ A2)) & (C2 != (zero_zero @ A))))))))))). % 6.41/2.68 thf(fact_741_zero__less__divide__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ B2) @ (zero_zero @ A))))))))))). % 6.41/2.68 thf(fact_742_divide__strict__right__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((divide_divide @ A) @ A2) @ C2)) @ (((divide_divide @ A) @ B2) @ C2)))))))))). % 6.41/2.68 thf(fact_743_divide__strict__right__mono__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((divide_divide @ A) @ A2) @ C2)) @ (((divide_divide @ A) @ B2) @ C2)))))))))). % 6.41/2.68 thf(fact_744_frac__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y:A,X3:A,W2:A,Z2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => ((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_less @ A) @ (zero_zero @ A)) @ W2) => ((((ord_less_eq @ A) @ W2) @ Z2) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X3) @ Z2)) @ (((divide_divide @ A) @ Y) @ W2)))))))))))). % 6.41/2.68 thf(fact_745_frac__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A,W2:A,Z2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less @ A) @ X3) @ Y) => ((((ord_less @ A) @ (zero_zero @ A)) @ W2) => ((((ord_less_eq @ A) @ W2) @ Z2) => (((ord_less @ A) @ (((divide_divide @ A) @ X3) @ Z2)) @ (((divide_divide @ A) @ Y) @ W2)))))))))))). % 6.41/2.68 thf(fact_746_frac__less2, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A,W2:A,Z2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_less @ A) @ (zero_zero @ A)) @ W2) => ((((ord_less @ A) @ W2) @ Z2) => (((ord_less @ A) @ (((divide_divide @ A) @ X3) @ Z2)) @ (((divide_divide @ A) @ Y) @ W2)))))))))))). % 6.41/2.68 thf(fact_747_divide__le__cancel, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ C2)) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A2) @ B2)) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ A2)))))))))). % 6.41/2.68 thf(fact_748_divide__nonneg__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less @ A) @ Y) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_749_divide__nonneg__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X3) @ Y)))))))))). % 6.41/2.68 thf(fact_750_divide__nonpos__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ (zero_zero @ A)) => ((((ord_less @ A) @ Y) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ X3) @ Y)))))))))). % 6.41/2.68 thf(fact_751_divide__nonpos__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ Y) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_752_field__sum__of__halves, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A]: (((((plus_plus @ A) @ (((divide_divide @ A) @ X3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((divide_divide @ A) @ X3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) = X3))))))). % 6.41/2.68 thf(fact_753_half__gt__zero, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))). % 6.41/2.68 thf(fact_754_half__gt__zero__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))). % 6.41/2.68 thf(fact_755_field__less__half__sum, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (((ord_less @ A) @ X3) @ (((divide_divide @ A) @ (((plus_plus @ A) @ X3) @ Y)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))). % 6.41/2.68 thf(fact_756_power__not__zero, axiom, (! [A:$tType]: (((semiri2026040879449505780visors @ A) => (! [A2:A,N2:nat]: (((A2 != (zero_zero @ A)) => ((((power_power @ A) @ A2) @ N2) != (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_757_zero__le__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ N2))))))))). % 6.41/2.68 thf(fact_758_power__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B2:A,N2:nat]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N2)) @ (((power_power @ A) @ B2) @ N2)))))))))). % 6.41/2.68 thf(fact_759_zero__less__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ N2))))))))). % 6.41/2.68 thf(fact_760_nat__power__less__imp__less, axiom, (! [I:nat,M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ I) => ((((ord_less @ nat) @ (((power_power @ nat) @ I) @ M)) @ (((power_power @ nat) @ I) @ N2)) => (((ord_less @ nat) @ M) @ N2)))))). % 6.41/2.68 thf(fact_761_power__less__imp__less__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat,B2:A]: (((((ord_less @ A) @ (((power_power @ A) @ A2) @ N2)) @ (((power_power @ A) @ B2) @ N2)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_762_power__le__imp__le__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat,B2:A]: (((((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ (suc @ N2))) @ (((power_power @ A) @ B2) @ (suc @ N2))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_763_power__inject__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat,B2:A]: ((((((power_power @ A) @ A2) @ (suc @ N2)) = (((power_power @ A) @ B2) @ (suc @ N2))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (A2 = B2)))))))))). % 6.41/2.68 thf(fact_764_zero__power, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((power_power @ A) @ (zero_zero @ A)) @ N2) = (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_765_power__gt__expt, axiom, (! [N2:nat,K:nat]: (((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ N2) => (((ord_less @ nat) @ K) @ (((power_power @ nat) @ N2) @ K)))))). % 6.41/2.68 thf(fact_766_nat__one__le__power, axiom, (! [I:nat,N2:nat]: (((((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ I) => (((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ (((power_power @ nat) @ I) @ N2)))))). % 6.41/2.68 thf(fact_767_power__eq__iff__eq__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N2:nat,A2:A,B2:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((((power_power @ A) @ A2) @ N2) = (((power_power @ A) @ B2) @ N2)) = (A2 = B2))))))))))). % 6.41/2.68 thf(fact_768_power__eq__imp__eq__base, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat,B2:A]: ((((((power_power @ A) @ A2) @ N2) = (((power_power @ A) @ B2) @ N2)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (A2 = B2))))))))))). % 6.41/2.68 thf(fact_769_zero__power2, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((((power_power @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (zero_zero @ A)))))). % 6.41/2.68 thf(fact_770_less__exp, axiom, (! [N2:nat]: ((((ord_less @ nat) @ N2) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))))). % 6.41/2.68 thf(fact_771_power2__nat__le__imp__le, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ (((power_power @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ N2) => (((ord_less_eq @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_772_power2__nat__le__eq__le, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ (((power_power @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (((ord_less_eq @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_773_self__le__ge2__pow, axiom, (! [K:nat,M:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K) => (((ord_less_eq @ nat) @ M) @ (((power_power @ nat) @ K) @ M)))))). % 6.41/2.68 thf(fact_774_power2__le__imp__le, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((ord_less_eq @ A) @ X3) @ Y))))))))). % 6.41/2.68 thf(fact_775_power2__eq__imp__eq, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X3:A,Y:A]: ((((((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (X3 = Y)))))))))). % 6.41/2.68 thf(fact_776_zero__le__power2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))). % 6.41/2.68 thf(fact_777_power__strict__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B2:A,N2:nat]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((ord_less @ A) @ (((power_power @ A) @ A2) @ N2)) @ (((power_power @ A) @ B2) @ N2))))))))))). % 6.41/2.68 thf(fact_778_power2__less__0, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ (((power_power @ A) @ A2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_779_power2__less__imp__less, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((ord_less @ A) @ X3) @ Y))))))))). % 6.41/2.68 thf(fact_780_sum__power2__le__zero__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (zero_zero @ A)) = ((X3 = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_781_sum__power2__ge__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A,Y:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))). % 6.41/2.68 thf(fact_782_not__sum__power2__lt__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A,Y:A]: ((~ ((((ord_less @ A) @ (((plus_plus @ A) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_783_sum__power2__gt__zero__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) = ((X3 != (zero_zero @ A)) | (Y != (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_784_both__member__options__from__chilf__to__complete__tree, axiom, (! [X3:nat,Deg:nat,TreeList2:(list @ vEBT_VEBT),Mi:nat,Ma:nat,Summary:vEBT_VEBT]: (((((ord_less @ nat) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList2)) => ((((ord_less_eq @ nat) @ (one_one @ nat)) @ Deg) => (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList2) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ X3) @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) => ((vEBT_V8194947554948674370ptions @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi) @ Ma))) @ Deg) @ TreeList2) @ Summary)) @ X3))))))). % 6.41/2.68 thf(fact_785_add__self__div__2, axiom, (! [M:nat]: (((((divide_divide @ nat) @ (((plus_plus @ nat) @ M) @ M)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = M)))). % 6.41/2.68 thf(fact_786_div2__Suc__Suc, axiom, (! [M:nat]: (((((divide_divide @ nat) @ (suc @ (suc @ M))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (suc @ (((divide_divide @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))). % 6.41/2.68 thf(fact_787_both__member__options__from__complete__tree__to__child, axiom, (! [Deg:nat,Mi:nat,Ma:nat,TreeList2:(list @ vEBT_VEBT),Summary:vEBT_VEBT,X3:nat]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ Deg) => (((vEBT_V8194947554948674370ptions @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi) @ Ma))) @ Deg) @ TreeList2) @ Summary)) @ X3) => (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList2) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ X3) @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) | ((X3 = Mi) | (X3 = Ma)))))))). % 6.41/2.68 thf(fact_788_semiring__norm_I76_J, axiom, (! [N2:num]: ((((ord_less @ num) @ one2) @ (bit0 @ N2))))). % 6.41/2.68 thf(fact_789_div__less, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ N2) => ((((divide_divide @ nat) @ M) @ N2) = (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_790_div__by__Suc__0, axiom, (! [M:nat]: (((((divide_divide @ nat) @ M) @ (suc @ (zero_zero @ nat))) = M)))). % 6.41/2.68 thf(fact_791_set__n__deg__not__0, axiom, (! [TreeList2:(list @ vEBT_VEBT),N2:nat,M:nat]: (((! [X5:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X5) @ ((set2 @ vEBT_VEBT) @ TreeList2)) => ((vEBT_invar_vebt @ X5) @ N2)))) => ((((size_size @ (list @ vEBT_VEBT)) @ TreeList2) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) => (((ord_less_eq @ nat) @ (one_one @ nat)) @ N2)))))). % 6.41/2.68 thf(fact_792_Suc__n__div__2__gt__zero, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((ord_less @ nat) @ (zero_zero @ nat)) @ (((divide_divide @ nat) @ (suc @ N2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))). % 6.41/2.68 thf(fact_793_div__2__gt__zero, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ N2) => (((ord_less @ nat) @ (zero_zero @ nat)) @ (((divide_divide @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))). % 6.41/2.68 thf(fact_794_deg1Leaf, axiom, (! [T2:vEBT_VEBT]: ((((vEBT_invar_vebt @ T2) @ (one_one @ nat)) = (? [A4:$o,B4:$o]: ((T2 = ((vEBT_Leaf @ A4) @ B4)))))))). % 6.41/2.68 thf(fact_795_deg__1__Leaf, axiom, (! [T2:vEBT_VEBT]: ((((vEBT_invar_vebt @ T2) @ (one_one @ nat)) => (? [A5:$o,B5:$o]: ((T2 = ((vEBT_Leaf @ A5) @ B5)))))))). % 6.41/2.68 thf(fact_796_deg__1__Leafy, axiom, (! [T2:vEBT_VEBT,N2:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => ((N2 = (one_one @ nat)) => (? [A5:$o,B5:$o]: ((T2 = ((vEBT_Leaf @ A5) @ B5))))))))). % 6.41/2.68 thf(fact_797_div__by__1, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (one_one @ A)) = A2))))))). % 6.41/2.68 thf(fact_798_semiring__norm_I78_J, axiom, (! [M:num,N2:num]: (((((ord_less @ num) @ (bit0 @ M)) @ (bit0 @ N2)) = (((ord_less @ num) @ M) @ N2))))). % 6.41/2.68 thf(fact_799_semiring__norm_I75_J, axiom, (! [M:num]: ((~ ((((ord_less @ num) @ M) @ one2)))))). % 6.41/2.68 thf(fact_800_divide__eq__1__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A]: ((((((divide_divide @ A) @ A2) @ B2) = (one_one @ A)) = ((B2 != (zero_zero @ A)) & (A2 = B2))))))))). % 6.41/2.68 thf(fact_801_div__self, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ A2) = (one_one @ A))))))))). % 6.41/2.68 thf(fact_802_one__eq__divide__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A]: ((((one_one @ A) = (((divide_divide @ A) @ A2) @ B2)) = ((B2 != (zero_zero @ A)) & (A2 = B2))))))))). % 6.41/2.68 thf(fact_803_divide__self, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ A2) = (one_one @ A))))))))). % 6.41/2.68 thf(fact_804_divide__self__if, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: ((((A2 = (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ A2) = (zero_zero @ A))) & ((A2 != (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ A2) = (one_one @ A)))))))))). % 6.41/2.68 thf(fact_805_divide__eq__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A2:A]: ((((((divide_divide @ A) @ B2) @ A2) = (one_one @ A)) = ((A2 != (zero_zero @ A)) & (A2 = B2))))))))). % 6.41/2.68 thf(fact_806_eq__divide__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A2:A]: ((((one_one @ A) = (((divide_divide @ A) @ B2) @ A2)) = ((A2 != (zero_zero @ A)) & (A2 = B2))))))))). % 6.41/2.68 thf(fact_807_one__divide__eq__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: ((((((divide_divide @ A) @ (one_one @ A)) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_808_zero__eq__1__divide__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: ((((zero_zero @ A) = (((divide_divide @ A) @ (one_one @ A)) @ A2)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_809_power__inject__exp, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,M:nat,N2:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A2) => (((((power_power @ A) @ A2) @ M) = (((power_power @ A) @ A2) @ N2)) = (M = N2))))))))). % 6.41/2.68 thf(fact_810_less__one, axiom, (! [N2:nat]: (((((ord_less @ nat) @ N2) @ (one_one @ nat)) = (N2 = (zero_zero @ nat)))))). % 6.41/2.68 thf(fact_811_divide__le__0__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_812_zero__le__divide__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))). % 6.41/2.68 thf(fact_813_zero__less__divide__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))). % 6.41/2.68 thf(fact_814_less__divide__eq__1__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A2)) = (((ord_less @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_815_less__divide__eq__1__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A2)) = (((ord_less @ A) @ B2) @ A2))))))))). % 6.41/2.68 thf(fact_816_divide__less__eq__1__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ A2)) @ (one_one @ A)) = (((ord_less @ A) @ B2) @ A2))))))))). % 6.41/2.68 thf(fact_817_divide__less__eq__1__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ A2)) @ (one_one @ A)) = (((ord_less @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_818_divide__less__0__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_819_power__strict__increasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,X3:nat,Y:nat]: (((((ord_less @ A) @ (one_one @ A)) @ B2) => ((((ord_less @ A) @ (((power_power @ A) @ B2) @ X3)) @ (((power_power @ A) @ B2) @ Y)) = (((ord_less @ nat) @ X3) @ Y))))))))). % 6.41/2.68 thf(fact_820_le__divide__eq__1__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A2)) = (((ord_less_eq @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_821_le__divide__eq__1__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A2)) = (((ord_less_eq @ A) @ B2) @ A2))))))))). % 6.41/2.68 thf(fact_822_divide__le__eq__1__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ A2)) @ (one_one @ A)) = (((ord_less_eq @ A) @ B2) @ A2))))))))). % 6.41/2.68 thf(fact_823_divide__le__eq__1__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ A2)) @ (one_one @ A)) = (((ord_less_eq @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_824_one__add__one, axiom, (! [A:$tType]: (((numeral @ A) => ((((plus_plus @ A) @ (one_one @ A)) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit0 @ one2))))))). % 6.41/2.68 thf(fact_825_power__strict__decreasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,M:nat,N2:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ A) @ B2) @ (one_one @ A)) => ((((ord_less @ A) @ (((power_power @ A) @ B2) @ M)) @ (((power_power @ A) @ B2) @ N2)) = (((ord_less @ nat) @ N2) @ M)))))))))). % 6.41/2.68 thf(fact_826_power__increasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,X3:nat,Y:nat]: (((((ord_less @ A) @ (one_one @ A)) @ B2) => ((((ord_less_eq @ A) @ (((power_power @ A) @ B2) @ X3)) @ (((power_power @ A) @ B2) @ Y)) = (((ord_less_eq @ nat) @ X3) @ Y))))))))). % 6.41/2.68 thf(fact_827_Suc__1, axiom, ((suc @ (one_one @ nat)) = ((numeral_numeral @ nat) @ (bit0 @ one2)))). % 6.41/2.68 thf(fact_828_one__plus__numeral, axiom, (! [A:$tType]: (((numeral @ A) => (! [N2:num]: (((((plus_plus @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N2)) = ((numeral_numeral @ A) @ (((plus_plus @ num) @ one2) @ N2))))))))). % 6.41/2.68 thf(fact_829_numeral__plus__one, axiom, (! [A:$tType]: (((numeral @ A) => (! [N2:num]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ N2)) @ (one_one @ A)) = ((numeral_numeral @ A) @ (((plus_plus @ num) @ N2) @ one2))))))))). % 6.41/2.68 thf(fact_830_numeral__le__one__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N2:num]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ N2)) @ (one_one @ A)) = (((ord_less_eq @ num) @ N2) @ one2)))))))). % 6.41/2.68 thf(fact_831_one__less__numeral__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N2:num]: (((((ord_less @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N2)) = (((ord_less @ num) @ one2) @ N2)))))))). % 6.41/2.68 thf(fact_832_one__div__two__eq__zero, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => ((((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A)))))). % 6.41/2.68 thf(fact_833_power__decreasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,M:nat,N2:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ A) @ B2) @ (one_one @ A)) => ((((ord_less_eq @ A) @ (((power_power @ A) @ B2) @ M)) @ (((power_power @ A) @ B2) @ N2)) = (((ord_less_eq @ nat) @ N2) @ M)))))))))). % 6.41/2.68 thf(fact_834_one__reorient, axiom, (! [A:$tType]: (((one @ A) => (! [X3:A]: ((((one_one @ A) = X3) = (X3 = (one_one @ A))))))))). % 6.41/2.68 thf(fact_835_le__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (((ord_less_eq @ A) @ (one_one @ A)) @ (one_one @ A)))))). % 6.41/2.68 thf(fact_836_zero__neq__one, axiom, (! [A:$tType]: (((zero_neq_one @ A) => ((zero_zero @ A) != (one_one @ A)))))). % 6.41/2.68 thf(fact_837_less__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (one_one @ A)))))))). % 6.41/2.68 thf(fact_838_div__add__self2, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,A2:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ (one_one @ A)))))))))). % 6.41/2.68 thf(fact_839_div__add__self1, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,A2:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ B2) @ A2)) @ B2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ (one_one @ A)))))))))). % 6.41/2.68 thf(fact_840_zero__less__one__class_Ozero__le__one, axiom, (! [A:$tType]: (((zero_less_one @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (one_one @ A)))))). % 6.41/2.68 thf(fact_841_linordered__nonzero__semiring__class_Ozero__le__one, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (one_one @ A)))))). % 6.41/2.68 thf(fact_842_not__one__le__zero, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))). % 6.41/2.68 thf(fact_843_not__one__less__zero, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))). % 6.41/2.68 thf(fact_844_zero__less__one, axiom, (! [A:$tType]: (((zero_less_one @ A) => (((ord_less @ A) @ (zero_zero @ A)) @ (one_one @ A)))))). % 6.41/2.68 thf(fact_845_less__numeral__extra_I1_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (((ord_less @ A) @ (zero_zero @ A)) @ (one_one @ A)))))). % 6.41/2.68 thf(fact_846_one__le__numeral, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N2:num]: ((((ord_less_eq @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N2)))))))). % 6.41/2.68 thf(fact_847_not__numeral__less__one, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N2:num]: ((~ ((((ord_less @ A) @ ((numeral_numeral @ A) @ N2)) @ (one_one @ A)))))))))). % 6.41/2.68 thf(fact_848_add__mono1, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ (one_one @ A))) @ (((plus_plus @ A) @ B2) @ (one_one @ A)))))))))). % 6.41/2.68 thf(fact_849_less__add__one, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A]: ((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ A2) @ (one_one @ A))))))))). % 6.41/2.68 thf(fact_850_right__inverse__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,A2:A]: (((B2 != (zero_zero @ A)) => (((((divide_divide @ A) @ A2) @ B2) = (one_one @ A)) = (A2 = B2))))))))). % 6.41/2.68 thf(fact_851_one__plus__numeral__commute, axiom, (! [A:$tType]: (((numeral @ A) => (! [X3:num]: (((((plus_plus @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ X3)) = (((plus_plus @ A) @ ((numeral_numeral @ A) @ X3)) @ (one_one @ A))))))))). % 6.41/2.68 thf(fact_852_one__le__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat]: (((((ord_less_eq @ A) @ (one_one @ A)) @ A2) => (((ord_less_eq @ A) @ (one_one @ A)) @ (((power_power @ A) @ A2) @ N2))))))))). % 6.41/2.68 thf(fact_853_power__0, axiom, (! [A:$tType]: (((power @ A) => (! [A2:A]: (((((power_power @ A) @ A2) @ (zero_zero @ nat)) = (one_one @ A)))))))). % 6.41/2.68 thf(fact_854_div__eq__dividend__iff, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((((divide_divide @ nat) @ M) @ N2) = M) = (N2 = (one_one @ nat))))))). % 6.41/2.68 thf(fact_855_div__less__dividend, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (one_one @ nat)) @ N2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((ord_less @ nat) @ (((divide_divide @ nat) @ M) @ N2)) @ M)))))). % 6.41/2.68 thf(fact_856_One__nat__def, axiom, ((one_one @ nat) = (suc @ (zero_zero @ nat)))). % 6.41/2.68 thf(fact_857_Suc__eq__plus1, axiom, (suc = (^ [N:nat]: ((((plus_plus @ nat) @ N) @ (one_one @ nat)))))). % 6.41/2.68 thf(fact_858_plus__1__eq__Suc, axiom, (((plus_plus @ nat) @ (one_one @ nat)) = suc)). % 6.41/2.68 thf(fact_859_Suc__eq__plus1__left, axiom, (suc = ((plus_plus @ nat) @ (one_one @ nat)))). % 6.41/2.68 thf(fact_860_VEBT__internal_Ovalid_H_Osimps_I1_J, axiom, (! [Uu:$o,Uv:$o,D2:nat]: ((((vEBT_VEBT_valid @ ((vEBT_Leaf @ Uu) @ Uv)) @ D2) = (D2 = (one_one @ nat)))))). % 6.41/2.68 thf(fact_861_zero__less__two, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (one_one @ A)) @ (one_one @ A))))))). % 6.41/2.68 thf(fact_862_power__le__one, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ A2) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N2)) @ (one_one @ A)))))))))). % 6.41/2.68 thf(fact_863_less__divide__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ A2) @ B2)) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ B2) @ A2)))))))))). % 6.41/2.68 thf(fact_864_divide__less__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ A2)) @ (one_one @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ B2) @ A2)) | (((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ A2) @ B2)) | (A2 = (zero_zero @ A))))))))))). % 6.41/2.68 thf(fact_865_less__half__sum, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ A2) @ (((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (((plus_plus @ A) @ (one_one @ A)) @ (one_one @ A))))))))))). % 6.41/2.68 thf(fact_866_gt__half__sum, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ (((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (((plus_plus @ A) @ (one_one @ A)) @ (one_one @ A)))) @ B2)))))))). % 6.41/2.68 thf(fact_867_power__0__left, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N2:nat]: ((((N2 = (zero_zero @ nat)) => ((((power_power @ A) @ (zero_zero @ A)) @ N2) = (one_one @ A))) & ((N2 != (zero_zero @ nat)) => ((((power_power @ A) @ (zero_zero @ A)) @ N2) = (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_868_power__gt1, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A2) => (((ord_less @ A) @ (one_one @ A)) @ (((power_power @ A) @ A2) @ (suc @ N2)))))))))). % 6.41/2.68 thf(fact_869_power__less__imp__less__exp, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,M:nat,N2:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A2) => ((((ord_less @ A) @ (((power_power @ A) @ A2) @ M)) @ (((power_power @ A) @ A2) @ N2)) => (((ord_less @ nat) @ M) @ N2))))))))). % 6.41/2.68 thf(fact_870_power__strict__increasing, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N2:nat,N7:nat,A2:A]: (((((ord_less @ nat) @ N2) @ N7) => ((((ord_less @ A) @ (one_one @ A)) @ A2) => (((ord_less @ A) @ (((power_power @ A) @ A2) @ N2)) @ (((power_power @ A) @ A2) @ N7)))))))))). % 6.41/2.68 thf(fact_871_power__increasing, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N2:nat,N7:nat,A2:A]: (((((ord_less_eq @ nat) @ N2) @ N7) => ((((ord_less_eq @ A) @ (one_one @ A)) @ A2) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N2)) @ (((power_power @ A) @ A2) @ N7)))))))))). % 6.41/2.68 thf(fact_872_nat__induct__non__zero, axiom, (! [N2:nat,P:(nat > $o)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((P @ (one_one @ nat)) => ((! [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) => ((P @ N3) => (P @ (suc @ N3)))))) => (P @ N2))))))). % 6.41/2.68 thf(fact_873_vebt__member_Osimps_I1_J, axiom, (! [A2:$o,B2:$o,X3:nat]: ((((vEBT_vebt_member @ ((vEBT_Leaf @ A2) @ B2)) @ X3) = (((X3 = (zero_zero @ nat)) => A2) & ((X3 != (zero_zero @ nat)) => (((X3 = (one_one @ nat)) => B2) & (X3 = (one_one @ nat))))))))). % 6.41/2.68 thf(fact_874_VEBT__internal_Onaive__member_Osimps_I1_J, axiom, (! [A2:$o,B2:$o,X3:nat]: ((((vEBT_V5719532721284313246member @ ((vEBT_Leaf @ A2) @ B2)) @ X3) = (((X3 = (zero_zero @ nat)) => A2) & ((X3 != (zero_zero @ nat)) => (((X3 = (one_one @ nat)) => B2) & (X3 = (one_one @ nat))))))))). % 6.41/2.68 thf(fact_875_le__divide__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ A2) @ B2)) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B2) @ A2)))))))))). % 6.41/2.68 thf(fact_876_divide__le__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ A2)) @ (one_one @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ B2) @ A2)) | (((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ A2) @ B2)) | (A2 = (zero_zero @ A))))))))))). % 6.41/2.68 thf(fact_877_power__Suc__le__self, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ A2) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ (suc @ N2))) @ A2))))))))). % 6.41/2.68 thf(fact_878_power__Suc__less__one, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ A2) @ (one_one @ A)) => (((ord_less @ A) @ (((power_power @ A) @ A2) @ (suc @ N2))) @ (one_one @ A)))))))))). % 6.41/2.68 thf(fact_879_power__strict__decreasing, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N2:nat,N7:nat,A2:A]: (((((ord_less @ nat) @ N2) @ N7) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ A2) @ (one_one @ A)) => (((ord_less @ A) @ (((power_power @ A) @ A2) @ N7)) @ (((power_power @ A) @ A2) @ N2))))))))))). % 6.41/2.68 thf(fact_880_power__decreasing, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N2:nat,N7:nat,A2:A]: (((((ord_less_eq @ nat) @ N2) @ N7) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ A2) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N7)) @ (((power_power @ A) @ A2) @ N2))))))))))). % 6.41/2.68 thf(fact_881_power__le__imp__le__exp, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,M:nat,N2:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A2) => ((((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ M)) @ (((power_power @ A) @ A2) @ N2)) => (((ord_less_eq @ nat) @ M) @ N2))))))))). % 6.41/2.68 thf(fact_882_self__le__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat]: (((((ord_less_eq @ A) @ (one_one @ A)) @ A2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((ord_less_eq @ A) @ A2) @ (((power_power @ A) @ A2) @ N2)))))))))). % 6.41/2.68 thf(fact_883_one__less__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((ord_less @ A) @ (one_one @ A)) @ (((power_power @ A) @ A2) @ N2)))))))))). % 6.41/2.68 thf(fact_884_nat__1__add__1, axiom, ((((plus_plus @ nat) @ (one_one @ nat)) @ (one_one @ nat)) = ((numeral_numeral @ nat) @ (bit0 @ one2)))). % 6.41/2.68 thf(fact_885_vebt__maxt_Osimps_I1_J, axiom, (! [B2:$o,A2:$o]: (((B2 => ((vEBT_vebt_maxt @ ((vEBT_Leaf @ A2) @ B2)) = ((some @ nat) @ (one_one @ nat)))) & ((~ (B2)) => ((A2 => ((vEBT_vebt_maxt @ ((vEBT_Leaf @ A2) @ B2)) = ((some @ nat) @ (zero_zero @ nat)))) & ((~ (A2)) => ((vEBT_vebt_maxt @ ((vEBT_Leaf @ A2) @ B2)) = (none @ nat))))))))). % 6.41/2.68 thf(fact_886_vebt__mint_Osimps_I1_J, axiom, (! [A2:$o,B2:$o]: (((A2 => ((vEBT_vebt_mint @ ((vEBT_Leaf @ A2) @ B2)) = ((some @ nat) @ (zero_zero @ nat)))) & ((~ (A2)) => ((B2 => ((vEBT_vebt_mint @ ((vEBT_Leaf @ A2) @ B2)) = ((some @ nat) @ (one_one @ nat)))) & ((~ (B2)) => ((vEBT_vebt_mint @ ((vEBT_Leaf @ A2) @ B2)) = (none @ nat))))))))). % 6.41/2.68 thf(fact_887_div__le__mono, axiom, (! [M:nat,N2:nat,K:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => (((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ K)) @ (((divide_divide @ nat) @ N2) @ K)))))). % 6.41/2.68 thf(fact_888_div__le__dividend, axiom, (! [M:nat,N2:nat]: ((((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ N2)) @ M)))). % 6.41/2.68 thf(fact_889_ex__power__ivl2, axiom, (! [B2:nat,K:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ B2) => ((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K) => (? [N3:nat]: (((((ord_less @ nat) @ (((power_power @ nat) @ B2) @ N3)) @ K) & (((ord_less_eq @ nat) @ K) @ (((power_power @ nat) @ B2) @ (((plus_plus @ nat) @ N3) @ (one_one @ nat)))))))))))). % 6.41/2.68 thf(fact_890_ex__power__ivl1, axiom, (! [B2:nat,K:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ B2) => ((((ord_less_eq @ nat) @ (one_one @ nat)) @ K) => (? [N3:nat]: (((((ord_less_eq @ nat) @ (((power_power @ nat) @ B2) @ N3)) @ K) & (((ord_less @ nat) @ K) @ (((power_power @ nat) @ B2) @ (((plus_plus @ nat) @ N3) @ (one_one @ nat)))))))))))). % 6.41/2.68 thf(fact_891_Euclidean__Division_Odiv__eq__0__iff, axiom, (! [M:nat,N2:nat]: ((((((divide_divide @ nat) @ M) @ N2) = (zero_zero @ nat)) = ((((ord_less @ nat) @ M) @ N2) | (N2 = (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_892_Suc__div__le__mono, axiom, (! [M:nat,N2:nat]: ((((ord_less_eq @ nat) @ (((divide_divide @ nat) @ M) @ N2)) @ (((divide_divide @ nat) @ (suc @ M)) @ N2))))). % 6.41/2.68 thf(fact_893_vebt__maxt_Oelims, axiom, (! [X3:vEBT_VEBT,Y:(option @ nat)]: ((((vEBT_vebt_maxt @ X3) = Y) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => (~ (((B5 => (Y = ((some @ nat) @ (one_one @ nat)))) & ((~ (B5)) => ((A5 => (Y = ((some @ nat) @ (zero_zero @ nat)))) & ((~ (A5)) => (Y = (none @ nat))))))))))) => (((? [Uu2:nat,Uv2:(list @ vEBT_VEBT),Uw2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu2) @ Uv2) @ Uw2)))) => (Y != (none @ nat))) => (~ ((! [Mi2:nat,Ma2:nat]: (((? [Ux2:nat,Uy2:(list @ vEBT_VEBT),Uz2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ Ux2) @ Uy2) @ Uz2)))) => (Y != ((some @ nat) @ Ma2))))))))))))). % 6.41/2.68 thf(fact_894_vebt__mint_Oelims, axiom, (! [X3:vEBT_VEBT,Y:(option @ nat)]: ((((vEBT_vebt_mint @ X3) = Y) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => (~ (((A5 => (Y = ((some @ nat) @ (zero_zero @ nat)))) & ((~ (A5)) => ((B5 => (Y = ((some @ nat) @ (one_one @ nat)))) & ((~ (B5)) => (Y = (none @ nat))))))))))) => (((? [Uu2:nat,Uv2:(list @ vEBT_VEBT),Uw2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu2) @ Uv2) @ Uw2)))) => (Y != (none @ nat))) => (~ ((! [Mi2:nat]: (((? [Ma2:nat,Ux2:nat,Uy2:(list @ vEBT_VEBT),Uz2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ Ux2) @ Uy2) @ Uz2)))) => (Y != ((some @ nat) @ Mi2))))))))))))). % 6.41/2.68 thf(fact_895_div__greater__zero__iff, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((divide_divide @ nat) @ M) @ N2)) = ((((ord_less_eq @ nat) @ N2) @ M) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N2)))))). % 6.41/2.68 thf(fact_896_div__le__mono2, axiom, (! [M:nat,N2:nat,K:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less_eq @ nat) @ M) @ N2) => (((ord_less_eq @ nat) @ (((divide_divide @ nat) @ K) @ N2)) @ (((divide_divide @ nat) @ K) @ M))))))). % 6.41/2.68 thf(fact_897_bits__1__div__2, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => ((((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A)))))). % 6.41/2.68 thf(fact_898_nat__induct2, axiom, (! [P:(nat > $o),N2:nat]: (((P @ (zero_zero @ nat)) => ((P @ (one_one @ nat)) => ((! [N3:nat]: (((P @ N3) => (P @ (((plus_plus @ nat) @ N3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) => (P @ N2))))))). % 6.41/2.68 thf(fact_899_div__exp__eq, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A,M:nat,N2:nat]: (((((divide_divide @ A) @ (((divide_divide @ A) @ A2) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) = (((divide_divide @ A) @ A2) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ M) @ N2)))))))))). % 6.41/2.68 thf(fact_900_exp__add__not__zero__imp__right, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M:nat,N2:nat]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ M) @ N2)) != (zero_zero @ A)) => ((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2) != (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_901_exp__add__not__zero__imp__left, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M:nat,N2:nat]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ M) @ N2)) != (zero_zero @ A)) => ((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M) != (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_902_bit__concat__def, axiom, (vEBT_VEBT_bit_concat = (^ [H:nat,L2:nat,D5:nat]: ((((plus_plus @ nat) @ (((times_times @ nat) @ H) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ D5))) @ L2))))). % 6.41/2.68 thf(fact_903_low__inv, axiom, (! [X3:nat,N2:nat,Y:nat]: (((((ord_less @ nat) @ X3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) => (((vEBT_VEBT_low @ (((plus_plus @ nat) @ (((times_times @ nat) @ Y) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) @ X3)) @ N2) = X3))))). % 6.41/2.68 thf(fact_904_high__inv, axiom, (! [X3:nat,N2:nat,Y:nat]: (((((ord_less @ nat) @ X3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) => (((vEBT_VEBT_high @ (((plus_plus @ nat) @ (((times_times @ nat) @ Y) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) @ X3)) @ N2) = Y))))). % 6.41/2.68 thf(fact_905_enat__ord__number_I1_J, axiom, (! [M:num,N2:num]: (((((ord_less_eq @ extended_enat) @ ((numeral_numeral @ extended_enat) @ M)) @ ((numeral_numeral @ extended_enat) @ N2)) = (((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ M)) @ ((numeral_numeral @ nat) @ N2)))))). % 6.41/2.68 thf(fact_906_enat__ord__number_I2_J, axiom, (! [M:num,N2:num]: (((((ord_less @ extended_enat) @ ((numeral_numeral @ extended_enat) @ M)) @ ((numeral_numeral @ extended_enat) @ N2)) = (((ord_less @ nat) @ ((numeral_numeral @ nat) @ M)) @ ((numeral_numeral @ nat) @ N2)))))). % 6.41/2.68 thf(fact_907_mul__shift, axiom, (! [X3:nat,Y:nat,Z2:nat]: ((((((times_times @ nat) @ X3) @ Y) = Z2) = (((vEBT_VEBT_mul @ ((some @ nat) @ X3)) @ ((some @ nat) @ Y)) = ((some @ nat) @ Z2)))))). % 6.41/2.68 thf(fact_908_mul__def, axiom, (vEBT_VEBT_mul = ((vEBT_V2048590022279873568_shift @ nat) @ (times_times @ nat)))). % 6.41/2.68 thf(fact_909_i0__less, axiom, (! [N2:extended_enat]: (((((ord_less @ extended_enat) @ (zero_zero @ extended_enat)) @ N2) = (N2 != (zero_zero @ extended_enat)))))). % 6.41/2.68 thf(fact_910_mult__cancel__right, axiom, (! [A:$tType]: (((semiri6575147826004484403cancel @ A) => (! [A2:A,C2:A,B2:A]: ((((((times_times @ A) @ A2) @ C2) = (((times_times @ A) @ B2) @ C2)) = ((C2 = (zero_zero @ A)) | (A2 = B2))))))))). % 6.41/2.68 thf(fact_911_mult__cancel__left, axiom, (! [A:$tType]: (((semiri6575147826004484403cancel @ A) => (! [C2:A,A2:A,B2:A]: ((((((times_times @ A) @ C2) @ A2) = (((times_times @ A) @ C2) @ B2)) = ((C2 = (zero_zero @ A)) | (A2 = B2))))))))). % 6.41/2.68 thf(fact_912_mult__eq__0__iff, axiom, (! [A:$tType]: (((semiri3467727345109120633visors @ A) => (! [A2:A,B2:A]: ((((((times_times @ A) @ A2) @ B2) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_913_mult__zero__right, axiom, (! [A:$tType]: (((mult_zero @ A) => (! [A2:A]: (((((times_times @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))). % 6.41/2.68 thf(fact_914_mult__zero__left, axiom, (! [A:$tType]: (((mult_zero @ A) => (! [A2:A]: (((((times_times @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))). % 6.41/2.68 thf(fact_915_bits__div__by__0, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))). % 6.41/2.68 thf(fact_916_bits__div__0, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A]: (((((divide_divide @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))). % 6.41/2.68 thf(fact_917_mult_Oright__neutral, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A2:A]: (((((times_times @ A) @ A2) @ (one_one @ A)) = A2))))))). % 6.41/2.68 thf(fact_918_mult__1, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A2:A]: (((((times_times @ A) @ (one_one @ A)) @ A2) = A2))))))). % 6.41/2.68 thf(fact_919_half__negative__int__iff, axiom, (! [K:int]: (((((ord_less @ int) @ (((divide_divide @ int) @ K) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (zero_zero @ int)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))). % 6.41/2.68 thf(fact_920_times__divide__eq__left, axiom, (! [A:$tType]: (((field @ A) => (! [B2:A,C2:A,A2:A]: (((((times_times @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A2) = (((divide_divide @ A) @ (((times_times @ A) @ B2) @ A2)) @ C2)))))))). % 6.41/2.68 thf(fact_921_divide__divide__eq__left, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ C2) = (((divide_divide @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))). % 6.41/2.68 thf(fact_922_divide__divide__eq__right, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A,C2:A]: (((((divide_divide @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((times_times @ A) @ A2) @ C2)) @ B2)))))))). % 6.41/2.68 thf(fact_923_times__divide__eq__right, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.68 thf(fact_924_mult__cancel2, axiom, (! [M:nat,K:nat,N2:nat]: ((((((times_times @ nat) @ M) @ K) = (((times_times @ nat) @ N2) @ K)) = ((M = N2) | (K = (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_925_mult__cancel1, axiom, (! [K:nat,M:nat,N2:nat]: ((((((times_times @ nat) @ K) @ M) = (((times_times @ nat) @ K) @ N2)) = ((M = N2) | (K = (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_926_mult__0__right, axiom, (! [M:nat]: (((((times_times @ nat) @ M) @ (zero_zero @ nat)) = (zero_zero @ nat))))). % 6.41/2.68 thf(fact_927_mult__is__0, axiom, (! [M:nat,N2:nat]: ((((((times_times @ nat) @ M) @ N2) = (zero_zero @ nat)) = ((M = (zero_zero @ nat)) | (N2 = (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_928_nat__1__eq__mult__iff, axiom, (! [M:nat,N2:nat]: ((((one_one @ nat) = (((times_times @ nat) @ M) @ N2)) = ((M = (one_one @ nat)) & (N2 = (one_one @ nat))))))). % 6.41/2.68 thf(fact_929_nat__mult__eq__1__iff, axiom, (! [M:nat,N2:nat]: ((((((times_times @ nat) @ M) @ N2) = (one_one @ nat)) = ((M = (one_one @ nat)) & (N2 = (one_one @ nat))))))). % 6.41/2.68 thf(fact_930_mult__cancel__right2, axiom, (! [A:$tType]: (((ring_15535105094025558882visors @ A) => (! [A2:A,C2:A]: ((((((times_times @ A) @ A2) @ C2) = C2) = ((C2 = (zero_zero @ A)) | (A2 = (one_one @ A)))))))))). % 6.41/2.68 thf(fact_931_mult__cancel__right1, axiom, (! [A:$tType]: (((ring_15535105094025558882visors @ A) => (! [C2:A,B2:A]: (((C2 = (((times_times @ A) @ B2) @ C2)) = ((C2 = (zero_zero @ A)) | (B2 = (one_one @ A)))))))))). % 6.41/2.68 thf(fact_932_mult__cancel__left2, axiom, (! [A:$tType]: (((ring_15535105094025558882visors @ A) => (! [C2:A,A2:A]: ((((((times_times @ A) @ C2) @ A2) = C2) = ((C2 = (zero_zero @ A)) | (A2 = (one_one @ A)))))))))). % 6.41/2.68 thf(fact_933_mult__cancel__left1, axiom, (! [A:$tType]: (((ring_15535105094025558882visors @ A) => (! [C2:A,B2:A]: (((C2 = (((times_times @ A) @ C2) @ B2)) = ((C2 = (zero_zero @ A)) | (B2 = (one_one @ A)))))))))). % 6.41/2.68 thf(fact_934_sum__squares__eq__zero__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [X3:A,Y:A]: ((((((plus_plus @ A) @ (((times_times @ A) @ X3) @ X3)) @ (((times_times @ A) @ Y) @ Y)) = (zero_zero @ A)) = ((X3 = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_935_nonzero__mult__divide__mult__cancel__right2, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A2:A,B2:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ C2) @ B2)) = (((divide_divide @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_936_nonzero__mult__div__cancel__right, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [B2:A,A2:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ A2) @ B2)) @ B2) = A2)))))))). % 6.41/2.68 thf(fact_937_nonzero__mult__divide__mult__cancel__right, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A2:A,B2:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = (((divide_divide @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_938_nonzero__mult__divide__mult__cancel__left2, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A2:A,B2:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ B2) @ C2)) = (((divide_divide @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_939_nonzero__mult__div__cancel__left, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A2:A,B2:A]: (((A2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ A2) @ B2)) @ A2) = B2)))))))). % 6.41/2.68 thf(fact_940_nonzero__mult__divide__mult__cancel__left, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A2:A,B2:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (((divide_divide @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_941_mult__divide__mult__cancel__left__if, axiom, (! [A:$tType]: (((field @ A) => (! [C2:A,A2:A,B2:A]: ((((C2 = (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (zero_zero @ A))) & ((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (((divide_divide @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_942_div__mult__mult1__if, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,A2:A,B2:A]: ((((C2 = (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (zero_zero @ A))) & ((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (((divide_divide @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_943_div__mult__mult2, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,A2:A,B2:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = (((divide_divide @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_944_div__mult__mult1, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,A2:A,B2:A]: (((C2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (((divide_divide @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_945_distrib__right__numeral, axiom, (! [A:$tType]: ((((numeral @ A) & (semiring @ A)) => (! [A2:A,B2:A,V2:num]: (((((times_times @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ ((numeral_numeral @ A) @ V2)) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ ((numeral_numeral @ A) @ V2))) @ (((times_times @ A) @ B2) @ ((numeral_numeral @ A) @ V2)))))))))). % 6.41/2.68 thf(fact_946_distrib__left__numeral, axiom, (! [A:$tType]: ((((numeral @ A) & (semiring @ A)) => (! [V2:num,B2:A,C2:A]: (((((times_times @ A) @ ((numeral_numeral @ A) @ V2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ V2)) @ B2)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ V2)) @ C2))))))))). % 6.41/2.68 thf(fact_947_mult__eq__1__iff, axiom, (! [M:nat,N2:nat]: ((((((times_times @ nat) @ M) @ N2) = (suc @ (zero_zero @ nat))) = ((M = (suc @ (zero_zero @ nat))) & (N2 = (suc @ (zero_zero @ nat)))))))). % 6.41/2.68 thf(fact_948_one__eq__mult__iff, axiom, (! [M:nat,N2:nat]: ((((suc @ (zero_zero @ nat)) = (((times_times @ nat) @ M) @ N2)) = ((M = (suc @ (zero_zero @ nat))) & (N2 = (suc @ (zero_zero @ nat)))))))). % 6.41/2.68 thf(fact_949_mult__less__cancel2, axiom, (! [M:nat,K:nat,N2:nat]: (((((ord_less @ nat) @ (((times_times @ nat) @ M) @ K)) @ (((times_times @ nat) @ N2) @ K)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) & (((ord_less @ nat) @ M) @ N2)))))). % 6.41/2.68 thf(fact_950_nat__0__less__mult__iff, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((times_times @ nat) @ M) @ N2)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N2)))))). % 6.41/2.68 thf(fact_951_nat__mult__less__cancel__disj, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) & (((ord_less @ nat) @ M) @ N2)))))). % 6.41/2.68 thf(fact_952_mult__Suc__right, axiom, (! [M:nat,N2:nat]: (((((times_times @ nat) @ M) @ (suc @ N2)) = (((plus_plus @ nat) @ M) @ (((times_times @ nat) @ M) @ N2)))))). % 6.41/2.68 thf(fact_953_nat__mult__div__cancel__disj, axiom, (! [K:nat,M:nat,N2:nat]: ((((K = (zero_zero @ nat)) => ((((divide_divide @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)) = (zero_zero @ nat))) & ((K != (zero_zero @ nat)) => ((((divide_divide @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)) = (((divide_divide @ nat) @ M) @ N2))))))). % 6.41/2.68 thf(fact_954_divide__le__eq__numeral1_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,W2:num,A2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ ((numeral_numeral @ A) @ W2))) @ A2) = (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ A2) @ ((numeral_numeral @ A) @ W2)))))))))). % 6.41/2.68 thf(fact_955_le__divide__eq__numeral1_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,W2:num]: (((((ord_less_eq @ A) @ A2) @ (((divide_divide @ A) @ B2) @ ((numeral_numeral @ A) @ W2))) = (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ ((numeral_numeral @ A) @ W2))) @ B2)))))))). % 6.41/2.68 thf(fact_956_divide__eq__eq__numeral1_I1_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,W2:num,A2:A]: ((((((divide_divide @ A) @ B2) @ ((numeral_numeral @ A) @ W2)) = A2) = (((((numeral_numeral @ A) @ W2) != (zero_zero @ A)) => (B2 = (((times_times @ A) @ A2) @ ((numeral_numeral @ A) @ W2)))) & ((((numeral_numeral @ A) @ W2) = (zero_zero @ A)) => (A2 = (zero_zero @ A))))))))))). % 6.41/2.68 thf(fact_957_eq__divide__eq__numeral1_I1_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A,W2:num]: (((A2 = (((divide_divide @ A) @ B2) @ ((numeral_numeral @ A) @ W2))) = (((((numeral_numeral @ A) @ W2) != (zero_zero @ A)) => ((((times_times @ A) @ A2) @ ((numeral_numeral @ A) @ W2)) = B2)) & ((((numeral_numeral @ A) @ W2) = (zero_zero @ A)) => (A2 = (zero_zero @ A))))))))))). % 6.41/2.68 thf(fact_958_less__divide__eq__numeral1_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,W2:num]: (((((ord_less @ A) @ A2) @ (((divide_divide @ A) @ B2) @ ((numeral_numeral @ A) @ W2))) = (((ord_less @ A) @ (((times_times @ A) @ A2) @ ((numeral_numeral @ A) @ W2))) @ B2)))))))). % 6.41/2.68 thf(fact_959_divide__less__eq__numeral1_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,W2:num,A2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ ((numeral_numeral @ A) @ W2))) @ A2) = (((ord_less @ A) @ B2) @ (((times_times @ A) @ A2) @ ((numeral_numeral @ A) @ W2)))))))))). % 6.41/2.68 thf(fact_960_nonzero__divide__mult__cancel__left, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A]: (((A2 != (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ (((times_times @ A) @ A2) @ B2)) = (((divide_divide @ A) @ (one_one @ A)) @ B2))))))))). % 6.41/2.68 thf(fact_961_nonzero__divide__mult__cancel__right, axiom, (! [A:$tType]: (((field @ A) => (! [B2:A,A2:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ B2) @ (((times_times @ A) @ A2) @ B2)) = (((divide_divide @ A) @ (one_one @ A)) @ A2))))))))). % 6.41/2.68 thf(fact_962_div__mult__self1, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,A2:A,C2:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ (((times_times @ A) @ C2) @ B2))) @ B2) = (((plus_plus @ A) @ C2) @ (((divide_divide @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_963_div__mult__self2, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,A2:A,C2:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))) @ B2) = (((plus_plus @ A) @ C2) @ (((divide_divide @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_964_div__mult__self3, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,C2:A,A2:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ C2) @ B2)) @ A2)) @ B2) = (((plus_plus @ A) @ C2) @ (((divide_divide @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_965_div__mult__self4, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,C2:A,A2:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ C2)) @ A2)) @ B2) = (((plus_plus @ A) @ C2) @ (((divide_divide @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_966_one__le__mult__iff, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ (((times_times @ nat) @ M) @ N2)) = ((((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ M) & (((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ N2)))))). % 6.41/2.68 thf(fact_967_mult__le__cancel2, axiom, (! [M:nat,K:nat,N2:nat]: (((((ord_less_eq @ nat) @ (((times_times @ nat) @ M) @ K)) @ (((times_times @ nat) @ N2) @ K)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less_eq @ nat) @ M) @ N2)))))). % 6.41/2.68 thf(fact_968_nat__mult__le__cancel__disj, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less_eq @ nat) @ M) @ N2)))))). % 6.41/2.68 thf(fact_969_div__mult__self__is__m, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((divide_divide @ nat) @ (((times_times @ nat) @ M) @ N2)) @ N2) = M))))). % 6.41/2.68 thf(fact_970_div__mult__self1__is__m, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((divide_divide @ nat) @ (((times_times @ nat) @ N2) @ M)) @ N2) = M))))). % 6.41/2.68 thf(fact_971_ab__semigroup__mult__class_Omult__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_mult @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))). % 6.41/2.68 thf(fact_972_mult_Oassoc, axiom, (! [A:$tType]: (((semigroup_mult @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))). % 6.41/2.68 thf(fact_973_mult_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_mult @ A) => ((times_times @ A) = (^ [A4:A,B4:A]: ((((times_times @ A) @ B4) @ A4)))))))). % 6.41/2.68 thf(fact_974_mult_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_mult @ A) => (! [B2:A,A2:A,C2:A]: (((((times_times @ A) @ B2) @ (((times_times @ A) @ A2) @ C2)) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))). % 6.41/2.68 thf(fact_975_not__iless0, axiom, (! [N2:extended_enat]: ((~ ((((ord_less @ extended_enat) @ N2) @ (zero_zero @ extended_enat))))))). % 6.41/2.68 thf(fact_976_enat__less__induct, axiom, (! [P:(extended_enat > $o),N2:extended_enat]: (((! [N3:extended_enat]: (((! [M6:extended_enat]: (((((ord_less @ extended_enat) @ M6) @ N3) => (P @ M6)))) => (P @ N3)))) => (P @ N2))))). % 6.41/2.68 thf(fact_977_mult__right__cancel, axiom, (! [A:$tType]: (((semiri6575147826004484403cancel @ A) => (! [C2:A,A2:A,B2:A]: (((C2 != (zero_zero @ A)) => (((((times_times @ A) @ A2) @ C2) = (((times_times @ A) @ B2) @ C2)) = (A2 = B2))))))))). % 6.41/2.68 thf(fact_978_mult__left__cancel, axiom, (! [A:$tType]: (((semiri6575147826004484403cancel @ A) => (! [C2:A,A2:A,B2:A]: (((C2 != (zero_zero @ A)) => (((((times_times @ A) @ C2) @ A2) = (((times_times @ A) @ C2) @ B2)) = (A2 = B2))))))))). % 6.41/2.68 thf(fact_979_no__zero__divisors, axiom, (! [A:$tType]: (((semiri3467727345109120633visors @ A) => (! [A2:A,B2:A]: (((A2 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => ((((times_times @ A) @ A2) @ B2) != (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_980_divisors__zero, axiom, (! [A:$tType]: (((semiri3467727345109120633visors @ A) => (! [A2:A,B2:A]: ((((((times_times @ A) @ A2) @ B2) = (zero_zero @ A)) => ((A2 = (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_981_mult__not__zero, axiom, (! [A:$tType]: (((mult_zero @ A) => (! [A2:A,B2:A]: ((((((times_times @ A) @ A2) @ B2) != (zero_zero @ A)) => ((A2 != (zero_zero @ A)) & (B2 != (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_982_comm__monoid__mult__class_Omult__1, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A]: (((((times_times @ A) @ (one_one @ A)) @ A2) = A2))))))). % 6.41/2.68 thf(fact_983_mult_Ocomm__neutral, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A]: (((((times_times @ A) @ A2) @ (one_one @ A)) = A2))))))). % 6.41/2.68 thf(fact_984_combine__common__factor, axiom, (! [A:$tType]: (((semiring @ A) => (! [A2:A,E3:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((times_times @ A) @ A2) @ E3)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E3)) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ E3)) @ C2)))))))). % 6.41/2.68 thf(fact_985_distrib__right, axiom, (! [A:$tType]: (((semiring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2))))))))). % 6.41/2.68 thf(fact_986_distrib__left, axiom, (! [A:$tType]: (((semiring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ B2)) @ (((times_times @ A) @ A2) @ C2))))))))). % 6.41/2.68 thf(fact_987_comm__semiring__class_Odistrib, axiom, (! [A:$tType]: (((comm_semiring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2))))))))). % 6.41/2.68 thf(fact_988_ring__class_Oring__distribs_I1_J, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ B2)) @ (((times_times @ A) @ A2) @ C2))))))))). % 6.41/2.68 thf(fact_989_ring__class_Oring__distribs_I2_J, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2))))))))). % 6.41/2.68 thf(fact_990_crossproduct__noteq, axiom, (! [A:$tType]: (((semiri1453513574482234551roduct @ A) => (! [A2:A,B2:A,C2:A,D2:A]: ((((A2 != B2) & (C2 != D2)) = ((((plus_plus @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ D2)) != (((plus_plus @ A) @ (((times_times @ A) @ A2) @ D2)) @ (((times_times @ A) @ B2) @ C2)))))))))). % 6.41/2.68 thf(fact_991_crossproduct__eq, axiom, (! [A:$tType]: (((semiri1453513574482234551roduct @ A) => (! [W2:A,Y:A,X3:A,Z2:A]: ((((((plus_plus @ A) @ (((times_times @ A) @ W2) @ Y)) @ (((times_times @ A) @ X3) @ Z2)) = (((plus_plus @ A) @ (((times_times @ A) @ W2) @ Z2)) @ (((times_times @ A) @ X3) @ Y))) = ((W2 = X3) | (Y = Z2))))))))). % 6.41/2.68 thf(fact_992_times__divide__times__eq, axiom, (! [A:$tType]: (((field @ A) => (! [X3:A,Y:A,Z2:A,W2:A]: (((((times_times @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ (((divide_divide @ A) @ Z2) @ W2)) = (((divide_divide @ A) @ (((times_times @ A) @ X3) @ Z2)) @ (((times_times @ A) @ Y) @ W2))))))))). % 6.41/2.68 thf(fact_993_divide__divide__times__eq, axiom, (! [A:$tType]: (((field @ A) => (! [X3:A,Y:A,Z2:A,W2:A]: (((((divide_divide @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ (((divide_divide @ A) @ Z2) @ W2)) = (((divide_divide @ A) @ (((times_times @ A) @ X3) @ W2)) @ (((times_times @ A) @ Y) @ Z2))))))))). % 6.41/2.68 thf(fact_994_divide__divide__eq__left_H, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ C2) = (((divide_divide @ A) @ A2) @ (((times_times @ A) @ C2) @ B2))))))))). % 6.41/2.68 thf(fact_995_Suc__mult__cancel1, axiom, (! [K:nat,M:nat,N2:nat]: ((((((times_times @ nat) @ (suc @ K)) @ M) = (((times_times @ nat) @ (suc @ K)) @ N2)) = (M = N2))))). % 6.41/2.68 thf(fact_996_mult__0, axiom, (! [N2:nat]: (((((times_times @ nat) @ (zero_zero @ nat)) @ N2) = (zero_zero @ nat))))). % 6.41/2.68 thf(fact_997_nat__mult__eq__cancel__disj, axiom, (! [K:nat,M:nat,N2:nat]: ((((((times_times @ nat) @ K) @ M) = (((times_times @ nat) @ K) @ N2)) = ((K = (zero_zero @ nat)) | (M = N2)))))). % 6.41/2.68 thf(fact_998_mult__le__mono2, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (((times_times @ nat) @ K) @ I)) @ (((times_times @ nat) @ K) @ J)))))). % 6.41/2.68 thf(fact_999_mult__le__mono1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (((times_times @ nat) @ I) @ K)) @ (((times_times @ nat) @ J) @ K)))))). % 6.41/2.68 thf(fact_1000_mult__le__mono, axiom, (! [I:nat,J:nat,K:nat,L:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ K) @ L) => (((ord_less_eq @ nat) @ (((times_times @ nat) @ I) @ K)) @ (((times_times @ nat) @ J) @ L))))))). % 6.41/2.68 thf(fact_1001_le__square, axiom, (! [M:nat]: ((((ord_less_eq @ nat) @ M) @ (((times_times @ nat) @ M) @ M))))). % 6.41/2.68 thf(fact_1002_le__cube, axiom, (! [M:nat]: ((((ord_less_eq @ nat) @ M) @ (((times_times @ nat) @ M) @ (((times_times @ nat) @ M) @ M)))))). % 6.41/2.68 thf(fact_1003_add__mult__distrib2, axiom, (! [K:nat,M:nat,N2:nat]: (((((times_times @ nat) @ K) @ (((plus_plus @ nat) @ M) @ N2)) = (((plus_plus @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)))))). % 6.41/2.68 thf(fact_1004_add__mult__distrib, axiom, (! [M:nat,N2:nat,K:nat]: (((((times_times @ nat) @ (((plus_plus @ nat) @ M) @ N2)) @ K) = (((plus_plus @ nat) @ (((times_times @ nat) @ M) @ K)) @ (((times_times @ nat) @ N2) @ K)))))). % 6.41/2.68 thf(fact_1005_left__add__mult__distrib, axiom, (! [I:nat,U:nat,J:nat,K:nat]: (((((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ K)) = (((plus_plus @ nat) @ (((times_times @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ U)) @ K))))). % 6.41/2.68 thf(fact_1006_nat__mult__1, axiom, (! [N2:nat]: (((((times_times @ nat) @ (one_one @ nat)) @ N2) = N2)))). % 6.41/2.68 thf(fact_1007_nat__mult__1__right, axiom, (! [N2:nat]: (((((times_times @ nat) @ N2) @ (one_one @ nat)) = N2)))). % 6.41/2.68 thf(fact_1008_ordered__comm__semiring__class_Ocomm__mult__left__mono, axiom, (! [A:$tType]: (((ordere2520102378445227354miring @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)))))))))). % 6.41/2.68 thf(fact_1009_zero__le__mult__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))))))))))). % 6.41/2.68 thf(fact_1010_mult__nonneg__nonpos2, axiom, (! [A:$tType]: (((ordered_semiring_0 @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ B2) @ A2)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_1011_mult__nonpos__nonneg, axiom, (! [A:$tType]: (((ordered_semiring_0 @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ B2)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_1012_mult__nonneg__nonpos, axiom, (! [A:$tType]: (((ordered_semiring_0 @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ B2)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_1013_mult__nonneg__nonneg, axiom, (! [A:$tType]: (((ordered_semiring_0 @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_1014_split__mult__neg__le, axiom, (! [A:$tType]: (((ordered_semiring_0 @ A) => (! [A2:A,B2:A]: (((((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2))) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ B2)) @ (zero_zero @ A))))))))). % 6.41/2.68 thf(fact_1015_mult__le__0__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ B2)) @ (zero_zero @ A)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))))). % 6.41/2.68 thf(fact_1016_mult__right__mono, axiom, (! [A:$tType]: (((ordered_semiring @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))). % 6.41/2.68 thf(fact_1017_mult__right__mono__neg, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))). % 6.41/2.68 thf(fact_1018_mult__left__mono, axiom, (! [A:$tType]: (((ordered_semiring @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)))))))))). % 6.41/2.68 thf(fact_1019_mult__nonpos__nonpos, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_1020_mult__left__mono__neg, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)))))))))). % 6.41/2.68 thf(fact_1021_split__mult__pos__le, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A2:A,B2:A]: (((((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_1022_zero__le__square, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ A2)))))))). % 6.41/2.68 thf(fact_1023_mult__mono_H, axiom, (! [A:$tType]: (((ordered_semiring @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ D2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ D2)))))))))))). % 6.41/2.68 thf(fact_1024_mult__mono, axiom, (! [A:$tType]: (((ordered_semiring @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ D2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ D2)))))))))))). % 6.41/2.68 thf(fact_1025_mult__neg__neg, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_1026_not__square__less__zero, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ (((times_times @ A) @ A2) @ A2)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_1027_mult__less__0__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ A2) @ B2)) @ (zero_zero @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))))). % 6.41/2.68 thf(fact_1028_mult__neg__pos, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ B2)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_1029_mult__pos__neg, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ B2)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_1030_mult__pos__pos, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)))))))))). % 6.41/2.68 thf(fact_1031_mult__pos__neg2, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ B2) @ A2)) @ (zero_zero @ A)))))))))). % 6.41/2.68 thf(fact_1032_zero__less__mult__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ B2) @ (zero_zero @ A))))))))))). % 6.41/2.68 thf(fact_1033_zero__less__mult__pos, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ B2))))))))). % 6.41/2.68 thf(fact_1034_zero__less__mult__pos2, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ B2) @ A2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ B2))))))))). % 6.41/2.68 thf(fact_1035_mult__less__cancel__left__neg, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (((ord_less @ A) @ B2) @ A2))))))))). % 6.41/2.68 thf(fact_1036_mult__less__cancel__left__pos, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (((ord_less @ A) @ A2) @ B2))))))))). % 6.41/2.68 thf(fact_1037_mult__strict__left__mono__neg, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)))))))))). % 6.41/2.68 thf(fact_1038_mult__strict__left__mono, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)))))))))). % 6.41/2.68 thf(fact_1039_mult__less__cancel__left__disj, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) & (((ord_less @ A) @ A2) @ B2)) | ((((ord_less @ A) @ C2) @ (zero_zero @ A)) & (((ord_less @ A) @ B2) @ A2)))))))))). % 6.41/2.68 thf(fact_1040_mult__strict__right__mono__neg, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))). % 6.41/2.68 thf(fact_1041_mult__strict__right__mono, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))). % 6.41/2.68 thf(fact_1042_mult__less__cancel__right__disj, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) & (((ord_less @ A) @ A2) @ B2)) | ((((ord_less @ A) @ C2) @ (zero_zero @ A)) & (((ord_less @ A) @ B2) @ A2)))))))))). % 6.41/2.68 thf(fact_1043_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono, axiom, (! [A:$tType]: (((linord2810124833399127020strict @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)))))))))). % 6.41/2.68 thf(fact_1044_add__scale__eq__noteq, axiom, (! [A:$tType]: (((semiri1453513574482234551roduct @ A) => (! [R2:A,A2:A,B2:A,C2:A,D2:A]: (((R2 != (zero_zero @ A)) => (((A2 = B2) & (C2 != D2)) => ((((plus_plus @ A) @ A2) @ (((times_times @ A) @ R2) @ C2)) != (((plus_plus @ A) @ B2) @ (((times_times @ A) @ R2) @ D2))))))))))). % 6.41/2.68 thf(fact_1045_less__1__mult, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [M:A,N2:A]: (((((ord_less @ A) @ (one_one @ A)) @ M) => ((((ord_less @ A) @ (one_one @ A)) @ N2) => (((ord_less @ A) @ (one_one @ A)) @ (((times_times @ A) @ M) @ N2)))))))))). % 6.41/2.68 thf(fact_1046_frac__eq__eq, axiom, (! [A:$tType]: (((field @ A) => (! [Y:A,Z2:A,X3:A,W2:A]: (((Y != (zero_zero @ A)) => ((Z2 != (zero_zero @ A)) => (((((divide_divide @ A) @ X3) @ Y) = (((divide_divide @ A) @ W2) @ Z2)) = ((((times_times @ A) @ X3) @ Z2) = (((times_times @ A) @ W2) @ Y))))))))))). % 6.41/2.68 thf(fact_1047_divide__eq__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,C2:A,A2:A]: ((((((divide_divide @ A) @ B2) @ C2) = A2) = (((C2 != (zero_zero @ A)) => (B2 = (((times_times @ A) @ A2) @ C2))) & ((C2 = (zero_zero @ A)) => (A2 = (zero_zero @ A))))))))))). % 6.41/2.68 thf(fact_1048_eq__divide__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A,C2:A]: (((A2 = (((divide_divide @ A) @ B2) @ C2)) = (((C2 != (zero_zero @ A)) => ((((times_times @ A) @ A2) @ C2) = B2)) & ((C2 = (zero_zero @ A)) => (A2 = (zero_zero @ A))))))))))). % 6.41/2.68 thf(fact_1049_divide__eq__imp, axiom, (! [A:$tType]: (((division_ring @ A) => (! [C2:A,B2:A,A2:A]: (((C2 != (zero_zero @ A)) => ((B2 = (((times_times @ A) @ A2) @ C2)) => ((((divide_divide @ A) @ B2) @ C2) = A2))))))))). % 6.41/2.68 thf(fact_1050_eq__divide__imp, axiom, (! [A:$tType]: (((division_ring @ A) => (! [C2:A,A2:A,B2:A]: (((C2 != (zero_zero @ A)) => (((((times_times @ A) @ A2) @ C2) = B2) => (A2 = (((divide_divide @ A) @ B2) @ C2)))))))))). % 6.41/2.68 thf(fact_1051_nonzero__divide__eq__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [C2:A,B2:A,A2:A]: (((C2 != (zero_zero @ A)) => (((((divide_divide @ A) @ B2) @ C2) = A2) = (B2 = (((times_times @ A) @ A2) @ C2)))))))))). % 6.41/2.68 thf(fact_1052_nonzero__eq__divide__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [C2:A,A2:A,B2:A]: (((C2 != (zero_zero @ A)) => ((A2 = (((divide_divide @ A) @ B2) @ C2)) = ((((times_times @ A) @ A2) @ C2) = B2))))))))). % 6.41/2.68 thf(fact_1053_power__Suc, axiom, (! [A:$tType]: (((power @ A) => (! [A2:A,N2:nat]: (((((power_power @ A) @ A2) @ (suc @ N2)) = (((times_times @ A) @ A2) @ (((power_power @ A) @ A2) @ N2))))))))). % 6.41/2.68 thf(fact_1054_power__Suc2, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A2:A,N2:nat]: (((((power_power @ A) @ A2) @ (suc @ N2)) = (((times_times @ A) @ (((power_power @ A) @ A2) @ N2)) @ A2)))))))). % 6.41/2.68 thf(fact_1055_Suc__mult__less__cancel1, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less @ nat) @ (((times_times @ nat) @ (suc @ K)) @ M)) @ (((times_times @ nat) @ (suc @ K)) @ N2)) = (((ord_less @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_1056_power__add, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A2:A,M:nat,N2:nat]: (((((power_power @ A) @ A2) @ (((plus_plus @ nat) @ M) @ N2)) = (((times_times @ A) @ (((power_power @ A) @ A2) @ M)) @ (((power_power @ A) @ A2) @ N2))))))))). % 6.41/2.68 thf(fact_1057_mult__less__mono1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less @ nat) @ (((times_times @ nat) @ I) @ K)) @ (((times_times @ nat) @ J) @ K))))))). % 6.41/2.68 thf(fact_1058_mult__less__mono2, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less @ nat) @ (((times_times @ nat) @ K) @ I)) @ (((times_times @ nat) @ K) @ J))))))). % 6.41/2.68 thf(fact_1059_nat__mult__eq__cancel1, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((((times_times @ nat) @ K) @ M) = (((times_times @ nat) @ K) @ N2)) = (M = N2)))))). % 6.41/2.68 thf(fact_1060_nat__mult__less__cancel1, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((ord_less @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)) = (((ord_less @ nat) @ M) @ N2)))))). % 6.41/2.68 thf(fact_1061_Suc__mult__le__cancel1, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ (((times_times @ nat) @ (suc @ K)) @ M)) @ (((times_times @ nat) @ (suc @ K)) @ N2)) = (((ord_less_eq @ nat) @ M) @ N2))))). % 6.41/2.68 thf(fact_1062_mult__Suc, axiom, (! [M:nat,N2:nat]: (((((times_times @ nat) @ (suc @ M)) @ N2) = (((plus_plus @ nat) @ N2) @ (((times_times @ nat) @ M) @ N2)))))). % 6.41/2.68 thf(fact_1063_mult__eq__self__implies__10, axiom, (! [M:nat,N2:nat]: (((M = (((times_times @ nat) @ M) @ N2)) => ((N2 = (one_one @ nat)) | (M = (zero_zero @ nat))))))). % 6.41/2.68 thf(fact_1064_less__mult__imp__div__less, axiom, (! [M:nat,I:nat,N2:nat]: (((((ord_less @ nat) @ M) @ (((times_times @ nat) @ I) @ N2)) => (((ord_less @ nat) @ (((divide_divide @ nat) @ M) @ N2)) @ I))))). % 6.41/2.68 thf(fact_1065_times__div__less__eq__dividend, axiom, (! [N2:nat,M:nat]: ((((ord_less_eq @ nat) @ (((times_times @ nat) @ N2) @ (((divide_divide @ nat) @ M) @ N2))) @ M)))). % 6.41/2.68 thf(fact_1066_div__times__less__eq__dividend, axiom, (! [M:nat,N2:nat]: ((((ord_less_eq @ nat) @ (((times_times @ nat) @ (((divide_divide @ nat) @ M) @ N2)) @ N2)) @ M)))). % 6.41/2.68 thf(fact_1067_power__odd__eq, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A2:A,N2:nat]: (((((power_power @ A) @ A2) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) = (((times_times @ A) @ A2) @ (((power_power @ A) @ (((power_power @ A) @ A2) @ N2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_1068_Suc__double__not__eq__double, axiom, (! [M:nat,N2:nat]: (((suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) != (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))))). % 6.41/2.69 thf(fact_1069_double__not__eq__Suc__double, axiom, (! [M:nat,N2:nat]: (((((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M) != (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)))))). % 6.41/2.69 thf(fact_1070_mult__le__cancel__left, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A2) @ B2)) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ A2)))))))))). % 6.41/2.69 thf(fact_1071_mult__le__cancel__right, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A2) @ B2)) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ A2)))))))))). % 6.41/2.69 thf(fact_1072_mult__left__less__imp__less, axiom, (! [A:$tType]: (((linordered_semiring @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1073_mult__strict__mono, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ C2) @ D2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ D2)))))))))))). % 6.41/2.69 thf(fact_1074_mult__less__cancel__left, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A2) @ B2)) & ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ A2)))))))))). % 6.41/2.69 thf(fact_1075_mult__right__less__imp__less, axiom, (! [A:$tType]: (((linordered_semiring @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1076_mult__strict__mono_H, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ C2) @ D2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ D2)))))))))))). % 6.41/2.69 thf(fact_1077_mult__less__cancel__right, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A2) @ B2)) & ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ A2)))))))))). % 6.41/2.69 thf(fact_1078_mult__le__cancel__left__neg, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (((ord_less_eq @ A) @ B2) @ A2))))))))). % 6.41/2.69 thf(fact_1079_mult__le__cancel__left__pos, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (((ord_less_eq @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1080_mult__left__le__imp__le, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1081_mult__right__le__imp__le, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1082_mult__le__less__imp__less, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ C2) @ D2) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ D2)))))))))))). % 6.41/2.69 thf(fact_1083_mult__less__le__imp__less, axiom, (! [A:$tType]: (((linord8928482502909563296strict @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ D2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ D2)))))))))))). % 6.41/2.69 thf(fact_1084_mult__left__le, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [C2:A,A2:A]: (((((ord_less_eq @ A) @ C2) @ (one_one @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ A2))))))))). % 6.41/2.69 thf(fact_1085_mult__le__one, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (one_one @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ B2) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ B2)) @ (one_one @ A))))))))))). % 6.41/2.69 thf(fact_1086_mult__right__le__one__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => ((((ord_less_eq @ A) @ Y) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ X3) @ Y)) @ X3)))))))))). % 6.41/2.69 thf(fact_1087_mult__left__le__one__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => ((((ord_less_eq @ A) @ Y) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ Y) @ X3)) @ X3)))))))))). % 6.41/2.69 thf(fact_1088_sum__squares__le__zero__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ X3) @ X3)) @ (((times_times @ A) @ Y) @ Y))) @ (zero_zero @ A)) = ((X3 = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_1089_sum__squares__ge__zero, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [X3:A,Y:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (((times_times @ A) @ X3) @ X3)) @ (((times_times @ A) @ Y) @ Y))))))))). % 6.41/2.69 thf(fact_1090_sum__squares__gt__zero__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (((times_times @ A) @ X3) @ X3)) @ (((times_times @ A) @ Y) @ Y))) = ((X3 != (zero_zero @ A)) | (Y != (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_1091_not__sum__squares__lt__zero, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [X3:A,Y:A]: ((~ ((((ord_less @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ X3) @ X3)) @ (((times_times @ A) @ Y) @ Y))) @ (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_1092_divide__less__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A2) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ B2) @ (((times_times @ A) @ A2) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))))))). % 6.41/2.69 thf(fact_1093_less__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ (((times_times @ A) @ A2) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))))))). % 6.41/2.69 thf(fact_1094_neg__divide__less__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A2) = (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ B2))))))))). % 6.41/2.69 thf(fact_1095_neg__less__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2)) = (((ord_less @ A) @ B2) @ (((times_times @ A) @ A2) @ C2)))))))))). % 6.41/2.69 thf(fact_1096_pos__divide__less__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A2) = (((ord_less @ A) @ B2) @ (((times_times @ A) @ A2) @ C2)))))))))). % 6.41/2.69 thf(fact_1097_pos__less__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2)) = (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ B2))))))))). % 6.41/2.69 thf(fact_1098_mult__imp__div__pos__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y:A,X3:A,Z2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Y) => ((((ord_less @ A) @ X3) @ (((times_times @ A) @ Z2) @ Y)) => (((ord_less @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ Z2))))))))). % 6.41/2.69 thf(fact_1099_mult__imp__less__div__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y:A,Z2:A,X3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Y) => ((((ord_less @ A) @ (((times_times @ A) @ Z2) @ Y)) @ X3) => (((ord_less @ A) @ Z2) @ (((divide_divide @ A) @ X3) @ Y)))))))))). % 6.41/2.69 thf(fact_1100_divide__strict__left__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)) => (((ord_less @ A) @ (((divide_divide @ A) @ C2) @ A2)) @ (((divide_divide @ A) @ C2) @ B2))))))))))). % 6.41/2.69 thf(fact_1101_divide__strict__left__mono__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)) => (((ord_less @ A) @ (((divide_divide @ A) @ C2) @ A2)) @ (((divide_divide @ A) @ C2) @ B2))))))))))). % 6.41/2.69 thf(fact_1102_divide__eq__eq__numeral_I1_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,C2:A,W2:num]: ((((((divide_divide @ A) @ B2) @ C2) = ((numeral_numeral @ A) @ W2)) = (((C2 != (zero_zero @ A)) => (B2 = (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2))) & ((C2 = (zero_zero @ A)) => (((numeral_numeral @ A) @ W2) = (zero_zero @ A))))))))))). % 6.41/2.69 thf(fact_1103_eq__divide__eq__numeral_I1_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [W2:num,B2:A,C2:A]: (((((numeral_numeral @ A) @ W2) = (((divide_divide @ A) @ B2) @ C2)) = (((C2 != (zero_zero @ A)) => ((((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2) = B2)) & ((C2 = (zero_zero @ A)) => (((numeral_numeral @ A) @ W2) = (zero_zero @ A))))))))))). % 6.41/2.69 thf(fact_1104_add__divide__eq__if__simps_I2_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,A2:A,B2:A]: ((((Z2 = (zero_zero @ A)) => ((((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ Z2)) @ B2) = B2)) & ((Z2 != (zero_zero @ A)) => ((((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ Z2)) @ B2) = (((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ (((times_times @ A) @ B2) @ Z2))) @ Z2)))))))))). % 6.41/2.69 thf(fact_1105_add__divide__eq__if__simps_I1_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,A2:A,B2:A]: ((((Z2 = (zero_zero @ A)) => ((((plus_plus @ A) @ A2) @ (((divide_divide @ A) @ B2) @ Z2)) = A2)) & ((Z2 != (zero_zero @ A)) => ((((plus_plus @ A) @ A2) @ (((divide_divide @ A) @ B2) @ Z2)) = (((divide_divide @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ A2) @ Z2)) @ B2)) @ Z2)))))))))). % 6.41/2.69 thf(fact_1106_add__frac__eq, axiom, (! [A:$tType]: (((field @ A) => (! [Y:A,Z2:A,X3:A,W2:A]: (((Y != (zero_zero @ A)) => ((Z2 != (zero_zero @ A)) => ((((plus_plus @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ (((divide_divide @ A) @ W2) @ Z2)) = (((divide_divide @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ X3) @ Z2)) @ (((times_times @ A) @ W2) @ Y))) @ (((times_times @ A) @ Y) @ Z2))))))))))). % 6.41/2.69 thf(fact_1107_add__frac__num, axiom, (! [A:$tType]: (((field @ A) => (! [Y:A,X3:A,Z2:A]: (((Y != (zero_zero @ A)) => ((((plus_plus @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ Z2) = (((divide_divide @ A) @ (((plus_plus @ A) @ X3) @ (((times_times @ A) @ Z2) @ Y))) @ Y))))))))). % 6.41/2.69 thf(fact_1108_add__num__frac, axiom, (! [A:$tType]: (((field @ A) => (! [Y:A,Z2:A,X3:A]: (((Y != (zero_zero @ A)) => ((((plus_plus @ A) @ Z2) @ (((divide_divide @ A) @ X3) @ Y)) = (((divide_divide @ A) @ (((plus_plus @ A) @ X3) @ (((times_times @ A) @ Z2) @ Y))) @ Y))))))))). % 6.41/2.69 thf(fact_1109_add__divide__eq__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,X3:A,Y:A]: (((Z2 != (zero_zero @ A)) => ((((plus_plus @ A) @ X3) @ (((divide_divide @ A) @ Y) @ Z2)) = (((divide_divide @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ X3) @ Z2)) @ Y)) @ Z2))))))))). % 6.41/2.69 thf(fact_1110_divide__add__eq__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,X3:A,Y:A]: (((Z2 != (zero_zero @ A)) => ((((plus_plus @ A) @ (((divide_divide @ A) @ X3) @ Z2)) @ Y) = (((divide_divide @ A) @ (((plus_plus @ A) @ X3) @ (((times_times @ A) @ Y) @ Z2))) @ Z2))))))))). % 6.41/2.69 thf(fact_1111_power__gt1__lemma, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A2) => (((ord_less @ A) @ (one_one @ A)) @ (((times_times @ A) @ A2) @ (((power_power @ A) @ A2) @ N2)))))))))). % 6.41/2.69 thf(fact_1112_power__less__power__Suc, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A2) => (((ord_less @ A) @ (((power_power @ A) @ A2) @ N2)) @ (((times_times @ A) @ A2) @ (((power_power @ A) @ A2) @ N2)))))))))). % 6.41/2.69 thf(fact_1113_one__less__mult, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ N2) => ((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ M) => (((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ (((times_times @ nat) @ M) @ N2))))))). % 6.41/2.69 thf(fact_1114_n__less__m__mult__n, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ M) => (((ord_less @ nat) @ N2) @ (((times_times @ nat) @ M) @ N2))))))). % 6.41/2.69 thf(fact_1115_n__less__n__mult__m, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ M) => (((ord_less @ nat) @ N2) @ (((times_times @ nat) @ N2) @ M))))))). % 6.41/2.69 thf(fact_1116_nat__mult__le__cancel1, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((ord_less_eq @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)) = (((ord_less_eq @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_1117_nat__mult__div__cancel1, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((divide_divide @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)) = (((divide_divide @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_1118_div__less__iff__less__mult, axiom, (! [Q4:nat,M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ Q4) => ((((ord_less @ nat) @ (((divide_divide @ nat) @ M) @ Q4)) @ N2) = (((ord_less @ nat) @ M) @ (((times_times @ nat) @ N2) @ Q4))))))). % 6.41/2.69 thf(fact_1119_mult__le__cancel__left1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ (((times_times @ A) @ C2) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (one_one @ A)) @ B2)) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ (one_one @ A))))))))))). % 6.41/2.69 thf(fact_1120_mult__le__cancel__left2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,A2:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A2)) @ C2) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A2) @ (one_one @ A))) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ A2)))))))))). % 6.41/2.69 thf(fact_1121_mult__le__cancel__right1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ (((times_times @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (one_one @ A)) @ B2)) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ (one_one @ A))))))))))). % 6.41/2.69 thf(fact_1122_mult__le__cancel__right2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,C2:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ C2) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A2) @ (one_one @ A))) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ A2)))))))))). % 6.41/2.69 thf(fact_1123_mult__less__cancel__left1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,B2:A]: (((((ord_less @ A) @ C2) @ (((times_times @ A) @ C2) @ B2)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (one_one @ A)) @ B2)) & ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ (one_one @ A))))))))))). % 6.41/2.69 thf(fact_1124_mult__less__cancel__left2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,A2:A]: (((((ord_less @ A) @ (((times_times @ A) @ C2) @ A2)) @ C2) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A2) @ (one_one @ A))) & ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (one_one @ A)) @ A2)))))))))). % 6.41/2.69 thf(fact_1125_mult__less__cancel__right1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,B2:A]: (((((ord_less @ A) @ C2) @ (((times_times @ A) @ B2) @ C2)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (one_one @ A)) @ B2)) & ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ (one_one @ A))))))))))). % 6.41/2.69 thf(fact_1126_mult__less__cancel__right2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,C2:A]: (((((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ C2) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A2) @ (one_one @ A))) & ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (one_one @ A)) @ A2)))))))))). % 6.41/2.69 thf(fact_1127_field__le__mult__one__interval, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: (((! [Z3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Z3) => ((((ord_less @ A) @ Z3) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ Z3) @ X3)) @ Y))))) => (((ord_less_eq @ A) @ X3) @ Y)))))))). % 6.41/2.69 thf(fact_1128_divide__le__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A2) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ A2) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))))))). % 6.41/2.69 thf(fact_1129_le__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ A2) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))))))). % 6.41/2.69 thf(fact_1130_divide__left__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ C2) @ A2)) @ (((divide_divide @ A) @ C2) @ B2))))))))))). % 6.41/2.69 thf(fact_1131_neg__divide__le__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A2) = (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ B2))))))))). % 6.41/2.69 thf(fact_1132_neg__le__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2)) = (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ A2) @ C2)))))))))). % 6.41/2.69 thf(fact_1133_pos__divide__le__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A2) = (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ A2) @ C2)))))))))). % 6.41/2.69 thf(fact_1134_pos__le__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less_eq @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2)) = (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ B2))))))))). % 6.41/2.69 thf(fact_1135_mult__imp__div__pos__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y:A,X3:A,Z2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Y) => ((((ord_less_eq @ A) @ X3) @ (((times_times @ A) @ Z2) @ Y)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ Z2))))))))). % 6.41/2.69 thf(fact_1136_mult__imp__le__div__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y:A,Z2:A,X3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Y) => ((((ord_less_eq @ A) @ (((times_times @ A) @ Z2) @ Y)) @ X3) => (((ord_less_eq @ A) @ Z2) @ (((divide_divide @ A) @ X3) @ Y)))))))))). % 6.41/2.69 thf(fact_1137_divide__left__mono__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ C2) @ A2)) @ (((divide_divide @ A) @ C2) @ B2))))))))))). % 6.41/2.69 thf(fact_1138_convex__bound__le, axiom, (! [A:$tType]: (((linord6961819062388156250ring_1 @ A) => (! [X3:A,A2:A,Y:A,U:A,V2:A]: (((((ord_less_eq @ A) @ X3) @ A2) => ((((ord_less_eq @ A) @ Y) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ U) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ V2) => (((((plus_plus @ A) @ U) @ V2) = (one_one @ A)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ U) @ X3)) @ (((times_times @ A) @ V2) @ Y))) @ A2)))))))))))). % 6.41/2.69 thf(fact_1139_divide__less__eq__numeral_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,W2:num]: (((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ ((numeral_numeral @ A) @ W2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ B2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ W2))))))))))))). % 6.41/2.69 thf(fact_1140_less__divide__eq__numeral_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [W2:num,B2:A,C2:A]: (((((ord_less @ A) @ ((numeral_numeral @ A) @ W2)) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ ((numeral_numeral @ A) @ W2)) @ (zero_zero @ A))))))))))))). % 6.41/2.69 thf(fact_1141_power__Suc__less, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,N2:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ A2) @ (one_one @ A)) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ (((power_power @ A) @ A2) @ N2))) @ (((power_power @ A) @ A2) @ N2)))))))))). % 6.41/2.69 thf(fact_1142_mult__2, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [Z2:A]: (((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Z2) = (((plus_plus @ A) @ Z2) @ Z2)))))))). % 6.41/2.69 thf(fact_1143_mult__2__right, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [Z2:A]: (((((times_times @ A) @ Z2) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (((plus_plus @ A) @ Z2) @ Z2)))))))). % 6.41/2.69 thf(fact_1144_left__add__twice, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B2)) = (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2)) @ B2)))))))). % 6.41/2.69 thf(fact_1145_div__nat__eqI, axiom, (! [N2:nat,Q4:nat,M:nat]: (((((ord_less_eq @ nat) @ (((times_times @ nat) @ N2) @ Q4)) @ M) => ((((ord_less @ nat) @ M) @ (((times_times @ nat) @ N2) @ (suc @ Q4))) => ((((divide_divide @ nat) @ M) @ N2) = Q4)))))). % 6.41/2.69 thf(fact_1146_less__eq__div__iff__mult__less__eq, axiom, (! [Q4:nat,M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ Q4) => ((((ord_less_eq @ nat) @ M) @ (((divide_divide @ nat) @ N2) @ Q4)) = (((ord_less_eq @ nat) @ (((times_times @ nat) @ M) @ Q4)) @ N2)))))). % 6.41/2.69 thf(fact_1147_split__div, axiom, (! [P:(nat > $o),M:nat,N2:nat]: (((P @ (((divide_divide @ nat) @ M) @ N2)) = (((N2 = (zero_zero @ nat)) => (P @ (zero_zero @ nat))) & ((N2 != (zero_zero @ nat)) => (! [I3:nat,J3:nat]: (((((ord_less @ nat) @ J3) @ N2) => ((M = (((plus_plus @ nat) @ (((times_times @ nat) @ N2) @ I3)) @ J3)) => (P @ I3))))))))))). % 6.41/2.69 thf(fact_1148_dividend__less__div__times, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((ord_less @ nat) @ M) @ (((plus_plus @ nat) @ N2) @ (((times_times @ nat) @ (((divide_divide @ nat) @ M) @ N2)) @ N2))))))). % 6.41/2.69 thf(fact_1149_dividend__less__times__div, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((ord_less @ nat) @ M) @ (((plus_plus @ nat) @ N2) @ (((times_times @ nat) @ N2) @ (((divide_divide @ nat) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_1150_nat__bit__induct, axiom, (! [P:(nat > $o),N2:nat]: (((P @ (zero_zero @ nat)) => ((! [N3:nat]: (((P @ N3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) => (P @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N3)))))) => ((! [N3:nat]: (((P @ N3) => (P @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N3)))))) => (P @ N2))))))). % 6.41/2.69 thf(fact_1151_convex__bound__lt, axiom, (! [A:$tType]: (((linord715952674999750819strict @ A) => (! [X3:A,A2:A,Y:A,U:A,V2:A]: (((((ord_less @ A) @ X3) @ A2) => ((((ord_less @ A) @ Y) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ U) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ V2) => (((((plus_plus @ A) @ U) @ V2) = (one_one @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ U) @ X3)) @ (((times_times @ A) @ V2) @ Y))) @ A2)))))))))))). % 6.41/2.69 thf(fact_1152_divide__le__eq__numeral_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,W2:num]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ ((numeral_numeral @ A) @ W2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ W2))))))))))))). % 6.41/2.69 thf(fact_1153_le__divide__eq__numeral_I1_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [W2:num,B2:A,C2:A]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ W2)) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ ((numeral_numeral @ A) @ W2)) @ (zero_zero @ A))))))))))))). % 6.41/2.69 thf(fact_1154_split__div_H, axiom, (! [P:(nat > $o),M:nat,N2:nat]: (((P @ (((divide_divide @ nat) @ M) @ N2)) = (((N2 = (zero_zero @ nat)) & (P @ (zero_zero @ nat))) | (? [Q5:nat]: (((((ord_less_eq @ nat) @ (((times_times @ nat) @ N2) @ Q5)) @ M) & ((((ord_less @ nat) @ M) @ (((times_times @ nat) @ N2) @ (suc @ Q5))) & (P @ Q5)))))))))). % 6.41/2.69 thf(fact_1155_power2__sum, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [X3:A,Y:A]: (((((power_power @ A) @ (((plus_plus @ A) @ X3) @ Y)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((plus_plus @ A) @ (((plus_plus @ A) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (((times_times @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ X3)) @ Y))))))))). % 6.41/2.69 thf(fact_1156_zero__le__even__power_H, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,N2:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))))))))). % 6.41/2.69 thf(fact_1157_triangle__def, axiom, (nat_triangle = (^ [N:nat]: ((((divide_divide @ nat) @ (((times_times @ nat) @ N) @ (suc @ N))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))). % 6.41/2.69 thf(fact_1158_odd__0__le__power__imp__0__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,N2:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))). % 6.41/2.69 thf(fact_1159_odd__power__less__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,N2:nat]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((power_power @ A) @ A2) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)))) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1160_arith__geo__mean, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [U:A,X3:A,Y:A]: ((((((power_power @ A) @ U) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((times_times @ A) @ X3) @ Y)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((ord_less_eq @ A) @ U) @ (((divide_divide @ A) @ (((plus_plus @ A) @ X3) @ Y)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))))). % 6.41/2.69 thf(fact_1161_sum__squares__bound, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A]: ((((ord_less_eq @ A) @ (((times_times @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ X3)) @ Y)) @ (((plus_plus @ A) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_1162_set__bit__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se5668285175392031749et_bit @ A) @ (zero_zero @ nat)) @ A2) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))). % 6.41/2.69 thf(fact_1163_pos2, axiom, (((ord_less @ nat) @ (zero_zero @ nat)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))). % 6.41/2.69 thf(fact_1164_unset__bit__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se2638667681897837118et_bit @ A) @ (zero_zero @ nat)) @ A2) = (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_1165_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => ((((divide_divide @ A) @ A2) @ (((times_times @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ C2))))))))). % 6.41/2.69 thf(fact_1166_discrete, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => ((ord_less @ A) = (^ [A4:A]: (((ord_less_eq @ A) @ (((plus_plus @ A) @ A4) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_1167_unique__euclidean__semiring__numeral__class_Odiv__less, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ A2) @ B2) => ((((divide_divide @ A) @ A2) @ B2) = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_1168_div__positive, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_1169_zle__add1__eq__le, axiom, (! [W2:int,Z2:int]: (((((ord_less @ int) @ W2) @ (((plus_plus @ int) @ Z2) @ (one_one @ int))) = (((ord_less_eq @ int) @ W2) @ Z2))))). % 6.41/2.69 thf(fact_1170_set__bit__negative__int__iff, axiom, (! [N2:nat,K:int]: (((((ord_less @ int) @ (((bit_se5668285175392031749et_bit @ int) @ N2) @ K)) @ (zero_zero @ int)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))). % 6.41/2.69 thf(fact_1171_unset__bit__negative__int__iff, axiom, (! [N2:nat,K:int]: (((((ord_less @ int) @ (((bit_se2638667681897837118et_bit @ int) @ N2) @ K)) @ (zero_zero @ int)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))). % 6.41/2.69 thf(fact_1172_div__pos__pos__trivial, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((ord_less @ int) @ K) @ L) => ((((divide_divide @ int) @ K) @ L) = (zero_zero @ int))))))). % 6.41/2.69 thf(fact_1173_div__neg__neg__trivial, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ K) @ (zero_zero @ int)) => ((((ord_less @ int) @ L) @ K) => ((((divide_divide @ int) @ K) @ L) = (zero_zero @ int))))))). % 6.41/2.69 thf(fact_1174_nonneg1__imp__zdiv__pos__iff, axiom, (! [A2:int,B2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A2) => ((((ord_less @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ A2) @ B2)) = ((((ord_less_eq @ int) @ B2) @ A2) & (((ord_less @ int) @ (zero_zero @ int)) @ B2))))))). % 6.41/2.69 thf(fact_1175_pos__imp__zdiv__nonneg__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ A2) @ B2)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ A2)))))). % 6.41/2.69 thf(fact_1176_neg__imp__zdiv__nonneg__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ B2) @ (zero_zero @ int)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ A2) @ B2)) = (((ord_less_eq @ int) @ A2) @ (zero_zero @ int))))))). % 6.41/2.69 thf(fact_1177_int__one__le__iff__zero__less, axiom, (! [Z2:int]: (((((ord_less_eq @ int) @ (one_one @ int)) @ Z2) = (((ord_less @ int) @ (zero_zero @ int)) @ Z2))))). % 6.41/2.69 thf(fact_1178_pos__imp__zdiv__pos__iff, axiom, (! [K:int,I:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => ((((ord_less @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ I) @ K)) = (((ord_less_eq @ int) @ K) @ I)))))). % 6.41/2.69 thf(fact_1179_div__nonpos__pos__le0, axiom, (! [A2:int,B2:int]: (((((ord_less_eq @ int) @ A2) @ (zero_zero @ int)) => ((((ord_less @ int) @ (zero_zero @ int)) @ B2) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (zero_zero @ int))))))). % 6.41/2.69 thf(fact_1180_div__nonneg__neg__le0, axiom, (! [A2:int,B2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A2) => ((((ord_less @ int) @ B2) @ (zero_zero @ int)) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (zero_zero @ int))))))). % 6.41/2.69 thf(fact_1181_div__positive__int, axiom, (! [L:int,K:int]: (((((ord_less_eq @ int) @ L) @ K) => ((((ord_less @ int) @ (zero_zero @ int)) @ L) => (((ord_less @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ K) @ L))))))). % 6.41/2.69 thf(fact_1182_div__int__pos__iff, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((divide_divide @ int) @ K) @ L)) = ((K = (zero_zero @ int)) | ((L = (zero_zero @ int)) | (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ L)) | ((((ord_less @ int) @ K) @ (zero_zero @ int)) & (((ord_less @ int) @ L) @ (zero_zero @ int)))))))))). % 6.41/2.69 thf(fact_1183_zless__imp__add1__zle, axiom, (! [W2:int,Z2:int]: (((((ord_less @ int) @ W2) @ Z2) => (((ord_less_eq @ int) @ (((plus_plus @ int) @ W2) @ (one_one @ int))) @ Z2))))). % 6.41/2.69 thf(fact_1184_zdiv__mono2__neg, axiom, (! [A2:int,B7:int,B2:int]: (((((ord_less @ int) @ A2) @ (zero_zero @ int)) => ((((ord_less @ int) @ (zero_zero @ int)) @ B7) => ((((ord_less_eq @ int) @ B7) @ B2) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B7)) @ (((divide_divide @ int) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1185_zdiv__mono1__neg, axiom, (! [A2:int,A7:int,B2:int]: (((((ord_less_eq @ int) @ A2) @ A7) => ((((ord_less @ int) @ B2) @ (zero_zero @ int)) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A7) @ B2)) @ (((divide_divide @ int) @ A2) @ B2))))))). % 6.41/2.69 thf(fact_1186_int__div__pos__eq, axiom, (! [A2:int,B2:int,Q4:int,R2:int]: (((A2 = (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R2)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ R2) => ((((ord_less @ int) @ R2) @ B2) => ((((divide_divide @ int) @ A2) @ B2) = Q4))))))). % 6.41/2.69 thf(fact_1187_int__div__neg__eq, axiom, (! [A2:int,B2:int,Q4:int,R2:int]: (((A2 = (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R2)) => ((((ord_less_eq @ int) @ R2) @ (zero_zero @ int)) => ((((ord_less @ int) @ B2) @ R2) => ((((divide_divide @ int) @ A2) @ B2) = Q4))))))). % 6.41/2.69 thf(fact_1188_zdiv__eq__0__iff, axiom, (! [I:int,K:int]: ((((((divide_divide @ int) @ I) @ K) = (zero_zero @ int)) = ((K = (zero_zero @ int)) | (((((ord_less_eq @ int) @ (zero_zero @ int)) @ I) & (((ord_less @ int) @ I) @ K)) | ((((ord_less_eq @ int) @ I) @ (zero_zero @ int)) & (((ord_less @ int) @ K) @ I)))))))). % 6.41/2.69 thf(fact_1189_zdiv__mono2, axiom, (! [A2:int,B7:int,B2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A2) => ((((ord_less @ int) @ (zero_zero @ int)) @ B7) => ((((ord_less_eq @ int) @ B7) @ B2) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (((divide_divide @ int) @ A2) @ B7)))))))). % 6.41/2.69 thf(fact_1190_zdiv__mono1, axiom, (! [A2:int,A7:int,B2:int]: (((((ord_less_eq @ int) @ A2) @ A7) => ((((ord_less @ int) @ (zero_zero @ int)) @ B2) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (((divide_divide @ int) @ A7) @ B2))))))). % 6.41/2.69 thf(fact_1191_split__zdiv, axiom, (! [P:(int > $o),N2:int,K:int]: (((P @ (((divide_divide @ int) @ N2) @ K)) = (((K = (zero_zero @ int)) => (P @ (zero_zero @ int))) & (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (! [I3:int,J3:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ J3) & ((((ord_less @ int) @ J3) @ K) & (N2 = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I3)) @ J3)))) => (P @ I3))))) & ((((ord_less @ int) @ K) @ (zero_zero @ int)) => (! [I3:int,J3:int]: ((((((ord_less @ int) @ K) @ J3) & ((((ord_less_eq @ int) @ J3) @ (zero_zero @ int)) & (N2 = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I3)) @ J3)))) => (P @ I3))))))))))). % 6.41/2.69 thf(fact_1192_le__imp__0__less, axiom, (! [Z2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => (((ord_less @ int) @ (zero_zero @ int)) @ (((plus_plus @ int) @ (one_one @ int)) @ Z2)))))). % 6.41/2.69 thf(fact_1193_add1__zle__eq, axiom, (! [W2:int,Z2:int]: (((((ord_less_eq @ int) @ (((plus_plus @ int) @ W2) @ (one_one @ int))) @ Z2) = (((ord_less @ int) @ W2) @ Z2))))). % 6.41/2.69 thf(fact_1194_odd__less__0__iff, axiom, (! [Z2:int]: (((((ord_less @ int) @ (((plus_plus @ int) @ (((plus_plus @ int) @ (one_one @ int)) @ Z2)) @ Z2)) @ (zero_zero @ int)) = (((ord_less @ int) @ Z2) @ (zero_zero @ int)))))). % 6.41/2.69 thf(fact_1195_pos__zmult__eq__1__iff, axiom, (! [M:int,N2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ M) => (((((times_times @ int) @ M) @ N2) = (one_one @ int)) = ((M = (one_one @ int)) & (N2 = (one_one @ int)))))))). % 6.41/2.69 thf(fact_1196_zless__add1__eq, axiom, (! [W2:int,Z2:int]: (((((ord_less @ int) @ W2) @ (((plus_plus @ int) @ Z2) @ (one_one @ int))) = ((((ord_less @ int) @ W2) @ Z2) | (W2 = Z2)))))). % 6.41/2.69 thf(fact_1197_int__gr__induct, axiom, (! [K:int,I:int,P:(int > $o)]: (((((ord_less @ int) @ K) @ I) => ((P @ (((plus_plus @ int) @ K) @ (one_one @ int))) => ((! [I4:int]: (((((ord_less @ int) @ K) @ I4) => ((P @ I4) => (P @ (((plus_plus @ int) @ I4) @ (one_one @ int))))))) => (P @ I))))))). % 6.41/2.69 thf(fact_1198_pos__imp__zdiv__neg__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (zero_zero @ int)) = (((ord_less @ int) @ A2) @ (zero_zero @ int))))))). % 6.41/2.69 thf(fact_1199_neg__imp__zdiv__neg__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ B2) @ (zero_zero @ int)) => ((((ord_less @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (zero_zero @ int)) = (((ord_less @ int) @ (zero_zero @ int)) @ A2)))))). % 6.41/2.69 thf(fact_1200_int__div__less__self, axiom, (! [X3:int,K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ X3) => ((((ord_less @ int) @ (one_one @ int)) @ K) => (((ord_less @ int) @ (((divide_divide @ int) @ X3) @ K)) @ X3)))))). % 6.41/2.69 thf(fact_1201_div__neg__pos__less0, axiom, (! [A2:int,B2:int]: (((((ord_less @ int) @ A2) @ (zero_zero @ int)) => ((((ord_less @ int) @ (zero_zero @ int)) @ B2) => (((ord_less @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (zero_zero @ int))))))). % 6.41/2.69 thf(fact_1202_odd__nonzero, axiom, (! [Z2:int]: (((((plus_plus @ int) @ (((plus_plus @ int) @ (one_one @ int)) @ Z2)) @ Z2) != (zero_zero @ int))))). % 6.41/2.69 thf(fact_1203_times__int__code_I2_J, axiom, (! [L:int]: (((((times_times @ int) @ (zero_zero @ int)) @ L) = (zero_zero @ int))))). % 6.41/2.69 thf(fact_1204_times__int__code_I1_J, axiom, (! [K:int]: (((((times_times @ int) @ K) @ (zero_zero @ int)) = (zero_zero @ int))))). % 6.41/2.69 thf(fact_1205_bot__enat__def, axiom, ((bot_bot @ extended_enat) = (zero_zero @ extended_enat))). % 6.41/2.69 thf(fact_1206_incr__mult__lemma, axiom, (! [D2:int,P:(int > $o),K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D2) => ((! [X5:int]: (((P @ X5) => (P @ (((plus_plus @ int) @ X5) @ D2))))) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (! [X2:int]: (((P @ X2) => (P @ (((plus_plus @ int) @ X2) @ (((times_times @ int) @ K) @ D2)))))))))))). % 6.41/2.69 thf(fact_1207_zmult__zless__mono2, axiom, (! [I:int,J:int,K:int]: (((((ord_less @ int) @ I) @ J) => ((((ord_less @ int) @ (zero_zero @ int)) @ K) => (((ord_less @ int) @ (((times_times @ int) @ K) @ I)) @ (((times_times @ int) @ K) @ J))))))). % 6.41/2.69 thf(fact_1208_less__int__code_I1_J, axiom, (~ ((((ord_less @ int) @ (zero_zero @ int)) @ (zero_zero @ int))))). % 6.41/2.69 thf(fact_1209_q__pos__lemma, axiom, (! [B7:int,Q6:int,R3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((plus_plus @ int) @ (((times_times @ int) @ B7) @ Q6)) @ R3)) => ((((ord_less @ int) @ R3) @ B7) => ((((ord_less @ int) @ (zero_zero @ int)) @ B7) => (((ord_less_eq @ int) @ (zero_zero @ int)) @ Q6))))))). % 6.41/2.69 thf(fact_1210_zdiv__mono2__lemma, axiom, (! [B2:int,Q4:int,R2:int,B7:int,Q6:int,R3:int]: ((((((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R2) = (((plus_plus @ int) @ (((times_times @ int) @ B7) @ Q6)) @ R3)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((plus_plus @ int) @ (((times_times @ int) @ B7) @ Q6)) @ R3)) => ((((ord_less @ int) @ R3) @ B7) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ R2) => ((((ord_less @ int) @ (zero_zero @ int)) @ B7) => ((((ord_less_eq @ int) @ B7) @ B2) => (((ord_less_eq @ int) @ Q4) @ Q6)))))))))). % 6.41/2.69 thf(fact_1211_zdiv__mono2__neg__lemma, axiom, (! [B2:int,Q4:int,R2:int,B7:int,Q6:int,R3:int]: ((((((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R2) = (((plus_plus @ int) @ (((times_times @ int) @ B7) @ Q6)) @ R3)) => ((((ord_less @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ B7) @ Q6)) @ R3)) @ (zero_zero @ int)) => ((((ord_less @ int) @ R2) @ B2) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ R3) => ((((ord_less @ int) @ (zero_zero @ int)) @ B7) => ((((ord_less_eq @ int) @ B7) @ B2) => (((ord_less_eq @ int) @ Q6) @ Q4)))))))))). % 6.41/2.69 thf(fact_1212_unique__quotient__lemma, axiom, (! [B2:int,Q6:int,R3:int,Q4:int,R2:int]: (((((ord_less_eq @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q6)) @ R3)) @ (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R2)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ R3) => ((((ord_less @ int) @ R3) @ B2) => ((((ord_less @ int) @ R2) @ B2) => (((ord_less_eq @ int) @ Q6) @ Q4)))))))). % 6.41/2.69 thf(fact_1213_unique__quotient__lemma__neg, axiom, (! [B2:int,Q6:int,R3:int,Q4:int,R2:int]: (((((ord_less_eq @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q6)) @ R3)) @ (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R2)) => ((((ord_less_eq @ int) @ R2) @ (zero_zero @ int)) => ((((ord_less @ int) @ B2) @ R2) => ((((ord_less @ int) @ B2) @ R3) => (((ord_less_eq @ int) @ Q4) @ Q6)))))))). % 6.41/2.69 thf(fact_1214_enat__0__less__mult__iff, axiom, (! [M:extended_enat,N2:extended_enat]: (((((ord_less @ extended_enat) @ (zero_zero @ extended_enat)) @ (((times_times @ extended_enat) @ M) @ N2)) = ((((ord_less @ extended_enat) @ (zero_zero @ extended_enat)) @ M) & (((ord_less @ extended_enat) @ (zero_zero @ extended_enat)) @ N2)))))). % 6.41/2.69 thf(fact_1215_realpow__pos__nth2, axiom, (! [A2:real,N2:nat]: (((((ord_less @ real) @ (zero_zero @ real)) @ A2) => (? [R4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R4) & ((((power_power @ real) @ R4) @ (suc @ N2)) = A2)))))))). % 6.41/2.69 thf(fact_1216_realpow__pos__nth__unique, axiom, (! [N2:nat,A2:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ (zero_zero @ real)) @ A2) => (? [X5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X5) & (((((power_power @ real) @ X5) @ N2) = A2) & (! [Y5:real]: ((((((ord_less @ real) @ (zero_zero @ real)) @ Y5) & ((((power_power @ real) @ Y5) @ N2) = A2)) => (Y5 = X5))))))))))))). % 6.41/2.69 thf(fact_1217_realpow__pos__nth, axiom, (! [N2:nat,A2:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ (zero_zero @ real)) @ A2) => (? [R4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R4) & ((((power_power @ real) @ R4) @ N2) = A2))))))))). % 6.41/2.69 thf(fact_1218_mult__le__cancel__iff2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:A,X3:A,Y:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Z2) => ((((ord_less_eq @ A) @ (((times_times @ A) @ Z2) @ X3)) @ (((times_times @ A) @ Z2) @ Y)) = (((ord_less_eq @ A) @ X3) @ Y))))))))). % 6.41/2.69 thf(fact_1219_mult__le__cancel__iff1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:A,X3:A,Y:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Z2) => ((((ord_less_eq @ A) @ (((times_times @ A) @ X3) @ Z2)) @ (((times_times @ A) @ Y) @ Z2)) = (((ord_less_eq @ A) @ X3) @ Y))))))))). % 6.41/2.69 thf(fact_1220_divides__aux__eq, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [Q4:A,R2:A]: ((((unique5940410009612947441es_aux @ A) @ ((((product_Pair @ A) @ A) @ Q4) @ R2)) = (R2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1221_low__def, axiom, (vEBT_VEBT_low = (^ [X:nat,N:nat]: ((((modulo_modulo @ nat) @ X) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))))). % 6.41/2.69 thf(fact_1222_even__succ__div__exp, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A,N2:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((divide_divide @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ A2)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) = (((divide_divide @ A) @ A2) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2))))))))))). % 6.41/2.69 thf(fact_1223_set__decode__Suc, axiom, (! [N2:nat,X3:nat]: (((((member @ nat) @ (suc @ N2)) @ (nat_set_decode @ X3)) = (((member @ nat) @ N2) @ (nat_set_decode @ (((divide_divide @ nat) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))). % 6.41/2.69 thf(fact_1224_length__product, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B)]: ((((size_size @ (list @ ((product_prod @ A) @ B))) @ ((((product @ A) @ B) @ Xs) @ Ys)) = (((times_times @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ B)) @ Ys)))))). % 6.41/2.69 thf(fact_1225_mult__less__iff1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:A,X3:A,Y:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Z2) => ((((ord_less @ A) @ (((times_times @ A) @ X3) @ Z2)) @ (((times_times @ A) @ Y) @ Z2)) = (((ord_less @ A) @ X3) @ Y))))))))). % 6.41/2.69 thf(fact_1226_num_Osize__gen_I2_J, axiom, (! [X22:num]: (((size_num @ (bit0 @ X22)) = (((plus_plus @ nat) @ (size_num @ X22)) @ (suc @ (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_1227_nat__dvd__1__iff__1, axiom, (! [M:nat]: (((((dvd_dvd @ nat) @ M) @ (one_one @ nat)) = (M = (one_one @ nat)))))). % 6.41/2.69 thf(fact_1228_dvd__0__left__iff, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A2:A]: (((((dvd_dvd @ A) @ (zero_zero @ A)) @ A2) = (A2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1229_dvd__0__right, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A2:A]: ((((dvd_dvd @ A) @ A2) @ (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1230_dvd__add__triv__right__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ (((plus_plus @ A) @ B2) @ A2)) = (((dvd_dvd @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1231_dvd__add__triv__left__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B2)) = (((dvd_dvd @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1232_bits__mod__0, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A]: (((((modulo_modulo @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1233_mod__self, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A2:A]: (((((modulo_modulo @ A) @ A2) @ A2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1234_mod__by__0, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A2:A]: (((((modulo_modulo @ A) @ A2) @ (zero_zero @ A)) = A2))))))). % 6.41/2.69 thf(fact_1235_mod__0, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A2:A]: (((((modulo_modulo @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1236_dvd__1__left, axiom, (! [K:nat]: ((((dvd_dvd @ nat) @ (suc @ (zero_zero @ nat))) @ K)))). % 6.41/2.69 thf(fact_1237_dvd__1__iff__1, axiom, (! [M:nat]: (((((dvd_dvd @ nat) @ M) @ (suc @ (zero_zero @ nat))) = (M = (suc @ (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_1238_div__dvd__div, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ B2) => ((((dvd_dvd @ A) @ A2) @ C2) => ((((dvd_dvd @ A) @ (((divide_divide @ A) @ B2) @ A2)) @ (((divide_divide @ A) @ C2) @ A2)) = (((dvd_dvd @ A) @ B2) @ C2)))))))))). % 6.41/2.69 thf(fact_1239_not__real__square__gt__zero, axiom, (! [X3:real]: (((~ ((((ord_less @ real) @ (zero_zero @ real)) @ (((times_times @ real) @ X3) @ X3)))) = (X3 = (zero_zero @ real)))))). % 6.41/2.69 thf(fact_1240_mod__add__self2, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A2:A,B2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = (((modulo_modulo @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1241_mod__add__self1, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,A2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ B2) @ A2)) @ B2) = (((modulo_modulo @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1242_nat__mult__dvd__cancel__disj, axiom, (! [K:nat,M:nat,N2:nat]: (((((dvd_dvd @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)) = ((K = (zero_zero @ nat)) | (((dvd_dvd @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_1243_mod__less, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ N2) => ((((modulo_modulo @ nat) @ M) @ N2) = M))))). % 6.41/2.69 thf(fact_1244_dvd__mult__cancel__left, axiom, (! [A:$tType]: (((idom @ A) => (! [C2:A,A2:A,B2:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = ((C2 = (zero_zero @ A)) | (((dvd_dvd @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1245_dvd__mult__cancel__right, axiom, (! [A:$tType]: (((idom @ A) => (! [A2:A,C2:A,B2:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = ((C2 = (zero_zero @ A)) | (((dvd_dvd @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1246_dvd__times__left__cancel__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A,C2:A]: (((A2 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ B2)) @ (((times_times @ A) @ A2) @ C2)) = (((dvd_dvd @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_1247_dvd__times__right__cancel__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A,C2:A]: (((A2 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ (((times_times @ A) @ B2) @ A2)) @ (((times_times @ A) @ C2) @ A2)) = (((dvd_dvd @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_1248_unit__prod, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ B2)) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_1249_dvd__add__times__triv__right__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ (((plus_plus @ A) @ B2) @ (((times_times @ A) @ C2) @ A2))) = (((dvd_dvd @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1250_dvd__add__times__triv__left__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A2:A,C2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ (((plus_plus @ A) @ (((times_times @ A) @ C2) @ A2)) @ B2)) = (((dvd_dvd @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1251_mod__mult__self1__is__0, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,A2:A]: (((((modulo_modulo @ A) @ (((times_times @ A) @ B2) @ A2)) @ B2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1252_mod__mult__self2__is__0, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A2:A,B2:A]: (((((modulo_modulo @ A) @ (((times_times @ A) @ A2) @ B2)) @ B2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1253_bits__mod__by__1, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A]: (((((modulo_modulo @ A) @ A2) @ (one_one @ A)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1254_mod__by__1, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A2:A]: (((((modulo_modulo @ A) @ A2) @ (one_one @ A)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1255_dvd__div__mult__self, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ B2) => ((((times_times @ A) @ (((divide_divide @ A) @ B2) @ A2)) @ A2) = B2)))))))). % 6.41/2.69 thf(fact_1256_dvd__mult__div__cancel, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ B2) => ((((times_times @ A) @ A2) @ (((divide_divide @ A) @ B2) @ A2)) = B2)))))))). % 6.41/2.69 thf(fact_1257_unit__div, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((dvd_dvd @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_1258_unit__div__1__unit, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => (((dvd_dvd @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_1259_unit__div__1__div__1, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => ((((divide_divide @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) = A2)))))))). % 6.41/2.69 thf(fact_1260_div__add, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,A2:A,B2:A]: (((((dvd_dvd @ A) @ C2) @ A2) => ((((dvd_dvd @ A) @ C2) @ B2) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ C2)) @ (((divide_divide @ A) @ B2) @ C2))))))))))). % 6.41/2.69 thf(fact_1261_bits__mod__div__trivial, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A,B2:A]: (((((divide_divide @ A) @ (((modulo_modulo @ A) @ A2) @ B2)) @ B2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1262_mod__div__trivial, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A2:A,B2:A]: (((((divide_divide @ A) @ (((modulo_modulo @ A) @ A2) @ B2)) @ B2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1263_mod__mult__self4, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,C2:A,A2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ C2)) @ A2)) @ B2) = (((modulo_modulo @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1264_mod__mult__self3, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,B2:A,A2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ C2) @ B2)) @ A2)) @ B2) = (((modulo_modulo @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1265_mod__mult__self2, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A2:A,B2:A,C2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))) @ B2) = (((modulo_modulo @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1266_mod__mult__self1, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A2:A,C2:A,B2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ A2) @ (((times_times @ A) @ C2) @ B2))) @ B2) = (((modulo_modulo @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1267_dvd__imp__mod__0, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ B2) => ((((modulo_modulo @ A) @ B2) @ A2) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1268_mod__by__Suc__0, axiom, (! [M:nat]: (((((modulo_modulo @ nat) @ M) @ (suc @ (zero_zero @ nat))) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_1269_set__decode__zero, axiom, ((nat_set_decode @ (zero_zero @ nat)) = (bot_bot @ (set @ nat)))). % 6.41/2.69 thf(fact_1270_set__encode__inverse, axiom, (! [A3:(set @ nat)]: ((((finite_finite2 @ nat) @ A3) => ((nat_set_decode @ (nat_set_encode @ A3)) = A3))))). % 6.41/2.69 thf(fact_1271_unit__div__mult__self, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => ((((times_times @ A) @ (((divide_divide @ A) @ B2) @ A2)) @ A2) = B2)))))))). % 6.41/2.69 thf(fact_1272_unit__mult__div__div, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => ((((times_times @ A) @ B2) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) = (((divide_divide @ A) @ B2) @ A2))))))))). % 6.41/2.69 thf(fact_1273_even__Suc, axiom, (! [N2:nat]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ N2)) = (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))))))). % 6.41/2.69 thf(fact_1274_even__Suc__Suc__iff, axiom, (! [N2:nat]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ (suc @ N2))) = (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))))). % 6.41/2.69 thf(fact_1275_pow__divides__pow__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [N2:nat,A2:A,B2:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((dvd_dvd @ A) @ (((power_power @ A) @ A2) @ N2)) @ (((power_power @ A) @ B2) @ N2)) = (((dvd_dvd @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1276_Suc__mod__mult__self4, axiom, (! [N2:nat,K:nat,M:nat]: (((((modulo_modulo @ nat) @ (suc @ (((plus_plus @ nat) @ (((times_times @ nat) @ N2) @ K)) @ M))) @ N2) = (((modulo_modulo @ nat) @ (suc @ M)) @ N2))))). % 6.41/2.69 thf(fact_1277_Suc__mod__mult__self3, axiom, (! [K:nat,N2:nat,M:nat]: (((((modulo_modulo @ nat) @ (suc @ (((plus_plus @ nat) @ (((times_times @ nat) @ K) @ N2)) @ M))) @ N2) = (((modulo_modulo @ nat) @ (suc @ M)) @ N2))))). % 6.41/2.69 thf(fact_1278_Suc__mod__mult__self2, axiom, (! [M:nat,N2:nat,K:nat]: (((((modulo_modulo @ nat) @ (suc @ (((plus_plus @ nat) @ M) @ (((times_times @ nat) @ N2) @ K)))) @ N2) = (((modulo_modulo @ nat) @ (suc @ M)) @ N2))))). % 6.41/2.69 thf(fact_1279_Suc__mod__mult__self1, axiom, (! [M:nat,K:nat,N2:nat]: (((((modulo_modulo @ nat) @ (suc @ (((plus_plus @ nat) @ M) @ (((times_times @ nat) @ K) @ N2)))) @ N2) = (((modulo_modulo @ nat) @ (suc @ M)) @ N2))))). % 6.41/2.69 thf(fact_1280_even__add, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ A) @ A2) @ B2)) = ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2) = (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))))))))). % 6.41/2.69 thf(fact_1281_odd__add, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A2:A,B2:A]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ A) @ A2) @ B2)))) = ((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2))) != (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))))))))))). % 6.41/2.69 thf(fact_1282_odd__Suc__div__two, axiom, (! [N2:nat]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) => ((((divide_divide @ nat) @ (suc @ N2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (suc @ (((divide_divide @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))). % 6.41/2.69 thf(fact_1283_even__Suc__div__two, axiom, (! [N2:nat]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2) => ((((divide_divide @ nat) @ (suc @ N2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((divide_divide @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))). % 6.41/2.69 thf(fact_1284_mod2__Suc__Suc, axiom, (! [M:nat]: (((((modulo_modulo @ nat) @ (suc @ (suc @ M))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((modulo_modulo @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))). % 6.41/2.69 thf(fact_1285_Suc__times__numeral__mod__eq, axiom, (! [K:num,N2:nat]: (((((numeral_numeral @ nat) @ K) != (one_one @ nat)) => ((((modulo_modulo @ nat) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ K)) @ N2))) @ ((numeral_numeral @ nat) @ K)) = (one_one @ nat)))))). % 6.41/2.69 thf(fact_1286_set__decode__0, axiom, (! [X3:nat]: (((((member @ nat) @ (zero_zero @ nat)) @ (nat_set_decode @ X3)) = (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ X3))))))). % 6.41/2.69 thf(fact_1287_zero__le__power__eq__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,W2:num]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ ((numeral_numeral @ nat) @ W2))) = ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ W2)) | ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ W2)))) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))))). % 6.41/2.69 thf(fact_1288_power__less__zero__eq__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,W2:num]: (((((ord_less @ A) @ (((power_power @ A) @ A2) @ ((numeral_numeral @ nat) @ W2))) @ (zero_zero @ A)) = ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ W2)))) & (((ord_less @ A) @ A2) @ (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_1289_power__less__zero__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,N2:nat]: (((((ord_less @ A) @ (((power_power @ A) @ A2) @ N2)) @ (zero_zero @ A)) = ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) & (((ord_less @ A) @ A2) @ (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_1290_even__plus__one__iff, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A2:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ A) @ A2) @ (one_one @ A))) = (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2)))))))))). % 6.41/2.69 thf(fact_1291_not__mod__2__eq__1__eq__0, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A2:A]: ((((((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))) != (one_one @ A)) = ((((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1292_not__mod__2__eq__0__eq__1, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A2:A]: ((((((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))) != (zero_zero @ A)) = ((((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (one_one @ A))))))))). % 6.41/2.69 thf(fact_1293_not__mod2__eq__Suc__0__eq__0, axiom, (! [N2:nat]: ((((((modulo_modulo @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) != (suc @ (zero_zero @ nat))) = ((((modulo_modulo @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_1294_add__self__mod__2, axiom, (! [M:nat]: (((((modulo_modulo @ nat) @ (((plus_plus @ nat) @ M) @ M)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_1295_zero__less__power__eq__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,W2:num]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ ((numeral_numeral @ nat) @ W2))) = ((((numeral_numeral @ nat) @ W2) = (zero_zero @ nat)) | (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ W2)) & (A2 != (zero_zero @ A))) | ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ W2)))) & (((ord_less @ A) @ (zero_zero @ A)) @ A2))))))))))). % 6.41/2.69 thf(fact_1296_even__succ__div__two, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A2:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ (one_one @ A))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_1297_odd__succ__div__two, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A2:A]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2))) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ (one_one @ A))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_1298_even__succ__div__2, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2) => ((((divide_divide @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ A2)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_1299_even__power, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A2:A,N2:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((power_power @ A) @ A2) @ N2)) = ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))))))). % 6.41/2.69 thf(fact_1300_mod2__gr__0, axiom, (! [M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((modulo_modulo @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = ((((modulo_modulo @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (one_one @ nat)))))). % 6.41/2.69 thf(fact_1301_odd__two__times__div__two__succ, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A2:A]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2))) => ((((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ (one_one @ A)) = A2)))))))). % 6.41/2.69 thf(fact_1302_power__le__zero__eq__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,W2:num]: (((((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ ((numeral_numeral @ nat) @ W2))) @ (zero_zero @ A)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ ((numeral_numeral @ nat) @ W2)) & (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ W2)))) & (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))) | ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ W2)) & (A2 = (zero_zero @ A)))))))))))). % 6.41/2.69 thf(fact_1303_even__succ__mod__exp, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A,N2:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((modulo_modulo @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ A2)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) = (((plus_plus @ A) @ (one_one @ A)) @ (((modulo_modulo @ A) @ A2) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)))))))))))). % 6.41/2.69 thf(fact_1304_real__arch__pow, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ (one_one @ real)) @ X3) => (? [N3:nat]: ((((ord_less @ real) @ Y) @ (((power_power @ real) @ X3) @ N3)))))))). % 6.41/2.69 thf(fact_1305_real__arch__pow__inv, axiom, (! [Y:real,X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ Y) => ((((ord_less @ real) @ X3) @ (one_one @ real)) => (? [N3:nat]: ((((ord_less @ real) @ (((power_power @ real) @ X3) @ N3)) @ Y)))))))). % 6.41/2.69 thf(fact_1306_int__distrib_I2_J, axiom, (! [W2:int,Z1:int,Z22:int]: (((((times_times @ int) @ W2) @ (((plus_plus @ int) @ Z1) @ Z22)) = (((plus_plus @ int) @ (((times_times @ int) @ W2) @ Z1)) @ (((times_times @ int) @ W2) @ Z22)))))). % 6.41/2.69 thf(fact_1307_int__distrib_I1_J, axiom, (! [Z1:int,Z22:int,W2:int]: (((((times_times @ int) @ (((plus_plus @ int) @ Z1) @ Z22)) @ W2) = (((plus_plus @ int) @ (((times_times @ int) @ Z1) @ W2)) @ (((times_times @ int) @ Z22) @ W2)))))). % 6.41/2.69 thf(fact_1308_zdvd__period, axiom, (! [A2:int,D2:int,X3:int,T2:int,C2:int]: (((((dvd_dvd @ int) @ A2) @ D2) => ((((dvd_dvd @ int) @ A2) @ (((plus_plus @ int) @ X3) @ T2)) = (((dvd_dvd @ int) @ A2) @ (((plus_plus @ int) @ (((plus_plus @ int) @ X3) @ (((times_times @ int) @ C2) @ D2))) @ T2))))))). % 6.41/2.69 thf(fact_1309_zdvd__reduce, axiom, (! [K:int,N2:int,M:int]: (((((dvd_dvd @ int) @ K) @ (((plus_plus @ int) @ N2) @ (((times_times @ int) @ K) @ M))) = (((dvd_dvd @ int) @ K) @ N2))))). % 6.41/2.69 thf(fact_1310_less__eq__real__def, axiom, ((ord_less_eq @ real) = (^ [X:real,Y2:real]: (((((ord_less @ real) @ X) @ Y2) | (X = Y2)))))). % 6.41/2.69 thf(fact_1311_mod__eq__0__iff__dvd, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A2:A,B2:A]: ((((((modulo_modulo @ A) @ A2) @ B2) = (zero_zero @ A)) = (((dvd_dvd @ A) @ B2) @ A2)))))))). % 6.41/2.69 thf(fact_1312_dvd__eq__mod__eq__0, axiom, (! [A:$tType]: (((semidom_modulo @ A) => ((dvd_dvd @ A) = (^ [A4:A,B4:A]: (((((modulo_modulo @ A) @ B4) @ A4) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1313_mod__0__imp__dvd, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A2:A,B2:A]: ((((((modulo_modulo @ A) @ A2) @ B2) = (zero_zero @ A)) => (((dvd_dvd @ A) @ B2) @ A2)))))))). % 6.41/2.69 thf(fact_1314_dvd__refl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A]: ((((dvd_dvd @ A) @ A2) @ A2))))))). % 6.41/2.69 thf(fact_1315_dvd__trans, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ B2) => ((((dvd_dvd @ A) @ B2) @ C2) => (((dvd_dvd @ A) @ A2) @ C2))))))))). % 6.41/2.69 thf(fact_1316_dvd__mod__iff, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [C2:A,B2:A,A2:A]: (((((dvd_dvd @ A) @ C2) @ B2) => ((((dvd_dvd @ A) @ C2) @ (((modulo_modulo @ A) @ A2) @ B2)) = (((dvd_dvd @ A) @ C2) @ A2))))))))). % 6.41/2.69 thf(fact_1317_dvd__mod__imp__dvd, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [C2:A,A2:A,B2:A]: (((((dvd_dvd @ A) @ C2) @ (((modulo_modulo @ A) @ A2) @ B2)) => ((((dvd_dvd @ A) @ C2) @ B2) => (((dvd_dvd @ A) @ C2) @ A2))))))))). % 6.41/2.69 thf(fact_1318_int__ge__induct, axiom, (! [K:int,I:int,P:(int > $o)]: (((((ord_less_eq @ int) @ K) @ I) => ((P @ K) => ((! [I4:int]: (((((ord_less_eq @ int) @ K) @ I4) => ((P @ I4) => (P @ (((plus_plus @ int) @ I4) @ (one_one @ int))))))) => (P @ I))))))). % 6.41/2.69 thf(fact_1319_unit__imp__mod__eq__0, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [B2:A,A2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((modulo_modulo @ A) @ A2) @ B2) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1320_mod__greater__zero__iff__not__dvd, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((modulo_modulo @ nat) @ M) @ N2)) = (~ ((((dvd_dvd @ nat) @ N2) @ M))))))). % 6.41/2.69 thf(fact_1321_mod__add__right__eq, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A2:A,B2:A,C2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ A2) @ (((modulo_modulo @ A) @ B2) @ C2))) @ C2) = (((modulo_modulo @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_1322_mod__add__left__eq, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A2:A,C2:A,B2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ (((modulo_modulo @ A) @ A2) @ C2)) @ B2)) @ C2) = (((modulo_modulo @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_1323_mod__add__cong, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A2:A,C2:A,A7:A,B2:A,B7:A]: ((((((modulo_modulo @ A) @ A2) @ C2) = (((modulo_modulo @ A) @ A7) @ C2)) => (((((modulo_modulo @ A) @ B2) @ C2) = (((modulo_modulo @ A) @ B7) @ C2)) => ((((modulo_modulo @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((modulo_modulo @ A) @ (((plus_plus @ A) @ A7) @ B7)) @ C2)))))))))). % 6.41/2.69 thf(fact_1324_mod__add__eq, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A2:A,C2:A,B2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ (((modulo_modulo @ A) @ A2) @ C2)) @ (((modulo_modulo @ A) @ B2) @ C2))) @ C2) = (((modulo_modulo @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_1325_dvd__0__left, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A2:A]: (((((dvd_dvd @ A) @ (zero_zero @ A)) @ A2) => (A2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1326_dvd__field__iff, axiom, (! [A:$tType]: (((field @ A) => ((dvd_dvd @ A) = (^ [A4:A,B4:A]: (((A4 = (zero_zero @ A)) => (B4 = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_1327_mod__Suc__Suc__eq, axiom, (! [M:nat,N2:nat]: (((((modulo_modulo @ nat) @ (suc @ (suc @ (((modulo_modulo @ nat) @ M) @ N2)))) @ N2) = (((modulo_modulo @ nat) @ (suc @ (suc @ M))) @ N2))))). % 6.41/2.69 thf(fact_1328_mod__Suc__eq, axiom, (! [M:nat,N2:nat]: (((((modulo_modulo @ nat) @ (suc @ (((modulo_modulo @ nat) @ M) @ N2))) @ N2) = (((modulo_modulo @ nat) @ (suc @ M)) @ N2))))). % 6.41/2.69 thf(fact_1329_dvd__productE, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [P4:A,A2:A,B2:A]: (((((dvd_dvd @ A) @ P4) @ (((times_times @ A) @ A2) @ B2)) => (~ ((! [X5:A,Y4:A]: (((P4 = (((times_times @ A) @ X5) @ Y4)) => ((((dvd_dvd @ A) @ X5) @ A2) => (~ ((((dvd_dvd @ A) @ Y4) @ B2)))))))))))))))). % 6.41/2.69 thf(fact_1330_division__decomp, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ (((times_times @ A) @ B2) @ C2)) => (? [B9:A,C6:A]: (((A2 = (((times_times @ A) @ B9) @ C6)) & ((((dvd_dvd @ A) @ B9) @ B2) & (((dvd_dvd @ A) @ C6) @ C2)))))))))))). % 6.41/2.69 thf(fact_1331_dvdE, axiom, (! [A:$tType]: (((dvd @ A) => (! [B2:A,A2:A]: (((((dvd_dvd @ A) @ B2) @ A2) => (~ ((! [K2:A]: ((A2 != (((times_times @ A) @ B2) @ K2))))))))))))). % 6.41/2.69 thf(fact_1332_dvdI, axiom, (! [A:$tType]: (((dvd @ A) => (! [A2:A,B2:A,K:A]: (((A2 = (((times_times @ A) @ B2) @ K)) => (((dvd_dvd @ A) @ B2) @ A2)))))))). % 6.41/2.69 thf(fact_1333_dvd__def, axiom, (! [A:$tType]: (((dvd @ A) => ((dvd_dvd @ A) = (^ [B4:A,A4:A]: ((? [K3:A]: ((A4 = (((times_times @ A) @ B4) @ K3))))))))))). % 6.41/2.69 thf(fact_1334_dvd__mult, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A,C2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ C2) => (((dvd_dvd @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_1335_dvd__mult2, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ B2) => (((dvd_dvd @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_1336_dvd__mult__left, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2) => (((dvd_dvd @ A) @ A2) @ C2)))))))). % 6.41/2.69 thf(fact_1337_dvd__triv__left, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A,B2:A]: ((((dvd_dvd @ A) @ A2) @ (((times_times @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1338_mult__dvd__mono, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((dvd_dvd @ A) @ A2) @ B2) => ((((dvd_dvd @ A) @ C2) @ D2) => (((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ D2)))))))))). % 6.41/2.69 thf(fact_1339_dvd__mult__right, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2) => (((dvd_dvd @ A) @ B2) @ C2)))))))). % 6.41/2.69 thf(fact_1340_dvd__triv__right, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A,B2:A]: ((((dvd_dvd @ A) @ A2) @ (((times_times @ A) @ B2) @ A2)))))))). % 6.41/2.69 thf(fact_1341_one__dvd, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A]: ((((dvd_dvd @ A) @ (one_one @ A)) @ A2))))))). % 6.41/2.69 thf(fact_1342_unit__imp__dvd, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((dvd_dvd @ A) @ B2) @ A2)))))))). % 6.41/2.69 thf(fact_1343_dvd__unit__imp__unit, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ B2) => ((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((dvd_dvd @ A) @ A2) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_1344_dvd__add__right__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ B2) => ((((dvd_dvd @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2)) = (((dvd_dvd @ A) @ A2) @ C2))))))))). % 6.41/2.69 thf(fact_1345_dvd__add__left__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A2:A,C2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ C2) => ((((dvd_dvd @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2)) = (((dvd_dvd @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1346_dvd__add, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ B2) => ((((dvd_dvd @ A) @ A2) @ C2) => (((dvd_dvd @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2)))))))))). % 6.41/2.69 thf(fact_1347_dvd__div__eq__iff, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [C2:A,A2:A,B2:A]: (((((dvd_dvd @ A) @ C2) @ A2) => ((((dvd_dvd @ A) @ C2) @ B2) => (((((divide_divide @ A) @ A2) @ C2) = (((divide_divide @ A) @ B2) @ C2)) = (A2 = B2)))))))))). % 6.41/2.69 thf(fact_1348_dvd__div__eq__cancel, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A2:A,C2:A,B2:A]: ((((((divide_divide @ A) @ A2) @ C2) = (((divide_divide @ A) @ B2) @ C2)) => ((((dvd_dvd @ A) @ C2) @ A2) => ((((dvd_dvd @ A) @ C2) @ B2) => (A2 = B2)))))))))). % 6.41/2.69 thf(fact_1349_div__div__div__same, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [D2:A,B2:A,A2:A]: (((((dvd_dvd @ A) @ D2) @ B2) => ((((dvd_dvd @ A) @ B2) @ A2) => ((((divide_divide @ A) @ (((divide_divide @ A) @ A2) @ D2)) @ (((divide_divide @ A) @ B2) @ D2)) = (((divide_divide @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_1350_mod__less__eq__dividend, axiom, (! [M:nat,N2:nat]: ((((ord_less_eq @ nat) @ (((modulo_modulo @ nat) @ M) @ N2)) @ M)))). % 6.41/2.69 thf(fact_1351_gcd__nat_Oextremum__uniqueI, axiom, (! [A2:nat]: (((((dvd_dvd @ nat) @ (zero_zero @ nat)) @ A2) => (A2 = (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_1352_gcd__nat_Onot__eq__extremum, axiom, (! [A2:nat]: (((A2 != (zero_zero @ nat)) = ((((dvd_dvd @ nat) @ A2) @ (zero_zero @ nat)) & (A2 != (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_1353_gcd__nat_Oextremum__unique, axiom, (! [A2:nat]: (((((dvd_dvd @ nat) @ (zero_zero @ nat)) @ A2) = (A2 = (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_1354_gcd__nat_Oextremum__strict, axiom, (! [A2:nat]: ((~ (((((dvd_dvd @ nat) @ (zero_zero @ nat)) @ A2) & ((zero_zero @ nat) != A2))))))). % 6.41/2.69 thf(fact_1355_gcd__nat_Oextremum, axiom, (! [A2:nat]: ((((dvd_dvd @ nat) @ A2) @ (zero_zero @ nat))))). % 6.41/2.69 thf(fact_1356_zdvd__antisym__nonneg, axiom, (! [M:int,N2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ M) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N2) => ((((dvd_dvd @ int) @ M) @ N2) => ((((dvd_dvd @ int) @ N2) @ M) => (M = N2)))))))). % 6.41/2.69 thf(fact_1357_zdvd__mult__cancel, axiom, (! [K:int,M:int,N2:int]: (((((dvd_dvd @ int) @ (((times_times @ int) @ K) @ M)) @ (((times_times @ int) @ K) @ N2)) => ((K != (zero_zero @ int)) => (((dvd_dvd @ int) @ M) @ N2)))))). % 6.41/2.69 thf(fact_1358_even__iff__mod__2__eq__zero, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A2:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2) = ((((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1359_subset__decode__imp__le, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ (set @ nat)) @ (nat_set_decode @ M)) @ (nat_set_decode @ N2)) => (((ord_less_eq @ nat) @ M) @ N2))))). % 6.41/2.69 thf(fact_1360_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ (((modulo_modulo @ A) @ A2) @ B2)) @ A2)))))))). % 6.41/2.69 thf(fact_1361_unique__euclidean__semiring__numeral__class_Opos__mod__bound, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (((modulo_modulo @ A) @ A2) @ B2)) @ B2)))))))). % 6.41/2.69 thf(fact_1362_mod__eq__self__iff__div__eq__0, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [A2:A,B2:A]: ((((((modulo_modulo @ A) @ A2) @ B2) = A2) = ((((divide_divide @ A) @ A2) @ B2) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1363_mod__eqE, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A2:A,C2:A,B2:A]: ((((((modulo_modulo @ A) @ A2) @ C2) = (((modulo_modulo @ A) @ B2) @ C2)) => (~ ((! [D3:A]: ((B2 != (((plus_plus @ A) @ A2) @ (((times_times @ A) @ C2) @ D3)))))))))))))). % 6.41/2.69 thf(fact_1364_div__add1__eq, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [A2:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ (((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ C2)) @ (((divide_divide @ A) @ B2) @ C2))) @ (((divide_divide @ A) @ (((plus_plus @ A) @ (((modulo_modulo @ A) @ A2) @ C2)) @ (((modulo_modulo @ A) @ B2) @ C2))) @ C2))))))))). % 6.41/2.69 thf(fact_1365_not__is__unit__0, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (~ ((((dvd_dvd @ A) @ (zero_zero @ A)) @ (one_one @ A)))))))). % 6.41/2.69 thf(fact_1366_minf_I10_J, axiom, (! [B:$tType]: ((((plus @ B) & ((linorder @ B) & (dvd @ B))) => (! [D2:B,S:B]: ((? [Z3:B]: ((! [X2:B]: (((((ord_less @ B) @ X2) @ Z3) => ((~ ((((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X2) @ S)))) = (~ ((((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X2) @ S)))))))))))))))). % 6.41/2.69 thf(fact_1367_minf_I9_J, axiom, (! [B:$tType]: ((((plus @ B) & ((linorder @ B) & (dvd @ B))) => (! [D2:B,S:B]: ((? [Z3:B]: ((! [X2:B]: (((((ord_less @ B) @ X2) @ Z3) => ((((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X2) @ S)) = (((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X2) @ S)))))))))))))). % 6.41/2.69 thf(fact_1368_pinf_I10_J, axiom, (! [B:$tType]: ((((plus @ B) & ((linorder @ B) & (dvd @ B))) => (! [D2:B,S:B]: ((? [Z3:B]: ((! [X2:B]: (((((ord_less @ B) @ Z3) @ X2) => ((~ ((((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X2) @ S)))) = (~ ((((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X2) @ S)))))))))))))))). % 6.41/2.69 thf(fact_1369_pinf_I9_J, axiom, (! [B:$tType]: ((((plus @ B) & ((linorder @ B) & (dvd @ B))) => (! [D2:B,S:B]: ((? [Z3:B]: ((! [X2:B]: (((((ord_less @ B) @ Z3) @ X2) => ((((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X2) @ S)) = (((dvd_dvd @ B) @ D2) @ (((plus_plus @ B) @ X2) @ S)))))))))))))). % 6.41/2.69 thf(fact_1370_dvd__div__eq__0__iff, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [B2:A,A2:A]: (((((dvd_dvd @ A) @ B2) @ A2) => (((((divide_divide @ A) @ A2) @ B2) = (zero_zero @ A)) = (A2 = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_1371_unit__mult__right__cancel, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => (((((times_times @ A) @ B2) @ A2) = (((times_times @ A) @ C2) @ A2)) = (B2 = C2))))))))). % 6.41/2.69 thf(fact_1372_unit__mult__left__cancel, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => (((((times_times @ A) @ A2) @ B2) = (((times_times @ A) @ A2) @ C2)) = (B2 = C2))))))))). % 6.41/2.69 thf(fact_1373_mult__unit__dvd__iff_H, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2) = (((dvd_dvd @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_1374_dvd__mult__unit__iff_H, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A2:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ A2) @ (((times_times @ A) @ B2) @ C2)) = (((dvd_dvd @ A) @ A2) @ C2))))))))). % 6.41/2.69 thf(fact_1375_mult__unit__dvd__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A2:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2) = (((dvd_dvd @ A) @ A2) @ C2))))))))). % 6.41/2.69 thf(fact_1376_dvd__mult__unit__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A2:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ A2) @ (((times_times @ A) @ C2) @ B2)) = (((dvd_dvd @ A) @ A2) @ C2))))))))). % 6.41/2.69 thf(fact_1377_is__unit__mult__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ B2)) @ (one_one @ A)) = ((((dvd_dvd @ A) @ A2) @ (one_one @ A)) & (((dvd_dvd @ A) @ B2) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_1378_mod__Suc, axiom, (! [M:nat,N2:nat]: (((((suc @ (((modulo_modulo @ nat) @ M) @ N2)) = N2) => ((((modulo_modulo @ nat) @ (suc @ M)) @ N2) = (zero_zero @ nat))) & (((suc @ (((modulo_modulo @ nat) @ M) @ N2)) != N2) => ((((modulo_modulo @ nat) @ (suc @ M)) @ N2) = (suc @ (((modulo_modulo @ nat) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_1379_finite__set__decode, axiom, (! [N2:nat]: (((finite_finite2 @ nat) @ (nat_set_decode @ N2))))). % 6.41/2.69 thf(fact_1380_mod__induct, axiom, (! [P:(nat > $o),N2:nat,P4:nat,M:nat]: (((P @ N2) => ((((ord_less @ nat) @ N2) @ P4) => ((((ord_less @ nat) @ M) @ P4) => ((! [N3:nat]: (((((ord_less @ nat) @ N3) @ P4) => ((P @ N3) => (P @ (((modulo_modulo @ nat) @ (suc @ N3)) @ P4)))))) => (P @ M)))))))). % 6.41/2.69 thf(fact_1381_dvd__div__mult, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,B2:A,A2:A]: (((((dvd_dvd @ A) @ C2) @ B2) => ((((times_times @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ A2) = (((divide_divide @ A) @ (((times_times @ A) @ B2) @ A2)) @ C2))))))))). % 6.41/2.69 thf(fact_1382_div__mult__swap, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,B2:A,A2:A]: (((((dvd_dvd @ A) @ C2) @ B2) => ((((times_times @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2))))))))). % 6.41/2.69 thf(fact_1383_div__div__eq__right, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,B2:A,A2:A]: (((((dvd_dvd @ A) @ C2) @ B2) => ((((dvd_dvd @ A) @ B2) @ A2) => ((((divide_divide @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2)) = (((times_times @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ C2)))))))))). % 6.41/2.69 thf(fact_1384_dvd__div__mult2__eq, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,C2:A,A2:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ B2) @ C2)) @ A2) => ((((divide_divide @ A) @ A2) @ (((times_times @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ C2))))))))). % 6.41/2.69 thf(fact_1385_dvd__mult__imp__div, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,C2:A,B2:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ C2)) @ B2) => (((dvd_dvd @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_1386_div__mult__div__if__dvd, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A2:A,D2:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ A2) => ((((dvd_dvd @ A) @ D2) @ C2) => ((((times_times @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ (((divide_divide @ A) @ C2) @ D2)) = (((divide_divide @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ D2))))))))))). % 6.41/2.69 thf(fact_1387_gcd__nat__induct, axiom, (! [P:(nat > (nat > $o)),M:nat,N2:nat]: (((! [M3:nat]: (((P @ M3) @ (zero_zero @ nat)))) => ((! [M3:nat,N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) => (((P @ N3) @ (((modulo_modulo @ nat) @ M3) @ N3)) => ((P @ M3) @ N3))))) => ((P @ M) @ N2)))))). % 6.41/2.69 thf(fact_1388_mod__less__divisor, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((ord_less @ nat) @ (((modulo_modulo @ nat) @ M) @ N2)) @ N2))))). % 6.41/2.69 thf(fact_1389_dvd__div__unit__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A2:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ A2) @ (((divide_divide @ A) @ C2) @ B2)) = (((dvd_dvd @ A) @ A2) @ C2))))))))). % 6.41/2.69 thf(fact_1390_div__unit__dvd__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A2:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ C2) = (((dvd_dvd @ A) @ A2) @ C2))))))))). % 6.41/2.69 thf(fact_1391_unit__div__cancel, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => (((((divide_divide @ A) @ B2) @ A2) = (((divide_divide @ A) @ C2) @ A2)) = (B2 = C2))))))))). % 6.41/2.69 thf(fact_1392_div__plus__div__distrib__dvd__right, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,B2:A,A2:A]: (((((dvd_dvd @ A) @ C2) @ B2) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ C2)) @ (((divide_divide @ A) @ B2) @ C2)))))))))). % 6.41/2.69 thf(fact_1393_div__plus__div__distrib__dvd__left, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,A2:A,B2:A]: (((((dvd_dvd @ A) @ C2) @ A2) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ C2)) @ (((divide_divide @ A) @ B2) @ C2)))))))))). % 6.41/2.69 thf(fact_1394_mod__Suc__le__divisor, axiom, (! [M:nat,N2:nat]: ((((ord_less_eq @ nat) @ (((modulo_modulo @ nat) @ M) @ (suc @ N2))) @ N2)))). % 6.41/2.69 thf(fact_1395_mod__eq__0D, axiom, (! [M:nat,D2:nat]: ((((((modulo_modulo @ nat) @ M) @ D2) = (zero_zero @ nat)) => (? [Q2:nat]: ((M = (((times_times @ nat) @ D2) @ Q2)))))))). % 6.41/2.69 thf(fact_1396_le__imp__power__dvd, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [M:nat,N2:nat,A2:A]: (((((ord_less_eq @ nat) @ M) @ N2) => (((dvd_dvd @ A) @ (((power_power @ A) @ A2) @ M)) @ (((power_power @ A) @ A2) @ N2))))))))). % 6.41/2.69 thf(fact_1397_power__le__dvd, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A2:A,N2:nat,B2:A,M:nat]: (((((dvd_dvd @ A) @ (((power_power @ A) @ A2) @ N2)) @ B2) => ((((ord_less_eq @ nat) @ M) @ N2) => (((dvd_dvd @ A) @ (((power_power @ A) @ A2) @ M)) @ B2))))))))). % 6.41/2.69 thf(fact_1398_dvd__power__le, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [X3:A,Y:A,N2:nat,M:nat]: (((((dvd_dvd @ A) @ X3) @ Y) => ((((ord_less_eq @ nat) @ N2) @ M) => (((dvd_dvd @ A) @ (((power_power @ A) @ X3) @ N2)) @ (((power_power @ A) @ Y) @ M)))))))))). % 6.41/2.69 thf(fact_1399_nat__mod__eq__iff, axiom, (! [X3:nat,N2:nat,Y:nat]: ((((((modulo_modulo @ nat) @ X3) @ N2) = (((modulo_modulo @ nat) @ Y) @ N2)) = (? [Q1:nat,Q22:nat]: (((((plus_plus @ nat) @ X3) @ (((times_times @ nat) @ N2) @ Q1)) = (((plus_plus @ nat) @ Y) @ (((times_times @ nat) @ N2) @ Q22))))))))). % 6.41/2.69 thf(fact_1400_nat__dvd__not__less, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less @ nat) @ M) @ N2) => (~ ((((dvd_dvd @ nat) @ N2) @ M)))))))). % 6.41/2.69 thf(fact_1401_dvd__pos__nat, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((dvd_dvd @ nat) @ M) @ N2) => (((ord_less @ nat) @ (zero_zero @ nat)) @ M)))))). % 6.41/2.69 thf(fact_1402_zdvd__imp__le, axiom, (! [Z2:int,N2:int]: (((((dvd_dvd @ int) @ Z2) @ N2) => ((((ord_less @ int) @ (zero_zero @ int)) @ N2) => (((ord_less_eq @ int) @ Z2) @ N2)))))). % 6.41/2.69 thf(fact_1403_zdvd__not__zless, axiom, (! [M:int,N2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ M) => ((((ord_less @ int) @ M) @ N2) => (~ ((((dvd_dvd @ int) @ N2) @ M)))))))). % 6.41/2.69 thf(fact_1404_bezout__add__nat, axiom, (! [A2:nat,B2:nat]: ((? [D3:nat,X5:nat,Y4:nat]: (((((dvd_dvd @ nat) @ D3) @ A2) & ((((dvd_dvd @ nat) @ D3) @ B2) & (((((times_times @ nat) @ A2) @ X5) = (((plus_plus @ nat) @ (((times_times @ nat) @ B2) @ Y4)) @ D3)) | ((((times_times @ nat) @ B2) @ X5) = (((plus_plus @ nat) @ (((times_times @ nat) @ A2) @ Y4)) @ D3)))))))))). % 6.41/2.69 thf(fact_1405_bezout__lemma__nat, axiom, (! [D2:nat,A2:nat,B2:nat,X3:nat,Y:nat]: (((((dvd_dvd @ nat) @ D2) @ A2) => ((((dvd_dvd @ nat) @ D2) @ B2) => ((((((times_times @ nat) @ A2) @ X3) = (((plus_plus @ nat) @ (((times_times @ nat) @ B2) @ Y)) @ D2)) | ((((times_times @ nat) @ B2) @ X3) = (((plus_plus @ nat) @ (((times_times @ nat) @ A2) @ Y)) @ D2))) => (? [X5:nat,Y4:nat]: (((((dvd_dvd @ nat) @ D2) @ A2) & ((((dvd_dvd @ nat) @ D2) @ (((plus_plus @ nat) @ A2) @ B2)) & (((((times_times @ nat) @ A2) @ X5) = (((plus_plus @ nat) @ (((times_times @ nat) @ (((plus_plus @ nat) @ A2) @ B2)) @ Y4)) @ D2)) | ((((times_times @ nat) @ (((plus_plus @ nat) @ A2) @ B2)) @ X5) = (((plus_plus @ nat) @ (((times_times @ nat) @ A2) @ Y4)) @ D2))))))))))))). % 6.41/2.69 thf(fact_1406_parity__cases, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A2:A]: ((((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2) => ((((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))) != (zero_zero @ A))) => (~ (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2))) => ((((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))) != (one_one @ A)))))))))))). % 6.41/2.69 thf(fact_1407_mod2__eq__if, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A2:A]: ((((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2) => ((((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A))) & ((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2))) => ((((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (one_one @ A)))))))))). % 6.41/2.69 thf(fact_1408_unique__euclidean__semiring__numeral__class_Omod__less, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ A2) @ B2) => ((((modulo_modulo @ A) @ A2) @ B2) = A2))))))))). % 6.41/2.69 thf(fact_1409_unique__euclidean__semiring__numeral__class_Opos__mod__sign, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((modulo_modulo @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1410_cong__exp__iff__simps_I2_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N2:num,Q4:num]: ((((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ (bit0 @ N2))) @ ((numeral_numeral @ A) @ (bit0 @ Q4))) = (zero_zero @ A)) = ((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ N2)) @ ((numeral_numeral @ A) @ Q4)) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1411_cong__exp__iff__simps_I1_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N2:num]: (((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ N2)) @ ((numeral_numeral @ A) @ one2)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1412_mult__div__mod__eq, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [B2:A,A2:A]: (((((plus_plus @ A) @ (((times_times @ A) @ B2) @ (((divide_divide @ A) @ A2) @ B2))) @ (((modulo_modulo @ A) @ A2) @ B2)) = A2))))))). % 6.41/2.69 thf(fact_1413_mod__mult__div__eq, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ (((modulo_modulo @ A) @ A2) @ B2)) @ (((times_times @ A) @ B2) @ (((divide_divide @ A) @ A2) @ B2))) = A2))))))). % 6.41/2.69 thf(fact_1414_mod__div__mult__eq, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ (((modulo_modulo @ A) @ A2) @ B2)) @ (((times_times @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ B2)) = A2))))))). % 6.41/2.69 thf(fact_1415_div__mult__mod__eq, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ (((times_times @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ B2)) @ (((modulo_modulo @ A) @ A2) @ B2)) = A2))))))). % 6.41/2.69 thf(fact_1416_mod__div__decomp, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A2:A,B2:A]: ((A2 = (((plus_plus @ A) @ (((times_times @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ B2)) @ (((modulo_modulo @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1417_cancel__div__mod__rules_I1_J, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ B2)) @ (((modulo_modulo @ A) @ A2) @ B2))) @ C2) = (((plus_plus @ A) @ A2) @ C2)))))))). % 6.41/2.69 thf(fact_1418_cancel__div__mod__rules_I2_J, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [B2:A,A2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ (((divide_divide @ A) @ A2) @ B2))) @ (((modulo_modulo @ A) @ A2) @ B2))) @ C2) = (((plus_plus @ A) @ A2) @ C2)))))))). % 6.41/2.69 thf(fact_1419_div__mult1__eq, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [A2:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2))) @ (((divide_divide @ A) @ (((times_times @ A) @ A2) @ (((modulo_modulo @ A) @ B2) @ C2))) @ C2))))))))). % 6.41/2.69 thf(fact_1420_unit__dvdE, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => (~ (((A2 != (zero_zero @ A)) => (! [C3:A]: ((B2 != (((times_times @ A) @ A2) @ C3)))))))))))))). % 6.41/2.69 thf(fact_1421_unity__coeff__ex, axiom, (! [A:$tType]: ((((dvd @ A) & (semiring_0 @ A)) => (! [P:(A > $o),L:A]: (((? [X:A]: ((P @ (((times_times @ A) @ L) @ X)))) = (? [X:A]: (((((dvd_dvd @ A) @ L) @ (((plus_plus @ A) @ X) @ (zero_zero @ A))) & (P @ X))))))))))). % 6.41/2.69 thf(fact_1422_dvd__div__eq__mult, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A,C2:A]: (((A2 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ A2) @ B2) => (((((divide_divide @ A) @ B2) @ A2) = C2) = (B2 = (((times_times @ A) @ C2) @ A2))))))))))). % 6.41/2.69 thf(fact_1423_div__dvd__iff__mult, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A2:A,C2:A]: (((B2 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ B2) @ A2) => ((((dvd_dvd @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ C2) = (((dvd_dvd @ A) @ A2) @ (((times_times @ A) @ C2) @ B2))))))))))). % 6.41/2.69 thf(fact_1424_dvd__div__iff__mult, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,B2:A,A2:A]: (((C2 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ C2) @ B2) => ((((dvd_dvd @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2)) = (((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ C2)) @ B2)))))))))). % 6.41/2.69 thf(fact_1425_dvd__div__div__eq__mult, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,C2:A,B2:A,D2:A]: (((A2 != (zero_zero @ A)) => ((C2 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ A2) @ B2) => ((((dvd_dvd @ A) @ C2) @ D2) => (((((divide_divide @ A) @ B2) @ A2) = (((divide_divide @ A) @ D2) @ C2)) = ((((times_times @ A) @ B2) @ C2) = (((times_times @ A) @ A2) @ D2))))))))))))). % 6.41/2.69 thf(fact_1426_unit__div__eq__0__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((((divide_divide @ A) @ A2) @ B2) = (zero_zero @ A)) = (A2 = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_1427_is__unit__div__mult2__eq, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,C2:A,A2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ C2) @ (one_one @ A)) => ((((divide_divide @ A) @ A2) @ (((times_times @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ C2)))))))))). % 6.41/2.69 thf(fact_1428_unit__div__mult__swap, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,A2:A,B2:A]: (((((dvd_dvd @ A) @ C2) @ (one_one @ A)) => ((((times_times @ A) @ A2) @ (((divide_divide @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2))))))))). % 6.41/2.69 thf(fact_1429_unit__div__commute, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A2:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((times_times @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ C2) = (((divide_divide @ A) @ (((times_times @ A) @ A2) @ C2)) @ B2))))))))). % 6.41/2.69 thf(fact_1430_div__mult__unit2, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,B2:A,A2:A]: (((((dvd_dvd @ A) @ C2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ B2) @ A2) => ((((divide_divide @ A) @ A2) @ (((times_times @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ C2)))))))))). % 6.41/2.69 thf(fact_1431_unit__eq__div2, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A2:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((A2 = (((divide_divide @ A) @ C2) @ B2)) = ((((times_times @ A) @ A2) @ B2) = C2))))))))). % 6.41/2.69 thf(fact_1432_unit__eq__div1, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A2:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((((divide_divide @ A) @ A2) @ B2) = C2) = (A2 = (((times_times @ A) @ C2) @ B2)))))))))). % 6.41/2.69 thf(fact_1433_mod__le__divisor, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((ord_less_eq @ nat) @ (((modulo_modulo @ nat) @ M) @ N2)) @ N2))))). % 6.41/2.69 thf(fact_1434_is__unit__power__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,N2:nat]: (((((dvd_dvd @ A) @ (((power_power @ A) @ A2) @ N2)) @ (one_one @ A)) = ((((dvd_dvd @ A) @ A2) @ (one_one @ A)) | (N2 = (zero_zero @ nat)))))))))). % 6.41/2.69 thf(fact_1435_div__less__mono, axiom, (! [A3:nat,B3:nat,N2:nat]: (((((ord_less @ nat) @ A3) @ B3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((((modulo_modulo @ nat) @ A3) @ N2) = (zero_zero @ nat)) => (((((modulo_modulo @ nat) @ B3) @ N2) = (zero_zero @ nat)) => (((ord_less @ nat) @ (((divide_divide @ nat) @ A3) @ N2)) @ (((divide_divide @ nat) @ B3) @ N2))))))))). % 6.41/2.69 thf(fact_1436_mod__eq__nat1E, axiom, (! [M:nat,Q4:nat,N2:nat]: ((((((modulo_modulo @ nat) @ M) @ Q4) = (((modulo_modulo @ nat) @ N2) @ Q4)) => ((((ord_less_eq @ nat) @ N2) @ M) => (~ ((! [S2:nat]: ((M != (((plus_plus @ nat) @ N2) @ (((times_times @ nat) @ Q4) @ S2)))))))))))). % 6.41/2.69 thf(fact_1437_mod__eq__nat2E, axiom, (! [M:nat,Q4:nat,N2:nat]: ((((((modulo_modulo @ nat) @ M) @ Q4) = (((modulo_modulo @ nat) @ N2) @ Q4)) => ((((ord_less_eq @ nat) @ M) @ N2) => (~ ((! [S2:nat]: ((N2 != (((plus_plus @ nat) @ M) @ (((times_times @ nat) @ Q4) @ S2)))))))))))). % 6.41/2.69 thf(fact_1438_nat__mod__eq__lemma, axiom, (! [X3:nat,N2:nat,Y:nat]: ((((((modulo_modulo @ nat) @ X3) @ N2) = (((modulo_modulo @ nat) @ Y) @ N2)) => ((((ord_less_eq @ nat) @ Y) @ X3) => (? [Q2:nat]: ((X3 = (((plus_plus @ nat) @ Y) @ (((times_times @ nat) @ N2) @ Q2)))))))))). % 6.41/2.69 thf(fact_1439_dvd__imp__le, axiom, (! [K:nat,N2:nat]: (((((dvd_dvd @ nat) @ K) @ N2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((ord_less_eq @ nat) @ K) @ N2)))))). % 6.41/2.69 thf(fact_1440_div__mod__decomp, axiom, (! [A3:nat,N2:nat]: ((A3 = (((plus_plus @ nat) @ (((times_times @ nat) @ (((divide_divide @ nat) @ A3) @ N2)) @ N2)) @ (((modulo_modulo @ nat) @ A3) @ N2)))))). % 6.41/2.69 thf(fact_1441_mod__mult2__eq, axiom, (! [M:nat,N2:nat,Q4:nat]: (((((modulo_modulo @ nat) @ M) @ (((times_times @ nat) @ N2) @ Q4)) = (((plus_plus @ nat) @ (((times_times @ nat) @ N2) @ (((modulo_modulo @ nat) @ (((divide_divide @ nat) @ M) @ N2)) @ Q4))) @ (((modulo_modulo @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_1442_dvd__mult__cancel, axiom, (! [K:nat,M:nat,N2:nat]: (((((dvd_dvd @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((dvd_dvd @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_1443_nat__mult__dvd__cancel1, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((dvd_dvd @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)) = (((dvd_dvd @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_1444_bezout__add__strong__nat, axiom, (! [A2:nat,B2:nat]: (((A2 != (zero_zero @ nat)) => (? [D3:nat,X5:nat,Y4:nat]: (((((dvd_dvd @ nat) @ D3) @ A2) & ((((dvd_dvd @ nat) @ D3) @ B2) & ((((times_times @ nat) @ A2) @ X5) = (((plus_plus @ nat) @ (((times_times @ nat) @ B2) @ Y4)) @ D3)))))))))). % 6.41/2.69 thf(fact_1445_even__zero, axiom, (! [A:$tType]: (((semiring_parity @ A) => (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (zero_zero @ A)))))). % 6.41/2.69 thf(fact_1446_is__unitE, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => (~ (((A2 != (zero_zero @ A)) => (! [B5:A]: (((B5 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ B5) @ (one_one @ A)) => (((((divide_divide @ A) @ (one_one @ A)) @ A2) = B5) => (((((divide_divide @ A) @ (one_one @ A)) @ B5) = A2) => (((((times_times @ A) @ A2) @ B5) = (one_one @ A)) => ((((divide_divide @ A) @ C2) @ A2) != (((times_times @ A) @ C2) @ B5))))))))))))))))))). % 6.41/2.69 thf(fact_1447_is__unit__div__mult__cancel__left, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A]: (((A2 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((divide_divide @ A) @ A2) @ (((times_times @ A) @ A2) @ B2)) = (((divide_divide @ A) @ (one_one @ A)) @ B2)))))))))). % 6.41/2.69 thf(fact_1448_is__unit__div__mult__cancel__right, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A2:A,B2:A]: (((A2 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((divide_divide @ A) @ A2) @ (((times_times @ A) @ B2) @ A2)) = (((divide_divide @ A) @ (one_one @ A)) @ B2)))))))))). % 6.41/2.69 thf(fact_1449_odd__even__add, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A2:A,B2:A]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2))) => ((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) => (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_1450_dvd__power__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [X3:A,M:nat,N2:nat]: (((X3 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ (((power_power @ A) @ X3) @ M)) @ (((power_power @ A) @ X3) @ N2)) = ((((dvd_dvd @ A) @ X3) @ (one_one @ A)) | (((ord_less_eq @ nat) @ M) @ N2)))))))))). % 6.41/2.69 thf(fact_1451_dvd__power, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [N2:nat,X3:A]: ((((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) | (X3 = (one_one @ A))) => (((dvd_dvd @ A) @ X3) @ (((power_power @ A) @ X3) @ N2))))))))). % 6.41/2.69 thf(fact_1452_split__mod, axiom, (! [P:(nat > $o),M:nat,N2:nat]: (((P @ (((modulo_modulo @ nat) @ M) @ N2)) = (((N2 = (zero_zero @ nat)) => (P @ M)) & ((N2 != (zero_zero @ nat)) => (! [I3:nat,J3:nat]: (((((ord_less @ nat) @ J3) @ N2) => ((M = (((plus_plus @ nat) @ (((times_times @ nat) @ N2) @ I3)) @ J3)) => (P @ J3))))))))))). % 6.41/2.69 thf(fact_1453_dvd__mult__cancel1, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((dvd_dvd @ nat) @ (((times_times @ nat) @ M) @ N2)) @ M) = (N2 = (one_one @ nat))))))). % 6.41/2.69 thf(fact_1454_dvd__mult__cancel2, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((dvd_dvd @ nat) @ (((times_times @ nat) @ N2) @ M)) @ M) = (N2 = (one_one @ nat))))))). % 6.41/2.69 thf(fact_1455_power__dvd__imp__le, axiom, (! [I:nat,M:nat,N2:nat]: (((((dvd_dvd @ nat) @ (((power_power @ nat) @ I) @ M)) @ (((power_power @ nat) @ I) @ N2)) => ((((ord_less @ nat) @ (one_one @ nat)) @ I) => (((ord_less_eq @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_1456_unique__euclidean__semiring__numeral__class_Omod__mult2__eq, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => ((((modulo_modulo @ A) @ A2) @ (((times_times @ A) @ B2) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ B2) @ (((modulo_modulo @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ C2))) @ (((modulo_modulo @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_1457_product__nth, axiom, (! [A:$tType,B:$tType,N2:nat,Xs:(list @ A),Ys:(list @ B)]: (((((ord_less @ nat) @ N2) @ (((times_times @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ B)) @ Ys))) => ((((nth @ ((product_prod @ A) @ B)) @ ((((product @ A) @ B) @ Xs) @ Ys)) @ N2) = ((((product_Pair @ A) @ B) @ (((nth @ A) @ Xs) @ (((divide_divide @ nat) @ N2) @ ((size_size @ (list @ B)) @ Ys)))) @ (((nth @ B) @ Ys) @ (((modulo_modulo @ nat) @ N2) @ ((size_size @ (list @ B)) @ Ys))))))))). % 6.41/2.69 thf(fact_1458_power__mono__odd, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:nat,A2:A,B2:A]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) => ((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N2)) @ (((power_power @ A) @ B2) @ N2)))))))))). % 6.41/2.69 thf(fact_1459_Suc__times__mod__eq, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ M) => ((((modulo_modulo @ nat) @ (suc @ (((times_times @ nat) @ M) @ N2))) @ M) = (one_one @ nat)))))). % 6.41/2.69 thf(fact_1460_odd__pos, axiom, (! [N2:nat]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))). % 6.41/2.69 thf(fact_1461_dvd__power__iff__le, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K) => ((((dvd_dvd @ nat) @ (((power_power @ nat) @ K) @ M)) @ (((power_power @ nat) @ K) @ N2)) = (((ord_less_eq @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_1462_even__unset__bit__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,A2:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se2638667681897837118et_bit @ A) @ M) @ A2)) = ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2) | (M = (zero_zero @ nat)))))))))). % 6.41/2.69 thf(fact_1463_even__set__bit__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,A2:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5668285175392031749et_bit @ A) @ M) @ A2)) = ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2) & (M != (zero_zero @ nat)))))))))). % 6.41/2.69 thf(fact_1464_num_Osize__gen_I1_J, axiom, ((size_num @ one2) = (zero_zero @ nat))). % 6.41/2.69 thf(fact_1465_divmod__digit__0_I2_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ A) @ (((modulo_modulo @ A) @ A2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) @ B2) => ((((modulo_modulo @ A) @ A2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2)) = (((modulo_modulo @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_1466_bits__stable__imp__add__self, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A]: ((((((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = A2) => ((((plus_plus @ A) @ A2) @ (((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1467_div__exp__mod__exp__eq, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A,N2:nat,M:nat]: (((((modulo_modulo @ A) @ (((divide_divide @ A) @ A2) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)) = (((divide_divide @ A) @ (((modulo_modulo @ A) @ A2) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ N2) @ M)))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2))))))))). % 6.41/2.69 thf(fact_1468_oddE, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A2:A]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2))) => (~ ((! [B5:A]: ((A2 != (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B5)) @ (one_one @ A)))))))))))))). % 6.41/2.69 thf(fact_1469_zero__le__even__power, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:nat,A2:A]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ N2))))))))). % 6.41/2.69 thf(fact_1470_zero__le__odd__power, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:nat,A2:A]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ N2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2))))))))). % 6.41/2.69 thf(fact_1471_zero__le__power__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,N2:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ N2)) = ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2) | ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))))). % 6.41/2.69 thf(fact_1472_verit__le__mono__div, axiom, (! [A3:nat,B3:nat,N2:nat]: (((((ord_less @ nat) @ A3) @ B3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (((divide_divide @ nat) @ A3) @ N2)) @ ((((if @ nat) @ ((((modulo_modulo @ nat) @ B3) @ N2) = (zero_zero @ nat))) @ (one_one @ nat)) @ (zero_zero @ nat)))) @ (((divide_divide @ nat) @ B3) @ N2))))))). % 6.41/2.69 thf(fact_1473_even__set__encode__iff, axiom, (! [A3:(set @ nat)]: ((((finite_finite2 @ nat) @ A3) => ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (nat_set_encode @ A3)) = (~ ((((member @ nat) @ (zero_zero @ nat)) @ A3)))))))). % 6.41/2.69 thf(fact_1474_divmod__digit__0_I1_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ A) @ (((modulo_modulo @ A) @ A2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) @ B2) => ((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) = (((divide_divide @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_1475_zero__less__power__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,N2:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ A2) @ N2)) = ((N2 = (zero_zero @ nat)) | (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2) & (A2 != (zero_zero @ A))) | ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) & (((ord_less @ A) @ (zero_zero @ A)) @ A2))))))))))). % 6.41/2.69 thf(fact_1476_mod__double__modulus, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M:A,X3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ M) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => (((((modulo_modulo @ A) @ X3) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)) = (((modulo_modulo @ A) @ X3) @ M)) | ((((modulo_modulo @ A) @ X3) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)) = (((plus_plus @ A) @ (((modulo_modulo @ A) @ X3) @ M)) @ M))))))))))). % 6.41/2.69 thf(fact_1477_unset__bit__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,A2:A]: (((((bit_se2638667681897837118et_bit @ A) @ (suc @ N2)) @ A2) = (((plus_plus @ A) @ (((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se2638667681897837118et_bit @ A) @ N2) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))))). % 6.41/2.69 thf(fact_1478_set__bit__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,A2:A]: (((((bit_se5668285175392031749et_bit @ A) @ (suc @ N2)) @ A2) = (((plus_plus @ A) @ (((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5668285175392031749et_bit @ A) @ N2) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))))). % 6.41/2.69 thf(fact_1479_power__le__zero__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,N2:nat]: (((((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N2)) @ (zero_zero @ A)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) & (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) & (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))) | ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2) & (A2 = (zero_zero @ A)))))))))))). % 6.41/2.69 thf(fact_1480_divmod__digit__1_I1_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ B2) @ (((modulo_modulo @ A) @ A2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) => ((((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2)))) @ (one_one @ A)) = (((divide_divide @ A) @ A2) @ B2))))))))))). % 6.41/2.69 thf(fact_1481_flip__bit__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,A2:A]: (((((bit_se8732182000553998342ip_bit @ A) @ (suc @ N2)) @ A2) = (((plus_plus @ A) @ (((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se8732182000553998342ip_bit @ A) @ N2) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))))). % 6.41/2.69 thf(fact_1482_pos__eucl__rel__int__mult__2, axiom, (! [B2:int,A2:int,Q4:int,R2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ B2) => ((((eucl_rel_int @ A2) @ B2) @ ((((product_Pair @ int) @ int) @ Q4) @ R2)) => (((eucl_rel_int @ (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ A2))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ B2)) @ ((((product_Pair @ int) @ int) @ Q4) @ (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ R2))))))))). % 6.41/2.69 thf(fact_1483_signed__take__bit__Suc, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N2:nat,A2:A]: (((((bit_ri4674362597316999326ke_bit @ A) @ (suc @ N2)) @ A2) = (((plus_plus @ A) @ (((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_ri4674362597316999326ke_bit @ A) @ N2) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))))). % 6.41/2.69 thf(fact_1484_even__mult__exp__div__exp__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A,M:nat,N2:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ (((times_times @ A) @ A2) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2))) = ((((ord_less @ nat) @ N2) @ M) | (((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2) = (zero_zero @ A)) | ((((ord_less_eq @ nat) @ M) @ N2) & (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A2) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ N2) @ M)))))))))))))). % 6.41/2.69 thf(fact_1485_even__flip__bit__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,A2:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se8732182000553998342ip_bit @ A) @ M) @ A2)) = ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2) != (M = (zero_zero @ nat)))))))))). % 6.41/2.69 thf(fact_1486_one__mod__2__pow__eq, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [N2:nat]: (((((modulo_modulo @ A) @ (one_one @ A)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) = ((zero_neq_one_of_bool @ A) @ (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))))))). % 6.41/2.69 thf(fact_1487_set__encode__insert, axiom, (! [A3:(set @ nat),N2:nat]: ((((finite_finite2 @ nat) @ A3) => ((~ ((((member @ nat) @ N2) @ A3))) => ((nat_set_encode @ (((insert @ nat) @ N2) @ A3)) = (((plus_plus @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) @ (nat_set_encode @ A3)))))))). % 6.41/2.69 thf(fact_1488_insert__absorb2, axiom, (! [A:$tType,X3:A,A3:(set @ A)]: (((((insert @ A) @ X3) @ (((insert @ A) @ X3) @ A3)) = (((insert @ A) @ X3) @ A3))))). % 6.41/2.69 thf(fact_1489_insert__iff, axiom, (! [A:$tType,A2:A,B2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A3)) = ((A2 = B2) | (((member @ A) @ A2) @ A3)))))). % 6.41/2.69 thf(fact_1490_insertCI, axiom, (! [A:$tType,A2:A,B3:(set @ A),B2:A]: ((((~ ((((member @ A) @ A2) @ B3))) => (A2 = B2)) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B3)))))). % 6.41/2.69 thf(fact_1491_diff__self, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1492_diff__0__right, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))). % 6.41/2.69 thf(fact_1493_zero__diff, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A2:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1494_diff__zero, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ (zero_zero @ A)) = A2))))))). % 6.41/2.69 thf(fact_1495_cancel__comm__monoid__add__class_Odiff__cancel, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ A2) @ A2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1496_add__diff__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = A2))))))). % 6.41/2.69 thf(fact_1497_diff__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ B2) = A2))))))). % 6.41/2.69 thf(fact_1498_add__diff__cancel__left, axiom, (! [A:$tType]: (((cancel2418104881723323429up_add @ A) => (! [C2:A,A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2)) = (((minus_minus @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1499_add__diff__cancel__left_H, axiom, (! [A:$tType]: (((cancel2418104881723323429up_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ A2) = B2))))))). % 6.41/2.69 thf(fact_1500_add__diff__cancel__right, axiom, (! [A:$tType]: (((cancel2418104881723323429up_add @ A) => (! [A2:A,C2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1501_add__diff__cancel__right_H, axiom, (! [A:$tType]: (((cancel2418104881723323429up_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = A2))))))). % 6.41/2.69 thf(fact_1502_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))). % 6.41/2.69 thf(fact_1503_finite__insert, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: ((((finite_finite2 @ A) @ (((insert @ A) @ A2) @ A3)) = ((finite_finite2 @ A) @ A3))))). % 6.41/2.69 thf(fact_1504_insert__subset, axiom, (! [A:$tType,X3:A,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ B3) = ((((member @ A) @ X3) @ B3) & (((ord_less_eq @ (set @ A)) @ A3) @ B3)))))). % 6.41/2.69 thf(fact_1505_diff__Suc__Suc, axiom, (! [M:nat,N2:nat]: (((((minus_minus @ nat) @ (suc @ M)) @ (suc @ N2)) = (((minus_minus @ nat) @ M) @ N2))))). % 6.41/2.69 thf(fact_1506_Suc__diff__diff, axiom, (! [M:nat,N2:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ (suc @ M)) @ N2)) @ (suc @ K)) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ N2)) @ K))))). % 6.41/2.69 thf(fact_1507_diff__0__eq__0, axiom, (! [N2:nat]: (((((minus_minus @ nat) @ (zero_zero @ nat)) @ N2) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_1508_diff__self__eq__0, axiom, (! [M:nat]: (((((minus_minus @ nat) @ M) @ M) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_1509_diff__diff__cancel, axiom, (! [I:nat,N2:nat]: (((((ord_less_eq @ nat) @ I) @ N2) => ((((minus_minus @ nat) @ N2) @ (((minus_minus @ nat) @ N2) @ I)) = I))))). % 6.41/2.69 thf(fact_1510_diff__diff__left, axiom, (! [I:nat,J:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ K) = (((minus_minus @ nat) @ I) @ (((plus_plus @ nat) @ J) @ K)))))). % 6.41/2.69 thf(fact_1511_flip__bit__negative__int__iff, axiom, (! [N2:nat,K:int]: (((((ord_less @ int) @ (((bit_se8732182000553998342ip_bit @ int) @ N2) @ K)) @ (zero_zero @ int)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))). % 6.41/2.69 thf(fact_1512_of__bool__less__eq__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [P:$o,Q:$o]: (((((ord_less_eq @ A) @ ((zero_neq_one_of_bool @ A) @ P)) @ ((zero_neq_one_of_bool @ A) @ Q)) = (P => Q)))))))). % 6.41/2.69 thf(fact_1513_of__bool__eq_I1_J, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (((zero_neq_one_of_bool @ A) @ $false) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_1514_of__bool__eq__0__iff, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (! [P:$o]: (((((zero_neq_one_of_bool @ A) @ P) = (zero_zero @ A)) = (~ (P))))))))). % 6.41/2.69 thf(fact_1515_of__bool__less__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [P:$o,Q:$o]: (((((ord_less @ A) @ ((zero_neq_one_of_bool @ A) @ P)) @ ((zero_neq_one_of_bool @ A) @ Q)) = ((~ (P)) & Q)))))))). % 6.41/2.69 thf(fact_1516_of__bool__eq__1__iff, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (! [P:$o]: (((((zero_neq_one_of_bool @ A) @ P) = (one_one @ A)) = P))))))). % 6.41/2.69 thf(fact_1517_of__bool__eq_I2_J, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (((zero_neq_one_of_bool @ A) @ $true) = (one_one @ A)))))). % 6.41/2.69 thf(fact_1518_signed__take__bit__of__0, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N2:nat]: (((((bit_ri4674362597316999326ke_bit @ A) @ N2) @ (zero_zero @ A)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1519_diff__ge__0__iff__ge, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A2) @ B2)) = (((ord_less_eq @ A) @ B2) @ A2)))))))). % 6.41/2.69 thf(fact_1520_diff__gt__0__iff__gt, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A2) @ B2)) = (((ord_less @ A) @ B2) @ A2)))))))). % 6.41/2.69 thf(fact_1521_le__add__diff__inverse2, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((plus_plus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ B2) = A2)))))))). % 6.41/2.69 thf(fact_1522_le__add__diff__inverse, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((plus_plus @ A) @ B2) @ (((minus_minus @ A) @ A2) @ B2)) = A2)))))))). % 6.41/2.69 thf(fact_1523_diff__numeral__special_I9_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((minus_minus @ A) @ (one_one @ A)) @ (one_one @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_1524_diff__add__zero, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ A2) @ (((plus_plus @ A) @ A2) @ B2)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1525_div__diff, axiom, (! [A:$tType]: (((idom_modulo @ A) => (! [C2:A,A2:A,B2:A]: (((((dvd_dvd @ A) @ C2) @ A2) => ((((dvd_dvd @ A) @ C2) @ B2) => ((((divide_divide @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((minus_minus @ A) @ (((divide_divide @ A) @ A2) @ C2)) @ (((divide_divide @ A) @ B2) @ C2))))))))))). % 6.41/2.69 thf(fact_1526_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A2:A,A3:(set @ A),B2:A]: ((((((insert @ A) @ A2) @ A3) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))). % 6.41/2.69 thf(fact_1527_singleton__insert__inj__eq, axiom, (! [A:$tType,B2:A,A2:A,A3:(set @ A)]: ((((((insert @ A) @ B2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A2) @ A3)) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))). % 6.41/2.69 thf(fact_1528_zero__less__of__bool__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [P:$o]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((zero_neq_one_of_bool @ A) @ P)) = P))))))). % 6.41/2.69 thf(fact_1529_zero__less__diff, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N2) @ M)) = (((ord_less @ nat) @ M) @ N2))))). % 6.41/2.69 thf(fact_1530_diff__is__0__eq, axiom, (! [M:nat,N2:nat]: ((((((minus_minus @ nat) @ M) @ N2) = (zero_zero @ nat)) = (((ord_less_eq @ nat) @ M) @ N2))))). % 6.41/2.69 thf(fact_1531_diff__is__0__eq_H, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((minus_minus @ nat) @ M) @ N2) = (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_1532_of__bool__less__one__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [P:$o]: (((((ord_less @ A) @ ((zero_neq_one_of_bool @ A) @ P)) @ (one_one @ A)) = (~ (P))))))))). % 6.41/2.69 thf(fact_1533_of__bool__not__iff, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [P:$o]: ((((zero_neq_one_of_bool @ A) @ (~ (P))) = (((minus_minus @ A) @ (one_one @ A)) @ ((zero_neq_one_of_bool @ A) @ P))))))))). % 6.41/2.69 thf(fact_1534_Nat_Odiff__diff__right, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((minus_minus @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K)) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ J)))))). % 6.41/2.69 thf(fact_1535_Nat_Oadd__diff__assoc2, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((plus_plus @ nat) @ (((minus_minus @ nat) @ J) @ K)) @ I) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ J) @ I)) @ K)))))). % 6.41/2.69 thf(fact_1536_Nat_Oadd__diff__assoc, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((plus_plus @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K)) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ K)))))). % 6.41/2.69 thf(fact_1537_diff__Suc__1, axiom, (! [N2:nat]: (((((minus_minus @ nat) @ (suc @ N2)) @ (one_one @ nat)) = N2)))). % 6.41/2.69 thf(fact_1538_Suc__0__mod__eq, axiom, (! [N2:nat]: (((((modulo_modulo @ nat) @ (suc @ (zero_zero @ nat))) @ N2) = ((zero_neq_one_of_bool @ nat) @ (N2 != (suc @ (zero_zero @ nat)))))))). % 6.41/2.69 thf(fact_1539_mod__pos__pos__trivial, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((ord_less @ int) @ K) @ L) => ((((modulo_modulo @ int) @ K) @ L) = K)))))). % 6.41/2.69 thf(fact_1540_mod__neg__neg__trivial, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ K) @ (zero_zero @ int)) => ((((ord_less @ int) @ L) @ K) => ((((modulo_modulo @ int) @ K) @ L) = K)))))). % 6.41/2.69 thf(fact_1541_signed__take__bit__Suc__1, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N2:nat]: (((((bit_ri4674362597316999326ke_bit @ A) @ (suc @ N2)) @ (one_one @ A)) = (one_one @ A)))))))). % 6.41/2.69 thf(fact_1542_Suc__pred, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((suc @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat)))) = N2))))). % 6.41/2.69 thf(fact_1543_diff__Suc__diff__eq2, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((minus_minus @ nat) @ (suc @ (((minus_minus @ nat) @ J) @ K))) @ I) = (((minus_minus @ nat) @ (suc @ J)) @ (((plus_plus @ nat) @ K) @ I))))))). % 6.41/2.69 thf(fact_1544_diff__Suc__diff__eq1, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((minus_minus @ nat) @ I) @ (suc @ (((minus_minus @ nat) @ J) @ K))) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (suc @ J))))))). % 6.41/2.69 thf(fact_1545_signed__take__bit__Suc__bit0, axiom, (! [N2:nat,K:num]: (((((bit_ri4674362597316999326ke_bit @ int) @ (suc @ N2)) @ ((numeral_numeral @ int) @ (bit0 @ K))) = (((times_times @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N2) @ ((numeral_numeral @ int) @ K))) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))). % 6.41/2.69 thf(fact_1546_Suc__diff__1, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((suc @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))) = N2))))). % 6.41/2.69 thf(fact_1547_set__replicate, axiom, (! [A:$tType,N2:nat,X3:A]: (((N2 != (zero_zero @ nat)) => (((set2 @ A) @ (((replicate @ A) @ N2) @ X3)) = (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))). % 6.41/2.69 thf(fact_1548_even__diff, axiom, (! [A:$tType]: (((ring_parity @ A) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((minus_minus @ A) @ A2) @ B2)) = (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1549_of__bool__half__eq__0, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [B2:$o]: (((((divide_divide @ A) @ ((zero_neq_one_of_bool @ A) @ B2)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1550_odd__Suc__minus__one, axiom, (! [N2:nat]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) => ((suc @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat)))) = N2))))). % 6.41/2.69 thf(fact_1551_even__diff__nat, axiom, (! [M:nat,N2:nat]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ M) @ N2)) = ((((ord_less @ nat) @ M) @ N2) | (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ M) @ N2))))))). % 6.41/2.69 thf(fact_1552_semiring__parity__class_Oeven__mask__iff, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [N2:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) @ (one_one @ A))) = (N2 = (zero_zero @ nat))))))))). % 6.41/2.69 thf(fact_1553_one__div__2__pow__eq, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [N2:nat]: (((((divide_divide @ A) @ (one_one @ A)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) = ((zero_neq_one_of_bool @ A) @ (N2 = (zero_zero @ nat)))))))))). % 6.41/2.69 thf(fact_1554_bits__1__div__exp, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [N2:nat]: (((((divide_divide @ A) @ (one_one @ A)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) = ((zero_neq_one_of_bool @ A) @ (N2 = (zero_zero @ nat)))))))))). % 6.41/2.69 thf(fact_1555_flip__bit__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se8732182000553998342ip_bit @ A) @ (zero_zero @ nat)) @ A2) = (((plus_plus @ A) @ ((zero_neq_one_of_bool @ A) @ (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))). % 6.41/2.69 thf(fact_1556_mod__int__unique, axiom, (! [K:int,L:int,Q4:int,R2:int]: (((((eucl_rel_int @ K) @ L) @ ((((product_Pair @ int) @ int) @ Q4) @ R2)) => ((((modulo_modulo @ int) @ K) @ L) = R2))))). % 6.41/2.69 thf(fact_1557_gcd__nat_Oasym, axiom, (! [A2:nat,B2:nat]: ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) => (~ (((((dvd_dvd @ nat) @ B2) @ A2) & (B2 != A2)))))))). % 6.41/2.69 thf(fact_1558_gcd__nat_Orefl, axiom, (! [A2:nat]: ((((dvd_dvd @ nat) @ A2) @ A2)))). % 6.41/2.69 thf(fact_1559_gcd__nat_Otrans, axiom, (! [A2:nat,B2:nat,C2:nat]: (((((dvd_dvd @ nat) @ A2) @ B2) => ((((dvd_dvd @ nat) @ B2) @ C2) => (((dvd_dvd @ nat) @ A2) @ C2)))))). % 6.41/2.69 thf(fact_1560_gcd__nat_Oeq__iff, axiom, ((^ [Y3:nat,Z:nat]: ((Y3 = Z))) = (^ [A4:nat,B4:nat]: (((((dvd_dvd @ nat) @ A4) @ B4) & (((dvd_dvd @ nat) @ B4) @ A4)))))). % 6.41/2.69 thf(fact_1561_gcd__nat_Oirrefl, axiom, (! [A2:nat]: ((~ (((((dvd_dvd @ nat) @ A2) @ A2) & (A2 != A2))))))). % 6.41/2.69 thf(fact_1562_gcd__nat_Oantisym, axiom, (! [A2:nat,B2:nat]: (((((dvd_dvd @ nat) @ A2) @ B2) => ((((dvd_dvd @ nat) @ B2) @ A2) => (A2 = B2)))))). % 6.41/2.69 thf(fact_1563_gcd__nat_Ostrict__trans, axiom, (! [A2:nat,B2:nat,C2:nat]: ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) => (((((dvd_dvd @ nat) @ B2) @ C2) & (B2 != C2)) => ((((dvd_dvd @ nat) @ A2) @ C2) & (A2 != C2))))))). % 6.41/2.69 thf(fact_1564_gcd__nat_Ostrict__trans1, axiom, (! [A2:nat,B2:nat,C2:nat]: (((((dvd_dvd @ nat) @ A2) @ B2) => (((((dvd_dvd @ nat) @ B2) @ C2) & (B2 != C2)) => ((((dvd_dvd @ nat) @ A2) @ C2) & (A2 != C2))))))). % 6.41/2.69 thf(fact_1565_gcd__nat_Ostrict__trans2, axiom, (! [A2:nat,B2:nat,C2:nat]: ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) => ((((dvd_dvd @ nat) @ B2) @ C2) => ((((dvd_dvd @ nat) @ A2) @ C2) & (A2 != C2))))))). % 6.41/2.69 thf(fact_1566_gcd__nat_Ostrict__iff__not, axiom, (! [A2:nat,B2:nat]: ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) = ((((dvd_dvd @ nat) @ A2) @ B2) & (~ ((((dvd_dvd @ nat) @ B2) @ A2)))))))). % 6.41/2.69 thf(fact_1567_gcd__nat_Oorder__iff__strict, axiom, ((dvd_dvd @ nat) = (^ [A4:nat,B4:nat]: ((((((dvd_dvd @ nat) @ A4) @ B4) & (A4 != B4)) | (A4 = B4)))))). % 6.41/2.69 thf(fact_1568_gcd__nat_Ostrict__iff__order, axiom, (! [A2:nat,B2:nat]: ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) = ((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)))))). % 6.41/2.69 thf(fact_1569_gcd__nat_Ostrict__implies__order, axiom, (! [A2:nat,B2:nat]: ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) => (((dvd_dvd @ nat) @ A2) @ B2))))). % 6.41/2.69 thf(fact_1570_gcd__nat_Ostrict__implies__not__eq, axiom, (! [A2:nat,B2:nat]: ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) => (A2 != B2))))). % 6.41/2.69 thf(fact_1571_gcd__nat_Onot__eq__order__implies__strict, axiom, (! [A2:nat,B2:nat]: (((A2 != B2) => ((((dvd_dvd @ nat) @ A2) @ B2) => ((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2))))))). % 6.41/2.69 thf(fact_1572_dvd__antisym, axiom, (! [M:nat,N2:nat]: (((((dvd_dvd @ nat) @ M) @ N2) => ((((dvd_dvd @ nat) @ N2) @ M) => (M = N2)))))). % 6.41/2.69 thf(fact_1573_dvd__diff__nat, axiom, (! [K:nat,M:nat,N2:nat]: (((((dvd_dvd @ nat) @ K) @ M) => ((((dvd_dvd @ nat) @ K) @ N2) => (((dvd_dvd @ nat) @ K) @ (((minus_minus @ nat) @ M) @ N2))))))). % 6.41/2.69 thf(fact_1574_eucl__rel__int, axiom, (! [K:int,L:int]: ((((eucl_rel_int @ K) @ L) @ ((((product_Pair @ int) @ int) @ (((divide_divide @ int) @ K) @ L)) @ (((modulo_modulo @ int) @ K) @ L)))))). % 6.41/2.69 thf(fact_1575_mk__disjoint__insert, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => (? [B10:(set @ A)]: (((A3 = (((insert @ A) @ A2) @ B10)) & (~ ((((member @ A) @ A2) @ B10)))))))))). % 6.41/2.69 thf(fact_1576_insert__commute, axiom, (! [A:$tType,X3:A,Y:A,A3:(set @ A)]: (((((insert @ A) @ X3) @ (((insert @ A) @ Y) @ A3)) = (((insert @ A) @ Y) @ (((insert @ A) @ X3) @ A3)))))). % 6.41/2.69 thf(fact_1577_insert__eq__iff, axiom, (! [A:$tType,A2:A,A3:(set @ A),B2:A,B3:(set @ A)]: (((~ ((((member @ A) @ A2) @ A3))) => ((~ ((((member @ A) @ B2) @ B3))) => (((((insert @ A) @ A2) @ A3) = (((insert @ A) @ B2) @ B3)) = (((A2 = B2) => (A3 = B3)) & ((A2 != B2) => (? [C7:(set @ A)]: (((A3 = (((insert @ A) @ B2) @ C7)) & ((~ ((((member @ A) @ B2) @ C7))) & ((B3 = (((insert @ A) @ A2) @ C7)) & (~ ((((member @ A) @ A2) @ C7)))))))))))))))). % 6.41/2.69 thf(fact_1578_insert__absorb, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => ((((insert @ A) @ A2) @ A3) = A3))))). % 6.41/2.69 thf(fact_1579_insert__ident, axiom, (! [A:$tType,X3:A,A3:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X3) @ A3))) => ((~ ((((member @ A) @ X3) @ B3))) => (((((insert @ A) @ X3) @ A3) = (((insert @ A) @ X3) @ B3)) = (A3 = B3))))))). % 6.41/2.69 thf(fact_1580_Set_Oset__insert, axiom, (! [A:$tType,X3:A,A3:(set @ A)]: (((((member @ A) @ X3) @ A3) => (~ ((! [B10:(set @ A)]: (((A3 = (((insert @ A) @ X3) @ B10)) => (((member @ A) @ X3) @ B10)))))))))). % 6.41/2.69 thf(fact_1581_insertI2, axiom, (! [A:$tType,A2:A,B3:(set @ A),B2:A]: (((((member @ A) @ A2) @ B3) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B3)))))). % 6.41/2.69 thf(fact_1582_insertI1, axiom, (! [A:$tType,A2:A,B3:(set @ A)]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ B3))))). % 6.41/2.69 thf(fact_1583_insertE, axiom, (! [A:$tType,A2:A,B2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A3)) => ((A2 != B2) => (((member @ A) @ A2) @ A3)))))). % 6.41/2.69 thf(fact_1584_of__bool__eq__iff, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (! [P4:$o,Q4:$o]: (((((zero_neq_one_of_bool @ A) @ P4) = ((zero_neq_one_of_bool @ A) @ Q4)) = (P4 = Q4)))))))). % 6.41/2.69 thf(fact_1585_diff__commute, axiom, (! [I:nat,J:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ K) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ K)) @ J))))). % 6.41/2.69 thf(fact_1586_diff__right__commute, axiom, (! [A:$tType]: (((cancel2418104881723323429up_add @ A) => (! [A2:A,C2:A,B2:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ B2) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_1587_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A,D2:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C2) @ D2)) => ((A2 = B2) = (C2 = D2))))))))). % 6.41/2.69 thf(fact_1588_full__exhaustive__int_H_Ocases, axiom, (! [X3:((product_prod @ (((product_prod @ int) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ int) @ int))]: ((~ ((! [F4:(((product_prod @ int) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term)))),D3:int,I4:int]: ((X3 != ((((product_Pair @ (((product_prod @ int) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ int) @ int)) @ F4) @ ((((product_Pair @ int) @ int) @ D3) @ I4)))))))))). % 6.41/2.69 thf(fact_1589_exhaustive__int_H_Ocases, axiom, (! [X3:((product_prod @ (int > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ int) @ int))]: ((~ ((! [F4:(int > (option @ ((product_prod @ $o) @ (list @ code_term)))),D3:int,I4:int]: ((X3 != ((((product_Pair @ (int > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ int) @ int)) @ F4) @ ((((product_Pair @ int) @ int) @ D3) @ I4)))))))))). % 6.41/2.69 thf(fact_1590_small__lazy_H_Ocases, axiom, (! [X3:((product_prod @ int) @ int)]: ((~ ((! [D3:int,I4:int]: ((X3 != ((((product_Pair @ int) @ int) @ D3) @ I4))))))))). % 6.41/2.69 thf(fact_1591_unique__quotient, axiom, (! [A2:int,B2:int,Q4:int,R2:int,Q6:int,R3:int]: (((((eucl_rel_int @ A2) @ B2) @ ((((product_Pair @ int) @ int) @ Q4) @ R2)) => ((((eucl_rel_int @ A2) @ B2) @ ((((product_Pair @ int) @ int) @ Q6) @ R3)) => (Q4 = Q6)))))). % 6.41/2.69 thf(fact_1592_unique__remainder, axiom, (! [A2:int,B2:int,Q4:int,R2:int,Q6:int,R3:int]: (((((eucl_rel_int @ A2) @ B2) @ ((((product_Pair @ int) @ int) @ Q4) @ R2)) => ((((eucl_rel_int @ A2) @ B2) @ ((((product_Pair @ int) @ int) @ Q6) @ R3)) => (R2 = R3)))))). % 6.41/2.69 thf(fact_1593_of__bool__conj, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [P:$o,Q:$o]: ((((zero_neq_one_of_bool @ A) @ (P & Q)) = (((times_times @ A) @ ((zero_neq_one_of_bool @ A) @ P)) @ ((zero_neq_one_of_bool @ A) @ Q))))))))). % 6.41/2.69 thf(fact_1594_diff__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A,D2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ D2) @ C2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ (((minus_minus @ A) @ B2) @ D2)))))))))). % 6.41/2.69 thf(fact_1595_diff__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C2) @ A2)) @ (((minus_minus @ A) @ C2) @ B2))))))))). % 6.41/2.69 thf(fact_1596_diff__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ (((minus_minus @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_1597_diff__eq__diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A,C2:A,D2:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C2) @ D2)) => ((((ord_less_eq @ A) @ A2) @ B2) = (((ord_less_eq @ A) @ C2) @ D2))))))))). % 6.41/2.69 thf(fact_1598_eq__iff__diff__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [A4:A,B4:A]: (((((minus_minus @ A) @ A4) @ B4) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1599_diff__strict__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A,D2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ D2) @ C2) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ (((minus_minus @ A) @ B2) @ D2)))))))))). % 6.41/2.69 thf(fact_1600_diff__eq__diff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A,C2:A,D2:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C2) @ D2)) => ((((ord_less @ A) @ A2) @ B2) = (((ord_less @ A) @ C2) @ D2))))))))). % 6.41/2.69 thf(fact_1601_diff__strict__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less @ A) @ B2) @ A2) => (((ord_less @ A) @ (((minus_minus @ A) @ C2) @ A2)) @ (((minus_minus @ A) @ C2) @ B2))))))))). % 6.41/2.69 thf(fact_1602_diff__strict__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ (((minus_minus @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_1603_right__diff__distrib_H, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ A2) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((times_times @ A) @ A2) @ B2)) @ (((times_times @ A) @ A2) @ C2))))))))). % 6.41/2.69 thf(fact_1604_left__diff__distrib_H, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [B2:A,C2:A,A2:A]: (((((times_times @ A) @ (((minus_minus @ A) @ B2) @ C2)) @ A2) = (((minus_minus @ A) @ (((times_times @ A) @ B2) @ A2)) @ (((times_times @ A) @ C2) @ A2))))))))). % 6.41/2.69 thf(fact_1605_right__diff__distrib, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ A2) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((times_times @ A) @ A2) @ B2)) @ (((times_times @ A) @ A2) @ C2))))))))). % 6.41/2.69 thf(fact_1606_left__diff__distrib, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((minus_minus @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_1607_add__diff__add, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,C2:A,B2:A,D2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ (((plus_plus @ A) @ B2) @ D2)) = (((plus_plus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ (((minus_minus @ A) @ C2) @ D2))))))))). % 6.41/2.69 thf(fact_1608_group__cancel_Osub1, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,K:A,A2:A,B2:A]: (((A3 = (((plus_plus @ A) @ K) @ A2)) => ((((minus_minus @ A) @ A3) @ B2) = (((plus_plus @ A) @ K) @ (((minus_minus @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_1609_diff__eq__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: ((((((minus_minus @ A) @ A2) @ B2) = C2) = (A2 = (((plus_plus @ A) @ C2) @ B2))))))))). % 6.41/2.69 thf(fact_1610_eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,C2:A,B2:A]: (((A2 = (((minus_minus @ A) @ C2) @ B2)) = ((((plus_plus @ A) @ A2) @ B2) = C2)))))))). % 6.41/2.69 thf(fact_1611_add__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ A2) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_1612_diff__diff__eq2, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((minus_minus @ A) @ A2) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ B2)))))))). % 6.41/2.69 thf(fact_1613_diff__add__eq, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ C2)) @ B2)))))))). % 6.41/2.69 thf(fact_1614_diff__add__eq__diff__diff__swap, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((minus_minus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ B2)))))))). % 6.41/2.69 thf(fact_1615_add__implies__diff, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [C2:A,B2:A,A2:A]: ((((((plus_plus @ A) @ C2) @ B2) = A2) => (C2 = (((minus_minus @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1616_diff__diff__eq, axiom, (! [A:$tType]: (((cancel2418104881723323429up_add @ A) => (! [A2:A,B2:A,C2:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((minus_minus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_1617_diff__divide__distrib, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((minus_minus @ A) @ (((divide_divide @ A) @ A2) @ C2)) @ (((divide_divide @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_1618_singletonD, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => (B2 = A2))))). % 6.41/2.69 thf(fact_1619_singleton__iff, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = (B2 = A2))))). % 6.41/2.69 thf(fact_1620_doubleton__eq__iff, axiom, (! [A:$tType,A2:A,B2:A,C2:A,D2:A]: ((((((insert @ A) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C2) @ (((insert @ A) @ D2) @ (bot_bot @ (set @ A))))) = (((A2 = C2) & (B2 = D2)) | ((A2 = D2) & (B2 = C2))))))). % 6.41/2.69 thf(fact_1621_insert__not__empty, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((insert @ A) @ A2) @ A3) != (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_1622_singleton__inject, axiom, (! [A:$tType,A2:A,B2:A]: ((((((insert @ A) @ A2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) => (A2 = B2))))). % 6.41/2.69 thf(fact_1623_finite_OinsertI, axiom, (! [A:$tType,A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((finite_finite2 @ A) @ (((insert @ A) @ A2) @ A3)))))). % 6.41/2.69 thf(fact_1624_insert__subsetI, axiom, (! [A:$tType,X3:A,A3:(set @ A),X8:(set @ A)]: (((((member @ A) @ X3) @ A3) => ((((ord_less_eq @ (set @ A)) @ X8) @ A3) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ X8)) @ A3)))))). % 6.41/2.69 thf(fact_1625_insert__mono, axiom, (! [A:$tType,C5:(set @ A),D6:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ C5) @ D6) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A2) @ C5)) @ (((insert @ A) @ A2) @ D6)))))). % 6.41/2.69 thf(fact_1626_subset__insert, axiom, (! [A:$tType,X3:A,A3:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X3) @ A3))) => ((((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ B3)) = (((ord_less_eq @ (set @ A)) @ A3) @ B3)))))). % 6.41/2.69 thf(fact_1627_subset__insertI, axiom, (! [A:$tType,B3:(set @ A),A2:A]: ((((ord_less_eq @ (set @ A)) @ B3) @ (((insert @ A) @ A2) @ B3))))). % 6.41/2.69 thf(fact_1628_subset__insertI2, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),B2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ B2) @ B3)))))). % 6.41/2.69 thf(fact_1629_dvd__diff, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [X3:A,Y:A,Z2:A]: (((((dvd_dvd @ A) @ X3) @ Y) => ((((dvd_dvd @ A) @ X3) @ Z2) => (((dvd_dvd @ A) @ X3) @ (((minus_minus @ A) @ Y) @ Z2)))))))))). % 6.41/2.69 thf(fact_1630_zero__induct__lemma, axiom, (! [P:(nat > $o),K:nat,I:nat]: (((P @ K) => ((! [N3:nat]: (((P @ (suc @ N3)) => (P @ N3)))) => (P @ (((minus_minus @ nat) @ K) @ I))))))). % 6.41/2.69 thf(fact_1631_minus__nat_Odiff__0, axiom, (! [M:nat]: (((((minus_minus @ nat) @ M) @ (zero_zero @ nat)) = M)))). % 6.41/2.69 thf(fact_1632_diffs0__imp__equal, axiom, (! [M:nat,N2:nat]: ((((((minus_minus @ nat) @ M) @ N2) = (zero_zero @ nat)) => (((((minus_minus @ nat) @ N2) @ M) = (zero_zero @ nat)) => (M = N2)))))). % 6.41/2.69 thf(fact_1633_diff__less__mono2, axiom, (! [M:nat,N2:nat,L:nat]: (((((ord_less @ nat) @ M) @ N2) => ((((ord_less @ nat) @ M) @ L) => (((ord_less @ nat) @ (((minus_minus @ nat) @ L) @ N2)) @ (((minus_minus @ nat) @ L) @ M))))))). % 6.41/2.69 thf(fact_1634_less__imp__diff__less, axiom, (! [J:nat,K:nat,N2:nat]: (((((ord_less @ nat) @ J) @ K) => (((ord_less @ nat) @ (((minus_minus @ nat) @ J) @ N2)) @ K))))). % 6.41/2.69 thf(fact_1635_dvd__minus__self, axiom, (! [M:nat,N2:nat]: (((((dvd_dvd @ nat) @ M) @ (((minus_minus @ nat) @ N2) @ M)) = ((((ord_less @ nat) @ N2) @ M) | (((dvd_dvd @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_1636_eq__diff__iff, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N2) => (((((minus_minus @ nat) @ M) @ K) = (((minus_minus @ nat) @ N2) @ K)) = (M = N2))))))). % 6.41/2.69 thf(fact_1637_le__diff__iff, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N2) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N2) @ K)) = (((ord_less_eq @ nat) @ M) @ N2))))))). % 6.41/2.69 thf(fact_1638_Nat_Odiff__diff__eq, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N2) => ((((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N2) @ K)) = (((minus_minus @ nat) @ M) @ N2))))))). % 6.41/2.69 thf(fact_1639_diff__le__mono, axiom, (! [M:nat,N2:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ L)) @ (((minus_minus @ nat) @ N2) @ L)))))). % 6.41/2.69 thf(fact_1640_diff__le__self, axiom, (! [M:nat,N2:nat]: ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ N2)) @ M)))). % 6.41/2.69 thf(fact_1641_le__diff__iff_H, axiom, (! [A2:nat,C2:nat,B2:nat]: (((((ord_less_eq @ nat) @ A2) @ C2) => ((((ord_less_eq @ nat) @ B2) @ C2) => ((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ C2) @ A2)) @ (((minus_minus @ nat) @ C2) @ B2)) = (((ord_less_eq @ nat) @ B2) @ A2))))))). % 6.41/2.69 thf(fact_1642_diff__le__mono2, axiom, (! [M:nat,N2:nat,L:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ L) @ N2)) @ (((minus_minus @ nat) @ L) @ M)))))). % 6.41/2.69 thf(fact_1643_dvd__diffD, axiom, (! [K:nat,M:nat,N2:nat]: (((((dvd_dvd @ nat) @ K) @ (((minus_minus @ nat) @ M) @ N2)) => ((((dvd_dvd @ nat) @ K) @ N2) => ((((ord_less_eq @ nat) @ N2) @ M) => (((dvd_dvd @ nat) @ K) @ M))))))). % 6.41/2.69 thf(fact_1644_dvd__diffD1, axiom, (! [K:nat,M:nat,N2:nat]: (((((dvd_dvd @ nat) @ K) @ (((minus_minus @ nat) @ M) @ N2)) => ((((dvd_dvd @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ N2) @ M) => (((dvd_dvd @ nat) @ K) @ N2))))))). % 6.41/2.69 thf(fact_1645_less__eq__dvd__minus, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((dvd_dvd @ nat) @ M) @ N2) = (((dvd_dvd @ nat) @ M) @ (((minus_minus @ nat) @ N2) @ M))))))). % 6.41/2.69 thf(fact_1646_Nat_Odiff__cancel, axiom, (! [K:nat,M:nat,N2:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N2)) = (((minus_minus @ nat) @ M) @ N2))))). % 6.41/2.69 thf(fact_1647_diff__cancel2, axiom, (! [M:nat,K:nat,N2:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ (((plus_plus @ nat) @ N2) @ K)) = (((minus_minus @ nat) @ M) @ N2))))). % 6.41/2.69 thf(fact_1648_diff__add__inverse, axiom, (! [N2:nat,M:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ N2) @ M)) @ N2) = M)))). % 6.41/2.69 thf(fact_1649_diff__add__inverse2, axiom, (! [M:nat,N2:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ N2)) @ N2) = M)))). % 6.41/2.69 thf(fact_1650_diff__mult__distrib2, axiom, (! [K:nat,M:nat,N2:nat]: (((((times_times @ nat) @ K) @ (((minus_minus @ nat) @ M) @ N2)) = (((minus_minus @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)))))). % 6.41/2.69 thf(fact_1651_diff__mult__distrib, axiom, (! [M:nat,N2:nat,K:nat]: (((((times_times @ nat) @ (((minus_minus @ nat) @ M) @ N2)) @ K) = (((minus_minus @ nat) @ (((times_times @ nat) @ M) @ K)) @ (((times_times @ nat) @ N2) @ K)))))). % 6.41/2.69 thf(fact_1652_bezout1__nat, axiom, (! [A2:nat,B2:nat]: ((? [D3:nat,X5:nat,Y4:nat]: (((((dvd_dvd @ nat) @ D3) @ A2) & ((((dvd_dvd @ nat) @ D3) @ B2) & (((((minus_minus @ nat) @ (((times_times @ nat) @ A2) @ X5)) @ (((times_times @ nat) @ B2) @ Y4)) = D3) | ((((minus_minus @ nat) @ (((times_times @ nat) @ B2) @ X5)) @ (((times_times @ nat) @ A2) @ Y4)) = D3))))))))). % 6.41/2.69 thf(fact_1653_eucl__rel__int__by0, axiom, (! [K:int]: ((((eucl_rel_int @ K) @ (zero_zero @ int)) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ K))))). % 6.41/2.69 thf(fact_1654_div__int__unique, axiom, (! [K:int,L:int,Q4:int,R2:int]: (((((eucl_rel_int @ K) @ L) @ ((((product_Pair @ int) @ int) @ Q4) @ R2)) => ((((divide_divide @ int) @ K) @ L) = Q4))))). % 6.41/2.69 thf(fact_1655_zero__less__eq__of__bool, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [P:$o]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((zero_neq_one_of_bool @ A) @ P)))))))). % 6.41/2.69 thf(fact_1656_of__bool__less__eq__one, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [P:$o]: ((((ord_less_eq @ A) @ ((zero_neq_one_of_bool @ A) @ P)) @ (one_one @ A)))))))). % 6.41/2.69 thf(fact_1657_split__of__bool__asm, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (! [P:(A > $o),P4:$o]: (((P @ ((zero_neq_one_of_bool @ A) @ P4)) = (~ (((P4 & (~ ((P @ (one_one @ A))))) | ((~ (P4)) & (~ ((P @ (zero_zero @ A))))))))))))))). % 6.41/2.69 thf(fact_1658_split__of__bool, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (! [P:(A > $o),P4:$o]: (((P @ ((zero_neq_one_of_bool @ A) @ P4)) = ((P4 => (P @ (one_one @ A))) & ((~ (P4)) => (P @ (zero_zero @ A))))))))))). % 6.41/2.69 thf(fact_1659_of__bool__def, axiom, (! [A:$tType]: (((zero_neq_one @ A) => ((zero_neq_one_of_bool @ A) = (^ [P6:$o]: (((((if @ A) @ P6) @ (one_one @ A)) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1660_le__iff__diff__le__0, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ A4) @ B4)) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1661_less__iff__diff__less__0, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => ((ord_less @ A) = (^ [A4:A,B4:A]: ((((ord_less @ A) @ (((minus_minus @ A) @ A4) @ B4)) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1662_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ A2) @ B2) => (((((minus_minus @ A) @ B2) @ A2) = C2) = (B2 = (((plus_plus @ A) @ C2) @ A2))))))))))). % 6.41/2.69 thf(fact_1663_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((plus_plus @ A) @ A2) @ (((minus_minus @ A) @ B2) @ A2)) = B2)))))))). % 6.41/2.69 thf(fact_1664_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((minus_minus @ A) @ C2) @ (((minus_minus @ A) @ B2) @ A2)) = (((minus_minus @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ B2))))))))). % 6.41/2.69 thf(fact_1665_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((minus_minus @ A) @ (((plus_plus @ A) @ B2) @ C2)) @ A2) = (((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ A2)) @ C2))))))))). % 6.41/2.69 thf(fact_1666_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ A2)) @ C2) = (((minus_minus @ A) @ (((plus_plus @ A) @ B2) @ C2)) @ A2))))))))). % 6.41/2.69 thf(fact_1667_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((minus_minus @ A) @ (((plus_plus @ A) @ C2) @ B2)) @ A2) = (((plus_plus @ A) @ C2) @ (((minus_minus @ A) @ B2) @ A2)))))))))). % 6.41/2.69 thf(fact_1668_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((plus_plus @ A) @ C2) @ (((minus_minus @ A) @ B2) @ A2)) = (((minus_minus @ A) @ (((plus_plus @ A) @ C2) @ B2)) @ A2))))))))). % 6.41/2.69 thf(fact_1669_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ (((minus_minus @ A) @ B2) @ A2)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ B2))))))))). % 6.41/2.69 thf(fact_1670_le__add__diff, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ C2) @ (((minus_minus @ A) @ (((plus_plus @ A) @ B2) @ C2)) @ A2))))))))). % 6.41/2.69 thf(fact_1671_diff__add, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ A2)) @ A2) = B2)))))))). % 6.41/2.69 thf(fact_1672_le__diff__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ (((minus_minus @ A) @ C2) @ B2)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_1673_diff__le__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((ord_less_eq @ A) @ A2) @ (((plus_plus @ A) @ C2) @ B2))))))))). % 6.41/2.69 thf(fact_1674_add__le__add__imp__diff__le, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [I:A,K:A,N2:A,J:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ N2) => ((((ord_less_eq @ A) @ N2) @ (((plus_plus @ A) @ J) @ K)) => ((((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ N2) => ((((ord_less_eq @ A) @ N2) @ (((plus_plus @ A) @ J) @ K)) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ N2) @ K)) @ J))))))))))). % 6.41/2.69 thf(fact_1675_add__le__imp__le__diff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [I:A,K:A,N2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ N2) => (((ord_less_eq @ A) @ I) @ (((minus_minus @ A) @ N2) @ K))))))))). % 6.41/2.69 thf(fact_1676_less__diff__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less @ A) @ A2) @ (((minus_minus @ A) @ C2) @ B2)) = (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_1677_diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((ord_less @ A) @ A2) @ (((plus_plus @ A) @ C2) @ B2))))))))). % 6.41/2.69 thf(fact_1678_linordered__semidom__class_Oadd__diff__inverse, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A,B2:A]: (((~ ((((ord_less @ A) @ A2) @ B2))) => ((((plus_plus @ A) @ B2) @ (((minus_minus @ A) @ A2) @ B2)) = A2)))))))). % 6.41/2.69 thf(fact_1679_mult__diff__mult, axiom, (! [A:$tType]: (((ring @ A) => (! [X3:A,Y:A,A2:A,B2:A]: (((((minus_minus @ A) @ (((times_times @ A) @ X3) @ Y)) @ (((times_times @ A) @ A2) @ B2)) = (((plus_plus @ A) @ (((times_times @ A) @ X3) @ (((minus_minus @ A) @ Y) @ B2))) @ (((times_times @ A) @ (((minus_minus @ A) @ X3) @ A2)) @ B2))))))))). % 6.41/2.69 thf(fact_1680_square__diff__square__factored, axiom, (! [A:$tType]: (((comm_ring @ A) => (! [X3:A,Y:A]: (((((minus_minus @ A) @ (((times_times @ A) @ X3) @ X3)) @ (((times_times @ A) @ Y) @ Y)) = (((times_times @ A) @ (((plus_plus @ A) @ X3) @ Y)) @ (((minus_minus @ A) @ X3) @ Y))))))))). % 6.41/2.69 thf(fact_1681_eq__add__iff2, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,E3:A,C2:A,B2:A,D2:A]: ((((((plus_plus @ A) @ (((times_times @ A) @ A2) @ E3)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E3)) @ D2)) = (C2 = (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ B2) @ A2)) @ E3)) @ D2))))))))). % 6.41/2.69 thf(fact_1682_eq__add__iff1, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,E3:A,C2:A,B2:A,D2:A]: ((((((plus_plus @ A) @ (((times_times @ A) @ A2) @ E3)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E3)) @ D2)) = ((((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ E3)) @ C2) = D2)))))))). % 6.41/2.69 thf(fact_1683_finite_Ocases, axiom, (! [A:$tType,A2:(set @ A)]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => (~ ((! [A8:(set @ A)]: (((? [A5:A]: ((A2 = (((insert @ A) @ A5) @ A8)))) => (~ (((finite_finite2 @ A) @ A8))))))))))))). % 6.41/2.69 thf(fact_1684_finite_Osimps, axiom, (! [A:$tType]: (((finite_finite2 @ A) = (^ [A4:(set @ A)]: (((A4 = (bot_bot @ (set @ A))) | (? [A6:(set @ A),B4:A]: (((A4 = (((insert @ A) @ B4) @ A6)) & ((finite_finite2 @ A) @ A6))))))))))). % 6.41/2.69 thf(fact_1685_finite__induct, axiom, (! [A:$tType,F5:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F5) => ((P @ (bot_bot @ (set @ A))) => ((! [X5:A,F6:(set @ A)]: ((((finite_finite2 @ A) @ F6) => ((~ ((((member @ A) @ X5) @ F6))) => ((P @ F6) => (P @ (((insert @ A) @ X5) @ F6))))))) => (P @ F5))))))). % 6.41/2.69 thf(fact_1686_finite__ne__induct, axiom, (! [A:$tType,F5:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F5) => ((F5 != (bot_bot @ (set @ A))) => ((! [X5:A]: ((P @ (((insert @ A) @ X5) @ (bot_bot @ (set @ A)))))) => ((! [X5:A,F6:(set @ A)]: ((((finite_finite2 @ A) @ F6) => ((F6 != (bot_bot @ (set @ A))) => ((~ ((((member @ A) @ X5) @ F6))) => ((P @ F6) => (P @ (((insert @ A) @ X5) @ F6)))))))) => (P @ F5)))))))). % 6.41/2.69 thf(fact_1687_infinite__finite__induct, axiom, (! [A:$tType,P:((set @ A) > $o),A3:(set @ A)]: (((! [A8:(set @ A)]: (((~ (((finite_finite2 @ A) @ A8))) => (P @ A8)))) => ((P @ (bot_bot @ (set @ A))) => ((! [X5:A,F6:(set @ A)]: ((((finite_finite2 @ A) @ F6) => ((~ ((((member @ A) @ X5) @ F6))) => ((P @ F6) => (P @ (((insert @ A) @ X5) @ F6))))))) => (P @ A3))))))). % 6.41/2.69 thf(fact_1688_Euclidean__Division_Opos__mod__bound, axiom, (! [L:int,K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ L) => (((ord_less @ int) @ (((modulo_modulo @ int) @ K) @ L)) @ L))))). % 6.41/2.69 thf(fact_1689_neg__mod__bound, axiom, (! [L:int,K:int]: (((((ord_less @ int) @ L) @ (zero_zero @ int)) => (((ord_less @ int) @ L) @ (((modulo_modulo @ int) @ K) @ L)))))). % 6.41/2.69 thf(fact_1690_subset__singleton__iff, axiom, (! [A:$tType,X8:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ X8) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = ((X8 = (bot_bot @ (set @ A))) | (X8 = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))). % 6.41/2.69 thf(fact_1691_subset__singletonD, axiom, (! [A:$tType,A3:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) => ((A3 = (bot_bot @ (set @ A))) | (A3 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))). % 6.41/2.69 thf(fact_1692_dvd__minus__mod, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [B2:A,A2:A]: ((((dvd_dvd @ A) @ B2) @ (((minus_minus @ A) @ A2) @ (((modulo_modulo @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1693_diff__less__Suc, axiom, (! [M:nat,N2:nat]: ((((ord_less @ nat) @ (((minus_minus @ nat) @ M) @ N2)) @ (suc @ M))))). % 6.41/2.69 thf(fact_1694_Suc__diff__Suc, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ N2) @ M) => ((suc @ (((minus_minus @ nat) @ M) @ (suc @ N2))) = (((minus_minus @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_1695_diff__less, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((ord_less @ nat) @ (((minus_minus @ nat) @ M) @ N2)) @ M)))))). % 6.41/2.69 thf(fact_1696_Suc__diff__le, axiom, (! [N2:nat,M:nat]: (((((ord_less_eq @ nat) @ N2) @ M) => ((((minus_minus @ nat) @ (suc @ M)) @ N2) = (suc @ (((minus_minus @ nat) @ M) @ N2))))))). % 6.41/2.69 thf(fact_1697_diff__less__mono, axiom, (! [A2:nat,B2:nat,C2:nat]: (((((ord_less @ nat) @ A2) @ B2) => ((((ord_less_eq @ nat) @ C2) @ A2) => (((ord_less @ nat) @ (((minus_minus @ nat) @ A2) @ C2)) @ (((minus_minus @ nat) @ B2) @ C2))))))). % 6.41/2.69 thf(fact_1698_less__diff__iff, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ K) @ N2) => ((((ord_less @ nat) @ (((minus_minus @ nat) @ M) @ K)) @ (((minus_minus @ nat) @ N2) @ K)) = (((ord_less @ nat) @ M) @ N2))))))). % 6.41/2.69 thf(fact_1699_diff__add__0, axiom, (! [N2:nat,M:nat]: (((((minus_minus @ nat) @ N2) @ (((plus_plus @ nat) @ N2) @ M)) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_1700_add__diff__inverse__nat, axiom, (! [M:nat,N2:nat]: (((~ ((((ord_less @ nat) @ M) @ N2))) => ((((plus_plus @ nat) @ N2) @ (((minus_minus @ nat) @ M) @ N2)) = M))))). % 6.41/2.69 thf(fact_1701_less__diff__conv, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K)) = (((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ J))))). % 6.41/2.69 thf(fact_1702_Nat_Ole__imp__diff__is__add, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((((minus_minus @ nat) @ J) @ I) = K) = (J = (((plus_plus @ nat) @ K) @ I))))))). % 6.41/2.69 thf(fact_1703_Nat_Odiff__add__assoc2, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((minus_minus @ nat) @ (((plus_plus @ nat) @ J) @ I)) @ K) = (((plus_plus @ nat) @ (((minus_minus @ nat) @ J) @ K)) @ I)))))). % 6.41/2.69 thf(fact_1704_Nat_Odiff__add__assoc, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((minus_minus @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ K) = (((plus_plus @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K))))))). % 6.41/2.69 thf(fact_1705_Nat_Ole__diff__conv2, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((ord_less_eq @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K)) = (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ J)))))). % 6.41/2.69 thf(fact_1706_le__diff__conv, axiom, (! [J:nat,K:nat,I:nat]: (((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ J) @ K)) @ I) = (((ord_less_eq @ nat) @ J) @ (((plus_plus @ nat) @ I) @ K)))))). % 6.41/2.69 thf(fact_1707_diff__Suc__eq__diff__pred, axiom, (! [M:nat,N2:nat]: (((((minus_minus @ nat) @ M) @ (suc @ N2)) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ M) @ (one_one @ nat))) @ N2))))). % 6.41/2.69 thf(fact_1708_mod__geq, axiom, (! [M:nat,N2:nat]: (((~ ((((ord_less @ nat) @ M) @ N2))) => ((((modulo_modulo @ nat) @ M) @ N2) = (((modulo_modulo @ nat) @ (((minus_minus @ nat) @ M) @ N2)) @ N2)))))). % 6.41/2.69 thf(fact_1709_mod__if, axiom, ((modulo_modulo @ nat) = (^ [M2:nat,N:nat]: (((((if @ nat) @ (((ord_less @ nat) @ M2) @ N)) @ M2) @ (((modulo_modulo @ nat) @ (((minus_minus @ nat) @ M2) @ N)) @ N)))))). % 6.41/2.69 thf(fact_1710_le__mod__geq, axiom, (! [N2:nat,M:nat]: (((((ord_less_eq @ nat) @ N2) @ M) => ((((modulo_modulo @ nat) @ M) @ N2) = (((modulo_modulo @ nat) @ (((minus_minus @ nat) @ M) @ N2)) @ N2)))))). % 6.41/2.69 thf(fact_1711_mod__eq__dvd__iff__nat, axiom, (! [N2:nat,M:nat,Q4:nat]: (((((ord_less_eq @ nat) @ N2) @ M) => (((((modulo_modulo @ nat) @ M) @ Q4) = (((modulo_modulo @ nat) @ N2) @ Q4)) = (((dvd_dvd @ nat) @ Q4) @ (((minus_minus @ nat) @ M) @ N2))))))). % 6.41/2.69 thf(fact_1712_ordered__ring__class_Ole__add__iff2, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A2:A,E3:A,C2:A,B2:A,D2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ A2) @ E3)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E3)) @ D2)) = (((ord_less_eq @ A) @ C2) @ (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ B2) @ A2)) @ E3)) @ D2))))))))). % 6.41/2.69 thf(fact_1713_ordered__ring__class_Ole__add__iff1, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A2:A,E3:A,C2:A,B2:A,D2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ A2) @ E3)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E3)) @ D2)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ E3)) @ C2)) @ D2)))))))). % 6.41/2.69 thf(fact_1714_less__add__iff2, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A2:A,E3:A,C2:A,B2:A,D2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ A2) @ E3)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E3)) @ D2)) = (((ord_less @ A) @ C2) @ (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ B2) @ A2)) @ E3)) @ D2))))))))). % 6.41/2.69 thf(fact_1715_less__add__iff1, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A2:A,E3:A,C2:A,B2:A,D2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ A2) @ E3)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E3)) @ D2)) = (((ord_less @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ E3)) @ C2)) @ D2)))))))). % 6.41/2.69 thf(fact_1716_divide__diff__eq__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,X3:A,Y:A]: (((Z2 != (zero_zero @ A)) => ((((minus_minus @ A) @ (((divide_divide @ A) @ X3) @ Z2)) @ Y) = (((divide_divide @ A) @ (((minus_minus @ A) @ X3) @ (((times_times @ A) @ Y) @ Z2))) @ Z2))))))))). % 6.41/2.69 thf(fact_1717_diff__divide__eq__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,X3:A,Y:A]: (((Z2 != (zero_zero @ A)) => ((((minus_minus @ A) @ X3) @ (((divide_divide @ A) @ Y) @ Z2)) = (((divide_divide @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ X3) @ Z2)) @ Y)) @ Z2))))))))). % 6.41/2.69 thf(fact_1718_diff__frac__eq, axiom, (! [A:$tType]: (((field @ A) => (! [Y:A,Z2:A,X3:A,W2:A]: (((Y != (zero_zero @ A)) => ((Z2 != (zero_zero @ A)) => ((((minus_minus @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ (((divide_divide @ A) @ W2) @ Z2)) = (((divide_divide @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ X3) @ Z2)) @ (((times_times @ A) @ W2) @ Y))) @ (((times_times @ A) @ Y) @ Z2))))))))))). % 6.41/2.69 thf(fact_1719_add__divide__eq__if__simps_I4_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,A2:A,B2:A]: ((((Z2 = (zero_zero @ A)) => ((((minus_minus @ A) @ A2) @ (((divide_divide @ A) @ B2) @ Z2)) = A2)) & ((Z2 != (zero_zero @ A)) => ((((minus_minus @ A) @ A2) @ (((divide_divide @ A) @ B2) @ Z2)) = (((divide_divide @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ A2) @ Z2)) @ B2)) @ Z2)))))))))). % 6.41/2.69 thf(fact_1720_square__diff__one__factored, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X3:A]: (((((minus_minus @ A) @ (((times_times @ A) @ X3) @ X3)) @ (one_one @ A)) = (((times_times @ A) @ (((plus_plus @ A) @ X3) @ (one_one @ A))) @ (((minus_minus @ A) @ X3) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_1721_finite__ranking__induct, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [S3:(set @ B),P:((set @ B) > $o),F2:(B > A)]: ((((finite_finite2 @ B) @ S3) => ((P @ (bot_bot @ (set @ B))) => ((! [X5:B,S5:(set @ B)]: ((((finite_finite2 @ B) @ S5) => ((! [Y5:B]: (((((member @ B) @ Y5) @ S5) => (((ord_less_eq @ A) @ (F2 @ Y5)) @ (F2 @ X5))))) => ((P @ S5) => (P @ (((insert @ B) @ X5) @ S5))))))) => (P @ S3)))))))))). % 6.41/2.69 thf(fact_1722_finite__linorder__min__induct, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A3) => ((P @ (bot_bot @ (set @ A))) => ((! [B5:A,A8:(set @ A)]: ((((finite_finite2 @ A) @ A8) => ((! [X2:A]: (((((member @ A) @ X2) @ A8) => (((ord_less @ A) @ B5) @ X2)))) => ((P @ A8) => (P @ (((insert @ A) @ B5) @ A8))))))) => (P @ A3)))))))))). % 6.41/2.69 thf(fact_1723_finite__linorder__max__induct, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A3) => ((P @ (bot_bot @ (set @ A))) => ((! [B5:A,A8:(set @ A)]: ((((finite_finite2 @ A) @ A8) => ((! [X2:A]: (((((member @ A) @ X2) @ A8) => (((ord_less @ A) @ X2) @ B5)))) => ((P @ A8) => (P @ (((insert @ A) @ B5) @ A8))))))) => (P @ A3)))))))))). % 6.41/2.69 thf(fact_1724_inf__period_I4_J, axiom, (! [A:$tType]: ((((comm_ring @ A) & (dvd @ A)) => (! [D2:A,D6:A,T2:A]: (((((dvd_dvd @ A) @ D2) @ D6) => (! [X2:A,K4:A]: (((~ ((((dvd_dvd @ A) @ D2) @ (((plus_plus @ A) @ X2) @ T2)))) = (~ ((((dvd_dvd @ A) @ D2) @ (((plus_plus @ A) @ (((minus_minus @ A) @ X2) @ (((times_times @ A) @ K4) @ D6))) @ T2)))))))))))))). % 6.41/2.69 thf(fact_1725_inf__period_I3_J, axiom, (! [A:$tType]: ((((comm_ring @ A) & (dvd @ A)) => (! [D2:A,D6:A,T2:A]: (((((dvd_dvd @ A) @ D2) @ D6) => (! [X2:A,K4:A]: (((((dvd_dvd @ A) @ D2) @ (((plus_plus @ A) @ X2) @ T2)) = (((dvd_dvd @ A) @ D2) @ (((plus_plus @ A) @ (((minus_minus @ A) @ X2) @ (((times_times @ A) @ K4) @ D6))) @ T2)))))))))))). % 6.41/2.69 thf(fact_1726_finite__subset__induct, axiom, (! [A:$tType,F5:(set @ A),A3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F5) => ((((ord_less_eq @ (set @ A)) @ F5) @ A3) => ((P @ (bot_bot @ (set @ A))) => ((! [A5:A,F6:(set @ A)]: ((((finite_finite2 @ A) @ F6) => ((((member @ A) @ A5) @ A3) => ((~ ((((member @ A) @ A5) @ F6))) => ((P @ F6) => (P @ (((insert @ A) @ A5) @ F6)))))))) => (P @ F5)))))))). % 6.41/2.69 thf(fact_1727_finite__subset__induct_H, axiom, (! [A:$tType,F5:(set @ A),A3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F5) => ((((ord_less_eq @ (set @ A)) @ F5) @ A3) => ((P @ (bot_bot @ (set @ A))) => ((! [A5:A,F6:(set @ A)]: ((((finite_finite2 @ A) @ F6) => ((((member @ A) @ A5) @ A3) => ((((ord_less_eq @ (set @ A)) @ F6) @ A3) => ((~ ((((member @ A) @ A5) @ F6))) => ((P @ F6) => (P @ (((insert @ A) @ A5) @ F6))))))))) => (P @ F5)))))))). % 6.41/2.69 thf(fact_1728_minus__mult__div__eq__mod, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ A2) @ (((times_times @ A) @ B2) @ (((divide_divide @ A) @ A2) @ B2))) = (((modulo_modulo @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1729_minus__mod__eq__mult__div, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ A2) @ (((modulo_modulo @ A) @ A2) @ B2)) = (((times_times @ A) @ B2) @ (((divide_divide @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1730_minus__mod__eq__div__mult, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ A2) @ (((modulo_modulo @ A) @ A2) @ B2)) = (((times_times @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ B2)))))))). % 6.41/2.69 thf(fact_1731_minus__div__mult__eq__mod, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ A2) @ (((times_times @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ B2)) = (((modulo_modulo @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1732_eucl__rel__int__dividesI, axiom, (! [L:int,K:int,Q4:int]: (((L != (zero_zero @ int)) => ((K = (((times_times @ int) @ Q4) @ L)) => (((eucl_rel_int @ K) @ L) @ ((((product_Pair @ int) @ int) @ Q4) @ (zero_zero @ int)))))))). % 6.41/2.69 thf(fact_1733_Euclidean__Division_Opos__mod__sign, axiom, (! [L:int,K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ L) => (((ord_less_eq @ int) @ (zero_zero @ int)) @ (((modulo_modulo @ int) @ K) @ L)))))). % 6.41/2.69 thf(fact_1734_neg__mod__sign, axiom, (! [L:int,K:int]: (((((ord_less @ int) @ L) @ (zero_zero @ int)) => (((ord_less_eq @ int) @ (((modulo_modulo @ int) @ K) @ L)) @ (zero_zero @ int)))))). % 6.41/2.69 thf(fact_1735_neg__mod__conj, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ B2) @ (zero_zero @ int)) => ((((ord_less_eq @ int) @ (((modulo_modulo @ int) @ A2) @ B2)) @ (zero_zero @ int)) & (((ord_less @ int) @ B2) @ (((modulo_modulo @ int) @ A2) @ B2))))))). % 6.41/2.69 thf(fact_1736_pos__mod__conj, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((modulo_modulo @ int) @ A2) @ B2)) & (((ord_less @ int) @ (((modulo_modulo @ int) @ A2) @ B2)) @ B2)))))). % 6.41/2.69 thf(fact_1737_zmod__trivial__iff, axiom, (! [I:int,K:int]: ((((((modulo_modulo @ int) @ I) @ K) = I) = ((K = (zero_zero @ int)) | (((((ord_less_eq @ int) @ (zero_zero @ int)) @ I) & (((ord_less @ int) @ I) @ K)) | ((((ord_less_eq @ int) @ I) @ (zero_zero @ int)) & (((ord_less @ int) @ K) @ I)))))))). % 6.41/2.69 thf(fact_1738_mod__int__pos__iff, axiom, (! [K:int,L:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((modulo_modulo @ int) @ K) @ L)) = ((((dvd_dvd @ int) @ L) @ K) | (((L = (zero_zero @ int)) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ K)) | (((ord_less @ int) @ (zero_zero @ int)) @ L))))))). % 6.41/2.69 thf(fact_1739_diff__Suc__less, axiom, (! [N2:nat,I:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((ord_less @ nat) @ (((minus_minus @ nat) @ N2) @ (suc @ I))) @ N2))))). % 6.41/2.69 thf(fact_1740_zdiv__mono__strict, axiom, (! [A3:int,B3:int,N2:int]: (((((ord_less @ int) @ A3) @ B3) => ((((ord_less @ int) @ (zero_zero @ int)) @ N2) => (((((modulo_modulo @ int) @ A3) @ N2) = (zero_zero @ int)) => (((((modulo_modulo @ int) @ B3) @ N2) = (zero_zero @ int)) => (((ord_less @ int) @ (((divide_divide @ int) @ A3) @ N2)) @ (((divide_divide @ int) @ B3) @ N2))))))))). % 6.41/2.69 thf(fact_1741_nat__diff__split__asm, axiom, (! [P:(nat > $o),A2:nat,B2:nat]: (((P @ (((minus_minus @ nat) @ A2) @ B2)) = (~ ((((((ord_less @ nat) @ A2) @ B2) & (~ ((P @ (zero_zero @ nat))))) | (? [D5:nat]: (((A2 = (((plus_plus @ nat) @ B2) @ D5)) & (~ ((P @ D5))))))))))))). % 6.41/2.69 thf(fact_1742_nat__diff__split, axiom, (! [P:(nat > $o),A2:nat,B2:nat]: (((P @ (((minus_minus @ nat) @ A2) @ B2)) = (((((ord_less @ nat) @ A2) @ B2) => (P @ (zero_zero @ nat))) & (! [D5:nat]: (((A2 = (((plus_plus @ nat) @ B2) @ D5)) => (P @ D5))))))))). % 6.41/2.69 thf(fact_1743_less__diff__conv2, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((ord_less @ nat) @ (((minus_minus @ nat) @ J) @ K)) @ I) = (((ord_less @ nat) @ J) @ (((plus_plus @ nat) @ I) @ K))))))). % 6.41/2.69 thf(fact_1744_nat__diff__add__eq2, axiom, (! [I:nat,J:nat,U:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((minus_minus @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N2)) = (((minus_minus @ nat) @ M) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N2))))))). % 6.41/2.69 thf(fact_1745_nat__diff__add__eq1, axiom, (! [J:nat,I:nat,U:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ J) @ I) => ((((minus_minus @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N2)) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M)) @ N2)))))). % 6.41/2.69 thf(fact_1746_nat__le__add__iff2, axiom, (! [I:nat,J:nat,U:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N2)) = (((ord_less_eq @ nat) @ M) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N2))))))). % 6.41/2.69 thf(fact_1747_nat__le__add__iff1, axiom, (! [J:nat,I:nat,U:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ J) @ I) => ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N2)) = (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M)) @ N2)))))). % 6.41/2.69 thf(fact_1748_nat__eq__add__iff2, axiom, (! [I:nat,J:nat,U:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M) = (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N2)) = (M = (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N2))))))). % 6.41/2.69 thf(fact_1749_nat__eq__add__iff1, axiom, (! [J:nat,I:nat,U:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ J) @ I) => (((((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M) = (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N2)) = ((((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M) = N2)))))). % 6.41/2.69 thf(fact_1750_dvd__minus__add, axiom, (! [Q4:nat,N2:nat,R2:nat,M:nat]: (((((ord_less_eq @ nat) @ Q4) @ N2) => ((((ord_less_eq @ nat) @ Q4) @ (((times_times @ nat) @ R2) @ M)) => ((((dvd_dvd @ nat) @ M) @ (((minus_minus @ nat) @ N2) @ Q4)) = (((dvd_dvd @ nat) @ M) @ (((plus_plus @ nat) @ N2) @ (((minus_minus @ nat) @ (((times_times @ nat) @ R2) @ M)) @ Q4))))))))). % 6.41/2.69 thf(fact_1751_set__update__subset__insert, axiom, (! [A:$tType,Xs:(list @ A),I:nat,X3:A]: ((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ ((((list_update @ A) @ Xs) @ I) @ X3))) @ (((insert @ A) @ X3) @ ((set2 @ A) @ Xs)))))). % 6.41/2.69 thf(fact_1752_mod__nat__eqI, axiom, (! [R2:nat,N2:nat,M:nat]: (((((ord_less @ nat) @ R2) @ N2) => ((((ord_less_eq @ nat) @ R2) @ M) => ((((dvd_dvd @ nat) @ N2) @ (((minus_minus @ nat) @ M) @ R2)) => ((((modulo_modulo @ nat) @ M) @ N2) = R2))))))). % 6.41/2.69 thf(fact_1753_exp__div__exp__eq, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M:nat,N2:nat]: (((((divide_divide @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) = (((times_times @ A) @ ((zero_neq_one_of_bool @ A) @ (((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M) != (zero_zero @ A)) & (((ord_less_eq @ nat) @ N2) @ M)))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ M) @ N2)))))))))). % 6.41/2.69 thf(fact_1754_frac__le__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y:A,Z2:A,X3:A,W2:A]: (((Y != (zero_zero @ A)) => ((Z2 != (zero_zero @ A)) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ (((divide_divide @ A) @ W2) @ Z2)) = (((ord_less_eq @ A) @ (((divide_divide @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ X3) @ Z2)) @ (((times_times @ A) @ W2) @ Y))) @ (((times_times @ A) @ Y) @ Z2))) @ (zero_zero @ A))))))))))). % 6.41/2.69 thf(fact_1755_frac__less__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y:A,Z2:A,X3:A,W2:A]: (((Y != (zero_zero @ A)) => ((Z2 != (zero_zero @ A)) => ((((ord_less @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ (((divide_divide @ A) @ W2) @ Z2)) = (((ord_less @ A) @ (((divide_divide @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ X3) @ Z2)) @ (((times_times @ A) @ W2) @ Y))) @ (((times_times @ A) @ Y) @ Z2))) @ (zero_zero @ A))))))))))). % 6.41/2.69 thf(fact_1756_signed__take__bit__int__less__exp, axiom, (! [N2:nat,K:int]: ((((ord_less @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N2) @ K)) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2))))). % 6.41/2.69 thf(fact_1757_power__diff, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A2:A,N2:nat,M:nat]: (((A2 != (zero_zero @ A)) => ((((ord_less_eq @ nat) @ N2) @ M) => ((((power_power @ A) @ A2) @ (((minus_minus @ nat) @ M) @ N2)) = (((divide_divide @ A) @ (((power_power @ A) @ A2) @ M)) @ (((power_power @ A) @ A2) @ N2))))))))))). % 6.41/2.69 thf(fact_1758_mod__pos__neg__trivial, axiom, (! [K:int,L:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => ((((ord_less_eq @ int) @ (((plus_plus @ int) @ K) @ L)) @ (zero_zero @ int)) => ((((modulo_modulo @ int) @ K) @ L) = (((plus_plus @ int) @ K) @ L))))))). % 6.41/2.69 thf(fact_1759_Suc__diff__eq__diff__pred, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((minus_minus @ nat) @ (suc @ M)) @ N2) = (((minus_minus @ nat) @ M) @ (((minus_minus @ nat) @ N2) @ (one_one @ nat)))))))). % 6.41/2.69 thf(fact_1760_Suc__pred_H, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (N2 = (suc @ (((minus_minus @ nat) @ N2) @ (one_one @ nat)))))))). % 6.41/2.69 thf(fact_1761_div__geq, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((~ ((((ord_less @ nat) @ M) @ N2))) => ((((divide_divide @ nat) @ M) @ N2) = (suc @ (((divide_divide @ nat) @ (((minus_minus @ nat) @ M) @ N2)) @ N2)))))))). % 6.41/2.69 thf(fact_1762_div__if, axiom, ((divide_divide @ nat) = (^ [M2:nat,N:nat]: (((((if @ nat) @ ((((ord_less @ nat) @ M2) @ N) | (N = (zero_zero @ nat)))) @ (zero_zero @ nat)) @ (suc @ (((divide_divide @ nat) @ (((minus_minus @ nat) @ M2) @ N)) @ N))))))). % 6.41/2.69 thf(fact_1763_set__replicate__Suc, axiom, (! [A:$tType,N2:nat,X3:A]: ((((set2 @ A) @ (((replicate @ A) @ (suc @ N2)) @ X3)) = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))). % 6.41/2.69 thf(fact_1764_add__eq__if, axiom, ((plus_plus @ nat) = (^ [M2:nat,N:nat]: (((((if @ nat) @ (M2 = (zero_zero @ nat))) @ N) @ (suc @ (((plus_plus @ nat) @ (((minus_minus @ nat) @ M2) @ (one_one @ nat))) @ N))))))). % 6.41/2.69 thf(fact_1765_set__replicate__conv__if, axiom, (! [A:$tType,N2:nat,X3:A]: ((((N2 = (zero_zero @ nat)) => (((set2 @ A) @ (((replicate @ A) @ N2) @ X3)) = (bot_bot @ (set @ A)))) & ((N2 != (zero_zero @ nat)) => (((set2 @ A) @ (((replicate @ A) @ N2) @ X3)) = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))). % 6.41/2.69 thf(fact_1766_nat__less__add__iff2, axiom, (! [I:nat,J:nat,U:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N2)) = (((ord_less @ nat) @ M) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N2))))))). % 6.41/2.69 thf(fact_1767_nat__less__add__iff1, axiom, (! [J:nat,I:nat,U:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ J) @ I) => ((((ord_less @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N2)) = (((ord_less @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M)) @ N2)))))). % 6.41/2.69 thf(fact_1768_mult__eq__if, axiom, ((times_times @ nat) = (^ [M2:nat,N:nat]: (((((if @ nat) @ (M2 = (zero_zero @ nat))) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ N) @ (((times_times @ nat) @ (((minus_minus @ nat) @ M2) @ (one_one @ nat))) @ N))))))). % 6.41/2.69 thf(fact_1769_scaling__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [U:A,V2:A,R2:A,S:A]: (((((ord_less_eq @ A) @ U) @ V2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ R2) => ((((ord_less_eq @ A) @ R2) @ S) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ U) @ (((divide_divide @ A) @ (((times_times @ A) @ R2) @ (((minus_minus @ A) @ V2) @ U))) @ S))) @ V2)))))))))). % 6.41/2.69 thf(fact_1770_exp__not__zero__imp__exp__diff__not__zero, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [N2:nat,M:nat]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2) != (zero_zero @ A)) => ((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ N2) @ M)) != (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1771_signed__take__bit__int__greater__eq__self__iff, axiom, (! [K:int,N2:nat]: (((((ord_less_eq @ int) @ K) @ (((bit_ri4674362597316999326ke_bit @ int) @ N2) @ K)) = (((ord_less @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)))))). % 6.41/2.69 thf(fact_1772_signed__take__bit__int__less__self__iff, axiom, (! [N2:nat,K:int]: (((((ord_less @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N2) @ K)) @ K) = (((ord_less_eq @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)) @ K))))). % 6.41/2.69 thf(fact_1773_power__diff__power__eq, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A2:A,N2:nat,M:nat]: (((A2 != (zero_zero @ A)) => (((((ord_less_eq @ nat) @ N2) @ M) => ((((divide_divide @ A) @ (((power_power @ A) @ A2) @ M)) @ (((power_power @ A) @ A2) @ N2)) = (((power_power @ A) @ A2) @ (((minus_minus @ nat) @ M) @ N2)))) & ((~ ((((ord_less_eq @ nat) @ N2) @ M))) => ((((divide_divide @ A) @ (((power_power @ A) @ A2) @ M)) @ (((power_power @ A) @ A2) @ N2)) = (((divide_divide @ A) @ (one_one @ A)) @ (((power_power @ A) @ A2) @ (((minus_minus @ nat) @ N2) @ M))))))))))))). % 6.41/2.69 thf(fact_1774_split__zmod, axiom, (! [P:(int > $o),N2:int,K:int]: (((P @ (((modulo_modulo @ int) @ N2) @ K)) = (((K = (zero_zero @ int)) => (P @ N2)) & (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (! [I3:int,J3:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ J3) & ((((ord_less @ int) @ J3) @ K) & (N2 = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I3)) @ J3)))) => (P @ J3))))) & ((((ord_less @ int) @ K) @ (zero_zero @ int)) => (! [I3:int,J3:int]: ((((((ord_less @ int) @ K) @ J3) & ((((ord_less_eq @ int) @ J3) @ (zero_zero @ int)) & (N2 = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I3)) @ J3)))) => (P @ J3))))))))))). % 6.41/2.69 thf(fact_1775_int__mod__neg__eq, axiom, (! [A2:int,B2:int,Q4:int,R2:int]: (((A2 = (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R2)) => ((((ord_less_eq @ int) @ R2) @ (zero_zero @ int)) => ((((ord_less @ int) @ B2) @ R2) => ((((modulo_modulo @ int) @ A2) @ B2) = R2))))))). % 6.41/2.69 thf(fact_1776_int__mod__pos__eq, axiom, (! [A2:int,B2:int,Q4:int,R2:int]: (((A2 = (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R2)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ R2) => ((((ord_less @ int) @ R2) @ B2) => ((((modulo_modulo @ int) @ A2) @ B2) = R2))))))). % 6.41/2.69 thf(fact_1777_power__eq__if, axiom, (! [A:$tType]: (((power @ A) => ((power_power @ A) = (^ [P6:A,M2:nat]: (((((if @ A) @ (M2 = (zero_zero @ nat))) @ (one_one @ A)) @ (((times_times @ A) @ P6) @ (((power_power @ A) @ P6) @ (((minus_minus @ nat) @ M2) @ (one_one @ nat)))))))))))). % 6.41/2.69 thf(fact_1778_power__minus__mult, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [N2:nat,A2:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((times_times @ A) @ (((power_power @ A) @ A2) @ (((minus_minus @ nat) @ N2) @ (one_one @ nat)))) @ A2) = (((power_power @ A) @ A2) @ N2))))))))). % 6.41/2.69 thf(fact_1779_diff__le__diff__pow, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ M) @ N2)) @ (((minus_minus @ nat) @ (((power_power @ nat) @ K) @ M)) @ (((power_power @ nat) @ K) @ N2))))))). % 6.41/2.69 thf(fact_1780_le__div__geq, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less_eq @ nat) @ N2) @ M) => ((((divide_divide @ nat) @ M) @ N2) = (suc @ (((divide_divide @ nat) @ (((minus_minus @ nat) @ M) @ N2)) @ N2)))))))). % 6.41/2.69 thf(fact_1781_bits__induct, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [P:(A > $o),A2:A]: (((! [A5:A]: ((((((divide_divide @ A) @ A5) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = A5) => (P @ A5)))) => ((! [A5:A,B5:$o]: (((P @ A5) => (((((divide_divide @ A) @ (((plus_plus @ A) @ ((zero_neq_one_of_bool @ A) @ B5)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A5))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = A5) => (P @ (((plus_plus @ A) @ ((zero_neq_one_of_bool @ A) @ B5)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A5))))))) => (P @ A2))))))))). % 6.41/2.69 thf(fact_1782_verit__le__mono__div__int, axiom, (! [A3:int,B3:int,N2:int]: (((((ord_less @ int) @ A3) @ B3) => ((((ord_less @ int) @ (zero_zero @ int)) @ N2) => (((ord_less_eq @ int) @ (((plus_plus @ int) @ (((divide_divide @ int) @ A3) @ N2)) @ ((((if @ int) @ ((((modulo_modulo @ int) @ B3) @ N2) = (zero_zero @ int))) @ (one_one @ int)) @ (zero_zero @ int)))) @ (((divide_divide @ int) @ B3) @ N2))))))). % 6.41/2.69 thf(fact_1783_split__pos__lemma, axiom, (! [K:int,P:(int > (int > $o)),N2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (((P @ (((divide_divide @ int) @ N2) @ K)) @ (((modulo_modulo @ int) @ N2) @ K)) = (! [I3:int,J3:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ J3) & ((((ord_less @ int) @ J3) @ K) & (N2 = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I3)) @ J3)))) => ((P @ I3) @ J3))))))))). % 6.41/2.69 thf(fact_1784_split__neg__lemma, axiom, (! [K:int,P:(int > (int > $o)),N2:int]: (((((ord_less @ int) @ K) @ (zero_zero @ int)) => (((P @ (((divide_divide @ int) @ N2) @ K)) @ (((modulo_modulo @ int) @ N2) @ K)) = (! [I3:int,J3:int]: ((((((ord_less @ int) @ K) @ J3) & ((((ord_less_eq @ int) @ J3) @ (zero_zero @ int)) & (N2 = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I3)) @ J3)))) => ((P @ I3) @ J3))))))))). % 6.41/2.69 thf(fact_1785_int__power__div__base, axiom, (! [M:nat,K:int]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less @ int) @ (zero_zero @ int)) @ K) => ((((divide_divide @ int) @ (((power_power @ int) @ K) @ M)) @ K) = (((power_power @ int) @ K) @ (((minus_minus @ nat) @ M) @ (suc @ (zero_zero @ nat)))))))))). % 6.41/2.69 thf(fact_1786_set__decode__plus__power__2, axiom, (! [N2:nat,Z2:nat]: (((~ ((((member @ nat) @ N2) @ (nat_set_decode @ Z2)))) => ((nat_set_decode @ (((plus_plus @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) @ Z2)) = (((insert @ nat) @ N2) @ (nat_set_decode @ Z2))))))). % 6.41/2.69 thf(fact_1787_exp__mod__exp, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [M:nat,N2:nat]: (((((modulo_modulo @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) = (((times_times @ A) @ ((zero_neq_one_of_bool @ A) @ (((ord_less @ nat) @ M) @ N2))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M))))))))). % 6.41/2.69 thf(fact_1788_power2__diff, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [X3:A,Y:A]: (((((power_power @ A) @ (((minus_minus @ A) @ X3) @ Y)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((minus_minus @ A) @ (((plus_plus @ A) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (((times_times @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ X3)) @ Y))))))))). % 6.41/2.69 thf(fact_1789_mult__exp__mod__exp__eq, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M:nat,N2:nat,A2:A]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((modulo_modulo @ A) @ (((times_times @ A) @ A2) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) = (((times_times @ A) @ (((modulo_modulo @ A) @ A2) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ N2) @ M)))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)))))))))). % 6.41/2.69 thf(fact_1790_even__mod__4__div__2, axiom, (! [N2:nat]: ((((((modulo_modulo @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ one2)))) = (suc @ (zero_zero @ nat))) => (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((divide_divide @ nat) @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat)))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))). % 6.41/2.69 thf(fact_1791_eucl__rel__int__iff, axiom, (! [K:int,L:int,Q4:int,R2:int]: (((((eucl_rel_int @ K) @ L) @ ((((product_Pair @ int) @ int) @ Q4) @ R2)) = ((K = (((plus_plus @ int) @ (((times_times @ int) @ L) @ Q4)) @ R2)) & (((((ord_less @ int) @ (zero_zero @ int)) @ L) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ R2) & (((ord_less @ int) @ R2) @ L))) & ((~ ((((ord_less @ int) @ (zero_zero @ int)) @ L))) => (((((ord_less @ int) @ L) @ (zero_zero @ int)) => ((((ord_less @ int) @ L) @ R2) & (((ord_less_eq @ int) @ R2) @ (zero_zero @ int)))) & ((~ ((((ord_less @ int) @ L) @ (zero_zero @ int)))) => (Q4 = (zero_zero @ int))))))))))). % 6.41/2.69 thf(fact_1792_divmod__digit__1_I2_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ B2) @ (((modulo_modulo @ A) @ A2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) => ((((minus_minus @ A) @ (((modulo_modulo @ A) @ A2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) @ B2) = (((modulo_modulo @ A) @ A2) @ B2))))))))))). % 6.41/2.69 thf(fact_1793_even__mask__div__iff_H, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [M:nat,N2:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)) @ (one_one @ A))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2))) = (((ord_less_eq @ nat) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_1794_even__mask__div__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M:nat,N2:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)) @ (one_one @ A))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2))) = (((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2) = (zero_zero @ A)) | (((ord_less_eq @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_1795_divmod__step__eq, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [L:num,R2:A,Q4:A]: ((((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ L)) @ R2) => ((((unique1321980374590559556d_step @ A) @ L) @ ((((product_Pair @ A) @ A) @ Q4) @ R2)) = ((((product_Pair @ A) @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Q4)) @ (one_one @ A))) @ (((minus_minus @ A) @ R2) @ ((numeral_numeral @ A) @ L))))) & ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ L)) @ R2))) => ((((unique1321980374590559556d_step @ A) @ L) @ ((((product_Pair @ A) @ A) @ Q4) @ R2)) = ((((product_Pair @ A) @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Q4)) @ R2)))))))))). % 6.41/2.69 thf(fact_1796_inrange, axiom, (! [T2:vEBT_VEBT,N2:nat]: ((((vEBT_invar_vebt @ T2) @ N2) => (((ord_less_eq @ (set @ nat)) @ (vEBT_VEBT_set_vebt @ T2)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) @ (one_one @ nat)))))))). % 6.41/2.69 thf(fact_1797_Suc__if__eq, axiom, (! [A:$tType,F2:(nat > A),H2:(nat > A),G:A,N2:nat]: (((! [N3:nat]: (((F2 @ (suc @ N3)) = (H2 @ N3)))) => (((F2 @ (zero_zero @ nat)) = G) => (((N2 = (zero_zero @ nat)) => ((F2 @ N2) = G)) & ((N2 != (zero_zero @ nat)) => ((F2 @ N2) = (H2 @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))))))))))). % 6.41/2.69 thf(fact_1798_Divides_Oadjust__div__eq, axiom, (! [Q4:int,R2:int]: (((adjust_div @ ((((product_Pair @ int) @ int) @ Q4) @ R2)) = (((plus_plus @ int) @ Q4) @ ((zero_neq_one_of_bool @ int) @ (R2 != (zero_zero @ int)))))))). % 6.41/2.69 thf(fact_1799_signed__take__bit__rec, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((bit_ri4674362597316999326ke_bit @ A) = (^ [N:nat,A4:A]: (((((if @ A) @ (N = (zero_zero @ nat))) @ ((uminus_uminus @ A) @ (((modulo_modulo @ A) @ A4) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ (((plus_plus @ A) @ (((modulo_modulo @ A) @ A4) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_ri4674362597316999326ke_bit @ A) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ (((divide_divide @ A) @ A4) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))))). % 6.41/2.69 thf(fact_1800_neg__eucl__rel__int__mult__2, axiom, (! [B2:int,A2:int,Q4:int,R2:int]: (((((ord_less_eq @ int) @ B2) @ (zero_zero @ int)) => ((((eucl_rel_int @ (((plus_plus @ int) @ A2) @ (one_one @ int))) @ B2) @ ((((product_Pair @ int) @ int) @ Q4) @ R2)) => (((eucl_rel_int @ (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ A2))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ B2)) @ ((((product_Pair @ int) @ int) @ Q4) @ (((minus_minus @ int) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ R2)) @ (one_one @ int))))))))). % 6.41/2.69 thf(fact_1801_diff__shunt__var, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A,Y:A]: ((((((minus_minus @ A) @ X3) @ Y) = (bot_bot @ A)) = (((ord_less_eq @ A) @ X3) @ Y)))))))). % 6.41/2.69 thf(fact_1802_take__bit__rec, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se2584673776208193580ke_bit @ A) = (^ [N:nat,A4:A]: (((((if @ A) @ (N = (zero_zero @ nat))) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (((times_times @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ (((divide_divide @ A) @ A4) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((modulo_modulo @ A) @ A4) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))). % 6.41/2.69 thf(fact_1803_add_Oinverse__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ A2)) = A2))))))). % 6.41/2.69 thf(fact_1804_neg__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((uminus_uminus @ A) @ A2) = ((uminus_uminus @ A) @ B2)) = (A2 = B2)))))))). % 6.41/2.69 thf(fact_1805_Diff__cancel, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ A3) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_1806_empty__Diff, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_1807_Diff__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = A3)))). % 6.41/2.69 thf(fact_1808_finite__Diff2, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => (((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = ((finite_finite2 @ A) @ A3)))))). % 6.41/2.69 thf(fact_1809_finite__Diff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A3) @ B3)))))). % 6.41/2.69 thf(fact_1810_Compl__subset__Compl__iff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A3)) @ ((uminus_uminus @ (set @ A)) @ B3)) = (((ord_less_eq @ (set @ A)) @ B3) @ A3))))). % 6.41/2.69 thf(fact_1811_Compl__anti__mono, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B3)) @ ((uminus_uminus @ (set @ A)) @ A3)))))). % 6.41/2.69 thf(fact_1812_Diff__insert0, axiom, (! [A:$tType,X3:A,A3:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X3) @ A3))) => ((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ B3)) = (((minus_minus @ (set @ A)) @ A3) @ B3)))))). % 6.41/2.69 thf(fact_1813_insert__Diff1, axiom, (! [A:$tType,X3:A,B3:(set @ A),A3:(set @ A)]: (((((member @ A) @ X3) @ B3) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ B3) = (((minus_minus @ (set @ A)) @ A3) @ B3)))))). % 6.41/2.69 thf(fact_1814_compl__le__compl__iff, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X3)) @ ((uminus_uminus @ A) @ Y)) = (((ord_less_eq @ A) @ Y) @ X3)))))))). % 6.41/2.69 thf(fact_1815_neg__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1816_neg__equal__zero, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A2:A]: (((((uminus_uminus @ A) @ A2) = A2) = (A2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1817_equal__neg__zero, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A2:A]: (((A2 = ((uminus_uminus @ A) @ A2)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1818_neg__equal__0__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((uminus_uminus @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1819_neg__0__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((zero_zero @ A) = ((uminus_uminus @ A) @ A2)) = ((zero_zero @ A) = A2)))))))). % 6.41/2.69 thf(fact_1820_add_Oinverse__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (((uminus_uminus @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_1821_compl__less__compl__iff, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ X3)) @ ((uminus_uminus @ A) @ Y)) = (((ord_less @ A) @ Y) @ X3)))))))). % 6.41/2.69 thf(fact_1822_neg__less__iff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1823_mult__minus__left, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,B2:A]: (((((times_times @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = ((uminus_uminus @ A) @ (((times_times @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1824_minus__mult__minus, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,B2:A]: (((((times_times @ A) @ ((uminus_uminus @ A) @ A2)) @ ((uminus_uminus @ A) @ B2)) = (((times_times @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1825_mult__minus__right, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,B2:A]: (((((times_times @ A) @ A2) @ ((uminus_uminus @ A) @ B2)) = ((uminus_uminus @ A) @ (((times_times @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1826_minus__add__distrib, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A2) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ ((uminus_uminus @ A) @ B2))))))))). % 6.41/2.69 thf(fact_1827_minus__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ (((plus_plus @ A) @ A2) @ B2)) = B2))))))). % 6.41/2.69 thf(fact_1828_add__minus__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ B2)) = B2))))))). % 6.41/2.69 thf(fact_1829_minus__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((minus_minus @ A) @ A2) @ B2)) = (((minus_minus @ A) @ B2) @ A2)))))))). % 6.41/2.69 thf(fact_1830_dvd__minus__iff, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [X3:A,Y:A]: (((((dvd_dvd @ A) @ X3) @ ((uminus_uminus @ A) @ Y)) = (((dvd_dvd @ A) @ X3) @ Y)))))))). % 6.41/2.69 thf(fact_1831_minus__dvd__iff, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [X3:A,Y:A]: (((((dvd_dvd @ A) @ ((uminus_uminus @ A) @ X3)) @ Y) = (((dvd_dvd @ A) @ X3) @ Y)))))))). % 6.41/2.69 thf(fact_1832_Diff__eq__empty__iff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((((minus_minus @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))) = (((ord_less_eq @ (set @ A)) @ A3) @ B3))))). % 6.41/2.69 thf(fact_1833_atLeastAtMost__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,L:A,U:A]: (((((member @ A) @ I) @ (((set_or1337092689740270186AtMost @ A) @ L) @ U)) = ((((ord_less_eq @ A) @ L) @ I) & (((ord_less_eq @ A) @ I) @ U))))))))). % 6.41/2.69 thf(fact_1834_Icc__eq__Icc, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,H2:A,L3:A,H3:A]: ((((((set_or1337092689740270186AtMost @ A) @ L) @ H2) = (((set_or1337092689740270186AtMost @ A) @ L3) @ H3)) = (((L = L3) & (H2 = H3)) | ((~ ((((ord_less_eq @ A) @ L) @ H2))) & (~ ((((ord_less_eq @ A) @ L3) @ H3)))))))))))). % 6.41/2.69 thf(fact_1835_insert__Diff__single, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((insert @ A) @ A2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = (((insert @ A) @ A2) @ A3))))). % 6.41/2.69 thf(fact_1836_finite__Diff__insert, axiom, (! [A:$tType,A3:(set @ A),A2:A,B3:(set @ A)]: ((((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B3))) = ((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A3) @ B3)))))). % 6.41/2.69 thf(fact_1837_take__bit__of__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ N2) @ (zero_zero @ A)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1838_finite__atLeastAtMost, axiom, (! [L:nat,U:nat]: (((finite_finite2 @ nat) @ (((set_or1337092689740270186AtMost @ nat) @ L) @ U))))). % 6.41/2.69 thf(fact_1839_neg__less__eq__nonneg, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))). % 6.41/2.69 thf(fact_1840_less__eq__neg__nonpos, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1841_neg__le__0__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))). % 6.41/2.69 thf(fact_1842_neg__0__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1843_less__neg__neg, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1844_neg__less__pos, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))). % 6.41/2.69 thf(fact_1845_neg__0__less__iff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1846_neg__less__0__iff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))). % 6.41/2.69 thf(fact_1847_add_Oright__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1848_ab__left__minus, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1849_verit__minus__simplify_I3_J, axiom, (! [B:$tType]: (((group_add @ B) => (! [B2:B]: (((((minus_minus @ B) @ (zero_zero @ B)) @ B2) = ((uminus_uminus @ B) @ B2)))))))). % 6.41/2.69 thf(fact_1850_diff__0, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A2) = ((uminus_uminus @ A) @ A2)))))))). % 6.41/2.69 thf(fact_1851_add__neg__numeral__simps_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num,N2:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))) = ((uminus_uminus @ A) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N2)))))))))). % 6.41/2.69 thf(fact_1852_uminus__add__conv__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,B2:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = (((minus_minus @ A) @ B2) @ A2)))))))). % 6.41/2.69 thf(fact_1853_diff__minus__eq__add, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((minus_minus @ A) @ A2) @ ((uminus_uminus @ A) @ B2)) = (((plus_plus @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1854_divide__minus1, axiom, (! [A:$tType]: (((field @ A) => (! [X3:A]: (((((divide_divide @ A) @ X3) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ X3)))))))). % 6.41/2.69 thf(fact_1855_atLeastatMost__empty__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A]: ((((((set_or1337092689740270186AtMost @ A) @ A2) @ B2) = (bot_bot @ (set @ A))) = (~ ((((ord_less_eq @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_1856_atLeastatMost__empty__iff2, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A]: ((((bot_bot @ (set @ A)) = (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (~ ((((ord_less_eq @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_1857_atLeastatMost__subset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D2)) = ((~ ((((ord_less_eq @ A) @ A2) @ B2))) | ((((ord_less_eq @ A) @ C2) @ A2) & (((ord_less_eq @ A) @ B2) @ D2)))))))))). % 6.41/2.69 thf(fact_1858_atLeastatMost__empty, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => ((((set_or1337092689740270186AtMost @ A) @ A2) @ B2) = (bot_bot @ (set @ A)))))))))). % 6.41/2.69 thf(fact_1859_infinite__Icc__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A]: (((~ (((finite_finite2 @ A) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)))) = (((ord_less @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1860_subset__Compl__singleton, axiom, (! [A:$tType,A3:(set @ A),B2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ ((uminus_uminus @ (set @ A)) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))) = (~ ((((member @ A) @ B2) @ A3))))))). % 6.41/2.69 thf(fact_1861_take__bit__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se2584673776208193580ke_bit @ A) @ (zero_zero @ nat)) @ A2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1862_atLeastAtMost__singleton, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: (((((set_or1337092689740270186AtMost @ A) @ A2) @ A2) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))))))))). % 6.41/2.69 thf(fact_1863_atLeastAtMost__singleton__iff, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C2:A]: ((((((set_or1337092689740270186AtMost @ A) @ A2) @ B2) = (((insert @ A) @ C2) @ (bot_bot @ (set @ A)))) = ((A2 = B2) & (B2 = C2))))))))). % 6.41/2.69 thf(fact_1864_take__bit__Suc__1, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N2:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ (suc @ N2)) @ (one_one @ A)) = (one_one @ A)))))))). % 6.41/2.69 thf(fact_1865_add__neg__numeral__special_I8_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((plus_plus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (one_one @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_1866_add__neg__numeral__special_I7_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((plus_plus @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_1867_diff__numeral__special_I12_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((minus_minus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ A) @ (one_one @ A))) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_1868_mod__minus1__right, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A2:A]: (((((modulo_modulo @ A) @ A2) @ ((uminus_uminus @ A) @ (one_one @ A))) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1869_take__bit__of__1__eq__0__iff, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N2:nat]: ((((((bit_se2584673776208193580ke_bit @ A) @ N2) @ (one_one @ A)) = (zero_zero @ A)) = (N2 = (zero_zero @ nat))))))))). % 6.41/2.69 thf(fact_1870_semiring__norm_I168_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [V2:num,W2:num,Y:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ Y)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ V2) @ W2)))) @ Y)))))))). % 6.41/2.69 thf(fact_1871_neg__numeral__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num,N2:num]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))) = (((ord_less_eq @ num) @ N2) @ M)))))))). % 6.41/2.69 thf(fact_1872_neg__numeral__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num,N2:num]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))) = (((ord_less @ num) @ N2) @ M)))))))). % 6.41/2.69 thf(fact_1873_zle__diff1__eq, axiom, (! [W2:int,Z2:int]: (((((ord_less_eq @ int) @ W2) @ (((minus_minus @ int) @ Z2) @ (one_one @ int))) = (((ord_less @ int) @ W2) @ Z2))))). % 6.41/2.69 thf(fact_1874_take__bit__of__Suc__0, axiom, (! [N2:nat]: (((((bit_se2584673776208193580ke_bit @ nat) @ N2) @ (suc @ (zero_zero @ nat))) = ((zero_neq_one_of_bool @ nat) @ (((ord_less @ nat) @ (zero_zero @ nat)) @ N2)))))). % 6.41/2.69 thf(fact_1875_not__neg__one__le__neg__numeral__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: (((~ ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))))) = (M != one2)))))))). % 6.41/2.69 thf(fact_1876_divide__le__eq__numeral1_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,W2:num,A2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))) @ A2) = (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))) @ B2)))))))). % 6.41/2.69 thf(fact_1877_le__divide__eq__numeral1_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,W2:num]: (((((ord_less_eq @ A) @ A2) @ (((divide_divide @ A) @ B2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))) = (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ A2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))))))))))). % 6.41/2.69 thf(fact_1878_divide__eq__eq__numeral1_I2_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,W2:num,A2:A]: ((((((divide_divide @ A) @ B2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) = A2) = (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)) != (zero_zero @ A)) => (B2 = (((times_times @ A) @ A2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))))) & ((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)) = (zero_zero @ A)) => (A2 = (zero_zero @ A))))))))))). % 6.41/2.69 thf(fact_1879_eq__divide__eq__numeral1_I2_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A,W2:num]: (((A2 = (((divide_divide @ A) @ B2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))) = (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)) != (zero_zero @ A)) => ((((times_times @ A) @ A2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) = B2)) & ((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)) = (zero_zero @ A)) => (A2 = (zero_zero @ A))))))))))). % 6.41/2.69 thf(fact_1880_neg__numeral__less__neg__one__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ (one_one @ A))) = (M != one2)))))))). % 6.41/2.69 thf(fact_1881_less__divide__eq__numeral1_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,W2:num]: (((((ord_less @ A) @ A2) @ (((divide_divide @ A) @ B2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))) = (((ord_less @ A) @ B2) @ (((times_times @ A) @ A2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))))))))))). % 6.41/2.69 thf(fact_1882_divide__less__eq__numeral1_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,W2:num,A2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))) @ A2) = (((ord_less @ A) @ (((times_times @ A) @ A2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))) @ B2)))))))). % 6.41/2.69 thf(fact_1883_take__bit__of__1, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ N2) @ (one_one @ A)) = ((zero_neq_one_of_bool @ A) @ (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))))))). % 6.41/2.69 thf(fact_1884_add__neg__numeral__special_I9_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((plus_plus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))). % 6.41/2.69 thf(fact_1885_even__take__bit__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,A2:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2)) = ((N2 = (zero_zero @ nat)) | (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2))))))))). % 6.41/2.69 thf(fact_1886_signed__take__bit__Suc__minus__bit0, axiom, (! [N2:nat,K:num]: (((((bit_ri4674362597316999326ke_bit @ int) @ (suc @ N2)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ K)))) = (((times_times @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N2) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K)))) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))). % 6.41/2.69 thf(fact_1887_take__bit__Suc__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se2584673776208193580ke_bit @ A) @ (suc @ (zero_zero @ nat))) @ A2) = (((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_1888_signed__take__bit__0, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A2:A]: (((((bit_ri4674362597316999326ke_bit @ A) @ (zero_zero @ nat)) @ A2) = ((uminus_uminus @ A) @ (((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_1889_take__bit__of__exp, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M:nat,N2:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ M) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) = (((times_times @ A) @ ((zero_neq_one_of_bool @ A) @ (((ord_less @ nat) @ N2) @ M))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2))))))))). % 6.41/2.69 thf(fact_1890_take__bit__of__2, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N2:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ N2) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (((times_times @ A) @ ((zero_neq_one_of_bool @ A) @ (((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_1891_minus__int__code_I2_J, axiom, (! [L:int]: (((((minus_minus @ int) @ (zero_zero @ int)) @ L) = ((uminus_uminus @ int) @ L))))). % 6.41/2.69 thf(fact_1892_compl__mono, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y)) @ ((uminus_uminus @ A) @ X3))))))))). % 6.41/2.69 thf(fact_1893_compl__le__swap1, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [Y:A,X3:A]: (((((ord_less_eq @ A) @ Y) @ ((uminus_uminus @ A) @ X3)) => (((ord_less_eq @ A) @ X3) @ ((uminus_uminus @ A) @ Y))))))))). % 6.41/2.69 thf(fact_1894_compl__le__swap2, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [Y:A,X3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y)) @ X3) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X3)) @ Y)))))))). % 6.41/2.69 thf(fact_1895_equation__minus__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((A2 = ((uminus_uminus @ A) @ B2)) = (B2 = ((uminus_uminus @ A) @ A2))))))))). % 6.41/2.69 thf(fact_1896_minus__equation__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((uminus_uminus @ A) @ A2) = B2) = (((uminus_uminus @ A) @ B2) = A2)))))))). % 6.41/2.69 thf(fact_1897_compl__less__swap1, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [Y:A,X3:A]: (((((ord_less @ A) @ Y) @ ((uminus_uminus @ A) @ X3)) => (((ord_less @ A) @ X3) @ ((uminus_uminus @ A) @ Y))))))))). % 6.41/2.69 thf(fact_1898_compl__less__swap2, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [Y:A,X3:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ Y)) @ X3) => (((ord_less @ A) @ ((uminus_uminus @ A) @ X3)) @ Y)))))))). % 6.41/2.69 thf(fact_1899_Compl__insert, axiom, (! [A:$tType,X3:A,A3:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) = (((minus_minus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A3)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))). % 6.41/2.69 thf(fact_1900_take__bit__add, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N2:nat,A2:A,B2:A]: (((((bit_se2584673776208193580ke_bit @ A) @ N2) @ (((plus_plus @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2)) @ (((bit_se2584673776208193580ke_bit @ A) @ N2) @ B2))) = (((bit_se2584673776208193580ke_bit @ A) @ N2) @ (((plus_plus @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1901_take__bit__tightened, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,A2:A,B2:A,M:nat]: ((((((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2) = (((bit_se2584673776208193580ke_bit @ A) @ N2) @ B2)) => ((((ord_less_eq @ nat) @ M) @ N2) => ((((bit_se2584673776208193580ke_bit @ A) @ M) @ A2) = (((bit_se2584673776208193580ke_bit @ A) @ M) @ B2)))))))))). % 6.41/2.69 thf(fact_1902_take__bit__nat__less__eq__self, axiom, (! [N2:nat,M:nat]: ((((ord_less_eq @ nat) @ (((bit_se2584673776208193580ke_bit @ nat) @ N2) @ M)) @ M)))). % 6.41/2.69 thf(fact_1903_take__bit__tightened__less__eq__nat, axiom, (! [M:nat,N2:nat,Q4:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => (((ord_less_eq @ nat) @ (((bit_se2584673776208193580ke_bit @ nat) @ M) @ Q4)) @ (((bit_se2584673776208193580ke_bit @ nat) @ N2) @ Q4)))))). % 6.41/2.69 thf(fact_1904_le__imp__neg__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2))))))))). % 6.41/2.69 thf(fact_1905_minus__le__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ A2)))))))). % 6.41/2.69 thf(fact_1906_le__minus__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ B2)) = (((ord_less_eq @ A) @ B2) @ ((uminus_uminus @ A) @ A2))))))))). % 6.41/2.69 thf(fact_1907_minus__less__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ A2)))))))). % 6.41/2.69 thf(fact_1908_less__minus__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ ((uminus_uminus @ A) @ B2)) = (((ord_less @ A) @ B2) @ ((uminus_uminus @ A) @ A2))))))))). % 6.41/2.69 thf(fact_1909_verit__negate__coefficient_I2_J, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2))))))))). % 6.41/2.69 thf(fact_1910_square__eq__iff, axiom, (! [A:$tType]: (((idom @ A) => (! [A2:A,B2:A]: ((((((times_times @ A) @ A2) @ A2) = (((times_times @ A) @ B2) @ B2)) = ((A2 = B2) | (A2 = ((uminus_uminus @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_1911_minus__mult__commute, axiom, (! [A:$tType]: (((ring @ A) => (! [A2:A,B2:A]: (((((times_times @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = (((times_times @ A) @ A2) @ ((uminus_uminus @ A) @ B2))))))))). % 6.41/2.69 thf(fact_1912_is__num__normalize_I8_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A2) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2))))))))). % 6.41/2.69 thf(fact_1913_add_Oinverse__distrib__swap, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A2) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2))))))))). % 6.41/2.69 thf(fact_1914_group__cancel_Oneg1, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,K:A,A2:A]: (((A3 = (((plus_plus @ A) @ K) @ A2)) => (((uminus_uminus @ A) @ A3) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ K)) @ ((uminus_uminus @ A) @ A2)))))))))). % 6.41/2.69 thf(fact_1915_minus__diff__commute, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [B2:A,A2:A]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ B2)) @ A2) = (((minus_minus @ A) @ ((uminus_uminus @ A) @ A2)) @ B2)))))))). % 6.41/2.69 thf(fact_1916_minus__divide__right, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((divide_divide @ A) @ A2) @ B2)) = (((divide_divide @ A) @ A2) @ ((uminus_uminus @ A) @ B2))))))))). % 6.41/2.69 thf(fact_1917_minus__divide__divide, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A]: (((((divide_divide @ A) @ ((uminus_uminus @ A) @ A2)) @ ((uminus_uminus @ A) @ B2)) = (((divide_divide @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_1918_minus__divide__left, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A]: ((((uminus_uminus @ A) @ (((divide_divide @ A) @ A2) @ B2)) = (((divide_divide @ A) @ ((uminus_uminus @ A) @ A2)) @ B2)))))))). % 6.41/2.69 thf(fact_1919_Diff__infinite__finite, axiom, (! [A:$tType,T4:(set @ A),S3:(set @ A)]: ((((finite_finite2 @ A) @ T4) => ((~ (((finite_finite2 @ A) @ S3))) => (~ (((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ S3) @ T4))))))))). % 6.41/2.69 thf(fact_1920_double__diff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C5) => ((((minus_minus @ (set @ A)) @ B3) @ (((minus_minus @ (set @ A)) @ C5) @ A3)) = A3)))))). % 6.41/2.69 thf(fact_1921_Diff__subset, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ A3)))). % 6.41/2.69 thf(fact_1922_Diff__mono, axiom, (! [A:$tType,A3:(set @ A),C5:(set @ A),D6:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ C5) => ((((ord_less_eq @ (set @ A)) @ D6) @ B3) => (((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ (((minus_minus @ (set @ A)) @ C5) @ D6))))))). % 6.41/2.69 thf(fact_1923_insert__Diff__if, axiom, (! [A:$tType,X3:A,B3:(set @ A),A3:(set @ A)]: ((((((member @ A) @ X3) @ B3) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ B3) = (((minus_minus @ (set @ A)) @ A3) @ B3))) & ((~ ((((member @ A) @ X3) @ B3))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ B3) = (((insert @ A) @ X3) @ (((minus_minus @ (set @ A)) @ A3) @ B3)))))))). % 6.41/2.69 thf(fact_1924_minus__int__code_I1_J, axiom, (! [K:int]: (((((minus_minus @ int) @ K) @ (zero_zero @ int)) = K)))). % 6.41/2.69 thf(fact_1925_uminus__int__code_I1_J, axiom, (((uminus_uminus @ int) @ (zero_zero @ int)) = (zero_zero @ int))). % 6.41/2.69 thf(fact_1926_int__distrib_I4_J, axiom, (! [W2:int,Z1:int,Z22:int]: (((((times_times @ int) @ W2) @ (((minus_minus @ int) @ Z1) @ Z22)) = (((minus_minus @ int) @ (((times_times @ int) @ W2) @ Z1)) @ (((times_times @ int) @ W2) @ Z22)))))). % 6.41/2.69 thf(fact_1927_int__distrib_I3_J, axiom, (! [Z1:int,Z22:int,W2:int]: (((((times_times @ int) @ (((minus_minus @ int) @ Z1) @ Z22)) @ W2) = (((minus_minus @ int) @ (((times_times @ int) @ Z1) @ W2)) @ (((times_times @ int) @ Z22) @ W2)))))). % 6.41/2.69 thf(fact_1928_psubset__imp__ex__mem, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B3) => (? [B5:A]: ((((member @ A) @ B5) @ (((minus_minus @ (set @ A)) @ B3) @ A3)))))))). % 6.41/2.69 thf(fact_1929_zdvd__zdiffD, axiom, (! [K:int,M:int,N2:int]: (((((dvd_dvd @ int) @ K) @ (((minus_minus @ int) @ M) @ N2)) => ((((dvd_dvd @ int) @ K) @ N2) => (((dvd_dvd @ int) @ K) @ M)))))). % 6.41/2.69 thf(fact_1930_zmod__minus1, axiom, (! [B2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((modulo_modulo @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ B2) = (((minus_minus @ int) @ B2) @ (one_one @ int))))))). % 6.41/2.69 thf(fact_1931_take__bit__minus__small__eq, axiom, (! [K:int,N2:nat]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => ((((ord_less_eq @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)) => ((((bit_se2584673776208193580ke_bit @ int) @ N2) @ ((uminus_uminus @ int) @ K)) = (((minus_minus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)) @ K))))))). % 6.41/2.69 thf(fact_1932_infinite__Icc, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ (((finite_finite2 @ A) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2))))))))))). % 6.41/2.69 thf(fact_1933_atLeastAtMost__singleton_H, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((A2 = B2) => ((((set_or1337092689740270186AtMost @ A) @ A2) @ B2) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))))). % 6.41/2.69 thf(fact_1934_take__bit__tightened__less__eq__int, axiom, (! [M:nat,N2:nat,K:int]: (((((ord_less_eq @ nat) @ M) @ N2) => (((ord_less_eq @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ M) @ K)) @ (((bit_se2584673776208193580ke_bit @ int) @ N2) @ K)))))). % 6.41/2.69 thf(fact_1935_take__bit__int__greater__self__iff, axiom, (! [K:int,N2:nat]: (((((ord_less @ int) @ K) @ (((bit_se2584673776208193580ke_bit @ int) @ N2) @ K)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))). % 6.41/2.69 thf(fact_1936_not__take__bit__negative, axiom, (! [N2:nat,K:int]: ((~ ((((ord_less @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N2) @ K)) @ (zero_zero @ int))))))). % 6.41/2.69 thf(fact_1937_signed__take__bit__eq__iff__take__bit__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N2:nat,A2:A,B2:A]: ((((((bit_ri4674362597316999326ke_bit @ A) @ N2) @ A2) = (((bit_ri4674362597316999326ke_bit @ A) @ N2) @ B2)) = ((((bit_se2584673776208193580ke_bit @ A) @ (suc @ N2)) @ A2) = (((bit_se2584673776208193580ke_bit @ A) @ (suc @ N2)) @ B2))))))))). % 6.41/2.69 thf(fact_1938_signed__take__bit__take__bit, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [M:nat,N2:nat,A2:A]: (((((bit_ri4674362597316999326ke_bit @ A) @ M) @ (((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2)) = (((((if @ (A > A)) @ (((ord_less_eq @ nat) @ N2) @ M)) @ ((bit_se2584673776208193580ke_bit @ A) @ N2)) @ ((bit_ri4674362597316999326ke_bit @ A) @ M)) @ A2)))))))). % 6.41/2.69 thf(fact_1939_ex__nat__less, axiom, (! [N2:nat,P:(nat > $o)]: (((? [M2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) & (P @ M2)))) = (? [X:nat]: (((((member @ nat) @ X) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2)) & (P @ X)))))))). % 6.41/2.69 thf(fact_1940_all__nat__less, axiom, (! [N2:nat,P:(nat > $o)]: (((! [M2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => (P @ M2)))) = (! [X:nat]: (((((member @ nat) @ X) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2)) => (P @ X)))))))). % 6.41/2.69 thf(fact_1941_neg__numeral__le__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num,N2:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((numeral_numeral @ A) @ N2)))))))). % 6.41/2.69 thf(fact_1942_not__numeral__le__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num,N2:num]: ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))))))))))). % 6.41/2.69 thf(fact_1943_zero__neq__neg__numeral, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [N2:num]: (((zero_zero @ A) != ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))))))))). % 6.41/2.69 thf(fact_1944_not__numeral__less__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num,N2:num]: ((~ ((((ord_less @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))))))))))). % 6.41/2.69 thf(fact_1945_neg__numeral__less__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num,N2:num]: ((((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((numeral_numeral @ A) @ N2)))))))). % 6.41/2.69 thf(fact_1946_le__minus__one__simps_I2_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (one_one @ A)))))). % 6.41/2.69 thf(fact_1947_le__minus__one__simps_I4_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_1948_zero__neq__neg__one, axiom, (! [A:$tType]: (((ring_char_0 @ A) => ((zero_zero @ A) != ((uminus_uminus @ A) @ (one_one @ A))))))). % 6.41/2.69 thf(fact_1949_add__eq__0__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (zero_zero @ A)) = (B2 = ((uminus_uminus @ A) @ A2))))))))). % 6.41/2.69 thf(fact_1950_ab__group__add__class_Oab__left__minus, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1951_add_Oinverse__unique, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: ((((((plus_plus @ A) @ A2) @ B2) = (zero_zero @ A)) => (((uminus_uminus @ A) @ A2) = B2)))))))). % 6.41/2.69 thf(fact_1952_eq__neg__iff__add__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((A2 = ((uminus_uminus @ A) @ B2)) = ((((plus_plus @ A) @ A2) @ B2) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1953_neg__eq__iff__add__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A]: (((((uminus_uminus @ A) @ A2) = B2) = ((((plus_plus @ A) @ A2) @ B2) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_1954_less__minus__one__simps_I4_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_1955_less__minus__one__simps_I2_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((ord_less @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (one_one @ A)))))). % 6.41/2.69 thf(fact_1956_nonzero__minus__divide__right, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,A2:A]: (((B2 != (zero_zero @ A)) => (((uminus_uminus @ A) @ (((divide_divide @ A) @ A2) @ B2)) = (((divide_divide @ A) @ A2) @ ((uminus_uminus @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_1957_nonzero__minus__divide__divide, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,A2:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ ((uminus_uminus @ A) @ A2)) @ ((uminus_uminus @ A) @ B2)) = (((divide_divide @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_1958_square__eq__1__iff, axiom, (! [A:$tType]: (((ring_15535105094025558882visors @ A) => (! [X3:A]: ((((((times_times @ A) @ X3) @ X3) = (one_one @ A)) = ((X3 = (one_one @ A)) | (X3 = ((uminus_uminus @ A) @ (one_one @ A))))))))))). % 6.41/2.69 thf(fact_1959_group__cancel_Osub2, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [B3:A,K:A,B2:A,A2:A]: (((B3 = (((plus_plus @ A) @ K) @ B2)) => ((((minus_minus @ A) @ A2) @ B3) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ K)) @ (((minus_minus @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_1960_diff__conv__add__uminus, axiom, (! [A:$tType]: (((group_add @ A) => ((minus_minus @ A) = (^ [A4:A,B4:A]: ((((plus_plus @ A) @ A4) @ ((uminus_uminus @ A) @ B4))))))))). % 6.41/2.69 thf(fact_1961_ab__group__add__class_Oab__diff__conv__add__uminus, axiom, (! [A:$tType]: (((ab_group_add @ A) => ((minus_minus @ A) = (^ [A4:A,B4:A]: ((((plus_plus @ A) @ A4) @ ((uminus_uminus @ A) @ B4))))))))). % 6.41/2.69 thf(fact_1962_take__bit__unset__bit__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,M:nat,A2:A]: ((((((ord_less_eq @ nat) @ N2) @ M) => ((((bit_se2584673776208193580ke_bit @ A) @ N2) @ (((bit_se2638667681897837118et_bit @ A) @ M) @ A2)) = (((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2))) & ((~ ((((ord_less_eq @ nat) @ N2) @ M))) => ((((bit_se2584673776208193580ke_bit @ A) @ N2) @ (((bit_se2638667681897837118et_bit @ A) @ M) @ A2)) = (((bit_se2638667681897837118et_bit @ A) @ M) @ (((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2))))))))))). % 6.41/2.69 thf(fact_1963_take__bit__set__bit__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,M:nat,A2:A]: ((((((ord_less_eq @ nat) @ N2) @ M) => ((((bit_se2584673776208193580ke_bit @ A) @ N2) @ (((bit_se5668285175392031749et_bit @ A) @ M) @ A2)) = (((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2))) & ((~ ((((ord_less_eq @ nat) @ N2) @ M))) => ((((bit_se2584673776208193580ke_bit @ A) @ N2) @ (((bit_se5668285175392031749et_bit @ A) @ M) @ A2)) = (((bit_se5668285175392031749et_bit @ A) @ M) @ (((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2))))))))))). % 6.41/2.69 thf(fact_1964_take__bit__flip__bit__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,M:nat,A2:A]: ((((((ord_less_eq @ nat) @ N2) @ M) => ((((bit_se2584673776208193580ke_bit @ A) @ N2) @ (((bit_se8732182000553998342ip_bit @ A) @ M) @ A2)) = (((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2))) & ((~ ((((ord_less_eq @ nat) @ N2) @ M))) => ((((bit_se2584673776208193580ke_bit @ A) @ N2) @ (((bit_se8732182000553998342ip_bit @ A) @ M) @ A2)) = (((bit_se8732182000553998342ip_bit @ A) @ M) @ (((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2))))))))))). % 6.41/2.69 thf(fact_1965_dvd__neg__div, axiom, (! [A:$tType]: (((idom_divide @ A) => (! [B2:A,A2:A]: (((((dvd_dvd @ A) @ B2) @ A2) => ((((divide_divide @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = ((uminus_uminus @ A) @ (((divide_divide @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_1966_dvd__div__neg, axiom, (! [A:$tType]: (((idom_divide @ A) => (! [B2:A,A2:A]: (((((dvd_dvd @ A) @ B2) @ A2) => ((((divide_divide @ A) @ A2) @ ((uminus_uminus @ A) @ B2)) = ((uminus_uminus @ A) @ (((divide_divide @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_1967_subset__Compl__self__eq, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ ((uminus_uminus @ (set @ A)) @ A3)) = (A3 = (bot_bot @ (set @ A))))))). % 6.41/2.69 thf(fact_1968_Diff__insert__absorb, axiom, (! [A:$tType,X3:A,A3:(set @ A)]: (((~ ((((member @ A) @ X3) @ A3))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = A3))))). % 6.41/2.69 thf(fact_1969_Diff__insert2, axiom, (! [A:$tType,A3:(set @ A),A2:A,B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B3)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) @ B3))))). % 6.41/2.69 thf(fact_1970_insert__Diff, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => ((((insert @ A) @ A2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = A3))))). % 6.41/2.69 thf(fact_1971_Diff__insert, axiom, (! [A:$tType,A3:(set @ A),A2:A,B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B3)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))))))). % 6.41/2.69 thf(fact_1972_subset__Diff__insert, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),X3:A,C5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ (((minus_minus @ (set @ A)) @ B3) @ (((insert @ A) @ X3) @ C5))) = ((((ord_less_eq @ (set @ A)) @ A3) @ (((minus_minus @ (set @ A)) @ B3) @ C5)) & (~ ((((member @ A) @ X3) @ A3)))))))). % 6.41/2.69 thf(fact_1973_real__0__less__add__iff, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ (((plus_plus @ real) @ X3) @ Y)) = (((ord_less @ real) @ ((uminus_uminus @ real) @ X3)) @ Y))))). % 6.41/2.69 thf(fact_1974_real__add__less__0__iff, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ (((plus_plus @ real) @ X3) @ Y)) @ (zero_zero @ real)) = (((ord_less @ real) @ Y) @ ((uminus_uminus @ real) @ X3)))))). % 6.41/2.69 thf(fact_1975_zminus1__lemma, axiom, (! [A2:int,B2:int,Q4:int,R2:int]: (((((eucl_rel_int @ A2) @ B2) @ ((((product_Pair @ int) @ int) @ Q4) @ R2)) => ((B2 != (zero_zero @ int)) => (((eucl_rel_int @ ((uminus_uminus @ int) @ A2)) @ B2) @ ((((product_Pair @ int) @ int) @ ((((if @ int) @ (R2 = (zero_zero @ int))) @ ((uminus_uminus @ int) @ Q4)) @ (((minus_minus @ int) @ ((uminus_uminus @ int) @ Q4)) @ (one_one @ int)))) @ ((((if @ int) @ (R2 = (zero_zero @ int))) @ (zero_zero @ int)) @ (((minus_minus @ int) @ B2) @ R2))))))))). % 6.41/2.69 thf(fact_1976_int__le__induct, axiom, (! [I:int,K:int,P:(int > $o)]: (((((ord_less_eq @ int) @ I) @ K) => ((P @ K) => ((! [I4:int]: (((((ord_less_eq @ int) @ I4) @ K) => ((P @ I4) => (P @ (((minus_minus @ int) @ I4) @ (one_one @ int))))))) => (P @ I))))))). % 6.41/2.69 thf(fact_1977_int__less__induct, axiom, (! [I:int,K:int,P:(int > $o)]: (((((ord_less @ int) @ I) @ K) => ((P @ (((minus_minus @ int) @ K) @ (one_one @ int))) => ((! [I4:int]: (((((ord_less @ int) @ I4) @ K) => ((P @ I4) => (P @ (((minus_minus @ int) @ I4) @ (one_one @ int))))))) => (P @ I))))))). % 6.41/2.69 thf(fact_1978_pos__zmult__eq__1__iff__lemma, axiom, (! [M:int,N2:int]: ((((((times_times @ int) @ M) @ N2) = (one_one @ int)) => ((M = (one_one @ int)) | (M = ((uminus_uminus @ int) @ (one_one @ int)))))))). % 6.41/2.69 thf(fact_1979_zmult__eq__1__iff, axiom, (! [M:int,N2:int]: ((((((times_times @ int) @ M) @ N2) = (one_one @ int)) = (((M = (one_one @ int)) & (N2 = (one_one @ int))) | ((M = ((uminus_uminus @ int) @ (one_one @ int))) & (N2 = ((uminus_uminus @ int) @ (one_one @ int))))))))). % 6.41/2.69 thf(fact_1980_signed__take__bit__eq__take__bit__shift, axiom, ((bit_ri4674362597316999326ke_bit @ int) = (^ [N:nat,K3:int]: ((((minus_minus @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ (suc @ N)) @ (((plus_plus @ int) @ K3) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)))) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)))))). % 6.41/2.69 thf(fact_1981_take__bit__Suc__minus__bit0, axiom, (! [N2:nat,K:num]: (((((bit_se2584673776208193580ke_bit @ int) @ (suc @ N2)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ K)))) = (((times_times @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N2) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K)))) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))). % 6.41/2.69 thf(fact_1982_take__bit__int__less__eq, axiom, (! [N2:nat,K:int]: (((((ord_less_eq @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)) @ K) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((ord_less_eq @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N2) @ K)) @ (((minus_minus @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)))))))). % 6.41/2.69 thf(fact_1983_atLeastatMost__psubset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D2)) = (((~ ((((ord_less_eq @ A) @ A2) @ B2))) | ((((ord_less_eq @ A) @ C2) @ A2) & ((((ord_less_eq @ A) @ B2) @ D2) & ((((ord_less @ A) @ C2) @ A2) | (((ord_less @ A) @ B2) @ D2))))) & (((ord_less_eq @ A) @ C2) @ D2))))))))). % 6.41/2.69 thf(fact_1984_take__bit__signed__take__bit, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [M:nat,N2:nat,A2:A]: (((((ord_less_eq @ nat) @ M) @ (suc @ N2)) => ((((bit_se2584673776208193580ke_bit @ A) @ M) @ (((bit_ri4674362597316999326ke_bit @ A) @ N2) @ A2)) = (((bit_se2584673776208193580ke_bit @ A) @ M) @ A2))))))))). % 6.41/2.69 thf(fact_1985_not__zero__le__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:num]: ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))))))))))). % 6.41/2.69 thf(fact_1986_neg__numeral__le__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))) @ (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1987_neg__numeral__less__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:num]: ((((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))) @ (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_1988_not__zero__less__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:num]: ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))))))))))). % 6.41/2.69 thf(fact_1989_le__minus__one__simps_I1_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (zero_zero @ A)))))). % 6.41/2.69 thf(fact_1990_le__minus__one__simps_I3_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_1991_less__minus__one__simps_I3_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_1992_less__minus__one__simps_I1_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((ord_less @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (zero_zero @ A)))))). % 6.41/2.69 thf(fact_1993_neg__numeral__le__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ (one_one @ A)))))))). % 6.41/2.69 thf(fact_1994_neg__one__le__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((numeral_numeral @ A) @ M)))))))). % 6.41/2.69 thf(fact_1995_neg__numeral__le__neg__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_1996_not__numeral__le__neg__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))))). % 6.41/2.69 thf(fact_1997_not__one__le__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((~ ((((ord_less_eq @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))))))))))). % 6.41/2.69 thf(fact_1998_not__neg__one__less__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((~ ((((ord_less @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))))))))))). % 6.41/2.69 thf(fact_1999_not__one__less__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((~ ((((ord_less @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))))))))))). % 6.41/2.69 thf(fact_2000_not__numeral__less__neg__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((~ ((((ord_less @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))))). % 6.41/2.69 thf(fact_2001_neg__one__less__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((((ord_less @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((numeral_numeral @ A) @ M)))))))). % 6.41/2.69 thf(fact_2002_neg__numeral__less__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:num]: ((((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ (one_one @ A)))))))). % 6.41/2.69 thf(fact_2003_eq__minus__divide__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A,C2:A]: (((A2 = ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) = (((C2 != (zero_zero @ A)) => ((((times_times @ A) @ A2) @ C2) = ((uminus_uminus @ A) @ B2))) & ((C2 = (zero_zero @ A)) => (A2 = (zero_zero @ A))))))))))). % 6.41/2.69 thf(fact_2004_minus__divide__eq__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,C2:A,A2:A]: (((((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2)) = A2) = (((C2 != (zero_zero @ A)) => (((uminus_uminus @ A) @ B2) = (((times_times @ A) @ A2) @ C2))) & ((C2 = (zero_zero @ A)) => (A2 = (zero_zero @ A))))))))))). % 6.41/2.69 thf(fact_2005_nonzero__neg__divide__eq__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,A2:A,C2:A]: (((B2 != (zero_zero @ A)) => ((((uminus_uminus @ A) @ (((divide_divide @ A) @ A2) @ B2)) = C2) = (((uminus_uminus @ A) @ A2) = (((times_times @ A) @ C2) @ B2)))))))))). % 6.41/2.69 thf(fact_2006_nonzero__neg__divide__eq__eq2, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,C2:A,A2:A]: (((B2 != (zero_zero @ A)) => ((C2 = ((uminus_uminus @ A) @ (((divide_divide @ A) @ A2) @ B2))) = ((((times_times @ A) @ C2) @ B2) = ((uminus_uminus @ A) @ A2)))))))))). % 6.41/2.69 thf(fact_2007_divide__eq__minus__1__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A]: ((((((divide_divide @ A) @ A2) @ B2) = ((uminus_uminus @ A) @ (one_one @ A))) = ((B2 != (zero_zero @ A)) & (A2 = ((uminus_uminus @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_2008_finite__empty__induct, axiom, (! [A:$tType,A3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A3) => ((P @ A3) => ((! [A5:A,A8:(set @ A)]: ((((finite_finite2 @ A) @ A8) => ((((member @ A) @ A5) @ A8) => ((P @ A8) => (P @ (((minus_minus @ (set @ A)) @ A8) @ (((insert @ A) @ A5) @ (bot_bot @ (set @ A)))))))))) => (P @ (bot_bot @ (set @ A))))))))). % 6.41/2.69 thf(fact_2009_infinite__coinduct, axiom, (! [A:$tType,X8:((set @ A) > $o),A3:(set @ A)]: (((X8 @ A3) => ((! [A8:(set @ A)]: (((X8 @ A8) => (? [X2:A]: (((((member @ A) @ X2) @ A8) & ((X8 @ (((minus_minus @ (set @ A)) @ A8) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))) | (~ (((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A8) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))))))))))) => (~ (((finite_finite2 @ A) @ A3)))))))). % 6.41/2.69 thf(fact_2010_infinite__remove, axiom, (! [A:$tType,S3:(set @ A),A2:A]: (((~ (((finite_finite2 @ A) @ S3))) => (~ (((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ S3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))))). % 6.41/2.69 thf(fact_2011_Diff__single__insert, axiom, (! [A:$tType,A3:(set @ A),X3:A,B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) @ B3) => (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ B3)))))). % 6.41/2.69 thf(fact_2012_subset__insert__iff, axiom, (! [A:$tType,A3:(set @ A),X3:A,B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ B3)) = (((((member @ A) @ X3) @ A3) => (((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) @ B3)) & ((~ ((((member @ A) @ X3) @ A3))) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))))). % 6.41/2.69 thf(fact_2013_take__bit__Suc__minus__1__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N2:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ (suc @ N2)) @ ((uminus_uminus @ A) @ (one_one @ A))) = (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (suc @ N2))) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_2014_atLeast0__atMost__Suc, axiom, (! [N2:nat]: (((((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N2)) = (((insert @ nat) @ (suc @ N2)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2)))))). % 6.41/2.69 thf(fact_2015_mod__pos__geq, axiom, (! [L:int,K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ L) => ((((ord_less_eq @ int) @ L) @ K) => ((((modulo_modulo @ int) @ K) @ L) = (((modulo_modulo @ int) @ (((minus_minus @ int) @ K) @ L)) @ L))))))). % 6.41/2.69 thf(fact_2016_Icc__eq__insert__lb__nat, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((set_or1337092689740270186AtMost @ nat) @ M) @ N2) = (((insert @ nat) @ M) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ N2))))))). % 6.41/2.69 thf(fact_2017_atLeastAtMostSuc__conv, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N2)) => ((((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N2)) = (((insert @ nat) @ (suc @ N2)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))). % 6.41/2.69 thf(fact_2018_atLeastAtMost__insertL, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((insert @ nat) @ M) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ N2)) = (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_2019_plusinfinity, axiom, (! [D2:int,P5:(int > $o),P:(int > $o)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D2) => ((! [X5:int,K2:int]: (((P5 @ X5) = (P5 @ (((minus_minus @ int) @ X5) @ (((times_times @ int) @ K2) @ D2)))))) => ((? [Z4:int]: ((! [X5:int]: (((((ord_less @ int) @ Z4) @ X5) => ((P @ X5) = (P5 @ X5))))))) => ((? [X_12:int]: ((P5 @ X_12))) => (? [X_1:int]: ((P @ X_1)))))))))). % 6.41/2.69 thf(fact_2020_minusinfinity, axiom, (! [D2:int,P1:(int > $o),P:(int > $o)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D2) => ((! [X5:int,K2:int]: (((P1 @ X5) = (P1 @ (((minus_minus @ int) @ X5) @ (((times_times @ int) @ K2) @ D2)))))) => ((? [Z4:int]: ((! [X5:int]: (((((ord_less @ int) @ X5) @ Z4) => ((P @ X5) = (P1 @ X5))))))) => ((? [X_12:int]: ((P1 @ X_12))) => (? [X_1:int]: ((P @ X_1)))))))))). % 6.41/2.69 thf(fact_2021_int__induct, axiom, (! [P:(int > $o),K:int,I:int]: (((P @ K) => ((! [I4:int]: (((((ord_less_eq @ int) @ K) @ I4) => ((P @ I4) => (P @ (((plus_plus @ int) @ I4) @ (one_one @ int))))))) => ((! [I4:int]: (((((ord_less_eq @ int) @ I4) @ K) => ((P @ I4) => (P @ (((minus_minus @ int) @ I4) @ (one_one @ int))))))) => (P @ I))))))). % 6.41/2.69 thf(fact_2022_subset__eq__atLeast0__atMost__finite, axiom, (! [N7:(set @ nat),N2:nat]: (((((ord_less_eq @ (set @ nat)) @ N7) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2)) => ((finite_finite2 @ nat) @ N7))))). % 6.41/2.69 thf(fact_2023_take__bit__int__less__exp, axiom, (! [N2:nat,K:int]: ((((ord_less @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N2) @ K)) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2))))). % 6.41/2.69 thf(fact_2024_pos__minus__divide__less__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) @ A2) = (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A2) @ C2)))))))))). % 6.41/2.69 thf(fact_2025_pos__less__minus__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less @ A) @ A2) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) = (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ ((uminus_uminus @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_2026_neg__minus__divide__less__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) @ A2) = (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ ((uminus_uminus @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_2027_neg__less__minus__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less @ A) @ A2) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) = (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A2) @ C2)))))))))). % 6.41/2.69 thf(fact_2028_minus__divide__less__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) @ A2) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A2) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ ((uminus_uminus @ A) @ B2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))))))). % 6.41/2.69 thf(fact_2029_less__minus__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ ((uminus_uminus @ A) @ B2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A2) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))))))). % 6.41/2.69 thf(fact_2030_divide__eq__eq__numeral_I2_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,C2:A,W2:num]: ((((((divide_divide @ A) @ B2) @ C2) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) = (((C2 != (zero_zero @ A)) => (B2 = (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2))) & ((C2 = (zero_zero @ A)) => (((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)) = (zero_zero @ A))))))))))). % 6.41/2.69 thf(fact_2031_eq__divide__eq__numeral_I2_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [W2:num,B2:A,C2:A]: (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)) = (((divide_divide @ A) @ B2) @ C2)) = (((C2 != (zero_zero @ A)) => ((((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2) = B2)) & ((C2 = (zero_zero @ A)) => (((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)) = (zero_zero @ A))))))))))). % 6.41/2.69 thf(fact_2032_minus__divide__add__eq__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,X3:A,Y:A]: (((Z2 != (zero_zero @ A)) => ((((plus_plus @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ X3) @ Z2))) @ Y) = (((divide_divide @ A) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ X3)) @ (((times_times @ A) @ Y) @ Z2))) @ Z2))))))))). % 6.41/2.69 thf(fact_2033_add__divide__eq__if__simps_I3_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,A2:A,B2:A]: ((((Z2 = (zero_zero @ A)) => ((((plus_plus @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ A2) @ Z2))) @ B2) = B2)) & ((Z2 != (zero_zero @ A)) => ((((plus_plus @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ A2) @ Z2))) @ B2) = (((divide_divide @ A) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ A2)) @ (((times_times @ A) @ B2) @ Z2))) @ Z2)))))))))). % 6.41/2.69 thf(fact_2034_add__divide__eq__if__simps_I6_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,A2:A,B2:A]: ((((Z2 = (zero_zero @ A)) => ((((minus_minus @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ A2) @ Z2))) @ B2) = ((uminus_uminus @ A) @ B2))) & ((Z2 != (zero_zero @ A)) => ((((minus_minus @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ A2) @ Z2))) @ B2) = (((divide_divide @ A) @ (((minus_minus @ A) @ ((uminus_uminus @ A) @ A2)) @ (((times_times @ A) @ B2) @ Z2))) @ Z2)))))))))). % 6.41/2.69 thf(fact_2035_add__divide__eq__if__simps_I5_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,A2:A,B2:A]: ((((Z2 = (zero_zero @ A)) => ((((minus_minus @ A) @ (((divide_divide @ A) @ A2) @ Z2)) @ B2) = ((uminus_uminus @ A) @ B2))) & ((Z2 != (zero_zero @ A)) => ((((minus_minus @ A) @ (((divide_divide @ A) @ A2) @ Z2)) @ B2) = (((divide_divide @ A) @ (((minus_minus @ A) @ A2) @ (((times_times @ A) @ B2) @ Z2))) @ Z2)))))))))). % 6.41/2.69 thf(fact_2036_minus__divide__diff__eq__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,X3:A,Y:A]: (((Z2 != (zero_zero @ A)) => ((((minus_minus @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ X3) @ Z2))) @ Y) = (((divide_divide @ A) @ (((minus_minus @ A) @ ((uminus_uminus @ A) @ X3)) @ (((times_times @ A) @ Y) @ Z2))) @ Z2))))))))). % 6.41/2.69 thf(fact_2037_remove__induct, axiom, (! [A:$tType,P:((set @ A) > $o),B3:(set @ A)]: (((P @ (bot_bot @ (set @ A))) => (((~ (((finite_finite2 @ A) @ B3))) => (P @ B3)) => ((! [A8:(set @ A)]: ((((finite_finite2 @ A) @ A8) => ((A8 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ A8) @ B3) => ((! [X2:A]: (((((member @ A) @ X2) @ A8) => (P @ (((minus_minus @ (set @ A)) @ A8) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))))) => (P @ A8))))))) => (P @ B3))))))). % 6.41/2.69 thf(fact_2038_finite__remove__induct, axiom, (! [A:$tType,B3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ B3) => ((P @ (bot_bot @ (set @ A))) => ((! [A8:(set @ A)]: ((((finite_finite2 @ A) @ A8) => ((A8 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ A8) @ B3) => ((! [X2:A]: (((((member @ A) @ X2) @ A8) => (P @ (((minus_minus @ (set @ A)) @ A8) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))))) => (P @ A8))))))) => (P @ B3))))))). % 6.41/2.69 thf(fact_2039_signed__take__bit__int__greater__self__iff, axiom, (! [K:int,N2:nat]: (((((ord_less @ int) @ K) @ (((bit_ri4674362597316999326ke_bit @ int) @ N2) @ K)) = (((ord_less @ int) @ K) @ ((uminus_uminus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2))))))). % 6.41/2.69 thf(fact_2040_finite__induct__select, axiom, (! [A:$tType,S3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ S3) => ((P @ (bot_bot @ (set @ A))) => ((! [T5:(set @ A)]: (((((ord_less @ (set @ A)) @ T5) @ S3) => ((P @ T5) => (? [X2:A]: (((((member @ A) @ X2) @ (((minus_minus @ (set @ A)) @ S3) @ T5)) & (P @ (((insert @ A) @ X2) @ T5))))))))) => (P @ S3))))))). % 6.41/2.69 thf(fact_2041_psubset__insert__iff, axiom, (! [A:$tType,A3:(set @ A),X3:A,B3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ B3)) = (((((member @ A) @ X3) @ B3) => (((ord_less @ (set @ A)) @ A3) @ B3)) & ((~ ((((member @ A) @ X3) @ B3))) => (((((member @ A) @ X3) @ A3) => (((ord_less @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) @ B3)) & ((~ ((((member @ A) @ X3) @ A3))) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))))))). % 6.41/2.69 thf(fact_2042_decr__mult__lemma, axiom, (! [D2:int,P:(int > $o),K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D2) => ((! [X5:int]: (((P @ X5) => (P @ (((minus_minus @ int) @ X5) @ D2))))) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (! [X2:int]: (((P @ X2) => (P @ (((minus_minus @ int) @ X2) @ (((times_times @ int) @ K) @ D2)))))))))))). % 6.41/2.69 thf(fact_2043_verit__less__mono__div__int2, axiom, (! [A3:int,B3:int,N2:int]: (((((ord_less_eq @ int) @ A3) @ B3) => ((((ord_less @ int) @ (zero_zero @ int)) @ ((uminus_uminus @ int) @ N2)) => (((ord_less_eq @ int) @ (((divide_divide @ int) @ B3) @ N2)) @ (((divide_divide @ int) @ A3) @ N2))))))). % 6.41/2.69 thf(fact_2044_div__eq__minus1, axiom, (! [B2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((divide_divide @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ B2) = ((uminus_uminus @ int) @ (one_one @ int))))))). % 6.41/2.69 thf(fact_2045_take__bit__Suc__bit0, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N2:nat,K:num]: (((((bit_se2584673776208193580ke_bit @ A) @ (suc @ N2)) @ ((numeral_numeral @ A) @ (bit0 @ K))) = (((times_times @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ N2) @ ((numeral_numeral @ A) @ K))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_2046_take__bit__nat__eq__self, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) => ((((bit_se2584673776208193580ke_bit @ nat) @ N2) @ M) = M))))). % 6.41/2.69 thf(fact_2047_take__bit__nat__less__exp, axiom, (! [N2:nat,M:nat]: ((((ord_less @ nat) @ (((bit_se2584673776208193580ke_bit @ nat) @ N2) @ M)) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))))). % 6.41/2.69 thf(fact_2048_take__bit__nat__eq__self__iff, axiom, (! [N2:nat,M:nat]: ((((((bit_se2584673776208193580ke_bit @ nat) @ N2) @ M) = M) = (((ord_less @ nat) @ M) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)))))). % 6.41/2.69 thf(fact_2049_take__bit__int__less__self__iff, axiom, (! [N2:nat,K:int]: (((((ord_less @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N2) @ K)) @ K) = (((ord_less_eq @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)) @ K))))). % 6.41/2.69 thf(fact_2050_take__bit__int__greater__eq__self__iff, axiom, (! [K:int,N2:nat]: (((((ord_less_eq @ int) @ K) @ (((bit_se2584673776208193580ke_bit @ int) @ N2) @ K)) = (((ord_less @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)))))). % 6.41/2.69 thf(fact_2051_le__minus__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ ((uminus_uminus @ A) @ B2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A2) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))))))). % 6.41/2.69 thf(fact_2052_minus__divide__le__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) @ A2) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A2) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ ((uminus_uminus @ A) @ B2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))))))). % 6.41/2.69 thf(fact_2053_neg__le__minus__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A2) @ C2)))))))))). % 6.41/2.69 thf(fact_2054_neg__minus__divide__le__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) @ A2) = (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ ((uminus_uminus @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_2055_pos__le__minus__divide__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) = (((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ C2)) @ ((uminus_uminus @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_2056_pos__minus__divide__le__eq, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,B2:A,A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ B2) @ C2))) @ A2) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ (((times_times @ A) @ A2) @ C2)))))))))). % 6.41/2.69 thf(fact_2057_divide__less__eq__numeral_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,W2:num]: (((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ B2) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))))))))))))). % 6.41/2.69 thf(fact_2058_less__divide__eq__numeral_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [W2:num,B2:A,C2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ (zero_zero @ A))))))))))))). % 6.41/2.69 thf(fact_2059_signed__take__bit__int__less__eq, axiom, (! [N2:nat,K:int]: (((((ord_less_eq @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)) @ K) => (((ord_less_eq @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N2) @ K)) @ (((minus_minus @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (suc @ N2)))))))). % 6.41/2.69 thf(fact_2060_neg__one__power__add__eq__neg__one__power__diff, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [K:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ N2) => ((((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (((plus_plus @ nat) @ N2) @ K)) = (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (((minus_minus @ nat) @ N2) @ K)))))))))). % 6.41/2.69 thf(fact_2061_signed__take__bit__int__eq__self, axiom, (! [N2:nat,K:int]: (((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2))) @ K) => ((((ord_less @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)) => ((((bit_ri4674362597316999326ke_bit @ int) @ N2) @ K) = K)))))). % 6.41/2.69 thf(fact_2062_signed__take__bit__int__eq__self__iff, axiom, (! [N2:nat,K:int]: ((((((bit_ri4674362597316999326ke_bit @ int) @ N2) @ K) = K) = ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2))) @ K) & (((ord_less @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2))))))). % 6.41/2.69 thf(fact_2063_take__bit__eq__0__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,A2:A]: ((((((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2) = (zero_zero @ A)) = (((dvd_dvd @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) @ A2)))))))). % 6.41/2.69 thf(fact_2064_take__bit__nat__less__self__iff, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (((bit_se2584673776208193580ke_bit @ nat) @ N2) @ M)) @ M) = (((ord_less_eq @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) @ M))))). % 6.41/2.69 thf(fact_2065_take__bit__int__eq__self, axiom, (! [K:int,N2:nat]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((ord_less @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)) => ((((bit_se2584673776208193580ke_bit @ int) @ N2) @ K) = K)))))). % 6.41/2.69 thf(fact_2066_take__bit__int__eq__self__iff, axiom, (! [N2:nat,K:int]: ((((((bit_se2584673776208193580ke_bit @ int) @ N2) @ K) = K) = ((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) & (((ord_less @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2))))))). % 6.41/2.69 thf(fact_2067_divide__le__eq__numeral_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,C2:A,W2:num]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ C2)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2)))))))))))))). % 6.41/2.69 thf(fact_2068_le__divide__eq__numeral_I2_J, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [W2:num,B2:A,C2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ (((divide_divide @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2)) @ B2)) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ (((times_times @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ C2))) & ((~ ((((ord_less @ A) @ C2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ (zero_zero @ A))))))))))))). % 6.41/2.69 thf(fact_2069_square__le__1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ X3) => ((((ord_less_eq @ A) @ X3) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_2070_div__pos__geq, axiom, (! [L:int,K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ L) => ((((ord_less_eq @ int) @ L) @ K) => ((((divide_divide @ int) @ K) @ L) = (((plus_plus @ int) @ (((divide_divide @ int) @ (((minus_minus @ int) @ K) @ L)) @ L)) @ (one_one @ int)))))))). % 6.41/2.69 thf(fact_2071_div__pos__neg__trivial, axiom, (! [K:int,L:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => ((((ord_less_eq @ int) @ (((plus_plus @ int) @ K) @ L)) @ (zero_zero @ int)) => ((((divide_divide @ int) @ K) @ L) = ((uminus_uminus @ int) @ (one_one @ int)))))))). % 6.41/2.69 thf(fact_2072_take__bit__int__greater__eq, axiom, (! [K:int,N2:nat]: (((((ord_less @ int) @ K) @ (zero_zero @ int)) => (((ord_less_eq @ int) @ (((plus_plus @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2))) @ (((bit_se2584673776208193580ke_bit @ int) @ N2) @ K)))))). % 6.41/2.69 thf(fact_2073_power__minus1__odd, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N2:nat]: (((((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) = ((uminus_uminus @ A) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_2074_take__bit__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,A2:A]: (((((bit_se2584673776208193580ke_bit @ A) @ (suc @ N2)) @ A2) = (((plus_plus @ A) @ (((times_times @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ N2) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_2075_signed__take__bit__int__greater__eq, axiom, (! [K:int,N2:nat]: (((((ord_less @ int) @ K) @ ((uminus_uminus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2))) => (((ord_less_eq @ int) @ (((plus_plus @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (suc @ N2)))) @ (((bit_ri4674362597316999326ke_bit @ int) @ N2) @ K)))))). % 6.41/2.69 thf(fact_2076_stable__imp__take__bit__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A,N2:nat]: ((((((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = A2) => (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2) => ((((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2) = (zero_zero @ A))) & ((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2))) => ((((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2) = (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) @ (one_one @ A)))))))))))). % 6.41/2.69 thf(fact_2077_signed__take__bit__Suc__minus__bit1, axiom, (! [N2:nat,K:num]: (((((bit_ri4674362597316999326ke_bit @ int) @ (suc @ N2)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ K)))) = (((plus_plus @ int) @ (((times_times @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N2) @ (((minus_minus @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K))) @ (one_one @ int)))) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (one_one @ int)))))). % 6.41/2.69 thf(fact_2078_concat__bit__Suc, axiom, (! [N2:nat,K:int,L:int]: (((((bit_concat_bit @ (suc @ N2)) @ K) @ L) = (((plus_plus @ int) @ (((modulo_modulo @ int) @ K) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_concat_bit @ N2) @ (((divide_divide @ int) @ K) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ L))))))). % 6.41/2.69 thf(fact_2079_Bolzano, axiom, (! [A2:real,B2:real,P:(real > (real > $o))]: (((((ord_less_eq @ real) @ A2) @ B2) => ((! [A5:real,B5:real,C3:real]: ((((P @ A5) @ B5) => (((P @ B5) @ C3) => ((((ord_less_eq @ real) @ A5) @ B5) => ((((ord_less_eq @ real) @ B5) @ C3) => ((P @ A5) @ C3))))))) => ((! [X5:real]: (((((ord_less_eq @ real) @ A2) @ X5) => ((((ord_less_eq @ real) @ X5) @ B2) => (? [D4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D4) & (! [A5:real,B5:real]: ((((((ord_less_eq @ real) @ A5) @ X5) & ((((ord_less_eq @ real) @ X5) @ B5) & (((ord_less @ real) @ (((minus_minus @ real) @ B5) @ A5)) @ D4))) => ((P @ A5) @ B5))))))))))) => ((P @ A2) @ B2))))))). % 6.41/2.69 thf(fact_2080_signed__take__bit__Suc__bit1, axiom, (! [N2:nat,K:num]: (((((bit_ri4674362597316999326ke_bit @ int) @ (suc @ N2)) @ ((numeral_numeral @ int) @ (bit1 @ K))) = (((plus_plus @ int) @ (((times_times @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N2) @ ((numeral_numeral @ int) @ K))) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (one_one @ int)))))). % 6.41/2.69 thf(fact_2081_and__int_Oelims, axiom, (! [X3:int,Xa2:int,Y:int]: ((((((bit_se5824344872417868541ns_and @ int) @ X3) @ Xa2) = Y) => ((((((member @ int) @ X3) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ Xa2) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))) => (Y = ((uminus_uminus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ X3))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Xa2)))))))) & ((~ (((((member @ int) @ X3) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ Xa2) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))))) => (Y = (((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ X3))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Xa2)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ (((divide_divide @ int) @ X3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ Xa2) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))))))). % 6.41/2.69 thf(fact_2082_and__int_Osimps, axiom, ((bit_se5824344872417868541ns_and @ int) = (^ [K3:int,L2:int]: (((((if @ int) @ ((((member @ int) @ K3) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ L2) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))))) @ ((uminus_uminus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K3))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ L2))))))) @ (((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K3))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ L2)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ (((divide_divide @ int) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L2) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_2083_Diff__idemp, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ B3) = (((minus_minus @ (set @ A)) @ A3) @ B3))))). % 6.41/2.69 thf(fact_2084_Diff__iff, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = ((((member @ A) @ C2) @ A3) & (~ ((((member @ A) @ C2) @ B3)))))))). % 6.41/2.69 thf(fact_2085_DiffI, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ A3) => ((~ ((((member @ A) @ C2) @ B3))) => (((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A3) @ B3))))))). % 6.41/2.69 thf(fact_2086_ComplI, axiom, (! [A:$tType,C2:A,A3:(set @ A)]: (((~ ((((member @ A) @ C2) @ A3))) => (((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A3)))))). % 6.41/2.69 thf(fact_2087_Compl__iff, axiom, (! [A:$tType,C2:A,A3:(set @ A)]: (((((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A3)) = (~ ((((member @ A) @ C2) @ A3))))))). % 6.41/2.69 thf(fact_2088_Compl__eq__Compl__iff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((uminus_uminus @ (set @ A)) @ A3) = ((uminus_uminus @ (set @ A)) @ B3)) = (A3 = B3))))). % 6.41/2.69 thf(fact_2089_and__zero__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se5824344872417868541ns_and @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_2090_zero__and__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se5824344872417868541ns_and @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_2091_bit_Oconj__zero__left, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X3:A]: (((((bit_se5824344872417868541ns_and @ A) @ (zero_zero @ A)) @ X3) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_2092_bit_Oconj__zero__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X3:A]: (((((bit_se5824344872417868541ns_and @ A) @ X3) @ (zero_zero @ A)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_2093_concat__bit__0, axiom, (! [K:int,L:int]: (((((bit_concat_bit @ (zero_zero @ nat)) @ K) @ L) = L)))). % 6.41/2.69 thf(fact_2094_semiring__norm_I80_J, axiom, (! [M:num,N2:num]: (((((ord_less @ num) @ (bit1 @ M)) @ (bit1 @ N2)) = (((ord_less @ num) @ M) @ N2))))). % 6.41/2.69 thf(fact_2095_and__negative__int__iff, axiom, (! [K:int,L:int]: (((((ord_less @ int) @ (((bit_se5824344872417868541ns_and @ int) @ K) @ L)) @ (zero_zero @ int)) = ((((ord_less @ int) @ K) @ (zero_zero @ int)) & (((ord_less @ int) @ L) @ (zero_zero @ int))))))). % 6.41/2.69 thf(fact_2096_semiring__norm_I81_J, axiom, (! [M:num,N2:num]: (((((ord_less @ num) @ (bit1 @ M)) @ (bit0 @ N2)) = (((ord_less @ num) @ M) @ N2))))). % 6.41/2.69 thf(fact_2097_semiring__norm_I77_J, axiom, (! [N2:num]: ((((ord_less @ num) @ one2) @ (bit1 @ N2))))). % 6.41/2.69 thf(fact_2098_concat__bit__negative__iff, axiom, (! [N2:nat,K:int,L:int]: (((((ord_less @ int) @ (((bit_concat_bit @ N2) @ K) @ L)) @ (zero_zero @ int)) = (((ord_less @ int) @ L) @ (zero_zero @ int)))))). % 6.41/2.69 thf(fact_2099_semiring__norm_I79_J, axiom, (! [M:num,N2:num]: (((((ord_less @ num) @ (bit0 @ M)) @ (bit1 @ N2)) = (((ord_less_eq @ num) @ M) @ N2))))). % 6.41/2.69 thf(fact_2100_semiring__norm_I74_J, axiom, (! [M:num,N2:num]: (((((ord_less_eq @ num) @ (bit1 @ M)) @ (bit0 @ N2)) = (((ord_less @ num) @ M) @ N2))))). % 6.41/2.69 thf(fact_2101_and__numerals_I5_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X3:num]: (((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ (bit0 @ X3))) @ (one_one @ A)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_2102_and__numerals_I1_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [Y:num]: (((((bit_se5824344872417868541ns_and @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ Y))) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_2103_div__Suc__eq__div__add3, axiom, (! [M:nat,N2:nat]: (((((divide_divide @ nat) @ M) @ (suc @ (suc @ (suc @ N2)))) = (((divide_divide @ nat) @ M) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ N2)))))). % 6.41/2.69 thf(fact_2104_Suc__div__eq__add3__div__numeral, axiom, (! [M:nat,V2:num]: (((((divide_divide @ nat) @ (suc @ (suc @ (suc @ M)))) @ ((numeral_numeral @ nat) @ V2)) = (((divide_divide @ nat) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ M)) @ ((numeral_numeral @ nat) @ V2)))))). % 6.41/2.69 thf(fact_2105_mod__Suc__eq__mod__add3, axiom, (! [M:nat,N2:nat]: (((((modulo_modulo @ nat) @ M) @ (suc @ (suc @ (suc @ N2)))) = (((modulo_modulo @ nat) @ M) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ N2)))))). % 6.41/2.69 thf(fact_2106_Suc__mod__eq__add3__mod__numeral, axiom, (! [M:nat,V2:num]: (((((modulo_modulo @ nat) @ (suc @ (suc @ (suc @ M)))) @ ((numeral_numeral @ nat) @ V2)) = (((modulo_modulo @ nat) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ M)) @ ((numeral_numeral @ nat) @ V2)))))). % 6.41/2.69 thf(fact_2107_and__numerals_I7_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X3:num,Y:num]: (((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ (bit1 @ X3))) @ ((numeral_numeral @ A) @ (bit1 @ Y))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ X3)) @ ((numeral_numeral @ A) @ Y))))))))))). % 6.41/2.69 thf(fact_2108_DiffD2, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) => (~ ((((member @ A) @ C2) @ B3))))))). % 6.41/2.69 thf(fact_2109_DiffD1, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) => (((member @ A) @ C2) @ A3))))). % 6.41/2.69 thf(fact_2110_DiffE, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) => (~ (((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B3)))))))). % 6.41/2.69 thf(fact_2111_ComplD, axiom, (! [A:$tType,C2:A,A3:(set @ A)]: (((((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A3)) => (~ ((((member @ A) @ C2) @ A3))))))). % 6.41/2.69 thf(fact_2112_double__complement, axiom, (! [A:$tType,A3:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A3)) = A3)))). % 6.41/2.69 thf(fact_2113_concat__bit__assoc, axiom, (! [N2:nat,K:int,M:nat,L:int,R2:int]: (((((bit_concat_bit @ N2) @ K) @ (((bit_concat_bit @ M) @ L) @ R2)) = (((bit_concat_bit @ (((plus_plus @ nat) @ M) @ N2)) @ (((bit_concat_bit @ N2) @ K) @ L)) @ R2))))). % 6.41/2.69 thf(fact_2114_xor__num_Ocases, axiom, (! [X3:((product_prod @ num) @ num)]: (((X3 != ((((product_Pair @ num) @ num) @ one2) @ one2)) => ((! [N3:num]: ((X3 != ((((product_Pair @ num) @ num) @ one2) @ (bit0 @ N3))))) => ((! [N3:num]: ((X3 != ((((product_Pair @ num) @ num) @ one2) @ (bit1 @ N3))))) => ((! [M3:num]: ((X3 != ((((product_Pair @ num) @ num) @ (bit0 @ M3)) @ one2)))) => ((! [M3:num,N3:num]: ((X3 != ((((product_Pair @ num) @ num) @ (bit0 @ M3)) @ (bit0 @ N3))))) => ((! [M3:num,N3:num]: ((X3 != ((((product_Pair @ num) @ num) @ (bit0 @ M3)) @ (bit1 @ N3))))) => ((! [M3:num]: ((X3 != ((((product_Pair @ num) @ num) @ (bit1 @ M3)) @ one2)))) => ((! [M3:num,N3:num]: ((X3 != ((((product_Pair @ num) @ num) @ (bit1 @ M3)) @ (bit0 @ N3))))) => (~ ((! [M3:num,N3:num]: ((X3 != ((((product_Pair @ num) @ num) @ (bit1 @ M3)) @ (bit1 @ N3)))))))))))))))))). % 6.41/2.69 thf(fact_2115_and__less__eq, axiom, (! [L:int,K:int]: (((((ord_less @ int) @ L) @ (zero_zero @ int)) => (((ord_less_eq @ int) @ (((bit_se5824344872417868541ns_and @ int) @ K) @ L)) @ K))))). % 6.41/2.69 thf(fact_2116_AND__upper1_H_H, axiom, (! [Y:int,Z2:int,Ya:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y) => ((((ord_less @ int) @ Y) @ Z2) => (((ord_less @ int) @ (((bit_se5824344872417868541ns_and @ int) @ Y) @ Ya)) @ Z2)))))). % 6.41/2.69 thf(fact_2117_AND__upper2_H_H, axiom, (! [Y:int,Z2:int,X3:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y) => ((((ord_less @ int) @ Y) @ Z2) => (((ord_less @ int) @ (((bit_se5824344872417868541ns_and @ int) @ X3) @ Y)) @ Z2)))))). % 6.41/2.69 thf(fact_2118_numeral__Bit1, axiom, (! [A:$tType]: (((numeral @ A) => (! [N2:num]: ((((numeral_numeral @ A) @ (bit1 @ N2)) = (((plus_plus @ A) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ N2)) @ ((numeral_numeral @ A) @ N2))) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_2119_simp__from__to, axiom, ((set_or1337092689740270186AtMost @ int) = (^ [I3:int,J3:int]: (((((if @ (set @ int)) @ (((ord_less @ int) @ J3) @ I3)) @ (bot_bot @ (set @ int))) @ (((insert @ int) @ I3) @ (((set_or1337092689740270186AtMost @ int) @ (((plus_plus @ int) @ I3) @ (one_one @ int))) @ J3))))))). % 6.41/2.69 thf(fact_2120_eval__nat__numeral_I3_J, axiom, (! [N2:num]: ((((numeral_numeral @ nat) @ (bit1 @ N2)) = (suc @ ((numeral_numeral @ nat) @ (bit0 @ N2))))))). % 6.41/2.69 thf(fact_2121_cong__exp__iff__simps_I3_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N2:num,Q4:num]: (((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ (bit1 @ N2))) @ ((numeral_numeral @ A) @ (bit0 @ Q4))) != (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_2122_numeral__3__eq__3, axiom, (((numeral_numeral @ nat) @ (bit1 @ one2)) = (suc @ (suc @ (suc @ (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_2123_Suc3__eq__add__3, axiom, (! [N2:nat]: (((suc @ (suc @ (suc @ N2))) = (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ N2))))). % 6.41/2.69 thf(fact_2124_num_Osize_I6_J, axiom, (! [X32:num]: ((((size_size @ num) @ (bit1 @ X32)) = (((plus_plus @ nat) @ ((size_size @ num) @ X32)) @ (suc @ (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_2125_num_Osize__gen_I3_J, axiom, (! [X32:num]: (((size_num @ (bit1 @ X32)) = (((plus_plus @ nat) @ (size_num @ X32)) @ (suc @ (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_2126_cong__exp__iff__simps_I11_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M:num,Q4:num]: ((((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ (bit1 @ M))) @ ((numeral_numeral @ A) @ (bit0 @ Q4))) = (((modulo_modulo @ A) @ ((numeral_numeral @ A) @ one2)) @ ((numeral_numeral @ A) @ (bit0 @ Q4)))) = ((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ Q4)) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2127_cong__exp__iff__simps_I7_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [Q4:num,N2:num]: ((((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ one2)) @ ((numeral_numeral @ A) @ (bit0 @ Q4))) = (((modulo_modulo @ A) @ ((numeral_numeral @ A) @ (bit1 @ N2))) @ ((numeral_numeral @ A) @ (bit0 @ Q4)))) = ((((modulo_modulo @ A) @ ((numeral_numeral @ A) @ N2)) @ ((numeral_numeral @ A) @ Q4)) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2128_periodic__finite__ex, axiom, (! [D2:int,P:(int > $o)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D2) => ((! [X5:int,K2:int]: (((P @ X5) = (P @ (((minus_minus @ int) @ X5) @ (((times_times @ int) @ K2) @ D2)))))) => ((? [X6:int]: ((P @ X6))) = (? [X:int]: (((((member @ int) @ X) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D2)) & (P @ X)))))))))). % 6.41/2.69 thf(fact_2129_bset_I3_J, axiom, (! [D6:int,T2:int,B3:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D6) => ((((member @ int) @ (((minus_minus @ int) @ T2) @ (one_one @ int))) @ B3) => (! [X2:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B3) => (X2 != (((plus_plus @ int) @ Xb2) @ Xa3)))))))) => ((X2 = T2) => ((((minus_minus @ int) @ X2) @ D6) = T2)))))))))). % 6.41/2.69 thf(fact_2130_bset_I4_J, axiom, (! [D6:int,T2:int,B3:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D6) => ((((member @ int) @ T2) @ B3) => (! [X2:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B3) => (X2 != (((plus_plus @ int) @ Xb2) @ Xa3)))))))) => ((X2 != T2) => ((((minus_minus @ int) @ X2) @ D6) != T2)))))))))). % 6.41/2.69 thf(fact_2131_bset_I5_J, axiom, (! [D6:int,B3:(set @ int),T2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D6) => (! [X2:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B3) => (X2 != (((plus_plus @ int) @ Xb2) @ Xa3)))))))) => ((((ord_less @ int) @ X2) @ T2) => (((ord_less @ int) @ (((minus_minus @ int) @ X2) @ D6)) @ T2))))))))). % 6.41/2.69 thf(fact_2132_bset_I7_J, axiom, (! [D6:int,T2:int,B3:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D6) => ((((member @ int) @ T2) @ B3) => (! [X2:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B3) => (X2 != (((plus_plus @ int) @ Xb2) @ Xa3)))))))) => ((((ord_less @ int) @ T2) @ X2) => (((ord_less @ int) @ T2) @ (((minus_minus @ int) @ X2) @ D6))))))))))). % 6.41/2.69 thf(fact_2133_aset_I3_J, axiom, (! [D6:int,T2:int,A3:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D6) => ((((member @ int) @ (((plus_plus @ int) @ T2) @ (one_one @ int))) @ A3) => (! [X2:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A3) => (X2 != (((minus_minus @ int) @ Xb2) @ Xa3)))))))) => ((X2 = T2) => ((((plus_plus @ int) @ X2) @ D6) = T2)))))))))). % 6.41/2.69 thf(fact_2134_aset_I4_J, axiom, (! [D6:int,T2:int,A3:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D6) => ((((member @ int) @ T2) @ A3) => (! [X2:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A3) => (X2 != (((minus_minus @ int) @ Xb2) @ Xa3)))))))) => ((X2 != T2) => ((((plus_plus @ int) @ X2) @ D6) != T2)))))))))). % 6.41/2.69 thf(fact_2135_aset_I5_J, axiom, (! [D6:int,T2:int,A3:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D6) => ((((member @ int) @ T2) @ A3) => (! [X2:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A3) => (X2 != (((minus_minus @ int) @ Xb2) @ Xa3)))))))) => ((((ord_less @ int) @ X2) @ T2) => (((ord_less @ int) @ (((plus_plus @ int) @ X2) @ D6)) @ T2)))))))))). % 6.41/2.69 thf(fact_2136_aset_I7_J, axiom, (! [D6:int,A3:(set @ int),T2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D6) => (! [X2:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A3) => (X2 != (((minus_minus @ int) @ Xb2) @ Xa3)))))))) => ((((ord_less @ int) @ T2) @ X2) => (((ord_less @ int) @ T2) @ (((plus_plus @ int) @ X2) @ D6)))))))))). % 6.41/2.69 thf(fact_2137_Suc__div__eq__add3__div, axiom, (! [M:nat,N2:nat]: (((((divide_divide @ nat) @ (suc @ (suc @ (suc @ M)))) @ N2) = (((divide_divide @ nat) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ M)) @ N2))))). % 6.41/2.69 thf(fact_2138_Suc__mod__eq__add3__mod, axiom, (! [M:nat,N2:nat]: (((((modulo_modulo @ nat) @ (suc @ (suc @ (suc @ M)))) @ N2) = (((modulo_modulo @ nat) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ M)) @ N2))))). % 6.41/2.69 thf(fact_2139_aset_I8_J, axiom, (! [D6:int,A3:(set @ int),T2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D6) => (! [X2:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A3) => (X2 != (((minus_minus @ int) @ Xb2) @ Xa3)))))))) => ((((ord_less_eq @ int) @ T2) @ X2) => (((ord_less_eq @ int) @ T2) @ (((plus_plus @ int) @ X2) @ D6)))))))))). % 6.41/2.69 thf(fact_2140_aset_I6_J, axiom, (! [D6:int,T2:int,A3:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D6) => ((((member @ int) @ (((plus_plus @ int) @ T2) @ (one_one @ int))) @ A3) => (! [X2:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A3) => (X2 != (((minus_minus @ int) @ Xb2) @ Xa3)))))))) => ((((ord_less_eq @ int) @ X2) @ T2) => (((ord_less_eq @ int) @ (((plus_plus @ int) @ X2) @ D6)) @ T2)))))))))). % 6.41/2.69 thf(fact_2141_bset_I8_J, axiom, (! [D6:int,T2:int,B3:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D6) => ((((member @ int) @ (((minus_minus @ int) @ T2) @ (one_one @ int))) @ B3) => (! [X2:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B3) => (X2 != (((plus_plus @ int) @ Xb2) @ Xa3)))))))) => ((((ord_less_eq @ int) @ T2) @ X2) => (((ord_less_eq @ int) @ T2) @ (((minus_minus @ int) @ X2) @ D6))))))))))). % 6.41/2.69 thf(fact_2142_bset_I6_J, axiom, (! [D6:int,B3:(set @ int),T2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D6) => (! [X2:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B3) => (X2 != (((plus_plus @ int) @ Xb2) @ Xa3)))))))) => ((((ord_less_eq @ int) @ X2) @ T2) => (((ord_less_eq @ int) @ (((minus_minus @ int) @ X2) @ D6)) @ T2))))))))). % 6.41/2.69 thf(fact_2143_cppi, axiom, (! [D6:int,P:(int > $o),P5:(int > $o),A3:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D6) => ((? [Z4:int]: ((! [X5:int]: (((((ord_less @ int) @ Z4) @ X5) => ((P @ X5) = (P5 @ X5))))))) => ((! [X5:int]: (((! [Xa:int]: (((((member @ int) @ Xa) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ A3) => (X5 != (((minus_minus @ int) @ Xb3) @ Xa)))))))) => ((P @ X5) => (P @ (((plus_plus @ int) @ X5) @ D6)))))) => ((! [X5:int,K2:int]: (((P5 @ X5) = (P5 @ (((minus_minus @ int) @ X5) @ (((times_times @ int) @ K2) @ D6)))))) => ((? [X6:int]: ((P @ X6))) = ((? [X:int]: (((((member @ int) @ X) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) & (P5 @ X)))) | (? [X:int]: (((((member @ int) @ X) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) & (? [Y2:int]: (((((member @ int) @ Y2) @ A3) & (P @ (((minus_minus @ int) @ Y2) @ X))))))))))))))))). % 6.41/2.69 thf(fact_2144_cpmi, axiom, (! [D6:int,P:(int > $o),P5:(int > $o),B3:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D6) => ((? [Z4:int]: ((! [X5:int]: (((((ord_less @ int) @ X5) @ Z4) => ((P @ X5) = (P5 @ X5))))))) => ((! [X5:int]: (((! [Xa:int]: (((((member @ int) @ Xa) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ B3) => (X5 != (((plus_plus @ int) @ Xb3) @ Xa)))))))) => ((P @ X5) => (P @ (((minus_minus @ int) @ X5) @ D6)))))) => ((! [X5:int,K2:int]: (((P5 @ X5) = (P5 @ (((minus_minus @ int) @ X5) @ (((times_times @ int) @ K2) @ D6)))))) => ((? [X6:int]: ((P @ X6))) = ((? [X:int]: (((((member @ int) @ X) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) & (P5 @ X)))) | (? [X:int]: (((((member @ int) @ X) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D6)) & (? [Y2:int]: (((((member @ int) @ Y2) @ B3) & (P @ (((plus_plus @ int) @ Y2) @ X))))))))))))))))). % 6.41/2.69 thf(fact_2145_take__bit__Suc__bit1, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N2:nat,K:num]: (((((bit_se2584673776208193580ke_bit @ A) @ (suc @ N2)) @ ((numeral_numeral @ A) @ (bit1 @ K))) = (((plus_plus @ A) @ (((times_times @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ N2) @ ((numeral_numeral @ A) @ K))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_2146_odd__mod__4__div__2, axiom, (! [N2:nat]: ((((((modulo_modulo @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ one2)))) = ((numeral_numeral @ nat) @ (bit1 @ one2))) => (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((divide_divide @ nat) @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat)))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_2147_mod__exhaust__less__4, axiom, (! [M:nat]: ((((((modulo_modulo @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ one2)))) = (zero_zero @ nat)) | (((((modulo_modulo @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ one2)))) = (one_one @ nat)) | (((((modulo_modulo @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ one2)))) = ((numeral_numeral @ nat) @ (bit0 @ one2))) | ((((modulo_modulo @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ one2)))) = ((numeral_numeral @ nat) @ (bit1 @ one2))))))))). % 6.41/2.69 thf(fact_2148_divmod__algorithm__code_I7_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M:num,N2:num]: ((((((ord_less_eq @ num) @ M) @ N2) => ((((unique8689654367752047608divmod @ A) @ (bit0 @ M)) @ (bit1 @ N2)) = ((((product_Pair @ A) @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ (bit0 @ M))))) & ((~ ((((ord_less_eq @ num) @ M) @ N2))) => ((((unique8689654367752047608divmod @ A) @ (bit0 @ M)) @ (bit1 @ N2)) = (((unique1321980374590559556d_step @ A) @ (bit1 @ N2)) @ (((unique8689654367752047608divmod @ A) @ (bit0 @ M)) @ (bit0 @ (bit1 @ N2))))))))))))). % 6.41/2.69 thf(fact_2149_divmod__algorithm__code_I8_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M:num,N2:num]: ((((((ord_less @ num) @ M) @ N2) => ((((unique8689654367752047608divmod @ A) @ (bit1 @ M)) @ (bit1 @ N2)) = ((((product_Pair @ A) @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ (bit1 @ M))))) & ((~ ((((ord_less @ num) @ M) @ N2))) => ((((unique8689654367752047608divmod @ A) @ (bit1 @ M)) @ (bit1 @ N2)) = (((unique1321980374590559556d_step @ A) @ (bit1 @ N2)) @ (((unique8689654367752047608divmod @ A) @ (bit1 @ M)) @ (bit0 @ (bit1 @ N2))))))))))))). % 6.41/2.69 thf(fact_2150_take__bit__Suc__minus__bit1, axiom, (! [N2:nat,K:num]: (((((bit_se2584673776208193580ke_bit @ int) @ (suc @ N2)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ K)))) = (((plus_plus @ int) @ (((times_times @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N2) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (inc @ K))))) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (one_one @ int)))))). % 6.41/2.69 thf(fact_2151_and__int_Opsimps, axiom, (! [K:int,L:int]: (((((accp @ ((product_prod @ int) @ int)) @ bit_and_int_rel) @ ((((product_Pair @ int) @ int) @ K) @ L)) => ((((((member @ int) @ K) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ L) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))) => ((((bit_se5824344872417868541ns_and @ int) @ K) @ L) = ((uminus_uminus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ L)))))))) & ((~ (((((member @ int) @ K) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ L) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))))) => ((((bit_se5824344872417868541ns_and @ int) @ K) @ L) = (((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ L)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ (((divide_divide @ int) @ K) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))))))). % 6.41/2.69 thf(fact_2152_and__int_Opelims, axiom, (! [X3:int,Xa2:int,Y:int]: ((((((bit_se5824344872417868541ns_and @ int) @ X3) @ Xa2) = Y) => ((((accp @ ((product_prod @ int) @ int)) @ bit_and_int_rel) @ ((((product_Pair @ int) @ int) @ X3) @ Xa2)) => (~ ((((((((member @ int) @ X3) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ Xa2) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))) => (Y = ((uminus_uminus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ X3))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Xa2)))))))) & ((~ (((((member @ int) @ X3) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ Xa2) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))))) => (Y = (((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ X3))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Xa2)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ (((divide_divide @ int) @ X3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ Xa2) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))) => (~ ((((accp @ ((product_prod @ int) @ int)) @ bit_and_int_rel) @ ((((product_Pair @ int) @ int) @ X3) @ Xa2)))))))))))). % 6.41/2.69 thf(fact_2153_finite__atLeastAtMost__int, axiom, (! [L:int,U:int]: (((finite_finite2 @ int) @ (((set_or1337092689740270186AtMost @ int) @ L) @ U))))). % 6.41/2.69 thf(fact_2154_and__nat__numerals_I1_J, axiom, (! [Y:num]: (((((bit_se5824344872417868541ns_and @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ (bit0 @ Y))) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_2155_and__nat__numerals_I3_J, axiom, (! [X3:num]: (((((bit_se5824344872417868541ns_and @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ X3))) @ (suc @ (zero_zero @ nat))) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_2156_and__nat__numerals_I2_J, axiom, (! [Y:num]: (((((bit_se5824344872417868541ns_and @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ (bit1 @ Y))) = (one_one @ nat))))). % 6.41/2.69 thf(fact_2157_and__nat__numerals_I4_J, axiom, (! [X3:num]: (((((bit_se5824344872417868541ns_and @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ X3))) @ (suc @ (zero_zero @ nat))) = (one_one @ nat))))). % 6.41/2.69 thf(fact_2158_divmod__algorithm__code_I2_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M:num]: (((((unique8689654367752047608divmod @ A) @ M) @ one2) = ((((product_Pair @ A) @ A) @ ((numeral_numeral @ A) @ M)) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2159_add__neg__numeral__special_I6_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (inc @ M)))))))))). % 6.41/2.69 thf(fact_2160_add__neg__numeral__special_I5_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [N2:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (inc @ N2)))))))))). % 6.41/2.69 thf(fact_2161_and__Suc__0__eq, axiom, (! [N2:nat]: (((((bit_se5824344872417868541ns_and @ nat) @ N2) @ (suc @ (zero_zero @ nat))) = (((modulo_modulo @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))). % 6.41/2.69 thf(fact_2162_Suc__0__and__eq, axiom, (! [N2:nat]: (((((bit_se5824344872417868541ns_and @ nat) @ (suc @ (zero_zero @ nat))) @ N2) = (((modulo_modulo @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))). % 6.41/2.69 thf(fact_2163_divmod__algorithm__code_I3_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N2:num]: (((((unique8689654367752047608divmod @ A) @ one2) @ (bit0 @ N2)) = ((((product_Pair @ A) @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ one2))))))))). % 6.41/2.69 thf(fact_2164_divmod__algorithm__code_I4_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N2:num]: (((((unique8689654367752047608divmod @ A) @ one2) @ (bit1 @ N2)) = ((((product_Pair @ A) @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ one2))))))))). % 6.41/2.69 thf(fact_2165_numeral__inc, axiom, (! [A:$tType]: (((numeral @ A) => (! [X3:num]: ((((numeral_numeral @ A) @ (inc @ X3)) = (((plus_plus @ A) @ ((numeral_numeral @ A) @ X3)) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_2166_divmod__int__def, axiom, ((unique8689654367752047608divmod @ int) = (^ [M2:num,N:num]: (((((product_Pair @ int) @ int) @ (((divide_divide @ int) @ ((numeral_numeral @ int) @ M2)) @ ((numeral_numeral @ int) @ N))) @ (((modulo_modulo @ int) @ ((numeral_numeral @ int) @ M2)) @ ((numeral_numeral @ int) @ N))))))). % 6.41/2.69 thf(fact_2167_divmod__def, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => ((unique8689654367752047608divmod @ A) = (^ [M2:num,N:num]: (((((product_Pair @ A) @ A) @ (((divide_divide @ A) @ ((numeral_numeral @ A) @ M2)) @ ((numeral_numeral @ A) @ N))) @ (((modulo_modulo @ A) @ ((numeral_numeral @ A) @ M2)) @ ((numeral_numeral @ A) @ N)))))))))). % 6.41/2.69 thf(fact_2168_divmod_H__nat__def, axiom, ((unique8689654367752047608divmod @ nat) = (^ [M2:num,N:num]: (((((product_Pair @ nat) @ nat) @ (((divide_divide @ nat) @ ((numeral_numeral @ nat) @ M2)) @ ((numeral_numeral @ nat) @ N))) @ (((modulo_modulo @ nat) @ ((numeral_numeral @ nat) @ M2)) @ ((numeral_numeral @ nat) @ N))))))). % 6.41/2.69 thf(fact_2169_and__nat__unfold, axiom, ((bit_se5824344872417868541ns_and @ nat) = (^ [M2:nat,N:nat]: (((((if @ nat) @ ((M2 = (zero_zero @ nat)) | (N = (zero_zero @ nat)))) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((modulo_modulo @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((modulo_modulo @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ nat) @ (((divide_divide @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_2170_and__nat__rec, axiom, ((bit_se5824344872417868541ns_and @ nat) = (^ [M2:nat,N:nat]: ((((plus_plus @ nat) @ ((zero_neq_one_of_bool @ nat) @ ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M2))) & (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ nat) @ (((divide_divide @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_2171_and__int_Opinduct, axiom, (! [A0:int,A12:int,P:(int > (int > $o))]: (((((accp @ ((product_prod @ int) @ int)) @ bit_and_int_rel) @ ((((product_Pair @ int) @ int) @ A0) @ A12)) => ((! [K2:int,L4:int]: (((((accp @ ((product_prod @ int) @ int)) @ bit_and_int_rel) @ ((((product_Pair @ int) @ int) @ K2) @ L4)) => (((~ (((((member @ int) @ K2) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ L4) @ (((insert @ int) @ (zero_zero @ int)) @ (((insert @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))))) => ((P @ (((divide_divide @ int) @ K2) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L4) @ ((numeral_numeral @ int) @ (bit0 @ one2))))) => ((P @ K2) @ L4))))) => ((P @ A0) @ A12)))))). % 6.41/2.69 thf(fact_2172_divmod__divmod__step, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => ((unique8689654367752047608divmod @ A) = (^ [M2:num,N:num]: (((((if @ ((product_prod @ A) @ A)) @ (((ord_less @ num) @ M2) @ N)) @ ((((product_Pair @ A) @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ M2))) @ (((unique1321980374590559556d_step @ A) @ N) @ (((unique8689654367752047608divmod @ A) @ M2) @ (bit0 @ N))))))))))). % 6.41/2.69 thf(fact_2173_upto_Opinduct, axiom, (! [A0:int,A12:int,P:(int > (int > $o))]: (((((accp @ ((product_prod @ int) @ int)) @ upto_rel) @ ((((product_Pair @ int) @ int) @ A0) @ A12)) => ((! [I4:int,J2:int]: (((((accp @ ((product_prod @ int) @ int)) @ upto_rel) @ ((((product_Pair @ int) @ int) @ I4) @ J2)) => (((((ord_less_eq @ int) @ I4) @ J2) => ((P @ (((plus_plus @ int) @ I4) @ (one_one @ int))) @ J2)) => ((P @ I4) @ J2))))) => ((P @ A0) @ A12)))))). % 6.41/2.69 thf(fact_2174_signed__take__bit__eq__take__bit__minus, axiom, ((bit_ri4674362597316999326ke_bit @ int) = (^ [N:nat,K3:int]: ((((minus_minus @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ (suc @ N)) @ K3)) @ (((times_times @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (suc @ N))) @ ((zero_neq_one_of_bool @ int) @ (((bit_se5641148757651400278ts_bit @ int) @ K3) @ N)))))))). % 6.41/2.69 thf(fact_2175_take__bit__numeral__bit1, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [L:num,K:num]: (((((bit_se2584673776208193580ke_bit @ A) @ ((numeral_numeral @ nat) @ L)) @ ((numeral_numeral @ A) @ (bit1 @ K))) = (((plus_plus @ A) @ (((times_times @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ (pred_numeral @ L)) @ ((numeral_numeral @ A) @ K))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_2176_of__int__less__neg__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:int,X3:num,N2:nat]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ A2)) @ (((power_power @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X3))) @ N2)) = (((ord_less @ int) @ A2) @ (((power_power @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ X3))) @ N2))))))))). % 6.41/2.69 thf(fact_2177_neg__numeral__power__less__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:num,N2:nat,A2:int]: (((((ord_less @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X3))) @ N2)) @ ((ring_1_of_int @ A) @ A2)) = (((ord_less @ int) @ (((power_power @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ X3))) @ N2)) @ A2)))))))). % 6.41/2.69 thf(fact_2178_of__int__le__neg__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:int,X3:num,N2:nat]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ A2)) @ (((power_power @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X3))) @ N2)) = (((ord_less_eq @ int) @ A2) @ (((power_power @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ X3))) @ N2))))))))). % 6.41/2.69 thf(fact_2179_neg__numeral__power__le__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:num,N2:nat,A2:int]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X3))) @ N2)) @ ((ring_1_of_int @ A) @ A2)) = (((ord_less_eq @ int) @ (((power_power @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ X3))) @ N2)) @ A2)))))))). % 6.41/2.69 thf(fact_2180_of__int__eq__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [W2:int,Z2:int]: (((((ring_1_of_int @ A) @ W2) = ((ring_1_of_int @ A) @ Z2)) = (W2 = Z2)))))))). % 6.41/2.69 thf(fact_2181_bit__0__eq, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (((bit_se5641148757651400278ts_bit @ A) @ (zero_zero @ A)) = (bot_bot @ (nat > $o))))))). % 6.41/2.69 thf(fact_2182_of__int__0, axiom, (! [A:$tType]: (((ring_1 @ A) => (((ring_1_of_int @ A) @ (zero_zero @ int)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_2183_of__int__0__eq__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z2:int]: ((((zero_zero @ A) = ((ring_1_of_int @ A) @ Z2)) = (Z2 = (zero_zero @ int))))))))). % 6.41/2.69 thf(fact_2184_of__int__eq__0__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z2:int]: (((((ring_1_of_int @ A) @ Z2) = (zero_zero @ A)) = (Z2 = (zero_zero @ int))))))))). % 6.41/2.69 thf(fact_2185_of__int__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [W2:int,Z2:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ W2)) @ ((ring_1_of_int @ A) @ Z2)) = (((ord_less_eq @ int) @ W2) @ Z2)))))))). % 6.41/2.69 thf(fact_2186_of__int__eq__numeral__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z2:int,N2:num]: (((((ring_1_of_int @ A) @ Z2) = ((numeral_numeral @ A) @ N2)) = (Z2 = ((numeral_numeral @ int) @ N2))))))))). % 6.41/2.69 thf(fact_2187_of__int__numeral, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [K:num]: ((((ring_1_of_int @ A) @ ((numeral_numeral @ int) @ K)) = ((numeral_numeral @ A) @ K)))))))). % 6.41/2.69 thf(fact_2188_of__int__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [W2:int,Z2:int]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ W2)) @ ((ring_1_of_int @ A) @ Z2)) = (((ord_less @ int) @ W2) @ Z2)))))))). % 6.41/2.69 thf(fact_2189_of__int__eq__1__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z2:int]: (((((ring_1_of_int @ A) @ Z2) = (one_one @ A)) = (Z2 = (one_one @ int))))))))). % 6.41/2.69 thf(fact_2190_of__int__1, axiom, (! [A:$tType]: (((ring_1 @ A) => (((ring_1_of_int @ A) @ (one_one @ int)) = (one_one @ A)))))). % 6.41/2.69 thf(fact_2191_of__int__mult, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [W2:int,Z2:int]: ((((ring_1_of_int @ A) @ (((times_times @ int) @ W2) @ Z2)) = (((times_times @ A) @ ((ring_1_of_int @ A) @ W2)) @ ((ring_1_of_int @ A) @ Z2))))))))). % 6.41/2.69 thf(fact_2192_of__int__add, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [W2:int,Z2:int]: ((((ring_1_of_int @ A) @ (((plus_plus @ int) @ W2) @ Z2)) = (((plus_plus @ A) @ ((ring_1_of_int @ A) @ W2)) @ ((ring_1_of_int @ A) @ Z2))))))))). % 6.41/2.69 thf(fact_2193_of__int__minus, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z2:int]: ((((ring_1_of_int @ A) @ ((uminus_uminus @ int) @ Z2)) = ((uminus_uminus @ A) @ ((ring_1_of_int @ A) @ Z2))))))))). % 6.41/2.69 thf(fact_2194_of__int__diff, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [W2:int,Z2:int]: ((((ring_1_of_int @ A) @ (((minus_minus @ int) @ W2) @ Z2)) = (((minus_minus @ A) @ ((ring_1_of_int @ A) @ W2)) @ ((ring_1_of_int @ A) @ Z2))))))))). % 6.41/2.69 thf(fact_2195_pred__numeral__simps_I1_J, axiom, ((pred_numeral @ one2) = (zero_zero @ nat))). % 6.41/2.69 thf(fact_2196_eq__numeral__Suc, axiom, (! [K:num,N2:nat]: (((((numeral_numeral @ nat) @ K) = (suc @ N2)) = ((pred_numeral @ K) = N2))))). % 6.41/2.69 thf(fact_2197_Suc__eq__numeral, axiom, (! [N2:nat,K:num]: ((((suc @ N2) = ((numeral_numeral @ nat) @ K)) = (N2 = (pred_numeral @ K)))))). % 6.41/2.69 thf(fact_2198_of__int__power__eq__of__int__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [X3:int,B2:int,W2:nat]: (((((ring_1_of_int @ A) @ X3) = (((power_power @ A) @ ((ring_1_of_int @ A) @ B2)) @ W2)) = (X3 = (((power_power @ int) @ B2) @ W2))))))))). % 6.41/2.69 thf(fact_2199_of__int__eq__of__int__power__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [B2:int,W2:nat,X3:int]: ((((((power_power @ A) @ ((ring_1_of_int @ A) @ B2)) @ W2) = ((ring_1_of_int @ A) @ X3)) = ((((power_power @ int) @ B2) @ W2) = X3)))))))). % 6.41/2.69 thf(fact_2200_of__int__power, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z2:int,N2:nat]: ((((ring_1_of_int @ A) @ (((power_power @ int) @ Z2) @ N2)) = (((power_power @ A) @ ((ring_1_of_int @ A) @ Z2)) @ N2)))))))). % 6.41/2.69 thf(fact_2201_of__int__of__bool, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [P:$o]: ((((ring_1_of_int @ A) @ ((zero_neq_one_of_bool @ int) @ P)) = ((zero_neq_one_of_bool @ A) @ P)))))))). % 6.41/2.69 thf(fact_2202_bit__numeral__Bit0__Suc__iff, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M:num,N2:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((numeral_numeral @ A) @ (bit0 @ M))) @ (suc @ N2)) = (((bit_se5641148757651400278ts_bit @ A) @ ((numeral_numeral @ A) @ M)) @ N2)))))))). % 6.41/2.69 thf(fact_2203_bit__numeral__Bit1__Suc__iff, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M:num,N2:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((numeral_numeral @ A) @ (bit1 @ M))) @ (suc @ N2)) = (((bit_se5641148757651400278ts_bit @ A) @ ((numeral_numeral @ A) @ M)) @ N2)))))))). % 6.41/2.69 thf(fact_2204_less__numeral__Suc, axiom, (! [K:num,N2:nat]: (((((ord_less @ nat) @ ((numeral_numeral @ nat) @ K)) @ (suc @ N2)) = (((ord_less @ nat) @ (pred_numeral @ K)) @ N2))))). % 6.41/2.69 thf(fact_2205_less__Suc__numeral, axiom, (! [N2:nat,K:num]: (((((ord_less @ nat) @ (suc @ N2)) @ ((numeral_numeral @ nat) @ K)) = (((ord_less @ nat) @ N2) @ (pred_numeral @ K)))))). % 6.41/2.69 thf(fact_2206_le__numeral__Suc, axiom, (! [K:num,N2:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ K)) @ (suc @ N2)) = (((ord_less_eq @ nat) @ (pred_numeral @ K)) @ N2))))). % 6.41/2.69 thf(fact_2207_le__Suc__numeral, axiom, (! [N2:nat,K:num]: (((((ord_less_eq @ nat) @ (suc @ N2)) @ ((numeral_numeral @ nat) @ K)) = (((ord_less_eq @ nat) @ N2) @ (pred_numeral @ K)))))). % 6.41/2.69 thf(fact_2208_diff__Suc__numeral, axiom, (! [N2:nat,K:num]: (((((minus_minus @ nat) @ (suc @ N2)) @ ((numeral_numeral @ nat) @ K)) = (((minus_minus @ nat) @ N2) @ (pred_numeral @ K)))))). % 6.41/2.69 thf(fact_2209_diff__numeral__Suc, axiom, (! [K:num,N2:nat]: (((((minus_minus @ nat) @ ((numeral_numeral @ nat) @ K)) @ (suc @ N2)) = (((minus_minus @ nat) @ (pred_numeral @ K)) @ N2))))). % 6.41/2.69 thf(fact_2210_signed__take__bit__negative__iff, axiom, (! [N2:nat,K:int]: (((((ord_less @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N2) @ K)) @ (zero_zero @ int)) = (((bit_se5641148757651400278ts_bit @ int) @ K) @ N2))))). % 6.41/2.69 thf(fact_2211_of__int__0__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z2)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2)))))))). % 6.41/2.69 thf(fact_2212_of__int__le__0__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (zero_zero @ A)) = (((ord_less_eq @ int) @ Z2) @ (zero_zero @ int))))))))). % 6.41/2.69 thf(fact_2213_of__int__less__0__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (zero_zero @ A)) = (((ord_less @ int) @ Z2) @ (zero_zero @ int))))))))). % 6.41/2.69 thf(fact_2214_of__int__0__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z2)) = (((ord_less @ int) @ (zero_zero @ int)) @ Z2)))))))). % 6.41/2.69 thf(fact_2215_of__int__le__numeral__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int,N2:num]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z2)) @ ((numeral_numeral @ A) @ N2)) = (((ord_less_eq @ int) @ Z2) @ ((numeral_numeral @ int) @ N2))))))))). % 6.41/2.69 thf(fact_2216_of__int__numeral__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:num,Z2:int]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ N2)) @ ((ring_1_of_int @ A) @ Z2)) = (((ord_less_eq @ int) @ ((numeral_numeral @ int) @ N2)) @ Z2)))))))). % 6.41/2.69 thf(fact_2217_of__int__less__numeral__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int,N2:num]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ Z2)) @ ((numeral_numeral @ A) @ N2)) = (((ord_less @ int) @ Z2) @ ((numeral_numeral @ int) @ N2))))))))). % 6.41/2.69 thf(fact_2218_of__int__numeral__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:num,Z2:int]: (((((ord_less @ A) @ ((numeral_numeral @ A) @ N2)) @ ((ring_1_of_int @ A) @ Z2)) = (((ord_less @ int) @ ((numeral_numeral @ int) @ N2)) @ Z2)))))))). % 6.41/2.69 thf(fact_2219_of__int__le__1__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A)) = (((ord_less_eq @ int) @ Z2) @ (one_one @ int))))))))). % 6.41/2.69 thf(fact_2220_of__int__1__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less_eq @ A) @ (one_one @ A)) @ ((ring_1_of_int @ A) @ Z2)) = (((ord_less_eq @ int) @ (one_one @ int)) @ Z2)))))))). % 6.41/2.69 thf(fact_2221_of__int__less__1__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A)) = (((ord_less @ int) @ Z2) @ (one_one @ int))))))))). % 6.41/2.69 thf(fact_2222_of__int__1__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less @ A) @ (one_one @ A)) @ ((ring_1_of_int @ A) @ Z2)) = (((ord_less @ int) @ (one_one @ int)) @ Z2)))))))). % 6.41/2.69 thf(fact_2223_of__int__le__of__int__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [B2:int,W2:nat,X3:int]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B2)) @ W2)) @ ((ring_1_of_int @ A) @ X3)) = (((ord_less_eq @ int) @ (((power_power @ int) @ B2) @ W2)) @ X3)))))))). % 6.41/2.69 thf(fact_2224_of__int__power__le__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:int,B2:int,W2:nat]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ X3)) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B2)) @ W2)) = (((ord_less_eq @ int) @ X3) @ (((power_power @ int) @ B2) @ W2))))))))). % 6.41/2.69 thf(fact_2225_of__int__eq__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Y:int,X3:num,N2:nat]: (((((ring_1_of_int @ A) @ Y) = (((power_power @ A) @ ((numeral_numeral @ A) @ X3)) @ N2)) = (Y = (((power_power @ int) @ ((numeral_numeral @ int) @ X3)) @ N2))))))))). % 6.41/2.69 thf(fact_2226_numeral__power__eq__of__int__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [X3:num,N2:nat,Y:int]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ X3)) @ N2) = ((ring_1_of_int @ A) @ Y)) = ((((power_power @ int) @ ((numeral_numeral @ int) @ X3)) @ N2) = Y)))))))). % 6.41/2.69 thf(fact_2227_of__int__less__of__int__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [B2:int,W2:nat,X3:int]: (((((ord_less @ A) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B2)) @ W2)) @ ((ring_1_of_int @ A) @ X3)) = (((ord_less @ int) @ (((power_power @ int) @ B2) @ W2)) @ X3)))))))). % 6.41/2.69 thf(fact_2228_of__int__power__less__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:int,B2:int,W2:nat]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ X3)) @ (((power_power @ A) @ ((ring_1_of_int @ A) @ B2)) @ W2)) = (((ord_less @ int) @ X3) @ (((power_power @ int) @ B2) @ W2))))))))). % 6.41/2.69 thf(fact_2229_bit__minus__numeral__Bit0__Suc__iff, axiom, (! [W2:num,N2:nat]: (((((bit_se5641148757651400278ts_bit @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ W2)))) @ (suc @ N2)) = (((bit_se5641148757651400278ts_bit @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ W2))) @ N2))))). % 6.41/2.69 thf(fact_2230_bit__minus__numeral__Bit1__Suc__iff, axiom, (! [W2:num,N2:nat]: (((((bit_se5641148757651400278ts_bit @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ W2)))) @ (suc @ N2)) = (~ ((((bit_se5641148757651400278ts_bit @ int) @ ((numeral_numeral @ int) @ W2)) @ N2))))))). % 6.41/2.69 thf(fact_2231_bit__0, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A]: (((((bit_se5641148757651400278ts_bit @ A) @ A2) @ (zero_zero @ nat)) = (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2)))))))))). % 6.41/2.69 thf(fact_2232_bit__mod__2__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A,N2:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ (((modulo_modulo @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ N2) = ((N2 = (zero_zero @ nat)) & (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2))))))))))). % 6.41/2.69 thf(fact_2233_numeral__power__le__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:num,N2:nat,A2:int]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ X3)) @ N2)) @ ((ring_1_of_int @ A) @ A2)) = (((ord_less_eq @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X3)) @ N2)) @ A2)))))))). % 6.41/2.69 thf(fact_2234_of__int__le__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:int,X3:num,N2:nat]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ A2)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ X3)) @ N2)) = (((ord_less_eq @ int) @ A2) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X3)) @ N2))))))))). % 6.41/2.69 thf(fact_2235_numeral__power__less__of__int__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:num,N2:nat,A2:int]: (((((ord_less @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ X3)) @ N2)) @ ((ring_1_of_int @ A) @ A2)) = (((ord_less @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X3)) @ N2)) @ A2)))))))). % 6.41/2.69 thf(fact_2236_of__int__less__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:int,X3:num,N2:nat]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ A2)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ X3)) @ N2)) = (((ord_less @ int) @ A2) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X3)) @ N2))))))))). % 6.41/2.69 thf(fact_2237_of__int__eq__neg__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Y:int,X3:num,N2:nat]: (((((ring_1_of_int @ A) @ Y) = (((power_power @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X3))) @ N2)) = (Y = (((power_power @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ X3))) @ N2))))))))). % 6.41/2.69 thf(fact_2238_neg__numeral__power__eq__of__int__cancel__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [X3:num,N2:nat,Y:int]: ((((((power_power @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X3))) @ N2) = ((ring_1_of_int @ A) @ Y)) = ((((power_power @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ X3))) @ N2) = Y)))))))). % 6.41/2.69 thf(fact_2239_bit__disjunctive__add__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A,B2:A,N2:nat]: (((! [N3:nat]: (((~ ((((bit_se5641148757651400278ts_bit @ A) @ A2) @ N3))) | (~ ((((bit_se5641148757651400278ts_bit @ A) @ B2) @ N3)))))) => ((((bit_se5641148757651400278ts_bit @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ N2) = ((((bit_se5641148757651400278ts_bit @ A) @ A2) @ N2) | (((bit_se5641148757651400278ts_bit @ A) @ B2) @ N2)))))))))). % 6.41/2.69 thf(fact_2240_ex__le__of__int, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X3:A]: ((? [Z3:int]: ((((ord_less_eq @ A) @ X3) @ ((ring_1_of_int @ A) @ Z3)))))))))). % 6.41/2.69 thf(fact_2241_ex__less__of__int, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X3:A]: ((? [Z3:int]: ((((ord_less @ A) @ X3) @ ((ring_1_of_int @ A) @ Z3)))))))))). % 6.41/2.69 thf(fact_2242_ex__of__int__less, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X3:A]: ((? [Z3:int]: ((((ord_less @ A) @ ((ring_1_of_int @ A) @ Z3)) @ X3))))))))). % 6.41/2.69 thf(fact_2243_mult__of__int__commute, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X3:int,Y:A]: (((((times_times @ A) @ ((ring_1_of_int @ A) @ X3)) @ Y) = (((times_times @ A) @ Y) @ ((ring_1_of_int @ A) @ X3))))))))). % 6.41/2.69 thf(fact_2244_not__bit__1__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat]: ((~ ((((bit_se5641148757651400278ts_bit @ A) @ (one_one @ A)) @ (suc @ N2)))))))))). % 6.41/2.69 thf(fact_2245_bit__1__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [N2:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ (one_one @ A)) @ N2) = (N2 = (zero_zero @ nat))))))))). % 6.41/2.69 thf(fact_2246_bit__take__bit__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,A2:A,N2:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ M) @ A2)) @ N2) = ((((ord_less @ nat) @ N2) @ M) & (((bit_se5641148757651400278ts_bit @ A) @ A2) @ N2))))))))). % 6.41/2.69 thf(fact_2247_bit__of__bool__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [B2:$o,N2:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((zero_neq_one_of_bool @ A) @ B2)) @ N2) = (B2 & (N2 = (zero_zero @ nat)))))))))). % 6.41/2.69 thf(fact_2248_numeral__eq__Suc, axiom, ((numeral_numeral @ nat) = (^ [K3:num]: ((suc @ (pred_numeral @ K3)))))). % 6.41/2.69 thf(fact_2249_of__int__nonneg, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z2))))))))). % 6.41/2.69 thf(fact_2250_of__int__pos, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ Z2) => (((ord_less @ A) @ (zero_zero @ A)) @ ((ring_1_of_int @ A) @ Z2))))))))). % 6.41/2.69 thf(fact_2251_floor__exists, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X3:A]: ((? [Z3:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z3)) @ X3) & (((ord_less @ A) @ X3) @ ((ring_1_of_int @ A) @ (((plus_plus @ int) @ Z3) @ (one_one @ int))))))))))))). % 6.41/2.69 thf(fact_2252_floor__exists1, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X3:A]: ((? [X5:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ X5)) @ X3) & ((((ord_less @ A) @ X3) @ ((ring_1_of_int @ A) @ (((plus_plus @ int) @ X5) @ (one_one @ int)))) & (! [Y5:int]: ((((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Y5)) @ X3) & (((ord_less @ A) @ X3) @ ((ring_1_of_int @ A) @ (((plus_plus @ int) @ Y5) @ (one_one @ int))))) => (Y5 = X5)))))))))))))). % 6.41/2.69 thf(fact_2253_of__int__neg__numeral, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [K:num]: ((((ring_1_of_int @ A) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ K))))))))). % 6.41/2.69 thf(fact_2254_bit__imp__take__bit__positive, axiom, (! [N2:nat,M:nat,K:int]: (((((ord_less @ nat) @ N2) @ M) => ((((bit_se5641148757651400278ts_bit @ int) @ K) @ N2) => (((ord_less @ int) @ (zero_zero @ int)) @ (((bit_se2584673776208193580ke_bit @ int) @ M) @ K))))))). % 6.41/2.69 thf(fact_2255_int__le__real__less, axiom, ((ord_less_eq @ int) = (^ [N:int,M2:int]: ((((ord_less @ real) @ ((ring_1_of_int @ real) @ N)) @ (((plus_plus @ real) @ ((ring_1_of_int @ real) @ M2)) @ (one_one @ real))))))). % 6.41/2.69 thf(fact_2256_int__less__real__le, axiom, ((ord_less @ int) = (^ [N:int,M2:int]: ((((ord_less_eq @ real) @ (((plus_plus @ real) @ ((ring_1_of_int @ real) @ N)) @ (one_one @ real))) @ ((ring_1_of_int @ real) @ M2)))))). % 6.41/2.69 thf(fact_2257_bit__concat__bit__iff, axiom, (! [M:nat,K:int,L:int,N2:nat]: (((((bit_se5641148757651400278ts_bit @ int) @ (((bit_concat_bit @ M) @ K) @ L)) @ N2) = (((((ord_less @ nat) @ N2) @ M) & (((bit_se5641148757651400278ts_bit @ int) @ K) @ N2)) | ((((ord_less_eq @ nat) @ M) @ N2) & (((bit_se5641148757651400278ts_bit @ int) @ L) @ (((minus_minus @ nat) @ N2) @ M)))))))). % 6.41/2.69 thf(fact_2258_exp__eq__0__imp__not__bit, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [N2:nat,A2:A]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2) = (zero_zero @ A)) => (~ ((((bit_se5641148757651400278ts_bit @ A) @ A2) @ N2)))))))))). % 6.41/2.69 thf(fact_2259_bit__Suc, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A,N2:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ A2) @ (suc @ N2)) = (((bit_se5641148757651400278ts_bit @ A) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ N2)))))))). % 6.41/2.69 thf(fact_2260_int__bit__bound, axiom, (! [K:int]: ((~ ((! [N3:nat]: (((! [M6:nat]: (((((ord_less_eq @ nat) @ N3) @ M6) => ((((bit_se5641148757651400278ts_bit @ int) @ K) @ M6) = (((bit_se5641148757651400278ts_bit @ int) @ K) @ N3))))) => (~ (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) => ((((bit_se5641148757651400278ts_bit @ int) @ K) @ (((minus_minus @ nat) @ N3) @ (one_one @ nat))) = (~ ((((bit_se5641148757651400278ts_bit @ int) @ K) @ N3))))))))))))))). % 6.41/2.69 thf(fact_2261_and__exp__eq__0__iff__not__bit, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A,N2:nat]: ((((((bit_se5824344872417868541ns_and @ A) @ A2) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) = (zero_zero @ A)) = (~ ((((bit_se5641148757651400278ts_bit @ A) @ A2) @ N2)))))))))). % 6.41/2.69 thf(fact_2262_even__bit__succ__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A,N2:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2) => ((((bit_se5641148757651400278ts_bit @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ A2)) @ N2) = ((((bit_se5641148757651400278ts_bit @ A) @ A2) @ N2) | (N2 = (zero_zero @ nat))))))))))). % 6.41/2.69 thf(fact_2263_odd__bit__iff__bit__pred, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A2:A,N2:nat]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2))) => ((((bit_se5641148757651400278ts_bit @ A) @ A2) @ N2) = ((((bit_se5641148757651400278ts_bit @ A) @ (((minus_minus @ A) @ A2) @ (one_one @ A))) @ N2) | (N2 = (zero_zero @ nat))))))))))). % 6.41/2.69 thf(fact_2264_bit__sum__mult__2__cases, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A,B2:A,N2:nat]: (((! [J2:nat]: ((~ ((((bit_se5641148757651400278ts_bit @ A) @ A2) @ (suc @ J2)))))) => ((((bit_se5641148757651400278ts_bit @ A) @ (((plus_plus @ A) @ A2) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) @ N2) = (((N2 = (zero_zero @ nat)) => (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2)))) & ((N2 != (zero_zero @ nat)) => (((bit_se5641148757651400278ts_bit @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2)) @ N2))))))))))). % 6.41/2.69 thf(fact_2265_bit__rec, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => ((bit_se5641148757651400278ts_bit @ A) = (^ [A4:A,N:nat]: ((((N = (zero_zero @ nat)) => (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A4)))) & ((N != (zero_zero @ nat)) => (((bit_se5641148757651400278ts_bit @ A) @ (((divide_divide @ A) @ A4) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((minus_minus @ nat) @ N) @ (one_one @ nat)))))))))))). % 6.41/2.69 thf(fact_2266_take__bit__Suc__from__most, axiom, (! [N2:nat,K:int]: (((((bit_se2584673776208193580ke_bit @ int) @ (suc @ N2)) @ K) = (((plus_plus @ int) @ (((times_times @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)) @ ((zero_neq_one_of_bool @ int) @ (((bit_se5641148757651400278ts_bit @ int) @ K) @ N2)))) @ (((bit_se2584673776208193580ke_bit @ int) @ N2) @ K)))))). % 6.41/2.69 thf(fact_2267_round__unique, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Y:int]: (((((ord_less @ A) @ (((minus_minus @ A) @ X3) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ ((ring_1_of_int @ A) @ Y)) => ((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Y)) @ (((plus_plus @ A) @ X3) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) => (((archimedean_round @ A) @ X3) = Y))))))))). % 6.41/2.69 thf(fact_2268_of__int__round__gt, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: ((((ord_less @ A) @ (((minus_minus @ A) @ X3) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ ((ring_1_of_int @ A) @ ((archimedean_round @ A) @ X3))))))))). % 6.41/2.69 thf(fact_2269_of__int__round__ge, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ X3) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ ((ring_1_of_int @ A) @ ((archimedean_round @ A) @ X3))))))))). % 6.41/2.69 thf(fact_2270_of__int__round__le, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ ((archimedean_round @ A) @ X3))) @ (((plus_plus @ A) @ X3) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_2271_divmod__BitM__2__eq, axiom, (! [M:num]: (((((unique8689654367752047608divmod @ int) @ (bitM @ M)) @ (bit0 @ one2)) = ((((product_Pair @ int) @ int) @ (((minus_minus @ int) @ ((numeral_numeral @ int) @ M)) @ (one_one @ int))) @ (one_one @ int)))))). % 6.41/2.69 thf(fact_2272_mask__numeral, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:num]: ((((bit_se2239418461657761734s_mask @ A) @ ((numeral_numeral @ nat) @ N2)) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((bit_se2239418461657761734s_mask @ A) @ (pred_numeral @ N2))))))))))). % 6.41/2.69 thf(fact_2273_neg__numeral__le__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V2:num,X3:A]: (((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V2))) @ ((archimedean_ceiling @ A) @ X3)) = (((ord_less @ A) @ (((minus_minus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) @ (one_one @ A))) @ X3)))))))). % 6.41/2.69 thf(fact_2274_mask__nat__positive__iff, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((bit_se2239418461657761734s_mask @ nat) @ N2)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))). % 6.41/2.69 thf(fact_2275_ceiling__zero, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (((archimedean_ceiling @ A) @ (zero_zero @ A)) = (zero_zero @ int)))))). % 6.41/2.69 thf(fact_2276_mask__eq__0__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat]: (((((bit_se2239418461657761734s_mask @ A) @ N2) = (zero_zero @ A)) = (N2 = (zero_zero @ nat))))))))). % 6.41/2.69 thf(fact_2277_mask__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (((bit_se2239418461657761734s_mask @ A) @ (zero_zero @ nat)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_2278_round__0, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (((archimedean_round @ A) @ (zero_zero @ A)) = (zero_zero @ int)))))). % 6.41/2.69 thf(fact_2279_mask__Suc__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (((bit_se2239418461657761734s_mask @ A) @ (suc @ (zero_zero @ nat))) = (one_one @ A)))))). % 6.41/2.69 thf(fact_2280_ceiling__add__of__int, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Z2:int]: ((((archimedean_ceiling @ A) @ (((plus_plus @ A) @ X3) @ ((ring_1_of_int @ A) @ Z2))) = (((plus_plus @ int) @ ((archimedean_ceiling @ A) @ X3)) @ Z2)))))))). % 6.41/2.69 thf(fact_2281_ceiling__le__zero, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X3)) @ (zero_zero @ int)) = (((ord_less_eq @ A) @ X3) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2282_zero__less__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less @ int) @ (zero_zero @ int)) @ ((archimedean_ceiling @ A) @ X3)) = (((ord_less @ A) @ (zero_zero @ A)) @ X3)))))))). % 6.41/2.69 thf(fact_2283_ceiling__le__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,V2:num]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X3)) @ ((numeral_numeral @ int) @ V2)) = (((ord_less_eq @ A) @ X3) @ ((numeral_numeral @ A) @ V2))))))))). % 6.41/2.69 thf(fact_2284_ceiling__less__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less @ int) @ ((archimedean_ceiling @ A) @ X3)) @ (one_one @ int)) = (((ord_less_eq @ A) @ X3) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2285_one__le__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less_eq @ int) @ (one_one @ int)) @ ((archimedean_ceiling @ A) @ X3)) = (((ord_less @ A) @ (zero_zero @ A)) @ X3)))))))). % 6.41/2.69 thf(fact_2286_numeral__less__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V2:num,X3:A]: (((((ord_less @ int) @ ((numeral_numeral @ int) @ V2)) @ ((archimedean_ceiling @ A) @ X3)) = (((ord_less @ A) @ ((numeral_numeral @ A) @ V2)) @ X3)))))))). % 6.41/2.69 thf(fact_2287_ceiling__le__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X3)) @ (one_one @ int)) = (((ord_less_eq @ A) @ X3) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_2288_one__less__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less @ int) @ (one_one @ int)) @ ((archimedean_ceiling @ A) @ X3)) = (((ord_less @ A) @ (one_one @ A)) @ X3)))))))). % 6.41/2.69 thf(fact_2289_ceiling__add__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,V2:num]: ((((archimedean_ceiling @ A) @ (((plus_plus @ A) @ X3) @ ((numeral_numeral @ A) @ V2))) = (((plus_plus @ int) @ ((archimedean_ceiling @ A) @ X3)) @ ((numeral_numeral @ int) @ V2))))))))). % 6.41/2.69 thf(fact_2290_ceiling__add__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: ((((archimedean_ceiling @ A) @ (((plus_plus @ A) @ X3) @ (one_one @ A))) = (((plus_plus @ int) @ ((archimedean_ceiling @ A) @ X3)) @ (one_one @ int))))))))). % 6.41/2.69 thf(fact_2291_ceiling__less__zero, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less @ int) @ ((archimedean_ceiling @ A) @ X3)) @ (zero_zero @ int)) = (((ord_less_eq @ A) @ X3) @ ((uminus_uminus @ A) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_2292_zero__le__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((archimedean_ceiling @ A) @ X3)) = (((ord_less @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ X3)))))))). % 6.41/2.69 thf(fact_2293_ceiling__less__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,V2:num]: (((((ord_less @ int) @ ((archimedean_ceiling @ A) @ X3)) @ ((numeral_numeral @ int) @ V2)) = (((ord_less_eq @ A) @ X3) @ (((minus_minus @ A) @ ((numeral_numeral @ A) @ V2)) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_2294_numeral__le__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V2:num,X3:A]: (((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ V2)) @ ((archimedean_ceiling @ A) @ X3)) = (((ord_less @ A) @ (((minus_minus @ A) @ ((numeral_numeral @ A) @ V2)) @ (one_one @ A))) @ X3)))))))). % 6.41/2.69 thf(fact_2295_ceiling__le__neg__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,V2:num]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X3)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V2))) = (((ord_less_eq @ A) @ X3) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2)))))))))). % 6.41/2.69 thf(fact_2296_neg__numeral__less__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V2:num,X3:A]: (((((ord_less @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V2))) @ ((archimedean_ceiling @ A) @ X3)) = (((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) @ X3)))))))). % 6.41/2.69 thf(fact_2297_ceiling__less__neg__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,V2:num]: (((((ord_less @ int) @ ((archimedean_ceiling @ A) @ X3)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V2))) = (((ord_less_eq @ A) @ X3) @ (((minus_minus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_2298_less__eq__mask, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ N2) @ ((bit_se2239418461657761734s_mask @ nat) @ N2))))). % 6.41/2.69 thf(fact_2299_bit__Suc__0__iff, axiom, (! [N2:nat]: (((((bit_se5641148757651400278ts_bit @ nat) @ (suc @ (zero_zero @ nat))) @ N2) = (N2 = (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_2300_not__bit__Suc__0__Suc, axiom, (! [N2:nat]: ((~ ((((bit_se5641148757651400278ts_bit @ nat) @ (suc @ (zero_zero @ nat))) @ (suc @ N2))))))). % 6.41/2.69 thf(fact_2301_ceiling__mono, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Y:A,X3:A]: (((((ord_less_eq @ A) @ Y) @ X3) => (((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ Y)) @ ((archimedean_ceiling @ A) @ X3))))))))). % 6.41/2.69 thf(fact_2302_le__of__int__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ X3) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ X3))))))))). % 6.41/2.69 thf(fact_2303_ceiling__less__cancel, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Y:A]: (((((ord_less @ int) @ ((archimedean_ceiling @ A) @ X3)) @ ((archimedean_ceiling @ A) @ Y)) => (((ord_less @ A) @ X3) @ Y)))))))). % 6.41/2.69 thf(fact_2304_not__mask__negative__int, axiom, (! [N2:nat]: ((~ ((((ord_less @ int) @ ((bit_se2239418461657761734s_mask @ int) @ N2)) @ (zero_zero @ int))))))). % 6.41/2.69 thf(fact_2305_not__bit__Suc__0__numeral, axiom, (! [N2:num]: ((~ ((((bit_se5641148757651400278ts_bit @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ N2))))))). % 6.41/2.69 thf(fact_2306_ceiling__le__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Z2:int]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X3)) @ Z2) = (((ord_less_eq @ A) @ X3) @ ((ring_1_of_int @ A) @ Z2))))))))). % 6.41/2.69 thf(fact_2307_ceiling__le, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,A2:int]: (((((ord_less_eq @ A) @ X3) @ ((ring_1_of_int @ A) @ A2)) => (((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X3)) @ A2)))))))). % 6.41/2.69 thf(fact_2308_less__ceiling__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z2:int,X3:A]: (((((ord_less @ int) @ Z2) @ ((archimedean_ceiling @ A) @ X3)) = (((ord_less @ A) @ ((ring_1_of_int @ A) @ Z2)) @ X3)))))))). % 6.41/2.69 thf(fact_2309_ceiling__add__le, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Y:A]: ((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ (((plus_plus @ A) @ X3) @ Y))) @ (((plus_plus @ int) @ ((archimedean_ceiling @ A) @ X3)) @ ((archimedean_ceiling @ A) @ Y))))))))). % 6.41/2.69 thf(fact_2310_round__mono, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => (((ord_less_eq @ int) @ ((archimedean_round @ A) @ X3)) @ ((archimedean_round @ A) @ Y))))))))). % 6.41/2.69 thf(fact_2311_less__mask, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ N2) => (((ord_less @ nat) @ N2) @ ((bit_se2239418461657761734s_mask @ nat) @ N2)))))). % 6.41/2.69 thf(fact_2312_eval__nat__numeral_I2_J, axiom, (! [N2:num]: ((((numeral_numeral @ nat) @ (bit0 @ N2)) = (suc @ ((numeral_numeral @ nat) @ (bitM @ N2))))))). % 6.41/2.69 thf(fact_2313_of__int__ceiling__le__add__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [R2:A]: ((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ R2))) @ (((plus_plus @ A) @ R2) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_2314_of__int__ceiling__diff__one__le, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [R2:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ R2))) @ (one_one @ A))) @ R2))))))). % 6.41/2.69 thf(fact_2315_ceiling__split, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [P:(int > $o),T2:A]: (((P @ ((archimedean_ceiling @ A) @ T2)) = (! [I3:int]: ((((((ord_less @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ I3)) @ (one_one @ A))) @ T2) & (((ord_less_eq @ A) @ T2) @ ((ring_1_of_int @ A) @ I3))) => (P @ I3))))))))))). % 6.41/2.69 thf(fact_2316_ceiling__eq__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,A2:int]: (((((archimedean_ceiling @ A) @ X3) = A2) = ((((ord_less @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ A2)) @ (one_one @ A))) @ X3) & (((ord_less_eq @ A) @ X3) @ ((ring_1_of_int @ A) @ A2)))))))))). % 6.41/2.69 thf(fact_2317_ceiling__unique, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z2:int,X3:A]: (((((ord_less @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A))) @ X3) => ((((ord_less_eq @ A) @ X3) @ ((ring_1_of_int @ A) @ Z2)) => (((archimedean_ceiling @ A) @ X3) = Z2))))))))). % 6.41/2.69 thf(fact_2318_ceiling__correct, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ X3))) @ (one_one @ A))) @ X3) & (((ord_less_eq @ A) @ X3) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ X3)))))))))). % 6.41/2.69 thf(fact_2319_mult__ceiling__le, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ (((times_times @ A) @ A2) @ B2))) @ (((times_times @ int) @ ((archimedean_ceiling @ A) @ A2)) @ ((archimedean_ceiling @ A) @ B2))))))))))). % 6.41/2.69 thf(fact_2320_ceiling__less__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Z2:int]: (((((ord_less @ int) @ ((archimedean_ceiling @ A) @ X3)) @ Z2) = (((ord_less_eq @ A) @ X3) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_2321_le__ceiling__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z2:int,X3:A]: (((((ord_less_eq @ int) @ Z2) @ ((archimedean_ceiling @ A) @ X3)) = (((ord_less @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A))) @ X3)))))))). % 6.41/2.69 thf(fact_2322_ceiling__divide__upper, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Q4:A,P4:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Q4) => (((ord_less_eq @ A) @ P4) @ (((times_times @ A) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ (((divide_divide @ A) @ P4) @ Q4)))) @ Q4))))))))). % 6.41/2.69 thf(fact_2323_Suc__mask__eq__exp, axiom, (! [N2:nat]: (((suc @ ((bit_se2239418461657761734s_mask @ nat) @ N2)) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))))). % 6.41/2.69 thf(fact_2324_mask__nat__less__exp, axiom, (! [N2:nat]: ((((ord_less @ nat) @ ((bit_se2239418461657761734s_mask @ nat) @ N2)) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))))). % 6.41/2.69 thf(fact_2325_semiring__bit__operations__class_Oeven__mask__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((bit_se2239418461657761734s_mask @ A) @ N2)) = (N2 = (zero_zero @ nat))))))))). % 6.41/2.69 thf(fact_2326_ceiling__divide__lower, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Q4:A,P4:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Q4) => (((ord_less @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ (((divide_divide @ A) @ P4) @ Q4)))) @ (one_one @ A))) @ Q4)) @ P4)))))))). % 6.41/2.69 thf(fact_2327_ceiling__eq, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [N2:int,X3:A]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ N2)) @ X3) => ((((ord_less_eq @ A) @ X3) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ N2)) @ (one_one @ A))) => (((archimedean_ceiling @ A) @ X3) = (((plus_plus @ int) @ N2) @ (one_one @ int))))))))))). % 6.41/2.69 thf(fact_2328_Suc__0__xor__eq, axiom, (! [N2:nat]: (((((bit_se5824344971392196577ns_xor @ nat) @ (suc @ (zero_zero @ nat))) @ N2) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ N2) @ ((zero_neq_one_of_bool @ nat) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)))) @ ((zero_neq_one_of_bool @ nat) @ (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))))))))). % 6.41/2.69 thf(fact_2329_xor__Suc__0__eq, axiom, (! [N2:nat]: (((((bit_se5824344971392196577ns_xor @ nat) @ N2) @ (suc @ (zero_zero @ nat))) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ N2) @ ((zero_neq_one_of_bool @ nat) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)))) @ ((zero_neq_one_of_bool @ nat) @ (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))))))))). % 6.41/2.69 thf(fact_2330_horner__sum__of__bool__2__less, axiom, (! [Bs:(list @ $o)]: ((((ord_less @ int) @ (((((groups4207007520872428315er_sum @ $o) @ int) @ (zero_neq_one_of_bool @ int)) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Bs)) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ ((size_size @ (list @ $o)) @ Bs)))))). % 6.41/2.69 thf(fact_2331_dbl__dec__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl_dec @ A) @ (zero_zero @ A)) = ((uminus_uminus @ A) @ (one_one @ A))))))). % 6.41/2.69 thf(fact_2332_the__elem__eq, axiom, (! [A:$tType,X3:A]: ((((the_elem @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = X3)))). % 6.41/2.69 thf(fact_2333_vebt__member_Oelims_I1_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat,Y:$o]: (((((vEBT_vebt_member @ X3) @ Xa2) = Y) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => (Y = (~ ((((Xa2 = (zero_zero @ nat)) => A5) & ((Xa2 != (zero_zero @ nat)) => (((Xa2 = (one_one @ nat)) => B5) & (Xa2 = (one_one @ nat))))))))))) => (((? [Uu2:nat,Uv2:(list @ vEBT_VEBT),Uw2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu2) @ Uv2) @ Uw2)))) => Y) => (((? [V3:((product_prod @ nat) @ nat),Uy2:(list @ vEBT_VEBT),Uz2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V3)) @ (zero_zero @ nat)) @ Uy2) @ Uz2)))) => Y) => (((? [V3:((product_prod @ nat) @ nat),Vb2:(list @ vEBT_VEBT),Vc2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V3)) @ (suc @ (zero_zero @ nat))) @ Vb2) @ Vc2)))) => Y) => (~ ((! [Mi2:nat,Ma2:nat,Va2:nat,TreeList3:(list @ vEBT_VEBT)]: (((? [Summary2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ (suc @ Va2))) @ TreeList3) @ Summary2)))) => (Y = (~ (((Xa2 != Mi2) => ((Xa2 != Ma2) => ((~ ((((ord_less @ nat) @ Xa2) @ Mi2))) & ((~ ((((ord_less @ nat) @ Xa2) @ Mi2))) => ((~ ((((ord_less @ nat) @ Ma2) @ Xa2))) & ((~ ((((ord_less @ nat) @ Ma2) @ Xa2))) => (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_vebt_member @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3))))))))))))))))))))))))). % 6.41/2.69 thf(fact_2334_set__vebt_H__def, axiom, (vEBT_VEBT_set_vebt = (^ [T3:vEBT_VEBT]: (((collect @ nat) @ (vEBT_vebt_member @ T3)))))). % 6.41/2.69 thf(fact_2335_finite__Collect__conjI, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((finite_finite2 @ A) @ ((collect @ A) @ P)) | ((finite_finite2 @ A) @ ((collect @ A) @ Q))) => ((finite_finite2 @ A) @ ((collect @ A) @ (^ [X:A]: (((P @ X) & (Q @ X)))))))))). % 6.41/2.69 thf(fact_2336_finite__Collect__disjI, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((finite_finite2 @ A) @ ((collect @ A) @ (^ [X:A]: (((P @ X) | (Q @ X)))))) = (((finite_finite2 @ A) @ ((collect @ A) @ P)) & ((finite_finite2 @ A) @ ((collect @ A) @ Q))))))). % 6.41/2.69 thf(fact_2337_bit_Oxor__self, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X3:A]: (((((bit_se5824344971392196577ns_xor @ A) @ X3) @ X3) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_2338_xor__self__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se5824344971392196577ns_xor @ A) @ A2) @ A2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_2339_xor_Oleft__neutral, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se5824344971392196577ns_xor @ A) @ (zero_zero @ A)) @ A2) = A2))))))). % 6.41/2.69 thf(fact_2340_xor_Oright__neutral, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se5824344971392196577ns_xor @ A) @ A2) @ (zero_zero @ A)) = A2))))))). % 6.41/2.69 thf(fact_2341_finite__Collect__bounded__ex, axiom, (! [B:$tType,A:$tType,P:(A > $o),Q:(B > (A > $o))]: ((((finite_finite2 @ A) @ ((collect @ A) @ P)) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [X:B]: ((? [Y2:A]: (((P @ Y2) & ((Q @ X) @ Y2)))))))) = (! [Y2:A]: (((P @ Y2) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [X:B]: (((Q @ X) @ Y2))))))))))))). % 6.41/2.69 thf(fact_2342_finite__Collect__subsets, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((finite_finite2 @ (set @ A)) @ ((collect @ (set @ A)) @ (^ [B6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B6) @ A3))))))))). % 6.41/2.69 thf(fact_2343_singleton__conv2, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ ((^ [Y3:A,Z:A]: ((Y3 = Z))) @ A2)) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))). % 6.41/2.69 thf(fact_2344_singleton__conv, axiom, (! [A:$tType,A2:A]: ((((collect @ A) @ (^ [X:A]: ((X = A2)))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))). % 6.41/2.69 thf(fact_2345_finite__Collect__less__nat, axiom, (! [K:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N:nat]: ((((ord_less @ nat) @ N) @ K)))))))). % 6.41/2.69 thf(fact_2346_finite__Collect__le__nat, axiom, (! [K:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N:nat]: ((((ord_less_eq @ nat) @ N) @ K)))))))). % 6.41/2.69 thf(fact_2347_finite__interval__int1, axiom, (! [A2:int,B2:int]: (((finite_finite2 @ int) @ ((collect @ int) @ (^ [I3:int]: (((((ord_less_eq @ int) @ A2) @ I3) & (((ord_less_eq @ int) @ I3) @ B2))))))))). % 6.41/2.69 thf(fact_2348_finite__interval__int4, axiom, (! [A2:int,B2:int]: (((finite_finite2 @ int) @ ((collect @ int) @ (^ [I3:int]: (((((ord_less @ int) @ A2) @ I3) & (((ord_less @ int) @ I3) @ B2))))))))). % 6.41/2.69 thf(fact_2349_finite__interval__int2, axiom, (! [A2:int,B2:int]: (((finite_finite2 @ int) @ ((collect @ int) @ (^ [I3:int]: (((((ord_less_eq @ int) @ A2) @ I3) & (((ord_less @ int) @ I3) @ B2))))))))). % 6.41/2.69 thf(fact_2350_finite__interval__int3, axiom, (! [A2:int,B2:int]: (((finite_finite2 @ int) @ ((collect @ int) @ (^ [I3:int]: (((((ord_less @ int) @ A2) @ I3) & (((ord_less_eq @ int) @ I3) @ B2))))))))). % 6.41/2.69 thf(fact_2351_xor__nat__numerals_I4_J, axiom, (! [X3:num]: (((((bit_se5824344971392196577ns_xor @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ X3))) @ (suc @ (zero_zero @ nat))) = ((numeral_numeral @ nat) @ (bit0 @ X3)))))). % 6.41/2.69 thf(fact_2352_xor__nat__numerals_I3_J, axiom, (! [X3:num]: (((((bit_se5824344971392196577ns_xor @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ X3))) @ (suc @ (zero_zero @ nat))) = ((numeral_numeral @ nat) @ (bit1 @ X3)))))). % 6.41/2.69 thf(fact_2353_xor__nat__numerals_I2_J, axiom, (! [Y:num]: (((((bit_se5824344971392196577ns_xor @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ (bit1 @ Y))) = ((numeral_numeral @ nat) @ (bit0 @ Y)))))). % 6.41/2.69 thf(fact_2354_xor__nat__numerals_I1_J, axiom, (! [Y:num]: (((((bit_se5824344971392196577ns_xor @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ (bit0 @ Y))) = ((numeral_numeral @ nat) @ (bit1 @ Y)))))). % 6.41/2.69 thf(fact_2355_xor__numerals_I6_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X3:num,Y:num]: (((((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ (bit1 @ X3))) @ ((numeral_numeral @ A) @ (bit0 @ Y))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ X3)) @ ((numeral_numeral @ A) @ Y))))))))))). % 6.41/2.69 thf(fact_2356_xor__numerals_I4_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X3:num,Y:num]: (((((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ (bit0 @ X3))) @ ((numeral_numeral @ A) @ (bit1 @ Y))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ X3)) @ ((numeral_numeral @ A) @ Y))))))))))). % 6.41/2.69 thf(fact_2357_empty__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (^ [X:A]: ($false))))))). % 6.41/2.69 thf(fact_2358_Collect__conv__if2, axiom, (! [A:$tType,P:(A > $o),A2:A]: ((((P @ A2) => (((collect @ A) @ (^ [X:A]: (((A2 = X) & (P @ X))))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A2))) => (((collect @ A) @ (^ [X:A]: (((A2 = X) & (P @ X))))) = (bot_bot @ (set @ A)))))))). % 6.41/2.69 thf(fact_2359_Collect__conv__if, axiom, (! [A:$tType,P:(A > $o),A2:A]: ((((P @ A2) => (((collect @ A) @ (^ [X:A]: (((X = A2) & (P @ X))))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) & ((~ ((P @ A2))) => (((collect @ A) @ (^ [X:A]: (((X = A2) & (P @ X))))) = (bot_bot @ (set @ A)))))))). % 6.41/2.69 thf(fact_2360_pred__subset__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (A > (B > $o))) @ (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ R)))) @ (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ S3)))) = (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S3))))). % 6.41/2.69 thf(fact_2361_strict__subset__divisors__dvd, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A,B2:A]: (((((ord_less @ (set @ A)) @ ((collect @ A) @ (^ [C4:A]: ((((dvd_dvd @ A) @ C4) @ A2))))) @ ((collect @ A) @ (^ [C4:A]: ((((dvd_dvd @ A) @ C4) @ B2))))) = ((((dvd_dvd @ A) @ A2) @ B2) & (~ ((((dvd_dvd @ A) @ B2) @ A2))))))))))). % 6.41/2.69 thf(fact_2362_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: ((((^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ R))) = (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ S3)))) = (R = S3))))). % 6.41/2.69 thf(fact_2363_subset__divisors__dvd, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [C4:A]: ((((dvd_dvd @ A) @ C4) @ A2))))) @ ((collect @ A) @ (^ [C4:A]: ((((dvd_dvd @ A) @ C4) @ B2))))) = (((dvd_dvd @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_2364_less__set__def, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((((ord_less @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ A6)))) @ (^ [X:A]: ((((member @ A) @ X) @ B6)))))))))). % 6.41/2.69 thf(fact_2365_Collect__subset, axiom, (! [A:$tType,A3:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A3) & (P @ X)))))) @ A3)))). % 6.41/2.69 thf(fact_2366_less__eq__set__def, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((((ord_less_eq @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ A6)))) @ (^ [X:A]: ((((member @ A) @ X) @ B6)))))))))). % 6.41/2.69 thf(fact_2367_prop__restrict, axiom, (! [A:$tType,X3:A,Z5:(set @ A),X8:(set @ A),P:(A > $o)]: (((((member @ A) @ X3) @ Z5) => ((((ord_less_eq @ (set @ A)) @ Z5) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ X8) & (P @ X)))))) => (P @ X3)))))). % 6.41/2.69 thf(fact_2368_Collect__restrict, axiom, (! [A:$tType,X8:(set @ A),P:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ X8) & (P @ X)))))) @ X8)))). % 6.41/2.69 thf(fact_2369_pred__subset__eq, axiom, (! [A:$tType,R:(set @ A),S3:(set @ A)]: (((((ord_less_eq @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ R)))) @ (^ [X:A]: ((((member @ A) @ X) @ S3)))) = (((ord_less_eq @ (set @ A)) @ R) @ S3))))). % 6.41/2.69 thf(fact_2370_lambda__one, axiom, (! [A:$tType]: (((monoid_mult @ A) => ((^ [X:A]: (X)) = ((times_times @ A) @ (one_one @ A))))))). % 6.41/2.69 thf(fact_2371_lambda__zero, axiom, (! [A:$tType]: (((mult_zero @ A) => ((^ [H:A]: ((zero_zero @ A))) = ((times_times @ A) @ (zero_zero @ A))))))). % 6.41/2.69 thf(fact_2372_uminus__set__def, axiom, (! [A:$tType]: (((uminus_uminus @ (set @ A)) = (^ [A6:(set @ A)]: (((collect @ A) @ ((uminus_uminus @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ A6))))))))))). % 6.41/2.69 thf(fact_2373_Collect__neg__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((collect @ A) @ (^ [X:A]: ((~ ((P @ X)))))) = ((uminus_uminus @ (set @ A)) @ ((collect @ A) @ P)))))). % 6.41/2.69 thf(fact_2374_Compl__eq, axiom, (! [A:$tType]: (((uminus_uminus @ (set @ A)) = (^ [A6:(set @ A)]: (((collect @ A) @ (^ [X:A]: ((~ ((((member @ A) @ X) @ A6)))))))))))). % 6.41/2.69 thf(fact_2375_set__diff__eq, axiom, (! [A:$tType]: (((minus_minus @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A6) & (~ ((((member @ A) @ X) @ B6))))))))))))). % 6.41/2.69 thf(fact_2376_insert__compr, axiom, (! [A:$tType]: (((insert @ A) = (^ [A4:A,B6:(set @ A)]: (((collect @ A) @ (^ [X:A]: (((X = A4) | (((member @ A) @ X) @ B6))))))))))). % 6.41/2.69 thf(fact_2377_minus__set__def, axiom, (! [A:$tType]: (((minus_minus @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((collect @ A) @ (((minus_minus @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ A6)))) @ (^ [X:A]: ((((member @ A) @ X) @ B6))))))))))). % 6.41/2.69 thf(fact_2378_insert__Collect, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((insert @ A) @ A2) @ ((collect @ A) @ P)) = ((collect @ A) @ (^ [U2:A]: (((U2 != A2) => (P @ U2))))))))). % 6.41/2.69 thf(fact_2379_not__finite__existsD, axiom, (! [A:$tType,P:(A > $o)]: (((~ (((finite_finite2 @ A) @ ((collect @ A) @ P)))) => (? [X_1:A]: ((P @ X_1))))))). % 6.41/2.69 thf(fact_2380_pigeonhole__infinite__rel, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B),R:(A > (B > $o))]: (((~ (((finite_finite2 @ A) @ A3))) => (((finite_finite2 @ B) @ B3) => ((! [X5:A]: (((((member @ A) @ X5) @ A3) => (? [Xa:B]: (((((member @ B) @ Xa) @ B3) & ((R @ X5) @ Xa))))))) => (? [X5:B]: (((((member @ B) @ X5) @ B3) & (~ (((finite_finite2 @ A) @ ((collect @ A) @ (^ [A4:A]: (((((member @ A) @ A4) @ A3) & ((R @ A4) @ X5))))))))))))))))). % 6.41/2.69 thf(fact_2381_finite__image__set2, axiom, (! [A:$tType,B:$tType,C:$tType,P:(A > $o),Q:(B > $o),F2:(A > (B > C))]: ((((finite_finite2 @ A) @ ((collect @ A) @ P)) => (((finite_finite2 @ B) @ ((collect @ B) @ Q)) => ((finite_finite2 @ C) @ ((collect @ C) @ (^ [Uu3:C]: ((? [X:A,Y2:B]: (((Uu3 = ((F2 @ X) @ Y2)) & ((P @ X) & (Q @ Y2)))))))))))))). % 6.41/2.69 thf(fact_2382_finite__image__set, axiom, (! [A:$tType,B:$tType,P:(A > $o),F2:(A > B)]: ((((finite_finite2 @ A) @ ((collect @ A) @ P)) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [Uu3:B]: ((? [X:A]: (((Uu3 = (F2 @ X)) & (P @ X)))))))))))). % 6.41/2.69 thf(fact_2383_bot__empty__eq2, axiom, (! [B:$tType,A:$tType]: (((bot_bot @ (A > (B > $o))) = (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ (bot_bot @ (set @ ((product_prod @ A) @ B)))))))))). % 6.41/2.69 thf(fact_2384_finite__M__bounded__by__nat, axiom, (! [P:(nat > $o),I:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [K3:nat]: (((P @ K3) & (((ord_less @ nat) @ K3) @ I))))))))). % 6.41/2.69 thf(fact_2385_finite__less__ub, axiom, (! [F2:(nat > nat),U:nat]: (((! [N3:nat]: ((((ord_less_eq @ nat) @ N3) @ (F2 @ N3)))) => ((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N:nat]: ((((ord_less_eq @ nat) @ (F2 @ N)) @ U))))))))). % 6.41/2.69 thf(fact_2386_numeral__code_I2_J, axiom, (! [A:$tType]: (((numeral @ A) => (! [N2:num]: ((((numeral_numeral @ A) @ (bit0 @ N2)) = (((plus_plus @ A) @ ((numeral_numeral @ A) @ N2)) @ ((numeral_numeral @ A) @ N2))))))))). % 6.41/2.69 thf(fact_2387_set__vebt__def, axiom, (vEBT_set_vebt = (^ [T3:vEBT_VEBT]: (((collect @ nat) @ (vEBT_V8194947554948674370ptions @ T3)))))). % 6.41/2.69 thf(fact_2388_finite__divisors__int, axiom, (! [I:int]: (((I != (zero_zero @ int)) => ((finite_finite2 @ int) @ ((collect @ int) @ (^ [D5:int]: ((((dvd_dvd @ int) @ D5) @ I))))))))). % 6.41/2.69 thf(fact_2389_numeral__code_I3_J, axiom, (! [A:$tType]: (((numeral @ A) => (! [N2:num]: ((((numeral_numeral @ A) @ (bit1 @ N2)) = (((plus_plus @ A) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ N2)) @ ((numeral_numeral @ A) @ N2))) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_2390_finite__divisors__nat, axiom, (! [M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [D5:nat]: ((((dvd_dvd @ nat) @ D5) @ M))))))))). % 6.41/2.69 thf(fact_2391_set__conv__nth, axiom, (! [A:$tType]: (((set2 @ A) = (^ [Xs2:(list @ A)]: (((collect @ A) @ (^ [Uu3:A]: ((? [I3:nat]: (((Uu3 = (((nth @ A) @ Xs2) @ I3)) & (((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs2)))))))))))))). % 6.41/2.69 thf(fact_2392_finite__lists__length__eq, axiom, (! [A:$tType,A3:(set @ A),N2:nat]: ((((finite_finite2 @ A) @ A3) => ((finite_finite2 @ (list @ A)) @ ((collect @ (list @ A)) @ (^ [Xs2:(list @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs2)) @ A3) & (((size_size @ (list @ A)) @ Xs2) = N2)))))))))). % 6.41/2.69 thf(fact_2393_finite__lists__length__le, axiom, (! [A:$tType,A3:(set @ A),N2:nat]: ((((finite_finite2 @ A) @ A3) => ((finite_finite2 @ (list @ A)) @ ((collect @ (list @ A)) @ (^ [Xs2:(list @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs2)) @ A3) & (((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs2)) @ N2)))))))))). % 6.41/2.69 thf(fact_2394_vebt__buildup_Osimps_I3_J, axiom, (! [Va:nat]: ((((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ (suc @ Va))) => ((vEBT_vebt_buildup @ (suc @ (suc @ Va))) = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ (suc @ Va))) @ (((replicate @ vEBT_VEBT) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((divide_divide @ nat) @ (suc @ (suc @ Va))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (vEBT_vebt_buildup @ (((divide_divide @ nat) @ (suc @ (suc @ Va))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ (vEBT_vebt_buildup @ (((divide_divide @ nat) @ (suc @ (suc @ Va))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) & ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ (suc @ Va))))) => ((vEBT_vebt_buildup @ (suc @ (suc @ Va))) = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ (suc @ Va))) @ (((replicate @ vEBT_VEBT) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ (((divide_divide @ nat) @ (suc @ (suc @ Va))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ (vEBT_vebt_buildup @ (((divide_divide @ nat) @ (suc @ (suc @ Va))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ (vEBT_vebt_buildup @ (suc @ (((divide_divide @ nat) @ (suc @ (suc @ Va))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))). % 6.41/2.69 thf(fact_2395_vebt__buildup_Oelims, axiom, (! [X3:nat,Y:vEBT_VEBT]: ((((vEBT_vebt_buildup @ X3) = Y) => (((X3 = (zero_zero @ nat)) => (Y != ((vEBT_Leaf @ $false) @ $false))) => (((X3 = (suc @ (zero_zero @ nat))) => (Y != ((vEBT_Leaf @ $false) @ $false))) => (~ ((! [Va2:nat]: (((X3 = (suc @ (suc @ Va2))) => (~ ((((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ (suc @ Va2))) => (Y = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ (suc @ Va2))) @ (((replicate @ vEBT_VEBT) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (vEBT_vebt_buildup @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ (vEBT_vebt_buildup @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) & ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ (suc @ Va2))))) => (Y = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ (suc @ Va2))) @ (((replicate @ vEBT_VEBT) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ (vEBT_vebt_buildup @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ (vEBT_vebt_buildup @ (suc @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))))))))))))). % 6.41/2.69 thf(fact_2396_VEBT__internal_Onaive__member_Osimps_I3_J, axiom, (! [Uy:(option @ ((product_prod @ nat) @ nat)),V2:nat,TreeList2:(list @ vEBT_VEBT),S:vEBT_VEBT,X3:nat]: ((((vEBT_V5719532721284313246member @ ((((vEBT_Node @ Uy) @ (suc @ V2)) @ TreeList2) @ S)) @ X3) = (((((ord_less @ nat) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ (suc @ V2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList2)) => ((vEBT_V5719532721284313246member @ (((nth @ vEBT_VEBT) @ TreeList2) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ (suc @ V2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ X3) @ (((divide_divide @ nat) @ (suc @ V2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ (suc @ V2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList2))))))). % 6.41/2.69 thf(fact_2397_dbl__dec__def, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((neg_numeral_dbl_dec @ A) = (^ [X:A]: ((((minus_minus @ A) @ (((plus_plus @ A) @ X) @ X)) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_2398_VEBT__internal_Omembermima_Osimps_I5_J, axiom, (! [V2:nat,TreeList2:(list @ vEBT_VEBT),Vd2:vEBT_VEBT,X3:nat]: ((((vEBT_VEBT_membermima @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ V2)) @ TreeList2) @ Vd2)) @ X3) = (((((ord_less @ nat) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ (suc @ V2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList2)) => ((vEBT_VEBT_membermima @ (((nth @ vEBT_VEBT) @ TreeList2) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ (suc @ V2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ X3) @ (((divide_divide @ nat) @ (suc @ V2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ (suc @ V2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList2))))))). % 6.41/2.69 thf(fact_2399_vebt__member_Osimps_I5_J, axiom, (! [Mi:nat,Ma:nat,Va:nat,TreeList2:(list @ vEBT_VEBT),Summary:vEBT_VEBT,X3:nat]: ((((vEBT_vebt_member @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi) @ Ma))) @ (suc @ (suc @ Va))) @ TreeList2) @ Summary)) @ X3) = ((X3 != Mi) => ((X3 != Ma) => ((~ ((((ord_less @ nat) @ X3) @ Mi))) & ((~ ((((ord_less @ nat) @ X3) @ Mi))) => ((~ ((((ord_less @ nat) @ Ma) @ X3))) & ((~ ((((ord_less @ nat) @ Ma) @ X3))) => (((((ord_less @ nat) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ (suc @ (suc @ Va))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList2)) => ((vEBT_vebt_member @ (((nth @ vEBT_VEBT) @ TreeList2) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ (suc @ (suc @ Va))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ X3) @ (((divide_divide @ nat) @ (suc @ (suc @ Va))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ (suc @ (suc @ Va))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList2))))))))))))). % 6.41/2.69 thf(fact_2400_VEBT__internal_Omembermima_Osimps_I4_J, axiom, (! [Mi:nat,Ma:nat,V2:nat,TreeList2:(list @ vEBT_VEBT),Vc:vEBT_VEBT,X3:nat]: ((((vEBT_VEBT_membermima @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi) @ Ma))) @ (suc @ V2)) @ TreeList2) @ Vc)) @ X3) = ((X3 = Mi) | ((X3 = Ma) | (((((ord_less @ nat) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ (suc @ V2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList2)) => ((vEBT_VEBT_membermima @ (((nth @ vEBT_VEBT) @ TreeList2) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ (suc @ V2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ X3) @ (((divide_divide @ nat) @ (suc @ V2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ X3) @ (((divide_divide @ nat) @ (suc @ V2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList2))))))))). % 6.41/2.69 thf(fact_2401_VEBT__internal_Onaive__member_Oelims_I1_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat,Y:$o]: (((((vEBT_V5719532721284313246member @ X3) @ Xa2) = Y) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => (Y = (~ ((((Xa2 = (zero_zero @ nat)) => A5) & ((Xa2 != (zero_zero @ nat)) => (((Xa2 = (one_one @ nat)) => B5) & (Xa2 = (one_one @ nat))))))))))) => (((? [Uu2:(option @ ((product_prod @ nat) @ nat)),Uv2:(list @ vEBT_VEBT),Uw2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ Uu2) @ (zero_zero @ nat)) @ Uv2) @ Uw2)))) => Y) => (~ ((! [Uy2:(option @ ((product_prod @ nat) @ nat)),V3:nat,TreeList3:(list @ vEBT_VEBT)]: (((? [S2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ Uy2) @ (suc @ V3)) @ TreeList3) @ S2)))) => (Y = (~ ((((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_V5719532721284313246member @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3))))))))))))))))). % 6.41/2.69 thf(fact_2402_VEBT__internal_Onaive__member_Oelims_I2_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: ((((vEBT_V5719532721284313246member @ X3) @ Xa2) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => (~ ((((Xa2 = (zero_zero @ nat)) => A5) & ((Xa2 != (zero_zero @ nat)) => (((Xa2 = (one_one @ nat)) => B5) & (Xa2 = (one_one @ nat)))))))))) => (~ ((! [Uy2:(option @ ((product_prod @ nat) @ nat)),V3:nat,TreeList3:(list @ vEBT_VEBT)]: (((? [S2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ Uy2) @ (suc @ V3)) @ TreeList3) @ S2)))) => (~ ((((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_V5719532721284313246member @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3))))))))))))))). % 6.41/2.69 thf(fact_2403_VEBT__internal_Onaive__member_Oelims_I3_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: (((~ (((vEBT_V5719532721284313246member @ X3) @ Xa2))) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => (((Xa2 = (zero_zero @ nat)) => A5) & ((Xa2 != (zero_zero @ nat)) => (((Xa2 = (one_one @ nat)) => B5) & (Xa2 = (one_one @ nat)))))))) => ((! [Uu2:(option @ ((product_prod @ nat) @ nat)),Uv2:(list @ vEBT_VEBT),Uw2:vEBT_VEBT]: ((X3 != ((((vEBT_Node @ Uu2) @ (zero_zero @ nat)) @ Uv2) @ Uw2)))) => (~ ((! [Uy2:(option @ ((product_prod @ nat) @ nat)),V3:nat,TreeList3:(list @ vEBT_VEBT)]: (((? [S2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ Uy2) @ (suc @ V3)) @ TreeList3) @ S2)))) => (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_V5719532721284313246member @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)))))))))))))). % 6.41/2.69 thf(fact_2404_VEBT__internal_Omembermima_Oelims_I2_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: ((((vEBT_VEBT_membermima @ X3) @ Xa2) => ((! [Mi2:nat,Ma2:nat]: (((? [Va3:(list @ vEBT_VEBT),Vb2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (zero_zero @ nat)) @ Va3) @ Vb2)))) => (~ (((Xa2 = Mi2) | (Xa2 = Ma2))))))) => ((! [Mi2:nat,Ma2:nat,V3:nat,TreeList3:(list @ vEBT_VEBT)]: (((? [Vc2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ V3)) @ TreeList3) @ Vc2)))) => (~ (((Xa2 = Mi2) | ((Xa2 = Ma2) | (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_VEBT_membermima @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)))))))))) => (~ ((! [V3:nat,TreeList3:(list @ vEBT_VEBT)]: (((? [Vd:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ V3)) @ TreeList3) @ Vd)))) => (~ ((((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_VEBT_membermima @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)))))))))))))))). % 6.41/2.69 thf(fact_2405_xor__nat__unfold, axiom, ((bit_se5824344971392196577ns_xor @ nat) = (^ [M2:nat,N:nat]: (((((if @ nat) @ (M2 = (zero_zero @ nat))) @ N) @ ((((if @ nat) @ (N = (zero_zero @ nat))) @ M2) @ (((plus_plus @ nat) @ (((modulo_modulo @ nat) @ (((plus_plus @ nat) @ (((modulo_modulo @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((modulo_modulo @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se5824344971392196577ns_xor @ nat) @ (((divide_divide @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))). % 6.41/2.69 thf(fact_2406_xor__nat__rec, axiom, ((bit_se5824344971392196577ns_xor @ nat) = (^ [M2:nat,N:nat]: ((((plus_plus @ nat) @ ((zero_neq_one_of_bool @ nat) @ ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M2))) != (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se5824344971392196577ns_xor @ nat) @ (((divide_divide @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_2407_one__xor__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se5824344971392196577ns_xor @ A) @ (one_one @ A)) @ A2) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ ((zero_neq_one_of_bool @ A) @ (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2)))) @ ((zero_neq_one_of_bool @ A) @ (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2)))))))))))). % 6.41/2.69 thf(fact_2408_xor__one__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se5824344971392196577ns_xor @ A) @ A2) @ (one_one @ A)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ ((zero_neq_one_of_bool @ A) @ (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2)))) @ ((zero_neq_one_of_bool @ A) @ (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2)))))))))))). % 6.41/2.69 thf(fact_2409_vebt__member_Oelims_I2_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: ((((vEBT_vebt_member @ X3) @ Xa2) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => (~ ((((Xa2 = (zero_zero @ nat)) => A5) & ((Xa2 != (zero_zero @ nat)) => (((Xa2 = (one_one @ nat)) => B5) & (Xa2 = (one_one @ nat)))))))))) => (~ ((! [Mi2:nat,Ma2:nat,Va2:nat,TreeList3:(list @ vEBT_VEBT)]: (((? [Summary2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ (suc @ Va2))) @ TreeList3) @ Summary2)))) => (~ (((Xa2 != Mi2) => ((Xa2 != Ma2) => ((~ ((((ord_less @ nat) @ Xa2) @ Mi2))) & ((~ ((((ord_less @ nat) @ Xa2) @ Mi2))) => ((~ ((((ord_less @ nat) @ Ma2) @ Xa2))) & ((~ ((((ord_less @ nat) @ Ma2) @ Xa2))) => (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_vebt_member @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3))))))))))))))))))))). % 6.41/2.69 thf(fact_2410_VEBT__internal_Omembermima_Oelims_I3_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: (((~ (((vEBT_VEBT_membermima @ X3) @ Xa2))) => ((! [Uu2:$o,Uv2:$o]: ((X3 != ((vEBT_Leaf @ Uu2) @ Uv2)))) => ((! [Ux2:(list @ vEBT_VEBT),Uy2:vEBT_VEBT]: ((X3 != ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (zero_zero @ nat)) @ Ux2) @ Uy2)))) => ((! [Mi2:nat,Ma2:nat]: (((? [Va3:(list @ vEBT_VEBT),Vb2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (zero_zero @ nat)) @ Va3) @ Vb2)))) => ((Xa2 = Mi2) | (Xa2 = Ma2))))) => ((! [Mi2:nat,Ma2:nat,V3:nat,TreeList3:(list @ vEBT_VEBT)]: (((? [Vc2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ V3)) @ TreeList3) @ Vc2)))) => ((Xa2 = Mi2) | ((Xa2 = Ma2) | (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_VEBT_membermima @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)))))))) => (~ ((! [V3:nat,TreeList3:(list @ vEBT_VEBT)]: (((? [Vd:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ V3)) @ TreeList3) @ Vd)))) => (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_VEBT_membermima @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)))))))))))))))). % 6.41/2.69 thf(fact_2411_VEBT__internal_Omembermima_Oelims_I1_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat,Y:$o]: (((((vEBT_VEBT_membermima @ X3) @ Xa2) = Y) => (((? [Uu2:$o,Uv2:$o]: ((X3 = ((vEBT_Leaf @ Uu2) @ Uv2)))) => Y) => (((? [Ux2:(list @ vEBT_VEBT),Uy2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (zero_zero @ nat)) @ Ux2) @ Uy2)))) => Y) => ((! [Mi2:nat,Ma2:nat]: (((? [Va3:(list @ vEBT_VEBT),Vb2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (zero_zero @ nat)) @ Va3) @ Vb2)))) => (Y = (~ (((Xa2 = Mi2) | (Xa2 = Ma2)))))))) => ((! [Mi2:nat,Ma2:nat,V3:nat,TreeList3:(list @ vEBT_VEBT)]: (((? [Vc2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ V3)) @ TreeList3) @ Vc2)))) => (Y = (~ (((Xa2 = Mi2) | ((Xa2 = Ma2) | (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_VEBT_membermima @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3))))))))))) => (~ ((! [V3:nat,TreeList3:(list @ vEBT_VEBT)]: (((? [Vd:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ V3)) @ TreeList3) @ Vd)))) => (Y = (~ ((((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_VEBT_membermima @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3))))))))))))))))))). % 6.41/2.69 thf(fact_2412_bit__horner__sum__bit__iff, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [Bs:(list @ $o),N2:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ (((((groups4207007520872428315er_sum @ $o) @ A) @ (zero_neq_one_of_bool @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Bs)) @ N2) = ((((ord_less @ nat) @ N2) @ ((size_size @ (list @ $o)) @ Bs)) & (((nth @ $o) @ Bs) @ N2))))))))). % 6.41/2.69 thf(fact_2413_vebt__member_Oelims_I3_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: (((~ (((vEBT_vebt_member @ X3) @ Xa2))) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => (((Xa2 = (zero_zero @ nat)) => A5) & ((Xa2 != (zero_zero @ nat)) => (((Xa2 = (one_one @ nat)) => B5) & (Xa2 = (one_one @ nat)))))))) => ((! [Uu2:nat,Uv2:(list @ vEBT_VEBT),Uw2:vEBT_VEBT]: ((X3 != ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu2) @ Uv2) @ Uw2)))) => ((! [V3:((product_prod @ nat) @ nat),Uy2:(list @ vEBT_VEBT),Uz2:vEBT_VEBT]: ((X3 != ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V3)) @ (zero_zero @ nat)) @ Uy2) @ Uz2)))) => ((! [V3:((product_prod @ nat) @ nat),Vb2:(list @ vEBT_VEBT),Vc2:vEBT_VEBT]: ((X3 != ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V3)) @ (suc @ (zero_zero @ nat))) @ Vb2) @ Vc2)))) => (~ ((! [Mi2:nat,Ma2:nat,Va2:nat,TreeList3:(list @ vEBT_VEBT)]: (((? [Summary2:vEBT_VEBT]: ((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ (suc @ Va2))) @ TreeList3) @ Summary2)))) => ((Xa2 != Mi2) => ((Xa2 != Ma2) => ((~ ((((ord_less @ nat) @ Xa2) @ Mi2))) & ((~ ((((ord_less @ nat) @ Xa2) @ Mi2))) => ((~ ((((ord_less @ nat) @ Ma2) @ Xa2))) & ((~ ((((ord_less @ nat) @ Ma2) @ Xa2))) => (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_vebt_member @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)))))))))))))))))))))). % 6.41/2.69 thf(fact_2414_of__int__code__if, axiom, (! [A:$tType]: (((ring_1 @ A) => ((ring_1_of_int @ A) = (^ [K3:int]: (((((if @ A) @ (K3 = (zero_zero @ int))) @ (zero_zero @ A)) @ ((((if @ A) @ (((ord_less @ int) @ K3) @ (zero_zero @ int))) @ ((uminus_uminus @ A) @ ((ring_1_of_int @ A) @ ((uminus_uminus @ int) @ K3)))) @ ((((if @ A) @ ((((modulo_modulo @ int) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2))) = (zero_zero @ int))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((ring_1_of_int @ A) @ (((divide_divide @ int) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((ring_1_of_int @ A) @ (((divide_divide @ int) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))) @ (one_one @ A)))))))))))). % 6.41/2.69 thf(fact_2415_finite__nth__roots, axiom, (! [N2:nat,C2:complex]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((finite_finite2 @ complex) @ ((collect @ complex) @ (^ [Z6:complex]: (((((power_power @ complex) @ Z6) @ N2) = C2))))))))). % 6.41/2.69 thf(fact_2416_finite__roots__unity, axiom, (! [A:$tType]: ((((real_V8999393235501362500lgebra @ A) & (idom @ A)) => (! [N2:nat]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ N2) => ((finite_finite2 @ A) @ ((collect @ A) @ (^ [Z6:A]: (((((power_power @ A) @ Z6) @ N2) = (one_one @ A))))))))))))). % 6.41/2.69 thf(fact_2417_prod_Ofinite__Collect__op, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [I6:(set @ B),X3:(B > A),Y:(B > A)]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [I3:B]: (((((member @ B) @ I3) @ I6) & ((X3 @ I3) != (one_one @ A))))))) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [I3:B]: (((((member @ B) @ I3) @ I6) & ((Y @ I3) != (one_one @ A))))))) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [I3:B]: (((((member @ B) @ I3) @ I6) & ((((times_times @ A) @ (X3 @ I3)) @ (Y @ I3)) != (one_one @ A))))))))))))))). % 6.41/2.69 thf(fact_2418_vebt__member_Opelims_I1_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat,Y:$o]: (((((vEBT_vebt_member @ X3) @ Xa2) = Y) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_vebt_member_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ X3) @ Xa2)) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => ((Y = (((Xa2 = (zero_zero @ nat)) => A5) & ((Xa2 != (zero_zero @ nat)) => (((Xa2 = (one_one @ nat)) => B5) & (Xa2 = (one_one @ nat)))))) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_vebt_member_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((vEBT_Leaf @ A5) @ B5)) @ Xa2)))))))) => ((! [Uu2:nat,Uv2:(list @ vEBT_VEBT),Uw2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu2) @ Uv2) @ Uw2)) => ((~ (Y)) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_vebt_member_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu2) @ Uv2) @ Uw2)) @ Xa2)))))))) => ((! [V3:((product_prod @ nat) @ nat),Uy2:(list @ vEBT_VEBT),Uz2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V3)) @ (zero_zero @ nat)) @ Uy2) @ Uz2)) => ((~ (Y)) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_vebt_member_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V3)) @ (zero_zero @ nat)) @ Uy2) @ Uz2)) @ Xa2)))))))) => ((! [V3:((product_prod @ nat) @ nat),Vb2:(list @ vEBT_VEBT),Vc2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V3)) @ (suc @ (zero_zero @ nat))) @ Vb2) @ Vc2)) => ((~ (Y)) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_vebt_member_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V3)) @ (suc @ (zero_zero @ nat))) @ Vb2) @ Vc2)) @ Xa2)))))))) => (~ ((! [Mi2:nat,Ma2:nat,Va2:nat,TreeList3:(list @ vEBT_VEBT),Summary2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ (suc @ Va2))) @ TreeList3) @ Summary2)) => ((Y = ((Xa2 != Mi2) => ((Xa2 != Ma2) => ((~ ((((ord_less @ nat) @ Xa2) @ Mi2))) & ((~ ((((ord_less @ nat) @ Xa2) @ Mi2))) => ((~ ((((ord_less @ nat) @ Ma2) @ Xa2))) & ((~ ((((ord_less @ nat) @ Ma2) @ Xa2))) => (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_vebt_member @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)))))))))) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_vebt_member_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ (suc @ Va2))) @ TreeList3) @ Summary2)) @ Xa2))))))))))))))))))). % 6.41/2.69 thf(fact_2419_vebt__member_Opelims_I3_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: (((~ (((vEBT_vebt_member @ X3) @ Xa2))) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_vebt_member_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ X3) @ Xa2)) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_vebt_member_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((vEBT_Leaf @ A5) @ B5)) @ Xa2)) => (((Xa2 = (zero_zero @ nat)) => A5) & ((Xa2 != (zero_zero @ nat)) => (((Xa2 = (one_one @ nat)) => B5) & (Xa2 = (one_one @ nat))))))))) => ((! [Uu2:nat,Uv2:(list @ vEBT_VEBT),Uw2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu2) @ Uv2) @ Uw2)) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_vebt_member_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu2) @ Uv2) @ Uw2)) @ Xa2))))))) => ((! [V3:((product_prod @ nat) @ nat),Uy2:(list @ vEBT_VEBT),Uz2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V3)) @ (zero_zero @ nat)) @ Uy2) @ Uz2)) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_vebt_member_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V3)) @ (zero_zero @ nat)) @ Uy2) @ Uz2)) @ Xa2))))))) => ((! [V3:((product_prod @ nat) @ nat),Vb2:(list @ vEBT_VEBT),Vc2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V3)) @ (suc @ (zero_zero @ nat))) @ Vb2) @ Vc2)) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_vebt_member_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ V3)) @ (suc @ (zero_zero @ nat))) @ Vb2) @ Vc2)) @ Xa2))))))) => (~ ((! [Mi2:nat,Ma2:nat,Va2:nat,TreeList3:(list @ vEBT_VEBT),Summary2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ (suc @ Va2))) @ TreeList3) @ Summary2)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_vebt_member_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ (suc @ Va2))) @ TreeList3) @ Summary2)) @ Xa2)) => ((Xa2 != Mi2) => ((Xa2 != Ma2) => ((~ ((((ord_less @ nat) @ Xa2) @ Mi2))) & ((~ ((((ord_less @ nat) @ Xa2) @ Mi2))) => ((~ ((((ord_less @ nat) @ Ma2) @ Xa2))) & ((~ ((((ord_less @ nat) @ Ma2) @ Xa2))) => (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_vebt_member @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)))))))))))))))))))))))). % 6.41/2.69 thf(fact_2420_xor__negative__int__iff, axiom, (! [K:int,L:int]: (((((ord_less @ int) @ (((bit_se5824344971392196577ns_xor @ int) @ K) @ L)) @ (zero_zero @ int)) = ((((ord_less @ int) @ K) @ (zero_zero @ int)) != (((ord_less @ int) @ L) @ (zero_zero @ int))))))). % 6.41/2.69 thf(fact_2421_bot2E, axiom, (! [A:$tType,B:$tType,X3:A,Y:B]: ((~ ((((bot_bot @ (A > (B > $o))) @ X3) @ Y)))))). % 6.41/2.69 thf(fact_2422_XOR__upper, axiom, (! [X3:int,N2:nat,Y:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X3) => ((((ord_less @ int) @ X3) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)) => ((((ord_less @ int) @ Y) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)) => (((ord_less @ int) @ (((bit_se5824344971392196577ns_xor @ int) @ X3) @ Y)) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)))))))). % 6.41/2.69 thf(fact_2423_vebt__member_Opelims_I2_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: ((((vEBT_vebt_member @ X3) @ Xa2) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_vebt_member_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ X3) @ Xa2)) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_vebt_member_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((vEBT_Leaf @ A5) @ B5)) @ Xa2)) => (~ ((((Xa2 = (zero_zero @ nat)) => A5) & ((Xa2 != (zero_zero @ nat)) => (((Xa2 = (one_one @ nat)) => B5) & (Xa2 = (one_one @ nat))))))))))) => (~ ((! [Mi2:nat,Ma2:nat,Va2:nat,TreeList3:(list @ vEBT_VEBT),Summary2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ (suc @ Va2))) @ TreeList3) @ Summary2)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_vebt_member_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ (suc @ Va2))) @ TreeList3) @ Summary2)) @ Xa2)) => (~ (((Xa2 != Mi2) => ((Xa2 != Ma2) => ((~ ((((ord_less @ nat) @ Xa2) @ Mi2))) & ((~ ((((ord_less @ nat) @ Xa2) @ Mi2))) => ((~ ((((ord_less @ nat) @ Ma2) @ Xa2))) & ((~ ((((ord_less @ nat) @ Ma2) @ Xa2))) => (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_vebt_member @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3))))))))))))))))))))))). % 6.41/2.69 thf(fact_2424_sum_Ofinite__Collect__op, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [I6:(set @ B),X3:(B > A),Y:(B > A)]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [I3:B]: (((((member @ B) @ I3) @ I6) & ((X3 @ I3) != (zero_zero @ A))))))) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [I3:B]: (((((member @ B) @ I3) @ I6) & ((Y @ I3) != (zero_zero @ A))))))) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [I3:B]: (((((member @ B) @ I3) @ I6) & ((((plus_plus @ A) @ (X3 @ I3)) @ (Y @ I3)) != (zero_zero @ A))))))))))))))). % 6.41/2.69 thf(fact_2425_VEBT__internal_Onaive__member_Opelims_I1_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat,Y:$o]: (((((vEBT_V5719532721284313246member @ X3) @ Xa2) = Y) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V5765760719290551771er_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ X3) @ Xa2)) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => ((Y = (((Xa2 = (zero_zero @ nat)) => A5) & ((Xa2 != (zero_zero @ nat)) => (((Xa2 = (one_one @ nat)) => B5) & (Xa2 = (one_one @ nat)))))) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V5765760719290551771er_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((vEBT_Leaf @ A5) @ B5)) @ Xa2)))))))) => ((! [Uu2:(option @ ((product_prod @ nat) @ nat)),Uv2:(list @ vEBT_VEBT),Uw2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ Uu2) @ (zero_zero @ nat)) @ Uv2) @ Uw2)) => ((~ (Y)) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V5765760719290551771er_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ Uu2) @ (zero_zero @ nat)) @ Uv2) @ Uw2)) @ Xa2)))))))) => (~ ((! [Uy2:(option @ ((product_prod @ nat) @ nat)),V3:nat,TreeList3:(list @ vEBT_VEBT),S2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ Uy2) @ (suc @ V3)) @ TreeList3) @ S2)) => ((Y = (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_V5719532721284313246member @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)))) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V5765760719290551771er_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ Uy2) @ (suc @ V3)) @ TreeList3) @ S2)) @ Xa2))))))))))))))))). % 6.41/2.69 thf(fact_2426_VEBT__internal_Onaive__member_Opelims_I2_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: ((((vEBT_V5719532721284313246member @ X3) @ Xa2) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V5765760719290551771er_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ X3) @ Xa2)) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V5765760719290551771er_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((vEBT_Leaf @ A5) @ B5)) @ Xa2)) => (~ ((((Xa2 = (zero_zero @ nat)) => A5) & ((Xa2 != (zero_zero @ nat)) => (((Xa2 = (one_one @ nat)) => B5) & (Xa2 = (one_one @ nat))))))))))) => (~ ((! [Uy2:(option @ ((product_prod @ nat) @ nat)),V3:nat,TreeList3:(list @ vEBT_VEBT),S2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ Uy2) @ (suc @ V3)) @ TreeList3) @ S2)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V5765760719290551771er_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ Uy2) @ (suc @ V3)) @ TreeList3) @ S2)) @ Xa2)) => (~ ((((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_V5719532721284313246member @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3))))))))))))))))). % 6.41/2.69 thf(fact_2427_VEBT__internal_Onaive__member_Opelims_I3_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: (((~ (((vEBT_V5719532721284313246member @ X3) @ Xa2))) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V5765760719290551771er_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ X3) @ Xa2)) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V5765760719290551771er_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((vEBT_Leaf @ A5) @ B5)) @ Xa2)) => (((Xa2 = (zero_zero @ nat)) => A5) & ((Xa2 != (zero_zero @ nat)) => (((Xa2 = (one_one @ nat)) => B5) & (Xa2 = (one_one @ nat))))))))) => ((! [Uu2:(option @ ((product_prod @ nat) @ nat)),Uv2:(list @ vEBT_VEBT),Uw2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ Uu2) @ (zero_zero @ nat)) @ Uv2) @ Uw2)) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V5765760719290551771er_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ Uu2) @ (zero_zero @ nat)) @ Uv2) @ Uw2)) @ Xa2))))))) => (~ ((! [Uy2:(option @ ((product_prod @ nat) @ nat)),V3:nat,TreeList3:(list @ vEBT_VEBT),S2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ Uy2) @ (suc @ V3)) @ TreeList3) @ S2)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V5765760719290551771er_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ Uy2) @ (suc @ V3)) @ TreeList3) @ S2)) @ Xa2)) => (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_V5719532721284313246member @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)))))))))))))))). % 6.41/2.69 thf(fact_2428_VEBT__internal_Omembermima_Opelims_I3_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: (((~ (((vEBT_VEBT_membermima @ X3) @ Xa2))) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ X3) @ Xa2)) => ((! [Uu2:$o,Uv2:$o]: (((X3 = ((vEBT_Leaf @ Uu2) @ Uv2)) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((vEBT_Leaf @ Uu2) @ Uv2)) @ Xa2))))))) => ((! [Ux2:(list @ vEBT_VEBT),Uy2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (zero_zero @ nat)) @ Ux2) @ Uy2)) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (zero_zero @ nat)) @ Ux2) @ Uy2)) @ Xa2))))))) => ((! [Mi2:nat,Ma2:nat,Va3:(list @ vEBT_VEBT),Vb2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (zero_zero @ nat)) @ Va3) @ Vb2)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (zero_zero @ nat)) @ Va3) @ Vb2)) @ Xa2)) => ((Xa2 = Mi2) | (Xa2 = Ma2)))))) => ((! [Mi2:nat,Ma2:nat,V3:nat,TreeList3:(list @ vEBT_VEBT),Vc2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ V3)) @ TreeList3) @ Vc2)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ V3)) @ TreeList3) @ Vc2)) @ Xa2)) => ((Xa2 = Mi2) | ((Xa2 = Ma2) | (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_VEBT_membermima @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3))))))))) => (~ ((! [V3:nat,TreeList3:(list @ vEBT_VEBT),Vd:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ V3)) @ TreeList3) @ Vd)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ V3)) @ TreeList3) @ Vd)) @ Xa2)) => (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_VEBT_membermima @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)))))))))))))))))). % 6.41/2.69 thf(fact_2429_VEBT__internal_Omembermima_Opelims_I1_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat,Y:$o]: (((((vEBT_VEBT_membermima @ X3) @ Xa2) = Y) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ X3) @ Xa2)) => ((! [Uu2:$o,Uv2:$o]: (((X3 = ((vEBT_Leaf @ Uu2) @ Uv2)) => ((~ (Y)) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((vEBT_Leaf @ Uu2) @ Uv2)) @ Xa2)))))))) => ((! [Ux2:(list @ vEBT_VEBT),Uy2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (zero_zero @ nat)) @ Ux2) @ Uy2)) => ((~ (Y)) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (zero_zero @ nat)) @ Ux2) @ Uy2)) @ Xa2)))))))) => ((! [Mi2:nat,Ma2:nat,Va3:(list @ vEBT_VEBT),Vb2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (zero_zero @ nat)) @ Va3) @ Vb2)) => ((Y = ((Xa2 = Mi2) | (Xa2 = Ma2))) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (zero_zero @ nat)) @ Va3) @ Vb2)) @ Xa2)))))))) => ((! [Mi2:nat,Ma2:nat,V3:nat,TreeList3:(list @ vEBT_VEBT),Vc2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ V3)) @ TreeList3) @ Vc2)) => ((Y = ((Xa2 = Mi2) | ((Xa2 = Ma2) | (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_VEBT_membermima @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)))))) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ V3)) @ TreeList3) @ Vc2)) @ Xa2)))))))) => (~ ((! [V3:nat,TreeList3:(list @ vEBT_VEBT),Vd:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ V3)) @ TreeList3) @ Vd)) => ((Y = (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_VEBT_membermima @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)))) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ V3)) @ TreeList3) @ Vd)) @ Xa2))))))))))))))))))). % 6.41/2.69 thf(fact_2430_VEBT__internal_Omembermima_Opelims_I2_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: ((((vEBT_VEBT_membermima @ X3) @ Xa2) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ X3) @ Xa2)) => ((! [Mi2:nat,Ma2:nat,Va3:(list @ vEBT_VEBT),Vb2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (zero_zero @ nat)) @ Va3) @ Vb2)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (zero_zero @ nat)) @ Va3) @ Vb2)) @ Xa2)) => (~ (((Xa2 = Mi2) | (Xa2 = Ma2)))))))) => ((! [Mi2:nat,Ma2:nat,V3:nat,TreeList3:(list @ vEBT_VEBT),Vc2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ V3)) @ TreeList3) @ Vc2)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ (suc @ V3)) @ TreeList3) @ Vc2)) @ Xa2)) => (~ (((Xa2 = Mi2) | ((Xa2 = Ma2) | (((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_VEBT_membermima @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3))))))))))) => (~ ((! [V3:nat,TreeList3:(list @ vEBT_VEBT),Vd:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ V3)) @ TreeList3) @ Vd)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_V4351362008482014158ma_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ V3)) @ TreeList3) @ Vd)) @ Xa2)) => (~ ((((((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)) => ((vEBT_VEBT_membermima @ (((nth @ vEBT_VEBT) @ TreeList3) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((vEBT_VEBT_low @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((ord_less @ nat) @ ((vEBT_VEBT_high @ Xa2) @ (((divide_divide @ nat) @ (suc @ V3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((size_size @ (list @ vEBT_VEBT)) @ TreeList3)))))))))))))))))). % 6.41/2.69 thf(fact_2431_arcosh__1, axiom, (! [A:$tType]: (((ln @ A) => (((arcosh @ A) @ (one_one @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_2432_arsinh__0, axiom, (! [A:$tType]: (((ln @ A) => (((arsinh @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_2433_artanh__0, axiom, (! [A:$tType]: ((((real_V3459762299906320749_field @ A) & (ln @ A)) => (((artanh @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_2434_vebt__buildup_Opelims, axiom, (! [X3:nat,Y:vEBT_VEBT]: ((((vEBT_vebt_buildup @ X3) = Y) => ((((accp @ nat) @ vEBT_v4011308405150292612up_rel) @ X3) => (((X3 = (zero_zero @ nat)) => ((Y = ((vEBT_Leaf @ $false) @ $false)) => (~ ((((accp @ nat) @ vEBT_v4011308405150292612up_rel) @ (zero_zero @ nat)))))) => (((X3 = (suc @ (zero_zero @ nat))) => ((Y = ((vEBT_Leaf @ $false) @ $false)) => (~ ((((accp @ nat) @ vEBT_v4011308405150292612up_rel) @ (suc @ (zero_zero @ nat))))))) => (~ ((! [Va2:nat]: (((X3 = (suc @ (suc @ Va2))) => ((((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ (suc @ Va2))) => (Y = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ (suc @ Va2))) @ (((replicate @ vEBT_VEBT) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (vEBT_vebt_buildup @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ (vEBT_vebt_buildup @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) & ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ (suc @ Va2))))) => (Y = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ (suc @ (suc @ Va2))) @ (((replicate @ vEBT_VEBT) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (suc @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ (vEBT_vebt_buildup @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ (vEBT_vebt_buildup @ (suc @ (((divide_divide @ nat) @ (suc @ (suc @ Va2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))) => (~ ((((accp @ nat) @ vEBT_v4011308405150292612up_rel) @ (suc @ (suc @ Va2)))))))))))))))))). % 6.41/2.69 thf(fact_2435_divmod__algorithm__code_I6_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M:num,N2:num]: (((((unique8689654367752047608divmod @ A) @ (bit1 @ M)) @ (bit0 @ N2)) = (((((product_case_prod @ A) @ A) @ ((product_prod @ A) @ A)) @ (^ [Q5:A,R5:A]: (((((product_Pair @ A) @ A) @ Q5) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ R5)) @ (one_one @ A)))))) @ (((unique8689654367752047608divmod @ A) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_2436_artanh__def, axiom, (! [A:$tType]: ((((real_V3459762299906320749_field @ A) & (ln @ A)) => ((artanh @ A) = (^ [X:A]: ((((divide_divide @ A) @ ((ln_ln @ A) @ (((divide_divide @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ X)) @ (((minus_minus @ A) @ (one_one @ A)) @ X)))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_2437_ln__inj__iff, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (zero_zero @ real)) @ Y) => ((((ln_ln @ real) @ X3) = ((ln_ln @ real) @ Y)) = (X3 = Y))))))). % 6.41/2.69 thf(fact_2438_ln__less__cancel__iff, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (zero_zero @ real)) @ Y) => ((((ord_less @ real) @ ((ln_ln @ real) @ X3)) @ ((ln_ln @ real) @ Y)) = (((ord_less @ real) @ X3) @ Y))))))). % 6.41/2.69 thf(fact_2439_case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,F2:(B > (C > A)),A2:B,B2:C]: (((((((product_case_prod @ B) @ C) @ A) @ F2) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((F2 @ A2) @ B2))))). % 6.41/2.69 thf(fact_2440_ln__le__cancel__iff, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (zero_zero @ real)) @ Y) => ((((ord_less_eq @ real) @ ((ln_ln @ real) @ X3)) @ ((ln_ln @ real) @ Y)) = (((ord_less_eq @ real) @ X3) @ Y))))))). % 6.41/2.69 thf(fact_2441_ln__less__zero__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ ((ln_ln @ real) @ X3)) @ (zero_zero @ real)) = (((ord_less @ real) @ X3) @ (one_one @ real))))))). % 6.41/2.69 thf(fact_2442_ln__gt__zero__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (zero_zero @ real)) @ ((ln_ln @ real) @ X3)) = (((ord_less @ real) @ (one_one @ real)) @ X3)))))). % 6.41/2.69 thf(fact_2443_ln__eq__zero__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ln_ln @ real) @ X3) = (zero_zero @ real)) = (X3 = (one_one @ real))))))). % 6.41/2.69 thf(fact_2444_ln__one, axiom, (! [A:$tType]: (((ln @ A) => (((ln_ln @ A) @ (one_one @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_2445_ln__le__zero__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ ((ln_ln @ real) @ X3)) @ (zero_zero @ real)) = (((ord_less_eq @ real) @ X3) @ (one_one @ real))))))). % 6.41/2.69 thf(fact_2446_ln__ge__zero__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ ((ln_ln @ real) @ X3)) = (((ord_less_eq @ real) @ (one_one @ real)) @ X3)))))). % 6.41/2.69 thf(fact_2447_divmod__algorithm__code_I5_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M:num,N2:num]: (((((unique8689654367752047608divmod @ A) @ (bit0 @ M)) @ (bit0 @ N2)) = (((((product_case_prod @ A) @ A) @ ((product_prod @ A) @ A)) @ (^ [Q5:A,R5:A]: (((((product_Pair @ A) @ A) @ Q5) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ R5))))) @ (((unique8689654367752047608divmod @ A) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_2448_prod_Ocase__distrib, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,H2:(C > D),F2:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((H2 @ (((((product_case_prod @ A) @ B) @ C) @ F2) @ Prod)) = (((((product_case_prod @ A) @ B) @ D) @ (^ [X15:A,X24:B]: ((H2 @ ((F2 @ X15) @ X24))))) @ Prod))))). % 6.41/2.69 thf(fact_2449_old_Oprod_Ocase, axiom, (! [A:$tType,C:$tType,B:$tType,F2:(A > (B > C)),X1:A,X22:B]: (((((((product_case_prod @ A) @ B) @ C) @ F2) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = ((F2 @ X1) @ X22))))). % 6.41/2.69 thf(fact_2450_split__cong, axiom, (! [C:$tType,B:$tType,A:$tType,Q4:((product_prod @ A) @ B),F2:(A > (B > C)),G:(A > (B > C)),P4:((product_prod @ A) @ B)]: (((! [X5:A,Y4:B]: (((((((product_Pair @ A) @ B) @ X5) @ Y4) = Q4) => (((F2 @ X5) @ Y4) = ((G @ X5) @ Y4))))) => ((P4 = Q4) => ((((((product_case_prod @ A) @ B) @ C) @ F2) @ P4) = (((((product_case_prod @ A) @ B) @ C) @ G) @ Q4))))))). % 6.41/2.69 thf(fact_2451_cond__case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F2:(A > (B > C)),G:(((product_prod @ A) @ B) > C)]: (((! [X5:A,Y4:B]: ((((F2 @ X5) @ Y4) = (G @ ((((product_Pair @ A) @ B) @ X5) @ Y4))))) => (((((product_case_prod @ A) @ B) @ C) @ F2) = G))))). % 6.41/2.69 thf(fact_2452_case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F2:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ (^ [X:A,Y2:B]: ((F2 @ ((((product_Pair @ A) @ B) @ X) @ Y2))))) = F2)))). % 6.41/2.69 thf(fact_2453_case__prodE2, axiom, (! [B:$tType,A:$tType,C:$tType,Q:(A > $o),P:(B > (C > A)),Z2:((product_prod @ B) @ C)]: (((Q @ (((((product_case_prod @ B) @ C) @ A) @ P) @ Z2)) => (~ ((! [X5:B,Y4:C]: (((Z2 = ((((product_Pair @ B) @ C) @ X5) @ Y4)) => (~ ((Q @ ((P @ X5) @ Y4))))))))))))). % 6.41/2.69 thf(fact_2454_ln__less__self, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((ord_less @ real) @ ((ln_ln @ real) @ X3)) @ X3))))). % 6.41/2.69 thf(fact_2455_ln__bound, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((ord_less_eq @ real) @ ((ln_ln @ real) @ X3)) @ X3))))). % 6.41/2.69 thf(fact_2456_ln__gt__zero__imp__gt__one, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ ((ln_ln @ real) @ X3)) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((ord_less @ real) @ (one_one @ real)) @ X3)))))). % 6.41/2.69 thf(fact_2457_ln__less__zero, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ (one_one @ real)) => (((ord_less @ real) @ ((ln_ln @ real) @ X3)) @ (zero_zero @ real))))))). % 6.41/2.69 thf(fact_2458_ln__gt__zero, axiom, (! [X3:real]: (((((ord_less @ real) @ (one_one @ real)) @ X3) => (((ord_less @ real) @ (zero_zero @ real)) @ ((ln_ln @ real) @ X3)))))). % 6.41/2.69 thf(fact_2459_ln__div, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (zero_zero @ real)) @ Y) => (((ln_ln @ real) @ (((divide_divide @ real) @ X3) @ Y)) = (((minus_minus @ real) @ ((ln_ln @ real) @ X3)) @ ((ln_ln @ real) @ Y)))))))). % 6.41/2.69 thf(fact_2460_ln__2__less__1, axiom, (((ord_less @ real) @ ((ln_ln @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) @ (one_one @ real))). % 6.41/2.69 thf(fact_2461_ln__ge__zero__imp__ge__one, axiom, (! [X3:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ ((ln_ln @ real) @ X3)) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((ord_less_eq @ real) @ (one_one @ real)) @ X3)))))). % 6.41/2.69 thf(fact_2462_ln__diff__le, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (zero_zero @ real)) @ Y) => (((ord_less_eq @ real) @ (((minus_minus @ real) @ ((ln_ln @ real) @ X3)) @ ((ln_ln @ real) @ Y))) @ (((divide_divide @ real) @ (((minus_minus @ real) @ X3) @ Y)) @ Y))))))). % 6.41/2.69 thf(fact_2463_ln__mult, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (zero_zero @ real)) @ Y) => (((ln_ln @ real) @ (((times_times @ real) @ X3) @ Y)) = (((plus_plus @ real) @ ((ln_ln @ real) @ X3)) @ ((ln_ln @ real) @ Y)))))))). % 6.41/2.69 thf(fact_2464_ln__eq__minus__one, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ln_ln @ real) @ X3) = (((minus_minus @ real) @ X3) @ (one_one @ real))) => (X3 = (one_one @ real))))))). % 6.41/2.69 thf(fact_2465_ln__le__minus__one, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((ord_less_eq @ real) @ ((ln_ln @ real) @ X3)) @ (((minus_minus @ real) @ X3) @ (one_one @ real))))))). % 6.41/2.69 thf(fact_2466_ln__add__one__self__le__self2, axiom, (! [X3:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ X3) => (((ord_less_eq @ real) @ ((ln_ln @ real) @ (((plus_plus @ real) @ (one_one @ real)) @ X3))) @ X3))))). % 6.41/2.69 thf(fact_2467_ln__one__minus__pos__upper__bound, axiom, (! [X3:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ (one_one @ real)) => (((ord_less_eq @ real) @ ((ln_ln @ real) @ (((minus_minus @ real) @ (one_one @ real)) @ X3))) @ ((uminus_uminus @ real) @ X3))))))). % 6.41/2.69 thf(fact_2468_divmod__step__nat__def, axiom, ((unique1321980374590559556d_step @ nat) = (^ [L2:num]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [Q5:nat,R5:nat]: (((((if @ ((product_prod @ nat) @ nat)) @ (((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ L2)) @ R5)) @ ((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Q5)) @ (one_one @ nat))) @ (((minus_minus @ nat) @ R5) @ ((numeral_numeral @ nat) @ L2)))) @ ((((product_Pair @ nat) @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Q5)) @ R5))))))))). % 6.41/2.69 thf(fact_2469_divmod__step__int__def, axiom, ((unique1321980374590559556d_step @ int) = (^ [L2:num]: (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [Q5:int,R5:int]: (((((if @ ((product_prod @ int) @ int)) @ (((ord_less_eq @ int) @ ((numeral_numeral @ int) @ L2)) @ R5)) @ ((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Q5)) @ (one_one @ int))) @ (((minus_minus @ int) @ R5) @ ((numeral_numeral @ int) @ L2)))) @ ((((product_Pair @ int) @ int) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Q5)) @ R5))))))))). % 6.41/2.69 thf(fact_2470_divmod__step__def, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => ((unique1321980374590559556d_step @ A) = (^ [L2:num]: (((((product_case_prod @ A) @ A) @ ((product_prod @ A) @ A)) @ (^ [Q5:A,R5:A]: (((((if @ ((product_prod @ A) @ A)) @ (((ord_less_eq @ A) @ ((numeral_numeral @ A) @ L2)) @ R5)) @ ((((product_Pair @ A) @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Q5)) @ (one_one @ A))) @ (((minus_minus @ A) @ R5) @ ((numeral_numeral @ A) @ L2)))) @ ((((product_Pair @ A) @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Q5)) @ R5)))))))))))). % 6.41/2.69 thf(fact_2471_divmod__nat__if, axiom, (divmod_nat = (^ [M2:nat,N:nat]: (((((if @ ((product_prod @ nat) @ nat)) @ ((N = (zero_zero @ nat)) | (((ord_less @ nat) @ M2) @ N))) @ ((((product_Pair @ nat) @ nat) @ (zero_zero @ nat)) @ M2)) @ (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [Q5:nat]: ((((product_Pair @ nat) @ nat) @ (suc @ Q5))))) @ ((divmod_nat @ (((minus_minus @ nat) @ M2) @ N)) @ N))))))). % 6.41/2.69 thf(fact_2472_tanh__ln__real, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((tanh @ real) @ ((ln_ln @ real) @ X3)) = (((divide_divide @ real) @ (((minus_minus @ real) @ (((power_power @ real) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ real))) @ (((plus_plus @ real) @ (((power_power @ real) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ real)))))))). % 6.41/2.69 thf(fact_2473_Sum__Icc__nat, axiom, (! [M:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X:nat]: (X))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (((divide_divide @ nat) @ (((minus_minus @ nat) @ (((times_times @ nat) @ N2) @ (((plus_plus @ nat) @ N2) @ (one_one @ nat)))) @ (((times_times @ nat) @ M) @ (((minus_minus @ nat) @ M) @ (one_one @ nat))))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))). % 6.41/2.69 thf(fact_2474_arith__series__nat, axiom, (! [A2:nat,D2:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [I3:nat]: ((((plus_plus @ nat) @ A2) @ (((times_times @ nat) @ I3) @ D2))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2)) = (((divide_divide @ nat) @ (((times_times @ nat) @ (suc @ N2)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ A2)) @ (((times_times @ nat) @ N2) @ D2)))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))). % 6.41/2.69 thf(fact_2475_case__prod__Pair__iden, axiom, (! [B:$tType,A:$tType,P4:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ ((product_prod @ A) @ B)) @ ((product_Pair @ A) @ B)) @ P4) = P4)))). % 6.41/2.69 thf(fact_2476_round__unique_H, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,N2:int]: (((((ord_less @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ X3) @ ((ring_1_of_int @ A) @ N2)))) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) => (((archimedean_round @ A) @ X3) = N2)))))))). % 6.41/2.69 thf(fact_2477_abs__abs, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A]: ((((abs_abs @ A) @ ((abs_abs @ A) @ A2)) = ((abs_abs @ A) @ A2)))))))). % 6.41/2.69 thf(fact_2478_abs__idempotent, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: ((((abs_abs @ A) @ ((abs_abs @ A) @ A2)) = ((abs_abs @ A) @ A2)))))))). % 6.41/2.69 thf(fact_2479_case__prodI, axiom, (! [A:$tType,B:$tType,F2:(A > (B > $o)),A2:A,B2:B]: ((((F2 @ A2) @ B2) => (((((product_case_prod @ A) @ B) @ $o) @ F2) @ ((((product_Pair @ A) @ B) @ A2) @ B2)))))). % 6.41/2.69 thf(fact_2480_case__prodI2, axiom, (! [B:$tType,A:$tType,P4:((product_prod @ A) @ B),C2:(A > (B > $o))]: (((! [A5:A,B5:B]: (((P4 = ((((product_Pair @ A) @ B) @ A5) @ B5)) => ((C2 @ A5) @ B5)))) => (((((product_case_prod @ A) @ B) @ $o) @ C2) @ P4))))). % 6.41/2.69 thf(fact_2481_mem__case__prodI, axiom, (! [A:$tType,B:$tType,C:$tType,Z2:A,C2:(B > (C > (set @ A))),A2:B,B2:C]: (((((member @ A) @ Z2) @ ((C2 @ A2) @ B2)) => (((member @ A) @ Z2) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C2) @ ((((product_Pair @ B) @ C) @ A2) @ B2))))))). % 6.41/2.69 thf(fact_2482_mem__case__prodI2, axiom, (! [C:$tType,B:$tType,A:$tType,P4:((product_prod @ A) @ B),Z2:C,C2:(A > (B > (set @ C)))]: (((! [A5:A,B5:B]: (((P4 = ((((product_Pair @ A) @ B) @ A5) @ B5)) => (((member @ C) @ Z2) @ ((C2 @ A5) @ B5))))) => (((member @ C) @ Z2) @ (((((product_case_prod @ A) @ B) @ (set @ C)) @ C2) @ P4)))))). % 6.41/2.69 thf(fact_2483_case__prodI2_H, axiom, (! [A:$tType,B:$tType,C:$tType,P4:((product_prod @ A) @ B),C2:(A > (B > (C > $o))),X3:C]: (((! [A5:A,B5:B]: (((((((product_Pair @ A) @ B) @ A5) @ B5) = P4) => (((C2 @ A5) @ B5) @ X3)))) => ((((((product_case_prod @ A) @ B) @ (C > $o)) @ C2) @ P4) @ X3))))). % 6.41/2.69 thf(fact_2484_abs__0__eq, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: ((((zero_zero @ A) = ((abs_abs @ A) @ A2)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2485_abs__eq__0, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: (((((abs_abs @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2486_abs__zero, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (((abs_abs @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_2487_abs__0, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((abs_abs @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_2488_abs__mult__self__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((times_times @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ A2)) = (((times_times @ A) @ A2) @ A2)))))))). % 6.41/2.69 thf(fact_2489_abs__1, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((abs_abs @ A) @ (one_one @ A)) = (one_one @ A)))))). % 6.41/2.69 thf(fact_2490_abs__add__abs, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A,B2:A]: ((((abs_abs @ A) @ (((plus_plus @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2))) = (((plus_plus @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2))))))))). % 6.41/2.69 thf(fact_2491_abs__divide, axiom, (! [A:$tType]: (((field_abs_sgn @ A) => (! [A2:A,B2:A]: ((((abs_abs @ A) @ (((divide_divide @ A) @ A2) @ B2)) = (((divide_divide @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2))))))))). % 6.41/2.69 thf(fact_2492_abs__minus, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A]: ((((abs_abs @ A) @ ((uminus_uminus @ A) @ A2)) = ((abs_abs @ A) @ A2)))))))). % 6.41/2.69 thf(fact_2493_abs__minus__cancel, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: ((((abs_abs @ A) @ ((uminus_uminus @ A) @ A2)) = ((abs_abs @ A) @ A2)))))))). % 6.41/2.69 thf(fact_2494_dvd__abs__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:A,K:A]: (((((dvd_dvd @ A) @ M) @ ((abs_abs @ A) @ K)) = (((dvd_dvd @ A) @ M) @ K)))))))). % 6.41/2.69 thf(fact_2495_abs__dvd__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M:A,K:A]: (((((dvd_dvd @ A) @ ((abs_abs @ A) @ M)) @ K) = (((dvd_dvd @ A) @ M) @ K)))))))). % 6.41/2.69 thf(fact_2496_of__int__abs, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:int]: ((((ring_1_of_int @ A) @ ((abs_abs @ int) @ X3)) = ((abs_abs @ A) @ ((ring_1_of_int @ A) @ X3))))))))). % 6.41/2.69 thf(fact_2497_abs__bool__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [P:$o]: ((((abs_abs @ A) @ ((zero_neq_one_of_bool @ A) @ P)) = ((zero_neq_one_of_bool @ A) @ P)))))))). % 6.41/2.69 thf(fact_2498_zdvd1__eq, axiom, (! [X3:int]: (((((dvd_dvd @ int) @ X3) @ (one_one @ int)) = (((abs_abs @ int) @ X3) = (one_one @ int)))))). % 6.41/2.69 thf(fact_2499_tanh__0, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (((tanh @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_2500_tanh__real__less__iff, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ ((tanh @ real) @ X3)) @ ((tanh @ real) @ Y)) = (((ord_less @ real) @ X3) @ Y))))). % 6.41/2.69 thf(fact_2501_sum_Oneutral__const, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B)]: ((((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [Uu3:B]: ((zero_zero @ A)))) @ A3) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_2502_of__int__sum, axiom, (! [A:$tType,B:$tType]: (((ring_1 @ A) => (! [F2:(B > int),A3:(set @ B)]: ((((ring_1_of_int @ A) @ ((((groups7311177749621191930dd_sum @ B) @ int) @ F2) @ A3)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X:B]: (((ring_1_of_int @ A) @ (F2 @ X))))) @ A3)))))))). % 6.41/2.69 thf(fact_2503_abs__of__nonneg, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((abs_abs @ A) @ A2) = A2)))))))). % 6.41/2.69 thf(fact_2504_abs__le__self__iff, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ A2) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))). % 6.41/2.69 thf(fact_2505_abs__le__zero__iff, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2506_zero__less__abs__iff, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((abs_abs @ A) @ A2)) = (A2 != (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2507_sum_Oempty, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [G:(B > A)]: ((((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (bot_bot @ (set @ B))) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_2508_sum_Oinfinite, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),G:(B > A)]: (((~ (((finite_finite2 @ B) @ A3))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2509_sum__eq__0__iff, axiom, (! [A:$tType,B:$tType]: (((canoni5634975068530333245id_add @ A) => (! [F5:(set @ B),F2:(B > A)]: ((((finite_finite2 @ B) @ F5) => ((((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ F5) = (zero_zero @ A)) = (! [X:B]: (((((member @ B) @ X) @ F5) => ((F2 @ X) = (zero_zero @ A))))))))))))). % 6.41/2.69 thf(fact_2510_artanh__minus__real, axiom, (! [X3:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ X3)) @ (one_one @ real)) => (((artanh @ real) @ ((uminus_uminus @ real) @ X3)) = ((uminus_uminus @ real) @ ((artanh @ real) @ X3))))))). % 6.41/2.69 thf(fact_2511_zabs__less__one__iff, axiom, (! [Z2:int]: (((((ord_less @ int) @ ((abs_abs @ int) @ Z2)) @ (one_one @ int)) = (Z2 = (zero_zero @ int)))))). % 6.41/2.69 thf(fact_2512_tanh__real__pos__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ ((tanh @ real) @ X3)) = (((ord_less @ real) @ (zero_zero @ real)) @ X3))))). % 6.41/2.69 thf(fact_2513_tanh__real__neg__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ ((tanh @ real) @ X3)) @ (zero_zero @ real)) = (((ord_less @ real) @ X3) @ (zero_zero @ real)))))). % 6.41/2.69 thf(fact_2514_sum_Odelta, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ B),A2:B,B2:(B > A)]: ((((finite_finite2 @ B) @ S3) => (((((member @ B) @ A2) @ S3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A2)) @ (B2 @ K3)) @ (zero_zero @ A))))) @ S3) = (B2 @ A2))) & ((~ ((((member @ B) @ A2) @ S3))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A2)) @ (B2 @ K3)) @ (zero_zero @ A))))) @ S3) = (zero_zero @ A))))))))))). % 6.41/2.69 thf(fact_2515_sum_Odelta_H, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ B),A2:B,B2:(B > A)]: ((((finite_finite2 @ B) @ S3) => (((((member @ B) @ A2) @ S3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (A2 = K3)) @ (B2 @ K3)) @ (zero_zero @ A))))) @ S3) = (B2 @ A2))) & ((~ ((((member @ B) @ A2) @ S3))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (A2 = K3)) @ (B2 @ K3)) @ (zero_zero @ A))))) @ S3) = (zero_zero @ A))))))))))). % 6.41/2.69 thf(fact_2516_sum__abs, axiom, (! [B:$tType,A:$tType]: (((ordere166539214618696060dd_abs @ B) => (! [F2:(A > B),A3:(set @ A)]: ((((ord_less_eq @ B) @ ((abs_abs @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F2) @ A3))) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ (^ [I3:A]: (((abs_abs @ B) @ (F2 @ I3))))) @ A3)))))))). % 6.41/2.69 thf(fact_2517_divide__le__0__abs__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ A2) @ ((abs_abs @ A) @ B2))) @ (zero_zero @ A)) = ((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_2518_zero__le__divide__abs__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ A2) @ ((abs_abs @ A) @ B2))) = ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) | (B2 = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_2519_abs__of__nonpos, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (zero_zero @ A)) => (((abs_abs @ A) @ A2) = ((uminus_uminus @ A) @ A2))))))))). % 6.41/2.69 thf(fact_2520_sum_Oinsert, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),X3:B,G:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((~ ((((member @ B) @ X3) @ A3))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((insert @ B) @ X3) @ A3)) = (((plus_plus @ A) @ (G @ X3)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3))))))))))). % 6.41/2.69 thf(fact_2521_sum__abs__ge__zero, axiom, (! [B:$tType,A:$tType]: (((ordere166539214618696060dd_abs @ B) => (! [F2:(A > B),A3:(set @ A)]: ((((ord_less_eq @ B) @ (zero_zero @ B)) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ (^ [I3:A]: (((abs_abs @ B) @ (F2 @ I3))))) @ A3)))))))). % 6.41/2.69 thf(fact_2522_zero__less__power__abs__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,N2:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ ((abs_abs @ A) @ A2)) @ N2)) = ((A2 != (zero_zero @ A)) | (N2 = (zero_zero @ nat)))))))))). % 6.41/2.69 thf(fact_2523_sum_Ocl__ivl__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [N2:nat,M:nat,G:(nat > A)]: ((((((ord_less @ nat) @ (suc @ N2)) @ M) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N2))) = (zero_zero @ A))) & ((~ ((((ord_less @ nat) @ (suc @ N2)) @ M))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N2))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))) @ (G @ (suc @ N2)))))))))))). % 6.41/2.69 thf(fact_2524_sum__zero__power, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:(set @ nat),C2:(nat > A)]: ((((((finite_finite2 @ nat) @ A3) & (((member @ nat) @ (zero_zero @ nat)) @ A3)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ (zero_zero @ A)) @ I3))))) @ A3) = (C2 @ (zero_zero @ nat)))) & ((~ ((((finite_finite2 @ nat) @ A3) & (((member @ nat) @ (zero_zero @ nat)) @ A3)))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ (zero_zero @ A)) @ I3))))) @ A3) = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_2525_sum__zero__power_H, axiom, (! [A:$tType]: (((field @ A) => (! [A3:(set @ nat),C2:(nat > A),D2:(nat > A)]: ((((((finite_finite2 @ nat) @ A3) & (((member @ nat) @ (zero_zero @ nat)) @ A3)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((divide_divide @ A) @ (((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ (zero_zero @ A)) @ I3))) @ (D2 @ I3))))) @ A3) = (((divide_divide @ A) @ (C2 @ (zero_zero @ nat))) @ (D2 @ (zero_zero @ nat))))) & ((~ ((((finite_finite2 @ nat) @ A3) & (((member @ nat) @ (zero_zero @ nat)) @ A3)))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((divide_divide @ A) @ (((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ (zero_zero @ A)) @ I3))) @ (D2 @ I3))))) @ A3) = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_2526_sum_Onot__neutral__contains__not__neutral, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [G:(B > A),A3:(set @ B)]: (((((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3) != (zero_zero @ A)) => (~ ((! [A5:B]: (((((member @ B) @ A5) @ A3) => ((G @ A5) = (zero_zero @ A)))))))))))))). % 6.41/2.69 thf(fact_2527_sum_Oneutral, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),G:(B > A)]: (((! [X5:B]: (((((member @ B) @ X5) @ A3) => ((G @ X5) = (zero_zero @ A))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2528_abs__le__D1, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ B2) => (((ord_less_eq @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_2529_abs__ge__self, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ ((abs_abs @ A) @ A2)))))))). % 6.41/2.69 thf(fact_2530_abs__eq__0__iff, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A]: (((((abs_abs @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2531_abs__mult, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A,B2:A]: ((((abs_abs @ A) @ (((times_times @ A) @ A2) @ B2)) = (((times_times @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2))))))))). % 6.41/2.69 thf(fact_2532_abs__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((abs_abs @ A) @ (one_one @ A)) = (one_one @ A)))))). % 6.41/2.69 thf(fact_2533_abs__minus__commute, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A,B2:A]: ((((abs_abs @ A) @ (((minus_minus @ A) @ A2) @ B2)) = ((abs_abs @ A) @ (((minus_minus @ A) @ B2) @ A2))))))))). % 6.41/2.69 thf(fact_2534_abs__eq__iff, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [X3:A,Y:A]: (((((abs_abs @ A) @ X3) = ((abs_abs @ A) @ Y)) = ((X3 = Y) | (X3 = ((uminus_uminus @ A) @ Y)))))))))). % 6.41/2.69 thf(fact_2535_dvd__if__abs__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [L:A,K:A]: (((((abs_abs @ A) @ L) = ((abs_abs @ A) @ K)) => (((dvd_dvd @ A) @ L) @ K)))))))). % 6.41/2.69 thf(fact_2536_mem__case__prodE, axiom, (! [B:$tType,A:$tType,C:$tType,Z2:A,C2:(B > (C > (set @ A))),P4:((product_prod @ B) @ C)]: (((((member @ A) @ Z2) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C2) @ P4)) => (~ ((! [X5:B,Y4:C]: (((P4 = ((((product_Pair @ B) @ C) @ X5) @ Y4)) => (~ ((((member @ A) @ Z2) @ ((C2 @ X5) @ Y4))))))))))))). % 6.41/2.69 thf(fact_2537_zdvd__antisym__abs, axiom, (! [A2:int,B2:int]: (((((dvd_dvd @ int) @ A2) @ B2) => ((((dvd_dvd @ int) @ B2) @ A2) => (((abs_abs @ int) @ A2) = ((abs_abs @ int) @ B2))))))). % 6.41/2.69 thf(fact_2538_case__prodD, axiom, (! [A:$tType,B:$tType,F2:(A > (B > $o)),A2:A,B2:B]: (((((((product_case_prod @ A) @ B) @ $o) @ F2) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) => ((F2 @ A2) @ B2))))). % 6.41/2.69 thf(fact_2539_case__prodE, axiom, (! [A:$tType,B:$tType,C2:(A > (B > $o)),P4:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ $o) @ C2) @ P4) => (~ ((! [X5:A,Y4:B]: (((P4 = ((((product_Pair @ A) @ B) @ X5) @ Y4)) => (~ (((C2 @ X5) @ Y4)))))))))))). % 6.41/2.69 thf(fact_2540_sum__mono, axiom, (! [A:$tType,B:$tType]: (((ordere6911136660526730532id_add @ A) => (! [K5:(set @ B),F2:(B > A),G:(B > A)]: (((! [I4:B]: (((((member @ B) @ I4) @ K5) => (((ord_less_eq @ A) @ (F2 @ I4)) @ (G @ I4))))) => (((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ K5)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ K5))))))))). % 6.41/2.69 thf(fact_2541_sum_Odistrib, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [G:(B > A),H2:(B > A),A3:(set @ B)]: ((((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X:B]: ((((plus_plus @ A) @ (G @ X)) @ (H2 @ X))))) @ A3) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ A3))))))))). % 6.41/2.69 thf(fact_2542_sum_Oswap__restrict, axiom, (! [A:$tType,B:$tType,C:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),B3:(set @ C),G:(B > (C > A)),R:(B > (C > $o))]: ((((finite_finite2 @ B) @ A3) => (((finite_finite2 @ C) @ B3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X:B]: (((((groups7311177749621191930dd_sum @ C) @ A) @ (G @ X)) @ ((collect @ C) @ (^ [Y2:C]: (((((member @ C) @ Y2) @ B3) & ((R @ X) @ Y2))))))))) @ A3) = ((((groups7311177749621191930dd_sum @ C) @ A) @ (^ [Y2:C]: (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X:B]: (((G @ X) @ Y2)))) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ A3) & ((R @ X) @ Y2))))))))) @ B3)))))))))). % 6.41/2.69 thf(fact_2543_case__prodD_H, axiom, (! [B:$tType,A:$tType,C:$tType,R:(A > (B > (C > $o))),A2:A,B2:B,C2:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ R) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ C2) => (((R @ A2) @ B2) @ C2))))). % 6.41/2.69 thf(fact_2544_case__prodE_H, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(A > (B > (C > $o))),P4:((product_prod @ A) @ B),Z2:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ C2) @ P4) @ Z2) => (~ ((! [X5:A,Y4:B]: (((P4 = ((((product_Pair @ A) @ B) @ X5) @ Y4)) => (~ ((((C2 @ X5) @ Y4) @ Z2)))))))))))). % 6.41/2.69 thf(fact_2545_sum__nonneg, axiom, (! [A:$tType,B:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:(set @ B),F2:(B > A)]: (((! [X5:B]: (((((member @ B) @ X5) @ A3) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ X5))))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ A3))))))))). % 6.41/2.69 thf(fact_2546_sum__nonpos, axiom, (! [B:$tType,A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:(set @ B),F2:(B > A)]: (((! [X5:B]: (((((member @ B) @ X5) @ A3) => (((ord_less_eq @ A) @ (F2 @ X5)) @ (zero_zero @ A))))) => (((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ A3)) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2547_sum__mono__inv, axiom, (! [A:$tType,I7:$tType]: (((ordere8940638589300402666id_add @ A) => (! [F2:(I7 > A),I6:(set @ I7),G:(I7 > A),I:I7]: (((((((groups7311177749621191930dd_sum @ I7) @ A) @ F2) @ I6) = ((((groups7311177749621191930dd_sum @ I7) @ A) @ G) @ I6)) => ((! [I4:I7]: (((((member @ I7) @ I4) @ I6) => (((ord_less_eq @ A) @ (F2 @ I4)) @ (G @ I4))))) => ((((member @ I7) @ I) @ I6) => (((finite_finite2 @ I7) @ I6) => ((F2 @ I) = (G @ I)))))))))))). % 6.41/2.69 thf(fact_2548_abs__ge__zero, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((abs_abs @ A) @ A2)))))))). % 6.41/2.69 thf(fact_2549_abs__not__less__zero, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ ((abs_abs @ A) @ A2)) @ (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_2550_abs__of__pos, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((abs_abs @ A) @ A2) = A2)))))))). % 6.41/2.69 thf(fact_2551_abs__triangle__ineq, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((plus_plus @ A) @ A2) @ B2))) @ (((plus_plus @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2))))))))). % 6.41/2.69 thf(fact_2552_abs__mult__less, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,C2:A,B2:A,D2:A]: (((((ord_less @ A) @ ((abs_abs @ A) @ A2)) @ C2) => ((((ord_less @ A) @ ((abs_abs @ A) @ B2)) @ D2) => (((ord_less @ A) @ (((times_times @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2))) @ (((times_times @ A) @ C2) @ D2)))))))))). % 6.41/2.69 thf(fact_2553_abs__triangle__ineq2, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2))) @ ((abs_abs @ A) @ (((minus_minus @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_2554_abs__triangle__ineq3, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2)))) @ ((abs_abs @ A) @ (((minus_minus @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_2555_abs__triangle__ineq2__sym, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2))) @ ((abs_abs @ A) @ (((minus_minus @ A) @ B2) @ A2))))))))). % 6.41/2.69 thf(fact_2556_nonzero__abs__divide, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A2:A]: (((B2 != (zero_zero @ A)) => (((abs_abs @ A) @ (((divide_divide @ A) @ A2) @ B2)) = (((divide_divide @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_2557_abs__ge__minus__self, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ ((abs_abs @ A) @ A2)))))))). % 6.41/2.69 thf(fact_2558_abs__le__iff, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ B2) = ((((ord_less_eq @ A) @ A2) @ B2) & (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ B2))))))))). % 6.41/2.69 thf(fact_2559_abs__le__D2, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ B2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ B2)))))))). % 6.41/2.69 thf(fact_2560_abs__leI, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ B2))))))))). % 6.41/2.69 thf(fact_2561_abs__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ ((abs_abs @ A) @ A2)) @ B2) = ((((ord_less @ A) @ A2) @ B2) & (((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ B2))))))))). % 6.41/2.69 thf(fact_2562_sum__cong__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ nat),F2:(nat > A),G:(nat > A)]: (((~ ((((member @ nat) @ (zero_zero @ nat)) @ A3))) => ((! [X5:nat]: (((((member @ nat) @ (suc @ X5)) @ A3) => ((F2 @ (suc @ X5)) = (G @ (suc @ X5)))))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ A3) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ A3)))))))))). % 6.41/2.69 thf(fact_2563_Collect__case__prod__mono, axiom, (! [B:$tType,A:$tType,A3:(A > (B > $o)),B3:(A > (B > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ A3) @ B3) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A3))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ B3))))))). % 6.41/2.69 thf(fact_2564_abs__zmult__eq__1, axiom, (! [M:int,N2:int]: (((((abs_abs @ int) @ (((times_times @ int) @ M) @ N2)) = (one_one @ int)) => (((abs_abs @ int) @ M) = (one_one @ int)))))). % 6.41/2.69 thf(fact_2565_sum_Ointer__filter, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),G:(B > A),P:(B > $o)]: ((((finite_finite2 @ B) @ A3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ A3) & (P @ X)))))) = ((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X:B]: (((((if @ A) @ (P @ X)) @ (G @ X)) @ (zero_zero @ A))))) @ A3))))))))). % 6.41/2.69 thf(fact_2566_sum__subtractf__nat, axiom, (! [A:$tType,A3:(set @ A),G:(A > nat),F2:(A > nat)]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((ord_less_eq @ nat) @ (G @ X5)) @ (F2 @ X5))))) => (((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [X:A]: ((((minus_minus @ nat) @ (F2 @ X)) @ (G @ X))))) @ A3) = (((minus_minus @ nat) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ A3)) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ G) @ A3))))))). % 6.41/2.69 thf(fact_2567_convex__sum__bound__le, axiom, (! [A:$tType,B:$tType]: (((linordered_idom @ B) => (! [I6:(set @ A),X3:(A > B),A2:(A > B),B2:B,Delta:B]: (((! [I4:A]: (((((member @ A) @ I4) @ I6) => (((ord_less_eq @ B) @ (zero_zero @ B)) @ (X3 @ I4))))) => ((((((groups7311177749621191930dd_sum @ A) @ B) @ X3) @ I6) = (one_one @ B)) => ((! [I4:A]: (((((member @ A) @ I4) @ I6) => (((ord_less_eq @ B) @ ((abs_abs @ B) @ (((minus_minus @ B) @ (A2 @ I4)) @ B2))) @ Delta)))) => (((ord_less_eq @ B) @ ((abs_abs @ B) @ (((minus_minus @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ (^ [I3:A]: ((((times_times @ B) @ (A2 @ I3)) @ (X3 @ I3))))) @ I6)) @ B2))) @ Delta)))))))))). % 6.41/2.69 thf(fact_2568_infinite__int__iff__unbounded__le, axiom, (! [S3:(set @ int)]: (((~ (((finite_finite2 @ int) @ S3))) = (! [M2:int]: ((? [N:int]: (((((ord_less_eq @ int) @ M2) @ ((abs_abs @ int) @ N)) & (((member @ int) @ N) @ S3)))))))))). % 6.41/2.69 thf(fact_2569_infinite__int__iff__unbounded, axiom, (! [S3:(set @ int)]: (((~ (((finite_finite2 @ int) @ S3))) = (! [M2:int]: ((? [N:int]: (((((ord_less @ int) @ M2) @ ((abs_abs @ int) @ N)) & (((member @ int) @ N) @ S3)))))))))). % 6.41/2.69 thf(fact_2570_sum_Oshift__bounds__cl__Suc__ivl, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (suc @ N2))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_2571_sum_Oshift__bounds__cl__nat__ivl, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,K:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ (((plus_plus @ nat) @ N2) @ K))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (((plus_plus @ nat) @ I3) @ K))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_2572_tanh__real__lt__1, axiom, (! [X3:real]: ((((ord_less @ real) @ ((tanh @ real) @ X3)) @ (one_one @ real))))). % 6.41/2.69 thf(fact_2573_sum__le__included, axiom, (! [B:$tType,A:$tType,C:$tType]: (((ordere6911136660526730532id_add @ A) => (! [S:(set @ B),T2:(set @ C),G:(C > A),I:(C > B),F2:(B > A)]: ((((finite_finite2 @ B) @ S) => (((finite_finite2 @ C) @ T2) => ((! [X5:C]: (((((member @ C) @ X5) @ T2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (G @ X5))))) => ((! [X5:B]: (((((member @ B) @ X5) @ S) => (? [Xa:C]: (((((member @ C) @ Xa) @ T2) & (((I @ Xa) = X5) & (((ord_less_eq @ A) @ (F2 @ X5)) @ (G @ Xa))))))))) => (((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ S)) @ ((((groups7311177749621191930dd_sum @ C) @ A) @ G) @ T2)))))))))))). % 6.41/2.69 thf(fact_2574_sum__nonneg__eq__0__iff, axiom, (! [A:$tType,B:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:(set @ B),F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((! [X5:B]: (((((member @ B) @ X5) @ A3) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ X5))))) => ((((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ A3) = (zero_zero @ A)) = (! [X:B]: (((((member @ B) @ X) @ A3) => ((F2 @ X) = (zero_zero @ A)))))))))))))). % 6.41/2.69 thf(fact_2575_sum__strict__mono__ex1, axiom, (! [A:$tType,I7:$tType]: (((ordere8940638589300402666id_add @ A) => (! [A3:(set @ I7),F2:(I7 > A),G:(I7 > A)]: ((((finite_finite2 @ I7) @ A3) => ((! [X5:I7]: (((((member @ I7) @ X5) @ A3) => (((ord_less_eq @ A) @ (F2 @ X5)) @ (G @ X5))))) => ((? [X2:I7]: (((((member @ I7) @ X2) @ A3) & (((ord_less @ A) @ (F2 @ X2)) @ (G @ X2))))) => (((ord_less @ A) @ ((((groups7311177749621191930dd_sum @ I7) @ A) @ F2) @ A3)) @ ((((groups7311177749621191930dd_sum @ I7) @ A) @ G) @ A3))))))))))). % 6.41/2.69 thf(fact_2576_sum_Orelated, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [R:(A > (A > $o)),S3:(set @ B),H2:(B > A),G:(B > A)]: ((((R @ (zero_zero @ A)) @ (zero_zero @ A)) => ((! [X16:A,Y15:A,X23:A,Y23:A]: (((((R @ X16) @ X23) & ((R @ Y15) @ Y23)) => ((R @ (((plus_plus @ A) @ X16) @ Y15)) @ (((plus_plus @ A) @ X23) @ Y23))))) => (((finite_finite2 @ B) @ S3) => ((! [X5:B]: (((((member @ B) @ X5) @ S3) => ((R @ (H2 @ X5)) @ (G @ X5))))) => ((R @ ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ S3)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ S3)))))))))))). % 6.41/2.69 thf(fact_2577_dense__eq0__I, axiom, (! [A:$tType]: ((((ordere166539214618696060dd_abs @ A) & (dense_linorder @ A)) => (! [X3:A]: (((! [E:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ X3)) @ E)))) => (X3 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2578_abs__mult__pos, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => ((((times_times @ A) @ ((abs_abs @ A) @ Y)) @ X3) = ((abs_abs @ A) @ (((times_times @ A) @ Y) @ X3)))))))))). % 6.41/2.69 thf(fact_2579_abs__eq__mult, axiom, (! [A:$tType]: (((ordered_ring_abs @ A) => (! [A2:A,B2:A]: (((((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) | (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))) & ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) | (((ord_less_eq @ A) @ B2) @ (zero_zero @ A)))) => (((abs_abs @ A) @ (((times_times @ A) @ A2) @ B2)) = (((times_times @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_2580_sum__strict__mono, axiom, (! [A:$tType,B:$tType]: (((strict7427464778891057005id_add @ A) => (! [A3:(set @ B),F2:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((A3 != (bot_bot @ (set @ B))) => ((! [X5:B]: (((((member @ B) @ X5) @ A3) => (((ord_less @ A) @ (F2 @ X5)) @ (G @ X5))))) => (((ord_less @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ A3)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3))))))))))). % 6.41/2.69 thf(fact_2581_abs__minus__le__zero, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((abs_abs @ A) @ A2))) @ (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_2582_abs__eq__iff_H, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [A2:A,B2:A]: (((((abs_abs @ A) @ A2) = B2) = ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) & ((A2 = B2) | (A2 = ((uminus_uminus @ A) @ B2))))))))))). % 6.41/2.69 thf(fact_2583_eq__abs__iff_H, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [A2:A,B2:A]: (((A2 = ((abs_abs @ A) @ B2)) = ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & ((B2 = A2) | (B2 = ((uminus_uminus @ A) @ A2))))))))))). % 6.41/2.69 thf(fact_2584_zero__le__power__abs, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,N2:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ ((abs_abs @ A) @ A2)) @ N2)))))))). % 6.41/2.69 thf(fact_2585_abs__div__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [Y:A,X3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Y) => ((((divide_divide @ A) @ ((abs_abs @ A) @ X3)) @ Y) = ((abs_abs @ A) @ (((divide_divide @ A) @ X3) @ Y)))))))))). % 6.41/2.69 thf(fact_2586_abs__if__raw, axiom, (! [A:$tType]: (((abs_if @ A) => ((abs_abs @ A) = (^ [A4:A]: (((((if @ A) @ (((ord_less @ A) @ A4) @ (zero_zero @ A))) @ ((uminus_uminus @ A) @ A4)) @ A4)))))))). % 6.41/2.69 thf(fact_2587_abs__of__neg, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => (((abs_abs @ A) @ A2) = ((uminus_uminus @ A) @ A2))))))))). % 6.41/2.69 thf(fact_2588_abs__if, axiom, (! [A:$tType]: (((abs_if @ A) => ((abs_abs @ A) = (^ [A4:A]: (((((if @ A) @ (((ord_less @ A) @ A4) @ (zero_zero @ A))) @ ((uminus_uminus @ A) @ A4)) @ A4)))))))). % 6.41/2.69 thf(fact_2589_abs__diff__triangle__ineq, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A,B2:A,C2:A,D2:A]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (((plus_plus @ A) @ C2) @ D2)))) @ (((plus_plus @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ A2) @ C2))) @ ((abs_abs @ A) @ (((minus_minus @ A) @ B2) @ D2)))))))))). % 6.41/2.69 thf(fact_2590_abs__triangle__ineq4, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ A2) @ B2))) @ (((plus_plus @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2))))))))). % 6.41/2.69 thf(fact_2591_abs__diff__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A,A2:A,R2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ X3) @ A2))) @ R2) = ((((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ R2)) @ X3) & (((ord_less_eq @ A) @ X3) @ (((plus_plus @ A) @ A2) @ R2)))))))))). % 6.41/2.69 thf(fact_2592_sum_Oinsert__if, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),X3:B,G:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((((member @ B) @ X3) @ A3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((insert @ B) @ X3) @ A3)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3))) & ((~ ((((member @ B) @ X3) @ A3))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((insert @ B) @ X3) @ A3)) = (((plus_plus @ A) @ (G @ X3)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3)))))))))))). % 6.41/2.69 thf(fact_2593_abs__diff__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A,A2:A,R2:A]: (((((ord_less @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ X3) @ A2))) @ R2) = ((((ord_less @ A) @ (((minus_minus @ A) @ A2) @ R2)) @ X3) & (((ord_less @ A) @ X3) @ (((plus_plus @ A) @ A2) @ R2)))))))))). % 6.41/2.69 thf(fact_2594_sum_Oreindex__bij__witness__not__neutral, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comm_monoid_add @ A) => (! [S6:(set @ B),T6:(set @ C),S3:(set @ B),I:(C > B),J:(B > C),T4:(set @ C),G:(B > A),H2:(C > A)]: ((((finite_finite2 @ B) @ S6) => (((finite_finite2 @ C) @ T6) => ((! [A5:B]: (((((member @ B) @ A5) @ (((minus_minus @ (set @ B)) @ S3) @ S6)) => ((I @ (J @ A5)) = A5)))) => ((! [A5:B]: (((((member @ B) @ A5) @ (((minus_minus @ (set @ B)) @ S3) @ S6)) => (((member @ C) @ (J @ A5)) @ (((minus_minus @ (set @ C)) @ T4) @ T6))))) => ((! [B5:C]: (((((member @ C) @ B5) @ (((minus_minus @ (set @ C)) @ T4) @ T6)) => ((J @ (I @ B5)) = B5)))) => ((! [B5:C]: (((((member @ C) @ B5) @ (((minus_minus @ (set @ C)) @ T4) @ T6)) => (((member @ B) @ (I @ B5)) @ (((minus_minus @ (set @ B)) @ S3) @ S6))))) => ((! [A5:B]: (((((member @ B) @ A5) @ S6) => ((G @ A5) = (zero_zero @ A))))) => ((! [B5:C]: (((((member @ C) @ B5) @ T6) => ((H2 @ B5) = (zero_zero @ A))))) => ((! [A5:B]: (((((member @ B) @ A5) @ S3) => ((H2 @ (J @ A5)) = (G @ A5))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ S3) = ((((groups7311177749621191930dd_sum @ C) @ A) @ H2) @ T4))))))))))))))))). % 6.41/2.69 thf(fact_2595_sum__eq__Suc0__iff, axiom, (! [A:$tType,A3:(set @ A),F2:(A > nat)]: ((((finite_finite2 @ A) @ A3) => ((((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ A3) = (suc @ (zero_zero @ nat))) = (? [X:A]: (((((member @ A) @ X) @ A3) & (((F2 @ X) = (suc @ (zero_zero @ nat))) & (! [Y2:A]: (((((member @ A) @ Y2) @ A3) => ((X != Y2) => ((F2 @ Y2) = (zero_zero @ nat))))))))))))))). % 6.41/2.69 thf(fact_2596_sum__SucD, axiom, (! [A:$tType,F2:(A > nat),A3:(set @ A),N2:nat]: (((((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ A3) = (suc @ N2)) => (? [X5:A]: (((((member @ A) @ X5) @ A3) & (((ord_less @ nat) @ (zero_zero @ nat)) @ (F2 @ X5))))))))). % 6.41/2.69 thf(fact_2597_sum__eq__1__iff, axiom, (! [A:$tType,A3:(set @ A),F2:(A > nat)]: ((((finite_finite2 @ A) @ A3) => ((((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ A3) = (one_one @ nat)) = (? [X:A]: (((((member @ A) @ X) @ A3) & (((F2 @ X) = (one_one @ nat)) & (! [Y2:A]: (((((member @ A) @ Y2) @ A3) => ((X != Y2) => ((F2 @ Y2) = (zero_zero @ nat))))))))))))))). % 6.41/2.69 thf(fact_2598_abs__real__def, axiom, ((abs_abs @ real) = (^ [A4:real]: (((((if @ real) @ (((ord_less @ real) @ A4) @ (zero_zero @ real))) @ ((uminus_uminus @ real) @ A4)) @ A4))))). % 6.41/2.69 thf(fact_2599_zabs__def, axiom, ((abs_abs @ int) = (^ [I3:int]: (((((if @ int) @ (((ord_less @ int) @ I3) @ (zero_zero @ int))) @ ((uminus_uminus @ int) @ I3)) @ I3))))). % 6.41/2.69 thf(fact_2600_sum__nonneg__leq__bound, axiom, (! [B:$tType,A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [S:(set @ B),F2:(B > A),B3:A,I:B]: ((((finite_finite2 @ B) @ S) => ((! [I4:B]: (((((member @ B) @ I4) @ S) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ I4))))) => ((((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ S) = B3) => ((((member @ B) @ I) @ S) => (((ord_less_eq @ A) @ (F2 @ I)) @ B3))))))))))). % 6.41/2.69 thf(fact_2601_sum__nonneg__0, axiom, (! [B:$tType,A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [S:(set @ B),F2:(B > A),I:B]: ((((finite_finite2 @ B) @ S) => ((! [I4:B]: (((((member @ B) @ I4) @ S) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ I4))))) => ((((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ S) = (zero_zero @ A)) => ((((member @ B) @ I) @ S) => ((F2 @ I) = (zero_zero @ A)))))))))))). % 6.41/2.69 thf(fact_2602_dvd__imp__le__int, axiom, (! [I:int,D2:int]: (((I != (zero_zero @ int)) => ((((dvd_dvd @ int) @ D2) @ I) => (((ord_less_eq @ int) @ ((abs_abs @ int) @ D2)) @ ((abs_abs @ int) @ I))))))). % 6.41/2.69 thf(fact_2603_abs__mod__less, axiom, (! [L:int,K:int]: (((L != (zero_zero @ int)) => (((ord_less @ int) @ ((abs_abs @ int) @ (((modulo_modulo @ int) @ K) @ L))) @ ((abs_abs @ int) @ L)))))). % 6.41/2.69 thf(fact_2604_sum_Osetdiff__irrelevant, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A3) @ ((collect @ B) @ (^ [X:B]: (((G @ X) = (zero_zero @ A))))))) = ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3))))))))). % 6.41/2.69 thf(fact_2605_sum__power__add, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X3:A,M:nat,I6:(set @ nat)]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((power_power @ A) @ X3) @ (((plus_plus @ nat) @ M) @ I3))))) @ I6) = (((times_times @ A) @ (((power_power @ A) @ X3) @ M)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X3)) @ I6))))))))). % 6.41/2.69 thf(fact_2606_sum_OatLeastAtMost__rev, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N2:nat,M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ N2) @ M)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ N2)) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ N2) @ M))))))))). % 6.41/2.69 thf(fact_2607_tanh__real__gt__neg1, axiom, (! [X3:real]: ((((ord_less @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ ((tanh @ real) @ X3))))). % 6.41/2.69 thf(fact_2608_sum__pos2, axiom, (! [A:$tType,B:$tType]: (((ordere6911136660526730532id_add @ A) => (! [I6:(set @ B),I:B,F2:(B > A)]: ((((finite_finite2 @ B) @ I6) => ((((member @ B) @ I) @ I6) => ((((ord_less @ A) @ (zero_zero @ A)) @ (F2 @ I)) => ((! [I4:B]: (((((member @ B) @ I4) @ I6) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ I4))))) => (((ord_less @ A) @ (zero_zero @ A)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ I6)))))))))))). % 6.41/2.69 thf(fact_2609_sum__pos, axiom, (! [A:$tType,B:$tType]: (((ordere6911136660526730532id_add @ A) => (! [I6:(set @ B),F2:(B > A)]: ((((finite_finite2 @ B) @ I6) => ((I6 != (bot_bot @ (set @ B))) => ((! [I4:B]: (((((member @ B) @ I4) @ I6) => (((ord_less @ A) @ (zero_zero @ A)) @ (F2 @ I4))))) => (((ord_less @ A) @ (zero_zero @ A)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ I6))))))))))). % 6.41/2.69 thf(fact_2610_abs__add__one__gt__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A]: ((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (one_one @ A)) @ ((abs_abs @ A) @ X3))))))))). % 6.41/2.69 thf(fact_2611_sum_Omono__neutral__cong__right, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [T4:(set @ B),S3:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ T4) => ((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [X5:B]: (((((member @ B) @ X5) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((G @ X5) = (zero_zero @ A))))) => ((! [X5:B]: (((((member @ B) @ X5) @ S3) => ((G @ X5) = (H2 @ X5))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ T4) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ S3)))))))))))). % 6.41/2.69 thf(fact_2612_sum_Omono__neutral__cong__left, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [T4:(set @ B),S3:(set @ B),H2:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ T4) => ((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [X5:B]: (((((member @ B) @ X5) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((H2 @ X5) = (zero_zero @ A))))) => ((! [X5:B]: (((((member @ B) @ X5) @ S3) => ((G @ X5) = (H2 @ X5))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ S3) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ T4)))))))))))). % 6.41/2.69 thf(fact_2613_sum_Omono__neutral__right, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [T4:(set @ B),S3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ T4) => ((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [X5:B]: (((((member @ B) @ X5) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((G @ X5) = (zero_zero @ A))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ T4) = ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ S3))))))))))). % 6.41/2.69 thf(fact_2614_sum_Omono__neutral__left, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [T4:(set @ B),S3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ T4) => ((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [X5:B]: (((((member @ B) @ X5) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((G @ X5) = (zero_zero @ A))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ S3) = ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ T4))))))))))). % 6.41/2.69 thf(fact_2615_sum_Osame__carrierI, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [C5:(set @ B),A3:(set @ B),B3:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ C5) => ((((ord_less_eq @ (set @ B)) @ A3) @ C5) => ((((ord_less_eq @ (set @ B)) @ B3) @ C5) => ((! [A5:B]: (((((member @ B) @ A5) @ (((minus_minus @ (set @ B)) @ C5) @ A3)) => ((G @ A5) = (zero_zero @ A))))) => ((! [B5:B]: (((((member @ B) @ B5) @ (((minus_minus @ (set @ B)) @ C5) @ B3)) => ((H2 @ B5) = (zero_zero @ A))))) => ((((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ C5) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ C5)) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ B3)))))))))))))). % 6.41/2.69 thf(fact_2616_sum_Osame__carrier, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [C5:(set @ B),A3:(set @ B),B3:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ C5) => ((((ord_less_eq @ (set @ B)) @ A3) @ C5) => ((((ord_less_eq @ (set @ B)) @ B3) @ C5) => ((! [A5:B]: (((((member @ B) @ A5) @ (((minus_minus @ (set @ B)) @ C5) @ A3)) => ((G @ A5) = (zero_zero @ A))))) => ((! [B5:B]: (((((member @ B) @ B5) @ (((minus_minus @ (set @ B)) @ C5) @ B3)) => ((H2 @ B5) = (zero_zero @ A))))) => ((((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ B3)) = (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ C5) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ C5)))))))))))))). % 6.41/2.69 thf(fact_2617_sum_Osubset__diff, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [B3:(set @ B),A3:(set @ B),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ B3) @ A3) => (((finite_finite2 @ B) @ A3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A3) @ B3))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ B3))))))))))). % 6.41/2.69 thf(fact_2618_sum__diff, axiom, (! [A:$tType,B:$tType]: (((ab_group_add @ A) => (! [A3:(set @ B),B3:(set @ B),F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((((ord_less_eq @ (set @ B)) @ B3) @ A3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ (((minus_minus @ (set @ B)) @ A3) @ B3)) = (((minus_minus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ A3)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ B3))))))))))). % 6.41/2.69 thf(fact_2619_of__int__leD, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:int,X3:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ ((ring_1_of_int @ A) @ N2))) @ X3) => ((N2 = (zero_zero @ int)) | (((ord_less_eq @ A) @ (one_one @ A)) @ X3))))))))). % 6.41/2.69 thf(fact_2620_sum__diff__nat, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A),F2:(A > nat)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => (((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = (((minus_minus @ nat) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ A3)) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ B3)))))))). % 6.41/2.69 thf(fact_2621_of__int__lessD, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:int,X3:A]: (((((ord_less @ A) @ ((abs_abs @ A) @ ((ring_1_of_int @ A) @ N2))) @ X3) => ((N2 = (zero_zero @ int)) | (((ord_less @ A) @ (one_one @ A)) @ X3))))))))). % 6.41/2.69 thf(fact_2622_sum__diff1__nat, axiom, (! [A:$tType,A2:A,A3:(set @ A),F2:(A > nat)]: ((((((member @ A) @ A2) @ A3) => (((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = (((minus_minus @ nat) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ A3)) @ (F2 @ A2)))) & ((~ ((((member @ A) @ A2) @ A3))) => (((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = ((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ A3))))))). % 6.41/2.69 thf(fact_2623_sum__shift__lb__Suc0__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [F2:(nat > A),K:nat]: ((((F2 @ (zero_zero @ nat)) = (zero_zero @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ K)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ K)))))))))). % 6.41/2.69 thf(fact_2624_sum_OatLeast0__atMost__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N2))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2))) @ (G @ (suc @ N2)))))))))). % 6.41/2.69 thf(fact_2625_sum_OatLeast__Suc__atMost, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (((plus_plus @ A) @ (G @ M)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ N2))))))))))). % 6.41/2.69 thf(fact_2626_sum_Onat__ivl__Suc_H, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ (suc @ N2)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N2))) = (((plus_plus @ A) @ (G @ (suc @ N2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))))). % 6.41/2.69 thf(fact_2627_round__diff__minimal, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z2:A,M:int]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ Z2) @ ((ring_1_of_int @ A) @ ((archimedean_round @ A) @ Z2))))) @ ((abs_abs @ A) @ (((minus_minus @ A) @ Z2) @ ((ring_1_of_int @ A) @ M)))))))))). % 6.41/2.69 thf(fact_2628_zdvd__mult__cancel1, axiom, (! [M:int,N2:int]: (((M != (zero_zero @ int)) => ((((dvd_dvd @ int) @ (((times_times @ int) @ M) @ N2)) @ M) = (((abs_abs @ int) @ N2) = (one_one @ int))))))). % 6.41/2.69 thf(fact_2629_sum_OSuc__reindex__ivl, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))) @ (G @ (suc @ N2))) = (((plus_plus @ A) @ (G @ M)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))))). % 6.41/2.69 thf(fact_2630_sum__Suc__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [M:nat,N2:nat,F2:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ (suc @ N2)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((minus_minus @ A) @ (F2 @ (suc @ I3))) @ (F2 @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (((minus_minus @ A) @ (F2 @ (suc @ N2))) @ (F2 @ M)))))))))). % 6.41/2.69 thf(fact_2631_fun__cong__unused__0, axiom, (! [A:$tType,B:$tType,C:$tType]: (((zero @ B) => (! [F2:((A > B) > C),G:C]: (((F2 = (^ [X:(A > B)]: (G))) => ((F2 @ (^ [X:A]: ((zero_zero @ B)))) = G)))))))). % 6.41/2.69 thf(fact_2632_sum__mono2, axiom, (! [A:$tType,B:$tType]: (((ordere6911136660526730532id_add @ A) => (! [B3:(set @ B),A3:(set @ B),F2:(B > A)]: ((((finite_finite2 @ B) @ B3) => ((((ord_less_eq @ (set @ B)) @ A3) @ B3) => ((! [B5:B]: (((((member @ B) @ B5) @ (((minus_minus @ (set @ B)) @ B3) @ A3)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ B5))))) => (((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ A3)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ B3))))))))))). % 6.41/2.69 thf(fact_2633_abs__le__square__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ X3)) @ ((abs_abs @ A) @ Y)) = (((ord_less_eq @ A) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_2634_sum_Oremove, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),X3:B,G:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((((member @ B) @ X3) @ A3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3) = (((plus_plus @ A) @ (G @ X3)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ X3) @ (bot_bot @ (set @ B))))))))))))))). % 6.41/2.69 thf(fact_2635_sum_Oinsert__remove, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),G:(B > A),X3:B]: ((((finite_finite2 @ B) @ A3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((insert @ B) @ X3) @ A3)) = (((plus_plus @ A) @ (G @ X3)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ X3) @ (bot_bot @ (set @ B)))))))))))))). % 6.41/2.69 thf(fact_2636_sum__diff1, axiom, (! [A:$tType,B:$tType]: (((ab_group_add @ A) => (! [A3:(set @ B),A2:B,F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((((member @ B) @ A2) @ A3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B))))) = (((minus_minus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ A3)) @ (F2 @ A2)))) & ((~ ((((member @ B) @ A2) @ A3))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B))))) = ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ A3))))))))))). % 6.41/2.69 thf(fact_2637_sum_Oub__add__nat, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N2:nat,G:(nat > A),P4:nat]: (((((ord_less_eq @ nat) @ M) @ (((plus_plus @ nat) @ N2) @ (one_one @ nat))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (((plus_plus @ nat) @ N2) @ P4))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ N2) @ (one_one @ nat))) @ (((plus_plus @ nat) @ N2) @ P4)))))))))))). % 6.41/2.69 thf(fact_2638_sum_Odelta__remove, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ B),A2:B,B2:(B > A),C2:(B > A)]: ((((finite_finite2 @ B) @ S3) => (((((member @ B) @ A2) @ S3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A2)) @ (B2 @ K3)) @ (C2 @ K3))))) @ S3) = (((plus_plus @ A) @ (B2 @ A2)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ C2) @ (((minus_minus @ (set @ B)) @ S3) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B)))))))) & ((~ ((((member @ B) @ A2) @ S3))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A2)) @ (B2 @ K3)) @ (C2 @ K3))))) @ S3) = ((((groups7311177749621191930dd_sum @ B) @ A) @ C2) @ (((minus_minus @ (set @ B)) @ S3) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B))))))))))))))). % 6.41/2.69 thf(fact_2639_sum__count__set, axiom, (! [A:$tType,Xs:(list @ A),X8:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ X8) => (((finite_finite2 @ A) @ X8) => (((((groups7311177749621191930dd_sum @ A) @ nat) @ ((count_list @ A) @ Xs)) @ X8) = ((size_size @ (list @ A)) @ Xs))))))). % 6.41/2.69 thf(fact_2640_sum__strict__mono2, axiom, (! [B:$tType,A:$tType]: (((ordere8940638589300402666id_add @ B) => (! [B3:(set @ A),A3:(set @ A),B2:A,F2:(A > B)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ B2) @ (((minus_minus @ (set @ A)) @ B3) @ A3)) => ((((ord_less @ B) @ (zero_zero @ B)) @ (F2 @ B2)) => ((! [X5:A]: (((((member @ A) @ X5) @ B3) => (((ord_less_eq @ B) @ (zero_zero @ B)) @ (F2 @ X5))))) => (((ord_less @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F2) @ A3)) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F2) @ B3))))))))))))). % 6.41/2.69 thf(fact_2641_member__le__sum, axiom, (! [B:$tType,C:$tType]: ((((ordere6911136660526730532id_add @ B) & (semiring_1 @ B)) => (! [I:C,A3:(set @ C),F2:(C > B)]: (((((member @ C) @ I) @ A3) => ((! [X5:C]: (((((member @ C) @ X5) @ (((minus_minus @ (set @ C)) @ A3) @ (((insert @ C) @ I) @ (bot_bot @ (set @ C))))) => (((ord_less_eq @ B) @ (zero_zero @ B)) @ (F2 @ X5))))) => (((finite_finite2 @ C) @ A3) => (((ord_less_eq @ B) @ (F2 @ I)) @ ((((groups7311177749621191930dd_sum @ C) @ B) @ F2) @ A3))))))))))). % 6.41/2.69 thf(fact_2642_power2__le__iff__abs__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Y:A,X3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => ((((ord_less_eq @ A) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (((ord_less_eq @ A) @ ((abs_abs @ A) @ X3)) @ Y))))))))). % 6.41/2.69 thf(fact_2643_abs__sqrt__wlog, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [P:(A > (A > $o)),X3:A]: (((! [X5:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X5) => ((P @ X5) @ (((power_power @ A) @ X5) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) => ((P @ ((abs_abs @ A) @ X3)) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_2644_abs__square__le__1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A]: (((((ord_less_eq @ A) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ A)) = (((ord_less_eq @ A) @ ((abs_abs @ A) @ X3)) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_2645_abs__square__less__1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A]: (((((ord_less @ A) @ (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ A)) = (((ord_less @ A) @ ((abs_abs @ A) @ X3)) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_2646_power__mono__even, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:nat,A2:A,B2:A]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2) => ((((ord_less_eq @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2)) => (((ord_less_eq @ A) @ (((power_power @ A) @ A2) @ N2)) @ (((power_power @ A) @ B2) @ N2)))))))))). % 6.41/2.69 thf(fact_2647_nat__intermed__int__val, axiom, (! [M:nat,N2:nat,F2:(nat > int),K:int]: (((! [I4:nat]: ((((((ord_less_eq @ nat) @ M) @ I4) & (((ord_less @ nat) @ I4) @ N2)) => (((ord_less_eq @ int) @ ((abs_abs @ int) @ (((minus_minus @ int) @ (F2 @ (suc @ I4))) @ (F2 @ I4)))) @ (one_one @ int))))) => ((((ord_less_eq @ nat) @ M) @ N2) => ((((ord_less_eq @ int) @ (F2 @ M)) @ K) => ((((ord_less_eq @ int) @ K) @ (F2 @ N2)) => (? [I4:nat]: (((((ord_less_eq @ nat) @ M) @ I4) & ((((ord_less_eq @ nat) @ I4) @ N2) & ((F2 @ I4) = K)))))))))))). % 6.41/2.69 thf(fact_2648_incr__lemma, axiom, (! [D2:int,Z2:int,X3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D2) => (((ord_less @ int) @ Z2) @ (((plus_plus @ int) @ X3) @ (((times_times @ int) @ (((plus_plus @ int) @ ((abs_abs @ int) @ (((minus_minus @ int) @ X3) @ Z2))) @ (one_one @ int))) @ D2))))))). % 6.41/2.69 thf(fact_2649_decr__lemma, axiom, (! [D2:int,X3:int,Z2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D2) => (((ord_less @ int) @ (((minus_minus @ int) @ X3) @ (((times_times @ int) @ (((plus_plus @ int) @ ((abs_abs @ int) @ (((minus_minus @ int) @ X3) @ Z2))) @ (one_one @ int))) @ D2))) @ Z2))))). % 6.41/2.69 thf(fact_2650_sum__natinterval__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [M:nat,N2:nat,F2:(nat > A)]: ((((((ord_less_eq @ nat) @ M) @ N2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((minus_minus @ A) @ (F2 @ K3)) @ (F2 @ (((plus_plus @ nat) @ K3) @ (one_one @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (((minus_minus @ A) @ (F2 @ M)) @ (F2 @ (((plus_plus @ nat) @ N2) @ (one_one @ nat)))))) & ((~ ((((ord_less_eq @ nat) @ M) @ N2))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((minus_minus @ A) @ (F2 @ K3)) @ (F2 @ (((plus_plus @ nat) @ K3) @ (one_one @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_2651_sum__telescope_H_H, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [M:nat,N2:nat,F2:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((minus_minus @ A) @ (F2 @ K3)) @ (F2 @ (((minus_minus @ nat) @ K3) @ (one_one @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ N2)) = (((minus_minus @ A) @ (F2 @ N2)) @ (F2 @ M)))))))))). % 6.41/2.69 thf(fact_2652_mask__eq__sum__exp, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [N2:nat]: (((((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) @ (one_one @ A)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ ((collect @ nat) @ (^ [Q5:nat]: ((((ord_less @ nat) @ Q5) @ N2)))))))))))). % 6.41/2.69 thf(fact_2653_nat__ivt__aux, axiom, (! [N2:nat,F2:(nat > int),K:int]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ N2) => (((ord_less_eq @ int) @ ((abs_abs @ int) @ (((minus_minus @ int) @ (F2 @ (suc @ I4))) @ (F2 @ I4)))) @ (one_one @ int))))) => ((((ord_less_eq @ int) @ (F2 @ (zero_zero @ nat))) @ K) => ((((ord_less_eq @ int) @ K) @ (F2 @ N2)) => (? [I4:nat]: (((((ord_less_eq @ nat) @ I4) @ N2) & ((F2 @ I4) = K)))))))))). % 6.41/2.69 thf(fact_2654_divmod__nat__def, axiom, (divmod_nat = (^ [M2:nat,N:nat]: (((((product_Pair @ nat) @ nat) @ (((divide_divide @ nat) @ M2) @ N)) @ (((modulo_modulo @ nat) @ M2) @ N)))))). % 6.41/2.69 thf(fact_2655_sum__gp__multiplied, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [M:nat,N2:nat,X3:A]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((times_times @ A) @ (((minus_minus @ A) @ (one_one @ A)) @ X3)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X3)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))) = (((minus_minus @ A) @ (((power_power @ A) @ X3) @ M)) @ (((power_power @ A) @ X3) @ (suc @ N2))))))))))). % 6.41/2.69 thf(fact_2656_sum_Oin__pairs, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_2657_nat0__intermed__int__val, axiom, (! [N2:nat,F2:(nat > int),K:int]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ N2) => (((ord_less_eq @ int) @ ((abs_abs @ int) @ (((minus_minus @ int) @ (F2 @ (((plus_plus @ nat) @ I4) @ (one_one @ nat)))) @ (F2 @ I4)))) @ (one_one @ int))))) => ((((ord_less_eq @ int) @ (F2 @ (zero_zero @ nat))) @ K) => ((((ord_less_eq @ int) @ K) @ (F2 @ N2)) => (? [I4:nat]: (((((ord_less_eq @ nat) @ I4) @ N2) & ((F2 @ I4) = K)))))))))). % 6.41/2.69 thf(fact_2658_mask__eq__sum__exp__nat, axiom, (! [N2:nat]: (((((minus_minus @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) @ (suc @ (zero_zero @ nat))) = ((((groups7311177749621191930dd_sum @ nat) @ nat) @ ((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ ((collect @ nat) @ (^ [Q5:nat]: ((((ord_less @ nat) @ Q5) @ N2))))))))). % 6.41/2.69 thf(fact_2659_gauss__sum__nat, axiom, (! [N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X:nat]: (X))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2)) = (((divide_divide @ nat) @ (((times_times @ nat) @ N2) @ (suc @ N2))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))). % 6.41/2.69 thf(fact_2660_of__int__round__abs__le, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ ((archimedean_round @ A) @ X3))) @ X3))) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_2661_listrel1p__def, axiom, (! [A:$tType]: (((listrel1p @ A) = (^ [R5:(A > (A > $o)),Xs2:(list @ A),Ys3:(list @ A)]: ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs2) @ Ys3)) @ ((listrel1 @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ R5)))))))))). % 6.41/2.69 thf(fact_2662_lemma__interval, axiom, (! [A2:real,X3:real,B2:real]: (((((ord_less @ real) @ A2) @ X3) => ((((ord_less @ real) @ X3) @ B2) => (? [D3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D3) & (! [Y5:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ (((minus_minus @ real) @ X3) @ Y5))) @ D3) => ((((ord_less_eq @ real) @ A2) @ Y5) & (((ord_less_eq @ real) @ Y5) @ B2))))))))))))). % 6.41/2.69 thf(fact_2663_sum__gp, axiom, (! [A:$tType]: ((((division_ring @ A) & (comm_ring @ A)) => (! [N2:nat,M:nat,X3:A]: ((((((ord_less @ nat) @ N2) @ M) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X3)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (zero_zero @ A))) & ((~ ((((ord_less @ nat) @ N2) @ M))) => (((X3 = (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X3)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = ((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ N2) @ (one_one @ nat))) @ M)))) & ((X3 != (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X3)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (((divide_divide @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ X3) @ M)) @ (((power_power @ A) @ X3) @ (suc @ N2)))) @ (((minus_minus @ A) @ (one_one @ A)) @ X3))))))))))))). % 6.41/2.69 thf(fact_2664_gauss__sum__from__Suc__0, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (semiring_1_of_nat @ A)) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ N2)) = (((divide_divide @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (one_one @ A)))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_2665_lemma__interval__lt, axiom, (! [A2:real,X3:real,B2:real]: (((((ord_less @ real) @ A2) @ X3) => ((((ord_less @ real) @ X3) @ B2) => (? [D3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D3) & (! [Y5:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ (((minus_minus @ real) @ X3) @ Y5))) @ D3) => ((((ord_less @ real) @ A2) @ Y5) & (((ord_less @ real) @ Y5) @ B2))))))))))))). % 6.41/2.69 thf(fact_2666_arctan__double, axiom, (! [X3:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ X3)) @ (one_one @ real)) => ((((times_times @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ (arctan @ X3)) = (arctan @ (((divide_divide @ real) @ (((times_times @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ X3)) @ (((minus_minus @ real) @ (one_one @ real)) @ (((power_power @ real) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_2667_of__nat__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat,N2:nat]: (((((semiring_1_of_nat @ A) @ M) = ((semiring_1_of_nat @ A) @ N2)) = (M = N2)))))))). % 6.41/2.69 thf(fact_2668_split__part, axiom, (! [B:$tType,A:$tType,P:$o,Q:(A > (B > $o))]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [A4:A,B4:B]: ((P & ((Q @ A4) @ B4))))) = (^ [Ab:((product_prod @ A) @ B)]: ((P & (((((product_case_prod @ A) @ B) @ $o) @ Q) @ Ab)))))))). % 6.41/2.69 thf(fact_2669_int__eq__iff__numeral, axiom, (! [M:nat,V2:num]: (((((semiring_1_of_nat @ int) @ M) = ((numeral_numeral @ int) @ V2)) = (M = ((numeral_numeral @ nat) @ V2)))))). % 6.41/2.69 thf(fact_2670_abs__of__nat, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:nat]: ((((abs_abs @ A) @ ((semiring_1_of_nat @ A) @ N2)) = ((semiring_1_of_nat @ A) @ N2)))))))). % 6.41/2.69 thf(fact_2671_negative__eq__positive, axiom, (! [N2:nat,M:nat]: (((((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N2)) = ((semiring_1_of_nat @ int) @ M)) = ((N2 = (zero_zero @ nat)) & (M = (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_2672_of__int__of__nat__eq, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N2:nat]: ((((ring_1_of_int @ A) @ ((semiring_1_of_nat @ int) @ N2)) = ((semiring_1_of_nat @ A) @ N2)))))))). % 6.41/2.69 thf(fact_2673_negative__zle, axiom, (! [N2:nat,M:nat]: ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N2))) @ ((semiring_1_of_nat @ int) @ M))))). % 6.41/2.69 thf(fact_2674_int__dvd__int__iff, axiom, (! [M:nat,N2:nat]: (((((dvd_dvd @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N2)) = (((dvd_dvd @ nat) @ M) @ N2))))). % 6.41/2.69 thf(fact_2675_of__nat__eq__0__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [M:nat]: (((((semiring_1_of_nat @ A) @ M) = (zero_zero @ A)) = (M = (zero_zero @ nat))))))))). % 6.41/2.69 thf(fact_2676_of__nat__0__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N2:nat]: ((((zero_zero @ A) = ((semiring_1_of_nat @ A) @ N2)) = ((zero_zero @ nat) = N2)))))))). % 6.41/2.69 thf(fact_2677_of__nat__0, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((semiring_1_of_nat @ A) @ (zero_zero @ nat)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_2678_of__nat__less__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:nat,N2:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N2)) = (((ord_less @ nat) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_2679_of__nat__le__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:nat,N2:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N2)) = (((ord_less_eq @ nat) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_2680_of__nat__add, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [M:nat,N2:nat]: ((((semiring_1_of_nat @ A) @ (((plus_plus @ nat) @ M) @ N2)) = (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N2))))))))). % 6.41/2.69 thf(fact_2681_of__nat__mult, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [M:nat,N2:nat]: ((((semiring_1_of_nat @ A) @ (((times_times @ nat) @ M) @ N2)) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N2))))))))). % 6.41/2.69 thf(fact_2682_of__nat__1, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((semiring_1_of_nat @ A) @ (one_one @ nat)) = (one_one @ A)))))). % 6.41/2.69 thf(fact_2683_of__nat__1__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N2:nat]: ((((one_one @ A) = ((semiring_1_of_nat @ A) @ N2)) = (N2 = (one_one @ nat))))))))). % 6.41/2.69 thf(fact_2684_of__nat__eq__1__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N2:nat]: (((((semiring_1_of_nat @ A) @ N2) = (one_one @ A)) = (N2 = (one_one @ nat))))))))). % 6.41/2.69 thf(fact_2685_negative__zless, axiom, (! [N2:nat,M:nat]: ((((ord_less @ int) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (suc @ N2)))) @ ((semiring_1_of_nat @ int) @ M))))). % 6.41/2.69 thf(fact_2686_arctan__less__zero__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ (arctan @ X3)) @ (zero_zero @ real)) = (((ord_less @ real) @ X3) @ (zero_zero @ real)))))). % 6.41/2.69 thf(fact_2687_zero__less__arctan__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ (arctan @ X3)) = (((ord_less @ real) @ (zero_zero @ real)) @ X3))))). % 6.41/2.69 thf(fact_2688_of__nat__of__bool, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [P:$o]: ((((semiring_1_of_nat @ A) @ ((zero_neq_one_of_bool @ nat) @ P)) = ((zero_neq_one_of_bool @ A) @ P)))))))). % 6.41/2.69 thf(fact_2689_of__nat__sum, axiom, (! [A:$tType,B:$tType]: (((semiring_1 @ A) => (! [F2:(B > nat),A3:(set @ B)]: ((((semiring_1_of_nat @ A) @ ((((groups7311177749621191930dd_sum @ B) @ nat) @ F2) @ A3)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X:B]: (((semiring_1_of_nat @ A) @ (F2 @ X))))) @ A3)))))))). % 6.41/2.69 thf(fact_2690_of__nat__le__0__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)) = (M = (zero_zero @ nat))))))))). % 6.41/2.69 thf(fact_2691_of__nat__Suc, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [M:nat]: ((((semiring_1_of_nat @ A) @ (suc @ M)) = (((plus_plus @ A) @ (one_one @ A)) @ ((semiring_1_of_nat @ A) @ M))))))))). % 6.41/2.69 thf(fact_2692_real__of__nat__less__numeral__iff, axiom, (! [N2:nat,W2:num]: (((((ord_less @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ ((numeral_numeral @ real) @ W2)) = (((ord_less @ nat) @ N2) @ ((numeral_numeral @ nat) @ W2)))))). % 6.41/2.69 thf(fact_2693_numeral__less__real__of__nat__iff, axiom, (! [W2:num,N2:nat]: (((((ord_less @ real) @ ((numeral_numeral @ real) @ W2)) @ ((semiring_1_of_nat @ real) @ N2)) = (((ord_less @ nat) @ ((numeral_numeral @ nat) @ W2)) @ N2))))). % 6.41/2.69 thf(fact_2694_numeral__le__real__of__nat__iff, axiom, (! [N2:num,M:nat]: (((((ord_less_eq @ real) @ ((numeral_numeral @ real) @ N2)) @ ((semiring_1_of_nat @ real) @ M)) = (((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ N2)) @ M))))). % 6.41/2.69 thf(fact_2695_of__nat__0__less__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N2:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N2)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N2)))))))). % 6.41/2.69 thf(fact_2696_of__nat__power__less__of__nat__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X3:nat,B2:nat,W2:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ X3)) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W2)) = (((ord_less @ nat) @ X3) @ (((power_power @ nat) @ B2) @ W2))))))))). % 6.41/2.69 thf(fact_2697_of__nat__less__of__nat__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:nat,W2:nat,X3:nat]: (((((ord_less @ A) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W2)) @ ((semiring_1_of_nat @ A) @ X3)) = (((ord_less @ nat) @ (((power_power @ nat) @ B2) @ W2)) @ X3)))))))). % 6.41/2.69 thf(fact_2698_of__nat__power__le__of__nat__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X3:nat,B2:nat,W2:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ X3)) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W2)) = (((ord_less_eq @ nat) @ X3) @ (((power_power @ nat) @ B2) @ W2))))))))). % 6.41/2.69 thf(fact_2699_of__nat__le__of__nat__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:nat,W2:nat,X3:nat]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ B2)) @ W2)) @ ((semiring_1_of_nat @ A) @ X3)) = (((ord_less_eq @ nat) @ (((power_power @ nat) @ B2) @ W2)) @ X3)))))))). % 6.41/2.69 thf(fact_2700_of__nat__zero__less__power__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X3:nat,N2:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((power_power @ A) @ ((semiring_1_of_nat @ A) @ X3)) @ N2)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ X3) | (N2 = (zero_zero @ nat)))))))))). % 6.41/2.69 thf(fact_2701_numeral__power__less__of__nat__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [I:num,N2:nat,X3:nat]: (((((ord_less @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ I)) @ N2)) @ ((semiring_1_of_nat @ A) @ X3)) = (((ord_less @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ I)) @ N2)) @ X3)))))))). % 6.41/2.69 thf(fact_2702_of__nat__less__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X3:nat,I:num,N2:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ X3)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ I)) @ N2)) = (((ord_less @ nat) @ X3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ I)) @ N2))))))))). % 6.41/2.69 thf(fact_2703_of__nat__le__numeral__power__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X3:nat,I:num,N2:nat]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ X3)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ I)) @ N2)) = (((ord_less_eq @ nat) @ X3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ I)) @ N2))))))))). % 6.41/2.69 thf(fact_2704_numeral__power__le__of__nat__cancel__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [I:num,N2:nat,X3:nat]: (((((ord_less_eq @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ I)) @ N2)) @ ((semiring_1_of_nat @ A) @ X3)) = (((ord_less_eq @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ I)) @ N2)) @ X3)))))))). % 6.41/2.69 thf(fact_2705_prod_Odisc__eq__case, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [Uu3:A,Uv3:B]: ($true))) @ Prod)))). % 6.41/2.69 thf(fact_2706_int__sum, axiom, (! [B:$tType,F2:(B > nat),A3:(set @ B)]: ((((semiring_1_of_nat @ int) @ ((((groups7311177749621191930dd_sum @ B) @ nat) @ F2) @ A3)) = ((((groups7311177749621191930dd_sum @ B) @ int) @ (^ [X:B]: (((semiring_1_of_nat @ int) @ (F2 @ X))))) @ A3))))). % 6.41/2.69 thf(fact_2707_real__arch__simple, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X3:A]: ((? [N3:nat]: ((((ord_less_eq @ A) @ X3) @ ((semiring_1_of_nat @ A) @ N3)))))))))). % 6.41/2.69 thf(fact_2708_reals__Archimedean2, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X3:A]: ((? [N3:nat]: ((((ord_less @ A) @ X3) @ ((semiring_1_of_nat @ A) @ N3)))))))))). % 6.41/2.69 thf(fact_2709_mult__of__nat__commute, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [X3:nat,Y:A]: (((((times_times @ A) @ ((semiring_1_of_nat @ A) @ X3)) @ Y) = (((times_times @ A) @ Y) @ ((semiring_1_of_nat @ A) @ X3))))))))). % 6.41/2.69 thf(fact_2710_int__cases2, axiom, (! [Z2:int]: (((! [N3:nat]: ((Z2 != ((semiring_1_of_nat @ int) @ N3)))) => (~ ((! [N3:nat]: ((Z2 != ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N3))))))))))). % 6.41/2.69 thf(fact_2711_arctan__less__iff, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ (arctan @ X3)) @ (arctan @ Y)) = (((ord_less @ real) @ X3) @ Y))))). % 6.41/2.69 thf(fact_2712_arctan__monotone, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ X3) @ Y) => (((ord_less @ real) @ (arctan @ X3)) @ (arctan @ Y)))))). % 6.41/2.69 thf(fact_2713_int__diff__cases, axiom, (! [Z2:int]: ((~ ((! [M3:nat,N3:nat]: ((Z2 != (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ M3)) @ ((semiring_1_of_nat @ int) @ N3)))))))))). % 6.41/2.69 thf(fact_2714_of__nat__less__of__int__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:nat,X3:int]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ ((ring_1_of_int @ A) @ X3)) = (((ord_less @ int) @ ((semiring_1_of_nat @ int) @ N2)) @ X3)))))))). % 6.41/2.69 thf(fact_2715_of__nat__0__le__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N2:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N2)))))))). % 6.41/2.69 thf(fact_2716_of__nat__less__0__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:nat]: ((~ ((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_2717_of__nat__neq__0, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N2:nat]: ((((semiring_1_of_nat @ A) @ (suc @ N2)) != (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_2718_of__nat__less__imp__less, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:nat,N2:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N2)) => (((ord_less @ nat) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_2719_less__imp__of__nat__less, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ N2) => (((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N2))))))))). % 6.41/2.69 thf(fact_2720_of__nat__mono, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [I:nat,J:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ I)) @ ((semiring_1_of_nat @ A) @ J))))))))). % 6.41/2.69 thf(fact_2721_int__ops_I1_J, axiom, (((semiring_1_of_nat @ int) @ (zero_zero @ nat)) = (zero_zero @ int))). % 6.41/2.69 thf(fact_2722_nat__int__comparison_I2_J, axiom, ((ord_less @ nat) = (^ [A4:nat,B4:nat]: ((((ord_less @ int) @ ((semiring_1_of_nat @ int) @ A4)) @ ((semiring_1_of_nat @ int) @ B4)))))). % 6.41/2.69 thf(fact_2723_int__of__nat__induct, axiom, (! [P:(int > $o),Z2:int]: (((! [N3:nat]: ((P @ ((semiring_1_of_nat @ int) @ N3)))) => ((! [N3:nat]: ((P @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (suc @ N3)))))) => (P @ Z2)))))). % 6.41/2.69 thf(fact_2724_int__cases, axiom, (! [Z2:int]: (((! [N3:nat]: ((Z2 != ((semiring_1_of_nat @ int) @ N3)))) => (~ ((! [N3:nat]: ((Z2 != ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (suc @ N3)))))))))))). % 6.41/2.69 thf(fact_2725_zle__int, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N2)) = (((ord_less_eq @ nat) @ M) @ N2))))). % 6.41/2.69 thf(fact_2726_nat__int__comparison_I3_J, axiom, ((ord_less_eq @ nat) = (^ [A4:nat,B4:nat]: ((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ A4)) @ ((semiring_1_of_nat @ int) @ B4)))))). % 6.41/2.69 thf(fact_2727_zero__le__imp__eq__int, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (? [N3:nat]: ((K = ((semiring_1_of_nat @ int) @ N3)))))))). % 6.41/2.69 thf(fact_2728_nonneg__int__cases, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (~ ((! [N3:nat]: ((K != ((semiring_1_of_nat @ int) @ N3)))))))))). % 6.41/2.69 thf(fact_2729_zadd__int__left, axiom, (! [M:nat,N2:nat,Z2:int]: (((((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ M)) @ (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ N2)) @ Z2)) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ M) @ N2))) @ Z2))))). % 6.41/2.69 thf(fact_2730_int__ops_I5_J, axiom, (! [A2:nat,B2:nat]: ((((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ A2) @ B2)) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ A2)) @ ((semiring_1_of_nat @ int) @ B2)))))). % 6.41/2.69 thf(fact_2731_int__plus, axiom, (! [N2:nat,M:nat]: ((((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ N2) @ M)) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ N2)) @ ((semiring_1_of_nat @ int) @ M)))))). % 6.41/2.69 thf(fact_2732_zle__iff__zadd, axiom, ((ord_less_eq @ int) = (^ [W3:int,Z6:int]: ((? [N:nat]: ((Z6 = (((plus_plus @ int) @ W3) @ ((semiring_1_of_nat @ int) @ N))))))))). % 6.41/2.69 thf(fact_2733_not__int__zless__negative, axiom, (! [N2:nat,M:nat]: ((~ ((((ord_less @ int) @ ((semiring_1_of_nat @ int) @ N2)) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ M)))))))). % 6.41/2.69 thf(fact_2734_nat__less__as__int, axiom, ((ord_less @ nat) = (^ [A4:nat,B4:nat]: ((((ord_less @ int) @ ((semiring_1_of_nat @ int) @ A4)) @ ((semiring_1_of_nat @ int) @ B4)))))). % 6.41/2.69 thf(fact_2735_nat__leq__as__int, axiom, ((ord_less_eq @ nat) = (^ [A4:nat,B4:nat]: ((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ A4)) @ ((semiring_1_of_nat @ int) @ B4)))))). % 6.41/2.69 thf(fact_2736_ex__less__of__nat__mult, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X3) => (? [N3:nat]: ((((ord_less @ A) @ Y) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N3)) @ X3))))))))))). % 6.41/2.69 thf(fact_2737_of__nat__diff, axiom, (! [A:$tType]: (((semiring_1_cancel @ A) => (! [N2:nat,M:nat]: (((((ord_less_eq @ nat) @ N2) @ M) => (((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ M) @ N2)) = (((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N2)))))))))). % 6.41/2.69 thf(fact_2738_reals__Archimedean3, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (! [Y5:real]: ((? [N3:nat]: ((((ord_less @ real) @ Y5) @ (((times_times @ real) @ ((semiring_1_of_nat @ real) @ N3)) @ X3)))))))))). % 6.41/2.69 thf(fact_2739_int__cases4, axiom, (! [M:int]: (((! [N3:nat]: ((M != ((semiring_1_of_nat @ int) @ N3)))) => (~ ((! [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) => (M != ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N3)))))))))))). % 6.41/2.69 thf(fact_2740_int__zle__neg, axiom, (! [N2:nat,M:nat]: (((((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ N2)) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ M))) = ((N2 = (zero_zero @ nat)) & (M = (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_2741_int__ops_I4_J, axiom, (! [A2:nat]: ((((semiring_1_of_nat @ int) @ (suc @ A2)) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ A2)) @ (one_one @ int)))))). % 6.41/2.69 thf(fact_2742_int__Suc, axiom, (! [N2:nat]: ((((semiring_1_of_nat @ int) @ (suc @ N2)) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ N2)) @ (one_one @ int)))))). % 6.41/2.69 thf(fact_2743_zless__iff__Suc__zadd, axiom, ((ord_less @ int) = (^ [W3:int,Z6:int]: ((? [N:nat]: ((Z6 = (((plus_plus @ int) @ W3) @ ((semiring_1_of_nat @ int) @ (suc @ N)))))))))). % 6.41/2.69 thf(fact_2744_negative__zle__0, axiom, (! [N2:nat]: ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N2))) @ (zero_zero @ int))))). % 6.41/2.69 thf(fact_2745_nonpos__int__cases, axiom, (! [K:int]: (((((ord_less_eq @ int) @ K) @ (zero_zero @ int)) => (~ ((! [N3:nat]: ((K != ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N3))))))))))). % 6.41/2.69 thf(fact_2746_sum__roots__unity, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (one_one @ nat)) @ N2) => (((((groups7311177749621191930dd_sum @ complex) @ complex) @ (^ [X:complex]: (X))) @ ((collect @ complex) @ (^ [Z6:complex]: (((((power_power @ complex) @ Z6) @ N2) = (one_one @ complex)))))) = (zero_zero @ complex)))))). % 6.41/2.69 thf(fact_2747_sum__nth__roots, axiom, (! [N2:nat,C2:complex]: (((((ord_less @ nat) @ (one_one @ nat)) @ N2) => (((((groups7311177749621191930dd_sum @ complex) @ complex) @ (^ [X:complex]: (X))) @ ((collect @ complex) @ (^ [Z6:complex]: (((((power_power @ complex) @ Z6) @ N2) = C2))))) = (zero_zero @ complex)))))). % 6.41/2.69 thf(fact_2748_mod__mult2__eq_H, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A2:A,M:nat,N2:nat]: (((((modulo_modulo @ A) @ A2) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N2))) = (((plus_plus @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (((modulo_modulo @ A) @ (((divide_divide @ A) @ A2) @ ((semiring_1_of_nat @ A) @ M))) @ ((semiring_1_of_nat @ A) @ N2)))) @ (((modulo_modulo @ A) @ A2) @ ((semiring_1_of_nat @ A) @ M)))))))))). % 6.41/2.69 thf(fact_2749_pos__int__cases, axiom, (! [K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (~ ((! [N3:nat]: (((K = ((semiring_1_of_nat @ int) @ N3)) => (~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N3)))))))))))). % 6.41/2.69 thf(fact_2750_zero__less__imp__eq__int, axiom, (! [K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (? [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) & (K = ((semiring_1_of_nat @ int) @ N3))))))))). % 6.41/2.69 thf(fact_2751_int__cases3, axiom, (! [K:int]: (((K != (zero_zero @ int)) => ((! [N3:nat]: (((K = ((semiring_1_of_nat @ int) @ N3)) => (~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N3)))))) => (~ ((! [N3:nat]: (((K = ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N3))) => (~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N3))))))))))))). % 6.41/2.69 thf(fact_2752_nat__less__real__le, axiom, ((ord_less @ nat) = (^ [N:nat,M2:nat]: ((((ord_less_eq @ real) @ (((plus_plus @ real) @ ((semiring_1_of_nat @ real) @ N)) @ (one_one @ real))) @ ((semiring_1_of_nat @ real) @ M2)))))). % 6.41/2.69 thf(fact_2753_nat__le__real__less, axiom, ((ord_less_eq @ nat) = (^ [N:nat,M2:nat]: ((((ord_less @ real) @ ((semiring_1_of_nat @ real) @ N)) @ (((plus_plus @ real) @ ((semiring_1_of_nat @ real) @ M2)) @ (one_one @ real))))))). % 6.41/2.69 thf(fact_2754_zmult__zless__mono2__lemma, axiom, (! [I:int,J:int,K:nat]: (((((ord_less @ int) @ I) @ J) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less @ int) @ (((times_times @ int) @ ((semiring_1_of_nat @ int) @ K)) @ I)) @ (((times_times @ int) @ ((semiring_1_of_nat @ int) @ K)) @ J))))))). % 6.41/2.69 thf(fact_2755_not__zle__0__negative, axiom, (! [N2:nat]: ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (suc @ N2))))))))). % 6.41/2.69 thf(fact_2756_negative__zless__0, axiom, (! [N2:nat]: ((((ord_less @ int) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (suc @ N2)))) @ (zero_zero @ int))))). % 6.41/2.69 thf(fact_2757_negD, axiom, (! [X3:int]: (((((ord_less @ int) @ X3) @ (zero_zero @ int)) => (? [N3:nat]: ((X3 = ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (suc @ N3)))))))))). % 6.41/2.69 thf(fact_2758_int__ops_I6_J, axiom, (! [A2:nat,B2:nat]: ((((((ord_less @ int) @ ((semiring_1_of_nat @ int) @ A2)) @ ((semiring_1_of_nat @ int) @ B2)) => (((semiring_1_of_nat @ int) @ (((minus_minus @ nat) @ A2) @ B2)) = (zero_zero @ int))) & ((~ ((((ord_less @ int) @ ((semiring_1_of_nat @ int) @ A2)) @ ((semiring_1_of_nat @ int) @ B2)))) => (((semiring_1_of_nat @ int) @ (((minus_minus @ nat) @ A2) @ B2)) = (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ A2)) @ ((semiring_1_of_nat @ int) @ B2)))))))). % 6.41/2.69 thf(fact_2759_nat__approx__posE, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [E3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E3) => (~ ((! [N3:nat]: ((~ ((((ord_less @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ ((semiring_1_of_nat @ A) @ (suc @ N3)))) @ E3)))))))))))))). % 6.41/2.69 thf(fact_2760_of__nat__less__two__power, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:nat]: ((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)))))))). % 6.41/2.69 thf(fact_2761_inverse__of__nat__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [N2:nat,M:nat]: (((((ord_less_eq @ nat) @ N2) @ M) => ((N2 != (zero_zero @ nat)) => (((ord_less_eq @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ ((semiring_1_of_nat @ A) @ M))) @ (((divide_divide @ A) @ (one_one @ A)) @ ((semiring_1_of_nat @ A) @ N2))))))))))). % 6.41/2.69 thf(fact_2762_real__archimedian__rdiv__eq__0, axiom, (! [X3:real,C2:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ C2) => ((! [M3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M3) => (((ord_less_eq @ real) @ (((times_times @ real) @ ((semiring_1_of_nat @ real) @ M3)) @ X3)) @ C2)))) => (X3 = (zero_zero @ real)))))))). % 6.41/2.69 thf(fact_2763_neg__int__cases, axiom, (! [K:int]: (((((ord_less @ int) @ K) @ (zero_zero @ int)) => (~ ((! [N3:nat]: (((K = ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N3))) => (~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N3)))))))))))). % 6.41/2.69 thf(fact_2764_zdiff__int__split, axiom, (! [P:(int > $o),X3:nat,Y:nat]: (((P @ ((semiring_1_of_nat @ int) @ (((minus_minus @ nat) @ X3) @ Y))) = (((((ord_less_eq @ nat) @ Y) @ X3) => (P @ (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ X3)) @ ((semiring_1_of_nat @ int) @ Y)))) & ((((ord_less @ nat) @ X3) @ Y) => (P @ (zero_zero @ int)))))))). % 6.41/2.69 thf(fact_2765_ln__realpow, axiom, (! [X3:real,N2:nat]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((ln_ln @ real) @ (((power_power @ real) @ X3) @ N2)) = (((times_times @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ ((ln_ln @ real) @ X3))))))). % 6.41/2.69 thf(fact_2766_arctan__add, axiom, (! [X3:real,Y:real]: (((((ord_less_eq @ real) @ ((abs_abs @ real) @ X3)) @ (one_one @ real)) => ((((ord_less @ real) @ ((abs_abs @ real) @ Y)) @ (one_one @ real)) => ((((plus_plus @ real) @ (arctan @ X3)) @ (arctan @ Y)) = (arctan @ (((divide_divide @ real) @ (((plus_plus @ real) @ X3) @ Y)) @ (((minus_minus @ real) @ (one_one @ real)) @ (((times_times @ real) @ X3) @ Y)))))))))). % 6.41/2.69 thf(fact_2767_double__gauss__sum, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [N2:nat]: (((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (semiring_1_of_nat @ A)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2))) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_2768_double__arith__series, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A2:A,D2:A,N2:nat]: (((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ A2) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ I3)) @ D2))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2))) = (((times_times @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (one_one @ A))) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ D2)))))))))). % 6.41/2.69 thf(fact_2769_gauss__sum, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (semiring_1_of_nat @ A)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2)) = (((divide_divide @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (one_one @ A)))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_2770_arith__series, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A2:A,D2:A,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ A2) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ I3)) @ D2))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2)) = (((divide_divide @ A) @ (((times_times @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (one_one @ A))) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ D2)))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_2771_double__gauss__sum__from__Suc__0, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [N2:nat]: (((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (semiring_1_of_nat @ A)) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ N2))) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_2772_sum__gp__offset, axiom, (! [A:$tType]: ((((division_ring @ A) & (comm_ring @ A)) => (! [X3:A,M:nat,N2:nat]: ((((X3 = (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X3)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (((plus_plus @ nat) @ M) @ N2))) = (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (one_one @ A)))) & ((X3 != (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X3)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (((plus_plus @ nat) @ M) @ N2))) = (((divide_divide @ A) @ (((times_times @ A) @ (((power_power @ A) @ X3) @ M)) @ (((minus_minus @ A) @ (one_one @ A)) @ (((power_power @ A) @ X3) @ (suc @ N2))))) @ (((minus_minus @ A) @ (one_one @ A)) @ X3))))))))))). % 6.41/2.69 thf(fact_2773_of__nat__code__if, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((semiring_1_of_nat @ A) = (^ [N:nat]: (((((if @ A) @ (N = (zero_zero @ nat))) @ (zero_zero @ A)) @ (((((product_case_prod @ nat) @ nat) @ A) @ (^ [M2:nat,Q5:nat]: (((((if @ A) @ (Q5 = (zero_zero @ nat))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ A) @ M2))) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ A) @ M2))) @ (one_one @ A)))))) @ ((divmod_nat @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))). % 6.41/2.69 thf(fact_2774_monoseq__arctan__series, axiom, (! [X3:real]: (((((ord_less_eq @ real) @ ((abs_abs @ real) @ X3)) @ (one_one @ real)) => ((topological_monoseq @ real) @ (^ [N:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ (one_one @ real)) @ ((semiring_1_of_nat @ real) @ (((plus_plus @ nat) @ (((times_times @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ nat))))) @ (((power_power @ real) @ X3) @ (((plus_plus @ nat) @ (((times_times @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ nat))))))))))). % 6.41/2.69 thf(fact_2775_lemma__termdiff3, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [H2:A,Z2:A,K5:real,N2:nat]: (((H2 != (zero_zero @ A)) => ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ Z2)) @ K5) => ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((plus_plus @ A) @ Z2) @ H2))) @ K5) => (((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ (((divide_divide @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ (((plus_plus @ A) @ Z2) @ H2)) @ N2)) @ (((power_power @ A) @ Z2) @ N2))) @ H2)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (((power_power @ A) @ Z2) @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat)))))))) @ (((times_times @ real) @ (((times_times @ real) @ (((times_times @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ ((semiring_1_of_nat @ real) @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat)))))) @ (((power_power @ real) @ K5) @ (((minus_minus @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ ((real_V7770717601297561774m_norm @ A) @ H2)))))))))))). % 6.41/2.69 thf(fact_2776_ln__series, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ ((numeral_numeral @ real) @ (bit0 @ one2))) => (((ln_ln @ real) @ X3) = ((suminf @ real) @ (^ [N:nat]: ((((times_times @ real) @ (((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ N)) @ (((divide_divide @ real) @ (one_one @ real)) @ ((semiring_1_of_nat @ real) @ (((plus_plus @ nat) @ N) @ (one_one @ nat)))))) @ (((power_power @ real) @ (((minus_minus @ real) @ X3) @ (one_one @ real))) @ (suc @ N)))))))))))). % 6.41/2.69 thf(fact_2777_lemma__termdiff2, axiom, (! [A:$tType]: (((field @ A) => (! [H2:A,Z2:A,N2:nat]: (((H2 != (zero_zero @ A)) => ((((minus_minus @ A) @ (((divide_divide @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ (((plus_plus @ A) @ Z2) @ H2)) @ N2)) @ (((power_power @ A) @ Z2) @ N2))) @ H2)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (((power_power @ A) @ Z2) @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat)))))) = (((times_times @ A) @ H2) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [P6:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [Q5:nat]: ((((times_times @ A) @ (((power_power @ A) @ (((plus_plus @ A) @ Z2) @ H2)) @ Q5)) @ (((power_power @ A) @ Z2) @ (((minus_minus @ nat) @ (((minus_minus @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ Q5)))))) @ ((set_ord_lessThan @ nat) @ (((minus_minus @ nat) @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat)))) @ P6)))))) @ ((set_ord_lessThan @ nat) @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat)))))))))))))). % 6.41/2.69 thf(fact_2778_arctan__series, axiom, (! [X3:real]: (((((ord_less_eq @ real) @ ((abs_abs @ real) @ X3)) @ (one_one @ real)) => ((arctan @ X3) = ((suminf @ real) @ (^ [K3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ K3)) @ (((times_times @ real) @ (((divide_divide @ real) @ (one_one @ real)) @ ((semiring_1_of_nat @ real) @ (((plus_plus @ nat) @ (((times_times @ nat) @ K3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ nat))))) @ (((power_power @ real) @ X3) @ (((plus_plus @ nat) @ (((times_times @ nat) @ K3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ nat))))))))))))). % 6.41/2.69 thf(fact_2779_lessThan__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_lessThan @ A) @ K)) = (((ord_less @ A) @ I) @ K)))))))). % 6.41/2.69 thf(fact_2780_finite__lessThan, axiom, (! [K:nat]: (((finite_finite2 @ nat) @ ((set_ord_lessThan @ nat) @ K))))). % 6.41/2.69 thf(fact_2781_lessThan__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_lessThan @ A) @ X3)) @ ((set_ord_lessThan @ A) @ Y)) = (((ord_less_eq @ A) @ X3) @ Y)))))))). % 6.41/2.69 thf(fact_2782_lessThan__0, axiom, (((set_ord_lessThan @ nat) @ (zero_zero @ nat)) = (bot_bot @ (set @ nat)))). % 6.41/2.69 thf(fact_2783_single__Diff__lessThan, axiom, (! [A:$tType]: (((preorder @ A) => (! [K:A]: (((((minus_minus @ (set @ A)) @ (((insert @ A) @ K) @ (bot_bot @ (set @ A)))) @ ((set_ord_lessThan @ A) @ K)) = (((insert @ A) @ K) @ (bot_bot @ (set @ A)))))))))). % 6.41/2.69 thf(fact_2784_sum_OlessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (suc @ N2))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ N2))) @ (G @ N2))))))))). % 6.41/2.69 thf(fact_2785_powser__zero, axiom, (! [A:$tType]: (((real_V2822296259951069270ebra_1 @ A) => (! [F2:(nat > A)]: ((((suminf @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ N)) @ (((power_power @ A) @ (zero_zero @ A)) @ N))))) = (F2 @ (zero_zero @ nat))))))))). % 6.41/2.69 thf(fact_2786_int__int__eq, axiom, (! [M:nat,N2:nat]: (((((semiring_1_of_nat @ int) @ M) = ((semiring_1_of_nat @ int) @ N2)) = (M = N2))))). % 6.41/2.69 thf(fact_2787_lessThan__non__empty, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X3:A]: ((((set_ord_lessThan @ A) @ X3) != (bot_bot @ (set @ A))))))))). % 6.41/2.69 thf(fact_2788_infinite__Iio, axiom, (! [A:$tType]: ((((linorder @ A) & (no_bot @ A)) => (! [A2:A]: ((~ (((finite_finite2 @ A) @ ((set_ord_lessThan @ A) @ A2)))))))))). % 6.41/2.69 thf(fact_2789_lessThan__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_ord_lessThan @ A) = (^ [U2:A]: (((collect @ A) @ (^ [X:A]: ((((ord_less @ A) @ X) @ U2))))))))))). % 6.41/2.69 thf(fact_2790_Iio__eq__empty__iff, axiom, (! [A:$tType]: ((((linorder @ A) & (order_bot @ A)) => (! [N2:A]: (((((set_ord_lessThan @ A) @ N2) = (bot_bot @ (set @ A))) = (N2 = (bot_bot @ A))))))))). % 6.41/2.69 thf(fact_2791_lessThan__strict__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [M:A,N2:A]: (((((ord_less @ (set @ A)) @ ((set_ord_lessThan @ A) @ M)) @ ((set_ord_lessThan @ A) @ N2)) = (((ord_less @ A) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_2792_lessThan__Suc, axiom, (! [K:nat]: ((((set_ord_lessThan @ nat) @ (suc @ K)) = (((insert @ nat) @ K) @ ((set_ord_lessThan @ nat) @ K)))))). % 6.41/2.69 thf(fact_2793_lessThan__empty__iff, axiom, (! [N2:nat]: (((((set_ord_lessThan @ nat) @ N2) = (bot_bot @ (set @ nat))) = (N2 = (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_2794_finite__nat__iff__bounded, axiom, ((finite_finite2 @ nat) = (^ [S7:(set @ nat)]: ((? [K3:nat]: ((((ord_less_eq @ (set @ nat)) @ S7) @ ((set_ord_lessThan @ nat) @ K3)))))))). % 6.41/2.69 thf(fact_2795_finite__nat__bounded, axiom, (! [S3:(set @ nat)]: ((((finite_finite2 @ nat) @ S3) => (? [K2:nat]: ((((ord_less_eq @ (set @ nat)) @ S3) @ ((set_ord_lessThan @ nat) @ K2)))))))). % 6.41/2.69 thf(fact_2796_sum_Onat__diff__reindex, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ N2) @ (suc @ I3)))))) @ ((set_ord_lessThan @ nat) @ N2)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ N2))))))))). % 6.41/2.69 thf(fact_2797_sum__diff__distrib, axiom, (! [A:$tType]: (((ord @ A) => (! [Q:(A > nat),P:(A > nat),N2:A]: (((! [X5:A]: ((((ord_less_eq @ nat) @ (Q @ X5)) @ (P @ X5)))) => ((((minus_minus @ nat) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ P) @ ((set_ord_lessThan @ A) @ N2))) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ Q) @ ((set_ord_lessThan @ A) @ N2))) = ((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [X:A]: ((((minus_minus @ nat) @ (P @ X)) @ (Q @ X))))) @ ((set_ord_lessThan @ A) @ N2)))))))))). % 6.41/2.69 thf(fact_2798_lemma__NBseq__def2, axiom, (! [A:$tType,B:$tType]: (((real_V822414075346904944vector @ B) => (! [X8:(A > B)]: (((? [K6:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ K6) & (! [N:A]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ B) @ (X8 @ N))) @ K6)))))) = (? [N5:nat]: ((! [N:A]: ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ B) @ (X8 @ N))) @ ((semiring_1_of_nat @ real) @ (suc @ N5)))))))))))))). % 6.41/2.69 thf(fact_2799_lemma__NBseq__def, axiom, (! [A:$tType,B:$tType]: (((real_V822414075346904944vector @ B) => (! [X8:(A > B)]: (((? [K6:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ K6) & (! [N:A]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ B) @ (X8 @ N))) @ K6)))))) = (? [N5:nat]: ((! [N:A]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ B) @ (X8 @ N))) @ ((semiring_1_of_nat @ real) @ (suc @ N5)))))))))))))). % 6.41/2.69 thf(fact_2800_sum_OlessThan__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (suc @ N2))) = (((plus_plus @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ ((set_ord_lessThan @ nat) @ N2)))))))))). % 6.41/2.69 thf(fact_2801_sum__lessThan__telescope, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [F2:(nat > A),M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [N:nat]: ((((minus_minus @ A) @ (F2 @ (suc @ N))) @ (F2 @ N))))) @ ((set_ord_lessThan @ nat) @ M)) = (((minus_minus @ A) @ (F2 @ M)) @ (F2 @ (zero_zero @ nat)))))))))). % 6.41/2.69 thf(fact_2802_sum__lessThan__telescope_H, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [F2:(nat > A),M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [N:nat]: ((((minus_minus @ A) @ (F2 @ N)) @ (F2 @ (suc @ N)))))) @ ((set_ord_lessThan @ nat) @ M)) = (((minus_minus @ A) @ (F2 @ (zero_zero @ nat))) @ (F2 @ M))))))))). % 6.41/2.69 thf(fact_2803_sum_OatLeast1__atMost__eq, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ N2)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((G @ (suc @ K3))))) @ ((set_ord_lessThan @ nat) @ N2))))))))). % 6.41/2.69 thf(fact_2804_lemma__termdiff1, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [Z2:A,H2:A,M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [P6:nat]: ((((minus_minus @ A) @ (((times_times @ A) @ (((power_power @ A) @ (((plus_plus @ A) @ Z2) @ H2)) @ (((minus_minus @ nat) @ M) @ P6))) @ (((power_power @ A) @ Z2) @ P6))) @ (((power_power @ A) @ Z2) @ M))))) @ ((set_ord_lessThan @ nat) @ M)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [P6:nat]: ((((times_times @ A) @ (((power_power @ A) @ Z2) @ P6)) @ (((minus_minus @ A) @ (((power_power @ A) @ (((plus_plus @ A) @ Z2) @ H2)) @ (((minus_minus @ nat) @ M) @ P6))) @ (((power_power @ A) @ Z2) @ (((minus_minus @ nat) @ M) @ P6))))))) @ ((set_ord_lessThan @ nat) @ M))))))))). % 6.41/2.69 thf(fact_2805_diff__power__eq__sum, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X3:A,N2:nat,Y:A]: (((((minus_minus @ A) @ (((power_power @ A) @ X3) @ (suc @ N2))) @ (((power_power @ A) @ Y) @ (suc @ N2))) = (((times_times @ A) @ (((minus_minus @ A) @ X3) @ Y)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [P6:nat]: ((((times_times @ A) @ (((power_power @ A) @ X3) @ P6)) @ (((power_power @ A) @ Y) @ (((minus_minus @ nat) @ N2) @ P6)))))) @ ((set_ord_lessThan @ nat) @ (suc @ N2))))))))))). % 6.41/2.69 thf(fact_2806_power__diff__sumr2, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X3:A,N2:nat,Y:A]: (((((minus_minus @ A) @ (((power_power @ A) @ X3) @ N2)) @ (((power_power @ A) @ Y) @ N2)) = (((times_times @ A) @ (((minus_minus @ A) @ X3) @ Y)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (((power_power @ A) @ Y) @ (((minus_minus @ nat) @ N2) @ (suc @ I3)))) @ (((power_power @ A) @ X3) @ I3))))) @ ((set_ord_lessThan @ nat) @ N2)))))))))). % 6.41/2.69 thf(fact_2807_real__sum__nat__ivl__bounded2, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N2:nat,F2:(nat > A),K5:A,K:nat]: (((! [P7:nat]: (((((ord_less @ nat) @ P7) @ N2) => (((ord_less_eq @ A) @ (F2 @ P7)) @ K5)))) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ K5) => (((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ ((set_ord_lessThan @ nat) @ (((minus_minus @ nat) @ N2) @ K)))) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ K5)))))))))). % 6.41/2.69 thf(fact_2808_one__diff__power__eq_H, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X3:A,N2:nat]: (((((minus_minus @ A) @ (one_one @ A)) @ (((power_power @ A) @ X3) @ N2)) = (((times_times @ A) @ (((minus_minus @ A) @ (one_one @ A)) @ X3)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((power_power @ A) @ X3) @ (((minus_minus @ nat) @ N2) @ (suc @ I3)))))) @ ((set_ord_lessThan @ nat) @ N2)))))))))). % 6.41/2.69 thf(fact_2809_sum__split__even__odd, axiom, (! [F2:(nat > real),G:(nat > real),N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: (((((if @ real) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3)) @ (F2 @ I3)) @ (G @ I3))))) @ ((set_ord_lessThan @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: ((F2 @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))))) @ ((set_ord_lessThan @ nat) @ N2))) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: ((G @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3)) @ (one_one @ nat)))))) @ ((set_ord_lessThan @ nat) @ N2))))))). % 6.41/2.69 thf(fact_2810_norm__le__zero__iff, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X3:A]: (((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ (zero_zero @ real)) = (X3 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2811_zero__less__norm__iff, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X3:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ ((real_V7770717601297561774m_norm @ A) @ X3)) = (X3 != (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2812_suminf__geometric, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [C2:A]: (((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ C2)) @ (one_one @ real)) => (((suminf @ A) @ ((power_power @ A) @ C2)) = (((divide_divide @ A) @ (one_one @ A)) @ (((minus_minus @ A) @ (one_one @ A)) @ C2)))))))))). % 6.41/2.69 thf(fact_2813_norm__zero, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (((real_V7770717601297561774m_norm @ A) @ (zero_zero @ A)) = (zero_zero @ real)))))). % 6.41/2.69 thf(fact_2814_norm__eq__zero, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X3:A]: (((((real_V7770717601297561774m_norm @ A) @ X3) = (zero_zero @ real)) = (X3 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2815_suminf__zero, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topological_t2_space @ A)) => (((suminf @ A) @ (^ [N:nat]: ((zero_zero @ A)))) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_2816_norm__not__less__zero, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X3:A]: ((~ ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ (zero_zero @ real)))))))))). % 6.41/2.69 thf(fact_2817_norm__uminus__minus, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X3:A,Y:A]: ((((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ ((uminus_uminus @ A) @ X3)) @ Y)) = ((real_V7770717601297561774m_norm @ A) @ (((plus_plus @ A) @ X3) @ Y))))))))). % 6.41/2.69 thf(fact_2818_nonzero__norm__divide, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [B2:A,A2:A]: (((B2 != (zero_zero @ A)) => (((real_V7770717601297561774m_norm @ A) @ (((divide_divide @ A) @ A2) @ B2)) = (((divide_divide @ real) @ ((real_V7770717601297561774m_norm @ A) @ A2)) @ ((real_V7770717601297561774m_norm @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_2819_power__eq__imp__eq__norm, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [W2:A,N2:nat,Z2:A]: ((((((power_power @ A) @ W2) @ N2) = (((power_power @ A) @ Z2) @ N2)) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((real_V7770717601297561774m_norm @ A) @ W2) = ((real_V7770717601297561774m_norm @ A) @ Z2)))))))))). % 6.41/2.69 thf(fact_2820_norm__mult__less, axiom, (! [A:$tType]: (((real_V4412858255891104859lgebra @ A) => (! [X3:A,R2:real,Y:A,S:real]: (((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ R2) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ Y)) @ S) => (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((times_times @ A) @ X3) @ Y))) @ (((times_times @ real) @ R2) @ S)))))))))). % 6.41/2.69 thf(fact_2821_norm__triangle__lt, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X3:A,Y:A,E3:real]: (((((ord_less @ real) @ (((plus_plus @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ ((real_V7770717601297561774m_norm @ A) @ Y))) @ E3) => (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((plus_plus @ A) @ X3) @ Y))) @ E3)))))))). % 6.41/2.69 thf(fact_2822_norm__add__less, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X3:A,R2:real,Y:A,S:real]: (((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ R2) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ Y)) @ S) => (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((plus_plus @ A) @ X3) @ Y))) @ (((plus_plus @ real) @ R2) @ S)))))))))). % 6.41/2.69 thf(fact_2823_norm__triangle__mono, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [A2:A,R2:real,B2:A,S:real]: (((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ A2)) @ R2) => ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ B2)) @ S) => (((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((plus_plus @ A) @ A2) @ B2))) @ (((plus_plus @ real) @ R2) @ S)))))))))). % 6.41/2.69 thf(fact_2824_norm__triangle__ineq, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X3:A,Y:A]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((plus_plus @ A) @ X3) @ Y))) @ (((plus_plus @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ ((real_V7770717601297561774m_norm @ A) @ Y))))))))). % 6.41/2.69 thf(fact_2825_norm__triangle__le, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X3:A,Y:A,E3:real]: (((((ord_less_eq @ real) @ (((plus_plus @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ ((real_V7770717601297561774m_norm @ A) @ Y))) @ E3) => (((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((plus_plus @ A) @ X3) @ Y))) @ E3)))))))). % 6.41/2.69 thf(fact_2826_norm__add__leD, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [A2:A,B2:A,C2:real]: (((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((plus_plus @ A) @ A2) @ B2))) @ C2) => (((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ B2)) @ (((plus_plus @ real) @ ((real_V7770717601297561774m_norm @ A) @ A2)) @ C2))))))))). % 6.41/2.69 thf(fact_2827_norm__diff__triangle__less, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X3:A,Y:A,E1:real,Z2:A,E22:real]: (((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ X3) @ Y))) @ E1) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ Y) @ Z2))) @ E22) => (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ X3) @ Z2))) @ (((plus_plus @ real) @ E1) @ E22)))))))))). % 6.41/2.69 thf(fact_2828_norm__diff__ineq, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ real) @ (((minus_minus @ real) @ ((real_V7770717601297561774m_norm @ A) @ A2)) @ ((real_V7770717601297561774m_norm @ A) @ B2))) @ ((real_V7770717601297561774m_norm @ A) @ (((plus_plus @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_2829_suminf__finite, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topological_t2_space @ A)) => (! [N7:(set @ nat),F2:(nat > A)]: ((((finite_finite2 @ nat) @ N7) => ((! [N3:nat]: (((~ ((((member @ nat) @ N3) @ N7))) => ((F2 @ N3) = (zero_zero @ A))))) => (((suminf @ A) @ F2) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ N7)))))))))). % 6.41/2.69 thf(fact_2830_power__eq__1__iff, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [W2:A,N2:nat]: ((((((power_power @ A) @ W2) @ N2) = (one_one @ A)) => ((((real_V7770717601297561774m_norm @ A) @ W2) = (one_one @ real)) | (N2 = (zero_zero @ nat)))))))))). % 6.41/2.69 thf(fact_2831_norm__diff__triangle__ineq, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [A2:A,B2:A,C2:A,D2:A]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (((plus_plus @ A) @ C2) @ D2)))) @ (((plus_plus @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ A2) @ C2))) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ B2) @ D2)))))))))). % 6.41/2.69 thf(fact_2832_sum__bounds__lt__plus1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [F2:(nat > A),Mm:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((F2 @ (suc @ K3))))) @ ((set_ord_lessThan @ nat) @ Mm)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ Mm))))))))). % 6.41/2.69 thf(fact_2833_sumr__cos__zero__one, axiom, (! [N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (cos_coeff @ M2)) @ (((power_power @ real) @ (zero_zero @ real)) @ M2))))) @ ((set_ord_lessThan @ nat) @ (suc @ N2))) = (one_one @ real))))). % 6.41/2.69 thf(fact_2834_pi__series, axiom, ((((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ (bit0 @ one2)))) = ((suminf @ real) @ (^ [K3:nat]: ((((divide_divide @ real) @ (((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ K3)) @ (one_one @ real))) @ ((semiring_1_of_nat @ real) @ (((plus_plus @ nat) @ (((times_times @ nat) @ K3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ nat))))))))). % 6.41/2.69 thf(fact_2835_summable__arctan__series, axiom, (! [X3:real]: (((((ord_less_eq @ real) @ ((abs_abs @ real) @ X3)) @ (one_one @ real)) => ((summable @ real) @ (^ [K3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ K3)) @ (((times_times @ real) @ (((divide_divide @ real) @ (one_one @ real)) @ ((semiring_1_of_nat @ real) @ (((plus_plus @ nat) @ (((times_times @ nat) @ K3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ nat))))) @ (((power_power @ real) @ X3) @ (((plus_plus @ nat) @ (((times_times @ nat) @ K3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ nat)))))))))))). % 6.41/2.69 thf(fact_2836_ceiling__log__nat__eq__powr__iff, axiom, (! [B2:nat,K:nat,N2:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ B2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((archimedean_ceiling @ real) @ ((log @ ((semiring_1_of_nat @ real) @ B2)) @ ((semiring_1_of_nat @ real) @ K))) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ N2)) @ (one_one @ int))) = ((((ord_less @ nat) @ (((power_power @ nat) @ B2) @ N2)) @ K) & (((ord_less_eq @ nat) @ K) @ (((power_power @ nat) @ B2) @ (((plus_plus @ nat) @ N2) @ (one_one @ nat))))))))))). % 6.41/2.69 thf(fact_2837_geometric__deriv__sums, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [Z2:A]: (((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ Z2)) @ (one_one @ real)) => (((sums @ A) @ (^ [N:nat]: ((((times_times @ A) @ ((semiring_1_of_nat @ A) @ (suc @ N))) @ (((power_power @ A) @ Z2) @ N))))) @ (((divide_divide @ A) @ (one_one @ A)) @ (((power_power @ A) @ (((minus_minus @ A) @ (one_one @ A)) @ Z2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))). % 6.41/2.69 thf(fact_2838_summable__zero, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topolo4958980785337419405_space @ A)) => ((summable @ A) @ (^ [N:nat]: ((zero_zero @ A)))))))). % 6.41/2.69 thf(fact_2839_summable__single, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topolo4958980785337419405_space @ A)) => (! [I:nat,F2:(nat > A)]: (((summable @ A) @ (^ [R5:nat]: (((((if @ A) @ (R5 = I)) @ (F2 @ R5)) @ (zero_zero @ A))))))))))). % 6.41/2.69 thf(fact_2840_sums__zero, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topolo4958980785337419405_space @ A)) => (((sums @ A) @ (^ [N:nat]: ((zero_zero @ A)))) @ (zero_zero @ A)))))). % 6.41/2.69 thf(fact_2841_summable__iff__shift, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),K:nat]: ((((summable @ A) @ (^ [N:nat]: ((F2 @ (((plus_plus @ nat) @ N) @ K))))) = ((summable @ A) @ F2)))))))). % 6.41/2.69 thf(fact_2842_cos__coeff__0, axiom, ((cos_coeff @ (zero_zero @ nat)) = (one_one @ real))). % 6.41/2.69 thf(fact_2843_summable__cmult__iff, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [C2:A,F2:(nat > A)]: ((((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ C2) @ (F2 @ N))))) = ((C2 = (zero_zero @ A)) | ((summable @ A) @ F2))))))))). % 6.41/2.69 thf(fact_2844_summable__divide__iff, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(nat > A),C2:A]: ((((summable @ A) @ (^ [N:nat]: ((((divide_divide @ A) @ (F2 @ N)) @ C2)))) = ((C2 = (zero_zero @ A)) | ((summable @ A) @ F2))))))))). % 6.41/2.69 thf(fact_2845_summable__If__finite, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topolo4958980785337419405_space @ A)) => (! [P:(nat > $o),F2:(nat > A)]: ((((finite_finite2 @ nat) @ ((collect @ nat) @ P)) => ((summable @ A) @ (^ [R5:nat]: (((((if @ A) @ (P @ R5)) @ (F2 @ R5)) @ (zero_zero @ A)))))))))))). % 6.41/2.69 thf(fact_2846_summable__If__finite__set, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topolo4958980785337419405_space @ A)) => (! [A3:(set @ nat),F2:(nat > A)]: ((((finite_finite2 @ nat) @ A3) => ((summable @ A) @ (^ [R5:nat]: (((((if @ A) @ (((member @ nat) @ R5) @ A3)) @ (F2 @ R5)) @ (zero_zero @ A)))))))))))). % 6.41/2.69 thf(fact_2847_log__eq__one, axiom, (! [A2:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ A2) => ((A2 != (one_one @ real)) => (((log @ A2) @ A2) = (one_one @ real))))))). % 6.41/2.69 thf(fact_2848_log__less__cancel__iff, axiom, (! [A2:real,X3:real,Y:real]: (((((ord_less @ real) @ (one_one @ real)) @ A2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (zero_zero @ real)) @ Y) => ((((ord_less @ real) @ ((log @ A2) @ X3)) @ ((log @ A2) @ Y)) = (((ord_less @ real) @ X3) @ Y)))))))). % 6.41/2.69 thf(fact_2849_log__less__one__cancel__iff, axiom, (! [A2:real,X3:real]: (((((ord_less @ real) @ (one_one @ real)) @ A2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ ((log @ A2) @ X3)) @ (one_one @ real)) = (((ord_less @ real) @ X3) @ A2))))))). % 6.41/2.69 thf(fact_2850_one__less__log__cancel__iff, axiom, (! [A2:real,X3:real]: (((((ord_less @ real) @ (one_one @ real)) @ A2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (one_one @ real)) @ ((log @ A2) @ X3)) = (((ord_less @ real) @ A2) @ X3))))))). % 6.41/2.69 thf(fact_2851_log__less__zero__cancel__iff, axiom, (! [A2:real,X3:real]: (((((ord_less @ real) @ (one_one @ real)) @ A2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ ((log @ A2) @ X3)) @ (zero_zero @ real)) = (((ord_less @ real) @ X3) @ (one_one @ real)))))))). % 6.41/2.69 thf(fact_2852_zero__less__log__cancel__iff, axiom, (! [A2:real,X3:real]: (((((ord_less @ real) @ (one_one @ real)) @ A2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (zero_zero @ real)) @ ((log @ A2) @ X3)) = (((ord_less @ real) @ (one_one @ real)) @ X3))))))). % 6.41/2.69 thf(fact_2853_log__le__cancel__iff, axiom, (! [A2:real,X3:real,Y:real]: (((((ord_less @ real) @ (one_one @ real)) @ A2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (zero_zero @ real)) @ Y) => ((((ord_less_eq @ real) @ ((log @ A2) @ X3)) @ ((log @ A2) @ Y)) = (((ord_less_eq @ real) @ X3) @ Y)))))))). % 6.41/2.69 thf(fact_2854_log__le__one__cancel__iff, axiom, (! [A2:real,X3:real]: (((((ord_less @ real) @ (one_one @ real)) @ A2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ ((log @ A2) @ X3)) @ (one_one @ real)) = (((ord_less_eq @ real) @ X3) @ A2))))))). % 6.41/2.69 thf(fact_2855_one__le__log__cancel__iff, axiom, (! [A2:real,X3:real]: (((((ord_less @ real) @ (one_one @ real)) @ A2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ (one_one @ real)) @ ((log @ A2) @ X3)) = (((ord_less_eq @ real) @ A2) @ X3))))))). % 6.41/2.69 thf(fact_2856_log__le__zero__cancel__iff, axiom, (! [A2:real,X3:real]: (((((ord_less @ real) @ (one_one @ real)) @ A2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ ((log @ A2) @ X3)) @ (zero_zero @ real)) = (((ord_less_eq @ real) @ X3) @ (one_one @ real)))))))). % 6.41/2.69 thf(fact_2857_zero__le__log__cancel__iff, axiom, (! [A2:real,X3:real]: (((((ord_less @ real) @ (one_one @ real)) @ A2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ ((log @ A2) @ X3)) = (((ord_less_eq @ real) @ (one_one @ real)) @ X3))))))). % 6.41/2.69 thf(fact_2858_powser__sums__zero__iff, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [A2:(nat > A),X3:A]: (((((sums @ A) @ (^ [N:nat]: ((((times_times @ A) @ (A2 @ N)) @ (((power_power @ A) @ (zero_zero @ A)) @ N))))) @ X3) = ((A2 @ (zero_zero @ nat)) = X3)))))))). % 6.41/2.69 thf(fact_2859_summable__geometric__iff, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [C2:A]: ((((summable @ A) @ ((power_power @ A) @ C2)) = (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ C2)) @ (one_one @ real))))))))). % 6.41/2.69 thf(fact_2860_log__pow__cancel, axiom, (! [A2:real,B2:nat]: (((((ord_less @ real) @ (zero_zero @ real)) @ A2) => ((A2 != (one_one @ real)) => (((log @ A2) @ (((power_power @ real) @ A2) @ B2)) = ((semiring_1_of_nat @ real) @ B2))))))). % 6.41/2.69 thf(fact_2861_sums__0, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topolo4958980785337419405_space @ A)) => (! [F2:(nat > A)]: (((! [N3:nat]: (((F2 @ N3) = (zero_zero @ A)))) => (((sums @ A) @ F2) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2862_sums__le, axiom, (! [A:$tType]: ((((ordere6911136660526730532id_add @ A) & (topolo1944317154257567458pology @ A)) => (! [F2:(nat > A),G:(nat > A),S:A,T2:A]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (F2 @ N3)) @ (G @ N3)))) => ((((sums @ A) @ F2) @ S) => ((((sums @ A) @ G) @ T2) => (((ord_less_eq @ A) @ S) @ T2)))))))))). % 6.41/2.69 thf(fact_2863_sums__single, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topolo4958980785337419405_space @ A)) => (! [I:nat,F2:(nat > A)]: ((((sums @ A) @ (^ [R5:nat]: (((((if @ A) @ (R5 = I)) @ (F2 @ R5)) @ (zero_zero @ A))))) @ (F2 @ I)))))))). % 6.41/2.69 thf(fact_2864_sums__add, axiom, (! [A:$tType]: ((((topolo5987344860129210374id_add @ A) & (topological_t2_space @ A)) => (! [F2:(nat > A),A2:A,G:(nat > A),B2:A]: (((((sums @ A) @ F2) @ A2) => ((((sums @ A) @ G) @ B2) => (((sums @ A) @ (^ [N:nat]: ((((plus_plus @ A) @ (F2 @ N)) @ (G @ N))))) @ (((plus_plus @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_2865_summable__comparison__test, axiom, (! [A:$tType]: (((real_Vector_banach @ A) => (! [F2:(nat > A),G:(nat > real)]: (((? [N8:nat]: ((! [N3:nat]: (((((ord_less_eq @ nat) @ N8) @ N3) => (((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (F2 @ N3))) @ (G @ N3))))))) => (((summable @ real) @ G) => ((summable @ A) @ F2))))))))). % 6.41/2.69 thf(fact_2866_summable__comparison__test_H, axiom, (! [A:$tType]: (((real_Vector_banach @ A) => (! [G:(nat > real),N7:nat,F2:(nat > A)]: ((((summable @ real) @ G) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ N7) @ N3) => (((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (F2 @ N3))) @ (G @ N3))))) => ((summable @ A) @ F2))))))))). % 6.41/2.69 thf(fact_2867_summable__const__iff, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [C2:A]: ((((summable @ A) @ (^ [Uu3:nat]: (C2))) = (C2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2868_summable__add, axiom, (! [A:$tType]: ((((topolo5987344860129210374id_add @ A) & (topological_t2_space @ A)) => (! [F2:(nat > A),G:(nat > A)]: ((((summable @ A) @ F2) => (((summable @ A) @ G) => ((summable @ A) @ (^ [N:nat]: ((((plus_plus @ A) @ (F2 @ N)) @ (G @ N))))))))))))). % 6.41/2.69 thf(fact_2869_summable__Suc__iff, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A)]: ((((summable @ A) @ (^ [N:nat]: ((F2 @ (suc @ N))))) = ((summable @ A) @ F2)))))))). % 6.41/2.69 thf(fact_2870_summable__ignore__initial__segment, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),K:nat]: ((((summable @ A) @ F2) => ((summable @ A) @ (^ [N:nat]: ((F2 @ (((plus_plus @ nat) @ N) @ K)))))))))))). % 6.41/2.69 thf(fact_2871_powser__insidea, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [F2:(nat > A),X3:A,Z2:A]: ((((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ N)) @ (((power_power @ A) @ X3) @ N))))) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ Z2)) @ ((real_V7770717601297561774m_norm @ A) @ X3)) => ((summable @ real) @ (^ [N:nat]: (((real_V7770717601297561774m_norm @ A) @ (((times_times @ A) @ (F2 @ N)) @ (((power_power @ A) @ Z2) @ N)))))))))))))). % 6.41/2.69 thf(fact_2872_suminf__le, axiom, (! [A:$tType]: ((((ordere6911136660526730532id_add @ A) & (topolo1944317154257567458pology @ A)) => (! [F2:(nat > A),G:(nat > A)]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (F2 @ N3)) @ (G @ N3)))) => (((summable @ A) @ F2) => (((summable @ A) @ G) => (((ord_less_eq @ A) @ ((suminf @ A) @ F2)) @ ((suminf @ A) @ G))))))))))). % 6.41/2.69 thf(fact_2873_summable__finite, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topolo4958980785337419405_space @ A)) => (! [N7:(set @ nat),F2:(nat > A)]: ((((finite_finite2 @ nat) @ N7) => ((! [N3:nat]: (((~ ((((member @ nat) @ N3) @ N7))) => ((F2 @ N3) = (zero_zero @ A))))) => ((summable @ A) @ F2))))))))). % 6.41/2.69 thf(fact_2874_sums__mult__iff, axiom, (! [A:$tType]: ((((field @ A) & (real_V4412858255891104859lgebra @ A)) => (! [C2:A,F2:(nat > A),D2:A]: (((C2 != (zero_zero @ A)) => ((((sums @ A) @ (^ [N:nat]: ((((times_times @ A) @ C2) @ (F2 @ N))))) @ (((times_times @ A) @ C2) @ D2)) = (((sums @ A) @ F2) @ D2))))))))). % 6.41/2.69 thf(fact_2875_sums__mult2__iff, axiom, (! [A:$tType]: ((((field @ A) & (real_V4412858255891104859lgebra @ A)) => (! [C2:A,F2:(nat > A),D2:A]: (((C2 != (zero_zero @ A)) => ((((sums @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ N)) @ C2)))) @ (((times_times @ A) @ D2) @ C2)) = (((sums @ A) @ F2) @ D2))))))))). % 6.41/2.69 thf(fact_2876_summable__mult__D, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [C2:A,F2:(nat > A)]: ((((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ C2) @ (F2 @ N))))) => ((C2 != (zero_zero @ A)) => ((summable @ A) @ F2))))))))). % 6.41/2.69 thf(fact_2877_summable__zero__power, axiom, (! [A:$tType]: ((((comm_ring_1 @ A) & (topolo4958980785337419405_space @ A)) => ((summable @ A) @ ((power_power @ A) @ (zero_zero @ A))))))). % 6.41/2.69 thf(fact_2878_pi__not__less__zero, axiom, (~ ((((ord_less @ real) @ pi) @ (zero_zero @ real))))). % 6.41/2.69 thf(fact_2879_pi__gt__zero, axiom, (((ord_less @ real) @ (zero_zero @ real)) @ pi)). % 6.41/2.69 thf(fact_2880_suminf__add, axiom, (! [A:$tType]: ((((topolo5987344860129210374id_add @ A) & (topological_t2_space @ A)) => (! [F2:(nat > A),G:(nat > A)]: ((((summable @ A) @ F2) => (((summable @ A) @ G) => ((((plus_plus @ A) @ ((suminf @ A) @ F2)) @ ((suminf @ A) @ G)) = ((suminf @ A) @ (^ [N:nat]: ((((plus_plus @ A) @ (F2 @ N)) @ (G @ N)))))))))))))). % 6.41/2.69 thf(fact_2881_suminf__eq__zero__iff, axiom, (! [A:$tType]: ((((ordere6911136660526730532id_add @ A) & (topolo1944317154257567458pology @ A)) => (! [F2:(nat > A)]: ((((summable @ A) @ F2) => ((! [N3:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ N3)))) => ((((suminf @ A) @ F2) = (zero_zero @ A)) = (! [N:nat]: (((F2 @ N) = (zero_zero @ A))))))))))))). % 6.41/2.69 thf(fact_2882_suminf__nonneg, axiom, (! [A:$tType]: ((((ordere6911136660526730532id_add @ A) & (topolo1944317154257567458pology @ A)) => (! [F2:(nat > A)]: ((((summable @ A) @ F2) => ((! [N3:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ N3)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((suminf @ A) @ F2)))))))))). % 6.41/2.69 thf(fact_2883_suminf__pos, axiom, (! [A:$tType]: ((((ordere6911136660526730532id_add @ A) & (topolo1944317154257567458pology @ A)) => (! [F2:(nat > A)]: ((((summable @ A) @ F2) => ((! [N3:nat]: ((((ord_less @ A) @ (zero_zero @ A)) @ (F2 @ N3)))) => (((ord_less @ A) @ (zero_zero @ A)) @ ((suminf @ A) @ F2)))))))))). % 6.41/2.69 thf(fact_2884_sums__mult__D, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [C2:A,F2:(nat > A),A2:A]: (((((sums @ A) @ (^ [N:nat]: ((((times_times @ A) @ C2) @ (F2 @ N))))) @ A2) => ((C2 != (zero_zero @ A)) => (((sums @ A) @ F2) @ (((divide_divide @ A) @ A2) @ C2)))))))))). % 6.41/2.69 thf(fact_2885_sums__Suc__imp, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),S:A]: ((((F2 @ (zero_zero @ nat)) = (zero_zero @ A)) => ((((sums @ A) @ (^ [N:nat]: ((F2 @ (suc @ N))))) @ S) => (((sums @ A) @ F2) @ S))))))))). % 6.41/2.69 thf(fact_2886_sums__Suc, axiom, (! [A:$tType]: ((((topolo5987344860129210374id_add @ A) & (topological_t2_space @ A)) => (! [F2:(nat > A),L:A]: (((((sums @ A) @ (^ [N:nat]: ((F2 @ (suc @ N))))) @ L) => (((sums @ A) @ F2) @ (((plus_plus @ A) @ L) @ (F2 @ (zero_zero @ nat))))))))))). % 6.41/2.69 thf(fact_2887_sums__Suc__iff, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),S:A]: (((((sums @ A) @ (^ [N:nat]: ((F2 @ (suc @ N))))) @ S) = (((sums @ A) @ F2) @ (((plus_plus @ A) @ S) @ (F2 @ (zero_zero @ nat))))))))))). % 6.41/2.69 thf(fact_2888_sums__zero__iff__shift, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [N2:nat,F2:(nat > A),S:A]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ N2) => ((F2 @ I4) = (zero_zero @ A))))) => ((((sums @ A) @ (^ [I3:nat]: ((F2 @ (((plus_plus @ nat) @ I3) @ N2))))) @ S) = (((sums @ A) @ F2) @ S))))))))). % 6.41/2.69 thf(fact_2889_summable__0__powser, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [F2:(nat > A)]: (((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ N)) @ (((power_power @ A) @ (zero_zero @ A)) @ N))))))))))). % 6.41/2.69 thf(fact_2890_summable__zero__power_H, axiom, (! [A:$tType]: ((((ring_1 @ A) & (topolo4958980785337419405_space @ A)) => (! [F2:(nat > A)]: (((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ N)) @ (((power_power @ A) @ (zero_zero @ A)) @ N))))))))))). % 6.41/2.69 thf(fact_2891_summable__powser__split__head, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [F2:(nat > A),Z2:A]: ((((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ (suc @ N))) @ (((power_power @ A) @ Z2) @ N))))) = ((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ N)) @ (((power_power @ A) @ Z2) @ N)))))))))))). % 6.41/2.69 thf(fact_2892_powser__split__head_I3_J, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V8999393235501362500lgebra @ A)) => (! [F2:(nat > A),Z2:A]: ((((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ N)) @ (((power_power @ A) @ Z2) @ N))))) => ((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ (suc @ N))) @ (((power_power @ A) @ Z2) @ N)))))))))))). % 6.41/2.69 thf(fact_2893_summable__powser__ignore__initial__segment, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [F2:(nat > A),M:nat,Z2:A]: ((((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ (((plus_plus @ nat) @ N) @ M))) @ (((power_power @ A) @ Z2) @ N))))) = ((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ N)) @ (((power_power @ A) @ Z2) @ N)))))))))))). % 6.41/2.69 thf(fact_2894_sums__finite, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topolo4958980785337419405_space @ A)) => (! [N7:(set @ nat),F2:(nat > A)]: ((((finite_finite2 @ nat) @ N7) => ((! [N3:nat]: (((~ ((((member @ nat) @ N3) @ N7))) => ((F2 @ N3) = (zero_zero @ A))))) => (((sums @ A) @ F2) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ N7)))))))))). % 6.41/2.69 thf(fact_2895_sums__If__finite, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topolo4958980785337419405_space @ A)) => (! [P:(nat > $o),F2:(nat > A)]: ((((finite_finite2 @ nat) @ ((collect @ nat) @ P)) => (((sums @ A) @ (^ [R5:nat]: (((((if @ A) @ (P @ R5)) @ (F2 @ R5)) @ (zero_zero @ A))))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ ((collect @ nat) @ P)))))))))). % 6.41/2.69 thf(fact_2896_sums__If__finite__set, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topolo4958980785337419405_space @ A)) => (! [A3:(set @ nat),F2:(nat > A)]: ((((finite_finite2 @ nat) @ A3) => (((sums @ A) @ (^ [R5:nat]: (((((if @ A) @ (((member @ nat) @ R5) @ A3)) @ (F2 @ R5)) @ (zero_zero @ A))))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ A3))))))))). % 6.41/2.69 thf(fact_2897_summable__norm__comparison__test, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),G:(nat > real)]: (((? [N8:nat]: ((! [N3:nat]: (((((ord_less_eq @ nat) @ N8) @ N3) => (((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (F2 @ N3))) @ (G @ N3))))))) => (((summable @ real) @ G) => ((summable @ real) @ (^ [N:nat]: (((real_V7770717601297561774m_norm @ A) @ (F2 @ N))))))))))))). % 6.41/2.69 thf(fact_2898_summable__rabs__comparison__test, axiom, (! [F2:(nat > real),G:(nat > real)]: (((? [N8:nat]: ((! [N3:nat]: (((((ord_less_eq @ nat) @ N8) @ N3) => (((ord_less_eq @ real) @ ((abs_abs @ real) @ (F2 @ N3))) @ (G @ N3))))))) => (((summable @ real) @ G) => ((summable @ real) @ (^ [N:nat]: (((abs_abs @ real) @ (F2 @ N)))))))))). % 6.41/2.69 thf(fact_2899_suminf__pos__iff, axiom, (! [A:$tType]: ((((ordere6911136660526730532id_add @ A) & (topolo1944317154257567458pology @ A)) => (! [F2:(nat > A)]: ((((summable @ A) @ F2) => ((! [N3:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ N3)))) => ((((ord_less @ A) @ (zero_zero @ A)) @ ((suminf @ A) @ F2)) = (? [I3:nat]: ((((ord_less @ A) @ (zero_zero @ A)) @ (F2 @ I3))))))))))))). % 6.41/2.69 thf(fact_2900_suminf__pos2, axiom, (! [A:$tType]: ((((ordere6911136660526730532id_add @ A) & (topolo1944317154257567458pology @ A)) => (! [F2:(nat > A),I:nat]: ((((summable @ A) @ F2) => ((! [N3:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ N3)))) => ((((ord_less @ A) @ (zero_zero @ A)) @ (F2 @ I)) => (((ord_less @ A) @ (zero_zero @ A)) @ ((suminf @ A) @ F2))))))))))). % 6.41/2.69 thf(fact_2901_suminf__le__const, axiom, (! [A:$tType]: ((((ordere6911136660526730532id_add @ A) & (topolo1944317154257567458pology @ A)) => (! [F2:(nat > A),X3:A]: ((((summable @ A) @ F2) => ((! [N3:nat]: ((((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ ((set_ord_lessThan @ nat) @ N3))) @ X3))) => (((ord_less_eq @ A) @ ((suminf @ A) @ F2)) @ X3))))))))). % 6.41/2.69 thf(fact_2902_powser__sums__if, axiom, (! [A:$tType]: ((((ring_1 @ A) & (topolo4958980785337419405_space @ A)) => (! [M:nat,Z2:A]: ((((sums @ A) @ (^ [N:nat]: ((((times_times @ A) @ ((((if @ A) @ (N = M)) @ (one_one @ A)) @ (zero_zero @ A))) @ (((power_power @ A) @ Z2) @ N))))) @ (((power_power @ A) @ Z2) @ M)))))))). % 6.41/2.69 thf(fact_2903_powser__sums__zero, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [A2:(nat > A)]: ((((sums @ A) @ (^ [N:nat]: ((((times_times @ A) @ (A2 @ N)) @ (((power_power @ A) @ (zero_zero @ A)) @ N))))) @ (A2 @ (zero_zero @ nat))))))))). % 6.41/2.69 thf(fact_2904_log__base__change, axiom, (! [A2:real,B2:real,X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ A2) => ((A2 != (one_one @ real)) => (((log @ B2) @ X3) = (((divide_divide @ real) @ ((log @ A2) @ X3)) @ ((log @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_2905_log__of__power__eq, axiom, (! [M:nat,B2:real,N2:nat]: (((((semiring_1_of_nat @ real) @ M) = (((power_power @ real) @ B2) @ N2)) => ((((ord_less @ real) @ (one_one @ real)) @ B2) => (((semiring_1_of_nat @ real) @ N2) = ((log @ B2) @ ((semiring_1_of_nat @ real) @ M)))))))). % 6.41/2.69 thf(fact_2906_less__log__of__power, axiom, (! [B2:real,N2:nat,M:real]: (((((ord_less @ real) @ (((power_power @ real) @ B2) @ N2)) @ M) => ((((ord_less @ real) @ (one_one @ real)) @ B2) => (((ord_less @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ ((log @ B2) @ M))))))). % 6.41/2.69 thf(fact_2907_powser__inside, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V8999393235501362500lgebra @ A)) => (! [F2:(nat > A),X3:A,Z2:A]: ((((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ N)) @ (((power_power @ A) @ X3) @ N))))) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ Z2)) @ ((real_V7770717601297561774m_norm @ A) @ X3)) => ((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ N)) @ (((power_power @ A) @ Z2) @ N))))))))))))). % 6.41/2.69 thf(fact_2908_sums__iff__shift, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),N2:nat,S:A]: (((((sums @ A) @ (^ [I3:nat]: ((F2 @ (((plus_plus @ nat) @ I3) @ N2))))) @ S) = (((sums @ A) @ F2) @ (((plus_plus @ A) @ S) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ ((set_ord_lessThan @ nat) @ N2))))))))))). % 6.41/2.69 thf(fact_2909_summableI__nonneg__bounded, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & ((ordere6911136660526730532id_add @ A) & (topolo1944317154257567458pology @ A))) => (! [F2:(nat > A),X3:A]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ N3)))) => ((! [N3:nat]: ((((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ ((set_ord_lessThan @ nat) @ N3))) @ X3))) => ((summable @ A) @ F2))))))))). % 6.41/2.69 thf(fact_2910_sums__split__initial__segment, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),S:A,N2:nat]: (((((sums @ A) @ F2) @ S) => (((sums @ A) @ (^ [I3:nat]: ((F2 @ (((plus_plus @ nat) @ I3) @ N2))))) @ (((minus_minus @ A) @ S) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ ((set_ord_lessThan @ nat) @ N2))))))))))). % 6.41/2.69 thf(fact_2911_sums__iff__shift_H, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),N2:nat,S:A]: (((((sums @ A) @ (^ [I3:nat]: ((F2 @ (((plus_plus @ nat) @ I3) @ N2))))) @ (((minus_minus @ A) @ S) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ ((set_ord_lessThan @ nat) @ N2)))) = (((sums @ A) @ F2) @ S)))))))). % 6.41/2.69 thf(fact_2912_complete__algebra__summable__geometric, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => (! [X3:A]: (((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ (one_one @ real)) => ((summable @ A) @ ((power_power @ A) @ X3))))))))). % 6.41/2.69 thf(fact_2913_summable__geometric, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [C2:A]: (((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ C2)) @ (one_one @ real)) => ((summable @ A) @ ((power_power @ A) @ C2))))))))). % 6.41/2.69 thf(fact_2914_sums__If__finite__set_H, axiom, (! [A:$tType]: ((((topolo1287966508704411220up_add @ A) & (topological_t2_space @ A)) => (! [G:(nat > A),S3:A,A3:(set @ nat),S6:A,F2:(nat > A)]: (((((sums @ A) @ G) @ S3) => (((finite_finite2 @ nat) @ A3) => ((S6 = (((plus_plus @ A) @ S3) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [N:nat]: ((((minus_minus @ A) @ (F2 @ N)) @ (G @ N))))) @ A3))) => (((sums @ A) @ (^ [N:nat]: (((((if @ A) @ (((member @ nat) @ N) @ A3)) @ (F2 @ N)) @ (G @ N))))) @ S6)))))))))). % 6.41/2.69 thf(fact_2915_suminf__split__head, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A)]: ((((summable @ A) @ F2) => (((suminf @ A) @ (^ [N:nat]: ((F2 @ (suc @ N))))) = (((minus_minus @ A) @ ((suminf @ A) @ F2)) @ (F2 @ (zero_zero @ nat))))))))))). % 6.41/2.69 thf(fact_2916_pi__less__4, axiom, (((ord_less @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ (bit0 @ one2))))). % 6.41/2.69 thf(fact_2917_sum__le__suminf, axiom, (! [A:$tType]: ((((ordere6911136660526730532id_add @ A) & (topolo1944317154257567458pology @ A)) => (! [F2:(nat > A),I6:(set @ nat)]: ((((summable @ A) @ F2) => (((finite_finite2 @ nat) @ I6) => ((! [N3:nat]: (((((member @ nat) @ N3) @ ((uminus_uminus @ (set @ nat)) @ I6)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ N3))))) => (((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ I6)) @ ((suminf @ A) @ F2))))))))))). % 6.41/2.69 thf(fact_2918_log__mult, axiom, (! [A2:real,X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ A2) => ((A2 != (one_one @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (zero_zero @ real)) @ Y) => (((log @ A2) @ (((times_times @ real) @ X3) @ Y)) = (((plus_plus @ real) @ ((log @ A2) @ X3)) @ ((log @ A2) @ Y)))))))))). % 6.41/2.69 thf(fact_2919_log__divide, axiom, (! [A2:real,X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ A2) => ((A2 != (one_one @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (zero_zero @ real)) @ Y) => (((log @ A2) @ (((divide_divide @ real) @ X3) @ Y)) = (((minus_minus @ real) @ ((log @ A2) @ X3)) @ ((log @ A2) @ Y)))))))))). % 6.41/2.69 thf(fact_2920_le__log__of__power, axiom, (! [B2:real,N2:nat,M:real]: (((((ord_less_eq @ real) @ (((power_power @ real) @ B2) @ N2)) @ M) => ((((ord_less @ real) @ (one_one @ real)) @ B2) => (((ord_less_eq @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ ((log @ B2) @ M))))))). % 6.41/2.69 thf(fact_2921_log__base__pow, axiom, (! [A2:real,N2:nat,X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ A2) => (((log @ (((power_power @ real) @ A2) @ N2)) @ X3) = (((divide_divide @ real) @ ((log @ A2) @ X3)) @ ((semiring_1_of_nat @ real) @ N2))))))). % 6.41/2.69 thf(fact_2922_log__nat__power, axiom, (! [X3:real,B2:real,N2:nat]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((log @ B2) @ (((power_power @ real) @ X3) @ N2)) = (((times_times @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ ((log @ B2) @ X3))))))). % 6.41/2.69 thf(fact_2923_suminf__split__initial__segment, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),K:nat]: ((((summable @ A) @ F2) => (((suminf @ A) @ F2) = (((plus_plus @ A) @ ((suminf @ A) @ (^ [N:nat]: ((F2 @ (((plus_plus @ nat) @ N) @ K)))))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ ((set_ord_lessThan @ nat) @ K))))))))))). % 6.41/2.69 thf(fact_2924_suminf__minus__initial__segment, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),K:nat]: ((((summable @ A) @ F2) => (((suminf @ A) @ (^ [N:nat]: ((F2 @ (((plus_plus @ nat) @ N) @ K))))) = (((minus_minus @ A) @ ((suminf @ A) @ F2)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ ((set_ord_lessThan @ nat) @ K))))))))))). % 6.41/2.69 thf(fact_2925_pi__half__less__two, axiom, (((ord_less @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) @ ((numeral_numeral @ real) @ (bit0 @ one2)))). % 6.41/2.69 thf(fact_2926_sum__less__suminf, axiom, (! [A:$tType]: ((((ordere8940638589300402666id_add @ A) & (topolo1944317154257567458pology @ A)) => (! [F2:(nat > A),N2:nat]: ((((summable @ A) @ F2) => ((! [M3:nat]: (((((ord_less_eq @ nat) @ N2) @ M3) => (((ord_less @ A) @ (zero_zero @ A)) @ (F2 @ M3))))) => (((ord_less @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ ((set_ord_lessThan @ nat) @ N2))) @ ((suminf @ A) @ F2)))))))))). % 6.41/2.69 thf(fact_2927_powser__split__head_I1_J, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V8999393235501362500lgebra @ A)) => (! [F2:(nat > A),Z2:A]: ((((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ N)) @ (((power_power @ A) @ Z2) @ N))))) => (((suminf @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ N)) @ (((power_power @ A) @ Z2) @ N))))) = (((plus_plus @ A) @ (F2 @ (zero_zero @ nat))) @ (((times_times @ A) @ ((suminf @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ (suc @ N))) @ (((power_power @ A) @ Z2) @ N)))))) @ Z2)))))))))). % 6.41/2.69 thf(fact_2928_powser__split__head_I2_J, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V8999393235501362500lgebra @ A)) => (! [F2:(nat > A),Z2:A]: ((((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ N)) @ (((power_power @ A) @ Z2) @ N))))) => ((((times_times @ A) @ ((suminf @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ (suc @ N))) @ (((power_power @ A) @ Z2) @ N)))))) @ Z2) = (((minus_minus @ A) @ ((suminf @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F2 @ N)) @ (((power_power @ A) @ Z2) @ N)))))) @ (F2 @ (zero_zero @ nat))))))))))). % 6.41/2.69 thf(fact_2929_log__of__power__less, axiom, (! [M:nat,B2:real,N2:nat]: (((((ord_less @ real) @ ((semiring_1_of_nat @ real) @ M)) @ (((power_power @ real) @ B2) @ N2)) => ((((ord_less @ real) @ (one_one @ real)) @ B2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((ord_less @ real) @ ((log @ B2) @ ((semiring_1_of_nat @ real) @ M))) @ ((semiring_1_of_nat @ real) @ N2)))))))). % 6.41/2.69 thf(fact_2930_log__eq__div__ln__mult__log, axiom, (! [A2:real,B2:real,X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ A2) => ((A2 != (one_one @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ B2) => ((B2 != (one_one @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((log @ A2) @ X3) = (((times_times @ real) @ (((divide_divide @ real) @ ((ln_ln @ real) @ B2)) @ ((ln_ln @ real) @ A2))) @ ((log @ B2) @ X3))))))))))). % 6.41/2.69 thf(fact_2931_summable__partial__sum__bound, axiom, (! [A:$tType]: (((real_Vector_banach @ A) => (! [F2:(nat > A),E3:real]: ((((summable @ A) @ F2) => ((((ord_less @ real) @ (zero_zero @ real)) @ E3) => (~ ((! [N9:nat]: ((~ ((! [M6:nat]: (((((ord_less_eq @ nat) @ N9) @ M6) => (! [N6:nat]: ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ (((set_or1337092689740270186AtMost @ nat) @ M6) @ N6)))) @ E3)))))))))))))))))))). % 6.41/2.69 thf(fact_2932_suminf__exist__split, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [R2:real,F2:(nat > A)]: (((((ord_less @ real) @ (zero_zero @ real)) @ R2) => (((summable @ A) @ F2) => (? [N9:nat]: ((! [N6:nat]: (((((ord_less_eq @ nat) @ N9) @ N6) => (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ ((suminf @ A) @ (^ [I3:nat]: ((F2 @ (((plus_plus @ nat) @ I3) @ N6))))))) @ R2)))))))))))))). % 6.41/2.69 thf(fact_2933_summable__power__series, axiom, (! [F2:(nat > real),Z2:real]: (((! [I4:nat]: ((((ord_less_eq @ real) @ (F2 @ I4)) @ (one_one @ real)))) => ((! [I4:nat]: ((((ord_less_eq @ real) @ (zero_zero @ real)) @ (F2 @ I4)))) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ Z2) => ((((ord_less @ real) @ Z2) @ (one_one @ real)) => ((summable @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (F2 @ I3)) @ (((power_power @ real) @ Z2) @ I3)))))))))))). % 6.41/2.69 thf(fact_2934_Abel__lemma, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [R2:real,R0:real,A2:(nat > A),M7:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ R2) => ((((ord_less @ real) @ R2) @ R0) => ((! [N3:nat]: ((((ord_less_eq @ real) @ (((times_times @ real) @ ((real_V7770717601297561774m_norm @ A) @ (A2 @ N3))) @ (((power_power @ real) @ R0) @ N3))) @ M7))) => ((summable @ real) @ (^ [N:nat]: ((((times_times @ real) @ ((real_V7770717601297561774m_norm @ A) @ (A2 @ N))) @ (((power_power @ real) @ R2) @ N)))))))))))))). % 6.41/2.69 thf(fact_2935_pi__half__gt__zero, axiom, (((ord_less @ real) @ (zero_zero @ real)) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))). % 6.41/2.69 thf(fact_2936_m2pi__less__pi, axiom, (((ord_less @ real) @ ((uminus_uminus @ real) @ (((times_times @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ pi))) @ pi)). % 6.41/2.69 thf(fact_2937_summable__ratio__test, axiom, (! [A:$tType]: (((real_Vector_banach @ A) => (! [C2:real,N7:nat,F2:(nat > A)]: (((((ord_less @ real) @ C2) @ (one_one @ real)) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ N7) @ N3) => (((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (F2 @ (suc @ N3)))) @ (((times_times @ real) @ C2) @ ((real_V7770717601297561774m_norm @ A) @ (F2 @ N3))))))) => ((summable @ A) @ F2))))))))). % 6.41/2.69 thf(fact_2938_sum__less__suminf2, axiom, (! [A:$tType]: ((((ordere8940638589300402666id_add @ A) & (topolo1944317154257567458pology @ A)) => (! [F2:(nat > A),N2:nat,I:nat]: ((((summable @ A) @ F2) => ((! [M3:nat]: (((((ord_less_eq @ nat) @ N2) @ M3) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ M3))))) => ((((ord_less_eq @ nat) @ N2) @ I) => ((((ord_less @ A) @ (zero_zero @ A)) @ (F2 @ I)) => (((ord_less @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ ((set_ord_lessThan @ nat) @ N2))) @ ((suminf @ A) @ F2)))))))))))). % 6.41/2.69 thf(fact_2939_arctan__ubound, axiom, (! [Y:real]: ((((ord_less @ real) @ (arctan @ Y)) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))))). % 6.41/2.69 thf(fact_2940_geometric__sums, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [C2:A]: (((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ C2)) @ (one_one @ real)) => (((sums @ A) @ ((power_power @ A) @ C2)) @ (((divide_divide @ A) @ (one_one @ A)) @ (((minus_minus @ A) @ (one_one @ A)) @ C2)))))))))). % 6.41/2.69 thf(fact_2941_power__half__series, axiom, (((sums @ real) @ (^ [N:nat]: ((((power_power @ real) @ (((divide_divide @ real) @ (one_one @ real)) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) @ (suc @ N))))) @ (one_one @ real))). % 6.41/2.69 thf(fact_2942_log__of__power__le, axiom, (! [M:nat,B2:real,N2:nat]: (((((ord_less_eq @ real) @ ((semiring_1_of_nat @ real) @ M)) @ (((power_power @ real) @ B2) @ N2)) => ((((ord_less @ real) @ (one_one @ real)) @ B2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((ord_less_eq @ real) @ ((log @ B2) @ ((semiring_1_of_nat @ real) @ M))) @ ((semiring_1_of_nat @ real) @ N2)))))))). % 6.41/2.69 thf(fact_2943_minus__pi__half__less__zero, axiom, (((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ (zero_zero @ real))). % 6.41/2.69 thf(fact_2944_arctan__lbound, axiom, (! [Y:real]: ((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ (arctan @ Y))))). % 6.41/2.69 thf(fact_2945_arctan__bounded, axiom, (! [Y:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ (arctan @ Y)) & (((ord_less @ real) @ (arctan @ Y)) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))))))). % 6.41/2.69 thf(fact_2946_less__log2__of__power, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) @ M) => (((ord_less @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ real) @ M))))))). % 6.41/2.69 thf(fact_2947_le__log2__of__power, axiom, (! [N2:nat,M:nat]: (((((ord_less_eq @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) @ M) => (((ord_less_eq @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ real) @ M))))))). % 6.41/2.69 thf(fact_2948_log2__of__power__less, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((ord_less @ real) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ real) @ M))) @ ((semiring_1_of_nat @ real) @ N2))))))). % 6.41/2.69 thf(fact_2949_log2__of__power__le, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => (((ord_less_eq @ real) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ real) @ M))) @ ((semiring_1_of_nat @ real) @ N2))))))). % 6.41/2.69 thf(fact_2950_sum__pos__lt__pair, axiom, (! [F2:(nat > real),K:nat]: ((((summable @ real) @ F2) => ((! [D3:nat]: ((((ord_less @ real) @ (zero_zero @ real)) @ (((plus_plus @ real) @ (F2 @ (((plus_plus @ nat) @ K) @ (((times_times @ nat) @ (suc @ (suc @ (zero_zero @ nat)))) @ D3)))) @ (F2 @ (((plus_plus @ nat) @ K) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (suc @ (suc @ (zero_zero @ nat)))) @ D3)) @ (one_one @ nat)))))))) => (((ord_less @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ F2) @ ((set_ord_lessThan @ nat) @ K))) @ ((suminf @ real) @ F2))))))). % 6.41/2.69 thf(fact_2951_ceiling__log__nat__eq__if, axiom, (! [B2:nat,N2:nat,K:nat]: (((((ord_less @ nat) @ (((power_power @ nat) @ B2) @ N2)) @ K) => ((((ord_less_eq @ nat) @ K) @ (((power_power @ nat) @ B2) @ (((plus_plus @ nat) @ N2) @ (one_one @ nat)))) => ((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ B2) => (((archimedean_ceiling @ real) @ ((log @ ((semiring_1_of_nat @ real) @ B2)) @ ((semiring_1_of_nat @ real) @ K))) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ N2)) @ (one_one @ int))))))))). % 6.41/2.69 thf(fact_2952_ceiling__log2__div2, axiom, (! [N2:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2) => (((archimedean_ceiling @ real) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ real) @ N2))) = (((plus_plus @ int) @ ((archimedean_ceiling @ real) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ real) @ (((plus_plus @ nat) @ (((divide_divide @ nat) @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ nat)))))) @ (one_one @ int))))))). % 6.41/2.69 thf(fact_2953_sin__cos__npi, axiom, (! [N2:nat]: ((((sin @ real) @ (((divide_divide @ real) @ (((times_times @ real) @ ((semiring_1_of_nat @ real) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)))) @ pi)) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) = (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ N2))))). % 6.41/2.69 thf(fact_2954_diffs__equiv, axiom, (! [A:$tType]: ((((real_V822414075346904944vector @ A) & (ring_1 @ A)) => (! [C2:(nat > A),X3:A]: ((((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (((diffs @ A) @ C2) @ N)) @ (((power_power @ A) @ X3) @ N))))) => (((sums @ A) @ (^ [N:nat]: ((((times_times @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (C2 @ N))) @ (((power_power @ A) @ X3) @ (((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat)))))))) @ ((suminf @ A) @ (^ [N:nat]: ((((times_times @ A) @ (((diffs @ A) @ C2) @ N)) @ (((power_power @ A) @ X3) @ N))))))))))))). % 6.41/2.69 thf(fact_2955_cos__pi__eq__zero, axiom, (! [M:nat]: ((((cos @ real) @ (((divide_divide @ real) @ (((times_times @ real) @ pi) @ ((semiring_1_of_nat @ real) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M))))) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) = (zero_zero @ real))))). % 6.41/2.69 thf(fact_2956_ceiling__log__eq__powr__iff, axiom, (! [X3:real,B2:real,K:nat]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (one_one @ real)) @ B2) => ((((archimedean_ceiling @ real) @ ((log @ B2) @ X3)) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ K)) @ (one_one @ int))) = ((((ord_less @ real) @ (((powr @ real) @ B2) @ ((semiring_1_of_nat @ real) @ K))) @ X3) & (((ord_less_eq @ real) @ X3) @ (((powr @ real) @ B2) @ ((semiring_1_of_nat @ real) @ (((plus_plus @ nat) @ K) @ (one_one @ nat)))))))))))). % 6.41/2.69 thf(fact_2957_floor__log__nat__eq__powr__iff, axiom, (! [B2:nat,K:nat,N2:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ B2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((archim6421214686448440834_floor @ real) @ ((log @ ((semiring_1_of_nat @ real) @ B2)) @ ((semiring_1_of_nat @ real) @ K))) = ((semiring_1_of_nat @ int) @ N2)) = ((((ord_less_eq @ nat) @ (((power_power @ nat) @ B2) @ N2)) @ K) & (((ord_less @ nat) @ K) @ (((power_power @ nat) @ B2) @ (((plus_plus @ nat) @ N2) @ (one_one @ nat))))))))))). % 6.41/2.69 thf(fact_2958_sin__zero, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => (((sin @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_2959_powr__eq__0__iff, axiom, (! [A:$tType]: (((ln @ A) => (! [W2:A,Z2:A]: ((((((powr @ A) @ W2) @ Z2) = (zero_zero @ A)) = (W2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2960_powr__0, axiom, (! [A:$tType]: (((ln @ A) => (! [Z2:A]: (((((powr @ A) @ (zero_zero @ A)) @ Z2) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_2961_cos__zero, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => (((cos @ A) @ (zero_zero @ A)) = (one_one @ A)))))). % 6.41/2.69 thf(fact_2962_powr__zero__eq__one, axiom, (! [A:$tType]: (((ln @ A) => (! [X3:A]: ((((X3 = (zero_zero @ A)) => ((((powr @ A) @ X3) @ (zero_zero @ A)) = (zero_zero @ A))) & ((X3 != (zero_zero @ A)) => ((((powr @ A) @ X3) @ (zero_zero @ A)) = (one_one @ A)))))))))). % 6.41/2.69 thf(fact_2963_floor__zero, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (((archim6421214686448440834_floor @ A) @ (zero_zero @ A)) = (zero_zero @ int)))))). % 6.41/2.69 thf(fact_2964_powr__gt__zero, axiom, (! [X3:real,A2:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ (((powr @ real) @ X3) @ A2)) = (X3 != (zero_zero @ real)))))). % 6.41/2.69 thf(fact_2965_powr__less__cancel__iff, axiom, (! [X3:real,A2:real,B2:real]: (((((ord_less @ real) @ (one_one @ real)) @ X3) => ((((ord_less @ real) @ (((powr @ real) @ X3) @ A2)) @ (((powr @ real) @ X3) @ B2)) = (((ord_less @ real) @ A2) @ B2)))))). % 6.41/2.69 thf(fact_2966_powr__eq__one__iff, axiom, (! [A2:real,X3:real]: (((((ord_less @ real) @ (one_one @ real)) @ A2) => (((((powr @ real) @ A2) @ X3) = (one_one @ real)) = (X3 = (zero_zero @ real))))))). % 6.41/2.69 thf(fact_2967_powr__le__cancel__iff, axiom, (! [X3:real,A2:real,B2:real]: (((((ord_less @ real) @ (one_one @ real)) @ X3) => ((((ord_less_eq @ real) @ (((powr @ real) @ X3) @ A2)) @ (((powr @ real) @ X3) @ B2)) = (((ord_less_eq @ real) @ A2) @ B2)))))). % 6.41/2.69 thf(fact_2968_zero__le__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((archim6421214686448440834_floor @ A) @ X3)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ X3)))))))). % 6.41/2.69 thf(fact_2969_sin__cos__squared__add3, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((plus_plus @ A) @ (((times_times @ A) @ ((cos @ A) @ X3)) @ ((cos @ A) @ X3))) @ (((times_times @ A) @ ((sin @ A) @ X3)) @ ((sin @ A) @ X3))) = (one_one @ A)))))))). % 6.41/2.69 thf(fact_2970_floor__less__zero, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ (zero_zero @ int)) = (((ord_less @ A) @ X3) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2971_numeral__le__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V2:num,X3:A]: (((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ V2)) @ ((archim6421214686448440834_floor @ A) @ X3)) = (((ord_less_eq @ A) @ ((numeral_numeral @ A) @ V2)) @ X3)))))))). % 6.41/2.69 thf(fact_2972_zero__less__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less @ int) @ (zero_zero @ int)) @ ((archim6421214686448440834_floor @ A) @ X3)) = (((ord_less_eq @ A) @ (one_one @ A)) @ X3)))))))). % 6.41/2.69 thf(fact_2973_floor__le__zero, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ (zero_zero @ int)) = (((ord_less @ A) @ X3) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_2974_floor__less__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,V2:num]: (((((ord_less @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ ((numeral_numeral @ int) @ V2)) = (((ord_less @ A) @ X3) @ ((numeral_numeral @ A) @ V2))))))))). % 6.41/2.69 thf(fact_2975_one__le__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less_eq @ int) @ (one_one @ int)) @ ((archim6421214686448440834_floor @ A) @ X3)) = (((ord_less_eq @ A) @ (one_one @ A)) @ X3)))))))). % 6.41/2.69 thf(fact_2976_floor__less__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ (one_one @ int)) = (((ord_less @ A) @ X3) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_2977_powr__log__cancel, axiom, (! [A2:real,X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ A2) => ((A2 != (one_one @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((powr @ real) @ A2) @ ((log @ A2) @ X3)) = X3))))))). % 6.41/2.69 thf(fact_2978_log__powr__cancel, axiom, (! [A2:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ A2) => ((A2 != (one_one @ real)) => (((log @ A2) @ (((powr @ real) @ A2) @ Y)) = Y)))))). % 6.41/2.69 thf(fact_2979_numeral__less__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V2:num,X3:A]: (((((ord_less @ int) @ ((numeral_numeral @ int) @ V2)) @ ((archim6421214686448440834_floor @ A) @ X3)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ V2)) @ (one_one @ A))) @ X3)))))))). % 6.41/2.69 thf(fact_2980_floor__le__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,V2:num]: (((((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ ((numeral_numeral @ int) @ V2)) = (((ord_less @ A) @ X3) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ V2)) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_2981_one__less__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less @ int) @ (one_one @ int)) @ ((archim6421214686448440834_floor @ A) @ X3)) = (((ord_less_eq @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ X3)))))))). % 6.41/2.69 thf(fact_2982_floor__le__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ (one_one @ int)) = (((ord_less @ A) @ X3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_2983_neg__numeral__le__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V2:num,X3:A]: (((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V2))) @ ((archim6421214686448440834_floor @ A) @ X3)) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) @ X3)))))))). % 6.41/2.69 thf(fact_2984_floor__less__neg__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,V2:num]: (((((ord_less @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V2))) = (((ord_less @ A) @ X3) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2)))))))))). % 6.41/2.69 thf(fact_2985_sin__cos__squared__add2, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((plus_plus @ A) @ (((power_power @ A) @ ((cos @ A) @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ ((sin @ A) @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (one_one @ A)))))))). % 6.41/2.69 thf(fact_2986_sin__cos__squared__add, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((plus_plus @ A) @ (((power_power @ A) @ ((sin @ A) @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ ((cos @ A) @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (one_one @ A)))))))). % 6.41/2.69 thf(fact_2987_neg__numeral__less__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V2:num,X3:A]: (((((ord_less @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V2))) @ ((archim6421214686448440834_floor @ A) @ X3)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) @ (one_one @ A))) @ X3)))))))). % 6.41/2.69 thf(fact_2988_floor__le__neg__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,V2:num]: (((((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V2))) = (((ord_less @ A) @ X3) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_2989_cos__one__sin__zero, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((cos @ A) @ X3) = (one_one @ A)) => (((sin @ A) @ X3) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_2990_sin__add, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,Y:A]: ((((sin @ A) @ (((plus_plus @ A) @ X3) @ Y)) = (((plus_plus @ A) @ (((times_times @ A) @ ((sin @ A) @ X3)) @ ((cos @ A) @ Y))) @ (((times_times @ A) @ ((cos @ A) @ X3)) @ ((sin @ A) @ Y)))))))))). % 6.41/2.69 thf(fact_2991_cos__diff, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,Y:A]: ((((cos @ A) @ (((minus_minus @ A) @ X3) @ Y)) = (((plus_plus @ A) @ (((times_times @ A) @ ((cos @ A) @ X3)) @ ((cos @ A) @ Y))) @ (((times_times @ A) @ ((sin @ A) @ X3)) @ ((sin @ A) @ Y)))))))))). % 6.41/2.69 thf(fact_2992_cos__add, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,Y:A]: ((((cos @ A) @ (((plus_plus @ A) @ X3) @ Y)) = (((minus_minus @ A) @ (((times_times @ A) @ ((cos @ A) @ X3)) @ ((cos @ A) @ Y))) @ (((times_times @ A) @ ((sin @ A) @ X3)) @ ((sin @ A) @ Y)))))))))). % 6.41/2.69 thf(fact_2993_sin__zero__norm__cos__one, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((sin @ A) @ X3) = (zero_zero @ A)) => (((real_V7770717601297561774m_norm @ A) @ ((cos @ A) @ X3)) = (one_one @ real))))))))). % 6.41/2.69 thf(fact_2994_sincos__principal__value, axiom, (! [X3:real]: ((? [Y4:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ pi)) @ Y4) & ((((ord_less_eq @ real) @ Y4) @ pi) & ((((sin @ real) @ Y4) = ((sin @ real) @ X3)) & (((cos @ real) @ Y4) = ((cos @ real) @ X3)))))))))). % 6.41/2.69 thf(fact_2995_floor__mono, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => (((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ ((archim6421214686448440834_floor @ A) @ Y))))))))). % 6.41/2.69 thf(fact_2996_of__int__floor__le, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ ((archim6421214686448440834_floor @ A) @ X3))) @ X3))))))). % 6.41/2.69 thf(fact_2997_floor__less__cancel, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Y:A]: (((((ord_less @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ ((archim6421214686448440834_floor @ A) @ Y)) => (((ord_less @ A) @ X3) @ Y)))))))). % 6.41/2.69 thf(fact_2998_powr__non__neg, axiom, (! [A2:real,X3:real]: ((~ ((((ord_less @ real) @ (((powr @ real) @ A2) @ X3)) @ (zero_zero @ real))))))). % 6.41/2.69 thf(fact_2999_powr__less__mono2__neg, axiom, (! [A2:real,X3:real,Y:real]: (((((ord_less @ real) @ A2) @ (zero_zero @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ Y) => (((ord_less @ real) @ (((powr @ real) @ Y) @ A2)) @ (((powr @ real) @ X3) @ A2)))))))). % 6.41/2.69 thf(fact_3000_powr__less__cancel, axiom, (! [X3:real,A2:real,B2:real]: (((((ord_less @ real) @ (((powr @ real) @ X3) @ A2)) @ (((powr @ real) @ X3) @ B2)) => ((((ord_less @ real) @ (one_one @ real)) @ X3) => (((ord_less @ real) @ A2) @ B2)))))). % 6.41/2.69 thf(fact_3001_powr__less__mono, axiom, (! [A2:real,B2:real,X3:real]: (((((ord_less @ real) @ A2) @ B2) => ((((ord_less @ real) @ (one_one @ real)) @ X3) => (((ord_less @ real) @ (((powr @ real) @ X3) @ A2)) @ (((powr @ real) @ X3) @ B2))))))). % 6.41/2.69 thf(fact_3002_le__floor__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z2:int,X3:A]: (((((ord_less_eq @ int) @ Z2) @ ((archim6421214686448440834_floor @ A) @ X3)) = (((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z2)) @ X3)))))))). % 6.41/2.69 thf(fact_3003_floor__less__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Z2:int]: (((((ord_less @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ Z2) = (((ord_less @ A) @ X3) @ ((ring_1_of_int @ A) @ Z2))))))))). % 6.41/2.69 thf(fact_3004_powr__mono2_H, axiom, (! [A2:real,X3:real,Y:real]: (((((ord_less_eq @ real) @ A2) @ (zero_zero @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ X3) @ Y) => (((ord_less_eq @ real) @ (((powr @ real) @ Y) @ A2)) @ (((powr @ real) @ X3) @ A2)))))))). % 6.41/2.69 thf(fact_3005_powr__less__mono2, axiom, (! [A2:real,X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ A2) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ Y) => (((ord_less @ real) @ (((powr @ real) @ X3) @ A2)) @ (((powr @ real) @ Y) @ A2)))))))). % 6.41/2.69 thf(fact_3006_le__floor__add, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Y:A]: ((((ord_less_eq @ int) @ (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ ((archim6421214686448440834_floor @ A) @ Y))) @ ((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X3) @ Y))))))))). % 6.41/2.69 thf(fact_3007_gr__one__powr, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ (one_one @ real)) @ X3) => ((((ord_less @ real) @ (zero_zero @ real)) @ Y) => (((ord_less @ real) @ (one_one @ real)) @ (((powr @ real) @ X3) @ Y))))))). % 6.41/2.69 thf(fact_3008_powr__inj, axiom, (! [A2:real,X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ A2) => ((A2 != (one_one @ real)) => (((((powr @ real) @ A2) @ X3) = (((powr @ real) @ A2) @ Y)) = (X3 = Y))))))). % 6.41/2.69 thf(fact_3009_int__add__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z2:int,X3:A]: (((((plus_plus @ int) @ Z2) @ ((archim6421214686448440834_floor @ A) @ X3)) = ((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ Z2)) @ X3))))))))). % 6.41/2.69 thf(fact_3010_floor__add__int, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Z2:int]: (((((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ Z2) = ((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X3) @ ((ring_1_of_int @ A) @ Z2)))))))))). % 6.41/2.69 thf(fact_3011_sin__gt__zero, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ pi) => (((ord_less @ real) @ (zero_zero @ real)) @ ((sin @ real) @ X3))))))). % 6.41/2.69 thf(fact_3012_sin__times__sin, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [W2:A,Z2:A]: (((((times_times @ A) @ ((sin @ A) @ W2)) @ ((sin @ A) @ Z2)) = (((divide_divide @ A) @ (((minus_minus @ A) @ ((cos @ A) @ (((minus_minus @ A) @ W2) @ Z2))) @ ((cos @ A) @ (((plus_plus @ A) @ W2) @ Z2)))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_3013_sin__times__cos, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [W2:A,Z2:A]: (((((times_times @ A) @ ((sin @ A) @ W2)) @ ((cos @ A) @ Z2)) = (((divide_divide @ A) @ (((plus_plus @ A) @ ((sin @ A) @ (((plus_plus @ A) @ W2) @ Z2))) @ ((sin @ A) @ (((minus_minus @ A) @ W2) @ Z2)))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_3014_cos__times__sin, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [W2:A,Z2:A]: (((((times_times @ A) @ ((cos @ A) @ W2)) @ ((sin @ A) @ Z2)) = (((divide_divide @ A) @ (((minus_minus @ A) @ ((sin @ A) @ (((plus_plus @ A) @ W2) @ Z2))) @ ((sin @ A) @ (((minus_minus @ A) @ W2) @ Z2)))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_3015_sin__plus__sin, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [W2:A,Z2:A]: (((((plus_plus @ A) @ ((sin @ A) @ W2)) @ ((sin @ A) @ Z2)) = (((times_times @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((sin @ A) @ (((divide_divide @ A) @ (((plus_plus @ A) @ W2) @ Z2)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))) @ ((cos @ A) @ (((divide_divide @ A) @ (((minus_minus @ A) @ W2) @ Z2)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))). % 6.41/2.69 thf(fact_3016_sin__diff__sin, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [W2:A,Z2:A]: (((((minus_minus @ A) @ ((sin @ A) @ W2)) @ ((sin @ A) @ Z2)) = (((times_times @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((sin @ A) @ (((divide_divide @ A) @ (((minus_minus @ A) @ W2) @ Z2)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))) @ ((cos @ A) @ (((divide_divide @ A) @ (((plus_plus @ A) @ W2) @ Z2)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))). % 6.41/2.69 thf(fact_3017_cos__diff__cos, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [W2:A,Z2:A]: (((((minus_minus @ A) @ ((cos @ A) @ W2)) @ ((cos @ A) @ Z2)) = (((times_times @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((sin @ A) @ (((divide_divide @ A) @ (((plus_plus @ A) @ W2) @ Z2)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))) @ ((sin @ A) @ (((divide_divide @ A) @ (((minus_minus @ A) @ Z2) @ W2)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))). % 6.41/2.69 thf(fact_3018_powr__add, axiom, (! [A:$tType]: ((((real_V3459762299906320749_field @ A) & (ln @ A)) => (! [X3:A,A2:A,B2:A]: (((((powr @ A) @ X3) @ (((plus_plus @ A) @ A2) @ B2)) = (((times_times @ A) @ (((powr @ A) @ X3) @ A2)) @ (((powr @ A) @ X3) @ B2))))))))). % 6.41/2.69 thf(fact_3019_one__add__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ (one_one @ int)) = ((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X3) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_3020_floor__log__eq__powr__iff, axiom, (! [X3:real,B2:real,K:int]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (one_one @ real)) @ B2) => ((((archim6421214686448440834_floor @ real) @ ((log @ B2) @ X3)) = K) = ((((ord_less_eq @ real) @ (((powr @ real) @ B2) @ ((ring_1_of_int @ real) @ K))) @ X3) & (((ord_less @ real) @ X3) @ (((powr @ real) @ B2) @ ((ring_1_of_int @ real) @ (((plus_plus @ int) @ K) @ (one_one @ int)))))))))))). % 6.41/2.69 thf(fact_3021_powr__realpow, axiom, (! [X3:real,N2:nat]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((powr @ real) @ X3) @ ((semiring_1_of_nat @ real) @ N2)) = (((power_power @ real) @ X3) @ N2)))))). % 6.41/2.69 thf(fact_3022_less__log__iff, axiom, (! [B2:real,X3:real,Y:real]: (((((ord_less @ real) @ (one_one @ real)) @ B2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ Y) @ ((log @ B2) @ X3)) = (((ord_less @ real) @ (((powr @ real) @ B2) @ Y)) @ X3))))))). % 6.41/2.69 thf(fact_3023_log__less__iff, axiom, (! [B2:real,X3:real,Y:real]: (((((ord_less @ real) @ (one_one @ real)) @ B2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ ((log @ B2) @ X3)) @ Y) = (((ord_less @ real) @ X3) @ (((powr @ real) @ B2) @ Y)))))))). % 6.41/2.69 thf(fact_3024_less__powr__iff, axiom, (! [B2:real,X3:real,Y:real]: (((((ord_less @ real) @ (one_one @ real)) @ B2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ (((powr @ real) @ B2) @ Y)) = (((ord_less @ real) @ ((log @ B2) @ X3)) @ Y))))))). % 6.41/2.69 thf(fact_3025_powr__less__iff, axiom, (! [B2:real,X3:real,Y:real]: (((((ord_less @ real) @ (one_one @ real)) @ B2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ (((powr @ real) @ B2) @ Y)) @ X3) = (((ord_less @ real) @ Y) @ ((log @ B2) @ X3)))))))). % 6.41/2.69 thf(fact_3026_cos__mono__less__eq, axiom, (! [X3:real,Y:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ X3) @ pi) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ Y) => ((((ord_less_eq @ real) @ Y) @ pi) => ((((ord_less @ real) @ ((cos @ real) @ X3)) @ ((cos @ real) @ Y)) = (((ord_less @ real) @ Y) @ X3))))))))). % 6.41/2.69 thf(fact_3027_cos__monotone__0__pi, axiom, (! [Y:real,X3:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ Y) => ((((ord_less @ real) @ Y) @ X3) => ((((ord_less_eq @ real) @ X3) @ pi) => (((ord_less @ real) @ ((cos @ real) @ X3)) @ ((cos @ real) @ Y)))))))). % 6.41/2.69 thf(fact_3028_sin__eq__0__pi, axiom, (! [X3:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ pi)) @ X3) => ((((ord_less @ real) @ X3) @ pi) => ((((sin @ real) @ X3) = (zero_zero @ real)) => (X3 = (zero_zero @ real)))))))). % 6.41/2.69 thf(fact_3029_real__of__int__floor__add__one__gt, axiom, (! [R2:real]: ((((ord_less @ real) @ R2) @ (((plus_plus @ real) @ ((ring_1_of_int @ real) @ ((archim6421214686448440834_floor @ real) @ R2))) @ (one_one @ real)))))). % 6.41/2.69 thf(fact_3030_floor__eq, axiom, (! [N2:int,X3:real]: (((((ord_less @ real) @ ((ring_1_of_int @ real) @ N2)) @ X3) => ((((ord_less @ real) @ X3) @ (((plus_plus @ real) @ ((ring_1_of_int @ real) @ N2)) @ (one_one @ real))) => (((archim6421214686448440834_floor @ real) @ X3) = N2)))))). % 6.41/2.69 thf(fact_3031_sin__zero__pi__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ X3)) @ pi) => ((((sin @ real) @ X3) = (zero_zero @ real)) = (X3 = (zero_zero @ real))))))). % 6.41/2.69 thf(fact_3032_real__of__int__floor__gt__diff__one, axiom, (! [R2:real]: ((((ord_less @ real) @ (((minus_minus @ real) @ R2) @ (one_one @ real))) @ ((ring_1_of_int @ real) @ ((archim6421214686448440834_floor @ real) @ R2)))))). % 6.41/2.69 thf(fact_3033_diffs__def, axiom, (! [A:$tType]: (((ring_1 @ A) => ((diffs @ A) = (^ [C4:(nat > A),N:nat]: ((((times_times @ A) @ ((semiring_1_of_nat @ A) @ (suc @ N))) @ (C4 @ (suc @ N)))))))))). % 6.41/2.69 thf(fact_3034_sin__expansion__lemma, axiom, (! [X3:real,M:nat]: ((((sin @ real) @ (((plus_plus @ real) @ X3) @ (((divide_divide @ real) @ (((times_times @ real) @ ((semiring_1_of_nat @ real) @ (suc @ M))) @ pi)) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) = ((cos @ real) @ (((plus_plus @ real) @ X3) @ (((divide_divide @ real) @ (((times_times @ real) @ ((semiring_1_of_nat @ real) @ M)) @ pi)) @ ((numeral_numeral @ real) @ (bit0 @ one2))))))))). % 6.41/2.69 thf(fact_3035_floor__unique, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z2:int,X3:A]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z2)) @ X3) => ((((ord_less @ A) @ X3) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A))) => (((archim6421214686448440834_floor @ A) @ X3) = Z2))))))))). % 6.41/2.69 thf(fact_3036_floor__eq__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,A2:int]: (((((archim6421214686448440834_floor @ A) @ X3) = A2) = ((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ A2)) @ X3) & (((ord_less @ A) @ X3) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ A2)) @ (one_one @ A))))))))))). % 6.41/2.69 thf(fact_3037_floor__split, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [P:(int > $o),T2:A]: (((P @ ((archim6421214686448440834_floor @ A) @ T2)) = (! [I3:int]: ((((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ I3)) @ T2) & (((ord_less @ A) @ T2) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ I3)) @ (one_one @ A)))) => (P @ I3))))))))))). % 6.41/2.69 thf(fact_3038_le__mult__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ int) @ (((times_times @ int) @ ((archim6421214686448440834_floor @ A) @ A2)) @ ((archim6421214686448440834_floor @ A) @ B2))) @ ((archim6421214686448440834_floor @ A) @ (((times_times @ A) @ A2) @ B2))))))))))). % 6.41/2.69 thf(fact_3039_less__floor__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z2:int,X3:A]: (((((ord_less @ int) @ Z2) @ ((archim6421214686448440834_floor @ A) @ X3)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A))) @ X3)))))))). % 6.41/2.69 thf(fact_3040_floor__le__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Z2:int]: (((((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ Z2) = (((ord_less @ A) @ X3) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_3041_floor__correct, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ ((archim6421214686448440834_floor @ A) @ X3))) @ X3) & (((ord_less @ A) @ X3) @ ((ring_1_of_int @ A) @ (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ (one_one @ int))))))))))). % 6.41/2.69 thf(fact_3042_cos__expansion__lemma, axiom, (! [X3:real,M:nat]: ((((cos @ real) @ (((plus_plus @ real) @ X3) @ (((divide_divide @ real) @ (((times_times @ real) @ ((semiring_1_of_nat @ real) @ (suc @ M))) @ pi)) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) = ((uminus_uminus @ real) @ ((sin @ real) @ (((plus_plus @ real) @ X3) @ (((divide_divide @ real) @ (((times_times @ real) @ ((semiring_1_of_nat @ real) @ M)) @ pi)) @ ((numeral_numeral @ real) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_3043_powr__neg__one, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((powr @ real) @ X3) @ ((uminus_uminus @ real) @ (one_one @ real))) = (((divide_divide @ real) @ (one_one @ real)) @ X3)))))). % 6.41/2.69 thf(fact_3044_sin__gt__zero__02, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ ((numeral_numeral @ real) @ (bit0 @ one2))) => (((ord_less @ real) @ (zero_zero @ real)) @ ((sin @ real) @ X3))))))). % 6.41/2.69 thf(fact_3045_le__log__iff, axiom, (! [B2:real,X3:real,Y:real]: (((((ord_less @ real) @ (one_one @ real)) @ B2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ Y) @ ((log @ B2) @ X3)) = (((ord_less_eq @ real) @ (((powr @ real) @ B2) @ Y)) @ X3))))))). % 6.41/2.69 thf(fact_3046_log__le__iff, axiom, (! [B2:real,X3:real,Y:real]: (((((ord_less @ real) @ (one_one @ real)) @ B2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ ((log @ B2) @ X3)) @ Y) = (((ord_less_eq @ real) @ X3) @ (((powr @ real) @ B2) @ Y)))))))). % 6.41/2.69 thf(fact_3047_le__powr__iff, axiom, (! [B2:real,X3:real,Y:real]: (((((ord_less @ real) @ (one_one @ real)) @ B2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ X3) @ (((powr @ real) @ B2) @ Y)) = (((ord_less_eq @ real) @ ((log @ B2) @ X3)) @ Y))))))). % 6.41/2.69 thf(fact_3048_powr__le__iff, axiom, (! [B2:real,X3:real,Y:real]: (((((ord_less @ real) @ (one_one @ real)) @ B2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ (((powr @ real) @ B2) @ Y)) @ X3) = (((ord_less_eq @ real) @ Y) @ ((log @ B2) @ X3)))))))). % 6.41/2.69 thf(fact_3049_cos__two__less__zero, axiom, (((ord_less @ real) @ ((cos @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) @ (zero_zero @ real))). % 6.41/2.69 thf(fact_3050_floor__eq2, axiom, (! [N2:int,X3:real]: (((((ord_less_eq @ real) @ ((ring_1_of_int @ real) @ N2)) @ X3) => ((((ord_less @ real) @ X3) @ (((plus_plus @ real) @ ((ring_1_of_int @ real) @ N2)) @ (one_one @ real))) => (((archim6421214686448440834_floor @ real) @ X3) = N2)))))). % 6.41/2.69 thf(fact_3051_cos__monotone__minus__pi__0, axiom, (! [Y:real,X3:real]: (((((ord_less_eq @ real) @ ((uminus_uminus @ real) @ pi)) @ Y) => ((((ord_less @ real) @ Y) @ X3) => ((((ord_less_eq @ real) @ X3) @ (zero_zero @ real)) => (((ord_less @ real) @ ((cos @ real) @ Y)) @ ((cos @ real) @ X3)))))))). % 6.41/2.69 thf(fact_3052_floor__divide__lower, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Q4:A,P4:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Q4) => (((ord_less_eq @ A) @ (((times_times @ A) @ ((ring_1_of_int @ A) @ ((archim6421214686448440834_floor @ A) @ (((divide_divide @ A) @ P4) @ Q4)))) @ Q4)) @ P4)))))))). % 6.41/2.69 thf(fact_3053_sincos__total__2pi, axiom, (! [X3:real,Y:real]: ((((((plus_plus @ real) @ (((power_power @ real) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ real) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (one_one @ real)) => (~ ((! [T7:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ T7) => ((((ord_less @ real) @ T7) @ (((times_times @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ pi)) => ((X3 = ((cos @ real) @ T7)) => (Y != ((sin @ real) @ T7))))))))))))). % 6.41/2.69 thf(fact_3054_ln__powr__bound, axiom, (! [X3:real,A2:real]: (((((ord_less_eq @ real) @ (one_one @ real)) @ X3) => ((((ord_less @ real) @ (zero_zero @ real)) @ A2) => (((ord_less_eq @ real) @ ((ln_ln @ real) @ X3)) @ (((divide_divide @ real) @ (((powr @ real) @ X3) @ A2)) @ A2))))))). % 6.41/2.69 thf(fact_3055_ln__powr__bound2, axiom, (! [X3:real,A2:real]: (((((ord_less @ real) @ (one_one @ real)) @ X3) => ((((ord_less @ real) @ (zero_zero @ real)) @ A2) => (((ord_less_eq @ real) @ (((powr @ real) @ ((ln_ln @ real) @ X3)) @ A2)) @ (((times_times @ real) @ (((powr @ real) @ A2) @ A2)) @ X3))))))). % 6.41/2.69 thf(fact_3056_log__add__eq__powr, axiom, (! [B2:real,X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ B2) => ((B2 != (one_one @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((plus_plus @ real) @ ((log @ B2) @ X3)) @ Y) = ((log @ B2) @ (((times_times @ real) @ X3) @ (((powr @ real) @ B2) @ Y)))))))))). % 6.41/2.69 thf(fact_3057_add__log__eq__powr, axiom, (! [B2:real,X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ B2) => ((B2 != (one_one @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((plus_plus @ real) @ Y) @ ((log @ B2) @ X3)) = ((log @ B2) @ (((times_times @ real) @ (((powr @ real) @ B2) @ Y)) @ X3))))))))). % 6.41/2.69 thf(fact_3058_minus__log__eq__powr, axiom, (! [B2:real,X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ B2) => ((B2 != (one_one @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((minus_minus @ real) @ Y) @ ((log @ B2) @ X3)) = ((log @ B2) @ (((divide_divide @ real) @ (((powr @ real) @ B2) @ Y)) @ X3))))))))). % 6.41/2.69 thf(fact_3059_sin__pi__divide__n__ge__0, axiom, (! [N2:nat]: (((N2 != (zero_zero @ nat)) => (((ord_less_eq @ real) @ (zero_zero @ real)) @ ((sin @ real) @ (((divide_divide @ real) @ pi) @ ((semiring_1_of_nat @ real) @ N2)))))))). % 6.41/2.69 thf(fact_3060_cos__times__cos, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [W2:A,Z2:A]: (((((times_times @ A) @ ((cos @ A) @ W2)) @ ((cos @ A) @ Z2)) = (((divide_divide @ A) @ (((plus_plus @ A) @ ((cos @ A) @ (((minus_minus @ A) @ W2) @ Z2))) @ ((cos @ A) @ (((plus_plus @ A) @ W2) @ Z2)))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_3061_cos__plus__cos, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [W2:A,Z2:A]: (((((plus_plus @ A) @ ((cos @ A) @ W2)) @ ((cos @ A) @ Z2)) = (((times_times @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((cos @ A) @ (((divide_divide @ A) @ (((plus_plus @ A) @ W2) @ Z2)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))) @ ((cos @ A) @ (((divide_divide @ A) @ (((minus_minus @ A) @ W2) @ Z2)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))). % 6.41/2.69 thf(fact_3062_floor__divide__upper, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Q4:A,P4:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Q4) => (((ord_less @ A) @ P4) @ (((times_times @ A) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ ((archim6421214686448440834_floor @ A) @ (((divide_divide @ A) @ P4) @ Q4)))) @ (one_one @ A))) @ Q4))))))))). % 6.41/2.69 thf(fact_3063_round__def, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => ((archimedean_round @ A) = (^ [X:A]: (((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))). % 6.41/2.69 thf(fact_3064_sin__gt__zero2, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => (((ord_less @ real) @ (zero_zero @ real)) @ ((sin @ real) @ X3))))))). % 6.41/2.69 thf(fact_3065_sin__lt__zero, axiom, (! [X3:real]: (((((ord_less @ real) @ pi) @ X3) => ((((ord_less @ real) @ X3) @ (((times_times @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ pi)) => (((ord_less @ real) @ ((sin @ real) @ X3)) @ (zero_zero @ real))))))). % 6.41/2.69 thf(fact_3066_cos__double__less__one, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ ((numeral_numeral @ real) @ (bit0 @ one2))) => (((ord_less @ real) @ ((cos @ real) @ (((times_times @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ X3))) @ (one_one @ real))))))). % 6.41/2.69 thf(fact_3067_cos__gt__zero, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => (((ord_less @ real) @ (zero_zero @ real)) @ ((cos @ real) @ X3))))))). % 6.41/2.69 thf(fact_3068_log__minus__eq__powr, axiom, (! [B2:real,X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ B2) => ((B2 != (one_one @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((minus_minus @ real) @ ((log @ B2) @ X3)) @ Y) = ((log @ B2) @ (((times_times @ real) @ X3) @ (((powr @ real) @ B2) @ ((uminus_uminus @ real) @ Y))))))))))). % 6.41/2.69 thf(fact_3069_termdiff__converges, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,K5:real,C2:(nat > A)]: (((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ K5) => ((! [X5:A]: (((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X5)) @ K5) => ((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (C2 @ N)) @ (((power_power @ A) @ X5) @ N)))))))) => ((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (((diffs @ A) @ C2) @ N)) @ (((power_power @ A) @ X3) @ N))))))))))))). % 6.41/2.69 thf(fact_3070_powr__neg__numeral, axiom, (! [X3:real,N2:num]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((powr @ real) @ X3) @ ((uminus_uminus @ real) @ ((numeral_numeral @ real) @ N2))) = (((divide_divide @ real) @ (one_one @ real)) @ (((power_power @ real) @ X3) @ ((numeral_numeral @ nat) @ N2)))))))). % 6.41/2.69 thf(fact_3071_sin__le__zero, axiom, (! [X3:real]: (((((ord_less_eq @ real) @ pi) @ X3) => ((((ord_less @ real) @ X3) @ (((times_times @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ pi)) => (((ord_less_eq @ real) @ ((sin @ real) @ X3)) @ (zero_zero @ real))))))). % 6.41/2.69 thf(fact_3072_sin__less__zero, axiom, (! [X3:real]: (((((ord_less @ real) @ (((divide_divide @ real) @ ((uminus_uminus @ real) @ pi)) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) @ X3) => ((((ord_less @ real) @ X3) @ (zero_zero @ real)) => (((ord_less @ real) @ ((sin @ real) @ X3)) @ (zero_zero @ real))))))). % 6.41/2.69 thf(fact_3073_sin__mono__less__eq, axiom, (! [X3:real,Y:real]: (((((ord_less_eq @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ X3) => ((((ord_less_eq @ real) @ X3) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => ((((ord_less_eq @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ Y) => ((((ord_less_eq @ real) @ Y) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => ((((ord_less @ real) @ ((sin @ real) @ X3)) @ ((sin @ real) @ Y)) = (((ord_less @ real) @ X3) @ Y))))))))). % 6.41/2.69 thf(fact_3074_sin__monotone__2pi, axiom, (! [Y:real,X3:real]: (((((ord_less_eq @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ Y) => ((((ord_less @ real) @ Y) @ X3) => ((((ord_less_eq @ real) @ X3) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => (((ord_less @ real) @ ((sin @ real) @ Y)) @ ((sin @ real) @ X3)))))))). % 6.41/2.69 thf(fact_3075_cos__gt__zero__pi, axiom, (! [X3:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ X3) => ((((ord_less @ real) @ X3) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => (((ord_less @ real) @ (zero_zero @ real)) @ ((cos @ real) @ X3))))))). % 6.41/2.69 thf(fact_3076_sin__pi__divide__n__gt__0, axiom, (! [N2:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2) => (((ord_less @ real) @ (zero_zero @ real)) @ ((sin @ real) @ (((divide_divide @ real) @ pi) @ ((semiring_1_of_nat @ real) @ N2)))))))). % 6.41/2.69 thf(fact_3077_floor__log2__div2, axiom, (! [N2:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2) => (((archim6421214686448440834_floor @ real) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ real) @ N2))) = (((plus_plus @ int) @ ((archim6421214686448440834_floor @ real) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ real) @ (((divide_divide @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) @ (one_one @ int))))))). % 6.41/2.69 thf(fact_3078_floor__log__nat__eq__if, axiom, (! [B2:nat,N2:nat,K:nat]: (((((ord_less_eq @ nat) @ (((power_power @ nat) @ B2) @ N2)) @ K) => ((((ord_less @ nat) @ K) @ (((power_power @ nat) @ B2) @ (((plus_plus @ nat) @ N2) @ (one_one @ nat)))) => ((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ B2) => (((archim6421214686448440834_floor @ real) @ ((log @ ((semiring_1_of_nat @ real) @ B2)) @ ((semiring_1_of_nat @ real) @ K))) = ((semiring_1_of_nat @ int) @ N2)))))))). % 6.41/2.69 thf(fact_3079_Maclaurin__minus__cos__expansion, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ X3) @ (zero_zero @ real)) => (? [T7:real]: (((((ord_less @ real) @ X3) @ T7) & ((((ord_less @ real) @ T7) @ (zero_zero @ real)) & (((cos @ real) @ X3) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (cos_coeff @ M2)) @ (((power_power @ real) @ X3) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ (((divide_divide @ real) @ ((cos @ real) @ (((plus_plus @ real) @ T7) @ (((times_times @ real) @ (((times_times @ real) @ (((divide_divide @ real) @ (one_one @ real)) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) @ ((semiring_1_of_nat @ real) @ N2))) @ pi)))) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ real) @ X3) @ N2))))))))))))). % 6.41/2.69 thf(fact_3080_Maclaurin__cos__expansion2, axiom, (! [X3:real,N2:nat]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (? [T7:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ T7) & ((((ord_less @ real) @ T7) @ X3) & (((cos @ real) @ X3) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (cos_coeff @ M2)) @ (((power_power @ real) @ X3) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ (((divide_divide @ real) @ ((cos @ real) @ (((plus_plus @ real) @ T7) @ (((times_times @ real) @ (((times_times @ real) @ (((divide_divide @ real) @ (one_one @ real)) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) @ ((semiring_1_of_nat @ real) @ N2))) @ pi)))) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ real) @ X3) @ N2))))))))))))). % 6.41/2.69 thf(fact_3081_sin__paired, axiom, (! [X3:real]: ((((sums @ real) @ (^ [N:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ N)) @ ((semiring_char_0_fact @ real) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ (one_one @ nat))))) @ (((power_power @ real) @ X3) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ (one_one @ nat))))))) @ ((sin @ real) @ X3))))). % 6.41/2.69 thf(fact_3082_tan__double, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((cos @ A) @ X3) != (zero_zero @ A)) => ((((cos @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ X3)) != (zero_zero @ A)) => (((tan @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ X3)) = (((divide_divide @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((tan @ A) @ X3))) @ (((minus_minus @ A) @ (one_one @ A)) @ (((power_power @ A) @ ((tan @ A) @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))))). % 6.41/2.69 thf(fact_3083_arcosh__def, axiom, (! [A:$tType]: (((ln @ A) => ((arcosh @ A) = (^ [X:A]: (((ln_ln @ A) @ (((plus_plus @ A) @ X) @ (((powr @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ A))) @ ((real_Vector_of_real @ A) @ (((divide_divide @ real) @ (one_one @ real)) @ ((numeral_numeral @ real) @ (bit0 @ one2)))))))))))))). % 6.41/2.69 thf(fact_3084_tan__zero, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (((tan @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_3085_of__real__eq__0__iff, axiom, (! [A:$tType]: (((real_V2191834092415804123ebra_1 @ A) => (! [X3:real]: (((((real_Vector_of_real @ A) @ X3) = (zero_zero @ A)) = (X3 = (zero_zero @ real))))))))). % 6.41/2.69 thf(fact_3086_of__real__0, axiom, (! [A:$tType]: (((real_V2191834092415804123ebra_1 @ A) => (((real_Vector_of_real @ A) @ (zero_zero @ real)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_3087_fact__0, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (((semiring_char_0_fact @ A) @ (zero_zero @ nat)) = (one_one @ A)))))). % 6.41/2.69 thf(fact_3088_of__real__add, axiom, (! [A:$tType]: (((real_V2191834092415804123ebra_1 @ A) => (! [X3:real,Y:real]: ((((real_Vector_of_real @ A) @ (((plus_plus @ real) @ X3) @ Y)) = (((plus_plus @ A) @ ((real_Vector_of_real @ A) @ X3)) @ ((real_Vector_of_real @ A) @ Y))))))))). % 6.41/2.69 thf(fact_3089_fact__Suc__0, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (((semiring_char_0_fact @ A) @ (suc @ (zero_zero @ nat))) = (one_one @ A)))))). % 6.41/2.69 thf(fact_3090_fact__Suc, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N2:nat]: ((((semiring_char_0_fact @ A) @ (suc @ N2)) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ (suc @ N2))) @ ((semiring_char_0_fact @ A) @ N2))))))))). % 6.41/2.69 thf(fact_3091_sin__of__real__pi, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => (((sin @ A) @ ((real_Vector_of_real @ A) @ pi)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_3092_norm__of__real__add1, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [X3:real]: ((((real_V7770717601297561774m_norm @ A) @ (((plus_plus @ A) @ ((real_Vector_of_real @ A) @ X3)) @ (one_one @ A))) = ((abs_abs @ real) @ (((plus_plus @ real) @ X3) @ (one_one @ real)))))))))). % 6.41/2.69 thf(fact_3093_norm__of__real__addn, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [X3:real,B2:num]: ((((real_V7770717601297561774m_norm @ A) @ (((plus_plus @ A) @ ((real_Vector_of_real @ A) @ X3)) @ ((numeral_numeral @ A) @ B2))) = ((abs_abs @ real) @ (((plus_plus @ real) @ X3) @ ((numeral_numeral @ real) @ B2)))))))))). % 6.41/2.69 thf(fact_3094_cos__of__real__pi__half, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & ((real_V7773925162809079976_field @ A) & (real_V2822296259951069270ebra_1 @ A))) => (((cos @ A) @ (((divide_divide @ A) @ ((real_Vector_of_real @ A) @ pi)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_3095_fact__nonzero, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (semiri3467727345109120633visors @ A)) => (! [N2:nat]: ((((semiring_char_0_fact @ A) @ N2) != (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_3096_fact__ge__zero, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N2:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((semiring_char_0_fact @ A) @ N2)))))))). % 6.41/2.69 thf(fact_3097_fact__not__neg, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N2:nat]: ((~ ((((ord_less @ A) @ ((semiring_char_0_fact @ A) @ N2)) @ (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_3098_fact__gt__zero, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N2:nat]: ((((ord_less @ A) @ (zero_zero @ A)) @ ((semiring_char_0_fact @ A) @ N2)))))))). % 6.41/2.69 thf(fact_3099_fact__ge__1, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N2:nat]: ((((ord_less_eq @ A) @ (one_one @ A)) @ ((semiring_char_0_fact @ A) @ N2)))))))). % 6.41/2.69 thf(fact_3100_fact__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => (((ord_less_eq @ A) @ ((semiring_char_0_fact @ A) @ M)) @ ((semiring_char_0_fact @ A) @ N2))))))))). % 6.41/2.69 thf(fact_3101_fact__dvd, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N2:nat,M:nat]: (((((ord_less_eq @ nat) @ N2) @ M) => (((dvd_dvd @ A) @ ((semiring_char_0_fact @ A) @ N2)) @ ((semiring_char_0_fact @ A) @ M))))))))). % 6.41/2.69 thf(fact_3102_fact__less__mono, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less @ nat) @ M) @ N2) => (((ord_less @ A) @ ((semiring_char_0_fact @ A) @ M)) @ ((semiring_char_0_fact @ A) @ N2)))))))))). % 6.41/2.69 thf(fact_3103_fact__fact__dvd__fact, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [K:nat,N2:nat]: ((((dvd_dvd @ A) @ (((times_times @ A) @ ((semiring_char_0_fact @ A) @ K)) @ ((semiring_char_0_fact @ A) @ N2))) @ ((semiring_char_0_fact @ A) @ (((plus_plus @ nat) @ K) @ N2))))))))). % 6.41/2.69 thf(fact_3104_fact__mod, axiom, (! [A:$tType]: ((((linordered_semidom @ A) & (semidom_modulo @ A)) => (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((modulo_modulo @ A) @ ((semiring_char_0_fact @ A) @ N2)) @ ((semiring_char_0_fact @ A) @ M)) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3105_fact__le__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N2:nat]: ((((ord_less_eq @ A) @ ((semiring_char_0_fact @ A) @ N2)) @ ((semiring_1_of_nat @ A) @ (((power_power @ nat) @ N2) @ N2))))))))). % 6.41/2.69 thf(fact_3106_norm__less__p1, axiom, (! [A:$tType]: (((real_V2822296259951069270ebra_1 @ A) => (! [X3:A]: ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ ((real_V7770717601297561774m_norm @ A) @ (((plus_plus @ A) @ ((real_Vector_of_real @ A) @ ((real_V7770717601297561774m_norm @ A) @ X3))) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_3107_choose__dvd, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [K:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ N2) => (((dvd_dvd @ A) @ (((times_times @ A) @ ((semiring_char_0_fact @ A) @ K)) @ ((semiring_char_0_fact @ A) @ (((minus_minus @ nat) @ N2) @ K)))) @ ((semiring_char_0_fact @ A) @ N2))))))))). % 6.41/2.69 thf(fact_3108_fact__num__eq__if, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => ((semiring_char_0_fact @ A) = (^ [M2:nat]: (((((if @ A) @ (M2 = (zero_zero @ nat))) @ (one_one @ A)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ M2)) @ ((semiring_char_0_fact @ A) @ (((minus_minus @ nat) @ M2) @ (one_one @ nat)))))))))))). % 6.41/2.69 thf(fact_3109_fact__reduce, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((semiring_char_0_fact @ A) @ N2) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ ((semiring_char_0_fact @ A) @ (((minus_minus @ nat) @ N2) @ (one_one @ nat)))))))))))). % 6.41/2.69 thf(fact_3110_tan__gt__zero, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => (((ord_less @ real) @ (zero_zero @ real)) @ ((tan @ real) @ X3))))))). % 6.41/2.69 thf(fact_3111_lemma__tan__total, axiom, (! [Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ Y) => (? [X5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X5) & ((((ord_less @ real) @ X5) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) & (((ord_less @ real) @ Y) @ ((tan @ real) @ X5)))))))))). % 6.41/2.69 thf(fact_3112_lemma__tan__total1, axiom, (! [Y:real]: ((? [X5:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ X5) & ((((ord_less @ real) @ X5) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) & (((tan @ real) @ X5) = Y)))))))). % 6.41/2.69 thf(fact_3113_tan__mono__lt__eq, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ X3) => ((((ord_less @ real) @ X3) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => ((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ Y) => ((((ord_less @ real) @ Y) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => ((((ord_less @ real) @ ((tan @ real) @ X3)) @ ((tan @ real) @ Y)) = (((ord_less @ real) @ X3) @ Y))))))))). % 6.41/2.69 thf(fact_3114_tan__monotone_H, axiom, (! [Y:real,X3:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ Y) => ((((ord_less @ real) @ Y) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => ((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ X3) => ((((ord_less @ real) @ X3) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => ((((ord_less @ real) @ Y) @ X3) = (((ord_less @ real) @ ((tan @ real) @ Y)) @ ((tan @ real) @ X3)))))))))). % 6.41/2.69 thf(fact_3115_tan__monotone, axiom, (! [Y:real,X3:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ Y) => ((((ord_less @ real) @ Y) @ X3) => ((((ord_less @ real) @ X3) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => (((ord_less @ real) @ ((tan @ real) @ Y)) @ ((tan @ real) @ X3)))))))). % 6.41/2.69 thf(fact_3116_tan__total, axiom, (! [Y:real]: ((? [X5:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ X5) & ((((ord_less @ real) @ X5) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) & ((((tan @ real) @ X5) = Y) & (! [Y5:real]: ((((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ Y5) & ((((ord_less @ real) @ Y5) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) & (((tan @ real) @ Y5) = Y))) => (Y5 = X5)))))))))))). % 6.41/2.69 thf(fact_3117_add__tan__eq, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,Y:A]: (((((cos @ A) @ X3) != (zero_zero @ A)) => ((((cos @ A) @ Y) != (zero_zero @ A)) => ((((plus_plus @ A) @ ((tan @ A) @ X3)) @ ((tan @ A) @ Y)) = (((divide_divide @ A) @ ((sin @ A) @ (((plus_plus @ A) @ X3) @ Y))) @ (((times_times @ A) @ ((cos @ A) @ X3)) @ ((cos @ A) @ Y)))))))))))). % 6.41/2.69 thf(fact_3118_tan__total__pos, axiom, (! [Y:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ Y) => (? [X5:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ X5) & ((((ord_less @ real) @ X5) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) & (((tan @ real) @ X5) = Y))))))))). % 6.41/2.69 thf(fact_3119_tan__pos__pi2__le, axiom, (! [X3:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => (((ord_less_eq @ real) @ (zero_zero @ real)) @ ((tan @ real) @ X3))))))). % 6.41/2.69 thf(fact_3120_tan__less__zero, axiom, (! [X3:real]: (((((ord_less @ real) @ (((divide_divide @ real) @ ((uminus_uminus @ real) @ pi)) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) @ X3) => ((((ord_less @ real) @ X3) @ (zero_zero @ real)) => (((ord_less @ real) @ ((tan @ real) @ X3)) @ (zero_zero @ real))))))). % 6.41/2.69 thf(fact_3121_tan__mono__le__eq, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ X3) => ((((ord_less @ real) @ X3) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => ((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ Y) => ((((ord_less @ real) @ Y) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => ((((ord_less_eq @ real) @ ((tan @ real) @ X3)) @ ((tan @ real) @ Y)) = (((ord_less_eq @ real) @ X3) @ Y))))))))). % 6.41/2.69 thf(fact_3122_tan__mono__le, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ X3) => ((((ord_less_eq @ real) @ X3) @ Y) => ((((ord_less @ real) @ Y) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => (((ord_less_eq @ real) @ ((tan @ real) @ X3)) @ ((tan @ real) @ Y)))))))). % 6.41/2.69 thf(fact_3123_tan__bound__pi2, axiom, (! [X3:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ X3)) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ (bit0 @ one2))))) => (((ord_less @ real) @ ((abs_abs @ real) @ ((tan @ real) @ X3))) @ (one_one @ real)))))). % 6.41/2.69 thf(fact_3124_arctan, axiom, (! [Y:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ (arctan @ Y)) & ((((ord_less @ real) @ (arctan @ Y)) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) & (((tan @ real) @ (arctan @ Y)) = Y)))))). % 6.41/2.69 thf(fact_3125_arctan__tan, axiom, (! [X3:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ X3) => ((((ord_less @ real) @ X3) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => ((arctan @ ((tan @ real) @ X3)) = X3)))))). % 6.41/2.69 thf(fact_3126_arctan__unique, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ X3) => ((((ord_less @ real) @ X3) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => ((((tan @ real) @ X3) = Y) => ((arctan @ Y) = X3))))))). % 6.41/2.69 thf(fact_3127_Maclaurin__zero, axiom, (! [A:$tType]: (((zero @ A) => (! [X3:real,N2:nat,Diff:(nat > (A > real))]: (((X3 = (zero_zero @ real)) => ((N2 != (zero_zero @ nat)) => (((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ M2) @ (zero_zero @ A))) @ ((semiring_char_0_fact @ real) @ M2))) @ (((power_power @ real) @ X3) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2)) = ((Diff @ (zero_zero @ nat)) @ (zero_zero @ A))))))))))). % 6.41/2.69 thf(fact_3128_Maclaurin__lemma, axiom, (! [H2:real,F2:(real > real),J:(nat > real),N2:nat]: (((((ord_less @ real) @ (zero_zero @ real)) @ H2) => (? [B10:real]: (((F2 @ H2) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ (J @ M2)) @ ((semiring_char_0_fact @ real) @ M2))) @ (((power_power @ real) @ H2) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ B10) @ (((divide_divide @ real) @ (((power_power @ real) @ H2) @ N2)) @ ((semiring_char_0_fact @ real) @ N2))))))))))). % 6.41/2.69 thf(fact_3129_tan__add, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,Y:A]: (((((cos @ A) @ X3) != (zero_zero @ A)) => ((((cos @ A) @ Y) != (zero_zero @ A)) => ((((cos @ A) @ (((plus_plus @ A) @ X3) @ Y)) != (zero_zero @ A)) => (((tan @ A) @ (((plus_plus @ A) @ X3) @ Y)) = (((divide_divide @ A) @ (((plus_plus @ A) @ ((tan @ A) @ X3)) @ ((tan @ A) @ Y))) @ (((minus_minus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((tan @ A) @ X3)) @ ((tan @ A) @ Y)))))))))))))). % 6.41/2.69 thf(fact_3130_tan__diff, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,Y:A]: (((((cos @ A) @ X3) != (zero_zero @ A)) => ((((cos @ A) @ Y) != (zero_zero @ A)) => ((((cos @ A) @ (((minus_minus @ A) @ X3) @ Y)) != (zero_zero @ A)) => (((tan @ A) @ (((minus_minus @ A) @ X3) @ Y)) = (((divide_divide @ A) @ (((minus_minus @ A) @ ((tan @ A) @ X3)) @ ((tan @ A) @ Y))) @ (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((tan @ A) @ X3)) @ ((tan @ A) @ Y)))))))))))))). % 6.41/2.69 thf(fact_3131_lemma__tan__add1, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,Y:A]: (((((cos @ A) @ X3) != (zero_zero @ A)) => ((((cos @ A) @ Y) != (zero_zero @ A)) => ((((minus_minus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((tan @ A) @ X3)) @ ((tan @ A) @ Y))) = (((divide_divide @ A) @ ((cos @ A) @ (((plus_plus @ A) @ X3) @ Y))) @ (((times_times @ A) @ ((cos @ A) @ X3)) @ ((cos @ A) @ Y)))))))))))). % 6.41/2.69 thf(fact_3132_minus__sin__cos__eq, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: ((((uminus_uminus @ A) @ ((sin @ A) @ X3)) = ((cos @ A) @ (((plus_plus @ A) @ X3) @ (((divide_divide @ A) @ ((real_Vector_of_real @ A) @ pi)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))). % 6.41/2.69 thf(fact_3133_tan__total__pi4, axiom, (! [X3:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ X3)) @ (one_one @ real)) => (? [Z3:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ (bit0 @ one2)))))) @ Z3) & ((((ord_less @ real) @ Z3) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ (bit0 @ one2))))) & (((tan @ real) @ Z3) = X3))))))))). % 6.41/2.69 thf(fact_3134_tan__half, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => ((tan @ A) = (^ [X:A]: ((((divide_divide @ A) @ ((sin @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ X))) @ (((plus_plus @ A) @ ((cos @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ X))) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_3135_arsinh__def, axiom, (! [A:$tType]: (((ln @ A) => ((arsinh @ A) = (^ [X:A]: (((ln_ln @ A) @ (((plus_plus @ A) @ X) @ (((powr @ A) @ (((plus_plus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ A))) @ ((real_Vector_of_real @ A) @ (((divide_divide @ real) @ (one_one @ real)) @ ((numeral_numeral @ real) @ (bit0 @ one2)))))))))))))). % 6.41/2.69 thf(fact_3136_Maclaurin__sin__expansion3, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (? [T7:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ T7) & ((((ord_less @ real) @ T7) @ X3) & (((sin @ real) @ X3) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (sin_coeff @ M2)) @ (((power_power @ real) @ X3) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ (((divide_divide @ real) @ ((sin @ real) @ (((plus_plus @ real) @ T7) @ (((times_times @ real) @ (((times_times @ real) @ (((divide_divide @ real) @ (one_one @ real)) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) @ ((semiring_1_of_nat @ real) @ N2))) @ pi)))) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ real) @ X3) @ N2))))))))))))). % 6.41/2.69 thf(fact_3137_Maclaurin__sin__expansion4, axiom, (! [X3:real,N2:nat]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (? [T7:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ T7) & ((((ord_less_eq @ real) @ T7) @ X3) & (((sin @ real) @ X3) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (sin_coeff @ M2)) @ (((power_power @ real) @ X3) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ (((divide_divide @ real) @ ((sin @ real) @ (((plus_plus @ real) @ T7) @ (((times_times @ real) @ (((times_times @ real) @ (((divide_divide @ real) @ (one_one @ real)) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) @ ((semiring_1_of_nat @ real) @ N2))) @ pi)))) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ real) @ X3) @ N2)))))))))))). % 6.41/2.69 thf(fact_3138_sin__coeff__def, axiom, (sin_coeff = (^ [N:nat]: (((((if @ real) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ (zero_zero @ real)) @ (((divide_divide @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ (((divide_divide @ nat) @ (((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat)))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((semiring_char_0_fact @ real) @ N))))))). % 6.41/2.69 thf(fact_3139_Maclaurin__exp__lt, axiom, (! [X3:real,N2:nat]: (((X3 != (zero_zero @ real)) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (? [T7:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ ((abs_abs @ real) @ T7)) & ((((ord_less @ real) @ ((abs_abs @ real) @ T7)) @ ((abs_abs @ real) @ X3)) & (((exp @ real) @ X3) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((divide_divide @ real) @ (((power_power @ real) @ X3) @ M2)) @ ((semiring_char_0_fact @ real) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ (((divide_divide @ real) @ ((exp @ real) @ T7)) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ real) @ X3) @ N2))))))))))))). % 6.41/2.69 thf(fact_3140_round__altdef, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => ((archimedean_round @ A) = (^ [X:A]: (((((if @ int) @ (((ord_less_eq @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ ((archimedean_frac @ A) @ X))) @ ((archimedean_ceiling @ A) @ X)) @ ((archim6421214686448440834_floor @ A) @ X))))))))). % 6.41/2.69 thf(fact_3141_exp__less__mono, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ X3) @ Y) => (((ord_less @ real) @ ((exp @ real) @ X3)) @ ((exp @ real) @ Y)))))). % 6.41/2.69 thf(fact_3142_exp__less__cancel__iff, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ ((exp @ real) @ X3)) @ ((exp @ real) @ Y)) = (((ord_less @ real) @ X3) @ Y))))). % 6.41/2.69 thf(fact_3143_exp__zero, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => (((exp @ A) @ (zero_zero @ A)) = (one_one @ A)))))). % 6.41/2.69 thf(fact_3144_frac__of__int, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z2:int]: ((((archimedean_frac @ A) @ ((ring_1_of_int @ A) @ Z2)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_3145_sin__coeff__0, axiom, ((sin_coeff @ (zero_zero @ nat)) = (zero_zero @ real))). % 6.41/2.69 thf(fact_3146_one__less__exp__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ (one_one @ real)) @ ((exp @ real) @ X3)) = (((ord_less @ real) @ (zero_zero @ real)) @ X3))))). % 6.41/2.69 thf(fact_3147_exp__less__one__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ ((exp @ real) @ X3)) @ (one_one @ real)) = (((ord_less @ real) @ X3) @ (zero_zero @ real)))))). % 6.41/2.69 thf(fact_3148_exp__ln__iff, axiom, (! [X3:real]: (((((exp @ real) @ ((ln_ln @ real) @ X3)) = X3) = (((ord_less @ real) @ (zero_zero @ real)) @ X3))))). % 6.41/2.69 thf(fact_3149_exp__ln, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((exp @ real) @ ((ln_ln @ real) @ X3)) = X3))))). % 6.41/2.69 thf(fact_3150_fact__mono__nat, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => (((ord_less_eq @ nat) @ ((semiring_char_0_fact @ nat) @ M)) @ ((semiring_char_0_fact @ nat) @ N2)))))). % 6.41/2.69 thf(fact_3151_fact__ge__self, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ N2) @ ((semiring_char_0_fact @ nat) @ N2))))). % 6.41/2.69 thf(fact_3152_exp__less__cancel, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ ((exp @ real) @ X3)) @ ((exp @ real) @ Y)) => (((ord_less @ real) @ X3) @ Y))))). % 6.41/2.69 thf(fact_3153_exp__not__eq__zero, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => (! [X3:A]: ((((exp @ A) @ X3) != (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_3154_fact__less__mono__nat, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less @ nat) @ M) @ N2) => (((ord_less @ nat) @ ((semiring_char_0_fact @ nat) @ M)) @ ((semiring_char_0_fact @ nat) @ N2))))))). % 6.41/2.69 thf(fact_3155_exp__total, axiom, (! [Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ Y) => (? [X5:real]: ((((exp @ real) @ X5) = Y))))))). % 6.41/2.69 thf(fact_3156_exp__gt__zero, axiom, (! [X3:real]: ((((ord_less @ real) @ (zero_zero @ real)) @ ((exp @ real) @ X3))))). % 6.41/2.69 thf(fact_3157_not__exp__less__zero, axiom, (! [X3:real]: ((~ ((((ord_less @ real) @ ((exp @ real) @ X3)) @ (zero_zero @ real))))))). % 6.41/2.69 thf(fact_3158_mult__exp__exp, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,Y:A]: (((((times_times @ A) @ ((exp @ A) @ X3)) @ ((exp @ A) @ Y)) = ((exp @ A) @ (((plus_plus @ A) @ X3) @ Y))))))))). % 6.41/2.69 thf(fact_3159_exp__add__commuting, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => (! [X3:A,Y:A]: ((((((times_times @ A) @ X3) @ Y) = (((times_times @ A) @ Y) @ X3)) => (((exp @ A) @ (((plus_plus @ A) @ X3) @ Y)) = (((times_times @ A) @ ((exp @ A) @ X3)) @ ((exp @ A) @ Y)))))))))). % 6.41/2.69 thf(fact_3160_frac__ge__0, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((archimedean_frac @ A) @ X3)))))))). % 6.41/2.69 thf(fact_3161_frac__lt__1, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: ((((ord_less @ A) @ ((archimedean_frac @ A) @ X3)) @ (one_one @ A)))))))). % 6.41/2.69 thf(fact_3162_frac__1__eq, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: ((((archimedean_frac @ A) @ (((plus_plus @ A) @ X3) @ (one_one @ A))) = ((archimedean_frac @ A) @ X3)))))))). % 6.41/2.69 thf(fact_3163_fact__ge__Suc__0__nat, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ ((semiring_char_0_fact @ nat) @ N2))))). % 6.41/2.69 thf(fact_3164_exp__gt__one, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((ord_less @ real) @ (one_one @ real)) @ ((exp @ real) @ X3)))))). % 6.41/2.69 thf(fact_3165_dvd__fact, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ M) => ((((ord_less_eq @ nat) @ M) @ N2) => (((dvd_dvd @ nat) @ M) @ ((semiring_char_0_fact @ nat) @ N2))))))). % 6.41/2.69 thf(fact_3166_fact__diff__Suc, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ N2) @ (suc @ M)) => (((semiring_char_0_fact @ nat) @ (((minus_minus @ nat) @ (suc @ M)) @ N2)) = (((times_times @ nat) @ (((minus_minus @ nat) @ (suc @ M)) @ N2)) @ ((semiring_char_0_fact @ nat) @ (((minus_minus @ nat) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_3167_ln__ge__iff, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ Y) @ ((ln_ln @ real) @ X3)) = (((ord_less_eq @ real) @ ((exp @ real) @ Y)) @ X3)))))). % 6.41/2.69 thf(fact_3168_fact__div__fact__le__pow, axiom, (! [R2:nat,N2:nat]: (((((ord_less_eq @ nat) @ R2) @ N2) => (((ord_less_eq @ nat) @ (((divide_divide @ nat) @ ((semiring_char_0_fact @ nat) @ N2)) @ ((semiring_char_0_fact @ nat) @ (((minus_minus @ nat) @ N2) @ R2)))) @ (((power_power @ nat) @ N2) @ R2)))))). % 6.41/2.69 thf(fact_3169_powr__def, axiom, (! [A:$tType]: (((ln @ A) => ((powr @ A) = (^ [X:A,A4:A]: (((((if @ A) @ (X = (zero_zero @ A))) @ (zero_zero @ A)) @ ((exp @ A) @ (((times_times @ A) @ A4) @ ((ln_ln @ A) @ X))))))))))). % 6.41/2.69 thf(fact_3170_exp__divide__power__eq, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [N2:nat,X3:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((power_power @ A) @ ((exp @ A) @ (((divide_divide @ A) @ X3) @ ((semiring_1_of_nat @ A) @ N2)))) @ N2) = ((exp @ A) @ X3))))))))). % 6.41/2.69 thf(fact_3171_frac__eq, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((archimedean_frac @ A) @ X3) = X3) = ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) & (((ord_less @ A) @ X3) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_3172_frac__add, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Y:A]: ((((((ord_less @ A) @ (((plus_plus @ A) @ ((archimedean_frac @ A) @ X3)) @ ((archimedean_frac @ A) @ Y))) @ (one_one @ A)) => (((archimedean_frac @ A) @ (((plus_plus @ A) @ X3) @ Y)) = (((plus_plus @ A) @ ((archimedean_frac @ A) @ X3)) @ ((archimedean_frac @ A) @ Y)))) & ((~ ((((ord_less @ A) @ (((plus_plus @ A) @ ((archimedean_frac @ A) @ X3)) @ ((archimedean_frac @ A) @ Y))) @ (one_one @ A)))) => (((archimedean_frac @ A) @ (((plus_plus @ A) @ X3) @ Y)) = (((minus_minus @ A) @ (((plus_plus @ A) @ ((archimedean_frac @ A) @ X3)) @ ((archimedean_frac @ A) @ Y))) @ (one_one @ A))))))))))). % 6.41/2.69 thf(fact_3173_tanh__altdef, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => ((tanh @ A) = (^ [X:A]: ((((divide_divide @ A) @ (((minus_minus @ A) @ ((exp @ A) @ X)) @ ((exp @ A) @ ((uminus_uminus @ A) @ X)))) @ (((plus_plus @ A) @ ((exp @ A) @ X)) @ ((exp @ A) @ ((uminus_uminus @ A) @ X))))))))))). % 6.41/2.69 thf(fact_3174_sin__coeff__Suc, axiom, (! [N2:nat]: (((sin_coeff @ (suc @ N2)) = (((divide_divide @ real) @ (cos_coeff @ N2)) @ ((semiring_1_of_nat @ real) @ (suc @ N2))))))). % 6.41/2.69 thf(fact_3175_floor__add, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Y:A]: ((((((ord_less @ A) @ (((plus_plus @ A) @ ((archimedean_frac @ A) @ X3)) @ ((archimedean_frac @ A) @ Y))) @ (one_one @ A)) => (((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X3) @ Y)) = (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ ((archim6421214686448440834_floor @ A) @ Y)))) & ((~ ((((ord_less @ A) @ (((plus_plus @ A) @ ((archimedean_frac @ A) @ X3)) @ ((archimedean_frac @ A) @ Y))) @ (one_one @ A)))) => (((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X3) @ Y)) = (((plus_plus @ int) @ (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ ((archim6421214686448440834_floor @ A) @ Y))) @ (one_one @ int))))))))))). % 6.41/2.69 thf(fact_3176_cos__coeff__Suc, axiom, (! [N2:nat]: (((cos_coeff @ (suc @ N2)) = (((divide_divide @ real) @ ((uminus_uminus @ real) @ (sin_coeff @ N2))) @ ((semiring_1_of_nat @ real) @ (suc @ N2))))))). % 6.41/2.69 thf(fact_3177_exp__ge__one__plus__x__over__n__power__n, axiom, (! [N2:nat,X3:real]: (((((ord_less_eq @ real) @ ((uminus_uminus @ real) @ ((semiring_1_of_nat @ real) @ N2))) @ X3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((ord_less_eq @ real) @ (((power_power @ real) @ (((plus_plus @ real) @ (one_one @ real)) @ (((divide_divide @ real) @ X3) @ ((semiring_1_of_nat @ real) @ N2)))) @ N2)) @ ((exp @ real) @ X3))))))). % 6.41/2.69 thf(fact_3178_exp__ge__one__minus__x__over__n__power__n, axiom, (! [X3:real,N2:nat]: (((((ord_less_eq @ real) @ X3) @ ((semiring_1_of_nat @ real) @ N2)) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((ord_less_eq @ real) @ (((power_power @ real) @ (((minus_minus @ real) @ (one_one @ real)) @ (((divide_divide @ real) @ X3) @ ((semiring_1_of_nat @ real) @ N2)))) @ N2)) @ ((exp @ real) @ ((uminus_uminus @ real) @ X3)))))))). % 6.41/2.69 thf(fact_3179_log__base__10__eq2, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((log @ ((numeral_numeral @ real) @ (bit0 @ (bit1 @ (bit0 @ one2))))) @ X3) = (((times_times @ real) @ ((log @ ((numeral_numeral @ real) @ (bit0 @ (bit1 @ (bit0 @ one2))))) @ ((exp @ real) @ (one_one @ real)))) @ ((ln_ln @ real) @ X3))))))). % 6.41/2.69 thf(fact_3180_log__base__10__eq1, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((log @ ((numeral_numeral @ real) @ (bit0 @ (bit1 @ (bit0 @ one2))))) @ X3) = (((times_times @ real) @ (((divide_divide @ real) @ ((ln_ln @ real) @ ((exp @ real) @ (one_one @ real)))) @ ((ln_ln @ real) @ ((numeral_numeral @ real) @ (bit0 @ (bit1 @ (bit0 @ one2))))))) @ ((ln_ln @ real) @ X3))))))). % 6.41/2.69 thf(fact_3181_sin__tan, axiom, (! [X3:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ X3)) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => (((sin @ real) @ X3) = (((divide_divide @ real) @ ((tan @ real) @ X3)) @ (sqrt @ (((plus_plus @ real) @ (one_one @ real)) @ (((power_power @ real) @ ((tan @ real) @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_3182_cos__tan, axiom, (! [X3:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ X3)) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => (((cos @ real) @ X3) = (((divide_divide @ real) @ (one_one @ real)) @ (sqrt @ (((plus_plus @ real) @ (one_one @ real)) @ (((power_power @ real) @ ((tan @ real) @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_3183_monoseq__def, axiom, (! [A:$tType]: (((order @ A) => ((topological_monoseq @ A) = (^ [X6:(nat > A)]: (((! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => (((ord_less_eq @ A) @ (X6 @ M2)) @ (X6 @ N))))) | (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => (((ord_less_eq @ A) @ (X6 @ N)) @ (X6 @ M2))))))))))))). % 6.41/2.69 thf(fact_3184_monoI2, axiom, (! [A:$tType]: (((order @ A) => (! [X8:(nat > A)]: (((! [M3:nat,N3:nat]: (((((ord_less_eq @ nat) @ M3) @ N3) => (((ord_less_eq @ A) @ (X8 @ N3)) @ (X8 @ M3))))) => ((topological_monoseq @ A) @ X8)))))))). % 6.41/2.69 thf(fact_3185_monoI1, axiom, (! [A:$tType]: (((order @ A) => (! [X8:(nat > A)]: (((! [M3:nat,N3:nat]: (((((ord_less_eq @ nat) @ M3) @ N3) => (((ord_less_eq @ A) @ (X8 @ M3)) @ (X8 @ N3))))) => ((topological_monoseq @ A) @ X8)))))))). % 6.41/2.69 thf(fact_3186_real__sqrt__less__iff, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ (sqrt @ X3)) @ (sqrt @ Y)) = (((ord_less @ real) @ X3) @ Y))))). % 6.41/2.69 thf(fact_3187_real__sqrt__lt__0__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ (sqrt @ X3)) @ (zero_zero @ real)) = (((ord_less @ real) @ X3) @ (zero_zero @ real)))))). % 6.41/2.69 thf(fact_3188_real__sqrt__gt__0__iff, axiom, (! [Y:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ (sqrt @ Y)) = (((ord_less @ real) @ (zero_zero @ real)) @ Y))))). % 6.41/2.69 thf(fact_3189_real__sqrt__lt__1__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ (sqrt @ X3)) @ (one_one @ real)) = (((ord_less @ real) @ X3) @ (one_one @ real)))))). % 6.41/2.69 thf(fact_3190_real__sqrt__gt__1__iff, axiom, (! [Y:real]: (((((ord_less @ real) @ (one_one @ real)) @ (sqrt @ Y)) = (((ord_less @ real) @ (one_one @ real)) @ Y))))). % 6.41/2.69 thf(fact_3191_real__sqrt__less__mono, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ X3) @ Y) => (((ord_less @ real) @ (sqrt @ X3)) @ (sqrt @ Y)))))). % 6.41/2.69 thf(fact_3192_real__sqrt__gt__zero, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((ord_less @ real) @ (zero_zero @ real)) @ (sqrt @ X3)))))). % 6.41/2.69 thf(fact_3193_sqrt2__less__2, axiom, (((ord_less @ real) @ (sqrt @ ((numeral_numeral @ real) @ (bit0 @ one2)))) @ ((numeral_numeral @ real) @ (bit0 @ one2)))). % 6.41/2.69 thf(fact_3194_real__less__rsqrt, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ (((power_power @ real) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ Y) => (((ord_less @ real) @ X3) @ (sqrt @ Y)))))). % 6.41/2.69 thf(fact_3195_lemma__real__divide__sqrt__less, axiom, (! [U:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ U) => (((ord_less @ real) @ (((divide_divide @ real) @ U) @ (sqrt @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ U))))). % 6.41/2.69 thf(fact_3196_real__less__lsqrt, axiom, (! [X3:real,Y:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ Y) => ((((ord_less @ real) @ X3) @ (((power_power @ real) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) => (((ord_less @ real) @ (sqrt @ X3)) @ Y))))))). % 6.41/2.69 thf(fact_3197_ln__sqrt, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((ln_ln @ real) @ (sqrt @ X3)) = (((divide_divide @ real) @ ((ln_ln @ real) @ X3)) @ ((numeral_numeral @ real) @ (bit0 @ one2)))))))). % 6.41/2.69 thf(fact_3198_arsinh__real__aux, axiom, (! [X3:real]: ((((ord_less @ real) @ (zero_zero @ real)) @ (((plus_plus @ real) @ X3) @ (sqrt @ (((plus_plus @ real) @ (((power_power @ real) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ real)))))))). % 6.41/2.69 thf(fact_3199_real__sqrt__sum__squares__less, axiom, (! [X3:real,U:real,Y:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ X3)) @ (((divide_divide @ real) @ U) @ (sqrt @ ((numeral_numeral @ real) @ (bit0 @ one2))))) => ((((ord_less @ real) @ ((abs_abs @ real) @ Y)) @ (((divide_divide @ real) @ U) @ (sqrt @ ((numeral_numeral @ real) @ (bit0 @ one2))))) => (((ord_less @ real) @ (sqrt @ (((plus_plus @ real) @ (((power_power @ real) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ real) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ U)))))). % 6.41/2.69 thf(fact_3200_sqrt__sum__squares__half__less, axiom, (! [X3:real,U:real,Y:real]: (((((ord_less @ real) @ X3) @ (((divide_divide @ real) @ U) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => ((((ord_less @ real) @ Y) @ (((divide_divide @ real) @ U) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ Y) => (((ord_less @ real) @ (sqrt @ (((plus_plus @ real) @ (((power_power @ real) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ real) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ U)))))))). % 6.41/2.69 thf(fact_3201_monoseq__Suc, axiom, (! [A:$tType]: (((order @ A) => ((topological_monoseq @ A) = (^ [X6:(nat > A)]: (((! [N:nat]: ((((ord_less_eq @ A) @ (X6 @ N)) @ (X6 @ (suc @ N))))) | (! [N:nat]: ((((ord_less_eq @ A) @ (X6 @ (suc @ N))) @ (X6 @ N)))))))))))). % 6.41/2.69 thf(fact_3202_mono__SucI2, axiom, (! [A:$tType]: (((order @ A) => (! [X8:(nat > A)]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (X8 @ (suc @ N3))) @ (X8 @ N3)))) => ((topological_monoseq @ A) @ X8)))))))). % 6.41/2.69 thf(fact_3203_mono__SucI1, axiom, (! [A:$tType]: (((order @ A) => (! [X8:(nat > A)]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (X8 @ N3)) @ (X8 @ (suc @ N3))))) => ((topological_monoseq @ A) @ X8)))))))). % 6.41/2.69 thf(fact_3204_pochhammer__double, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [Z2:A,N2:nat]: (((((comm_s3205402744901411588hammer @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Z2)) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) = (((times_times @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)))) @ (((comm_s3205402744901411588hammer @ A) @ Z2) @ N2))) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ Z2) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ N2))))))))). % 6.41/2.69 thf(fact_3205_of__nat__code, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((semiring_1_of_nat @ A) = (^ [N:nat]: (((((semiri8178284476397505188at_aux @ A) @ (^ [I3:A]: ((((plus_plus @ A) @ I3) @ (one_one @ A))))) @ N) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3206_gchoose__row__sum__weighted, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [R2:A,M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((gbinomial @ A) @ R2) @ K3)) @ (((minus_minus @ A) @ (((divide_divide @ A) @ R2) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ ((semiring_1_of_nat @ A) @ K3)))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ M)) = (((times_times @ A) @ (((divide_divide @ A) @ ((semiring_1_of_nat @ A) @ (suc @ M))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((gbinomial @ A) @ R2) @ (suc @ M)))))))))). % 6.41/2.69 thf(fact_3207_central__binomial__lower__bound, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((ord_less_eq @ real) @ (((divide_divide @ real) @ (((power_power @ real) @ ((numeral_numeral @ real) @ (bit0 @ (bit0 @ one2)))) @ N2)) @ (((times_times @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ real) @ N2)))) @ ((semiring_1_of_nat @ real) @ ((binomial @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) @ N2))))))). % 6.41/2.69 thf(fact_3208_complex__unimodular__polar, axiom, (! [Z2:complex]: (((((real_V7770717601297561774m_norm @ complex) @ Z2) = (one_one @ real)) => (~ ((! [T7:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ T7) => ((((ord_less @ real) @ T7) @ (((times_times @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ pi)) => (Z2 != ((complex2 @ ((cos @ real) @ T7)) @ ((sin @ real) @ T7))))))))))))). % 6.41/2.69 thf(fact_3209_binomial__Suc__n, axiom, (! [N2:nat]: ((((binomial @ (suc @ N2)) @ N2) = (suc @ N2))))). % 6.41/2.69 thf(fact_3210_gbinomial__0_I2_J, axiom, (! [B:$tType]: ((((semiring_char_0 @ B) & (semidom_divide @ B)) => (! [K:nat]: (((((gbinomial @ B) @ (zero_zero @ B)) @ (suc @ K)) = (zero_zero @ B)))))))). % 6.41/2.69 thf(fact_3211_binomial__1, axiom, (! [N2:nat]: ((((binomial @ N2) @ (suc @ (zero_zero @ nat))) = N2)))). % 6.41/2.69 thf(fact_3212_binomial__0__Suc, axiom, (! [K:nat]: ((((binomial @ (zero_zero @ nat)) @ (suc @ K)) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_3213_binomial__eq__0__iff, axiom, (! [N2:nat,K:nat]: (((((binomial @ N2) @ K) = (zero_zero @ nat)) = (((ord_less @ nat) @ N2) @ K))))). % 6.41/2.69 thf(fact_3214_gbinomial__0_I1_J, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (semidom_divide @ A)) => (! [A2:A]: (((((gbinomial @ A) @ A2) @ (zero_zero @ nat)) = (one_one @ A)))))))). % 6.41/2.69 thf(fact_3215_binomial__Suc__Suc, axiom, (! [N2:nat,K:nat]: ((((binomial @ (suc @ N2)) @ (suc @ K)) = (((plus_plus @ nat) @ ((binomial @ N2) @ K)) @ ((binomial @ N2) @ (suc @ K))))))). % 6.41/2.69 thf(fact_3216_pochhammer__0, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A2:A]: (((((comm_s3205402744901411588hammer @ A) @ A2) @ (zero_zero @ nat)) = (one_one @ A)))))))). % 6.41/2.69 thf(fact_3217_gbinomial__Suc0, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (semidom_divide @ A)) => (! [A2:A]: (((((gbinomial @ A) @ A2) @ (suc @ (zero_zero @ nat))) = A2))))))). % 6.41/2.69 thf(fact_3218_pochhammer__Suc0, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A2:A]: (((((comm_s3205402744901411588hammer @ A) @ A2) @ (suc @ (zero_zero @ nat))) = A2))))))). % 6.41/2.69 thf(fact_3219_binomial__n__0, axiom, (! [N2:nat]: ((((binomial @ N2) @ (zero_zero @ nat)) = (one_one @ nat))))). % 6.41/2.69 thf(fact_3220_zero__less__binomial__iff, axiom, (! [N2:nat,K:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((binomial @ N2) @ K)) = (((ord_less_eq @ nat) @ K) @ N2))))). % 6.41/2.69 thf(fact_3221_binomial__eq__0, axiom, (! [N2:nat,K:nat]: (((((ord_less @ nat) @ N2) @ K) => (((binomial @ N2) @ K) = (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_3222_Suc__times__binomial, axiom, (! [K:nat,N2:nat]: (((((times_times @ nat) @ (suc @ K)) @ ((binomial @ (suc @ N2)) @ (suc @ K))) = (((times_times @ nat) @ (suc @ N2)) @ ((binomial @ N2) @ K)))))). % 6.41/2.69 thf(fact_3223_Suc__times__binomial__eq, axiom, (! [N2:nat,K:nat]: (((((times_times @ nat) @ (suc @ N2)) @ ((binomial @ N2) @ K)) = (((times_times @ nat) @ ((binomial @ (suc @ N2)) @ (suc @ K))) @ (suc @ K)))))). % 6.41/2.69 thf(fact_3224_binomial__symmetric, axiom, (! [K:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ N2) => (((binomial @ N2) @ K) = ((binomial @ N2) @ (((minus_minus @ nat) @ N2) @ K))))))). % 6.41/2.69 thf(fact_3225_choose__mult__lemma, axiom, (! [M:nat,R2:nat,K:nat]: (((((times_times @ nat) @ ((binomial @ (((plus_plus @ nat) @ (((plus_plus @ nat) @ M) @ R2)) @ K)) @ (((plus_plus @ nat) @ M) @ K))) @ ((binomial @ (((plus_plus @ nat) @ M) @ K)) @ K)) = (((times_times @ nat) @ ((binomial @ (((plus_plus @ nat) @ (((plus_plus @ nat) @ M) @ R2)) @ K)) @ K)) @ ((binomial @ (((plus_plus @ nat) @ M) @ R2)) @ M)))))). % 6.41/2.69 thf(fact_3226_binomial__le__pow, axiom, (! [R2:nat,N2:nat]: (((((ord_less_eq @ nat) @ R2) @ N2) => (((ord_less_eq @ nat) @ ((binomial @ N2) @ R2)) @ (((power_power @ nat) @ N2) @ R2)))))). % 6.41/2.69 thf(fact_3227_pochhammer__pos, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X3:A,N2:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ X3) => (((ord_less @ A) @ (zero_zero @ A)) @ (((comm_s3205402744901411588hammer @ A) @ X3) @ N2))))))))). % 6.41/2.69 thf(fact_3228_pochhammer__neq__0__mono, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A2:A,M:nat,N2:nat]: ((((((comm_s3205402744901411588hammer @ A) @ A2) @ M) != (zero_zero @ A)) => ((((ord_less_eq @ nat) @ N2) @ M) => ((((comm_s3205402744901411588hammer @ A) @ A2) @ N2) != (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_3229_pochhammer__eq__0__mono, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A2:A,N2:nat,M:nat]: ((((((comm_s3205402744901411588hammer @ A) @ A2) @ N2) = (zero_zero @ A)) => ((((ord_less_eq @ nat) @ N2) @ M) => ((((comm_s3205402744901411588hammer @ A) @ A2) @ M) = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_3230_zero__less__binomial, axiom, (! [K:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ N2) => (((ord_less @ nat) @ (zero_zero @ nat)) @ ((binomial @ N2) @ K)))))). % 6.41/2.69 thf(fact_3231_Suc__times__binomial__add, axiom, (! [A2:nat,B2:nat]: (((((times_times @ nat) @ (suc @ A2)) @ ((binomial @ (suc @ (((plus_plus @ nat) @ A2) @ B2))) @ (suc @ A2))) = (((times_times @ nat) @ (suc @ B2)) @ ((binomial @ (suc @ (((plus_plus @ nat) @ A2) @ B2))) @ A2)))))). % 6.41/2.69 thf(fact_3232_choose__mult, axiom, (! [K:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ M) => ((((ord_less_eq @ nat) @ M) @ N2) => ((((times_times @ nat) @ ((binomial @ N2) @ M)) @ ((binomial @ M) @ K)) = (((times_times @ nat) @ ((binomial @ N2) @ K)) @ ((binomial @ (((minus_minus @ nat) @ N2) @ K)) @ (((minus_minus @ nat) @ M) @ K))))))))). % 6.41/2.69 thf(fact_3233_binomial__Suc__Suc__eq__times, axiom, (! [N2:nat,K:nat]: ((((binomial @ (suc @ N2)) @ (suc @ K)) = (((divide_divide @ nat) @ (((times_times @ nat) @ (suc @ N2)) @ ((binomial @ N2) @ K))) @ (suc @ K)))))). % 6.41/2.69 thf(fact_3234_gbinomial__pochhammer_H, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((gbinomial @ A) = (^ [A4:A,K3:nat]: ((((divide_divide @ A) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ (((minus_minus @ A) @ A4) @ ((semiring_1_of_nat @ A) @ K3))) @ (one_one @ A))) @ K3)) @ ((semiring_char_0_fact @ A) @ K3))))))))). % 6.41/2.69 thf(fact_3235_gbinomial__Suc__Suc, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A2:A,K:nat]: (((((gbinomial @ A) @ (((plus_plus @ A) @ A2) @ (one_one @ A))) @ (suc @ K)) = (((plus_plus @ A) @ (((gbinomial @ A) @ A2) @ K)) @ (((gbinomial @ A) @ A2) @ (suc @ K)))))))))). % 6.41/2.69 thf(fact_3236_gbinomial__of__nat__symmetric, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ N2) => ((((gbinomial @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ K) = (((gbinomial @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (((minus_minus @ nat) @ N2) @ K)))))))))). % 6.41/2.69 thf(fact_3237_pochhammer__nonneg, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [X3:A,N2:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ X3) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((comm_s3205402744901411588hammer @ A) @ X3) @ N2))))))))). % 6.41/2.69 thf(fact_3238_of__nat__aux_Osimps_I2_J, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [Inc:(A > A),N2:nat,I:A]: ((((((semiri8178284476397505188at_aux @ A) @ Inc) @ (suc @ N2)) @ I) = ((((semiri8178284476397505188at_aux @ A) @ Inc) @ N2) @ (Inc @ I))))))))). % 6.41/2.69 thf(fact_3239_of__nat__aux_Osimps_I1_J, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [Inc:(A > A),I:A]: ((((((semiri8178284476397505188at_aux @ A) @ Inc) @ (zero_zero @ nat)) @ I) = I))))))). % 6.41/2.69 thf(fact_3240_pochhammer__0__left, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [N2:nat]: ((((N2 = (zero_zero @ nat)) => ((((comm_s3205402744901411588hammer @ A) @ (zero_zero @ A)) @ N2) = (one_one @ A))) & ((N2 != (zero_zero @ nat)) => ((((comm_s3205402744901411588hammer @ A) @ (zero_zero @ A)) @ N2) = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_3241_binomial__absorption, axiom, (! [K:nat,N2:nat]: (((((times_times @ nat) @ (suc @ K)) @ ((binomial @ N2) @ (suc @ K))) = (((times_times @ nat) @ N2) @ ((binomial @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))) @ K)))))). % 6.41/2.69 thf(fact_3242_gbinomial__addition__formula, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A2:A,K:nat]: (((((gbinomial @ A) @ A2) @ (suc @ K)) = (((plus_plus @ A) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A2) @ (one_one @ A))) @ (suc @ K))) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A2) @ (one_one @ A))) @ K))))))))). % 6.41/2.69 thf(fact_3243_gbinomial__ge__n__over__k__pow__k, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [K:nat,A2:A]: (((((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ K)) @ A2) => (((ord_less_eq @ A) @ (((power_power @ A) @ (((divide_divide @ A) @ A2) @ ((semiring_1_of_nat @ A) @ K))) @ K)) @ (((gbinomial @ A) @ A2) @ K))))))))). % 6.41/2.69 thf(fact_3244_gbinomial__mult__1_H, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A2:A,K:nat]: (((((times_times @ A) @ (((gbinomial @ A) @ A2) @ K)) @ A2) = (((plus_plus @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ K)) @ (((gbinomial @ A) @ A2) @ K))) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ (suc @ K))) @ (((gbinomial @ A) @ A2) @ (suc @ K))))))))))). % 6.41/2.69 thf(fact_3245_gbinomial__mult__1, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A2:A,K:nat]: (((((times_times @ A) @ A2) @ (((gbinomial @ A) @ A2) @ K)) = (((plus_plus @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ K)) @ (((gbinomial @ A) @ A2) @ K))) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ (suc @ K))) @ (((gbinomial @ A) @ A2) @ (suc @ K))))))))))). % 6.41/2.69 thf(fact_3246_binomial__fact__lemma, axiom, (! [K:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ N2) => ((((times_times @ nat) @ (((times_times @ nat) @ ((semiring_char_0_fact @ nat) @ K)) @ ((semiring_char_0_fact @ nat) @ (((minus_minus @ nat) @ N2) @ K)))) @ ((binomial @ N2) @ K)) = ((semiring_char_0_fact @ nat) @ N2)))))). % 6.41/2.69 thf(fact_3247_pochhammer__rec, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A2:A,N2:nat]: (((((comm_s3205402744901411588hammer @ A) @ A2) @ (suc @ N2)) = (((times_times @ A) @ A2) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ A2) @ (one_one @ A))) @ N2))))))))). % 6.41/2.69 thf(fact_3248_pochhammer__rec_H, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [Z2:A,N2:nat]: (((((comm_s3205402744901411588hammer @ A) @ Z2) @ (suc @ N2)) = (((times_times @ A) @ (((plus_plus @ A) @ Z2) @ ((semiring_1_of_nat @ A) @ N2))) @ (((comm_s3205402744901411588hammer @ A) @ Z2) @ N2))))))))). % 6.41/2.69 thf(fact_3249_pochhammer__Suc, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A2:A,N2:nat]: (((((comm_s3205402744901411588hammer @ A) @ A2) @ (suc @ N2)) = (((times_times @ A) @ (((comm_s3205402744901411588hammer @ A) @ A2) @ N2)) @ (((plus_plus @ A) @ A2) @ ((semiring_1_of_nat @ A) @ N2)))))))))). % 6.41/2.69 thf(fact_3250_pochhammer__eq__0__iff, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A2:A,N2:nat]: ((((((comm_s3205402744901411588hammer @ A) @ A2) @ N2) = (zero_zero @ A)) = (? [K3:nat]: (((((ord_less @ nat) @ K3) @ N2) & (A2 = ((uminus_uminus @ A) @ ((semiring_1_of_nat @ A) @ K3))))))))))))). % 6.41/2.69 thf(fact_3251_pochhammer__of__nat__eq__0__iff, axiom, (! [A:$tType]: ((((ring_char_0 @ A) & (idom @ A)) => (! [N2:nat,K:nat]: ((((((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ ((semiring_1_of_nat @ A) @ N2))) @ K) = (zero_zero @ A)) = (((ord_less @ nat) @ N2) @ K)))))))). % 6.41/2.69 thf(fact_3252_pochhammer__of__nat__eq__0__lemma, axiom, (! [A:$tType]: (((idom @ A) => (! [N2:nat,K:nat]: (((((ord_less @ nat) @ N2) @ K) => ((((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ ((semiring_1_of_nat @ A) @ N2))) @ K) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3253_pochhammer__of__nat__eq__0__lemma_H, axiom, (! [A:$tType]: ((((ring_char_0 @ A) & (idom @ A)) => (! [K:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ N2) => ((((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ ((semiring_1_of_nat @ A) @ N2))) @ K) != (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3254_pochhammer__product_H, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [Z2:A,N2:nat,M:nat]: (((((comm_s3205402744901411588hammer @ A) @ Z2) @ (((plus_plus @ nat) @ N2) @ M)) = (((times_times @ A) @ (((comm_s3205402744901411588hammer @ A) @ Z2) @ N2)) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ Z2) @ ((semiring_1_of_nat @ A) @ N2))) @ M))))))))). % 6.41/2.69 thf(fact_3255_binomial__ge__n__over__k__pow__k, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [K:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ N2) => (((ord_less_eq @ A) @ (((power_power @ A) @ (((divide_divide @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ ((semiring_1_of_nat @ A) @ K))) @ K)) @ ((semiring_1_of_nat @ A) @ ((binomial @ N2) @ K)))))))))). % 6.41/2.69 thf(fact_3256_binomial__maximum_H, axiom, (! [N2:nat,K:nat]: ((((ord_less_eq @ nat) @ ((binomial @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) @ K)) @ ((binomial @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) @ N2))))). % 6.41/2.69 thf(fact_3257_binomial__mono, axiom, (! [K:nat,K7:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ K7) => ((((ord_less_eq @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K7)) @ N2) => (((ord_less_eq @ nat) @ ((binomial @ N2) @ K)) @ ((binomial @ N2) @ K7))))))). % 6.41/2.69 thf(fact_3258_binomial__antimono, axiom, (! [K:nat,K7:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ K7) => ((((ord_less_eq @ nat) @ (((divide_divide @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ K) => ((((ord_less_eq @ nat) @ K7) @ N2) => (((ord_less_eq @ nat) @ ((binomial @ N2) @ K7)) @ ((binomial @ N2) @ K)))))))). % 6.41/2.69 thf(fact_3259_binomial__maximum, axiom, (! [N2:nat,K:nat]: ((((ord_less_eq @ nat) @ ((binomial @ N2) @ K)) @ ((binomial @ N2) @ (((divide_divide @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))). % 6.41/2.69 thf(fact_3260_binomial__le__pow2, axiom, (! [N2:nat,K:nat]: ((((ord_less_eq @ nat) @ ((binomial @ N2) @ K)) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))))). % 6.41/2.69 thf(fact_3261_choose__reduce__nat, axiom, (! [N2:nat,K:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((binomial @ N2) @ K) = (((plus_plus @ nat) @ ((binomial @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))) @ (((minus_minus @ nat) @ K) @ (one_one @ nat)))) @ ((binomial @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))) @ K)))))))). % 6.41/2.69 thf(fact_3262_times__binomial__minus1__eq, axiom, (! [K:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((times_times @ nat) @ K) @ ((binomial @ N2) @ K)) = (((times_times @ nat) @ N2) @ ((binomial @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))) @ (((minus_minus @ nat) @ K) @ (one_one @ nat))))))))). % 6.41/2.69 thf(fact_3263_Suc__times__gbinomial, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,A2:A]: (((((times_times @ A) @ ((semiring_1_of_nat @ A) @ (suc @ K))) @ (((gbinomial @ A) @ (((plus_plus @ A) @ A2) @ (one_one @ A))) @ (suc @ K))) = (((times_times @ A) @ (((plus_plus @ A) @ A2) @ (one_one @ A))) @ (((gbinomial @ A) @ A2) @ K))))))))). % 6.41/2.69 thf(fact_3264_gbinomial__absorption, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,A2:A]: (((((times_times @ A) @ ((semiring_1_of_nat @ A) @ (suc @ K))) @ (((gbinomial @ A) @ A2) @ (suc @ K))) = (((times_times @ A) @ A2) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A2) @ (one_one @ A))) @ K))))))))). % 6.41/2.69 thf(fact_3265_binomial__altdef__nat, axiom, (! [K:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ N2) => (((binomial @ N2) @ K) = (((divide_divide @ nat) @ ((semiring_char_0_fact @ nat) @ N2)) @ (((times_times @ nat) @ ((semiring_char_0_fact @ nat) @ K)) @ ((semiring_char_0_fact @ nat) @ (((minus_minus @ nat) @ N2) @ K))))))))). % 6.41/2.69 thf(fact_3266_gbinomial__trinomial__revision, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,M:nat,A2:A]: (((((ord_less_eq @ nat) @ K) @ M) => ((((times_times @ A) @ (((gbinomial @ A) @ A2) @ M)) @ (((gbinomial @ A) @ ((semiring_1_of_nat @ A) @ M)) @ K)) = (((times_times @ A) @ (((gbinomial @ A) @ A2) @ K)) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A2) @ ((semiring_1_of_nat @ A) @ K))) @ (((minus_minus @ nat) @ M) @ K))))))))))). % 6.41/2.69 thf(fact_3267_pochhammer__product, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [M:nat,N2:nat,Z2:A]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((comm_s3205402744901411588hammer @ A) @ Z2) @ N2) = (((times_times @ A) @ (((comm_s3205402744901411588hammer @ A) @ Z2) @ M)) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ Z2) @ ((semiring_1_of_nat @ A) @ M))) @ (((minus_minus @ nat) @ N2) @ M))))))))))). % 6.41/2.69 thf(fact_3268_binomial__strict__antimono, axiom, (! [K:nat,K7:nat,N2:nat]: (((((ord_less @ nat) @ K) @ K7) => ((((ord_less_eq @ nat) @ N2) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K)) => ((((ord_less_eq @ nat) @ K7) @ N2) => (((ord_less @ nat) @ ((binomial @ N2) @ K7)) @ ((binomial @ N2) @ K)))))))). % 6.41/2.69 thf(fact_3269_binomial__strict__mono, axiom, (! [K:nat,K7:nat,N2:nat]: (((((ord_less @ nat) @ K) @ K7) => ((((ord_less_eq @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K7)) @ N2) => (((ord_less @ nat) @ ((binomial @ N2) @ K)) @ ((binomial @ N2) @ K7))))))). % 6.41/2.69 thf(fact_3270_binomial__less__binomial__Suc, axiom, (! [K:nat,N2:nat]: (((((ord_less @ nat) @ K) @ (((divide_divide @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) => (((ord_less @ nat) @ ((binomial @ N2) @ K)) @ ((binomial @ N2) @ (suc @ K))))))). % 6.41/2.69 thf(fact_3271_central__binomial__odd, axiom, (! [N2:nat]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) => (((binomial @ N2) @ (suc @ (((divide_divide @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) = ((binomial @ N2) @ (((divide_divide @ nat) @ N2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))). % 6.41/2.69 thf(fact_3272_binomial__addition__formula, axiom, (! [N2:nat,K:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((binomial @ N2) @ (suc @ K)) = (((plus_plus @ nat) @ ((binomial @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))) @ (suc @ K))) @ ((binomial @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))) @ K))))))). % 6.41/2.69 thf(fact_3273_binomial__fact, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ N2) => (((semiring_1_of_nat @ A) @ ((binomial @ N2) @ K)) = (((divide_divide @ A) @ ((semiring_char_0_fact @ A) @ N2)) @ (((times_times @ A) @ ((semiring_char_0_fact @ A) @ K)) @ ((semiring_char_0_fact @ A) @ (((minus_minus @ nat) @ N2) @ K)))))))))))). % 6.41/2.69 thf(fact_3274_fact__binomial, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ N2) => ((((times_times @ A) @ ((semiring_char_0_fact @ A) @ K)) @ ((semiring_1_of_nat @ A) @ ((binomial @ N2) @ K))) = (((divide_divide @ A) @ ((semiring_char_0_fact @ A) @ N2)) @ ((semiring_char_0_fact @ A) @ (((minus_minus @ nat) @ N2) @ K))))))))))). % 6.41/2.69 thf(fact_3275_gbinomial__rec, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A2:A,K:nat]: (((((gbinomial @ A) @ (((plus_plus @ A) @ A2) @ (one_one @ A))) @ (suc @ K)) = (((times_times @ A) @ (((gbinomial @ A) @ A2) @ K)) @ (((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ (one_one @ A))) @ ((semiring_1_of_nat @ A) @ (suc @ K))))))))))). % 6.41/2.69 thf(fact_3276_gbinomial__factors, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A2:A,K:nat]: (((((gbinomial @ A) @ (((plus_plus @ A) @ A2) @ (one_one @ A))) @ (suc @ K)) = (((times_times @ A) @ (((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ (one_one @ A))) @ ((semiring_1_of_nat @ A) @ (suc @ K)))) @ (((gbinomial @ A) @ A2) @ K))))))))). % 6.41/2.69 thf(fact_3277_pochhammer__absorb__comp, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [R2:A,K:nat]: (((((times_times @ A) @ (((minus_minus @ A) @ R2) @ ((semiring_1_of_nat @ A) @ K))) @ (((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ R2)) @ K)) = (((times_times @ A) @ R2) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ R2)) @ (one_one @ A))) @ K))))))))). % 6.41/2.69 thf(fact_3278_gbinomial__minus, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A2:A,K:nat]: (((((gbinomial @ A) @ ((uminus_uminus @ A) @ A2)) @ K) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ K)) @ (((gbinomial @ A) @ (((minus_minus @ A) @ (((plus_plus @ A) @ A2) @ ((semiring_1_of_nat @ A) @ K))) @ (one_one @ A))) @ K))))))))). % 6.41/2.69 thf(fact_3279_gbinomial__reduce__nat, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,A2:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((gbinomial @ A) @ A2) @ K) = (((plus_plus @ A) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A2) @ (one_one @ A))) @ (((minus_minus @ nat) @ K) @ (one_one @ nat)))) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A2) @ (one_one @ A))) @ K)))))))))). % 6.41/2.69 thf(fact_3280_pochhammer__minus, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [B2:A,K:nat]: (((((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ B2)) @ K) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ K)) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ ((semiring_1_of_nat @ A) @ K))) @ (one_one @ A))) @ K))))))))). % 6.41/2.69 thf(fact_3281_pochhammer__minus_H, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [B2:A,K:nat]: (((((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ ((semiring_1_of_nat @ A) @ K))) @ (one_one @ A))) @ K) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ K)) @ (((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ B2)) @ K))))))))). % 6.41/2.69 thf(fact_3282_gbinomial__sum__up__index, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J3:nat]: ((((gbinomial @ A) @ ((semiring_1_of_nat @ A) @ J3)) @ K)))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2)) = (((gbinomial @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (one_one @ A))) @ (((plus_plus @ nat) @ K) @ (one_one @ nat)))))))))). % 6.41/2.69 thf(fact_3283_gbinomial__absorption_H, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,A2:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((gbinomial @ A) @ A2) @ K) = (((times_times @ A) @ (((divide_divide @ A) @ A2) @ ((semiring_1_of_nat @ A) @ K))) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A2) @ (one_one @ A))) @ (((minus_minus @ nat) @ K) @ (one_one @ nat)))))))))))). % 6.41/2.69 thf(fact_3284_binomial__code, axiom, (binomial = (^ [N:nat,K3:nat]: (((((if @ nat) @ (((ord_less @ nat) @ N) @ K3)) @ (zero_zero @ nat)) @ ((((if @ nat) @ (((ord_less @ nat) @ N) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K3))) @ ((binomial @ N) @ (((minus_minus @ nat) @ N) @ K3))) @ (((divide_divide @ nat) @ (((((set_fo6178422350223883121st_nat @ nat) @ (times_times @ nat)) @ (((plus_plus @ nat) @ (((minus_minus @ nat) @ N) @ K3)) @ (one_one @ nat))) @ N) @ (one_one @ nat))) @ ((semiring_char_0_fact @ nat) @ K3)))))))). % 6.41/2.69 thf(fact_3285_gbinomial__code, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((gbinomial @ A) = (^ [A4:A,K3:nat]: (((((if @ A) @ (K3 = (zero_zero @ nat))) @ (one_one @ A)) @ (((divide_divide @ A) @ (((((set_fo6178422350223883121st_nat @ A) @ (^ [L2:nat]: (((times_times @ A) @ (((minus_minus @ A) @ A4) @ ((semiring_1_of_nat @ A) @ L2)))))) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ K3) @ (one_one @ nat))) @ (one_one @ A))) @ ((semiring_char_0_fact @ A) @ K3)))))))))). % 6.41/2.69 thf(fact_3286_pochhammer__times__pochhammer__half, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [Z2:A,N2:nat]: (((((times_times @ A) @ (((comm_s3205402744901411588hammer @ A) @ Z2) @ (suc @ N2))) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ Z2) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ (suc @ N2))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [K3:nat]: ((((plus_plus @ A) @ Z2) @ (((divide_divide @ A) @ ((semiring_1_of_nat @ A) @ K3)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) @ (one_one @ nat))))))))))). % 6.41/2.69 thf(fact_3287_pochhammer__code, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => ((comm_s3205402744901411588hammer @ A) = (^ [A4:A,N:nat]: (((((if @ A) @ (N = (zero_zero @ nat))) @ (one_one @ A)) @ (((((set_fo6178422350223883121st_nat @ A) @ (^ [O:nat]: (((times_times @ A) @ (((plus_plus @ A) @ A4) @ ((semiring_1_of_nat @ A) @ O)))))) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_3288_gbinomial__partial__row__sum, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A2:A,M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((gbinomial @ A) @ A2) @ K3)) @ (((minus_minus @ A) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ ((semiring_1_of_nat @ A) @ K3)))))) @ ((set_ord_atMost @ nat) @ M)) = (((times_times @ A) @ (((divide_divide @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (one_one @ A))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((gbinomial @ A) @ A2) @ (((plus_plus @ nat) @ M) @ (one_one @ nat))))))))))). % 6.41/2.69 thf(fact_3289_atMost__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_atMost @ A) @ K)) = (((ord_less_eq @ A) @ I) @ K)))))))). % 6.41/2.69 thf(fact_3290_finite__atMost, axiom, (! [K:nat]: (((finite_finite2 @ nat) @ ((set_ord_atMost @ nat) @ K))))). % 6.41/2.69 thf(fact_3291_of__nat__prod, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_1 @ A) => (! [F2:(B > nat),A3:(set @ B)]: ((((semiring_1_of_nat @ A) @ ((((groups7121269368397514597t_prod @ B) @ nat) @ F2) @ A3)) = ((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X:B]: (((semiring_1_of_nat @ A) @ (F2 @ X))))) @ A3)))))))). % 6.41/2.69 thf(fact_3292_of__int__prod, axiom, (! [A:$tType,B:$tType]: (((comm_ring_1 @ A) => (! [F2:(B > int),A3:(set @ B)]: ((((ring_1_of_int @ A) @ ((((groups7121269368397514597t_prod @ B) @ int) @ F2) @ A3)) = ((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X:B]: (((ring_1_of_int @ A) @ (F2 @ X))))) @ A3)))))))). % 6.41/2.69 thf(fact_3293_prod__zero__iff, axiom, (! [A:$tType,B:$tType]: (((semidom @ A) => (! [A3:(set @ B),F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3) = (zero_zero @ A)) = (? [X:B]: (((((member @ B) @ X) @ A3) & ((F2 @ X) = (zero_zero @ A))))))))))))). % 6.41/2.69 thf(fact_3294_prod_Oempty, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [G:(B > A)]: ((((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (bot_bot @ (set @ B))) = (one_one @ A)))))))). % 6.41/2.69 thf(fact_3295_prod_Oinfinite, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),G:(B > A)]: (((~ (((finite_finite2 @ B) @ A3))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A3) = (one_one @ A))))))))). % 6.41/2.69 thf(fact_3296_atMost__subset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ X3)) @ ((set_ord_atMost @ A) @ Y)) = (((ord_less_eq @ A) @ X3) @ Y)))))))). % 6.41/2.69 thf(fact_3297_dvd__prod__eqI, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_1 @ A) => (! [A3:(set @ B),A2:B,B2:A,F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((((member @ B) @ A2) @ A3) => ((B2 = (F2 @ A2)) => (((dvd_dvd @ A) @ B2) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3))))))))))). % 6.41/2.69 thf(fact_3298_dvd__prodI, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_1 @ A) => (! [A3:(set @ B),A2:B,F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((((member @ B) @ A2) @ A3) => (((dvd_dvd @ A) @ (F2 @ A2)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3)))))))))). % 6.41/2.69 thf(fact_3299_prod_Odelta, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),A2:B,B2:(B > A)]: ((((finite_finite2 @ B) @ S3) => (((((member @ B) @ A2) @ S3) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A2)) @ (B2 @ K3)) @ (one_one @ A))))) @ S3) = (B2 @ A2))) & ((~ ((((member @ B) @ A2) @ S3))) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A2)) @ (B2 @ K3)) @ (one_one @ A))))) @ S3) = (one_one @ A))))))))))). % 6.41/2.69 thf(fact_3300_prod_Odelta_H, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),A2:B,B2:(B > A)]: ((((finite_finite2 @ B) @ S3) => (((((member @ B) @ A2) @ S3) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (A2 = K3)) @ (B2 @ K3)) @ (one_one @ A))))) @ S3) = (B2 @ A2))) & ((~ ((((member @ B) @ A2) @ S3))) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (A2 = K3)) @ (B2 @ K3)) @ (one_one @ A))))) @ S3) = (one_one @ A))))))))))). % 6.41/2.69 thf(fact_3301_prod_Oinsert, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),X3:B,G:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((~ ((((member @ B) @ X3) @ A3))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((insert @ B) @ X3) @ A3)) = (((times_times @ A) @ (G @ X3)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A3))))))))))). % 6.41/2.69 thf(fact_3302_Icc__subset__Iic__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [L:A,H2:A,H3:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ H2)) @ ((set_ord_atMost @ A) @ H3)) = ((~ ((((ord_less_eq @ A) @ L) @ H2))) | (((ord_less_eq @ A) @ H2) @ H3))))))))). % 6.41/2.69 thf(fact_3303_sum_OatMost__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ (suc @ N2))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ N2))) @ (G @ (suc @ N2)))))))))). % 6.41/2.69 thf(fact_3304_prod_OlessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (suc @ N2))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ N2))) @ (G @ N2))))))))). % 6.41/2.69 thf(fact_3305_prod_OatMost__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ (suc @ N2))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ N2))) @ (G @ (suc @ N2)))))))))). % 6.41/2.69 thf(fact_3306_atMost__0, axiom, (((set_ord_atMost @ nat) @ (zero_zero @ nat)) = (((insert @ nat) @ (zero_zero @ nat)) @ (bot_bot @ (set @ nat))))). % 6.41/2.69 thf(fact_3307_prod_Ocl__ivl__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [N2:nat,M:nat,G:(nat > A)]: ((((((ord_less @ nat) @ (suc @ N2)) @ M) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N2))) = (one_one @ A))) & ((~ ((((ord_less @ nat) @ (suc @ N2)) @ M))) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N2))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))) @ (G @ (suc @ N2)))))))))))). % 6.41/2.69 thf(fact_3308_not__empty__eq__Iic__eq__empty, axiom, (! [A:$tType]: (((preorder @ A) => (! [H2:A]: (((bot_bot @ (set @ A)) != ((set_ord_atMost @ A) @ H2)))))))). % 6.41/2.69 thf(fact_3309_infinite__Iic, axiom, (! [A:$tType]: ((((linorder @ A) & (no_bot @ A)) => (! [A2:A]: ((~ (((finite_finite2 @ A) @ ((set_ord_atMost @ A) @ A2)))))))))). % 6.41/2.69 thf(fact_3310_prod_Oswap__restrict, axiom, (! [A:$tType,B:$tType,C:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),B3:(set @ C),G:(B > (C > A)),R:(B > (C > $o))]: ((((finite_finite2 @ B) @ A3) => (((finite_finite2 @ C) @ B3) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X:B]: (((((groups7121269368397514597t_prod @ C) @ A) @ (G @ X)) @ ((collect @ C) @ (^ [Y2:C]: (((((member @ C) @ Y2) @ B3) & ((R @ X) @ Y2))))))))) @ A3) = ((((groups7121269368397514597t_prod @ C) @ A) @ (^ [Y2:C]: (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X:B]: (((G @ X) @ Y2)))) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ A3) & ((R @ X) @ Y2))))))))) @ B3)))))))))). % 6.41/2.69 thf(fact_3311_prod_OatMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ (suc @ N2))) = (((times_times @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ ((set_ord_atMost @ nat) @ N2)))))))))). % 6.41/2.69 thf(fact_3312_atMost__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_ord_atMost @ A) = (^ [U2:A]: (((collect @ A) @ (^ [X:A]: ((((ord_less_eq @ A) @ X) @ U2))))))))))). % 6.41/2.69 thf(fact_3313_prod_Onested__swap_H, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:(nat > (nat > A)),N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (A2 @ I3)) @ ((set_ord_lessThan @ nat) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [J3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: (((A2 @ I3) @ J3)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J3)) @ N2))))) @ ((set_ord_lessThan @ nat) @ N2))))))))). % 6.41/2.69 thf(fact_3314_prod__mono, axiom, (! [A:$tType,B:$tType]: (((linordered_semidom @ A) => (! [A3:(set @ B),F2:(B > A),G:(B > A)]: (((! [I4:B]: (((((member @ B) @ I4) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ I4)) & (((ord_less_eq @ A) @ (F2 @ I4)) @ (G @ I4)))))) => (((ord_less_eq @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A3))))))))). % 6.41/2.69 thf(fact_3315_prod__nonneg, axiom, (! [A:$tType,B:$tType]: (((linordered_semidom @ A) => (! [A3:(set @ B),F2:(B > A)]: (((! [X5:B]: (((((member @ B) @ X5) @ A3) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ X5))))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3))))))))). % 6.41/2.69 thf(fact_3316_prod__pos, axiom, (! [A:$tType,B:$tType]: (((linordered_semidom @ A) => (! [A3:(set @ B),F2:(B > A)]: (((! [X5:B]: (((((member @ B) @ X5) @ A3) => (((ord_less @ A) @ (zero_zero @ A)) @ (F2 @ X5))))) => (((ord_less @ A) @ (zero_zero @ A)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3))))))))). % 6.41/2.69 thf(fact_3317_prod__ge__1, axiom, (! [A:$tType,B:$tType]: (((linord181362715937106298miring @ A) => (! [A3:(set @ B),F2:(B > A)]: (((! [X5:B]: (((((member @ B) @ X5) @ A3) => (((ord_less_eq @ A) @ (one_one @ A)) @ (F2 @ X5))))) => (((ord_less_eq @ A) @ (one_one @ A)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3))))))))). % 6.41/2.69 thf(fact_3318_prod__zero, axiom, (! [B:$tType,A:$tType]: (((comm_semiring_1 @ A) => (! [A3:(set @ B),F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((? [X2:B]: (((((member @ B) @ X2) @ A3) & ((F2 @ X2) = (zero_zero @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3) = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_3319_prod_OatMost__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ N2)) = (((times_times @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ ((set_ord_lessThan @ nat) @ N2)))))))))). % 6.41/2.69 thf(fact_3320_atMost__atLeast0, axiom, ((set_ord_atMost @ nat) = ((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)))). % 6.41/2.69 thf(fact_3321_lessThan__Suc__atMost, axiom, (! [K:nat]: ((((set_ord_lessThan @ nat) @ (suc @ K)) = ((set_ord_atMost @ nat) @ K))))). % 6.41/2.69 thf(fact_3322_atMost__Suc, axiom, (! [K:nat]: ((((set_ord_atMost @ nat) @ (suc @ K)) = (((insert @ nat) @ (suc @ K)) @ ((set_ord_atMost @ nat) @ K)))))). % 6.41/2.69 thf(fact_3323_prod_Ointer__filter, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),G:(B > A),P:(B > $o)]: ((((finite_finite2 @ B) @ A3) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ A3) & (P @ X)))))) = ((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X:B]: (((((if @ A) @ (P @ X)) @ (G @ X)) @ (one_one @ A))))) @ A3))))))))). % 6.41/2.69 thf(fact_3324_not__Iic__le__Icc, axiom, (! [A:$tType]: (((no_bot @ A) => (! [H2:A,L3:A,H3:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ H2)) @ (((set_or1337092689740270186AtMost @ A) @ L3) @ H3)))))))))). % 6.41/2.69 thf(fact_3325_prod_Oshift__bounds__cl__Suc__ivl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (suc @ N2))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_3326_prod_Oshift__bounds__cl__nat__ivl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,K:nat,N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ (((plus_plus @ nat) @ N2) @ K))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (((plus_plus @ nat) @ I3) @ K))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_3327_finite__nat__iff__bounded__le, axiom, ((finite_finite2 @ nat) = (^ [S7:(set @ nat)]: ((? [K3:nat]: ((((ord_less_eq @ (set @ nat)) @ S7) @ ((set_ord_atMost @ nat) @ K3)))))))). % 6.41/2.69 thf(fact_3328_prod__le__1, axiom, (! [B:$tType,A:$tType]: (((linord181362715937106298miring @ A) => (! [A3:(set @ B),F2:(B > A)]: (((! [X5:B]: (((((member @ B) @ X5) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ X5)) & (((ord_less_eq @ A) @ (F2 @ X5)) @ (one_one @ A)))))) => (((ord_less_eq @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3)) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_3329_prod_Orelated, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [R:(A > (A > $o)),S3:(set @ B),H2:(B > A),G:(B > A)]: ((((R @ (one_one @ A)) @ (one_one @ A)) => ((! [X16:A,Y15:A,X23:A,Y23:A]: (((((R @ X16) @ X23) & ((R @ Y15) @ Y23)) => ((R @ (((times_times @ A) @ X16) @ Y15)) @ (((times_times @ A) @ X23) @ Y23))))) => (((finite_finite2 @ B) @ S3) => ((! [X5:B]: (((((member @ B) @ X5) @ S3) => ((R @ (H2 @ X5)) @ (G @ X5))))) => ((R @ ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ S3)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S3)))))))))))). % 6.41/2.69 thf(fact_3330_prod_Oinsert__if, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),X3:B,G:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((((member @ B) @ X3) @ A3) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((insert @ B) @ X3) @ A3)) = ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A3))) & ((~ ((((member @ B) @ X3) @ A3))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((insert @ B) @ X3) @ A3)) = (((times_times @ A) @ (G @ X3)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A3)))))))))))). % 6.41/2.69 thf(fact_3331_prod__dvd__prod__subset, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [B3:(set @ B),A3:(set @ B),F2:(B > A)]: ((((finite_finite2 @ B) @ B3) => ((((ord_less_eq @ (set @ B)) @ A3) @ B3) => (((dvd_dvd @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ B3)))))))))). % 6.41/2.69 thf(fact_3332_prod__dvd__prod__subset2, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_1 @ A) => (! [B3:(set @ B),A3:(set @ B),F2:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ B3) => ((((ord_less_eq @ (set @ B)) @ A3) @ B3) => ((! [A5:B]: (((((member @ B) @ A5) @ A3) => (((dvd_dvd @ A) @ (F2 @ A5)) @ (G @ A5))))) => (((dvd_dvd @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ B3))))))))))). % 6.41/2.69 thf(fact_3333_prod_Oreindex__bij__witness__not__neutral, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comm_monoid_mult @ A) => (! [S6:(set @ B),T6:(set @ C),S3:(set @ B),I:(C > B),J:(B > C),T4:(set @ C),G:(B > A),H2:(C > A)]: ((((finite_finite2 @ B) @ S6) => (((finite_finite2 @ C) @ T6) => ((! [A5:B]: (((((member @ B) @ A5) @ (((minus_minus @ (set @ B)) @ S3) @ S6)) => ((I @ (J @ A5)) = A5)))) => ((! [A5:B]: (((((member @ B) @ A5) @ (((minus_minus @ (set @ B)) @ S3) @ S6)) => (((member @ C) @ (J @ A5)) @ (((minus_minus @ (set @ C)) @ T4) @ T6))))) => ((! [B5:C]: (((((member @ C) @ B5) @ (((minus_minus @ (set @ C)) @ T4) @ T6)) => ((J @ (I @ B5)) = B5)))) => ((! [B5:C]: (((((member @ C) @ B5) @ (((minus_minus @ (set @ C)) @ T4) @ T6)) => (((member @ B) @ (I @ B5)) @ (((minus_minus @ (set @ B)) @ S3) @ S6))))) => ((! [A5:B]: (((((member @ B) @ A5) @ S6) => ((G @ A5) = (one_one @ A))))) => ((! [B5:C]: (((((member @ C) @ B5) @ T6) => ((H2 @ B5) = (one_one @ A))))) => ((! [A5:B]: (((((member @ B) @ A5) @ S3) => ((H2 @ (J @ A5)) = (G @ A5))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S3) = ((((groups7121269368397514597t_prod @ C) @ A) @ H2) @ T4))))))))))))))))). % 6.41/2.69 thf(fact_3334_prod_Oin__pairs__0, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))))))) @ ((set_ord_atMost @ nat) @ N2))))))))). % 6.41/2.69 thf(fact_3335_prod_Osetdiff__irrelevant, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A3) @ ((collect @ B) @ (^ [X:B]: (((G @ X) = (one_one @ A))))))) = ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A3))))))))). % 6.41/2.69 thf(fact_3336_exp__sum, axiom, (! [B:$tType,A:$tType]: ((((comm_monoid_mult @ B) & ((real_Vector_banach @ B) & (real_V2822296259951069270ebra_1 @ B))) => (! [I6:(set @ A),F2:(A > B)]: ((((finite_finite2 @ A) @ I6) => (((exp @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F2) @ I6)) = ((((groups7121269368397514597t_prod @ A) @ B) @ (^ [X:A]: (((exp @ B) @ (F2 @ X))))) @ I6))))))))). % 6.41/2.69 thf(fact_3337_prod_Onat__diff__reindex, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ N2) @ (suc @ I3)))))) @ ((set_ord_lessThan @ nat) @ N2)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ N2))))))))). % 6.41/2.69 thf(fact_3338_Iic__subset__Iio__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ A2)) @ ((set_ord_lessThan @ A) @ B2)) = (((ord_less @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_3339_prod_OatLeastAtMost__rev, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N2:nat,M:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ N2) @ M)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ N2)) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ N2) @ M))))))))). % 6.41/2.69 thf(fact_3340_prod_Ozero__middle, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [P4:nat,K:nat,G:(nat > A),H2:(nat > A)]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ P4) => ((((ord_less_eq @ nat) @ K) @ P4) => (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [J3:nat]: (((((if @ A) @ (((ord_less @ nat) @ J3) @ K)) @ (G @ J3)) @ ((((if @ A) @ (J3 = K)) @ (one_one @ A)) @ (H2 @ (((minus_minus @ nat) @ J3) @ (suc @ (zero_zero @ nat))))))))) @ ((set_ord_atMost @ nat) @ P4)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [J3:nat]: (((((if @ A) @ (((ord_less @ nat) @ J3) @ K)) @ (G @ J3)) @ (H2 @ J3))))) @ ((set_ord_atMost @ nat) @ (((minus_minus @ nat) @ P4) @ (suc @ (zero_zero @ nat)))))))))))))). % 6.41/2.69 thf(fact_3341_less__1__prod2, axiom, (! [B:$tType,A:$tType]: (((linordered_idom @ B) => (! [I6:(set @ A),I:A,F2:(A > B)]: ((((finite_finite2 @ A) @ I6) => ((((member @ A) @ I) @ I6) => ((((ord_less @ B) @ (one_one @ B)) @ (F2 @ I)) => ((! [I4:A]: (((((member @ A) @ I4) @ I6) => (((ord_less_eq @ B) @ (one_one @ B)) @ (F2 @ I4))))) => (((ord_less @ B) @ (one_one @ B)) @ ((((groups7121269368397514597t_prod @ A) @ B) @ F2) @ I6)))))))))))). % 6.41/2.69 thf(fact_3342_sum__choose__upper, axiom, (! [M:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [K3:nat]: (((binomial @ K3) @ M)))) @ ((set_ord_atMost @ nat) @ N2)) = ((binomial @ (suc @ N2)) @ (suc @ M)))))). % 6.41/2.69 thf(fact_3343_less__1__prod, axiom, (! [B:$tType,A:$tType]: (((linordered_idom @ B) => (! [I6:(set @ A),F2:(A > B)]: ((((finite_finite2 @ A) @ I6) => ((I6 != (bot_bot @ (set @ A))) => ((! [I4:A]: (((((member @ A) @ I4) @ I6) => (((ord_less @ B) @ (one_one @ B)) @ (F2 @ I4))))) => (((ord_less @ B) @ (one_one @ B)) @ ((((groups7121269368397514597t_prod @ A) @ B) @ F2) @ I6))))))))))). % 6.41/2.69 thf(fact_3344_prod_Osubset__diff, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [B3:(set @ B),A3:(set @ B),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ B3) @ A3) => (((finite_finite2 @ B) @ A3) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A3) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A3) @ B3))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ B3))))))))))). % 6.41/2.69 thf(fact_3345_prod_Omono__neutral__cong__right, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [T4:(set @ B),S3:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ T4) => ((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [X5:B]: (((((member @ B) @ X5) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((G @ X5) = (one_one @ A))))) => ((! [X5:B]: (((((member @ B) @ X5) @ S3) => ((G @ X5) = (H2 @ X5))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ T4) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ S3)))))))))))). % 6.41/2.69 thf(fact_3346_prod_Omono__neutral__cong__left, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [T4:(set @ B),S3:(set @ B),H2:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ T4) => ((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [X5:B]: (((((member @ B) @ X5) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((H2 @ X5) = (one_one @ A))))) => ((! [X5:B]: (((((member @ B) @ X5) @ S3) => ((G @ X5) = (H2 @ X5))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S3) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ T4)))))))))))). % 6.41/2.69 thf(fact_3347_prod_Omono__neutral__right, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [T4:(set @ B),S3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ T4) => ((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [X5:B]: (((((member @ B) @ X5) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((G @ X5) = (one_one @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ T4) = ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S3))))))))))). % 6.41/2.69 thf(fact_3348_prod_Omono__neutral__left, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [T4:(set @ B),S3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ T4) => ((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [X5:B]: (((((member @ B) @ X5) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((G @ X5) = (one_one @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S3) = ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ T4))))))))))). % 6.41/2.69 thf(fact_3349_prod_Osame__carrierI, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [C5:(set @ B),A3:(set @ B),B3:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ C5) => ((((ord_less_eq @ (set @ B)) @ A3) @ C5) => ((((ord_less_eq @ (set @ B)) @ B3) @ C5) => ((! [A5:B]: (((((member @ B) @ A5) @ (((minus_minus @ (set @ B)) @ C5) @ A3)) => ((G @ A5) = (one_one @ A))))) => ((! [B5:B]: (((((member @ B) @ B5) @ (((minus_minus @ (set @ B)) @ C5) @ B3)) => ((H2 @ B5) = (one_one @ A))))) => ((((((groups7121269368397514597t_prod @ B) @ A) @ G) @ C5) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ C5)) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A3) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ B3)))))))))))))). % 6.41/2.69 thf(fact_3350_prod_Osame__carrier, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [C5:(set @ B),A3:(set @ B),B3:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ C5) => ((((ord_less_eq @ (set @ B)) @ A3) @ C5) => ((((ord_less_eq @ (set @ B)) @ B3) @ C5) => ((! [A5:B]: (((((member @ B) @ A5) @ (((minus_minus @ (set @ B)) @ C5) @ A3)) => ((G @ A5) = (one_one @ A))))) => ((! [B5:B]: (((((member @ B) @ B5) @ (((minus_minus @ (set @ B)) @ C5) @ B3)) => ((H2 @ B5) = (one_one @ A))))) => ((((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A3) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ B3)) = (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ C5) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ C5)))))))))))))). % 6.41/2.69 thf(fact_3351_prod_OatLeast0__atMost__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N2))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2))) @ (G @ (suc @ N2)))))))))). % 6.41/2.69 thf(fact_3352_prod_OatLeast__Suc__atMost, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (((times_times @ A) @ (G @ M)) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ N2))))))))))). % 6.41/2.69 thf(fact_3353_prod_Onat__ivl__Suc_H, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ (suc @ N2)) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (suc @ N2))) = (((times_times @ A) @ (G @ (suc @ N2))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))))). % 6.41/2.69 thf(fact_3354_prod_OlessThan__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (suc @ N2))) = (((times_times @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ ((set_ord_lessThan @ nat) @ N2)))))))))). % 6.41/2.69 thf(fact_3355_prod_OSuc__reindex__ivl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))) @ (G @ (suc @ N2))) = (((times_times @ A) @ (G @ M)) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))))). % 6.41/2.69 thf(fact_3356_sum_OatMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ (suc @ N2))) = (((plus_plus @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ ((set_ord_atMost @ nat) @ N2)))))))))). % 6.41/2.69 thf(fact_3357_sum__telescope, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [F2:(nat > A),I:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((minus_minus @ A) @ (F2 @ I3)) @ (F2 @ (suc @ I3)))))) @ ((set_ord_atMost @ nat) @ I)) = (((minus_minus @ A) @ (F2 @ (zero_zero @ nat))) @ (F2 @ (suc @ I)))))))))). % 6.41/2.69 thf(fact_3358_polyfun__eq__coeffs, axiom, (! [A:$tType]: ((((real_V8999393235501362500lgebra @ A) & (idom @ A)) => (! [C2:(nat > A),N2:nat,D2:(nat > A)]: (((! [X:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ X) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (D2 @ I3)) @ (((power_power @ A) @ X) @ I3))))) @ ((set_ord_atMost @ nat) @ N2))))) = (! [I3:nat]: (((((ord_less_eq @ nat) @ I3) @ N2) => ((C2 @ I3) = (D2 @ I3)))))))))))). % 6.41/2.69 thf(fact_3359_prod_OatLeast1__atMost__eq, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ N2)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [K3:nat]: ((G @ (suc @ K3))))) @ ((set_ord_lessThan @ nat) @ N2))))))))). % 6.41/2.69 thf(fact_3360_bounded__imp__summable, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & ((linord2810124833399127020strict @ A) & (topolo1944317154257567458pology @ A))) => (! [A2:(nat > A),B3:A]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (A2 @ N3)))) => ((! [N3:nat]: ((((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ A2) @ ((set_ord_atMost @ nat) @ N3))) @ B3))) => ((summable @ A) @ A2))))))))). % 6.41/2.69 thf(fact_3361_sum_Onested__swap_H, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:(nat > (nat > A)),N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (A2 @ I3)) @ ((set_ord_lessThan @ nat) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J3:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: (((A2 @ I3) @ J3)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J3)) @ N2))))) @ ((set_ord_lessThan @ nat) @ N2))))))))). % 6.41/2.69 thf(fact_3362_prod__mono__strict, axiom, (! [A:$tType,B:$tType]: (((linordered_semidom @ A) => (! [A3:(set @ B),F2:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((! [I4:B]: (((((member @ B) @ I4) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ I4)) & (((ord_less @ A) @ (F2 @ I4)) @ (G @ I4)))))) => ((A3 != (bot_bot @ (set @ B))) => (((ord_less @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A3))))))))))). % 6.41/2.69 thf(fact_3363_even__prod__iff, axiom, (! [A:$tType,B:$tType]: (((semiring_parity @ A) => (! [A3:(set @ B),F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3)) = (? [X:B]: (((((member @ B) @ X) @ A3) & (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (F2 @ X))))))))))))). % 6.41/2.69 thf(fact_3364_sum__choose__lower, axiom, (! [R2:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [K3:nat]: (((binomial @ (((plus_plus @ nat) @ R2) @ K3)) @ K3)))) @ ((set_ord_atMost @ nat) @ N2)) = ((binomial @ (suc @ (((plus_plus @ nat) @ R2) @ N2))) @ N2))))). % 6.41/2.69 thf(fact_3365_prod_Oremove, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),X3:B,G:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((((member @ B) @ X3) @ A3) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A3) = (((times_times @ A) @ (G @ X3)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ X3) @ (bot_bot @ (set @ B))))))))))))))). % 6.41/2.69 thf(fact_3366_prod_Oinsert__remove, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),G:(B > A),X3:B]: ((((finite_finite2 @ B) @ A3) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((insert @ B) @ X3) @ A3)) = (((times_times @ A) @ (G @ X3)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ X3) @ (bot_bot @ (set @ B)))))))))))))). % 6.41/2.69 thf(fact_3367_choose__rising__sum_I1_J, axiom, (! [N2:nat,M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [J3:nat]: (((binomial @ (((plus_plus @ nat) @ N2) @ J3)) @ N2)))) @ ((set_ord_atMost @ nat) @ M)) = ((binomial @ (((plus_plus @ nat) @ (((plus_plus @ nat) @ N2) @ M)) @ (one_one @ nat))) @ (((plus_plus @ nat) @ N2) @ (one_one @ nat))))))). % 6.41/2.69 thf(fact_3368_choose__rising__sum_I2_J, axiom, (! [N2:nat,M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [J3:nat]: (((binomial @ (((plus_plus @ nat) @ N2) @ J3)) @ N2)))) @ ((set_ord_atMost @ nat) @ M)) = ((binomial @ (((plus_plus @ nat) @ (((plus_plus @ nat) @ N2) @ M)) @ (one_one @ nat))) @ M))))). % 6.41/2.69 thf(fact_3369_prod_Oub__add__nat, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N2:nat,G:(nat > A),P4:nat]: (((((ord_less_eq @ nat) @ M) @ (((plus_plus @ nat) @ N2) @ (one_one @ nat))) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ (((plus_plus @ nat) @ N2) @ P4))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ N2) @ (one_one @ nat))) @ (((plus_plus @ nat) @ N2) @ P4)))))))))))). % 6.41/2.69 thf(fact_3370_prod_Odelta__remove, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),A2:B,B2:(B > A),C2:(B > A)]: ((((finite_finite2 @ B) @ S3) => (((((member @ B) @ A2) @ S3) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A2)) @ (B2 @ K3)) @ (C2 @ K3))))) @ S3) = (((times_times @ A) @ (B2 @ A2)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ C2) @ (((minus_minus @ (set @ B)) @ S3) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B)))))))) & ((~ ((((member @ B) @ A2) @ S3))) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A2)) @ (B2 @ K3)) @ (C2 @ K3))))) @ S3) = ((((groups7121269368397514597t_prod @ B) @ A) @ C2) @ (((minus_minus @ (set @ B)) @ S3) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B))))))))))))))). % 6.41/2.69 thf(fact_3371_fold__atLeastAtMost__nat_Osimps, axiom, (! [A:$tType]: (((set_fo6178422350223883121st_nat @ A) = (^ [F3:(nat > (A > A)),A4:nat,B4:nat,Acc2:A]: (((((if @ A) @ (((ord_less @ nat) @ B4) @ A4)) @ Acc2) @ (((((set_fo6178422350223883121st_nat @ A) @ F3) @ (((plus_plus @ nat) @ A4) @ (one_one @ nat))) @ B4) @ ((F3 @ A4) @ Acc2))))))))). % 6.41/2.69 thf(fact_3372_fold__atLeastAtMost__nat_Oelims, axiom, (! [A:$tType,X3:(nat > (A > A)),Xa2:nat,Xb:nat,Xc:A,Y:A]: ((((((((set_fo6178422350223883121st_nat @ A) @ X3) @ Xa2) @ Xb) @ Xc) = Y) => (((((ord_less @ nat) @ Xb) @ Xa2) => (Y = Xc)) & ((~ ((((ord_less @ nat) @ Xb) @ Xa2))) => (Y = (((((set_fo6178422350223883121st_nat @ A) @ X3) @ (((plus_plus @ nat) @ Xa2) @ (one_one @ nat))) @ Xb) @ ((X3 @ Xa2) @ Xc))))))))). % 6.41/2.69 thf(fact_3373_polyfun__eq__0, axiom, (! [A:$tType]: ((((real_V8999393235501362500lgebra @ A) & (idom @ A)) => (! [C2:(nat > A),N2:nat]: (((! [X:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ X) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)) = (zero_zero @ A)))) = (! [I3:nat]: (((((ord_less_eq @ nat) @ I3) @ N2) => ((C2 @ I3) = (zero_zero @ A)))))))))))). % 6.41/2.69 thf(fact_3374_zero__polynom__imp__zero__coeffs, axiom, (! [A:$tType]: ((((ab_semigroup_mult @ A) & (real_V8999393235501362500lgebra @ A)) => (! [C2:(nat > A),N2:nat,K:nat]: (((! [W:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ W) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)) = (zero_zero @ A)))) => ((((ord_less_eq @ nat) @ K) @ N2) => ((C2 @ K) = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_3375_sum_OatMost__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ N2)) = (((plus_plus @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ ((set_ord_lessThan @ nat) @ N2)))))))))). % 6.41/2.69 thf(fact_3376_sum__up__index__split, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [F2:(nat > A),M:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ ((set_ord_atMost @ nat) @ (((plus_plus @ nat) @ M) @ N2))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ ((set_ord_atMost @ nat) @ M))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (((plus_plus @ nat) @ M) @ N2))))))))))). % 6.41/2.69 thf(fact_3377_atLeast1__atMost__eq__remove0, axiom, (! [N2:nat]: (((((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ N2) = (((minus_minus @ (set @ nat)) @ ((set_ord_atMost @ nat) @ N2)) @ (((insert @ nat) @ (zero_zero @ nat)) @ (bot_bot @ (set @ nat)))))))). % 6.41/2.69 thf(fact_3378_gbinomial__parallel__sum, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A2:A,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((gbinomial @ A) @ (((plus_plus @ A) @ A2) @ ((semiring_1_of_nat @ A) @ K3))) @ K3)))) @ ((set_ord_atMost @ nat) @ N2)) = (((gbinomial @ A) @ (((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ ((semiring_1_of_nat @ A) @ N2))) @ (one_one @ A))) @ N2)))))))). % 6.41/2.69 thf(fact_3379_sum_Otriangle__reindex__eq, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > (nat > A)),N2:nat]: ((((((groups7311177749621191930dd_sum @ ((product_prod @ nat) @ nat)) @ A) @ ((((product_case_prod @ nat) @ nat) @ A) @ G)) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [I3:nat,J3:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I3) @ J3)) @ N2)))))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: (((G @ I3) @ (((minus_minus @ nat) @ K3) @ I3))))) @ ((set_ord_atMost @ nat) @ K3))))) @ ((set_ord_atMost @ nat) @ N2))))))))). % 6.41/2.69 thf(fact_3380_fact__eq__fact__times, axiom, (! [N2:nat,M:nat]: (((((ord_less_eq @ nat) @ N2) @ M) => (((semiring_char_0_fact @ nat) @ M) = (((times_times @ nat) @ ((semiring_char_0_fact @ nat) @ N2)) @ ((((groups7121269368397514597t_prod @ nat) @ nat) @ (^ [X:nat]: (X))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ N2)) @ M)))))))). % 6.41/2.69 thf(fact_3381_prod__mono2, axiom, (! [B:$tType,A:$tType]: (((linordered_idom @ B) => (! [B3:(set @ A),A3:(set @ A),F2:(A > B)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((! [B5:A]: (((((member @ A) @ B5) @ (((minus_minus @ (set @ A)) @ B3) @ A3)) => (((ord_less_eq @ B) @ (one_one @ B)) @ (F2 @ B5))))) => ((! [A5:A]: (((((member @ A) @ A5) @ A3) => (((ord_less_eq @ B) @ (zero_zero @ B)) @ (F2 @ A5))))) => (((ord_less_eq @ B) @ ((((groups7121269368397514597t_prod @ A) @ B) @ F2) @ A3)) @ ((((groups7121269368397514597t_prod @ A) @ B) @ F2) @ B3)))))))))))). % 6.41/2.69 thf(fact_3382_prod__diff1, axiom, (! [A:$tType,B:$tType]: (((semidom_divide @ A) => (! [A3:(set @ B),F2:(B > A),A2:B]: ((((finite_finite2 @ B) @ A3) => (((F2 @ A2) != (zero_zero @ A)) => (((((member @ B) @ A2) @ A3) => (((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B))))) = (((divide_divide @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3)) @ (F2 @ A2)))) & ((~ ((((member @ B) @ A2) @ A3))) => (((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B))))) = ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3)))))))))))). % 6.41/2.69 thf(fact_3383_sum__choose__diagonal, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => (((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [K3:nat]: (((binomial @ (((minus_minus @ nat) @ N2) @ K3)) @ (((minus_minus @ nat) @ M) @ K3))))) @ ((set_ord_atMost @ nat) @ M)) = ((binomial @ (suc @ N2)) @ M)))))). % 6.41/2.69 thf(fact_3384_vandermonde, axiom, (! [M:nat,N2:nat,R2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [K3:nat]: ((((times_times @ nat) @ ((binomial @ M) @ K3)) @ ((binomial @ N2) @ (((minus_minus @ nat) @ R2) @ K3)))))) @ ((set_ord_atMost @ nat) @ R2)) = ((binomial @ (((plus_plus @ nat) @ M) @ N2)) @ R2))))). % 6.41/2.69 thf(fact_3385_sum__gp__basic, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X3:A,N2:nat]: (((((times_times @ A) @ (((minus_minus @ A) @ (one_one @ A)) @ X3)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X3)) @ ((set_ord_atMost @ nat) @ N2))) = (((minus_minus @ A) @ (one_one @ A)) @ (((power_power @ A) @ X3) @ (suc @ N2)))))))))). % 6.41/2.69 thf(fact_3386_polyfun__finite__roots, axiom, (! [A:$tType]: ((((real_V8999393235501362500lgebra @ A) & (idom @ A)) => (! [C2:(nat > A),N2:nat]: ((((finite_finite2 @ A) @ ((collect @ A) @ (^ [X:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ X) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)) = (zero_zero @ A)))))) = (? [I3:nat]: (((((ord_less_eq @ nat) @ I3) @ N2) & ((C2 @ I3) != (zero_zero @ A)))))))))))). % 6.41/2.69 thf(fact_3387_polyfun__roots__finite, axiom, (! [A:$tType]: ((((real_V8999393235501362500lgebra @ A) & (idom @ A)) => (! [C2:(nat > A),K:nat,N2:nat]: ((((C2 @ K) != (zero_zero @ A)) => ((((ord_less_eq @ nat) @ K) @ N2) => ((finite_finite2 @ A) @ ((collect @ A) @ (^ [Z6:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ Z6) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)) = (zero_zero @ A)))))))))))))). % 6.41/2.69 thf(fact_3388_pochhammer__Suc__prod, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A2:A,N2:nat]: (((((comm_s3205402744901411588hammer @ A) @ A2) @ (suc @ N2)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ A2) @ ((semiring_1_of_nat @ A) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2))))))))). % 6.41/2.69 thf(fact_3389_polyfun__linear__factor__root, axiom, (! [A:$tType]: (((idom @ A) => (! [C2:(nat > A),A2:A,N2:nat]: (((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ A2) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)) = (zero_zero @ A)) => (~ ((! [B5:(nat > A)]: ((~ ((! [Z4:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ Z4) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)) = (((times_times @ A) @ (((minus_minus @ A) @ Z4) @ A2)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (B5 @ I3)) @ (((power_power @ A) @ Z4) @ I3))))) @ ((set_ord_lessThan @ nat) @ N2))))))))))))))))))). % 6.41/2.69 thf(fact_3390_polyfun__linear__factor, axiom, (! [A:$tType]: (((idom @ A) => (! [C2:(nat > A),N2:nat,A2:A]: ((? [B5:(nat > A)]: ((! [Z4:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ Z4) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)) = (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ Z4) @ A2)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (B5 @ I3)) @ (((power_power @ A) @ Z4) @ I3))))) @ ((set_ord_lessThan @ nat) @ N2)))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ A2) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)))))))))))))). % 6.41/2.69 thf(fact_3391_pochhammer__prod__rev, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => ((comm_s3205402744901411588hammer @ A) = (^ [A4:A,N:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ A4) @ ((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ N) @ I3)))))) @ (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ N))))))))). % 6.41/2.69 thf(fact_3392_sum__power__shift, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [M:nat,N2:nat,X3:A]: (((((ord_less_eq @ nat) @ M) @ N2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X3)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (((times_times @ A) @ (((power_power @ A) @ X3) @ M)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X3)) @ ((set_ord_atMost @ nat) @ (((minus_minus @ nat) @ N2) @ M)))))))))))). % 6.41/2.69 thf(fact_3393_sum_Otriangle__reindex, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > (nat > A)),N2:nat]: ((((((groups7311177749621191930dd_sum @ ((product_prod @ nat) @ nat)) @ A) @ ((((product_case_prod @ nat) @ nat) @ A) @ G)) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [I3:nat,J3:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ I3) @ J3)) @ N2)))))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: (((G @ I3) @ (((minus_minus @ nat) @ K3) @ I3))))) @ ((set_ord_atMost @ nat) @ K3))))) @ ((set_ord_lessThan @ nat) @ N2))))))))). % 6.41/2.69 thf(fact_3394_fact__div__fact, axiom, (! [N2:nat,M:nat]: (((((ord_less_eq @ nat) @ N2) @ M) => ((((divide_divide @ nat) @ ((semiring_char_0_fact @ nat) @ M)) @ ((semiring_char_0_fact @ nat) @ N2)) = ((((groups7121269368397514597t_prod @ nat) @ nat) @ (^ [X:nat]: (X))) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ N2) @ (one_one @ nat))) @ M))))))). % 6.41/2.69 thf(fact_3395_binomial, axiom, (! [A2:nat,B2:nat,N2:nat]: (((((power_power @ nat) @ (((plus_plus @ nat) @ A2) @ B2)) @ N2) = ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [K3:nat]: ((((times_times @ nat) @ (((times_times @ nat) @ ((semiring_1_of_nat @ nat) @ ((binomial @ N2) @ K3))) @ (((power_power @ nat) @ A2) @ K3))) @ (((power_power @ nat) @ B2) @ (((minus_minus @ nat) @ N2) @ K3)))))) @ ((set_ord_atMost @ nat) @ N2)))))). % 6.41/2.69 thf(fact_3396_prod_Oin__pairs, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))))))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_3397_sum__atLeastAtMost__code, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [F2:(nat > A),A2:nat,B2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ (((set_or1337092689740270186AtMost @ nat) @ A2) @ B2)) = (((((set_fo6178422350223883121st_nat @ A) @ (^ [A4:nat]: (((plus_plus @ A) @ (F2 @ A4))))) @ A2) @ B2) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3398_sum_Oin__pairs__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))))))) @ ((set_ord_atMost @ nat) @ N2))))))))). % 6.41/2.69 thf(fact_3399_polynomial__product, axiom, (! [A:$tType]: (((idom @ A) => (! [M:nat,A2:(nat > A),N2:nat,B2:(nat > A),X3:A]: (((! [I4:nat]: (((((ord_less @ nat) @ M) @ I4) => ((A2 @ I4) = (zero_zero @ A))))) => ((! [J2:nat]: (((((ord_less @ nat) @ N2) @ J2) => ((B2 @ J2) = (zero_zero @ A))))) => ((((times_times @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (A2 @ I3)) @ (((power_power @ A) @ X3) @ I3))))) @ ((set_ord_atMost @ nat) @ M))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J3:nat]: ((((times_times @ A) @ (B2 @ J3)) @ (((power_power @ A) @ X3) @ J3))))) @ ((set_ord_atMost @ nat) @ N2))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [R5:nat]: ((((times_times @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (A2 @ K3)) @ (B2 @ (((minus_minus @ nat) @ R5) @ K3)))))) @ ((set_ord_atMost @ nat) @ R5))) @ (((power_power @ A) @ X3) @ R5))))) @ ((set_ord_atMost @ nat) @ (((plus_plus @ nat) @ M) @ N2)))))))))))). % 6.41/2.69 thf(fact_3400_pochhammer__Suc__prod__rev, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A2:A,N2:nat]: (((((comm_s3205402744901411588hammer @ A) @ A2) @ (suc @ N2)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ A2) @ ((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ N2) @ I3)))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2))))))))). % 6.41/2.69 thf(fact_3401_polyfun__eq__const, axiom, (! [A:$tType]: ((((real_V8999393235501362500lgebra @ A) & (idom @ A)) => (! [C2:(nat > A),N2:nat,K:A]: (((! [X:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ X) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)) = K))) = (((C2 @ (zero_zero @ nat)) = K) & (! [X:nat]: (((((member @ nat) @ X) @ (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ N2)) => ((C2 @ X) = (zero_zero @ A))))))))))))). % 6.41/2.69 thf(fact_3402_binomial__ring, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A2:A,B2:A,N2:nat]: (((((power_power @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ N2) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ ((binomial @ N2) @ K3))) @ (((power_power @ A) @ A2) @ K3))) @ (((power_power @ A) @ B2) @ (((minus_minus @ nat) @ N2) @ K3)))))) @ ((set_ord_atMost @ nat) @ N2))))))))). % 6.41/2.69 thf(fact_3403_pochhammer__binomial__sum, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [A2:A,B2:A,N2:nat]: (((((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ N2) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ ((binomial @ N2) @ K3))) @ (((comm_s3205402744901411588hammer @ A) @ A2) @ K3))) @ (((comm_s3205402744901411588hammer @ A) @ B2) @ (((minus_minus @ nat) @ N2) @ K3)))))) @ ((set_ord_atMost @ nat) @ N2))))))))). % 6.41/2.69 thf(fact_3404_polynomial__product__nat, axiom, (! [M:nat,A2:(nat > nat),N2:nat,B2:(nat > nat),X3:nat]: (((! [I4:nat]: (((((ord_less @ nat) @ M) @ I4) => ((A2 @ I4) = (zero_zero @ nat))))) => ((! [J2:nat]: (((((ord_less @ nat) @ N2) @ J2) => ((B2 @ J2) = (zero_zero @ nat))))) => ((((times_times @ nat) @ ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [I3:nat]: ((((times_times @ nat) @ (A2 @ I3)) @ (((power_power @ nat) @ X3) @ I3))))) @ ((set_ord_atMost @ nat) @ M))) @ ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [J3:nat]: ((((times_times @ nat) @ (B2 @ J3)) @ (((power_power @ nat) @ X3) @ J3))))) @ ((set_ord_atMost @ nat) @ N2))) = ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [R5:nat]: ((((times_times @ nat) @ ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [K3:nat]: ((((times_times @ nat) @ (A2 @ K3)) @ (B2 @ (((minus_minus @ nat) @ R5) @ K3)))))) @ ((set_ord_atMost @ nat) @ R5))) @ (((power_power @ nat) @ X3) @ R5))))) @ ((set_ord_atMost @ nat) @ (((plus_plus @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_3405_gbinomial__Suc, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (semidom_divide @ A)) => (! [A2:A,K:nat]: (((((gbinomial @ A) @ A2) @ (suc @ K)) = (((divide_divide @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((minus_minus @ A) @ A2) @ ((semiring_1_of_nat @ A) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ K))) @ ((semiring_char_0_fact @ A) @ (suc @ K)))))))))). % 6.41/2.69 thf(fact_3406_sum_Ozero__middle, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [P4:nat,K:nat,G:(nat > A),H2:(nat > A)]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ P4) => ((((ord_less_eq @ nat) @ K) @ P4) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J3:nat]: (((((if @ A) @ (((ord_less @ nat) @ J3) @ K)) @ (G @ J3)) @ ((((if @ A) @ (J3 = K)) @ (zero_zero @ A)) @ (H2 @ (((minus_minus @ nat) @ J3) @ (suc @ (zero_zero @ nat))))))))) @ ((set_ord_atMost @ nat) @ P4)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J3:nat]: (((((if @ A) @ (((ord_less @ nat) @ J3) @ K)) @ (G @ J3)) @ (H2 @ J3))))) @ ((set_ord_atMost @ nat) @ (((minus_minus @ nat) @ P4) @ (suc @ (zero_zero @ nat)))))))))))))). % 6.41/2.69 thf(fact_3407_gbinomial__partial__sum__poly, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [M:nat,A2:A,X3:A,Y:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((times_times @ A) @ (((gbinomial @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ M)) @ A2)) @ K3)) @ (((power_power @ A) @ X3) @ K3))) @ (((power_power @ A) @ Y) @ (((minus_minus @ nat) @ M) @ K3)))))) @ ((set_ord_atMost @ nat) @ M)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((times_times @ A) @ (((gbinomial @ A) @ ((uminus_uminus @ A) @ A2)) @ K3)) @ (((power_power @ A) @ ((uminus_uminus @ A) @ X3)) @ K3))) @ (((power_power @ A) @ (((plus_plus @ A) @ X3) @ Y)) @ (((minus_minus @ nat) @ M) @ K3)))))) @ ((set_ord_atMost @ nat) @ M))))))))). % 6.41/2.69 thf(fact_3408_root__polyfun, axiom, (! [A:$tType]: (((idom @ A) => (! [N2:nat,Z2:A,A2:A]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ N2) => (((((power_power @ A) @ Z2) @ N2) = A2) = (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ ((((if @ A) @ (I3 = (zero_zero @ nat))) @ ((uminus_uminus @ A) @ A2)) @ ((((if @ A) @ (I3 = N2)) @ (one_one @ A)) @ (zero_zero @ A)))) @ (((power_power @ A) @ Z2) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)) = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_3409_sum__gp0, axiom, (! [A:$tType]: ((((division_ring @ A) & (comm_ring @ A)) => (! [X3:A,N2:nat]: ((((X3 = (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X3)) @ ((set_ord_atMost @ nat) @ N2)) = ((semiring_1_of_nat @ A) @ (((plus_plus @ nat) @ N2) @ (one_one @ nat))))) & ((X3 != (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X3)) @ ((set_ord_atMost @ nat) @ N2)) = (((divide_divide @ A) @ (((minus_minus @ A) @ (one_one @ A)) @ (((power_power @ A) @ X3) @ (suc @ N2)))) @ (((minus_minus @ A) @ (one_one @ A)) @ X3))))))))))). % 6.41/2.69 thf(fact_3410_choose__alternating__linear__sum, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [N2:nat]: (((N2 != (one_one @ nat)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ I3)) @ ((semiring_1_of_nat @ A) @ I3))) @ ((semiring_1_of_nat @ A) @ ((binomial @ N2) @ I3)))))) @ ((set_ord_atMost @ nat) @ N2)) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3411_gbinomial__sum__nat__pow2, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((divide_divide @ A) @ (((gbinomial @ A) @ ((semiring_1_of_nat @ A) @ (((plus_plus @ nat) @ M) @ K3))) @ K3)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ K3))))) @ ((set_ord_atMost @ nat) @ M)) = (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M)))))))). % 6.41/2.69 thf(fact_3412_gbinomial__partial__sum__poly__xpos, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [M:nat,A2:A,X3:A,Y:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((times_times @ A) @ (((gbinomial @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ M)) @ A2)) @ K3)) @ (((power_power @ A) @ X3) @ K3))) @ (((power_power @ A) @ Y) @ (((minus_minus @ nat) @ M) @ K3)))))) @ ((set_ord_atMost @ nat) @ M)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((times_times @ A) @ (((gbinomial @ A) @ (((minus_minus @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ K3)) @ A2)) @ (one_one @ A))) @ K3)) @ (((power_power @ A) @ X3) @ K3))) @ (((power_power @ A) @ (((plus_plus @ A) @ X3) @ Y)) @ (((minus_minus @ nat) @ M) @ K3)))))) @ ((set_ord_atMost @ nat) @ M))))))))). % 6.41/2.69 thf(fact_3413_polyfun__diff__alt, axiom, (! [A:$tType]: (((idom @ A) => (! [N2:nat,A2:(nat > A),X3:A,Y:A]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ N2) => ((((minus_minus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (A2 @ I3)) @ (((power_power @ A) @ X3) @ I3))))) @ ((set_ord_atMost @ nat) @ N2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (A2 @ I3)) @ (((power_power @ A) @ Y) @ I3))))) @ ((set_ord_atMost @ nat) @ N2))) = (((times_times @ A) @ (((minus_minus @ A) @ X3) @ Y)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J3:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (((times_times @ A) @ (A2 @ (((plus_plus @ nat) @ (((plus_plus @ nat) @ J3) @ K3)) @ (one_one @ nat)))) @ (((power_power @ A) @ Y) @ K3))) @ (((power_power @ A) @ X3) @ J3))))) @ ((set_ord_lessThan @ nat) @ (((minus_minus @ nat) @ N2) @ J3)))))) @ ((set_ord_lessThan @ nat) @ N2))))))))))). % 6.41/2.69 thf(fact_3414_binomial__r__part__sum, axiom, (! [M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (binomial @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)) @ (one_one @ nat)))) @ ((set_ord_atMost @ nat) @ M)) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M)))))). % 6.41/2.69 thf(fact_3415_choose__alternating__sum, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ I3)) @ ((semiring_1_of_nat @ A) @ ((binomial @ N2) @ I3)))))) @ ((set_ord_atMost @ nat) @ N2)) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3416_polyfun__extremal__lemma, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [E3:real,C2:(nat > A),N2:nat]: (((((ord_less @ real) @ (zero_zero @ real)) @ E3) => (? [M8:real]: ((! [Z4:A]: (((((ord_less_eq @ real) @ M8) @ ((real_V7770717601297561774m_norm @ A) @ Z4)) => (((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ Z4) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)))) @ (((times_times @ real) @ E3) @ (((power_power @ real) @ ((real_V7770717601297561774m_norm @ A) @ Z4)) @ (suc @ N2)))))))))))))))). % 6.41/2.69 thf(fact_3417_polyfun__diff, axiom, (! [A:$tType]: (((idom @ A) => (! [N2:nat,A2:(nat > A),X3:A,Y:A]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ N2) => ((((minus_minus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (A2 @ I3)) @ (((power_power @ A) @ X3) @ I3))))) @ ((set_ord_atMost @ nat) @ N2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (A2 @ I3)) @ (((power_power @ A) @ Y) @ I3))))) @ ((set_ord_atMost @ nat) @ N2))) = (((times_times @ A) @ (((minus_minus @ A) @ X3) @ Y)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J3:nat]: ((((times_times @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (A2 @ I3)) @ (((power_power @ A) @ Y) @ (((minus_minus @ nat) @ (((minus_minus @ nat) @ I3) @ J3)) @ (one_one @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J3)) @ N2))) @ (((power_power @ A) @ X3) @ J3))))) @ ((set_ord_lessThan @ nat) @ N2))))))))))). % 6.41/2.69 thf(fact_3418_gbinomial__r__part__sum, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ ((gbinomial @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ A) @ M))) @ (one_one @ A)))) @ ((set_ord_atMost @ nat) @ M)) = (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M))))))))). % 6.41/2.69 thf(fact_3419_choose__odd__sum, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: (((((if @ A) @ (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3)))) @ ((semiring_1_of_nat @ A) @ ((binomial @ N2) @ I3))) @ (zero_zero @ A))))) @ ((set_ord_atMost @ nat) @ N2))) = (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2))))))))). % 6.41/2.69 thf(fact_3420_choose__even__sum, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: (((((if @ A) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3)) @ ((semiring_1_of_nat @ A) @ ((binomial @ N2) @ I3))) @ (zero_zero @ A))))) @ ((set_ord_atMost @ nat) @ N2))) = (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2))))))))). % 6.41/2.69 thf(fact_3421_sin__x__sin__y, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,Y:A]: ((((sums @ A) @ (^ [P6:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [N:nat]: (((((if @ A) @ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ P6) & (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))) @ (((times_times @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ ((ring_1_of_int @ real) @ (((times_times @ int) @ (((power_power @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (((divide_divide @ nat) @ P6) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((semiring_1_of_nat @ int) @ ((binomial @ P6) @ N))))) @ ((semiring_char_0_fact @ real) @ P6)))) @ (((power_power @ A) @ X3) @ N))) @ (((power_power @ A) @ Y) @ (((minus_minus @ nat) @ P6) @ N)))) @ (zero_zero @ A))))) @ ((set_ord_atMost @ nat) @ P6))))) @ (((times_times @ A) @ ((sin @ A) @ X3)) @ ((sin @ A) @ Y))))))))). % 6.41/2.69 thf(fact_3422_sums__cos__x__plus__y, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,Y:A]: ((((sums @ A) @ (^ [P6:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [N:nat]: (((((if @ A) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ P6)) @ (((times_times @ A) @ (((real_V8093663219630862766scaleR @ A) @ (((divide_divide @ real) @ ((ring_1_of_int @ real) @ (((times_times @ int) @ (((power_power @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (((divide_divide @ nat) @ P6) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((semiring_1_of_nat @ int) @ ((binomial @ P6) @ N))))) @ ((semiring_char_0_fact @ real) @ P6))) @ (((power_power @ A) @ X3) @ N))) @ (((power_power @ A) @ Y) @ (((minus_minus @ nat) @ P6) @ N)))) @ (zero_zero @ A))))) @ ((set_ord_atMost @ nat) @ P6))))) @ ((cos @ A) @ (((plus_plus @ A) @ X3) @ Y))))))))). % 6.41/2.69 thf(fact_3423_cos__x__cos__y, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,Y:A]: ((((sums @ A) @ (^ [P6:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [N:nat]: (((((if @ A) @ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ P6) & (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) @ (((times_times @ A) @ (((real_V8093663219630862766scaleR @ A) @ (((divide_divide @ real) @ ((ring_1_of_int @ real) @ (((times_times @ int) @ (((power_power @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (((divide_divide @ nat) @ P6) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((semiring_1_of_nat @ int) @ ((binomial @ P6) @ N))))) @ ((semiring_char_0_fact @ real) @ P6))) @ (((power_power @ A) @ X3) @ N))) @ (((power_power @ A) @ Y) @ (((minus_minus @ nat) @ P6) @ N)))) @ (zero_zero @ A))))) @ ((set_ord_atMost @ nat) @ P6))))) @ (((times_times @ A) @ ((cos @ A) @ X3)) @ ((cos @ A) @ Y))))))))). % 6.41/2.69 thf(fact_3424_cot__less__zero, axiom, (! [X3:real]: (((((ord_less @ real) @ (((divide_divide @ real) @ ((uminus_uminus @ real) @ pi)) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) @ X3) => ((((ord_less @ real) @ X3) @ (zero_zero @ real)) => (((ord_less @ real) @ ((cot @ real) @ X3)) @ (zero_zero @ real))))))). % 6.41/2.69 thf(fact_3425_int__ge__less__than2__def, axiom, (int_ge_less_than2 = (^ [D5:int]: (((collect @ ((product_prod @ int) @ int)) @ ((((product_case_prod @ int) @ int) @ $o) @ (^ [Z7:int,Z6:int]: (((((ord_less_eq @ int) @ D5) @ Z6) & (((ord_less @ int) @ Z7) @ Z6)))))))))). % 6.41/2.69 thf(fact_3426_of__nat__id, axiom, ((semiring_1_of_nat @ nat) = (^ [N:nat]: (N)))). % 6.41/2.69 thf(fact_3427_scaleR__cancel__right, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [A2:real,X3:A,B2:real]: ((((((real_V8093663219630862766scaleR @ A) @ A2) @ X3) = (((real_V8093663219630862766scaleR @ A) @ B2) @ X3)) = ((A2 = B2) | (X3 = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_3428_scaleR__zero__right, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [A2:real]: (((((real_V8093663219630862766scaleR @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_3429_cot__zero, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (((cot @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_3430_scaleR__zero__left, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [X3:A]: (((((real_V8093663219630862766scaleR @ A) @ (zero_zero @ real)) @ X3) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_3431_scaleR__eq__0__iff, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [A2:real,X3:A]: ((((((real_V8093663219630862766scaleR @ A) @ A2) @ X3) = (zero_zero @ A)) = ((A2 = (zero_zero @ real)) | (X3 = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_3432_scaleR__eq__iff, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [B2:A,U:real,A2:A]: ((((((plus_plus @ A) @ B2) @ (((real_V8093663219630862766scaleR @ A) @ U) @ A2)) = (((plus_plus @ A) @ A2) @ (((real_V8093663219630862766scaleR @ A) @ U) @ B2))) = ((A2 = B2) | (U = (one_one @ real)))))))))). % 6.41/2.69 thf(fact_3433_prod__eq__1__iff, axiom, (! [A:$tType,A3:(set @ A),F2:(A > nat)]: ((((finite_finite2 @ A) @ A3) => ((((((groups7121269368397514597t_prod @ A) @ nat) @ F2) @ A3) = (one_one @ nat)) = (! [X:A]: (((((member @ A) @ X) @ A3) => ((F2 @ X) = (one_one @ nat)))))))))). % 6.41/2.69 thf(fact_3434_scaleR__collapse, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [U:real,A2:A]: (((((plus_plus @ A) @ (((real_V8093663219630862766scaleR @ A) @ (((minus_minus @ real) @ (one_one @ real)) @ U)) @ A2)) @ (((real_V8093663219630862766scaleR @ A) @ U) @ A2)) = A2))))))). % 6.41/2.69 thf(fact_3435_prod__pos__nat__iff, axiom, (! [A:$tType,A3:(set @ A),F2:(A > nat)]: ((((finite_finite2 @ A) @ A3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ ((((groups7121269368397514597t_prod @ A) @ nat) @ F2) @ A3)) = (! [X:A]: (((((member @ A) @ X) @ A3) => (((ord_less @ nat) @ (zero_zero @ nat)) @ (F2 @ X)))))))))). % 6.41/2.69 thf(fact_3436_scaleR__half__double, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [A2:A]: (((((real_V8093663219630862766scaleR @ A) @ (((divide_divide @ real) @ (one_one @ real)) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) @ (((plus_plus @ A) @ A2) @ A2)) = A2))))))). % 6.41/2.69 thf(fact_3437_int__prod, axiom, (! [B:$tType,F2:(B > nat),A3:(set @ B)]: ((((semiring_1_of_nat @ int) @ ((((groups7121269368397514597t_prod @ B) @ nat) @ F2) @ A3)) = ((((groups7121269368397514597t_prod @ B) @ int) @ (^ [X:B]: (((semiring_1_of_nat @ int) @ (F2 @ X))))) @ A3))))). % 6.41/2.69 thf(fact_3438_scaleR__right__distrib, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [A2:real,X3:A,Y:A]: (((((real_V8093663219630862766scaleR @ A) @ A2) @ (((plus_plus @ A) @ X3) @ Y)) = (((plus_plus @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ X3)) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ Y))))))))). % 6.41/2.69 thf(fact_3439_scaleR__right__imp__eq, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [X3:A,A2:real,B2:real]: (((X3 != (zero_zero @ A)) => (((((real_V8093663219630862766scaleR @ A) @ A2) @ X3) = (((real_V8093663219630862766scaleR @ A) @ B2) @ X3)) => (A2 = B2))))))))). % 6.41/2.69 thf(fact_3440_scaleR__left__distrib, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [A2:real,B2:real,X3:A]: (((((real_V8093663219630862766scaleR @ A) @ (((plus_plus @ real) @ A2) @ B2)) @ X3) = (((plus_plus @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ X3)) @ (((real_V8093663219630862766scaleR @ A) @ B2) @ X3))))))))). % 6.41/2.69 thf(fact_3441_scaleR__left_Oadd, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [X3:real,Y:real,Xa2:A]: (((((real_V8093663219630862766scaleR @ A) @ (((plus_plus @ real) @ X3) @ Y)) @ Xa2) = (((plus_plus @ A) @ (((real_V8093663219630862766scaleR @ A) @ X3) @ Xa2)) @ (((real_V8093663219630862766scaleR @ A) @ Y) @ Xa2))))))))). % 6.41/2.69 thf(fact_3442_scaleR__right__mono__neg, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [B2:real,A2:real,C2:A]: (((((ord_less_eq @ real) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ C2)) @ (((real_V8093663219630862766scaleR @ A) @ B2) @ C2)))))))))). % 6.41/2.69 thf(fact_3443_scaleR__right__mono, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [A2:real,B2:real,X3:A]: (((((ord_less_eq @ real) @ A2) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => (((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ X3)) @ (((real_V8093663219630862766scaleR @ A) @ B2) @ X3)))))))))). % 6.41/2.69 thf(fact_3444_scaleR__le__cancel__left, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ B2)) = (((((ord_less @ real) @ (zero_zero @ real)) @ C2) => (((ord_less_eq @ A) @ A2) @ B2)) & ((((ord_less @ real) @ C2) @ (zero_zero @ real)) => (((ord_less_eq @ A) @ B2) @ A2)))))))))). % 6.41/2.69 thf(fact_3445_scaleR__le__cancel__left__neg, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,A2:A,B2:A]: (((((ord_less @ real) @ C2) @ (zero_zero @ real)) => ((((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ B2)) = (((ord_less_eq @ A) @ B2) @ A2))))))))). % 6.41/2.69 thf(fact_3446_scaleR__le__cancel__left__pos, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,A2:A,B2:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ C2) => ((((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ B2)) = (((ord_less_eq @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_3447_scaleR__left__mono__neg, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [B2:A,A2:A,C2:real]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ real) @ C2) @ (zero_zero @ real)) => (((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ B2)))))))))). % 6.41/2.69 thf(fact_3448_scaleR__left__mono, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [X3:A,Y:A,A2:real]: (((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ A2) => (((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ X3)) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ Y)))))))))). % 6.41/2.69 thf(fact_3449_eq__vector__fraction__iff, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [X3:A,U:real,V2:real,A2:A]: (((X3 = (((real_V8093663219630862766scaleR @ A) @ (((divide_divide @ real) @ U) @ V2)) @ A2)) = (((V2 = (zero_zero @ real)) => (X3 = (zero_zero @ A))) & ((V2 != (zero_zero @ real)) => ((((real_V8093663219630862766scaleR @ A) @ V2) @ X3) = (((real_V8093663219630862766scaleR @ A) @ U) @ A2))))))))))). % 6.41/2.69 thf(fact_3450_vector__fraction__eq__iff, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [U:real,V2:real,A2:A,X3:A]: ((((((real_V8093663219630862766scaleR @ A) @ (((divide_divide @ real) @ U) @ V2)) @ A2) = X3) = (((V2 = (zero_zero @ real)) => (X3 = (zero_zero @ A))) & ((V2 != (zero_zero @ real)) => ((((real_V8093663219630862766scaleR @ A) @ U) @ A2) = (((real_V8093663219630862766scaleR @ A) @ V2) @ X3))))))))))). % 6.41/2.69 thf(fact_3451_Real__Vector__Spaces_Ole__add__iff1, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [A2:real,E3:A,C2:A,B2:real,D2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ E3)) @ C2)) @ (((plus_plus @ A) @ (((real_V8093663219630862766scaleR @ A) @ B2) @ E3)) @ D2)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ (((real_V8093663219630862766scaleR @ A) @ (((minus_minus @ real) @ A2) @ B2)) @ E3)) @ C2)) @ D2)))))))). % 6.41/2.69 thf(fact_3452_Real__Vector__Spaces_Ole__add__iff2, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [A2:real,E3:A,C2:A,B2:real,D2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ E3)) @ C2)) @ (((plus_plus @ A) @ (((real_V8093663219630862766scaleR @ A) @ B2) @ E3)) @ D2)) = (((ord_less_eq @ A) @ C2) @ (((plus_plus @ A) @ (((real_V8093663219630862766scaleR @ A) @ (((minus_minus @ real) @ B2) @ A2)) @ E3)) @ D2))))))))). % 6.41/2.69 thf(fact_3453_scaleR__le__0__iff, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [A2:real,B2:A]: (((((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ B2)) @ (zero_zero @ A)) = (((((ord_less @ real) @ (zero_zero @ real)) @ A2) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))) | (((((ord_less @ real) @ A2) @ (zero_zero @ real)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)) | (A2 = (zero_zero @ real))))))))))). % 6.41/2.69 thf(fact_3454_zero__le__scaleR__iff, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [A2:real,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ B2)) = (((((ord_less @ real) @ (zero_zero @ real)) @ A2) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)) | (((((ord_less @ real) @ A2) @ (zero_zero @ real)) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A))) | (A2 = (zero_zero @ real))))))))))). % 6.41/2.69 thf(fact_3455_scaleR__nonpos__nonpos, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [A2:real,B2:A]: (((((ord_less_eq @ real) @ A2) @ (zero_zero @ real)) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_3456_scaleR__nonpos__nonneg, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [A2:real,X3:A]: (((((ord_less_eq @ real) @ A2) @ (zero_zero @ real)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => (((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ X3)) @ (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_3457_scaleR__nonneg__nonpos, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [A2:real,X3:A]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ A2) => ((((ord_less_eq @ A) @ X3) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ X3)) @ (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_3458_scaleR__nonneg__nonneg, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [A2:real,X3:A]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ X3)))))))))). % 6.41/2.69 thf(fact_3459_split__scaleR__pos__le, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [A2:real,B2:A]: (((((((ord_less_eq @ real) @ (zero_zero @ real)) @ A2) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less_eq @ real) @ A2) @ (zero_zero @ real)) & (((ord_less_eq @ A) @ B2) @ (zero_zero @ A)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_3460_split__scaleR__neg__le, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [A2:real,X3:A]: (((((((ord_less_eq @ real) @ (zero_zero @ real)) @ A2) & (((ord_less_eq @ A) @ X3) @ (zero_zero @ A))) | ((((ord_less_eq @ real) @ A2) @ (zero_zero @ real)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ X3))) => (((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ X3)) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3461_scaleR__mono_H, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [A2:real,B2:real,C2:A,D2:A]: (((((ord_less_eq @ real) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ D2) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ C2)) @ (((real_V8093663219630862766scaleR @ A) @ B2) @ D2)))))))))))). % 6.41/2.69 thf(fact_3462_scaleR__mono, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [A2:real,B2:real,X3:A,Y:A]: (((((ord_less_eq @ real) @ A2) @ B2) => ((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => (((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ X3)) @ (((real_V8093663219630862766scaleR @ A) @ B2) @ Y)))))))))))). % 6.41/2.69 thf(fact_3463_scaleR__left__le__one__le, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [X3:A,A2:real]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less_eq @ real) @ A2) @ (one_one @ real)) => (((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ X3)) @ X3))))))))). % 6.41/2.69 thf(fact_3464_scaleR__2, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [X3:A]: (((((real_V8093663219630862766scaleR @ A) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ X3) = (((plus_plus @ A) @ X3) @ X3)))))))). % 6.41/2.69 thf(fact_3465_prod__int__plus__eq, axiom, (! [I:nat,J:nat]: ((((((groups7121269368397514597t_prod @ nat) @ int) @ (semiring_1_of_nat @ int)) @ (((set_or1337092689740270186AtMost @ nat) @ I) @ (((plus_plus @ nat) @ I) @ J))) = ((((groups7121269368397514597t_prod @ int) @ int) @ (^ [X:int]: (X))) @ (((set_or1337092689740270186AtMost @ int) @ ((semiring_1_of_nat @ int) @ I)) @ ((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ I) @ J)))))))). % 6.41/2.69 thf(fact_3466_ln__prod, axiom, (! [A:$tType,I6:(set @ A),F2:(A > real)]: ((((finite_finite2 @ A) @ I6) => ((! [I4:A]: (((((member @ A) @ I4) @ I6) => (((ord_less @ real) @ (zero_zero @ real)) @ (F2 @ I4))))) => (((ln_ln @ real) @ ((((groups7121269368397514597t_prod @ A) @ real) @ F2) @ I6)) = ((((groups7311177749621191930dd_sum @ A) @ real) @ (^ [X:A]: (((ln_ln @ real) @ (F2 @ X))))) @ I6))))))). % 6.41/2.69 thf(fact_3467_prod_Otriangle__reindex__eq, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > (nat > A)),N2:nat]: ((((((groups7121269368397514597t_prod @ ((product_prod @ nat) @ nat)) @ A) @ ((((product_case_prod @ nat) @ nat) @ A) @ G)) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [I3:nat,J3:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I3) @ J3)) @ N2)))))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [K3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: (((G @ I3) @ (((minus_minus @ nat) @ K3) @ I3))))) @ ((set_ord_atMost @ nat) @ K3))))) @ ((set_ord_atMost @ nat) @ N2))))))))). % 6.41/2.69 thf(fact_3468_prod_Otriangle__reindex, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > (nat > A)),N2:nat]: ((((((groups7121269368397514597t_prod @ ((product_prod @ nat) @ nat)) @ A) @ ((((product_case_prod @ nat) @ nat) @ A) @ G)) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [I3:nat,J3:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ I3) @ J3)) @ N2)))))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [K3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: (((G @ I3) @ (((minus_minus @ nat) @ K3) @ I3))))) @ ((set_ord_atMost @ nat) @ K3))))) @ ((set_ord_lessThan @ nat) @ N2))))))))). % 6.41/2.69 thf(fact_3469_cot__gt__zero, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => (((ord_less @ real) @ (zero_zero @ real)) @ ((cot @ real) @ X3))))))). % 6.41/2.69 thf(fact_3470_int__ge__less__than__def, axiom, (int_ge_less_than = (^ [D5:int]: (((collect @ ((product_prod @ int) @ int)) @ ((((product_case_prod @ int) @ int) @ $o) @ (^ [Z7:int,Z6:int]: (((((ord_less_eq @ int) @ D5) @ Z7) & (((ord_less @ int) @ Z7) @ Z6)))))))))). % 6.41/2.69 thf(fact_3471_exp__first__two__terms, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => ((exp @ A) = (^ [X:A]: ((((plus_plus @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ X)) @ ((suminf @ A) @ (^ [N:nat]: ((((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ ((semiring_char_0_fact @ real) @ (((plus_plus @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ (((power_power @ A) @ X) @ (((plus_plus @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))))))). % 6.41/2.69 thf(fact_3472_arctan__def, axiom, (arctan = (^ [Y2:real]: (((the @ real) @ (^ [X:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ X) & ((((ord_less @ real) @ X) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) & (((tan @ real) @ X) = Y2)))))))))). % 6.41/2.69 thf(fact_3473_modulo__int__unfold, axiom, (! [L:int,K:int,N2:nat,M:nat]: (((((((sgn_sgn @ int) @ L) = (zero_zero @ int)) | ((((sgn_sgn @ int) @ K) = (zero_zero @ int)) | (N2 = (zero_zero @ nat)))) => ((((modulo_modulo @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((semiring_1_of_nat @ int) @ M))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ ((semiring_1_of_nat @ int) @ N2))) = (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((semiring_1_of_nat @ int) @ M)))) & ((~ (((((sgn_sgn @ int) @ L) = (zero_zero @ int)) | ((((sgn_sgn @ int) @ K) = (zero_zero @ int)) | (N2 = (zero_zero @ nat)))))) => (((((sgn_sgn @ int) @ K) = ((sgn_sgn @ int) @ L)) => ((((modulo_modulo @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((semiring_1_of_nat @ int) @ M))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ ((semiring_1_of_nat @ int) @ N2))) = (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ ((semiring_1_of_nat @ int) @ (((modulo_modulo @ nat) @ M) @ N2))))) & ((((sgn_sgn @ int) @ K) != ((sgn_sgn @ int) @ L)) => ((((modulo_modulo @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((semiring_1_of_nat @ int) @ M))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ ((semiring_1_of_nat @ int) @ N2))) = (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ (((times_times @ nat) @ N2) @ ((zero_neq_one_of_bool @ nat) @ (~ ((((dvd_dvd @ nat) @ N2) @ M))))))) @ ((semiring_1_of_nat @ int) @ (((modulo_modulo @ nat) @ M) @ N2)))))))))))). % 6.41/2.69 thf(fact_3474_powr__int, axiom, (! [X3:real,I:int]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ I) => ((((powr @ real) @ X3) @ ((ring_1_of_int @ real) @ I)) = (((power_power @ real) @ X3) @ (nat2 @ I)))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ I))) => ((((powr @ real) @ X3) @ ((ring_1_of_int @ real) @ I)) = (((divide_divide @ real) @ (one_one @ real)) @ (((power_power @ real) @ X3) @ (nat2 @ ((uminus_uminus @ int) @ I))))))))))). % 6.41/2.69 thf(fact_3475_inverse__inverse__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: ((((inverse_inverse @ A) @ ((inverse_inverse @ A) @ A2)) = A2))))))). % 6.41/2.69 thf(fact_3476_inverse__eq__iff__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A]: (((((inverse_inverse @ A) @ A2) = ((inverse_inverse @ A) @ B2)) = (A2 = B2)))))))). % 6.41/2.69 thf(fact_3477_sgn__sgn, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A]: ((((sgn_sgn @ A) @ ((sgn_sgn @ A) @ A2)) = ((sgn_sgn @ A) @ A2)))))))). % 6.41/2.69 thf(fact_3478_inverse__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (((inverse_inverse @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_3479_inverse__nonzero__iff__nonzero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((((inverse_inverse @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3480_inverse__mult__distrib, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A]: ((((inverse_inverse @ A) @ (((times_times @ A) @ A2) @ B2)) = (((times_times @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2))))))))). % 6.41/2.69 thf(fact_3481_inverse__1, axiom, (! [A:$tType]: (((division_ring @ A) => (((inverse_inverse @ A) @ (one_one @ A)) = (one_one @ A)))))). % 6.41/2.69 thf(fact_3482_inverse__eq__1__iff, axiom, (! [A:$tType]: (((field @ A) => (! [X3:A]: (((((inverse_inverse @ A) @ X3) = (one_one @ A)) = (X3 = (one_one @ A))))))))). % 6.41/2.69 thf(fact_3483_inverse__divide, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A]: ((((inverse_inverse @ A) @ (((divide_divide @ A) @ A2) @ B2)) = (((divide_divide @ A) @ B2) @ A2)))))))). % 6.41/2.69 thf(fact_3484_inverse__minus__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: ((((inverse_inverse @ A) @ ((uminus_uminus @ A) @ A2)) = ((uminus_uminus @ A) @ ((inverse_inverse @ A) @ A2))))))))). % 6.41/2.69 thf(fact_3485_sgn__0, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((sgn_sgn @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_3486_sgn__zero, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (((sgn_sgn @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_3487_sgn__1, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((sgn_sgn @ A) @ (one_one @ A)) = (one_one @ A)))))). % 6.41/2.69 thf(fact_3488_sgn__divide, axiom, (! [A:$tType]: (((field_abs_sgn @ A) => (! [A2:A,B2:A]: ((((sgn_sgn @ A) @ (((divide_divide @ A) @ A2) @ B2)) = (((divide_divide @ A) @ ((sgn_sgn @ A) @ A2)) @ ((sgn_sgn @ A) @ B2))))))))). % 6.41/2.69 thf(fact_3489_idom__abs__sgn__class_Osgn__minus, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A]: ((((sgn_sgn @ A) @ ((uminus_uminus @ A) @ A2)) = ((uminus_uminus @ A) @ ((sgn_sgn @ A) @ A2))))))))). % 6.41/2.69 thf(fact_3490_abs__inverse, axiom, (! [A:$tType]: (((field_abs_sgn @ A) => (! [A2:A]: ((((abs_abs @ A) @ ((inverse_inverse @ A) @ A2)) = ((inverse_inverse @ A) @ ((abs_abs @ A) @ A2))))))))). % 6.41/2.69 thf(fact_3491_inverse__sgn, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: ((((inverse_inverse @ A) @ ((sgn_sgn @ A) @ A2)) = ((sgn_sgn @ A) @ A2)))))))). % 6.41/2.69 thf(fact_3492_sgn__inverse, axiom, (! [A:$tType]: (((field_abs_sgn @ A) => (! [A2:A]: ((((sgn_sgn @ A) @ ((inverse_inverse @ A) @ A2)) = ((inverse_inverse @ A) @ ((sgn_sgn @ A) @ A2))))))))). % 6.41/2.69 thf(fact_3493_nat__int, axiom, (! [N2:nat]: (((nat2 @ ((semiring_1_of_nat @ int) @ N2)) = N2)))). % 6.41/2.69 thf(fact_3494_inverse__nonnegative__iff__nonnegative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))). % 6.41/2.69 thf(fact_3495_inverse__nonpositive__iff__nonpositive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3496_inverse__positive__iff__positive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))). % 6.41/2.69 thf(fact_3497_inverse__negative__iff__negative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3498_inverse__less__iff__less__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => ((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((ord_less @ A) @ B2) @ A2)))))))))). % 6.41/2.69 thf(fact_3499_inverse__less__iff__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((ord_less @ A) @ B2) @ A2)))))))))). % 6.41/2.69 thf(fact_3500_sgn__greater, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((sgn_sgn @ A) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))). % 6.41/2.69 thf(fact_3501_sgn__less, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((ord_less @ A) @ ((sgn_sgn @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3502_divide__sgn, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((divide_divide @ A) @ A2) @ ((sgn_sgn @ A) @ B2)) = (((times_times @ A) @ A2) @ ((sgn_sgn @ A) @ B2))))))))). % 6.41/2.69 thf(fact_3503_nat__numeral, axiom, (! [K:num]: (((nat2 @ ((numeral_numeral @ int) @ K)) = ((numeral_numeral @ nat) @ K))))). % 6.41/2.69 thf(fact_3504_nat__of__bool, axiom, (! [P:$o]: (((nat2 @ ((zero_neq_one_of_bool @ int) @ P)) = ((zero_neq_one_of_bool @ nat) @ P))))). % 6.41/2.69 thf(fact_3505_inverse__le__iff__le__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((ord_less_eq @ A) @ B2) @ A2)))))))))). % 6.41/2.69 thf(fact_3506_inverse__le__iff__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((ord_less_eq @ A) @ B2) @ A2)))))))))). % 6.41/2.69 thf(fact_3507_right__inverse, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => ((((times_times @ A) @ A2) @ ((inverse_inverse @ A) @ A2)) = (one_one @ A))))))))). % 6.41/2.69 thf(fact_3508_left__inverse, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => ((((times_times @ A) @ ((inverse_inverse @ A) @ A2)) @ A2) = (one_one @ A))))))))). % 6.41/2.69 thf(fact_3509_sgn__pos, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((sgn_sgn @ A) @ A2) = (one_one @ A))))))))). % 6.41/2.69 thf(fact_3510_abs__sgn__eq__1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => (((abs_abs @ A) @ ((sgn_sgn @ A) @ A2)) = (one_one @ A))))))))). % 6.41/2.69 thf(fact_3511_nat__1, axiom, ((nat2 @ (one_one @ int)) = (suc @ (zero_zero @ nat)))). % 6.41/2.69 thf(fact_3512_sgn__mult__self__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((times_times @ A) @ ((sgn_sgn @ A) @ A2)) @ ((sgn_sgn @ A) @ A2)) = ((zero_neq_one_of_bool @ A) @ (A2 != (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_3513_nat__0__iff, axiom, (! [I:int]: ((((nat2 @ I) = (zero_zero @ nat)) = (((ord_less_eq @ int) @ I) @ (zero_zero @ int)))))). % 6.41/2.69 thf(fact_3514_nat__le__0, axiom, (! [Z2:int]: (((((ord_less_eq @ int) @ Z2) @ (zero_zero @ int)) => ((nat2 @ Z2) = (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_3515_zless__nat__conj, axiom, (! [W2:int,Z2:int]: (((((ord_less @ nat) @ (nat2 @ W2)) @ (nat2 @ Z2)) = ((((ord_less @ int) @ (zero_zero @ int)) @ Z2) & (((ord_less @ int) @ W2) @ Z2)))))). % 6.41/2.69 thf(fact_3516_nat__neg__numeral, axiom, (! [K:num]: (((nat2 @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K))) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_3517_sgn__abs, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A]: ((((abs_abs @ A) @ ((sgn_sgn @ A) @ A2)) = ((zero_neq_one_of_bool @ A) @ (A2 != (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_3518_idom__abs__sgn__class_Oabs__sgn, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A]: ((((sgn_sgn @ A) @ ((abs_abs @ A) @ A2)) = ((zero_neq_one_of_bool @ A) @ (A2 != (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_3519_nat__zminus__int, axiom, (! [N2:nat]: (((nat2 @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ N2))) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_3520_int__nat__eq, axiom, (! [Z2:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => (((semiring_1_of_nat @ int) @ (nat2 @ Z2)) = Z2)) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2))) => (((semiring_1_of_nat @ int) @ (nat2 @ Z2)) = (zero_zero @ int))))))). % 6.41/2.69 thf(fact_3521_sgn__mult__dvd__iff, axiom, (! [R2:int,L:int,K:int]: (((((dvd_dvd @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ R2)) @ L)) @ K) = ((((dvd_dvd @ int) @ L) @ K) & ((R2 = (zero_zero @ int)) => (K = (zero_zero @ int)))))))). % 6.41/2.69 thf(fact_3522_mult__sgn__dvd__iff, axiom, (! [L:int,R2:int,K:int]: (((((dvd_dvd @ int) @ (((times_times @ int) @ L) @ ((sgn_sgn @ int) @ R2))) @ K) = ((((dvd_dvd @ int) @ L) @ K) & ((R2 = (zero_zero @ int)) => (K = (zero_zero @ int)))))))). % 6.41/2.69 thf(fact_3523_dvd__sgn__mult__iff, axiom, (! [L:int,R2:int,K:int]: (((((dvd_dvd @ int) @ L) @ (((times_times @ int) @ ((sgn_sgn @ int) @ R2)) @ K)) = ((((dvd_dvd @ int) @ L) @ K) | (R2 = (zero_zero @ int))))))). % 6.41/2.69 thf(fact_3524_dvd__mult__sgn__iff, axiom, (! [L:int,K:int,R2:int]: (((((dvd_dvd @ int) @ L) @ (((times_times @ int) @ K) @ ((sgn_sgn @ int) @ R2))) = ((((dvd_dvd @ int) @ L) @ K) | (R2 = (zero_zero @ int))))))). % 6.41/2.69 thf(fact_3525_sgn__neg, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => (((sgn_sgn @ A) @ A2) = ((uminus_uminus @ A) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_3526_zero__less__nat__eq, axiom, (! [Z2:int]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (nat2 @ Z2)) = (((ord_less @ int) @ (zero_zero @ int)) @ Z2))))). % 6.41/2.69 thf(fact_3527_of__nat__nat, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => (((semiring_1_of_nat @ A) @ (nat2 @ Z2)) = ((ring_1_of_int @ A) @ Z2))))))))). % 6.41/2.69 thf(fact_3528_sgn__of__nat, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:nat]: ((((sgn_sgn @ A) @ ((semiring_1_of_nat @ A) @ N2)) = ((zero_neq_one_of_bool @ A) @ (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))))))). % 6.41/2.69 thf(fact_3529_diff__nat__numeral, axiom, (! [V2:num,V4:num]: (((((minus_minus @ nat) @ ((numeral_numeral @ nat) @ V2)) @ ((numeral_numeral @ nat) @ V4)) = (nat2 @ (((minus_minus @ int) @ ((numeral_numeral @ int) @ V2)) @ ((numeral_numeral @ int) @ V4))))))). % 6.41/2.69 thf(fact_3530_numeral__power__eq__nat__cancel__iff, axiom, (! [X3:num,N2:nat,Y:int]: ((((((power_power @ nat) @ ((numeral_numeral @ nat) @ X3)) @ N2) = (nat2 @ Y)) = ((((power_power @ int) @ ((numeral_numeral @ int) @ X3)) @ N2) = Y))))). % 6.41/2.69 thf(fact_3531_nat__eq__numeral__power__cancel__iff, axiom, (! [Y:int,X3:num,N2:nat]: ((((nat2 @ Y) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ X3)) @ N2)) = (Y = (((power_power @ int) @ ((numeral_numeral @ int) @ X3)) @ N2)))))). % 6.41/2.69 thf(fact_3532_nat__abs__dvd__iff, axiom, (! [K:int,N2:nat]: (((((dvd_dvd @ nat) @ (nat2 @ ((abs_abs @ int) @ K))) @ N2) = (((dvd_dvd @ int) @ K) @ ((semiring_1_of_nat @ int) @ N2)))))). % 6.41/2.69 thf(fact_3533_dvd__nat__abs__iff, axiom, (! [N2:nat,K:int]: (((((dvd_dvd @ nat) @ N2) @ (nat2 @ ((abs_abs @ int) @ K))) = (((dvd_dvd @ int) @ ((semiring_1_of_nat @ int) @ N2)) @ K))))). % 6.41/2.69 thf(fact_3534_nat__ceiling__le__eq, axiom, (! [X3:real,A2:nat]: (((((ord_less_eq @ nat) @ (nat2 @ ((archimedean_ceiling @ real) @ X3))) @ A2) = (((ord_less_eq @ real) @ X3) @ ((semiring_1_of_nat @ real) @ A2)))))). % 6.41/2.69 thf(fact_3535_one__less__nat__eq, axiom, (! [Z2:int]: (((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ (nat2 @ Z2)) = (((ord_less @ int) @ (one_one @ int)) @ Z2))))). % 6.41/2.69 thf(fact_3536_nat__numeral__diff__1, axiom, (! [V2:num]: (((((minus_minus @ nat) @ ((numeral_numeral @ nat) @ V2)) @ (one_one @ nat)) = (nat2 @ (((minus_minus @ int) @ ((numeral_numeral @ int) @ V2)) @ (one_one @ int))))))). % 6.41/2.69 thf(fact_3537_nat__less__numeral__power__cancel__iff, axiom, (! [A2:int,X3:num,N2:nat]: (((((ord_less @ nat) @ (nat2 @ A2)) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ X3)) @ N2)) = (((ord_less @ int) @ A2) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X3)) @ N2)))))). % 6.41/2.69 thf(fact_3538_numeral__power__less__nat__cancel__iff, axiom, (! [X3:num,N2:nat,A2:int]: (((((ord_less @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ X3)) @ N2)) @ (nat2 @ A2)) = (((ord_less @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X3)) @ N2)) @ A2))))). % 6.41/2.69 thf(fact_3539_nat__le__numeral__power__cancel__iff, axiom, (! [A2:int,X3:num,N2:nat]: (((((ord_less_eq @ nat) @ (nat2 @ A2)) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ X3)) @ N2)) = (((ord_less_eq @ int) @ A2) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X3)) @ N2)))))). % 6.41/2.69 thf(fact_3540_numeral__power__le__nat__cancel__iff, axiom, (! [X3:num,N2:nat,A2:int]: (((((ord_less_eq @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ X3)) @ N2)) @ (nat2 @ A2)) = (((ord_less_eq @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ X3)) @ N2)) @ A2))))). % 6.41/2.69 thf(fact_3541_nonzero__norm__inverse, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => (((real_V7770717601297561774m_norm @ A) @ ((inverse_inverse @ A) @ A2)) = ((inverse_inverse @ real) @ ((real_V7770717601297561774m_norm @ A) @ A2)))))))))). % 6.41/2.69 thf(fact_3542_sgn__zero__iff, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X3:A]: (((((sgn_sgn @ A) @ X3) = (zero_zero @ A)) = (X3 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3543_sgn__eq__0__iff, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A]: (((((sgn_sgn @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3544_sgn__0__0, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((sgn_sgn @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3545_sgn__mult, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A,B2:A]: ((((sgn_sgn @ A) @ (((times_times @ A) @ A2) @ B2)) = (((times_times @ A) @ ((sgn_sgn @ A) @ A2)) @ ((sgn_sgn @ A) @ B2))))))))). % 6.41/2.69 thf(fact_3546_same__sgn__sgn__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [B2:A,A2:A]: (((((sgn_sgn @ A) @ B2) = ((sgn_sgn @ A) @ A2)) => (((sgn_sgn @ A) @ (((plus_plus @ A) @ A2) @ B2)) = ((sgn_sgn @ A) @ A2))))))))). % 6.41/2.69 thf(fact_3547_mult__commute__imp__mult__inverse__commute, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Y:A,X3:A]: ((((((times_times @ A) @ Y) @ X3) = (((times_times @ A) @ X3) @ Y)) => ((((times_times @ A) @ ((inverse_inverse @ A) @ Y)) @ X3) = (((times_times @ A) @ X3) @ ((inverse_inverse @ A) @ Y)))))))))). % 6.41/2.69 thf(fact_3548_inverse__eq__imp__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A]: (((((inverse_inverse @ A) @ A2) = ((inverse_inverse @ A) @ B2)) => (A2 = B2)))))))). % 6.41/2.69 thf(fact_3549_nonzero__imp__inverse__nonzero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => (((inverse_inverse @ A) @ A2) != (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3550_nonzero__inverse__inverse__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => (((inverse_inverse @ A) @ ((inverse_inverse @ A) @ A2)) = A2)))))))). % 6.41/2.69 thf(fact_3551_nonzero__inverse__eq__imp__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A]: (((((inverse_inverse @ A) @ A2) = ((inverse_inverse @ A) @ B2)) => ((A2 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => (A2 = B2)))))))))). % 6.41/2.69 thf(fact_3552_inverse__zero__imp__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((((inverse_inverse @ A) @ A2) = (zero_zero @ A)) => (A2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3553_field__class_Ofield__inverse__zero, axiom, (! [A:$tType]: (((field @ A) => (((inverse_inverse @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_3554_nonzero__inverse__scaleR__distrib, axiom, (! [A:$tType]: (((real_V5047593784448816457lgebra @ A) => (! [A2:real,X3:A]: (((A2 != (zero_zero @ real)) => ((X3 != (zero_zero @ A)) => (((inverse_inverse @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ X3)) = (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ A2)) @ ((inverse_inverse @ A) @ X3))))))))))). % 6.41/2.69 thf(fact_3555_norm__inverse__le__norm, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [R2:real,X3:A]: (((((ord_less_eq @ real) @ R2) @ ((real_V7770717601297561774m_norm @ A) @ X3)) => ((((ord_less @ real) @ (zero_zero @ real)) @ R2) => (((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ ((inverse_inverse @ A) @ X3))) @ ((inverse_inverse @ real) @ R2)))))))))). % 6.41/2.69 thf(fact_3556_positive__imp__inverse__positive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2))))))))). % 6.41/2.69 thf(fact_3557_negative__imp__inverse__negative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => (((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3558_inverse__positive__imp__positive, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2)) => ((A2 != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ A2))))))))). % 6.41/2.69 thf(fact_3559_inverse__negative__imp__negative, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A)) => ((A2 != (zero_zero @ A)) => (((ord_less @ A) @ A2) @ (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_3560_less__imp__inverse__less__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A2)))))))))). % 6.41/2.69 thf(fact_3561_inverse__less__imp__less__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ A2))))))))). % 6.41/2.69 thf(fact_3562_less__imp__inverse__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A2)))))))))). % 6.41/2.69 thf(fact_3563_inverse__less__imp__less, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ B2) @ A2))))))))). % 6.41/2.69 thf(fact_3564_nonzero__inverse__mult__distrib, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A]: (((A2 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => (((inverse_inverse @ A) @ (((times_times @ A) @ A2) @ B2)) = (((times_times @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A2))))))))))). % 6.41/2.69 thf(fact_3565_nonzero__inverse__minus__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => (((inverse_inverse @ A) @ ((uminus_uminus @ A) @ A2)) = ((uminus_uminus @ A) @ ((inverse_inverse @ A) @ A2)))))))))). % 6.41/2.69 thf(fact_3566_inverse__unique, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A]: ((((((times_times @ A) @ A2) @ B2) = (one_one @ A)) => (((inverse_inverse @ A) @ A2) = B2)))))))). % 6.41/2.69 thf(fact_3567_divide__inverse__commute, axiom, (! [A:$tType]: (((field @ A) => ((divide_divide @ A) = (^ [A4:A,B4:A]: ((((times_times @ A) @ ((inverse_inverse @ A) @ B4)) @ A4)))))))). % 6.41/2.69 thf(fact_3568_divide__inverse, axiom, (! [A:$tType]: (((division_ring @ A) => ((divide_divide @ A) = (^ [A4:A,B4:A]: ((((times_times @ A) @ A4) @ ((inverse_inverse @ A) @ B4))))))))). % 6.41/2.69 thf(fact_3569_field__class_Ofield__divide__inverse, axiom, (! [A:$tType]: (((field @ A) => ((divide_divide @ A) = (^ [A4:A,B4:A]: ((((times_times @ A) @ A4) @ ((inverse_inverse @ A) @ B4))))))))). % 6.41/2.69 thf(fact_3570_inverse__eq__divide, axiom, (! [A:$tType]: (((division_ring @ A) => ((inverse_inverse @ A) = ((divide_divide @ A) @ (one_one @ A))))))). % 6.41/2.69 thf(fact_3571_power__mult__power__inverse__commute, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,M:nat,N2:nat]: (((((times_times @ A) @ (((power_power @ A) @ X3) @ M)) @ (((power_power @ A) @ ((inverse_inverse @ A) @ X3)) @ N2)) = (((times_times @ A) @ (((power_power @ A) @ ((inverse_inverse @ A) @ X3)) @ N2)) @ (((power_power @ A) @ X3) @ M))))))))). % 6.41/2.69 thf(fact_3572_power__mult__inverse__distrib, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,M:nat]: (((((times_times @ A) @ (((power_power @ A) @ X3) @ M)) @ ((inverse_inverse @ A) @ X3)) = (((times_times @ A) @ ((inverse_inverse @ A) @ X3)) @ (((power_power @ A) @ X3) @ M))))))))). % 6.41/2.69 thf(fact_3573_mult__inverse__of__nat__commute, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Xa2:nat,X3:A]: (((((times_times @ A) @ ((inverse_inverse @ A) @ ((semiring_1_of_nat @ A) @ Xa2))) @ X3) = (((times_times @ A) @ X3) @ ((inverse_inverse @ A) @ ((semiring_1_of_nat @ A) @ Xa2)))))))))). % 6.41/2.69 thf(fact_3574_nonzero__abs__inverse, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => (((abs_abs @ A) @ ((inverse_inverse @ A) @ A2)) = ((inverse_inverse @ A) @ ((abs_abs @ A) @ A2)))))))))). % 6.41/2.69 thf(fact_3575_sgn__not__eq__imp, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [B2:A,A2:A]: (((((sgn_sgn @ A) @ B2) != ((sgn_sgn @ A) @ A2)) => ((((sgn_sgn @ A) @ A2) != (zero_zero @ A)) => ((((sgn_sgn @ A) @ B2) != (zero_zero @ A)) => (((sgn_sgn @ A) @ A2) = ((uminus_uminus @ A) @ ((sgn_sgn @ A) @ B2)))))))))))). % 6.41/2.69 thf(fact_3576_mult__inverse__of__int__commute, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Xa2:int,X3:A]: (((((times_times @ A) @ ((inverse_inverse @ A) @ ((ring_1_of_int @ A) @ Xa2))) @ X3) = (((times_times @ A) @ X3) @ ((inverse_inverse @ A) @ ((ring_1_of_int @ A) @ Xa2)))))))))). % 6.41/2.69 thf(fact_3577_sgn__minus__1, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((sgn_sgn @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ (one_one @ A))))))). % 6.41/2.69 thf(fact_3578_nat__zero__as__int, axiom, ((zero_zero @ nat) = (nat2 @ (zero_zero @ int)))). % 6.41/2.69 thf(fact_3579_mult__sgn__abs, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A]: (((((times_times @ A) @ ((sgn_sgn @ A) @ X3)) @ ((abs_abs @ A) @ X3)) = X3))))))). % 6.41/2.69 thf(fact_3580_sgn__mult__abs, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A]: (((((times_times @ A) @ ((sgn_sgn @ A) @ A2)) @ ((abs_abs @ A) @ A2)) = A2))))))). % 6.41/2.69 thf(fact_3581_abs__mult__sgn, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (! [A2:A]: (((((times_times @ A) @ ((abs_abs @ A) @ A2)) @ ((sgn_sgn @ A) @ A2)) = A2))))))). % 6.41/2.69 thf(fact_3582_linordered__idom__class_Oabs__sgn, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((abs_abs @ A) = (^ [K3:A]: ((((times_times @ A) @ K3) @ ((sgn_sgn @ A) @ K3))))))))). % 6.41/2.69 thf(fact_3583_nat__mono, axiom, (! [X3:int,Y:int]: (((((ord_less_eq @ int) @ X3) @ Y) => (((ord_less_eq @ nat) @ (nat2 @ X3)) @ (nat2 @ Y)))))). % 6.41/2.69 thf(fact_3584_int__sgnE, axiom, (! [K:int]: ((~ ((! [N3:nat,L4:int]: ((K != (((times_times @ int) @ ((sgn_sgn @ int) @ L4)) @ ((semiring_1_of_nat @ int) @ N3)))))))))). % 6.41/2.69 thf(fact_3585_same__sgn__abs__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [B2:A,A2:A]: (((((sgn_sgn @ A) @ B2) = ((sgn_sgn @ A) @ A2)) => (((abs_abs @ A) @ (((plus_plus @ A) @ A2) @ B2)) = (((plus_plus @ A) @ ((abs_abs @ A) @ A2)) @ ((abs_abs @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_3586_ex__nat, axiom, ((^ [P2:(nat > $o)]: ((? [X7:nat]: ((P2 @ X7))))) = (^ [P3:(nat > $o)]: ((? [X:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) & (P3 @ (nat2 @ X))))))))). % 6.41/2.69 thf(fact_3587_all__nat, axiom, ((^ [P2:(nat > $o)]: ((! [X7:nat]: ((P2 @ X7))))) = (^ [P3:(nat > $o)]: ((! [X:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X) => (P3 @ (nat2 @ X))))))))). % 6.41/2.69 thf(fact_3588_eq__nat__nat__iff, axiom, (! [Z2:int,Z8:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z8) => (((nat2 @ Z2) = (nat2 @ Z8)) = (Z2 = Z8))))))). % 6.41/2.69 thf(fact_3589_le__imp__inverse__le__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A2)))))))))). % 6.41/2.69 thf(fact_3590_inverse__le__imp__le__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ A2))))))))). % 6.41/2.69 thf(fact_3591_le__imp__inverse__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A2)))))))))). % 6.41/2.69 thf(fact_3592_inverse__le__imp__le, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less_eq @ A) @ B2) @ A2))))))))). % 6.41/2.69 thf(fact_3593_inverse__le__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ X3)) @ (one_one @ A)) = ((((ord_less_eq @ A) @ X3) @ (zero_zero @ A)) | (((ord_less_eq @ A) @ (one_one @ A)) @ X3))))))))). % 6.41/2.69 thf(fact_3594_one__less__inverse__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A]: (((((ord_less @ A) @ (one_one @ A)) @ ((inverse_inverse @ A) @ X3)) = ((((ord_less @ A) @ (zero_zero @ A)) @ X3) & (((ord_less @ A) @ X3) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_3595_one__less__inverse, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ A2) @ (one_one @ A)) => (((ord_less @ A) @ (one_one @ A)) @ ((inverse_inverse @ A) @ A2)))))))))). % 6.41/2.69 thf(fact_3596_field__class_Ofield__inverse, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => ((((times_times @ A) @ ((inverse_inverse @ A) @ A2)) @ A2) = (one_one @ A))))))))). % 6.41/2.69 thf(fact_3597_division__ring__inverse__add, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A]: (((A2 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => ((((plus_plus @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((times_times @ A) @ (((times_times @ A) @ ((inverse_inverse @ A) @ A2)) @ (((plus_plus @ A) @ A2) @ B2))) @ ((inverse_inverse @ A) @ B2))))))))))). % 6.41/2.69 thf(fact_3598_inverse__add, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A,B2:A]: (((A2 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => ((((plus_plus @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((times_times @ A) @ (((times_times @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ ((inverse_inverse @ A) @ A2))) @ ((inverse_inverse @ A) @ B2))))))))))). % 6.41/2.69 thf(fact_3599_division__ring__inverse__diff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A]: (((A2 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => ((((minus_minus @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((times_times @ A) @ (((times_times @ A) @ ((inverse_inverse @ A) @ A2)) @ (((minus_minus @ A) @ B2) @ A2))) @ ((inverse_inverse @ A) @ B2))))))))))). % 6.41/2.69 thf(fact_3600_nonzero__inverse__eq__divide, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A]: (((A2 != (zero_zero @ A)) => (((inverse_inverse @ A) @ A2) = (((divide_divide @ A) @ (one_one @ A)) @ A2))))))))). % 6.41/2.69 thf(fact_3601_sgn__1__pos, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((sgn_sgn @ A) @ A2) = (one_one @ A)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))). % 6.41/2.69 thf(fact_3602_abs__sgn__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: ((((A2 = (zero_zero @ A)) => (((abs_abs @ A) @ ((sgn_sgn @ A) @ A2)) = (zero_zero @ A))) & ((A2 != (zero_zero @ A)) => (((abs_abs @ A) @ ((sgn_sgn @ A) @ A2)) = (one_one @ A)))))))))). % 6.41/2.69 thf(fact_3603_nat__mono__iff, axiom, (! [Z2:int,W2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ Z2) => ((((ord_less @ nat) @ (nat2 @ W2)) @ (nat2 @ Z2)) = (((ord_less @ int) @ W2) @ Z2)))))). % 6.41/2.69 thf(fact_3604_of__nat__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [R2:A]: ((((ord_less_eq @ A) @ R2) @ ((semiring_1_of_nat @ A) @ (nat2 @ ((archimedean_ceiling @ A) @ R2)))))))))). % 6.41/2.69 thf(fact_3605_zless__nat__eq__int__zless, axiom, (! [M:nat,Z2:int]: (((((ord_less @ nat) @ M) @ (nat2 @ Z2)) = (((ord_less @ int) @ ((semiring_1_of_nat @ int) @ M)) @ Z2))))). % 6.41/2.69 thf(fact_3606_nat__le__iff, axiom, (! [X3:int,N2:nat]: (((((ord_less_eq @ nat) @ (nat2 @ X3)) @ N2) = (((ord_less_eq @ int) @ X3) @ ((semiring_1_of_nat @ int) @ N2)))))). % 6.41/2.69 thf(fact_3607_nat__0__le, axiom, (! [Z2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => (((semiring_1_of_nat @ int) @ (nat2 @ Z2)) = Z2))))). % 6.41/2.69 thf(fact_3608_int__eq__iff, axiom, (! [M:nat,Z2:int]: (((((semiring_1_of_nat @ int) @ M) = Z2) = ((M = (nat2 @ Z2)) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2)))))). % 6.41/2.69 thf(fact_3609_nat__int__add, axiom, (! [A2:nat,B2:nat]: (((nat2 @ (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ A2)) @ ((semiring_1_of_nat @ int) @ B2))) = (((plus_plus @ nat) @ A2) @ B2))))). % 6.41/2.69 thf(fact_3610_nat__abs__mult__distrib, axiom, (! [W2:int,Z2:int]: (((nat2 @ ((abs_abs @ int) @ (((times_times @ int) @ W2) @ Z2))) = (((times_times @ nat) @ (nat2 @ ((abs_abs @ int) @ W2))) @ (nat2 @ ((abs_abs @ int) @ Z2))))))). % 6.41/2.69 thf(fact_3611_nat__plus__as__int, axiom, ((plus_plus @ nat) = (^ [A4:nat,B4:nat]: ((nat2 @ (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ A4)) @ ((semiring_1_of_nat @ int) @ B4))))))). % 6.41/2.69 thf(fact_3612_inverse__less__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)) => (((ord_less @ A) @ B2) @ A2)) & ((((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ B2)) @ (zero_zero @ A)) => (((ord_less @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_3613_inverse__le__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)) => (((ord_less_eq @ A) @ B2) @ A2)) & ((((ord_less_eq @ A) @ (((times_times @ A) @ A2) @ B2)) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_3614_one__le__inverse__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A]: (((((ord_less_eq @ A) @ (one_one @ A)) @ ((inverse_inverse @ A) @ X3)) = ((((ord_less @ A) @ (zero_zero @ A)) @ X3) & (((ord_less_eq @ A) @ X3) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_3615_inverse__less__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ X3)) @ (one_one @ A)) = ((((ord_less_eq @ A) @ X3) @ (zero_zero @ A)) | (((ord_less @ A) @ (one_one @ A)) @ X3))))))))). % 6.41/2.69 thf(fact_3616_one__le__inverse, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ A2) @ (one_one @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ ((inverse_inverse @ A) @ A2)))))))))). % 6.41/2.69 thf(fact_3617_inverse__diff__inverse, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A2:A,B2:A]: (((A2 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => ((((minus_minus @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = ((uminus_uminus @ A) @ (((times_times @ A) @ (((times_times @ A) @ ((inverse_inverse @ A) @ A2)) @ (((minus_minus @ A) @ A2) @ B2))) @ ((inverse_inverse @ A) @ B2)))))))))))). % 6.41/2.69 thf(fact_3618_reals__Archimedean, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X3) => (? [N3:nat]: ((((ord_less @ A) @ ((inverse_inverse @ A) @ ((semiring_1_of_nat @ A) @ (suc @ N3)))) @ X3)))))))))). % 6.41/2.69 thf(fact_3619_sgn__if, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((sgn_sgn @ A) = (^ [X:A]: (((((if @ A) @ (X = (zero_zero @ A))) @ (zero_zero @ A)) @ ((((if @ A) @ (((ord_less @ A) @ (zero_zero @ A)) @ X)) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))))). % 6.41/2.69 thf(fact_3620_sgn__1__neg, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((sgn_sgn @ A) @ A2) = ((uminus_uminus @ A) @ (one_one @ A))) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3621_of__nat__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [R2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ R2) => (((ord_less_eq @ A) @ ((semiring_1_of_nat @ A) @ (nat2 @ ((archim6421214686448440834_floor @ A) @ R2)))) @ R2)))))))). % 6.41/2.69 thf(fact_3622_zsgn__def, axiom, ((sgn_sgn @ int) = (^ [I3:int]: (((((if @ int) @ (I3 = (zero_zero @ int))) @ (zero_zero @ int)) @ ((((if @ int) @ (((ord_less @ int) @ (zero_zero @ int)) @ I3)) @ (one_one @ int)) @ ((uminus_uminus @ int) @ (one_one @ int)))))))). % 6.41/2.69 thf(fact_3623_powr__real__of__int, axiom, (! [X3:real,N2:int]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ N2) => ((((powr @ real) @ X3) @ ((ring_1_of_int @ real) @ N2)) = (((power_power @ real) @ X3) @ (nat2 @ N2)))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N2))) => ((((powr @ real) @ X3) @ ((ring_1_of_int @ real) @ N2)) = ((inverse_inverse @ real) @ (((power_power @ real) @ X3) @ (nat2 @ ((uminus_uminus @ int) @ N2))))))))))). % 6.41/2.69 thf(fact_3624_real__vector__eq__affinity, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [M:real,Y:A,X3:A,C2:A]: (((M != (zero_zero @ real)) => ((Y = (((plus_plus @ A) @ (((real_V8093663219630862766scaleR @ A) @ M) @ X3)) @ C2)) = ((((minus_minus @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ M)) @ Y)) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ M)) @ C2)) = X3))))))))). % 6.41/2.69 thf(fact_3625_real__vector__affinity__eq, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [M:real,X3:A,C2:A,Y:A]: (((M != (zero_zero @ real)) => (((((plus_plus @ A) @ (((real_V8093663219630862766scaleR @ A) @ M) @ X3)) @ C2) = Y) = (X3 = (((minus_minus @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ M)) @ Y)) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ M)) @ C2))))))))))). % 6.41/2.69 thf(fact_3626_pos__divideR__le__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,B2:A,A2:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ C2) => ((((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2)) @ A2) = (((ord_less_eq @ A) @ B2) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)))))))))). % 6.41/2.69 thf(fact_3627_pos__le__divideR__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,A2:A,B2:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ C2) => ((((ord_less_eq @ A) @ A2) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2)) = (((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)) @ B2))))))))). % 6.41/2.69 thf(fact_3628_neg__divideR__le__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,B2:A,A2:A]: (((((ord_less @ real) @ C2) @ (zero_zero @ real)) => ((((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2)) @ A2) = (((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)) @ B2))))))))). % 6.41/2.69 thf(fact_3629_neg__le__divideR__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,A2:A,B2:A]: (((((ord_less @ real) @ C2) @ (zero_zero @ real)) => ((((ord_less_eq @ A) @ A2) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2)) = (((ord_less_eq @ A) @ B2) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)))))))))). % 6.41/2.69 thf(fact_3630_pos__divideR__less__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,B2:A,A2:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ C2) => ((((ord_less @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2)) @ A2) = (((ord_less @ A) @ B2) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)))))))))). % 6.41/2.69 thf(fact_3631_pos__less__divideR__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,A2:A,B2:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ C2) => ((((ord_less @ A) @ A2) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2)) = (((ord_less @ A) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)) @ B2))))))))). % 6.41/2.69 thf(fact_3632_neg__divideR__less__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,B2:A,A2:A]: (((((ord_less @ real) @ C2) @ (zero_zero @ real)) => ((((ord_less @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2)) @ A2) = (((ord_less @ A) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)) @ B2))))))))). % 6.41/2.69 thf(fact_3633_neg__less__divideR__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,A2:A,B2:A]: (((((ord_less @ real) @ C2) @ (zero_zero @ real)) => ((((ord_less @ A) @ A2) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2)) = (((ord_less @ A) @ B2) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)))))))))). % 6.41/2.69 thf(fact_3634_nat__less__eq__zless, axiom, (! [W2:int,Z2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W2) => ((((ord_less @ nat) @ (nat2 @ W2)) @ (nat2 @ Z2)) = (((ord_less @ int) @ W2) @ Z2)))))). % 6.41/2.69 thf(fact_3635_nat__le__eq__zle, axiom, (! [W2:int,Z2:int]: ((((((ord_less @ int) @ (zero_zero @ int)) @ W2) | (((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2)) => ((((ord_less_eq @ nat) @ (nat2 @ W2)) @ (nat2 @ Z2)) = (((ord_less_eq @ int) @ W2) @ Z2)))))). % 6.41/2.69 thf(fact_3636_norm__sgn, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X3:A]: ((((X3 = (zero_zero @ A)) => (((real_V7770717601297561774m_norm @ A) @ ((sgn_sgn @ A) @ X3)) = (zero_zero @ real))) & ((X3 != (zero_zero @ A)) => (((real_V7770717601297561774m_norm @ A) @ ((sgn_sgn @ A) @ X3)) = (one_one @ real)))))))))). % 6.41/2.69 thf(fact_3637_nat__eq__iff, axiom, (! [W2:int,M:nat]: ((((nat2 @ W2) = M) = (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W2) => (W2 = ((semiring_1_of_nat @ int) @ M))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ W2))) => (M = (zero_zero @ nat)))))))). % 6.41/2.69 thf(fact_3638_nat__eq__iff2, axiom, (! [M:nat,W2:int]: (((M = (nat2 @ W2)) = (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W2) => (W2 = ((semiring_1_of_nat @ int) @ M))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ W2))) => (M = (zero_zero @ nat)))))))). % 6.41/2.69 thf(fact_3639_split__nat, axiom, (! [P:(nat > $o),I:int]: (((P @ (nat2 @ I)) = ((! [N:nat]: (((I = ((semiring_1_of_nat @ int) @ N)) => (P @ N)))) & ((((ord_less @ int) @ I) @ (zero_zero @ int)) => (P @ (zero_zero @ nat)))))))). % 6.41/2.69 thf(fact_3640_le__mult__nat__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ nat) @ (((times_times @ nat) @ (nat2 @ ((archim6421214686448440834_floor @ A) @ A2))) @ (nat2 @ ((archim6421214686448440834_floor @ A) @ B2)))) @ (nat2 @ ((archim6421214686448440834_floor @ A) @ (((times_times @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_3641_le__nat__iff, axiom, (! [K:int,N2:nat]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => ((((ord_less_eq @ nat) @ N2) @ (nat2 @ K)) = (((ord_less_eq @ int) @ ((semiring_1_of_nat @ int) @ N2)) @ K)))))). % 6.41/2.69 thf(fact_3642_nat__add__distrib, axiom, (! [Z2:int,Z8:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z8) => ((nat2 @ (((plus_plus @ int) @ Z2) @ Z8)) = (((plus_plus @ nat) @ (nat2 @ Z2)) @ (nat2 @ Z8)))))))). % 6.41/2.69 thf(fact_3643_nat__mult__distrib, axiom, (! [Z2:int,Z8:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => ((nat2 @ (((times_times @ int) @ Z2) @ Z8)) = (((times_times @ nat) @ (nat2 @ Z2)) @ (nat2 @ Z8))))))). % 6.41/2.69 thf(fact_3644_Suc__as__int, axiom, (suc = (^ [A4:nat]: ((nat2 @ (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ A4)) @ (one_one @ int))))))). % 6.41/2.69 thf(fact_3645_forall__pos__mono__1, axiom, (! [P:(real > $o),E3:real]: (((! [D3:real,E:real]: (((((ord_less @ real) @ D3) @ E) => ((P @ D3) => (P @ E))))) => ((! [N3:nat]: ((P @ ((inverse_inverse @ real) @ ((semiring_1_of_nat @ real) @ (suc @ N3)))))) => ((((ord_less @ real) @ (zero_zero @ real)) @ E3) => (P @ E3))))))). % 6.41/2.69 thf(fact_3646_forall__pos__mono, axiom, (! [P:(real > $o),E3:real]: (((! [D3:real,E:real]: (((((ord_less @ real) @ D3) @ E) => ((P @ D3) => (P @ E))))) => ((! [N3:nat]: (((N3 != (zero_zero @ nat)) => (P @ ((inverse_inverse @ real) @ ((semiring_1_of_nat @ real) @ N3)))))) => ((((ord_less @ real) @ (zero_zero @ real)) @ E3) => (P @ E3))))))). % 6.41/2.69 thf(fact_3647_real__arch__inverse, axiom, (! [E3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E3) = (? [N:nat]: (((N != (zero_zero @ nat)) & ((((ord_less @ real) @ (zero_zero @ real)) @ ((inverse_inverse @ real) @ ((semiring_1_of_nat @ real) @ N))) & (((ord_less @ real) @ ((inverse_inverse @ real) @ ((semiring_1_of_nat @ real) @ N))) @ E3))))))))). % 6.41/2.69 thf(fact_3648_nat__diff__distrib, axiom, (! [Z8:int,Z2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z8) => ((((ord_less_eq @ int) @ Z8) @ Z2) => ((nat2 @ (((minus_minus @ int) @ Z2) @ Z8)) = (((minus_minus @ nat) @ (nat2 @ Z2)) @ (nat2 @ Z8)))))))). % 6.41/2.69 thf(fact_3649_nat__diff__distrib_H, axiom, (! [X3:int,Y:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X3) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y) => ((nat2 @ (((minus_minus @ int) @ X3) @ Y)) = (((minus_minus @ nat) @ (nat2 @ X3)) @ (nat2 @ Y)))))))). % 6.41/2.69 thf(fact_3650_nat__abs__triangle__ineq, axiom, (! [K:int,L:int]: ((((ord_less_eq @ nat) @ (nat2 @ ((abs_abs @ int) @ (((plus_plus @ int) @ K) @ L)))) @ (((plus_plus @ nat) @ (nat2 @ ((abs_abs @ int) @ K))) @ (nat2 @ ((abs_abs @ int) @ L))))))). % 6.41/2.69 thf(fact_3651_nat__floor__neg, axiom, (! [X3:real]: (((((ord_less_eq @ real) @ X3) @ (zero_zero @ real)) => ((nat2 @ ((archim6421214686448440834_floor @ real) @ X3)) = (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_3652_nat__power__eq, axiom, (! [Z2:int,N2:nat]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => ((nat2 @ (((power_power @ int) @ Z2) @ N2)) = (((power_power @ nat) @ (nat2 @ Z2)) @ N2)))))). % 6.41/2.69 thf(fact_3653_ln__inverse, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((ln_ln @ real) @ ((inverse_inverse @ real) @ X3)) = ((uminus_uminus @ real) @ ((ln_ln @ real) @ X3))))))). % 6.41/2.69 thf(fact_3654_floor__eq3, axiom, (! [N2:nat,X3:real]: (((((ord_less @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ X3) => ((((ord_less @ real) @ X3) @ ((semiring_1_of_nat @ real) @ (suc @ N2))) => ((nat2 @ ((archim6421214686448440834_floor @ real) @ X3)) = N2)))))). % 6.41/2.69 thf(fact_3655_le__nat__floor, axiom, (! [X3:nat,A2:real]: (((((ord_less_eq @ real) @ ((semiring_1_of_nat @ real) @ X3)) @ A2) => (((ord_less_eq @ nat) @ X3) @ (nat2 @ ((archim6421214686448440834_floor @ real) @ A2))))))). % 6.41/2.69 thf(fact_3656_the__elem__def, axiom, (! [A:$tType]: (((the_elem @ A) = (^ [X6:(set @ A)]: (((the @ A) @ (^ [X:A]: ((X6 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))))))). % 6.41/2.69 thf(fact_3657_divide__int__def, axiom, ((divide_divide @ int) = (^ [K3:int,L2:int]: (((((if @ int) @ (L2 = (zero_zero @ int))) @ (zero_zero @ int)) @ ((((if @ int) @ (((sgn_sgn @ int) @ K3) = ((sgn_sgn @ int) @ L2))) @ ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ (nat2 @ ((abs_abs @ int) @ K3))) @ (nat2 @ ((abs_abs @ int) @ L2))))) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ (((divide_divide @ nat) @ (nat2 @ ((abs_abs @ int) @ K3))) @ (nat2 @ ((abs_abs @ int) @ L2)))) @ ((zero_neq_one_of_bool @ nat) @ (~ ((((dvd_dvd @ int) @ L2) @ K3))))))))))))). % 6.41/2.69 thf(fact_3658_ex__inverse__of__nat__less, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X3) => (? [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) & (((ord_less @ A) @ ((inverse_inverse @ A) @ ((semiring_1_of_nat @ A) @ N3))) @ X3))))))))))). % 6.41/2.69 thf(fact_3659_power__diff__conv__inverse, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,M:nat,N2:nat]: (((X3 != (zero_zero @ A)) => ((((ord_less_eq @ nat) @ M) @ N2) => ((((power_power @ A) @ X3) @ (((minus_minus @ nat) @ N2) @ M)) = (((times_times @ A) @ (((power_power @ A) @ X3) @ N2)) @ (((power_power @ A) @ ((inverse_inverse @ A) @ X3)) @ M))))))))))). % 6.41/2.69 thf(fact_3660_nat__2, axiom, ((nat2 @ ((numeral_numeral @ int) @ (bit0 @ one2))) = (suc @ (suc @ (zero_zero @ nat))))). % 6.41/2.69 thf(fact_3661_pos__le__minus__divideR__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,A2:A,B2:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ C2) => ((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2))) = (((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)) @ ((uminus_uminus @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_3662_pos__minus__divideR__le__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,B2:A,A2:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ C2) => ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2))) @ A2) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)))))))))). % 6.41/2.69 thf(fact_3663_neg__le__minus__divideR__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,A2:A,B2:A]: (((((ord_less @ real) @ C2) @ (zero_zero @ real)) => ((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2))) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)))))))))). % 6.41/2.69 thf(fact_3664_neg__minus__divideR__le__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,B2:A,A2:A]: (((((ord_less @ real) @ C2) @ (zero_zero @ real)) => ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2))) @ A2) = (((ord_less_eq @ A) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)) @ ((uminus_uminus @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_3665_pos__less__minus__divideR__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,A2:A,B2:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ C2) => ((((ord_less @ A) @ A2) @ ((uminus_uminus @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2))) = (((ord_less @ A) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)) @ ((uminus_uminus @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_3666_pos__minus__divideR__less__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,B2:A,A2:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ C2) => ((((ord_less @ A) @ ((uminus_uminus @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2))) @ A2) = (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)))))))))). % 6.41/2.69 thf(fact_3667_neg__less__minus__divideR__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,A2:A,B2:A]: (((((ord_less @ real) @ C2) @ (zero_zero @ real)) => ((((ord_less @ A) @ A2) @ ((uminus_uminus @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2))) = (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)))))))))). % 6.41/2.69 thf(fact_3668_neg__minus__divideR__less__eq, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,B2:A,A2:A]: (((((ord_less @ real) @ C2) @ (zero_zero @ real)) => ((((ord_less @ A) @ ((uminus_uminus @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ C2)) @ B2))) @ A2) = (((ord_less @ A) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ A2)) @ ((uminus_uminus @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_3669_Suc__nat__eq__nat__zadd1, axiom, (! [Z2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => ((suc @ (nat2 @ Z2)) = (nat2 @ (((plus_plus @ int) @ (one_one @ int)) @ Z2))))))). % 6.41/2.69 thf(fact_3670_nat__less__iff, axiom, (! [W2:int,M:nat]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ W2) => ((((ord_less @ nat) @ (nat2 @ W2)) @ M) = (((ord_less @ int) @ W2) @ ((semiring_1_of_nat @ int) @ M))))))). % 6.41/2.69 thf(fact_3671_nat__mult__distrib__neg, axiom, (! [Z2:int,Z8:int]: (((((ord_less_eq @ int) @ Z2) @ (zero_zero @ int)) => ((nat2 @ (((times_times @ int) @ Z2) @ Z8)) = (((times_times @ nat) @ (nat2 @ ((uminus_uminus @ int) @ Z2))) @ (nat2 @ ((uminus_uminus @ int) @ Z8)))))))). % 6.41/2.69 thf(fact_3672_nat__abs__int__diff, axiom, (! [A2:nat,B2:nat]: ((((((ord_less_eq @ nat) @ A2) @ B2) => ((nat2 @ ((abs_abs @ int) @ (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ A2)) @ ((semiring_1_of_nat @ int) @ B2)))) = (((minus_minus @ nat) @ B2) @ A2))) & ((~ ((((ord_less_eq @ nat) @ A2) @ B2))) => ((nat2 @ ((abs_abs @ int) @ (((minus_minus @ int) @ ((semiring_1_of_nat @ int) @ A2)) @ ((semiring_1_of_nat @ int) @ B2)))) = (((minus_minus @ nat) @ A2) @ B2))))))). % 6.41/2.69 thf(fact_3673_log__inverse, axiom, (! [A2:real,X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ A2) => ((A2 != (one_one @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((log @ A2) @ ((inverse_inverse @ real) @ X3)) = ((uminus_uminus @ real) @ ((log @ A2) @ X3))))))))). % 6.41/2.69 thf(fact_3674_floor__eq4, axiom, (! [N2:nat,X3:real]: (((((ord_less_eq @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ X3) => ((((ord_less @ real) @ X3) @ ((semiring_1_of_nat @ real) @ (suc @ N2))) => ((nat2 @ ((archim6421214686448440834_floor @ real) @ X3)) = N2)))))). % 6.41/2.69 thf(fact_3675_diff__nat__eq__if, axiom, (! [Z8:int,Z2:int]: ((((((ord_less @ int) @ Z8) @ (zero_zero @ int)) => ((((minus_minus @ nat) @ (nat2 @ Z2)) @ (nat2 @ Z8)) = (nat2 @ Z2))) & ((~ ((((ord_less @ int) @ Z8) @ (zero_zero @ int)))) => ((((minus_minus @ nat) @ (nat2 @ Z2)) @ (nat2 @ Z8)) = ((((if @ nat) @ (((ord_less @ int) @ (((minus_minus @ int) @ Z2) @ Z8)) @ (zero_zero @ int))) @ (zero_zero @ nat)) @ (nat2 @ (((minus_minus @ int) @ Z2) @ Z8))))))))). % 6.41/2.69 thf(fact_3676_of__int__of__nat, axiom, (! [A:$tType]: (((ring_1 @ A) => ((ring_1_of_int @ A) = (^ [K3:int]: (((((if @ A) @ (((ord_less @ int) @ K3) @ (zero_zero @ int))) @ ((uminus_uminus @ A) @ ((semiring_1_of_nat @ A) @ (nat2 @ ((uminus_uminus @ int) @ K3))))) @ ((semiring_1_of_nat @ A) @ (nat2 @ K3)))))))))). % 6.41/2.69 thf(fact_3677_nat__dvd__iff, axiom, (! [Z2:int,M:nat]: (((((dvd_dvd @ nat) @ (nat2 @ Z2)) @ M) = (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => (((dvd_dvd @ int) @ Z2) @ ((semiring_1_of_nat @ int) @ M))) & ((~ ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2))) => (M = (zero_zero @ nat)))))))). % 6.41/2.69 thf(fact_3678_eucl__rel__int__remainderI, axiom, (! [R2:int,L:int,K:int,Q4:int]: (((((sgn_sgn @ int) @ R2) = ((sgn_sgn @ int) @ L)) => ((((ord_less @ int) @ ((abs_abs @ int) @ R2)) @ ((abs_abs @ int) @ L)) => ((K = (((plus_plus @ int) @ (((times_times @ int) @ Q4) @ L)) @ R2)) => (((eucl_rel_int @ K) @ L) @ ((((product_Pair @ int) @ int) @ Q4) @ R2)))))))). % 6.41/2.69 thf(fact_3679_plus__inverse__ge__2, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((ord_less_eq @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ (((plus_plus @ real) @ X3) @ ((inverse_inverse @ real) @ X3))))))). % 6.41/2.69 thf(fact_3680_real__inv__sqrt__pow2, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((power_power @ real) @ ((inverse_inverse @ real) @ (sqrt @ X3))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = ((inverse_inverse @ real) @ X3)))))). % 6.41/2.69 thf(fact_3681_eucl__rel__int_Ocases, axiom, (! [A12:int,A23:int,A32:((product_prod @ int) @ int)]: (((((eucl_rel_int @ A12) @ A23) @ A32) => (((A23 = (zero_zero @ int)) => (A32 != ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ A12))) => ((! [Q2:int]: (((A32 = ((((product_Pair @ int) @ int) @ Q2) @ (zero_zero @ int))) => ((A23 != (zero_zero @ int)) => (A12 != (((times_times @ int) @ Q2) @ A23)))))) => (~ ((! [R4:int,Q2:int]: (((A32 = ((((product_Pair @ int) @ int) @ Q2) @ R4)) => ((((sgn_sgn @ int) @ R4) = ((sgn_sgn @ int) @ A23)) => ((((ord_less @ int) @ ((abs_abs @ int) @ R4)) @ ((abs_abs @ int) @ A23)) => (A12 != (((plus_plus @ int) @ (((times_times @ int) @ Q2) @ A23)) @ R4))))))))))))))). % 6.41/2.69 thf(fact_3682_eucl__rel__int_Osimps, axiom, (eucl_rel_int = (^ [A1:int,A22:int,A33:((product_prod @ int) @ int)]: (((? [K3:int]: (((A1 = K3) & ((A22 = (zero_zero @ int)) & (A33 = ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ K3)))))) | ((? [L2:int,K3:int,Q5:int]: (((A1 = K3) & ((A22 = L2) & ((A33 = ((((product_Pair @ int) @ int) @ Q5) @ (zero_zero @ int))) & ((L2 != (zero_zero @ int)) & (K3 = (((times_times @ int) @ Q5) @ L2)))))))) | (? [R5:int,L2:int,K3:int,Q5:int]: (((A1 = K3) & ((A22 = L2) & ((A33 = ((((product_Pair @ int) @ int) @ Q5) @ R5)) & ((((sgn_sgn @ int) @ R5) = ((sgn_sgn @ int) @ L2)) & ((((ord_less @ int) @ ((abs_abs @ int) @ R5)) @ ((abs_abs @ int) @ L2)) & (K3 = (((plus_plus @ int) @ (((times_times @ int) @ Q5) @ L2)) @ R5))))))))))))))). % 6.41/2.69 thf(fact_3683_exp__series__add__commuting, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => (! [X3:A,Y:A,N2:nat]: ((((((times_times @ A) @ X3) @ Y) = (((times_times @ A) @ Y) @ X3)) => ((((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ A) @ (((plus_plus @ A) @ X3) @ Y)) @ N2)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ ((semiring_char_0_fact @ real) @ I3))) @ (((power_power @ A) @ X3) @ I3))) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ ((semiring_char_0_fact @ real) @ (((minus_minus @ nat) @ N2) @ I3)))) @ (((power_power @ A) @ Y) @ (((minus_minus @ nat) @ N2) @ I3))))))) @ ((set_ord_atMost @ nat) @ N2)))))))))). % 6.41/2.69 thf(fact_3684_exp__first__term, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => ((exp @ A) = (^ [X:A]: ((((plus_plus @ A) @ (one_one @ A)) @ ((suminf @ A) @ (^ [N:nat]: ((((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ ((semiring_char_0_fact @ real) @ (suc @ N)))) @ (((power_power @ A) @ X) @ (suc @ N)))))))))))))). % 6.41/2.69 thf(fact_3685_tan__sec, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((cos @ A) @ X3) != (zero_zero @ A)) => ((((plus_plus @ A) @ (one_one @ A)) @ (((power_power @ A) @ ((tan @ A) @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (((power_power @ A) @ ((inverse_inverse @ A) @ ((cos @ A) @ X3))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_3686_exp__first__terms, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => (! [K:nat]: (((exp @ A) = (^ [X:A]: ((((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [N:nat]: ((((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ ((semiring_char_0_fact @ real) @ N))) @ (((power_power @ A) @ X) @ N))))) @ ((set_ord_lessThan @ nat) @ K))) @ ((suminf @ A) @ (^ [N:nat]: ((((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ ((semiring_char_0_fact @ real) @ (((plus_plus @ nat) @ N) @ K)))) @ (((power_power @ A) @ X) @ (((plus_plus @ nat) @ N) @ K)))))))))))))))). % 6.41/2.69 thf(fact_3687_divide__int__unfold, axiom, (! [L:int,K:int,N2:nat,M:nat]: (((((((sgn_sgn @ int) @ L) = (zero_zero @ int)) | ((((sgn_sgn @ int) @ K) = (zero_zero @ int)) | (N2 = (zero_zero @ nat)))) => ((((divide_divide @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((semiring_1_of_nat @ int) @ M))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ ((semiring_1_of_nat @ int) @ N2))) = (zero_zero @ int))) & ((~ (((((sgn_sgn @ int) @ L) = (zero_zero @ int)) | ((((sgn_sgn @ int) @ K) = (zero_zero @ int)) | (N2 = (zero_zero @ nat)))))) => (((((sgn_sgn @ int) @ K) = ((sgn_sgn @ int) @ L)) => ((((divide_divide @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((semiring_1_of_nat @ int) @ M))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ ((semiring_1_of_nat @ int) @ N2))) = ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ M) @ N2)))) & ((((sgn_sgn @ int) @ K) != ((sgn_sgn @ int) @ L)) => ((((divide_divide @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((semiring_1_of_nat @ int) @ M))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ ((semiring_1_of_nat @ int) @ N2))) = ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ (((divide_divide @ nat) @ M) @ N2)) @ ((zero_neq_one_of_bool @ nat) @ (~ ((((dvd_dvd @ nat) @ N2) @ M))))))))))))))). % 6.41/2.69 thf(fact_3688_old_Orec__prod__def, axiom, (! [T:$tType,B:$tType,A:$tType]: (((((product_rec_prod @ A) @ B) @ T) = (^ [F12:(A > (B > T)),X:((product_prod @ A) @ B)]: (((the @ T) @ (((((product_rec_set_prod @ A) @ B) @ T) @ F12) @ X)))))))). % 6.41/2.69 thf(fact_3689_sinh__converges, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => (! [X3:A]: ((((sums @ A) @ (^ [N:nat]: (((((if @ A) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ (zero_zero @ A)) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ ((semiring_char_0_fact @ real) @ N))) @ (((power_power @ A) @ X3) @ N)))))) @ ((sinh @ A) @ X3)))))))). % 6.41/2.69 thf(fact_3690_cosh__converges, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => (! [X3:A]: ((((sums @ A) @ (^ [N:nat]: (((((if @ A) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ (((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ ((semiring_char_0_fact @ real) @ N))) @ (((power_power @ A) @ X3) @ N))) @ (zero_zero @ A))))) @ ((cosh @ A) @ X3)))))))). % 6.41/2.69 thf(fact_3691_sinh__ln__real, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((sinh @ real) @ ((ln_ln @ real) @ X3)) = (((divide_divide @ real) @ (((minus_minus @ real) @ X3) @ ((inverse_inverse @ real) @ X3))) @ ((numeral_numeral @ real) @ (bit0 @ one2)))))))). % 6.41/2.69 thf(fact_3692_arcsin__lt__bounded, axiom, (! [Y:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ Y) => ((((ord_less @ real) @ Y) @ (one_one @ real)) => ((((ord_less @ real) @ ((uminus_uminus @ real) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) @ (arcsin @ Y)) & (((ord_less @ real) @ (arcsin @ Y)) @ (((divide_divide @ real) @ pi) @ ((numeral_numeral @ real) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_3693_sinh__0, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => (((sinh @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_3694_sinh__real__less__iff, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ ((sinh @ real) @ X3)) @ ((sinh @ real) @ Y)) = (((ord_less @ real) @ X3) @ Y))))). % 6.41/2.69 thf(fact_3695_The__split__eq, axiom, (! [A:$tType,B:$tType,X3:A,Y:B]: ((((the @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X9:A,Y6:B]: (((X3 = X9) & (Y = Y6)))))) = ((((product_Pair @ A) @ B) @ X3) @ Y))))). % 6.41/2.69 thf(fact_3696_cosh__0, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => (((cosh @ A) @ (zero_zero @ A)) = (one_one @ A)))))). % 6.41/2.69 thf(fact_3697_sinh__real__neg__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ ((sinh @ real) @ X3)) @ (zero_zero @ real)) = (((ord_less @ real) @ X3) @ (zero_zero @ real)))))). % 6.41/2.69 thf(fact_3698_sinh__real__pos__iff, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ ((sinh @ real) @ X3)) = (((ord_less @ real) @ (zero_zero @ real)) @ X3))))). % 6.41/2.69 thf(fact_3699_sinh__plus__cosh, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => (! [X3:A]: (((((plus_plus @ A) @ ((sinh @ A) @ X3)) @ ((cosh @ A) @ X3)) = ((exp @ A) @ X3)))))))). % 6.41/2.69 thf(fact_3700_cosh__plus__sinh, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => (! [X3:A]: (((((plus_plus @ A) @ ((cosh @ A) @ X3)) @ ((sinh @ A) @ X3)) = ((exp @ A) @ X3)))))))). % 6.41/2.69 thf(fact_3701_sinh__less__cosh__real, axiom, (! [X3:real]: ((((ord_less @ real) @ ((sinh @ real) @ X3)) @ ((cosh @ real) @ X3))))). % 6.41/2.69 thf(fact_3702_sinh__add, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,Y:A]: ((((sinh @ A) @ (((plus_plus @ A) @ X3) @ Y)) = (((plus_plus @ A) @ (((times_times @ A) @ ((sinh @ A) @ X3)) @ ((cosh @ A) @ Y))) @ (((times_times @ A) @ ((cosh @ A) @ X3)) @ ((sinh @ A) @ Y)))))))))). % 6.41/2.69 thf(fact_3703_cosh__add, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,Y:A]: ((((cosh @ A) @ (((plus_plus @ A) @ X3) @ Y)) = (((plus_plus @ A) @ (((times_times @ A) @ ((cosh @ A) @ X3)) @ ((cosh @ A) @ Y))) @ (((times_times @ A) @ ((sinh @ A) @ X3)) @ ((sinh @ A) @ Y)))))))))). % 6.41/2.69 thf(fact_3704_cosh__real__pos, axiom, (! [X3:real]: ((((ord_less @ real) @ (zero_zero @ real)) @ ((cosh @ real) @ X3))))). % 6.41/2.69 thf(fact_3705_cosh__real__strict__mono, axiom, (! [X3:real,Y:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ Y) => (((ord_less @ real) @ ((cosh @ real) @ X3)) @ ((cosh @ real) @ Y))))))). % 6.41/2.69 thf(fact_3706_cosh__real__nonneg__less__iff, axiom, (! [X3:real,Y:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ Y) => ((((ord_less @ real) @ ((cosh @ real) @ X3)) @ ((cosh @ real) @ Y)) = (((ord_less @ real) @ X3) @ Y))))))). % 6.41/2.69 thf(fact_3707_cosh__real__nonpos__less__iff, axiom, (! [X3:real,Y:real]: (((((ord_less_eq @ real) @ X3) @ (zero_zero @ real)) => ((((ord_less_eq @ real) @ Y) @ (zero_zero @ real)) => ((((ord_less @ real) @ ((cosh @ real) @ X3)) @ ((cosh @ real) @ Y)) = (((ord_less @ real) @ Y) @ X3))))))). % 6.41/2.69 thf(fact_3708_cosh__square__eq, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((power_power @ A) @ ((cosh @ A) @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((plus_plus @ A) @ (((power_power @ A) @ ((sinh @ A) @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_3709_cosh__double, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: ((((cosh @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ X3)) = (((plus_plus @ A) @ (((power_power @ A) @ ((cosh @ A) @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ ((sinh @ A) @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_3710_sgn__real__def, axiom, ((sgn_sgn @ real) = (^ [A4:real]: (((((if @ real) @ (A4 = (zero_zero @ real))) @ (zero_zero @ real)) @ ((((if @ real) @ (((ord_less @ real) @ (zero_zero @ real)) @ A4)) @ (one_one @ real)) @ ((uminus_uminus @ real) @ (one_one @ real)))))))). % 6.41/2.69 thf(fact_3711_arcsin__less__arcsin, axiom, (! [X3:real,Y:real]: (((((ord_less_eq @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ X3) => ((((ord_less @ real) @ X3) @ Y) => ((((ord_less_eq @ real) @ Y) @ (one_one @ real)) => (((ord_less @ real) @ (arcsin @ X3)) @ (arcsin @ Y)))))))). % 6.41/2.69 thf(fact_3712_arcsin__less__mono, axiom, (! [X3:real,Y:real]: (((((ord_less_eq @ real) @ ((abs_abs @ real) @ X3)) @ (one_one @ real)) => ((((ord_less_eq @ real) @ ((abs_abs @ real) @ Y)) @ (one_one @ real)) => ((((ord_less @ real) @ (arcsin @ X3)) @ (arcsin @ Y)) = (((ord_less @ real) @ X3) @ Y))))))). % 6.41/2.69 thf(fact_3713_sgn__power__injE, axiom, (! [A2:real,N2:nat,X3:real,B2:real]: ((((((times_times @ real) @ ((sgn_sgn @ real) @ A2)) @ (((power_power @ real) @ ((abs_abs @ real) @ A2)) @ N2)) = X3) => ((X3 = (((times_times @ real) @ ((sgn_sgn @ real) @ B2)) @ (((power_power @ real) @ ((abs_abs @ real) @ B2)) @ N2))) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (A2 = B2))))))). % 6.41/2.69 thf(fact_3714_cos__arcsin__nonzero, axiom, (! [X3:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ X3) => ((((ord_less @ real) @ X3) @ (one_one @ real)) => (((cos @ real) @ (arcsin @ X3)) != (zero_zero @ real))))))). % 6.41/2.69 thf(fact_3715_tanh__add, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,Y:A]: (((((cosh @ A) @ X3) != (zero_zero @ A)) => ((((cosh @ A) @ Y) != (zero_zero @ A)) => (((tanh @ A) @ (((plus_plus @ A) @ X3) @ Y)) = (((divide_divide @ A) @ (((plus_plus @ A) @ ((tanh @ A) @ X3)) @ ((tanh @ A) @ Y))) @ (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((tanh @ A) @ X3)) @ ((tanh @ A) @ Y))))))))))))). % 6.41/2.69 thf(fact_3716_sinh__zero__iff, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((sinh @ A) @ X3) = (zero_zero @ A)) = (((member @ A) @ ((exp @ A) @ X3)) @ (((insert @ A) @ (one_one @ A)) @ (((insert @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (bot_bot @ (set @ A)))))))))))). % 6.41/2.69 thf(fact_3717_floor__real__def, axiom, ((archim6421214686448440834_floor @ real) = (^ [X:real]: (((the @ int) @ (^ [Z6:int]: (((((ord_less_eq @ real) @ ((ring_1_of_int @ real) @ Z6)) @ X) & (((ord_less @ real) @ X) @ ((ring_1_of_int @ real) @ (((plus_plus @ int) @ Z6) @ (one_one @ int)))))))))))). % 6.41/2.69 thf(fact_3718_cosh__field__def, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => ((cosh @ A) = (^ [Z6:A]: ((((divide_divide @ A) @ (((plus_plus @ A) @ ((exp @ A) @ Z6)) @ ((exp @ A) @ ((uminus_uminus @ A) @ Z6)))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_3719_cosh__zero__iff, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((cosh @ A) @ X3) = (zero_zero @ A)) = ((((power_power @ A) @ ((exp @ A) @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = ((uminus_uminus @ A) @ (one_one @ A)))))))))). % 6.41/2.69 thf(fact_3720_cosh__def, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V2822296259951069270ebra_1 @ A)) => ((cosh @ A) = (^ [X:A]: ((((real_V8093663219630862766scaleR @ A) @ ((inverse_inverse @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) @ (((plus_plus @ A) @ ((exp @ A) @ X)) @ ((exp @ A) @ ((uminus_uminus @ A) @ X))))))))))). % 6.41/2.69 thf(fact_3721_cosh__ln__real, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((cosh @ real) @ ((ln_ln @ real) @ X3)) = (((divide_divide @ real) @ (((plus_plus @ real) @ X3) @ ((inverse_inverse @ real) @ X3))) @ ((numeral_numeral @ real) @ (bit0 @ one2)))))))). % 6.41/2.69 thf(fact_3722_floor__rat__def, axiom, ((archim6421214686448440834_floor @ rat) = (^ [X:rat]: (((the @ int) @ (^ [Z6:int]: (((((ord_less_eq @ rat) @ ((ring_1_of_int @ rat) @ Z6)) @ X) & (((ord_less @ rat) @ X) @ ((ring_1_of_int @ rat) @ (((plus_plus @ int) @ Z6) @ (one_one @ int)))))))))))). % 6.41/2.69 thf(fact_3723_sum__diff1_H__aux, axiom, (! [B:$tType,A:$tType]: (((ab_group_add @ B) => (! [F5:(set @ A),I6:(set @ A),F2:(A > B),I:A]: ((((finite_finite2 @ A) @ F5) => ((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [I3:A]: (((((member @ A) @ I3) @ I6) & ((F2 @ I3) != (zero_zero @ B))))))) @ F5) => (((((member @ A) @ I) @ I6) => (((((groups1027152243600224163dd_sum @ A) @ B) @ F2) @ (((minus_minus @ (set @ A)) @ I6) @ (((insert @ A) @ I) @ (bot_bot @ (set @ A))))) = (((minus_minus @ B) @ ((((groups1027152243600224163dd_sum @ A) @ B) @ F2) @ I6)) @ (F2 @ I)))) & ((~ ((((member @ A) @ I) @ I6))) => (((((groups1027152243600224163dd_sum @ A) @ B) @ F2) @ (((minus_minus @ (set @ A)) @ I6) @ (((insert @ A) @ I) @ (bot_bot @ (set @ A))))) = ((((groups1027152243600224163dd_sum @ A) @ B) @ F2) @ I6)))))))))))). % 6.41/2.69 thf(fact_3724_VEBT_Osize_I3_J, axiom, (! [X11:(option @ ((product_prod @ nat) @ nat)),X12:nat,X13:(list @ vEBT_VEBT),X14:vEBT_VEBT]: ((((size_size @ vEBT_VEBT) @ ((((vEBT_Node @ X11) @ X12) @ X13) @ X14)) = (((plus_plus @ nat) @ (((plus_plus @ nat) @ (((size_list @ vEBT_VEBT) @ (size_size @ vEBT_VEBT)) @ X13)) @ ((size_size @ vEBT_VEBT) @ X14))) @ (suc @ (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_3725_sum__power2, axiom, (! [K:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ ((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ K)) = (((minus_minus @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K)) @ (one_one @ nat)))))). % 6.41/2.69 thf(fact_3726_finite__atLeastLessThan, axiom, (! [L:nat,U:nat]: (((finite_finite2 @ nat) @ (((set_or7035219750837199246ssThan @ nat) @ L) @ U))))). % 6.41/2.69 thf(fact_3727_atLeastLessThan__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,L:A,U:A]: (((((member @ A) @ I) @ (((set_or7035219750837199246ssThan @ A) @ L) @ U)) = ((((ord_less_eq @ A) @ L) @ I) & (((ord_less @ A) @ I) @ U))))))))). % 6.41/2.69 thf(fact_3728_atLeastLessThan__empty, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((set_or7035219750837199246ssThan @ A) @ A2) @ B2) = (bot_bot @ (set @ A)))))))))). % 6.41/2.69 thf(fact_3729_ivl__subset, axiom, (! [A:$tType]: (((linorder @ A) => (! [I:A,J:A,M:A,N2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ I) @ J)) @ (((set_or7035219750837199246ssThan @ A) @ M) @ N2)) = ((((ord_less_eq @ A) @ J) @ I) | ((((ord_less_eq @ A) @ M) @ I) & (((ord_less_eq @ A) @ J) @ N2)))))))))). % 6.41/2.69 thf(fact_3730_atLeastLessThan__empty__iff2, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A]: ((((bot_bot @ (set @ A)) = (((set_or7035219750837199246ssThan @ A) @ A2) @ B2)) = (~ ((((ord_less @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_3731_atLeastLessThan__empty__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A]: ((((((set_or7035219750837199246ssThan @ A) @ A2) @ B2) = (bot_bot @ (set @ A))) = (~ ((((ord_less @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_3732_infinite__Ico__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A]: (((~ (((finite_finite2 @ A) @ (((set_or7035219750837199246ssThan @ A) @ A2) @ B2)))) = (((ord_less @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_3733_ivl__diff, axiom, (! [A:$tType]: (((linorder @ A) => (! [I:A,N2:A,M:A]: (((((ord_less_eq @ A) @ I) @ N2) => ((((minus_minus @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ I) @ M)) @ (((set_or7035219750837199246ssThan @ A) @ I) @ N2)) = (((set_or7035219750837199246ssThan @ A) @ N2) @ M))))))))). % 6.41/2.69 thf(fact_3734_sum_Oempty_H, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (! [P4:(B > A)]: ((((((groups1027152243600224163dd_sum @ B) @ A) @ P4) @ (bot_bot @ (set @ B))) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_3735_sum_Oeq__sum, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [I6:(set @ B),P4:(B > A)]: ((((finite_finite2 @ B) @ I6) => (((((groups1027152243600224163dd_sum @ B) @ A) @ P4) @ I6) = ((((groups7311177749621191930dd_sum @ B) @ A) @ P4) @ I6))))))))). % 6.41/2.69 thf(fact_3736_atLeastLessThan__singleton, axiom, (! [M:nat]: (((((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ M)) = (((insert @ nat) @ M) @ (bot_bot @ (set @ nat))))))). % 6.41/2.69 thf(fact_3737_sum_Oinsert_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [I6:(set @ B),P4:(B > A),I:B]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ I6) & ((P4 @ X) != (zero_zero @ A))))))) => (((((member @ B) @ I) @ I6) => (((((groups1027152243600224163dd_sum @ B) @ A) @ P4) @ (((insert @ B) @ I) @ I6)) = ((((groups1027152243600224163dd_sum @ B) @ A) @ P4) @ I6))) & ((~ ((((member @ B) @ I) @ I6))) => (((((groups1027152243600224163dd_sum @ B) @ A) @ P4) @ (((insert @ B) @ I) @ I6)) = (((plus_plus @ A) @ (P4 @ I)) @ ((((groups1027152243600224163dd_sum @ B) @ A) @ P4) @ I6)))))))))))). % 6.41/2.69 thf(fact_3738_sum_Oop__ivl__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [N2:nat,M:nat,G:(nat > A)]: ((((((ord_less @ nat) @ N2) @ M) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ N2))) = (zero_zero @ A))) & ((~ ((((ord_less @ nat) @ N2) @ M))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ N2))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))) @ (G @ N2))))))))))). % 6.41/2.69 thf(fact_3739_prod_Oop__ivl__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [N2:nat,M:nat,G:(nat > A)]: ((((((ord_less @ nat) @ N2) @ M) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ N2))) = (one_one @ A))) & ((~ ((((ord_less @ nat) @ N2) @ M))) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ N2))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))) @ (G @ N2))))))))))). % 6.41/2.69 thf(fact_3740_sgn__rat__def, axiom, ((sgn_sgn @ rat) = (^ [A4:rat]: (((((if @ rat) @ (A4 = (zero_zero @ rat))) @ (zero_zero @ rat)) @ ((((if @ rat) @ (((ord_less @ rat) @ (zero_zero @ rat)) @ A4)) @ (one_one @ rat)) @ ((uminus_uminus @ rat) @ (one_one @ rat)))))))). % 6.41/2.69 thf(fact_3741_abs__rat__def, axiom, ((abs_abs @ rat) = (^ [A4:rat]: (((((if @ rat) @ (((ord_less @ rat) @ A4) @ (zero_zero @ rat))) @ ((uminus_uminus @ rat) @ A4)) @ A4))))). % 6.41/2.69 thf(fact_3742_atLeastLessThan__eq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ C2) @ D2) => (((((set_or7035219750837199246ssThan @ A) @ A2) @ B2) = (((set_or7035219750837199246ssThan @ A) @ C2) @ D2)) = ((A2 = C2) & (B2 = D2))))))))))). % 6.41/2.69 thf(fact_3743_atLeastLessThan__inj_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: ((((((set_or7035219750837199246ssThan @ A) @ A2) @ B2) = (((set_or7035219750837199246ssThan @ A) @ C2) @ D2)) => ((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ C2) @ D2) => (A2 = C2)))))))))). % 6.41/2.69 thf(fact_3744_atLeastLessThan__inj_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: ((((((set_or7035219750837199246ssThan @ A) @ A2) @ B2) = (((set_or7035219750837199246ssThan @ A) @ C2) @ D2)) => ((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ C2) @ D2) => (B2 = D2)))))))))). % 6.41/2.69 thf(fact_3745_less__eq__rat__def, axiom, ((ord_less_eq @ rat) = (^ [X:rat,Y2:rat]: (((((ord_less @ rat) @ X) @ Y2) | (X = Y2)))))). % 6.41/2.69 thf(fact_3746_obtain__pos__sum, axiom, (! [R2:rat]: (((((ord_less @ rat) @ (zero_zero @ rat)) @ R2) => (~ ((! [S2:rat]: (((((ord_less @ rat) @ (zero_zero @ rat)) @ S2) => (! [T7:rat]: (((((ord_less @ rat) @ (zero_zero @ rat)) @ T7) => (R2 != (((plus_plus @ rat) @ S2) @ T7)))))))))))))). % 6.41/2.69 thf(fact_3747_sum_Onon__neutral_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [G:(B > A),I6:(set @ B)]: ((((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ I6) & ((G @ X) != (zero_zero @ A))))))) = ((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ I6)))))))). % 6.41/2.69 thf(fact_3748_atLeastLessThan__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ A2) @ B2)) @ (((set_or7035219750837199246ssThan @ A) @ C2) @ D2)) => ((((ord_less_eq @ A) @ B2) @ A2) | ((((ord_less_eq @ A) @ C2) @ A2) & (((ord_less_eq @ A) @ B2) @ D2)))))))))). % 6.41/2.69 thf(fact_3749_infinite__Ico, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ (((finite_finite2 @ A) @ (((set_or7035219750837199246ssThan @ A) @ A2) @ B2))))))))))). % 6.41/2.69 thf(fact_3750_ex__nat__less__eq, axiom, (! [N2:nat,P:(nat > $o)]: (((? [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) & (P @ M2)))) = (? [X:nat]: (((((member @ nat) @ X) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2)) & (P @ X)))))))). % 6.41/2.69 thf(fact_3751_all__nat__less__eq, axiom, (! [N2:nat,P:(nat > $o)]: (((! [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (P @ M2)))) = (! [X:nat]: (((((member @ nat) @ X) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2)) => (P @ X)))))))). % 6.41/2.69 thf(fact_3752_atLeastLessThanSuc__atLeastAtMost, axiom, (! [L:nat,U:nat]: (((((set_or7035219750837199246ssThan @ nat) @ L) @ (suc @ U)) = (((set_or1337092689740270186AtMost @ nat) @ L) @ U))))). % 6.41/2.69 thf(fact_3753_lessThan__atLeast0, axiom, ((set_ord_lessThan @ nat) = ((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)))). % 6.41/2.69 thf(fact_3754_atLeastLessThan0, axiom, (! [M:nat]: (((((set_or7035219750837199246ssThan @ nat) @ M) @ (zero_zero @ nat)) = (bot_bot @ (set @ nat)))))). % 6.41/2.69 thf(fact_3755_sum_Oshift__bounds__Suc__ivl, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ (suc @ N2))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_3756_sum_Oshift__bounds__nat__ivl, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,K:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ (((plus_plus @ nat) @ N2) @ K))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (((plus_plus @ nat) @ I3) @ K))))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_3757_prod_Oshift__bounds__Suc__ivl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ (suc @ N2))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_3758_prod_Oshift__bounds__nat__ivl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,K:nat,N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ (((plus_plus @ nat) @ N2) @ K))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (((plus_plus @ nat) @ I3) @ K))))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_3759_sum_Odistrib__triv_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [I6:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ I6) => (((((groups1027152243600224163dd_sum @ B) @ A) @ (^ [I3:B]: ((((plus_plus @ A) @ (G @ I3)) @ (H2 @ I3))))) @ I6) = (((plus_plus @ A) @ ((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ I6)) @ ((((groups1027152243600224163dd_sum @ B) @ A) @ H2) @ I6)))))))))). % 6.41/2.69 thf(fact_3760_sum_Oivl__cong, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (comm_monoid_add @ A)) => (! [A2:B,C2:B,B2:B,D2:B,G:(B > A),H2:(B > A)]: (((A2 = C2) => ((B2 = D2) => ((! [X5:B]: (((((ord_less_eq @ B) @ C2) @ X5) => ((((ord_less @ B) @ X5) @ D2) => ((G @ X5) = (H2 @ X5)))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((set_or7035219750837199246ssThan @ B) @ A2) @ B2)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ (((set_or7035219750837199246ssThan @ B) @ C2) @ D2)))))))))))). % 6.41/2.69 thf(fact_3761_prod_Oivl__cong, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (comm_monoid_mult @ A)) => (! [A2:B,C2:B,B2:B,D2:B,G:(B > A),H2:(B > A)]: (((A2 = C2) => ((B2 = D2) => ((! [X5:B]: (((((ord_less_eq @ B) @ C2) @ X5) => ((((ord_less @ B) @ X5) @ D2) => ((G @ X5) = (H2 @ X5)))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((set_or7035219750837199246ssThan @ B) @ A2) @ B2)) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ (((set_or7035219750837199246ssThan @ B) @ C2) @ D2)))))))))))). % 6.41/2.69 thf(fact_3762_sum_OatLeastLessThan__concat, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N2:nat,P4:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((ord_less_eq @ nat) @ N2) @ P4) => ((((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N2) @ P4))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ P4))))))))))). % 6.41/2.69 thf(fact_3763_sum__diff__nat__ivl, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [M:nat,N2:nat,P4:nat,F2:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((ord_less_eq @ nat) @ N2) @ P4) => ((((minus_minus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ P4))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ (((set_or7035219750837199246ssThan @ nat) @ N2) @ P4))))))))))). % 6.41/2.69 thf(fact_3764_size__list__estimation, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Y:nat,F2:(A > nat)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => ((((ord_less @ nat) @ Y) @ (F2 @ X3)) => (((ord_less @ nat) @ Y) @ (((size_list @ A) @ F2) @ Xs))))))). % 6.41/2.69 thf(fact_3765_size__list__pointwise, axiom, (! [A:$tType,Xs:(list @ A),F2:(A > nat),G:(A > nat)]: (((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ nat) @ (F2 @ X5)) @ (G @ X5))))) => (((ord_less_eq @ nat) @ (((size_list @ A) @ F2) @ Xs)) @ (((size_list @ A) @ G) @ Xs)))))). % 6.41/2.69 thf(fact_3766_size__list__estimation_H, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Y:nat,F2:(A > nat)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => ((((ord_less_eq @ nat) @ Y) @ (F2 @ X3)) => (((ord_less_eq @ nat) @ Y) @ (((size_list @ A) @ F2) @ Xs))))))). % 6.41/2.69 thf(fact_3767_prod_OatLeastLessThan__concat, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N2:nat,P4:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((ord_less_eq @ nat) @ N2) @ P4) => ((((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N2) @ P4))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ P4))))))))))). % 6.41/2.69 thf(fact_3768_atLeast0__lessThan__Suc, axiom, (! [N2:nat]: (((((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N2)) = (((insert @ nat) @ N2) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2)))))). % 6.41/2.69 thf(fact_3769_subset__eq__atLeast0__lessThan__finite, axiom, (! [N7:(set @ nat),N2:nat]: (((((ord_less_eq @ (set @ nat)) @ N7) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2)) => ((finite_finite2 @ nat) @ N7))))). % 6.41/2.69 thf(fact_3770_sum_Omono__neutral__left_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ B),T4:(set @ B),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [X5:B]: (((((member @ B) @ X5) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((G @ X5) = (zero_zero @ A))))) => (((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ S3) = ((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ T4)))))))))). % 6.41/2.69 thf(fact_3771_sum_Omono__neutral__right_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ B),T4:(set @ B),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [X5:B]: (((((member @ B) @ X5) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((G @ X5) = (zero_zero @ A))))) => (((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ T4) = ((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ S3)))))))))). % 6.41/2.69 thf(fact_3772_sum_Omono__neutral__cong__left_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ B),T4:(set @ B),H2:(B > A),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [I4:B]: (((((member @ B) @ I4) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((H2 @ I4) = (zero_zero @ A))))) => ((! [X5:B]: (((((member @ B) @ X5) @ S3) => ((G @ X5) = (H2 @ X5))))) => (((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ S3) = ((((groups1027152243600224163dd_sum @ B) @ A) @ H2) @ T4))))))))))). % 6.41/2.69 thf(fact_3773_sum_Omono__neutral__cong__right_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ B),T4:(set @ B),G:(B > A),H2:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [X5:B]: (((((member @ B) @ X5) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((G @ X5) = (zero_zero @ A))))) => ((! [X5:B]: (((((member @ B) @ X5) @ S3) => ((G @ X5) = (H2 @ X5))))) => (((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ T4) = ((((groups1027152243600224163dd_sum @ B) @ A) @ H2) @ S3))))))))))). % 6.41/2.69 thf(fact_3774_atLeastLessThan__subseteq__atLeastAtMost__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ A2) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D2)) = ((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ A2) & (((ord_less_eq @ A) @ B2) @ D2)))))))))). % 6.41/2.69 thf(fact_3775_atLeastAtMost__subseteq__atLeastLessThan__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) @ (((set_or7035219750837199246ssThan @ A) @ C2) @ D2)) = ((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ A2) & (((ord_less @ A) @ B2) @ D2)))))))))). % 6.41/2.69 thf(fact_3776_sum__shift__lb__Suc0__0__upt, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [F2:(nat > A),K:nat]: ((((F2 @ (zero_zero @ nat)) = (zero_zero @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ (zero_zero @ nat))) @ K)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ K)))))))))). % 6.41/2.69 thf(fact_3777_sum_OatLeast0__lessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N2))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2))) @ (G @ N2))))))))). % 6.41/2.69 thf(fact_3778_sum_OatLeast__Suc__lessThan, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N2:nat,G:(nat > A)]: (((((ord_less @ nat) @ M) @ N2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2)) = (((plus_plus @ A) @ (G @ M)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ N2))))))))))). % 6.41/2.69 thf(fact_3779_sum_OatLeastLessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:nat,B2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ A2) @ B2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ A2) @ (suc @ B2))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ A2) @ B2))) @ (G @ B2)))))))))). % 6.41/2.69 thf(fact_3780_sum_Odistrib_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [I6:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ I6) & ((G @ X) != (zero_zero @ A))))))) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ I6) & ((H2 @ X) != (zero_zero @ A))))))) => (((((groups1027152243600224163dd_sum @ B) @ A) @ (^ [I3:B]: ((((plus_plus @ A) @ (G @ I3)) @ (H2 @ I3))))) @ I6) = (((plus_plus @ A) @ ((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ I6)) @ ((((groups1027152243600224163dd_sum @ B) @ A) @ H2) @ I6))))))))))). % 6.41/2.69 thf(fact_3781_atLeastLessThan__eq__atLeastAtMost__diff, axiom, (! [A:$tType]: (((order @ A) => ((set_or7035219750837199246ssThan @ A) = (^ [A4:A,B4:A]: ((((minus_minus @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A4) @ B4)) @ (((insert @ A) @ B4) @ (bot_bot @ (set @ A))))))))))). % 6.41/2.69 thf(fact_3782_prod_OatLeast0__lessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N2))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2))) @ (G @ N2))))))))). % 6.41/2.69 thf(fact_3783_prod_OatLeast__Suc__lessThan, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N2:nat,G:(nat > A)]: (((((ord_less @ nat) @ M) @ N2) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2)) = (((times_times @ A) @ (G @ M)) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ N2))))))))))). % 6.41/2.69 thf(fact_3784_prod_OatLeastLessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:nat,B2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ A2) @ B2) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ A2) @ (suc @ B2))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ A2) @ B2))) @ (G @ B2)))))))))). % 6.41/2.69 thf(fact_3785_sum_OG__def, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (((groups1027152243600224163dd_sum @ B) @ A) = (^ [P6:(B > A),I8:(set @ B)]: (((((if @ A) @ ((finite_finite2 @ B) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ I8) & ((P6 @ X) != (zero_zero @ A)))))))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ P6) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ I8) & ((P6 @ X) != (zero_zero @ A)))))))) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_3786_sum_Olast__plus, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (((plus_plus @ A) @ (G @ N2)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))))))))))). % 6.41/2.69 thf(fact_3787_prod_Olast__plus, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (((times_times @ A) @ (G @ N2)) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))))))))))). % 6.41/2.69 thf(fact_3788_sum__Suc__diff_H, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [M:nat,N2:nat,F2:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((minus_minus @ A) @ (F2 @ (suc @ I3))) @ (F2 @ I3))))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2)) = (((minus_minus @ A) @ (F2 @ N2)) @ (F2 @ M)))))))))). % 6.41/2.69 thf(fact_3789_atLeastLessThanSuc, axiom, (! [M:nat,N2:nat]: ((((((ord_less_eq @ nat) @ M) @ N2) => ((((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ N2)) = (((insert @ nat) @ N2) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2)))) & ((~ ((((ord_less_eq @ nat) @ M) @ N2))) => ((((set_or7035219750837199246ssThan @ nat) @ M) @ (suc @ N2)) = (bot_bot @ (set @ nat)))))))). % 6.41/2.69 thf(fact_3790_sum_OatLeastLessThan__rev, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N2:nat,M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N2) @ M)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ N2)) @ (suc @ I3)))))) @ (((set_or7035219750837199246ssThan @ nat) @ N2) @ M))))))))). % 6.41/2.69 thf(fact_3791_sum_Onested__swap, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A2:(nat > (nat > A)),N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (A2 @ I3)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J3:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: (((A2 @ I3) @ J3)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J3)) @ N2))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2))))))))). % 6.41/2.69 thf(fact_3792_prod_OatLeastLessThan__rev, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N2:nat,M:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N2) @ M)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ N2)) @ (suc @ I3)))))) @ (((set_or7035219750837199246ssThan @ nat) @ N2) @ M))))))))). % 6.41/2.69 thf(fact_3793_prod_Onested__swap, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A2:(nat > (nat > A)),N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (A2 @ I3)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [J3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: (((A2 @ I3) @ J3)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J3)) @ N2))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2))))))))). % 6.41/2.69 thf(fact_3794_sum_Onat__group, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),K:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [M2:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((times_times @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ M2) @ K)) @ K)))))) @ ((set_ord_lessThan @ nat) @ N2)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (((times_times @ nat) @ N2) @ K)))))))))). % 6.41/2.69 thf(fact_3795_prod_Onat__group, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),K:nat,N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [M2:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((times_times @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ M2) @ K)) @ K)))))) @ ((set_ord_lessThan @ nat) @ N2)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (((times_times @ nat) @ N2) @ K)))))))))). % 6.41/2.69 thf(fact_3796_prod__Suc__Suc__fact, axiom, (! [N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ nat) @ suc) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ (zero_zero @ nat))) @ N2)) = ((semiring_char_0_fact @ nat) @ N2))))). % 6.41/2.69 thf(fact_3797_prod__Suc__fact, axiom, (! [N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ nat) @ suc) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2)) = ((semiring_char_0_fact @ nat) @ N2))))). % 6.41/2.69 thf(fact_3798_sum_Ohead__if, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [N2:nat,M:nat,G:(nat > A)]: ((((((ord_less @ nat) @ N2) @ M) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (zero_zero @ A))) & ((~ ((((ord_less @ nat) @ N2) @ M))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))) @ (G @ N2))))))))))). % 6.41/2.69 thf(fact_3799_prod_Ohead__if, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [N2:nat,M:nat,G:(nat > A)]: ((((((ord_less @ nat) @ N2) @ M) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (one_one @ A))) & ((~ ((((ord_less @ nat) @ N2) @ M))) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))) @ (G @ N2))))))))))). % 6.41/2.69 thf(fact_3800_fact__prod__Suc, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => ((semiring_char_0_fact @ A) = (^ [N:nat]: (((semiring_1_of_nat @ A) @ ((((groups7121269368397514597t_prod @ nat) @ nat) @ suc) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)))))))))). % 6.41/2.69 thf(fact_3801_sum_OatLeastLessThan__rev__at__least__Suc__atMost, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N2:nat,M:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N2) @ M)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ N2)) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ N2)) @ M))))))))). % 6.41/2.69 thf(fact_3802_prod_OatLeastLessThan__rev__at__least__Suc__atMost, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N2:nat,M:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N2) @ M)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M) @ N2)) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ N2)) @ M))))))))). % 6.41/2.69 thf(fact_3803_pochhammer__prod, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => ((comm_s3205402744901411588hammer @ A) = (^ [A4:A,N:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ A4) @ ((semiring_1_of_nat @ A) @ I3))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))))))))). % 6.41/2.69 thf(fact_3804_atLeastLessThan__nat__numeral, axiom, (! [M:nat,K:num]: ((((((ord_less_eq @ nat) @ M) @ (pred_numeral @ K)) => ((((set_or7035219750837199246ssThan @ nat) @ M) @ ((numeral_numeral @ nat) @ K)) = (((insert @ nat) @ (pred_numeral @ K)) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ (pred_numeral @ K))))) & ((~ ((((ord_less_eq @ nat) @ M) @ (pred_numeral @ K)))) => ((((set_or7035219750837199246ssThan @ nat) @ M) @ ((numeral_numeral @ nat) @ K)) = (bot_bot @ (set @ nat)))))))). % 6.41/2.69 thf(fact_3805_fact__prod__rev, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => ((semiring_char_0_fact @ A) = (^ [N:nat]: (((semiring_1_of_nat @ A) @ ((((groups7121269368397514597t_prod @ nat) @ nat) @ ((minus_minus @ nat) @ N)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)))))))))). % 6.41/2.69 thf(fact_3806_summable__Cauchy, axiom, (! [A:$tType]: (((real_Vector_banach @ A) => ((summable @ A) = (^ [F3:(nat > A)]: ((! [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) => (? [N5:nat]: ((! [M2:nat]: (((((ord_less_eq @ nat) @ N5) @ M2) => (! [N:nat]: ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F3) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)))) @ E4)))))))))))))))))). % 6.41/2.69 thf(fact_3807_sums__group, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topolo4958980785337419405_space @ A)) => (! [F2:(nat > A),S:A,K:nat]: (((((sums @ A) @ F2) @ S) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((sums @ A) @ (^ [N:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ (((set_or7035219750837199246ssThan @ nat) @ (((times_times @ nat) @ N) @ K)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ N) @ K)) @ K)))))) @ S))))))))). % 6.41/2.69 thf(fact_3808_atLeast1__lessThan__eq__remove0, axiom, (! [N2:nat]: (((((set_or7035219750837199246ssThan @ nat) @ (suc @ (zero_zero @ nat))) @ N2) = (((minus_minus @ (set @ nat)) @ ((set_ord_lessThan @ nat) @ N2)) @ (((insert @ nat) @ (zero_zero @ nat)) @ (bot_bot @ (set @ nat)))))))). % 6.41/2.69 thf(fact_3809_fact__split, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [K:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ N2) => (((semiring_char_0_fact @ A) @ N2) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ ((((groups7121269368397514597t_prod @ nat) @ nat) @ suc) @ (((set_or7035219750837199246ssThan @ nat) @ (((minus_minus @ nat) @ N2) @ K)) @ N2)))) @ ((semiring_char_0_fact @ A) @ (((minus_minus @ nat) @ N2) @ K))))))))))). % 6.41/2.69 thf(fact_3810_binomial__altdef__of__nat, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,N2:nat]: (((((ord_less_eq @ nat) @ K) @ N2) => (((semiring_1_of_nat @ A) @ ((binomial @ N2) @ K)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((divide_divide @ A) @ ((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ N2) @ I3))) @ ((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ K) @ I3)))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ K)))))))))). % 6.41/2.69 thf(fact_3811_gbinomial__altdef__of__nat, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((gbinomial @ A) = (^ [A4:A,K3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((divide_divide @ A) @ (((minus_minus @ A) @ A4) @ ((semiring_1_of_nat @ A) @ I3))) @ ((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ K3) @ I3)))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ K3))))))))). % 6.41/2.69 thf(fact_3812_gbinomial__mult__fact_H, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A2:A,K:nat]: (((((times_times @ A) @ (((gbinomial @ A) @ A2) @ K)) @ ((semiring_char_0_fact @ A) @ K)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((minus_minus @ A) @ A2) @ ((semiring_1_of_nat @ A) @ I3))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ K))))))))). % 6.41/2.69 thf(fact_3813_gbinomial__mult__fact, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,A2:A]: (((((times_times @ A) @ ((semiring_char_0_fact @ A) @ K)) @ (((gbinomial @ A) @ A2) @ K)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((minus_minus @ A) @ A2) @ ((semiring_1_of_nat @ A) @ I3))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ K))))))))). % 6.41/2.69 thf(fact_3814_gbinomial__prod__rev, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (semidom_divide @ A)) => ((gbinomial @ A) = (^ [A4:A,K3:nat]: ((((divide_divide @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((minus_minus @ A) @ A4) @ ((semiring_1_of_nat @ A) @ I3))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ K3))) @ ((semiring_char_0_fact @ A) @ K3))))))))). % 6.41/2.69 thf(fact_3815_sum__diff1_H, axiom, (! [B:$tType,A:$tType]: (((ab_group_add @ B) => (! [I6:(set @ A),F2:(A > B),I:A]: ((((finite_finite2 @ A) @ ((collect @ A) @ (^ [I3:A]: (((((member @ A) @ I3) @ I6) & ((F2 @ I3) != (zero_zero @ B))))))) => (((((member @ A) @ I) @ I6) => (((((groups1027152243600224163dd_sum @ A) @ B) @ F2) @ (((minus_minus @ (set @ A)) @ I6) @ (((insert @ A) @ I) @ (bot_bot @ (set @ A))))) = (((minus_minus @ B) @ ((((groups1027152243600224163dd_sum @ A) @ B) @ F2) @ I6)) @ (F2 @ I)))) & ((~ ((((member @ A) @ I) @ I6))) => (((((groups1027152243600224163dd_sum @ A) @ B) @ F2) @ (((minus_minus @ (set @ A)) @ I6) @ (((insert @ A) @ I) @ (bot_bot @ (set @ A))))) = ((((groups1027152243600224163dd_sum @ A) @ B) @ F2) @ I6))))))))))). % 6.41/2.69 thf(fact_3816_horner__sum__eq__sum, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_1 @ A) => (((groups4207007520872428315er_sum @ B) @ A) = (^ [F3:(B > A),A4:A,Xs2:(list @ B)]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [N:nat]: ((((times_times @ A) @ (F3 @ (((nth @ B) @ Xs2) @ N))) @ (((power_power @ A) @ A4) @ N))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ B)) @ Xs2)))))))))). % 6.41/2.69 thf(fact_3817_Chebyshev__sum__upper, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:nat,A2:(nat > A),B2:(nat > A)]: (((! [I4:nat,J2:nat]: (((((ord_less_eq @ nat) @ I4) @ J2) => ((((ord_less @ nat) @ J2) @ N2) => (((ord_less_eq @ A) @ (A2 @ I4)) @ (A2 @ J2)))))) => ((! [I4:nat,J2:nat]: (((((ord_less_eq @ nat) @ I4) @ J2) => ((((ord_less @ nat) @ J2) @ N2) => (((ord_less_eq @ A) @ (B2 @ J2)) @ (B2 @ I4)))))) => (((ord_less_eq @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((times_times @ A) @ (A2 @ K3)) @ (B2 @ K3))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2)))) @ (((times_times @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ A2) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ B2) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2)))))))))))). % 6.41/2.69 thf(fact_3818_Chebyshev__sum__upper__nat, axiom, (! [N2:nat,A2:(nat > nat),B2:(nat > nat)]: (((! [I4:nat,J2:nat]: (((((ord_less_eq @ nat) @ I4) @ J2) => ((((ord_less @ nat) @ J2) @ N2) => (((ord_less_eq @ nat) @ (A2 @ I4)) @ (A2 @ J2)))))) => ((! [I4:nat,J2:nat]: (((((ord_less_eq @ nat) @ I4) @ J2) => ((((ord_less @ nat) @ J2) @ N2) => (((ord_less_eq @ nat) @ (B2 @ J2)) @ (B2 @ I4)))))) => (((ord_less_eq @ nat) @ (((times_times @ nat) @ N2) @ ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [I3:nat]: ((((times_times @ nat) @ (A2 @ I3)) @ (B2 @ I3))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2)))) @ (((times_times @ nat) @ ((((groups7311177749621191930dd_sum @ nat) @ nat) @ A2) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2))) @ ((((groups7311177749621191930dd_sum @ nat) @ nat) @ B2) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2))))))))). % 6.41/2.69 thf(fact_3819_VEBT_Osize__gen_I1_J, axiom, (! [X11:(option @ ((product_prod @ nat) @ nat)),X12:nat,X13:(list @ vEBT_VEBT),X14:vEBT_VEBT]: (((vEBT_size_VEBT @ ((((vEBT_Node @ X11) @ X12) @ X13) @ X14)) = (((plus_plus @ nat) @ (((plus_plus @ nat) @ (((size_list @ vEBT_VEBT) @ vEBT_size_VEBT) @ X13)) @ (vEBT_size_VEBT @ X14))) @ (suc @ (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_3820_finite__atLeastLessThan__int, axiom, (! [L:int,U:int]: (((finite_finite2 @ int) @ (((set_or7035219750837199246ssThan @ int) @ L) @ U))))). % 6.41/2.69 thf(fact_3821_finite__atLeastZeroLessThan__int, axiom, (! [U:int]: (((finite_finite2 @ int) @ (((set_or7035219750837199246ssThan @ int) @ (zero_zero @ int)) @ U))))). % 6.41/2.69 thf(fact_3822_VEBT_Osize__gen_I2_J, axiom, (! [X21:$o,X222:$o]: (((vEBT_size_VEBT @ ((vEBT_Leaf @ X21) @ X222)) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_3823_rat__inverse__code, axiom, (! [P4:rat]: (((quotient_of @ ((inverse_inverse @ rat) @ P4)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A4:int,B4:int]: (((((if @ ((product_prod @ int) @ int)) @ (A4 = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ A4)) @ B4)) @ ((abs_abs @ int) @ A4)))))) @ (quotient_of @ P4)))))). % 6.41/2.69 thf(fact_3824_normalize__negative, axiom, (! [Q4:int,P4:int]: (((((ord_less @ int) @ Q4) @ (zero_zero @ int)) => ((normalize @ ((((product_Pair @ int) @ int) @ P4) @ Q4)) = (normalize @ ((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ P4)) @ ((uminus_uminus @ int) @ Q4)))))))). % 6.41/2.69 thf(fact_3825_Cauchy__iff2, axiom, ((topolo3814608138187158403Cauchy @ real) = (^ [X6:(nat > real)]: ((! [J3:nat]: ((? [M9:nat]: ((! [M2:nat]: (((((ord_less_eq @ nat) @ M9) @ M2) => (! [N:nat]: (((((ord_less_eq @ nat) @ M9) @ N) => (((ord_less @ real) @ ((abs_abs @ real) @ (((minus_minus @ real) @ (X6 @ M2)) @ (X6 @ N)))) @ ((inverse_inverse @ real) @ ((semiring_1_of_nat @ real) @ (suc @ J3)))))))))))))))))). % 6.41/2.69 thf(fact_3826_vebt__mint_Opelims, axiom, (! [X3:vEBT_VEBT,Y:(option @ nat)]: ((((vEBT_vebt_mint @ X3) = Y) => ((((accp @ vEBT_VEBT) @ vEBT_vebt_mint_rel) @ X3) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => (((A5 => (Y = ((some @ nat) @ (zero_zero @ nat)))) & ((~ (A5)) => ((B5 => (Y = ((some @ nat) @ (one_one @ nat)))) & ((~ (B5)) => (Y = (none @ nat)))))) => (~ ((((accp @ vEBT_VEBT) @ vEBT_vebt_mint_rel) @ ((vEBT_Leaf @ A5) @ B5)))))))) => ((! [Uu2:nat,Uv2:(list @ vEBT_VEBT),Uw2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu2) @ Uv2) @ Uw2)) => ((Y = (none @ nat)) => (~ ((((accp @ vEBT_VEBT) @ vEBT_vebt_mint_rel) @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu2) @ Uv2) @ Uw2)))))))) => (~ ((! [Mi2:nat,Ma2:nat,Ux2:nat,Uy2:(list @ vEBT_VEBT),Uz2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ Ux2) @ Uy2) @ Uz2)) => ((Y = ((some @ nat) @ Mi2)) => (~ ((((accp @ vEBT_VEBT) @ vEBT_vebt_mint_rel) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ Ux2) @ Uy2) @ Uz2))))))))))))))))). % 6.41/2.69 thf(fact_3827_quotient__of__number_I3_J, axiom, (! [K:num]: (((quotient_of @ ((numeral_numeral @ rat) @ K)) = ((((product_Pair @ int) @ int) @ ((numeral_numeral @ int) @ K)) @ (one_one @ int)))))). % 6.41/2.69 thf(fact_3828_normalize__denom__zero, axiom, (! [P4:int]: (((normalize @ ((((product_Pair @ int) @ int) @ P4) @ (zero_zero @ int))) = ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int)))))). % 6.41/2.69 thf(fact_3829_rat__one__code, axiom, ((quotient_of @ (one_one @ rat)) = ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (one_one @ int)))). % 6.41/2.69 thf(fact_3830_rat__zero__code, axiom, ((quotient_of @ (zero_zero @ rat)) = ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int)))). % 6.41/2.69 thf(fact_3831_quotient__of__number_I5_J, axiom, (! [K:num]: (((quotient_of @ ((uminus_uminus @ rat) @ ((numeral_numeral @ rat) @ K))) = ((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K))) @ (one_one @ int)))))). % 6.41/2.69 thf(fact_3832_quotient__of__number_I4_J, axiom, ((quotient_of @ ((uminus_uminus @ rat) @ (one_one @ rat))) = ((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (one_one @ int)))). % 6.41/2.69 thf(fact_3833_rat__divide__code, axiom, (! [P4:rat,Q4:rat]: (((quotient_of @ (((divide_divide @ rat) @ P4) @ Q4)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A4:int,C4:int]: ((((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [B4:int,D5:int]: ((normalize @ ((((product_Pair @ int) @ int) @ (((times_times @ int) @ A4) @ D5)) @ (((times_times @ int) @ C4) @ B4)))))) @ (quotient_of @ Q4))))) @ (quotient_of @ P4)))))). % 6.41/2.69 thf(fact_3834_rat__times__code, axiom, (! [P4:rat,Q4:rat]: (((quotient_of @ (((times_times @ rat) @ P4) @ Q4)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A4:int,C4:int]: ((((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [B4:int,D5:int]: ((normalize @ ((((product_Pair @ int) @ int) @ (((times_times @ int) @ A4) @ B4)) @ (((times_times @ int) @ C4) @ D5)))))) @ (quotient_of @ Q4))))) @ (quotient_of @ P4)))))). % 6.41/2.69 thf(fact_3835_quotient__of__div, axiom, (! [R2:rat,N2:int,D2:int]: ((((quotient_of @ R2) = ((((product_Pair @ int) @ int) @ N2) @ D2)) => (R2 = (((divide_divide @ rat) @ ((ring_1_of_int @ rat) @ N2)) @ ((ring_1_of_int @ rat) @ D2))))))). % 6.41/2.69 thf(fact_3836_rat__minus__code, axiom, (! [P4:rat,Q4:rat]: (((quotient_of @ (((minus_minus @ rat) @ P4) @ Q4)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A4:int,C4:int]: ((((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [B4:int,D5:int]: ((normalize @ ((((product_Pair @ int) @ int) @ (((minus_minus @ int) @ (((times_times @ int) @ A4) @ D5)) @ (((times_times @ int) @ B4) @ C4))) @ (((times_times @ int) @ C4) @ D5)))))) @ (quotient_of @ Q4))))) @ (quotient_of @ P4)))))). % 6.41/2.69 thf(fact_3837_quotient__of__denom__pos, axiom, (! [R2:rat,P4:int,Q4:int]: ((((quotient_of @ R2) = ((((product_Pair @ int) @ int) @ P4) @ Q4)) => (((ord_less @ int) @ (zero_zero @ int)) @ Q4))))). % 6.41/2.69 thf(fact_3838_rat__plus__code, axiom, (! [P4:rat,Q4:rat]: (((quotient_of @ (((plus_plus @ rat) @ P4) @ Q4)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A4:int,C4:int]: ((((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [B4:int,D5:int]: ((normalize @ ((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ A4) @ D5)) @ (((times_times @ int) @ B4) @ C4))) @ (((times_times @ int) @ C4) @ D5)))))) @ (quotient_of @ Q4))))) @ (quotient_of @ P4)))))). % 6.41/2.69 thf(fact_3839_normalize__denom__pos, axiom, (! [R2:((product_prod @ int) @ int),P4:int,Q4:int]: ((((normalize @ R2) = ((((product_Pair @ int) @ int) @ P4) @ Q4)) => (((ord_less @ int) @ (zero_zero @ int)) @ Q4))))). % 6.41/2.69 thf(fact_3840_normalize__crossproduct, axiom, (! [Q4:int,S:int,P4:int,R2:int]: (((Q4 != (zero_zero @ int)) => ((S != (zero_zero @ int)) => (((normalize @ ((((product_Pair @ int) @ int) @ P4) @ Q4)) = (normalize @ ((((product_Pair @ int) @ int) @ R2) @ S))) => ((((times_times @ int) @ P4) @ S) = (((times_times @ int) @ R2) @ Q4)))))))). % 6.41/2.69 thf(fact_3841_rat__uminus__code, axiom, (! [P4:rat]: (((quotient_of @ ((uminus_uminus @ rat) @ P4)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A4:int]: ((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ A4))))) @ (quotient_of @ P4)))))). % 6.41/2.69 thf(fact_3842_rat__less__code, axiom, ((ord_less @ rat) = (^ [P6:rat,Q5:rat]: ((((((product_case_prod @ int) @ int) @ $o) @ (^ [A4:int,C4:int]: ((((((product_case_prod @ int) @ int) @ $o) @ (^ [B4:int,D5:int]: ((((ord_less @ int) @ (((times_times @ int) @ A4) @ D5)) @ (((times_times @ int) @ C4) @ B4))))) @ (quotient_of @ Q5))))) @ (quotient_of @ P6)))))). % 6.41/2.69 thf(fact_3843_rat__abs__code, axiom, (! [P4:rat]: (((quotient_of @ ((abs_abs @ rat) @ P4)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A4:int]: ((((product_Pair @ int) @ int) @ ((abs_abs @ int) @ A4))))) @ (quotient_of @ P4)))))). % 6.41/2.69 thf(fact_3844_Cauchy__iff, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => ((topolo3814608138187158403Cauchy @ A) = (^ [X6:(nat > A)]: ((! [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) => (? [M9:nat]: ((! [M2:nat]: (((((ord_less_eq @ nat) @ M9) @ M2) => (! [N:nat]: (((((ord_less_eq @ nat) @ M9) @ N) => (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ (X6 @ M2)) @ (X6 @ N)))) @ E4))))))))))))))))))). % 6.41/2.69 thf(fact_3845_CauchyI, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X8:(nat > A)]: (((! [E:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E) => (? [M10:nat]: ((! [M3:nat]: (((((ord_less_eq @ nat) @ M10) @ M3) => (! [N3:nat]: (((((ord_less_eq @ nat) @ M10) @ N3) => (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ (X8 @ M3)) @ (X8 @ N3)))) @ E)))))))))))) => ((topolo3814608138187158403Cauchy @ A) @ X8)))))))). % 6.41/2.69 thf(fact_3846_CauchyD, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X8:(nat > A),E3:real]: ((((topolo3814608138187158403Cauchy @ A) @ X8) => ((((ord_less @ real) @ (zero_zero @ real)) @ E3) => (? [M8:nat]: ((! [M6:nat]: (((((ord_less_eq @ nat) @ M8) @ M6) => (! [N6:nat]: (((((ord_less_eq @ nat) @ M8) @ N6) => (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ (X8 @ M6)) @ (X8 @ N6)))) @ E3))))))))))))))))). % 6.41/2.69 thf(fact_3847_vebt__maxt_Opelims, axiom, (! [X3:vEBT_VEBT,Y:(option @ nat)]: ((((vEBT_vebt_maxt @ X3) = Y) => ((((accp @ vEBT_VEBT) @ vEBT_vebt_maxt_rel) @ X3) => ((! [A5:$o,B5:$o]: (((X3 = ((vEBT_Leaf @ A5) @ B5)) => (((B5 => (Y = ((some @ nat) @ (one_one @ nat)))) & ((~ (B5)) => ((A5 => (Y = ((some @ nat) @ (zero_zero @ nat)))) & ((~ (A5)) => (Y = (none @ nat)))))) => (~ ((((accp @ vEBT_VEBT) @ vEBT_vebt_maxt_rel) @ ((vEBT_Leaf @ A5) @ B5)))))))) => ((! [Uu2:nat,Uv2:(list @ vEBT_VEBT),Uw2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu2) @ Uv2) @ Uw2)) => ((Y = (none @ nat)) => (~ ((((accp @ vEBT_VEBT) @ vEBT_vebt_maxt_rel) @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uu2) @ Uv2) @ Uw2)))))))) => (~ ((! [Mi2:nat,Ma2:nat,Ux2:nat,Uy2:(list @ vEBT_VEBT),Uz2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ Ux2) @ Uy2) @ Uz2)) => ((Y = ((some @ nat) @ Ma2)) => (~ ((((accp @ vEBT_VEBT) @ vEBT_vebt_maxt_rel) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ Mi2) @ Ma2))) @ Ux2) @ Uy2) @ Uz2))))))))))))))))). % 6.41/2.69 thf(fact_3848_quotient__of__int, axiom, (! [A2:int]: (((quotient_of @ (of_int @ A2)) = ((((product_Pair @ int) @ int) @ A2) @ (one_one @ int)))))). % 6.41/2.69 thf(fact_3849_VEBT__internal_OminNull_Opelims_I1_J, axiom, (! [X3:vEBT_VEBT,Y:$o]: ((((vEBT_VEBT_minNull @ X3) = Y) => ((((accp @ vEBT_VEBT) @ vEBT_V6963167321098673237ll_rel) @ X3) => (((X3 = ((vEBT_Leaf @ $false) @ $false)) => (Y => (~ ((((accp @ vEBT_VEBT) @ vEBT_V6963167321098673237ll_rel) @ ((vEBT_Leaf @ $false) @ $false)))))) => ((! [Uv2:$o]: (((X3 = ((vEBT_Leaf @ $true) @ Uv2)) => ((~ (Y)) => (~ ((((accp @ vEBT_VEBT) @ vEBT_V6963167321098673237ll_rel) @ ((vEBT_Leaf @ $true) @ Uv2)))))))) => ((! [Uu2:$o]: (((X3 = ((vEBT_Leaf @ Uu2) @ $true)) => ((~ (Y)) => (~ ((((accp @ vEBT_VEBT) @ vEBT_V6963167321098673237ll_rel) @ ((vEBT_Leaf @ Uu2) @ $true)))))))) => ((! [Uw2:nat,Ux2:(list @ vEBT_VEBT),Uy2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uw2) @ Ux2) @ Uy2)) => (Y => (~ ((((accp @ vEBT_VEBT) @ vEBT_V6963167321098673237ll_rel) @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uw2) @ Ux2) @ Uy2)))))))) => (~ ((! [Uz2:((product_prod @ nat) @ nat),Va3:nat,Vb2:(list @ vEBT_VEBT),Vc2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ Uz2)) @ Va3) @ Vb2) @ Vc2)) => ((~ (Y)) => (~ ((((accp @ vEBT_VEBT) @ vEBT_V6963167321098673237ll_rel) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ Uz2)) @ Va3) @ Vb2) @ Vc2))))))))))))))))))). % 6.41/2.69 thf(fact_3850_VEBT__internal_OminNull_Opelims_I2_J, axiom, (! [X3:vEBT_VEBT]: (((vEBT_VEBT_minNull @ X3) => ((((accp @ vEBT_VEBT) @ vEBT_V6963167321098673237ll_rel) @ X3) => (((X3 = ((vEBT_Leaf @ $false) @ $false)) => (~ ((((accp @ vEBT_VEBT) @ vEBT_V6963167321098673237ll_rel) @ ((vEBT_Leaf @ $false) @ $false))))) => (~ ((! [Uw2:nat,Ux2:(list @ vEBT_VEBT),Uy2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uw2) @ Ux2) @ Uy2)) => (~ ((((accp @ vEBT_VEBT) @ vEBT_V6963167321098673237ll_rel) @ ((((vEBT_Node @ (none @ ((product_prod @ nat) @ nat))) @ Uw2) @ Ux2) @ Uy2))))))))))))))). % 6.41/2.69 thf(fact_3851_VEBT__internal_OminNull_Opelims_I3_J, axiom, (! [X3:vEBT_VEBT]: (((~ ((vEBT_VEBT_minNull @ X3))) => ((((accp @ vEBT_VEBT) @ vEBT_V6963167321098673237ll_rel) @ X3) => ((! [Uv2:$o]: (((X3 = ((vEBT_Leaf @ $true) @ Uv2)) => (~ ((((accp @ vEBT_VEBT) @ vEBT_V6963167321098673237ll_rel) @ ((vEBT_Leaf @ $true) @ Uv2))))))) => ((! [Uu2:$o]: (((X3 = ((vEBT_Leaf @ Uu2) @ $true)) => (~ ((((accp @ vEBT_VEBT) @ vEBT_V6963167321098673237ll_rel) @ ((vEBT_Leaf @ Uu2) @ $true))))))) => (~ ((! [Uz2:((product_prod @ nat) @ nat),Va3:nat,Vb2:(list @ vEBT_VEBT),Vc2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ Uz2)) @ Va3) @ Vb2) @ Vc2)) => (~ ((((accp @ vEBT_VEBT) @ vEBT_V6963167321098673237ll_rel) @ ((((vEBT_Node @ ((some @ ((product_prod @ nat) @ nat)) @ Uz2)) @ Va3) @ Vb2) @ Vc2)))))))))))))))). % 6.41/2.69 thf(fact_3852_Frct__code__post_I5_J, axiom, (! [K:num]: (((frct @ ((((product_Pair @ int) @ int) @ (one_one @ int)) @ ((numeral_numeral @ int) @ K))) = (((divide_divide @ rat) @ (one_one @ rat)) @ ((numeral_numeral @ rat) @ K)))))). % 6.41/2.69 thf(fact_3853_is__singleton__the__elem, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A6:(set @ A)]: ((A6 = (((insert @ A) @ ((the_elem @ A) @ A6)) @ (bot_bot @ (set @ A)))))))))). % 6.41/2.69 thf(fact_3854_length__subseqs, axiom, (! [A:$tType,Xs:(list @ A)]: ((((size_size @ (list @ (list @ A))) @ ((subseqs @ A) @ Xs)) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((size_size @ (list @ A)) @ Xs)))))). % 6.41/2.69 thf(fact_3855_divmod__step__integer__def, axiom, ((unique1321980374590559556d_step @ code_integer) = (^ [L2:num]: (((((product_case_prod @ code_integer) @ code_integer) @ ((product_prod @ code_integer) @ code_integer)) @ (^ [Q5:code_integer,R5:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (((ord_less_eq @ code_integer) @ ((numeral_numeral @ code_integer) @ L2)) @ R5)) @ ((((product_Pair @ code_integer) @ code_integer) @ (((plus_plus @ code_integer) @ (((times_times @ code_integer) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))) @ Q5)) @ (one_one @ code_integer))) @ (((minus_minus @ code_integer) @ R5) @ ((numeral_numeral @ code_integer) @ L2)))) @ ((((product_Pair @ code_integer) @ code_integer) @ (((times_times @ code_integer) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))) @ Q5)) @ R5))))))))). % 6.41/2.69 thf(fact_3856_is__singletonI, axiom, (! [A:$tType,X3:A]: (((is_singleton @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))). % 6.41/2.69 thf(fact_3857_divmod__integer_H__def, axiom, ((unique8689654367752047608divmod @ code_integer) = (^ [M2:num,N:num]: (((((product_Pair @ code_integer) @ code_integer) @ (((divide_divide @ code_integer) @ ((numeral_numeral @ code_integer) @ M2)) @ ((numeral_numeral @ code_integer) @ N))) @ (((modulo_modulo @ code_integer) @ ((numeral_numeral @ code_integer) @ M2)) @ ((numeral_numeral @ code_integer) @ N))))))). % 6.41/2.69 thf(fact_3858_sgn__integer__code, axiom, ((sgn_sgn @ code_integer) = (^ [K3:code_integer]: (((((if @ code_integer) @ (K3 = (zero_zero @ code_integer))) @ (zero_zero @ code_integer)) @ ((((if @ code_integer) @ (((ord_less @ code_integer) @ K3) @ (zero_zero @ code_integer))) @ ((uminus_uminus @ code_integer) @ (one_one @ code_integer))) @ (one_one @ code_integer))))))). % 6.41/2.69 thf(fact_3859_exhaustive__integer_H_Ocases, axiom, (! [X3:((product_prod @ (code_integer > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_integer) @ code_integer))]: ((~ ((! [F4:(code_integer > (option @ ((product_prod @ $o) @ (list @ code_term)))),D3:code_integer,I4:code_integer]: ((X3 != ((((product_Pair @ (code_integer > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_integer) @ code_integer)) @ F4) @ ((((product_Pair @ code_integer) @ code_integer) @ D3) @ I4)))))))))). % 6.41/2.69 thf(fact_3860_full__exhaustive__integer_H_Ocases, axiom, (! [X3:((product_prod @ (((product_prod @ code_integer) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_integer) @ code_integer))]: ((~ ((! [F4:(((product_prod @ code_integer) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term)))),D3:code_integer,I4:code_integer]: ((X3 != ((((product_Pair @ (((product_prod @ code_integer) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_integer) @ code_integer)) @ F4) @ ((((product_Pair @ code_integer) @ code_integer) @ D3) @ I4)))))))))). % 6.41/2.69 thf(fact_3861_subseqs__refl, axiom, (! [A:$tType,Xs:(list @ A)]: ((((member @ (list @ A)) @ Xs) @ ((set2 @ (list @ A)) @ ((subseqs @ A) @ Xs)))))). % 6.41/2.69 thf(fact_3862_is__singletonI_H, axiom, (! [A:$tType,A3:(set @ A)]: (((A3 != (bot_bot @ (set @ A))) => ((! [X5:A,Y4:A]: (((((member @ A) @ X5) @ A3) => ((((member @ A) @ Y4) @ A3) => (X5 = Y4))))) => ((is_singleton @ A) @ A3)))))). % 6.41/2.69 thf(fact_3863_zero__natural_Orsp, axiom, ((zero_zero @ nat) = (zero_zero @ nat))). % 6.41/2.69 thf(fact_3864_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A6:(set @ A)]: ((? [X:A]: ((A6 = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))))). % 6.41/2.69 thf(fact_3865_is__singletonE, axiom, (! [A:$tType,A3:(set @ A)]: ((((is_singleton @ A) @ A3) => (~ ((! [X5:A]: ((A3 != (((insert @ A) @ X5) @ (bot_bot @ (set @ A)))))))))))). % 6.41/2.69 thf(fact_3866_Frct__code__post_I1_J, axiom, (! [A2:int]: (((frct @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ A2)) = (zero_zero @ rat))))). % 6.41/2.69 thf(fact_3867_Frct__code__post_I2_J, axiom, (! [A2:int]: (((frct @ ((((product_Pair @ int) @ int) @ A2) @ (zero_zero @ int))) = (zero_zero @ rat))))). % 6.41/2.69 thf(fact_3868_Frct__code__post_I8_J, axiom, (! [A2:int,B2:int]: (((frct @ ((((product_Pair @ int) @ int) @ A2) @ ((uminus_uminus @ int) @ B2))) = ((uminus_uminus @ rat) @ (frct @ ((((product_Pair @ int) @ int) @ A2) @ B2))))))). % 6.41/2.69 thf(fact_3869_Frct__code__post_I7_J, axiom, (! [A2:int,B2:int]: (((frct @ ((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ A2)) @ B2)) = ((uminus_uminus @ rat) @ (frct @ ((((product_Pair @ int) @ int) @ A2) @ B2))))))). % 6.41/2.69 thf(fact_3870_Frct__code__post_I3_J, axiom, ((frct @ ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (one_one @ int))) = (one_one @ rat))). % 6.41/2.69 thf(fact_3871_Frct__code__post_I4_J, axiom, (! [K:num]: (((frct @ ((((product_Pair @ int) @ int) @ ((numeral_numeral @ int) @ K)) @ (one_one @ int))) = ((numeral_numeral @ rat) @ K))))). % 6.41/2.69 thf(fact_3872_Frct__code__post_I6_J, axiom, (! [K:num,L:num]: (((frct @ ((((product_Pair @ int) @ int) @ ((numeral_numeral @ int) @ K)) @ ((numeral_numeral @ int) @ L))) = (((divide_divide @ rat) @ ((numeral_numeral @ rat) @ K)) @ ((numeral_numeral @ rat) @ L)))))). % 6.41/2.69 thf(fact_3873_integer__of__int__code, axiom, (code_integer_of_int = (^ [K3:int]: (((((if @ code_integer) @ (((ord_less @ int) @ K3) @ (zero_zero @ int))) @ ((uminus_uminus @ code_integer) @ (code_integer_of_int @ ((uminus_uminus @ int) @ K3)))) @ ((((if @ code_integer) @ (K3 = (zero_zero @ int))) @ (zero_zero @ code_integer)) @ ((((if @ code_integer) @ ((((modulo_modulo @ int) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2))) = (zero_zero @ int))) @ (((times_times @ code_integer) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))) @ (code_integer_of_int @ (((divide_divide @ int) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))) @ (((plus_plus @ code_integer) @ (((times_times @ code_integer) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))) @ (code_integer_of_int @ (((divide_divide @ int) @ K3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))) @ (one_one @ code_integer))))))))). % 6.41/2.69 thf(fact_3874_length__mul__elem, axiom, (! [A:$tType,Xs:(list @ (list @ A)),N2:nat]: (((! [X5:(list @ A)]: (((((member @ (list @ A)) @ X5) @ ((set2 @ (list @ A)) @ Xs)) => (((size_size @ (list @ A)) @ X5) = N2)))) => (((size_size @ (list @ A)) @ ((concat @ A) @ Xs)) = (((times_times @ nat) @ ((size_size @ (list @ (list @ A))) @ Xs)) @ N2)))))). % 6.41/2.69 thf(fact_3875_even__sum__iff, axiom, (! [A:$tType,B:$tType]: (((semiring_parity @ A) => (! [A3:(set @ B),F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ A3)) = (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((finite_card @ B) @ ((collect @ B) @ (^ [A4:B]: (((((member @ B) @ A4) @ A3) & (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (F2 @ A4)))))))))))))))))). % 6.41/2.69 thf(fact_3876_pair__leqI2, axiom, (! [A2:nat,B2:nat,S:nat,T2:nat]: (((((ord_less_eq @ nat) @ A2) @ B2) => ((((ord_less_eq @ nat) @ S) @ T2) => (((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ A2) @ S)) @ ((((product_Pair @ nat) @ nat) @ B2) @ T2))) @ fun_pair_leq)))))). % 6.41/2.69 thf(fact_3877_card__Collect__less__nat, axiom, (! [N2:nat]: ((((finite_card @ nat) @ ((collect @ nat) @ (^ [I3:nat]: ((((ord_less @ nat) @ I3) @ N2))))) = N2)))). % 6.41/2.69 thf(fact_3878_card__atMost, axiom, (! [U:nat]: ((((finite_card @ nat) @ ((set_ord_atMost @ nat) @ U)) = (suc @ U))))). % 6.41/2.69 thf(fact_3879_card__Collect__le__nat, axiom, (! [N2:nat]: ((((finite_card @ nat) @ ((collect @ nat) @ (^ [I3:nat]: ((((ord_less_eq @ nat) @ I3) @ N2))))) = (suc @ N2))))). % 6.41/2.69 thf(fact_3880_card_Oempty, axiom, (! [A:$tType]: ((((finite_card @ A) @ (bot_bot @ (set @ A))) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_3881_card_Oinfinite, axiom, (! [A:$tType,A3:(set @ A)]: (((~ (((finite_finite2 @ A) @ A3))) => (((finite_card @ A) @ A3) = (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_3882_card__atLeastAtMost, axiom, (! [L:nat,U:nat]: ((((finite_card @ nat) @ (((set_or1337092689740270186AtMost @ nat) @ L) @ U)) = (((minus_minus @ nat) @ (suc @ U)) @ L))))). % 6.41/2.69 thf(fact_3883_card__0__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((((finite_card @ A) @ A3) = (zero_zero @ nat)) = (A3 = (bot_bot @ (set @ A)))))))). % 6.41/2.69 thf(fact_3884_card__insert__disjoint, axiom, (! [A:$tType,A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((~ ((((member @ A) @ X3) @ A3))) => (((finite_card @ A) @ (((insert @ A) @ X3) @ A3)) = (suc @ ((finite_card @ A) @ A3)))))))). % 6.41/2.69 thf(fact_3885_abs__integer__code, axiom, ((abs_abs @ code_integer) = (^ [K3:code_integer]: (((((if @ code_integer) @ (((ord_less @ code_integer) @ K3) @ (zero_zero @ code_integer))) @ ((uminus_uminus @ code_integer) @ K3)) @ K3))))). % 6.41/2.69 thf(fact_3886_less__integer__code_I1_J, axiom, (~ ((((ord_less @ code_integer) @ (zero_zero @ code_integer)) @ (zero_zero @ code_integer))))). % 6.41/2.69 thf(fact_3887_less__integer_Oabs__eq, axiom, (! [Xa2:int,X3:int]: (((((ord_less @ code_integer) @ (code_integer_of_int @ Xa2)) @ (code_integer_of_int @ X3)) = (((ord_less @ int) @ Xa2) @ X3))))). % 6.41/2.69 thf(fact_3888_n__subsets, axiom, (! [A:$tType,A3:(set @ A),K:nat]: ((((finite_finite2 @ A) @ A3) => (((finite_card @ (set @ A)) @ ((collect @ (set @ A)) @ (^ [B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B6) @ A3) & (((finite_card @ A) @ B6) = K)))))) = ((binomial @ ((finite_card @ A) @ A3)) @ K)))))). % 6.41/2.69 thf(fact_3889_card__subset__eq, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((finite_card @ A) @ A3) = ((finite_card @ A) @ B3)) => (A3 = B3))))))). % 6.41/2.69 thf(fact_3890_infinite__arbitrarily__large, axiom, (! [A:$tType,A3:(set @ A),N2:nat]: (((~ (((finite_finite2 @ A) @ A3))) => (? [B10:(set @ A)]: ((((finite_finite2 @ A) @ B10) & ((((finite_card @ A) @ B10) = N2) & (((ord_less_eq @ (set @ A)) @ B10) @ A3))))))))). % 6.41/2.69 thf(fact_3891_card__le__if__inj__on__rel, axiom, (! [B:$tType,A:$tType,B3:(set @ A),A3:(set @ B),R2:(B > (A > $o))]: ((((finite_finite2 @ A) @ B3) => ((! [A5:B]: (((((member @ B) @ A5) @ A3) => (? [B11:A]: (((((member @ A) @ B11) @ B3) & ((R2 @ A5) @ B11))))))) => ((! [A13:B,A24:B,B5:A]: (((((member @ B) @ A13) @ A3) => ((((member @ B) @ A24) @ A3) => ((((member @ A) @ B5) @ B3) => (((R2 @ A13) @ B5) => (((R2 @ A24) @ B5) => (A13 = A24)))))))) => (((ord_less_eq @ nat) @ ((finite_card @ B) @ A3)) @ ((finite_card @ A) @ B3)))))))). % 6.41/2.69 thf(fact_3892_card__insert__le, axiom, (! [A:$tType,A3:(set @ A),X3:A]: ((((ord_less_eq @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ (((insert @ A) @ X3) @ A3)))))). % 6.41/2.69 thf(fact_3893_sum__multicount__gen, axiom, (! [A:$tType,B:$tType,S:(set @ A),T2:(set @ B),R:(A > (B > $o)),K:(B > nat)]: ((((finite_finite2 @ A) @ S) => (((finite_finite2 @ B) @ T2) => ((! [X5:B]: (((((member @ B) @ X5) @ T2) => (((finite_card @ A) @ ((collect @ A) @ (^ [I3:A]: (((((member @ A) @ I3) @ S) & ((R @ I3) @ X5)))))) = (K @ X5))))) => (((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [I3:A]: (((finite_card @ B) @ ((collect @ B) @ (^ [J3:B]: (((((member @ B) @ J3) @ T2) & ((R @ I3) @ J3))))))))) @ S) = ((((groups7311177749621191930dd_sum @ B) @ nat) @ K) @ T2)))))))). % 6.41/2.69 thf(fact_3894_card__lists__length__eq, axiom, (! [A:$tType,A3:(set @ A),N2:nat]: ((((finite_finite2 @ A) @ A3) => (((finite_card @ (list @ A)) @ ((collect @ (list @ A)) @ (^ [Xs2:(list @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs2)) @ A3) & (((size_size @ (list @ A)) @ Xs2) = N2)))))) = (((power_power @ nat) @ ((finite_card @ A) @ A3)) @ N2)))))). % 6.41/2.69 thf(fact_3895_card__eq__0__iff, axiom, (! [A:$tType,A3:(set @ A)]: (((((finite_card @ A) @ A3) = (zero_zero @ nat)) = ((A3 = (bot_bot @ (set @ A))) | (~ (((finite_finite2 @ A) @ A3)))))))). % 6.41/2.69 thf(fact_3896_card__ge__0__finite, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ A3)) => ((finite_finite2 @ A) @ A3))))). % 6.41/2.69 thf(fact_3897_card__insert__if, axiom, (! [A:$tType,A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => (((((member @ A) @ X3) @ A3) => (((finite_card @ A) @ (((insert @ A) @ X3) @ A3)) = ((finite_card @ A) @ A3))) & ((~ ((((member @ A) @ X3) @ A3))) => (((finite_card @ A) @ (((insert @ A) @ X3) @ A3)) = (suc @ ((finite_card @ A) @ A3))))))))). % 6.41/2.69 thf(fact_3898_card__Suc__eq__finite, axiom, (! [A:$tType,A3:(set @ A),K:nat]: (((((finite_card @ A) @ A3) = (suc @ K)) = (? [B4:A,B6:(set @ A)]: (((A3 = (((insert @ A) @ B4) @ B6)) & ((~ ((((member @ A) @ B4) @ B6))) & ((((finite_card @ A) @ B6) = K) & ((finite_finite2 @ A) @ B6)))))))))). % 6.41/2.69 thf(fact_3899_finite__if__finite__subsets__card__bdd, axiom, (! [A:$tType,F5:(set @ A),C5:nat]: (((! [G5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ G5) @ F5) => (((finite_finite2 @ A) @ G5) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ G5)) @ C5))))) => (((finite_finite2 @ A) @ F5) & (((ord_less_eq @ nat) @ ((finite_card @ A) @ F5)) @ C5)))))). % 6.41/2.69 thf(fact_3900_obtain__subset__with__card__n, axiom, (! [A:$tType,N2:nat,S3:(set @ A)]: (((((ord_less_eq @ nat) @ N2) @ ((finite_card @ A) @ S3)) => (~ ((! [T5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ T5) @ S3) => ((((finite_card @ A) @ T5) = N2) => (~ (((finite_finite2 @ A) @ T5))))))))))))). % 6.41/2.69 thf(fact_3901_card__seteq, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ B3)) @ ((finite_card @ A) @ A3)) => (A3 = B3))))))). % 6.41/2.69 thf(fact_3902_card__mono, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3))))))). % 6.41/2.69 thf(fact_3903_card__less__sym__Diff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ A) @ B3) => ((((ord_less @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3)) => (((ord_less @ nat) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A3) @ B3))) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ B3) @ A3))))))))). % 6.41/2.69 thf(fact_3904_card__le__sym__Diff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3)) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A3) @ B3))) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ B3) @ A3))))))))). % 6.41/2.69 thf(fact_3905_card__length, axiom, (! [A:$tType,Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((finite_card @ A) @ ((set2 @ A) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))). % 6.41/2.69 thf(fact_3906_card__1__singletonE, axiom, (! [A:$tType,A3:(set @ A)]: (((((finite_card @ A) @ A3) = (one_one @ nat)) => (~ ((! [X5:A]: ((A3 != (((insert @ A) @ X5) @ (bot_bot @ (set @ A)))))))))))). % 6.41/2.69 thf(fact_3907_psubset__card__mono, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less @ (set @ A)) @ A3) @ B3) => (((ord_less @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3))))))). % 6.41/2.69 thf(fact_3908_card__less, axiom, (! [M7:(set @ nat),I:nat]: (((((member @ nat) @ (zero_zero @ nat)) @ M7) => (((finite_card @ nat) @ ((collect @ nat) @ (^ [K3:nat]: (((((member @ nat) @ K3) @ M7) & (((ord_less @ nat) @ K3) @ (suc @ I))))))) != (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_3909_card__less__Suc, axiom, (! [M7:(set @ nat),I:nat]: (((((member @ nat) @ (zero_zero @ nat)) @ M7) => ((suc @ ((finite_card @ nat) @ ((collect @ nat) @ (^ [K3:nat]: (((((member @ nat) @ (suc @ K3)) @ M7) & (((ord_less @ nat) @ K3) @ I))))))) = ((finite_card @ nat) @ ((collect @ nat) @ (^ [K3:nat]: (((((member @ nat) @ K3) @ M7) & (((ord_less @ nat) @ K3) @ (suc @ I)))))))))))). % 6.41/2.69 thf(fact_3910_card__less__Suc2, axiom, (! [M7:(set @ nat),I:nat]: (((~ ((((member @ nat) @ (zero_zero @ nat)) @ M7))) => (((finite_card @ nat) @ ((collect @ nat) @ (^ [K3:nat]: (((((member @ nat) @ (suc @ K3)) @ M7) & (((ord_less @ nat) @ K3) @ I)))))) = ((finite_card @ nat) @ ((collect @ nat) @ (^ [K3:nat]: (((((member @ nat) @ K3) @ M7) & (((ord_less @ nat) @ K3) @ (suc @ I)))))))))))). % 6.41/2.69 thf(fact_3911_sum__Suc, axiom, (! [A:$tType,F2:(A > nat),A3:(set @ A)]: ((((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [X:A]: ((suc @ (F2 @ X))))) @ A3) = (((plus_plus @ nat) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ A3)) @ ((finite_card @ A) @ A3)))))). % 6.41/2.69 thf(fact_3912_subset__card__intvl__is__intvl, axiom, (! [A3:(set @ nat),K:nat]: (((((ord_less_eq @ (set @ nat)) @ A3) @ (((set_or7035219750837199246ssThan @ nat) @ K) @ (((plus_plus @ nat) @ K) @ ((finite_card @ nat) @ A3)))) => (A3 = (((set_or7035219750837199246ssThan @ nat) @ K) @ (((plus_plus @ nat) @ K) @ ((finite_card @ nat) @ A3)))))))). % 6.41/2.69 thf(fact_3913_sum__multicount, axiom, (! [A:$tType,B:$tType,S3:(set @ A),T4:(set @ B),R:(A > (B > $o)),K:nat]: ((((finite_finite2 @ A) @ S3) => (((finite_finite2 @ B) @ T4) => ((! [X5:B]: (((((member @ B) @ X5) @ T4) => (((finite_card @ A) @ ((collect @ A) @ (^ [I3:A]: (((((member @ A) @ I3) @ S3) & ((R @ I3) @ X5)))))) = K)))) => (((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [I3:A]: (((finite_card @ B) @ ((collect @ B) @ (^ [J3:B]: (((((member @ B) @ J3) @ T4) & ((R @ I3) @ J3))))))))) @ S3) = (((times_times @ nat) @ K) @ ((finite_card @ B) @ T4))))))))). % 6.41/2.69 thf(fact_3914_sum__bounded__below, axiom, (! [A:$tType,B:$tType]: ((((ordere6911136660526730532id_add @ A) & (semiring_1 @ A)) => (! [A3:(set @ B),K5:A,F2:(B > A)]: (((! [I4:B]: (((((member @ B) @ I4) @ A3) => (((ord_less_eq @ A) @ K5) @ (F2 @ I4))))) => (((ord_less_eq @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ ((finite_card @ B) @ A3))) @ K5)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ A3))))))))). % 6.41/2.69 thf(fact_3915_sum__bounded__above, axiom, (! [B:$tType,A:$tType]: ((((ordere6911136660526730532id_add @ A) & (semiring_1 @ A)) => (! [A3:(set @ B),F2:(B > A),K5:A]: (((! [I4:B]: (((((member @ B) @ I4) @ A3) => (((ord_less_eq @ A) @ (F2 @ I4)) @ K5)))) => (((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ A3)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ ((finite_card @ B) @ A3))) @ K5))))))))). % 6.41/2.69 thf(fact_3916_card__gt__0__iff, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ A3)) = ((A3 != (bot_bot @ (set @ A))) & ((finite_finite2 @ A) @ A3)))))). % 6.41/2.69 thf(fact_3917_card__Suc__eq, axiom, (! [A:$tType,A3:(set @ A),K:nat]: (((((finite_card @ A) @ A3) = (suc @ K)) = (? [B4:A,B6:(set @ A)]: (((A3 = (((insert @ A) @ B4) @ B6)) & ((~ ((((member @ A) @ B4) @ B6))) & ((((finite_card @ A) @ B6) = K) & ((K = (zero_zero @ nat)) => (B6 = (bot_bot @ (set @ A))))))))))))). % 6.41/2.69 thf(fact_3918_card__eq__SucD, axiom, (! [A:$tType,A3:(set @ A),K:nat]: (((((finite_card @ A) @ A3) = (suc @ K)) => (? [B5:A,B10:(set @ A)]: (((A3 = (((insert @ A) @ B5) @ B10)) & ((~ ((((member @ A) @ B5) @ B10))) & ((((finite_card @ A) @ B10) = K) & ((K = (zero_zero @ nat)) => (B10 = (bot_bot @ (set @ A))))))))))))). % 6.41/2.69 thf(fact_3919_card__1__singleton__iff, axiom, (! [A:$tType,A3:(set @ A)]: (((((finite_card @ A) @ A3) = (suc @ (zero_zero @ nat))) = (? [X:A]: ((A3 = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))). % 6.41/2.69 thf(fact_3920_card__le__Suc0__iff__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ A3)) @ (suc @ (zero_zero @ nat))) = (! [X:A]: (((((member @ A) @ X) @ A3) => (! [Y2:A]: (((((member @ A) @ Y2) @ A3) => (X = Y2)))))))))))). % 6.41/2.69 thf(fact_3921_card__le__Suc__iff, axiom, (! [A:$tType,N2:nat,A3:(set @ A)]: (((((ord_less_eq @ nat) @ (suc @ N2)) @ ((finite_card @ A) @ A3)) = (? [A4:A,B6:(set @ A)]: (((A3 = (((insert @ A) @ A4) @ B6)) & ((~ ((((member @ A) @ A4) @ B6))) & ((((ord_less_eq @ nat) @ N2) @ ((finite_card @ A) @ B6)) & ((finite_finite2 @ A) @ B6)))))))))). % 6.41/2.69 thf(fact_3922_card__Diff1__le, axiom, (! [A:$tType,A3:(set @ A),X3:A]: ((((ord_less_eq @ nat) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))) @ ((finite_card @ A) @ A3))))). % 6.41/2.69 thf(fact_3923_card__Diff__subset, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => (((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = (((minus_minus @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3)))))))). % 6.41/2.69 thf(fact_3924_card__psubset, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3)) => (((ord_less @ (set @ A)) @ A3) @ B3))))))). % 6.41/2.69 thf(fact_3925_diff__card__le__card__Diff, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => (((ord_less_eq @ nat) @ (((minus_minus @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3))) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A3) @ B3))))))). % 6.41/2.69 thf(fact_3926_card__lists__length__le, axiom, (! [A:$tType,A3:(set @ A),N2:nat]: ((((finite_finite2 @ A) @ A3) => (((finite_card @ (list @ A)) @ ((collect @ (list @ A)) @ (^ [Xs2:(list @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs2)) @ A3) & (((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs2)) @ N2)))))) = ((((groups7311177749621191930dd_sum @ nat) @ nat) @ ((power_power @ nat) @ ((finite_card @ A) @ A3))) @ ((set_ord_atMost @ nat) @ N2))))))). % 6.41/2.69 thf(fact_3927_card__roots__unity, axiom, (! [A:$tType]: ((((real_V8999393235501362500lgebra @ A) & (idom @ A)) => (! [N2:nat]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ N2) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ ((collect @ A) @ (^ [Z6:A]: (((((power_power @ A) @ Z6) @ N2) = (one_one @ A))))))) @ N2)))))))). % 6.41/2.69 thf(fact_3928_subset__eq__atLeast0__lessThan__card, axiom, (! [N7:(set @ nat),N2:nat]: (((((ord_less_eq @ (set @ nat)) @ N7) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2)) => (((ord_less_eq @ nat) @ ((finite_card @ nat) @ N7)) @ N2))))). % 6.41/2.69 thf(fact_3929_card__sum__le__nat__sum, axiom, (! [S3:(set @ nat)]: ((((ord_less_eq @ nat) @ ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X:nat]: (X))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((finite_card @ nat) @ S3)))) @ ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X:nat]: (X))) @ S3))))). % 6.41/2.69 thf(fact_3930_card__nth__roots, axiom, (! [C2:complex,N2:nat]: (((C2 != (zero_zero @ complex)) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((finite_card @ complex) @ ((collect @ complex) @ (^ [Z6:complex]: (((((power_power @ complex) @ Z6) @ N2) = C2))))) = N2)))))). % 6.41/2.69 thf(fact_3931_card__roots__unity__eq, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((finite_card @ complex) @ ((collect @ complex) @ (^ [Z6:complex]: (((((power_power @ complex) @ Z6) @ N2) = (one_one @ complex)))))) = N2))))). % 6.41/2.69 thf(fact_3932_card__2__iff, axiom, (! [A:$tType,S3:(set @ A)]: (((((finite_card @ A) @ S3) = ((numeral_numeral @ nat) @ (bit0 @ one2))) = (? [X:A,Y2:A]: (((S3 = (((insert @ A) @ X) @ (((insert @ A) @ Y2) @ (bot_bot @ (set @ A))))) & (X != Y2)))))))). % 6.41/2.69 thf(fact_3933_card__3__iff, axiom, (! [A:$tType,S3:(set @ A)]: (((((finite_card @ A) @ S3) = ((numeral_numeral @ nat) @ (bit1 @ one2))) = (? [X:A,Y2:A,Z6:A]: (((S3 = (((insert @ A) @ X) @ (((insert @ A) @ Y2) @ (((insert @ A) @ Z6) @ (bot_bot @ (set @ A)))))) & ((X != Y2) & ((Y2 != Z6) & (X != Z6)))))))))). % 6.41/2.69 thf(fact_3934_odd__card__imp__not__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((finite_card @ A) @ A3)))) => (A3 != (bot_bot @ (set @ A))))))). % 6.41/2.69 thf(fact_3935_card__Suc__Diff1, axiom, (! [A:$tType,A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => ((suc @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))) = ((finite_card @ A) @ A3))))))). % 6.41/2.69 thf(fact_3936_card_Oinsert__remove, axiom, (! [A:$tType,A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => (((finite_card @ A) @ (((insert @ A) @ X3) @ A3)) = (suc @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))))). % 6.41/2.69 thf(fact_3937_card_Oremove, axiom, (! [A:$tType,A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => (((finite_card @ A) @ A3) = (suc @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))). % 6.41/2.69 thf(fact_3938_card__Diff1__less, axiom, (! [A:$tType,A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => (((ord_less @ nat) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))) @ ((finite_card @ A) @ A3))))))). % 6.41/2.69 thf(fact_3939_card__Diff2__less, axiom, (! [A:$tType,A3:(set @ A),X3:A,Y:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => ((((member @ A) @ Y) @ A3) => (((ord_less @ nat) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) @ (((insert @ A) @ Y) @ (bot_bot @ (set @ A)))))) @ ((finite_card @ A) @ A3)))))))). % 6.41/2.69 thf(fact_3940_card__Diff1__less__iff, axiom, (! [A:$tType,A3:(set @ A),X3:A]: (((((ord_less @ nat) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))) @ ((finite_card @ A) @ A3)) = (((finite_finite2 @ A) @ A3) & (((member @ A) @ X3) @ A3)))))). % 6.41/2.69 thf(fact_3941_card__Diff__singleton__if, axiom, (! [A:$tType,X3:A,A3:(set @ A)]: ((((((member @ A) @ X3) @ A3) => (((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) = (((minus_minus @ nat) @ ((finite_card @ A) @ A3)) @ (one_one @ nat)))) & ((~ ((((member @ A) @ X3) @ A3))) => (((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) = ((finite_card @ A) @ A3))))))). % 6.41/2.69 thf(fact_3942_card__Diff__singleton, axiom, (! [A:$tType,X3:A,A3:(set @ A)]: (((((member @ A) @ X3) @ A3) => (((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) = (((minus_minus @ nat) @ ((finite_card @ A) @ A3)) @ (one_one @ nat))))))). % 6.41/2.69 thf(fact_3943_prod__le__power, axiom, (! [B:$tType,A:$tType]: (((linordered_semidom @ A) => (! [A3:(set @ B),F2:(B > A),N2:A,K:nat]: (((! [I4:B]: (((((member @ B) @ I4) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (F2 @ I4)) & (((ord_less_eq @ A) @ (F2 @ I4)) @ N2))))) => ((((ord_less_eq @ nat) @ ((finite_card @ B) @ A3)) @ K) => ((((ord_less_eq @ A) @ (one_one @ A)) @ N2) => (((ord_less_eq @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3)) @ (((power_power @ A) @ N2) @ K))))))))))). % 6.41/2.69 thf(fact_3944_sum__bounded__above__strict, axiom, (! [B:$tType,A:$tType]: ((((ordere8940638589300402666id_add @ A) & (semiring_1 @ A)) => (! [A3:(set @ B),F2:(B > A),K5:A]: (((! [I4:B]: (((((member @ B) @ I4) @ A3) => (((ord_less @ A) @ (F2 @ I4)) @ K5)))) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ ((finite_card @ B) @ A3)) => (((ord_less @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ A3)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ ((finite_card @ B) @ A3))) @ K5)))))))))). % 6.41/2.69 thf(fact_3945_sum__bounded__above__divide, axiom, (! [B:$tType,A:$tType]: (((linordered_field @ A) => (! [A3:(set @ B),F2:(B > A),K5:A]: (((! [I4:B]: (((((member @ B) @ I4) @ A3) => (((ord_less_eq @ A) @ (F2 @ I4)) @ (((divide_divide @ A) @ K5) @ ((semiring_1_of_nat @ A) @ ((finite_card @ B) @ A3))))))) => (((finite_finite2 @ B) @ A3) => ((A3 != (bot_bot @ (set @ B))) => (((ord_less_eq @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ A3)) @ K5)))))))))). % 6.41/2.69 thf(fact_3946_card__insert__le__m1, axiom, (! [A:$tType,N2:nat,Y:(set @ A),X3:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ Y)) @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ (((insert @ A) @ X3) @ Y))) @ N2)))))). % 6.41/2.69 thf(fact_3947_polyfun__roots__card, axiom, (! [A:$tType]: ((((real_V8999393235501362500lgebra @ A) & (idom @ A)) => (! [C2:(nat > A),K:nat,N2:nat]: ((((C2 @ K) != (zero_zero @ A)) => ((((ord_less_eq @ nat) @ K) @ N2) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ ((collect @ A) @ (^ [Z6:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ Z6) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)) = (zero_zero @ A))))))) @ N2))))))))). % 6.41/2.69 thf(fact_3948_prod__gen__delta, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),A2:B,B2:(B > A),C2:A]: ((((finite_finite2 @ B) @ S3) => (((((member @ B) @ A2) @ S3) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A2)) @ (B2 @ K3)) @ C2)))) @ S3) = (((times_times @ A) @ (B2 @ A2)) @ (((power_power @ A) @ C2) @ (((minus_minus @ nat) @ ((finite_card @ B) @ S3)) @ (one_one @ nat)))))) & ((~ ((((member @ B) @ A2) @ S3))) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K3:B]: (((((if @ A) @ (K3 = A2)) @ (B2 @ K3)) @ C2)))) @ S3) = (((power_power @ A) @ C2) @ ((finite_card @ B) @ S3)))))))))))). % 6.41/2.69 thf(fact_3949_polyfun__rootbound, axiom, (! [A:$tType]: ((((real_V8999393235501362500lgebra @ A) & (idom @ A)) => (! [C2:(nat > A),K:nat,N2:nat]: ((((C2 @ K) != (zero_zero @ A)) => ((((ord_less_eq @ nat) @ K) @ N2) => (((finite_finite2 @ A) @ ((collect @ A) @ (^ [Z6:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ Z6) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)) = (zero_zero @ A)))))) & (((ord_less_eq @ nat) @ ((finite_card @ A) @ ((collect @ A) @ (^ [Z6:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ Z6) @ I3))))) @ ((set_ord_atMost @ nat) @ N2)) = (zero_zero @ A))))))) @ N2)))))))))). % 6.41/2.69 thf(fact_3950_pair__leqI1, axiom, (! [A2:nat,B2:nat,S:nat,T2:nat]: (((((ord_less @ nat) @ A2) @ B2) => (((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ A2) @ S)) @ ((((product_Pair @ nat) @ nat) @ B2) @ T2))) @ fun_pair_leq))))). % 6.41/2.69 thf(fact_3951_card__lists__distinct__length__eq, axiom, (! [A:$tType,A3:(set @ A),K:nat]: ((((finite_finite2 @ A) @ A3) => ((((ord_less_eq @ nat) @ K) @ ((finite_card @ A) @ A3)) => (((finite_card @ (list @ A)) @ ((collect @ (list @ A)) @ (^ [Xs2:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) = K) & (((distinct @ A) @ Xs2) & (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs2)) @ A3))))))) = ((((groups7121269368397514597t_prod @ nat) @ nat) @ (^ [X:nat]: (X))) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ (((minus_minus @ nat) @ ((finite_card @ A) @ A3)) @ K)) @ (one_one @ nat))) @ ((finite_card @ A) @ A3))))))))). % 6.41/2.69 thf(fact_3952_card__lists__distinct__length__eq_H, axiom, (! [A:$tType,K:nat,A3:(set @ A)]: (((((ord_less @ nat) @ K) @ ((finite_card @ A) @ A3)) => (((finite_card @ (list @ A)) @ ((collect @ (list @ A)) @ (^ [Xs2:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) = K) & (((distinct @ A) @ Xs2) & (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs2)) @ A3))))))) = ((((groups7121269368397514597t_prod @ nat) @ nat) @ (^ [X:nat]: (X))) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ (((minus_minus @ nat) @ ((finite_card @ A) @ A3)) @ K)) @ (one_one @ nat))) @ ((finite_card @ A) @ A3)))))))). % 6.41/2.69 thf(fact_3953_set__n__lists, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((set2 @ (list @ A)) @ (((n_lists @ A) @ N2) @ Xs)) = ((collect @ (list @ A)) @ (^ [Ys3:(list @ A)]: (((((size_size @ (list @ A)) @ Ys3) = N2) & (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Ys3)) @ ((set2 @ A) @ Xs)))))))))). % 6.41/2.69 thf(fact_3954_int__of__integer__code, axiom, (code_int_of_integer = (^ [K3:code_integer]: (((((if @ int) @ (((ord_less @ code_integer) @ K3) @ (zero_zero @ code_integer))) @ ((uminus_uminus @ int) @ (code_int_of_integer @ ((uminus_uminus @ code_integer) @ K3)))) @ ((((if @ int) @ (K3 = (zero_zero @ code_integer))) @ (zero_zero @ int)) @ (((((product_case_prod @ code_integer) @ code_integer) @ int) @ (^ [L2:code_integer,J3:code_integer]: (((((if @ int) @ (J3 = (zero_zero @ code_integer))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (code_int_of_integer @ L2))) @ (((plus_plus @ int) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (code_int_of_integer @ L2))) @ (one_one @ int)))))) @ ((code_divmod_integer @ K3) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_3955_distinct__swap, axiom, (! [A:$tType,I:nat,Xs:(list @ A),J:nat]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((distinct @ A) @ ((((list_update @ A) @ ((((list_update @ A) @ Xs) @ I) @ (((nth @ A) @ Xs) @ J))) @ J) @ (((nth @ A) @ Xs) @ I))) = ((distinct @ A) @ Xs))))))). % 6.41/2.69 thf(fact_3956_finite__lists__distinct__length__eq, axiom, (! [A:$tType,A3:(set @ A),N2:nat]: ((((finite_finite2 @ A) @ A3) => ((finite_finite2 @ (list @ A)) @ ((collect @ (list @ A)) @ (^ [Xs2:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) = N2) & (((distinct @ A) @ Xs2) & (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs2)) @ A3))))))))))). % 6.41/2.69 thf(fact_3957_distinct__enumerate, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((distinct @ ((product_prod @ nat) @ A)) @ (((enumerate @ A) @ N2) @ Xs))))). % 6.41/2.69 thf(fact_3958_distinct__product, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B)]: ((((distinct @ A) @ Xs) => (((distinct @ B) @ Ys) => ((distinct @ ((product_prod @ A) @ B)) @ ((((product @ A) @ B) @ Xs) @ Ys))))))). % 6.41/2.69 thf(fact_3959_sorted__list__of__set_Odistinct__if__distinct__map, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: ((((distinct @ A) @ Xs) => ((distinct @ A) @ Xs)))))))). % 6.41/2.69 thf(fact_3960_distinct__zipI1, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B)]: ((((distinct @ A) @ Xs) => ((distinct @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys)))))). % 6.41/2.69 thf(fact_3961_distinct__zipI2, axiom, (! [B:$tType,A:$tType,Ys:(list @ A),Xs:(list @ B)]: ((((distinct @ A) @ Ys) => ((distinct @ ((product_prod @ B) @ A)) @ ((((zip @ B) @ A) @ Xs) @ Ys)))))). % 6.41/2.69 thf(fact_3962_finite__distinct__list, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (? [Xs3:(list @ A)]: (((((set2 @ A) @ Xs3) = A3) & ((distinct @ A) @ Xs3)))))))). % 6.41/2.69 thf(fact_3963_less__integer_Orep__eq, axiom, ((ord_less @ code_integer) = (^ [X:code_integer,Xa4:code_integer]: ((((ord_less @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa4)))))). % 6.41/2.69 thf(fact_3964_integer__less__iff, axiom, ((ord_less @ code_integer) = (^ [K3:code_integer,L2:code_integer]: ((((ord_less @ int) @ (code_int_of_integer @ K3)) @ (code_int_of_integer @ L2)))))). % 6.41/2.69 thf(fact_3965_subseqs__distinctD, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ A)]: (((((member @ (list @ A)) @ Ys) @ ((set2 @ (list @ A)) @ ((subseqs @ A) @ Xs))) => (((distinct @ A) @ Xs) => ((distinct @ A) @ Ys)))))). % 6.41/2.69 thf(fact_3966_distinct__card, axiom, (! [A:$tType,Xs:(list @ A)]: ((((distinct @ A) @ Xs) => (((finite_card @ A) @ ((set2 @ A) @ Xs)) = ((size_size @ (list @ A)) @ Xs)))))). % 6.41/2.69 thf(fact_3967_card__distinct, axiom, (! [A:$tType,Xs:(list @ A)]: (((((finite_card @ A) @ ((set2 @ A) @ Xs)) = ((size_size @ (list @ A)) @ Xs)) => ((distinct @ A) @ Xs))))). % 6.41/2.69 thf(fact_3968_distinct__conv__nth, axiom, (! [A:$tType]: (((distinct @ A) = (^ [Xs2:(list @ A)]: ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs2)) => (! [J3:nat]: (((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs2)) => ((I3 != J3) => ((((nth @ A) @ Xs2) @ I3) != (((nth @ A) @ Xs2) @ J3))))))))))))))). % 6.41/2.69 thf(fact_3969_nth__eq__iff__index__eq, axiom, (! [A:$tType,Xs:(list @ A),I:nat,J:nat]: ((((distinct @ A) @ Xs) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((((nth @ A) @ Xs) @ I) = (((nth @ A) @ Xs) @ J)) = (I = J)))))))). % 6.41/2.69 thf(fact_3970_distinct__Ex1, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((distinct @ A) @ Xs) => ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (? [X5:nat]: (((((ord_less @ nat) @ X5) @ ((size_size @ (list @ A)) @ Xs)) & (((((nth @ A) @ Xs) @ X5) = X3) & (! [Y5:nat]: ((((((ord_less @ nat) @ Y5) @ ((size_size @ (list @ A)) @ Xs)) & ((((nth @ A) @ Xs) @ Y5) = X3)) => (Y5 = X5))))))))))))). % 6.41/2.69 thf(fact_3971_length__n__lists__elem, axiom, (! [A:$tType,Ys:(list @ A),N2:nat,Xs:(list @ A)]: (((((member @ (list @ A)) @ Ys) @ ((set2 @ (list @ A)) @ (((n_lists @ A) @ N2) @ Xs))) => (((size_size @ (list @ A)) @ Ys) = N2))))). % 6.41/2.69 thf(fact_3972_distinct__list__update, axiom, (! [A:$tType,Xs:(list @ A),A2:A,I:nat]: ((((distinct @ A) @ Xs) => ((~ ((((member @ A) @ A2) @ (((minus_minus @ (set @ A)) @ ((set2 @ A) @ Xs)) @ (((insert @ A) @ (((nth @ A) @ Xs) @ I)) @ (bot_bot @ (set @ A))))))) => ((distinct @ A) @ ((((list_update @ A) @ Xs) @ I) @ A2))))))). % 6.41/2.69 thf(fact_3973_length__n__lists, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((size_size @ (list @ (list @ A))) @ (((n_lists @ A) @ N2) @ Xs)) = (((power_power @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N2))))). % 6.41/2.69 thf(fact_3974_set__update__distinct, axiom, (! [A:$tType,Xs:(list @ A),N2:nat,X3:A]: ((((distinct @ A) @ Xs) => ((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => (((set2 @ A) @ ((((list_update @ A) @ Xs) @ N2) @ X3)) = (((insert @ A) @ X3) @ (((minus_minus @ (set @ A)) @ ((set2 @ A) @ Xs)) @ (((insert @ A) @ (((nth @ A) @ Xs) @ N2)) @ (bot_bot @ (set @ A))))))))))). % 6.41/2.69 thf(fact_3975_divmod__integer__def, axiom, (code_divmod_integer = (^ [K3:code_integer,L2:code_integer]: (((((product_Pair @ code_integer) @ code_integer) @ (((divide_divide @ code_integer) @ K3) @ L2)) @ (((modulo_modulo @ code_integer) @ K3) @ L2)))))). % 6.41/2.69 thf(fact_3976_nat__of__integer__code, axiom, (code_nat_of_integer = (^ [K3:code_integer]: (((((if @ nat) @ (((ord_less_eq @ code_integer) @ K3) @ (zero_zero @ code_integer))) @ (zero_zero @ nat)) @ (((((product_case_prod @ code_integer) @ code_integer) @ nat) @ (^ [L2:code_integer,J3:code_integer]: (((((if @ nat) @ (J3 = (zero_zero @ code_integer))) @ (((plus_plus @ nat) @ (code_nat_of_integer @ L2)) @ (code_nat_of_integer @ L2))) @ (((plus_plus @ nat) @ (((plus_plus @ nat) @ (code_nat_of_integer @ L2)) @ (code_nat_of_integer @ L2))) @ (one_one @ nat)))))) @ ((code_divmod_integer @ K3) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))))))))). % 6.41/2.69 thf(fact_3977_distinct__union, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((distinct @ A) @ (((union @ A) @ Xs) @ Ys)) = ((distinct @ A) @ Ys))))). % 6.41/2.69 thf(fact_3978_bit__cut__integer__def, axiom, (code_bit_cut_integer = (^ [K3:code_integer]: (((((product_Pair @ code_integer) @ $o) @ (((divide_divide @ code_integer) @ K3) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2)))) @ (~ ((((dvd_dvd @ code_integer) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))) @ K3)))))))). % 6.41/2.69 thf(fact_3979_divmod__abs__def, axiom, (code_divmod_abs = (^ [K3:code_integer,L2:code_integer]: (((((product_Pair @ code_integer) @ code_integer) @ (((divide_divide @ code_integer) @ ((abs_abs @ code_integer) @ K3)) @ ((abs_abs @ code_integer) @ L2))) @ (((modulo_modulo @ code_integer) @ ((abs_abs @ code_integer) @ K3)) @ ((abs_abs @ code_integer) @ L2))))))). % 6.41/2.69 thf(fact_3980_nat__of__integer__non__positive, axiom, (! [K:code_integer]: (((((ord_less_eq @ code_integer) @ K) @ (zero_zero @ code_integer)) => ((code_nat_of_integer @ K) = (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_3981_nat__of__integer__code__post_I1_J, axiom, ((code_nat_of_integer @ (zero_zero @ code_integer)) = (zero_zero @ nat))). % 6.41/2.69 thf(fact_3982_divmod__abs__code_I6_J, axiom, (! [J:code_integer]: ((((code_divmod_abs @ (zero_zero @ code_integer)) @ J) = ((((product_Pair @ code_integer) @ code_integer) @ (zero_zero @ code_integer)) @ (zero_zero @ code_integer)))))). % 6.41/2.69 thf(fact_3983_divmod__abs__code_I5_J, axiom, (! [J:code_integer]: ((((code_divmod_abs @ J) @ (zero_zero @ code_integer)) = ((((product_Pair @ code_integer) @ code_integer) @ (zero_zero @ code_integer)) @ ((abs_abs @ code_integer) @ J)))))). % 6.41/2.69 thf(fact_3984_bit__cut__integer__code, axiom, (code_bit_cut_integer = (^ [K3:code_integer]: (((((if @ ((product_prod @ code_integer) @ $o)) @ (K3 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ $o) @ (zero_zero @ code_integer)) @ $false)) @ (((((product_case_prod @ code_integer) @ code_integer) @ ((product_prod @ code_integer) @ $o)) @ (^ [R5:code_integer,S8:code_integer]: (((((product_Pair @ code_integer) @ $o) @ ((((if @ code_integer) @ (((ord_less @ code_integer) @ (zero_zero @ code_integer)) @ K3)) @ R5) @ (((minus_minus @ code_integer) @ ((uminus_uminus @ code_integer) @ R5)) @ S8))) @ (S8 = (one_one @ code_integer)))))) @ ((code_divmod_abs @ K3) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))))))))). % 6.41/2.69 thf(fact_3985_divmod__integer__code, axiom, (code_divmod_integer = (^ [K3:code_integer,L2:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (K3 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (zero_zero @ code_integer)) @ (zero_zero @ code_integer))) @ ((((if @ ((product_prod @ code_integer) @ code_integer)) @ (((ord_less @ code_integer) @ (zero_zero @ code_integer)) @ L2)) @ ((((if @ ((product_prod @ code_integer) @ code_integer)) @ (((ord_less @ code_integer) @ (zero_zero @ code_integer)) @ K3)) @ ((code_divmod_abs @ K3) @ L2)) @ (((((product_case_prod @ code_integer) @ code_integer) @ ((product_prod @ code_integer) @ code_integer)) @ (^ [R5:code_integer,S8:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (S8 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ ((uminus_uminus @ code_integer) @ R5)) @ (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (((minus_minus @ code_integer) @ ((uminus_uminus @ code_integer) @ R5)) @ (one_one @ code_integer))) @ (((minus_minus @ code_integer) @ L2) @ S8)))))) @ ((code_divmod_abs @ K3) @ L2)))) @ ((((if @ ((product_prod @ code_integer) @ code_integer)) @ (L2 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (zero_zero @ code_integer)) @ K3)) @ (((((product_apsnd @ code_integer) @ code_integer) @ code_integer) @ (uminus_uminus @ code_integer)) @ ((((if @ ((product_prod @ code_integer) @ code_integer)) @ (((ord_less @ code_integer) @ K3) @ (zero_zero @ code_integer))) @ ((code_divmod_abs @ K3) @ L2)) @ (((((product_case_prod @ code_integer) @ code_integer) @ ((product_prod @ code_integer) @ code_integer)) @ (^ [R5:code_integer,S8:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (S8 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ ((uminus_uminus @ code_integer) @ R5)) @ (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (((minus_minus @ code_integer) @ ((uminus_uminus @ code_integer) @ R5)) @ (one_one @ code_integer))) @ (((minus_minus @ code_integer) @ ((uminus_uminus @ code_integer) @ L2)) @ S8)))))) @ ((code_divmod_abs @ K3) @ L2))))))))))). % 6.41/2.69 thf(fact_3986_card__Pow, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (((finite_card @ (set @ A)) @ ((pow2 @ A) @ A3)) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((finite_card @ A) @ A3))))))). % 6.41/2.69 thf(fact_3987_rec__nat__add__eq__if, axiom, (! [A:$tType,A2:A,F2:(nat > (A > A)),V2:num,N2:nat]: ((((((rec_nat @ A) @ A2) @ F2) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ V2)) @ N2)) = ((F2 @ (((plus_plus @ nat) @ (pred_numeral @ V2)) @ N2)) @ ((((rec_nat @ A) @ A2) @ F2) @ (((plus_plus @ nat) @ (pred_numeral @ V2)) @ N2))))))). % 6.41/2.69 thf(fact_3988_case__nat__add__eq__if, axiom, (! [A:$tType,A2:A,F2:(nat > A),V2:num,N2:nat]: ((((((case_nat @ A) @ A2) @ F2) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ V2)) @ N2)) = (F2 @ (((plus_plus @ nat) @ (pred_numeral @ V2)) @ N2)))))). % 6.41/2.69 thf(fact_3989_Pow__empty, axiom, (! [A:$tType]: ((((pow2 @ A) @ (bot_bot @ (set @ A))) = (((insert @ (set @ A)) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ (set @ A)))))))). % 6.41/2.69 thf(fact_3990_Pow__singleton__iff, axiom, (! [A:$tType,X8:(set @ A),Y7:(set @ A)]: (((((pow2 @ A) @ X8) = (((insert @ (set @ A)) @ Y7) @ (bot_bot @ (set @ (set @ A))))) = ((X8 = (bot_bot @ (set @ A))) & (Y7 = (bot_bot @ (set @ A)))))))). % 6.41/2.69 thf(fact_3991_PowI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((member @ (set @ A)) @ A3) @ ((pow2 @ A) @ B3)))))). % 6.41/2.69 thf(fact_3992_Pow__iff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((member @ (set @ A)) @ A3) @ ((pow2 @ A) @ B3)) = (((ord_less_eq @ (set @ A)) @ A3) @ B3))))). % 6.41/2.69 thf(fact_3993_apsnd__conv, axiom, (! [A:$tType,B:$tType,C:$tType,F2:(C > B),X3:A,Y:C]: (((((((product_apsnd @ C) @ B) @ A) @ F2) @ ((((product_Pair @ A) @ C) @ X3) @ Y)) = ((((product_Pair @ A) @ B) @ X3) @ (F2 @ Y)))))). % 6.41/2.69 thf(fact_3994_old_Onat_Osimps_I7_J, axiom, (! [T:$tType,F1:T,F22:(nat > (T > T)),Nat:nat]: ((((((rec_nat @ T) @ F1) @ F22) @ (suc @ Nat)) = ((F22 @ Nat) @ ((((rec_nat @ T) @ F1) @ F22) @ Nat)))))). % 6.41/2.69 thf(fact_3995_old_Onat_Osimps_I6_J, axiom, (! [T:$tType,F1:T,F22:(nat > (T > T))]: ((((((rec_nat @ T) @ F1) @ F22) @ (zero_zero @ nat)) = F1)))). % 6.41/2.69 thf(fact_3996_finite__Pow__iff, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ (set @ A)) @ ((pow2 @ A) @ A3)) = ((finite_finite2 @ A) @ A3))))). % 6.41/2.69 thf(fact_3997_Pow__bottom, axiom, (! [A:$tType,B3:(set @ A)]: ((((member @ (set @ A)) @ (bot_bot @ (set @ A))) @ ((pow2 @ A) @ B3))))). % 6.41/2.69 thf(fact_3998_nat_Ocase__distrib, axiom, (! [B:$tType,A:$tType,H2:(A > B),F1:A,F22:(nat > A),Nat:nat]: (((H2 @ ((((case_nat @ A) @ F1) @ F22) @ Nat)) = ((((case_nat @ B) @ (H2 @ F1)) @ (^ [X:nat]: ((H2 @ (F22 @ X))))) @ Nat))))). % 6.41/2.69 thf(fact_3999_Pow__not__empty, axiom, (! [A:$tType,A3:(set @ A)]: ((((pow2 @ A) @ A3) != (bot_bot @ (set @ (set @ A))))))). % 6.41/2.69 thf(fact_4000_Pow__top, axiom, (! [A:$tType,A3:(set @ A)]: ((((member @ (set @ A)) @ A3) @ ((pow2 @ A) @ A3))))). % 6.41/2.69 thf(fact_4001_Pow__mono, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ (set @ (set @ A))) @ ((pow2 @ A) @ A3)) @ ((pow2 @ A) @ B3)))))). % 6.41/2.69 thf(fact_4002_PowD, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((member @ (set @ A)) @ A3) @ ((pow2 @ A) @ B3)) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))). % 6.41/2.69 thf(fact_4003_Pow__def, axiom, (! [A:$tType]: (((pow2 @ A) = (^ [A6:(set @ A)]: (((collect @ (set @ A)) @ (^ [B6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B6) @ A6)))))))))). % 6.41/2.69 thf(fact_4004_Pow__Compl, axiom, (! [A:$tType,A3:(set @ A)]: ((((pow2 @ A) @ ((uminus_uminus @ (set @ A)) @ A3)) = ((collect @ (set @ A)) @ (^ [Uu3:(set @ A)]: ((? [B6:(set @ A)]: (((Uu3 = ((uminus_uminus @ (set @ A)) @ B6)) & (((member @ (set @ A)) @ A3) @ ((pow2 @ A) @ B6)))))))))))). % 6.41/2.69 thf(fact_4005_old_Onat_Osimps_I5_J, axiom, (! [A:$tType,F1:A,F22:(nat > A),X22:nat]: ((((((case_nat @ A) @ F1) @ F22) @ (suc @ X22)) = (F22 @ X22))))). % 6.41/2.69 thf(fact_4006_old_Onat_Osimps_I4_J, axiom, (! [A:$tType,F1:A,F22:(nat > A)]: ((((((case_nat @ A) @ F1) @ F22) @ (zero_zero @ nat)) = F1)))). % 6.41/2.69 thf(fact_4007_nat_Odisc__eq__case_I2_J, axiom, (! [Nat:nat]: (((Nat != (zero_zero @ nat)) = ((((case_nat @ $o) @ $false) @ (^ [Uu3:nat]: ($true))) @ Nat))))). % 6.41/2.69 thf(fact_4008_nat_Odisc__eq__case_I1_J, axiom, (! [Nat:nat]: (((Nat = (zero_zero @ nat)) = ((((case_nat @ $o) @ $true) @ (^ [Uu3:nat]: ($false))) @ Nat))))). % 6.41/2.69 thf(fact_4009_less__eq__nat_Osimps_I2_J, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N2) = ((((case_nat @ $o) @ $false) @ ((ord_less_eq @ nat) @ M)) @ N2))))). % 6.41/2.69 thf(fact_4010_diff__Suc, axiom, (! [M:nat,N2:nat]: (((((minus_minus @ nat) @ M) @ (suc @ N2)) = ((((case_nat @ nat) @ (zero_zero @ nat)) @ (^ [K3:nat]: (K3))) @ (((minus_minus @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_4011_Nitpick_Ocase__nat__unfold, axiom, (! [A:$tType]: (((case_nat @ A) = (^ [X:A,F3:(nat > A),N:nat]: (((((if @ A) @ (N = (zero_zero @ nat))) @ X) @ (F3 @ (((minus_minus @ nat) @ N) @ (one_one @ nat)))))))))). % 6.41/2.69 thf(fact_4012_binomial__def, axiom, (binomial = (^ [N:nat,K3:nat]: (((finite_card @ (set @ nat)) @ ((collect @ (set @ nat)) @ (^ [K6:(set @ nat)]: (((((member @ (set @ nat)) @ K6) @ ((pow2 @ nat) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))) & (((finite_card @ nat) @ K6) = K3)))))))))). % 6.41/2.69 thf(fact_4013_old_Orec__nat__def, axiom, (! [T:$tType]: (((rec_nat @ T) = (^ [F12:T,F23:(nat > (T > T)),X:nat]: (((the @ T) @ ((((rec_set_nat @ T) @ F12) @ F23) @ X)))))))). % 6.41/2.69 thf(fact_4014_rec__nat__0__imp, axiom, (! [A:$tType,F2:(nat > A),F1:A,F22:(nat > (A > A))]: (((F2 = (((rec_nat @ A) @ F1) @ F22)) => ((F2 @ (zero_zero @ nat)) = F1))))). % 6.41/2.69 thf(fact_4015_rec__nat__Suc__imp, axiom, (! [A:$tType,F2:(nat > A),F1:A,F22:(nat > (A > A)),N2:nat]: (((F2 = (((rec_nat @ A) @ F1) @ F22)) => ((F2 @ (suc @ N2)) = ((F22 @ N2) @ (F2 @ N2))))))). % 6.41/2.69 thf(fact_4016_nat_Osplit__sels_I1_J, axiom, (! [A:$tType,P:(A > $o),F1:A,F22:(nat > A),Nat:nat]: (((P @ ((((case_nat @ A) @ F1) @ F22) @ Nat)) = (((Nat = (zero_zero @ nat)) => (P @ F1)) & ((Nat = (suc @ (pred @ Nat))) => (P @ (F22 @ (pred @ Nat))))))))). % 6.41/2.69 thf(fact_4017_pred__def, axiom, (pred = (((case_nat @ nat) @ (zero_zero @ nat)) @ (^ [X24:nat]: (X24))))). % 6.41/2.69 thf(fact_4018_subset__Collect__iff, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B3) @ A3) => ((((ord_less_eq @ (set @ A)) @ B3) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A3) & (P @ X)))))) = (! [X:A]: (((((member @ A) @ X) @ B3) => (P @ X))))))))). % 6.41/2.69 thf(fact_4019_subset__CollectI, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A),Q:(A > $o),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ B3) @ A3) => ((! [X5:A]: (((((member @ A) @ X5) @ B3) => ((Q @ X5) => (P @ X5))))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ B3) & (Q @ X)))))) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A3) & (P @ X))))))))))). % 6.41/2.69 thf(fact_4020_nat_Osplit__sels_I2_J, axiom, (! [A:$tType,P:(A > $o),F1:A,F22:(nat > A),Nat:nat]: (((P @ ((((case_nat @ A) @ F1) @ F22) @ Nat)) = (~ ((((Nat = (zero_zero @ nat)) & (~ ((P @ F1)))) | ((Nat = (suc @ (pred @ Nat))) & (~ ((P @ (F22 @ (pred @ Nat))))))))))))). % 6.41/2.69 thf(fact_4021_signed__take__bit__code, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((bit_ri4674362597316999326ke_bit @ A) = (^ [N:nat,A4:A]: (((((if @ A) @ (((bit_se5641148757651400278ts_bit @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ (suc @ N)) @ A4)) @ N)) @ (((plus_plus @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ (suc @ N)) @ A4)) @ (((bit_se4730199178511100633sh_bit @ A) @ (suc @ N)) @ ((uminus_uminus @ A) @ (one_one @ A))))) @ (((bit_se2584673776208193580ke_bit @ A) @ (suc @ N)) @ A4))))))))). % 6.41/2.69 thf(fact_4022_wmin__insertI, axiom, (! [X3:((product_prod @ nat) @ nat),XS:(set @ ((product_prod @ nat) @ nat)),Y:((product_prod @ nat) @ nat),YS:(set @ ((product_prod @ nat) @ nat))]: (((((member @ ((product_prod @ nat) @ nat)) @ X3) @ XS) => ((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ X3) @ Y)) @ fun_pair_leq) => ((((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ XS) @ YS)) @ fun_min_weak) => (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ XS) @ (((insert @ ((product_prod @ nat) @ nat)) @ Y) @ YS))) @ fun_min_weak))))))). % 6.41/2.69 thf(fact_4023_wmax__insertI, axiom, (! [Y:((product_prod @ nat) @ nat),YS:(set @ ((product_prod @ nat) @ nat)),X3:((product_prod @ nat) @ nat),XS:(set @ ((product_prod @ nat) @ nat))]: (((((member @ ((product_prod @ nat) @ nat)) @ Y) @ YS) => ((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ X3) @ Y)) @ fun_pair_leq) => ((((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ XS) @ YS)) @ fun_max_weak) => (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ (((insert @ ((product_prod @ nat) @ nat)) @ X3) @ XS)) @ YS)) @ fun_max_weak))))))). % 6.41/2.69 thf(fact_4024_bezw__0, axiom, (! [X3:nat]: ((((bezw @ X3) @ (zero_zero @ nat)) = ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (zero_zero @ int)))))). % 6.41/2.69 thf(fact_4025_push__bit__negative__int__iff, axiom, (! [N2:nat,K:int]: (((((ord_less @ int) @ (((bit_se4730199178511100633sh_bit @ int) @ N2) @ K)) @ (zero_zero @ int)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))). % 6.41/2.69 thf(fact_4026_push__bit__eq__0__iff, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N2:nat,A2:A]: ((((((bit_se4730199178511100633sh_bit @ A) @ N2) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_4027_push__bit__of__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat]: (((((bit_se4730199178511100633sh_bit @ A) @ N2) @ (zero_zero @ A)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_4028_push__bit__push__bit, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,N2:nat,A2:A]: (((((bit_se4730199178511100633sh_bit @ A) @ M) @ (((bit_se4730199178511100633sh_bit @ A) @ N2) @ A2)) = (((bit_se4730199178511100633sh_bit @ A) @ (((plus_plus @ nat) @ M) @ N2)) @ A2)))))))). % 6.41/2.69 thf(fact_4029_push__bit__Suc__numeral, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,K:num]: (((((bit_se4730199178511100633sh_bit @ A) @ (suc @ N2)) @ ((numeral_numeral @ A) @ K)) = (((bit_se4730199178511100633sh_bit @ A) @ N2) @ ((numeral_numeral @ A) @ (bit0 @ K)))))))))). % 6.41/2.69 thf(fact_4030_push__bit__Suc__minus__numeral, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N2:nat,K:num]: (((((bit_se4730199178511100633sh_bit @ A) @ (suc @ N2)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ K))) = (((bit_se4730199178511100633sh_bit @ A) @ N2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit0 @ K))))))))))). % 6.41/2.69 thf(fact_4031_push__bit__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,A2:A]: (((((bit_se4730199178511100633sh_bit @ A) @ (suc @ N2)) @ A2) = (((bit_se4730199178511100633sh_bit @ A) @ N2) @ (((times_times @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_4032_push__bit__of__Suc__0, axiom, (! [N2:nat]: (((((bit_se4730199178511100633sh_bit @ nat) @ N2) @ (suc @ (zero_zero @ nat))) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))))). % 6.41/2.69 thf(fact_4033_even__push__bit__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,A2:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se4730199178511100633sh_bit @ A) @ N2) @ A2)) = ((N2 != (zero_zero @ nat)) | (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2))))))))). % 6.41/2.69 thf(fact_4034_wmin__emptyI, axiom, (! [X8:(set @ ((product_prod @ nat) @ nat))]: ((((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ X8) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat))))) @ fun_min_weak)))). % 6.41/2.69 thf(fact_4035_wmax__emptyI, axiom, (! [X8:(set @ ((product_prod @ nat) @ nat))]: ((((finite_finite2 @ ((product_prod @ nat) @ nat)) @ X8) => (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat)))) @ X8)) @ fun_max_weak))))). % 6.41/2.69 thf(fact_4036_push__bit__add, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,A2:A,B2:A]: (((((bit_se4730199178511100633sh_bit @ A) @ N2) @ (((plus_plus @ A) @ A2) @ B2)) = (((plus_plus @ A) @ (((bit_se4730199178511100633sh_bit @ A) @ N2) @ A2)) @ (((bit_se4730199178511100633sh_bit @ A) @ N2) @ B2))))))))). % 6.41/2.69 thf(fact_4037_push__bit__take__bit, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,N2:nat,A2:A]: (((((bit_se4730199178511100633sh_bit @ A) @ M) @ (((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2)) = (((bit_se2584673776208193580ke_bit @ A) @ (((plus_plus @ nat) @ M) @ N2)) @ (((bit_se4730199178511100633sh_bit @ A) @ M) @ A2))))))))). % 6.41/2.69 thf(fact_4038_bit__push__bit__iff__int, axiom, (! [M:nat,K:int,N2:nat]: (((((bit_se5641148757651400278ts_bit @ int) @ (((bit_se4730199178511100633sh_bit @ int) @ M) @ K)) @ N2) = ((((ord_less_eq @ nat) @ M) @ N2) & (((bit_se5641148757651400278ts_bit @ int) @ K) @ (((minus_minus @ nat) @ N2) @ M))))))). % 6.41/2.69 thf(fact_4039_bit__push__bit__iff__nat, axiom, (! [M:nat,Q4:nat,N2:nat]: (((((bit_se5641148757651400278ts_bit @ nat) @ (((bit_se4730199178511100633sh_bit @ nat) @ M) @ Q4)) @ N2) = ((((ord_less_eq @ nat) @ M) @ N2) & (((bit_se5641148757651400278ts_bit @ nat) @ Q4) @ (((minus_minus @ nat) @ N2) @ M))))))). % 6.41/2.69 thf(fact_4040_bit__iff__and__push__bit__not__eq__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se5641148757651400278ts_bit @ A) = (^ [A4:A,N:nat]: (((((bit_se5824344872417868541ns_and @ A) @ A4) @ (((bit_se4730199178511100633sh_bit @ A) @ N) @ (one_one @ A))) != (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_4041_take__bit__sum, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => ((bit_se2584673776208193580ke_bit @ A) = (^ [N:nat,A4:A]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K3:nat]: ((((bit_se4730199178511100633sh_bit @ A) @ K3) @ ((zero_neq_one_of_bool @ A) @ (((bit_se5641148757651400278ts_bit @ A) @ A4) @ K3)))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))))))))). % 6.41/2.69 thf(fact_4042_smax__insertI, axiom, (! [Y:((product_prod @ nat) @ nat),Y7:(set @ ((product_prod @ nat) @ nat)),X3:((product_prod @ nat) @ nat),X8:(set @ ((product_prod @ nat) @ nat))]: (((((member @ ((product_prod @ nat) @ nat)) @ Y) @ Y7) => ((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ X3) @ Y)) @ fun_pair_less) => ((((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ X8) @ Y7)) @ fun_max_strict) => (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ (((insert @ ((product_prod @ nat) @ nat)) @ X3) @ X8)) @ Y7)) @ fun_max_strict))))))). % 6.41/2.69 thf(fact_4043_smin__insertI, axiom, (! [X3:((product_prod @ nat) @ nat),XS:(set @ ((product_prod @ nat) @ nat)),Y:((product_prod @ nat) @ nat),YS:(set @ ((product_prod @ nat) @ nat))]: (((((member @ ((product_prod @ nat) @ nat)) @ X3) @ XS) => ((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ X3) @ Y)) @ fun_pair_less) => ((((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ XS) @ YS)) @ fun_min_strict) => (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ XS) @ (((insert @ ((product_prod @ nat) @ nat)) @ Y) @ YS))) @ fun_min_strict))))))). % 6.41/2.69 thf(fact_4044_prod__decode__aux_Osimps, axiom, (nat_prod_decode_aux = (^ [K3:nat,M2:nat]: (((((if @ ((product_prod @ nat) @ nat)) @ (((ord_less_eq @ nat) @ M2) @ K3)) @ ((((product_Pair @ nat) @ nat) @ M2) @ (((minus_minus @ nat) @ K3) @ M2))) @ ((nat_prod_decode_aux @ (suc @ K3)) @ (((minus_minus @ nat) @ M2) @ (suc @ K3)))))))). % 6.41/2.69 thf(fact_4045_prod__decode__aux_Oelims, axiom, (! [X3:nat,Xa2:nat,Y:((product_prod @ nat) @ nat)]: (((((nat_prod_decode_aux @ X3) @ Xa2) = Y) => (((((ord_less_eq @ nat) @ Xa2) @ X3) => (Y = ((((product_Pair @ nat) @ nat) @ Xa2) @ (((minus_minus @ nat) @ X3) @ Xa2)))) & ((~ ((((ord_less_eq @ nat) @ Xa2) @ X3))) => (Y = ((nat_prod_decode_aux @ (suc @ X3)) @ (((minus_minus @ nat) @ Xa2) @ (suc @ X3)))))))))). % 6.41/2.69 thf(fact_4046_smax__emptyI, axiom, (! [Y7:(set @ ((product_prod @ nat) @ nat))]: ((((finite_finite2 @ ((product_prod @ nat) @ nat)) @ Y7) => ((Y7 != (bot_bot @ (set @ ((product_prod @ nat) @ nat)))) => (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat)))) @ Y7)) @ fun_max_strict)))))). % 6.41/2.69 thf(fact_4047_smin__emptyI, axiom, (! [X8:(set @ ((product_prod @ nat) @ nat))]: (((X8 != (bot_bot @ (set @ ((product_prod @ nat) @ nat)))) => (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ X8) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat))))) @ fun_min_strict))))). % 6.41/2.69 thf(fact_4048_Suc__0__div__numeral, axiom, (! [K:num]: (((((divide_divide @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ K)) = (((product_fst @ nat) @ nat) @ (((unique8689654367752047608divmod @ nat) @ one2) @ K)))))). % 6.41/2.69 thf(fact_4049_Suc__0__mod__numeral, axiom, (! [K:num]: (((((modulo_modulo @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ K)) = (((product_snd @ nat) @ nat) @ (((unique8689654367752047608divmod @ nat) @ one2) @ K)))))). % 6.41/2.69 thf(fact_4050_finite__enumerate, axiom, (! [S3:(set @ nat)]: ((((finite_finite2 @ nat) @ S3) => (? [R4:(nat > nat)]: ((((((strict_mono_on @ nat) @ nat) @ R4) @ ((set_ord_lessThan @ nat) @ ((finite_card @ nat) @ S3))) & (! [N6:nat]: (((((ord_less @ nat) @ N6) @ ((finite_card @ nat) @ S3)) => (((member @ nat) @ (R4 @ N6)) @ S3))))))))))). % 6.41/2.69 thf(fact_4051_card__UNION, axiom, (! [A:$tType,A3:(set @ (set @ A))]: ((((finite_finite2 @ (set @ A)) @ A3) => ((! [X5:(set @ A)]: (((((member @ (set @ A)) @ X5) @ A3) => ((finite_finite2 @ A) @ X5)))) => (((finite_card @ A) @ ((complete_Sup_Sup @ (set @ A)) @ A3)) = (nat2 @ ((((groups7311177749621191930dd_sum @ (set @ (set @ A))) @ int) @ (^ [I8:(set @ (set @ A))]: ((((times_times @ int) @ (((power_power @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (((plus_plus @ nat) @ ((finite_card @ (set @ A)) @ I8)) @ (one_one @ nat)))) @ ((semiring_1_of_nat @ int) @ ((finite_card @ A) @ ((complete_Inf_Inf @ (set @ A)) @ I8))))))) @ ((collect @ (set @ (set @ A))) @ (^ [I8:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ I8) @ A3) & (I8 != (bot_bot @ (set @ (set @ A)))))))))))))))). % 6.41/2.69 thf(fact_4052_finite__Inter, axiom, (! [A:$tType,M7:(set @ (set @ A))]: (((? [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ M7) & ((finite_finite2 @ A) @ X2)))) => ((finite_finite2 @ A) @ ((complete_Inf_Inf @ (set @ A)) @ M7)))))). % 6.41/2.69 thf(fact_4053_fst__apsnd, axiom, (! [B:$tType,C:$tType,A:$tType,F2:(C > B),X3:((product_prod @ A) @ C)]: (((((product_fst @ A) @ B) @ (((((product_apsnd @ C) @ B) @ A) @ F2) @ X3)) = (((product_fst @ A) @ C) @ X3))))). % 6.41/2.69 thf(fact_4054_apsnd__eq__conv, axiom, (! [B:$tType,C:$tType,A:$tType,F2:(C > B),X3:((product_prod @ A) @ C),G:(C > B)]: ((((((((product_apsnd @ C) @ B) @ A) @ F2) @ X3) = (((((product_apsnd @ C) @ B) @ A) @ G) @ X3)) = ((F2 @ (((product_snd @ A) @ C) @ X3)) = (G @ (((product_snd @ A) @ C) @ X3))))))). % 6.41/2.69 thf(fact_4055_snd__apsnd, axiom, (! [A:$tType,C:$tType,B:$tType,F2:(C > A),X3:((product_prod @ B) @ C)]: (((((product_snd @ B) @ A) @ (((((product_apsnd @ C) @ A) @ B) @ F2) @ X3)) = (F2 @ (((product_snd @ B) @ C) @ X3)))))). % 6.41/2.69 thf(fact_4056_cSup__atLeastAtMost, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [Y:A,X3:A]: (((((ord_less_eq @ A) @ Y) @ X3) => (((complete_Sup_Sup @ A) @ (((set_or1337092689740270186AtMost @ A) @ Y) @ X3)) = X3)))))))). % 6.41/2.69 thf(fact_4057_Sup__atLeastAtMost, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => (((complete_Sup_Sup @ A) @ (((set_or1337092689740270186AtMost @ A) @ X3) @ Y)) = Y)))))))). % 6.41/2.69 thf(fact_4058_cSup__singleton, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X3:A]: ((((complete_Sup_Sup @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = X3))))))). % 6.41/2.69 thf(fact_4059_cInf__atLeastAtMost, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [Y:A,X3:A]: (((((ord_less_eq @ A) @ Y) @ X3) => (((complete_Inf_Inf @ A) @ (((set_or1337092689740270186AtMost @ A) @ Y) @ X3)) = Y)))))))). % 6.41/2.69 thf(fact_4060_Inf__atLeastAtMost, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => (((complete_Inf_Inf @ A) @ (((set_or1337092689740270186AtMost @ A) @ X3) @ Y)) = X3)))))))). % 6.41/2.69 thf(fact_4061_cInf__singleton, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X3:A]: ((((complete_Inf_Inf @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = X3))))))). % 6.41/2.69 thf(fact_4062_Sup__atLeastLessThan, axiom, (! [A:$tType]: ((((comple6319245703460814977attice @ A) & (dense_linorder @ A)) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (((complete_Sup_Sup @ A) @ (((set_or7035219750837199246ssThan @ A) @ X3) @ Y)) = Y)))))))). % 6.41/2.69 thf(fact_4063_cSup__atLeastLessThan, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (dense_linorder @ A)) => (! [Y:A,X3:A]: (((((ord_less @ A) @ Y) @ X3) => (((complete_Sup_Sup @ A) @ (((set_or7035219750837199246ssThan @ A) @ Y) @ X3)) = X3)))))))). % 6.41/2.69 thf(fact_4064_Inf__atLeastLessThan, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (((complete_Inf_Inf @ A) @ (((set_or7035219750837199246ssThan @ A) @ X3) @ Y)) = X3)))))))). % 6.41/2.69 thf(fact_4065_cInf__atLeastLessThan, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [Y:A,X3:A]: (((((ord_less @ A) @ Y) @ X3) => (((complete_Inf_Inf @ A) @ (((set_or7035219750837199246ssThan @ A) @ Y) @ X3)) = Y)))))))). % 6.41/2.69 thf(fact_4066_Inf__atMost, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X3:A]: ((((complete_Inf_Inf @ A) @ ((set_ord_atMost @ A) @ X3)) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4067_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))). % 6.41/2.69 thf(fact_4068_finite__Union, axiom, (! [A:$tType,A3:(set @ (set @ A))]: ((((finite_finite2 @ (set @ A)) @ A3) => ((! [M8:(set @ A)]: (((((member @ (set @ A)) @ M8) @ A3) => ((finite_finite2 @ A) @ M8)))) => ((finite_finite2 @ A) @ ((complete_Sup_Sup @ (set @ A)) @ A3))))))). % 6.41/2.69 thf(fact_4069_split__comp__eq, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,F2:(A > (B > C)),G:(D > A)]: (((^ [U2:((product_prod @ D) @ B)]: (((F2 @ (G @ (((product_fst @ D) @ B) @ U2))) @ (((product_snd @ D) @ B) @ U2)))) = ((((product_case_prod @ D) @ B) @ C) @ (^ [X:D]: ((F2 @ (G @ X))))))))). % 6.41/2.69 thf(fact_4070_case__prod__beta_H, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [F3:(A > (B > C)),X:((product_prod @ A) @ B)]: (((F3 @ (((product_fst @ A) @ B) @ X)) @ (((product_snd @ A) @ B) @ X)))))))). % 6.41/2.69 thf(fact_4071_case__prod__unfold, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [C4:(A > (B > C)),P6:((product_prod @ A) @ B)]: (((C4 @ (((product_fst @ A) @ B) @ P6)) @ (((product_snd @ A) @ B) @ P6)))))))). % 6.41/2.69 thf(fact_4072_The__case__prod, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o))]: ((((the @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ P)) = ((the @ ((product_prod @ A) @ B)) @ (^ [Xy:((product_prod @ A) @ B)]: (((P @ (((product_fst @ A) @ B) @ Xy)) @ (((product_snd @ A) @ B) @ Xy))))))))). % 6.41/2.69 thf(fact_4073_cSup__eq__maximum, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [Z2:A,X8:(set @ A)]: (((((member @ A) @ Z2) @ X8) => ((! [X5:A]: (((((member @ A) @ X5) @ X8) => (((ord_less_eq @ A) @ X5) @ Z2)))) => (((complete_Sup_Sup @ A) @ X8) = Z2))))))))). % 6.41/2.69 thf(fact_4074_cSup__eq, axiom, (! [A:$tType]: ((((condit1219197933456340205attice @ A) & (no_bot @ A)) => (! [X8:(set @ A),A2:A]: (((! [X5:A]: (((((member @ A) @ X5) @ X8) => (((ord_less_eq @ A) @ X5) @ A2)))) => ((! [Y4:A]: (((! [X2:A]: (((((member @ A) @ X2) @ X8) => (((ord_less_eq @ A) @ X2) @ Y4)))) => (((ord_less_eq @ A) @ A2) @ Y4)))) => (((complete_Sup_Sup @ A) @ X8) = A2))))))))). % 6.41/2.69 thf(fact_4075_cInf__eq__minimum, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [Z2:A,X8:(set @ A)]: (((((member @ A) @ Z2) @ X8) => ((! [X5:A]: (((((member @ A) @ X5) @ X8) => (((ord_less_eq @ A) @ Z2) @ X5)))) => (((complete_Inf_Inf @ A) @ X8) = Z2))))))))). % 6.41/2.69 thf(fact_4076_cInf__eq, axiom, (! [A:$tType]: ((((condit1219197933456340205attice @ A) & (no_top @ A)) => (! [X8:(set @ A),A2:A]: (((! [X5:A]: (((((member @ A) @ X5) @ X8) => (((ord_less_eq @ A) @ A2) @ X5)))) => ((! [Y4:A]: (((! [X2:A]: (((((member @ A) @ X2) @ X8) => (((ord_less_eq @ A) @ Y4) @ X2)))) => (((ord_less_eq @ A) @ Y4) @ A2)))) => (((complete_Inf_Inf @ A) @ X8) = A2))))))))). % 6.41/2.69 thf(fact_4077_snd__def, axiom, (! [B:$tType,A:$tType]: ((((product_snd @ A) @ B) = ((((product_case_prod @ A) @ B) @ B) @ (^ [X15:A,X24:B]: (X24))))))). % 6.41/2.69 thf(fact_4078_fst__def, axiom, (! [B:$tType,A:$tType]: ((((product_fst @ A) @ B) = ((((product_case_prod @ A) @ B) @ A) @ (^ [X15:A,X24:B]: (X15))))))). % 6.41/2.69 thf(fact_4079_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y3:((product_prod @ A) @ B),Z:((product_prod @ A) @ B)]: ((Y3 = Z))) = (^ [S8:((product_prod @ A) @ B),T3:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S8) = (((product_fst @ A) @ B) @ T3)) & ((((product_snd @ A) @ B) @ S8) = (((product_snd @ A) @ B) @ T3))))))))). % 6.41/2.69 thf(fact_4080_prod__eqI, axiom, (! [B:$tType,A:$tType,P4:((product_prod @ A) @ B),Q4:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P4) = (((product_fst @ A) @ B) @ Q4)) => (((((product_snd @ A) @ B) @ P4) = (((product_snd @ A) @ B) @ Q4)) => (P4 = Q4)))))). % 6.41/2.69 thf(fact_4081_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod2:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod2)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod2))) => (Prod = Prod2))))). % 6.41/2.69 thf(fact_4082_split__beta, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [F3:(A > (B > C)),Prod3:((product_prod @ A) @ B)]: (((F3 @ (((product_fst @ A) @ B) @ Prod3)) @ (((product_snd @ A) @ B) @ Prod3)))))))). % 6.41/2.69 thf(fact_4083_case__prod__beta, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ A) = (^ [F3:(B > (C > A)),P6:((product_prod @ B) @ C)]: (((F3 @ (((product_fst @ B) @ C) @ P6)) @ (((product_snd @ B) @ C) @ P6)))))))). % 6.41/2.69 thf(fact_4084_Product__Type_OCollect__case__prodD, axiom, (! [B:$tType,A:$tType,X3:((product_prod @ A) @ B),A3:(A > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ X3) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A3))) => ((A3 @ (((product_fst @ A) @ B) @ X3)) @ (((product_snd @ A) @ B) @ X3)))))). % 6.41/2.69 thf(fact_4085_prod_Osplit__sel__asm, axiom, (! [C:$tType,B:$tType,A:$tType,P:(C > $o),F2:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P @ (((((product_case_prod @ A) @ B) @ C) @ F2) @ Prod)) = (~ (((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) & (~ ((P @ ((F2 @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))))))). % 6.41/2.69 thf(fact_4086_prod_Osplit__sel, axiom, (! [C:$tType,B:$tType,A:$tType,P:(C > $o),F2:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P @ (((((product_case_prod @ A) @ B) @ C) @ F2) @ Prod)) = ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) => (P @ ((F2 @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))). % 6.41/2.69 thf(fact_4087_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X22:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = X1)))). % 6.41/2.69 thf(fact_4088_fst__eqD, axiom, (! [B:$tType,A:$tType,X3:A,Y:B,A2:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) = A2) => (X3 = A2))))). % 6.41/2.69 thf(fact_4089_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))). % 6.41/2.69 thf(fact_4090_surjective__pairing, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: ((T2 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T2)) @ (((product_snd @ A) @ B) @ T2)))))). % 6.41/2.69 thf(fact_4091_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X22:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X22)) = X22)))). % 6.41/2.69 thf(fact_4092_snd__eqD, axiom, (! [B:$tType,A:$tType,X3:B,Y:A,A2:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X3) @ Y)) = A2) => (Y = A2))))). % 6.41/2.69 thf(fact_4093_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),X3:A,Y:B,A2:((product_prod @ A) @ B)]: ((((P @ X3) @ Y) => ((A2 = ((((product_Pair @ A) @ B) @ X3) @ Y)) => ((P @ (((product_fst @ A) @ B) @ A2)) @ (((product_snd @ A) @ B) @ A2))))))). % 6.41/2.69 thf(fact_4094_cInf__eq__non__empty, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),A2:A]: (((X8 != (bot_bot @ (set @ A))) => ((! [X5:A]: (((((member @ A) @ X5) @ X8) => (((ord_less_eq @ A) @ A2) @ X5)))) => ((! [Y4:A]: (((! [X2:A]: (((((member @ A) @ X2) @ X8) => (((ord_less_eq @ A) @ Y4) @ X2)))) => (((ord_less_eq @ A) @ Y4) @ A2)))) => (((complete_Inf_Inf @ A) @ X8) = A2)))))))))). % 6.41/2.69 thf(fact_4095_cInf__greatest, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),Z2:A]: (((X8 != (bot_bot @ (set @ A))) => ((! [X5:A]: (((((member @ A) @ X5) @ X8) => (((ord_less_eq @ A) @ Z2) @ X5)))) => (((ord_less_eq @ A) @ Z2) @ ((complete_Inf_Inf @ A) @ X8)))))))))). % 6.41/2.69 thf(fact_4096_cInf__le__finite, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),X3:A]: ((((finite_finite2 @ A) @ X8) => ((((member @ A) @ X3) @ X8) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ X8)) @ X3))))))))). % 6.41/2.69 thf(fact_4097_cInf__lessD, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),Z2:A]: (((X8 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ ((complete_Inf_Inf @ A) @ X8)) @ Z2) => (? [X5:A]: (((((member @ A) @ X5) @ X8) & (((ord_less @ A) @ X5) @ Z2)))))))))))). % 6.41/2.69 thf(fact_4098_finite__imp__less__Inf, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),X3:A,A2:A]: ((((finite_finite2 @ A) @ X8) => ((((member @ A) @ X3) @ X8) => ((! [X5:A]: (((((member @ A) @ X5) @ X8) => (((ord_less @ A) @ A2) @ X5)))) => (((ord_less @ A) @ A2) @ ((complete_Inf_Inf @ A) @ X8))))))))))). % 6.41/2.69 thf(fact_4099_cSup__eq__non__empty, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),A2:A]: (((X8 != (bot_bot @ (set @ A))) => ((! [X5:A]: (((((member @ A) @ X5) @ X8) => (((ord_less_eq @ A) @ X5) @ A2)))) => ((! [Y4:A]: (((! [X2:A]: (((((member @ A) @ X2) @ X8) => (((ord_less_eq @ A) @ X2) @ Y4)))) => (((ord_less_eq @ A) @ A2) @ Y4)))) => (((complete_Sup_Sup @ A) @ X8) = A2)))))))))). % 6.41/2.69 thf(fact_4100_cSup__least, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),Z2:A]: (((X8 != (bot_bot @ (set @ A))) => ((! [X5:A]: (((((member @ A) @ X5) @ X8) => (((ord_less_eq @ A) @ X5) @ Z2)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ X8)) @ Z2))))))))). % 6.41/2.69 thf(fact_4101_le__cSup__finite, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),X3:A]: ((((finite_finite2 @ A) @ X8) => ((((member @ A) @ X3) @ X8) => (((ord_less_eq @ A) @ X3) @ ((complete_Sup_Sup @ A) @ X8)))))))))). % 6.41/2.69 thf(fact_4102_update__zip, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys:(list @ B),I:nat,Xy2:((product_prod @ A) @ B)]: ((((((list_update @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys)) @ I) @ Xy2) = ((((zip @ A) @ B) @ ((((list_update @ A) @ Xs) @ I) @ (((product_fst @ A) @ B) @ Xy2))) @ ((((list_update @ B) @ Ys) @ I) @ (((product_snd @ A) @ B) @ Xy2))))))). % 6.41/2.69 thf(fact_4103_less__cSupE, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [Y:A,X8:(set @ A)]: (((((ord_less @ A) @ Y) @ ((complete_Sup_Sup @ A) @ X8)) => ((X8 != (bot_bot @ (set @ A))) => (~ ((! [X5:A]: (((((member @ A) @ X5) @ X8) => (~ ((((ord_less @ A) @ Y) @ X5)))))))))))))))). % 6.41/2.69 thf(fact_4104_less__cSupD, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),Z2:A]: (((X8 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ Z2) @ ((complete_Sup_Sup @ A) @ X8)) => (? [X5:A]: (((((member @ A) @ X5) @ X8) & (((ord_less @ A) @ Z2) @ X5)))))))))))). % 6.41/2.69 thf(fact_4105_finite__imp__Sup__less, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),X3:A,A2:A]: ((((finite_finite2 @ A) @ X8) => ((((member @ A) @ X3) @ X8) => ((! [X5:A]: (((((member @ A) @ X5) @ X8) => (((ord_less @ A) @ X5) @ A2)))) => (((ord_less @ A) @ ((complete_Sup_Sup @ A) @ X8)) @ A2)))))))))). % 6.41/2.69 thf(fact_4106_card__Union__le__sum__card, axiom, (! [A:$tType,U3:(set @ (set @ A))]: ((((ord_less_eq @ nat) @ ((finite_card @ A) @ ((complete_Sup_Sup @ (set @ A)) @ U3))) @ ((((groups7311177749621191930dd_sum @ (set @ A)) @ nat) @ (finite_card @ A)) @ U3))))). % 6.41/2.69 thf(fact_4107_finite__UnionD, axiom, (! [A:$tType,A3:(set @ (set @ A))]: ((((finite_finite2 @ A) @ ((complete_Sup_Sup @ (set @ A)) @ A3)) => ((finite_finite2 @ (set @ A)) @ A3))))). % 6.41/2.69 thf(fact_4108_in__set__zip, axiom, (! [B:$tType,A:$tType,P4:((product_prod @ A) @ B),Xs:(list @ A),Ys:(list @ B)]: (((((member @ ((product_prod @ A) @ B)) @ P4) @ ((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys))) = (? [N:nat]: ((((((nth @ A) @ Xs) @ N) = (((product_fst @ A) @ B) @ P4)) & (((((nth @ B) @ Ys) @ N) = (((product_snd @ A) @ B) @ P4)) & ((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) & (((ord_less @ nat) @ N) @ ((size_size @ (list @ B)) @ Ys))))))))))). % 6.41/2.69 thf(fact_4109_divides__aux__def, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => ((unique5940410009612947441es_aux @ A) = (^ [Qr:((product_prod @ A) @ A)]: (((((product_snd @ A) @ A) @ Qr) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_4110_finite__less__Inf__iff, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),A2:A]: ((((finite_finite2 @ A) @ X8) => ((X8 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ A2) @ ((complete_Inf_Inf @ A) @ X8)) = (! [X:A]: (((((member @ A) @ X) @ X8) => (((ord_less @ A) @ A2) @ X))))))))))))). % 6.41/2.69 thf(fact_4111_finite__Sup__less__iff, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),A2:A]: ((((finite_finite2 @ A) @ X8) => ((X8 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ ((complete_Sup_Sup @ A) @ X8)) @ A2) = (! [X:A]: (((((member @ A) @ X) @ X8) => (((ord_less @ A) @ X) @ A2))))))))))))). % 6.41/2.69 thf(fact_4112_cInf__abs__ge, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (linordered_idom @ A)) => (! [S3:(set @ A),A2:A]: (((S3 != (bot_bot @ (set @ A))) => ((! [X5:A]: (((((member @ A) @ X5) @ S3) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ X5)) @ A2)))) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ ((complete_Inf_Inf @ A) @ S3))) @ A2))))))))). % 6.41/2.69 thf(fact_4113_cSup__abs__le, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (linordered_idom @ A)) => (! [S3:(set @ A),A2:A]: (((S3 != (bot_bot @ (set @ A))) => ((! [X5:A]: (((((member @ A) @ X5) @ S3) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ X5)) @ A2)))) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ ((complete_Sup_Sup @ A) @ S3))) @ A2))))))))). % 6.41/2.69 thf(fact_4114_card__Union__le__sum__card__weak, axiom, (! [A:$tType,U3:(set @ (set @ A))]: (((! [X5:(set @ A)]: (((((member @ (set @ A)) @ X5) @ U3) => ((finite_finite2 @ A) @ X5)))) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ ((complete_Sup_Sup @ (set @ A)) @ U3))) @ ((((groups7311177749621191930dd_sum @ (set @ A)) @ nat) @ (finite_card @ A)) @ U3)))))). % 6.41/2.69 thf(fact_4115_in__set__enumerate__eq, axiom, (! [A:$tType,P4:((product_prod @ nat) @ A),N2:nat,Xs:(list @ A)]: (((((member @ ((product_prod @ nat) @ A)) @ P4) @ ((set2 @ ((product_prod @ nat) @ A)) @ (((enumerate @ A) @ N2) @ Xs))) = ((((ord_less_eq @ nat) @ N2) @ (((product_fst @ nat) @ A) @ P4)) & ((((ord_less @ nat) @ (((product_fst @ nat) @ A) @ P4)) @ (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N2)) & ((((nth @ A) @ Xs) @ (((minus_minus @ nat) @ (((product_fst @ nat) @ A) @ P4)) @ N2)) = (((product_snd @ nat) @ A) @ P4)))))))). % 6.41/2.69 thf(fact_4116_cInf__asclose, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (linordered_idom @ A)) => (! [S3:(set @ A),L:A,E3:A]: (((S3 != (bot_bot @ (set @ A))) => ((! [X5:A]: (((((member @ A) @ X5) @ S3) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ X5) @ L))) @ E3)))) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ ((complete_Inf_Inf @ A) @ S3)) @ L))) @ E3))))))))). % 6.41/2.69 thf(fact_4117_cSup__asclose, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (linordered_idom @ A)) => (! [S3:(set @ A),L:A,E3:A]: (((S3 != (bot_bot @ (set @ A))) => ((! [X5:A]: (((((member @ A) @ X5) @ S3) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ X5) @ L))) @ E3)))) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ ((complete_Sup_Sup @ A) @ S3)) @ L))) @ E3))))))))). % 6.41/2.69 thf(fact_4118_finite__subset__Union, axiom, (! [A:$tType,A3:(set @ A),B12:(set @ (set @ A))]: ((((finite_finite2 @ A) @ A3) => ((((ord_less_eq @ (set @ A)) @ A3) @ ((complete_Sup_Sup @ (set @ A)) @ B12)) => (~ ((! [F7:(set @ (set @ A))]: ((((finite_finite2 @ (set @ A)) @ F7) => ((((ord_less_eq @ (set @ (set @ A))) @ F7) @ B12) => (~ ((((ord_less_eq @ (set @ A)) @ A3) @ ((complete_Sup_Sup @ (set @ A)) @ F7))))))))))))))). % 6.41/2.69 thf(fact_4119_ccpo__Sup__singleton, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [X3:A]: ((((complete_Sup_Sup @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = X3))))))). % 6.41/2.69 thf(fact_4120_Sup__empty, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (((complete_Sup_Sup @ A) @ (bot_bot @ (set @ A))) = (bot_bot @ A)))))). % 6.41/2.69 thf(fact_4121_ccSup__empty, axiom, (! [A:$tType]: (((counta3822494911875563373attice @ A) => (((complete_Sup_Sup @ A) @ (bot_bot @ (set @ A))) = (bot_bot @ A)))))). % 6.41/2.69 thf(fact_4122_Inf__eq__bot__iff, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [A3:(set @ A)]: (((((complete_Inf_Inf @ A) @ A3) = (bot_bot @ A)) = (! [X:A]: (((((ord_less @ A) @ (bot_bot @ A)) @ X) => (? [Y2:A]: (((((member @ A) @ Y2) @ A3) & (((ord_less @ A) @ Y2) @ X)))))))))))))). % 6.41/2.69 thf(fact_4123_Sup__bot__conv_I1_J, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A)]: (((((complete_Sup_Sup @ A) @ A3) = (bot_bot @ A)) = (! [X:A]: (((((member @ A) @ X) @ A3) => (X = (bot_bot @ A)))))))))))). % 6.41/2.69 thf(fact_4124_Sup__bot__conv_I2_J, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A)]: ((((bot_bot @ A) = ((complete_Sup_Sup @ A) @ A3)) = (! [X:A]: (((((member @ A) @ X) @ A3) => (X = (bot_bot @ A)))))))))))). % 6.41/2.69 thf(fact_4125_Sup__nat__empty, axiom, (((complete_Sup_Sup @ nat) @ (bot_bot @ (set @ nat))) = (zero_zero @ nat))). % 6.41/2.69 thf(fact_4126_Inf__nat__def1, axiom, (! [K5:(set @ nat)]: (((K5 != (bot_bot @ (set @ nat))) => (((member @ nat) @ ((complete_Inf_Inf @ nat) @ K5)) @ K5))))). % 6.41/2.69 thf(fact_4127_quotient__of__denom__pos_H, axiom, (! [R2:rat]: ((((ord_less @ int) @ (zero_zero @ int)) @ (((product_snd @ int) @ int) @ (quotient_of @ R2)))))). % 6.41/2.69 thf(fact_4128_bezw__non__0, axiom, (! [Y:nat,X3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ Y) => (((bezw @ X3) @ Y) = ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Y) @ (((modulo_modulo @ nat) @ X3) @ Y)))) @ (((minus_minus @ int) @ (((product_fst @ int) @ int) @ ((bezw @ Y) @ (((modulo_modulo @ nat) @ X3) @ Y)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Y) @ (((modulo_modulo @ nat) @ X3) @ Y)))) @ ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ X3) @ Y)))))))))). % 6.41/2.69 thf(fact_4129_bezw_Oelims, axiom, (! [X3:nat,Xa2:nat,Y:((product_prod @ int) @ int)]: (((((bezw @ X3) @ Xa2) = Y) => (((Xa2 = (zero_zero @ nat)) => (Y = ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (zero_zero @ int)))) & ((Xa2 != (zero_zero @ nat)) => (Y = ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Xa2) @ (((modulo_modulo @ nat) @ X3) @ Xa2)))) @ (((minus_minus @ int) @ (((product_fst @ int) @ int) @ ((bezw @ Xa2) @ (((modulo_modulo @ nat) @ X3) @ Xa2)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Xa2) @ (((modulo_modulo @ nat) @ X3) @ Xa2)))) @ ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ X3) @ Xa2)))))))))))). % 6.41/2.69 thf(fact_4130_bezw_Osimps, axiom, (bezw = (^ [X:nat,Y2:nat]: (((((if @ ((product_prod @ int) @ int)) @ (Y2 = (zero_zero @ nat))) @ ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Y2) @ (((modulo_modulo @ nat) @ X) @ Y2)))) @ (((minus_minus @ int) @ (((product_fst @ int) @ int) @ ((bezw @ Y2) @ (((modulo_modulo @ nat) @ X) @ Y2)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Y2) @ (((modulo_modulo @ nat) @ X) @ Y2)))) @ ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ X) @ Y2)))))))))). % 6.41/2.69 thf(fact_4131_rat__sgn__code, axiom, (! [P4:rat]: (((quotient_of @ ((sgn_sgn @ rat) @ P4)) = ((((product_Pair @ int) @ int) @ ((sgn_sgn @ int) @ (((product_fst @ int) @ int) @ (quotient_of @ P4)))) @ (one_one @ int)))))). % 6.41/2.69 thf(fact_4132_Sup__upper2, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [U:A,A3:(set @ A),V2:A]: (((((member @ A) @ U) @ A3) => ((((ord_less_eq @ A) @ V2) @ U) => (((ord_less_eq @ A) @ V2) @ ((complete_Sup_Sup @ A) @ A3)))))))))). % 6.41/2.69 thf(fact_4133_Sup__le__iff, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A),B2:A]: (((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A3)) @ B2) = (! [X:A]: (((((member @ A) @ X) @ A3) => (((ord_less_eq @ A) @ X) @ B2))))))))))). % 6.41/2.69 thf(fact_4134_Sup__upper, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X3:A,A3:(set @ A)]: (((((member @ A) @ X3) @ A3) => (((ord_less_eq @ A) @ X3) @ ((complete_Sup_Sup @ A) @ A3))))))))). % 6.41/2.69 thf(fact_4135_Sup__least, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A),Z2:A]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((ord_less_eq @ A) @ X5) @ Z2)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A3)) @ Z2)))))))). % 6.41/2.69 thf(fact_4136_Sup__mono, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A),B3:(set @ A)]: (((! [A5:A]: (((((member @ A) @ A5) @ A3) => (? [X2:A]: (((((member @ A) @ X2) @ B3) & (((ord_less_eq @ A) @ A5) @ X2))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A3)) @ ((complete_Sup_Sup @ A) @ B3))))))))). % 6.41/2.69 thf(fact_4137_Sup__eqI, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A),X3:A]: (((! [Y4:A]: (((((member @ A) @ Y4) @ A3) => (((ord_less_eq @ A) @ Y4) @ X3)))) => ((! [Y4:A]: (((! [Z4:A]: (((((member @ A) @ Z4) @ A3) => (((ord_less_eq @ A) @ Z4) @ Y4)))) => (((ord_less_eq @ A) @ X3) @ Y4)))) => (((complete_Sup_Sup @ A) @ A3) = X3))))))))). % 6.41/2.69 thf(fact_4138_less__Sup__iff, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [A2:A,S3:(set @ A)]: (((((ord_less @ A) @ A2) @ ((complete_Sup_Sup @ A) @ S3)) = (? [X:A]: (((((member @ A) @ X) @ S3) & (((ord_less @ A) @ A2) @ X))))))))))). % 6.41/2.69 thf(fact_4139_Inf__greatest, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A),Z2:A]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((ord_less_eq @ A) @ Z2) @ X5)))) => (((ord_less_eq @ A) @ Z2) @ ((complete_Inf_Inf @ A) @ A3))))))))). % 6.41/2.69 thf(fact_4140_le__Inf__iff, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [B2:A,A3:(set @ A)]: (((((ord_less_eq @ A) @ B2) @ ((complete_Inf_Inf @ A) @ A3)) = (! [X:A]: (((((member @ A) @ X) @ A3) => (((ord_less_eq @ A) @ B2) @ X))))))))))). % 6.41/2.69 thf(fact_4141_Inf__lower2, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [U:A,A3:(set @ A),V2:A]: (((((member @ A) @ U) @ A3) => ((((ord_less_eq @ A) @ U) @ V2) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A3)) @ V2))))))))). % 6.41/2.69 thf(fact_4142_Inf__lower, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X3:A,A3:(set @ A)]: (((((member @ A) @ X3) @ A3) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A3)) @ X3)))))))). % 6.41/2.69 thf(fact_4143_Inf__mono, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [B3:(set @ A),A3:(set @ A)]: (((! [B5:A]: (((((member @ A) @ B5) @ B3) => (? [X2:A]: (((((member @ A) @ X2) @ A3) & (((ord_less_eq @ A) @ X2) @ B5))))))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A3)) @ ((complete_Inf_Inf @ A) @ B3))))))))). % 6.41/2.69 thf(fact_4144_Inf__eqI, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A),X3:A]: (((! [I4:A]: (((((member @ A) @ I4) @ A3) => (((ord_less_eq @ A) @ X3) @ I4)))) => ((! [Y4:A]: (((! [I5:A]: (((((member @ A) @ I5) @ A3) => (((ord_less_eq @ A) @ Y4) @ I5)))) => (((ord_less_eq @ A) @ Y4) @ X3)))) => (((complete_Inf_Inf @ A) @ A3) = X3))))))))). % 6.41/2.69 thf(fact_4145_Inf__less__iff, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [S3:(set @ A),A2:A]: (((((ord_less @ A) @ ((complete_Inf_Inf @ A) @ S3)) @ A2) = (? [X:A]: (((((member @ A) @ X) @ S3) & (((ord_less @ A) @ X) @ A2))))))))))). % 6.41/2.69 thf(fact_4146_Union__empty__conv, axiom, (! [A:$tType,A3:(set @ (set @ A))]: (((((complete_Sup_Sup @ (set @ A)) @ A3) = (bot_bot @ (set @ A))) = (! [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ A3) => (X = (bot_bot @ (set @ A)))))))))). % 6.41/2.69 thf(fact_4147_empty__Union__conv, axiom, (! [A:$tType,A3:(set @ (set @ A))]: ((((bot_bot @ (set @ A)) = ((complete_Sup_Sup @ (set @ A)) @ A3)) = (! [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ A3) => (X = (bot_bot @ (set @ A)))))))))). % 6.41/2.69 thf(fact_4148_Union__least, axiom, (! [A:$tType,A3:(set @ (set @ A)),C5:(set @ A)]: (((! [X10:(set @ A)]: (((((member @ (set @ A)) @ X10) @ A3) => (((ord_less_eq @ (set @ A)) @ X10) @ C5)))) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ A3)) @ C5))))). % 6.41/2.69 thf(fact_4149_Union__upper, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ (set @ A))]: (((((member @ (set @ A)) @ B3) @ A3) => (((ord_less_eq @ (set @ A)) @ B3) @ ((complete_Sup_Sup @ (set @ A)) @ A3)))))). % 6.41/2.69 thf(fact_4150_Union__subsetI, axiom, (! [A:$tType,A3:(set @ (set @ A)),B3:(set @ (set @ A))]: (((! [X5:(set @ A)]: (((((member @ (set @ A)) @ X5) @ A3) => (? [Y5:(set @ A)]: (((((member @ (set @ A)) @ Y5) @ B3) & (((ord_less_eq @ (set @ A)) @ X5) @ Y5))))))) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ A3)) @ ((complete_Sup_Sup @ (set @ A)) @ B3)))))). % 6.41/2.69 thf(fact_4151_Inter__lower, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ (set @ A))]: (((((member @ (set @ A)) @ B3) @ A3) => (((ord_less_eq @ (set @ A)) @ ((complete_Inf_Inf @ (set @ A)) @ A3)) @ B3))))). % 6.41/2.69 thf(fact_4152_Inter__greatest, axiom, (! [A:$tType,A3:(set @ (set @ A)),C5:(set @ A)]: (((! [X10:(set @ A)]: (((((member @ (set @ A)) @ X10) @ A3) => (((ord_less_eq @ (set @ A)) @ C5) @ X10)))) => (((ord_less_eq @ (set @ A)) @ C5) @ ((complete_Inf_Inf @ (set @ A)) @ A3)))))). % 6.41/2.69 thf(fact_4153_Union__SetCompr__eq, axiom, (! [B:$tType,A:$tType,F2:(B > (set @ A)),P:(B > $o)]: ((((complete_Sup_Sup @ (set @ A)) @ ((collect @ (set @ A)) @ (^ [Uu3:(set @ A)]: ((? [X:B]: (((Uu3 = (F2 @ X)) & (P @ X)))))))) = ((collect @ A) @ (^ [A4:A]: ((? [X:B]: (((P @ X) & (((member @ A) @ A4) @ (F2 @ X)))))))))))). % 6.41/2.69 thf(fact_4154_le__Sup__iff, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [X3:A,A3:(set @ A)]: (((((ord_less_eq @ A) @ X3) @ ((complete_Sup_Sup @ A) @ A3)) = (! [Y2:A]: (((((ord_less @ A) @ Y2) @ X3) => (? [X:A]: (((((member @ A) @ X) @ A3) & (((ord_less @ A) @ Y2) @ X)))))))))))))). % 6.41/2.69 thf(fact_4155_Inf__le__iff, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [A3:(set @ A),X3:A]: (((((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A3)) @ X3) = (! [Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (? [X:A]: (((((member @ A) @ X) @ A3) & (((ord_less @ A) @ X) @ Y2)))))))))))))). % 6.41/2.69 thf(fact_4156_less__eq__Sup, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A),U:A]: (((! [V3:A]: (((((member @ A) @ V3) @ A3) => (((ord_less_eq @ A) @ U) @ V3)))) => ((A3 != (bot_bot @ (set @ A))) => (((ord_less_eq @ A) @ U) @ ((complete_Sup_Sup @ A) @ A3)))))))))). % 6.41/2.69 thf(fact_4157_Sup__subset__mono, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A3)) @ ((complete_Sup_Sup @ A) @ B3))))))))). % 6.41/2.69 thf(fact_4158_Inf__less__eq, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A),U:A]: (((! [V3:A]: (((((member @ A) @ V3) @ A3) => (((ord_less_eq @ A) @ V3) @ U)))) => ((A3 != (bot_bot @ (set @ A))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A3)) @ U))))))))). % 6.41/2.69 thf(fact_4159_Inf__superset__mono, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [B3:(set @ A),A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B3) @ A3) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A3)) @ ((complete_Inf_Inf @ A) @ B3))))))))). % 6.41/2.69 thf(fact_4160_Union__mono, axiom, (! [A:$tType,A3:(set @ (set @ A)),B3:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ A3) @ B3) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ A3)) @ ((complete_Sup_Sup @ (set @ A)) @ B3)))))). % 6.41/2.69 thf(fact_4161_Inter__anti__mono, axiom, (! [A:$tType,B3:(set @ (set @ A)),A3:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ B3) @ A3) => (((ord_less_eq @ (set @ A)) @ ((complete_Inf_Inf @ (set @ A)) @ A3)) @ ((complete_Inf_Inf @ (set @ A)) @ B3)))))). % 6.41/2.69 thf(fact_4162_Union__empty, axiom, (! [A:$tType]: ((((complete_Sup_Sup @ (set @ A)) @ (bot_bot @ (set @ (set @ A)))) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_4163_Inter__subset, axiom, (! [A:$tType,A3:(set @ (set @ A)),B3:(set @ A)]: (((! [X10:(set @ A)]: (((((member @ (set @ A)) @ X10) @ A3) => (((ord_less_eq @ (set @ A)) @ X10) @ B3)))) => ((A3 != (bot_bot @ (set @ (set @ A)))) => (((ord_less_eq @ (set @ A)) @ ((complete_Inf_Inf @ (set @ A)) @ A3)) @ B3)))))). % 6.41/2.69 thf(fact_4164_Inf__le__Sup, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A)]: (((A3 != (bot_bot @ (set @ A))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A3)) @ ((complete_Sup_Sup @ A) @ A3))))))))). % 6.41/2.69 thf(fact_4165_size__prod__simp, axiom, (! [B:$tType,A:$tType]: ((((basic_BNF_size_prod @ A) @ B) = (^ [F3:(A > nat),G2:(B > nat),P6:((product_prod @ A) @ B)]: ((((plus_plus @ nat) @ (((plus_plus @ nat) @ (F3 @ (((product_fst @ A) @ B) @ P6))) @ (G2 @ (((product_snd @ A) @ B) @ P6)))) @ (suc @ (zero_zero @ nat))))))))). % 6.41/2.69 thf(fact_4166_conjI__realizer, axiom, (! [A:$tType,B:$tType,P:(A > $o),P4:A,Q:(B > $o),Q4:B]: (((P @ P4) => ((Q @ Q4) => ((P @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P4) @ Q4))) & (Q @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P4) @ Q4))))))))). % 6.41/2.69 thf(fact_4167_exI__realizer, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Y:A,X3:B]: ((((P @ Y) @ X3) => ((P @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X3) @ Y))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X3) @ Y))))))). % 6.41/2.69 thf(fact_4168_bezw_Opelims, axiom, (! [X3:nat,Xa2:nat,Y:((product_prod @ int) @ int)]: (((((bezw @ X3) @ Xa2) = Y) => ((((accp @ ((product_prod @ nat) @ nat)) @ bezw_rel) @ ((((product_Pair @ nat) @ nat) @ X3) @ Xa2)) => (~ (((((Xa2 = (zero_zero @ nat)) => (Y = ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (zero_zero @ int)))) & ((Xa2 != (zero_zero @ nat)) => (Y = ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Xa2) @ (((modulo_modulo @ nat) @ X3) @ Xa2)))) @ (((minus_minus @ int) @ (((product_fst @ int) @ int) @ ((bezw @ Xa2) @ (((modulo_modulo @ nat) @ X3) @ Xa2)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Xa2) @ (((modulo_modulo @ nat) @ X3) @ Xa2)))) @ ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ X3) @ Xa2)))))))) => (~ ((((accp @ ((product_prod @ nat) @ nat)) @ bezw_rel) @ ((((product_Pair @ nat) @ nat) @ X3) @ Xa2)))))))))))). % 6.41/2.69 thf(fact_4169_prod__decode__aux_Opelims, axiom, (! [X3:nat,Xa2:nat,Y:((product_prod @ nat) @ nat)]: (((((nat_prod_decode_aux @ X3) @ Xa2) = Y) => ((((accp @ ((product_prod @ nat) @ nat)) @ nat_pr5047031295181774490ux_rel) @ ((((product_Pair @ nat) @ nat) @ X3) @ Xa2)) => (~ (((((((ord_less_eq @ nat) @ Xa2) @ X3) => (Y = ((((product_Pair @ nat) @ nat) @ Xa2) @ (((minus_minus @ nat) @ X3) @ Xa2)))) & ((~ ((((ord_less_eq @ nat) @ Xa2) @ X3))) => (Y = ((nat_prod_decode_aux @ (suc @ X3)) @ (((minus_minus @ nat) @ Xa2) @ (suc @ X3)))))) => (~ ((((accp @ ((product_prod @ nat) @ nat)) @ nat_pr5047031295181774490ux_rel) @ ((((product_Pair @ nat) @ nat) @ X3) @ Xa2)))))))))))). % 6.41/2.69 thf(fact_4170_normalize__def, axiom, (normalize = (^ [P6:((product_prod @ int) @ int)]: (((((if @ ((product_prod @ int) @ int)) @ (((ord_less @ int) @ (zero_zero @ int)) @ (((product_snd @ int) @ int) @ P6))) @ ((((product_Pair @ int) @ int) @ (((divide_divide @ int) @ (((product_fst @ int) @ int) @ P6)) @ (((gcd_gcd @ int) @ (((product_fst @ int) @ int) @ P6)) @ (((product_snd @ int) @ int) @ P6)))) @ (((divide_divide @ int) @ (((product_snd @ int) @ int) @ P6)) @ (((gcd_gcd @ int) @ (((product_fst @ int) @ int) @ P6)) @ (((product_snd @ int) @ int) @ P6))))) @ ((((if @ ((product_prod @ int) @ int)) @ ((((product_snd @ int) @ int) @ P6) = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (((divide_divide @ int) @ (((product_fst @ int) @ int) @ P6)) @ ((uminus_uminus @ int) @ (((gcd_gcd @ int) @ (((product_fst @ int) @ int) @ P6)) @ (((product_snd @ int) @ int) @ P6))))) @ (((divide_divide @ int) @ (((product_snd @ int) @ int) @ P6)) @ ((uminus_uminus @ int) @ (((gcd_gcd @ int) @ (((product_fst @ int) @ int) @ P6)) @ (((product_snd @ int) @ int) @ P6))))))))))). % 6.41/2.69 thf(fact_4171_strict__mono__on__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (((strict_mono_on @ A) @ B) = (^ [F3:(A > B),A6:(set @ A)]: ((! [R5:A,S8:A]: ((((((member @ A) @ R5) @ A6) & ((((member @ A) @ S8) @ A6) & (((ord_less @ A) @ R5) @ S8))) => (((ord_less @ B) @ (F3 @ R5)) @ (F3 @ S8)))))))))))). % 6.41/2.69 thf(fact_4172_strict__mono__onI, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (! [A3:(set @ A),F2:(A > B)]: (((! [R4:A,S2:A]: (((((member @ A) @ R4) @ A3) => ((((member @ A) @ S2) @ A3) => ((((ord_less @ A) @ R4) @ S2) => (((ord_less @ B) @ (F2 @ R4)) @ (F2 @ S2))))))) => ((((strict_mono_on @ A) @ B) @ F2) @ A3)))))))). % 6.41/2.69 thf(fact_4173_gcd__left__idem, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A]: (((((gcd_gcd @ A) @ A2) @ (((gcd_gcd @ A) @ A2) @ B2)) = (((gcd_gcd @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_4174_gcd__right__idem, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A]: (((((gcd_gcd @ A) @ (((gcd_gcd @ A) @ A2) @ B2)) @ B2) = (((gcd_gcd @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_4175_gcd__eq__0__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A]: ((((((gcd_gcd @ A) @ A2) @ B2) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) & (B2 = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_4176_gcd_Obottom__right__bottom, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A]: (((((gcd_gcd @ A) @ A2) @ (one_one @ A)) = (one_one @ A)))))))). % 6.41/2.69 thf(fact_4177_gcd_Obottom__left__bottom, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A]: (((((gcd_gcd @ A) @ (one_one @ A)) @ A2) = (one_one @ A)))))))). % 6.41/2.69 thf(fact_4178_gcd__add1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [M:A,N2:A]: (((((gcd_gcd @ A) @ (((plus_plus @ A) @ M) @ N2)) @ N2) = (((gcd_gcd @ A) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_4179_gcd__add2, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [M:A,N2:A]: (((((gcd_gcd @ A) @ M) @ (((plus_plus @ A) @ M) @ N2)) = (((gcd_gcd @ A) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_4180_gcd__neg2, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [A2:A,B2:A]: (((((gcd_gcd @ A) @ A2) @ ((uminus_uminus @ A) @ B2)) = (((gcd_gcd @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_4181_gcd__neg1, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [A2:A,B2:A]: (((((gcd_gcd @ A) @ ((uminus_uminus @ A) @ A2)) @ B2) = (((gcd_gcd @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_4182_gcd__exp, axiom, (! [A:$tType]: (((semiri6843258321239162965malize @ A) => (! [A2:A,N2:nat,B2:A]: (((((gcd_gcd @ A) @ (((power_power @ A) @ A2) @ N2)) @ (((power_power @ A) @ B2) @ N2)) = (((power_power @ A) @ (((gcd_gcd @ A) @ A2) @ B2)) @ N2)))))))). % 6.41/2.69 thf(fact_4183_gcd__dvd1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A]: ((((dvd_dvd @ A) @ (((gcd_gcd @ A) @ A2) @ B2)) @ A2))))))). % 6.41/2.69 thf(fact_4184_gcd__dvd2, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A]: ((((dvd_dvd @ A) @ (((gcd_gcd @ A) @ A2) @ B2)) @ B2))))))). % 6.41/2.69 thf(fact_4185_gcd__greatest__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ (((gcd_gcd @ A) @ B2) @ C2)) = ((((dvd_dvd @ A) @ A2) @ B2) & (((dvd_dvd @ A) @ A2) @ C2))))))))). % 6.41/2.69 thf(fact_4186_gcd__1__int, axiom, (! [M:int]: (((((gcd_gcd @ int) @ M) @ (one_one @ int)) = (one_one @ int))))). % 6.41/2.69 thf(fact_4187_gcd__neg2__int, axiom, (! [X3:int,Y:int]: (((((gcd_gcd @ int) @ X3) @ ((uminus_uminus @ int) @ Y)) = (((gcd_gcd @ int) @ X3) @ Y))))). % 6.41/2.69 thf(fact_4188_gcd__neg1__int, axiom, (! [X3:int,Y:int]: (((((gcd_gcd @ int) @ ((uminus_uminus @ int) @ X3)) @ Y) = (((gcd_gcd @ int) @ X3) @ Y))))). % 6.41/2.69 thf(fact_4189_abs__gcd__int, axiom, (! [X3:int,Y:int]: ((((abs_abs @ int) @ (((gcd_gcd @ int) @ X3) @ Y)) = (((gcd_gcd @ int) @ X3) @ Y))))). % 6.41/2.69 thf(fact_4190_gcd__abs1__int, axiom, (! [X3:int,Y:int]: (((((gcd_gcd @ int) @ ((abs_abs @ int) @ X3)) @ Y) = (((gcd_gcd @ int) @ X3) @ Y))))). % 6.41/2.69 thf(fact_4191_gcd__abs2__int, axiom, (! [X3:int,Y:int]: (((((gcd_gcd @ int) @ X3) @ ((abs_abs @ int) @ Y)) = (((gcd_gcd @ int) @ X3) @ Y))))). % 6.41/2.69 thf(fact_4192_gcd__neg__numeral__1, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [N2:num,A2:A]: (((((gcd_gcd @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))) @ A2) = (((gcd_gcd @ A) @ ((numeral_numeral @ A) @ N2)) @ A2)))))))). % 6.41/2.69 thf(fact_4193_gcd__neg__numeral__2, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [A2:A,N2:num]: (((((gcd_gcd @ A) @ A2) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))) = (((gcd_gcd @ A) @ A2) @ ((numeral_numeral @ A) @ N2))))))))). % 6.41/2.69 thf(fact_4194_is__unit__gcd__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ (((gcd_gcd @ A) @ A2) @ B2)) @ (one_one @ A)) = ((((gcd_gcd @ A) @ A2) @ B2) = (one_one @ A))))))))). % 6.41/2.69 thf(fact_4195_gcd__pos__int, axiom, (! [M:int,N2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ (((gcd_gcd @ int) @ M) @ N2)) = ((M != (zero_zero @ int)) | (N2 != (zero_zero @ int))))))). % 6.41/2.69 thf(fact_4196_gcd__neg__numeral__1__int, axiom, (! [N2:num,X3:int]: (((((gcd_gcd @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N2))) @ X3) = (((gcd_gcd @ int) @ ((numeral_numeral @ int) @ N2)) @ X3))))). % 6.41/2.69 thf(fact_4197_gcd__neg__numeral__2__int, axiom, (! [X3:int,N2:num]: (((((gcd_gcd @ int) @ X3) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N2))) = (((gcd_gcd @ int) @ X3) @ ((numeral_numeral @ int) @ N2)))))). % 6.41/2.69 thf(fact_4198_gcd__0__int, axiom, (! [X3:int]: (((((gcd_gcd @ int) @ X3) @ (zero_zero @ int)) = ((abs_abs @ int) @ X3))))). % 6.41/2.69 thf(fact_4199_gcd__0__left__int, axiom, (! [X3:int]: (((((gcd_gcd @ int) @ (zero_zero @ int)) @ X3) = ((abs_abs @ int) @ X3))))). % 6.41/2.69 thf(fact_4200_gcd__proj1__if__dvd__int, axiom, (! [X3:int,Y:int]: (((((dvd_dvd @ int) @ X3) @ Y) => ((((gcd_gcd @ int) @ X3) @ Y) = ((abs_abs @ int) @ X3)))))). % 6.41/2.69 thf(fact_4201_gcd__proj2__if__dvd__int, axiom, (! [Y:int,X3:int]: (((((dvd_dvd @ int) @ Y) @ X3) => ((((gcd_gcd @ int) @ X3) @ Y) = ((abs_abs @ int) @ Y)))))). % 6.41/2.69 thf(fact_4202_gcd__idem__int, axiom, (! [X3:int]: (((((gcd_gcd @ int) @ X3) @ X3) = ((abs_abs @ int) @ X3))))). % 6.41/2.69 thf(fact_4203_gcd__diff1, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [M:A,N2:A]: (((((gcd_gcd @ A) @ (((minus_minus @ A) @ M) @ N2)) @ N2) = (((gcd_gcd @ A) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_4204_gcd__diff2, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [N2:A,M:A]: (((((gcd_gcd @ A) @ (((minus_minus @ A) @ N2) @ M)) @ N2) = (((gcd_gcd @ A) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_4205_gcd_Oassoc, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A,C2:A]: (((((gcd_gcd @ A) @ (((gcd_gcd @ A) @ A2) @ B2)) @ C2) = (((gcd_gcd @ A) @ A2) @ (((gcd_gcd @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_4206_gcd_Ocommute, axiom, (! [A:$tType]: (((semiring_gcd @ A) => ((gcd_gcd @ A) = (^ [A4:A,B4:A]: ((((gcd_gcd @ A) @ B4) @ A4)))))))). % 6.41/2.69 thf(fact_4207_gcd_Oleft__commute, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [B2:A,A2:A,C2:A]: (((((gcd_gcd @ A) @ B2) @ (((gcd_gcd @ A) @ A2) @ C2)) = (((gcd_gcd @ A) @ A2) @ (((gcd_gcd @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_4208_gcd__red__int, axiom, ((gcd_gcd @ int) = (^ [X:int,Y2:int]: ((((gcd_gcd @ int) @ Y2) @ (((modulo_modulo @ int) @ X) @ Y2)))))). % 6.41/2.69 thf(fact_4209_gcd__mono, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,C2:A,B2:A,D2:A]: (((((dvd_dvd @ A) @ A2) @ C2) => ((((dvd_dvd @ A) @ B2) @ D2) => (((dvd_dvd @ A) @ (((gcd_gcd @ A) @ A2) @ B2)) @ (((gcd_gcd @ A) @ C2) @ D2)))))))))). % 6.41/2.69 thf(fact_4210_dvd__gcdD1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ (((gcd_gcd @ A) @ B2) @ C2)) => (((dvd_dvd @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_4211_dvd__gcdD2, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ (((gcd_gcd @ A) @ B2) @ C2)) => (((dvd_dvd @ A) @ A2) @ C2)))))))). % 6.41/2.69 thf(fact_4212_gcd__dvdI1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,C2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ C2) => (((dvd_dvd @ A) @ (((gcd_gcd @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_4213_gcd__dvdI2, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [B2:A,C2:A,A2:A]: (((((dvd_dvd @ A) @ B2) @ C2) => (((dvd_dvd @ A) @ (((gcd_gcd @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_4214_gcd__greatest, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [C2:A,A2:A,B2:A]: (((((dvd_dvd @ A) @ C2) @ A2) => ((((dvd_dvd @ A) @ C2) @ B2) => (((dvd_dvd @ A) @ C2) @ (((gcd_gcd @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_4215_gcd__dvd__antisym, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((dvd_dvd @ A) @ (((gcd_gcd @ A) @ A2) @ B2)) @ (((gcd_gcd @ A) @ C2) @ D2)) => ((((dvd_dvd @ A) @ (((gcd_gcd @ A) @ C2) @ D2)) @ (((gcd_gcd @ A) @ A2) @ B2)) => ((((gcd_gcd @ A) @ A2) @ B2) = (((gcd_gcd @ A) @ C2) @ D2)))))))))). % 6.41/2.69 thf(fact_4216_gcd__add__mult, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [M:A,K:A,N2:A]: (((((gcd_gcd @ A) @ M) @ (((plus_plus @ A) @ (((times_times @ A) @ K) @ M)) @ N2)) = (((gcd_gcd @ A) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_4217_gcd__dvd__prod, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A,K:A]: ((((dvd_dvd @ A) @ (((gcd_gcd @ A) @ A2) @ B2)) @ (((times_times @ A) @ K) @ B2)))))))). % 6.41/2.69 thf(fact_4218_gcd__ge__0__int, axiom, (! [X3:int,Y:int]: ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((gcd_gcd @ int) @ X3) @ Y))))). % 6.41/2.69 thf(fact_4219_bezout__int, axiom, (! [X3:int,Y:int]: ((? [U4:int,V3:int]: (((((plus_plus @ int) @ (((times_times @ int) @ U4) @ X3)) @ (((times_times @ int) @ V3) @ Y)) = (((gcd_gcd @ int) @ X3) @ Y))))))). % 6.41/2.69 thf(fact_4220_gcd__mult__distrib__int, axiom, (! [K:int,M:int,N2:int]: (((((times_times @ int) @ ((abs_abs @ int) @ K)) @ (((gcd_gcd @ int) @ M) @ N2)) = (((gcd_gcd @ int) @ (((times_times @ int) @ K) @ M)) @ (((times_times @ int) @ K) @ N2)))))). % 6.41/2.69 thf(fact_4221_gcd__mult__unit2, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => ((((gcd_gcd @ A) @ B2) @ (((times_times @ A) @ C2) @ A2)) = (((gcd_gcd @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_4222_gcd__mult__unit1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => ((((gcd_gcd @ A) @ (((times_times @ A) @ B2) @ A2)) @ C2) = (((gcd_gcd @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_4223_gcd__div__unit2, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => ((((gcd_gcd @ A) @ B2) @ (((divide_divide @ A) @ C2) @ A2)) = (((gcd_gcd @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_4224_gcd__div__unit1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A2:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => ((((gcd_gcd @ A) @ (((divide_divide @ A) @ B2) @ A2)) @ C2) = (((gcd_gcd @ A) @ B2) @ C2))))))))). % 6.41/2.69 thf(fact_4225_gcd__le2__int, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => (((ord_less_eq @ int) @ (((gcd_gcd @ int) @ A2) @ B2)) @ B2))))). % 6.41/2.69 thf(fact_4226_gcd__le1__int, axiom, (! [A2:int,B2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ A2) => (((ord_less_eq @ int) @ (((gcd_gcd @ int) @ A2) @ B2)) @ A2))))). % 6.41/2.69 thf(fact_4227_gcd__cases__int, axiom, (! [X3:int,Y:int,P:(int > $o)]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ X3) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y) => (P @ (((gcd_gcd @ int) @ X3) @ Y)))) => (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X3) => ((((ord_less_eq @ int) @ Y) @ (zero_zero @ int)) => (P @ (((gcd_gcd @ int) @ X3) @ ((uminus_uminus @ int) @ Y))))) => (((((ord_less_eq @ int) @ X3) @ (zero_zero @ int)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Y) => (P @ (((gcd_gcd @ int) @ ((uminus_uminus @ int) @ X3)) @ Y)))) => (((((ord_less_eq @ int) @ X3) @ (zero_zero @ int)) => ((((ord_less_eq @ int) @ Y) @ (zero_zero @ int)) => (P @ (((gcd_gcd @ int) @ ((uminus_uminus @ int) @ X3)) @ ((uminus_uminus @ int) @ Y))))) => (P @ (((gcd_gcd @ int) @ X3) @ Y))))))))). % 6.41/2.69 thf(fact_4228_gcd__unique__int, axiom, (! [D2:int,A2:int,B2:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ D2) & ((((dvd_dvd @ int) @ D2) @ A2) & ((((dvd_dvd @ int) @ D2) @ B2) & (! [E4:int]: ((((((dvd_dvd @ int) @ E4) @ A2) & (((dvd_dvd @ int) @ E4) @ B2)) => (((dvd_dvd @ int) @ E4) @ D2))))))) = (D2 = (((gcd_gcd @ int) @ A2) @ B2)))))). % 6.41/2.69 thf(fact_4229_gcd__non__0__int, axiom, (! [Y:int,X3:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ Y) => ((((gcd_gcd @ int) @ X3) @ Y) = (((gcd_gcd @ int) @ Y) @ (((modulo_modulo @ int) @ X3) @ Y))))))). % 6.41/2.69 thf(fact_4230_gcd__code__int, axiom, ((gcd_gcd @ int) = (^ [K3:int,L2:int]: (((abs_abs @ int) @ ((((if @ int) @ (L2 = (zero_zero @ int))) @ K3) @ (((gcd_gcd @ int) @ L2) @ (((modulo_modulo @ int) @ ((abs_abs @ int) @ K3)) @ ((abs_abs @ int) @ L2))))))))). % 6.41/2.69 thf(fact_4231_strict__mono__on__leD, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (preorder @ B)) => (! [F2:(A > B),A3:(set @ A),X3:A,Y:A]: ((((((strict_mono_on @ A) @ B) @ F2) @ A3) => ((((member @ A) @ X3) @ A3) => ((((member @ A) @ Y) @ A3) => ((((ord_less_eq @ A) @ X3) @ Y) => (((ord_less_eq @ B) @ (F2 @ X3)) @ (F2 @ Y)))))))))))). % 6.41/2.69 thf(fact_4232_strict__mono__onD, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (! [F2:(A > B),A3:(set @ A),R2:A,S:A]: ((((((strict_mono_on @ A) @ B) @ F2) @ A3) => ((((member @ A) @ R2) @ A3) => ((((member @ A) @ S) @ A3) => ((((ord_less @ A) @ R2) @ S) => (((ord_less @ B) @ (F2 @ R2)) @ (F2 @ S)))))))))))). % 6.41/2.69 thf(fact_4233_card__partition, axiom, (! [A:$tType,C5:(set @ (set @ A)),K:nat]: ((((finite_finite2 @ (set @ A)) @ C5) => (((finite_finite2 @ A) @ ((complete_Sup_Sup @ (set @ A)) @ C5)) => ((! [C3:(set @ A)]: (((((member @ (set @ A)) @ C3) @ C5) => (((finite_card @ A) @ C3) = K)))) => ((! [C1:(set @ A),C22:(set @ A)]: (((((member @ (set @ A)) @ C1) @ C5) => ((((member @ (set @ A)) @ C22) @ C5) => ((C1 != C22) => ((((inf_inf @ (set @ A)) @ C1) @ C22) = (bot_bot @ (set @ A)))))))) => ((((times_times @ nat) @ K) @ ((finite_card @ (set @ A)) @ C5)) = ((finite_card @ A) @ ((complete_Sup_Sup @ (set @ A)) @ C5)))))))))). % 6.41/2.69 thf(fact_4234_or__numerals_I4_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X3:num,Y:num]: (((((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ (bit0 @ X3))) @ ((numeral_numeral @ A) @ (bit1 @ Y))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ X3)) @ ((numeral_numeral @ A) @ Y))))))))))). % 6.41/2.69 thf(fact_4235_or__numerals_I6_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X3:num,Y:num]: (((((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ (bit1 @ X3))) @ ((numeral_numeral @ A) @ (bit0 @ Y))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ X3)) @ ((numeral_numeral @ A) @ Y))))))))))). % 6.41/2.69 thf(fact_4236_or__numerals_I7_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X3:num,Y:num]: (((((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ (bit1 @ X3))) @ ((numeral_numeral @ A) @ (bit1 @ Y))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ X3)) @ ((numeral_numeral @ A) @ Y))))))))))). % 6.41/2.69 thf(fact_4237_gcd__nat_Oeq__neutr__iff, axiom, (! [A2:nat,B2:nat]: ((((((gcd_gcd @ nat) @ A2) @ B2) = (zero_zero @ nat)) = ((A2 = (zero_zero @ nat)) & (B2 = (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_4238_gcd__nat_Oleft__neutral, axiom, (! [A2:nat]: (((((gcd_gcd @ nat) @ (zero_zero @ nat)) @ A2) = A2)))). % 6.41/2.69 thf(fact_4239_gcd__nat_Oneutr__eq__iff, axiom, (! [A2:nat,B2:nat]: ((((zero_zero @ nat) = (((gcd_gcd @ nat) @ A2) @ B2)) = ((A2 = (zero_zero @ nat)) & (B2 = (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_4240_gcd__nat_Oright__neutral, axiom, (! [A2:nat]: (((((gcd_gcd @ nat) @ A2) @ (zero_zero @ nat)) = A2)))). % 6.41/2.69 thf(fact_4241_gcd__0__nat, axiom, (! [X3:nat]: (((((gcd_gcd @ nat) @ X3) @ (zero_zero @ nat)) = X3)))). % 6.41/2.69 thf(fact_4242_gcd__0__left__nat, axiom, (! [X3:nat]: (((((gcd_gcd @ nat) @ (zero_zero @ nat)) @ X3) = X3)))). % 6.41/2.69 thf(fact_4243_Int__iff, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) = ((((member @ A) @ C2) @ A3) & (((member @ A) @ C2) @ B3)))))). % 6.41/2.69 thf(fact_4244_IntI, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ A3) => ((((member @ A) @ C2) @ B3) => (((member @ A) @ C2) @ (((inf_inf @ (set @ A)) @ A3) @ B3))))))). % 6.41/2.69 thf(fact_4245_gcd__1__nat, axiom, (! [M:nat]: (((((gcd_gcd @ nat) @ M) @ (one_one @ nat)) = (one_one @ nat))))). % 6.41/2.69 thf(fact_4246_gcd__proj2__if__dvd__nat, axiom, (! [Y:nat,X3:nat]: (((((dvd_dvd @ nat) @ Y) @ X3) => ((((gcd_gcd @ nat) @ X3) @ Y) = Y))))). % 6.41/2.69 thf(fact_4247_gcd__proj1__if__dvd__nat, axiom, (! [X3:nat,Y:nat]: (((((dvd_dvd @ nat) @ X3) @ Y) => ((((gcd_gcd @ nat) @ X3) @ Y) = X3))))). % 6.41/2.69 thf(fact_4248_gcd__nat_Obounded__iff, axiom, (! [A2:nat,B2:nat,C2:nat]: (((((dvd_dvd @ nat) @ A2) @ (((gcd_gcd @ nat) @ B2) @ C2)) = ((((dvd_dvd @ nat) @ A2) @ B2) & (((dvd_dvd @ nat) @ A2) @ C2)))))). % 6.41/2.69 thf(fact_4249_gcd__nat_Oabsorb2, axiom, (! [B2:nat,A2:nat]: (((((dvd_dvd @ nat) @ B2) @ A2) => ((((gcd_gcd @ nat) @ A2) @ B2) = B2))))). % 6.41/2.69 thf(fact_4250_gcd__nat_Oabsorb1, axiom, (! [A2:nat,B2:nat]: (((((dvd_dvd @ nat) @ A2) @ B2) => ((((gcd_gcd @ nat) @ A2) @ B2) = A2))))). % 6.41/2.69 thf(fact_4251_gcd__Suc__0, axiom, (! [M:nat]: (((((gcd_gcd @ nat) @ M) @ (suc @ (zero_zero @ nat))) = (suc @ (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_4252_gcd__pos__nat, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((gcd_gcd @ nat) @ M) @ N2)) = ((M != (zero_zero @ nat)) | (N2 != (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_4253_boolean__algebra_Oconj__zero__left, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A]: (((((inf_inf @ A) @ (bot_bot @ A)) @ X3) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4254_boolean__algebra_Oconj__zero__right, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A]: (((((inf_inf @ A) @ X3) @ (bot_bot @ A)) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4255_finite__Int, axiom, (! [A:$tType,F5:(set @ A),G6:(set @ A)]: (((((finite_finite2 @ A) @ F5) | ((finite_finite2 @ A) @ G6)) => ((finite_finite2 @ A) @ (((inf_inf @ (set @ A)) @ F5) @ G6)))))). % 6.41/2.69 thf(fact_4256_Int__subset__iff, axiom, (! [A:$tType,C5:(set @ A),A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C5) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) = ((((ord_less_eq @ (set @ A)) @ C5) @ A3) & (((ord_less_eq @ (set @ A)) @ C5) @ B3)))))). % 6.41/2.69 thf(fact_4257_Int__insert__left__if0, axiom, (! [A:$tType,A2:A,C5:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ A2) @ C5))) => ((((inf_inf @ (set @ A)) @ (((insert @ A) @ A2) @ B3)) @ C5) = (((inf_inf @ (set @ A)) @ B3) @ C5)))))). % 6.41/2.69 thf(fact_4258_Int__insert__left__if1, axiom, (! [A:$tType,A2:A,C5:(set @ A),B3:(set @ A)]: (((((member @ A) @ A2) @ C5) => ((((inf_inf @ (set @ A)) @ (((insert @ A) @ A2) @ B3)) @ C5) = (((insert @ A) @ A2) @ (((inf_inf @ (set @ A)) @ B3) @ C5))))))). % 6.41/2.69 thf(fact_4259_insert__inter__insert, axiom, (! [A:$tType,A2:A,A3:(set @ A),B3:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((insert @ A) @ A2) @ A3)) @ (((insert @ A) @ A2) @ B3)) = (((insert @ A) @ A2) @ (((inf_inf @ (set @ A)) @ A3) @ B3)))))). % 6.41/2.69 thf(fact_4260_Int__insert__right__if0, axiom, (! [A:$tType,A2:A,A3:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ A2) @ A3))) => ((((inf_inf @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B3)) = (((inf_inf @ (set @ A)) @ A3) @ B3)))))). % 6.41/2.69 thf(fact_4261_Int__insert__right__if1, axiom, (! [A:$tType,A2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ A2) @ A3) => ((((inf_inf @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B3)) = (((insert @ A) @ A2) @ (((inf_inf @ (set @ A)) @ A3) @ B3))))))). % 6.41/2.69 thf(fact_4262_or_Oright__neutral, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se1065995026697491101ons_or @ A) @ A2) @ (zero_zero @ A)) = A2))))))). % 6.41/2.69 thf(fact_4263_or_Oleft__neutral, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se1065995026697491101ons_or @ A) @ (zero_zero @ A)) @ A2) = A2))))))). % 6.41/2.69 thf(fact_4264_or__negative__int__iff, axiom, (! [K:int,L:int]: (((((ord_less @ int) @ (((bit_se1065995026697491101ons_or @ int) @ K) @ L)) @ (zero_zero @ int)) = ((((ord_less @ int) @ K) @ (zero_zero @ int)) | (((ord_less @ int) @ L) @ (zero_zero @ int))))))). % 6.41/2.69 thf(fact_4265_Pow__Int__eq, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((pow2 @ A) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) = (((inf_inf @ (set @ (set @ A))) @ ((pow2 @ A) @ A3)) @ ((pow2 @ A) @ B3)))))). % 6.41/2.69 thf(fact_4266_gcd__int__int__eq, axiom, (! [M:nat,N2:nat]: (((((gcd_gcd @ int) @ ((semiring_1_of_nat @ int) @ M)) @ ((semiring_1_of_nat @ int) @ N2)) = ((semiring_1_of_nat @ int) @ (((gcd_gcd @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_4267_boolean__algebra_Oconj__cancel__right, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A]: (((((inf_inf @ A) @ X3) @ ((uminus_uminus @ A) @ X3)) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4268_boolean__algebra_Oconj__cancel__left, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A]: (((((inf_inf @ A) @ ((uminus_uminus @ A) @ X3)) @ X3) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4269_inf__compl__bot__right, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A,Y:A]: (((((inf_inf @ A) @ X3) @ (((inf_inf @ A) @ Y) @ ((uminus_uminus @ A) @ X3))) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4270_inf__compl__bot__left2, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A,Y:A]: (((((inf_inf @ A) @ X3) @ (((inf_inf @ A) @ ((uminus_uminus @ A) @ X3)) @ Y)) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4271_inf__compl__bot__left1, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A,Y:A]: (((((inf_inf @ A) @ ((uminus_uminus @ A) @ X3)) @ (((inf_inf @ A) @ X3) @ Y)) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4272_insert__disjoint_I1_J, axiom, (! [A:$tType,A2:A,A3:(set @ A),B3:(set @ A)]: ((((((inf_inf @ (set @ A)) @ (((insert @ A) @ A2) @ A3)) @ B3) = (bot_bot @ (set @ A))) = ((~ ((((member @ A) @ A2) @ B3))) & ((((inf_inf @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A)))))))). % 6.41/2.69 thf(fact_4273_insert__disjoint_I2_J, axiom, (! [A:$tType,A2:A,A3:(set @ A),B3:(set @ A)]: ((((bot_bot @ (set @ A)) = (((inf_inf @ (set @ A)) @ (((insert @ A) @ A2) @ A3)) @ B3)) = ((~ ((((member @ A) @ A2) @ B3))) & ((bot_bot @ (set @ A)) = (((inf_inf @ (set @ A)) @ A3) @ B3))))))). % 6.41/2.69 thf(fact_4274_disjoint__insert_I1_J, axiom, (! [A:$tType,B3:(set @ A),A2:A,A3:(set @ A)]: ((((((inf_inf @ (set @ A)) @ B3) @ (((insert @ A) @ A2) @ A3)) = (bot_bot @ (set @ A))) = ((~ ((((member @ A) @ A2) @ B3))) & ((((inf_inf @ (set @ A)) @ B3) @ A3) = (bot_bot @ (set @ A)))))))). % 6.41/2.69 thf(fact_4275_disjoint__insert_I2_J, axiom, (! [A:$tType,A3:(set @ A),B2:A,B3:(set @ A)]: ((((bot_bot @ (set @ A)) = (((inf_inf @ (set @ A)) @ A3) @ (((insert @ A) @ B2) @ B3))) = ((~ ((((member @ A) @ B2) @ A3))) & ((bot_bot @ (set @ A)) = (((inf_inf @ (set @ A)) @ A3) @ B3))))))). % 6.41/2.69 thf(fact_4276_Diff__disjoint, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (((minus_minus @ (set @ A)) @ B3) @ A3)) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_4277_Compl__disjoint2, axiom, (! [A:$tType,A3:(set @ A)]: (((((inf_inf @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A3)) @ A3) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_4278_Compl__disjoint, axiom, (! [A:$tType,A3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ ((uminus_uminus @ (set @ A)) @ A3)) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_4279_Diff__Compl, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ ((uminus_uminus @ (set @ A)) @ B3)) = (((inf_inf @ (set @ A)) @ A3) @ B3))))). % 6.41/2.69 thf(fact_4280_or__nat__numerals_I4_J, axiom, (! [X3:num]: (((((bit_se1065995026697491101ons_or @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ X3))) @ (suc @ (zero_zero @ nat))) = ((numeral_numeral @ nat) @ (bit1 @ X3)))))). % 6.41/2.69 thf(fact_4281_or__nat__numerals_I2_J, axiom, (! [Y:num]: (((((bit_se1065995026697491101ons_or @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ (bit1 @ Y))) = ((numeral_numeral @ nat) @ (bit1 @ Y)))))). % 6.41/2.69 thf(fact_4282_gcd__nat__abs__right__eq, axiom, (! [N2:nat,K:int]: (((((gcd_gcd @ nat) @ N2) @ (nat2 @ ((abs_abs @ int) @ K))) = (nat2 @ (((gcd_gcd @ int) @ ((semiring_1_of_nat @ int) @ N2)) @ K)))))). % 6.41/2.69 thf(fact_4283_gcd__nat__abs__left__eq, axiom, (! [K:int,N2:nat]: (((((gcd_gcd @ nat) @ (nat2 @ ((abs_abs @ int) @ K))) @ N2) = (nat2 @ (((gcd_gcd @ int) @ K) @ ((semiring_1_of_nat @ int) @ N2))))))). % 6.41/2.69 thf(fact_4284_or__nat__numerals_I3_J, axiom, (! [X3:num]: (((((bit_se1065995026697491101ons_or @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ X3))) @ (suc @ (zero_zero @ nat))) = ((numeral_numeral @ nat) @ (bit1 @ X3)))))). % 6.41/2.69 thf(fact_4285_or__nat__numerals_I1_J, axiom, (! [Y:num]: (((((bit_se1065995026697491101ons_or @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ (bit0 @ Y))) = ((numeral_numeral @ nat) @ (bit1 @ Y)))))). % 6.41/2.69 thf(fact_4286_sum__of__bool__mult__eq, axiom, (! [A:$tType,B:$tType]: (((semiring_1 @ A) => (! [A3:(set @ B),P:(B > $o),F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X:B]: ((((times_times @ A) @ ((zero_neq_one_of_bool @ A) @ (P @ X))) @ (F2 @ X))))) @ A3) = ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ (((inf_inf @ (set @ B)) @ A3) @ ((collect @ B) @ P))))))))))). % 6.41/2.69 thf(fact_4287_sum__mult__of__bool__eq, axiom, (! [A:$tType,B:$tType]: (((semiring_1 @ A) => (! [A3:(set @ B),F2:(B > A),P:(B > $o)]: ((((finite_finite2 @ B) @ A3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X:B]: ((((times_times @ A) @ (F2 @ X)) @ ((zero_neq_one_of_bool @ A) @ (P @ X)))))) @ A3) = ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ (((inf_inf @ (set @ B)) @ A3) @ ((collect @ B) @ P))))))))))). % 6.41/2.69 thf(fact_4288_sum__of__bool__eq, axiom, (! [A:$tType,B:$tType]: (((semiring_1 @ A) => (! [A3:(set @ B),P:(B > $o)]: ((((finite_finite2 @ B) @ A3) => (((finite_finite2 @ B) @ A3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X:B]: (((zero_neq_one_of_bool @ A) @ (P @ X))))) @ A3) = ((semiring_1_of_nat @ A) @ ((finite_card @ B) @ (((inf_inf @ (set @ B)) @ A3) @ ((collect @ B) @ P))))))))))))). % 6.41/2.69 thf(fact_4289_gcd__mult__distrib__nat, axiom, (! [K:nat,M:nat,N2:nat]: (((((times_times @ nat) @ K) @ (((gcd_gcd @ nat) @ M) @ N2)) = (((gcd_gcd @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N2)))))). % 6.41/2.69 thf(fact_4290_gcd__red__nat, axiom, ((gcd_gcd @ nat) = (^ [X:nat,Y2:nat]: ((((gcd_gcd @ nat) @ Y2) @ (((modulo_modulo @ nat) @ X) @ Y2)))))). % 6.41/2.69 thf(fact_4291_gcd__unique__nat, axiom, (! [D2:nat,A2:nat,B2:nat]: ((((((dvd_dvd @ nat) @ D2) @ A2) & ((((dvd_dvd @ nat) @ D2) @ B2) & (! [E4:nat]: ((((((dvd_dvd @ nat) @ E4) @ A2) & (((dvd_dvd @ nat) @ E4) @ B2)) => (((dvd_dvd @ nat) @ E4) @ D2)))))) = (D2 = (((gcd_gcd @ nat) @ A2) @ B2)))))). % 6.41/2.69 thf(fact_4292_gcd__nat_Ostrict__coboundedI2, axiom, (! [B2:nat,C2:nat,A2:nat]: ((((((dvd_dvd @ nat) @ B2) @ C2) & (B2 != C2)) => ((((dvd_dvd @ nat) @ (((gcd_gcd @ nat) @ A2) @ B2)) @ C2) & ((((gcd_gcd @ nat) @ A2) @ B2) != C2)))))). % 6.41/2.69 thf(fact_4293_gcd__nat_Ostrict__coboundedI1, axiom, (! [A2:nat,C2:nat,B2:nat]: ((((((dvd_dvd @ nat) @ A2) @ C2) & (A2 != C2)) => ((((dvd_dvd @ nat) @ (((gcd_gcd @ nat) @ A2) @ B2)) @ C2) & ((((gcd_gcd @ nat) @ A2) @ B2) != C2)))))). % 6.41/2.69 thf(fact_4294_gcd__nat_Ostrict__order__iff, axiom, (! [A2:nat,B2:nat]: ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) = ((A2 = (((gcd_gcd @ nat) @ A2) @ B2)) & (A2 != B2)))))). % 6.41/2.69 thf(fact_4295_gcd__nat_Ostrict__boundedE, axiom, (! [A2:nat,B2:nat,C2:nat]: ((((((dvd_dvd @ nat) @ A2) @ (((gcd_gcd @ nat) @ B2) @ C2)) & (A2 != (((gcd_gcd @ nat) @ B2) @ C2))) => (~ ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) => (~ (((((dvd_dvd @ nat) @ A2) @ C2) & (A2 != C2))))))))))). % 6.41/2.69 thf(fact_4296_gcd__nat_OcoboundedI2, axiom, (! [B2:nat,C2:nat,A2:nat]: (((((dvd_dvd @ nat) @ B2) @ C2) => (((dvd_dvd @ nat) @ (((gcd_gcd @ nat) @ A2) @ B2)) @ C2))))). % 6.41/2.69 thf(fact_4297_gcd__nat_OcoboundedI1, axiom, (! [A2:nat,C2:nat,B2:nat]: (((((dvd_dvd @ nat) @ A2) @ C2) => (((dvd_dvd @ nat) @ (((gcd_gcd @ nat) @ A2) @ B2)) @ C2))))). % 6.41/2.69 thf(fact_4298_gcd__nat_Oabsorb__iff2, axiom, ((dvd_dvd @ nat) = (^ [B4:nat,A4:nat]: (((((gcd_gcd @ nat) @ A4) @ B4) = B4))))). % 6.41/2.69 thf(fact_4299_gcd__nat_Oabsorb__iff1, axiom, ((dvd_dvd @ nat) = (^ [A4:nat,B4:nat]: (((((gcd_gcd @ nat) @ A4) @ B4) = A4))))). % 6.41/2.69 thf(fact_4300_gcd__nat_Ocobounded2, axiom, (! [A2:nat,B2:nat]: ((((dvd_dvd @ nat) @ (((gcd_gcd @ nat) @ A2) @ B2)) @ B2)))). % 6.41/2.69 thf(fact_4301_gcd__nat_Ocobounded1, axiom, (! [A2:nat,B2:nat]: ((((dvd_dvd @ nat) @ (((gcd_gcd @ nat) @ A2) @ B2)) @ A2)))). % 6.41/2.69 thf(fact_4302_gcd__nat_Oorder__iff, axiom, ((dvd_dvd @ nat) = (^ [A4:nat,B4:nat]: ((A4 = (((gcd_gcd @ nat) @ A4) @ B4)))))). % 6.41/2.69 thf(fact_4303_gcd__nat_OboundedI, axiom, (! [A2:nat,B2:nat,C2:nat]: (((((dvd_dvd @ nat) @ A2) @ B2) => ((((dvd_dvd @ nat) @ A2) @ C2) => (((dvd_dvd @ nat) @ A2) @ (((gcd_gcd @ nat) @ B2) @ C2))))))). % 6.41/2.69 thf(fact_4304_gcd__nat_OboundedE, axiom, (! [A2:nat,B2:nat,C2:nat]: (((((dvd_dvd @ nat) @ A2) @ (((gcd_gcd @ nat) @ B2) @ C2)) => (~ (((((dvd_dvd @ nat) @ A2) @ B2) => (~ ((((dvd_dvd @ nat) @ A2) @ C2)))))))))). % 6.41/2.69 thf(fact_4305_gcd__nat_Oabsorb4, axiom, (! [B2:nat,A2:nat]: ((((((dvd_dvd @ nat) @ B2) @ A2) & (B2 != A2)) => ((((gcd_gcd @ nat) @ A2) @ B2) = B2))))). % 6.41/2.69 thf(fact_4306_gcd__nat_Oabsorb3, axiom, (! [A2:nat,B2:nat]: ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) => ((((gcd_gcd @ nat) @ A2) @ B2) = A2))))). % 6.41/2.69 thf(fact_4307_gcd__nat_OorderI, axiom, (! [A2:nat,B2:nat]: (((A2 = (((gcd_gcd @ nat) @ A2) @ B2)) => (((dvd_dvd @ nat) @ A2) @ B2))))). % 6.41/2.69 thf(fact_4308_gcd__nat_OorderE, axiom, (! [A2:nat,B2:nat]: (((((dvd_dvd @ nat) @ A2) @ B2) => (A2 = (((gcd_gcd @ nat) @ A2) @ B2)))))). % 6.41/2.69 thf(fact_4309_gcd__nat_Omono, axiom, (! [A2:nat,C2:nat,B2:nat,D2:nat]: (((((dvd_dvd @ nat) @ A2) @ C2) => ((((dvd_dvd @ nat) @ B2) @ D2) => (((dvd_dvd @ nat) @ (((gcd_gcd @ nat) @ A2) @ B2)) @ (((gcd_gcd @ nat) @ C2) @ D2))))))). % 6.41/2.69 thf(fact_4310_disjoint__iff__not__equal, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))) = (! [X:A]: (((((member @ A) @ X) @ A3) => (! [Y2:A]: (((((member @ A) @ Y2) @ B3) => (X != Y2))))))))))). % 6.41/2.69 thf(fact_4311_Int__empty__right, axiom, (! [A:$tType,A3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_4312_Int__empty__left, axiom, (! [A:$tType,B3:(set @ A)]: (((((inf_inf @ (set @ A)) @ (bot_bot @ (set @ A))) @ B3) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_4313_disjoint__iff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))) = (! [X:A]: (((((member @ A) @ X) @ A3) => (~ ((((member @ A) @ X) @ B3)))))))))). % 6.41/2.69 thf(fact_4314_Int__emptyI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (~ ((((member @ A) @ X5) @ B3)))))) => ((((inf_inf @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))))))). % 6.41/2.69 thf(fact_4315_Int__mono, axiom, (! [A:$tType,A3:(set @ A),C5:(set @ A),B3:(set @ A),D6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ C5) => ((((ord_less_eq @ (set @ A)) @ B3) @ D6) => (((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) @ (((inf_inf @ (set @ A)) @ C5) @ D6))))))). % 6.41/2.69 thf(fact_4316_Int__lower1, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) @ A3)))). % 6.41/2.69 thf(fact_4317_Int__lower2, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) @ B3)))). % 6.41/2.69 thf(fact_4318_Int__absorb1, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B3) @ A3) => ((((inf_inf @ (set @ A)) @ A3) @ B3) = B3))))). % 6.41/2.69 thf(fact_4319_Int__absorb2, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((inf_inf @ (set @ A)) @ A3) @ B3) = A3))))). % 6.41/2.69 thf(fact_4320_Int__greatest, axiom, (! [A:$tType,C5:(set @ A),A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C5) @ A3) => ((((ord_less_eq @ (set @ A)) @ C5) @ B3) => (((ord_less_eq @ (set @ A)) @ C5) @ (((inf_inf @ (set @ A)) @ A3) @ B3))))))). % 6.41/2.69 thf(fact_4321_Int__Collect__mono, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((! [X5:A]: (((((member @ A) @ X5) @ A3) => ((P @ X5) => (Q @ X5))))) => (((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ ((collect @ A) @ P))) @ (((inf_inf @ (set @ A)) @ B3) @ ((collect @ A) @ Q)))))))). % 6.41/2.69 thf(fact_4322_Collect__conj__eq, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((collect @ A) @ (^ [X:A]: (((P @ X) & (Q @ X))))) = (((inf_inf @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))). % 6.41/2.69 thf(fact_4323_Int__Collect, axiom, (! [A:$tType,X3:A,A3:(set @ A),P:(A > $o)]: (((((member @ A) @ X3) @ (((inf_inf @ (set @ A)) @ A3) @ ((collect @ A) @ P))) = ((((member @ A) @ X3) @ A3) & (P @ X3)))))). % 6.41/2.69 thf(fact_4324_Int__def, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A6) & (((member @ A) @ X) @ B6))))))))))). % 6.41/2.69 thf(fact_4325_or__eq__0__iff, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A,B2:A]: ((((((bit_se1065995026697491101ons_or @ A) @ A2) @ B2) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) & (B2 = (zero_zero @ A)))))))))). % 6.41/2.69 thf(fact_4326_bit_Odisj__zero__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X3:A]: (((((bit_se1065995026697491101ons_or @ A) @ X3) @ (zero_zero @ A)) = X3))))))). % 6.41/2.69 thf(fact_4327_Int__left__commute, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (((inf_inf @ (set @ A)) @ B3) @ C5)) = (((inf_inf @ (set @ A)) @ B3) @ (((inf_inf @ (set @ A)) @ A3) @ C5)))))). % 6.41/2.69 thf(fact_4328_Int__left__absorb, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) = (((inf_inf @ (set @ A)) @ A3) @ B3))))). % 6.41/2.69 thf(fact_4329_Int__commute, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((((inf_inf @ (set @ A)) @ B6) @ A6))))))). % 6.41/2.69 thf(fact_4330_Int__absorb, axiom, (! [A:$tType,A3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ A3) = A3)))). % 6.41/2.69 thf(fact_4331_Int__assoc, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) @ C5) = (((inf_inf @ (set @ A)) @ A3) @ (((inf_inf @ (set @ A)) @ B3) @ C5)))))). % 6.41/2.69 thf(fact_4332_IntD2, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) => (((member @ A) @ C2) @ B3))))). % 6.41/2.69 thf(fact_4333_IntD1, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) => (((member @ A) @ C2) @ A3))))). % 6.41/2.69 thf(fact_4334_IntE, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) => (~ (((((member @ A) @ C2) @ A3) => (~ ((((member @ A) @ C2) @ B3)))))))))). % 6.41/2.69 thf(fact_4335_Diff__Int__distrib2, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ C5) = (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ C5)) @ (((inf_inf @ (set @ A)) @ B3) @ C5)))))). % 6.41/2.69 thf(fact_4336_Diff__Int__distrib, axiom, (! [A:$tType,C5:(set @ A),A3:(set @ A),B3:(set @ A)]: (((((inf_inf @ (set @ A)) @ C5) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ C5) @ A3)) @ (((inf_inf @ (set @ A)) @ C5) @ B3)))))). % 6.41/2.69 thf(fact_4337_Diff__Diff__Int, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = (((inf_inf @ (set @ A)) @ A3) @ B3))))). % 6.41/2.69 thf(fact_4338_Diff__Int2, axiom, (! [A:$tType,A3:(set @ A),C5:(set @ A),B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ C5)) @ (((inf_inf @ (set @ A)) @ B3) @ C5)) = (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ C5)) @ B3))))). % 6.41/2.69 thf(fact_4339_Int__Diff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) @ C5) = (((inf_inf @ (set @ A)) @ A3) @ (((minus_minus @ (set @ A)) @ B3) @ C5)))))). % 6.41/2.69 thf(fact_4340_Int__insert__left, axiom, (! [A:$tType,A2:A,C5:(set @ A),B3:(set @ A)]: ((((((member @ A) @ A2) @ C5) => ((((inf_inf @ (set @ A)) @ (((insert @ A) @ A2) @ B3)) @ C5) = (((insert @ A) @ A2) @ (((inf_inf @ (set @ A)) @ B3) @ C5)))) & ((~ ((((member @ A) @ A2) @ C5))) => ((((inf_inf @ (set @ A)) @ (((insert @ A) @ A2) @ B3)) @ C5) = (((inf_inf @ (set @ A)) @ B3) @ C5))))))). % 6.41/2.69 thf(fact_4341_Int__insert__right, axiom, (! [A:$tType,A2:A,A3:(set @ A),B3:(set @ A)]: ((((((member @ A) @ A2) @ A3) => ((((inf_inf @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B3)) = (((insert @ A) @ A2) @ (((inf_inf @ (set @ A)) @ A3) @ B3)))) & ((~ ((((member @ A) @ A2) @ A3))) => ((((inf_inf @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B3)) = (((inf_inf @ (set @ A)) @ A3) @ B3))))))). % 6.41/2.69 thf(fact_4342_Union__Int__subset, axiom, (! [A:$tType,A3:(set @ (set @ A)),B3:(set @ (set @ A))]: ((((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ (((inf_inf @ (set @ (set @ A))) @ A3) @ B3))) @ (((inf_inf @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ A3)) @ ((complete_Sup_Sup @ (set @ A)) @ B3)))))). % 6.41/2.69 thf(fact_4343_Sup__inter__less__eq, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ (((inf_inf @ (set @ A)) @ A3) @ B3))) @ (((inf_inf @ A) @ ((complete_Sup_Sup @ A) @ A3)) @ ((complete_Sup_Sup @ A) @ B3))))))))). % 6.41/2.69 thf(fact_4344_inf__cancel__left2, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A,A2:A,B2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ ((uminus_uminus @ A) @ X3)) @ A2)) @ (((inf_inf @ A) @ X3) @ B2)) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4345_inf__cancel__left1, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A,A2:A,B2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X3) @ A2)) @ (((inf_inf @ A) @ ((uminus_uminus @ A) @ X3)) @ B2)) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4346_gcd__le1__nat, axiom, (! [A2:nat,B2:nat]: (((A2 != (zero_zero @ nat)) => (((ord_less_eq @ nat) @ (((gcd_gcd @ nat) @ A2) @ B2)) @ A2))))). % 6.41/2.69 thf(fact_4347_gcd__le2__nat, axiom, (! [B2:nat,A2:nat]: (((B2 != (zero_zero @ nat)) => (((ord_less_eq @ nat) @ (((gcd_gcd @ nat) @ A2) @ B2)) @ B2))))). % 6.41/2.69 thf(fact_4348_Sup__inf__eq__bot__iff, axiom, (! [A:$tType]: (((comple592849572758109894attice @ A) => (! [B3:(set @ A),A2:A]: ((((((inf_inf @ A) @ ((complete_Sup_Sup @ A) @ B3)) @ A2) = (bot_bot @ A)) = (! [X:A]: (((((member @ A) @ X) @ B3) => ((((inf_inf @ A) @ X) @ A2) = (bot_bot @ A)))))))))))). % 6.41/2.69 thf(fact_4349_disjunctive__add, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A,B2:A]: (((! [N3:nat]: (((~ ((((bit_se5641148757651400278ts_bit @ A) @ A2) @ N3))) | (~ ((((bit_se5641148757651400278ts_bit @ A) @ B2) @ N3)))))) => ((((plus_plus @ A) @ A2) @ B2) = (((bit_se1065995026697491101ons_or @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_4350_gcd__diff1__nat, axiom, (! [N2:nat,M:nat]: (((((ord_less_eq @ nat) @ N2) @ M) => ((((gcd_gcd @ nat) @ (((minus_minus @ nat) @ M) @ N2)) @ N2) = (((gcd_gcd @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_4351_gcd__diff2__nat, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((gcd_gcd @ nat) @ (((minus_minus @ nat) @ N2) @ M)) @ N2) = (((gcd_gcd @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_4352_gcd__nat_Oelims, axiom, (! [X3:nat,Xa2:nat,Y:nat]: ((((((gcd_gcd @ nat) @ X3) @ Xa2) = Y) => (((Xa2 = (zero_zero @ nat)) => (Y = X3)) & ((Xa2 != (zero_zero @ nat)) => (Y = (((gcd_gcd @ nat) @ Xa2) @ (((modulo_modulo @ nat) @ X3) @ Xa2))))))))). % 6.41/2.69 thf(fact_4353_gcd__nat_Osimps, axiom, ((gcd_gcd @ nat) = (^ [X:nat,Y2:nat]: (((((if @ nat) @ (Y2 = (zero_zero @ nat))) @ X) @ (((gcd_gcd @ nat) @ Y2) @ (((modulo_modulo @ nat) @ X) @ Y2))))))). % 6.41/2.69 thf(fact_4354_gcd__non__0__nat, axiom, (! [Y:nat,X3:nat]: (((Y != (zero_zero @ nat)) => ((((gcd_gcd @ nat) @ X3) @ Y) = (((gcd_gcd @ nat) @ Y) @ (((modulo_modulo @ nat) @ X3) @ Y))))))). % 6.41/2.69 thf(fact_4355_Int__Diff__disjoint, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_4356_Diff__triv, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))) => ((((minus_minus @ (set @ A)) @ A3) @ B3) = A3))))). % 6.41/2.69 thf(fact_4357_Union__disjoint, axiom, (! [A:$tType,C5:(set @ (set @ A)),A3:(set @ A)]: ((((((inf_inf @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ C5)) @ A3) = (bot_bot @ (set @ A))) = (! [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ C5) => ((((inf_inf @ (set @ A)) @ X) @ A3) = (bot_bot @ (set @ A)))))))))). % 6.41/2.69 thf(fact_4358_ivl__disj__int__two_I3_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ M)) @ (((set_or7035219750837199246ssThan @ A) @ M) @ U)) = (bot_bot @ (set @ A))))))))). % 6.41/2.69 thf(fact_4359_gcd__integer_Orep__eq, axiom, (! [X3:code_integer,Xa2:code_integer]: (((code_int_of_integer @ (((gcd_gcd @ code_integer) @ X3) @ Xa2)) = (((gcd_gcd @ int) @ (code_int_of_integer @ X3)) @ (code_int_of_integer @ Xa2)))))). % 6.41/2.69 thf(fact_4360_gcd__integer_Oabs__eq, axiom, (! [Xa2:int,X3:int]: (((((gcd_gcd @ code_integer) @ (code_integer_of_int @ Xa2)) @ (code_integer_of_int @ X3)) = (code_integer_of_int @ (((gcd_gcd @ int) @ Xa2) @ X3)))))). % 6.41/2.69 thf(fact_4361_Diff__eq, axiom, (! [A:$tType]: (((minus_minus @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((((inf_inf @ (set @ A)) @ A6) @ ((uminus_uminus @ (set @ A)) @ B6)))))))). % 6.41/2.69 thf(fact_4362_inf__shunt, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A,Y:A]: ((((((inf_inf @ A) @ X3) @ Y) = (bot_bot @ A)) = (((ord_less_eq @ A) @ X3) @ ((uminus_uminus @ A) @ Y))))))))). % 6.41/2.69 thf(fact_4363_finite__Inf__in, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((! [X5:A,Y4:A]: (((((member @ A) @ X5) @ A3) => ((((member @ A) @ Y4) @ A3) => (((member @ A) @ (((inf_inf @ A) @ X5) @ Y4)) @ A3))))) => (((member @ A) @ ((complete_Inf_Inf @ A) @ A3)) @ A3)))))))))). % 6.41/2.69 thf(fact_4364_bezout__nat, axiom, (! [A2:nat,B2:nat]: (((A2 != (zero_zero @ nat)) => (? [X5:nat,Y4:nat]: (((((times_times @ nat) @ A2) @ X5) = (((plus_plus @ nat) @ (((times_times @ nat) @ B2) @ Y4)) @ (((gcd_gcd @ nat) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_4365_bezout__gcd__nat_H, axiom, (! [B2:nat,A2:nat]: ((? [X5:nat,Y4:nat]: ((((((ord_less_eq @ nat) @ (((times_times @ nat) @ B2) @ Y4)) @ (((times_times @ nat) @ A2) @ X5)) & ((((minus_minus @ nat) @ (((times_times @ nat) @ A2) @ X5)) @ (((times_times @ nat) @ B2) @ Y4)) = (((gcd_gcd @ nat) @ A2) @ B2))) | ((((ord_less_eq @ nat) @ (((times_times @ nat) @ A2) @ Y4)) @ (((times_times @ nat) @ B2) @ X5)) & ((((minus_minus @ nat) @ (((times_times @ nat) @ B2) @ X5)) @ (((times_times @ nat) @ A2) @ Y4)) = (((gcd_gcd @ nat) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_4366_ivl__disj__int__two_I7_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ M)) @ (((set_or1337092689740270186AtMost @ A) @ M) @ U)) = (bot_bot @ (set @ A))))))))). % 6.41/2.69 thf(fact_4367_ivl__disj__int__one_I4_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ ((set_ord_lessThan @ A) @ L)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ U)) = (bot_bot @ (set @ A))))))))). % 6.41/2.69 thf(fact_4368_ivl__disj__int__one_I2_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ ((set_ord_lessThan @ A) @ L)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ U)) = (bot_bot @ (set @ A))))))))). % 6.41/2.69 thf(fact_4369_disjoint__eq__subset__Compl, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))) = (((ord_less_eq @ (set @ A)) @ A3) @ ((uminus_uminus @ (set @ A)) @ B3)))))). % 6.41/2.69 thf(fact_4370_insert__partition, axiom, (! [A:$tType,X3:(set @ A),F5:(set @ (set @ A))]: (((~ ((((member @ (set @ A)) @ X3) @ F5))) => ((! [X5:(set @ A)]: (((((member @ (set @ A)) @ X5) @ (((insert @ (set @ A)) @ X3) @ F5)) => (! [Xa3:(set @ A)]: (((((member @ (set @ A)) @ Xa3) @ (((insert @ (set @ A)) @ X3) @ F5)) => ((X5 != Xa3) => ((((inf_inf @ (set @ A)) @ X5) @ Xa3) = (bot_bot @ (set @ A)))))))))) => ((((inf_inf @ (set @ A)) @ X3) @ ((complete_Sup_Sup @ (set @ A)) @ F5)) = (bot_bot @ (set @ A)))))))). % 6.41/2.69 thf(fact_4371_sum_Ointer__restrict, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),G:(B > A),B3:(set @ B)]: ((((finite_finite2 @ B) @ A3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A3) @ B3)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X:B]: (((((if @ A) @ (((member @ B) @ X) @ B3)) @ (G @ X)) @ (zero_zero @ A))))) @ A3))))))))). % 6.41/2.69 thf(fact_4372_prod_Ointer__restrict, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),G:(B > A),B3:(set @ B)]: ((((finite_finite2 @ B) @ A3) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A3) @ B3)) = ((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X:B]: (((((if @ A) @ (((member @ B) @ X) @ B3)) @ (G @ X)) @ (one_one @ A))))) @ A3))))))))). % 6.41/2.69 thf(fact_4373_gcd__code__integer, axiom, ((gcd_gcd @ code_integer) = (^ [K3:code_integer,L2:code_integer]: (((abs_abs @ code_integer) @ ((((if @ code_integer) @ (L2 = (zero_zero @ code_integer))) @ K3) @ (((gcd_gcd @ code_integer) @ L2) @ (((modulo_modulo @ code_integer) @ ((abs_abs @ code_integer) @ K3)) @ ((abs_abs @ code_integer) @ L2))))))))). % 6.41/2.69 thf(fact_4374_bit_Ocomplement__unique, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A2:A,X3:A,Y:A]: ((((((bit_se5824344872417868541ns_and @ A) @ A2) @ X3) = (zero_zero @ A)) => (((((bit_se1065995026697491101ons_or @ A) @ A2) @ X3) = ((uminus_uminus @ A) @ (one_one @ A))) => (((((bit_se5824344872417868541ns_and @ A) @ A2) @ Y) = (zero_zero @ A)) => (((((bit_se1065995026697491101ons_or @ A) @ A2) @ Y) = ((uminus_uminus @ A) @ (one_one @ A))) => (X3 = Y))))))))))). % 6.41/2.69 thf(fact_4375_sum_Omono__neutral__cong, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [T4:(set @ B),S3:(set @ B),H2:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ T4) => (((finite_finite2 @ B) @ S3) => ((! [I4:B]: (((((member @ B) @ I4) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((H2 @ I4) = (zero_zero @ A))))) => ((! [I4:B]: (((((member @ B) @ I4) @ (((minus_minus @ (set @ B)) @ S3) @ T4)) => ((G @ I4) = (zero_zero @ A))))) => ((! [X5:B]: (((((member @ B) @ X5) @ (((inf_inf @ (set @ B)) @ S3) @ T4)) => ((G @ X5) = (H2 @ X5))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ S3) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ T4))))))))))))). % 6.41/2.69 thf(fact_4376_Iio__Int__singleton, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,K:A]: ((((((ord_less @ A) @ X3) @ K) => ((((inf_inf @ (set @ A)) @ ((set_ord_lessThan @ A) @ K)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) & ((~ ((((ord_less @ A) @ X3) @ K))) => ((((inf_inf @ (set @ A)) @ ((set_ord_lessThan @ A) @ K)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))))))))))). % 6.41/2.69 thf(fact_4377_sum_OInt__Diff, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),G:(B > A),B3:(set @ B)]: ((((finite_finite2 @ B) @ A3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A3) @ B3))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A3) @ B3))))))))))). % 6.41/2.69 thf(fact_4378_prod_OInt__Diff, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),G:(B > A),B3:(set @ B)]: ((((finite_finite2 @ B) @ A3) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A3) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A3) @ B3))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A3) @ B3))))))))))). % 6.41/2.69 thf(fact_4379_prod_Omono__neutral__cong, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [T4:(set @ B),S3:(set @ B),H2:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ T4) => (((finite_finite2 @ B) @ S3) => ((! [I4:B]: (((((member @ B) @ I4) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((H2 @ I4) = (one_one @ A))))) => ((! [I4:B]: (((((member @ B) @ I4) @ (((minus_minus @ (set @ B)) @ S3) @ T4)) => ((G @ I4) = (one_one @ A))))) => ((! [X5:B]: (((((member @ B) @ X5) @ (((inf_inf @ (set @ B)) @ S3) @ T4)) => ((G @ X5) = (H2 @ X5))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S3) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ T4))))))))))))). % 6.41/2.69 thf(fact_4380_card__Diff__subset__Int, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) => (((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = (((minus_minus @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ (((inf_inf @ (set @ A)) @ A3) @ B3)))))))). % 6.41/2.69 thf(fact_4381_gcd__int__def, axiom, ((gcd_gcd @ int) = (^ [X:int,Y2:int]: (((semiring_1_of_nat @ int) @ (((gcd_gcd @ nat) @ (nat2 @ ((abs_abs @ int) @ X))) @ (nat2 @ ((abs_abs @ int) @ Y2)))))))). % 6.41/2.69 thf(fact_4382_sum_OIf__cases, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),P:(B > $o),H2:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X:B]: (((((if @ A) @ (P @ X)) @ (H2 @ X)) @ (G @ X))))) @ A3) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ (((inf_inf @ (set @ B)) @ A3) @ ((collect @ B) @ P)))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A3) @ ((uminus_uminus @ (set @ B)) @ ((collect @ B) @ P))))))))))))). % 6.41/2.69 thf(fact_4383_prod_OIf__cases, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),P:(B > $o),H2:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X:B]: (((((if @ A) @ (P @ X)) @ (H2 @ X)) @ (G @ X))))) @ A3) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ (((inf_inf @ (set @ B)) @ A3) @ ((collect @ B) @ P)))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A3) @ ((uminus_uminus @ (set @ B)) @ ((collect @ B) @ P))))))))))))). % 6.41/2.69 thf(fact_4384_dvd__partition, axiom, (! [A:$tType,C5:(set @ (set @ A)),K:nat]: ((((finite_finite2 @ A) @ ((complete_Sup_Sup @ (set @ A)) @ C5)) => ((! [X5:(set @ A)]: (((((member @ (set @ A)) @ X5) @ C5) => (((dvd_dvd @ nat) @ K) @ ((finite_card @ A) @ X5))))) => ((! [X5:(set @ A)]: (((((member @ (set @ A)) @ X5) @ C5) => (! [Xa3:(set @ A)]: (((((member @ (set @ A)) @ Xa3) @ C5) => ((X5 != Xa3) => ((((inf_inf @ (set @ A)) @ X5) @ Xa3) = (bot_bot @ (set @ A)))))))))) => (((dvd_dvd @ nat) @ K) @ ((finite_card @ A) @ ((complete_Sup_Sup @ (set @ A)) @ C5))))))))). % 6.41/2.69 thf(fact_4385_sum__div__partition, axiom, (! [B:$tType,A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:(set @ B),F2:(B > A),B2:A]: ((((finite_finite2 @ B) @ A3) => ((((divide_divide @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ A3)) @ B2) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [A4:B]: ((((divide_divide @ A) @ (F2 @ A4)) @ B2)))) @ (((inf_inf @ (set @ B)) @ A3) @ ((collect @ B) @ (^ [A4:B]: ((((dvd_dvd @ A) @ B2) @ (F2 @ A4)))))))) @ (((divide_divide @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ (((inf_inf @ (set @ B)) @ A3) @ ((collect @ B) @ (^ [A4:B]: ((~ ((((dvd_dvd @ A) @ B2) @ (F2 @ A4)))))))))) @ B2)))))))))). % 6.41/2.69 thf(fact_4386_distinct__concat, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: ((((distinct @ (list @ A)) @ Xs) => ((! [Ys4:(list @ A)]: (((((member @ (list @ A)) @ Ys4) @ ((set2 @ (list @ A)) @ Xs)) => ((distinct @ A) @ Ys4)))) => ((! [Ys4:(list @ A),Zs:(list @ A)]: (((((member @ (list @ A)) @ Ys4) @ ((set2 @ (list @ A)) @ Xs)) => ((((member @ (list @ A)) @ Zs) @ ((set2 @ (list @ A)) @ Xs)) => ((Ys4 != Zs) => ((((inf_inf @ (set @ A)) @ ((set2 @ A) @ Ys4)) @ ((set2 @ A) @ Zs)) = (bot_bot @ (set @ A)))))))) => ((distinct @ A) @ ((concat @ A) @ Xs)))))))). % 6.41/2.69 thf(fact_4387_mask__Suc__exp, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat]: ((((bit_se2239418461657761734s_mask @ A) @ (suc @ N2)) = (((bit_se1065995026697491101ons_or @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2)) @ ((bit_se2239418461657761734s_mask @ A) @ N2))))))))). % 6.41/2.69 thf(fact_4388_bezw__aux, axiom, (! [X3:nat,Y:nat]: ((((semiring_1_of_nat @ int) @ (((gcd_gcd @ nat) @ X3) @ Y)) = (((plus_plus @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ ((bezw @ X3) @ Y))) @ ((semiring_1_of_nat @ int) @ X3))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ ((bezw @ X3) @ Y))) @ ((semiring_1_of_nat @ int) @ Y))))))). % 6.41/2.69 thf(fact_4389_one__or__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se1065995026697491101ons_or @ A) @ (one_one @ A)) @ A2) = (((plus_plus @ A) @ A2) @ ((zero_neq_one_of_bool @ A) @ (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2)))))))))). % 6.41/2.69 thf(fact_4390_or__one__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A]: (((((bit_se1065995026697491101ons_or @ A) @ A2) @ (one_one @ A)) = (((plus_plus @ A) @ A2) @ ((zero_neq_one_of_bool @ A) @ (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A2)))))))))). % 6.41/2.69 thf(fact_4391_mask__Suc__double, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat]: ((((bit_se2239418461657761734s_mask @ A) @ (suc @ N2)) = (((bit_se1065995026697491101ons_or @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((bit_se2239418461657761734s_mask @ A) @ N2)))))))))). % 6.41/2.69 thf(fact_4392_OR__upper, axiom, (! [X3:int,N2:nat,Y:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ X3) => ((((ord_less @ int) @ X3) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)) => ((((ord_less @ int) @ Y) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)) => (((ord_less @ int) @ (((bit_se1065995026697491101ons_or @ int) @ X3) @ Y)) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N2)))))))). % 6.41/2.69 thf(fact_4393_or__Suc__0__eq, axiom, (! [N2:nat]: (((((bit_se1065995026697491101ons_or @ nat) @ N2) @ (suc @ (zero_zero @ nat))) = (((plus_plus @ nat) @ N2) @ ((zero_neq_one_of_bool @ nat) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))))))). % 6.41/2.69 thf(fact_4394_Suc__0__or__eq, axiom, (! [N2:nat]: (((((bit_se1065995026697491101ons_or @ nat) @ (suc @ (zero_zero @ nat))) @ N2) = (((plus_plus @ nat) @ N2) @ ((zero_neq_one_of_bool @ nat) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))))))). % 6.41/2.69 thf(fact_4395_or__nat__rec, axiom, ((bit_se1065995026697491101ons_or @ nat) = (^ [M2:nat,N:nat]: ((((plus_plus @ nat) @ ((zero_neq_one_of_bool @ nat) @ ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M2))) | (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ nat) @ (((divide_divide @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))). % 6.41/2.69 thf(fact_4396_gcd__nat_Opelims, axiom, (! [X3:nat,Xa2:nat,Y:nat]: ((((((gcd_gcd @ nat) @ X3) @ Xa2) = Y) => ((((accp @ ((product_prod @ nat) @ nat)) @ gcd_nat_rel) @ ((((product_Pair @ nat) @ nat) @ X3) @ Xa2)) => (~ (((((Xa2 = (zero_zero @ nat)) => (Y = X3)) & ((Xa2 != (zero_zero @ nat)) => (Y = (((gcd_gcd @ nat) @ Xa2) @ (((modulo_modulo @ nat) @ X3) @ Xa2))))) => (~ ((((accp @ ((product_prod @ nat) @ nat)) @ gcd_nat_rel) @ ((((product_Pair @ nat) @ nat) @ X3) @ Xa2)))))))))))). % 6.41/2.69 thf(fact_4397_inf__bot__left, axiom, (! [A:$tType]: (((bounded_lattice_bot @ A) => (! [X3:A]: (((((inf_inf @ A) @ (bot_bot @ A)) @ X3) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4398_inf__bot__right, axiom, (! [A:$tType]: (((bounded_lattice_bot @ A) => (! [X3:A]: (((((inf_inf @ A) @ X3) @ (bot_bot @ A)) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4399_inf_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C2)) = ((((ord_less_eq @ A) @ A2) @ B2) & (((ord_less_eq @ A) @ A2) @ C2))))))))). % 6.41/2.69 thf(fact_4400_gcd__nat_Oidem, axiom, (! [A2:nat]: (((((gcd_gcd @ nat) @ A2) @ A2) = A2)))). % 6.41/2.69 thf(fact_4401_gcd__nat_Oleft__idem, axiom, (! [A2:nat,B2:nat]: (((((gcd_gcd @ nat) @ A2) @ (((gcd_gcd @ nat) @ A2) @ B2)) = (((gcd_gcd @ nat) @ A2) @ B2))))). % 6.41/2.69 thf(fact_4402_gcd__nat_Oright__idem, axiom, (! [A2:nat,B2:nat]: (((((gcd_gcd @ nat) @ (((gcd_gcd @ nat) @ A2) @ B2)) @ B2) = (((gcd_gcd @ nat) @ A2) @ B2))))). % 6.41/2.69 thf(fact_4403_gcd__idem__nat, axiom, (! [X3:nat]: (((((gcd_gcd @ nat) @ X3) @ X3) = X3)))). % 6.41/2.69 thf(fact_4404_le__inf__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X3:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X3) @ (((inf_inf @ A) @ Y) @ Z2)) = ((((ord_less_eq @ A) @ X3) @ Y) & (((ord_less_eq @ A) @ X3) @ Z2))))))))). % 6.41/2.69 thf(fact_4405_inf__set__def, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((collect @ A) @ (((inf_inf @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ A6)))) @ (^ [X:A]: ((((member @ A) @ X) @ B6))))))))))). % 6.41/2.69 thf(fact_4406_inf__Int__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: (((((inf_inf @ (A > (B > $o))) @ (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ R)))) @ (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ S3)))) = (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ (((inf_inf @ (set @ ((product_prod @ A) @ B))) @ R) @ S3)))))))). % 6.41/2.69 thf(fact_4407_inf_OcoboundedI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_4408_inf_OcoboundedI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ C2) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_4409_inf_Oabsorb__iff2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: (((((inf_inf @ A) @ A4) @ B4) = B4)))))))). % 6.41/2.69 thf(fact_4410_inf_Oabsorb__iff1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: (((((inf_inf @ A) @ A4) @ B4) = A4)))))))). % 6.41/2.69 thf(fact_4411_inf_Ocobounded2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ B2))))))). % 6.41/2.69 thf(fact_4412_inf_Ocobounded1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ A2))))))). % 6.41/2.69 thf(fact_4413_inf_Oorder__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: ((A4 = (((inf_inf @ A) @ A4) @ B4))))))))). % 6.41/2.69 thf(fact_4414_inf__greatest, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X3:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_less_eq @ A) @ X3) @ Z2) => (((ord_less_eq @ A) @ X3) @ (((inf_inf @ A) @ Y) @ Z2)))))))))). % 6.41/2.69 thf(fact_4415_inf_OboundedI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ A2) @ C2) => (((ord_less_eq @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C2)))))))))). % 6.41/2.69 thf(fact_4416_inf_OboundedE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C2)) => (~ (((((ord_less_eq @ A) @ A2) @ B2) => (~ ((((ord_less_eq @ A) @ A2) @ C2))))))))))))). % 6.41/2.69 thf(fact_4417_inf__absorb2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [Y:A,X3:A]: (((((ord_less_eq @ A) @ Y) @ X3) => ((((inf_inf @ A) @ X3) @ Y) = Y)))))))). % 6.41/2.69 thf(fact_4418_inf__absorb1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => ((((inf_inf @ A) @ X3) @ Y) = X3)))))))). % 6.41/2.69 thf(fact_4419_inf_Oabsorb2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((inf_inf @ A) @ A2) @ B2) = B2)))))))). % 6.41/2.69 thf(fact_4420_inf_Oabsorb1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((inf_inf @ A) @ A2) @ B2) = A2)))))))). % 6.41/2.69 thf(fact_4421_le__iff__inf, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [X:A,Y2:A]: (((((inf_inf @ A) @ X) @ Y2) = X)))))))). % 6.41/2.69 thf(fact_4422_inf__unique, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [F2:(A > (A > A)),X3:A,Y:A]: (((! [X5:A,Y4:A]: ((((ord_less_eq @ A) @ ((F2 @ X5) @ Y4)) @ X5))) => ((! [X5:A,Y4:A]: ((((ord_less_eq @ A) @ ((F2 @ X5) @ Y4)) @ Y4))) => ((! [X5:A,Y4:A,Z3:A]: (((((ord_less_eq @ A) @ X5) @ Y4) => ((((ord_less_eq @ A) @ X5) @ Z3) => (((ord_less_eq @ A) @ X5) @ ((F2 @ Y4) @ Z3)))))) => ((((inf_inf @ A) @ X3) @ Y) = ((F2 @ X3) @ Y))))))))))). % 6.41/2.69 thf(fact_4423_inf_OorderI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A]: (((A2 = (((inf_inf @ A) @ A2) @ B2)) => (((ord_less_eq @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_4424_inf_OorderE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (A2 = (((inf_inf @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_4425_le__infI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,X3:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ X3) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ X3)))))))). % 6.41/2.69 thf(fact_4426_le__infI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,X3:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ X3) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ X3)))))))). % 6.41/2.69 thf(fact_4427_inf__mono, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,C2:A,B2:A,D2:A]: (((((ord_less_eq @ A) @ A2) @ C2) => ((((ord_less_eq @ A) @ B2) @ D2) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ (((inf_inf @ A) @ C2) @ D2)))))))))). % 6.41/2.69 thf(fact_4428_le__infI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X3:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ X3) @ A2) => ((((ord_less_eq @ A) @ X3) @ B2) => (((ord_less_eq @ A) @ X3) @ (((inf_inf @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_4429_le__infE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X3:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ X3) @ (((inf_inf @ A) @ A2) @ B2)) => (~ (((((ord_less_eq @ A) @ X3) @ A2) => (~ ((((ord_less_eq @ A) @ X3) @ B2))))))))))))). % 6.41/2.69 thf(fact_4430_inf__le2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X3:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X3) @ Y)) @ Y))))))). % 6.41/2.69 thf(fact_4431_inf__le1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X3:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X3) @ Y)) @ X3))))))). % 6.41/2.69 thf(fact_4432_inf__sup__ord_I1_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X3:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X3) @ Y)) @ X3))))))). % 6.41/2.69 thf(fact_4433_inf__sup__ord_I2_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X3:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X3) @ Y)) @ Y))))))). % 6.41/2.69 thf(fact_4434_less__infI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,X3:A,B2:A]: (((((ord_less @ A) @ A2) @ X3) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ X3)))))))). % 6.41/2.69 thf(fact_4435_less__infI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,X3:A,A2:A]: (((((ord_less @ A) @ B2) @ X3) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ X3)))))))). % 6.41/2.69 thf(fact_4436_inf_Oabsorb3, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((inf_inf @ A) @ A2) @ B2) = A2)))))))). % 6.41/2.69 thf(fact_4437_inf_Oabsorb4, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => ((((inf_inf @ A) @ A2) @ B2) = B2)))))))). % 6.41/2.69 thf(fact_4438_inf_Ostrict__boundedE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C2)) => (~ (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ A2) @ C2))))))))))))). % 6.41/2.69 thf(fact_4439_inf_Ostrict__order__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less @ A) = (^ [A4:A,B4:A]: (((A4 = (((inf_inf @ A) @ A4) @ B4)) & (A4 != B4))))))))). % 6.41/2.69 thf(fact_4440_inf_Ostrict__coboundedI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less @ A) @ A2) @ C2) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_4441_inf_Ostrict__coboundedI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_4442_finite__inf__Sup, axiom, (! [A:$tType]: (((finite8700451911770168679attice @ A) => (! [A2:A,A3:(set @ A)]: (((((inf_inf @ A) @ A2) @ ((complete_Sup_Sup @ A) @ A3)) = ((complete_Sup_Sup @ A) @ ((collect @ A) @ (^ [Uu3:A]: ((? [B4:A]: (((Uu3 = (((inf_inf @ A) @ A2) @ B4)) & (((member @ A) @ B4) @ A3))))))))))))))). % 6.41/2.69 thf(fact_4443_or__nat__unfold, axiom, ((bit_se1065995026697491101ons_or @ nat) = (^ [M2:nat,N:nat]: (((((if @ nat) @ (M2 = (zero_zero @ nat))) @ N) @ ((((if @ nat) @ (N = (zero_zero @ nat))) @ M2) @ (((plus_plus @ nat) @ (((ord_max @ nat) @ (((modulo_modulo @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((modulo_modulo @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ nat) @ (((divide_divide @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))). % 6.41/2.69 thf(fact_4444_card__disjoint__shuffles, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((((inf_inf @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys)) = (bot_bot @ (set @ A))) => (((finite_card @ (list @ A)) @ (((shuffles @ A) @ Xs) @ Ys)) = ((binomial @ (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ A)) @ Ys))) @ ((size_size @ (list @ A)) @ Xs))))))). % 6.41/2.69 thf(fact_4445_distinct__product__lists, axiom, (! [A:$tType,Xss:(list @ (list @ A))]: (((! [X5:(list @ A)]: (((((member @ (list @ A)) @ X5) @ ((set2 @ (list @ A)) @ Xss)) => ((distinct @ A) @ X5)))) => ((distinct @ (list @ A)) @ ((product_lists @ A) @ Xss)))))). % 6.41/2.69 thf(fact_4446_max_Oabsorb1, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_max @ A) @ A2) @ B2) = A2)))))))). % 6.41/2.69 thf(fact_4447_max_Oabsorb2, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_max @ A) @ A2) @ B2) = B2)))))))). % 6.41/2.69 thf(fact_4448_max_Obounded__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less_eq @ A) @ (((ord_max @ A) @ B2) @ C2)) @ A2) = ((((ord_less_eq @ A) @ B2) @ A2) & (((ord_less_eq @ A) @ C2) @ A2))))))))). % 6.41/2.69 thf(fact_4449_max_Oabsorb3, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_max @ A) @ A2) @ B2) = A2)))))))). % 6.41/2.69 thf(fact_4450_max_Oabsorb4, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_max @ A) @ A2) @ B2) = B2)))))))). % 6.41/2.69 thf(fact_4451_max__less__iff__conj, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A,Z2:A]: (((((ord_less @ A) @ (((ord_max @ A) @ X3) @ Y)) @ Z2) = ((((ord_less @ A) @ X3) @ Z2) & (((ord_less @ A) @ Y) @ Z2))))))))). % 6.41/2.69 thf(fact_4452_max__bot2, axiom, (! [A:$tType]: (((order_bot @ A) => (! [X3:A]: (((((ord_max @ A) @ X3) @ (bot_bot @ A)) = X3))))))). % 6.41/2.69 thf(fact_4453_max__bot, axiom, (! [A:$tType]: (((order_bot @ A) => (! [X3:A]: (((((ord_max @ A) @ (bot_bot @ A)) @ X3) = X3))))))). % 6.41/2.69 thf(fact_4454_of__bool__or__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [P:$o,Q:$o]: ((((zero_neq_one_of_bool @ A) @ (P | Q)) = (((ord_max @ A) @ ((zero_neq_one_of_bool @ A) @ P)) @ ((zero_neq_one_of_bool @ A) @ Q))))))))). % 6.41/2.69 thf(fact_4455_max__Suc__Suc, axiom, (! [M:nat,N2:nat]: (((((ord_max @ nat) @ (suc @ M)) @ (suc @ N2)) = (suc @ (((ord_max @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_4456_max__0R, axiom, (! [N2:nat]: (((((ord_max @ nat) @ N2) @ (zero_zero @ nat)) = N2)))). % 6.41/2.69 thf(fact_4457_max__0L, axiom, (! [N2:nat]: (((((ord_max @ nat) @ (zero_zero @ nat)) @ N2) = N2)))). % 6.41/2.69 thf(fact_4458_max__nat_Oright__neutral, axiom, (! [A2:nat]: (((((ord_max @ nat) @ A2) @ (zero_zero @ nat)) = A2)))). % 6.41/2.69 thf(fact_4459_max__nat_Oneutr__eq__iff, axiom, (! [A2:nat,B2:nat]: ((((zero_zero @ nat) = (((ord_max @ nat) @ A2) @ B2)) = ((A2 = (zero_zero @ nat)) & (B2 = (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_4460_max__nat_Oleft__neutral, axiom, (! [A2:nat]: (((((ord_max @ nat) @ (zero_zero @ nat)) @ A2) = A2)))). % 6.41/2.69 thf(fact_4461_max__nat_Oeq__neutr__iff, axiom, (! [A2:nat,B2:nat]: ((((((ord_max @ nat) @ A2) @ B2) = (zero_zero @ nat)) = ((A2 = (zero_zero @ nat)) & (B2 = (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_4462_finite__shuffles, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((finite_finite2 @ (list @ A)) @ (((shuffles @ A) @ Xs) @ Ys))))). % 6.41/2.69 thf(fact_4463_max__number__of_I1_J, axiom, (! [A:$tType]: ((((numeral @ A) & (ord @ A)) => (! [U:num,V2:num]: ((((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V2)) => ((((ord_max @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V2)) = ((numeral_numeral @ A) @ V2))) & ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V2)))) => ((((ord_max @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V2)) = ((numeral_numeral @ A) @ U)))))))))). % 6.41/2.69 thf(fact_4464_max__0__1_I4_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X3:num]: (((((ord_max @ A) @ ((numeral_numeral @ A) @ X3)) @ (zero_zero @ A)) = ((numeral_numeral @ A) @ X3)))))))). % 6.41/2.69 thf(fact_4465_max__0__1_I3_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X3:num]: (((((ord_max @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ X3)) = ((numeral_numeral @ A) @ X3)))))))). % 6.41/2.69 thf(fact_4466_max__0__1_I1_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => ((((ord_max @ A) @ (zero_zero @ A)) @ (one_one @ A)) = (one_one @ A)))))). % 6.41/2.69 thf(fact_4467_max__0__1_I2_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => ((((ord_max @ A) @ (one_one @ A)) @ (zero_zero @ A)) = (one_one @ A)))))). % 6.41/2.69 thf(fact_4468_max__number__of_I2_J, axiom, (! [A:$tType]: ((((uminus @ A) & ((numeral @ A) & (ord @ A))) => (! [U:num,V2:num]: ((((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) => ((((ord_max @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2)))) & ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))))) => ((((ord_max @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) = ((numeral_numeral @ A) @ U)))))))))). % 6.41/2.69 thf(fact_4469_max__number__of_I3_J, axiom, (! [A:$tType]: ((((uminus @ A) & ((numeral @ A) & (ord @ A))) => (! [U:num,V2:num]: ((((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V2)) => ((((ord_max @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V2)) = ((numeral_numeral @ A) @ V2))) & ((~ ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V2)))) => ((((ord_max @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V2)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))))))))))). % 6.41/2.69 thf(fact_4470_max__number__of_I4_J, axiom, (! [A:$tType]: ((((uminus @ A) & ((numeral @ A) & (ord @ A))) => (! [U:num,V2:num]: ((((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) => ((((ord_max @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2)))) & ((~ ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))))) => ((((ord_max @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))))))))))). % 6.41/2.69 thf(fact_4471_max__Suc__numeral, axiom, (! [N2:nat,K:num]: (((((ord_max @ nat) @ (suc @ N2)) @ ((numeral_numeral @ nat) @ K)) = (suc @ (((ord_max @ nat) @ N2) @ (pred_numeral @ K))))))). % 6.41/2.69 thf(fact_4472_max__numeral__Suc, axiom, (! [K:num,N2:nat]: (((((ord_max @ nat) @ ((numeral_numeral @ nat) @ K)) @ (suc @ N2)) = (suc @ (((ord_max @ nat) @ (pred_numeral @ K)) @ N2)))))). % 6.41/2.69 thf(fact_4473_max_Ostrict__coboundedI2, axiom, (! [A:$tType]: (((linorder @ A) => (! [C2:A,B2:A,A2:A]: (((((ord_less @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ (((ord_max @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_4474_max_Ostrict__coboundedI1, axiom, (! [A:$tType]: (((linorder @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ C2) @ A2) => (((ord_less @ A) @ C2) @ (((ord_max @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_4475_max_Ostrict__order__iff, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less @ A) = (^ [B4:A,A4:A]: (((A4 = (((ord_max @ A) @ A4) @ B4)) & (A4 != B4))))))))). % 6.41/2.69 thf(fact_4476_max_Ostrict__boundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less @ A) @ (((ord_max @ A) @ B2) @ C2)) @ A2) => (~ (((((ord_less @ A) @ B2) @ A2) => (~ ((((ord_less @ A) @ C2) @ A2))))))))))))). % 6.41/2.69 thf(fact_4477_less__max__iff__disj, axiom, (! [A:$tType]: (((linorder @ A) => (! [Z2:A,X3:A,Y:A]: (((((ord_less @ A) @ Z2) @ (((ord_max @ A) @ X3) @ Y)) = ((((ord_less @ A) @ Z2) @ X3) | (((ord_less @ A) @ Z2) @ Y))))))))). % 6.41/2.69 thf(fact_4478_max_Omono, axiom, (! [A:$tType]: (((linorder @ A) => (! [C2:A,A2:A,D2:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ A2) => ((((ord_less_eq @ A) @ D2) @ B2) => (((ord_less_eq @ A) @ (((ord_max @ A) @ C2) @ D2)) @ (((ord_max @ A) @ A2) @ B2)))))))))). % 6.41/2.69 thf(fact_4479_max_OorderE, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => (A2 = (((ord_max @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_4480_max_OorderI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((A2 = (((ord_max @ A) @ A2) @ B2)) => (((ord_less_eq @ A) @ B2) @ A2)))))))). % 6.41/2.69 thf(fact_4481_max_OboundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less_eq @ A) @ (((ord_max @ A) @ B2) @ C2)) @ A2) => (~ (((((ord_less_eq @ A) @ B2) @ A2) => (~ ((((ord_less_eq @ A) @ C2) @ A2))))))))))))). % 6.41/2.69 thf(fact_4482_max_OboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ A2) => (((ord_less_eq @ A) @ (((ord_max @ A) @ B2) @ C2)) @ A2))))))))). % 6.41/2.69 thf(fact_4483_max_Oorder__iff, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: ((A4 = (((ord_max @ A) @ A4) @ B4))))))))). % 6.41/2.69 thf(fact_4484_max_Ocobounded1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ A2) @ (((ord_max @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_4485_max_Ocobounded2, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A2:A]: ((((ord_less_eq @ A) @ B2) @ (((ord_max @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_4486_le__max__iff__disj, axiom, (! [A:$tType]: (((linorder @ A) => (! [Z2:A,X3:A,Y:A]: (((((ord_less_eq @ A) @ Z2) @ (((ord_max @ A) @ X3) @ Y)) = ((((ord_less_eq @ A) @ Z2) @ X3) | (((ord_less_eq @ A) @ Z2) @ Y))))))))). % 6.41/2.69 thf(fact_4487_max_Oabsorb__iff1, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: (((((ord_max @ A) @ A4) @ B4) = A4)))))))). % 6.41/2.69 thf(fact_4488_max_Oabsorb__iff2, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: (((((ord_max @ A) @ A4) @ B4) = B4)))))))). % 6.41/2.69 thf(fact_4489_max_OcoboundedI1, axiom, (! [A:$tType]: (((linorder @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ A2) => (((ord_less_eq @ A) @ C2) @ (((ord_max @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_4490_max_OcoboundedI2, axiom, (! [A:$tType]: (((linorder @ A) => (! [C2:A,B2:A,A2:A]: (((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ (((ord_max @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_4491_max__def__raw, axiom, (! [A:$tType]: (((ord @ A) => ((ord_max @ A) = (^ [A4:A,B4:A]: (((((if @ A) @ (((ord_less_eq @ A) @ A4) @ B4)) @ B4) @ A4)))))))). % 6.41/2.69 thf(fact_4492_max__def, axiom, (! [A:$tType]: (((ord @ A) => ((ord_max @ A) = (^ [A4:A,B4:A]: (((((if @ A) @ (((ord_less_eq @ A) @ A4) @ B4)) @ B4) @ A4)))))))). % 6.41/2.69 thf(fact_4493_max__absorb1, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X3:A]: (((((ord_less_eq @ A) @ Y) @ X3) => ((((ord_max @ A) @ X3) @ Y) = X3)))))))). % 6.41/2.69 thf(fact_4494_max__absorb2, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_max @ A) @ X3) @ Y) = Y)))))))). % 6.41/2.69 thf(fact_4495_of__nat__max, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X3:nat,Y:nat]: ((((semiring_1_of_nat @ A) @ (((ord_max @ nat) @ X3) @ Y)) = (((ord_max @ A) @ ((semiring_1_of_nat @ A) @ X3)) @ ((semiring_1_of_nat @ A) @ Y))))))))). % 6.41/2.69 thf(fact_4496_shuffles__commutes, axiom, (! [A:$tType]: (((shuffles @ A) = (^ [Xs2:(list @ A),Ys3:(list @ A)]: ((((shuffles @ A) @ Ys3) @ Xs2))))))). % 6.41/2.69 thf(fact_4497_of__int__max, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:int,Y:int]: ((((ring_1_of_int @ A) @ (((ord_max @ int) @ X3) @ Y)) = (((ord_max @ A) @ ((ring_1_of_int @ A) @ X3)) @ ((ring_1_of_int @ A) @ Y))))))))). % 6.41/2.69 thf(fact_4498_nat__add__max__right, axiom, (! [M:nat,N2:nat,Q4:nat]: (((((plus_plus @ nat) @ M) @ (((ord_max @ nat) @ N2) @ Q4)) = (((ord_max @ nat) @ (((plus_plus @ nat) @ M) @ N2)) @ (((plus_plus @ nat) @ M) @ Q4)))))). % 6.41/2.69 thf(fact_4499_nat__add__max__left, axiom, (! [M:nat,N2:nat,Q4:nat]: (((((plus_plus @ nat) @ (((ord_max @ nat) @ M) @ N2)) @ Q4) = (((ord_max @ nat) @ (((plus_plus @ nat) @ M) @ Q4)) @ (((plus_plus @ nat) @ N2) @ Q4)))))). % 6.41/2.69 thf(fact_4500_max__add__distrib__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [X3:A,Y:A,Z2:A]: (((((plus_plus @ A) @ X3) @ (((ord_max @ A) @ Y) @ Z2)) = (((ord_max @ A) @ (((plus_plus @ A) @ X3) @ Y)) @ (((plus_plus @ A) @ X3) @ Z2))))))))). % 6.41/2.69 thf(fact_4501_max__add__distrib__left, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [X3:A,Y:A,Z2:A]: (((((plus_plus @ A) @ (((ord_max @ A) @ X3) @ Y)) @ Z2) = (((ord_max @ A) @ (((plus_plus @ A) @ X3) @ Z2)) @ (((plus_plus @ A) @ Y) @ Z2))))))))). % 6.41/2.69 thf(fact_4502_max__diff__distrib__left, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X3:A,Y:A,Z2:A]: (((((minus_minus @ A) @ (((ord_max @ A) @ X3) @ Y)) @ Z2) = (((ord_max @ A) @ (((minus_minus @ A) @ X3) @ Z2)) @ (((minus_minus @ A) @ Y) @ Z2))))))))). % 6.41/2.69 thf(fact_4503_nat__mult__max__left, axiom, (! [M:nat,N2:nat,Q4:nat]: (((((times_times @ nat) @ (((ord_max @ nat) @ M) @ N2)) @ Q4) = (((ord_max @ nat) @ (((times_times @ nat) @ M) @ Q4)) @ (((times_times @ nat) @ N2) @ Q4)))))). % 6.41/2.69 thf(fact_4504_nat__mult__max__right, axiom, (! [M:nat,N2:nat,Q4:nat]: (((((times_times @ nat) @ M) @ (((ord_max @ nat) @ N2) @ Q4)) = (((ord_max @ nat) @ (((times_times @ nat) @ M) @ N2)) @ (((times_times @ nat) @ M) @ Q4)))))). % 6.41/2.69 thf(fact_4505_nat__minus__add__max, axiom, (! [N2:nat,M:nat]: (((((plus_plus @ nat) @ (((minus_minus @ nat) @ N2) @ M)) @ M) = (((ord_max @ nat) @ N2) @ M))))). % 6.41/2.69 thf(fact_4506_length__shuffles, axiom, (! [A:$tType,Zs2:(list @ A),Xs:(list @ A),Ys:(list @ A)]: (((((member @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ Xs) @ Ys)) => (((size_size @ (list @ A)) @ Zs2) = (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ A)) @ Ys))))))). % 6.41/2.69 thf(fact_4507_max__Suc2, axiom, (! [M:nat,N2:nat]: (((((ord_max @ nat) @ M) @ (suc @ N2)) = ((((case_nat @ nat) @ (suc @ N2)) @ (^ [M4:nat]: ((suc @ (((ord_max @ nat) @ M4) @ N2))))) @ M))))). % 6.41/2.69 thf(fact_4508_max__Suc1, axiom, (! [N2:nat,M:nat]: (((((ord_max @ nat) @ (suc @ N2)) @ M) = ((((case_nat @ nat) @ (suc @ N2)) @ (^ [M4:nat]: ((suc @ (((ord_max @ nat) @ N2) @ M4))))) @ M))))). % 6.41/2.69 thf(fact_4509_Sup__insert__finite, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [S3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ S3) => (((S3 = (bot_bot @ (set @ A))) => (((complete_Sup_Sup @ A) @ (((insert @ A) @ X3) @ S3)) = X3)) & ((S3 != (bot_bot @ (set @ A))) => (((complete_Sup_Sup @ A) @ (((insert @ A) @ X3) @ S3)) = (((ord_max @ A) @ X3) @ ((complete_Sup_Sup @ A) @ S3)))))))))))). % 6.41/2.69 thf(fact_4510_distinct__disjoint__shuffles, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A)]: ((((distinct @ A) @ Xs) => (((distinct @ A) @ Ys) => (((((inf_inf @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys)) = (bot_bot @ (set @ A))) => ((((member @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ Xs) @ Ys)) => ((distinct @ A) @ Zs2)))))))). % 6.41/2.69 thf(fact_4511_in__set__product__lists__length, axiom, (! [A:$tType,Xs:(list @ A),Xss:(list @ (list @ A))]: (((((member @ (list @ A)) @ Xs) @ ((set2 @ (list @ A)) @ ((product_lists @ A) @ Xss))) => (((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ (list @ A))) @ Xss)))))). % 6.41/2.69 thf(fact_4512_mlex__eq, axiom, (! [A:$tType]: (((mlex_prod @ A) = (^ [F3:(A > nat),R6:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X:A,Y2:A]: (((((ord_less @ nat) @ (F3 @ X)) @ (F3 @ Y2)) | ((((ord_less_eq @ nat) @ (F3 @ X)) @ (F3 @ Y2)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y2)) @ R6))))))))))))). % 6.41/2.69 thf(fact_4513_divmod__integer__eq__cases, axiom, (code_divmod_integer = (^ [K3:code_integer,L2:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (K3 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (zero_zero @ code_integer)) @ (zero_zero @ code_integer))) @ ((((if @ ((product_prod @ code_integer) @ code_integer)) @ (L2 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (zero_zero @ code_integer)) @ K3)) @ (((((((comp @ code_integer) @ (((product_prod @ code_integer) @ code_integer) > ((product_prod @ code_integer) @ code_integer))) @ code_integer) @ (((((comp @ (code_integer > code_integer)) @ (((product_prod @ code_integer) @ code_integer) > ((product_prod @ code_integer) @ code_integer))) @ code_integer) @ (((product_apsnd @ code_integer) @ code_integer) @ code_integer)) @ (times_times @ code_integer))) @ (sgn_sgn @ code_integer)) @ L2) @ ((((if @ ((product_prod @ code_integer) @ code_integer)) @ (((sgn_sgn @ code_integer) @ K3) = ((sgn_sgn @ code_integer) @ L2))) @ ((code_divmod_abs @ K3) @ L2)) @ (((((product_case_prod @ code_integer) @ code_integer) @ ((product_prod @ code_integer) @ code_integer)) @ (^ [R5:code_integer,S8:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (S8 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ ((uminus_uminus @ code_integer) @ R5)) @ (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (((minus_minus @ code_integer) @ ((uminus_uminus @ code_integer) @ R5)) @ (one_one @ code_integer))) @ (((minus_minus @ code_integer) @ ((abs_abs @ code_integer) @ L2)) @ S8)))))) @ ((code_divmod_abs @ K3) @ L2)))))))))). % 6.41/2.69 thf(fact_4514_totally__bounded__Union, axiom, (! [A:$tType]: (((topolo7287701948861334536_space @ A) => (! [M7:(set @ (set @ A))]: ((((finite_finite2 @ (set @ A)) @ M7) => ((! [S5:(set @ A)]: (((((member @ (set @ A)) @ S5) @ M7) => ((topolo6688025880775521714ounded @ A) @ S5)))) => ((topolo6688025880775521714ounded @ A) @ ((complete_Sup_Sup @ (set @ A)) @ M7)))))))))). % 6.41/2.69 thf(fact_4515_nth__rotate1, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ ((rotate1 @ A) @ Xs)) @ N2) = (((nth @ A) @ Xs) @ (((modulo_modulo @ nat) @ (suc @ N2)) @ ((size_size @ (list @ A)) @ Xs)))))))). % 6.41/2.69 thf(fact_4516_set__rotate1, axiom, (! [A:$tType,Xs:(list @ A)]: ((((set2 @ A) @ ((rotate1 @ A) @ Xs)) = ((set2 @ A) @ Xs))))). % 6.41/2.69 thf(fact_4517_length__rotate1, axiom, (! [A:$tType,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ ((rotate1 @ A) @ Xs)) = ((size_size @ (list @ A)) @ Xs))))). % 6.41/2.69 thf(fact_4518_distinct1__rotate, axiom, (! [A:$tType,Xs:(list @ A)]: ((((distinct @ A) @ ((rotate1 @ A) @ Xs)) = ((distinct @ A) @ Xs))))). % 6.41/2.69 thf(fact_4519_totally__bounded__empty, axiom, (! [A:$tType]: (((topolo7287701948861334536_space @ A) => ((topolo6688025880775521714ounded @ A) @ (bot_bot @ (set @ A))))))). % 6.41/2.69 thf(fact_4520_rotate1__length01, axiom, (! [A:$tType,Xs:(list @ A)]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)) => (((rotate1 @ A) @ Xs) = Xs))))). % 6.41/2.69 thf(fact_4521_apsnd__compose, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,F2:(C > B),G:(D > C),X3:((product_prod @ A) @ D)]: (((((((product_apsnd @ C) @ B) @ A) @ F2) @ (((((product_apsnd @ D) @ C) @ A) @ G) @ X3)) = (((((product_apsnd @ D) @ B) @ A) @ (((((comp @ C) @ B) @ D) @ F2) @ G)) @ X3))))). % 6.41/2.69 thf(fact_4522_totally__bounded__subset, axiom, (! [A:$tType]: (((topolo7287701948861334536_space @ A) => (! [S3:(set @ A),T4:(set @ A)]: ((((topolo6688025880775521714ounded @ A) @ S3) => ((((ord_less_eq @ (set @ A)) @ T4) @ S3) => ((topolo6688025880775521714ounded @ A) @ T4))))))))). % 6.41/2.69 thf(fact_4523_sum__comp__morphism, axiom, (! [A:$tType,B:$tType,C:$tType]: ((((comm_monoid_add @ B) & (comm_monoid_add @ A)) => (! [H2:(B > A),G:(C > B),A3:(set @ C)]: ((((H2 @ (zero_zero @ B)) = (zero_zero @ A)) => ((! [X5:B,Y4:B]: (((H2 @ (((plus_plus @ B) @ X5) @ Y4)) = (((plus_plus @ A) @ (H2 @ X5)) @ (H2 @ Y4))))) => (((((groups7311177749621191930dd_sum @ C) @ A) @ (((((comp @ B) @ A) @ C) @ H2) @ G)) @ A3) = (H2 @ ((((groups7311177749621191930dd_sum @ C) @ B) @ G) @ A3))))))))))). % 6.41/2.69 thf(fact_4524_case__prod__comp, axiom, (! [D:$tType,A:$tType,C:$tType,B:$tType,F2:(D > (C > A)),G:(B > D),X3:((product_prod @ B) @ C)]: (((((((product_case_prod @ B) @ C) @ A) @ (((((comp @ D) @ (C > A)) @ B) @ F2) @ G)) @ X3) = ((F2 @ (G @ (((product_fst @ B) @ C) @ X3))) @ (((product_snd @ B) @ C) @ X3)))))). % 6.41/2.69 thf(fact_4525_mlex__leq, axiom, (! [A:$tType,F2:(A > nat),X3:A,Y:A,R:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ nat) @ (F2 @ X3)) @ (F2 @ Y)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ (((mlex_prod @ A) @ F2) @ R))))))). % 6.41/2.69 thf(fact_4526_mlex__less, axiom, (! [A:$tType,F2:(A > nat),X3:A,Y:A,R:(set @ ((product_prod @ A) @ A))]: (((((ord_less @ nat) @ (F2 @ X3)) @ (F2 @ Y)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ (((mlex_prod @ A) @ F2) @ R)))))). % 6.41/2.69 thf(fact_4527_mlex__iff, axiom, (! [A:$tType,X3:A,Y:A,F2:(A > nat),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ (((mlex_prod @ A) @ F2) @ R)) = ((((ord_less @ nat) @ (F2 @ X3)) @ (F2 @ Y)) | (((F2 @ X3) = (F2 @ Y)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ R))))))). % 6.41/2.69 thf(fact_4528_set__remove1__eq, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((distinct @ A) @ Xs) => (((set2 @ A) @ (((remove1 @ A) @ X3) @ Xs)) = (((minus_minus @ (set @ A)) @ ((set2 @ A) @ Xs)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))). % 6.41/2.69 thf(fact_4529_in__set__remove1, axiom, (! [A:$tType,A2:A,B2:A,Xs:(list @ A)]: (((A2 != B2) => ((((member @ A) @ A2) @ ((set2 @ A) @ (((remove1 @ A) @ B2) @ Xs))) = (((member @ A) @ A2) @ ((set2 @ A) @ Xs))))))). % 6.41/2.69 thf(fact_4530_fst__comp__apsnd, axiom, (! [C:$tType,B:$tType,A:$tType,F2:(B > C)]: (((((((comp @ ((product_prod @ A) @ C)) @ A) @ ((product_prod @ A) @ B)) @ ((product_fst @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F2)) = ((product_fst @ A) @ B))))). % 6.41/2.69 thf(fact_4531_snd__comp__apsnd, axiom, (! [C:$tType,B:$tType,A:$tType,F2:(B > C)]: (((((((comp @ ((product_prod @ A) @ C)) @ C) @ ((product_prod @ A) @ B)) @ ((product_snd @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F2)) = (((((comp @ B) @ C) @ ((product_prod @ A) @ B)) @ F2) @ ((product_snd @ A) @ B)))))). % 6.41/2.69 thf(fact_4532_fst__diag__fst, axiom, (! [B:$tType,A:$tType]: (((((((comp @ ((product_prod @ A) @ A)) @ A) @ ((product_prod @ A) @ B)) @ ((product_fst @ A) @ A)) @ (((((comp @ A) @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ B)) @ (^ [X:A]: (((((product_Pair @ A) @ A) @ X) @ X)))) @ ((product_fst @ A) @ B))) = ((product_fst @ A) @ B))))). % 6.41/2.69 thf(fact_4533_snd__diag__snd, axiom, (! [B:$tType,A:$tType]: (((((((comp @ ((product_prod @ B) @ B)) @ B) @ ((product_prod @ A) @ B)) @ ((product_snd @ B) @ B)) @ (((((comp @ B) @ ((product_prod @ B) @ B)) @ ((product_prod @ A) @ B)) @ (^ [X:B]: (((((product_Pair @ B) @ B) @ X) @ X)))) @ ((product_snd @ A) @ B))) = ((product_snd @ A) @ B))))). % 6.41/2.69 thf(fact_4534_card_Ocomp__fun__commute__on, axiom, ((((((comp @ nat) @ nat) @ nat) @ suc) @ suc) = (((((comp @ nat) @ nat) @ nat) @ suc) @ suc))). % 6.41/2.69 thf(fact_4535_fst__diag__snd, axiom, (! [B:$tType,A:$tType]: (((((((comp @ ((product_prod @ B) @ B)) @ B) @ ((product_prod @ A) @ B)) @ ((product_fst @ B) @ B)) @ (((((comp @ B) @ ((product_prod @ B) @ B)) @ ((product_prod @ A) @ B)) @ (^ [X:B]: (((((product_Pair @ B) @ B) @ X) @ X)))) @ ((product_snd @ A) @ B))) = ((product_snd @ A) @ B))))). % 6.41/2.69 thf(fact_4536_snd__diag__fst, axiom, (! [B:$tType,A:$tType]: (((((((comp @ ((product_prod @ A) @ A)) @ A) @ ((product_prod @ A) @ B)) @ ((product_snd @ A) @ A)) @ (((((comp @ A) @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ B)) @ (^ [X:A]: (((((product_Pair @ A) @ A) @ X) @ X)))) @ ((product_fst @ A) @ B))) = ((product_fst @ A) @ B))))). % 6.41/2.69 thf(fact_4537_distinct__remove1, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((distinct @ A) @ Xs) => ((distinct @ A) @ (((remove1 @ A) @ X3) @ Xs)))))). % 6.41/2.69 thf(fact_4538_notin__set__remove1, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Y:A]: (((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)))) => (~ ((((member @ A) @ X3) @ ((set2 @ A) @ (((remove1 @ A) @ Y) @ Xs))))))))). % 6.41/2.69 thf(fact_4539_remove1__idem, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)))) => ((((remove1 @ A) @ X3) @ Xs) = Xs))))). % 6.41/2.69 thf(fact_4540_remove1__commute, axiom, (! [A:$tType,X3:A,Y:A,Zs2:(list @ A)]: (((((remove1 @ A) @ X3) @ (((remove1 @ A) @ Y) @ Zs2)) = (((remove1 @ A) @ Y) @ (((remove1 @ A) @ X3) @ Zs2)))))). % 6.41/2.69 thf(fact_4541_set__remove1__subset, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ (((remove1 @ A) @ X3) @ Xs))) @ ((set2 @ A) @ Xs))))). % 6.41/2.69 thf(fact_4542_sum_OatLeast__Suc__atMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (suc @ N2))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_4543_sum_OatLeast__Suc__lessThan__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ (suc @ N2))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_4544_sum_OatLeastAtMost__shift__bounds, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,K:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ (((plus_plus @ nat) @ N2) @ K))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ K))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_4545_sum_OatLeastLessThan__shift__bounds, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,K:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ (((plus_plus @ nat) @ N2) @ K))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ K))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_4546_prod_OatLeast__Suc__atMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (suc @ N2))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_4547_prod_OatLeast__Suc__lessThan__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ (suc @ N2))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_4548_prod_OatLeastAtMost__shift__bounds, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,K:nat,N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ (((plus_plus @ nat) @ N2) @ K))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ K))) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_4549_prod_OatLeastLessThan__shift__bounds, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,K:nat,N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ (((plus_plus @ nat) @ N2) @ K))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ K))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_4550_prod_OUnion__comp, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [B3:(set @ (set @ B)),G:(B > A)]: (((! [X5:(set @ B)]: (((((member @ (set @ B)) @ X5) @ B3) => ((finite_finite2 @ B) @ X5)))) => ((! [A14:(set @ B)]: (((((member @ (set @ B)) @ A14) @ B3) => (! [A25:(set @ B)]: (((((member @ (set @ B)) @ A25) @ B3) => ((A14 != A25) => (! [X5:B]: (((((member @ B) @ X5) @ A14) => ((((member @ B) @ X5) @ A25) => ((G @ X5) = (one_one @ A))))))))))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ ((complete_Sup_Sup @ (set @ B)) @ B3)) = (((((((comp @ ((set @ B) > A)) @ ((set @ (set @ B)) > A)) @ (B > A)) @ ((groups7121269368397514597t_prod @ (set @ B)) @ A)) @ ((groups7121269368397514597t_prod @ B) @ A)) @ G) @ B3)))))))))). % 6.41/2.69 thf(fact_4551_prod_OUnion__disjoint, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [C5:(set @ (set @ B)),G:(B > A)]: (((! [X5:(set @ B)]: (((((member @ (set @ B)) @ X5) @ C5) => ((finite_finite2 @ B) @ X5)))) => ((! [X5:(set @ B)]: (((((member @ (set @ B)) @ X5) @ C5) => (! [Xa3:(set @ B)]: (((((member @ (set @ B)) @ Xa3) @ C5) => ((X5 != Xa3) => ((((inf_inf @ (set @ B)) @ X5) @ Xa3) = (bot_bot @ (set @ B)))))))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ ((complete_Sup_Sup @ (set @ B)) @ C5)) = (((((((comp @ ((set @ B) > A)) @ ((set @ (set @ B)) > A)) @ (B > A)) @ ((groups7121269368397514597t_prod @ (set @ B)) @ A)) @ ((groups7121269368397514597t_prod @ B) @ A)) @ G) @ C5)))))))))). % 6.41/2.69 thf(fact_4552_sum_OatLeast0__atMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N2))) = (((plus_plus @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2)))))))))). % 6.41/2.69 thf(fact_4553_sum_OatLeast0__lessThan__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N2))) = (((plus_plus @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2)))))))))). % 6.41/2.69 thf(fact_4554_prod_OatLeast0__atMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N2))) = (((times_times @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2)))))))))). % 6.41/2.69 thf(fact_4555_prod_OatLeast0__lessThan__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N2))) = (((times_times @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2)))))))))). % 6.41/2.69 thf(fact_4556_sum_OatLeastLessThan__shift__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N2) @ M)))))))))). % 6.41/2.69 thf(fact_4557_prod_OatLeastLessThan__shift__0, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N2) @ M)))))))))). % 6.41/2.69 thf(fact_4558_length__remove1, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (((size_size @ (list @ A)) @ (((remove1 @ A) @ X3) @ Xs)) = (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)))) & ((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)))) => (((size_size @ (list @ A)) @ (((remove1 @ A) @ X3) @ Xs)) = ((size_size @ (list @ A)) @ Xs))))))). % 6.41/2.69 thf(fact_4559_sum_OUnion__comp, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [B3:(set @ (set @ B)),G:(B > A)]: (((! [X5:(set @ B)]: (((((member @ (set @ B)) @ X5) @ B3) => ((finite_finite2 @ B) @ X5)))) => ((! [A14:(set @ B)]: (((((member @ (set @ B)) @ A14) @ B3) => (! [A25:(set @ B)]: (((((member @ (set @ B)) @ A25) @ B3) => ((A14 != A25) => (! [X5:B]: (((((member @ B) @ X5) @ A14) => ((((member @ B) @ X5) @ A25) => ((G @ X5) = (zero_zero @ A))))))))))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ ((complete_Sup_Sup @ (set @ B)) @ B3)) = (((((((comp @ ((set @ B) > A)) @ ((set @ (set @ B)) > A)) @ (B > A)) @ ((groups7311177749621191930dd_sum @ (set @ B)) @ A)) @ ((groups7311177749621191930dd_sum @ B) @ A)) @ G) @ B3)))))))))). % 6.41/2.69 thf(fact_4560_sum_OatLeast__atMost__pred__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N:nat]: ((((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (suc @ N2))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_4561_sum_OatLeast__lessThan__pred__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N:nat]: ((((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat))))))) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ (suc @ N2))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_4562_prod_OatLeast__atMost__pred__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N:nat]: ((((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M)) @ (suc @ N2))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_4563_prod_OatLeast__lessThan__pred__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M:nat,N2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N:nat]: ((((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat))))))) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M)) @ (suc @ N2))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))))))))). % 6.41/2.69 thf(fact_4564_sum_OatLeastAtMost__shift__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N2) @ M))))))))))). % 6.41/2.69 thf(fact_4565_prod_OatLeastAtMost__shift__0, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N2) @ M))))))))))). % 6.41/2.69 thf(fact_4566_sum_OUnion__disjoint, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [C5:(set @ (set @ B)),G:(B > A)]: (((! [X5:(set @ B)]: (((((member @ (set @ B)) @ X5) @ C5) => ((finite_finite2 @ B) @ X5)))) => ((! [X5:(set @ B)]: (((((member @ (set @ B)) @ X5) @ C5) => (! [Xa3:(set @ B)]: (((((member @ (set @ B)) @ Xa3) @ C5) => ((X5 != Xa3) => ((((inf_inf @ (set @ B)) @ X5) @ Xa3) = (bot_bot @ (set @ B)))))))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ ((complete_Sup_Sup @ (set @ B)) @ C5)) = (((((((comp @ ((set @ B) > A)) @ ((set @ (set @ B)) > A)) @ (B > A)) @ ((groups7311177749621191930dd_sum @ (set @ B)) @ A)) @ ((groups7311177749621191930dd_sum @ B) @ A)) @ G) @ C5)))))))))). % 6.41/2.69 thf(fact_4567_in__finite__psubset, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ A3) @ B3)) @ (finite_psubset @ A)) = ((((ord_less @ (set @ A)) @ A3) @ B3) & ((finite_finite2 @ A) @ B3)))))). % 6.41/2.69 thf(fact_4568_snd__fst__flip, axiom, (! [A:$tType,B:$tType]: ((((product_snd @ B) @ A) = (((((comp @ ((product_prod @ A) @ B)) @ A) @ ((product_prod @ B) @ A)) @ ((product_fst @ A) @ B)) @ ((((product_case_prod @ B) @ A) @ ((product_prod @ A) @ B)) @ (^ [X:B,Y2:A]: (((((product_Pair @ A) @ B) @ Y2) @ X))))))))). % 6.41/2.69 thf(fact_4569_fst__snd__flip, axiom, (! [B:$tType,A:$tType]: ((((product_fst @ A) @ B) = (((((comp @ ((product_prod @ B) @ A)) @ A) @ ((product_prod @ A) @ B)) @ ((product_snd @ B) @ A)) @ ((((product_case_prod @ A) @ B) @ ((product_prod @ B) @ A)) @ (^ [X:A,Y2:B]: (((((product_Pair @ B) @ A) @ Y2) @ X))))))))). % 6.41/2.69 thf(fact_4570_in__measure, axiom, (! [A:$tType,X3:A,Y:A,F2:(A > nat)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ ((measure @ A) @ F2)) = (((ord_less @ nat) @ (F2 @ X3)) @ (F2 @ Y)))))). % 6.41/2.69 thf(fact_4571_fstI, axiom, (! [B:$tType,A:$tType,X3:((product_prod @ A) @ B),Y:A,Z2:B]: (((X3 = ((((product_Pair @ A) @ B) @ Y) @ Z2)) => ((((product_fst @ A) @ B) @ X3) = Y))))). % 6.41/2.69 thf(fact_4572_sndI, axiom, (! [A:$tType,B:$tType,X3:((product_prod @ A) @ B),Y:A,Z2:B]: (((X3 = ((((product_Pair @ A) @ B) @ Y) @ Z2)) => ((((product_snd @ A) @ B) @ X3) = Z2))))). % 6.41/2.69 thf(fact_4573_finite__psubset__def, axiom, (! [A:$tType]: (((finite_psubset @ A) = ((collect @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_case_prod @ (set @ A)) @ (set @ A)) @ $o) @ (^ [A6:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A6) @ B6) & ((finite_finite2 @ A) @ B6)))))))))). % 6.41/2.69 thf(fact_4574_drop__bit__rec, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se4197421643247451524op_bit @ A) = (^ [N:nat,A4:A]: (((((if @ A) @ (N = (zero_zero @ nat))) @ A4) @ (((bit_se4197421643247451524op_bit @ A) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ (((divide_divide @ A) @ A4) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))). % 6.41/2.69 thf(fact_4575_root__powr__inverse, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((root @ N2) @ X3) = (((powr @ real) @ X3) @ (((divide_divide @ real) @ (one_one @ real)) @ ((semiring_1_of_nat @ real) @ N2))))))))). % 6.41/2.69 thf(fact_4576_image2__def, axiom, (! [A:$tType,B:$tType,C:$tType]: (((((bNF_Greatest_image2 @ C) @ A) @ B) = (^ [A6:(set @ C),F3:(C > A),G2:(C > B)]: (((collect @ ((product_prod @ A) @ B)) @ (^ [Uu3:((product_prod @ A) @ B)]: ((? [A4:C]: (((Uu3 = ((((product_Pair @ A) @ B) @ (F3 @ A4)) @ (G2 @ A4))) & (((member @ C) @ A4) @ A6))))))))))))). % 6.41/2.69 thf(fact_4577_drop__bit__negative__int__iff, axiom, (! [N2:nat,K:int]: (((((ord_less @ int) @ (((bit_se4197421643247451524op_bit @ int) @ N2) @ K)) @ (zero_zero @ int)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))). % 6.41/2.69 thf(fact_4578_drop__bit__of__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat]: (((((bit_se4197421643247451524op_bit @ A) @ N2) @ (zero_zero @ A)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_4579_drop__bit__drop__bit, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,N2:nat,A2:A]: (((((bit_se4197421643247451524op_bit @ A) @ M) @ (((bit_se4197421643247451524op_bit @ A) @ N2) @ A2)) = (((bit_se4197421643247451524op_bit @ A) @ (((plus_plus @ nat) @ M) @ N2)) @ A2)))))))). % 6.41/2.69 thf(fact_4580_real__root__Suc__0, axiom, (! [X3:real]: ((((root @ (suc @ (zero_zero @ nat))) @ X3) = X3)))). % 6.41/2.69 thf(fact_4581_root__0, axiom, (! [X3:real]: ((((root @ (zero_zero @ nat)) @ X3) = (zero_zero @ real))))). % 6.41/2.69 thf(fact_4582_real__root__eq__iff, axiom, (! [N2:nat,X3:real,Y:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((root @ N2) @ X3) = ((root @ N2) @ Y)) = (X3 = Y)))))). % 6.41/2.69 thf(fact_4583_drop__bit__of__bool, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,B2:$o]: (((((bit_se4197421643247451524op_bit @ A) @ N2) @ ((zero_neq_one_of_bool @ A) @ B2)) = ((zero_neq_one_of_bool @ A) @ ((N2 = (zero_zero @ nat)) & B2))))))))). % 6.41/2.69 thf(fact_4584_drop__bit__of__Suc__0, axiom, (! [N2:nat]: (((((bit_se4197421643247451524op_bit @ nat) @ N2) @ (suc @ (zero_zero @ nat))) = ((zero_neq_one_of_bool @ nat) @ (N2 = (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_4585_drop__bit__Suc__minus__bit0, axiom, (! [N2:nat,K:num]: (((((bit_se4197421643247451524op_bit @ int) @ (suc @ N2)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ K)))) = (((bit_se4197421643247451524op_bit @ int) @ N2) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K))))))). % 6.41/2.69 thf(fact_4586_drop__bit__Suc__bit0, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N2:nat,K:num]: (((((bit_se4197421643247451524op_bit @ A) @ (suc @ N2)) @ ((numeral_numeral @ A) @ (bit0 @ K))) = (((bit_se4197421643247451524op_bit @ A) @ N2) @ ((numeral_numeral @ A) @ K))))))))). % 6.41/2.69 thf(fact_4587_drop__bit__Suc__bit1, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N2:nat,K:num]: (((((bit_se4197421643247451524op_bit @ A) @ (suc @ N2)) @ ((numeral_numeral @ A) @ (bit1 @ K))) = (((bit_se4197421643247451524op_bit @ A) @ N2) @ ((numeral_numeral @ A) @ K))))))))). % 6.41/2.69 thf(fact_4588_real__root__eq__0__iff, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((root @ N2) @ X3) = (zero_zero @ real)) = (X3 = (zero_zero @ real))))))). % 6.41/2.69 thf(fact_4589_real__root__less__iff, axiom, (! [N2:nat,X3:real,Y:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ ((root @ N2) @ X3)) @ ((root @ N2) @ Y)) = (((ord_less @ real) @ X3) @ Y)))))). % 6.41/2.69 thf(fact_4590_real__root__le__iff, axiom, (! [N2:nat,X3:real,Y:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less_eq @ real) @ ((root @ N2) @ X3)) @ ((root @ N2) @ Y)) = (((ord_less_eq @ real) @ X3) @ Y)))))). % 6.41/2.69 thf(fact_4591_real__root__eq__1__iff, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((root @ N2) @ X3) = (one_one @ real)) = (X3 = (one_one @ real))))))). % 6.41/2.69 thf(fact_4592_real__root__one, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((root @ N2) @ (one_one @ real)) = (one_one @ real)))))). % 6.41/2.69 thf(fact_4593_drop__bit__of__1, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat]: (((((bit_se4197421643247451524op_bit @ A) @ N2) @ (one_one @ A)) = ((zero_neq_one_of_bool @ A) @ (N2 = (zero_zero @ nat)))))))))). % 6.41/2.69 thf(fact_4594_real__root__lt__0__iff, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ ((root @ N2) @ X3)) @ (zero_zero @ real)) = (((ord_less @ real) @ X3) @ (zero_zero @ real))))))). % 6.41/2.69 thf(fact_4595_real__root__gt__0__iff, axiom, (! [N2:nat,Y:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ (zero_zero @ real)) @ ((root @ N2) @ Y)) = (((ord_less @ real) @ (zero_zero @ real)) @ Y)))))). % 6.41/2.69 thf(fact_4596_real__root__le__0__iff, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less_eq @ real) @ ((root @ N2) @ X3)) @ (zero_zero @ real)) = (((ord_less_eq @ real) @ X3) @ (zero_zero @ real))))))). % 6.41/2.69 thf(fact_4597_real__root__ge__0__iff, axiom, (! [N2:nat,Y:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ ((root @ N2) @ Y)) = (((ord_less_eq @ real) @ (zero_zero @ real)) @ Y)))))). % 6.41/2.69 thf(fact_4598_real__root__lt__1__iff, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ ((root @ N2) @ X3)) @ (one_one @ real)) = (((ord_less @ real) @ X3) @ (one_one @ real))))))). % 6.41/2.69 thf(fact_4599_real__root__gt__1__iff, axiom, (! [N2:nat,Y:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ (one_one @ real)) @ ((root @ N2) @ Y)) = (((ord_less @ real) @ (one_one @ real)) @ Y)))))). % 6.41/2.69 thf(fact_4600_real__root__le__1__iff, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less_eq @ real) @ ((root @ N2) @ X3)) @ (one_one @ real)) = (((ord_less_eq @ real) @ X3) @ (one_one @ real))))))). % 6.41/2.69 thf(fact_4601_real__root__ge__1__iff, axiom, (! [N2:nat,Y:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less_eq @ real) @ (one_one @ real)) @ ((root @ N2) @ Y)) = (((ord_less_eq @ real) @ (one_one @ real)) @ Y)))))). % 6.41/2.69 thf(fact_4602_drop__bit__Suc__minus__bit1, axiom, (! [N2:nat,K:num]: (((((bit_se4197421643247451524op_bit @ int) @ (suc @ N2)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ K)))) = (((bit_se4197421643247451524op_bit @ int) @ N2) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (inc @ K)))))))). % 6.41/2.69 thf(fact_4603_real__root__pow__pos2, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ X3) => ((((power_power @ real) @ ((root @ N2) @ X3)) @ N2) = X3)))))). % 6.41/2.69 thf(fact_4604_take__bit__eq__self__iff__drop__bit__eq__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,A2:A]: ((((((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2) = A2) = ((((bit_se4197421643247451524op_bit @ A) @ N2) @ A2) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_4605_take__bit__drop__bit, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,N2:nat,A2:A]: (((((bit_se2584673776208193580ke_bit @ A) @ M) @ (((bit_se4197421643247451524op_bit @ A) @ N2) @ A2)) = (((bit_se4197421643247451524op_bit @ A) @ N2) @ (((bit_se2584673776208193580ke_bit @ A) @ (((plus_plus @ nat) @ M) @ N2)) @ A2))))))))). % 6.41/2.69 thf(fact_4606_bit__drop__bit__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,A2:A]: ((((bit_se5641148757651400278ts_bit @ A) @ (((bit_se4197421643247451524op_bit @ A) @ N2) @ A2)) = (((((comp @ nat) @ $o) @ nat) @ ((bit_se5641148757651400278ts_bit @ A) @ A2)) @ ((plus_plus @ nat) @ N2))))))))). % 6.41/2.69 thf(fact_4607_real__root__less__mono, axiom, (! [N2:nat,X3:real,Y:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ X3) @ Y) => (((ord_less @ real) @ ((root @ N2) @ X3)) @ ((root @ N2) @ Y))))))). % 6.41/2.69 thf(fact_4608_real__root__le__mono, axiom, (! [N2:nat,X3:real,Y:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less_eq @ real) @ X3) @ Y) => (((ord_less_eq @ real) @ ((root @ N2) @ X3)) @ ((root @ N2) @ Y))))))). % 6.41/2.69 thf(fact_4609_real__root__power, axiom, (! [N2:nat,X3:real,K:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((root @ N2) @ (((power_power @ real) @ X3) @ K)) = (((power_power @ real) @ ((root @ N2) @ X3)) @ K)))))). % 6.41/2.69 thf(fact_4610_real__root__abs, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((root @ N2) @ ((abs_abs @ real) @ X3)) = ((abs_abs @ real) @ ((root @ N2) @ X3))))))). % 6.41/2.69 thf(fact_4611_sgn__root, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((sgn_sgn @ real) @ ((root @ N2) @ X3)) = ((sgn_sgn @ real) @ X3)))))). % 6.41/2.69 thf(fact_4612_bits__ident, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,A2:A]: (((((plus_plus @ A) @ (((bit_se4730199178511100633sh_bit @ A) @ N2) @ (((bit_se4197421643247451524op_bit @ A) @ N2) @ A2))) @ (((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2)) = A2))))))). % 6.41/2.69 thf(fact_4613_image2__eqI, axiom, (! [A:$tType,C:$tType,B:$tType,B2:A,F2:(B > A),X3:B,C2:C,G:(B > C),A3:(set @ B)]: (((B2 = (F2 @ X3)) => ((C2 = (G @ X3)) => ((((member @ B) @ X3) @ A3) => (((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ B2) @ C2)) @ ((((((bNF_Greatest_image2 @ B) @ A) @ C) @ A3) @ F2) @ G)))))))). % 6.41/2.69 thf(fact_4614_real__root__gt__zero, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((ord_less @ real) @ (zero_zero @ real)) @ ((root @ N2) @ X3))))))). % 6.41/2.69 thf(fact_4615_real__root__strict__decreasing, axiom, (! [N2:nat,N7:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ nat) @ N2) @ N7) => ((((ord_less @ real) @ (one_one @ real)) @ X3) => (((ord_less @ real) @ ((root @ N7) @ X3)) @ ((root @ N2) @ X3)))))))). % 6.41/2.69 thf(fact_4616_root__abs__power, axiom, (! [N2:nat,Y:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((abs_abs @ real) @ ((root @ N2) @ (((power_power @ real) @ Y) @ N2))) = ((abs_abs @ real) @ Y)))))). % 6.41/2.69 thf(fact_4617_real__root__pos__pos, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (((ord_less_eq @ real) @ (zero_zero @ real)) @ ((root @ N2) @ X3))))))). % 6.41/2.69 thf(fact_4618_real__root__strict__increasing, axiom, (! [N2:nat,N7:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ nat) @ N2) @ N7) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ (one_one @ real)) => (((ord_less @ real) @ ((root @ N2) @ X3)) @ ((root @ N7) @ X3))))))))). % 6.41/2.69 thf(fact_4619_real__root__decreasing, axiom, (! [N2:nat,N7:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less_eq @ nat) @ N2) @ N7) => ((((ord_less_eq @ real) @ (one_one @ real)) @ X3) => (((ord_less_eq @ real) @ ((root @ N7) @ X3)) @ ((root @ N2) @ X3)))))))). % 6.41/2.69 thf(fact_4620_real__root__pow__pos, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((power_power @ real) @ ((root @ N2) @ X3)) @ N2) = X3)))))). % 6.41/2.69 thf(fact_4621_real__root__pos__unique, axiom, (! [N2:nat,Y:real,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ Y) => (((((power_power @ real) @ Y) @ N2) = X3) => (((root @ N2) @ X3) = Y))))))). % 6.41/2.69 thf(fact_4622_real__root__power__cancel, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ X3) => (((root @ N2) @ (((power_power @ real) @ X3) @ N2)) = X3)))))). % 6.41/2.69 thf(fact_4623_drop__bit__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N2:nat,A2:A]: (((((bit_se4197421643247451524op_bit @ A) @ (suc @ N2)) @ A2) = (((bit_se4197421643247451524op_bit @ A) @ N2) @ (((divide_divide @ A) @ A2) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))). % 6.41/2.69 thf(fact_4624_real__root__increasing, axiom, (! [N2:nat,N7:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less_eq @ nat) @ N2) @ N7) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less_eq @ real) @ X3) @ (one_one @ real)) => (((ord_less_eq @ real) @ ((root @ N2) @ X3)) @ ((root @ N7) @ X3))))))))). % 6.41/2.69 thf(fact_4625_root__sgn__power, axiom, (! [N2:nat,Y:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((root @ N2) @ (((times_times @ real) @ ((sgn_sgn @ real) @ Y)) @ (((power_power @ real) @ ((abs_abs @ real) @ Y)) @ N2))) = Y))))). % 6.41/2.69 thf(fact_4626_sgn__power__root, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((times_times @ real) @ ((sgn_sgn @ real) @ ((root @ N2) @ X3))) @ (((power_power @ real) @ ((abs_abs @ real) @ ((root @ N2) @ X3))) @ N2)) = X3))))). % 6.41/2.69 thf(fact_4627_ln__root, axiom, (! [N2:nat,B2:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ (zero_zero @ real)) @ B2) => (((ln_ln @ real) @ ((root @ N2) @ B2)) = (((divide_divide @ real) @ ((ln_ln @ real) @ B2)) @ ((semiring_1_of_nat @ real) @ N2)))))))). % 6.41/2.69 thf(fact_4628_log__root, axiom, (! [N2:nat,A2:real,B2:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ (zero_zero @ real)) @ A2) => (((log @ B2) @ ((root @ N2) @ A2)) = (((divide_divide @ real) @ ((log @ B2) @ A2)) @ ((semiring_1_of_nat @ real) @ N2)))))))). % 6.41/2.69 thf(fact_4629_log__base__root, axiom, (! [N2:nat,B2:real,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ (zero_zero @ real)) @ B2) => (((log @ ((root @ N2) @ B2)) @ X3) = (((times_times @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ ((log @ B2) @ X3)))))))). % 6.41/2.69 thf(fact_4630_split__root, axiom, (! [P:(real > $o),N2:nat,X3:real]: (((P @ ((root @ N2) @ X3)) = (((N2 = (zero_zero @ nat)) => (P @ (zero_zero @ real))) & ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (! [Y2:real]: ((((((times_times @ real) @ ((sgn_sgn @ real) @ Y2)) @ (((power_power @ real) @ ((abs_abs @ real) @ Y2)) @ N2)) = X3) => (P @ Y2)))))))))). % 6.41/2.69 thf(fact_4631_in__lex__prod, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A7:A,B7:B,R2:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Pair @ A) @ B) @ A7) @ B7))) @ ((((lex_prod @ A) @ B) @ R2) @ S)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A7)) @ R2) | ((A2 = A7) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B2) @ B7)) @ S))))))). % 6.41/2.69 thf(fact_4632_div__add__self1__no__field, axiom, (! [B:$tType,A:$tType]: ((((euclid4440199948858584721cancel @ A) & (field @ B)) => (! [X3:B,B2:A,A2:A]: ((((((nO_MATCH @ B) @ A) @ X3) @ B2) => ((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ B2) @ A2)) @ B2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ (one_one @ A))))))))))). % 6.41/2.69 thf(fact_4633_div__add__self2__no__field, axiom, (! [B:$tType,A:$tType]: ((((euclid4440199948858584721cancel @ A) & (field @ B)) => (! [X3:B,B2:A,A2:A]: ((((((nO_MATCH @ B) @ A) @ X3) @ B2) => ((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ B2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ B2)) @ (one_one @ A))))))))))). % 6.41/2.69 thf(fact_4634_scale__right__distrib__NO__MATCH, axiom, (! [A:$tType]: (((real_V4867850818363320053vector @ A) => (! [X3:A,Y:A,A2:real]: ((((((nO_MATCH @ A) @ real) @ (((divide_divide @ A) @ X3) @ Y)) @ A2) => ((((real_V8093663219630862766scaleR @ A) @ A2) @ (((plus_plus @ A) @ X3) @ Y)) = (((plus_plus @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ X3)) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ Y)))))))))). % 6.41/2.69 thf(fact_4635_distrib__right__NO__MATCH, axiom, (! [B:$tType,A:$tType]: (((semiring @ A) => (! [X3:B,Y:B,C2:A,A2:A,B2:A]: ((((((nO_MATCH @ B) @ A) @ (((divide_divide @ B) @ X3) @ Y)) @ C2) => ((((times_times @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))). % 6.41/2.69 thf(fact_4636_distrib__left__NO__MATCH, axiom, (! [B:$tType,A:$tType]: (((semiring @ A) => (! [X3:B,Y:B,A2:A,B2:A,C2:A]: ((((((nO_MATCH @ B) @ A) @ (((divide_divide @ B) @ X3) @ Y)) @ A2) => ((((times_times @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ A2) @ B2)) @ (((times_times @ A) @ A2) @ C2)))))))))). % 6.41/2.69 thf(fact_4637_left__diff__distrib__NO__MATCH, axiom, (! [B:$tType,A:$tType]: (((ring @ A) => (! [X3:B,Y:B,C2:A,A2:A,B2:A]: ((((((nO_MATCH @ B) @ A) @ (((divide_divide @ B) @ X3) @ Y)) @ C2) => ((((times_times @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C2) = (((minus_minus @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))). % 6.41/2.69 thf(fact_4638_right__diff__distrib__NO__MATCH, axiom, (! [B:$tType,A:$tType]: (((ring @ A) => (! [X3:B,Y:B,A2:A,B2:A,C2:A]: ((((((nO_MATCH @ B) @ A) @ (((divide_divide @ B) @ X3) @ Y)) @ A2) => ((((times_times @ A) @ A2) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((times_times @ A) @ A2) @ B2)) @ (((times_times @ A) @ A2) @ C2)))))))))). % 6.41/2.69 thf(fact_4639_lex__prod__def, axiom, (! [B:$tType,A:$tType]: ((((lex_prod @ A) @ B) = (^ [Ra:(set @ ((product_prod @ A) @ A)),Rb:(set @ ((product_prod @ B) @ B))]: (((collect @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_case_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ $o) @ ((((product_case_prod @ A) @ B) @ (((product_prod @ A) @ B) > $o)) @ (^ [A4:A,B4:B]: (((((product_case_prod @ A) @ B) @ $o) @ (^ [A9:A,B13:B]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ A9)) @ Ra) | ((A4 = A9) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B4) @ B13)) @ Rb))))))))))))))))). % 6.41/2.69 thf(fact_4640_scale__left__distrib__NO__MATCH, axiom, (! [C:$tType,A:$tType]: (((real_V4867850818363320053vector @ A) => (! [X3:A,Y:A,C2:C,A2:real,B2:real]: ((((((nO_MATCH @ A) @ C) @ (((divide_divide @ A) @ X3) @ Y)) @ C2) => ((((real_V8093663219630862766scaleR @ A) @ (((plus_plus @ real) @ A2) @ B2)) @ X3) = (((plus_plus @ A) @ (((real_V8093663219630862766scaleR @ A) @ A2) @ X3)) @ (((real_V8093663219630862766scaleR @ A) @ B2) @ X3)))))))))). % 6.41/2.69 thf(fact_4641_horner__sum__eq__sum__funpow, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_0 @ A) => (((groups4207007520872428315er_sum @ B) @ A) = (^ [F3:(B > A),A4:A,Xs2:(list @ B)]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [N:nat]: (((((compow @ (A > A)) @ N) @ ((times_times @ A) @ A4)) @ (F3 @ (((nth @ B) @ Xs2) @ N)))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ B)) @ Xs2)))))))))). % 6.41/2.69 thf(fact_4642_set__zip, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys:(list @ B)]: ((((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys)) = ((collect @ ((product_prod @ A) @ B)) @ (^ [Uu3:((product_prod @ A) @ B)]: ((? [I3:nat]: (((Uu3 = ((((product_Pair @ A) @ B) @ (((nth @ A) @ Xs) @ I3)) @ (((nth @ B) @ Ys) @ I3))) & (((ord_less @ nat) @ I3) @ (((ord_min @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ B)) @ Ys))))))))))))). % 6.41/2.69 thf(fact_4643_Fpow__Pow__finite, axiom, (! [A:$tType]: (((finite_Fpow @ A) = (^ [A6:(set @ A)]: ((((inf_inf @ (set @ (set @ A))) @ ((pow2 @ A) @ A6)) @ ((collect @ (set @ A)) @ (finite_finite2 @ A))))))))). % 6.41/2.69 thf(fact_4644_Suc__funpow, axiom, (! [N2:nat]: (((((compow @ (nat > nat)) @ N2) @ suc) = ((plus_plus @ nat) @ N2))))). % 6.41/2.69 thf(fact_4645_min_Oabsorb1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_min @ A) @ A2) @ B2) = A2)))))))). % 6.41/2.69 thf(fact_4646_min_Oabsorb2, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_min @ A) @ A2) @ B2) = B2)))))))). % 6.41/2.69 thf(fact_4647_min_Obounded__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ (((ord_min @ A) @ B2) @ C2)) = ((((ord_less_eq @ A) @ A2) @ B2) & (((ord_less_eq @ A) @ A2) @ C2))))))))). % 6.41/2.69 thf(fact_4648_min_Oabsorb3, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_min @ A) @ A2) @ B2) = A2)))))))). % 6.41/2.69 thf(fact_4649_min_Oabsorb4, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_min @ A) @ A2) @ B2) = B2)))))))). % 6.41/2.69 thf(fact_4650_min__less__iff__conj, axiom, (! [A:$tType]: (((linorder @ A) => (! [Z2:A,X3:A,Y:A]: (((((ord_less @ A) @ Z2) @ (((ord_min @ A) @ X3) @ Y)) = ((((ord_less @ A) @ Z2) @ X3) & (((ord_less @ A) @ Z2) @ Y))))))))). % 6.41/2.69 thf(fact_4651_min__bot2, axiom, (! [A:$tType]: (((order_bot @ A) => (! [X3:A]: (((((ord_min @ A) @ X3) @ (bot_bot @ A)) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4652_min__bot, axiom, (! [A:$tType]: (((order_bot @ A) => (! [X3:A]: (((((ord_min @ A) @ (bot_bot @ A)) @ X3) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4653_funpow__0, axiom, (! [A:$tType,F2:(A > A),X3:A]: ((((((compow @ (A > A)) @ (zero_zero @ nat)) @ F2) @ X3) = X3)))). % 6.41/2.69 thf(fact_4654_max__min__same_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((((ord_max @ A) @ X3) @ (((ord_min @ A) @ X3) @ Y)) = X3))))))). % 6.41/2.69 thf(fact_4655_max__min__same_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((((ord_max @ A) @ (((ord_min @ A) @ X3) @ Y)) @ X3) = X3))))))). % 6.41/2.69 thf(fact_4656_max__min__same_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((((ord_max @ A) @ (((ord_min @ A) @ X3) @ Y)) @ Y) = Y))))))). % 6.41/2.69 thf(fact_4657_max__min__same_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X3:A]: (((((ord_max @ A) @ Y) @ (((ord_min @ A) @ X3) @ Y)) = Y))))))). % 6.41/2.69 thf(fact_4658_min__Suc__Suc, axiom, (! [M:nat,N2:nat]: (((((ord_min @ nat) @ (suc @ M)) @ (suc @ N2)) = (suc @ (((ord_min @ nat) @ M) @ N2)))))). % 6.41/2.69 thf(fact_4659_min__0R, axiom, (! [N2:nat]: (((((ord_min @ nat) @ N2) @ (zero_zero @ nat)) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_4660_min__0L, axiom, (! [N2:nat]: (((((ord_min @ nat) @ (zero_zero @ nat)) @ N2) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_4661_min__number__of_I1_J, axiom, (! [A:$tType]: ((((numeral @ A) & (ord @ A)) => (! [U:num,V2:num]: ((((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V2)) => ((((ord_min @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V2)) = ((numeral_numeral @ A) @ U))) & ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V2)))) => ((((ord_min @ A) @ ((numeral_numeral @ A) @ U)) @ ((numeral_numeral @ A) @ V2)) = ((numeral_numeral @ A) @ V2)))))))))). % 6.41/2.69 thf(fact_4662_min__0__1_I4_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X3:num]: (((((ord_min @ A) @ ((numeral_numeral @ A) @ X3)) @ (zero_zero @ A)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_4663_min__0__1_I3_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X3:num]: (((((ord_min @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ X3)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_4664_min__0__1_I1_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => ((((ord_min @ A) @ (zero_zero @ A)) @ (one_one @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_4665_min__0__1_I2_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => ((((ord_min @ A) @ (one_one @ A)) @ (zero_zero @ A)) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_4666_min__number__of_I2_J, axiom, (! [A:$tType]: ((((uminus @ A) & ((numeral @ A) & (ord @ A))) => (! [U:num,V2:num]: ((((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) => ((((ord_min @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) = ((numeral_numeral @ A) @ U))) & ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))))) => ((((ord_min @ A) @ ((numeral_numeral @ A) @ U)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))))))))))). % 6.41/2.69 thf(fact_4667_min__number__of_I3_J, axiom, (! [A:$tType]: ((((uminus @ A) & ((numeral @ A) & (ord @ A))) => (! [U:num,V2:num]: ((((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V2)) => ((((ord_min @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V2)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U)))) & ((~ ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V2)))) => ((((ord_min @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((numeral_numeral @ A) @ V2)) = ((numeral_numeral @ A) @ V2)))))))))). % 6.41/2.69 thf(fact_4668_min__number__of_I4_J, axiom, (! [A:$tType]: ((((uminus @ A) & ((numeral @ A) & (ord @ A))) => (! [U:num,V2:num]: ((((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) => ((((ord_min @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U)))) & ((~ ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))))) => ((((ord_min @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ U))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))))))))))). % 6.41/2.69 thf(fact_4669_min__Suc__numeral, axiom, (! [N2:nat,K:num]: (((((ord_min @ nat) @ (suc @ N2)) @ ((numeral_numeral @ nat) @ K)) = (suc @ (((ord_min @ nat) @ N2) @ (pred_numeral @ K))))))). % 6.41/2.69 thf(fact_4670_min__numeral__Suc, axiom, (! [K:num,N2:nat]: (((((ord_min @ nat) @ ((numeral_numeral @ nat) @ K)) @ (suc @ N2)) = (suc @ (((ord_min @ nat) @ (pred_numeral @ K)) @ N2)))))). % 6.41/2.69 thf(fact_4671_zip__replicate, axiom, (! [A:$tType,B:$tType,I:nat,X3:A,J:nat,Y:B]: ((((((zip @ A) @ B) @ (((replicate @ A) @ I) @ X3)) @ (((replicate @ B) @ J) @ Y)) = (((replicate @ ((product_prod @ A) @ B)) @ (((ord_min @ nat) @ I) @ J)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)))))). % 6.41/2.69 thf(fact_4672_length__zip, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B)]: ((((size_size @ (list @ ((product_prod @ A) @ B))) @ ((((zip @ A) @ B) @ Xs) @ Ys)) = (((ord_min @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ B)) @ Ys)))))). % 6.41/2.69 thf(fact_4673_comp__funpow, axiom, (! [B:$tType,A:$tType,N2:nat,F2:(A > A)]: (((((compow @ ((B > A) > (B > A))) @ N2) @ ((((comp @ A) @ A) @ B) @ F2)) = ((((comp @ A) @ A) @ B) @ (((compow @ (A > A)) @ N2) @ F2)))))). % 6.41/2.69 thf(fact_4674_min_Ostrict__coboundedI2, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ (((ord_min @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_4675_min_Ostrict__coboundedI1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less @ A) @ A2) @ C2) => (((ord_less @ A) @ (((ord_min @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_4676_min_Ostrict__order__iff, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less @ A) = (^ [A4:A,B4:A]: (((A4 = (((ord_min @ A) @ A4) @ B4)) & (A4 != B4))))))))). % 6.41/2.69 thf(fact_4677_min_Ostrict__boundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ (((ord_min @ A) @ B2) @ C2)) => (~ (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ A2) @ C2))))))))))))). % 6.41/2.69 thf(fact_4678_min__less__iff__disj, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A,Z2:A]: (((((ord_less @ A) @ (((ord_min @ A) @ X3) @ Y)) @ Z2) = ((((ord_less @ A) @ X3) @ Z2) | (((ord_less @ A) @ Y) @ Z2))))))))). % 6.41/2.69 thf(fact_4679_min_Omono, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,C2:A,B2:A,D2:A]: (((((ord_less_eq @ A) @ A2) @ C2) => ((((ord_less_eq @ A) @ B2) @ D2) => (((ord_less_eq @ A) @ (((ord_min @ A) @ A2) @ B2)) @ (((ord_min @ A) @ C2) @ D2)))))))))). % 6.41/2.69 thf(fact_4680_min_OorderE, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (A2 = (((ord_min @ A) @ A2) @ B2))))))))). % 6.41/2.69 thf(fact_4681_min_OorderI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((A2 = (((ord_min @ A) @ A2) @ B2)) => (((ord_less_eq @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_4682_min_OboundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ (((ord_min @ A) @ B2) @ C2)) => (~ (((((ord_less_eq @ A) @ A2) @ B2) => (~ ((((ord_less_eq @ A) @ A2) @ C2))))))))))))). % 6.41/2.69 thf(fact_4683_min_OboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ A2) @ C2) => (((ord_less_eq @ A) @ A2) @ (((ord_min @ A) @ B2) @ C2)))))))))). % 6.41/2.69 thf(fact_4684_min_Oorder__iff, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: ((A4 = (((ord_min @ A) @ A4) @ B4))))))))). % 6.41/2.69 thf(fact_4685_min_Ocobounded1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ (((ord_min @ A) @ A2) @ B2)) @ A2))))))). % 6.41/2.69 thf(fact_4686_min_Ocobounded2, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ (((ord_min @ A) @ A2) @ B2)) @ B2))))))). % 6.41/2.69 thf(fact_4687_min_Oabsorb__iff1, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: (((((ord_min @ A) @ A4) @ B4) = A4)))))))). % 6.41/2.69 thf(fact_4688_min_Oabsorb__iff2, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: (((((ord_min @ A) @ A4) @ B4) = B4)))))))). % 6.41/2.69 thf(fact_4689_min_OcoboundedI1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ C2) => (((ord_less_eq @ A) @ (((ord_min @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_4690_min_OcoboundedI2, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ (((ord_min @ A) @ A2) @ B2)) @ C2)))))))). % 6.41/2.69 thf(fact_4691_min__le__iff__disj, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ (((ord_min @ A) @ X3) @ Y)) @ Z2) = ((((ord_less_eq @ A) @ X3) @ Z2) | (((ord_less_eq @ A) @ Y) @ Z2))))))))). % 6.41/2.69 thf(fact_4692_min__def__raw, axiom, (! [A:$tType]: (((ord @ A) => ((ord_min @ A) = (^ [A4:A,B4:A]: (((((if @ A) @ (((ord_less_eq @ A) @ A4) @ B4)) @ A4) @ B4)))))))). % 6.41/2.69 thf(fact_4693_min__def, axiom, (! [A:$tType]: (((ord @ A) => ((ord_min @ A) = (^ [A4:A,B4:A]: (((((if @ A) @ (((ord_less_eq @ A) @ A4) @ B4)) @ A4) @ B4)))))))). % 6.41/2.69 thf(fact_4694_min__absorb1, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_min @ A) @ X3) @ Y) = X3)))))))). % 6.41/2.69 thf(fact_4695_min__absorb2, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X3:A]: (((((ord_less_eq @ A) @ Y) @ X3) => ((((ord_min @ A) @ X3) @ Y) = Y)))))))). % 6.41/2.69 thf(fact_4696_of__nat__min, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X3:nat,Y:nat]: ((((semiring_1_of_nat @ A) @ (((ord_min @ nat) @ X3) @ Y)) = (((ord_min @ A) @ ((semiring_1_of_nat @ A) @ X3)) @ ((semiring_1_of_nat @ A) @ Y))))))))). % 6.41/2.69 thf(fact_4697_funpow__swap1, axiom, (! [A:$tType,F2:(A > A),N2:nat,X3:A]: (((F2 @ ((((compow @ (A > A)) @ N2) @ F2) @ X3)) = ((((compow @ (A > A)) @ N2) @ F2) @ (F2 @ X3)))))). % 6.41/2.69 thf(fact_4698_of__int__min, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:int,Y:int]: ((((ring_1_of_int @ A) @ (((ord_min @ int) @ X3) @ Y)) = (((ord_min @ A) @ ((ring_1_of_int @ A) @ X3)) @ ((ring_1_of_int @ A) @ Y))))))))). % 6.41/2.69 thf(fact_4699_min__add__distrib__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [X3:A,Y:A,Z2:A]: (((((plus_plus @ A) @ X3) @ (((ord_min @ A) @ Y) @ Z2)) = (((ord_min @ A) @ (((plus_plus @ A) @ X3) @ Y)) @ (((plus_plus @ A) @ X3) @ Z2))))))))). % 6.41/2.69 thf(fact_4700_min__add__distrib__left, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [X3:A,Y:A,Z2:A]: (((((plus_plus @ A) @ (((ord_min @ A) @ X3) @ Y)) @ Z2) = (((ord_min @ A) @ (((plus_plus @ A) @ X3) @ Z2)) @ (((plus_plus @ A) @ Y) @ Z2))))))))). % 6.41/2.69 thf(fact_4701_min__diff__distrib__left, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X3:A,Y:A,Z2:A]: (((((minus_minus @ A) @ (((ord_min @ A) @ X3) @ Y)) @ Z2) = (((ord_min @ A) @ (((minus_minus @ A) @ X3) @ Z2)) @ (((minus_minus @ A) @ Y) @ Z2))))))))). % 6.41/2.69 thf(fact_4702_min__diff, axiom, (! [M:nat,I:nat,N2:nat]: (((((ord_min @ nat) @ (((minus_minus @ nat) @ M) @ I)) @ (((minus_minus @ nat) @ N2) @ I)) = (((minus_minus @ nat) @ (((ord_min @ nat) @ M) @ N2)) @ I))))). % 6.41/2.69 thf(fact_4703_funpow__mult, axiom, (! [A:$tType,N2:nat,M:nat,F2:(A > A)]: (((((compow @ (A > A)) @ N2) @ (((compow @ (A > A)) @ M) @ F2)) = (((compow @ (A > A)) @ (((times_times @ nat) @ M) @ N2)) @ F2))))). % 6.41/2.69 thf(fact_4704_nat__mult__min__left, axiom, (! [M:nat,N2:nat,Q4:nat]: (((((times_times @ nat) @ (((ord_min @ nat) @ M) @ N2)) @ Q4) = (((ord_min @ nat) @ (((times_times @ nat) @ M) @ Q4)) @ (((times_times @ nat) @ N2) @ Q4)))))). % 6.41/2.69 thf(fact_4705_nat__mult__min__right, axiom, (! [M:nat,N2:nat,Q4:nat]: (((((times_times @ nat) @ M) @ (((ord_min @ nat) @ N2) @ Q4)) = (((ord_min @ nat) @ (((times_times @ nat) @ M) @ N2)) @ (((times_times @ nat) @ M) @ Q4)))))). % 6.41/2.69 thf(fact_4706_inf__nat__def, axiom, ((inf_inf @ nat) = (ord_min @ nat))). % 6.41/2.69 thf(fact_4707_funpow__Suc__right, axiom, (! [A:$tType,N2:nat,F2:(A > A)]: (((((compow @ (A > A)) @ (suc @ N2)) @ F2) = (((((comp @ A) @ A) @ A) @ (((compow @ (A > A)) @ N2) @ F2)) @ F2))))). % 6.41/2.69 thf(fact_4708_funpow_Osimps_I2_J, axiom, (! [A:$tType,N2:nat,F2:(A > A)]: (((((compow @ (A > A)) @ (suc @ N2)) @ F2) = (((((comp @ A) @ A) @ A) @ F2) @ (((compow @ (A > A)) @ N2) @ F2)))))). % 6.41/2.69 thf(fact_4709_funpow__add, axiom, (! [A:$tType,M:nat,N2:nat,F2:(A > A)]: (((((compow @ (A > A)) @ (((plus_plus @ nat) @ M) @ N2)) @ F2) = (((((comp @ A) @ A) @ A) @ (((compow @ (A > A)) @ M) @ F2)) @ (((compow @ (A > A)) @ N2) @ F2)))))). % 6.41/2.69 thf(fact_4710_minus__min__eq__max, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [X3:A,Y:A]: ((((uminus_uminus @ A) @ (((ord_min @ A) @ X3) @ Y)) = (((ord_max @ A) @ ((uminus_uminus @ A) @ X3)) @ ((uminus_uminus @ A) @ Y))))))))). % 6.41/2.69 thf(fact_4711_minus__max__eq__min, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [X3:A,Y:A]: ((((uminus_uminus @ A) @ (((ord_max @ A) @ X3) @ Y)) = (((ord_min @ A) @ ((uminus_uminus @ A) @ X3)) @ ((uminus_uminus @ A) @ Y))))))))). % 6.41/2.69 thf(fact_4712_empty__in__Fpow, axiom, (! [A:$tType,A3:(set @ A)]: ((((member @ (set @ A)) @ (bot_bot @ (set @ A))) @ ((finite_Fpow @ A) @ A3))))). % 6.41/2.69 thf(fact_4713_Fpow__not__empty, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_Fpow @ A) @ A3) != (bot_bot @ (set @ (set @ A))))))). % 6.41/2.69 thf(fact_4714_max__mult__distrib__left, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [P4:A,X3:A,Y:A]: ((((((ord_less_eq @ A) @ (zero_zero @ A)) @ P4) => ((((times_times @ A) @ P4) @ (((ord_max @ A) @ X3) @ Y)) = (((ord_max @ A) @ (((times_times @ A) @ P4) @ X3)) @ (((times_times @ A) @ P4) @ Y)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ P4))) => ((((times_times @ A) @ P4) @ (((ord_max @ A) @ X3) @ Y)) = (((ord_min @ A) @ (((times_times @ A) @ P4) @ X3)) @ (((times_times @ A) @ P4) @ Y))))))))))). % 6.41/2.69 thf(fact_4715_min__mult__distrib__left, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [P4:A,X3:A,Y:A]: ((((((ord_less_eq @ A) @ (zero_zero @ A)) @ P4) => ((((times_times @ A) @ P4) @ (((ord_min @ A) @ X3) @ Y)) = (((ord_min @ A) @ (((times_times @ A) @ P4) @ X3)) @ (((times_times @ A) @ P4) @ Y)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ P4))) => ((((times_times @ A) @ P4) @ (((ord_min @ A) @ X3) @ Y)) = (((ord_max @ A) @ (((times_times @ A) @ P4) @ X3)) @ (((times_times @ A) @ P4) @ Y))))))))))). % 6.41/2.69 thf(fact_4716_max__mult__distrib__right, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [P4:A,X3:A,Y:A]: ((((((ord_less_eq @ A) @ (zero_zero @ A)) @ P4) => ((((times_times @ A) @ (((ord_max @ A) @ X3) @ Y)) @ P4) = (((ord_max @ A) @ (((times_times @ A) @ X3) @ P4)) @ (((times_times @ A) @ Y) @ P4)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ P4))) => ((((times_times @ A) @ (((ord_max @ A) @ X3) @ Y)) @ P4) = (((ord_min @ A) @ (((times_times @ A) @ X3) @ P4)) @ (((times_times @ A) @ Y) @ P4))))))))))). % 6.41/2.69 thf(fact_4717_min__mult__distrib__right, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [P4:A,X3:A,Y:A]: ((((((ord_less_eq @ A) @ (zero_zero @ A)) @ P4) => ((((times_times @ A) @ (((ord_min @ A) @ X3) @ Y)) @ P4) = (((ord_min @ A) @ (((times_times @ A) @ X3) @ P4)) @ (((times_times @ A) @ Y) @ P4)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ P4))) => ((((times_times @ A) @ (((ord_min @ A) @ X3) @ Y)) @ P4) = (((ord_max @ A) @ (((times_times @ A) @ X3) @ P4)) @ (((times_times @ A) @ Y) @ P4))))))))))). % 6.41/2.69 thf(fact_4718_of__nat__def, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((semiring_1_of_nat @ A) = (^ [N:nat]: (((((compow @ (A > A)) @ N) @ ((plus_plus @ A) @ (one_one @ A))) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_4719_numeral__add__unfold__funpow, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [K:num,A2:A]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ K)) @ A2) = ((((compow @ (A > A)) @ ((numeral_numeral @ nat) @ K)) @ ((plus_plus @ A) @ (one_one @ A))) @ A2)))))))). % 6.41/2.69 thf(fact_4720_min__divide__distrib__right, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [P4:A,X3:A,Y:A]: ((((((ord_less_eq @ A) @ (zero_zero @ A)) @ P4) => ((((divide_divide @ A) @ (((ord_min @ A) @ X3) @ Y)) @ P4) = (((ord_min @ A) @ (((divide_divide @ A) @ X3) @ P4)) @ (((divide_divide @ A) @ Y) @ P4)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ P4))) => ((((divide_divide @ A) @ (((ord_min @ A) @ X3) @ Y)) @ P4) = (((ord_max @ A) @ (((divide_divide @ A) @ X3) @ P4)) @ (((divide_divide @ A) @ Y) @ P4))))))))))). % 6.41/2.69 thf(fact_4721_max__divide__distrib__right, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [P4:A,X3:A,Y:A]: ((((((ord_less_eq @ A) @ (zero_zero @ A)) @ P4) => ((((divide_divide @ A) @ (((ord_max @ A) @ X3) @ Y)) @ P4) = (((ord_max @ A) @ (((divide_divide @ A) @ X3) @ P4)) @ (((divide_divide @ A) @ Y) @ P4)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ P4))) => ((((divide_divide @ A) @ (((ord_max @ A) @ X3) @ Y)) @ P4) = (((ord_min @ A) @ (((divide_divide @ A) @ X3) @ P4)) @ (((divide_divide @ A) @ Y) @ P4))))))))))). % 6.41/2.69 thf(fact_4722_min__Suc2, axiom, (! [M:nat,N2:nat]: (((((ord_min @ nat) @ M) @ (suc @ N2)) = ((((case_nat @ nat) @ (zero_zero @ nat)) @ (^ [M4:nat]: ((suc @ (((ord_min @ nat) @ M4) @ N2))))) @ M))))). % 6.41/2.69 thf(fact_4723_min__Suc1, axiom, (! [N2:nat,M:nat]: (((((ord_min @ nat) @ (suc @ N2)) @ M) = ((((case_nat @ nat) @ (zero_zero @ nat)) @ (^ [M4:nat]: ((suc @ (((ord_min @ nat) @ N2) @ M4))))) @ M))))). % 6.41/2.69 thf(fact_4724_Inf__insert__finite, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [S3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ S3) => (((S3 = (bot_bot @ (set @ A))) => (((complete_Inf_Inf @ A) @ (((insert @ A) @ X3) @ S3)) = X3)) & ((S3 != (bot_bot @ (set @ A))) => (((complete_Inf_Inf @ A) @ (((insert @ A) @ X3) @ S3)) = (((ord_min @ A) @ X3) @ ((complete_Inf_Inf @ A) @ S3)))))))))))). % 6.41/2.69 thf(fact_4725_Fpow__mono, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ (set @ (set @ A))) @ ((finite_Fpow @ A) @ A3)) @ ((finite_Fpow @ A) @ B3)))))). % 6.41/2.69 thf(fact_4726_Fpow__def, axiom, (! [A:$tType]: (((finite_Fpow @ A) = (^ [A6:(set @ A)]: (((collect @ (set @ A)) @ (^ [X6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ X6) @ A6) & ((finite_finite2 @ A) @ X6))))))))))). % 6.41/2.69 thf(fact_4727_numeral__unfold__funpow, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((numeral_numeral @ A) = (^ [K3:num]: (((((compow @ (A > A)) @ ((numeral_numeral @ nat) @ K3)) @ ((plus_plus @ A) @ (one_one @ A))) @ (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_4728_relpowp__bot, axiom, (! [A:$tType,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((compow @ (A > (A > $o))) @ N2) @ (bot_bot @ (A > (A > $o)))) = (bot_bot @ (A > (A > $o)))))))). % 6.41/2.69 thf(fact_4729_relpowp__fun__conv, axiom, (! [A:$tType]: (((compow @ (A > (A > $o))) = (^ [N:nat,P3:(A > (A > $o)),X:A,Y2:A]: ((? [F3:(nat > A)]: ((((F3 @ (zero_zero @ nat)) = X) & (((F3 @ N) = Y2) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ N) => ((P3 @ (F3 @ I3)) @ (F3 @ (suc @ I3)))))))))))))))). % 6.41/2.69 thf(fact_4730_same__fst__def, axiom, (! [B:$tType,A:$tType]: ((((same_fst @ A) @ B) = (^ [P3:(A > $o),R6:(A > (set @ ((product_prod @ B) @ B)))]: (((collect @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_case_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ $o) @ ((((product_case_prod @ A) @ B) @ (((product_prod @ A) @ B) > $o)) @ (^ [X9:A,Y6:B]: (((((product_case_prod @ A) @ B) @ $o) @ (^ [X:A,Y2:B]: (((X9 = X) & ((P3 @ X) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y6) @ Y2)) @ (R6 @ X)))))))))))))))))). % 6.41/2.69 thf(fact_4731_same__fstI, axiom, (! [B:$tType,A:$tType,P:(A > $o),X3:A,Y8:B,Y:B,R:(A > (set @ ((product_prod @ B) @ B)))]: (((P @ X3) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y8) @ Y)) @ (R @ X3)) => (((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y8)) @ ((((product_Pair @ A) @ B) @ X3) @ Y))) @ ((((same_fst @ A) @ B) @ P) @ R))))))). % 6.41/2.69 thf(fact_4732_inf__int__def, axiom, ((inf_inf @ int) = (ord_min @ int))). % 6.41/2.69 thf(fact_4733_relpowp__Suc__E, axiom, (! [A:$tType,N2:nat,P:(A > (A > $o)),X3:A,Z2:A]: (((((((compow @ (A > (A > $o))) @ (suc @ N2)) @ P) @ X3) @ Z2) => (~ ((! [Y4:A]: (((((((compow @ (A > (A > $o))) @ N2) @ P) @ X3) @ Y4) => (~ (((P @ Y4) @ Z2)))))))))))). % 6.41/2.69 thf(fact_4734_relpowp__Suc__I, axiom, (! [A:$tType,N2:nat,P:(A > (A > $o)),X3:A,Y:A,Z2:A]: (((((((compow @ (A > (A > $o))) @ N2) @ P) @ X3) @ Y) => (((P @ Y) @ Z2) => (((((compow @ (A > (A > $o))) @ (suc @ N2)) @ P) @ X3) @ Z2)))))). % 6.41/2.69 thf(fact_4735_relpowp__Suc__D2, axiom, (! [A:$tType,N2:nat,P:(A > (A > $o)),X3:A,Z2:A]: (((((((compow @ (A > (A > $o))) @ (suc @ N2)) @ P) @ X3) @ Z2) => (? [Y4:A]: ((((P @ X3) @ Y4) & (((((compow @ (A > (A > $o))) @ N2) @ P) @ Y4) @ Z2)))))))). % 6.41/2.69 thf(fact_4736_relpowp__Suc__E2, axiom, (! [A:$tType,N2:nat,P:(A > (A > $o)),X3:A,Z2:A]: (((((((compow @ (A > (A > $o))) @ (suc @ N2)) @ P) @ X3) @ Z2) => (~ ((! [Y4:A]: ((((P @ X3) @ Y4) => (~ ((((((compow @ (A > (A > $o))) @ N2) @ P) @ Y4) @ Z2)))))))))))). % 6.41/2.69 thf(fact_4737_relpowp__Suc__I2, axiom, (! [A:$tType,P:(A > (A > $o)),X3:A,Y:A,N2:nat,Z2:A]: ((((P @ X3) @ Y) => ((((((compow @ (A > (A > $o))) @ N2) @ P) @ Y) @ Z2) => (((((compow @ (A > (A > $o))) @ (suc @ N2)) @ P) @ X3) @ Z2)))))). % 6.41/2.69 thf(fact_4738_relpowp__0__I, axiom, (! [A:$tType,P:(A > (A > $o)),X3:A]: ((((((compow @ (A > (A > $o))) @ (zero_zero @ nat)) @ P) @ X3) @ X3)))). % 6.41/2.69 thf(fact_4739_relpowp__0__E, axiom, (! [A:$tType,P:(A > (A > $o)),X3:A,Y:A]: (((((((compow @ (A > (A > $o))) @ (zero_zero @ nat)) @ P) @ X3) @ Y) => (X3 = Y))))). % 6.41/2.69 thf(fact_4740_relpowp_Osimps_I1_J, axiom, (! [A:$tType,R:(A > (A > $o))]: (((((compow @ (A > (A > $o))) @ (zero_zero @ nat)) @ R) = (^ [Y3:A,Z:A]: ((Y3 = Z))))))). % 6.41/2.69 thf(fact_4741_relpowp__E2, axiom, (! [A:$tType,N2:nat,P:(A > (A > $o)),X3:A,Z2:A]: (((((((compow @ (A > (A > $o))) @ N2) @ P) @ X3) @ Z2) => (((N2 = (zero_zero @ nat)) => (X3 != Z2)) => (~ ((! [Y4:A,M3:nat]: (((N2 = (suc @ M3)) => (((P @ X3) @ Y4) => (~ ((((((compow @ (A > (A > $o))) @ M3) @ P) @ Y4) @ Z2)))))))))))))). % 6.41/2.69 thf(fact_4742_relpowp__E, axiom, (! [A:$tType,N2:nat,P:(A > (A > $o)),X3:A,Z2:A]: (((((((compow @ (A > (A > $o))) @ N2) @ P) @ X3) @ Z2) => (((N2 = (zero_zero @ nat)) => (X3 != Z2)) => (~ ((! [Y4:A,M3:nat]: (((N2 = (suc @ M3)) => ((((((compow @ (A > (A > $o))) @ M3) @ P) @ X3) @ Y4) => (~ (((P @ Y4) @ Z2)))))))))))))). % 6.41/2.69 thf(fact_4743_Nat_Ofunpow__code__def, axiom, (! [A:$tType]: (((funpow @ A) = (compow @ (A > A)))))). % 6.41/2.69 thf(fact_4744_max__nat_Osemilattice__neutr__order__axioms, axiom, (((((semila1105856199041335345_order @ nat) @ (ord_max @ nat)) @ (zero_zero @ nat)) @ (^ [X:nat,Y2:nat]: ((((ord_less_eq @ nat) @ Y2) @ X)))) @ (^ [X:nat,Y2:nat]: ((((ord_less @ nat) @ Y2) @ X))))). % 6.41/2.69 thf(fact_4745_apsnd__apfst, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,F2:(C > B),G:(D > A),X3:((product_prod @ D) @ C)]: (((((((product_apsnd @ C) @ B) @ A) @ F2) @ (((((product_apfst @ D) @ A) @ C) @ G) @ X3)) = ((((product_Pair @ A) @ B) @ (G @ (((product_fst @ D) @ C) @ X3))) @ (F2 @ (((product_snd @ D) @ C) @ X3))))))). % 6.41/2.69 thf(fact_4746_apfst__conv, axiom, (! [C:$tType,A:$tType,B:$tType,F2:(C > A),X3:C,Y:B]: (((((((product_apfst @ C) @ A) @ B) @ F2) @ ((((product_Pair @ C) @ B) @ X3) @ Y)) = ((((product_Pair @ A) @ B) @ (F2 @ X3)) @ Y))))). % 6.41/2.69 thf(fact_4747_apfst__eq__conv, axiom, (! [A:$tType,B:$tType,C:$tType,F2:(C > A),X3:((product_prod @ C) @ B),G:(C > A)]: ((((((((product_apfst @ C) @ A) @ B) @ F2) @ X3) = (((((product_apfst @ C) @ A) @ B) @ G) @ X3)) = ((F2 @ (((product_fst @ C) @ B) @ X3)) = (G @ (((product_fst @ C) @ B) @ X3))))))). % 6.41/2.69 thf(fact_4748_fst__apfst, axiom, (! [A:$tType,B:$tType,C:$tType,F2:(C > A),X3:((product_prod @ C) @ B)]: (((((product_fst @ A) @ B) @ (((((product_apfst @ C) @ A) @ B) @ F2) @ X3)) = (F2 @ (((product_fst @ C) @ B) @ X3)))))). % 6.41/2.69 thf(fact_4749_snd__apfst, axiom, (! [B:$tType,A:$tType,C:$tType,F2:(C > B),X3:((product_prod @ C) @ A)]: (((((product_snd @ B) @ A) @ (((((product_apfst @ C) @ B) @ A) @ F2) @ X3)) = (((product_snd @ C) @ A) @ X3))))). % 6.41/2.69 thf(fact_4750_snd__comp__apfst, axiom, (! [C:$tType,B:$tType,A:$tType,F2:(A > C)]: (((((((comp @ ((product_prod @ C) @ B)) @ B) @ ((product_prod @ A) @ B)) @ ((product_snd @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F2)) = ((product_snd @ A) @ B))))). % 6.41/2.69 thf(fact_4751_fst__comp__apfst, axiom, (! [C:$tType,B:$tType,A:$tType,F2:(A > C)]: (((((((comp @ ((product_prod @ C) @ B)) @ C) @ ((product_prod @ A) @ B)) @ ((product_fst @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F2)) = (((((comp @ A) @ C) @ ((product_prod @ A) @ B)) @ F2) @ ((product_fst @ A) @ B)))))). % 6.41/2.69 thf(fact_4752_apfst__apsnd, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,F2:(C > A),G:(D > B),X3:((product_prod @ C) @ D)]: (((((((product_apfst @ C) @ A) @ B) @ F2) @ (((((product_apsnd @ D) @ B) @ C) @ G) @ X3)) = ((((product_Pair @ A) @ B) @ (F2 @ (((product_fst @ C) @ D) @ X3))) @ (G @ (((product_snd @ C) @ D) @ X3))))))). % 6.41/2.69 thf(fact_4753_apfst__compose, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F2:(C > A),G:(D > C),X3:((product_prod @ D) @ B)]: (((((((product_apfst @ C) @ A) @ B) @ F2) @ (((((product_apfst @ D) @ C) @ B) @ G) @ X3)) = (((((product_apfst @ D) @ A) @ B) @ (((((comp @ C) @ A) @ D) @ F2) @ G)) @ X3))))). % 6.41/2.69 thf(fact_4754_apsnd__apfst__commute, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,F2:(C > B),G:(D > A),P4:((product_prod @ D) @ C)]: (((((((product_apsnd @ C) @ B) @ A) @ F2) @ (((((product_apfst @ D) @ A) @ C) @ G) @ P4)) = (((((product_apfst @ D) @ A) @ B) @ G) @ (((((product_apsnd @ C) @ B) @ D) @ F2) @ P4)))))). % 6.41/2.69 thf(fact_4755_gcd__nat_Osemilattice__neutr__order__axioms, axiom, (((((semila1105856199041335345_order @ nat) @ (gcd_gcd @ nat)) @ (zero_zero @ nat)) @ (dvd_dvd @ nat)) @ (^ [M2:nat,N:nat]: (((((dvd_dvd @ nat) @ M2) @ N) & (M2 != N)))))). % 6.41/2.69 thf(fact_4756_apfst__convE, axiom, (! [C:$tType,A:$tType,B:$tType,Q4:((product_prod @ A) @ B),F2:(C > A),P4:((product_prod @ C) @ B)]: (((Q4 = (((((product_apfst @ C) @ A) @ B) @ F2) @ P4)) => (~ ((! [X5:C,Y4:B]: (((P4 = ((((product_Pair @ C) @ B) @ X5) @ Y4)) => (Q4 != ((((product_Pair @ A) @ B) @ (F2 @ X5)) @ Y4))))))))))). % 6.41/2.69 thf(fact_4757_set__removeAll, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((set2 @ A) @ (((removeAll @ A) @ X3) @ Xs)) = (((minus_minus @ (set @ A)) @ ((set2 @ A) @ Xs)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))). % 6.41/2.69 thf(fact_4758_times__int_Oabs__eq, axiom, (! [Xa2:((product_prod @ nat) @ nat),X3:((product_prod @ nat) @ nat)]: (((((times_times @ int) @ (abs_Integ @ Xa2)) @ (abs_Integ @ X3)) = (abs_Integ @ ((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X:nat,Y2:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X) @ U2)) @ (((times_times @ nat) @ Y2) @ V5))) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X) @ V5)) @ (((times_times @ nat) @ Y2) @ U2))))))))) @ Xa2) @ X3)))))). % 6.41/2.69 thf(fact_4759_removeAll__id, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)))) => ((((removeAll @ A) @ X3) @ Xs) = Xs))))). % 6.41/2.69 thf(fact_4760_distinct__removeAll, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((distinct @ A) @ Xs) => ((distinct @ A) @ (((removeAll @ A) @ X3) @ Xs)))))). % 6.41/2.69 thf(fact_4761_int_Oabs__induct, axiom, (! [P:(int > $o),X3:int]: (((! [Y4:((product_prod @ nat) @ nat)]: ((P @ (abs_Integ @ Y4)))) => (P @ X3))))). % 6.41/2.69 thf(fact_4762_eq__Abs__Integ, axiom, (! [Z2:int]: ((~ ((! [X5:nat,Y4:nat]: ((Z2 != (abs_Integ @ ((((product_Pair @ nat) @ nat) @ X5) @ Y4)))))))))). % 6.41/2.69 thf(fact_4763_nat_Oabs__eq, axiom, (! [X3:((product_prod @ nat) @ nat)]: (((nat2 @ (abs_Integ @ X3)) = (((((product_case_prod @ nat) @ nat) @ nat) @ (minus_minus @ nat)) @ X3))))). % 6.41/2.69 thf(fact_4764_length__removeAll__less__eq, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ (((removeAll @ A) @ X3) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))). % 6.41/2.69 thf(fact_4765_distinct__remove1__removeAll, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((distinct @ A) @ Xs) => ((((remove1 @ A) @ X3) @ Xs) = (((removeAll @ A) @ X3) @ Xs)))))). % 6.41/2.69 thf(fact_4766_zero__int__def, axiom, ((zero_zero @ int) = (abs_Integ @ ((((product_Pair @ nat) @ nat) @ (zero_zero @ nat)) @ (zero_zero @ nat))))). % 6.41/2.69 thf(fact_4767_int__def, axiom, ((semiring_1_of_nat @ int) = (^ [N:nat]: ((abs_Integ @ ((((product_Pair @ nat) @ nat) @ N) @ (zero_zero @ nat))))))). % 6.41/2.69 thf(fact_4768_length__removeAll__less, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (((ord_less @ nat) @ ((size_size @ (list @ A)) @ (((removeAll @ A) @ X3) @ Xs))) @ ((size_size @ (list @ A)) @ Xs)))))). % 6.41/2.69 thf(fact_4769_uminus__int_Oabs__eq, axiom, (! [X3:((product_prod @ nat) @ nat)]: ((((uminus_uminus @ int) @ (abs_Integ @ X3)) = (abs_Integ @ (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [X:nat,Y2:nat]: (((((product_Pair @ nat) @ nat) @ Y2) @ X)))) @ X3)))))). % 6.41/2.69 thf(fact_4770_one__int__def, axiom, ((one_one @ int) = (abs_Integ @ ((((product_Pair @ nat) @ nat) @ (one_one @ nat)) @ (zero_zero @ nat))))). % 6.41/2.69 thf(fact_4771_of__int_Oabs__eq, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X3:((product_prod @ nat) @ nat)]: ((((ring_1_of_int @ A) @ (abs_Integ @ X3)) = (((((product_case_prod @ nat) @ nat) @ A) @ (^ [I3:nat,J3:nat]: ((((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ I3)) @ ((semiring_1_of_nat @ A) @ J3))))) @ X3)))))))). % 6.41/2.69 thf(fact_4772_less__int_Oabs__eq, axiom, (! [Xa2:((product_prod @ nat) @ nat),X3:((product_prod @ nat) @ nat)]: (((((ord_less @ int) @ (abs_Integ @ Xa2)) @ (abs_Integ @ X3)) = ((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X:nat,Y2:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V5:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ X) @ V5)) @ (((plus_plus @ nat) @ U2) @ Y2)))))))) @ Xa2) @ X3))))). % 6.41/2.69 thf(fact_4773_less__eq__int_Oabs__eq, axiom, (! [Xa2:((product_prod @ nat) @ nat),X3:((product_prod @ nat) @ nat)]: (((((ord_less_eq @ int) @ (abs_Integ @ Xa2)) @ (abs_Integ @ X3)) = ((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X:nat,Y2:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V5:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ X) @ V5)) @ (((plus_plus @ nat) @ U2) @ Y2)))))))) @ Xa2) @ X3))))). % 6.41/2.69 thf(fact_4774_plus__int_Oabs__eq, axiom, (! [Xa2:((product_prod @ nat) @ nat),X3:((product_prod @ nat) @ nat)]: (((((plus_plus @ int) @ (abs_Integ @ Xa2)) @ (abs_Integ @ X3)) = (abs_Integ @ ((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X:nat,Y2:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X) @ U2)) @ (((plus_plus @ nat) @ Y2) @ V5)))))))) @ Xa2) @ X3)))))). % 6.41/2.69 thf(fact_4775_minus__int_Oabs__eq, axiom, (! [Xa2:((product_prod @ nat) @ nat),X3:((product_prod @ nat) @ nat)]: (((((minus_minus @ int) @ (abs_Integ @ Xa2)) @ (abs_Integ @ X3)) = (abs_Integ @ ((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X:nat,Y2:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X) @ V5)) @ (((plus_plus @ nat) @ Y2) @ U2)))))))) @ Xa2) @ X3)))))). % 6.41/2.69 thf(fact_4776_distinct__concat__iff, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: ((((distinct @ A) @ ((concat @ A) @ Xs)) = (((distinct @ (list @ A)) @ (((removeAll @ (list @ A)) @ (nil @ A)) @ Xs)) & ((! [Ys3:(list @ A)]: (((((member @ (list @ A)) @ Ys3) @ ((set2 @ (list @ A)) @ Xs)) => ((distinct @ A) @ Ys3)))) & (! [Ys3:(list @ A),Zs3:(list @ A)]: ((((((member @ (list @ A)) @ Ys3) @ ((set2 @ (list @ A)) @ Xs)) & ((((member @ (list @ A)) @ Zs3) @ ((set2 @ (list @ A)) @ Xs)) & (Ys3 != Zs3))) => ((((inf_inf @ (set @ A)) @ ((set2 @ A) @ Ys3)) @ ((set2 @ A) @ Zs3)) = (bot_bot @ (set @ A)))))))))))). % 6.41/2.69 thf(fact_4777_max__rpair__set, axiom, ((fun_reduction_pair @ (set @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))) @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))) @ fun_max_strict) @ fun_max_weak))). % 6.41/2.69 thf(fact_4778_min__rpair__set, axiom, ((fun_reduction_pair @ (set @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))) @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))) @ fun_min_strict) @ fun_min_weak))). % 6.41/2.69 thf(fact_4779_zip__Nil, axiom, (! [B:$tType,A:$tType,Ys:(list @ B)]: ((((((zip @ A) @ B) @ (nil @ A)) @ Ys) = (nil @ ((product_prod @ A) @ B)))))). % 6.41/2.69 thf(fact_4780_Nil__eq__zip__iff, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B)]: ((((nil @ ((product_prod @ A) @ B)) = ((((zip @ A) @ B) @ Xs) @ Ys)) = ((Xs = (nil @ A)) | (Ys = (nil @ B))))))). % 6.41/2.69 thf(fact_4781_zip__eq__Nil__iff, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B)]: (((((((zip @ A) @ B) @ Xs) @ Ys) = (nil @ ((product_prod @ A) @ B))) = ((Xs = (nil @ A)) | (Ys = (nil @ B))))))). % 6.41/2.69 thf(fact_4782_list__update__nonempty, axiom, (! [A:$tType,Xs:(list @ A),K:nat,X3:A]: (((((((list_update @ A) @ Xs) @ K) @ X3) = (nil @ A)) = (Xs = (nil @ A)))))). % 6.41/2.69 thf(fact_4783_concat__replicate__trivial, axiom, (! [A:$tType,I:nat]: ((((concat @ A) @ (((replicate @ (list @ A)) @ I) @ (nil @ A))) = (nil @ A))))). % 6.41/2.69 thf(fact_4784_Nil__in__shuffles, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((member @ (list @ A)) @ (nil @ A)) @ (((shuffles @ A) @ Xs) @ Ys)) = ((Xs = (nil @ A)) & (Ys = (nil @ A))))))). % 6.41/2.69 thf(fact_4785_enumerate__simps_I1_J, axiom, (! [A:$tType,N2:nat]: (((((enumerate @ A) @ N2) @ (nil @ A)) = (nil @ ((product_prod @ nat) @ A)))))). % 6.41/2.69 thf(fact_4786_rotate1__is__Nil__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((((rotate1 @ A) @ Xs) = (nil @ A)) = (Xs = (nil @ A)))))). % 6.41/2.69 thf(fact_4787_set__empty, axiom, (! [A:$tType,Xs:(list @ A)]: (((((set2 @ A) @ Xs) = (bot_bot @ (set @ A))) = (Xs = (nil @ A)))))). % 6.41/2.69 thf(fact_4788_set__empty2, axiom, (! [A:$tType,Xs:(list @ A)]: ((((bot_bot @ (set @ A)) = ((set2 @ A) @ Xs)) = (Xs = (nil @ A)))))). % 6.41/2.69 thf(fact_4789_length__0__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) = (zero_zero @ nat)) = (Xs = (nil @ A)))))). % 6.41/2.69 thf(fact_4790_empty__replicate, axiom, (! [A:$tType,N2:nat,X3:A]: ((((nil @ A) = (((replicate @ A) @ N2) @ X3)) = (N2 = (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_4791_replicate__empty, axiom, (! [A:$tType,N2:nat,X3:A]: ((((((replicate @ A) @ N2) @ X3) = (nil @ A)) = (N2 = (zero_zero @ nat)))))). % 6.41/2.69 thf(fact_4792_horner__sum__simps_I1_J, axiom, (! [B:$tType,A:$tType]: (((comm_semiring_0 @ A) => (! [F2:(B > A),A2:A]: (((((((groups4207007520872428315er_sum @ B) @ A) @ F2) @ A2) @ (nil @ B)) = (zero_zero @ A)))))))). % 6.41/2.69 thf(fact_4793_concat__eq__Nil__conv, axiom, (! [A:$tType,Xss:(list @ (list @ A))]: (((((concat @ A) @ Xss) = (nil @ A)) = (! [X:(list @ A)]: (((((member @ (list @ A)) @ X) @ ((set2 @ (list @ A)) @ Xss)) => (X = (nil @ A))))))))). % 6.41/2.69 thf(fact_4794_Nil__eq__concat__conv, axiom, (! [A:$tType,Xss:(list @ (list @ A))]: ((((nil @ A) = ((concat @ A) @ Xss)) = (! [X:(list @ A)]: (((((member @ (list @ A)) @ X) @ ((set2 @ (list @ A)) @ Xss)) => (X = (nil @ A))))))))). % 6.41/2.69 thf(fact_4795_length__greater__0__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs)) = (Xs != (nil @ A)))))). % 6.41/2.69 thf(fact_4796_removeAll_Osimps_I1_J, axiom, (! [A:$tType,X3:A]: (((((removeAll @ A) @ X3) @ (nil @ A)) = (nil @ A))))). % 6.41/2.69 thf(fact_4797_rotate1_Osimps_I1_J, axiom, (! [A:$tType]: ((((rotate1 @ A) @ (nil @ A)) = (nil @ A))))). % 6.41/2.69 thf(fact_4798_distinct_Osimps_I1_J, axiom, (! [A:$tType]: (((distinct @ A) @ (nil @ A))))). % 6.41/2.69 thf(fact_4799_concat_Osimps_I1_J, axiom, (! [A:$tType]: ((((concat @ A) @ (nil @ (list @ A))) = (nil @ A))))). % 6.41/2.69 thf(fact_4800_list__update_Osimps_I1_J, axiom, (! [A:$tType,I:nat,V2:A]: ((((((list_update @ A) @ (nil @ A)) @ I) @ V2) = (nil @ A))))). % 6.41/2.69 thf(fact_4801_list__update__code_I1_J, axiom, (! [A:$tType,I:nat,Y:A]: ((((((list_update @ A) @ (nil @ A)) @ I) @ Y) = (nil @ A))))). % 6.41/2.69 thf(fact_4802_product_Osimps_I1_J, axiom, (! [B:$tType,A:$tType,Uu:(list @ B)]: ((((((product @ A) @ B) @ (nil @ A)) @ Uu) = (nil @ ((product_prod @ A) @ B)))))). % 6.41/2.69 thf(fact_4803_zip_Osimps_I1_J, axiom, (! [B:$tType,A:$tType,Xs:(list @ A)]: ((((((zip @ A) @ B) @ Xs) @ (nil @ B)) = (nil @ ((product_prod @ A) @ B)))))). % 6.41/2.69 thf(fact_4804_Nil__in__shufflesI, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((Xs = (nil @ A)) => ((Ys = (nil @ A)) => (((member @ (list @ A)) @ (nil @ A)) @ (((shuffles @ A) @ Xs) @ Ys))))))). % 6.41/2.69 thf(fact_4805_shuffles_Osimps_I1_J, axiom, (! [A:$tType,Ys:(list @ A)]: (((((shuffles @ A) @ (nil @ A)) @ Ys) = (((insert @ (list @ A)) @ Ys) @ (bot_bot @ (set @ (list @ A)))))))). % 6.41/2.69 thf(fact_4806_shuffles_Osimps_I2_J, axiom, (! [A:$tType,Xs:(list @ A)]: (((((shuffles @ A) @ Xs) @ (nil @ A)) = (((insert @ (list @ A)) @ Xs) @ (bot_bot @ (set @ (list @ A)))))))). % 6.41/2.69 thf(fact_4807_remove1_Osimps_I1_J, axiom, (! [A:$tType,X3:A]: (((((remove1 @ A) @ X3) @ (nil @ A)) = (nil @ A))))). % 6.41/2.69 thf(fact_4808_gen__length__code_I1_J, axiom, (! [A:$tType,N2:nat]: (((((gen_length @ A) @ N2) @ (nil @ A)) = N2)))). % 6.41/2.69 thf(fact_4809_empty__set, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((set2 @ A) @ (nil @ A)))))). % 6.41/2.69 thf(fact_4810_list_Osize_I3_J, axiom, (! [A:$tType]: ((((size_size @ (list @ A)) @ (nil @ A)) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_4811_replicate__0, axiom, (! [A:$tType,X3:A]: (((((replicate @ A) @ (zero_zero @ nat)) @ X3) = (nil @ A))))). % 6.41/2.69 thf(fact_4812_not__listrel1__Nil, axiom, (! [A:$tType,Xs:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: ((~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ (nil @ A))) @ ((listrel1 @ A) @ R2))))))). % 6.41/2.69 thf(fact_4813_not__Nil__listrel1, axiom, (! [A:$tType,Xs:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: ((~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Xs)) @ ((listrel1 @ A) @ R2))))))). % 6.41/2.69 thf(fact_4814_list_Osize__gen_I1_J, axiom, (! [A:$tType,X3:(A > nat)]: (((((size_list @ A) @ X3) @ (nil @ A)) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_4815_find_Osimps_I1_J, axiom, (! [A:$tType,Uu:(A > $o)]: (((((find @ A) @ Uu) @ (nil @ A)) = (none @ A))))). % 6.41/2.69 thf(fact_4816_count__list_Osimps_I1_J, axiom, (! [A:$tType,Y:A]: (((((count_list @ A) @ (nil @ A)) @ Y) = (zero_zero @ nat))))). % 6.41/2.69 thf(fact_4817_Pow__set_I1_J, axiom, (! [A:$tType]: ((((pow2 @ A) @ ((set2 @ A) @ (nil @ A))) = (((insert @ (set @ A)) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ (set @ A)))))))). % 6.41/2.69 thf(fact_4818_less__eq__int_Orep__eq, axiom, ((ord_less_eq @ int) = (^ [X:int,Xa4:int]: (((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [Y2:nat,Z6:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V5:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ Y2) @ V5)) @ (((plus_plus @ nat) @ U2) @ Z6)))))))) @ (rep_Integ @ X)) @ (rep_Integ @ Xa4)))))). % 6.41/2.69 thf(fact_4819_less__int_Orep__eq, axiom, ((ord_less @ int) = (^ [X:int,Xa4:int]: (((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [Y2:nat,Z6:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V5:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ Y2) @ V5)) @ (((plus_plus @ nat) @ U2) @ Z6)))))))) @ (rep_Integ @ X)) @ (rep_Integ @ Xa4)))))). % 6.41/2.69 thf(fact_4820_prod__encode__def, axiom, (nat_prod_encode = ((((product_case_prod @ nat) @ nat) @ nat) @ (^ [M2:nat,N:nat]: ((((plus_plus @ nat) @ (nat_triangle @ (((plus_plus @ nat) @ M2) @ N))) @ M2)))))). % 6.41/2.69 thf(fact_4821_le__prod__encode__2, axiom, (! [B2:nat,A2:nat]: ((((ord_less_eq @ nat) @ B2) @ (nat_prod_encode @ ((((product_Pair @ nat) @ nat) @ A2) @ B2)))))). % 6.41/2.69 thf(fact_4822_le__prod__encode__1, axiom, (! [A2:nat,B2:nat]: ((((ord_less_eq @ nat) @ A2) @ (nat_prod_encode @ ((((product_Pair @ nat) @ nat) @ A2) @ B2)))))). % 6.41/2.69 thf(fact_4823_nat_Orep__eq, axiom, (nat2 = (^ [X:int]: ((((((product_case_prod @ nat) @ nat) @ nat) @ (minus_minus @ nat)) @ (rep_Integ @ X)))))). % 6.41/2.69 thf(fact_4824_of__int_Orep__eq, axiom, (! [A:$tType]: (((ring_1 @ A) => ((ring_1_of_int @ A) = (^ [X:int]: ((((((product_case_prod @ nat) @ nat) @ A) @ (^ [I3:nat,J3:nat]: ((((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ I3)) @ ((semiring_1_of_nat @ A) @ J3))))) @ (rep_Integ @ X))))))))). % 6.41/2.69 thf(fact_4825_prod__encode__prod__decode__aux, axiom, (! [K:nat,M:nat]: (((nat_prod_encode @ ((nat_prod_decode_aux @ K) @ M)) = (((plus_plus @ nat) @ (nat_triangle @ K)) @ M))))). % 6.41/2.69 thf(fact_4826_listset_Osimps_I1_J, axiom, (! [A:$tType]: ((((listset @ A) @ (nil @ (set @ A))) = (((insert @ (list @ A)) @ (nil @ A)) @ (bot_bot @ (set @ (list @ A)))))))). % 6.41/2.69 thf(fact_4827_uminus__int__def, axiom, ((uminus_uminus @ int) = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ int) @ rep_Integ) @ abs_Integ) @ ((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [X:nat,Y2:nat]: (((((product_Pair @ nat) @ nat) @ Y2) @ X))))))). % 6.41/2.69 thf(fact_4828_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B2:A,P4:((product_prod @ B) @ A)]: (((B2 = (((product_snd @ B) @ A) @ P4)) = (? [A4:B]: ((P4 = ((((product_Pair @ B) @ A) @ A4) @ B2)))))))). % 6.41/2.69 thf(fact_4829_prod_Osize__neq, axiom, (! [A:$tType,B:$tType,X3:((product_prod @ A) @ B)]: ((((size_size @ ((product_prod @ A) @ B)) @ X3) != (zero_zero @ nat))))). % 6.41/2.69 thf(fact_4830_sum_Osize__neq, axiom, (! [A:$tType,B:$tType,X3:((sum_sum @ A) @ B)]: ((((size_size @ ((sum_sum @ A) @ B)) @ X3) != (zero_zero @ nat))))). % 6.41/2.69 thf(fact_4831_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A2:A,P4:((product_prod @ A) @ B)]: (((A2 = (((product_fst @ A) @ B) @ P4)) = (? [B4:B]: ((P4 = ((((product_Pair @ A) @ B) @ A2) @ B4)))))))). % 6.41/2.69 thf(fact_4832_times__int__def, axiom, ((times_times @ int) = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (int > int)) @ rep_Integ) @ ((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ int) @ rep_Integ) @ abs_Integ)) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X:nat,Y2:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X) @ U2)) @ (((times_times @ nat) @ Y2) @ V5))) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X) @ V5)) @ (((times_times @ nat) @ Y2) @ U2)))))))))))). % 6.41/2.69 thf(fact_4833_minus__int__def, axiom, ((minus_minus @ int) = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (int > int)) @ rep_Integ) @ ((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ int) @ rep_Integ) @ abs_Integ)) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X:nat,Y2:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X) @ V5)) @ (((plus_plus @ nat) @ Y2) @ U2))))))))))). % 6.41/2.69 thf(fact_4834_plus__int__def, axiom, ((plus_plus @ int) = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (int > int)) @ rep_Integ) @ ((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ int) @ rep_Integ) @ abs_Integ)) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X:nat,Y2:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V5:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X) @ U2)) @ (((plus_plus @ nat) @ Y2) @ V5))))))))))). % 6.41/2.69 thf(fact_4835_Gcd__remove0__nat, axiom, (! [M7:(set @ nat)]: ((((finite_finite2 @ nat) @ M7) => (((gcd_Gcd @ nat) @ M7) = ((gcd_Gcd @ nat) @ (((minus_minus @ (set @ nat)) @ M7) @ (((insert @ nat) @ (zero_zero @ nat)) @ (bot_bot @ (set @ nat)))))))))). % 6.41/2.69 thf(fact_4836_add__neg__numeral__special_I4_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [N2:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((numeral_numeral @ A) @ N2)) = (((neg_numeral_sub @ A) @ N2) @ one2)))))))). % 6.41/2.69 thf(fact_4837_add__neg__numeral__special_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ (one_one @ A))) = (((neg_numeral_sub @ A) @ M) @ one2)))))))). % 6.41/2.69 thf(fact_4838_sub__num__simps_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((neg_numeral_sub @ A) @ one2) @ one2) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_4839_Gcd__empty, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (((gcd_Gcd @ A) @ (bot_bot @ (set @ A))) = (zero_zero @ A)))))). % 6.41/2.69 thf(fact_4840_Gcd__insert, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A2:A,A3:(set @ A)]: ((((gcd_Gcd @ A) @ (((insert @ A) @ A2) @ A3)) = (((gcd_gcd @ A) @ A2) @ ((gcd_Gcd @ A) @ A3))))))))). % 6.41/2.69 thf(fact_4841_semiring__norm_I167_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [V2:num,W2:num,Y:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V2))) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ W2)) @ Y)) = (((plus_plus @ A) @ (((neg_numeral_sub @ A) @ W2) @ V2)) @ Y)))))))). % 6.41/2.69 thf(fact_4842_semiring__norm_I166_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [V2:num,W2:num,Y:A]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ V2)) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W2))) @ Y)) = (((plus_plus @ A) @ (((neg_numeral_sub @ A) @ V2) @ W2)) @ Y)))))))). % 6.41/2.69 thf(fact_4843_add__neg__numeral__simps_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num,N2:num]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N2))) = (((neg_numeral_sub @ A) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_4844_add__neg__numeral__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num,N2:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ ((numeral_numeral @ A) @ N2)) = (((neg_numeral_sub @ A) @ N2) @ M)))))))). % 6.41/2.69 thf(fact_4845_Gcd__2, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A2:A,B2:A]: ((((gcd_Gcd @ A) @ (((insert @ A) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))) = (((gcd_gcd @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_4846_Gcd__0__iff, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A3:(set @ A)]: (((((gcd_Gcd @ A) @ A3) = (zero_zero @ A)) = (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ (zero_zero @ A)) @ (bot_bot @ (set @ A))))))))))). % 6.41/2.69 thf(fact_4847_add__neg__numeral__special_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num]: (((((plus_plus @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) = (((neg_numeral_sub @ A) @ one2) @ M)))))))). % 6.41/2.69 thf(fact_4848_add__neg__numeral__special_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M))) @ (one_one @ A)) = (((neg_numeral_sub @ A) @ one2) @ M)))))))). % 6.41/2.69 thf(fact_4849_Gcd__dvd, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => (((dvd_dvd @ A) @ ((gcd_Gcd @ A) @ A3)) @ A2)))))))). % 6.41/2.69 thf(fact_4850_dvd__GcdD, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [X3:A,A3:(set @ A),Y:A]: (((((dvd_dvd @ A) @ X3) @ ((gcd_Gcd @ A) @ A3)) => ((((member @ A) @ Y) @ A3) => (((dvd_dvd @ A) @ X3) @ Y))))))))). % 6.41/2.69 thf(fact_4851_dvd__Gcd__iff, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [X3:A,A3:(set @ A)]: (((((dvd_dvd @ A) @ X3) @ ((gcd_Gcd @ A) @ A3)) = (! [X:A]: (((((member @ A) @ X) @ A3) => (((dvd_dvd @ A) @ X3) @ X))))))))))). % 6.41/2.69 thf(fact_4852_Gcd__greatest, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A3:(set @ A),A2:A]: (((! [B5:A]: (((((member @ A) @ B5) @ A3) => (((dvd_dvd @ A) @ A2) @ B5)))) => (((dvd_dvd @ A) @ A2) @ ((gcd_Gcd @ A) @ A3))))))))). % 6.41/2.69 thf(fact_4853_Gcd__dvd__nat, axiom, (! [A2:nat,A3:(set @ nat)]: (((((member @ nat) @ A2) @ A3) => (((dvd_dvd @ nat) @ ((gcd_Gcd @ nat) @ A3)) @ A2))))). % 6.41/2.69 thf(fact_4854_Gcd__greatest__nat, axiom, (! [A3:(set @ nat),A2:nat]: (((! [B5:nat]: (((((member @ nat) @ B5) @ A3) => (((dvd_dvd @ nat) @ A2) @ B5)))) => (((dvd_dvd @ nat) @ A2) @ ((gcd_Gcd @ nat) @ A3)))))). % 6.41/2.69 thf(fact_4855_Gcd__1, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A3:(set @ A)]: (((((member @ A) @ (one_one @ A)) @ A3) => (((gcd_Gcd @ A) @ A3) = (one_one @ A))))))))). % 6.41/2.69 thf(fact_4856_Gcd__nat__eq__one, axiom, (! [N7:(set @ nat)]: (((((member @ nat) @ (one_one @ nat)) @ N7) => (((gcd_Gcd @ nat) @ N7) = (one_one @ nat)))))). % 6.41/2.69 thf(fact_4857_Gcd__eq__1__I, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A2:A,A3:(set @ A)]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => ((((member @ A) @ A2) @ A3) => (((gcd_Gcd @ A) @ A3) = (one_one @ A)))))))))). % 6.41/2.69 thf(fact_4858_Gcd__in, axiom, (! [A3:(set @ nat)]: (((! [A5:nat,B5:nat]: (((((member @ nat) @ A5) @ A3) => ((((member @ nat) @ B5) @ A3) => (((member @ nat) @ (((gcd_gcd @ nat) @ A5) @ B5)) @ A3))))) => ((A3 != (bot_bot @ (set @ nat))) => (((member @ nat) @ ((gcd_Gcd @ nat) @ A3)) @ A3)))))). % 6.41/2.69 thf(fact_4859_sub__non__negative, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:num,M:num]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((neg_numeral_sub @ A) @ N2) @ M)) = (((ord_less_eq @ num) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_4860_sub__non__positive, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:num,M:num]: (((((ord_less_eq @ A) @ (((neg_numeral_sub @ A) @ N2) @ M)) @ (zero_zero @ A)) = (((ord_less_eq @ num) @ N2) @ M)))))))). % 6.41/2.69 thf(fact_4861_sub__positive, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:num,M:num]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((neg_numeral_sub @ A) @ N2) @ M)) = (((ord_less @ num) @ M) @ N2)))))))). % 6.41/2.69 thf(fact_4862_sub__negative, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N2:num,M:num]: (((((ord_less @ A) @ (((neg_numeral_sub @ A) @ N2) @ M)) @ (zero_zero @ A)) = (((ord_less @ num) @ N2) @ M)))))))). % 6.41/2.69 thf(fact_4863_sub__BitM__One__eq, axiom, (! [N2:num]: (((((neg_numeral_sub @ int) @ (bitM @ N2)) @ one2) = (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((neg_numeral_sub @ int) @ N2) @ one2)))))). % 6.41/2.69 thf(fact_4864_semiring__char__def, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((semiri4206861660011772517g_char @ A) = (^ [Uu4:(itself @ A)]: (((gcd_Gcd @ nat) @ ((collect @ nat) @ (^ [N:nat]: ((((semiring_1_of_nat @ A) @ N) = (zero_zero @ A))))))))))))). % 6.41/2.69 thf(fact_4865_num__of__nat_Osimps_I2_J, axiom, (! [N2:nat]: ((((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((num_of_nat @ (suc @ N2)) = (inc @ (num_of_nat @ N2)))) & ((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2))) => ((num_of_nat @ (suc @ N2)) = one2)))))). % 6.41/2.69 thf(fact_4866_pred__nat__def, axiom, (pred_nat = ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [M2:nat,N:nat]: ((N = (suc @ M2)))))))). % 6.41/2.69 thf(fact_4867_abs__Gcd__eq, axiom, (! [K5:(set @ int)]: ((((abs_abs @ int) @ ((gcd_Gcd @ int) @ K5)) = ((gcd_Gcd @ int) @ K5))))). % 6.41/2.69 thf(fact_4868_Gcd__dvd__int, axiom, (! [A2:int,A3:(set @ int)]: (((((member @ int) @ A2) @ A3) => (((dvd_dvd @ int) @ ((gcd_Gcd @ int) @ A3)) @ A2))))). % 6.41/2.69 thf(fact_4869_Gcd__greatest__int, axiom, (! [A3:(set @ int),A2:int]: (((! [B5:int]: (((((member @ int) @ B5) @ A3) => (((dvd_dvd @ int) @ A2) @ B5)))) => (((dvd_dvd @ int) @ A2) @ ((gcd_Gcd @ int) @ A3)))))). % 6.41/2.69 thf(fact_4870_Gcd__int__greater__eq__0, axiom, (! [K5:(set @ int)]: ((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((gcd_Gcd @ int) @ K5))))). % 6.41/2.69 thf(fact_4871_num__of__nat_Osimps_I1_J, axiom, ((num_of_nat @ (zero_zero @ nat)) = one2)). % 6.41/2.69 thf(fact_4872_numeral__num__of__nat, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((numeral_numeral @ nat) @ (num_of_nat @ N2)) = N2))))). % 6.41/2.69 thf(fact_4873_num__of__nat__One, axiom, (! [N2:nat]: (((((ord_less_eq @ nat) @ N2) @ (one_one @ nat)) => ((num_of_nat @ N2) = one2))))). % 6.41/2.69 thf(fact_4874_numeral__num__of__nat__unfold, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N2:nat]: ((((N2 = (zero_zero @ nat)) => (((numeral_numeral @ A) @ (num_of_nat @ N2)) = (one_one @ A))) & ((N2 != (zero_zero @ nat)) => (((numeral_numeral @ A) @ (num_of_nat @ N2)) = ((semiring_1_of_nat @ A) @ N2)))))))))). % 6.41/2.69 thf(fact_4875_num__of__nat__double, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((num_of_nat @ (((plus_plus @ nat) @ N2) @ N2)) = (bit0 @ (num_of_nat @ N2))))))). % 6.41/2.69 thf(fact_4876_num__of__nat__plus__distrib, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((num_of_nat @ (((plus_plus @ nat) @ M) @ N2)) = (((plus_plus @ num) @ (num_of_nat @ M)) @ (num_of_nat @ N2)))))))). % 6.41/2.69 thf(fact_4877_prod_Oinsert_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [I6:(set @ B),P4:(B > A),I:B]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ I6) & ((P4 @ X) != (one_one @ A))))))) => (((((member @ B) @ I) @ I6) => (((((groups1962203154675924110t_prod @ B) @ A) @ P4) @ (((insert @ B) @ I) @ I6)) = ((((groups1962203154675924110t_prod @ B) @ A) @ P4) @ I6))) & ((~ ((((member @ B) @ I) @ I6))) => (((((groups1962203154675924110t_prod @ B) @ A) @ P4) @ (((insert @ B) @ I) @ I6)) = (((times_times @ A) @ (P4 @ I)) @ ((((groups1962203154675924110t_prod @ B) @ A) @ P4) @ I6)))))))))))). % 6.41/2.69 thf(fact_4878_sorted__list__of__set_Osorted__key__list__of__set__remove, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => (((linord4507533701916653071of_set @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) = (((remove1 @ A) @ X3) @ ((linord4507533701916653071of_set @ A) @ A3)))))))))). % 6.41/2.69 thf(fact_4879_VEBT__internal_Ovalid_H_Oelims_I3_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: (((~ (((vEBT_VEBT_valid @ X3) @ Xa2))) => (((? [Uu2:$o,Uv2:$o]: ((X3 = ((vEBT_Leaf @ Uu2) @ Uv2)))) => (Xa2 = (one_one @ nat))) => (~ ((! [Mima:(option @ ((product_prod @ nat) @ nat)),Deg2:nat,TreeList3:(list @ vEBT_VEBT),Summary2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ Mima) @ Deg2) @ TreeList3) @ Summary2)) => ((Deg2 = Xa2) & ((! [X5:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X5) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => ((vEBT_VEBT_valid @ X5) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) & (((vEBT_VEBT_valid @ Summary2) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) & ((((size_size @ (list @ vEBT_VEBT)) @ TreeList3) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((((case_option @ $o) @ ((product_prod @ nat) @ nat)) @ ((~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ Summary2) @ X6))))) & (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6)))))))))) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [Mi3:nat,Ma3:nat]: (((((ord_less_eq @ nat) @ Mi3) @ Ma3) & ((((ord_less @ nat) @ Ma3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg2)) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) => ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList3) @ I3)) @ X6))) = ((vEBT_V8194947554948674370ptions @ Summary2) @ I3))))) & (((Mi3 = Ma3) => (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6))))))))) & ((Mi3 != Ma3) => ((((vEBT_V5917875025757280293ildren @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ TreeList3) @ Ma3) & (! [X:nat]: (((((ord_less @ nat) @ X) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg2)) => ((((vEBT_V5917875025757280293ildren @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ TreeList3) @ X) => ((((ord_less @ nat) @ Mi3) @ X) & (((ord_less_eq @ nat) @ X) @ Ma3)))))))))))))))) @ Mima))))))))))))))). % 6.41/2.69 thf(fact_4880_ball__empty, axiom, (! [A:$tType,P:(A > $o),X2:A]: (((((member @ A) @ X2) @ (bot_bot @ (set @ A))) => (P @ X2))))). % 6.41/2.69 thf(fact_4881_sorted__list__of__set_Osorted__key__list__of__set__empty, axiom, (! [A:$tType]: (((linorder @ A) => (((linord4507533701916653071of_set @ A) @ (bot_bot @ (set @ A))) = (nil @ A)))))). % 6.41/2.69 thf(fact_4882_sorted__list__of__set_Ofold__insort__key_Oinfinite, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: (((~ (((finite_finite2 @ A) @ A3))) => (((linord4507533701916653071of_set @ A) @ A3) = (nil @ A))))))))). % 6.41/2.69 thf(fact_4883_sorted__list__of__set_Oset__sorted__key__list__of__set, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (((set2 @ A) @ ((linord4507533701916653071of_set @ A) @ A3)) = A3)))))))). % 6.41/2.69 thf(fact_4884_prod_Oempty_H, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [P4:(B > A)]: ((((((groups1962203154675924110t_prod @ B) @ A) @ P4) @ (bot_bot @ (set @ B))) = (one_one @ A)))))))). % 6.41/2.69 thf(fact_4885_sorted__list__of__set_Olength__sorted__key__list__of__set, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: ((((size_size @ (list @ A)) @ ((linord4507533701916653071of_set @ A) @ A3)) = ((finite_card @ A) @ A3)))))))). % 6.41/2.69 thf(fact_4886_prod_Oeq__sum, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [I6:(set @ B),P4:(B > A)]: ((((finite_finite2 @ B) @ I6) => (((((groups1962203154675924110t_prod @ B) @ A) @ P4) @ I6) = ((((groups7121269368397514597t_prod @ B) @ A) @ P4) @ I6))))))))). % 6.41/2.69 thf(fact_4887_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((((linord4507533701916653071of_set @ A) @ A3) = (nil @ A)) = (A3 = (bot_bot @ (set @ A))))))))))). % 6.41/2.69 thf(fact_4888_finite__set__of__finite__funs, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B),D2:B]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ B) @ B3) => ((finite_finite2 @ (A > B)) @ ((collect @ (A > B)) @ (^ [F3:(A > B)]: ((! [X:A]: ((((((member @ A) @ X) @ A3) => (((member @ B) @ (F3 @ X)) @ B3)) & ((~ ((((member @ A) @ X) @ A3))) => ((F3 @ X) = D2)))))))))))))). % 6.41/2.69 thf(fact_4889_option_Osimps_I5_J, axiom, (! [B:$tType,A:$tType,F1:B,F22:(A > B),X22:A]: (((((((case_option @ B) @ A) @ F1) @ F22) @ ((some @ A) @ X22)) = (F22 @ X22))))). % 6.41/2.69 thf(fact_4890_option_Osimps_I4_J, axiom, (! [A:$tType,B:$tType,F1:B,F22:(A > B)]: (((((((case_option @ B) @ A) @ F1) @ F22) @ (none @ A)) = F1)))). % 6.41/2.69 thf(fact_4891_Ball__Collect, axiom, (! [A:$tType]: (((ball @ A) = (^ [A6:(set @ A),P3:(A > $o)]: ((((ord_less_eq @ (set @ A)) @ A6) @ ((collect @ A) @ P3)))))))). % 6.41/2.69 thf(fact_4892_option_Ocase__distrib, axiom, (! [C:$tType,B:$tType,A:$tType,H2:(B > C),F1:B,F22:(A > B),Option:(option @ A)]: (((H2 @ (((((case_option @ B) @ A) @ F1) @ F22) @ Option)) = (((((case_option @ C) @ A) @ (H2 @ F1)) @ (^ [X:A]: ((H2 @ (F22 @ X))))) @ Option))))). % 6.41/2.69 thf(fact_4893_Ball__def, axiom, (! [A:$tType]: (((ball @ A) = (^ [A6:(set @ A),P3:(A > $o)]: ((! [X:A]: (((((member @ A) @ X) @ A6) => (P3 @ X)))))))))). % 6.41/2.69 thf(fact_4894_sorted__list__of__set_Osorted__key__list__of__set__inject, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),B3:(set @ A)]: (((((linord4507533701916653071of_set @ A) @ A3) = ((linord4507533701916653071of_set @ A) @ B3)) => (((finite_finite2 @ A) @ A3) => (((finite_finite2 @ A) @ B3) => (A3 = B3)))))))))). % 6.41/2.69 thf(fact_4895_sorted__list__of__set_Odistinct__sorted__key__list__of__set, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: (((distinct @ A) @ ((linord4507533701916653071of_set @ A) @ A3)))))))). % 6.41/2.69 thf(fact_4896_option_Odisc__eq__case_I2_J, axiom, (! [A:$tType,Option:(option @ A)]: (((Option != (none @ A)) = (((((case_option @ $o) @ A) @ $false) @ (^ [Uu3:A]: ($true))) @ Option))))). % 6.41/2.69 thf(fact_4897_option_Odisc__eq__case_I1_J, axiom, (! [A:$tType,Option:(option @ A)]: (((Option = (none @ A)) = (((((case_option @ $o) @ A) @ $true) @ (^ [Uu3:A]: ($false))) @ Option))))). % 6.41/2.69 thf(fact_4898_prod_Odistrib__triv_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [I6:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ I6) => (((((groups1962203154675924110t_prod @ B) @ A) @ (^ [I3:B]: ((((times_times @ A) @ (G @ I3)) @ (H2 @ I3))))) @ I6) = (((times_times @ A) @ ((((groups1962203154675924110t_prod @ B) @ A) @ G) @ I6)) @ ((((groups1962203154675924110t_prod @ B) @ A) @ H2) @ I6)))))))))). % 6.41/2.69 thf(fact_4899_case__optionE, axiom, (! [A:$tType,P:$o,Q:(A > $o),X3:(option @ A)]: (((((((case_option @ $o) @ A) @ P) @ Q) @ X3) => (((X3 = (none @ A)) => (~ (P))) => (~ ((! [Y4:A]: (((X3 = ((some @ A) @ Y4)) => (~ ((Q @ Y4))))))))))))). % 6.41/2.69 thf(fact_4900_Sup__eq__Inf, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => ((complete_Sup_Sup @ A) = (^ [A6:(set @ A)]: (((complete_Inf_Inf @ A) @ ((collect @ A) @ (^ [B4:A]: ((! [X:A]: (((((member @ A) @ X) @ A6) => (((ord_less_eq @ A) @ X) @ B4))))))))))))))). % 6.41/2.69 thf(fact_4901_Inf__eq__Sup, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => ((complete_Inf_Inf @ A) = (^ [A6:(set @ A)]: (((complete_Sup_Sup @ A) @ ((collect @ A) @ (^ [B4:A]: ((! [X:A]: (((((member @ A) @ X) @ A6) => (((ord_less_eq @ A) @ B4) @ X))))))))))))))). % 6.41/2.69 thf(fact_4902_option_Ocase__eq__if, axiom, (! [A:$tType,B:$tType]: ((((case_option @ B) @ A) = (^ [F12:B,F23:(A > B),Option3:(option @ A)]: (((((if @ B) @ (Option3 = (none @ A))) @ F12) @ (F23 @ ((the2 @ A) @ Option3))))))))). % 6.41/2.69 thf(fact_4903_prod_Omono__neutral__cong__right_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),T4:(set @ B),G:(B > A),H2:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [X5:B]: (((((member @ B) @ X5) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((G @ X5) = (one_one @ A))))) => ((! [X5:B]: (((((member @ B) @ X5) @ S3) => ((G @ X5) = (H2 @ X5))))) => (((((groups1962203154675924110t_prod @ B) @ A) @ G) @ T4) = ((((groups1962203154675924110t_prod @ B) @ A) @ H2) @ S3))))))))))). % 6.41/2.69 thf(fact_4904_prod_Omono__neutral__cong__left_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),T4:(set @ B),H2:(B > A),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [I4:B]: (((((member @ B) @ I4) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((H2 @ I4) = (one_one @ A))))) => ((! [X5:B]: (((((member @ B) @ X5) @ S3) => ((G @ X5) = (H2 @ X5))))) => (((((groups1962203154675924110t_prod @ B) @ A) @ G) @ S3) = ((((groups1962203154675924110t_prod @ B) @ A) @ H2) @ T4))))))))))). % 6.41/2.69 thf(fact_4905_prod_Omono__neutral__right_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),T4:(set @ B),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [X5:B]: (((((member @ B) @ X5) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((G @ X5) = (one_one @ A))))) => (((((groups1962203154675924110t_prod @ B) @ A) @ G) @ T4) = ((((groups1962203154675924110t_prod @ B) @ A) @ G) @ S3)))))))))). % 6.41/2.69 thf(fact_4906_prod_Omono__neutral__left_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),T4:(set @ B),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S3) @ T4) => ((! [X5:B]: (((((member @ B) @ X5) @ (((minus_minus @ (set @ B)) @ T4) @ S3)) => ((G @ X5) = (one_one @ A))))) => (((((groups1962203154675924110t_prod @ B) @ A) @ G) @ S3) = ((((groups1962203154675924110t_prod @ B) @ A) @ G) @ T4)))))))))). % 6.41/2.69 thf(fact_4907_prod_Odistrib_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [I6:(set @ B),G:(B > A),H2:(B > A)]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ I6) & ((G @ X) != (one_one @ A))))))) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ I6) & ((H2 @ X) != (one_one @ A))))))) => (((((groups1962203154675924110t_prod @ B) @ A) @ (^ [I3:B]: ((((times_times @ A) @ (G @ I3)) @ (H2 @ I3))))) @ I6) = (((times_times @ A) @ ((((groups1962203154675924110t_prod @ B) @ A) @ G) @ I6)) @ ((((groups1962203154675924110t_prod @ B) @ A) @ H2) @ I6))))))))))). % 6.41/2.69 thf(fact_4908_prod_OG__def, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (((groups1962203154675924110t_prod @ B) @ A) = (^ [P6:(B > A),I8:(set @ B)]: (((((if @ A) @ ((finite_finite2 @ B) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ I8) & ((P6 @ X) != (one_one @ A)))))))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ P6) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ I8) & ((P6 @ X) != (one_one @ A)))))))) @ (one_one @ A))))))))). % 6.41/2.69 thf(fact_4909_option_Osplit__sel__asm, axiom, (! [B:$tType,A:$tType,P:(B > $o),F1:B,F22:(A > B),Option:(option @ A)]: (((P @ (((((case_option @ B) @ A) @ F1) @ F22) @ Option)) = (~ ((((Option = (none @ A)) & (~ ((P @ F1)))) | ((Option = ((some @ A) @ ((the2 @ A) @ Option))) & (~ ((P @ (F22 @ ((the2 @ A) @ Option))))))))))))). % 6.41/2.69 thf(fact_4910_option_Osplit__sel, axiom, (! [B:$tType,A:$tType,P:(B > $o),F1:B,F22:(A > B),Option:(option @ A)]: (((P @ (((((case_option @ B) @ A) @ F1) @ F22) @ Option)) = (((Option = (none @ A)) => (P @ F1)) & ((Option = ((some @ A) @ ((the2 @ A) @ Option))) => (P @ (F22 @ ((the2 @ A) @ Option))))))))). % 6.41/2.69 thf(fact_4911_VEBT__internal_Ovalid_H_Osimps_I2_J, axiom, (! [Mima2:(option @ ((product_prod @ nat) @ nat)),Deg:nat,TreeList2:(list @ vEBT_VEBT),Summary:vEBT_VEBT,Deg4:nat]: ((((vEBT_VEBT_valid @ ((((vEBT_Node @ Mima2) @ Deg) @ TreeList2) @ Summary)) @ Deg4) = ((Deg = Deg4) & ((! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList2)) => ((vEBT_VEBT_valid @ X) @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) & (((vEBT_VEBT_valid @ Summary) @ (((minus_minus @ nat) @ Deg) @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) & ((((size_size @ (list @ vEBT_VEBT)) @ TreeList2) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ Deg) @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((((case_option @ $o) @ ((product_prod @ nat) @ nat)) @ ((~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ Summary) @ X6))))) & (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList2)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6)))))))))) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [Mi3:nat,Ma3:nat]: (((((ord_less_eq @ nat) @ Mi3) @ Ma3) & ((((ord_less @ nat) @ Ma3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg)) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ Deg) @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) => ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList2) @ I3)) @ X6))) = ((vEBT_V8194947554948674370ptions @ Summary) @ I3))))) & (((Mi3 = Ma3) => (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList2)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6))))))))) & ((Mi3 != Ma3) => ((((vEBT_V5917875025757280293ildren @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ TreeList2) @ Ma3) & (! [X:nat]: (((((ord_less @ nat) @ X) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg)) => ((((vEBT_V5917875025757280293ildren @ (((divide_divide @ nat) @ Deg) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ TreeList2) @ X) => ((((ord_less @ nat) @ Mi3) @ X) & (((ord_less_eq @ nat) @ X) @ Ma3)))))))))))))))) @ Mima2))))))))). % 6.41/2.69 thf(fact_4912_VEBT__internal_Ovalid_H_Oelims_I1_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat,Y:$o]: (((((vEBT_VEBT_valid @ X3) @ Xa2) = Y) => (((? [Uu2:$o,Uv2:$o]: ((X3 = ((vEBT_Leaf @ Uu2) @ Uv2)))) => (Y = (Xa2 != (one_one @ nat)))) => (~ ((! [Mima:(option @ ((product_prod @ nat) @ nat)),Deg2:nat,TreeList3:(list @ vEBT_VEBT),Summary2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ Mima) @ Deg2) @ TreeList3) @ Summary2)) => (Y = (~ (((Deg2 = Xa2) & ((! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => ((vEBT_VEBT_valid @ X) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) & (((vEBT_VEBT_valid @ Summary2) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) & ((((size_size @ (list @ vEBT_VEBT)) @ TreeList3) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((((case_option @ $o) @ ((product_prod @ nat) @ nat)) @ ((~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ Summary2) @ X6))))) & (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6)))))))))) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [Mi3:nat,Ma3:nat]: (((((ord_less_eq @ nat) @ Mi3) @ Ma3) & ((((ord_less @ nat) @ Ma3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg2)) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) => ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList3) @ I3)) @ X6))) = ((vEBT_V8194947554948674370ptions @ Summary2) @ I3))))) & (((Mi3 = Ma3) => (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6))))))))) & ((Mi3 != Ma3) => ((((vEBT_V5917875025757280293ildren @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ TreeList3) @ Ma3) & (! [X:nat]: (((((ord_less @ nat) @ X) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg2)) => ((((vEBT_V5917875025757280293ildren @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ TreeList3) @ X) => ((((ord_less @ nat) @ Mi3) @ X) & (((ord_less_eq @ nat) @ X) @ Ma3)))))))))))))))) @ Mima)))))))))))))))))). % 6.41/2.69 thf(fact_4913_VEBT__internal_Ovalid_H_Oelims_I2_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: ((((vEBT_VEBT_valid @ X3) @ Xa2) => (((? [Uu2:$o,Uv2:$o]: ((X3 = ((vEBT_Leaf @ Uu2) @ Uv2)))) => (Xa2 != (one_one @ nat))) => (~ ((! [Mima:(option @ ((product_prod @ nat) @ nat)),Deg2:nat,TreeList3:(list @ vEBT_VEBT),Summary2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ Mima) @ Deg2) @ TreeList3) @ Summary2)) => (~ (((Deg2 = Xa2) & ((! [X2:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X2) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => ((vEBT_VEBT_valid @ X2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) & (((vEBT_VEBT_valid @ Summary2) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) & ((((size_size @ (list @ vEBT_VEBT)) @ TreeList3) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((((case_option @ $o) @ ((product_prod @ nat) @ nat)) @ ((~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ Summary2) @ X6))))) & (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6)))))))))) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [Mi3:nat,Ma3:nat]: (((((ord_less_eq @ nat) @ Mi3) @ Ma3) & ((((ord_less @ nat) @ Ma3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg2)) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) => ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList3) @ I3)) @ X6))) = ((vEBT_V8194947554948674370ptions @ Summary2) @ I3))))) & (((Mi3 = Ma3) => (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6))))))))) & ((Mi3 != Ma3) => ((((vEBT_V5917875025757280293ildren @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ TreeList3) @ Ma3) & (! [X:nat]: (((((ord_less @ nat) @ X) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg2)) => ((((vEBT_V5917875025757280293ildren @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ TreeList3) @ X) => ((((ord_less @ nat) @ Mi3) @ X) & (((ord_less_eq @ nat) @ X) @ Ma3)))))))))))))))) @ Mima))))))))))))))))). % 6.41/2.69 thf(fact_4914_VEBT__internal_Ovalid_H_Opelims_I1_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat,Y:$o]: (((((vEBT_VEBT_valid @ X3) @ Xa2) = Y) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_VEBT_valid_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ X3) @ Xa2)) => ((! [Uu2:$o,Uv2:$o]: (((X3 = ((vEBT_Leaf @ Uu2) @ Uv2)) => ((Y = (Xa2 = (one_one @ nat))) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_VEBT_valid_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((vEBT_Leaf @ Uu2) @ Uv2)) @ Xa2)))))))) => (~ ((! [Mima:(option @ ((product_prod @ nat) @ nat)),Deg2:nat,TreeList3:(list @ vEBT_VEBT),Summary2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ Mima) @ Deg2) @ TreeList3) @ Summary2)) => ((Y = ((Deg2 = Xa2) & ((! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => ((vEBT_VEBT_valid @ X) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) & (((vEBT_VEBT_valid @ Summary2) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) & ((((size_size @ (list @ vEBT_VEBT)) @ TreeList3) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((((case_option @ $o) @ ((product_prod @ nat) @ nat)) @ ((~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ Summary2) @ X6))))) & (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6)))))))))) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [Mi3:nat,Ma3:nat]: (((((ord_less_eq @ nat) @ Mi3) @ Ma3) & ((((ord_less @ nat) @ Ma3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg2)) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) => ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList3) @ I3)) @ X6))) = ((vEBT_V8194947554948674370ptions @ Summary2) @ I3))))) & (((Mi3 = Ma3) => (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6))))))))) & ((Mi3 != Ma3) => ((((vEBT_V5917875025757280293ildren @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ TreeList3) @ Ma3) & (! [X:nat]: (((((ord_less @ nat) @ X) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg2)) => ((((vEBT_V5917875025757280293ildren @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ TreeList3) @ X) => ((((ord_less @ nat) @ Mi3) @ X) & (((ord_less_eq @ nat) @ X) @ Ma3)))))))))))))))) @ Mima)))))) => (~ ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_VEBT_valid_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ Mima) @ Deg2) @ TreeList3) @ Summary2)) @ Xa2)))))))))))))))). % 6.41/2.69 thf(fact_4915_VEBT__internal_Ovalid_H_Opelims_I2_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: ((((vEBT_VEBT_valid @ X3) @ Xa2) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_VEBT_valid_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ X3) @ Xa2)) => ((! [Uu2:$o,Uv2:$o]: (((X3 = ((vEBT_Leaf @ Uu2) @ Uv2)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_VEBT_valid_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((vEBT_Leaf @ Uu2) @ Uv2)) @ Xa2)) => (Xa2 != (one_one @ nat)))))) => (~ ((! [Mima:(option @ ((product_prod @ nat) @ nat)),Deg2:nat,TreeList3:(list @ vEBT_VEBT),Summary2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ Mima) @ Deg2) @ TreeList3) @ Summary2)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_VEBT_valid_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ Mima) @ Deg2) @ TreeList3) @ Summary2)) @ Xa2)) => (~ (((Deg2 = Xa2) & ((! [X2:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X2) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => ((vEBT_VEBT_valid @ X2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) & (((vEBT_VEBT_valid @ Summary2) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) & ((((size_size @ (list @ vEBT_VEBT)) @ TreeList3) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((((case_option @ $o) @ ((product_prod @ nat) @ nat)) @ ((~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ Summary2) @ X6))))) & (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6)))))))))) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [Mi3:nat,Ma3:nat]: (((((ord_less_eq @ nat) @ Mi3) @ Ma3) & ((((ord_less @ nat) @ Ma3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg2)) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) => ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList3) @ I3)) @ X6))) = ((vEBT_V8194947554948674370ptions @ Summary2) @ I3))))) & (((Mi3 = Ma3) => (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6))))))))) & ((Mi3 != Ma3) => ((((vEBT_V5917875025757280293ildren @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ TreeList3) @ Ma3) & (! [X:nat]: (((((ord_less @ nat) @ X) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg2)) => ((((vEBT_V5917875025757280293ildren @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ TreeList3) @ X) => ((((ord_less @ nat) @ Mi3) @ X) & (((ord_less_eq @ nat) @ X) @ Ma3)))))))))))))))) @ Mima))))))))))))))))))). % 6.41/2.69 thf(fact_4916_VEBT__internal_Ovalid_H_Opelims_I3_J, axiom, (! [X3:vEBT_VEBT,Xa2:nat]: (((~ (((vEBT_VEBT_valid @ X3) @ Xa2))) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_VEBT_valid_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ X3) @ Xa2)) => ((! [Uu2:$o,Uv2:$o]: (((X3 = ((vEBT_Leaf @ Uu2) @ Uv2)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_VEBT_valid_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((vEBT_Leaf @ Uu2) @ Uv2)) @ Xa2)) => (Xa2 = (one_one @ nat)))))) => (~ ((! [Mima:(option @ ((product_prod @ nat) @ nat)),Deg2:nat,TreeList3:(list @ vEBT_VEBT),Summary2:vEBT_VEBT]: (((X3 = ((((vEBT_Node @ Mima) @ Deg2) @ TreeList3) @ Summary2)) => ((((accp @ ((product_prod @ vEBT_VEBT) @ nat)) @ vEBT_VEBT_valid_rel) @ ((((product_Pair @ vEBT_VEBT) @ nat) @ ((((vEBT_Node @ Mima) @ Deg2) @ TreeList3) @ Summary2)) @ Xa2)) => ((Deg2 = Xa2) & ((! [X5:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X5) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => ((vEBT_VEBT_valid @ X5) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) & (((vEBT_VEBT_valid @ Summary2) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) & ((((size_size @ (list @ vEBT_VEBT)) @ TreeList3) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) & (((((case_option @ $o) @ ((product_prod @ nat) @ nat)) @ ((~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ Summary2) @ X6))))) & (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6)))))))))) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [Mi3:nat,Ma3:nat]: (((((ord_less_eq @ nat) @ Mi3) @ Ma3) & ((((ord_less @ nat) @ Ma3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg2)) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ Deg2) @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) => ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ TreeList3) @ I3)) @ X6))) = ((vEBT_V8194947554948674370ptions @ Summary2) @ I3))))) & (((Mi3 = Ma3) => (! [X:vEBT_VEBT]: (((((member @ vEBT_VEBT) @ X) @ ((set2 @ vEBT_VEBT) @ TreeList3)) => (~ ((? [X6:nat]: (((vEBT_V8194947554948674370ptions @ X) @ X6))))))))) & ((Mi3 != Ma3) => ((((vEBT_V5917875025757280293ildren @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ TreeList3) @ Ma3) & (! [X:nat]: (((((ord_less @ nat) @ X) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Deg2)) => ((((vEBT_V5917875025757280293ildren @ (((divide_divide @ nat) @ Deg2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ TreeList3) @ X) => ((((ord_less @ nat) @ Mi3) @ X) & (((ord_less_eq @ nat) @ X) @ Ma3)))))))))))))))) @ Mima))))))))))))))))). % 6.41/2.69 thf(fact_4917_Union__maximal__sets, axiom, (! [A:$tType,F8:(set @ (set @ A))]: ((((finite_finite2 @ (set @ A)) @ F8) => (((complete_Sup_Sup @ (set @ A)) @ ((collect @ (set @ A)) @ (^ [T8:(set @ A)]: (((((member @ (set @ A)) @ T8) @ F8) & (! [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ F8) => (~ ((((ord_less @ (set @ A)) @ T8) @ X))))))))))) = ((complete_Sup_Sup @ (set @ A)) @ F8)))))). % 6.41/2.69 thf(fact_4918_concat__injective, axiom, (! [A:$tType,Xs:(list @ (list @ A)),Ys:(list @ (list @ A))]: (((((concat @ A) @ Xs) = ((concat @ A) @ Ys)) => ((((size_size @ (list @ (list @ A))) @ Xs) = ((size_size @ (list @ (list @ A))) @ Ys)) => ((! [X5:((product_prod @ (list @ A)) @ (list @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ X5) @ ((set2 @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((zip @ (list @ A)) @ (list @ A)) @ Xs) @ Ys))) => (((((product_case_prod @ (list @ A)) @ (list @ A)) @ $o) @ (^ [Y2:(list @ A),Z6:(list @ A)]: ((((size_size @ (list @ A)) @ Y2) = ((size_size @ (list @ A)) @ Z6))))) @ X5)))) => (Xs = Ys))))))). % 6.41/2.69 thf(fact_4919_concat__eq__concat__iff, axiom, (! [A:$tType,Xs:(list @ (list @ A)),Ys:(list @ (list @ A))]: (((! [X5:((product_prod @ (list @ A)) @ (list @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ X5) @ ((set2 @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((zip @ (list @ A)) @ (list @ A)) @ Xs) @ Ys))) => (((((product_case_prod @ (list @ A)) @ (list @ A)) @ $o) @ (^ [Y2:(list @ A),Z6:(list @ A)]: ((((size_size @ (list @ A)) @ Y2) = ((size_size @ (list @ A)) @ Z6))))) @ X5)))) => ((((size_size @ (list @ (list @ A))) @ Xs) = ((size_size @ (list @ (list @ A))) @ Ys)) => ((((concat @ A) @ Xs) = ((concat @ A) @ Ys)) = (Xs = Ys))))))). % 6.41/2.69 thf(fact_4920_list__eq__iff__zip__eq, axiom, (! [A:$tType]: (((^ [Y3:(list @ A),Z:(list @ A)]: ((Y3 = Z))) = (^ [Xs2:(list @ A),Ys3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) = ((size_size @ (list @ A)) @ Ys3)) & (! [X:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ X) @ ((set2 @ ((product_prod @ A) @ A)) @ ((((zip @ A) @ A) @ Xs2) @ Ys3))) => (((((product_case_prod @ A) @ A) @ $o) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) @ X))))))))))). % 6.41/2.69 thf(fact_4921_Greatest__def, axiom, (! [A:$tType]: (((order @ A) => ((order_Greatest @ A) = (^ [P3:(A > $o)]: (((the @ A) @ (^ [X:A]: (((P3 @ X) & (! [Y2:A]: (((P3 @ Y2) => (((ord_less_eq @ A) @ Y2) @ X))))))))))))))). % 6.41/2.69 thf(fact_4922_ord_OLeast__def, axiom, (! [A:$tType]: (((least @ A) = (^ [Less_eq:(A > (A > $o)),P3:(A > $o)]: (((the @ A) @ (^ [X:A]: (((P3 @ X) & (! [Y2:A]: (((P3 @ Y2) => ((Less_eq @ X) @ Y2)))))))))))))). % 6.41/2.69 thf(fact_4923_listrel__iff__zip, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys:(list @ B),R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs) @ Ys)) @ (((listrel @ A) @ B) @ R2)) = ((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) & (! [X:((product_prod @ A) @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys))) => (((((product_case_prod @ A) @ B) @ $o) @ (^ [Y2:A,Z6:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y2) @ Z6)) @ R2)))) @ X))))))))). % 6.41/2.69 thf(fact_4924_ord_OLeast_Ocong, axiom, (! [A:$tType]: (((least @ A) = (least @ A))))). % 6.41/2.69 thf(fact_4925_listrel__Nil2, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs) @ (nil @ B))) @ (((listrel @ A) @ B) @ R2)) => (Xs = (nil @ A)))))). % 6.41/2.69 thf(fact_4926_listrel__Nil1, axiom, (! [A:$tType,B:$tType,Xs:(list @ B),R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ Xs)) @ (((listrel @ A) @ B) @ R2)) => (Xs = (nil @ B)))))). % 6.41/2.69 thf(fact_4927_listrel_ONil, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B))]: ((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ (nil @ B))) @ (((listrel @ A) @ B) @ R2))))). % 6.41/2.69 thf(fact_4928_listrel__eq__len, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B),R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs) @ Ys)) @ (((listrel @ A) @ B) @ R2)) => (((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)))))). % 6.41/2.69 thf(fact_4929_GreatestI__nat, axiom, (! [P:(nat > $o),K:nat,B2:nat]: (((P @ K) => ((! [Y4:nat]: (((P @ Y4) => (((ord_less_eq @ nat) @ Y4) @ B2)))) => (P @ ((order_Greatest @ nat) @ P))))))). % 6.41/2.69 thf(fact_4930_Greatest__le__nat, axiom, (! [P:(nat > $o),K:nat,B2:nat]: (((P @ K) => ((! [Y4:nat]: (((P @ Y4) => (((ord_less_eq @ nat) @ Y4) @ B2)))) => (((ord_less_eq @ nat) @ K) @ ((order_Greatest @ nat) @ P))))))). % 6.41/2.69 thf(fact_4931_GreatestI__ex__nat, axiom, (! [P:(nat > $o),B2:nat]: (((? [X_12:nat]: ((P @ X_12))) => ((! [Y4:nat]: (((P @ Y4) => (((ord_less_eq @ nat) @ Y4) @ B2)))) => (P @ ((order_Greatest @ nat) @ P))))))). % 6.41/2.69 thf(fact_4932_listrel__mono, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R2) @ S) => (((ord_less_eq @ (set @ ((product_prod @ (list @ A)) @ (list @ B)))) @ (((listrel @ A) @ B) @ R2)) @ (((listrel @ A) @ B) @ S)))))). % 6.41/2.69 thf(fact_4933_GreatestI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X3:A,Q:(A > $o)]: (((P @ X3) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ A) @ Y4) @ X3)))) => ((! [X5:A]: (((P @ X5) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ Y5) @ X5)))) => (Q @ X5))))) => (Q @ ((order_Greatest @ A) @ P))))))))))). % 6.41/2.69 thf(fact_4934_Greatest__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X3:A]: (((P @ X3) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ A) @ Y4) @ X3)))) => (((order_Greatest @ A) @ P) = X3))))))))). % 6.41/2.69 thf(fact_4935_listrel__iff__nth, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B),R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs) @ Ys)) @ (((listrel @ A) @ B) @ R2)) = ((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) & (! [N:nat]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ (((nth @ A) @ Xs) @ N)) @ (((nth @ B) @ Ys) @ N))) @ R2))))))))). % 6.41/2.69 thf(fact_4936_take__bit__numeral__numeral, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:num,N2:num]: (((((bit_se2584673776208193580ke_bit @ A) @ ((numeral_numeral @ nat) @ M)) @ ((numeral_numeral @ A) @ N2)) = (((((case_option @ A) @ num) @ (zero_zero @ A)) @ (numeral_numeral @ A)) @ ((bit_take_bit_num @ ((numeral_numeral @ nat) @ M)) @ N2))))))))). % 6.41/2.69 thf(fact_4937_sorted__list__of__set_Osorted__key__list__of__set__insert__remove, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => (((linord4507533701916653071of_set @ A) @ (((insert @ A) @ X3) @ A3)) = (((((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X))) @ X3) @ ((linord4507533701916653071of_set @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))))). % 6.41/2.69 thf(fact_4938_sorted__list__of__set__def, axiom, (! [A:$tType]: (((linorder @ A) => ((linord4507533701916653071of_set @ A) = (((linord144544945434240204of_set @ A) @ A) @ (^ [X:A]: (X)))))))). % 6.41/2.69 thf(fact_4939_take__bit__num__simps_I1_J, axiom, (! [M:num]: ((((bit_take_bit_num @ (zero_zero @ nat)) @ M) = (none @ num))))). % 6.41/2.69 thf(fact_4940_remove1__insort__key, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [X3:B,F2:(B > A),Xs:(list @ B)]: (((((remove1 @ B) @ X3) @ (((((linorder_insort_key @ B) @ A) @ F2) @ X3) @ Xs)) = Xs))))))). % 6.41/2.69 thf(fact_4941_length__insort, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),X3:B,Xs:(list @ B)]: ((((size_size @ (list @ B)) @ (((((linorder_insort_key @ B) @ A) @ F2) @ X3) @ Xs)) = (suc @ ((size_size @ (list @ B)) @ Xs))))))))). % 6.41/2.69 thf(fact_4942_take__bit__num__simps_I2_J, axiom, (! [N2:nat]: ((((bit_take_bit_num @ (suc @ N2)) @ one2) = ((some @ num) @ one2))))). % 6.41/2.69 thf(fact_4943_take__bit__num__simps_I3_J, axiom, (! [N2:nat,M:num]: ((((bit_take_bit_num @ (suc @ N2)) @ (bit0 @ M)) = (((((case_option @ (option @ num)) @ num) @ (none @ num)) @ (^ [Q5:num]: (((some @ num) @ (bit0 @ Q5))))) @ ((bit_take_bit_num @ N2) @ M)))))). % 6.41/2.69 thf(fact_4944_take__bit__num__simps_I4_J, axiom, (! [N2:nat,M:num]: ((((bit_take_bit_num @ (suc @ N2)) @ (bit1 @ M)) = ((some @ num) @ (((((case_option @ num) @ num) @ one2) @ bit1) @ ((bit_take_bit_num @ N2) @ M))))))). % 6.41/2.69 thf(fact_4945_sorted__list__of__set_Osorted__key__list__of__set__insert, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((~ ((((member @ A) @ X3) @ A3))) => (((linord4507533701916653071of_set @ A) @ (((insert @ A) @ X3) @ A3)) = (((((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X))) @ X3) @ ((linord4507533701916653071of_set @ A) @ A3))))))))))). % 6.41/2.69 thf(fact_4946_take__bit__num__simps_I6_J, axiom, (! [R2:num,M:num]: ((((bit_take_bit_num @ ((numeral_numeral @ nat) @ R2)) @ (bit0 @ M)) = (((((case_option @ (option @ num)) @ num) @ (none @ num)) @ (^ [Q5:num]: (((some @ num) @ (bit0 @ Q5))))) @ ((bit_take_bit_num @ (pred_numeral @ R2)) @ M)))))). % 6.41/2.69 thf(fact_4947_insort__not__Nil, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),A2:B,Xs:(list @ B)]: (((((((linorder_insort_key @ B) @ A) @ F2) @ A2) @ Xs) != (nil @ B)))))))). % 6.41/2.69 thf(fact_4948_insort__key__left__comm, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),X3:B,Y:B,Xs:(list @ B)]: ((((F2 @ X3) != (F2 @ Y)) => ((((((linorder_insort_key @ B) @ A) @ F2) @ Y) @ (((((linorder_insort_key @ B) @ A) @ F2) @ X3) @ Xs)) = (((((linorder_insort_key @ B) @ A) @ F2) @ X3) @ (((((linorder_insort_key @ B) @ A) @ F2) @ Y) @ Xs)))))))))). % 6.41/2.69 thf(fact_4949_insort__left__comm, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A,Xs:(list @ A)]: (((((((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X))) @ X3) @ (((((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X))) @ Y) @ Xs)) = (((((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X))) @ Y) @ (((((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X))) @ X3) @ Xs))))))))). % 6.41/2.69 thf(fact_4950_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y:A,X3:A]: (((((((comp @ (list @ A)) @ (list @ A)) @ (list @ A)) @ ((((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X))) @ Y)) @ ((((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X))) @ X3)) = (((((comp @ (list @ A)) @ (list @ A)) @ (list @ A)) @ ((((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X))) @ X3)) @ ((((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X))) @ Y))))))))). % 6.41/2.69 thf(fact_4951_Code__Abstract__Nat_Otake__bit__num__code_I2_J, axiom, (! [N2:nat,M:num]: ((((bit_take_bit_num @ N2) @ (bit0 @ M)) = ((((case_nat @ (option @ num)) @ (none @ num)) @ (^ [N:nat]: ((((((case_option @ (option @ num)) @ num) @ (none @ num)) @ (^ [Q5:num]: (((some @ num) @ (bit0 @ Q5))))) @ ((bit_take_bit_num @ N) @ M))))) @ N2))))). % 6.41/2.69 thf(fact_4952_Code__Abstract__Nat_Otake__bit__num__code_I1_J, axiom, (! [N2:nat]: ((((bit_take_bit_num @ N2) @ one2) = ((((case_nat @ (option @ num)) @ (none @ num)) @ (^ [N:nat]: (((some @ num) @ one2)))) @ N2))))). % 6.41/2.69 thf(fact_4953_set__insort__key, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),X3:B,Xs:(list @ B)]: ((((set2 @ B) @ (((((linorder_insort_key @ B) @ A) @ F2) @ X3) @ Xs)) = (((insert @ B) @ X3) @ ((set2 @ B) @ Xs))))))))). % 6.41/2.69 thf(fact_4954_distinct__insort, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),X3:B,Xs:(list @ B)]: ((((distinct @ B) @ (((((linorder_insort_key @ B) @ A) @ F2) @ X3) @ Xs)) = ((~ ((((member @ B) @ X3) @ ((set2 @ B) @ Xs)))) & ((distinct @ B) @ Xs))))))))). % 6.41/2.69 thf(fact_4955_Code__Abstract__Nat_Otake__bit__num__code_I3_J, axiom, (! [N2:nat,M:num]: ((((bit_take_bit_num @ N2) @ (bit1 @ M)) = ((((case_nat @ (option @ num)) @ (none @ num)) @ (^ [N:nat]: (((some @ num) @ (((((case_option @ num) @ num) @ one2) @ bit1) @ ((bit_take_bit_num @ N) @ M)))))) @ N2))))). % 6.41/2.69 thf(fact_4956_take__bit__num__eq__None__imp, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M:nat,N2:num]: (((((bit_take_bit_num @ M) @ N2) = (none @ num)) => ((((bit_se2584673776208193580ke_bit @ A) @ M) @ ((numeral_numeral @ A) @ N2)) = (zero_zero @ A))))))))). % 6.41/2.69 thf(fact_4957_sorted__list__of__set_Ofold__insort__key_Oremove, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => (((linord4507533701916653071of_set @ A) @ A3) = (((((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X))) @ X3) @ ((linord4507533701916653071of_set @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))))))))). % 6.41/2.69 thf(fact_4958_take__bit__num__def, axiom, (bit_take_bit_num = (^ [N:nat,M2:num]: (((((if @ (option @ num)) @ ((((bit_se2584673776208193580ke_bit @ nat) @ N) @ ((numeral_numeral @ nat) @ M2)) = (zero_zero @ nat))) @ (none @ num)) @ ((some @ num) @ (num_of_nat @ (((bit_se2584673776208193580ke_bit @ nat) @ N) @ ((numeral_numeral @ nat) @ M2))))))))). % 6.41/2.69 thf(fact_4959_Bit__Operations_Otake__bit__num__code, axiom, (bit_take_bit_num = (^ [N:nat,M2:num]: ((((((product_case_prod @ nat) @ num) @ (option @ num)) @ (^ [A4:nat,X:num]: (((((case_nat @ (option @ num)) @ (none @ num)) @ (^ [O:nat]: ((((((case_num @ (option @ num)) @ ((some @ num) @ one2)) @ (^ [P6:num]: ((((((case_option @ (option @ num)) @ num) @ (none @ num)) @ (^ [Q5:num]: (((some @ num) @ (bit0 @ Q5))))) @ ((bit_take_bit_num @ O) @ P6))))) @ (^ [P6:num]: (((some @ num) @ (((((case_option @ num) @ num) @ one2) @ bit1) @ ((bit_take_bit_num @ O) @ P6)))))) @ X)))) @ A4)))) @ ((((product_Pair @ nat) @ num) @ N) @ M2)))))). % 6.41/2.69 thf(fact_4960_set__nths, axiom, (! [A:$tType,Xs:(list @ A),I6:(set @ nat)]: ((((set2 @ A) @ (((nths @ A) @ Xs) @ I6)) = ((collect @ A) @ (^ [Uu3:A]: ((? [I3:nat]: (((Uu3 = (((nth @ A) @ Xs) @ I3)) & ((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) & (((member @ nat) @ I3) @ I6)))))))))))). % 6.41/2.69 thf(fact_4961_Rats__eq__int__div__nat, axiom, ((field_char_0_Rats @ real) = ((collect @ real) @ (^ [Uu3:real]: ((? [I3:int,N:nat]: (((Uu3 = (((divide_divide @ real) @ ((ring_1_of_int @ real) @ I3)) @ ((semiring_1_of_nat @ real) @ N))) & (N != (zero_zero @ nat)))))))))). % 6.41/2.69 thf(fact_4962_nths__nil, axiom, (! [A:$tType,A3:(set @ nat)]: (((((nths @ A) @ (nil @ A)) @ A3) = (nil @ A))))). % 6.41/2.69 thf(fact_4963_nths__empty, axiom, (! [A:$tType,Xs:(list @ A)]: (((((nths @ A) @ Xs) @ (bot_bot @ (set @ nat))) = (nil @ A))))). % 6.41/2.69 thf(fact_4964_Rats__infinite, axiom, (! [A:$tType]: (((field_char_0 @ A) => (~ (((finite_finite2 @ A) @ (field_char_0_Rats @ A)))))))). % 6.41/2.69 thf(fact_4965_distinct__nthsI, axiom, (! [A:$tType,Xs:(list @ A),I6:(set @ nat)]: ((((distinct @ A) @ Xs) => ((distinct @ A) @ (((nths @ A) @ Xs) @ I6)))))). % 6.41/2.69 thf(fact_4966_Rats__add, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A2:A,B2:A]: (((((member @ A) @ A2) @ (field_char_0_Rats @ A)) => ((((member @ A) @ B2) @ (field_char_0_Rats @ A)) => (((member @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (field_char_0_Rats @ A)))))))))). % 6.41/2.69 thf(fact_4967_Rats__dense__in__real, axiom, (! [X3:real,Y:real]: (((((ord_less @ real) @ X3) @ Y) => (? [X5:real]: (((((member @ real) @ X5) @ (field_char_0_Rats @ real)) & ((((ord_less @ real) @ X3) @ X5) & (((ord_less @ real) @ X5) @ Y))))))))). % 6.41/2.69 thf(fact_4968_Rats__no__bot__less, axiom, (! [X3:real]: ((? [X5:real]: (((((member @ real) @ X5) @ (field_char_0_Rats @ real)) & (((ord_less @ real) @ X5) @ X3))))))). % 6.41/2.69 thf(fact_4969_Rats__0, axiom, (! [A:$tType]: (((field_char_0 @ A) => (((member @ A) @ (zero_zero @ A)) @ (field_char_0_Rats @ A)))))). % 6.41/2.69 thf(fact_4970_notin__set__nthsI, axiom, (! [A:$tType,X3:A,Xs:(list @ A),I6:(set @ nat)]: (((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)))) => (~ ((((member @ A) @ X3) @ ((set2 @ A) @ (((nths @ A) @ Xs) @ I6))))))))). % 6.41/2.69 thf(fact_4971_in__set__nthsD, axiom, (! [A:$tType,X3:A,Xs:(list @ A),I6:(set @ nat)]: (((((member @ A) @ X3) @ ((set2 @ A) @ (((nths @ A) @ Xs) @ I6))) => (((member @ A) @ X3) @ ((set2 @ A) @ Xs)))))). % 6.41/2.69 thf(fact_4972_set__nths__subset, axiom, (! [A:$tType,Xs:(list @ A),I6:(set @ nat)]: ((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ (((nths @ A) @ Xs) @ I6))) @ ((set2 @ A) @ Xs))))). % 6.41/2.69 thf(fact_4973_nths__all, axiom, (! [A:$tType,Xs:(list @ A),I6:(set @ nat)]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ nat) @ I4) @ I6)))) => ((((nths @ A) @ Xs) @ I6) = Xs))))). % 6.41/2.69 thf(fact_4974_length__nths, axiom, (! [A:$tType,Xs:(list @ A),I6:(set @ nat)]: ((((size_size @ (list @ A)) @ (((nths @ A) @ Xs) @ I6)) = ((finite_card @ nat) @ ((collect @ nat) @ (^ [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) & (((member @ nat) @ I3) @ I6)))))))))). % 6.41/2.69 thf(fact_4975_Nats__altdef1, axiom, (! [A:$tType]: (((ring_1 @ A) => ((semiring_1_Nats @ A) = ((collect @ A) @ (^ [Uu3:A]: ((? [N:int]: (((Uu3 = ((ring_1_of_int @ A) @ N)) & (((ord_less_eq @ int) @ (zero_zero @ int)) @ N)))))))))))). % 6.41/2.69 thf(fact_4976_rat__floor__lemma, axiom, (! [A2:int,B2:int]: (((((ord_less_eq @ rat) @ ((ring_1_of_int @ rat) @ (((divide_divide @ int) @ A2) @ B2))) @ ((fract @ A2) @ B2)) & (((ord_less @ rat) @ ((fract @ A2) @ B2)) @ ((ring_1_of_int @ rat) @ (((plus_plus @ int) @ (((divide_divide @ int) @ A2) @ B2)) @ (one_one @ int)))))))). % 6.41/2.69 thf(fact_4977_image__minus__const__atLeastLessThan__nat, axiom, (! [C2:nat,Y:nat,X3:nat]: ((((((ord_less @ nat) @ C2) @ Y) => (((((image @ nat) @ nat) @ (^ [I3:nat]: ((((minus_minus @ nat) @ I3) @ C2)))) @ (((set_or7035219750837199246ssThan @ nat) @ X3) @ Y)) = (((set_or7035219750837199246ssThan @ nat) @ (((minus_minus @ nat) @ X3) @ C2)) @ (((minus_minus @ nat) @ Y) @ C2)))) & ((~ ((((ord_less @ nat) @ C2) @ Y))) => (((((ord_less @ nat) @ X3) @ Y) => (((((image @ nat) @ nat) @ (^ [I3:nat]: ((((minus_minus @ nat) @ I3) @ C2)))) @ (((set_or7035219750837199246ssThan @ nat) @ X3) @ Y)) = (((insert @ nat) @ (zero_zero @ nat)) @ (bot_bot @ (set @ nat))))) & ((~ ((((ord_less @ nat) @ X3) @ Y))) => (((((image @ nat) @ nat) @ (^ [I3:nat]: ((((minus_minus @ nat) @ I3) @ C2)))) @ (((set_or7035219750837199246ssThan @ nat) @ X3) @ Y)) = (bot_bot @ (set @ nat)))))))))). % 6.41/2.69 thf(fact_4978_image__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F2:(B > A),X3:B,A3:(set @ B)]: (((B2 = (F2 @ X3)) => ((((member @ B) @ X3) @ A3) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F2) @ A3))))))). % 6.41/2.69 thf(fact_4979_image__ident, axiom, (! [A:$tType,Y7:(set @ A)]: ((((((image @ A) @ A) @ (^ [X:A]: (X))) @ Y7) = Y7)))). % 6.41/2.69 thf(fact_4980_image__is__empty, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B)]: (((((((image @ B) @ A) @ F2) @ A3) = (bot_bot @ (set @ A))) = (A3 = (bot_bot @ (set @ B))))))). % 6.41/2.69 thf(fact_4981_empty__is__image, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B)]: ((((bot_bot @ (set @ A)) = ((((image @ B) @ A) @ F2) @ A3)) = (A3 = (bot_bot @ (set @ B))))))). % 6.41/2.69 thf(fact_4982_image__empty, axiom, (! [B:$tType,A:$tType,F2:(B > A)]: ((((((image @ B) @ A) @ F2) @ (bot_bot @ (set @ B))) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_4983_finite__imageI, axiom, (! [B:$tType,A:$tType,F5:(set @ A),H2:(A > B)]: ((((finite_finite2 @ A) @ F5) => ((finite_finite2 @ B) @ ((((image @ A) @ B) @ H2) @ F5)))))). % 6.41/2.69 thf(fact_4984_image__insert, axiom, (! [A:$tType,B:$tType,F2:(B > A),A2:B,B3:(set @ B)]: ((((((image @ B) @ A) @ F2) @ (((insert @ B) @ A2) @ B3)) = (((insert @ A) @ (F2 @ A2)) @ ((((image @ B) @ A) @ F2) @ B3)))))). % 6.41/2.69 thf(fact_4985_insert__image, axiom, (! [B:$tType,A:$tType,X3:A,A3:(set @ A),F2:(A > B)]: (((((member @ A) @ X3) @ A3) => ((((insert @ B) @ (F2 @ X3)) @ ((((image @ A) @ B) @ F2) @ A3)) = ((((image @ A) @ B) @ F2) @ A3)))))). % 6.41/2.69 thf(fact_4986_image__add__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ A)]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ (zero_zero @ A))) @ S3) = S3))))))). % 6.41/2.69 thf(fact_4987_image__add__atLeastAtMost, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [K:A,I:A,J:A]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ K)) @ (((set_or1337092689740270186AtMost @ A) @ I) @ J)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ K))))))))). % 6.41/2.69 thf(fact_4988_image__add__atLeastLessThan, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [K:A,I:A,J:A]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ K)) @ (((set_or7035219750837199246ssThan @ A) @ I) @ J)) = (((set_or7035219750837199246ssThan @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ K))))))))). % 6.41/2.69 thf(fact_4989_image__add__atMost, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [C2:A,A2:A]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ C2)) @ ((set_ord_atMost @ A) @ A2)) = ((set_ord_atMost @ A) @ (((plus_plus @ A) @ C2) @ A2))))))))). % 6.41/2.69 thf(fact_4990_image__Suc__atLeastAtMost, axiom, (! [I:nat,J:nat]: ((((((image @ nat) @ nat) @ suc) @ (((set_or1337092689740270186AtMost @ nat) @ I) @ J)) = (((set_or1337092689740270186AtMost @ nat) @ (suc @ I)) @ (suc @ J)))))). % 6.41/2.69 thf(fact_4991_image__Suc__atLeastLessThan, axiom, (! [I:nat,J:nat]: ((((((image @ nat) @ nat) @ suc) @ (((set_or7035219750837199246ssThan @ nat) @ I) @ J)) = (((set_or7035219750837199246ssThan @ nat) @ (suc @ I)) @ (suc @ J)))))). % 6.41/2.69 thf(fact_4992_SUP__bot__conv_I2_J, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [B3:(B > A),A3:(set @ B)]: ((((bot_bot @ A) = ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ B3) @ A3))) = (! [X:B]: (((((member @ B) @ X) @ A3) => ((B3 @ X) = (bot_bot @ A)))))))))))). % 6.41/2.69 thf(fact_4993_SUP__bot__conv_I1_J, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [B3:(B > A),A3:(set @ B)]: (((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ B3) @ A3)) = (bot_bot @ A)) = (! [X:B]: (((((member @ B) @ X) @ A3) => ((B3 @ X) = (bot_bot @ A)))))))))))). % 6.41/2.69 thf(fact_4994_SUP__bot, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B)]: ((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [X:B]: ((bot_bot @ A)))) @ A3)) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4995_ccSUP__bot, axiom, (! [B:$tType,A:$tType]: (((counta3822494911875563373attice @ A) => (! [A3:(set @ B)]: ((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [X:B]: ((bot_bot @ A)))) @ A3)) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_4996_SUP__const, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),F2:A]: (((A3 != (bot_bot @ (set @ B))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [I3:B]: (F2))) @ A3)) = F2)))))))). % 6.41/2.69 thf(fact_4997_ccSUP__const, axiom, (! [B:$tType,A:$tType]: (((counta3822494911875563373attice @ A) => (! [A3:(set @ B),F2:A]: (((A3 != (bot_bot @ (set @ B))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [I3:B]: (F2))) @ A3)) = F2)))))))). % 6.41/2.69 thf(fact_4998_cSUP__const, axiom, (! [B:$tType,A:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ B),C2:A]: (((A3 != (bot_bot @ (set @ B))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [X:B]: (C2))) @ A3)) = C2)))))))). % 6.41/2.69 thf(fact_4999_image__add__atLeastAtMost_H, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [K:A,I:A,J:A]: ((((((image @ A) @ A) @ (^ [N:A]: ((((plus_plus @ A) @ N) @ K)))) @ (((set_or1337092689740270186AtMost @ A) @ I) @ J)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ K))))))))). % 6.41/2.69 thf(fact_5000_INF__const, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),F2:A]: (((A3 != (bot_bot @ (set @ B))) => (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ (^ [I3:B]: (F2))) @ A3)) = F2)))))))). % 6.41/2.69 thf(fact_5001_ccINF__const, axiom, (! [B:$tType,A:$tType]: (((counta3822494911875563373attice @ A) => (! [A3:(set @ B),F2:A]: (((A3 != (bot_bot @ (set @ B))) => (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ (^ [I3:B]: (F2))) @ A3)) = F2)))))))). % 6.41/2.69 thf(fact_5002_cINF__const, axiom, (! [B:$tType,A:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ B),C2:A]: (((A3 != (bot_bot @ (set @ B))) => (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ (^ [X:B]: (C2))) @ A3)) = C2)))))))). % 6.41/2.69 thf(fact_5003_image__add__atLeastLessThan_H, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [K:A,I:A,J:A]: ((((((image @ A) @ A) @ (^ [N:A]: ((((plus_plus @ A) @ N) @ K)))) @ (((set_or7035219750837199246ssThan @ A) @ I) @ J)) = (((set_or7035219750837199246ssThan @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ K))))))))). % 6.41/2.69 thf(fact_5004_INF__eq__bot__iff, axiom, (! [B:$tType,A:$tType]: (((comple5582772986160207858norder @ A) => (! [F2:(B > A),A3:(set @ B)]: (((((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3)) = (bot_bot @ A)) = (! [X:A]: (((((ord_less @ A) @ (bot_bot @ A)) @ X) => (? [Y2:B]: (((((member @ B) @ Y2) @ A3) & (((ord_less @ A) @ (F2 @ Y2)) @ X)))))))))))))). % 6.41/2.69 thf(fact_5005_ccSUP__empty, axiom, (! [B:$tType,A:$tType]: (((counta3822494911875563373attice @ A) => (! [F2:(B > A)]: ((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ (bot_bot @ (set @ B)))) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_5006_image__mult__atLeastAtMost, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [D2:A,A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ D2) => (((((image @ A) @ A) @ ((times_times @ A) @ D2)) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((times_times @ A) @ D2) @ A2)) @ (((times_times @ A) @ D2) @ B2)))))))))). % 6.41/2.69 thf(fact_5007_image__divide__atLeastAtMost, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [D2:A,A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ D2) => (((((image @ A) @ A) @ (^ [C4:A]: ((((divide_divide @ A) @ C4) @ D2)))) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((divide_divide @ A) @ A2) @ D2)) @ (((divide_divide @ A) @ B2) @ D2)))))))))). % 6.41/2.69 thf(fact_5008_less__rat, axiom, (! [B2:int,D2:int,A2:int,C2:int]: (((B2 != (zero_zero @ int)) => ((D2 != (zero_zero @ int)) => ((((ord_less @ rat) @ ((fract @ A2) @ B2)) @ ((fract @ C2) @ D2)) = (((ord_less @ int) @ (((times_times @ int) @ (((times_times @ int) @ A2) @ D2)) @ (((times_times @ int) @ B2) @ D2))) @ (((times_times @ int) @ (((times_times @ int) @ C2) @ B2)) @ (((times_times @ int) @ B2) @ D2))))))))). % 6.41/2.69 thf(fact_5009_setcompr__eq__image, axiom, (! [A:$tType,B:$tType,F2:(B > A),P:(B > $o)]: ((((collect @ A) @ (^ [Uu3:A]: ((? [X:B]: (((Uu3 = (F2 @ X)) & (P @ X))))))) = ((((image @ B) @ A) @ F2) @ ((collect @ B) @ P)))))). % 6.41/2.69 thf(fact_5010_Setcompr__eq__image, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B)]: ((((collect @ A) @ (^ [Uu3:A]: ((? [X:B]: (((Uu3 = (F2 @ X)) & (((member @ B) @ X) @ A3))))))) = ((((image @ B) @ A) @ F2) @ A3))))). % 6.41/2.69 thf(fact_5011_of__nat__in__Nats, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N2:nat]: ((((member @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (semiring_1_Nats @ A)))))))). % 6.41/2.69 thf(fact_5012_Nats__induct, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [X3:A,P:(A > $o)]: (((((member @ A) @ X3) @ (semiring_1_Nats @ A)) => ((! [N3:nat]: ((P @ ((semiring_1_of_nat @ A) @ N3)))) => (P @ X3))))))))). % 6.41/2.69 thf(fact_5013_Nats__cases, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [X3:A]: (((((member @ A) @ X3) @ (semiring_1_Nats @ A)) => (~ ((! [N3:nat]: ((X3 != ((semiring_1_of_nat @ A) @ N3))))))))))))). % 6.41/2.69 thf(fact_5014_image__Fpow__mono, axiom, (! [B:$tType,A:$tType,F2:(B > A),A3:(set @ B),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F2) @ A3)) @ B3) => (((ord_less_eq @ (set @ (set @ A))) @ ((((image @ (set @ B)) @ (set @ A)) @ (((image @ B) @ A) @ F2)) @ ((finite_Fpow @ B) @ A3))) @ ((finite_Fpow @ A) @ B3)))))). % 6.41/2.69 thf(fact_5015_imageI, axiom, (! [B:$tType,A:$tType,X3:A,A3:(set @ A),F2:(A > B)]: (((((member @ A) @ X3) @ A3) => (((member @ B) @ (F2 @ X3)) @ ((((image @ A) @ B) @ F2) @ A3)))))). % 6.41/2.69 thf(fact_5016_image__iff, axiom, (! [A:$tType,B:$tType,Z2:A,F2:(B > A),A3:(set @ B)]: (((((member @ A) @ Z2) @ ((((image @ B) @ A) @ F2) @ A3)) = (? [X:B]: (((((member @ B) @ X) @ A3) & (Z2 = (F2 @ X))))))))). % 6.41/2.69 thf(fact_5017_bex__imageD, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B),P:(A > $o)]: (((? [X2:A]: (((((member @ A) @ X2) @ ((((image @ B) @ A) @ F2) @ A3)) & (P @ X2)))) => (? [X5:B]: (((((member @ B) @ X5) @ A3) & (P @ (F2 @ X5))))))))). % 6.41/2.69 thf(fact_5018_image__cong, axiom, (! [B:$tType,A:$tType,M7:(set @ A),N7:(set @ A),F2:(A > B),G:(A > B)]: (((M7 = N7) => ((! [X5:A]: (((((member @ A) @ X5) @ N7) => ((F2 @ X5) = (G @ X5))))) => (((((image @ A) @ B) @ F2) @ M7) = ((((image @ A) @ B) @ G) @ N7))))))). % 6.41/2.69 thf(fact_5019_ball__imageD, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B),P:(A > $o)]: (((! [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F2) @ A3)) => (P @ X5)))) => (! [X2:B]: (((((member @ B) @ X2) @ A3) => (P @ (F2 @ X2))))))))). % 6.41/2.69 thf(fact_5020_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X3:A,A3:(set @ A),B2:B,F2:(A > B)]: (((((member @ A) @ X3) @ A3) => ((B2 = (F2 @ X3)) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ F2) @ A3))))))). % 6.41/2.69 thf(fact_5021_imageE, axiom, (! [A:$tType,B:$tType,B2:A,F2:(B > A),A3:(set @ B)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ F2) @ A3)) => (~ ((! [X5:B]: (((B2 = (F2 @ X5)) => (~ ((((member @ B) @ X5) @ A3)))))))))))). % 6.41/2.69 thf(fact_5022_image__image, axiom, (! [A:$tType,B:$tType,C:$tType,F2:(B > A),G:(C > B),A3:(set @ C)]: ((((((image @ B) @ A) @ F2) @ ((((image @ C) @ B) @ G) @ A3)) = ((((image @ C) @ A) @ (^ [X:C]: ((F2 @ (G @ X))))) @ A3))))). % 6.41/2.69 thf(fact_5023_Compr__image__eq, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B),P:(A > $o)]: ((((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ ((((image @ B) @ A) @ F2) @ A3)) & (P @ X))))) = ((((image @ B) @ A) @ F2) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ A3) & (P @ (F2 @ X))))))))))). % 6.41/2.69 thf(fact_5024_all__subset__image, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B),P:((set @ A) > $o)]: (((! [B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B6) @ ((((image @ B) @ A) @ F2) @ A3)) => (P @ B6)))) = (! [B6:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ B6) @ A3) => (P @ ((((image @ B) @ A) @ F2) @ B6))))))))). % 6.41/2.69 thf(fact_5025_subset__image__iff, axiom, (! [A:$tType,B:$tType,B3:(set @ A),F2:(B > A),A3:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B3) @ ((((image @ B) @ A) @ F2) @ A3)) = (? [AA:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ AA) @ A3) & (B3 = ((((image @ B) @ A) @ F2) @ AA))))))))). % 6.41/2.69 thf(fact_5026_image__subset__iff, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F2) @ A3)) @ B3) = (! [X:B]: (((((member @ B) @ X) @ A3) => (((member @ A) @ (F2 @ X)) @ B3)))))))). % 6.41/2.69 thf(fact_5027_subset__imageE, axiom, (! [A:$tType,B:$tType,B3:(set @ A),F2:(B > A),A3:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B3) @ ((((image @ B) @ A) @ F2) @ A3)) => (~ ((! [C8:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C8) @ A3) => (B3 != ((((image @ B) @ A) @ F2) @ C8))))))))))). % 6.41/2.69 thf(fact_5028_image__subsetI, axiom, (! [A:$tType,B:$tType,A3:(set @ A),F2:(A > B),B3:(set @ B)]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((member @ B) @ (F2 @ X5)) @ B3)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A3)) @ B3))))). % 6.41/2.69 thf(fact_5029_image__mono, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ A),F2:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A3)) @ ((((image @ A) @ B) @ F2) @ B3)))))). % 6.41/2.69 thf(fact_5030_image__Pow__surj, axiom, (! [B:$tType,A:$tType,F2:(B > A),A3:(set @ B),B3:(set @ A)]: (((((((image @ B) @ A) @ F2) @ A3) = B3) => (((((image @ (set @ B)) @ (set @ A)) @ (((image @ B) @ A) @ F2)) @ ((pow2 @ B) @ A3)) = ((pow2 @ A) @ B3)))))). % 6.41/2.69 thf(fact_5031_image__Collect__subsetI, axiom, (! [A:$tType,B:$tType,P:(A > $o),F2:(A > B),B3:(set @ B)]: (((! [X5:A]: (((P @ X5) => (((member @ B) @ (F2 @ X5)) @ B3)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ ((collect @ A) @ P))) @ B3))))). % 6.41/2.69 thf(fact_5032_image__Pow__mono, axiom, (! [B:$tType,A:$tType,F2:(B > A),A3:(set @ B),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F2) @ A3)) @ B3) => (((ord_less_eq @ (set @ (set @ A))) @ ((((image @ (set @ B)) @ (set @ A)) @ (((image @ B) @ A) @ F2)) @ ((pow2 @ B) @ A3))) @ ((pow2 @ A) @ B3)))))). % 6.41/2.69 thf(fact_5033_Nats__0, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((member @ A) @ (zero_zero @ A)) @ (semiring_1_Nats @ A)))))). % 6.41/2.69 thf(fact_5034_Nats__add, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [A2:A,B2:A]: (((((member @ A) @ A2) @ (semiring_1_Nats @ A)) => ((((member @ A) @ B2) @ (semiring_1_Nats @ A)) => (((member @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (semiring_1_Nats @ A)))))))))). % 6.41/2.69 thf(fact_5035_Nats__1, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((member @ A) @ (one_one @ A)) @ (semiring_1_Nats @ A)))))). % 6.41/2.69 thf(fact_5036_Nats__mult, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [A2:A,B2:A]: (((((member @ A) @ A2) @ (semiring_1_Nats @ A)) => ((((member @ A) @ B2) @ (semiring_1_Nats @ A)) => (((member @ A) @ (((times_times @ A) @ A2) @ B2)) @ (semiring_1_Nats @ A)))))))))). % 6.41/2.69 thf(fact_5037_Nats__numeral, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [W2:num]: ((((member @ A) @ ((numeral_numeral @ A) @ W2)) @ (semiring_1_Nats @ A)))))))). % 6.41/2.69 thf(fact_5038_UNION__singleton__eq__range, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B)]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [X:B]: ((((insert @ A) @ (F2 @ X)) @ (bot_bot @ (set @ A)))))) @ A3)) = ((((image @ B) @ A) @ F2) @ A3))))). % 6.41/2.69 thf(fact_5039_pigeonhole__infinite, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F2:(A > B)]: (((~ (((finite_finite2 @ A) @ A3))) => (((finite_finite2 @ B) @ ((((image @ A) @ B) @ F2) @ A3)) => (? [X5:A]: (((((member @ A) @ X5) @ A3) & (~ (((finite_finite2 @ A) @ ((collect @ A) @ (^ [A4:A]: (((((member @ A) @ A4) @ A3) & ((F2 @ A4) = (F2 @ X5))))))))))))))))). % 6.41/2.69 thf(fact_5040_SUP__eq, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),B3:(set @ C),F2:(B > A),G:(C > A)]: (((! [I4:B]: (((((member @ B) @ I4) @ A3) => (? [X2:C]: (((((member @ C) @ X2) @ B3) & (((ord_less_eq @ A) @ (F2 @ I4)) @ (G @ X2)))))))) => ((! [J2:C]: (((((member @ C) @ J2) @ B3) => (? [X2:B]: (((((member @ B) @ X2) @ A3) & (((ord_less_eq @ A) @ (G @ J2)) @ (F2 @ X2)))))))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3)) = ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ G) @ B3))))))))))). % 6.41/2.69 thf(fact_5041_INF__eq, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),B3:(set @ C),G:(C > A),F2:(B > A)]: (((! [I4:B]: (((((member @ B) @ I4) @ A3) => (? [X2:C]: (((((member @ C) @ X2) @ B3) & (((ord_less_eq @ A) @ (G @ X2)) @ (F2 @ I4)))))))) => ((! [J2:C]: (((((member @ C) @ J2) @ B3) => (? [X2:B]: (((((member @ B) @ X2) @ A3) & (((ord_less_eq @ A) @ (F2 @ X2)) @ (G @ J2)))))))) => (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3)) = ((complete_Inf_Inf @ A) @ ((((image @ C) @ A) @ G) @ B3))))))))))). % 6.41/2.69 thf(fact_5042_zero__notin__Suc__image, axiom, (! [A3:(set @ nat)]: ((~ ((((member @ nat) @ (zero_zero @ nat)) @ ((((image @ nat) @ nat) @ suc) @ A3))))))). % 6.41/2.69 thf(fact_5043_finite__surj, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B3:(set @ B),F2:(A > B)]: ((((finite_finite2 @ A) @ A3) => ((((ord_less_eq @ (set @ B)) @ B3) @ ((((image @ A) @ B) @ F2) @ A3)) => ((finite_finite2 @ B) @ B3)))))). % 6.41/2.69 thf(fact_5044_finite__subset__image, axiom, (! [A:$tType,B:$tType,B3:(set @ A),F2:(B > A),A3:(set @ B)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ ((((image @ B) @ A) @ F2) @ A3)) => (? [C8:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C8) @ A3) & (((finite_finite2 @ B) @ C8) & (B3 = ((((image @ B) @ A) @ F2) @ C8))))))))))). % 6.41/2.69 thf(fact_5045_ex__finite__subset__image, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B),P:((set @ A) > $o)]: (((? [B6:(set @ A)]: ((((finite_finite2 @ A) @ B6) & ((((ord_less_eq @ (set @ A)) @ B6) @ ((((image @ B) @ A) @ F2) @ A3)) & (P @ B6))))) = (? [B6:(set @ B)]: ((((finite_finite2 @ B) @ B6) & ((((ord_less_eq @ (set @ B)) @ B6) @ A3) & (P @ ((((image @ B) @ A) @ F2) @ B6)))))))))). % 6.41/2.69 thf(fact_5046_all__finite__subset__image, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B),P:((set @ A) > $o)]: (((! [B6:(set @ A)]: (((((finite_finite2 @ A) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ ((((image @ B) @ A) @ F2) @ A3))) => (P @ B6)))) = (! [B6:(set @ B)]: (((((finite_finite2 @ B) @ B6) & (((ord_less_eq @ (set @ B)) @ B6) @ A3)) => (P @ ((((image @ B) @ A) @ F2) @ B6))))))))). % 6.41/2.69 thf(fact_5047_translation__Int, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,S:(set @ A),T2:(set @ A)]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ (((inf_inf @ (set @ A)) @ S) @ T2)) = (((inf_inf @ (set @ A)) @ ((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ S)) @ ((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ T2))))))))). % 6.41/2.69 thf(fact_5048_SUP__eq__const, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [I6:(set @ B),F2:(B > A),X3:A]: (((I6 != (bot_bot @ (set @ B))) => ((! [I4:B]: (((((member @ B) @ I4) @ I6) => ((F2 @ I4) = X3)))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ I6)) = X3))))))))). % 6.41/2.69 thf(fact_5049_translation__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,S:(set @ A),T2:(set @ A)]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ (((minus_minus @ (set @ A)) @ S) @ T2)) = (((minus_minus @ (set @ A)) @ ((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ S)) @ ((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ T2))))))))). % 6.41/2.69 thf(fact_5050_INF__eq__const, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [I6:(set @ B),F2:(B > A),X3:A]: (((I6 != (bot_bot @ (set @ B))) => ((! [I4:B]: (((((member @ B) @ I4) @ I6) => ((F2 @ I4) = X3)))) => (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ I6)) = X3))))))))). % 6.41/2.69 thf(fact_5051_finite__image__absD, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [S3:(set @ A)]: ((((finite_finite2 @ A) @ ((((image @ A) @ A) @ (abs_abs @ A)) @ S3)) => ((finite_finite2 @ A) @ S3)))))))). % 6.41/2.69 thf(fact_5052_image__Int__subset, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B),B3:(set @ B)]: ((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F2) @ (((inf_inf @ (set @ B)) @ A3) @ B3))) @ (((inf_inf @ (set @ A)) @ ((((image @ B) @ A) @ F2) @ A3)) @ ((((image @ B) @ A) @ F2) @ B3)))))). % 6.41/2.69 thf(fact_5053_image__diff__subset, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B),B3:(set @ B)]: ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ ((((image @ B) @ A) @ F2) @ A3)) @ ((((image @ B) @ A) @ F2) @ B3))) @ ((((image @ B) @ A) @ F2) @ (((minus_minus @ (set @ B)) @ A3) @ B3)))))). % 6.41/2.69 thf(fact_5054_Rat__induct__pos, axiom, (! [P:(rat > $o),Q4:rat]: (((! [A5:int,B5:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B5) => (P @ ((fract @ A5) @ B5))))) => (P @ Q4))))). % 6.41/2.69 thf(fact_5055_translation__Compl, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A,T2:(set @ A)]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ ((uminus_uminus @ (set @ A)) @ T2)) = ((uminus_uminus @ (set @ A)) @ ((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ T2))))))))). % 6.41/2.69 thf(fact_5056_quotient__of__eq, axiom, (! [A2:int,B2:int,P4:int,Q4:int]: ((((quotient_of @ ((fract @ A2) @ B2)) = ((((product_Pair @ int) @ int) @ P4) @ Q4)) => (((fract @ P4) @ Q4) = ((fract @ A2) @ B2)))))). % 6.41/2.69 thf(fact_5057_SUP__eqI, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),F2:(B > A),X3:A]: (((! [I4:B]: (((((member @ B) @ I4) @ A3) => (((ord_less_eq @ A) @ (F2 @ I4)) @ X3)))) => ((! [Y4:A]: (((! [I5:B]: (((((member @ B) @ I5) @ A3) => (((ord_less_eq @ A) @ (F2 @ I5)) @ Y4)))) => (((ord_less_eq @ A) @ X3) @ Y4)))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3)) = X3))))))))). % 6.41/2.69 thf(fact_5058_SUP__mono, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),B3:(set @ C),F2:(B > A),G:(C > A)]: (((! [N3:B]: (((((member @ B) @ N3) @ A3) => (? [X2:C]: (((((member @ C) @ X2) @ B3) & (((ord_less_eq @ A) @ (F2 @ N3)) @ (G @ X2)))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ G) @ B3)))))))))). % 6.41/2.69 thf(fact_5059_SUP__least, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),F2:(B > A),U:A]: (((! [I4:B]: (((((member @ B) @ I4) @ A3) => (((ord_less_eq @ A) @ (F2 @ I4)) @ U)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ U)))))))). % 6.41/2.69 thf(fact_5060_SUP__mono_H, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [F2:(B > A),G:(B > A),A3:(set @ B)]: (((! [X5:B]: ((((ord_less_eq @ A) @ (F2 @ X5)) @ (G @ X5)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ G) @ A3)))))))))). % 6.41/2.69 thf(fact_5061_SUP__upper, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [I:B,A3:(set @ B),F2:(B > A)]: (((((member @ B) @ I) @ A3) => (((ord_less_eq @ A) @ (F2 @ I)) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3)))))))))). % 6.41/2.69 thf(fact_5062_SUP__le__iff, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [F2:(B > A),A3:(set @ B),U:A]: (((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ U) = (! [X:B]: (((((member @ B) @ X) @ A3) => (((ord_less_eq @ A) @ (F2 @ X)) @ U))))))))))). % 6.41/2.69 thf(fact_5063_SUP__upper2, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [I:B,A3:(set @ B),U:A,F2:(B > A)]: (((((member @ B) @ I) @ A3) => ((((ord_less_eq @ A) @ U) @ (F2 @ I)) => (((ord_less_eq @ A) @ U) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))))))))))). % 6.41/2.69 thf(fact_5064_less__SUP__iff, axiom, (! [A:$tType,B:$tType]: (((comple5582772986160207858norder @ A) => (! [A2:A,F2:(B > A),A3:(set @ B)]: (((((ord_less @ A) @ A2) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) = (? [X:B]: (((((member @ B) @ X) @ A3) & (((ord_less @ A) @ A2) @ (F2 @ X)))))))))))). % 6.41/2.69 thf(fact_5065_SUP__lessD, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [F2:(B > A),A3:(set @ B),Y:A,I:B]: (((((ord_less @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ Y) => ((((member @ B) @ I) @ A3) => (((ord_less @ A) @ (F2 @ I)) @ Y))))))))). % 6.41/2.69 thf(fact_5066_INF__eqI, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),X3:A,F2:(B > A)]: (((! [I4:B]: (((((member @ B) @ I4) @ A3) => (((ord_less_eq @ A) @ X3) @ (F2 @ I4))))) => ((! [Y4:A]: (((! [I5:B]: (((((member @ B) @ I5) @ A3) => (((ord_less_eq @ A) @ Y4) @ (F2 @ I5))))) => (((ord_less_eq @ A) @ Y4) @ X3)))) => (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3)) = X3))))))))). % 6.41/2.69 thf(fact_5067_INF__mono, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [B3:(set @ B),A3:(set @ C),F2:(C > A),G:(B > A)]: (((! [M3:B]: (((((member @ B) @ M3) @ B3) => (? [X2:C]: (((((member @ C) @ X2) @ A3) & (((ord_less_eq @ A) @ (F2 @ X2)) @ (G @ M3)))))))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ C) @ A) @ F2) @ A3))) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ G) @ B3)))))))))). % 6.41/2.69 thf(fact_5068_INF__lower, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [I:B,A3:(set @ B),F2:(B > A)]: (((((member @ B) @ I) @ A3) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ (F2 @ I))))))))). % 6.41/2.69 thf(fact_5069_INF__mono_H, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [F2:(B > A),G:(B > A),A3:(set @ B)]: (((! [X5:B]: ((((ord_less_eq @ A) @ (F2 @ X5)) @ (G @ X5)))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ G) @ A3)))))))))). % 6.41/2.69 thf(fact_5070_INF__lower2, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [I:B,A3:(set @ B),F2:(B > A),U:A]: (((((member @ B) @ I) @ A3) => ((((ord_less_eq @ A) @ (F2 @ I)) @ U) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ U))))))))). % 6.41/2.69 thf(fact_5071_le__INF__iff, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [U:A,F2:(B > A),A3:(set @ B)]: (((((ord_less_eq @ A) @ U) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) = (! [X:B]: (((((member @ B) @ X) @ A3) => (((ord_less_eq @ A) @ U) @ (F2 @ X)))))))))))). % 6.41/2.69 thf(fact_5072_INF__greatest, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),U:A,F2:(B > A)]: (((! [I4:B]: (((((member @ B) @ I4) @ A3) => (((ord_less_eq @ A) @ U) @ (F2 @ I4))))) => (((ord_less_eq @ A) @ U) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3)))))))))). % 6.41/2.69 thf(fact_5073_INF__less__iff, axiom, (! [A:$tType,B:$tType]: (((comple5582772986160207858norder @ A) => (! [F2:(B > A),A3:(set @ B),A2:A]: (((((ord_less @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ A2) = (? [X:B]: (((((member @ B) @ X) @ A3) & (((ord_less @ A) @ (F2 @ X)) @ A2))))))))))). % 6.41/2.69 thf(fact_5074_less__INF__D, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [Y:A,F2:(B > A),A3:(set @ B),I:B]: (((((ord_less @ A) @ Y) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) => ((((member @ B) @ I) @ A3) => (((ord_less @ A) @ Y) @ (F2 @ I)))))))))). % 6.41/2.69 thf(fact_5075_finite__conv__nat__seg__image, axiom, (! [A:$tType]: (((finite_finite2 @ A) = (^ [A6:(set @ A)]: ((? [N:nat,F3:(nat > A)]: ((A6 = ((((image @ nat) @ A) @ F3) @ ((collect @ nat) @ (^ [I3:nat]: ((((ord_less @ nat) @ I3) @ N)))))))))))))). % 6.41/2.69 thf(fact_5076_nat__seg__image__imp__finite, axiom, (! [A:$tType,A3:(set @ A),F2:(nat > A),N2:nat]: (((A3 = ((((image @ nat) @ A) @ F2) @ ((collect @ nat) @ (^ [I3:nat]: ((((ord_less @ nat) @ I3) @ N2)))))) => ((finite_finite2 @ A) @ A3))))). % 6.41/2.69 thf(fact_5077_image__constant, axiom, (! [A:$tType,B:$tType,X3:A,A3:(set @ A),C2:B]: (((((member @ A) @ X3) @ A3) => (((((image @ A) @ B) @ (^ [X:A]: (C2))) @ A3) = (((insert @ B) @ C2) @ (bot_bot @ (set @ B)))))))). % 6.41/2.69 thf(fact_5078_image__constant__conv, axiom, (! [B:$tType,A:$tType,A3:(set @ B),C2:A]: ((((A3 = (bot_bot @ (set @ B))) => (((((image @ B) @ A) @ (^ [X:B]: (C2))) @ A3) = (bot_bot @ (set @ A)))) & ((A3 != (bot_bot @ (set @ B))) => (((((image @ B) @ A) @ (^ [X:B]: (C2))) @ A3) = (((insert @ A) @ C2) @ (bot_bot @ (set @ A))))))))). % 6.41/2.69 thf(fact_5079_sum_Oimage__gen, axiom, (! [A:$tType,C:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ B),H2:(B > A),G:(B > C)]: ((((finite_finite2 @ B) @ S3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ S3) = ((((groups7311177749621191930dd_sum @ C) @ A) @ (^ [Y2:C]: (((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ S3) & ((G @ X) = Y2))))))))) @ ((((image @ B) @ C) @ G) @ S3)))))))))). % 6.41/2.69 thf(fact_5080_normalize__eq, axiom, (! [A2:int,B2:int,P4:int,Q4:int]: ((((normalize @ ((((product_Pair @ int) @ int) @ A2) @ B2)) = ((((product_Pair @ int) @ int) @ P4) @ Q4)) => (((fract @ P4) @ Q4) = ((fract @ A2) @ B2)))))). % 6.41/2.69 thf(fact_5081_prod_Oimage__gen, axiom, (! [A:$tType,C:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),H2:(B > A),G:(B > C)]: ((((finite_finite2 @ B) @ S3) => (((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ S3) = ((((groups7121269368397514597t_prod @ C) @ A) @ (^ [Y2:C]: (((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ S3) & ((G @ X) = Y2))))))))) @ ((((image @ B) @ C) @ G) @ S3)))))))))). % 6.41/2.69 thf(fact_5082_Gcd__mono, axiom, (! [A:$tType,B:$tType]: (((semiring_Gcd @ A) => (! [A3:(set @ B),F2:(B > A),G:(B > A)]: (((! [X5:B]: (((((member @ B) @ X5) @ A3) => (((dvd_dvd @ A) @ (F2 @ X5)) @ (G @ X5))))) => (((dvd_dvd @ A) @ ((gcd_Gcd @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ ((gcd_Gcd @ A) @ ((((image @ B) @ A) @ G) @ A3)))))))))). % 6.41/2.69 thf(fact_5083_the__elem__image__unique, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F2:(A > B),X3:A]: (((A3 != (bot_bot @ (set @ A))) => ((! [Y4:A]: (((((member @ A) @ Y4) @ A3) => ((F2 @ Y4) = (F2 @ X3))))) => (((the_elem @ B) @ ((((image @ A) @ B) @ F2) @ A3)) = (F2 @ X3))))))). % 6.41/2.69 thf(fact_5084_le__SUP__iff, axiom, (! [B:$tType,A:$tType]: (((comple5582772986160207858norder @ A) => (! [X3:A,F2:(B > A),A3:(set @ B)]: (((((ord_less_eq @ A) @ X3) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) = (! [Y2:A]: (((((ord_less @ A) @ Y2) @ X3) => (? [X:B]: (((((member @ B) @ X) @ A3) & (((ord_less @ A) @ Y2) @ (F2 @ X))))))))))))))). % 6.41/2.69 thf(fact_5085_INF__le__iff, axiom, (! [B:$tType,A:$tType]: (((comple5582772986160207858norder @ A) => (! [F2:(B > A),A3:(set @ B),X3:A]: (((((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ X3) = (! [Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => (? [X:B]: (((((member @ B) @ X) @ A3) & (((ord_less @ A) @ (F2 @ X)) @ Y2)))))))))))))). % 6.41/2.69 thf(fact_5086_SUP__eq__iff, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [I6:(set @ B),C2:A,F2:(B > A)]: (((I6 != (bot_bot @ (set @ B))) => ((! [I4:B]: (((((member @ B) @ I4) @ I6) => (((ord_less_eq @ A) @ C2) @ (F2 @ I4))))) => ((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ I6)) = C2) = (! [X:B]: (((((member @ B) @ X) @ I6) => ((F2 @ X) = C2))))))))))))). % 6.41/2.69 thf(fact_5087_cSUP__least, axiom, (! [B:$tType,A:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ B),F2:(B > A),M7:A]: (((A3 != (bot_bot @ (set @ B))) => ((! [X5:B]: (((((member @ B) @ X5) @ A3) => (((ord_less_eq @ A) @ (F2 @ X5)) @ M7)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ M7))))))))). % 6.41/2.69 thf(fact_5088_INF__eq__iff, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [I6:(set @ B),F2:(B > A),C2:A]: (((I6 != (bot_bot @ (set @ B))) => ((! [I4:B]: (((((member @ B) @ I4) @ I6) => (((ord_less_eq @ A) @ (F2 @ I4)) @ C2)))) => ((((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ I6)) = C2) = (! [X:B]: (((((member @ B) @ X) @ I6) => ((F2 @ X) = C2))))))))))))). % 6.41/2.69 thf(fact_5089_cINF__greatest, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ B),M:A,F2:(B > A)]: (((A3 != (bot_bot @ (set @ B))) => ((! [X5:B]: (((((member @ B) @ X5) @ A3) => (((ord_less_eq @ A) @ M) @ (F2 @ X5))))) => (((ord_less_eq @ A) @ M) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))))))))))). % 6.41/2.69 thf(fact_5090_card__image__le, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F2:(A > B)]: ((((finite_finite2 @ A) @ A3) => (((ord_less_eq @ nat) @ ((finite_card @ B) @ ((((image @ A) @ B) @ F2) @ A3))) @ ((finite_card @ A) @ A3)))))). % 6.41/2.69 thf(fact_5091_Nats__diff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A,B2:A]: (((((member @ A) @ A2) @ (semiring_1_Nats @ A)) => ((((member @ A) @ B2) @ (semiring_1_Nats @ A)) => ((((ord_less_eq @ A) @ B2) @ A2) => (((member @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ (semiring_1_Nats @ A))))))))))). % 6.41/2.69 thf(fact_5092_SUP__subset__mono, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),B3:(set @ B),F2:(B > A),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ A3) @ B3) => ((! [X5:B]: (((((member @ B) @ X5) @ A3) => (((ord_less_eq @ A) @ (F2 @ X5)) @ (G @ X5))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ G) @ B3))))))))))). % 6.41/2.69 thf(fact_5093_INF__superset__mono, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [B3:(set @ B),A3:(set @ B),F2:(B > A),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ B3) @ A3) => ((! [X5:B]: (((((member @ B) @ X5) @ B3) => (((ord_less_eq @ A) @ (F2 @ X5)) @ (G @ X5))))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ G) @ B3))))))))))). % 6.41/2.69 thf(fact_5094_SUP__empty, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [F2:(B > A)]: ((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ (bot_bot @ (set @ B)))) = (bot_bot @ A)))))))). % 6.41/2.69 thf(fact_5095_SUP__constant, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),C2:A]: ((((A3 = (bot_bot @ (set @ B))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [Y2:B]: (C2))) @ A3)) = (bot_bot @ A))) & ((A3 != (bot_bot @ (set @ B))) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [Y2:B]: (C2))) @ A3)) = C2))))))))). % 6.41/2.69 thf(fact_5096_sum_Ogroup, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [S3:(set @ B),T4:(set @ C),G:(B > C),H2:(B > A)]: ((((finite_finite2 @ B) @ S3) => (((finite_finite2 @ C) @ T4) => ((((ord_less_eq @ (set @ C)) @ ((((image @ B) @ C) @ G) @ S3)) @ T4) => (((((groups7311177749621191930dd_sum @ C) @ A) @ (^ [Y2:C]: (((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ S3) & ((G @ X) = Y2))))))))) @ T4) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H2) @ S3))))))))))). % 6.41/2.69 thf(fact_5097_INF__inf__const2, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [I6:(set @ B),F2:(B > A),X3:A]: (((I6 != (bot_bot @ (set @ B))) => (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ (^ [I3:B]: ((((inf_inf @ A) @ (F2 @ I3)) @ X3)))) @ I6)) = (((inf_inf @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ I6))) @ X3))))))))). % 6.41/2.69 thf(fact_5098_INF__inf__const1, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [I6:(set @ B),X3:A,F2:(B > A)]: (((I6 != (bot_bot @ (set @ B))) => (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ (^ [I3:B]: ((((inf_inf @ A) @ X3) @ (F2 @ I3))))) @ I6)) = (((inf_inf @ A) @ X3) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ I6))))))))))). % 6.41/2.69 thf(fact_5099_Nats__subset__Rats, axiom, (! [A:$tType]: (((field_char_0 @ A) => (((ord_less_eq @ (set @ A)) @ (semiring_1_Nats @ A)) @ (field_char_0_Rats @ A)))))). % 6.41/2.69 thf(fact_5100_prod_Ogroup, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S3:(set @ B),T4:(set @ C),G:(B > C),H2:(B > A)]: ((((finite_finite2 @ B) @ S3) => (((finite_finite2 @ C) @ T4) => ((((ord_less_eq @ (set @ C)) @ ((((image @ B) @ C) @ G) @ S3)) @ T4) => (((((groups7121269368397514597t_prod @ C) @ A) @ (^ [Y2:C]: (((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ ((collect @ B) @ (^ [X:B]: (((((member @ B) @ X) @ S3) & ((G @ X) = Y2))))))))) @ T4) = ((((groups7121269368397514597t_prod @ B) @ A) @ H2) @ S3))))))))))). % 6.41/2.69 thf(fact_5101_quotient__of__Fract, axiom, (! [A2:int,B2:int]: (((quotient_of @ ((fract @ A2) @ B2)) = (normalize @ ((((product_Pair @ int) @ int) @ A2) @ B2)))))). % 6.41/2.69 thf(fact_5102_sum_Oreindex__nontrivial, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),H2:(B > C),G:(C > A)]: ((((finite_finite2 @ B) @ A3) => ((! [X5:B,Y4:B]: (((((member @ B) @ X5) @ A3) => ((((member @ B) @ Y4) @ A3) => ((X5 != Y4) => (((H2 @ X5) = (H2 @ Y4)) => ((G @ (H2 @ X5)) = (zero_zero @ A)))))))) => (((((groups7311177749621191930dd_sum @ C) @ A) @ G) @ ((((image @ B) @ C) @ H2) @ A3)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ (((((comp @ C) @ A) @ B) @ G) @ H2)) @ A3)))))))))). % 6.41/2.69 thf(fact_5103_INF__le__SUP, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),F2:(B > A)]: (((A3 != (bot_bot @ (set @ B))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3)))))))))). % 6.41/2.69 thf(fact_5104_prod_Oreindex__nontrivial, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),H2:(B > C),G:(C > A)]: ((((finite_finite2 @ B) @ A3) => ((! [X5:B,Y4:B]: (((((member @ B) @ X5) @ A3) => ((((member @ B) @ Y4) @ A3) => ((X5 != Y4) => (((H2 @ X5) = (H2 @ Y4)) => ((G @ (H2 @ X5)) = (one_one @ A)))))))) => (((((groups7121269368397514597t_prod @ C) @ A) @ G) @ ((((image @ B) @ C) @ H2) @ A3)) = ((((groups7121269368397514597t_prod @ B) @ A) @ (((((comp @ C) @ A) @ B) @ G) @ H2)) @ A3)))))))))). % 6.41/2.69 thf(fact_5105_surj__card__le, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B),F2:(A > B)]: ((((finite_finite2 @ A) @ A3) => ((((ord_less_eq @ (set @ B)) @ B3) @ ((((image @ A) @ B) @ F2) @ A3)) => (((ord_less_eq @ nat) @ ((finite_card @ B) @ B3)) @ ((finite_card @ A) @ A3))))))). % 6.41/2.69 thf(fact_5106_scaleR__image__atLeastAtMost, axiom, (! [A:$tType]: (((real_V5355595471888546746vector @ A) => (! [C2:real,X3:A,Y:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ C2) => (((((image @ A) @ A) @ ((real_V8093663219630862766scaleR @ A) @ C2)) @ (((set_or1337092689740270186AtMost @ A) @ X3) @ Y)) = (((set_or1337092689740270186AtMost @ A) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ X3)) @ (((real_V8093663219630862766scaleR @ A) @ C2) @ Y)))))))))). % 6.41/2.69 thf(fact_5107_image__Suc__lessThan, axiom, (! [N2:nat]: ((((((image @ nat) @ nat) @ suc) @ ((set_ord_lessThan @ nat) @ N2)) = (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ N2))))). % 6.41/2.69 thf(fact_5108_image__Suc__atMost, axiom, (! [N2:nat]: ((((((image @ nat) @ nat) @ suc) @ ((set_ord_atMost @ nat) @ N2)) = (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ (suc @ N2)))))). % 6.41/2.69 thf(fact_5109_atLeast0__atMost__Suc__eq__insert__0, axiom, (! [N2:nat]: (((((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N2)) = (((insert @ nat) @ (zero_zero @ nat)) @ ((((image @ nat) @ nat) @ suc) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N2))))))). % 6.41/2.69 thf(fact_5110_atLeast0__lessThan__Suc__eq__insert__0, axiom, (! [N2:nat]: (((((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N2)) = (((insert @ nat) @ (zero_zero @ nat)) @ ((((image @ nat) @ nat) @ suc) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2))))))). % 6.41/2.69 thf(fact_5111_lessThan__Suc__eq__insert__0, axiom, (! [N2:nat]: ((((set_ord_lessThan @ nat) @ (suc @ N2)) = (((insert @ nat) @ (zero_zero @ nat)) @ ((((image @ nat) @ nat) @ suc) @ ((set_ord_lessThan @ nat) @ N2))))))). % 6.41/2.69 thf(fact_5112_atMost__Suc__eq__insert__0, axiom, (! [N2:nat]: ((((set_ord_atMost @ nat) @ (suc @ N2)) = (((insert @ nat) @ (zero_zero @ nat)) @ ((((image @ nat) @ nat) @ suc) @ ((set_ord_atMost @ nat) @ N2))))))). % 6.41/2.69 thf(fact_5113_Fract__less__zero__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less @ rat) @ ((fract @ A2) @ B2)) @ (zero_zero @ rat)) = (((ord_less @ int) @ A2) @ (zero_zero @ int))))))). % 6.41/2.69 thf(fact_5114_zero__less__Fract__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less @ rat) @ (zero_zero @ rat)) @ ((fract @ A2) @ B2)) = (((ord_less @ int) @ (zero_zero @ int)) @ A2)))))). % 6.41/2.69 thf(fact_5115_one__less__Fract__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less @ rat) @ (one_one @ rat)) @ ((fract @ A2) @ B2)) = (((ord_less @ int) @ B2) @ A2)))))). % 6.41/2.69 thf(fact_5116_Fract__less__one__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less @ rat) @ ((fract @ A2) @ B2)) @ (one_one @ rat)) = (((ord_less @ int) @ A2) @ B2)))))). % 6.41/2.69 thf(fact_5117_sum__image__le, axiom, (! [A:$tType,B:$tType,C:$tType]: (((ordere6911136660526730532id_add @ B) => (! [I6:(set @ C),G:(A > B),F2:(C > A)]: ((((finite_finite2 @ C) @ I6) => ((! [I4:C]: (((((member @ C) @ I4) @ I6) => (((ord_less_eq @ B) @ (zero_zero @ B)) @ (G @ (F2 @ I4)))))) => (((ord_less_eq @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ G) @ ((((image @ C) @ A) @ F2) @ I6))) @ ((((groups7311177749621191930dd_sum @ C) @ B) @ (((((comp @ A) @ B) @ C) @ G) @ F2)) @ I6)))))))))). % 6.41/2.69 thf(fact_5118_zero__le__Fract__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less_eq @ rat) @ (zero_zero @ rat)) @ ((fract @ A2) @ B2)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ A2)))))). % 6.41/2.69 thf(fact_5119_Fract__le__zero__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less_eq @ rat) @ ((fract @ A2) @ B2)) @ (zero_zero @ rat)) = (((ord_less_eq @ int) @ A2) @ (zero_zero @ int))))))). % 6.41/2.69 thf(fact_5120_Fract__le__one__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less_eq @ rat) @ ((fract @ A2) @ B2)) @ (one_one @ rat)) = (((ord_less_eq @ int) @ A2) @ B2)))))). % 6.41/2.69 thf(fact_5121_one__le__Fract__iff, axiom, (! [B2:int,A2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ B2) => ((((ord_less_eq @ rat) @ (one_one @ rat)) @ ((fract @ A2) @ B2)) = (((ord_less_eq @ int) @ B2) @ A2)))))). % 6.41/2.69 thf(fact_5122_image__mult__atLeastAtMost__if, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,X3:A,Y:A]: ((((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((((image @ A) @ A) @ ((times_times @ A) @ C2)) @ (((set_or1337092689740270186AtMost @ A) @ X3) @ Y)) = (((set_or1337092689740270186AtMost @ A) @ (((times_times @ A) @ C2) @ X3)) @ (((times_times @ A) @ C2) @ Y)))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less_eq @ A) @ X3) @ Y) => (((((image @ A) @ A) @ ((times_times @ A) @ C2)) @ (((set_or1337092689740270186AtMost @ A) @ X3) @ Y)) = (((set_or1337092689740270186AtMost @ A) @ (((times_times @ A) @ C2) @ Y)) @ (((times_times @ A) @ C2) @ X3)))) & ((~ ((((ord_less_eq @ A) @ X3) @ Y))) => (((((image @ A) @ A) @ ((times_times @ A) @ C2)) @ (((set_or1337092689740270186AtMost @ A) @ X3) @ Y)) = (bot_bot @ (set @ A))))))))))))). % 6.41/2.69 thf(fact_5123_image__mult__atLeastAtMost__if_H, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A,C2:A]: ((((((ord_less_eq @ A) @ X3) @ Y) => (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((((image @ A) @ A) @ (^ [X:A]: ((((times_times @ A) @ X) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ X3) @ Y)) = (((set_or1337092689740270186AtMost @ A) @ (((times_times @ A) @ X3) @ C2)) @ (((times_times @ A) @ Y) @ C2)))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((image @ A) @ A) @ (^ [X:A]: ((((times_times @ A) @ X) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ X3) @ Y)) = (((set_or1337092689740270186AtMost @ A) @ (((times_times @ A) @ Y) @ C2)) @ (((times_times @ A) @ X3) @ C2)))))) & ((~ ((((ord_less_eq @ A) @ X3) @ Y))) => (((((image @ A) @ A) @ (^ [X:A]: ((((times_times @ A) @ X) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ X3) @ Y)) = (bot_bot @ (set @ A))))))))))). % 6.41/2.69 thf(fact_5124_image__affinity__atLeastAtMost, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,M:A,C2:A]: (((((((set_or1337092689740270186AtMost @ A) @ A2) @ B2) = (bot_bot @ (set @ A))) => (((((image @ A) @ A) @ (^ [X:A]: ((((plus_plus @ A) @ (((times_times @ A) @ M) @ X)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (bot_bot @ (set @ A)))) & (((((set_or1337092689740270186AtMost @ A) @ A2) @ B2) != (bot_bot @ (set @ A))) => (((((ord_less_eq @ A) @ (zero_zero @ A)) @ M) => (((((image @ A) @ A) @ (^ [X:A]: ((((plus_plus @ A) @ (((times_times @ A) @ M) @ X)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ M) @ A2)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ M) @ B2)) @ C2)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ M))) => (((((image @ A) @ A) @ (^ [X:A]: ((((plus_plus @ A) @ (((times_times @ A) @ M) @ X)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ M) @ B2)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ M) @ A2)) @ C2))))))))))))). % 6.41/2.69 thf(fact_5125_image__affinity__atLeastAtMost__diff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,M:A,C2:A]: (((((((set_or1337092689740270186AtMost @ A) @ A2) @ B2) = (bot_bot @ (set @ A))) => (((((image @ A) @ A) @ (^ [X:A]: ((((minus_minus @ A) @ (((times_times @ A) @ M) @ X)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (bot_bot @ (set @ A)))) & (((((set_or1337092689740270186AtMost @ A) @ A2) @ B2) != (bot_bot @ (set @ A))) => (((((ord_less_eq @ A) @ (zero_zero @ A)) @ M) => (((((image @ A) @ A) @ (^ [X:A]: ((((minus_minus @ A) @ (((times_times @ A) @ M) @ X)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ M) @ A2)) @ C2)) @ (((minus_minus @ A) @ (((times_times @ A) @ M) @ B2)) @ C2)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ M))) => (((((image @ A) @ A) @ (^ [X:A]: ((((minus_minus @ A) @ (((times_times @ A) @ M) @ X)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ M) @ B2)) @ C2)) @ (((minus_minus @ A) @ (((times_times @ A) @ M) @ A2)) @ C2))))))))))))). % 6.41/2.69 thf(fact_5126_image__affinity__atLeastAtMost__div, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,M:A,C2:A]: (((((((set_or1337092689740270186AtMost @ A) @ A2) @ B2) = (bot_bot @ (set @ A))) => (((((image @ A) @ A) @ (^ [X:A]: ((((plus_plus @ A) @ (((divide_divide @ A) @ X) @ M)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (bot_bot @ (set @ A)))) & (((((set_or1337092689740270186AtMost @ A) @ A2) @ B2) != (bot_bot @ (set @ A))) => (((((ord_less_eq @ A) @ (zero_zero @ A)) @ M) => (((((image @ A) @ A) @ (^ [X:A]: ((((plus_plus @ A) @ (((divide_divide @ A) @ X) @ M)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ M)) @ C2)) @ (((plus_plus @ A) @ (((divide_divide @ A) @ B2) @ M)) @ C2)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ M))) => (((((image @ A) @ A) @ (^ [X:A]: ((((plus_plus @ A) @ (((divide_divide @ A) @ X) @ M)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ (((divide_divide @ A) @ B2) @ M)) @ C2)) @ (((plus_plus @ A) @ (((divide_divide @ A) @ A2) @ M)) @ C2))))))))))))). % 6.41/2.69 thf(fact_5127_image__affinity__atLeastAtMost__div__diff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,M:A,C2:A]: (((((((set_or1337092689740270186AtMost @ A) @ A2) @ B2) = (bot_bot @ (set @ A))) => (((((image @ A) @ A) @ (^ [X:A]: ((((minus_minus @ A) @ (((divide_divide @ A) @ X) @ M)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (bot_bot @ (set @ A)))) & (((((set_or1337092689740270186AtMost @ A) @ A2) @ B2) != (bot_bot @ (set @ A))) => (((((ord_less_eq @ A) @ (zero_zero @ A)) @ M) => (((((image @ A) @ A) @ (^ [X:A]: ((((minus_minus @ A) @ (((divide_divide @ A) @ X) @ M)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((minus_minus @ A) @ (((divide_divide @ A) @ A2) @ M)) @ C2)) @ (((minus_minus @ A) @ (((divide_divide @ A) @ B2) @ M)) @ C2)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ M))) => (((((image @ A) @ A) @ (^ [X:A]: ((((minus_minus @ A) @ (((divide_divide @ A) @ X) @ M)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((minus_minus @ A) @ (((divide_divide @ A) @ B2) @ M)) @ C2)) @ (((minus_minus @ A) @ (((divide_divide @ A) @ A2) @ M)) @ C2))))))))))))). % 6.41/2.69 thf(fact_5128_sum__fun__comp, axiom, (! [A:$tType,C:$tType,B:$tType]: (((semiring_1 @ C) => (! [S3:(set @ A),R:(set @ B),G:(A > B),F2:(B > C)]: ((((finite_finite2 @ A) @ S3) => (((finite_finite2 @ B) @ R) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ G) @ S3)) @ R) => (((((groups7311177749621191930dd_sum @ A) @ C) @ (^ [X:A]: ((F2 @ (G @ X))))) @ S3) = ((((groups7311177749621191930dd_sum @ B) @ C) @ (^ [Y2:B]: ((((times_times @ C) @ ((semiring_1_of_nat @ C) @ ((finite_card @ A) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ S3) & ((G @ X) = Y2)))))))) @ (F2 @ Y2))))) @ R))))))))))). % 6.41/2.69 thf(fact_5129_INF__nat__binary, axiom, (! [A:$tType]: (((counta3822494911875563373attice @ A) => (! [A3:A,B3:A]: (((((inf_inf @ A) @ A3) @ ((complete_Inf_Inf @ A) @ ((((image @ nat) @ A) @ (^ [X:nat]: (B3))) @ ((collect @ nat) @ ((ord_less @ nat) @ (zero_zero @ nat)))))) = (((inf_inf @ A) @ A3) @ B3)))))))). % 6.41/2.69 thf(fact_5130_empty__natural, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,F2:(A > C),G:(D > B)]: (((((((comp @ C) @ (set @ B)) @ A) @ (^ [Uu3:C]: ((bot_bot @ (set @ B))))) @ F2) = (((((comp @ (set @ D)) @ (set @ B)) @ A) @ (((image @ D) @ B) @ G)) @ (^ [Uu3:A]: ((bot_bot @ (set @ D))))))))). % 6.41/2.69 thf(fact_5131_listrel__def, axiom, (! [B:$tType,A:$tType]: ((((listrel @ A) @ B) = (^ [R5:(set @ ((product_prod @ A) @ B))]: (((collect @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_case_prod @ (list @ A)) @ (list @ B)) @ $o) @ (((listrelp @ A) @ B) @ (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ R5)))))))))))). % 6.41/2.69 thf(fact_5132_pair__imageI, axiom, (! [C:$tType,B:$tType,A:$tType,A2:A,B2:B,A3:(set @ ((product_prod @ A) @ B)),F2:(A > (B > C))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ A3) => (((member @ C) @ ((F2 @ A2) @ B2)) @ ((((image @ ((product_prod @ A) @ B)) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F2)) @ A3)))))). % 6.41/2.69 thf(fact_5133_Gcd__abs__eq, axiom, (! [K5:(set @ int)]: ((((gcd_Gcd @ int) @ ((((image @ int) @ int) @ (abs_abs @ int)) @ K5)) = ((gcd_Gcd @ int) @ K5))))). % 6.41/2.69 thf(fact_5134_finite__UN, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(A > (set @ B))]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ B) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ B3) @ A3))) = (! [X:A]: (((((member @ A) @ X) @ A3) => ((finite_finite2 @ B) @ (B3 @ X)))))))))). % 6.41/2.69 thf(fact_5135_UN__constant, axiom, (! [B:$tType,A:$tType,A3:(set @ B),C2:(set @ A)]: ((((A3 = (bot_bot @ (set @ B))) => (((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [Y2:B]: (C2))) @ A3)) = (bot_bot @ (set @ A)))) & ((A3 != (bot_bot @ (set @ B))) => (((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [Y2:B]: (C2))) @ A3)) = C2)))))). % 6.41/2.69 thf(fact_5136_finite__UN__I, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(A > (set @ B))]: ((((finite_finite2 @ A) @ A3) => ((! [A5:A]: (((((member @ A) @ A5) @ A3) => ((finite_finite2 @ B) @ (B3 @ A5))))) => ((finite_finite2 @ B) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ B3) @ A3)))))))). % 6.41/2.69 thf(fact_5137_finite__INT, axiom, (! [B:$tType,A:$tType,I6:(set @ A),A3:(A > (set @ B))]: (((? [X2:A]: (((((member @ A) @ X2) @ I6) & ((finite_finite2 @ B) @ (A3 @ X2))))) => ((finite_finite2 @ B) @ ((complete_Inf_Inf @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ A3) @ I6))))))). % 6.41/2.69 thf(fact_5138_Gcd__int__eq, axiom, (! [N7:(set @ nat)]: ((((gcd_Gcd @ int) @ ((((image @ nat) @ int) @ (semiring_1_of_nat @ int)) @ N7)) = ((semiring_1_of_nat @ int) @ ((gcd_Gcd @ nat) @ N7)))))). % 6.41/2.69 thf(fact_5139_UN__simps_I1_J, axiom, (! [A:$tType,B:$tType,C5:(set @ B),A2:A,B3:(B > (set @ A))]: ((((C5 = (bot_bot @ (set @ B))) => (((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [X:B]: ((((insert @ A) @ A2) @ (B3 @ X))))) @ C5)) = (bot_bot @ (set @ A)))) & ((C5 != (bot_bot @ (set @ B))) => (((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [X:B]: ((((insert @ A) @ A2) @ (B3 @ X))))) @ C5)) = (((insert @ A) @ A2) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B3) @ C5))))))))). % 6.41/2.69 thf(fact_5140_UN__singleton, axiom, (! [A:$tType,A3:(set @ A)]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image @ A) @ (set @ A)) @ (^ [X:A]: ((((insert @ A) @ X) @ (bot_bot @ (set @ A)))))) @ A3)) = A3)))). % 6.41/2.69 thf(fact_5141_set__concat, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: ((((set2 @ A) @ ((concat @ A) @ Xs)) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ (list @ A)) @ (set @ A)) @ (set2 @ A)) @ ((set2 @ (list @ A)) @ Xs))))))). % 6.41/2.69 thf(fact_5142_Gcd__nat__abs__eq, axiom, (! [K5:(set @ int)]: ((((gcd_Gcd @ nat) @ ((((image @ int) @ nat) @ (^ [K3:int]: ((nat2 @ ((abs_abs @ int) @ K3))))) @ K5)) = (nat2 @ ((gcd_Gcd @ int) @ K5)))))). % 6.41/2.69 thf(fact_5143_SUP__UN__eq2, axiom, (! [B:$tType,C:$tType,A:$tType,R2:(C > (set @ ((product_prod @ A) @ B))),S3:(set @ C)]: ((((complete_Sup_Sup @ (A > (B > $o))) @ ((((image @ C) @ (A > (B > $o))) @ (^ [I3:C,X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ (R2 @ I3))))) @ S3)) = (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ ((((image @ C) @ (set @ ((product_prod @ A) @ B))) @ R2) @ S3))))))))). % 6.41/2.69 thf(fact_5144_INF__INT__eq2, axiom, (! [B:$tType,C:$tType,A:$tType,R2:(C > (set @ ((product_prod @ A) @ B))),S3:(set @ C)]: ((((complete_Inf_Inf @ (A > (B > $o))) @ ((((image @ C) @ (A > (B > $o))) @ (^ [I3:C,X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ (R2 @ I3))))) @ S3)) = (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ ((complete_Inf_Inf @ (set @ ((product_prod @ A) @ B))) @ ((((image @ C) @ (set @ ((product_prod @ A) @ B))) @ R2) @ S3))))))))). % 6.41/2.69 thf(fact_5145_INF__Int__eq2, axiom, (! [B:$tType,A:$tType,S3:(set @ (set @ ((product_prod @ A) @ B)))]: ((((complete_Inf_Inf @ (A > (B > $o))) @ ((((image @ (set @ ((product_prod @ A) @ B))) @ (A > (B > $o))) @ (^ [I3:(set @ ((product_prod @ A) @ B)),X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ I3)))) @ S3)) = (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ ((complete_Inf_Inf @ (set @ ((product_prod @ A) @ B))) @ S3)))))))). % 6.41/2.69 thf(fact_5146_SUP__Sup__eq2, axiom, (! [B:$tType,A:$tType,S3:(set @ (set @ ((product_prod @ A) @ B)))]: ((((complete_Sup_Sup @ (A > (B > $o))) @ ((((image @ (set @ ((product_prod @ A) @ B))) @ (A > (B > $o))) @ (^ [I3:(set @ ((product_prod @ A) @ B)),X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ I3)))) @ S3)) = (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ S3)))))))). % 6.41/2.69 thf(fact_5147_Sup__SUP__eq2, axiom, (! [B:$tType,A:$tType]: (((complete_Sup_Sup @ (A > (B > $o))) = (^ [S7:(set @ (A > (B > $o))),X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ ((((image @ (((product_prod @ A) @ B) > $o)) @ (set @ ((product_prod @ A) @ B))) @ (collect @ ((product_prod @ A) @ B))) @ ((((image @ (A > (B > $o))) @ (((product_prod @ A) @ B) > $o)) @ (((product_case_prod @ A) @ B) @ $o)) @ S7)))))))))). % 6.41/2.69 thf(fact_5148_Inf__INT__eq2, axiom, (! [B:$tType,A:$tType]: (((complete_Inf_Inf @ (A > (B > $o))) = (^ [S7:(set @ (A > (B > $o))),X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ ((complete_Inf_Inf @ (set @ ((product_prod @ A) @ B))) @ ((((image @ (((product_prod @ A) @ B) > $o)) @ (set @ ((product_prod @ A) @ B))) @ (collect @ ((product_prod @ A) @ B))) @ ((((image @ (A > (B > $o))) @ (((product_prod @ A) @ B) > $o)) @ (((product_case_prod @ A) @ B) @ $o)) @ S7)))))))))). % 6.41/2.69 thf(fact_5149_UN__image__subset, axiom, (! [C:$tType,A:$tType,B:$tType,F2:(B > (set @ A)),G:(C > (set @ B)),X3:C,X8:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ F2) @ (G @ X3)))) @ X8) = (((ord_less_eq @ (set @ B)) @ (G @ X3)) @ ((collect @ B) @ (^ [X:B]: ((((ord_less_eq @ (set @ A)) @ (F2 @ X)) @ X8))))))))). % 6.41/2.69 thf(fact_5150_listrelp_ONil, axiom, (! [A:$tType,B:$tType,R2:(A > (B > $o))]: ((((((listrelp @ A) @ B) @ R2) @ (nil @ A)) @ (nil @ B))))). % 6.41/2.69 thf(fact_5151_None__notin__image__Some, axiom, (! [A:$tType,A3:(set @ A)]: ((~ ((((member @ (option @ A)) @ (none @ A)) @ ((((image @ A) @ (option @ A)) @ (some @ A)) @ A3))))))). % 6.41/2.69 thf(fact_5152_INF__filter__not__bot, axiom, (! [I7:$tType,A:$tType,B3:(set @ I7),F5:(I7 > (filter @ A))]: (((! [X10:(set @ I7)]: (((((ord_less_eq @ (set @ I7)) @ X10) @ B3) => (((finite_finite2 @ I7) @ X10) => (((complete_Inf_Inf @ (filter @ A)) @ ((((image @ I7) @ (filter @ A)) @ F5) @ X10)) != (bot_bot @ (filter @ A))))))) => (((complete_Inf_Inf @ (filter @ A)) @ ((((image @ I7) @ (filter @ A)) @ F5) @ B3)) != (bot_bot @ (filter @ A))))))). % 6.41/2.69 thf(fact_5153_finite__int__iff__bounded, axiom, ((finite_finite2 @ int) = (^ [S7:(set @ int)]: ((? [K3:int]: ((((ord_less_eq @ (set @ int)) @ ((((image @ int) @ int) @ (abs_abs @ int)) @ S7)) @ ((set_ord_lessThan @ int) @ K3)))))))). % 6.41/2.69 thf(fact_5154_finite__int__iff__bounded__le, axiom, ((finite_finite2 @ int) = (^ [S7:(set @ int)]: ((? [K3:int]: ((((ord_less_eq @ (set @ int)) @ ((((image @ int) @ int) @ (abs_abs @ int)) @ S7)) @ ((set_ord_atMost @ int) @ K3)))))))). % 6.41/2.69 thf(fact_5155_UN__empty2, axiom, (! [B:$tType,A:$tType,A3:(set @ B)]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [X:B]: ((bot_bot @ (set @ A))))) @ A3)) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_5156_UN__empty, axiom, (! [B:$tType,A:$tType,B3:(B > (set @ A))]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B3) @ (bot_bot @ (set @ B)))) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_5157_UNION__empty__conv_I1_J, axiom, (! [A:$tType,B:$tType,B3:(B > (set @ A)),A3:(set @ B)]: ((((bot_bot @ (set @ A)) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B3) @ A3))) = (! [X:B]: (((((member @ B) @ X) @ A3) => ((B3 @ X) = (bot_bot @ (set @ A)))))))))). % 6.41/2.69 thf(fact_5158_UNION__empty__conv_I2_J, axiom, (! [A:$tType,B:$tType,B3:(B > (set @ A)),A3:(set @ B)]: (((((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B3) @ A3)) = (bot_bot @ (set @ A))) = (! [X:B]: (((((member @ B) @ X) @ A3) => ((B3 @ X) = (bot_bot @ (set @ A)))))))))). % 6.41/2.69 thf(fact_5159_UN__subset__iff, axiom, (! [A:$tType,B:$tType,A3:(B > (set @ A)),I6:(set @ B),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ A3) @ I6))) @ B3) = (! [X:B]: (((((member @ B) @ X) @ I6) => (((ord_less_eq @ (set @ A)) @ (A3 @ X)) @ B3)))))))). % 6.41/2.69 thf(fact_5160_UN__upper, axiom, (! [B:$tType,A:$tType,A2:A,A3:(set @ A),B3:(A > (set @ B))]: (((((member @ A) @ A2) @ A3) => (((ord_less_eq @ (set @ B)) @ (B3 @ A2)) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ B3) @ A3))))))). % 6.41/2.69 thf(fact_5161_UN__least, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B3:(A > (set @ B)),C5:(set @ B)]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((ord_less_eq @ (set @ B)) @ (B3 @ X5)) @ C5)))) => (((ord_less_eq @ (set @ B)) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ B3) @ A3))) @ C5))))). % 6.41/2.69 thf(fact_5162_UN__mono, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ A),F2:(A > (set @ B)),G:(A > (set @ B))]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((ord_less_eq @ (set @ B)) @ (F2 @ X5)) @ (G @ X5))))) => (((ord_less_eq @ (set @ B)) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ F2) @ A3))) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ G) @ B3)))))))). % 6.41/2.69 thf(fact_5163_INT__subset__iff, axiom, (! [A:$tType,B:$tType,B3:(set @ A),A3:(B > (set @ A)),I6:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B3) @ ((complete_Inf_Inf @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ A3) @ I6))) = (! [X:B]: (((((member @ B) @ X) @ I6) => (((ord_less_eq @ (set @ A)) @ B3) @ (A3 @ X))))))))). % 6.41/2.69 thf(fact_5164_INT__anti__mono, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ A),F2:(A > (set @ B)),G:(A > (set @ B))]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((ord_less_eq @ (set @ B)) @ (F2 @ X5)) @ (G @ X5))))) => (((ord_less_eq @ (set @ B)) @ ((complete_Inf_Inf @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ F2) @ B3))) @ ((complete_Inf_Inf @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ G) @ A3)))))))). % 6.41/2.69 thf(fact_5165_INT__greatest, axiom, (! [B:$tType,A:$tType,A3:(set @ A),C5:(set @ B),B3:(A > (set @ B))]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((ord_less_eq @ (set @ B)) @ C5) @ (B3 @ X5))))) => (((ord_less_eq @ (set @ B)) @ C5) @ ((complete_Inf_Inf @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ B3) @ A3))))))). % 6.41/2.69 thf(fact_5166_INT__lower, axiom, (! [B:$tType,A:$tType,A2:A,A3:(set @ A),B3:(A > (set @ B))]: (((((member @ A) @ A2) @ A3) => (((ord_less_eq @ (set @ B)) @ ((complete_Inf_Inf @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ B3) @ A3))) @ (B3 @ A2)))))). % 6.41/2.69 thf(fact_5167_in__image__insert__iff, axiom, (! [A:$tType,B3:(set @ (set @ A)),X3:A,A3:(set @ A)]: (((! [C8:(set @ A)]: (((((member @ (set @ A)) @ C8) @ B3) => (~ ((((member @ A) @ X3) @ C8)))))) => ((((member @ (set @ A)) @ A3) @ ((((image @ (set @ A)) @ (set @ A)) @ ((insert @ A) @ X3)) @ B3)) = ((((member @ A) @ X3) @ A3) & (((member @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) @ B3))))))). % 6.41/2.69 thf(fact_5168_UN__extend__simps_I1_J, axiom, (! [A:$tType,B:$tType,C5:(set @ B),A2:A,B3:(B > (set @ A))]: ((((C5 = (bot_bot @ (set @ B))) => ((((insert @ A) @ A2) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B3) @ C5))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) & ((C5 != (bot_bot @ (set @ B))) => ((((insert @ A) @ A2) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B3) @ C5))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [X:B]: ((((insert @ A) @ A2) @ (B3 @ X))))) @ C5)))))))). % 6.41/2.69 thf(fact_5169_INT__extend__simps_I2_J, axiom, (! [C:$tType,D:$tType,C5:(set @ D),A3:(set @ C),B3:(D > (set @ C))]: ((((C5 = (bot_bot @ (set @ D))) => ((((inf_inf @ (set @ C)) @ A3) @ ((complete_Inf_Inf @ (set @ C)) @ ((((image @ D) @ (set @ C)) @ B3) @ C5))) = A3)) & ((C5 != (bot_bot @ (set @ D))) => ((((inf_inf @ (set @ C)) @ A3) @ ((complete_Inf_Inf @ (set @ C)) @ ((((image @ D) @ (set @ C)) @ B3) @ C5))) = ((complete_Inf_Inf @ (set @ C)) @ ((((image @ D) @ (set @ C)) @ (^ [X:D]: ((((inf_inf @ (set @ C)) @ A3) @ (B3 @ X))))) @ C5)))))))). % 6.41/2.69 thf(fact_5170_INT__extend__simps_I1_J, axiom, (! [B:$tType,A:$tType,C5:(set @ A),A3:(A > (set @ B)),B3:(set @ B)]: ((((C5 = (bot_bot @ (set @ A))) => ((((inf_inf @ (set @ B)) @ ((complete_Inf_Inf @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ A3) @ C5))) @ B3) = B3)) & ((C5 != (bot_bot @ (set @ A))) => ((((inf_inf @ (set @ B)) @ ((complete_Inf_Inf @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ A3) @ C5))) @ B3) = ((complete_Inf_Inf @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ (^ [X:A]: ((((inf_inf @ (set @ B)) @ (A3 @ X)) @ B3)))) @ C5)))))))). % 6.41/2.69 thf(fact_5171_Int__Inter__eq_I2_J, axiom, (! [A:$tType,B12:(set @ (set @ A)),A3:(set @ A)]: ((((B12 = (bot_bot @ (set @ (set @ A)))) => ((((inf_inf @ (set @ A)) @ ((complete_Inf_Inf @ (set @ A)) @ B12)) @ A3) = A3)) & ((B12 != (bot_bot @ (set @ (set @ A)))) => ((((inf_inf @ (set @ A)) @ ((complete_Inf_Inf @ (set @ A)) @ B12)) @ A3) = ((complete_Inf_Inf @ (set @ A)) @ ((((image @ (set @ A)) @ (set @ A)) @ (^ [B6:(set @ A)]: ((((inf_inf @ (set @ A)) @ B6) @ A3)))) @ B12)))))))). % 6.41/2.69 thf(fact_5172_Int__Inter__eq_I1_J, axiom, (! [A:$tType,B12:(set @ (set @ A)),A3:(set @ A)]: ((((B12 = (bot_bot @ (set @ (set @ A)))) => ((((inf_inf @ (set @ A)) @ A3) @ ((complete_Inf_Inf @ (set @ A)) @ B12)) = A3)) & ((B12 != (bot_bot @ (set @ (set @ A)))) => ((((inf_inf @ (set @ A)) @ A3) @ ((complete_Inf_Inf @ (set @ A)) @ B12)) = ((complete_Inf_Inf @ (set @ A)) @ ((((image @ (set @ A)) @ (set @ A)) @ ((inf_inf @ (set @ A)) @ A3)) @ B12)))))))). % 6.41/2.69 thf(fact_5173_Collect__split__mono__strong, axiom, (! [B:$tType,A:$tType,X8:(set @ A),A3:(set @ ((product_prod @ A) @ B)),Y7:(set @ B),P:(A > (B > $o)),Q:(A > (B > $o))]: (((X8 = ((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ A3)) => ((Y7 = ((((image @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ A3)) => ((! [X5:A]: (((((member @ A) @ X5) @ X8) => (! [Xa3:B]: (((((member @ B) @ Xa3) @ Y7) => (((P @ X5) @ Xa3) => ((Q @ X5) @ Xa3)))))))) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A3) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ P))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A3) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ Q)))))))))). % 6.41/2.69 thf(fact_5174_INT__extend__simps_I4_J, axiom, (! [G3:$tType,H4:$tType,C5:(set @ H4),A3:(set @ G3),B3:(H4 > (set @ G3))]: ((((C5 = (bot_bot @ (set @ H4))) => ((((minus_minus @ (set @ G3)) @ A3) @ ((complete_Sup_Sup @ (set @ G3)) @ ((((image @ H4) @ (set @ G3)) @ B3) @ C5))) = A3)) & ((C5 != (bot_bot @ (set @ H4))) => ((((minus_minus @ (set @ G3)) @ A3) @ ((complete_Sup_Sup @ (set @ G3)) @ ((((image @ H4) @ (set @ G3)) @ B3) @ C5))) = ((complete_Inf_Inf @ (set @ G3)) @ ((((image @ H4) @ (set @ G3)) @ (^ [X:H4]: ((((minus_minus @ (set @ G3)) @ A3) @ (B3 @ X))))) @ C5)))))))). % 6.41/2.69 thf(fact_5175_infinite__int__iff__infinite__nat__abs, axiom, (! [S3:(set @ int)]: (((~ (((finite_finite2 @ int) @ S3))) = (~ (((finite_finite2 @ nat) @ ((((image @ int) @ nat) @ (((((comp @ int) @ nat) @ int) @ nat2) @ (abs_abs @ int))) @ S3)))))))). % 6.41/2.69 thf(fact_5176_UN__le__add__shift__strict, axiom, (! [A:$tType,M7:(nat > (set @ A)),K:nat,N2:nat]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ (^ [I3:nat]: ((M7 @ (((plus_plus @ nat) @ I3) @ K))))) @ ((set_ord_lessThan @ nat) @ N2))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ M7) @ (((set_or7035219750837199246ssThan @ nat) @ K) @ (((plus_plus @ nat) @ N2) @ K)))))))). % 6.41/2.69 thf(fact_5177_UN__le__add__shift, axiom, (! [A:$tType,M7:(nat > (set @ A)),K:nat,N2:nat]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ (^ [I3:nat]: ((M7 @ (((plus_plus @ nat) @ I3) @ K))))) @ ((set_ord_atMost @ nat) @ N2))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ M7) @ (((set_or1337092689740270186AtMost @ nat) @ K) @ (((plus_plus @ nat) @ N2) @ K)))))))). % 6.41/2.69 thf(fact_5178_subset__subseqs, axiom, (! [A:$tType,X8:(set @ A),Xs:(list @ A)]: (((((ord_less_eq @ (set @ A)) @ X8) @ ((set2 @ A) @ Xs)) => (((member @ (set @ A)) @ X8) @ ((((image @ (list @ A)) @ (set @ A)) @ (set2 @ A)) @ ((set2 @ (list @ A)) @ ((subseqs @ A) @ Xs)))))))). % 6.41/2.69 thf(fact_5179_subseqs__powset, axiom, (! [A:$tType,Xs:(list @ A)]: ((((((image @ (list @ A)) @ (set @ A)) @ (set2 @ A)) @ ((set2 @ (list @ A)) @ ((subseqs @ A) @ Xs))) = ((pow2 @ A) @ ((set2 @ A) @ Xs)))))). % 6.41/2.69 thf(fact_5180_conj__subset__def, axiom, (! [A:$tType,A3:(set @ A),P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ A3) @ ((collect @ A) @ (^ [X:A]: (((P @ X) & (Q @ X)))))) = ((((ord_less_eq @ (set @ A)) @ A3) @ ((collect @ A) @ P)) & (((ord_less_eq @ (set @ A)) @ A3) @ ((collect @ A) @ Q))))))). % 6.41/2.69 thf(fact_5181_finite__Inf__Sup, axiom, (! [A:$tType]: (((finite8700451911770168679attice @ A) => (! [A3:(set @ (set @ A))]: ((((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ (set @ A)) @ A) @ (complete_Sup_Sup @ A)) @ A3))) @ ((complete_Sup_Sup @ A) @ ((((image @ (set @ A)) @ A) @ (complete_Inf_Inf @ A)) @ ((collect @ (set @ A)) @ (^ [Uu3:(set @ A)]: ((? [F3:((set @ A) > A)]: (((Uu3 = ((((image @ (set @ A)) @ A) @ F3) @ A3)) & (! [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ A3) => (((member @ A) @ (F3 @ X)) @ X))))))))))))))))))). % 6.41/2.69 thf(fact_5182_Inf__Sup__le, axiom, (! [A:$tType]: (((comple592849572758109894attice @ A) => (! [A3:(set @ (set @ A))]: ((((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ (set @ A)) @ A) @ (complete_Sup_Sup @ A)) @ A3))) @ ((complete_Sup_Sup @ A) @ ((((image @ (set @ A)) @ A) @ (complete_Inf_Inf @ A)) @ ((collect @ (set @ A)) @ (^ [Uu3:(set @ A)]: ((? [F3:((set @ A) > A)]: (((Uu3 = ((((image @ (set @ A)) @ A) @ F3) @ A3)) & (! [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ A3) => (((member @ A) @ (F3 @ X)) @ X))))))))))))))))))). % 6.41/2.69 thf(fact_5183_Sup__Inf__le, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ (set @ A))]: ((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ (set @ A)) @ A) @ (complete_Inf_Inf @ A)) @ ((collect @ (set @ A)) @ (^ [Uu3:(set @ A)]: ((? [F3:((set @ A) > A)]: (((Uu3 = ((((image @ (set @ A)) @ A) @ F3) @ A3)) & (! [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ A3) => (((member @ A) @ (F3 @ X)) @ X))))))))))))) @ ((complete_Inf_Inf @ A) @ ((((image @ (set @ A)) @ A) @ (complete_Sup_Sup @ A)) @ A3))))))))). % 6.41/2.69 thf(fact_5184_sum_OUNION__disjoint, axiom, (! [A:$tType,C:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [I6:(set @ B),A3:(B > (set @ C)),G:(C > A)]: ((((finite_finite2 @ B) @ I6) => ((! [X5:B]: (((((member @ B) @ X5) @ I6) => ((finite_finite2 @ C) @ (A3 @ X5))))) => ((! [X5:B]: (((((member @ B) @ X5) @ I6) => (! [Xa3:B]: (((((member @ B) @ Xa3) @ I6) => ((X5 != Xa3) => ((((inf_inf @ (set @ C)) @ (A3 @ X5)) @ (A3 @ Xa3)) = (bot_bot @ (set @ C)))))))))) => (((((groups7311177749621191930dd_sum @ C) @ A) @ G) @ ((complete_Sup_Sup @ (set @ C)) @ ((((image @ B) @ (set @ C)) @ A3) @ I6))) = ((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X:B]: (((((groups7311177749621191930dd_sum @ C) @ A) @ G) @ (A3 @ X))))) @ I6))))))))))). % 6.41/2.69 thf(fact_5185_prod_OUNION__disjoint, axiom, (! [A:$tType,C:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [I6:(set @ B),A3:(B > (set @ C)),G:(C > A)]: ((((finite_finite2 @ B) @ I6) => ((! [X5:B]: (((((member @ B) @ X5) @ I6) => ((finite_finite2 @ C) @ (A3 @ X5))))) => ((! [X5:B]: (((((member @ B) @ X5) @ I6) => (! [Xa3:B]: (((((member @ B) @ Xa3) @ I6) => ((X5 != Xa3) => ((((inf_inf @ (set @ C)) @ (A3 @ X5)) @ (A3 @ Xa3)) = (bot_bot @ (set @ C)))))))))) => (((((groups7121269368397514597t_prod @ C) @ A) @ G) @ ((complete_Sup_Sup @ (set @ C)) @ ((((image @ B) @ (set @ C)) @ A3) @ I6))) = ((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X:B]: (((((groups7121269368397514597t_prod @ C) @ A) @ G) @ (A3 @ X))))) @ I6))))))))))). % 6.41/2.69 thf(fact_5186_card__UN__le, axiom, (! [B:$tType,A:$tType,I6:(set @ A),A3:(A > (set @ B))]: ((((finite_finite2 @ A) @ I6) => (((ord_less_eq @ nat) @ ((finite_card @ B) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ A3) @ I6)))) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [I3:A]: (((finite_card @ B) @ (A3 @ I3))))) @ I6)))))). % 6.41/2.69 thf(fact_5187_listrelp__listrel__eq, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B))]: (((((listrelp @ A) @ B) @ (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ R2)))) = (^ [X:(list @ A),Y2:(list @ B)]: ((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ X) @ Y2)) @ (((listrel @ A) @ B) @ R2)))))))). % 6.41/2.69 thf(fact_5188_Gcd__int__def, axiom, ((gcd_Gcd @ int) = (^ [K6:(set @ int)]: (((semiring_1_of_nat @ int) @ ((gcd_Gcd @ nat) @ ((((image @ int) @ nat) @ (((((comp @ int) @ nat) @ int) @ nat2) @ (abs_abs @ int))) @ K6))))))). % 6.41/2.69 thf(fact_5189_card__UN__disjoint, axiom, (! [B:$tType,A:$tType,I6:(set @ A),A3:(A > (set @ B))]: ((((finite_finite2 @ A) @ I6) => ((! [X5:A]: (((((member @ A) @ X5) @ I6) => ((finite_finite2 @ B) @ (A3 @ X5))))) => ((! [X5:A]: (((((member @ A) @ X5) @ I6) => (! [Xa3:A]: (((((member @ A) @ Xa3) @ I6) => ((X5 != Xa3) => ((((inf_inf @ (set @ B)) @ (A3 @ X5)) @ (A3 @ Xa3)) = (bot_bot @ (set @ B)))))))))) => (((finite_card @ B) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ A3) @ I6))) = ((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [I3:A]: (((finite_card @ B) @ (A3 @ I3))))) @ I6)))))))). % 6.41/2.69 thf(fact_5190_finite__mono__strict__prefix__implies__finite__fixpoint, axiom, (! [A:$tType,F2:(nat > (set @ A)),S3:(set @ A)]: (((! [I4:nat]: ((((ord_less_eq @ (set @ A)) @ (F2 @ I4)) @ S3))) => (((finite_finite2 @ A) @ S3) => ((? [N8:nat]: (((! [N3:nat]: (((((ord_less_eq @ nat) @ N3) @ N8) => (! [M3:nat]: (((((ord_less_eq @ nat) @ M3) @ N8) => ((((ord_less @ nat) @ M3) @ N3) => (((ord_less @ (set @ A)) @ (F2 @ M3)) @ (F2 @ N3))))))))) & (! [N3:nat]: (((((ord_less_eq @ nat) @ N8) @ N3) => ((F2 @ N8) = (F2 @ N3)))))))) => ((F2 @ ((finite_card @ A) @ S3)) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ F2) @ (top_top @ (set @ nat))))))))))). % 6.41/2.69 thf(fact_5191_length__remdups__concat, axiom, (! [A:$tType,Xss:(list @ (list @ A))]: ((((size_size @ (list @ A)) @ ((remdups @ A) @ ((concat @ A) @ Xss))) = ((finite_card @ A) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ (list @ A)) @ (set @ A)) @ (set2 @ A)) @ ((set2 @ (list @ A)) @ Xss)))))))). % 6.41/2.69 thf(fact_5192_INT__simps_I4_J, axiom, (! [G3:$tType,H4:$tType,C5:(set @ H4),A3:(set @ G3),B3:(H4 > (set @ G3))]: ((((C5 = (bot_bot @ (set @ H4))) => (((complete_Inf_Inf @ (set @ G3)) @ ((((image @ H4) @ (set @ G3)) @ (^ [X:H4]: ((((minus_minus @ (set @ G3)) @ A3) @ (B3 @ X))))) @ C5)) = (top_top @ (set @ G3)))) & ((C5 != (bot_bot @ (set @ H4))) => (((complete_Inf_Inf @ (set @ G3)) @ ((((image @ H4) @ (set @ G3)) @ (^ [X:H4]: ((((minus_minus @ (set @ G3)) @ A3) @ (B3 @ X))))) @ C5)) = (((minus_minus @ (set @ G3)) @ A3) @ ((complete_Sup_Sup @ (set @ G3)) @ ((((image @ H4) @ (set @ G3)) @ B3) @ C5))))))))). % 6.41/2.69 thf(fact_5193_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X:D]: ((top_top @ C)))))))). % 6.41/2.69 thf(fact_5194_UNIV__I, axiom, (! [A:$tType,X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))). % 6.41/2.69 thf(fact_5195_finite__option__UNIV, axiom, (! [A:$tType]: ((((finite_finite2 @ (option @ A)) @ (top_top @ (set @ (option @ A)))) = ((finite_finite2 @ A) @ (top_top @ (set @ A))))))). % 6.41/2.69 thf(fact_5196_finite__Plus__UNIV__iff, axiom, (! [A:$tType,B:$tType]: ((((finite_finite2 @ ((sum_sum @ A) @ B)) @ (top_top @ (set @ ((sum_sum @ A) @ B)))) = (((finite_finite2 @ A) @ (top_top @ (set @ A))) & ((finite_finite2 @ B) @ (top_top @ (set @ B)))))))). % 6.41/2.69 thf(fact_5197_Int__UNIV, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A3) @ B3) = (top_top @ (set @ A))) = ((A3 = (top_top @ (set @ A))) & (B3 = (top_top @ (set @ A)))))))). % 6.41/2.69 thf(fact_5198_range__mult, axiom, (! [A2:real]: ((((A2 = (zero_zero @ real)) => (((((image @ real) @ real) @ ((times_times @ real) @ A2)) @ (top_top @ (set @ real))) = (((insert @ real) @ (zero_zero @ real)) @ (bot_bot @ (set @ real))))) & ((A2 != (zero_zero @ real)) => (((((image @ real) @ real) @ ((times_times @ real) @ A2)) @ (top_top @ (set @ real))) = (top_top @ (set @ real)))))))). % 6.41/2.69 thf(fact_5199_max__top, axiom, (! [A:$tType]: (((order_top @ A) => (! [X3:A]: (((((ord_max @ A) @ (top_top @ A)) @ X3) = (top_top @ A)))))))). % 6.41/2.69 thf(fact_5200_max__top2, axiom, (! [A:$tType]: (((order_top @ A) => (! [X3:A]: (((((ord_max @ A) @ X3) @ (top_top @ A)) = (top_top @ A)))))))). % 6.41/2.69 thf(fact_5201_min__top, axiom, (! [A:$tType]: (((order_top @ A) => (! [X3:A]: (((((ord_min @ A) @ (top_top @ A)) @ X3) = X3))))))). % 6.41/2.69 thf(fact_5202_min__top2, axiom, (! [A:$tType]: (((order_top @ A) => (! [X3:A]: (((((ord_min @ A) @ X3) @ (top_top @ A)) = X3))))))). % 6.41/2.69 thf(fact_5203_Pow__UNIV, axiom, (! [A:$tType]: ((((pow2 @ A) @ (top_top @ (set @ A))) = (top_top @ (set @ (set @ A))))))). % 6.41/2.69 thf(fact_5204_remdups__eq__nil__right__iff, axiom, (! [A:$tType,X3:(list @ A)]: ((((nil @ A) = ((remdups @ A) @ X3)) = (X3 = (nil @ A)))))). % 6.41/2.69 thf(fact_5205_remdups__eq__nil__iff, axiom, (! [A:$tType,X3:(list @ A)]: (((((remdups @ A) @ X3) = (nil @ A)) = (X3 = (nil @ A)))))). % 6.41/2.69 thf(fact_5206_set__remdups, axiom, (! [A:$tType,Xs:(list @ A)]: ((((set2 @ A) @ ((remdups @ A) @ Xs)) = ((set2 @ A) @ Xs))))). % 6.41/2.69 thf(fact_5207_length__remdups__eq, axiom, (! [A:$tType,Xs:(list @ A)]: (((((size_size @ (list @ A)) @ ((remdups @ A) @ Xs)) = ((size_size @ (list @ A)) @ Xs)) = (((remdups @ A) @ Xs) = Xs))))). % 6.41/2.69 thf(fact_5208_remdups__id__iff__distinct, axiom, (! [A:$tType,Xs:(list @ A)]: (((((remdups @ A) @ Xs) = Xs) = ((distinct @ A) @ Xs))))). % 6.41/2.69 thf(fact_5209_distinct__remdups, axiom, (! [A:$tType,Xs:(list @ A)]: (((distinct @ A) @ ((remdups @ A) @ Xs))))). % 6.41/2.69 thf(fact_5210_Collect__const, axiom, (! [A:$tType,P:$o]: (((P => (((collect @ A) @ (^ [S8:A]: (P))) = (top_top @ (set @ A)))) & ((~ (P)) => (((collect @ A) @ (^ [S8:A]: (P))) = (bot_bot @ (set @ A)))))))). % 6.41/2.69 thf(fact_5211_finite__Collect__not, axiom, (! [A:$tType,P:(A > $o)]: ((((finite_finite2 @ A) @ ((collect @ A) @ P)) => (((finite_finite2 @ A) @ ((collect @ A) @ (^ [X:A]: ((~ ((P @ X))))))) = ((finite_finite2 @ A) @ (top_top @ (set @ A)))))))). % 6.41/2.69 thf(fact_5212_range__abs__Nats, axiom, (((((image @ int) @ int) @ (abs_abs @ int)) @ (top_top @ (set @ int))) = (semiring_1_Nats @ int))). % 6.41/2.69 thf(fact_5213_Collect__const__case__prod, axiom, (! [B:$tType,A:$tType,P:$o]: (((P => (((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A4:A,B4:B]: (P)))) = (top_top @ (set @ ((product_prod @ A) @ B))))) & ((~ (P)) => (((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A4:A,B4:B]: (P)))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))))). % 6.41/2.69 thf(fact_5214_range__add, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))))). % 6.41/2.69 thf(fact_5215_surj__plus, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A2:A]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))))). % 6.41/2.69 thf(fact_5216_Sup__eq__top__iff, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [A3:(set @ A)]: (((((complete_Sup_Sup @ A) @ A3) = (top_top @ A)) = (! [X:A]: (((((ord_less @ A) @ X) @ (top_top @ A)) => (? [Y2:A]: (((((member @ A) @ Y2) @ A3) & (((ord_less @ A) @ X) @ Y2)))))))))))))). % 6.41/2.69 thf(fact_5217_boolean__algebra_Ocompl__one, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (((uminus_uminus @ A) @ (top_top @ A)) = (bot_bot @ A)))))). % 6.41/2.69 thf(fact_5218_boolean__algebra_Ocompl__zero, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (((uminus_uminus @ A) @ (bot_bot @ A)) = (top_top @ A)))))). % 6.41/2.69 thf(fact_5219_ball__UNIV, axiom, (! [A:$tType,P:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ (top_top @ (set @ A))) => (P @ X)))) = (! [X6:A]: ((P @ X6))))))). % 6.41/2.69 thf(fact_5220_Inf__UNIV, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (((complete_Inf_Inf @ A) @ (top_top @ (set @ A))) = (bot_bot @ A)))))). % 6.41/2.69 thf(fact_5221_ccInf__empty, axiom, (! [A:$tType]: (((counta3822494911875563373attice @ A) => (((complete_Inf_Inf @ A) @ (bot_bot @ (set @ A))) = (top_top @ A)))))). % 6.41/2.69 thf(fact_5222_Inf__empty, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (((complete_Inf_Inf @ A) @ (bot_bot @ (set @ A))) = (top_top @ A)))))). % 6.41/2.69 thf(fact_5223_Diff__UNIV, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (top_top @ (set @ A))) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_5224_surj__fn, axiom, (! [A:$tType,F2:(A > A),N2:nat]: (((((((image @ A) @ A) @ F2) @ (top_top @ (set @ A))) = (top_top @ (set @ A))) => (((((image @ A) @ A) @ (((compow @ (A > A)) @ N2) @ F2)) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))). % 6.41/2.69 thf(fact_5225_finite__compl, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ A) @ ((uminus_uminus @ (set @ A)) @ A3)) = ((finite_finite2 @ A) @ (top_top @ (set @ A)))))))). % 6.41/2.69 thf(fact_5226_Gcd__UNIV, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (((gcd_Gcd @ A) @ (top_top @ (set @ A))) = (one_one @ A)))))). % 6.41/2.69 thf(fact_5227_length__remdups__leq, axiom, (! [A:$tType,Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ ((remdups @ A) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))). % 6.41/2.69 thf(fact_5228_SUP__eq__top__iff, axiom, (! [B:$tType,A:$tType]: (((comple5582772986160207858norder @ A) => (! [F2:(B > A),A3:(set @ B)]: (((((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3)) = (top_top @ A)) = (! [X:A]: (((((ord_less @ A) @ X) @ (top_top @ A)) => (? [Y2:B]: (((((member @ B) @ Y2) @ A3) & (((ord_less @ A) @ X) @ (F2 @ Y2))))))))))))))). % 6.41/2.69 thf(fact_5229_range__constant, axiom, (! [B:$tType,A:$tType,X3:A]: ((((((image @ B) @ A) @ (^ [Uu3:B]: (X3))) @ (top_top @ (set @ B))) = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))). % 6.41/2.69 thf(fact_5230_ccINF__empty, axiom, (! [B:$tType,A:$tType]: (((counta3822494911875563373attice @ A) => (! [F2:(B > A)]: ((((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ (bot_bot @ (set @ B)))) = (top_top @ A)))))))). % 6.41/2.69 thf(fact_5231_INT__constant, axiom, (! [B:$tType,A:$tType,A3:(set @ B),C2:(set @ A)]: ((((A3 = (bot_bot @ (set @ B))) => (((complete_Inf_Inf @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [Y2:B]: (C2))) @ A3)) = (top_top @ (set @ A)))) & ((A3 != (bot_bot @ (set @ B))) => (((complete_Inf_Inf @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [Y2:B]: (C2))) @ A3)) = C2)))))). % 6.41/2.69 thf(fact_5232_Inf__atMostLessThan, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X3:A]: (((((ord_less @ A) @ (top_top @ A)) @ X3) => (((complete_Inf_Inf @ A) @ ((set_ord_lessThan @ A) @ X3)) = (bot_bot @ A))))))))). % 6.41/2.69 thf(fact_5233_INT__simps_I1_J, axiom, (! [A:$tType,B:$tType,C5:(set @ A),A3:(A > (set @ B)),B3:(set @ B)]: ((((C5 = (bot_bot @ (set @ A))) => (((complete_Inf_Inf @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ (^ [X:A]: ((((inf_inf @ (set @ B)) @ (A3 @ X)) @ B3)))) @ C5)) = (top_top @ (set @ B)))) & ((C5 != (bot_bot @ (set @ A))) => (((complete_Inf_Inf @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ (^ [X:A]: ((((inf_inf @ (set @ B)) @ (A3 @ X)) @ B3)))) @ C5)) = (((inf_inf @ (set @ B)) @ ((complete_Inf_Inf @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ A3) @ C5))) @ B3))))))). % 6.41/2.69 thf(fact_5234_INT__simps_I2_J, axiom, (! [C:$tType,D:$tType,C5:(set @ D),A3:(set @ C),B3:(D > (set @ C))]: ((((C5 = (bot_bot @ (set @ D))) => (((complete_Inf_Inf @ (set @ C)) @ ((((image @ D) @ (set @ C)) @ (^ [X:D]: ((((inf_inf @ (set @ C)) @ A3) @ (B3 @ X))))) @ C5)) = (top_top @ (set @ C)))) & ((C5 != (bot_bot @ (set @ D))) => (((complete_Inf_Inf @ (set @ C)) @ ((((image @ D) @ (set @ C)) @ (^ [X:D]: ((((inf_inf @ (set @ C)) @ A3) @ (B3 @ X))))) @ C5)) = (((inf_inf @ (set @ C)) @ A3) @ ((complete_Inf_Inf @ (set @ C)) @ ((((image @ D) @ (set @ C)) @ B3) @ C5))))))))). % 6.41/2.69 thf(fact_5235_INT__simps_I3_J, axiom, (! [E2:$tType,F:$tType,C5:(set @ E2),A3:(E2 > (set @ F)),B3:(set @ F)]: ((((C5 = (bot_bot @ (set @ E2))) => (((complete_Inf_Inf @ (set @ F)) @ ((((image @ E2) @ (set @ F)) @ (^ [X:E2]: ((((minus_minus @ (set @ F)) @ (A3 @ X)) @ B3)))) @ C5)) = (top_top @ (set @ F)))) & ((C5 != (bot_bot @ (set @ E2))) => (((complete_Inf_Inf @ (set @ F)) @ ((((image @ E2) @ (set @ F)) @ (^ [X:E2]: ((((minus_minus @ (set @ F)) @ (A3 @ X)) @ B3)))) @ C5)) = (((minus_minus @ (set @ F)) @ ((complete_Inf_Inf @ (set @ F)) @ ((((image @ E2) @ (set @ F)) @ A3) @ C5))) @ B3))))))). % 6.41/2.69 thf(fact_5236_UNIV__option__conv, axiom, (! [A:$tType]: (((top_top @ (set @ (option @ A))) = (((insert @ (option @ A)) @ (none @ A)) @ ((((image @ A) @ (option @ A)) @ (some @ A)) @ (top_top @ (set @ A)))))))). % 6.41/2.69 thf(fact_5237_rangeI, axiom, (! [A:$tType,B:$tType,F2:(B > A),X3:B]: ((((member @ A) @ (F2 @ X3)) @ ((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))))))). % 6.41/2.69 thf(fact_5238_range__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F2:(B > A),X3:B]: (((B2 = (F2 @ X3)) => (((member @ A) @ B2) @ ((((image @ B) @ A) @ F2) @ (top_top @ (set @ B)))))))). % 6.41/2.69 thf(fact_5239_rangeE, axiom, (! [A:$tType,B:$tType,B2:A,F2:(B > A)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ F2) @ (top_top @ (set @ B)))) => (~ ((! [X5:B]: ((B2 != (F2 @ X5)))))))))). % 6.41/2.69 thf(fact_5240_range__composition, axiom, (! [A:$tType,C:$tType,B:$tType,F2:(C > A),G:(B > C)]: ((((((image @ B) @ A) @ (^ [X:B]: ((F2 @ (G @ X))))) @ (top_top @ (set @ B))) = ((((image @ C) @ A) @ F2) @ ((((image @ B) @ C) @ G) @ (top_top @ (set @ B)))))))). % 6.41/2.69 thf(fact_5241_INF__filter__bot__base, axiom, (! [B:$tType,A:$tType,I6:(set @ A),F5:(A > (filter @ B))]: (((! [I4:A]: (((((member @ A) @ I4) @ I6) => (! [J2:A]: (((((member @ A) @ J2) @ I6) => (? [X2:A]: (((((member @ A) @ X2) @ I6) & (((ord_less_eq @ (filter @ B)) @ (F5 @ X2)) @ (((inf_inf @ (filter @ B)) @ (F5 @ I4)) @ (F5 @ J2)))))))))))) => ((((complete_Inf_Inf @ (filter @ B)) @ ((((image @ A) @ (filter @ B)) @ F5) @ I6)) = (bot_bot @ (filter @ B))) = (? [X:A]: (((((member @ A) @ X) @ I6) & ((F5 @ X) = (bot_bot @ (filter @ B))))))))))). % 6.41/2.69 thf(fact_5242_remdups_Osimps_I1_J, axiom, (! [A:$tType]: ((((remdups @ A) @ (nil @ A)) = (nil @ A))))). % 6.41/2.69 thf(fact_5243_Int__UNIV__right, axiom, (! [A:$tType,A3:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (top_top @ (set @ A))) = A3)))). % 6.41/2.69 thf(fact_5244_Int__UNIV__left, axiom, (! [A:$tType,B3:(set @ A)]: (((((inf_inf @ (set @ A)) @ (top_top @ (set @ A))) @ B3) = B3)))). % 6.41/2.69 thf(fact_5245_remdups__remdups, axiom, (! [A:$tType,Xs:(list @ A)]: ((((remdups @ A) @ ((remdups @ A) @ Xs)) = ((remdups @ A) @ Xs))))). % 6.41/2.69 thf(fact_5246_UNIV__eq__I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X5:A]: ((((member @ A) @ X5) @ A3))) => ((top_top @ (set @ A)) = A3))))). % 6.41/2.69 thf(fact_5247_UNIV__witness, axiom, (! [A:$tType]: ((? [X5:A]: ((((member @ A) @ X5) @ (top_top @ (set @ A)))))))). % 6.41/2.69 thf(fact_5248_top_Oextremum__strict, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ (top_top @ A)) @ A2))))))))). % 6.41/2.69 thf(fact_5249_top_Onot__eq__extremum, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: (((A2 != (top_top @ A)) = (((ord_less @ A) @ A2) @ (top_top @ A))))))))). % 6.41/2.69 thf(fact_5250_UNIV__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (^ [X:A]: ($true))))))). % 6.41/2.69 thf(fact_5251_subset__UNIV, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ (top_top @ (set @ A)))))). % 6.41/2.69 thf(fact_5252_top_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) => (A2 = (top_top @ A))))))))). % 6.41/2.69 thf(fact_5253_top_Oextremum__unique, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) = (A2 = (top_top @ A))))))))). % 6.41/2.69 thf(fact_5254_top__greatest, axiom, (! [A:$tType]: (((order_top @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ (top_top @ A)))))))). % 6.41/2.69 thf(fact_5255_distinct__remdups__id, axiom, (! [A:$tType,Xs:(list @ A)]: ((((distinct @ A) @ Xs) => (((remdups @ A) @ Xs) = Xs))))). % 6.41/2.69 thf(fact_5256_empty__not__UNIV, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) != (top_top @ (set @ A)))))). % 6.41/2.69 thf(fact_5257_atLeastAtMost__eq__UNIV__iff, axiom, (! [A:$tType]: (((bounded_lattice @ A) => (! [X3:A,Y:A]: ((((((set_or1337092689740270186AtMost @ A) @ X3) @ Y) = (top_top @ (set @ A))) = ((X3 = (bot_bot @ A)) & (Y = (top_top @ A)))))))))). % 6.41/2.69 thf(fact_5258_insert__UNIV, axiom, (! [A:$tType,X3:A]: (((((insert @ A) @ X3) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))). % 6.41/2.69 thf(fact_5259_Inf__filter__not__bot, axiom, (! [A:$tType,B3:(set @ (filter @ A))]: (((! [X10:(set @ (filter @ A))]: (((((ord_less_eq @ (set @ (filter @ A))) @ X10) @ B3) => (((finite_finite2 @ (filter @ A)) @ X10) => (((complete_Inf_Inf @ (filter @ A)) @ X10) != (bot_bot @ (filter @ A))))))) => (((complete_Inf_Inf @ (filter @ A)) @ B3) != (bot_bot @ (filter @ A))))))). % 6.41/2.69 thf(fact_5260_infinite__UNIV__int, axiom, (~ (((finite_finite2 @ int) @ (top_top @ (set @ int)))))). % 6.41/2.69 thf(fact_5261_nat__not__finite, axiom, (~ (((finite_finite2 @ nat) @ (top_top @ (set @ nat)))))). % 6.41/2.69 thf(fact_5262_infinite__UNIV__nat, axiom, (~ (((finite_finite2 @ nat) @ (top_top @ (set @ nat)))))). % 6.41/2.69 thf(fact_5263_Finite__Set_Ofinite__set, axiom, (! [A:$tType]: ((((finite_finite2 @ (set @ A)) @ (top_top @ (set @ (set @ A)))) = ((finite_finite2 @ A) @ (top_top @ (set @ A))))))). % 6.41/2.69 thf(fact_5264_finite__Prod__UNIV, axiom, (! [B:$tType,A:$tType]: ((((finite_finite2 @ A) @ (top_top @ (set @ A))) => (((finite_finite2 @ B) @ (top_top @ (set @ B))) => ((finite_finite2 @ ((product_prod @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B))))))))). % 6.41/2.69 thf(fact_5265_finite__prod, axiom, (! [A:$tType,B:$tType]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = (((finite_finite2 @ A) @ (top_top @ (set @ A))) & ((finite_finite2 @ B) @ (top_top @ (set @ B)))))))). % 6.41/2.69 thf(fact_5266_finite__fun__UNIVD2, axiom, (! [A:$tType,B:$tType]: ((((finite_finite2 @ (A > B)) @ (top_top @ (set @ (A > B)))) => ((finite_finite2 @ B) @ (top_top @ (set @ B))))))). % 6.41/2.69 thf(fact_5267_finite__UNIV, axiom, (! [A:$tType]: (((finite_finite @ A) => ((finite_finite2 @ A) @ (top_top @ (set @ A))))))). % 6.41/2.69 thf(fact_5268_ex__new__if__finite, axiom, (! [A:$tType,A3:(set @ A)]: (((~ (((finite_finite2 @ A) @ (top_top @ (set @ A))))) => (((finite_finite2 @ A) @ A3) => (? [A5:A]: ((~ ((((member @ A) @ A5) @ A3)))))))))). % 6.41/2.69 thf(fact_5269_infinite__UNIV__char__0, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (~ (((finite_finite2 @ A) @ (top_top @ (set @ A))))))))). % 6.41/2.69 thf(fact_5270_infinite__UNIV__listI, axiom, (! [A:$tType]: ((~ (((finite_finite2 @ (list @ A)) @ (top_top @ (set @ (list @ A))))))))). % 6.41/2.69 thf(fact_5271_range__subsetD, axiom, (! [B:$tType,A:$tType,F2:(B > A),B3:(set @ A),I:B]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F2) @ (top_top @ (set @ B)))) @ B3) => (((member @ A) @ (F2 @ I)) @ B3))))). % 6.41/2.69 thf(fact_5272_perfect__space__class_OUNIV__not__singleton, axiom, (! [A:$tType]: (((topolo8386298272705272623_space @ A) => (! [X3:A]: (((top_top @ (set @ A)) != (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))). % 6.41/2.69 thf(fact_5273_bot__finite__def, axiom, (! [A:$tType]: (((finite_lattice @ A) => ((bot_bot @ A) = ((complete_Inf_Inf @ A) @ (top_top @ (set @ A)))))))). % 6.41/2.69 thf(fact_5274_finite__range__Some, axiom, (! [A:$tType]: ((((finite_finite2 @ (option @ A)) @ ((((image @ A) @ (option @ A)) @ (some @ A)) @ (top_top @ (set @ A)))) = ((finite_finite2 @ A) @ (top_top @ (set @ A))))))). % 6.41/2.69 thf(fact_5275_not__UNIV__le__Icc, axiom, (! [A:$tType]: (((no_top @ A) => (! [L:A,H2:A]: ((~ ((((ord_less_eq @ (set @ A)) @ (top_top @ (set @ A))) @ (((set_or1337092689740270186AtMost @ A) @ L) @ H2)))))))))). % 6.41/2.69 thf(fact_5276_card__eq__UNIV__imp__eq__UNIV, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ (top_top @ (set @ A))) => ((((finite_card @ A) @ A3) = ((finite_card @ A) @ (top_top @ (set @ A)))) => (A3 = (top_top @ (set @ A)))))))). % 6.41/2.69 thf(fact_5277_not__UNIV__le__Iic, axiom, (! [A:$tType]: (((no_top @ A) => (! [H2:A]: ((~ ((((ord_less_eq @ (set @ A)) @ (top_top @ (set @ A))) @ ((set_ord_atMost @ A) @ H2)))))))))). % 6.41/2.69 thf(fact_5278_notin__range__Some, axiom, (! [A:$tType,X3:(option @ A)]: (((~ ((((member @ (option @ A)) @ X3) @ ((((image @ A) @ (option @ A)) @ (some @ A)) @ (top_top @ (set @ A)))))) = (X3 = (none @ A)))))). % 6.41/2.69 thf(fact_5279_Inter__UNIV, axiom, (! [A:$tType]: ((((complete_Inf_Inf @ (set @ A)) @ (top_top @ (set @ (set @ A)))) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_5280_Compl__UNIV__eq, axiom, (! [A:$tType]: ((((uminus_uminus @ (set @ A)) @ (top_top @ (set @ A))) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_5281_Compl__empty__eq, axiom, (! [A:$tType]: ((((uminus_uminus @ (set @ A)) @ (bot_bot @ (set @ A))) = (top_top @ (set @ A)))))). % 6.41/2.69 thf(fact_5282_int__in__range__abs, axiom, (! [N2:nat]: ((((member @ int) @ ((semiring_1_of_nat @ int) @ N2)) @ ((((image @ int) @ int) @ (abs_abs @ int)) @ (top_top @ (set @ int))))))). % 6.41/2.69 thf(fact_5283_Compl__eq__Diff__UNIV, axiom, (! [A:$tType]: (((uminus_uminus @ (set @ A)) = ((minus_minus @ (set @ A)) @ (top_top @ (set @ A))))))). % 6.41/2.69 thf(fact_5284_Inter__empty, axiom, (! [A:$tType]: ((((complete_Inf_Inf @ (set @ A)) @ (bot_bot @ (set @ (set @ A)))) = (top_top @ (set @ A)))))). % 6.41/2.69 thf(fact_5285_full__SetCompr__eq, axiom, (! [A:$tType,B:$tType,F2:(B > A)]: ((((collect @ A) @ (^ [U2:A]: ((? [X:B]: ((U2 = (F2 @ X))))))) = ((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))))))). % 6.41/2.69 thf(fact_5286_finite__range__imageI, axiom, (! [C:$tType,A:$tType,B:$tType,G:(B > A),F2:(A > C)]: ((((finite_finite2 @ A) @ ((((image @ B) @ A) @ G) @ (top_top @ (set @ B)))) => ((finite_finite2 @ C) @ ((((image @ B) @ C) @ (^ [X:B]: ((F2 @ (G @ X))))) @ (top_top @ (set @ B)))))))). % 6.41/2.69 thf(fact_5287_remove1__remdups, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((distinct @ A) @ Xs) => ((((remove1 @ A) @ X3) @ ((remdups @ A) @ Xs)) = ((remdups @ A) @ (((remove1 @ A) @ X3) @ Xs))))))). % 6.41/2.69 thf(fact_5288_finite__fun__UNIVD1, axiom, (! [B:$tType,A:$tType]: ((((finite_finite2 @ (A > B)) @ (top_top @ (set @ (A > B)))) => ((((finite_card @ B) @ (top_top @ (set @ B))) != (suc @ (zero_zero @ nat))) => ((finite_finite2 @ A) @ (top_top @ (set @ A)))))))). % 6.41/2.69 thf(fact_5289_range__eq__singletonD, axiom, (! [B:$tType,A:$tType,F2:(B > A),A2:A,X3:B]: (((((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => ((F2 @ X3) = A2))))). % 6.41/2.69 thf(fact_5290_INF__empty, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [F2:(B > A)]: ((((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ (bot_bot @ (set @ B)))) = (top_top @ A)))))))). % 6.41/2.69 thf(fact_5291_INF__constant, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),C2:A]: ((((A3 = (bot_bot @ (set @ B))) => (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ (^ [Y2:B]: (C2))) @ A3)) = (top_top @ A))) & ((A3 != (bot_bot @ (set @ B))) => (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ (^ [Y2:B]: (C2))) @ A3)) = C2))))))))). % 6.41/2.69 thf(fact_5292_Sup__finite__empty, axiom, (! [A:$tType]: (((finite_lattice @ A) => (((complete_Sup_Sup @ A) @ (bot_bot @ (set @ A))) = ((complete_Inf_Inf @ A) @ (top_top @ (set @ A)))))))). % 6.41/2.69 thf(fact_5293_Inf__finite__empty, axiom, (! [A:$tType]: (((finite_lattice @ A) => (((complete_Inf_Inf @ A) @ (bot_bot @ (set @ A))) = ((complete_Sup_Sup @ A) @ (top_top @ (set @ A)))))))). % 6.41/2.69 thf(fact_5294_surj__Compl__image__subset, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B)]: (((((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ ((((image @ B) @ A) @ F2) @ A3))) @ ((((image @ B) @ A) @ F2) @ ((uminus_uminus @ (set @ B)) @ A3))))))). % 6.41/2.69 thf(fact_5295_length__remdups__card__conv, axiom, (! [A:$tType,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ ((remdups @ A) @ Xs)) = ((finite_card @ A) @ ((set2 @ A) @ Xs)))))). % 6.41/2.69 thf(fact_5296_Nats__def, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((semiring_1_Nats @ A) = ((((image @ nat) @ A) @ (semiring_1_of_nat @ A)) @ (top_top @ (set @ nat)))))))). % 6.41/2.69 thf(fact_5297_Collect__ex__eq, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o))]: ((((collect @ A) @ (^ [X:A]: ((? [X6:B]: (((P @ X) @ X6)))))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [Y2:B]: (((collect @ A) @ (^ [X:A]: (((P @ X) @ Y2))))))) @ (top_top @ (set @ B)))))))). % 6.41/2.69 thf(fact_5298_INT__empty, axiom, (! [B:$tType,A:$tType,B3:(B > (set @ A))]: ((((complete_Inf_Inf @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B3) @ (bot_bot @ (set @ B)))) = (top_top @ (set @ A)))))). % 6.41/2.69 thf(fact_5299_UN__UN__finite__eq, axiom, (! [A:$tType,A3:(nat > (set @ A))]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ (^ [N:nat]: (((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A3) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)))))) @ (top_top @ (set @ nat)))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A3) @ (top_top @ (set @ nat)))))))). % 6.41/2.69 thf(fact_5300_inf__top_Osemilattice__neutr__order__axioms, axiom, (! [A:$tType]: (((bounde4346867609351753570nf_top @ A) => (((((semila1105856199041335345_order @ A) @ (inf_inf @ A)) @ (top_top @ A)) @ (ord_less_eq @ A)) @ (ord_less @ A)))))). % 6.41/2.69 thf(fact_5301_finite__UNIV__card__ge__0, axiom, (! [A:$tType]: ((((finite_finite2 @ A) @ (top_top @ (set @ A))) => (((ord_less @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ (top_top @ (set @ A)))))))). % 6.41/2.69 thf(fact_5302_UN__finite__subset, axiom, (! [A:$tType,A3:(nat > (set @ A)),C5:(set @ A)]: (((! [N3:nat]: ((((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A3) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N3)))) @ C5))) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A3) @ (top_top @ (set @ nat))))) @ C5))))). % 6.41/2.69 thf(fact_5303_UNIV__nat__eq, axiom, ((top_top @ (set @ nat)) = (((insert @ nat) @ (zero_zero @ nat)) @ ((((image @ nat) @ nat) @ suc) @ (top_top @ (set @ nat)))))). % 6.41/2.69 thf(fact_5304_UN__finite2__eq, axiom, (! [A:$tType,A3:(nat > (set @ A)),B3:(nat > (set @ A)),K:nat]: (((! [N3:nat]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A3) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N3))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ B3) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ N3) @ K))))))) => (((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A3) @ (top_top @ (set @ nat)))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ B3) @ (top_top @ (set @ nat))))))))). % 6.41/2.69 thf(fact_5305_INT__extend__simps_I3_J, axiom, (! [F:$tType,E2:$tType,C5:(set @ E2),A3:(E2 > (set @ F)),B3:(set @ F)]: ((((C5 = (bot_bot @ (set @ E2))) => ((((minus_minus @ (set @ F)) @ ((complete_Inf_Inf @ (set @ F)) @ ((((image @ E2) @ (set @ F)) @ A3) @ C5))) @ B3) = (((minus_minus @ (set @ F)) @ (top_top @ (set @ F))) @ B3))) & ((C5 != (bot_bot @ (set @ E2))) => ((((minus_minus @ (set @ F)) @ ((complete_Inf_Inf @ (set @ F)) @ ((((image @ E2) @ (set @ F)) @ A3) @ C5))) @ B3) = ((complete_Inf_Inf @ (set @ F)) @ ((((image @ E2) @ (set @ F)) @ (^ [X:E2]: ((((minus_minus @ (set @ F)) @ (A3 @ X)) @ B3)))) @ C5)))))))). % 6.41/2.69 thf(fact_5306_card__range__greater__zero, axiom, (! [A:$tType,B:$tType,F2:(B > A)]: ((((finite_finite2 @ A) @ ((((image @ B) @ A) @ F2) @ (top_top @ (set @ B)))) => (((ord_less @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ ((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))))))))). % 6.41/2.69 thf(fact_5307_UN__finite2__subset, axiom, (! [A:$tType,A3:(nat > (set @ A)),B3:(nat > (set @ A)),K:nat]: (((! [N3:nat]: ((((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A3) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N3)))) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ B3) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ N3) @ K))))))) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ A3) @ (top_top @ (set @ nat))))) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ B3) @ (top_top @ (set @ nat))))))))). % 6.41/2.69 thf(fact_5308_range__mod, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((((image @ nat) @ nat) @ (^ [M2:nat]: ((((modulo_modulo @ nat) @ M2) @ N2)))) @ (top_top @ (set @ nat))) = (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2)))))). % 6.41/2.69 thf(fact_5309_cclfp__def, axiom, (! [A:$tType]: (((counta3822494911875563373attice @ A) => ((order_532582986084564980_cclfp @ A) = (^ [F3:(A > A)]: (((complete_Sup_Sup @ A) @ ((((image @ nat) @ A) @ (^ [I3:nat]: (((((compow @ (A > A)) @ I3) @ F3) @ (bot_bot @ A))))) @ (top_top @ (set @ nat))))))))))). % 6.41/2.69 thf(fact_5310_root__def, axiom, (root = (^ [N:nat,X:real]: (((((if @ real) @ (N = (zero_zero @ nat))) @ (zero_zero @ real)) @ (((((the_inv_into @ real) @ real) @ (top_top @ (set @ real))) @ (^ [Y2:real]: ((((times_times @ real) @ ((sgn_sgn @ real) @ Y2)) @ (((power_power @ real) @ ((abs_abs @ real) @ Y2)) @ N))))) @ X)))))). % 6.41/2.69 thf(fact_5311_Id__on__def, axiom, (! [A:$tType]: (((id_on @ A) = (^ [A6:(set @ A)]: (((complete_Sup_Sup @ (set @ ((product_prod @ A) @ A))) @ ((((image @ A) @ (set @ ((product_prod @ A) @ A))) @ (^ [X:A]: ((((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ X)) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))) @ A6)))))))). % 6.41/2.69 thf(fact_5312_Id__onI, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ ((id_on @ A) @ A3)))))). % 6.41/2.69 thf(fact_5313_Id__on__empty, axiom, (! [A:$tType]: ((((id_on @ A) @ (bot_bot @ (set @ A))) = (bot_bot @ (set @ ((product_prod @ A) @ A))))))). % 6.41/2.69 thf(fact_5314_top__empty__eq2, axiom, (! [B:$tType,A:$tType]: (((top_top @ (A > (B > $o))) = (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ (top_top @ (set @ ((product_prod @ A) @ B)))))))))). % 6.41/2.69 thf(fact_5315_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))). % 6.41/2.69 thf(fact_5316_less__filter__def, axiom, (! [A:$tType]: (((ord_less @ (filter @ A)) = (^ [F9:(filter @ A),F10:(filter @ A)]: (((((ord_less_eq @ (filter @ A)) @ F9) @ F10) & (~ ((((ord_less_eq @ (filter @ A)) @ F10) @ F9)))))))))). % 6.41/2.69 thf(fact_5317_Id__on__iff, axiom, (! [A:$tType,X3:A,Y:A,A3:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ ((id_on @ A) @ A3)) = ((X3 = Y) & (((member @ A) @ X3) @ A3)))))). % 6.41/2.69 thf(fact_5318_Id__on__eqI, axiom, (! [A:$tType,A2:A,B2:A,A3:(set @ A)]: (((A2 = B2) => ((((member @ A) @ A2) @ A3) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((id_on @ A) @ A3))))))). % 6.41/2.69 thf(fact_5319_Id__onE, axiom, (! [A:$tType,C2:((product_prod @ A) @ A),A3:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ C2) @ ((id_on @ A) @ A3)) => (~ ((! [X5:A]: (((((member @ A) @ X5) @ A3) => (C2 != ((((product_Pair @ A) @ A) @ X5) @ X5))))))))))). % 6.41/2.69 thf(fact_5320_these__insert__Some, axiom, (! [A:$tType,X3:A,A3:(set @ (option @ A))]: ((((these @ A) @ (((insert @ (option @ A)) @ ((some @ A) @ X3)) @ A3)) = (((insert @ A) @ X3) @ ((these @ A) @ A3)))))). % 6.41/2.69 thf(fact_5321_DERIV__even__real__root, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2) => ((((ord_less @ real) @ X3) @ (zero_zero @ real)) => ((((has_field_derivative @ real) @ (root @ N2)) @ ((inverse_inverse @ real) @ (((times_times @ real) @ ((uminus_uminus @ real) @ ((semiring_1_of_nat @ real) @ N2))) @ (((power_power @ real) @ ((root @ N2) @ X3)) @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat))))))) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))))))))). % 6.41/2.69 thf(fact_5322_DERIV__real__root__generic, axiom, (! [N2:nat,X3:real,D6:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((X3 != (zero_zero @ real)) => (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (D6 = ((inverse_inverse @ real) @ (((times_times @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ (((power_power @ real) @ ((root @ N2) @ X3)) @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat))))))))) => (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2) => ((((ord_less @ real) @ X3) @ (zero_zero @ real)) => (D6 = ((uminus_uminus @ real) @ ((inverse_inverse @ real) @ (((times_times @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ (((power_power @ real) @ ((root @ N2) @ X3)) @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat)))))))))) => (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) => (D6 = ((inverse_inverse @ real) @ (((times_times @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ (((power_power @ real) @ ((root @ N2) @ X3)) @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat)))))))) => ((((has_field_derivative @ real) @ (root @ N2)) @ D6) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))))))))))). % 6.41/2.69 thf(fact_5323_top1I, axiom, (! [A:$tType,X3:A]: (((top_top @ (A > $o)) @ X3)))). % 6.41/2.69 thf(fact_5324_top2I, axiom, (! [A:$tType,B:$tType,X3:A,Y:B]: ((((top_top @ (A > (B > $o))) @ X3) @ Y)))). % 6.41/2.69 thf(fact_5325_these__empty, axiom, (! [A:$tType]: ((((these @ A) @ (bot_bot @ (set @ (option @ A)))) = (bot_bot @ (set @ A)))))). % 6.41/2.69 thf(fact_5326_at__within__empty, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [A2:A]: (((((topolo174197925503356063within @ A) @ A2) @ (bot_bot @ (set @ A))) = (bot_bot @ (filter @ A))))))))). % 6.41/2.69 thf(fact_5327_these__image__Some__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((these @ A) @ ((((image @ A) @ (option @ A)) @ (some @ A)) @ A3)) = A3)))). % 6.41/2.69 thf(fact_5328_these__insert__None, axiom, (! [A:$tType,A3:(set @ (option @ A))]: ((((these @ A) @ (((insert @ (option @ A)) @ (none @ A)) @ A3)) = ((these @ A) @ A3))))). % 6.41/2.69 thf(fact_5329_DERIV__at__within__shift__lemma, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),Y:A,Z2:A,X3:A,S3:(set @ A)]: ((((((has_field_derivative @ A) @ F2) @ Y) @ (((topolo174197925503356063within @ A) @ (((plus_plus @ A) @ Z2) @ X3)) @ ((((image @ A) @ A) @ ((plus_plus @ A) @ Z2)) @ S3))) => ((((has_field_derivative @ A) @ (((((comp @ A) @ A) @ A) @ F2) @ ((plus_plus @ A) @ Z2))) @ Y) @ (((topolo174197925503356063within @ A) @ X3) @ S3))))))))). % 6.41/2.69 thf(fact_5330_DERIV__at__within__shift, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),Y:A,Z2:A,X3:A,S3:(set @ A)]: ((((((has_field_derivative @ A) @ F2) @ Y) @ (((topolo174197925503356063within @ A) @ (((plus_plus @ A) @ Z2) @ X3)) @ ((((image @ A) @ A) @ ((plus_plus @ A) @ Z2)) @ S3))) = ((((has_field_derivative @ A) @ (^ [X:A]: ((F2 @ (((plus_plus @ A) @ Z2) @ X))))) @ Y) @ (((topolo174197925503356063within @ A) @ X3) @ S3))))))))). % 6.41/2.69 thf(fact_5331_DERIV__neg__dec__right, axiom, (! [F2:(real > real),L:real,X3:real]: ((((((has_field_derivative @ real) @ F2) @ L) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) => ((((ord_less @ real) @ L) @ (zero_zero @ real)) => (? [D3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D3) & (! [H5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ H5) => ((((ord_less @ real) @ H5) @ D3) => (((ord_less @ real) @ (F2 @ (((plus_plus @ real) @ X3) @ H5))) @ (F2 @ X3)))))))))))))). % 6.41/2.69 thf(fact_5332_DERIV__pos__inc__right, axiom, (! [F2:(real > real),L:real,X3:real]: ((((((has_field_derivative @ real) @ F2) @ L) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) => ((((ord_less @ real) @ (zero_zero @ real)) @ L) => (? [D3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D3) & (! [H5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ H5) => ((((ord_less @ real) @ H5) @ D3) => (((ord_less @ real) @ (F2 @ X3)) @ (F2 @ (((plus_plus @ real) @ X3) @ H5))))))))))))))). % 6.41/2.69 thf(fact_5333_DERIV__pos__inc__left, axiom, (! [F2:(real > real),L:real,X3:real]: ((((((has_field_derivative @ real) @ F2) @ L) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) => ((((ord_less @ real) @ (zero_zero @ real)) @ L) => (? [D3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D3) & (! [H5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ H5) => ((((ord_less @ real) @ H5) @ D3) => (((ord_less @ real) @ (F2 @ (((minus_minus @ real) @ X3) @ H5))) @ (F2 @ X3)))))))))))))). % 6.41/2.69 thf(fact_5334_DERIV__neg__dec__left, axiom, (! [F2:(real > real),L:real,X3:real]: ((((((has_field_derivative @ real) @ F2) @ L) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) => ((((ord_less @ real) @ L) @ (zero_zero @ real)) => (? [D3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D3) & (! [H5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ H5) => ((((ord_less @ real) @ H5) @ D3) => (((ord_less @ real) @ (F2 @ X3)) @ (F2 @ (((minus_minus @ real) @ X3) @ H5))))))))))))))). % 6.41/2.69 thf(fact_5335_DERIV__neg__imp__decreasing, axiom, (! [A2:real,B2:real,F2:(real > real)]: (((((ord_less @ real) @ A2) @ B2) => ((! [X5:real]: (((((ord_less_eq @ real) @ A2) @ X5) => ((((ord_less_eq @ real) @ X5) @ B2) => (? [Y5:real]: ((((((has_field_derivative @ real) @ F2) @ Y5) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))) & (((ord_less @ real) @ Y5) @ (zero_zero @ real))))))))) => (((ord_less @ real) @ (F2 @ B2)) @ (F2 @ A2))))))). % 6.41/2.69 thf(fact_5336_DERIV__pos__imp__increasing, axiom, (! [A2:real,B2:real,F2:(real > real)]: (((((ord_less @ real) @ A2) @ B2) => ((! [X5:real]: (((((ord_less_eq @ real) @ A2) @ X5) => ((((ord_less_eq @ real) @ X5) @ B2) => (? [Y5:real]: ((((((has_field_derivative @ real) @ F2) @ Y5) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))) & (((ord_less @ real) @ (zero_zero @ real)) @ Y5)))))))) => (((ord_less @ real) @ (F2 @ A2)) @ (F2 @ B2))))))). % 6.41/2.69 thf(fact_5337_DERIV__subset, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),F11:A,X3:A,S:(set @ A),T2:(set @ A)]: ((((((has_field_derivative @ A) @ F2) @ F11) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => ((((ord_less_eq @ (set @ A)) @ T2) @ S) => ((((has_field_derivative @ A) @ F2) @ F11) @ (((topolo174197925503356063within @ A) @ X3) @ T2)))))))))). % 6.41/2.69 thf(fact_5338_has__field__derivative__subset, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),Y:A,X3:A,S:(set @ A),T2:(set @ A)]: ((((((has_field_derivative @ A) @ F2) @ Y) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => ((((ord_less_eq @ (set @ A)) @ T2) @ S) => ((((has_field_derivative @ A) @ F2) @ Y) @ (((topolo174197925503356063within @ A) @ X3) @ T2)))))))))). % 6.41/2.69 thf(fact_5339_DERIV__shift, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),Y:A,X3:A,Z2:A]: ((((((has_field_derivative @ A) @ F2) @ Y) @ (((topolo174197925503356063within @ A) @ (((plus_plus @ A) @ X3) @ Z2)) @ (top_top @ (set @ A)))) = ((((has_field_derivative @ A) @ (^ [X:A]: ((F2 @ (((plus_plus @ A) @ X) @ Z2))))) @ Y) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A))))))))))). % 6.41/2.69 thf(fact_5340_at__discrete, axiom, (! [A:$tType]: (((topolo8865339358273720382pology @ A) => ((topolo174197925503356063within @ A) = (^ [X:A,S7:(set @ A)]: ((bot_bot @ (filter @ A))))))))). % 6.41/2.69 thf(fact_5341_DERIV__inverse_H, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),D6:A,X3:A,S:(set @ A)]: ((((((has_field_derivative @ A) @ F2) @ D6) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((F2 @ X3) != (zero_zero @ A)) => ((((has_field_derivative @ A) @ (^ [X:A]: (((inverse_inverse @ A) @ (F2 @ X))))) @ ((uminus_uminus @ A) @ (((times_times @ A) @ (((times_times @ A) @ ((inverse_inverse @ A) @ (F2 @ X3))) @ D6)) @ ((inverse_inverse @ A) @ (F2 @ X3))))) @ (((topolo174197925503356063within @ A) @ X3) @ S)))))))))). % 6.41/2.69 thf(fact_5342_has__real__derivative__pos__inc__right, axiom, (! [F2:(real > real),L:real,X3:real,S3:(set @ real)]: ((((((has_field_derivative @ real) @ F2) @ L) @ (((topolo174197925503356063within @ real) @ X3) @ S3)) => ((((ord_less @ real) @ (zero_zero @ real)) @ L) => (? [D3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D3) & (! [H5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ H5) => ((((member @ real) @ (((plus_plus @ real) @ X3) @ H5)) @ S3) => ((((ord_less @ real) @ H5) @ D3) => (((ord_less @ real) @ (F2 @ X3)) @ (F2 @ (((plus_plus @ real) @ X3) @ H5)))))))))))))))). % 6.41/2.69 thf(fact_5343_has__real__derivative__neg__dec__right, axiom, (! [F2:(real > real),L:real,X3:real,S3:(set @ real)]: ((((((has_field_derivative @ real) @ F2) @ L) @ (((topolo174197925503356063within @ real) @ X3) @ S3)) => ((((ord_less @ real) @ L) @ (zero_zero @ real)) => (? [D3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D3) & (! [H5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ H5) => ((((member @ real) @ (((plus_plus @ real) @ X3) @ H5)) @ S3) => ((((ord_less @ real) @ H5) @ D3) => (((ord_less @ real) @ (F2 @ (((plus_plus @ real) @ X3) @ H5))) @ (F2 @ X3))))))))))))))). % 6.41/2.69 thf(fact_5344_has__real__derivative__neg__dec__left, axiom, (! [F2:(real > real),L:real,X3:real,S3:(set @ real)]: ((((((has_field_derivative @ real) @ F2) @ L) @ (((topolo174197925503356063within @ real) @ X3) @ S3)) => ((((ord_less @ real) @ L) @ (zero_zero @ real)) => (? [D3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D3) & (! [H5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ H5) => ((((member @ real) @ (((minus_minus @ real) @ X3) @ H5)) @ S3) => ((((ord_less @ real) @ H5) @ D3) => (((ord_less @ real) @ (F2 @ X3)) @ (F2 @ (((minus_minus @ real) @ X3) @ H5)))))))))))))))). % 6.41/2.69 thf(fact_5345_has__real__derivative__pos__inc__left, axiom, (! [F2:(real > real),L:real,X3:real,S3:(set @ real)]: ((((((has_field_derivative @ real) @ F2) @ L) @ (((topolo174197925503356063within @ real) @ X3) @ S3)) => ((((ord_less @ real) @ (zero_zero @ real)) @ L) => (? [D3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D3) & (! [H5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ H5) => ((((member @ real) @ (((minus_minus @ real) @ X3) @ H5)) @ S3) => ((((ord_less @ real) @ H5) @ D3) => (((ord_less @ real) @ (F2 @ (((minus_minus @ real) @ X3) @ H5))) @ (F2 @ X3))))))))))))))). % 6.41/2.69 thf(fact_5346_DERIV__const, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [K:A,F5:(filter @ A)]: (((((has_field_derivative @ A) @ (^ [X:A]: (K))) @ (zero_zero @ A)) @ F5))))))). % 6.41/2.69 thf(fact_5347_field__differentiable__add, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),F11:A,F5:(filter @ A),G:(A > A),G7:A]: ((((((has_field_derivative @ A) @ F2) @ F11) @ F5) => (((((has_field_derivative @ A) @ G) @ G7) @ F5) => ((((has_field_derivative @ A) @ (^ [Z6:A]: ((((plus_plus @ A) @ (F2 @ Z6)) @ (G @ Z6))))) @ (((plus_plus @ A) @ F11) @ G7)) @ F5))))))))). % 6.41/2.69 thf(fact_5348_DERIV__add, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),D6:A,X3:A,S:(set @ A),G:(A > A),E5:A]: ((((((has_field_derivative @ A) @ F2) @ D6) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((((has_field_derivative @ A) @ G) @ E5) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => ((((has_field_derivative @ A) @ (^ [X:A]: ((((plus_plus @ A) @ (F2 @ X)) @ (G @ X))))) @ (((plus_plus @ A) @ D6) @ E5)) @ (((topolo174197925503356063within @ A) @ X3) @ S)))))))))). % 6.41/2.69 thf(fact_5349_DERIV__mult_H, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),D6:A,X3:A,S:(set @ A),G:(A > A),E5:A]: ((((((has_field_derivative @ A) @ F2) @ D6) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((((has_field_derivative @ A) @ G) @ E5) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => ((((has_field_derivative @ A) @ (^ [X:A]: ((((times_times @ A) @ (F2 @ X)) @ (G @ X))))) @ (((plus_plus @ A) @ (((times_times @ A) @ (F2 @ X3)) @ E5)) @ (((times_times @ A) @ D6) @ (G @ X3)))) @ (((topolo174197925503356063within @ A) @ X3) @ S)))))))))). % 6.41/2.69 thf(fact_5350_DERIV__mult, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),Da:A,X3:A,S:(set @ A),G:(A > A),Db:A]: ((((((has_field_derivative @ A) @ F2) @ Da) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((((has_field_derivative @ A) @ G) @ Db) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => ((((has_field_derivative @ A) @ (^ [X:A]: ((((times_times @ A) @ (F2 @ X)) @ (G @ X))))) @ (((plus_plus @ A) @ (((times_times @ A) @ Da) @ (G @ X3))) @ (((times_times @ A) @ Db) @ (F2 @ X3)))) @ (((topolo174197925503356063within @ A) @ X3) @ S)))))))))). % 6.41/2.69 thf(fact_5351_DERIV__divide, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),D6:A,X3:A,S:(set @ A),G:(A > A),E5:A]: ((((((has_field_derivative @ A) @ F2) @ D6) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((((has_field_derivative @ A) @ G) @ E5) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((G @ X3) != (zero_zero @ A)) => ((((has_field_derivative @ A) @ (^ [X:A]: ((((divide_divide @ A) @ (F2 @ X)) @ (G @ X))))) @ (((divide_divide @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ D6) @ (G @ X3))) @ (((times_times @ A) @ (F2 @ X3)) @ E5))) @ (((times_times @ A) @ (G @ X3)) @ (G @ X3)))) @ (((topolo174197925503356063within @ A) @ X3) @ S))))))))))). % 6.41/2.69 thf(fact_5352_in__these__eq, axiom, (! [A:$tType,X3:A,A3:(set @ (option @ A))]: (((((member @ A) @ X3) @ ((these @ A) @ A3)) = (((member @ (option @ A)) @ ((some @ A) @ X3)) @ A3))))). % 6.41/2.69 thf(fact_5353_at__neq__bot, axiom, (! [A:$tType]: (((topolo8386298272705272623_space @ A) => (! [A2:A]: (((((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))) != (bot_bot @ (filter @ A))))))))). % 6.41/2.69 thf(fact_5354_at__le, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [S:(set @ A),T2:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ S) @ T2) => (((ord_less_eq @ (filter @ A)) @ (((topolo174197925503356063within @ A) @ X3) @ S)) @ (((topolo174197925503356063within @ A) @ X3) @ T2))))))))). % 6.41/2.69 thf(fact_5355_trivial__limit__at__left__real, axiom, (! [A:$tType]: ((((dense_order @ A) & ((no_bot @ A) & (topolo1944317154257567458pology @ A))) => (! [X3:A]: (((((topolo174197925503356063within @ A) @ X3) @ ((set_ord_lessThan @ A) @ X3)) != (bot_bot @ (filter @ A))))))))). % 6.41/2.69 thf(fact_5356_MVT2, axiom, (! [A2:real,B2:real,F2:(real > real),F11:(real > real)]: (((((ord_less @ real) @ A2) @ B2) => ((! [X5:real]: (((((ord_less_eq @ real) @ A2) @ X5) => ((((ord_less_eq @ real) @ X5) @ B2) => ((((has_field_derivative @ real) @ F2) @ (F11 @ X5)) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))))))) => (? [Z3:real]: (((((ord_less @ real) @ A2) @ Z3) & ((((ord_less @ real) @ Z3) @ B2) & ((((minus_minus @ real) @ (F2 @ B2)) @ (F2 @ A2)) = (((times_times @ real) @ (((minus_minus @ real) @ B2) @ A2)) @ (F11 @ Z3)))))))))))). % 6.41/2.69 thf(fact_5357_DERIV__local__const, axiom, (! [F2:(real > real),L:real,X3:real,D2:real]: ((((((has_field_derivative @ real) @ F2) @ L) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) => ((((ord_less @ real) @ (zero_zero @ real)) @ D2) => ((! [Y4:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ (((minus_minus @ real) @ X3) @ Y4))) @ D2) => ((F2 @ X3) = (F2 @ Y4))))) => (L = (zero_zero @ real)))))))). % 6.41/2.69 thf(fact_5358_DERIV__ln, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((has_field_derivative @ real) @ (ln_ln @ real)) @ ((inverse_inverse @ real) @ X3)) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))))))). % 6.41/2.69 thf(fact_5359_DERIV__cos__add, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [K:A,Xa2:A]: (((((has_field_derivative @ A) @ (^ [X:A]: (((cos @ A) @ (((plus_plus @ A) @ X) @ K))))) @ ((uminus_uminus @ A) @ ((sin @ A) @ (((plus_plus @ A) @ Xa2) @ K)))) @ (((topolo174197925503356063within @ A) @ Xa2) @ (top_top @ (set @ A)))))))))). % 6.41/2.69 thf(fact_5360_DERIV__power__Suc, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),D6:A,X3:A,S:(set @ A),N2:nat]: ((((((has_field_derivative @ A) @ F2) @ D6) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => ((((has_field_derivative @ A) @ (^ [X:A]: ((((power_power @ A) @ (F2 @ X)) @ (suc @ N2))))) @ (((times_times @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ ((semiring_1_of_nat @ A) @ N2))) @ (((times_times @ A) @ D6) @ (((power_power @ A) @ (F2 @ X3)) @ N2)))) @ (((topolo174197925503356063within @ A) @ X3) @ S))))))))). % 6.41/2.69 thf(fact_5361_DERIV__inverse, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [X3:A,S:(set @ A)]: (((X3 != (zero_zero @ A)) => ((((has_field_derivative @ A) @ (inverse_inverse @ A)) @ ((uminus_uminus @ A) @ (((power_power @ A) @ ((inverse_inverse @ A) @ X3)) @ (suc @ (suc @ (zero_zero @ nat)))))) @ (((topolo174197925503356063within @ A) @ X3) @ S))))))))). % 6.41/2.69 thf(fact_5362_DERIV__power, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),D6:A,X3:A,S:(set @ A),N2:nat]: ((((((has_field_derivative @ A) @ F2) @ D6) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => ((((has_field_derivative @ A) @ (^ [X:A]: ((((power_power @ A) @ (F2 @ X)) @ N2)))) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (((times_times @ A) @ D6) @ (((power_power @ A) @ (F2 @ X3)) @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat))))))) @ (((topolo174197925503356063within @ A) @ X3) @ S))))))))). % 6.41/2.69 thf(fact_5363_DERIV__local__max, axiom, (! [F2:(real > real),L:real,X3:real,D2:real]: ((((((has_field_derivative @ real) @ F2) @ L) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) => ((((ord_less @ real) @ (zero_zero @ real)) @ D2) => ((! [Y4:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ (((minus_minus @ real) @ X3) @ Y4))) @ D2) => (((ord_less_eq @ real) @ (F2 @ Y4)) @ (F2 @ X3))))) => (L = (zero_zero @ real)))))))). % 6.41/2.69 thf(fact_5364_DERIV__local__min, axiom, (! [F2:(real > real),L:real,X3:real,D2:real]: ((((((has_field_derivative @ real) @ F2) @ L) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) => ((((ord_less @ real) @ (zero_zero @ real)) @ D2) => ((! [Y4:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ (((minus_minus @ real) @ X3) @ Y4))) @ D2) => (((ord_less_eq @ real) @ (F2 @ X3)) @ (F2 @ Y4))))) => (L = (zero_zero @ real)))))))). % 6.41/2.69 thf(fact_5365_DERIV__ln__divide, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((has_field_derivative @ real) @ (ln_ln @ real)) @ (((divide_divide @ real) @ (one_one @ real)) @ X3)) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))))))). % 6.41/2.69 thf(fact_5366_DERIV__pow, axiom, (! [N2:nat,X3:real,S:(set @ real)]: (((((has_field_derivative @ real) @ (^ [X:real]: ((((power_power @ real) @ X) @ N2)))) @ (((times_times @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ (((power_power @ real) @ X3) @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat)))))) @ (((topolo174197925503356063within @ real) @ X3) @ S))))). % 6.41/2.69 thf(fact_5367_at__within__Icc__at, axiom, (! [A:$tType]: (((topolo2564578578187576103pology @ A) => (! [A2:A,X3:A,B2:A]: (((((ord_less @ A) @ A2) @ X3) => ((((ord_less @ A) @ X3) @ B2) => ((((topolo174197925503356063within @ A) @ X3) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))))))))))). % 6.41/2.69 thf(fact_5368_at__within__Icc__at__left, axiom, (! [A:$tType]: (((topolo2564578578187576103pology @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((topolo174197925503356063within @ A) @ B2) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (((topolo174197925503356063within @ A) @ B2) @ ((set_ord_lessThan @ A) @ B2)))))))))). % 6.41/2.69 thf(fact_5369_trivial__limit__at__left__bot, axiom, (! [A:$tType]: ((((order_bot @ A) & (topolo1944317154257567458pology @ A)) => ((((topolo174197925503356063within @ A) @ (bot_bot @ A)) @ ((set_ord_lessThan @ A) @ (bot_bot @ A))) = (bot_bot @ (filter @ A))))))). % 6.41/2.69 thf(fact_5370_DERIV__quotient, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),D2:A,X3:A,S:(set @ A),G:(A > A),E3:A]: ((((((has_field_derivative @ A) @ F2) @ D2) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((((has_field_derivative @ A) @ G) @ E3) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((G @ X3) != (zero_zero @ A)) => ((((has_field_derivative @ A) @ (^ [Y2:A]: ((((divide_divide @ A) @ (F2 @ Y2)) @ (G @ Y2))))) @ (((divide_divide @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ D2) @ (G @ X3))) @ (((times_times @ A) @ E3) @ (F2 @ X3)))) @ (((power_power @ A) @ (G @ X3)) @ (suc @ (suc @ (zero_zero @ nat)))))) @ (((topolo174197925503356063within @ A) @ X3) @ S))))))))))). % 6.41/2.69 thf(fact_5371_DERIV__inverse__fun, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),D2:A,X3:A,S:(set @ A)]: ((((((has_field_derivative @ A) @ F2) @ D2) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((F2 @ X3) != (zero_zero @ A)) => ((((has_field_derivative @ A) @ (^ [X:A]: (((inverse_inverse @ A) @ (F2 @ X))))) @ ((uminus_uminus @ A) @ (((times_times @ A) @ D2) @ ((inverse_inverse @ A) @ (((power_power @ A) @ (F2 @ X3)) @ (suc @ (suc @ (zero_zero @ nat)))))))) @ (((topolo174197925503356063within @ A) @ X3) @ S)))))))))). % 6.41/2.69 thf(fact_5372_termdiffs__sums__strong, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [K5:real,C2:(nat > A),F2:(A > A),F11:A,Z2:A]: (((! [Z3:A]: (((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ Z3)) @ K5) => (((sums @ A) @ (^ [N:nat]: ((((times_times @ A) @ (C2 @ N)) @ (((power_power @ A) @ Z3) @ N))))) @ (F2 @ Z3))))) => (((((has_field_derivative @ A) @ F2) @ F11) @ (((topolo174197925503356063within @ A) @ Z2) @ (top_top @ (set @ A)))) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ Z2)) @ K5) => (((sums @ A) @ (^ [N:nat]: ((((times_times @ A) @ (((diffs @ A) @ C2) @ N)) @ (((power_power @ A) @ Z2) @ N))))) @ F11)))))))))). % 6.41/2.69 thf(fact_5373_has__real__derivative__powr, axiom, (! [Z2:real,R2:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ Z2) => ((((has_field_derivative @ real) @ (^ [Z6:real]: ((((powr @ real) @ Z6) @ R2)))) @ (((times_times @ real) @ R2) @ (((powr @ real) @ Z2) @ (((minus_minus @ real) @ R2) @ (one_one @ real))))) @ (((topolo174197925503356063within @ real) @ Z2) @ (top_top @ (set @ real)))))))). % 6.41/2.69 thf(fact_5374_termdiffs__strong_H, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [K5:real,C2:(nat > A),Z2:A]: (((! [Z3:A]: (((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ Z3)) @ K5) => ((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (C2 @ N)) @ (((power_power @ A) @ Z3) @ N)))))))) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ Z2)) @ K5) => ((((has_field_derivative @ A) @ (^ [Z6:A]: (((suminf @ A) @ (^ [N:nat]: ((((times_times @ A) @ (C2 @ N)) @ (((power_power @ A) @ Z6) @ N)))))))) @ ((suminf @ A) @ (^ [N:nat]: ((((times_times @ A) @ (((diffs @ A) @ C2) @ N)) @ (((power_power @ A) @ Z2) @ N)))))) @ (((topolo174197925503356063within @ A) @ Z2) @ (top_top @ (set @ A)))))))))))). % 6.41/2.69 thf(fact_5375_termdiffs__strong, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [C2:(nat > A),K5:A,X3:A]: ((((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (C2 @ N)) @ (((power_power @ A) @ K5) @ N))))) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ ((real_V7770717601297561774m_norm @ A) @ K5)) => ((((has_field_derivative @ A) @ (^ [X:A]: (((suminf @ A) @ (^ [N:nat]: ((((times_times @ A) @ (C2 @ N)) @ (((power_power @ A) @ X) @ N)))))))) @ ((suminf @ A) @ (^ [N:nat]: ((((times_times @ A) @ (((diffs @ A) @ C2) @ N)) @ (((power_power @ A) @ X3) @ N)))))) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))))))))))). % 6.41/2.69 thf(fact_5376_termdiffs, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [C2:(nat > A),K5:A,X3:A]: ((((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (C2 @ N)) @ (((power_power @ A) @ K5) @ N))))) => (((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (((diffs @ A) @ C2) @ N)) @ (((power_power @ A) @ K5) @ N))))) => (((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (((diffs @ A) @ ((diffs @ A) @ C2)) @ N)) @ (((power_power @ A) @ K5) @ N))))) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ ((real_V7770717601297561774m_norm @ A) @ K5)) => ((((has_field_derivative @ A) @ (^ [X:A]: (((suminf @ A) @ (^ [N:nat]: ((((times_times @ A) @ (C2 @ N)) @ (((power_power @ A) @ X) @ N)))))))) @ ((suminf @ A) @ (^ [N:nat]: ((((times_times @ A) @ (((diffs @ A) @ C2) @ N)) @ (((power_power @ A) @ X3) @ N)))))) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))))))))))))). % 6.41/2.69 thf(fact_5377_DERIV__log, axiom, (! [X3:real,B2:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((has_field_derivative @ real) @ (log @ B2)) @ (((divide_divide @ real) @ (one_one @ real)) @ (((times_times @ real) @ ((ln_ln @ real) @ B2)) @ X3))) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))))))). % 6.41/2.69 thf(fact_5378_DERIV__fun__powr, axiom, (! [G:(real > real),M:real,X3:real,R2:real]: ((((((has_field_derivative @ real) @ G) @ M) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) => ((((ord_less @ real) @ (zero_zero @ real)) @ (G @ X3)) => ((((has_field_derivative @ real) @ (^ [X:real]: ((((powr @ real) @ (G @ X)) @ R2)))) @ (((times_times @ real) @ (((times_times @ real) @ R2) @ (((powr @ real) @ (G @ X3)) @ (((minus_minus @ real) @ R2) @ ((semiring_1_of_nat @ real) @ (one_one @ nat)))))) @ M)) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real))))))))). % 6.41/2.69 thf(fact_5379_DERIV__powr, axiom, (! [G:(real > real),M:real,X3:real,F2:(real > real),R2:real]: ((((((has_field_derivative @ real) @ G) @ M) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) => ((((ord_less @ real) @ (zero_zero @ real)) @ (G @ X3)) => (((((has_field_derivative @ real) @ F2) @ R2) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) => ((((has_field_derivative @ real) @ (^ [X:real]: ((((powr @ real) @ (G @ X)) @ (F2 @ X))))) @ (((times_times @ real) @ (((powr @ real) @ (G @ X3)) @ (F2 @ X3))) @ (((plus_plus @ real) @ (((times_times @ real) @ R2) @ ((ln_ln @ real) @ (G @ X3)))) @ (((divide_divide @ real) @ (((times_times @ real) @ M) @ (F2 @ X3))) @ (G @ X3))))) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))))))))). % 6.41/2.69 thf(fact_5380_DERIV__tan, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((cos @ A) @ X3) != (zero_zero @ A)) => ((((has_field_derivative @ A) @ (tan @ A)) @ ((inverse_inverse @ A) @ (((power_power @ A) @ ((cos @ A) @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A))))))))))). % 6.41/2.69 thf(fact_5381_arcosh__real__has__field__derivative, axiom, (! [X3:real,A3:(set @ real)]: (((((ord_less @ real) @ (one_one @ real)) @ X3) => ((((has_field_derivative @ real) @ (arcosh @ real)) @ (((divide_divide @ real) @ (one_one @ real)) @ (sqrt @ (((minus_minus @ real) @ (((power_power @ real) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ real))))) @ (((topolo174197925503356063within @ real) @ X3) @ A3)))))). % 6.41/2.69 thf(fact_5382_DERIV__real__sqrt, axiom, (! [X3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((has_field_derivative @ real) @ sqrt) @ (((divide_divide @ real) @ ((inverse_inverse @ real) @ (sqrt @ X3))) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))))))). % 6.41/2.69 thf(fact_5383_DERIV__cot, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((sin @ A) @ X3) != (zero_zero @ A)) => ((((has_field_derivative @ A) @ (cot @ A)) @ ((uminus_uminus @ A) @ ((inverse_inverse @ A) @ (((power_power @ A) @ ((sin @ A) @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A))))))))))). % 6.41/2.69 thf(fact_5384_Option_Othese__def, axiom, (! [A:$tType]: (((these @ A) = (^ [A6:(set @ (option @ A))]: (((((image @ (option @ A)) @ A) @ (the2 @ A)) @ ((collect @ (option @ A)) @ (^ [X:(option @ A)]: (((((member @ (option @ A)) @ X) @ A6) & (X != (none @ A))))))))))))). % 6.41/2.69 thf(fact_5385_has__field__derivative__tanh, axiom, (! [A10:$tType]: ((((real_Vector_banach @ A10) & (real_V3459762299906320749_field @ A10)) => (! [G:(A10 > A10),X3:A10,Db:A10,S:(set @ A10)]: (((((cosh @ A10) @ (G @ X3)) != (zero_zero @ A10)) => (((((has_field_derivative @ A10) @ G) @ Db) @ (((topolo174197925503356063within @ A10) @ X3) @ S)) => ((((has_field_derivative @ A10) @ (^ [X:A10]: (((tanh @ A10) @ (G @ X))))) @ (((times_times @ A10) @ (((minus_minus @ A10) @ (one_one @ A10)) @ (((power_power @ A10) @ ((tanh @ A10) @ (G @ X3))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ Db)) @ (((topolo174197925503356063within @ A10) @ X3) @ S)))))))))). % 6.41/2.69 thf(fact_5386_DERIV__real__sqrt__generic, axiom, (! [X3:real,D6:real]: (((X3 != (zero_zero @ real)) => (((((ord_less @ real) @ (zero_zero @ real)) @ X3) => (D6 = (((divide_divide @ real) @ ((inverse_inverse @ real) @ (sqrt @ X3))) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) => (((((ord_less @ real) @ X3) @ (zero_zero @ real)) => (D6 = (((divide_divide @ real) @ ((uminus_uminus @ real) @ ((inverse_inverse @ real) @ (sqrt @ X3)))) @ ((numeral_numeral @ real) @ (bit0 @ one2))))) => ((((has_field_derivative @ real) @ sqrt) @ D6) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))))))))). % 6.41/2.69 thf(fact_5387_artanh__real__has__field__derivative, axiom, (! [X3:real,A3:(set @ real)]: (((((ord_less @ real) @ ((abs_abs @ real) @ X3)) @ (one_one @ real)) => ((((has_field_derivative @ real) @ (artanh @ real)) @ (((divide_divide @ real) @ (one_one @ real)) @ (((minus_minus @ real) @ (one_one @ real)) @ (((power_power @ real) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))) @ (((topolo174197925503356063within @ real) @ X3) @ A3)))))). % 6.41/2.69 thf(fact_5388_DERIV__real__root, axiom, (! [N2:nat,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((ord_less @ real) @ (zero_zero @ real)) @ X3) => ((((has_field_derivative @ real) @ (root @ N2)) @ ((inverse_inverse @ real) @ (((times_times @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ (((power_power @ real) @ ((root @ N2) @ X3)) @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat))))))) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real))))))))). % 6.41/2.69 thf(fact_5389_DERIV__arcsin, axiom, (! [X3:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ X3) => ((((ord_less @ real) @ X3) @ (one_one @ real)) => ((((has_field_derivative @ real) @ arcsin) @ ((inverse_inverse @ real) @ (sqrt @ (((minus_minus @ real) @ (one_one @ real)) @ (((power_power @ real) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real))))))))). % 6.41/2.69 thf(fact_5390_these__not__empty__eq, axiom, (! [A:$tType,B3:(set @ (option @ A))]: (((((these @ A) @ B3) != (bot_bot @ (set @ A))) = ((B3 != (bot_bot @ (set @ (option @ A)))) & (B3 != (((insert @ (option @ A)) @ (none @ A)) @ (bot_bot @ (set @ (option @ A)))))))))). % 6.41/2.69 thf(fact_5391_these__empty__eq, axiom, (! [A:$tType,B3:(set @ (option @ A))]: (((((these @ A) @ B3) = (bot_bot @ (set @ A))) = ((B3 = (bot_bot @ (set @ (option @ A)))) | (B3 = (((insert @ (option @ A)) @ (none @ A)) @ (bot_bot @ (set @ (option @ A)))))))))). % 6.41/2.69 thf(fact_5392_Maclaurin__all__le__objl, axiom, (! [Diff:(nat > (real > real)),F2:(real > real),X3:real,N2:nat]: (((((Diff @ (zero_zero @ nat)) = F2) & (! [M3:nat,X5:real]: (((((has_field_derivative @ real) @ (Diff @ M3)) @ ((Diff @ (suc @ M3)) @ X5)) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real))))))) => (? [T7:real]: (((((ord_less_eq @ real) @ ((abs_abs @ real) @ T7)) @ ((abs_abs @ real) @ X3)) & ((F2 @ X3) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ M2) @ (zero_zero @ real))) @ ((semiring_char_0_fact @ real) @ M2))) @ (((power_power @ real) @ X3) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ N2) @ T7)) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ real) @ X3) @ N2))))))))))). % 6.41/2.69 thf(fact_5393_Maclaurin__all__le, axiom, (! [Diff:(nat > (real > real)),F2:(real > real),X3:real,N2:nat]: ((((Diff @ (zero_zero @ nat)) = F2) => ((! [M3:nat,X5:real]: (((((has_field_derivative @ real) @ (Diff @ M3)) @ ((Diff @ (suc @ M3)) @ X5)) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))))) => (? [T7:real]: (((((ord_less_eq @ real) @ ((abs_abs @ real) @ T7)) @ ((abs_abs @ real) @ X3)) & ((F2 @ X3) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ M2) @ (zero_zero @ real))) @ ((semiring_char_0_fact @ real) @ M2))) @ (((power_power @ real) @ X3) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ N2) @ T7)) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ real) @ X3) @ N2)))))))))))). % 6.41/2.69 thf(fact_5394_Some__image__these__eq, axiom, (! [A:$tType,A3:(set @ (option @ A))]: ((((((image @ A) @ (option @ A)) @ (some @ A)) @ ((these @ A) @ A3)) = ((collect @ (option @ A)) @ (^ [X:(option @ A)]: (((((member @ (option @ A)) @ X) @ A3) & (X != (none @ A)))))))))). % 6.41/2.69 thf(fact_5395_DERIV__odd__real__root, axiom, (! [N2:nat,X3:real]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) => ((X3 != (zero_zero @ real)) => ((((has_field_derivative @ real) @ (root @ N2)) @ ((inverse_inverse @ real) @ (((times_times @ real) @ ((semiring_1_of_nat @ real) @ N2)) @ (((power_power @ real) @ ((root @ N2) @ X3)) @ (((minus_minus @ nat) @ N2) @ (suc @ (zero_zero @ nat))))))) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real))))))))). % 6.41/2.69 thf(fact_5396_Maclaurin__minus, axiom, (! [H2:real,N2:nat,Diff:(nat > (real > real)),F2:(real > real)]: (((((ord_less @ real) @ H2) @ (zero_zero @ real)) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((Diff @ (zero_zero @ nat)) = F2) => ((! [M3:nat,T7:real]: ((((((ord_less @ nat) @ M3) @ N2) & ((((ord_less_eq @ real) @ H2) @ T7) & (((ord_less_eq @ real) @ T7) @ (zero_zero @ real)))) => ((((has_field_derivative @ real) @ (Diff @ M3)) @ ((Diff @ (suc @ M3)) @ T7)) @ (((topolo174197925503356063within @ real) @ T7) @ (top_top @ (set @ real))))))) => (? [T7:real]: (((((ord_less @ real) @ H2) @ T7) & ((((ord_less @ real) @ T7) @ (zero_zero @ real)) & ((F2 @ H2) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ M2) @ (zero_zero @ real))) @ ((semiring_char_0_fact @ real) @ M2))) @ (((power_power @ real) @ H2) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ N2) @ T7)) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ real) @ H2) @ N2))))))))))))))). % 6.41/2.69 thf(fact_5397_Maclaurin2, axiom, (! [H2:real,Diff:(nat > (real > real)),F2:(real > real),N2:nat]: (((((ord_less @ real) @ (zero_zero @ real)) @ H2) => (((Diff @ (zero_zero @ nat)) = F2) => ((! [M3:nat,T7:real]: ((((((ord_less @ nat) @ M3) @ N2) & ((((ord_less_eq @ real) @ (zero_zero @ real)) @ T7) & (((ord_less_eq @ real) @ T7) @ H2))) => ((((has_field_derivative @ real) @ (Diff @ M3)) @ ((Diff @ (suc @ M3)) @ T7)) @ (((topolo174197925503356063within @ real) @ T7) @ (top_top @ (set @ real))))))) => (? [T7:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ T7) & ((((ord_less_eq @ real) @ T7) @ H2) & ((F2 @ H2) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ M2) @ (zero_zero @ real))) @ ((semiring_char_0_fact @ real) @ M2))) @ (((power_power @ real) @ H2) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ N2) @ T7)) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ real) @ H2) @ N2)))))))))))))). % 6.41/2.69 thf(fact_5398_Maclaurin, axiom, (! [H2:real,N2:nat,Diff:(nat > (real > real)),F2:(real > real)]: (((((ord_less @ real) @ (zero_zero @ real)) @ H2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((Diff @ (zero_zero @ nat)) = F2) => ((! [M3:nat,T7:real]: ((((((ord_less @ nat) @ M3) @ N2) & ((((ord_less_eq @ real) @ (zero_zero @ real)) @ T7) & (((ord_less_eq @ real) @ T7) @ H2))) => ((((has_field_derivative @ real) @ (Diff @ M3)) @ ((Diff @ (suc @ M3)) @ T7)) @ (((topolo174197925503356063within @ real) @ T7) @ (top_top @ (set @ real))))))) => (? [T7:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ T7) & ((((ord_less @ real) @ T7) @ H2) & ((F2 @ H2) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ M2) @ (zero_zero @ real))) @ ((semiring_char_0_fact @ real) @ M2))) @ (((power_power @ real) @ H2) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ N2) @ T7)) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ real) @ H2) @ N2))))))))))))))). % 6.41/2.69 thf(fact_5399_Maclaurin__all__lt, axiom, (! [Diff:(nat > (real > real)),F2:(real > real),N2:nat,X3:real]: ((((Diff @ (zero_zero @ nat)) = F2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((X3 != (zero_zero @ real)) => ((! [M3:nat,X5:real]: (((((has_field_derivative @ real) @ (Diff @ M3)) @ ((Diff @ (suc @ M3)) @ X5)) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))))) => (? [T7:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ ((abs_abs @ real) @ T7)) & ((((ord_less @ real) @ ((abs_abs @ real) @ T7)) @ ((abs_abs @ real) @ X3)) & ((F2 @ X3) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ M2) @ (zero_zero @ real))) @ ((semiring_char_0_fact @ real) @ M2))) @ (((power_power @ real) @ X3) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ N2) @ T7)) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ real) @ X3) @ N2))))))))))))))). % 6.41/2.69 thf(fact_5400_Maclaurin__bi__le, axiom, (! [Diff:(nat > (real > real)),F2:(real > real),N2:nat,X3:real]: ((((Diff @ (zero_zero @ nat)) = F2) => ((! [M3:nat,T7:real]: ((((((ord_less @ nat) @ M3) @ N2) & (((ord_less_eq @ real) @ ((abs_abs @ real) @ T7)) @ ((abs_abs @ real) @ X3))) => ((((has_field_derivative @ real) @ (Diff @ M3)) @ ((Diff @ (suc @ M3)) @ T7)) @ (((topolo174197925503356063within @ real) @ T7) @ (top_top @ (set @ real))))))) => (? [T7:real]: (((((ord_less_eq @ real) @ ((abs_abs @ real) @ T7)) @ ((abs_abs @ real) @ X3)) & ((F2 @ X3) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ M2) @ (zero_zero @ real))) @ ((semiring_char_0_fact @ real) @ M2))) @ (((power_power @ real) @ X3) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ N2) @ T7)) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ real) @ X3) @ N2)))))))))))). % 6.41/2.69 thf(fact_5401_Taylor, axiom, (! [N2:nat,Diff:(nat > (real > real)),F2:(real > real),A2:real,B2:real,C2:real,X3:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((Diff @ (zero_zero @ nat)) = F2) => ((! [M3:nat,T7:real]: ((((((ord_less @ nat) @ M3) @ N2) & ((((ord_less_eq @ real) @ A2) @ T7) & (((ord_less_eq @ real) @ T7) @ B2))) => ((((has_field_derivative @ real) @ (Diff @ M3)) @ ((Diff @ (suc @ M3)) @ T7)) @ (((topolo174197925503356063within @ real) @ T7) @ (top_top @ (set @ real))))))) => ((((ord_less_eq @ real) @ A2) @ C2) => ((((ord_less_eq @ real) @ C2) @ B2) => ((((ord_less_eq @ real) @ A2) @ X3) => ((((ord_less_eq @ real) @ X3) @ B2) => ((X3 != C2) => (? [T7:real]: ((((((ord_less @ real) @ X3) @ C2) => ((((ord_less @ real) @ X3) @ T7) & (((ord_less @ real) @ T7) @ C2))) & (((~ ((((ord_less @ real) @ X3) @ C2))) => ((((ord_less @ real) @ C2) @ T7) & (((ord_less @ real) @ T7) @ X3))) & ((F2 @ X3) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ M2) @ C2)) @ ((semiring_char_0_fact @ real) @ M2))) @ (((power_power @ real) @ (((minus_minus @ real) @ X3) @ C2)) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ N2) @ T7)) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ real) @ (((minus_minus @ real) @ X3) @ C2)) @ N2))))))))))))))))))). % 6.41/2.69 thf(fact_5402_Taylor__up, axiom, (! [N2:nat,Diff:(nat > (real > real)),F2:(real > real),A2:real,B2:real,C2:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((Diff @ (zero_zero @ nat)) = F2) => ((! [M3:nat,T7:real]: ((((((ord_less @ nat) @ M3) @ N2) & ((((ord_less_eq @ real) @ A2) @ T7) & (((ord_less_eq @ real) @ T7) @ B2))) => ((((has_field_derivative @ real) @ (Diff @ M3)) @ ((Diff @ (suc @ M3)) @ T7)) @ (((topolo174197925503356063within @ real) @ T7) @ (top_top @ (set @ real))))))) => ((((ord_less_eq @ real) @ A2) @ C2) => ((((ord_less @ real) @ C2) @ B2) => (? [T7:real]: (((((ord_less @ real) @ C2) @ T7) & ((((ord_less @ real) @ T7) @ B2) & ((F2 @ B2) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ M2) @ C2)) @ ((semiring_char_0_fact @ real) @ M2))) @ (((power_power @ real) @ (((minus_minus @ real) @ B2) @ C2)) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ N2) @ T7)) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ real) @ (((minus_minus @ real) @ B2) @ C2)) @ N2)))))))))))))))). % 6.41/2.69 thf(fact_5403_Taylor__down, axiom, (! [N2:nat,Diff:(nat > (real > real)),F2:(real > real),A2:real,B2:real,C2:real]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((Diff @ (zero_zero @ nat)) = F2) => ((! [M3:nat,T7:real]: ((((((ord_less @ nat) @ M3) @ N2) & ((((ord_less_eq @ real) @ A2) @ T7) & (((ord_less_eq @ real) @ T7) @ B2))) => ((((has_field_derivative @ real) @ (Diff @ M3)) @ ((Diff @ (suc @ M3)) @ T7)) @ (((topolo174197925503356063within @ real) @ T7) @ (top_top @ (set @ real))))))) => ((((ord_less @ real) @ A2) @ C2) => ((((ord_less_eq @ real) @ C2) @ B2) => (? [T7:real]: (((((ord_less @ real) @ A2) @ T7) & ((((ord_less @ real) @ T7) @ C2) & ((F2 @ A2) = (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [M2:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ M2) @ C2)) @ ((semiring_char_0_fact @ real) @ M2))) @ (((power_power @ real) @ (((minus_minus @ real) @ A2) @ C2)) @ M2))))) @ ((set_ord_lessThan @ nat) @ N2))) @ (((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ N2) @ T7)) @ ((semiring_char_0_fact @ real) @ N2))) @ (((power_power @ real) @ (((minus_minus @ real) @ A2) @ C2)) @ N2)))))))))))))))). % 6.41/2.69 thf(fact_5404_Maclaurin__lemma2, axiom, (! [N2:nat,H2:real,Diff:(nat > (real > real)),K:nat,B3:real]: (((! [M3:nat,T7:real]: ((((((ord_less @ nat) @ M3) @ N2) & ((((ord_less_eq @ real) @ (zero_zero @ real)) @ T7) & (((ord_less_eq @ real) @ T7) @ H2))) => ((((has_field_derivative @ real) @ (Diff @ M3)) @ ((Diff @ (suc @ M3)) @ T7)) @ (((topolo174197925503356063within @ real) @ T7) @ (top_top @ (set @ real))))))) => ((N2 = (suc @ K)) => (! [M6:nat,T9:real]: ((((((ord_less @ nat) @ M6) @ N2) & ((((ord_less_eq @ real) @ (zero_zero @ real)) @ T9) & (((ord_less_eq @ real) @ T9) @ H2))) => ((((has_field_derivative @ real) @ (^ [U2:real]: ((((minus_minus @ real) @ ((Diff @ M6) @ U2)) @ (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [P6:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ (((plus_plus @ nat) @ M6) @ P6)) @ (zero_zero @ real))) @ ((semiring_char_0_fact @ real) @ P6))) @ (((power_power @ real) @ U2) @ P6))))) @ ((set_ord_lessThan @ nat) @ (((minus_minus @ nat) @ N2) @ M6)))) @ (((times_times @ real) @ B3) @ (((divide_divide @ real) @ (((power_power @ real) @ U2) @ (((minus_minus @ nat) @ N2) @ M6))) @ ((semiring_char_0_fact @ real) @ (((minus_minus @ nat) @ N2) @ M6))))))))) @ (((minus_minus @ real) @ ((Diff @ (suc @ M6)) @ T9)) @ (((plus_plus @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [P6:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ ((Diff @ (((plus_plus @ nat) @ (suc @ M6)) @ P6)) @ (zero_zero @ real))) @ ((semiring_char_0_fact @ real) @ P6))) @ (((power_power @ real) @ T9) @ P6))))) @ ((set_ord_lessThan @ nat) @ (((minus_minus @ nat) @ N2) @ (suc @ M6))))) @ (((times_times @ real) @ B3) @ (((divide_divide @ real) @ (((power_power @ real) @ T9) @ (((minus_minus @ nat) @ N2) @ (suc @ M6)))) @ ((semiring_char_0_fact @ real) @ (((minus_minus @ nat) @ N2) @ (suc @ M6)))))))) @ (((topolo174197925503356063within @ real) @ T9) @ (top_top @ (set @ real)))))))))))). % 6.41/2.69 thf(fact_5405_DERIV__arctan__series, axiom, (! [X3:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ X3)) @ (one_one @ real)) => ((((has_field_derivative @ real) @ (^ [X9:real]: (((suminf @ real) @ (^ [K3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ K3)) @ (((times_times @ real) @ (((divide_divide @ real) @ (one_one @ real)) @ ((semiring_1_of_nat @ real) @ (((plus_plus @ nat) @ (((times_times @ nat) @ K3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ nat))))) @ (((power_power @ real) @ X9) @ (((plus_plus @ nat) @ (((times_times @ nat) @ K3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ nat))))))))))) @ ((suminf @ real) @ (^ [K3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ K3)) @ (((power_power @ real) @ X3) @ (((times_times @ nat) @ K3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))))))). % 6.41/2.69 thf(fact_5406_DERIV__arccos, axiom, (! [X3:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ X3) => ((((ord_less @ real) @ X3) @ (one_one @ real)) => ((((has_field_derivative @ real) @ arccos) @ ((inverse_inverse @ real) @ ((uminus_uminus @ real) @ (sqrt @ (((minus_minus @ real) @ (one_one @ real)) @ (((power_power @ real) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real))))))))). % 6.41/2.69 thf(fact_5407_DERIV__power__series_H, axiom, (! [R:real,F2:(nat > real),X0:real]: (((! [X5:real]: (((((member @ real) @ X5) @ (((set_or5935395276787703475ssThan @ real) @ ((uminus_uminus @ real) @ R)) @ R)) => ((summable @ real) @ (^ [N:nat]: ((((times_times @ real) @ (((times_times @ real) @ (F2 @ N)) @ ((semiring_1_of_nat @ real) @ (suc @ N)))) @ (((power_power @ real) @ X5) @ N)))))))) => ((((member @ real) @ X0) @ (((set_or5935395276787703475ssThan @ real) @ ((uminus_uminus @ real) @ R)) @ R)) => ((((ord_less @ real) @ (zero_zero @ real)) @ R) => ((((has_field_derivative @ real) @ (^ [X:real]: (((suminf @ real) @ (^ [N:nat]: ((((times_times @ real) @ (F2 @ N)) @ (((power_power @ real) @ X) @ (suc @ N))))))))) @ ((suminf @ real) @ (^ [N:nat]: ((((times_times @ real) @ (((times_times @ real) @ (F2 @ N)) @ ((semiring_1_of_nat @ real) @ (suc @ N)))) @ (((power_power @ real) @ X0) @ N)))))) @ (((topolo174197925503356063within @ real) @ X0) @ (top_top @ (set @ real)))))))))). % 6.41/2.69 thf(fact_5408_has__derivative__arcsin, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [G:(A > real),X3:A,G7:(A > real),S:(set @ A)]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ (G @ X3)) => ((((ord_less @ real) @ (G @ X3)) @ (one_one @ real)) => ((((((has_derivative @ A) @ real) @ G) @ G7) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((((has_derivative @ A) @ real) @ (^ [X:A]: ((arcsin @ (G @ X))))) @ (^ [X:A]: ((((times_times @ real) @ (G7 @ X)) @ ((inverse_inverse @ real) @ (sqrt @ (((minus_minus @ real) @ (one_one @ real)) @ (((power_power @ real) @ (G @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))) @ (((topolo174197925503356063within @ A) @ X3) @ S))))))))))). % 6.41/2.69 thf(fact_5409_greaterThanLessThan__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,L:A,U:A]: (((((member @ A) @ I) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) = ((((ord_less @ A) @ L) @ I) & (((ord_less @ A) @ I) @ U))))))))). % 6.41/2.69 thf(fact_5410_greaterThanLessThan__empty__iff2, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A]: ((((bot_bot @ (set @ A)) = (((set_or5935395276787703475ssThan @ A) @ A2) @ B2)) = (((ord_less_eq @ A) @ B2) @ A2)))))))). % 6.41/2.69 thf(fact_5411_greaterThanLessThan__empty__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A]: ((((((set_or5935395276787703475ssThan @ A) @ A2) @ B2) = (bot_bot @ (set @ A))) = (((ord_less_eq @ A) @ B2) @ A2)))))))). % 6.41/2.69 thf(fact_5412_greaterThanLessThan__empty, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,K:A]: (((((ord_less_eq @ A) @ L) @ K) => ((((set_or5935395276787703475ssThan @ A) @ K) @ L) = (bot_bot @ (set @ A)))))))))). % 6.41/2.69 thf(fact_5413_infinite__Ioo__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A]: (((~ (((finite_finite2 @ A) @ (((set_or5935395276787703475ssThan @ A) @ A2) @ B2)))) = (((ord_less @ A) @ A2) @ B2)))))))). % 6.41/2.69 thf(fact_5414_cSup__greaterThanLessThan, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (dense_linorder @ A)) => (! [Y:A,X3:A]: (((((ord_less @ A) @ Y) @ X3) => (((complete_Sup_Sup @ A) @ (((set_or5935395276787703475ssThan @ A) @ Y) @ X3)) = X3)))))))). % 6.41/2.69 thf(fact_5415_Sup__greaterThanLessThan, axiom, (! [A:$tType]: ((((comple6319245703460814977attice @ A) & (dense_linorder @ A)) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (((complete_Sup_Sup @ A) @ (((set_or5935395276787703475ssThan @ A) @ X3) @ Y)) = Y)))))))). % 6.41/2.69 thf(fact_5416_cInf__greaterThanLessThan, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (dense_linorder @ A)) => (! [Y:A,X3:A]: (((((ord_less @ A) @ Y) @ X3) => (((complete_Inf_Inf @ A) @ (((set_or5935395276787703475ssThan @ A) @ Y) @ X3)) = Y)))))))). % 6.41/2.69 thf(fact_5417_Inf__greaterThanLessThan, axiom, (! [A:$tType]: ((((comple6319245703460814977attice @ A) & (dense_linorder @ A)) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (((complete_Inf_Inf @ A) @ (((set_or5935395276787703475ssThan @ A) @ X3) @ Y)) = X3)))))))). % 6.41/2.69 thf(fact_5418_has__derivative__subset, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),F11:(A > B),X3:A,S:(set @ A),T2:(set @ A)]: (((((((has_derivative @ A) @ B) @ F2) @ F11) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => ((((ord_less_eq @ (set @ A)) @ T2) @ S) => (((((has_derivative @ A) @ B) @ F2) @ F11) @ (((topolo174197925503356063within @ A) @ X3) @ T2)))))))))). % 6.41/2.69 thf(fact_5419_has__derivative__scaleR, axiom, (! [C:$tType,D:$tType]: ((((real_V822414075346904944vector @ D) & (real_V822414075346904944vector @ C)) => (! [F2:(D > real),F11:(D > real),X3:D,S:(set @ D),G:(D > C),G7:(D > C)]: (((((((has_derivative @ D) @ real) @ F2) @ F11) @ (((topolo174197925503356063within @ D) @ X3) @ S)) => ((((((has_derivative @ D) @ C) @ G) @ G7) @ (((topolo174197925503356063within @ D) @ X3) @ S)) => (((((has_derivative @ D) @ C) @ (^ [X:D]: ((((real_V8093663219630862766scaleR @ C) @ (F2 @ X)) @ (G @ X))))) @ (^ [H:D]: ((((plus_plus @ C) @ (((real_V8093663219630862766scaleR @ C) @ (F2 @ X3)) @ (G7 @ H))) @ (((real_V8093663219630862766scaleR @ C) @ (F11 @ H)) @ (G @ X3)))))) @ (((topolo174197925503356063within @ D) @ X3) @ S)))))))))). % 6.41/2.69 thf(fact_5420_has__derivative__add, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),F11:(A > B),F5:(filter @ A),G:(A > B),G7:(A > B)]: (((((((has_derivative @ A) @ B) @ F2) @ F11) @ F5) => ((((((has_derivative @ A) @ B) @ G) @ G7) @ F5) => (((((has_derivative @ A) @ B) @ (^ [X:A]: ((((plus_plus @ B) @ (F2 @ X)) @ (G @ X))))) @ (^ [X:A]: ((((plus_plus @ B) @ (F11 @ X)) @ (G7 @ X))))) @ F5))))))))). % 6.41/2.69 thf(fact_5421_has__derivative__const, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [C2:B,F5:(filter @ A)]: ((((((has_derivative @ A) @ B) @ (^ [X:A]: (C2))) @ (^ [X:A]: ((zero_zero @ B)))) @ F5))))))). % 6.41/2.69 thf(fact_5422_infinite__Ioo, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ (((finite_finite2 @ A) @ (((set_or5935395276787703475ssThan @ A) @ A2) @ B2))))))))))). % 6.41/2.69 thf(fact_5423_has__derivative__zero__unique, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F5:(A > B),X3:A]: (((((((has_derivative @ A) @ B) @ (^ [X:A]: ((zero_zero @ B)))) @ F5) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))) => (F5 = (^ [H:A]: ((zero_zero @ B))))))))))). % 6.41/2.69 thf(fact_5424_has__derivative__mult, axiom, (! [A:$tType,D:$tType]: ((((real_V822414075346904944vector @ D) & (real_V4412858255891104859lgebra @ A)) => (! [F2:(D > A),F11:(D > A),X3:D,S:(set @ D),G:(D > A),G7:(D > A)]: (((((((has_derivative @ D) @ A) @ F2) @ F11) @ (((topolo174197925503356063within @ D) @ X3) @ S)) => ((((((has_derivative @ D) @ A) @ G) @ G7) @ (((topolo174197925503356063within @ D) @ X3) @ S)) => (((((has_derivative @ D) @ A) @ (^ [X:D]: ((((times_times @ A) @ (F2 @ X)) @ (G @ X))))) @ (^ [H:D]: ((((plus_plus @ A) @ (((times_times @ A) @ (F2 @ X3)) @ (G7 @ H))) @ (((times_times @ A) @ (F11 @ H)) @ (G @ X3)))))) @ (((topolo174197925503356063within @ D) @ X3) @ S)))))))))). % 6.41/2.70 thf(fact_5425_has__derivative__in__compose2, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((real_V822414075346904944vector @ C) & ((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B))) => (! [T2:(set @ A),G:(A > B),G7:(A > (A > B)),F2:(C > A),S:(set @ C),X3:C,F11:(C > A)]: (((! [X5:A]: (((((member @ A) @ X5) @ T2) => (((((has_derivative @ A) @ B) @ G) @ (G7 @ X5)) @ (((topolo174197925503356063within @ A) @ X5) @ T2))))) => ((((ord_less_eq @ (set @ A)) @ ((((image @ C) @ A) @ F2) @ S)) @ T2) => ((((member @ C) @ X3) @ S) => ((((((has_derivative @ C) @ A) @ F2) @ F11) @ (((topolo174197925503356063within @ C) @ X3) @ S)) => (((((has_derivative @ C) @ B) @ (^ [X:C]: ((G @ (F2 @ X))))) @ (^ [Y2:C]: (((G7 @ (F2 @ X3)) @ (F11 @ Y2))))) @ (((topolo174197925503356063within @ C) @ X3) @ S)))))))))))). % 6.41/2.70 thf(fact_5426_greaterThanLessThan__subseteq__greaterThanLessThan, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ A2) @ B2)) @ (((set_or5935395276787703475ssThan @ A) @ C2) @ D2)) = ((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ A2) & (((ord_less_eq @ A) @ B2) @ D2)))))))))). % 6.41/2.70 thf(fact_5427_ivl__disj__int__two_I4_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M)) @ (((set_or5935395276787703475ssThan @ A) @ M) @ U)) = (bot_bot @ (set @ A))))))))). % 6.41/2.70 thf(fact_5428_ivl__disj__int__two_I5_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ M)) @ (((set_or1337092689740270186AtMost @ A) @ M) @ U)) = (bot_bot @ (set @ A))))))))). % 6.41/2.70 thf(fact_5429_ivl__disj__int__two_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ M)) @ (((set_or7035219750837199246ssThan @ A) @ M) @ U)) = (bot_bot @ (set @ A))))))))). % 6.41/2.70 thf(fact_5430_ivl__disj__int__one_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ ((set_ord_atMost @ A) @ L)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) = (bot_bot @ (set @ A))))))))). % 6.41/2.70 thf(fact_5431_greaterThanLessThan__subseteq__atLeastAtMost__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ A2) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D2)) = ((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ A2) & (((ord_less_eq @ A) @ B2) @ D2)))))))))). % 6.41/2.70 thf(fact_5432_greaterThanLessThan__subseteq__atLeastLessThan__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ A2) @ B2)) @ (((set_or7035219750837199246ssThan @ A) @ C2) @ D2)) = ((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ A2) & (((ord_less_eq @ A) @ B2) @ D2)))))))))). % 6.41/2.70 thf(fact_5433_atLeastAtMost__diff__ends, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((((minus_minus @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) @ (((insert @ A) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))) = (((set_or5935395276787703475ssThan @ A) @ A2) @ B2)))))))). % 6.41/2.70 thf(fact_5434_has__derivative__divide_H, axiom, (! [A:$tType,C:$tType]: ((((real_V822414075346904944vector @ C) & (real_V3459762299906320749_field @ A)) => (! [F2:(C > A),F11:(C > A),X3:C,S3:(set @ C),G:(C > A),G7:(C > A)]: (((((((has_derivative @ C) @ A) @ F2) @ F11) @ (((topolo174197925503356063within @ C) @ X3) @ S3)) => ((((((has_derivative @ C) @ A) @ G) @ G7) @ (((topolo174197925503356063within @ C) @ X3) @ S3)) => (((G @ X3) != (zero_zero @ A)) => (((((has_derivative @ C) @ A) @ (^ [X:C]: ((((divide_divide @ A) @ (F2 @ X)) @ (G @ X))))) @ (^ [H:C]: ((((divide_divide @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ (F11 @ H)) @ (G @ X3))) @ (((times_times @ A) @ (F2 @ X3)) @ (G7 @ H)))) @ (((times_times @ A) @ (G @ X3)) @ (G @ X3)))))) @ (((topolo174197925503356063within @ C) @ X3) @ S3))))))))))). % 6.41/2.70 thf(fact_5435_arccos__less__arccos, axiom, (! [X3:real,Y:real]: (((((ord_less_eq @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ X3) => ((((ord_less @ real) @ X3) @ Y) => ((((ord_less_eq @ real) @ Y) @ (one_one @ real)) => (((ord_less @ real) @ (arccos @ Y)) @ (arccos @ X3)))))))). % 6.41/2.70 thf(fact_5436_arccos__less__mono, axiom, (! [X3:real,Y:real]: (((((ord_less_eq @ real) @ ((abs_abs @ real) @ X3)) @ (one_one @ real)) => ((((ord_less_eq @ real) @ ((abs_abs @ real) @ Y)) @ (one_one @ real)) => ((((ord_less @ real) @ (arccos @ X3)) @ (arccos @ Y)) = (((ord_less @ real) @ Y) @ X3))))))). % 6.41/2.70 thf(fact_5437_has__derivative__inverse, axiom, (! [A:$tType,C:$tType]: ((((real_V822414075346904944vector @ C) & (real_V8999393235501362500lgebra @ A)) => (! [F2:(C > A),X3:C,F11:(C > A),S3:(set @ C)]: ((((F2 @ X3) != (zero_zero @ A)) => ((((((has_derivative @ C) @ A) @ F2) @ F11) @ (((topolo174197925503356063within @ C) @ X3) @ S3)) => (((((has_derivative @ C) @ A) @ (^ [X:C]: (((inverse_inverse @ A) @ (F2 @ X))))) @ (^ [H:C]: (((uminus_uminus @ A) @ (((times_times @ A) @ (((times_times @ A) @ ((inverse_inverse @ A) @ (F2 @ X3))) @ (F11 @ H))) @ ((inverse_inverse @ A) @ (F2 @ X3))))))) @ (((topolo174197925503356063within @ C) @ X3) @ S3)))))))))). % 6.41/2.70 thf(fact_5438_has__derivative__inverse_H, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [X3:A,S3:(set @ A)]: (((X3 != (zero_zero @ A)) => (((((has_derivative @ A) @ A) @ (inverse_inverse @ A)) @ (^ [H:A]: (((uminus_uminus @ A) @ (((times_times @ A) @ (((times_times @ A) @ ((inverse_inverse @ A) @ X3)) @ H)) @ ((inverse_inverse @ A) @ X3)))))) @ (((topolo174197925503356063within @ A) @ X3) @ S3))))))))). % 6.41/2.70 thf(fact_5439_DERIV__isconst3, axiom, (! [A2:real,B2:real,X3:real,Y:real,F2:(real > real)]: (((((ord_less @ real) @ A2) @ B2) => ((((member @ real) @ X3) @ (((set_or5935395276787703475ssThan @ real) @ A2) @ B2)) => ((((member @ real) @ Y) @ (((set_or5935395276787703475ssThan @ real) @ A2) @ B2)) => ((! [X5:real]: (((((member @ real) @ X5) @ (((set_or5935395276787703475ssThan @ real) @ A2) @ B2)) => ((((has_field_derivative @ real) @ F2) @ (zero_zero @ real)) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real))))))) => ((F2 @ X3) = (F2 @ Y))))))))). % 6.41/2.70 thf(fact_5440_arccos__lt__bounded, axiom, (! [Y:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ Y) => ((((ord_less @ real) @ Y) @ (one_one @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ (arccos @ Y)) & (((ord_less @ real) @ (arccos @ Y)) @ pi))))))). % 6.41/2.70 thf(fact_5441_has__derivative__ln, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [G:(A > real),X3:A,G7:(A > real),S:(set @ A)]: (((((ord_less @ real) @ (zero_zero @ real)) @ (G @ X3)) => ((((((has_derivative @ A) @ real) @ G) @ G7) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((((has_derivative @ A) @ real) @ (^ [X:A]: (((ln_ln @ real) @ (G @ X))))) @ (^ [X:A]: ((((times_times @ real) @ (G7 @ X)) @ ((inverse_inverse @ real) @ (G @ X3)))))) @ (((topolo174197925503356063within @ A) @ X3) @ S)))))))))). % 6.41/2.70 thf(fact_5442_sin__arccos__nonzero, axiom, (! [X3:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ X3) => ((((ord_less @ real) @ X3) @ (one_one @ real)) => (((sin @ real) @ (arccos @ X3)) != (zero_zero @ real))))))). % 6.41/2.70 thf(fact_5443_has__derivative__arccos, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [G:(A > real),X3:A,G7:(A > real),S:(set @ A)]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ (G @ X3)) => ((((ord_less @ real) @ (G @ X3)) @ (one_one @ real)) => ((((((has_derivative @ A) @ real) @ G) @ G7) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((((has_derivative @ A) @ real) @ (^ [X:A]: ((arccos @ (G @ X))))) @ (^ [X:A]: ((((times_times @ real) @ (G7 @ X)) @ ((inverse_inverse @ real) @ ((uminus_uminus @ real) @ (sqrt @ (((minus_minus @ real) @ (one_one @ real)) @ (((power_power @ real) @ (G @ X3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))) @ (((topolo174197925503356063within @ A) @ X3) @ S))))))))))). % 6.41/2.70 thf(fact_5444_has__derivative__divide, axiom, (! [A:$tType,C:$tType]: ((((real_V822414075346904944vector @ C) & (real_V8999393235501362500lgebra @ A)) => (! [F2:(C > A),F11:(C > A),X3:C,S3:(set @ C),G:(C > A),G7:(C > A)]: (((((((has_derivative @ C) @ A) @ F2) @ F11) @ (((topolo174197925503356063within @ C) @ X3) @ S3)) => ((((((has_derivative @ C) @ A) @ G) @ G7) @ (((topolo174197925503356063within @ C) @ X3) @ S3)) => (((G @ X3) != (zero_zero @ A)) => (((((has_derivative @ C) @ A) @ (^ [X:C]: ((((divide_divide @ A) @ (F2 @ X)) @ (G @ X))))) @ (^ [H:C]: ((((plus_plus @ A) @ (((times_times @ A) @ ((uminus_uminus @ A) @ (F2 @ X3))) @ (((times_times @ A) @ (((times_times @ A) @ ((inverse_inverse @ A) @ (G @ X3))) @ (G7 @ H))) @ ((inverse_inverse @ A) @ (G @ X3))))) @ (((divide_divide @ A) @ (F11 @ H)) @ (G @ X3)))))) @ (((topolo174197925503356063within @ C) @ X3) @ S3))))))))))). % 6.41/2.70 thf(fact_5445_has__derivative__prod, axiom, (! [B:$tType,I7:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V3459762299906320749_field @ B)) => (! [I6:(set @ I7),F2:(I7 > (A > B)),F11:(I7 > (A > B)),X3:A,S3:(set @ A)]: (((! [I4:I7]: (((((member @ I7) @ I4) @ I6) => (((((has_derivative @ A) @ B) @ (F2 @ I4)) @ (F11 @ I4)) @ (((topolo174197925503356063within @ A) @ X3) @ S3))))) => (((((has_derivative @ A) @ B) @ (^ [X:A]: (((((groups7121269368397514597t_prod @ I7) @ B) @ (^ [I3:I7]: (((F2 @ I3) @ X)))) @ I6)))) @ (^ [Y2:A]: (((((groups7311177749621191930dd_sum @ I7) @ B) @ (^ [I3:I7]: ((((times_times @ B) @ ((F11 @ I3) @ Y2)) @ ((((groups7121269368397514597t_prod @ I7) @ B) @ (^ [J3:I7]: (((F2 @ J3) @ X3)))) @ (((minus_minus @ (set @ I7)) @ I6) @ (((insert @ I7) @ I3) @ (bot_bot @ (set @ I7))))))))) @ I6)))) @ (((topolo174197925503356063within @ A) @ X3) @ S3))))))))). % 6.41/2.70 thf(fact_5446_has__derivative__powr, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [G:(A > real),G7:(A > real),X3:A,X8:(set @ A),F2:(A > real),F11:(A > real)]: (((((((has_derivative @ A) @ real) @ G) @ G7) @ (((topolo174197925503356063within @ A) @ X3) @ X8)) => ((((((has_derivative @ A) @ real) @ F2) @ F11) @ (((topolo174197925503356063within @ A) @ X3) @ X8)) => ((((ord_less @ real) @ (zero_zero @ real)) @ (G @ X3)) => ((((member @ A) @ X3) @ X8) => (((((has_derivative @ A) @ real) @ (^ [X:A]: ((((powr @ real) @ (G @ X)) @ (F2 @ X))))) @ (^ [H:A]: ((((times_times @ real) @ (((powr @ real) @ (G @ X3)) @ (F2 @ X3))) @ (((plus_plus @ real) @ (((times_times @ real) @ (F11 @ H)) @ ((ln_ln @ real) @ (G @ X3)))) @ (((divide_divide @ real) @ (((times_times @ real) @ (G7 @ H)) @ (F2 @ X3))) @ (G @ X3))))))) @ (((topolo174197925503356063within @ A) @ X3) @ X8)))))))))))). % 6.41/2.70 thf(fact_5447_has__derivative__real__sqrt, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [G:(A > real),X3:A,G7:(A > real),S:(set @ A)]: (((((ord_less @ real) @ (zero_zero @ real)) @ (G @ X3)) => ((((((has_derivative @ A) @ real) @ G) @ G7) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((((has_derivative @ A) @ real) @ (^ [X:A]: ((sqrt @ (G @ X))))) @ (^ [X:A]: ((((times_times @ real) @ (G7 @ X)) @ (((divide_divide @ real) @ ((inverse_inverse @ real) @ (sqrt @ (G @ X3)))) @ ((numeral_numeral @ real) @ (bit0 @ one2))))))) @ (((topolo174197925503356063within @ A) @ X3) @ S)))))))))). % 6.41/2.70 thf(fact_5448_termdiffs__aux, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [C2:(nat > A),K5:A,X3:A]: ((((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (((diffs @ A) @ ((diffs @ A) @ C2)) @ N)) @ (((power_power @ A) @ K5) @ N))))) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ ((real_V7770717601297561774m_norm @ A) @ K5)) => (((((filterlim @ A) @ A) @ (^ [H:A]: (((suminf @ A) @ (^ [N:nat]: ((((times_times @ A) @ (C2 @ N)) @ (((minus_minus @ A) @ (((divide_divide @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ (((plus_plus @ A) @ X3) @ H)) @ N)) @ (((power_power @ A) @ X3) @ N))) @ H)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (((power_power @ A) @ X3) @ (((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat))))))))))))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A)))))))))))). % 6.41/2.70 thf(fact_5449_isCont__powser, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [C2:(nat > A),K5:A,X3:A]: ((((summable @ A) @ (^ [N:nat]: ((((times_times @ A) @ (C2 @ N)) @ (((power_power @ A) @ K5) @ N))))) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ ((real_V7770717601297561774m_norm @ A) @ K5)) => ((((topolo3448309680560233919inuous @ A) @ A) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))) @ (^ [X:A]: (((suminf @ A) @ (^ [N:nat]: ((((times_times @ A) @ (C2 @ N)) @ (((power_power @ A) @ X) @ N)))))))))))))))). % 6.41/2.70 thf(fact_5450_isCont__powser_H, axiom, (! [Aa:$tType,A:$tType]: ((((topological_t2_space @ A) & ((real_Vector_banach @ Aa) & (real_V3459762299906320749_field @ Aa))) => (! [A2:A,F2:(A > Aa),C2:(nat > Aa),K5:Aa]: ((((((topolo3448309680560233919inuous @ A) @ Aa) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ F2) => (((summable @ Aa) @ (^ [N:nat]: ((((times_times @ Aa) @ (C2 @ N)) @ (((power_power @ Aa) @ K5) @ N))))) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ Aa) @ (F2 @ A2))) @ ((real_V7770717601297561774m_norm @ Aa) @ K5)) => ((((topolo3448309680560233919inuous @ A) @ Aa) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ (^ [X:A]: (((suminf @ Aa) @ (^ [N:nat]: ((((times_times @ Aa) @ (C2 @ N)) @ (((power_power @ Aa) @ (F2 @ X)) @ N))))))))))))))))). % 6.41/2.70 thf(fact_5451_finite__greaterThanLessThan, axiom, (! [L:nat,U:nat]: (((finite_finite2 @ nat) @ (((set_or5935395276787703475ssThan @ nat) @ L) @ U))))). % 6.41/2.70 thf(fact_5452_finite__greaterThanLessThan__int, axiom, (! [L:int,U:int]: (((finite_finite2 @ int) @ (((set_or5935395276787703475ssThan @ int) @ L) @ U))))). % 6.41/2.70 thf(fact_5453_card__greaterThanLessThan, axiom, (! [L:nat,U:nat]: ((((finite_card @ nat) @ (((set_or5935395276787703475ssThan @ nat) @ L) @ U)) = (((minus_minus @ nat) @ U) @ (suc @ L)))))). % 6.41/2.70 thf(fact_5454_tendsto__mult__left__iff, axiom, (! [A:$tType,B:$tType]: ((((field @ A) & (topolo4211221413907600880p_mult @ A)) => (! [C2:A,F2:(B > A),L:A,F5:(filter @ B)]: (((C2 != (zero_zero @ A)) => ((((((filterlim @ B) @ A) @ (^ [X:B]: ((((times_times @ A) @ C2) @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ (((times_times @ A) @ C2) @ L))) @ F5) = (((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ F5))))))))). % 6.41/2.70 thf(fact_5455_tendsto__mult__right__iff, axiom, (! [A:$tType,B:$tType]: ((((field @ A) & (topolo4211221413907600880p_mult @ A)) => (! [C2:A,F2:(B > A),L:A,F5:(filter @ B)]: (((C2 != (zero_zero @ A)) => ((((((filterlim @ B) @ A) @ (^ [X:B]: ((((times_times @ A) @ (F2 @ X)) @ C2)))) @ ((topolo7230453075368039082e_nhds @ A) @ (((times_times @ A) @ L) @ C2))) @ F5) = (((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ F5))))))))). % 6.41/2.70 thf(fact_5456_power__tendsto__0__iff, axiom, (! [A:$tType,N2:nat,F2:(A > real),F5:(filter @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((((filterlim @ A) @ real) @ (^ [X:A]: ((((power_power @ real) @ (F2 @ X)) @ N2)))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ F5) = (((((filterlim @ A) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ F5)))))). % 6.41/2.70 thf(fact_5457_isCont__Pair, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((topological_t2_space @ A) & ((topolo4958980785337419405_space @ B) & (topolo4958980785337419405_space @ C))) => (! [A2:A,F2:(A > B),G:(A > C)]: ((((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ F2) => (((((topolo3448309680560233919inuous @ A) @ C) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ G) => ((((topolo3448309680560233919inuous @ A) @ ((product_prod @ B) @ C)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ (^ [X:A]: (((((product_Pair @ B) @ C) @ (F2 @ X)) @ (G @ X))))))))))))). % 6.41/2.70 thf(fact_5458_isCont__LIM__compose2, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & ((topolo4958980785337419405_space @ B) & (topolo4958980785337419405_space @ C))) => (! [A2:A,F2:(A > B),G:(B > C),L:C]: ((((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ F2) => ((((((filterlim @ B) @ C) @ G) @ ((topolo7230453075368039082e_nhds @ C) @ L)) @ (((topolo174197925503356063within @ B) @ (F2 @ A2)) @ (top_top @ (set @ B)))) => ((? [D4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D4) & (! [X5:A]: ((((X5 != A2) & (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ X5) @ A2))) @ D4)) => ((F2 @ X5) != (F2 @ A2)))))))) => (((((filterlim @ A) @ C) @ (^ [X:A]: ((G @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ C) @ L)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))))))))))))). % 6.41/2.70 thf(fact_5459_isCont__iff, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (topolo4958980785337419405_space @ B)) => (! [X3:A,F2:(A > B)]: ((((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))) @ F2) = (((((filterlim @ A) @ B) @ (^ [H:A]: ((F2 @ (((plus_plus @ A) @ X3) @ H))))) @ ((topolo7230453075368039082e_nhds @ B) @ (F2 @ X3))) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5460_LIM__not__zero, axiom, (! [Aa:$tType,A:$tType]: ((((topolo8386298272705272623_space @ A) & ((zero @ Aa) & (topological_t2_space @ Aa))) => (! [K:Aa,A2:A]: (((K != (zero_zero @ Aa)) => (~ ((((((filterlim @ A) @ Aa) @ (^ [X:A]: (K))) @ ((topolo7230453075368039082e_nhds @ Aa) @ (zero_zero @ Aa))) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))))))))))))). % 6.41/2.70 thf(fact_5461_tendsto__within__subset, axiom, (! [B:$tType,A:$tType]: (((topolo4958980785337419405_space @ A) => (! [F2:(A > B),L:(filter @ B),X3:A,S3:(set @ A),T4:(set @ A)]: (((((((filterlim @ A) @ B) @ F2) @ L) @ (((topolo174197925503356063within @ A) @ X3) @ S3)) => ((((ord_less_eq @ (set @ A)) @ T4) @ S3) => (((((filterlim @ A) @ B) @ F2) @ L) @ (((topolo174197925503356063within @ A) @ X3) @ T4)))))))))). % 6.41/2.70 thf(fact_5462_tendsto__arcosh, axiom, (! [B:$tType,F2:(B > real),A2:real,F5:(filter @ B)]: (((((((filterlim @ B) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ A2)) @ F5) => ((((ord_less @ real) @ (one_one @ real)) @ A2) => (((((filterlim @ B) @ real) @ (^ [X:B]: (((arcosh @ real) @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ real) @ ((arcosh @ real) @ A2))) @ F5)))))). % 6.41/2.70 thf(fact_5463_tendsto__const__iff, axiom, (! [B:$tType,A:$tType]: (((topological_t2_space @ A) => (! [F5:(filter @ B),A2:A,B2:A]: (((F5 != (bot_bot @ (filter @ B))) => ((((((filterlim @ B) @ A) @ (^ [X:B]: (A2))) @ ((topolo7230453075368039082e_nhds @ A) @ B2)) @ F5) = (A2 = B2))))))))). % 6.41/2.70 thf(fact_5464_tendsto__unique, axiom, (! [B:$tType,A:$tType]: (((topological_t2_space @ A) => (! [F5:(filter @ B),F2:(B > A),A2:A,B2:A]: (((F5 != (bot_bot @ (filter @ B))) => ((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ F5) => ((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ B2)) @ F5) => (A2 = B2)))))))))). % 6.41/2.70 thf(fact_5465_tendsto__bot, axiom, (! [A:$tType,B:$tType]: (((topolo4958980785337419405_space @ A) => (! [F2:(B > A),A2:A]: ((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ (bot_bot @ (filter @ B))))))))). % 6.41/2.70 thf(fact_5466_nhds__neq__bot, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [A2:A]: ((((topolo7230453075368039082e_nhds @ A) @ A2) != (bot_bot @ (filter @ A))))))))). % 6.41/2.70 thf(fact_5467_continuous__bot, axiom, (! [B:$tType,A:$tType]: ((((topological_t2_space @ A) & (topolo4958980785337419405_space @ B)) => (! [F2:(A > B)]: (((((topolo3448309680560233919inuous @ A) @ B) @ (bot_bot @ (filter @ A))) @ F2))))))). % 6.41/2.70 thf(fact_5468_continuous__trivial__limit, axiom, (! [B:$tType,A:$tType]: ((((topological_t2_space @ A) & (topolo4958980785337419405_space @ B)) => (! [Net:(filter @ A),F2:(A > B)]: (((Net = (bot_bot @ (filter @ A))) => ((((topolo3448309680560233919inuous @ A) @ B) @ Net) @ F2)))))))). % 6.41/2.70 thf(fact_5469_LIM__offset__zero__cancel, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (topolo4958980785337419405_space @ B)) => (! [F2:(A > B),A2:A,L5:B]: (((((((filterlim @ A) @ B) @ (^ [H:A]: ((F2 @ (((plus_plus @ A) @ A2) @ H))))) @ ((topolo7230453075368039082e_nhds @ B) @ L5)) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A)))) => (((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L5)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5470_LIM__offset__zero, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (topolo4958980785337419405_space @ B)) => (! [F2:(A > B),L5:B,A2:A]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L5)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) => (((((filterlim @ A) @ B) @ (^ [H:A]: ((F2 @ (((plus_plus @ A) @ A2) @ H))))) @ ((topolo7230453075368039082e_nhds @ B) @ L5)) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5471_LIM__isCont__iff, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (topolo4958980785337419405_space @ B)) => (! [F2:(A > B),A2:A]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ (F2 @ A2))) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) = (((((filterlim @ A) @ B) @ (^ [H:A]: ((F2 @ (((plus_plus @ A) @ A2) @ H))))) @ ((topolo7230453075368039082e_nhds @ B) @ (F2 @ A2))) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5472_tendsto__null__power, axiom, (! [B:$tType,A:$tType]: (((real_V2822296259951069270ebra_1 @ B) => (! [F2:(A > B),F5:(filter @ A),N2:nat]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ F5) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((((filterlim @ A) @ B) @ (^ [X:A]: ((((power_power @ B) @ (F2 @ X)) @ N2)))) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ F5))))))))). % 6.41/2.70 thf(fact_5473_LIM__offset, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (topolo4958980785337419405_space @ B)) => (! [F2:(A > B),L5:B,A2:A,K:A]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L5)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) => (((((filterlim @ A) @ B) @ (^ [X:A]: ((F2 @ (((plus_plus @ A) @ X) @ K))))) @ ((topolo7230453075368039082e_nhds @ B) @ L5)) @ (((topolo174197925503356063within @ A) @ (((minus_minus @ A) @ A2) @ K)) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5474_tendsto__Pair, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((topolo4958980785337419405_space @ B) & (topolo4958980785337419405_space @ C)) => (! [F2:(A > B),A2:B,F5:(filter @ A),G:(A > C),B2:C]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ A2)) @ F5) => ((((((filterlim @ A) @ C) @ G) @ ((topolo7230453075368039082e_nhds @ C) @ B2)) @ F5) => (((((filterlim @ A) @ ((product_prod @ B) @ C)) @ (^ [X:A]: (((((product_Pair @ B) @ C) @ (F2 @ X)) @ (G @ X))))) @ ((topolo7230453075368039082e_nhds @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ A2) @ B2))) @ F5))))))))). % 6.41/2.70 thf(fact_5475_continuous__Pair, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((topological_t2_space @ A) & ((topolo4958980785337419405_space @ B) & (topolo4958980785337419405_space @ C))) => (! [F5:(filter @ A),F2:(A > B),G:(A > C)]: ((((((topolo3448309680560233919inuous @ A) @ B) @ F5) @ F2) => (((((topolo3448309680560233919inuous @ A) @ C) @ F5) @ G) => ((((topolo3448309680560233919inuous @ A) @ ((product_prod @ B) @ C)) @ F5) @ (^ [X:A]: (((((product_Pair @ B) @ C) @ (F2 @ X)) @ (G @ X))))))))))))). % 6.41/2.70 thf(fact_5476_Lim__transform__eq, axiom, (! [A:$tType,B:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(B > A),G:(B > A),F5:(filter @ B),A2:A]: (((((((filterlim @ B) @ A) @ (^ [X:B]: ((((minus_minus @ A) @ (F2 @ X)) @ (G @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ F5) => ((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ F5) = (((((filterlim @ B) @ A) @ G) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ F5))))))))). % 6.41/2.70 thf(fact_5477_LIM__zero__cancel, axiom, (! [B:$tType,A:$tType]: (((real_V822414075346904944vector @ B) => (! [F2:(A > B),L:B,F5:(filter @ A)]: (((((((filterlim @ A) @ B) @ (^ [X:A]: ((((minus_minus @ B) @ (F2 @ X)) @ L)))) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ F5) => (((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L)) @ F5)))))))). % 6.41/2.70 thf(fact_5478_Lim__transform2, axiom, (! [A:$tType,B:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(B > A),A2:A,F5:(filter @ B),G:(B > A)]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ F5) => ((((((filterlim @ B) @ A) @ (^ [X:B]: ((((minus_minus @ A) @ (F2 @ X)) @ (G @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ F5) => (((((filterlim @ B) @ A) @ G) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ F5))))))))). % 6.41/2.70 thf(fact_5479_Lim__transform, axiom, (! [A:$tType,B:$tType]: (((real_V822414075346904944vector @ A) => (! [G:(B > A),A2:A,F5:(filter @ B),F2:(B > A)]: (((((((filterlim @ B) @ A) @ G) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ F5) => ((((((filterlim @ B) @ A) @ (^ [X:B]: ((((minus_minus @ A) @ (F2 @ X)) @ (G @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ F5) => (((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ F5))))))))). % 6.41/2.70 thf(fact_5480_LIM__zero__iff, axiom, (! [B:$tType,A:$tType]: (((real_V822414075346904944vector @ B) => (! [F2:(A > B),L:B,F5:(filter @ A)]: (((((((filterlim @ A) @ B) @ (^ [X:A]: ((((minus_minus @ B) @ (F2 @ X)) @ L)))) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ F5) = (((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L)) @ F5)))))))). % 6.41/2.70 thf(fact_5481_LIM__zero, axiom, (! [B:$tType,A:$tType]: (((real_V822414075346904944vector @ B) => (! [F2:(A > B),L:B,F5:(filter @ A)]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L)) @ F5) => (((((filterlim @ A) @ B) @ (^ [X:A]: ((((minus_minus @ B) @ (F2 @ X)) @ L)))) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ F5)))))))). % 6.41/2.70 thf(fact_5482_tendsto__mult__zero, axiom, (! [A:$tType,D:$tType]: (((real_V4412858255891104859lgebra @ A) => (! [F2:(D > A),F5:(filter @ D),G:(D > A)]: (((((((filterlim @ D) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ F5) => ((((((filterlim @ D) @ A) @ G) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ F5) => (((((filterlim @ D) @ A) @ (^ [X:D]: ((((times_times @ A) @ (F2 @ X)) @ (G @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ F5))))))))). % 6.41/2.70 thf(fact_5483_tendsto__mult__left__zero, axiom, (! [A:$tType,D:$tType]: (((real_V4412858255891104859lgebra @ A) => (! [F2:(D > A),F5:(filter @ D),C2:A]: (((((((filterlim @ D) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ F5) => (((((filterlim @ D) @ A) @ (^ [X:D]: ((((times_times @ A) @ (F2 @ X)) @ C2)))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ F5)))))))). % 6.41/2.70 thf(fact_5484_tendsto__mult__right__zero, axiom, (! [A:$tType,D:$tType]: (((real_V4412858255891104859lgebra @ A) => (! [F2:(D > A),F5:(filter @ D),C2:A]: (((((((filterlim @ D) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ F5) => (((((filterlim @ D) @ A) @ (^ [X:D]: ((((times_times @ A) @ C2) @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ F5)))))))). % 6.41/2.70 thf(fact_5485_tendsto__add__zero, axiom, (! [B:$tType,D:$tType]: (((topolo6943815403480290642id_add @ B) => (! [F2:(D > B),F5:(filter @ D),G:(D > B)]: (((((((filterlim @ D) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ F5) => ((((((filterlim @ D) @ B) @ G) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ F5) => (((((filterlim @ D) @ B) @ (^ [X:D]: ((((plus_plus @ B) @ (F2 @ X)) @ (G @ X))))) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ F5))))))))). % 6.41/2.70 thf(fact_5486_tendsto__inverse, axiom, (! [A:$tType,B:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [F2:(B > A),A2:A,F5:(filter @ B)]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ F5) => ((A2 != (zero_zero @ A)) => (((((filterlim @ B) @ A) @ (^ [X:B]: (((inverse_inverse @ A) @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ ((inverse_inverse @ A) @ A2))) @ F5))))))))). % 6.41/2.70 thf(fact_5487_tendsto__norm__zero, axiom, (! [B:$tType,A:$tType]: (((real_V822414075346904944vector @ B) => (! [F2:(A > B),F5:(filter @ A)]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ F5) => (((((filterlim @ A) @ real) @ (^ [X:A]: (((real_V7770717601297561774m_norm @ B) @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ F5)))))))). % 6.41/2.70 thf(fact_5488_tendsto__norm__zero__iff, axiom, (! [B:$tType,A:$tType]: (((real_V822414075346904944vector @ B) => (! [F2:(A > B),F5:(filter @ A)]: (((((((filterlim @ A) @ real) @ (^ [X:A]: (((real_V7770717601297561774m_norm @ B) @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ F5) = (((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ F5)))))))). % 6.41/2.70 thf(fact_5489_tendsto__norm__zero__cancel, axiom, (! [B:$tType,A:$tType]: (((real_V822414075346904944vector @ B) => (! [F2:(A > B),F5:(filter @ A)]: (((((((filterlim @ A) @ real) @ (^ [X:A]: (((real_V7770717601297561774m_norm @ B) @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ F5) => (((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ F5)))))))). % 6.41/2.70 thf(fact_5490_tendsto__sgn, axiom, (! [A:$tType,B:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(B > A),L:A,F5:(filter @ B)]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ F5) => ((L != (zero_zero @ A)) => (((((filterlim @ B) @ A) @ (^ [X:B]: (((sgn_sgn @ A) @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ ((sgn_sgn @ A) @ L))) @ F5))))))))). % 6.41/2.70 thf(fact_5491_tendsto__divide__zero, axiom, (! [A:$tType,B:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(B > A),F5:(filter @ B),C2:A]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ F5) => (((((filterlim @ B) @ A) @ (^ [X:B]: ((((divide_divide @ A) @ (F2 @ X)) @ C2)))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ F5)))))))). % 6.41/2.70 thf(fact_5492_tendsto__divide, axiom, (! [A:$tType,B:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(B > A),A2:A,F5:(filter @ B),G:(B > A),B2:A]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ F5) => ((((((filterlim @ B) @ A) @ G) @ ((topolo7230453075368039082e_nhds @ A) @ B2)) @ F5) => ((B2 != (zero_zero @ A)) => (((((filterlim @ B) @ A) @ (^ [X:B]: ((((divide_divide @ A) @ (F2 @ X)) @ (G @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ (((divide_divide @ A) @ A2) @ B2))) @ F5)))))))))). % 6.41/2.70 thf(fact_5493_continuous__add, axiom, (! [B:$tType,D:$tType]: ((((topological_t2_space @ D) & (topolo6943815403480290642id_add @ B)) => (! [F5:(filter @ D),F2:(D > B),G:(D > B)]: ((((((topolo3448309680560233919inuous @ D) @ B) @ F5) @ F2) => (((((topolo3448309680560233919inuous @ D) @ B) @ F5) @ G) => ((((topolo3448309680560233919inuous @ D) @ B) @ F5) @ (^ [X:D]: ((((plus_plus @ B) @ (F2 @ X)) @ (G @ X))))))))))))). % 6.41/2.70 thf(fact_5494_tendsto__add, axiom, (! [A:$tType,B:$tType]: (((topolo6943815403480290642id_add @ A) => (! [F2:(B > A),A2:A,F5:(filter @ B),G:(B > A),B2:A]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ F5) => ((((((filterlim @ B) @ A) @ G) @ ((topolo7230453075368039082e_nhds @ A) @ B2)) @ F5) => (((((filterlim @ B) @ A) @ (^ [X:B]: ((((plus_plus @ A) @ (F2 @ X)) @ (G @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ (((plus_plus @ A) @ A2) @ B2))) @ F5))))))))). % 6.41/2.70 thf(fact_5495_tendsto__add__const__iff, axiom, (! [A:$tType,B:$tType]: (((topolo1633459387980952147up_add @ A) => (! [C2:A,F2:(B > A),D2:A,F5:(filter @ B)]: (((((((filterlim @ B) @ A) @ (^ [X:B]: ((((plus_plus @ A) @ C2) @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ (((plus_plus @ A) @ C2) @ D2))) @ F5) = (((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ D2)) @ F5)))))))). % 6.41/2.70 thf(fact_5496_tendsto__tan, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [F2:(A > A),A2:A,F5:(filter @ A)]: (((((((filterlim @ A) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ F5) => ((((cos @ A) @ A2) != (zero_zero @ A)) => (((((filterlim @ A) @ A) @ (^ [X:A]: (((tan @ A) @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ ((tan @ A) @ A2))) @ F5))))))))). % 6.41/2.70 thf(fact_5497_tendsto__null__sum, axiom, (! [C:$tType,B:$tType,A:$tType]: (((topolo5987344860129210374id_add @ C) => (! [I6:(set @ B),F2:(A > (B > C)),F5:(filter @ A)]: (((! [I4:B]: (((((member @ B) @ I4) @ I6) => (((((filterlim @ A) @ C) @ (^ [X:A]: (((F2 @ X) @ I4)))) @ ((topolo7230453075368039082e_nhds @ C) @ (zero_zero @ C))) @ F5)))) => (((((filterlim @ A) @ C) @ (^ [I3:A]: (((((groups7311177749621191930dd_sum @ B) @ C) @ (F2 @ I3)) @ I6)))) @ ((topolo7230453075368039082e_nhds @ C) @ (zero_zero @ C))) @ F5)))))))). % 6.41/2.70 thf(fact_5498_tendsto__tanh, axiom, (! [A:$tType,C:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [F2:(C > A),A2:A,F5:(filter @ C)]: (((((((filterlim @ C) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ F5) => ((((cosh @ A) @ A2) != (zero_zero @ A)) => (((((filterlim @ C) @ A) @ (^ [X:C]: (((tanh @ A) @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ ((tanh @ A) @ A2))) @ F5))))))))). % 6.41/2.70 thf(fact_5499_tendsto__cot, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [F2:(A > A),A2:A,F5:(filter @ A)]: (((((((filterlim @ A) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ F5) => ((((sin @ A) @ A2) != (zero_zero @ A)) => (((((filterlim @ A) @ A) @ (^ [X:A]: (((cot @ A) @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ ((cot @ A) @ A2))) @ F5))))))))). % 6.41/2.70 thf(fact_5500_tendsto__log, axiom, (! [A:$tType,F2:(A > real),A2:real,F5:(filter @ A),G:(A > real),B2:real]: (((((((filterlim @ A) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ A2)) @ F5) => ((((((filterlim @ A) @ real) @ G) @ ((topolo7230453075368039082e_nhds @ real) @ B2)) @ F5) => ((((ord_less @ real) @ (zero_zero @ real)) @ A2) => ((A2 != (one_one @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ B2) => (((((filterlim @ A) @ real) @ (^ [X:A]: (((log @ (F2 @ X)) @ (G @ X))))) @ ((topolo7230453075368039082e_nhds @ real) @ ((log @ A2) @ B2))) @ F5))))))))). % 6.41/2.70 thf(fact_5501_atLeastSucLessThan__greaterThanLessThan, axiom, (! [L:nat,U:nat]: (((((set_or7035219750837199246ssThan @ nat) @ (suc @ L)) @ U) = (((set_or5935395276787703475ssThan @ nat) @ L) @ U))))). % 6.41/2.70 thf(fact_5502_tendsto__artanh, axiom, (! [A:$tType,F2:(A > real),A2:real,F5:(filter @ A)]: (((((((filterlim @ A) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ A2)) @ F5) => ((((ord_less @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ A2) => ((((ord_less @ real) @ A2) @ (one_one @ real)) => (((((filterlim @ A) @ real) @ (^ [X:A]: (((artanh @ real) @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ real) @ ((artanh @ real) @ A2))) @ F5))))))). % 6.41/2.70 thf(fact_5503_LIM__offset__zero__iff, axiom, (! [C:$tType,D:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & ((topolo4958980785337419405_space @ D) & (zero @ C))) => (! [A2:A,F2:(A > D),L5:D]: ((((((nO_MATCH @ C) @ A) @ (zero_zero @ C)) @ A2) => ((((((filterlim @ A) @ D) @ F2) @ ((topolo7230453075368039082e_nhds @ D) @ L5)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) = (((((filterlim @ A) @ D) @ (^ [H:A]: ((F2 @ (((plus_plus @ A) @ A2) @ H))))) @ ((topolo7230453075368039082e_nhds @ D) @ L5)) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A)))))))))))). % 6.41/2.70 thf(fact_5504_IVT, axiom, (! [A:$tType,B:$tType]: ((((topolo1944317154257567458pology @ B) & (topolo8458572112393995274pology @ A)) => (! [F2:(A > B),A2:A,Y:B,B2:A]: (((((ord_less_eq @ B) @ (F2 @ A2)) @ Y) => ((((ord_less_eq @ B) @ Y) @ (F2 @ B2)) => ((((ord_less_eq @ A) @ A2) @ B2) => ((! [X5:A]: ((((((ord_less_eq @ A) @ A2) @ X5) & (((ord_less_eq @ A) @ X5) @ B2)) => ((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ X5) @ (top_top @ (set @ A)))) @ F2)))) => (? [X5:A]: (((((ord_less_eq @ A) @ A2) @ X5) & ((((ord_less_eq @ A) @ X5) @ B2) & ((F2 @ X5) = Y))))))))))))))). % 6.41/2.70 thf(fact_5505_IVT2, axiom, (! [A:$tType,B:$tType]: ((((topolo1944317154257567458pology @ B) & (topolo8458572112393995274pology @ A)) => (! [F2:(A > B),B2:A,Y:B,A2:A]: (((((ord_less_eq @ B) @ (F2 @ B2)) @ Y) => ((((ord_less_eq @ B) @ Y) @ (F2 @ A2)) => ((((ord_less_eq @ A) @ A2) @ B2) => ((! [X5:A]: ((((((ord_less_eq @ A) @ A2) @ X5) & (((ord_less_eq @ A) @ X5) @ B2)) => ((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ X5) @ (top_top @ (set @ A)))) @ F2)))) => (? [X5:A]: (((((ord_less_eq @ A) @ A2) @ X5) & ((((ord_less_eq @ A) @ X5) @ B2) & ((F2 @ X5) = Y))))))))))))))). % 6.41/2.70 thf(fact_5506_LIM__equal2, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (topolo4958980785337419405_space @ B)) => (! [R:real,A2:A,F2:(A > B),G:(A > B),L:B]: (((((ord_less @ real) @ (zero_zero @ real)) @ R) => ((! [X5:A]: (((X5 != A2) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ X5) @ A2))) @ R) => ((F2 @ X5) = (G @ X5)))))) => ((((((filterlim @ A) @ B) @ G) @ ((topolo7230453075368039082e_nhds @ B) @ L)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) => (((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))))))))))))). % 6.41/2.70 thf(fact_5507_LIM__eq, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),L5:B,A2:A]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L5)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) = (! [R5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R5) => (? [S8:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ S8) & (! [X:A]: ((((X != A2) & (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ X) @ A2))) @ S8)) => (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ B) @ (((minus_minus @ B) @ (F2 @ X)) @ L5))) @ R5))))))))))))))))). % 6.41/2.70 thf(fact_5508_LIM__I, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [A2:A,F2:(A > B),L5:B]: (((! [R4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R4) => (? [S9:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ S9) & (! [X5:A]: ((((X5 != A2) & (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ X5) @ A2))) @ S9)) => (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ B) @ (((minus_minus @ B) @ (F2 @ X5)) @ L5))) @ R4)))))))))) => (((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L5)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5509_LIM__D, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),L5:B,A2:A,R2:real]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L5)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) => ((((ord_less @ real) @ (zero_zero @ real)) @ R2) => (? [S2:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ S2) & (! [X2:A]: ((((X2 != A2) & (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ X2) @ A2))) @ S2)) => (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ B) @ (((minus_minus @ B) @ (F2 @ X2)) @ L5))) @ R2))))))))))))))). % 6.41/2.70 thf(fact_5510_DERIV__LIM__iff, axiom, (! [A:$tType]: ((((inverse @ A) & (real_V822414075346904944vector @ A)) => (! [F2:(A > A),A2:A,D6:A]: (((((((filterlim @ A) @ A) @ (^ [H:A]: ((((divide_divide @ A) @ (((minus_minus @ A) @ (F2 @ (((plus_plus @ A) @ A2) @ H))) @ (F2 @ A2))) @ H)))) @ ((topolo7230453075368039082e_nhds @ A) @ D6)) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A)))) = (((((filterlim @ A) @ A) @ (^ [X:A]: ((((divide_divide @ A) @ (((minus_minus @ A) @ (F2 @ X)) @ (F2 @ A2))) @ (((minus_minus @ A) @ X) @ A2))))) @ ((topolo7230453075368039082e_nhds @ A) @ D6)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5511_LIM__fun__less__zero, axiom, (! [F2:(real > real),L:real,C2:real]: (((((((filterlim @ real) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ L)) @ (((topolo174197925503356063within @ real) @ C2) @ (top_top @ (set @ real)))) => ((((ord_less @ real) @ L) @ (zero_zero @ real)) => (? [R4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R4) & (! [X2:real]: ((((X2 != C2) & (((ord_less @ real) @ ((abs_abs @ real) @ (((minus_minus @ real) @ C2) @ X2))) @ R4)) => (((ord_less @ real) @ (F2 @ X2)) @ (zero_zero @ real))))))))))))). % 6.41/2.70 thf(fact_5512_LIM__fun__not__zero, axiom, (! [F2:(real > real),L:real,C2:real]: (((((((filterlim @ real) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ L)) @ (((topolo174197925503356063within @ real) @ C2) @ (top_top @ (set @ real)))) => ((L != (zero_zero @ real)) => (? [R4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R4) & (! [X2:real]: ((((X2 != C2) & (((ord_less @ real) @ ((abs_abs @ real) @ (((minus_minus @ real) @ C2) @ X2))) @ R4)) => ((F2 @ X2) != (zero_zero @ real))))))))))))). % 6.41/2.70 thf(fact_5513_LIM__fun__gt__zero, axiom, (! [F2:(real > real),L:real,C2:real]: (((((((filterlim @ real) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ L)) @ (((topolo174197925503356063within @ real) @ C2) @ (top_top @ (set @ real)))) => ((((ord_less @ real) @ (zero_zero @ real)) @ L) => (? [R4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R4) & (! [X2:real]: ((((X2 != C2) & (((ord_less @ real) @ ((abs_abs @ real) @ (((minus_minus @ real) @ C2) @ X2))) @ R4)) => (((ord_less @ real) @ (zero_zero @ real)) @ (F2 @ X2))))))))))))). % 6.41/2.70 thf(fact_5514_LIM__compose2, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & ((topolo4958980785337419405_space @ B) & (topolo4958980785337419405_space @ C))) => (! [F2:(A > B),B2:B,A2:A,G:(B > C),C2:C]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ B2)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) => ((((((filterlim @ B) @ C) @ G) @ ((topolo7230453075368039082e_nhds @ C) @ C2)) @ (((topolo174197925503356063within @ B) @ B2) @ (top_top @ (set @ B)))) => ((? [D4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D4) & (! [X5:A]: ((((X5 != A2) & (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ X5) @ A2))) @ D4)) => ((F2 @ X5) != B2))))))) => (((((filterlim @ A) @ C) @ (^ [X:A]: ((G @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ C) @ C2)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))))))))))))). % 6.41/2.70 thf(fact_5515_continuous__at__within__divide, axiom, (! [B:$tType,A:$tType]: ((((topological_t2_space @ A) & (real_V3459762299906320749_field @ B)) => (! [A2:A,S:(set @ A),F2:(A > B),G:(A > B)]: ((((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ S)) @ F2) => (((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ S)) @ G) => (((G @ A2) != (zero_zero @ B)) => ((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ S)) @ (^ [X:A]: ((((divide_divide @ B) @ (F2 @ X)) @ (G @ X)))))))))))))). % 6.41/2.70 thf(fact_5516_isCont__add, axiom, (! [B:$tType,A:$tType]: ((((topological_t2_space @ A) & (topolo6943815403480290642id_add @ B)) => (! [A2:A,F2:(A > B),G:(A > B)]: ((((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ F2) => (((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ G) => ((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ (^ [X:A]: ((((plus_plus @ B) @ (F2 @ X)) @ (G @ X))))))))))))). % 6.41/2.70 thf(fact_5517_continuous__at__within__inverse, axiom, (! [B:$tType,A:$tType]: ((((topological_t2_space @ A) & (real_V8999393235501362500lgebra @ B)) => (! [A2:A,S:(set @ A),F2:(A > B)]: ((((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ S)) @ F2) => (((F2 @ A2) != (zero_zero @ B)) => ((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ S)) @ (^ [X:A]: (((inverse_inverse @ B) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5518_continuous__at__within__sgn, axiom, (! [B:$tType,A:$tType]: ((((topological_t2_space @ A) & (real_V822414075346904944vector @ B)) => (! [A2:A,S:(set @ A),F2:(A > B)]: ((((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ S)) @ F2) => (((F2 @ A2) != (zero_zero @ B)) => ((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ S)) @ (^ [X:A]: (((sgn_sgn @ B) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5519_DERIV__D, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),D6:A,X3:A]: ((((((has_field_derivative @ A) @ F2) @ D6) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))) => (((((filterlim @ A) @ A) @ (^ [H:A]: ((((divide_divide @ A) @ (((minus_minus @ A) @ (F2 @ (((plus_plus @ A) @ X3) @ H))) @ (F2 @ X3))) @ H)))) @ ((topolo7230453075368039082e_nhds @ A) @ D6)) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5520_DERIV__def, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),D6:A,X3:A]: ((((((has_field_derivative @ A) @ F2) @ D6) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))) = (((((filterlim @ A) @ A) @ (^ [H:A]: ((((divide_divide @ A) @ (((minus_minus @ A) @ (F2 @ (((plus_plus @ A) @ X3) @ H))) @ (F2 @ X3))) @ H)))) @ ((topolo7230453075368039082e_nhds @ A) @ D6)) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5521_lim__exp__minus__1, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (((((filterlim @ A) @ A) @ (^ [Z6:A]: ((((divide_divide @ A) @ (((minus_minus @ A) @ ((exp @ A) @ Z6)) @ (one_one @ A))) @ Z6)))) @ ((topolo7230453075368039082e_nhds @ A) @ (one_one @ A))) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A)))))))). % 6.41/2.70 thf(fact_5522_lemma__termdiff4, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [K:real,F2:(A > B),K5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ K) => ((! [H6:A]: (((H6 != (zero_zero @ A)) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ H6)) @ K) => (((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ B) @ (F2 @ H6))) @ (((times_times @ real) @ K5) @ ((real_V7770717601297561774m_norm @ A) @ H6))))))) => (((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A)))))))))))). % 6.41/2.70 thf(fact_5523_isCont__eq__Lb, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [A2:real,B2:real,F2:(real > A)]: (((((ord_less_eq @ real) @ A2) @ B2) => ((! [X5:real]: ((((((ord_less_eq @ real) @ A2) @ X5) & (((ord_less_eq @ real) @ X5) @ B2)) => ((((topolo3448309680560233919inuous @ real) @ A) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))) @ F2)))) => (? [M8:A]: (((! [X2:real]: ((((((ord_less_eq @ real) @ A2) @ X2) & (((ord_less_eq @ real) @ X2) @ B2)) => (((ord_less_eq @ A) @ M8) @ (F2 @ X2))))) & (? [X5:real]: (((((ord_less_eq @ real) @ A2) @ X5) & ((((ord_less_eq @ real) @ X5) @ B2) & ((F2 @ X5) = M8)))))))))))))))). % 6.41/2.70 thf(fact_5524_isCont__eq__Ub, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [A2:real,B2:real,F2:(real > A)]: (((((ord_less_eq @ real) @ A2) @ B2) => ((! [X5:real]: ((((((ord_less_eq @ real) @ A2) @ X5) & (((ord_less_eq @ real) @ X5) @ B2)) => ((((topolo3448309680560233919inuous @ real) @ A) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))) @ F2)))) => (? [M8:A]: (((! [X2:real]: ((((((ord_less_eq @ real) @ A2) @ X2) & (((ord_less_eq @ real) @ X2) @ B2)) => (((ord_less_eq @ A) @ (F2 @ X2)) @ M8)))) & (? [X5:real]: (((((ord_less_eq @ real) @ A2) @ X5) & ((((ord_less_eq @ real) @ X5) @ B2) & ((F2 @ X5) = M8)))))))))))))))). % 6.41/2.70 thf(fact_5525_isCont__bounded, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [A2:real,B2:real,F2:(real > A)]: (((((ord_less_eq @ real) @ A2) @ B2) => ((! [X5:real]: ((((((ord_less_eq @ real) @ A2) @ X5) & (((ord_less_eq @ real) @ X5) @ B2)) => ((((topolo3448309680560233919inuous @ real) @ A) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))) @ F2)))) => (? [M8:A]: ((! [X2:real]: ((((((ord_less_eq @ real) @ A2) @ X2) & (((ord_less_eq @ real) @ X2) @ B2)) => (((ord_less_eq @ A) @ (F2 @ X2)) @ M8)))))))))))))). % 6.41/2.70 thf(fact_5526_isCont__inverse__function2, axiom, (! [A2:real,X3:real,B2:real,G:(real > real),F2:(real > real)]: (((((ord_less @ real) @ A2) @ X3) => ((((ord_less @ real) @ X3) @ B2) => ((! [Z3:real]: (((((ord_less_eq @ real) @ A2) @ Z3) => ((((ord_less_eq @ real) @ Z3) @ B2) => ((G @ (F2 @ Z3)) = Z3))))) => ((! [Z3:real]: (((((ord_less_eq @ real) @ A2) @ Z3) => ((((ord_less_eq @ real) @ Z3) @ B2) => ((((topolo3448309680560233919inuous @ real) @ real) @ (((topolo174197925503356063within @ real) @ Z3) @ (top_top @ (set @ real)))) @ F2))))) => ((((topolo3448309680560233919inuous @ real) @ real) @ (((topolo174197925503356063within @ real) @ (F2 @ X3)) @ (top_top @ (set @ real)))) @ G)))))))). % 6.41/2.70 thf(fact_5527_field__has__derivative__at, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),D6:A,X3:A]: (((((((has_derivative @ A) @ A) @ F2) @ ((times_times @ A) @ D6)) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))) = (((((filterlim @ A) @ A) @ (^ [H:A]: ((((divide_divide @ A) @ (((minus_minus @ A) @ (F2 @ (((plus_plus @ A) @ X3) @ H))) @ (F2 @ X3))) @ H)))) @ ((topolo7230453075368039082e_nhds @ A) @ D6)) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5528_isCont__divide, axiom, (! [B:$tType,A:$tType]: ((((topological_t2_space @ A) & (real_V3459762299906320749_field @ B)) => (! [A2:A,F2:(A > B),G:(A > B)]: ((((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ F2) => (((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ G) => (((G @ A2) != (zero_zero @ B)) => ((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ (^ [X:A]: ((((divide_divide @ B) @ (F2 @ X)) @ (G @ X)))))))))))))). % 6.41/2.70 thf(fact_5529_isCont__sgn, axiom, (! [B:$tType,A:$tType]: ((((topological_t2_space @ A) & (real_V822414075346904944vector @ B)) => (! [A2:A,F2:(A > B)]: ((((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ F2) => (((F2 @ A2) != (zero_zero @ B)) => ((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ (^ [X:A]: (((sgn_sgn @ B) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5530_filterlim__at__to__0, axiom, (! [B:$tType,A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(A > B),F5:(filter @ B),A2:A]: (((((((filterlim @ A) @ B) @ F2) @ F5) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) = (((((filterlim @ A) @ B) @ (^ [X:A]: ((F2 @ (((plus_plus @ A) @ X) @ A2))))) @ F5) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5531_continuous__within__tan, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,S:(set @ A),F2:(A > A)]: ((((((topolo3448309680560233919inuous @ A) @ A) @ (((topolo174197925503356063within @ A) @ X3) @ S)) @ F2) => ((((cos @ A) @ (F2 @ X3)) != (zero_zero @ A)) => ((((topolo3448309680560233919inuous @ A) @ A) @ (((topolo174197925503356063within @ A) @ X3) @ S)) @ (^ [X:A]: (((tan @ A) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5532_nth__sorted__list__of__set__greaterThanLessThan, axiom, (! [N2:nat,J:nat,I:nat]: (((((ord_less @ nat) @ N2) @ (((minus_minus @ nat) @ J) @ (suc @ I))) => ((((nth @ nat) @ ((linord4507533701916653071of_set @ nat) @ (((set_or5935395276787703475ssThan @ nat) @ I) @ J))) @ N2) = (suc @ (((plus_plus @ nat) @ I) @ N2))))))). % 6.41/2.70 thf(fact_5533_continuous__within__cot, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A,S:(set @ A),F2:(A > A)]: ((((((topolo3448309680560233919inuous @ A) @ A) @ (((topolo174197925503356063within @ A) @ X3) @ S)) @ F2) => ((((sin @ A) @ (F2 @ X3)) != (zero_zero @ A)) => ((((topolo3448309680560233919inuous @ A) @ A) @ (((topolo174197925503356063within @ A) @ X3) @ S)) @ (^ [X:A]: (((cot @ A) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5534_continuous__at__within__tanh, axiom, (! [A:$tType,C:$tType]: ((((topological_t2_space @ C) & ((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A))) => (! [X3:C,A3:(set @ C),F2:(C > A)]: ((((((topolo3448309680560233919inuous @ C) @ A) @ (((topolo174197925503356063within @ C) @ X3) @ A3)) @ F2) => ((((cosh @ A) @ (F2 @ X3)) != (zero_zero @ A)) => ((((topolo3448309680560233919inuous @ C) @ A) @ (((topolo174197925503356063within @ C) @ X3) @ A3)) @ (^ [X:C]: (((tanh @ A) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5535_isCont__has__Ub, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [A2:real,B2:real,F2:(real > A)]: (((((ord_less_eq @ real) @ A2) @ B2) => ((! [X5:real]: ((((((ord_less_eq @ real) @ A2) @ X5) & (((ord_less_eq @ real) @ X5) @ B2)) => ((((topolo3448309680560233919inuous @ real) @ A) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))) @ F2)))) => (? [M8:A]: (((! [X2:real]: ((((((ord_less_eq @ real) @ A2) @ X2) & (((ord_less_eq @ real) @ X2) @ B2)) => (((ord_less_eq @ A) @ (F2 @ X2)) @ M8)))) & (! [N8:A]: (((((ord_less @ A) @ N8) @ M8) => (? [X5:real]: (((((ord_less_eq @ real) @ A2) @ X5) & ((((ord_less_eq @ real) @ X5) @ B2) & (((ord_less @ A) @ N8) @ (F2 @ X5)))))))))))))))))))). % 6.41/2.70 thf(fact_5536_isCont__tan, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((cos @ A) @ X3) != (zero_zero @ A)) => ((((topolo3448309680560233919inuous @ A) @ A) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))) @ (tan @ A))))))))). % 6.41/2.70 thf(fact_5537_filterlim__shift__iff, axiom, (! [B:$tType,A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(A > B),D2:A,F5:(filter @ B),A2:A]: (((((((filterlim @ A) @ B) @ (((((comp @ A) @ B) @ A) @ F2) @ ((plus_plus @ A) @ D2))) @ F5) @ (((topolo174197925503356063within @ A) @ (((minus_minus @ A) @ A2) @ D2)) @ (top_top @ (set @ A)))) = (((((filterlim @ A) @ B) @ F2) @ F5) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5538_filterlim__shift, axiom, (! [B:$tType,A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(A > B),F5:(filter @ B),A2:A,D2:A]: (((((((filterlim @ A) @ B) @ F2) @ F5) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) => (((((filterlim @ A) @ B) @ (((((comp @ A) @ B) @ A) @ F2) @ ((plus_plus @ A) @ D2))) @ F5) @ (((topolo174197925503356063within @ A) @ (((minus_minus @ A) @ A2) @ D2)) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5539_isCont__cot, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((sin @ A) @ X3) != (zero_zero @ A)) => ((((topolo3448309680560233919inuous @ A) @ A) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))) @ (cot @ A))))))))). % 6.41/2.70 thf(fact_5540_isCont__tanh, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((cosh @ A) @ X3) != (zero_zero @ A)) => ((((topolo3448309680560233919inuous @ A) @ A) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))) @ (tanh @ A))))))))). % 6.41/2.70 thf(fact_5541_powser__limit__0, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [S:real,A2:(nat > A),F2:(A > A)]: (((((ord_less @ real) @ (zero_zero @ real)) @ S) => ((! [X5:A]: (((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X5)) @ S) => (((sums @ A) @ (^ [N:nat]: ((((times_times @ A) @ (A2 @ N)) @ (((power_power @ A) @ X5) @ N))))) @ (F2 @ X5))))) => (((((filterlim @ A) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ (A2 @ (zero_zero @ nat)))) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A)))))))))))). % 6.41/2.70 thf(fact_5542_powser__limit__0__strong, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [S:real,A2:(nat > A),F2:(A > A)]: (((((ord_less @ real) @ (zero_zero @ real)) @ S) => ((! [X5:A]: (((X5 != (zero_zero @ A)) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X5)) @ S) => (((sums @ A) @ (^ [N:nat]: ((((times_times @ A) @ (A2 @ N)) @ (((power_power @ A) @ X5) @ N))))) @ (F2 @ X5)))))) => (((((filterlim @ A) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ (A2 @ (zero_zero @ nat)))) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A)))))))))))). % 6.41/2.70 thf(fact_5543_lemma__termdiff5, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_Vector_banach @ B)) => (! [K:real,F2:(nat > real),G:(A > (nat > B))]: (((((ord_less @ real) @ (zero_zero @ real)) @ K) => (((summable @ real) @ F2) => ((! [H6:A,N3:nat]: (((H6 != (zero_zero @ A)) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ H6)) @ K) => (((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ B) @ ((G @ H6) @ N3))) @ (((times_times @ real) @ (F2 @ N3)) @ ((real_V7770717601297561774m_norm @ A) @ H6))))))) => (((((filterlim @ A) @ B) @ (^ [H:A]: (((suminf @ B) @ (G @ H))))) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A))))))))))))). % 6.41/2.70 thf(fact_5544_isCont__tan_H, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [A2:A,F2:(A > A)]: ((((((topolo3448309680560233919inuous @ A) @ A) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ F2) => ((((cos @ A) @ (F2 @ A2)) != (zero_zero @ A)) => ((((topolo3448309680560233919inuous @ A) @ A) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ (^ [X:A]: (((tan @ A) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5545_isCont__arcosh, axiom, (! [X3:real]: (((((ord_less @ real) @ (one_one @ real)) @ X3) => ((((topolo3448309680560233919inuous @ real) @ real) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) @ (arcosh @ real)))))). % 6.41/2.70 thf(fact_5546_continuous__at__within__log, axiom, (! [A:$tType]: (((topological_t2_space @ A) => (! [A2:A,S:(set @ A),F2:(A > real),G:(A > real)]: ((((((topolo3448309680560233919inuous @ A) @ real) @ (((topolo174197925503356063within @ A) @ A2) @ S)) @ F2) => (((((topolo3448309680560233919inuous @ A) @ real) @ (((topolo174197925503356063within @ A) @ A2) @ S)) @ G) => ((((ord_less @ real) @ (zero_zero @ real)) @ (F2 @ A2)) => (((F2 @ A2) != (one_one @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ (G @ A2)) => ((((topolo3448309680560233919inuous @ A) @ real) @ (((topolo174197925503356063within @ A) @ A2) @ S)) @ (^ [X:A]: (((log @ (F2 @ X)) @ (G @ X)))))))))))))))). % 6.41/2.70 thf(fact_5547_isCont__cot_H, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [A2:A,F2:(A > A)]: ((((((topolo3448309680560233919inuous @ A) @ A) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ F2) => ((((sin @ A) @ (F2 @ A2)) != (zero_zero @ A)) => ((((topolo3448309680560233919inuous @ A) @ A) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ (^ [X:A]: (((cot @ A) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5548_DERIV__inverse__function, axiom, (! [F2:(real > real),D6:real,G:(real > real),X3:real,A2:real,B2:real]: ((((((has_field_derivative @ real) @ F2) @ D6) @ (((topolo174197925503356063within @ real) @ (G @ X3)) @ (top_top @ (set @ real)))) => ((D6 != (zero_zero @ real)) => ((((ord_less @ real) @ A2) @ X3) => ((((ord_less @ real) @ X3) @ B2) => ((! [Y4:real]: (((((ord_less @ real) @ A2) @ Y4) => ((((ord_less @ real) @ Y4) @ B2) => ((F2 @ (G @ Y4)) = Y4))))) => (((((topolo3448309680560233919inuous @ real) @ real) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) @ G) => ((((has_field_derivative @ real) @ G) @ ((inverse_inverse @ real) @ D6)) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real))))))))))))). % 6.41/2.70 thf(fact_5549_isCont__arccos, axiom, (! [X3:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ X3) => ((((ord_less @ real) @ X3) @ (one_one @ real)) => ((((topolo3448309680560233919inuous @ real) @ real) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) @ arccos)))))). % 6.41/2.70 thf(fact_5550_isCont__arcsin, axiom, (! [X3:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ X3) => ((((ord_less @ real) @ X3) @ (one_one @ real)) => ((((topolo3448309680560233919inuous @ real) @ real) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) @ arcsin)))))). % 6.41/2.70 thf(fact_5551_LIM__less__bound, axiom, (! [B2:real,X3:real,F2:(real > real)]: (((((ord_less @ real) @ B2) @ X3) => ((! [X5:real]: (((((member @ real) @ X5) @ (((set_or5935395276787703475ssThan @ real) @ B2) @ X3)) => (((ord_less_eq @ real) @ (zero_zero @ real)) @ (F2 @ X5))))) => (((((topolo3448309680560233919inuous @ real) @ real) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) @ F2) => (((ord_less_eq @ real) @ (zero_zero @ real)) @ (F2 @ X3)))))))). % 6.41/2.70 thf(fact_5552_isCont__log, axiom, (! [A:$tType]: (((topological_t2_space @ A) => (! [A2:A,F2:(A > real),G:(A > real)]: ((((((topolo3448309680560233919inuous @ A) @ real) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ F2) => (((((topolo3448309680560233919inuous @ A) @ real) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ G) => ((((ord_less @ real) @ (zero_zero @ real)) @ (F2 @ A2)) => (((F2 @ A2) != (one_one @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ (G @ A2)) => ((((topolo3448309680560233919inuous @ A) @ real) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ (^ [X:A]: (((log @ (F2 @ X)) @ (G @ X)))))))))))))))). % 6.41/2.70 thf(fact_5553_isCont__artanh, axiom, (! [X3:real]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ X3) => ((((ord_less @ real) @ X3) @ (one_one @ real)) => ((((topolo3448309680560233919inuous @ real) @ real) @ (((topolo174197925503356063within @ real) @ X3) @ (top_top @ (set @ real)))) @ (artanh @ real))))))). % 6.41/2.70 thf(fact_5554_isCont__inverse__function, axiom, (! [D2:real,X3:real,G:(real > real),F2:(real > real)]: (((((ord_less @ real) @ (zero_zero @ real)) @ D2) => ((! [Z3:real]: (((((ord_less_eq @ real) @ ((abs_abs @ real) @ (((minus_minus @ real) @ Z3) @ X3))) @ D2) => ((G @ (F2 @ Z3)) = Z3)))) => ((! [Z3:real]: (((((ord_less_eq @ real) @ ((abs_abs @ real) @ (((minus_minus @ real) @ Z3) @ X3))) @ D2) => ((((topolo3448309680560233919inuous @ real) @ real) @ (((topolo174197925503356063within @ real) @ Z3) @ (top_top @ (set @ real)))) @ F2)))) => ((((topolo3448309680560233919inuous @ real) @ real) @ (((topolo174197925503356063within @ real) @ (F2 @ X3)) @ (top_top @ (set @ real)))) @ G))))))). % 6.41/2.70 thf(fact_5555_GMVT_H, axiom, (! [A2:real,B2:real,F2:(real > real),G:(real > real),G7:(real > real),F11:(real > real)]: (((((ord_less @ real) @ A2) @ B2) => ((! [Z3:real]: (((((ord_less_eq @ real) @ A2) @ Z3) => ((((ord_less_eq @ real) @ Z3) @ B2) => ((((topolo3448309680560233919inuous @ real) @ real) @ (((topolo174197925503356063within @ real) @ Z3) @ (top_top @ (set @ real)))) @ F2))))) => ((! [Z3:real]: (((((ord_less_eq @ real) @ A2) @ Z3) => ((((ord_less_eq @ real) @ Z3) @ B2) => ((((topolo3448309680560233919inuous @ real) @ real) @ (((topolo174197925503356063within @ real) @ Z3) @ (top_top @ (set @ real)))) @ G))))) => ((! [Z3:real]: (((((ord_less @ real) @ A2) @ Z3) => ((((ord_less @ real) @ Z3) @ B2) => ((((has_field_derivative @ real) @ G) @ (G7 @ Z3)) @ (((topolo174197925503356063within @ real) @ Z3) @ (top_top @ (set @ real)))))))) => ((! [Z3:real]: (((((ord_less @ real) @ A2) @ Z3) => ((((ord_less @ real) @ Z3) @ B2) => ((((has_field_derivative @ real) @ F2) @ (F11 @ Z3)) @ (((topolo174197925503356063within @ real) @ Z3) @ (top_top @ (set @ real)))))))) => (? [C3:real]: (((((ord_less @ real) @ A2) @ C3) & ((((ord_less @ real) @ C3) @ B2) & ((((times_times @ real) @ (((minus_minus @ real) @ (F2 @ B2)) @ (F2 @ A2))) @ (G7 @ C3)) = (((times_times @ real) @ (((minus_minus @ real) @ (G @ B2)) @ (G @ A2))) @ (F11 @ C3))))))))))))))). % 6.41/2.70 thf(fact_5556_summable__Leibniz_I2_J, axiom, (! [A2:(nat > real)]: (((((((filterlim @ nat) @ real) @ A2) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)) => (((topological_monoseq @ real) @ A2) => ((((ord_less @ real) @ (zero_zero @ real)) @ (A2 @ (zero_zero @ nat))) => (! [N6:nat]: ((((member @ real) @ ((suminf @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3)))))) @ (((set_or1337092689740270186AtMost @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))) @ ((set_ord_lessThan @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N6)))) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))) @ ((set_ord_lessThan @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N6)) @ (one_one @ nat)))))))))))))). % 6.41/2.70 thf(fact_5557_summable__Leibniz_I3_J, axiom, (! [A2:(nat > real)]: (((((((filterlim @ nat) @ real) @ A2) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)) => (((topological_monoseq @ real) @ A2) => ((((ord_less @ real) @ (A2 @ (zero_zero @ nat))) @ (zero_zero @ real)) => (! [N6:nat]: ((((member @ real) @ ((suminf @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3)))))) @ (((set_or1337092689740270186AtMost @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))) @ ((set_ord_lessThan @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N6)) @ (one_one @ nat))))) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))) @ ((set_ord_lessThan @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N6))))))))))))). % 6.41/2.70 thf(fact_5558_summable__Leibniz_H_I5_J, axiom, (! [A2:(nat > real)]: (((((((filterlim @ nat) @ real) @ A2) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)) => ((! [N3:nat]: ((((ord_less_eq @ real) @ (zero_zero @ real)) @ (A2 @ N3)))) => ((! [N3:nat]: ((((ord_less_eq @ real) @ (A2 @ (suc @ N3))) @ (A2 @ N3)))) => (((((filterlim @ nat) @ real) @ (^ [N:nat]: (((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))) @ ((set_ord_lessThan @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ (one_one @ nat))))))) @ ((topolo7230453075368039082e_nhds @ real) @ ((suminf @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))))) @ (at_top @ nat)))))))). % 6.41/2.70 thf(fact_5559_trivial__limit__sequentially, axiom, ((at_top @ nat) != (bot_bot @ (filter @ nat)))). % 6.41/2.70 thf(fact_5560_tendsto__zero__mult__left__iff, axiom, (! [A:$tType]: ((((field @ A) & (topolo4211221413907600880p_mult @ A)) => (! [C2:A,A2:(nat > A)]: (((C2 != (zero_zero @ A)) => ((((((filterlim @ nat) @ A) @ (^ [N:nat]: ((((times_times @ A) @ C2) @ (A2 @ N))))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_top @ nat)) = (((((filterlim @ nat) @ A) @ A2) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_top @ nat)))))))))). % 6.41/2.70 thf(fact_5561_tendsto__zero__mult__right__iff, axiom, (! [A:$tType]: ((((field @ A) & (topolo4211221413907600880p_mult @ A)) => (! [C2:A,A2:(nat > A)]: (((C2 != (zero_zero @ A)) => ((((((filterlim @ nat) @ A) @ (^ [N:nat]: ((((times_times @ A) @ (A2 @ N)) @ C2)))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_top @ nat)) = (((((filterlim @ nat) @ A) @ A2) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_top @ nat)))))))))). % 6.41/2.70 thf(fact_5562_tendsto__zero__divide__iff, axiom, (! [A:$tType]: ((((field @ A) & (topolo4211221413907600880p_mult @ A)) => (! [C2:A,A2:(nat > A)]: (((C2 != (zero_zero @ A)) => ((((((filterlim @ nat) @ A) @ (^ [N:nat]: ((((divide_divide @ A) @ (A2 @ N)) @ C2)))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_top @ nat)) = (((((filterlim @ nat) @ A) @ A2) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_top @ nat)))))))))). % 6.41/2.70 thf(fact_5563_filterlim__Suc, axiom, (((((filterlim @ nat) @ nat) @ suc) @ (at_top @ nat)) @ (at_top @ nat))). % 6.41/2.70 thf(fact_5564_filterlim__sequentially__Suc, axiom, (! [A:$tType,F2:(nat > A),F5:(filter @ A)]: (((((((filterlim @ nat) @ A) @ (^ [X:nat]: ((F2 @ (suc @ X))))) @ F5) @ (at_top @ nat)) = (((((filterlim @ nat) @ A) @ F2) @ F5) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5565_trivial__limit__at__top__linorder, axiom, (! [A:$tType]: (((linorder @ A) => ((at_top @ A) != (bot_bot @ (filter @ A))))))). % 6.41/2.70 thf(fact_5566_approx__from__above__dense__linorder, axiom, (! [A:$tType]: ((((dense_linorder @ A) & ((topolo3112930676232923870pology @ A) & (topolo1944317154257567458pology @ A))) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (? [U4:(nat > A)]: (((! [N6:nat]: ((((ord_less @ A) @ X3) @ (U4 @ N6)))) & (((((filterlim @ nat) @ A) @ U4) @ ((topolo7230453075368039082e_nhds @ A) @ X3)) @ (at_top @ nat)))))))))))). % 6.41/2.70 thf(fact_5567_approx__from__below__dense__linorder, axiom, (! [A:$tType]: ((((dense_linorder @ A) & ((topolo3112930676232923870pology @ A) & (topolo1944317154257567458pology @ A))) => (! [Y:A,X3:A]: (((((ord_less @ A) @ Y) @ X3) => (? [U4:(nat > A)]: (((! [N6:nat]: ((((ord_less @ A) @ (U4 @ N6)) @ X3))) & (((((filterlim @ nat) @ A) @ U4) @ ((topolo7230453075368039082e_nhds @ A) @ X3)) @ (at_top @ nat)))))))))))). % 6.41/2.70 thf(fact_5568_LIMSEQ__Suc, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [F2:(nat > A),L:A]: (((((((filterlim @ nat) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ (at_top @ nat)) => (((((filterlim @ nat) @ A) @ (^ [N:nat]: ((F2 @ (suc @ N))))) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5569_LIMSEQ__imp__Suc, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [F2:(nat > A),L:A]: (((((((filterlim @ nat) @ A) @ (^ [N:nat]: ((F2 @ (suc @ N))))) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ (at_top @ nat)) => (((((filterlim @ nat) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5570_LIMSEQ__offset, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [F2:(nat > A),K:nat,A2:A]: (((((((filterlim @ nat) @ A) @ (^ [N:nat]: ((F2 @ (((plus_plus @ nat) @ N) @ K))))) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ (at_top @ nat)) => (((((filterlim @ nat) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5571_LIMSEQ__ignore__initial__segment, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [F2:(nat > A),A2:A,K:nat]: (((((((filterlim @ nat) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ (at_top @ nat)) => (((((filterlim @ nat) @ A) @ (^ [N:nat]: ((F2 @ (((plus_plus @ nat) @ N) @ K))))) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5572_lim__mono, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [N7:nat,X8:(nat > A),Y7:(nat > A),X3:A,Y:A]: (((! [N3:nat]: (((((ord_less_eq @ nat) @ N7) @ N3) => (((ord_less_eq @ A) @ (X8 @ N3)) @ (Y7 @ N3))))) => ((((((filterlim @ nat) @ A) @ X8) @ ((topolo7230453075368039082e_nhds @ A) @ X3)) @ (at_top @ nat)) => ((((((filterlim @ nat) @ A) @ Y7) @ ((topolo7230453075368039082e_nhds @ A) @ Y)) @ (at_top @ nat)) => (((ord_less_eq @ A) @ X3) @ Y)))))))))). % 6.41/2.70 thf(fact_5573_LIMSEQ__le, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [X8:(nat > A),X3:A,Y7:(nat > A),Y:A]: (((((((filterlim @ nat) @ A) @ X8) @ ((topolo7230453075368039082e_nhds @ A) @ X3)) @ (at_top @ nat)) => ((((((filterlim @ nat) @ A) @ Y7) @ ((topolo7230453075368039082e_nhds @ A) @ Y)) @ (at_top @ nat)) => ((? [N8:nat]: ((! [N3:nat]: (((((ord_less_eq @ nat) @ N8) @ N3) => (((ord_less_eq @ A) @ (X8 @ N3)) @ (Y7 @ N3))))))) => (((ord_less_eq @ A) @ X3) @ Y)))))))))). % 6.41/2.70 thf(fact_5574_Lim__bounded, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [F2:(nat > A),L:A,M7:nat,C5:A]: (((((((filterlim @ nat) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ (at_top @ nat)) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ M7) @ N3) => (((ord_less_eq @ A) @ (F2 @ N3)) @ C5)))) => (((ord_less_eq @ A) @ L) @ C5))))))))). % 6.41/2.70 thf(fact_5575_Lim__bounded2, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [F2:(nat > A),L:A,N7:nat,C5:A]: (((((((filterlim @ nat) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ (at_top @ nat)) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ N7) @ N3) => (((ord_less_eq @ A) @ C5) @ (F2 @ N3))))) => (((ord_less_eq @ A) @ C5) @ L))))))))). % 6.41/2.70 thf(fact_5576_LIMSEQ__le__const, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [X8:(nat > A),X3:A,A2:A]: (((((((filterlim @ nat) @ A) @ X8) @ ((topolo7230453075368039082e_nhds @ A) @ X3)) @ (at_top @ nat)) => ((? [N8:nat]: ((! [N3:nat]: (((((ord_less_eq @ nat) @ N8) @ N3) => (((ord_less_eq @ A) @ A2) @ (X8 @ N3))))))) => (((ord_less_eq @ A) @ A2) @ X3))))))))). % 6.41/2.70 thf(fact_5577_LIMSEQ__le__const2, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [X8:(nat > A),X3:A,A2:A]: (((((((filterlim @ nat) @ A) @ X8) @ ((topolo7230453075368039082e_nhds @ A) @ X3)) @ (at_top @ nat)) => ((? [N8:nat]: ((! [N3:nat]: (((((ord_less_eq @ nat) @ N8) @ N3) => (((ord_less_eq @ A) @ (X8 @ N3)) @ A2)))))) => (((ord_less_eq @ A) @ X3) @ A2))))))))). % 6.41/2.70 thf(fact_5578_Sup__lim, axiom, (! [A:$tType]: ((((comple5582772986160207858norder @ A) & (topolo1944317154257567458pology @ A)) => (! [B2:(nat > A),S:(set @ A),A2:A]: (((! [N3:nat]: ((((member @ A) @ (B2 @ N3)) @ S))) => ((((((filterlim @ nat) @ A) @ B2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ (at_top @ nat)) => (((ord_less_eq @ A) @ A2) @ ((complete_Sup_Sup @ A) @ S)))))))))). % 6.41/2.70 thf(fact_5579_Inf__lim, axiom, (! [A:$tType]: ((((comple5582772986160207858norder @ A) & (topolo1944317154257567458pology @ A)) => (! [B2:(nat > A),S:(set @ A),A2:A]: (((! [N3:nat]: ((((member @ A) @ (B2 @ N3)) @ S))) => ((((((filterlim @ nat) @ A) @ B2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ (at_top @ nat)) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ S)) @ A2))))))))). % 6.41/2.70 thf(fact_5580_Inf__as__limit, axiom, (! [A:$tType]: ((((comple5582772986160207858norder @ A) & ((topolo3112930676232923870pology @ A) & (topolo1944317154257567458pology @ A))) => (! [A3:(set @ A)]: (((A3 != (bot_bot @ (set @ A))) => (? [U4:(nat > A)]: (((! [N6:nat]: ((((member @ A) @ (U4 @ N6)) @ A3))) & (((((filterlim @ nat) @ A) @ U4) @ ((topolo7230453075368039082e_nhds @ A) @ ((complete_Inf_Inf @ A) @ A3))) @ (at_top @ nat)))))))))))). % 6.41/2.70 thf(fact_5581_summable__LIMSEQ__zero, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A)]: ((((summable @ A) @ F2) => (((((filterlim @ nat) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5582_mult__nat__left__at__top, axiom, (! [C2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ C2) => (((((filterlim @ nat) @ nat) @ ((times_times @ nat) @ C2)) @ (at_top @ nat)) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5583_mult__nat__right__at__top, axiom, (! [C2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ C2) => (((((filterlim @ nat) @ nat) @ (^ [X:nat]: ((((times_times @ nat) @ X) @ C2)))) @ (at_top @ nat)) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5584_monoseq__le, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [A2:(nat > A),X3:A]: ((((topological_monoseq @ A) @ A2) => ((((((filterlim @ nat) @ A) @ A2) @ ((topolo7230453075368039082e_nhds @ A) @ X3)) @ (at_top @ nat)) => (((! [N6:nat]: ((((ord_less_eq @ A) @ (A2 @ N6)) @ X3))) & (! [M6:nat,N6:nat]: (((((ord_less_eq @ nat) @ M6) @ N6) => (((ord_less_eq @ A) @ (A2 @ M6)) @ (A2 @ N6)))))) | ((! [N6:nat]: ((((ord_less_eq @ A) @ X3) @ (A2 @ N6)))) & (! [M6:nat,N6:nat]: (((((ord_less_eq @ nat) @ M6) @ N6) => (((ord_less_eq @ A) @ (A2 @ N6)) @ (A2 @ M6))))))))))))))). % 6.41/2.70 thf(fact_5585_lim__const__over__n, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [A2:A]: ((((((filterlim @ nat) @ A) @ (^ [N:nat]: ((((divide_divide @ A) @ A2) @ ((semiring_1_of_nat @ A) @ N))))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_top @ nat)))))))). % 6.41/2.70 thf(fact_5586_lim__inverse__n, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (((((filterlim @ nat) @ A) @ (^ [N:nat]: (((inverse_inverse @ A) @ ((semiring_1_of_nat @ A) @ N))))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5587_LIMSEQ__linear, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [X8:(nat > A),X3:A,L:nat]: (((((((filterlim @ nat) @ A) @ X8) @ ((topolo7230453075368039082e_nhds @ A) @ X3)) @ (at_top @ nat)) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ L) => (((((filterlim @ nat) @ A) @ (^ [N:nat]: ((X8 @ (((times_times @ nat) @ N) @ L))))) @ ((topolo7230453075368039082e_nhds @ A) @ X3)) @ (at_top @ nat)))))))))). % 6.41/2.70 thf(fact_5588_telescope__summable_H, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),C2:A]: (((((((filterlim @ nat) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ C2)) @ (at_top @ nat)) => ((summable @ A) @ (^ [N:nat]: ((((minus_minus @ A) @ (F2 @ N)) @ (F2 @ (suc @ N))))))))))))). % 6.41/2.70 thf(fact_5589_telescope__summable, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),C2:A]: (((((((filterlim @ nat) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ C2)) @ (at_top @ nat)) => ((summable @ A) @ (^ [N:nat]: ((((minus_minus @ A) @ (F2 @ (suc @ N))) @ (F2 @ N)))))))))))). % 6.41/2.70 thf(fact_5590_nested__sequence__unique, axiom, (! [F2:(nat > real),G:(nat > real)]: (((! [N3:nat]: ((((ord_less_eq @ real) @ (F2 @ N3)) @ (F2 @ (suc @ N3))))) => ((! [N3:nat]: ((((ord_less_eq @ real) @ (G @ (suc @ N3))) @ (G @ N3)))) => ((! [N3:nat]: ((((ord_less_eq @ real) @ (F2 @ N3)) @ (G @ N3)))) => ((((((filterlim @ nat) @ real) @ (^ [N:nat]: ((((minus_minus @ real) @ (F2 @ N)) @ (G @ N))))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)) => (? [L4:real]: (((! [N6:nat]: ((((ord_less_eq @ real) @ (F2 @ N6)) @ L4))) & ((((((filterlim @ nat) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ L4)) @ (at_top @ nat)) & ((! [N6:nat]: ((((ord_less_eq @ real) @ L4) @ (G @ N6)))) & (((((filterlim @ nat) @ real) @ G) @ ((topolo7230453075368039082e_nhds @ real) @ L4)) @ (at_top @ nat)))))))))))))). % 6.41/2.70 thf(fact_5591_LIMSEQ__inverse__zero, axiom, (! [X8:(nat > real)]: (((! [R4:real]: ((? [N8:nat]: ((! [N3:nat]: (((((ord_less_eq @ nat) @ N8) @ N3) => (((ord_less @ real) @ R4) @ (X8 @ N3))))))))) => (((((filterlim @ nat) @ real) @ (^ [N:nat]: (((inverse_inverse @ real) @ (X8 @ N))))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5592_LIMSEQ__root__const, axiom, (! [C2:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ C2) => (((((filterlim @ nat) @ real) @ (^ [N:nat]: (((root @ N) @ C2)))) @ ((topolo7230453075368039082e_nhds @ real) @ (one_one @ real))) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5593_LIMSEQ__inverse__real__of__nat, axiom, (((((filterlim @ nat) @ real) @ (^ [N:nat]: (((inverse_inverse @ real) @ ((semiring_1_of_nat @ real) @ (suc @ N)))))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat))). % 6.41/2.70 thf(fact_5594_LIMSEQ__inverse__real__of__nat__add, axiom, (! [R2:real]: ((((((filterlim @ nat) @ real) @ (^ [N:nat]: ((((plus_plus @ real) @ R2) @ ((inverse_inverse @ real) @ ((semiring_1_of_nat @ real) @ (suc @ N))))))) @ ((topolo7230453075368039082e_nhds @ real) @ R2)) @ (at_top @ nat))))). % 6.41/2.70 thf(fact_5595_increasing__LIMSEQ, axiom, (! [F2:(nat > real),L:real]: (((! [N3:nat]: ((((ord_less_eq @ real) @ (F2 @ N3)) @ (F2 @ (suc @ N3))))) => ((! [N3:nat]: ((((ord_less_eq @ real) @ (F2 @ N3)) @ L))) => ((! [E:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E) => (? [N6:nat]: ((((ord_less_eq @ real) @ L) @ (((plus_plus @ real) @ (F2 @ N6)) @ E))))))) => (((((filterlim @ nat) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ L)) @ (at_top @ nat)))))))). % 6.41/2.70 thf(fact_5596_lim__1__over__n, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (((((filterlim @ nat) @ A) @ (^ [N:nat]: ((((divide_divide @ A) @ (one_one @ A)) @ ((semiring_1_of_nat @ A) @ N))))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5597_LIMSEQ__Suc__n__over__n, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (((((filterlim @ nat) @ A) @ (^ [N:nat]: ((((divide_divide @ A) @ ((semiring_1_of_nat @ A) @ (suc @ N))) @ ((semiring_1_of_nat @ A) @ N))))) @ ((topolo7230453075368039082e_nhds @ A) @ (one_one @ A))) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5598_LIMSEQ__n__over__Suc__n, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (((((filterlim @ nat) @ A) @ (^ [N:nat]: ((((divide_divide @ A) @ ((semiring_1_of_nat @ A) @ N)) @ ((semiring_1_of_nat @ A) @ (suc @ N)))))) @ ((topolo7230453075368039082e_nhds @ A) @ (one_one @ A))) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5599_LIMSEQ__realpow__zero, axiom, (! [X3:real]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ X3) => ((((ord_less @ real) @ X3) @ (one_one @ real)) => (((((filterlim @ nat) @ real) @ ((power_power @ real) @ X3)) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat))))))). % 6.41/2.70 thf(fact_5600_telescope__sums_H, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),C2:A]: (((((((filterlim @ nat) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ C2)) @ (at_top @ nat)) => (((sums @ A) @ (^ [N:nat]: ((((minus_minus @ A) @ (F2 @ N)) @ (F2 @ (suc @ N)))))) @ (((minus_minus @ A) @ (F2 @ (zero_zero @ nat))) @ C2))))))))). % 6.41/2.70 thf(fact_5601_telescope__sums, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),C2:A]: (((((((filterlim @ nat) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ C2)) @ (at_top @ nat)) => (((sums @ A) @ (^ [N:nat]: ((((minus_minus @ A) @ (F2 @ (suc @ N))) @ (F2 @ N))))) @ (((minus_minus @ A) @ C2) @ (F2 @ (zero_zero @ nat))))))))))). % 6.41/2.70 thf(fact_5602_LIMSEQ__divide__realpow__zero, axiom, (! [X3:real,A2:real]: (((((ord_less @ real) @ (one_one @ real)) @ X3) => (((((filterlim @ nat) @ real) @ (^ [N:nat]: ((((divide_divide @ real) @ A2) @ (((power_power @ real) @ X3) @ N))))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5603_LIMSEQ__abs__realpow__zero2, axiom, (! [C2:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ C2)) @ (one_one @ real)) => (((((filterlim @ nat) @ real) @ ((power_power @ real) @ C2)) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5604_LIMSEQ__abs__realpow__zero, axiom, (! [C2:real]: (((((ord_less @ real) @ ((abs_abs @ real) @ C2)) @ (one_one @ real)) => (((((filterlim @ nat) @ real) @ ((power_power @ real) @ ((abs_abs @ real) @ C2))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5605_LIMSEQ__inverse__realpow__zero, axiom, (! [X3:real]: (((((ord_less @ real) @ (one_one @ real)) @ X3) => (((((filterlim @ nat) @ real) @ (^ [N:nat]: (((inverse_inverse @ real) @ (((power_power @ real) @ X3) @ N))))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5606_sums__def_H, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topolo4958980785337419405_space @ A)) => ((sums @ A) = (^ [F3:(nat > A),S8:A]: ((((((filterlim @ nat) @ A) @ (^ [N:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ F3) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N))))) @ ((topolo7230453075368039082e_nhds @ A) @ S8)) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5607_root__test__convergence, axiom, (! [A:$tType]: (((real_Vector_banach @ A) => (! [F2:(nat > A),X3:real]: (((((((filterlim @ nat) @ real) @ (^ [N:nat]: (((root @ N) @ ((real_V7770717601297561774m_norm @ A) @ (F2 @ N)))))) @ ((topolo7230453075368039082e_nhds @ real) @ X3)) @ (at_top @ nat)) => ((((ord_less @ real) @ X3) @ (one_one @ real)) => ((summable @ A) @ F2))))))))). % 6.41/2.70 thf(fact_5608_LIMSEQ__inverse__real__of__nat__add__minus, axiom, (! [R2:real]: ((((((filterlim @ nat) @ real) @ (^ [N:nat]: ((((plus_plus @ real) @ R2) @ ((uminus_uminus @ real) @ ((inverse_inverse @ real) @ ((semiring_1_of_nat @ real) @ (suc @ N)))))))) @ ((topolo7230453075368039082e_nhds @ real) @ R2)) @ (at_top @ nat))))). % 6.41/2.70 thf(fact_5609_LIMSEQ__D, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X8:(nat > A),L5:A,R2:real]: (((((((filterlim @ nat) @ A) @ X8) @ ((topolo7230453075368039082e_nhds @ A) @ L5)) @ (at_top @ nat)) => ((((ord_less @ real) @ (zero_zero @ real)) @ R2) => (? [No:nat]: ((! [N6:nat]: (((((ord_less_eq @ nat) @ No) @ N6) => (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ (X8 @ N6)) @ L5))) @ R2)))))))))))))). % 6.41/2.70 thf(fact_5610_LIMSEQ__I, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X8:(nat > A),L5:A]: (((! [R4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R4) => (? [No2:nat]: ((! [N3:nat]: (((((ord_less_eq @ nat) @ No2) @ N3) => (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ (X8 @ N3)) @ L5))) @ R4))))))))) => (((((filterlim @ nat) @ A) @ X8) @ ((topolo7230453075368039082e_nhds @ A) @ L5)) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5611_LIMSEQ__iff, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X8:(nat > A),L5:A]: (((((((filterlim @ nat) @ A) @ X8) @ ((topolo7230453075368039082e_nhds @ A) @ L5)) @ (at_top @ nat)) = (! [R5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R5) => (? [No3:nat]: ((! [N:nat]: (((((ord_less_eq @ nat) @ No3) @ N) => (((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ (X8 @ N)) @ L5))) @ R5)))))))))))))))). % 6.41/2.70 thf(fact_5612_LIMSEQ__power__zero, axiom, (! [A:$tType]: (((real_V2822296259951069270ebra_1 @ A) => (! [X3:A]: (((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ (one_one @ real)) => (((((filterlim @ nat) @ A) @ ((power_power @ A) @ X3)) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5613_tendsto__at__iff__sequentially, axiom, (! [C:$tType,A:$tType]: ((((topolo3112930676232923870pology @ A) & (topolo4958980785337419405_space @ C)) => (! [F2:(A > C),A2:C,X3:A,S:(set @ A)]: (((((((filterlim @ A) @ C) @ F2) @ ((topolo7230453075368039082e_nhds @ C) @ A2)) @ (((topolo174197925503356063within @ A) @ X3) @ S)) = (! [X6:(nat > A)]: (((! [I3:nat]: ((((member @ A) @ (X6 @ I3)) @ (((minus_minus @ (set @ A)) @ S) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))) => ((((((filterlim @ nat) @ A) @ X6) @ ((topolo7230453075368039082e_nhds @ A) @ X3)) @ (at_top @ nat)) => (((((filterlim @ nat) @ C) @ (((((comp @ A) @ C) @ nat) @ F2) @ X6)) @ ((topolo7230453075368039082e_nhds @ C) @ A2)) @ (at_top @ nat))))))))))))). % 6.41/2.70 thf(fact_5614_tendsto__power__zero, axiom, (! [A:$tType,B:$tType]: (((real_V2822296259951069270ebra_1 @ A) => (! [F2:(B > nat),F5:(filter @ B),X3:A]: (((((((filterlim @ B) @ nat) @ F2) @ (at_top @ nat)) @ F5) => ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ (one_one @ real)) => (((((filterlim @ B) @ A) @ (^ [Y2:B]: ((((power_power @ A) @ X3) @ (F2 @ Y2))))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ F5))))))))). % 6.41/2.70 thf(fact_5615_LIMSEQ__inverse__real__of__nat__add__minus__mult, axiom, (! [R2:real]: ((((((filterlim @ nat) @ real) @ (^ [N:nat]: ((((times_times @ real) @ R2) @ (((plus_plus @ real) @ (one_one @ real)) @ ((uminus_uminus @ real) @ ((inverse_inverse @ real) @ ((semiring_1_of_nat @ real) @ (suc @ N))))))))) @ ((topolo7230453075368039082e_nhds @ real) @ R2)) @ (at_top @ nat))))). % 6.41/2.70 thf(fact_5616_LIMSEQ__norm__0, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A)]: (((! [N3:nat]: ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (F2 @ N3))) @ (((divide_divide @ real) @ (one_one @ real)) @ ((semiring_1_of_nat @ real) @ (suc @ N3)))))) => (((((filterlim @ nat) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5617_field__derivative__lim__unique, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),Df:A,Z2:A,S:(nat > A),A2:A]: ((((((has_field_derivative @ A) @ F2) @ Df) @ (((topolo174197925503356063within @ A) @ Z2) @ (top_top @ (set @ A)))) => ((((((filterlim @ nat) @ A) @ S) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_top @ nat)) => ((! [N3:nat]: (((S @ N3) != (zero_zero @ A)))) => ((((((filterlim @ nat) @ A) @ (^ [N:nat]: ((((divide_divide @ A) @ (((minus_minus @ A) @ (F2 @ (((plus_plus @ A) @ Z2) @ (S @ N)))) @ (F2 @ Z2))) @ (S @ N))))) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ (at_top @ nat)) => (Df = A2))))))))))). % 6.41/2.70 thf(fact_5618_powser__times__n__limit__0, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V8999393235501362500lgebra @ A)) => (! [X3:A]: (((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ X3)) @ (one_one @ real)) => (((((filterlim @ nat) @ A) @ (^ [N:nat]: ((((times_times @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (((power_power @ A) @ X3) @ N))))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5619_lim__n__over__pown, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [X3:A]: (((((ord_less @ real) @ (one_one @ real)) @ ((real_V7770717601297561774m_norm @ A) @ X3)) => (((((filterlim @ nat) @ A) @ (^ [N:nat]: ((((divide_divide @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (((power_power @ A) @ X3) @ N))))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5620_summable, axiom, (! [A2:(nat > real)]: (((((((filterlim @ nat) @ real) @ A2) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)) => ((! [N3:nat]: ((((ord_less_eq @ real) @ (zero_zero @ real)) @ (A2 @ N3)))) => ((! [N3:nat]: ((((ord_less_eq @ real) @ (A2 @ (suc @ N3))) @ (A2 @ N3)))) => ((summable @ real) @ (^ [N:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ N)) @ (A2 @ N))))))))))). % 6.41/2.70 thf(fact_5621_zeroseq__arctan__series, axiom, (! [X3:real]: (((((ord_less_eq @ real) @ ((abs_abs @ real) @ X3)) @ (one_one @ real)) => (((((filterlim @ nat) @ real) @ (^ [N:nat]: ((((times_times @ real) @ (((divide_divide @ real) @ (one_one @ real)) @ ((semiring_1_of_nat @ real) @ (((plus_plus @ nat) @ (((times_times @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ nat))))) @ (((power_power @ real) @ X3) @ (((plus_plus @ nat) @ (((times_times @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ nat))))))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5622_summable__Leibniz_H_I3_J, axiom, (! [A2:(nat > real)]: (((((((filterlim @ nat) @ real) @ A2) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)) => ((! [N3:nat]: ((((ord_less_eq @ real) @ (zero_zero @ real)) @ (A2 @ N3)))) => ((! [N3:nat]: ((((ord_less_eq @ real) @ (A2 @ (suc @ N3))) @ (A2 @ N3)))) => (((((filterlim @ nat) @ real) @ (^ [N:nat]: (((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))) @ ((set_ord_lessThan @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))))) @ ((topolo7230453075368039082e_nhds @ real) @ ((suminf @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))))) @ (at_top @ nat)))))))). % 6.41/2.70 thf(fact_5623_summable__Leibniz_H_I2_J, axiom, (! [A2:(nat > real),N2:nat]: (((((((filterlim @ nat) @ real) @ A2) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)) => ((! [N3:nat]: ((((ord_less_eq @ real) @ (zero_zero @ real)) @ (A2 @ N3)))) => ((! [N3:nat]: ((((ord_less_eq @ real) @ (A2 @ (suc @ N3))) @ (A2 @ N3)))) => (((ord_less_eq @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))) @ ((set_ord_lessThan @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)))) @ ((suminf @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3)))))))))))). % 6.41/2.70 thf(fact_5624_sums__alternating__upper__lower, axiom, (! [A2:(nat > real)]: (((! [N3:nat]: ((((ord_less_eq @ real) @ (A2 @ (suc @ N3))) @ (A2 @ N3)))) => ((! [N3:nat]: ((((ord_less_eq @ real) @ (zero_zero @ real)) @ (A2 @ N3)))) => ((((((filterlim @ nat) @ real) @ A2) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)) => (? [L4:real]: (((! [N6:nat]: ((((ord_less_eq @ real) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))) @ ((set_ord_lessThan @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N6)))) @ L4))) & ((((((filterlim @ nat) @ real) @ (^ [N:nat]: (((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))) @ ((set_ord_lessThan @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))))) @ ((topolo7230453075368039082e_nhds @ real) @ L4)) @ (at_top @ nat)) & ((! [N6:nat]: ((((ord_less_eq @ real) @ L4) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))) @ ((set_ord_lessThan @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N6)) @ (one_one @ nat))))))) & (((((filterlim @ nat) @ real) @ (^ [N:nat]: (((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))) @ ((set_ord_lessThan @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ (one_one @ nat))))))) @ ((topolo7230453075368039082e_nhds @ real) @ L4)) @ (at_top @ nat))))))))))))). % 6.41/2.70 thf(fact_5625_summable__Leibniz_I5_J, axiom, (! [A2:(nat > real)]: (((((((filterlim @ nat) @ real) @ A2) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)) => (((topological_monoseq @ real) @ A2) => (((((filterlim @ nat) @ real) @ (^ [N:nat]: (((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))) @ ((set_ord_lessThan @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ (one_one @ nat))))))) @ ((topolo7230453075368039082e_nhds @ real) @ ((suminf @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))))) @ (at_top @ nat))))))). % 6.41/2.70 thf(fact_5626_summable__Leibniz_H_I4_J, axiom, (! [A2:(nat > real),N2:nat]: (((((((filterlim @ nat) @ real) @ A2) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)) => ((! [N3:nat]: ((((ord_less_eq @ real) @ (zero_zero @ real)) @ (A2 @ N3)))) => ((! [N3:nat]: ((((ord_less_eq @ real) @ (A2 @ (suc @ N3))) @ (A2 @ N3)))) => (((ord_less_eq @ real) @ ((suminf @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3)))))) @ ((((groups7311177749621191930dd_sum @ nat) @ real) @ (^ [I3:nat]: ((((times_times @ real) @ (((power_power @ real) @ ((uminus_uminus @ real) @ (one_one @ real))) @ I3)) @ (A2 @ I3))))) @ ((set_ord_lessThan @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2)) @ (one_one @ nat))))))))))). % 6.41/2.70 thf(fact_5627_has__derivative__at2, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),F11:(A > B),X3:A]: (((((((has_derivative @ A) @ B) @ F2) @ F11) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))) = ((((real_V3181309239436604168linear @ A) @ B) @ F11) & (((((filterlim @ A) @ B) @ (^ [Y2:A]: ((((real_V8093663219630862766scaleR @ B) @ (((divide_divide @ real) @ (one_one @ real)) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ Y2) @ X3)))) @ (((minus_minus @ B) @ (F2 @ Y2)) @ (((plus_plus @ B) @ (F2 @ X3)) @ (F11 @ (((minus_minus @ A) @ Y2) @ X3)))))))) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))))))))))). % 6.41/2.70 thf(fact_5628_summable__Cauchy_H, axiom, (! [A:$tType]: (((real_Vector_banach @ A) => (! [F2:(nat > A),G:(nat > real)]: (((((eventually @ nat) @ (^ [M2:nat]: ((! [N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => (((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)))) @ (G @ M2)))))))) @ (at_top @ nat)) => ((((((filterlim @ nat) @ real) @ G) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)) => ((summable @ A) @ F2))))))))). % 6.41/2.70 thf(fact_5629_has__derivative__at, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),D6:(A > B),X3:A]: (((((((has_derivative @ A) @ B) @ F2) @ D6) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))) = ((((real_V3181309239436604168linear @ A) @ B) @ D6) & (((((filterlim @ A) @ real) @ (^ [H:A]: ((((divide_divide @ real) @ ((real_V7770717601297561774m_norm @ B) @ (((minus_minus @ B) @ (((minus_minus @ B) @ (F2 @ (((plus_plus @ A) @ X3) @ H))) @ (F2 @ X3))) @ (D6 @ H)))) @ ((real_V7770717601297561774m_norm @ A) @ H))))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A)))))))))))). % 6.41/2.70 thf(fact_5630_eventually__const, axiom, (! [A:$tType,F5:(filter @ A),P:$o]: (((F5 != (bot_bot @ (filter @ A))) => ((((eventually @ A) @ (^ [X:A]: (P))) @ F5) = P))))). % 6.41/2.70 thf(fact_5631_eventually__sequentially__Suc, axiom, (! [P:(nat > $o)]: (((((eventually @ nat) @ (^ [I3:nat]: ((P @ (suc @ I3))))) @ (at_top @ nat)) = (((eventually @ nat) @ P) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5632_eventually__sequentially__seg, axiom, (! [P:(nat > $o),K:nat]: (((((eventually @ nat) @ (^ [N:nat]: ((P @ (((plus_plus @ nat) @ N) @ K))))) @ (at_top @ nat)) = (((eventually @ nat) @ P) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5633_eventually__at__top__dense, axiom, (! [A:$tType]: ((((linorder @ A) & (no_top @ A)) => (! [P:(A > $o)]: (((((eventually @ A) @ P) @ (at_top @ A)) = (? [N5:A]: ((! [N:A]: (((((ord_less @ A) @ N5) @ N) => (P @ N))))))))))))). % 6.41/2.70 thf(fact_5634_eventually__at__top__linorderI, axiom, (! [A:$tType]: (((linorder @ A) => (! [C2:A,P:(A > $o)]: (((! [X5:A]: (((((ord_less_eq @ A) @ C2) @ X5) => (P @ X5)))) => (((eventually @ A) @ P) @ (at_top @ A))))))))). % 6.41/2.70 thf(fact_5635_eventually__at__top__linorder, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o)]: (((((eventually @ A) @ P) @ (at_top @ A)) = (? [N5:A]: ((! [N:A]: (((((ord_less_eq @ A) @ N5) @ N) => (P @ N))))))))))))). % 6.41/2.70 thf(fact_5636_eventually__sequentiallyI, axiom, (! [C2:nat,P:(nat > $o)]: (((! [X5:nat]: (((((ord_less_eq @ nat) @ C2) @ X5) => (P @ X5)))) => (((eventually @ nat) @ P) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5637_eventually__sequentially, axiom, (! [P:(nat > $o)]: (((((eventually @ nat) @ P) @ (at_top @ nat)) = (? [N5:nat]: ((! [N:nat]: (((((ord_less_eq @ nat) @ N5) @ N) => (P @ N)))))))))). % 6.41/2.70 thf(fact_5638_eventually__bot, axiom, (! [A:$tType,P:(A > $o)]: ((((eventually @ A) @ P) @ (bot_bot @ (filter @ A)))))). % 6.41/2.70 thf(fact_5639_eventually__happens, axiom, (! [A:$tType,P:(A > $o),Net:(filter @ A)]: (((((eventually @ A) @ P) @ Net) => ((Net = (bot_bot @ (filter @ A))) | (? [X_1:A]: ((P @ X_1)))))))). % 6.41/2.70 thf(fact_5640_eventually__happens_H, axiom, (! [A:$tType,F5:(filter @ A),P:(A > $o)]: (((F5 != (bot_bot @ (filter @ A))) => ((((eventually @ A) @ P) @ F5) => (? [X_1:A]: ((P @ X_1)))))))). % 6.41/2.70 thf(fact_5641_False__imp__not__eventually, axiom, (! [A:$tType,P:(A > $o),Net:(filter @ A)]: (((! [X5:A]: ((~ ((P @ X5))))) => ((Net != (bot_bot @ (filter @ A))) => (~ ((((eventually @ A) @ P) @ Net)))))))). % 6.41/2.70 thf(fact_5642_eventually__const__iff, axiom, (! [A:$tType,P:$o,F5:(filter @ A)]: (((((eventually @ A) @ (^ [X:A]: (P))) @ F5) = (P | (F5 = (bot_bot @ (filter @ A)))))))). % 6.41/2.70 thf(fact_5643_trivial__limit__def, axiom, (! [A:$tType,F5:(filter @ A)]: (((F5 = (bot_bot @ (filter @ A))) = (((eventually @ A) @ (^ [X:A]: ($false))) @ F5))))). % 6.41/2.70 thf(fact_5644_bounded__linear__zero, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (((real_V3181309239436604168linear @ A) @ B) @ (^ [X:A]: ((zero_zero @ B)))))))). % 6.41/2.70 thf(fact_5645_bounded__linear__add, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),G:(A > B)]: (((((real_V3181309239436604168linear @ A) @ B) @ F2) => ((((real_V3181309239436604168linear @ A) @ B) @ G) => (((real_V3181309239436604168linear @ A) @ B) @ (^ [X:A]: ((((plus_plus @ B) @ (F2 @ X)) @ (G @ X))))))))))))). % 6.41/2.70 thf(fact_5646_eventually__ex, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),F5:(filter @ A)]: (((((eventually @ A) @ (^ [X:A]: ((? [X6:B]: (((P @ X) @ X6)))))) @ F5) = (? [Y9:(A > B)]: ((((eventually @ A) @ (^ [X:A]: (((P @ X) @ (Y9 @ X))))) @ F5))))))). % 6.41/2.70 thf(fact_5647_eventually__ge__at__top, axiom, (! [A:$tType]: (((linorder @ A) => (! [C2:A]: ((((eventually @ A) @ ((ord_less_eq @ A) @ C2)) @ (at_top @ A)))))))). % 6.41/2.70 thf(fact_5648_eventually__gt__at__top, axiom, (! [A:$tType]: ((((linorder @ A) & (no_top @ A)) => (! [C2:A]: ((((eventually @ A) @ ((ord_less @ A) @ C2)) @ (at_top @ A)))))))). % 6.41/2.70 thf(fact_5649_eventually__ball__finite__distrib, axiom, (! [B:$tType,A:$tType,A3:(set @ A),P:(B > (A > $o)),Net:(filter @ B)]: ((((finite_finite2 @ A) @ A3) => ((((eventually @ B) @ (^ [X:B]: ((! [Y2:A]: (((((member @ A) @ Y2) @ A3) => ((P @ X) @ Y2))))))) @ Net) = (! [X:A]: (((((member @ A) @ X) @ A3) => (((eventually @ B) @ (^ [Y2:B]: (((P @ Y2) @ X)))) @ Net))))))))). % 6.41/2.70 thf(fact_5650_eventually__ball__finite, axiom, (! [A:$tType,B:$tType,A3:(set @ A),P:(B > (A > $o)),Net:(filter @ B)]: ((((finite_finite2 @ A) @ A3) => ((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((eventually @ B) @ (^ [Y2:B]: (((P @ Y2) @ X5)))) @ Net)))) => (((eventually @ B) @ (^ [X:B]: ((! [Y2:A]: (((((member @ A) @ Y2) @ A3) => ((P @ X) @ Y2))))))) @ Net)))))). % 6.41/2.70 thf(fact_5651_le__sequentially, axiom, (! [F5:(filter @ nat)]: (((((ord_less_eq @ (filter @ nat)) @ F5) @ (at_top @ nat)) = (! [N5:nat]: ((((eventually @ nat) @ ((ord_less_eq @ nat) @ N5)) @ F5))))))). % 6.41/2.70 thf(fact_5652_sequentially__offset, axiom, (! [P:(nat > $o),K:nat]: (((((eventually @ nat) @ P) @ (at_top @ nat)) => (((eventually @ nat) @ (^ [I3:nat]: ((P @ (((plus_plus @ nat) @ I3) @ K))))) @ (at_top @ nat)))))). % 6.41/2.70 thf(fact_5653_LIM__at__top__divide, axiom, (! [A:$tType,F2:(A > real),A2:real,F5:(filter @ A),G:(A > real)]: (((((((filterlim @ A) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ A2)) @ F5) => ((((ord_less @ real) @ (zero_zero @ real)) @ A2) => ((((((filterlim @ A) @ real) @ G) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ F5) => ((((eventually @ A) @ (^ [X:A]: ((((ord_less @ real) @ (zero_zero @ real)) @ (G @ X))))) @ F5) => (((((filterlim @ A) @ real) @ (^ [X:A]: ((((divide_divide @ real) @ (F2 @ X)) @ (G @ X))))) @ (at_top @ real)) @ F5)))))))). % 6.41/2.70 thf(fact_5654_filterlim__inverse__at__top, axiom, (! [A:$tType,F2:(A > real),F5:(filter @ A)]: (((((((filterlim @ A) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ F5) => ((((eventually @ A) @ (^ [X:A]: ((((ord_less @ real) @ (zero_zero @ real)) @ (F2 @ X))))) @ F5) => (((((filterlim @ A) @ real) @ (^ [X:A]: (((inverse_inverse @ real) @ (F2 @ X))))) @ (at_top @ real)) @ F5)))))). % 6.41/2.70 thf(fact_5655_filterlim__inverse__at__top__iff, axiom, (! [A:$tType,F2:(A > real),F5:(filter @ A)]: (((((eventually @ A) @ (^ [X:A]: ((((ord_less @ real) @ (zero_zero @ real)) @ (F2 @ X))))) @ F5) => ((((((filterlim @ A) @ real) @ (^ [X:A]: (((inverse_inverse @ real) @ (F2 @ X))))) @ (at_top @ real)) @ F5) = (((((filterlim @ A) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ F5)))))). % 6.41/2.70 thf(fact_5656_filterlim__at__top__iff__inverse__0, axiom, (! [A:$tType,F2:(A > real),F5:(filter @ A)]: (((((eventually @ A) @ (^ [X:A]: ((((ord_less @ real) @ (zero_zero @ real)) @ (F2 @ X))))) @ F5) => ((((((filterlim @ A) @ real) @ F2) @ (at_top @ real)) @ F5) = (((((filterlim @ A) @ real) @ (((((comp @ real) @ real) @ A) @ (inverse_inverse @ real)) @ F2)) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ F5)))))). % 6.41/2.70 thf(fact_5657_eventually__nhds__top, axiom, (! [A:$tType]: ((((order_top @ A) & (topolo1944317154257567458pology @ A)) => (! [B2:A,P:(A > $o)]: (((((ord_less @ A) @ B2) @ (top_top @ A)) => ((((eventually @ A) @ P) @ ((topolo7230453075368039082e_nhds @ A) @ (top_top @ A))) = (? [B4:A]: (((((ord_less @ A) @ B4) @ (top_top @ A)) & (! [Z6:A]: (((((ord_less @ A) @ B4) @ Z6) => (P @ Z6))))))))))))))). % 6.41/2.70 thf(fact_5658_filterlim__at__top__at__top, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (linorder @ B)) => (! [Q:(A > $o),F2:(A > B),P:(B > $o),G:(B > A)]: (((! [X5:A,Y4:A]: (((Q @ X5) => ((Q @ Y4) => ((((ord_less_eq @ A) @ X5) @ Y4) => (((ord_less_eq @ B) @ (F2 @ X5)) @ (F2 @ Y4))))))) => ((! [X5:B]: (((P @ X5) => ((F2 @ (G @ X5)) = X5)))) => ((! [X5:B]: (((P @ X5) => (Q @ (G @ X5))))) => ((((eventually @ A) @ Q) @ (at_top @ A)) => ((((eventually @ B) @ P) @ (at_top @ B)) => (((((filterlim @ A) @ B) @ F2) @ (at_top @ B)) @ (at_top @ A))))))))))))). % 6.41/2.70 thf(fact_5659_eventually__at__left__field, axiom, (! [A:$tType]: ((((linordered_field @ A) & (topolo1944317154257567458pology @ A)) => (! [P:(A > $o),X3:A]: (((((eventually @ A) @ P) @ (((topolo174197925503356063within @ A) @ X3) @ ((set_ord_lessThan @ A) @ X3))) = (? [B4:A]: (((((ord_less @ A) @ B4) @ X3) & (! [Y2:A]: (((((ord_less @ A) @ B4) @ Y2) => ((((ord_less @ A) @ Y2) @ X3) => (P @ Y2))))))))))))))). % 6.41/2.70 thf(fact_5660_eventually__at__left, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [Y:A,X3:A,P:(A > $o)]: (((((ord_less @ A) @ Y) @ X3) => ((((eventually @ A) @ P) @ (((topolo174197925503356063within @ A) @ X3) @ ((set_ord_lessThan @ A) @ X3))) = (? [B4:A]: (((((ord_less @ A) @ B4) @ X3) & (! [Y2:A]: (((((ord_less @ A) @ B4) @ Y2) => ((((ord_less @ A) @ Y2) @ X3) => (P @ Y2)))))))))))))))). % 6.41/2.70 thf(fact_5661_tendsto__sandwich, axiom, (! [A:$tType,B:$tType]: (((topolo2564578578187576103pology @ A) => (! [F2:(B > A),G:(B > A),Net:(filter @ B),H2:(B > A),C2:A]: (((((eventually @ B) @ (^ [N:B]: ((((ord_less_eq @ A) @ (F2 @ N)) @ (G @ N))))) @ Net) => ((((eventually @ B) @ (^ [N:B]: ((((ord_less_eq @ A) @ (G @ N)) @ (H2 @ N))))) @ Net) => ((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ C2)) @ Net) => ((((((filterlim @ B) @ A) @ H2) @ ((topolo7230453075368039082e_nhds @ A) @ C2)) @ Net) => (((((filterlim @ B) @ A) @ G) @ ((topolo7230453075368039082e_nhds @ A) @ C2)) @ Net))))))))))). % 6.41/2.70 thf(fact_5662_order__tendsto__iff, axiom, (! [B:$tType,A:$tType]: (((topolo2564578578187576103pology @ A) => (! [F2:(B > A),X3:A,F5:(filter @ B)]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ X3)) @ F5) = ((! [L2:A]: (((((ord_less @ A) @ L2) @ X3) => (((eventually @ B) @ (^ [X:B]: ((((ord_less @ A) @ L2) @ (F2 @ X))))) @ F5)))) & (! [U2:A]: (((((ord_less @ A) @ X3) @ U2) => (((eventually @ B) @ (^ [X:B]: ((((ord_less @ A) @ (F2 @ X)) @ U2)))) @ F5)))))))))))). % 6.41/2.70 thf(fact_5663_order__tendstoI, axiom, (! [A:$tType,B:$tType]: (((topolo2564578578187576103pology @ A) => (! [Y:A,F2:(B > A),F5:(filter @ B)]: (((! [A5:A]: (((((ord_less @ A) @ A5) @ Y) => (((eventually @ B) @ (^ [X:B]: ((((ord_less @ A) @ A5) @ (F2 @ X))))) @ F5)))) => ((! [A5:A]: (((((ord_less @ A) @ Y) @ A5) => (((eventually @ B) @ (^ [X:B]: ((((ord_less @ A) @ (F2 @ X)) @ A5)))) @ F5)))) => (((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ Y)) @ F5))))))))). % 6.41/2.70 thf(fact_5664_order__tendstoD_I1_J, axiom, (! [A:$tType,B:$tType]: (((topolo2564578578187576103pology @ A) => (! [F2:(B > A),Y:A,F5:(filter @ B),A2:A]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ Y)) @ F5) => ((((ord_less @ A) @ A2) @ Y) => (((eventually @ B) @ (^ [X:B]: ((((ord_less @ A) @ A2) @ (F2 @ X))))) @ F5))))))))). % 6.41/2.70 thf(fact_5665_order__tendstoD_I2_J, axiom, (! [A:$tType,B:$tType]: (((topolo2564578578187576103pology @ A) => (! [F2:(B > A),Y:A,F5:(filter @ B),A2:A]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ Y)) @ F5) => ((((ord_less @ A) @ Y) @ A2) => (((eventually @ B) @ (^ [X:B]: ((((ord_less @ A) @ (F2 @ X)) @ A2)))) @ F5))))))))). % 6.41/2.70 thf(fact_5666_eventually__all__ge__at__top, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o)]: (((((eventually @ A) @ P) @ (at_top @ A)) => (((eventually @ A) @ (^ [X:A]: ((! [Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => (P @ Y2))))))) @ (at_top @ A))))))))). % 6.41/2.70 thf(fact_5667_filterlim__at__top, axiom, (! [A:$tType,B:$tType]: (((linorder @ B) => (! [F2:(A > B),F5:(filter @ A)]: (((((((filterlim @ A) @ B) @ F2) @ (at_top @ B)) @ F5) = (! [Z9:B]: ((((eventually @ A) @ (^ [X:A]: ((((ord_less_eq @ B) @ Z9) @ (F2 @ X))))) @ F5)))))))))). % 6.41/2.70 thf(fact_5668_filterlim__at__top__ge, axiom, (! [A:$tType,B:$tType]: (((linorder @ B) => (! [F2:(A > B),F5:(filter @ A),C2:B]: (((((((filterlim @ A) @ B) @ F2) @ (at_top @ B)) @ F5) = (! [Z9:B]: (((((ord_less_eq @ B) @ C2) @ Z9) => (((eventually @ A) @ (^ [X:A]: ((((ord_less_eq @ B) @ Z9) @ (F2 @ X))))) @ F5))))))))))). % 6.41/2.70 thf(fact_5669_filterlim__at__top__mono, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),F5:(filter @ B),G:(B > A)]: (((((((filterlim @ B) @ A) @ F2) @ (at_top @ A)) @ F5) => ((((eventually @ B) @ (^ [X:B]: ((((ord_less_eq @ A) @ (F2 @ X)) @ (G @ X))))) @ F5) => (((((filterlim @ B) @ A) @ G) @ (at_top @ A)) @ F5))))))))). % 6.41/2.70 thf(fact_5670_filterlim__at__top__dense, axiom, (! [A:$tType,B:$tType]: (((unboun7993243217541854897norder @ B) => (! [F2:(A > B),F5:(filter @ A)]: (((((((filterlim @ A) @ B) @ F2) @ (at_top @ B)) @ F5) = (! [Z9:B]: ((((eventually @ A) @ (^ [X:A]: ((((ord_less @ B) @ Z9) @ (F2 @ X))))) @ F5)))))))))). % 6.41/2.70 thf(fact_5671_bounded__linear_Otendsto__zero, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),G:(C > A),F5:(filter @ C)]: (((((real_V3181309239436604168linear @ A) @ B) @ F2) => ((((((filterlim @ C) @ A) @ G) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ F5) => (((((filterlim @ C) @ B) @ (^ [X:C]: ((F2 @ (G @ X))))) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ F5))))))))). % 6.41/2.70 thf(fact_5672_eventually__Inf__base, axiom, (! [A:$tType,B3:(set @ (filter @ A)),P:(A > $o)]: (((B3 != (bot_bot @ (set @ (filter @ A)))) => ((! [F6:(filter @ A)]: (((((member @ (filter @ A)) @ F6) @ B3) => (! [G5:(filter @ A)]: (((((member @ (filter @ A)) @ G5) @ B3) => (? [X2:(filter @ A)]: (((((member @ (filter @ A)) @ X2) @ B3) & (((ord_less_eq @ (filter @ A)) @ X2) @ (((inf_inf @ (filter @ A)) @ F6) @ G5))))))))))) => ((((eventually @ A) @ P) @ ((complete_Inf_Inf @ (filter @ A)) @ B3)) = (? [X:(filter @ A)]: (((((member @ (filter @ A)) @ X) @ B3) & (((eventually @ A) @ P) @ X)))))))))). % 6.41/2.70 thf(fact_5673_eventually__INF__finite, axiom, (! [B:$tType,A:$tType,A3:(set @ A),P:(B > $o),F5:(A > (filter @ B))]: ((((finite_finite2 @ A) @ A3) => ((((eventually @ B) @ P) @ ((complete_Inf_Inf @ (filter @ B)) @ ((((image @ A) @ (filter @ B)) @ F5) @ A3))) = (? [Q7:(A > (B > $o))]: (((! [X:A]: (((((member @ A) @ X) @ A3) => (((eventually @ B) @ (Q7 @ X)) @ (F5 @ X))))) & (! [Y2:B]: (((! [X:A]: (((((member @ A) @ X) @ A3) => ((Q7 @ X) @ Y2)))) => (P @ Y2)))))))))))). % 6.41/2.70 thf(fact_5674_eventually__at__leftI, axiom, (! [A:$tType]: (((topolo2564578578187576103pology @ A) => (! [A2:A,B2:A,P:(A > $o)]: (((! [X5:A]: (((((member @ A) @ X5) @ (((set_or5935395276787703475ssThan @ A) @ A2) @ B2)) => (P @ X5)))) => ((((ord_less @ A) @ A2) @ B2) => (((eventually @ A) @ P) @ (((topolo174197925503356063within @ A) @ B2) @ ((set_ord_lessThan @ A) @ B2))))))))))). % 6.41/2.70 thf(fact_5675_has__derivative__within__singleton__iff, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),G:(A > B),X3:A]: (((((((has_derivative @ A) @ B) @ F2) @ G) @ (((topolo174197925503356063within @ A) @ X3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) = (((real_V3181309239436604168linear @ A) @ B) @ G)))))))). % 6.41/2.70 thf(fact_5676_eventually__at__to__0, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [P:(A > $o),A2:A]: (((((eventually @ A) @ P) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) = (((eventually @ A) @ (^ [X:A]: ((P @ (((plus_plus @ A) @ X) @ A2))))) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5677_decreasing__tendsto, axiom, (! [A:$tType,B:$tType]: (((topolo2564578578187576103pology @ A) => (! [L:A,F2:(B > A),F5:(filter @ B)]: (((((eventually @ B) @ (^ [N:B]: ((((ord_less_eq @ A) @ L) @ (F2 @ N))))) @ F5) => ((! [X5:A]: (((((ord_less @ A) @ L) @ X5) => (((eventually @ B) @ (^ [N:B]: ((((ord_less @ A) @ (F2 @ N)) @ X5)))) @ F5)))) => (((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ F5))))))))). % 6.41/2.70 thf(fact_5678_increasing__tendsto, axiom, (! [A:$tType,B:$tType]: (((topolo2564578578187576103pology @ A) => (! [F2:(B > A),L:A,F5:(filter @ B)]: (((((eventually @ B) @ (^ [N:B]: ((((ord_less_eq @ A) @ (F2 @ N)) @ L)))) @ F5) => ((! [X5:A]: (((((ord_less @ A) @ X5) @ L) => (((eventually @ B) @ (^ [N:B]: ((((ord_less @ A) @ X5) @ (F2 @ N))))) @ F5)))) => (((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ F5))))))))). % 6.41/2.70 thf(fact_5679_filterlim__at__top__gt, axiom, (! [A:$tType,B:$tType]: (((unboun7993243217541854897norder @ B) => (! [F2:(A > B),F5:(filter @ A),C2:B]: (((((((filterlim @ A) @ B) @ F2) @ (at_top @ B)) @ F5) = (! [Z9:B]: (((((ord_less @ B) @ C2) @ Z9) => (((eventually @ A) @ (^ [X:A]: ((((ord_less_eq @ B) @ Z9) @ (F2 @ X))))) @ F5))))))))))). % 6.41/2.70 thf(fact_5680_tendsto__le, axiom, (! [B:$tType,A:$tType]: (((topolo1944317154257567458pology @ A) => (! [F5:(filter @ B),F2:(B > A),X3:A,G:(B > A),Y:A]: (((F5 != (bot_bot @ (filter @ B))) => ((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ X3)) @ F5) => ((((((filterlim @ B) @ A) @ G) @ ((topolo7230453075368039082e_nhds @ A) @ Y)) @ F5) => ((((eventually @ B) @ (^ [X:B]: ((((ord_less_eq @ A) @ (G @ X)) @ (F2 @ X))))) @ F5) => (((ord_less_eq @ A) @ Y) @ X3))))))))))). % 6.41/2.70 thf(fact_5681_tendsto__lowerbound, axiom, (! [B:$tType,A:$tType]: (((topolo1944317154257567458pology @ A) => (! [F2:(B > A),X3:A,F5:(filter @ B),A2:A]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ X3)) @ F5) => ((((eventually @ B) @ (^ [I3:B]: ((((ord_less_eq @ A) @ A2) @ (F2 @ I3))))) @ F5) => ((F5 != (bot_bot @ (filter @ B))) => (((ord_less_eq @ A) @ A2) @ X3)))))))))). % 6.41/2.70 thf(fact_5682_tendsto__upperbound, axiom, (! [B:$tType,A:$tType]: (((topolo1944317154257567458pology @ A) => (! [F2:(B > A),X3:A,F5:(filter @ B),A2:A]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ X3)) @ F5) => ((((eventually @ B) @ (^ [I3:B]: ((((ord_less_eq @ A) @ (F2 @ I3)) @ A2)))) @ F5) => ((F5 != (bot_bot @ (filter @ B))) => (((ord_less_eq @ A) @ X3) @ A2)))))))))). % 6.41/2.70 thf(fact_5683_eventually__INF, axiom, (! [A:$tType,B:$tType,P:(A > $o),F5:(B > (filter @ A)),B3:(set @ B)]: (((((eventually @ A) @ P) @ ((complete_Inf_Inf @ (filter @ A)) @ ((((image @ B) @ (filter @ A)) @ F5) @ B3))) = (? [X6:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ X6) @ B3) & (((finite_finite2 @ B) @ X6) & (((eventually @ A) @ P) @ ((complete_Inf_Inf @ (filter @ A)) @ ((((image @ B) @ (filter @ A)) @ F5) @ X6))))))))))). % 6.41/2.70 thf(fact_5684_filterlim__pow__at__top, axiom, (! [A:$tType,N2:nat,F2:(A > real),F5:(filter @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((((filterlim @ A) @ real) @ F2) @ (at_top @ real)) @ F5) => (((((filterlim @ A) @ real) @ (^ [X:A]: ((((power_power @ real) @ (F2 @ X)) @ N2)))) @ (at_top @ real)) @ F5)))))). % 6.41/2.70 thf(fact_5685_bounded__linear_Opos__bounded, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B)]: (((((real_V3181309239436604168linear @ A) @ B) @ F2) => (? [K8:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ K8) & (! [X2:A]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ B) @ (F2 @ X2))) @ (((times_times @ real) @ ((real_V7770717601297561774m_norm @ A) @ X2)) @ K8)))))))))))))). % 6.41/2.70 thf(fact_5686_bounded__linear__intro, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),K5:real]: (((! [X5:A,Y4:A]: (((F2 @ (((plus_plus @ A) @ X5) @ Y4)) = (((plus_plus @ B) @ (F2 @ X5)) @ (F2 @ Y4))))) => ((! [R4:real,X5:A]: (((F2 @ (((real_V8093663219630862766scaleR @ A) @ R4) @ X5)) = (((real_V8093663219630862766scaleR @ B) @ R4) @ (F2 @ X5))))) => ((! [X5:A]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ B) @ (F2 @ X5))) @ (((times_times @ real) @ ((real_V7770717601297561774m_norm @ A) @ X5)) @ K5)))) => (((real_V3181309239436604168linear @ A) @ B) @ F2)))))))))). % 6.41/2.70 thf(fact_5687_tendsto__imp__filterlim__at__left, axiom, (! [B:$tType,A:$tType]: (((topolo2564578578187576103pology @ B) => (! [F2:(A > B),L5:B,F5:(filter @ A)]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L5)) @ F5) => ((((eventually @ A) @ (^ [X:A]: ((((ord_less @ B) @ (F2 @ X)) @ L5)))) @ F5) => (((((filterlim @ A) @ B) @ F2) @ (((topolo174197925503356063within @ B) @ L5) @ ((set_ord_lessThan @ B) @ L5))) @ F5))))))))). % 6.41/2.70 thf(fact_5688_eventually__Inf, axiom, (! [A:$tType,P:(A > $o),B3:(set @ (filter @ A))]: (((((eventually @ A) @ P) @ ((complete_Inf_Inf @ (filter @ A)) @ B3)) = (? [X6:(set @ (filter @ A))]: (((((ord_less_eq @ (set @ (filter @ A))) @ X6) @ B3) & (((finite_finite2 @ (filter @ A)) @ X6) & (((eventually @ A) @ P) @ ((complete_Inf_Inf @ (filter @ A)) @ X6)))))))))). % 6.41/2.70 thf(fact_5689_filterlim__at__top__mult__tendsto__pos, axiom, (! [A:$tType,F2:(A > real),C2:real,F5:(filter @ A),G:(A > real)]: (((((((filterlim @ A) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ C2)) @ F5) => ((((ord_less @ real) @ (zero_zero @ real)) @ C2) => ((((((filterlim @ A) @ real) @ G) @ (at_top @ real)) @ F5) => (((((filterlim @ A) @ real) @ (^ [X:A]: ((((times_times @ real) @ (G @ X)) @ (F2 @ X))))) @ (at_top @ real)) @ F5))))))). % 6.41/2.70 thf(fact_5690_filterlim__tendsto__pos__mult__at__top, axiom, (! [A:$tType,F2:(A > real),C2:real,F5:(filter @ A),G:(A > real)]: (((((((filterlim @ A) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ C2)) @ F5) => ((((ord_less @ real) @ (zero_zero @ real)) @ C2) => ((((((filterlim @ A) @ real) @ G) @ (at_top @ real)) @ F5) => (((((filterlim @ A) @ real) @ (^ [X:A]: ((((times_times @ real) @ (F2 @ X)) @ (G @ X))))) @ (at_top @ real)) @ F5))))))). % 6.41/2.70 thf(fact_5691_tendsto__neg__powr, axiom, (! [A:$tType,S:real,F2:(A > real),F5:(filter @ A)]: (((((ord_less @ real) @ S) @ (zero_zero @ real)) => ((((((filterlim @ A) @ real) @ F2) @ (at_top @ real)) @ F5) => (((((filterlim @ A) @ real) @ (^ [X:A]: ((((powr @ real) @ (F2 @ X)) @ S)))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ F5)))))). % 6.41/2.70 thf(fact_5692_filterlim__at__top__at__left, axiom, (! [B:$tType,A:$tType]: ((((topolo1944317154257567458pology @ A) & (linorder @ B)) => (! [Q:(A > $o),F2:(A > B),P:(B > $o),G:(B > A),A2:A]: (((! [X5:A,Y4:A]: (((Q @ X5) => ((Q @ Y4) => ((((ord_less_eq @ A) @ X5) @ Y4) => (((ord_less_eq @ B) @ (F2 @ X5)) @ (F2 @ Y4))))))) => ((! [X5:B]: (((P @ X5) => ((F2 @ (G @ X5)) = X5)))) => ((! [X5:B]: (((P @ X5) => (Q @ (G @ X5))))) => ((((eventually @ A) @ Q) @ (((topolo174197925503356063within @ A) @ A2) @ ((set_ord_lessThan @ A) @ A2))) => ((! [B5:A]: (((Q @ B5) => (((ord_less @ A) @ B5) @ A2)))) => ((((eventually @ B) @ P) @ (at_top @ B)) => (((((filterlim @ A) @ B) @ F2) @ (at_top @ B)) @ (((topolo174197925503356063within @ A) @ A2) @ ((set_ord_lessThan @ A) @ A2))))))))))))))). % 6.41/2.70 thf(fact_5693_eventually__INF__base, axiom, (! [B:$tType,A:$tType,B3:(set @ A),F5:(A > (filter @ B)),P:(B > $o)]: (((B3 != (bot_bot @ (set @ A))) => ((! [A5:A]: (((((member @ A) @ A5) @ B3) => (! [B5:A]: (((((member @ A) @ B5) @ B3) => (? [X2:A]: (((((member @ A) @ X2) @ B3) & (((ord_less_eq @ (filter @ B)) @ (F5 @ X2)) @ (((inf_inf @ (filter @ B)) @ (F5 @ A5)) @ (F5 @ B5)))))))))))) => ((((eventually @ B) @ P) @ ((complete_Inf_Inf @ (filter @ B)) @ ((((image @ A) @ (filter @ B)) @ F5) @ B3))) = (? [X:A]: (((((member @ A) @ X) @ B3) & (((eventually @ B) @ P) @ (F5 @ X))))))))))). % 6.41/2.70 thf(fact_5694_tendsto__0__le, axiom, (! [B:$tType,C:$tType,A:$tType]: ((((real_V822414075346904944vector @ C) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),F5:(filter @ A),G:(A > C),K5:real]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ F5) => ((((eventually @ A) @ (^ [X:A]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ C) @ (G @ X))) @ (((times_times @ real) @ ((real_V7770717601297561774m_norm @ B) @ (F2 @ X))) @ K5))))) @ F5) => (((((filterlim @ A) @ C) @ G) @ ((topolo7230453075368039082e_nhds @ C) @ (zero_zero @ C))) @ F5))))))))). % 6.41/2.70 thf(fact_5695_filterlim__at__withinI, axiom, (! [A:$tType,B:$tType]: (((topolo4958980785337419405_space @ A) => (! [F2:(B > A),C2:A,F5:(filter @ B),A3:(set @ A)]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ C2)) @ F5) => ((((eventually @ B) @ (^ [X:B]: ((((member @ A) @ (F2 @ X)) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ C2) @ (bot_bot @ (set @ A)))))))) @ F5) => (((((filterlim @ B) @ A) @ F2) @ (((topolo174197925503356063within @ A) @ C2) @ A3)) @ F5))))))))). % 6.41/2.70 thf(fact_5696_tendsto__powr_H, axiom, (! [A:$tType,F2:(A > real),A2:real,F5:(filter @ A),G:(A > real),B2:real]: (((((((filterlim @ A) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ A2)) @ F5) => ((((((filterlim @ A) @ real) @ G) @ ((topolo7230453075368039082e_nhds @ real) @ B2)) @ F5) => (((A2 != (zero_zero @ real)) | ((((ord_less @ real) @ (zero_zero @ real)) @ B2) & (((eventually @ A) @ (^ [X:A]: ((((ord_less_eq @ real) @ (zero_zero @ real)) @ (F2 @ X))))) @ F5))) => (((((filterlim @ A) @ real) @ (^ [X:A]: ((((powr @ real) @ (F2 @ X)) @ (G @ X))))) @ ((topolo7230453075368039082e_nhds @ real) @ (((powr @ real) @ A2) @ B2))) @ F5))))))). % 6.41/2.70 thf(fact_5697_tendsto__powr2, axiom, (! [A:$tType,F2:(A > real),A2:real,F5:(filter @ A),G:(A > real),B2:real]: (((((((filterlim @ A) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ A2)) @ F5) => ((((((filterlim @ A) @ real) @ G) @ ((topolo7230453075368039082e_nhds @ real) @ B2)) @ F5) => ((((eventually @ A) @ (^ [X:A]: ((((ord_less_eq @ real) @ (zero_zero @ real)) @ (F2 @ X))))) @ F5) => ((((ord_less @ real) @ (zero_zero @ real)) @ B2) => (((((filterlim @ A) @ real) @ (^ [X:A]: ((((powr @ real) @ (F2 @ X)) @ (G @ X))))) @ ((topolo7230453075368039082e_nhds @ real) @ (((powr @ real) @ A2) @ B2))) @ F5)))))))). % 6.41/2.70 thf(fact_5698_tendsto__zero__powrI, axiom, (! [A:$tType,F2:(A > real),F5:(filter @ A),G:(A > real),B2:real]: (((((((filterlim @ A) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ F5) => ((((((filterlim @ A) @ real) @ G) @ ((topolo7230453075368039082e_nhds @ real) @ B2)) @ F5) => ((((eventually @ A) @ (^ [X:A]: ((((ord_less_eq @ real) @ (zero_zero @ real)) @ (F2 @ X))))) @ F5) => ((((ord_less @ real) @ (zero_zero @ real)) @ B2) => (((((filterlim @ A) @ real) @ (^ [X:A]: ((((powr @ real) @ (F2 @ X)) @ (G @ X))))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ F5)))))))). % 6.41/2.70 thf(fact_5699_DERIV__neg__imp__decreasing__at__top, axiom, (! [B2:real,F2:(real > real),Flim:real]: (((! [X5:real]: (((((ord_less_eq @ real) @ B2) @ X5) => (? [Y5:real]: ((((((has_field_derivative @ real) @ F2) @ Y5) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))) & (((ord_less @ real) @ Y5) @ (zero_zero @ real)))))))) => ((((((filterlim @ real) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ Flim)) @ (at_top @ real)) => (((ord_less @ real) @ Flim) @ (F2 @ B2))))))). % 6.41/2.70 thf(fact_5700_has__derivative__at__within, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),F11:(A > B),X3:A,S:(set @ A)]: (((((((has_derivative @ A) @ B) @ F2) @ F11) @ (((topolo174197925503356063within @ A) @ X3) @ S)) = ((((real_V3181309239436604168linear @ A) @ B) @ F11) & (((((filterlim @ A) @ B) @ (^ [Y2:A]: ((((real_V8093663219630862766scaleR @ B) @ ((inverse_inverse @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ Y2) @ X3)))) @ (((minus_minus @ B) @ (((minus_minus @ B) @ (F2 @ Y2)) @ (F2 @ X3))) @ (F11 @ (((minus_minus @ A) @ Y2) @ X3))))))) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ (((topolo174197925503356063within @ A) @ X3) @ S)))))))))). % 6.41/2.70 thf(fact_5701_has__derivativeI, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F11:(A > B),X3:A,F2:(A > B),S:(set @ A)]: (((((real_V3181309239436604168linear @ A) @ B) @ F11) => ((((((filterlim @ A) @ B) @ (^ [Y2:A]: ((((real_V8093663219630862766scaleR @ B) @ ((inverse_inverse @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ Y2) @ X3)))) @ (((minus_minus @ B) @ (((minus_minus @ B) @ (F2 @ Y2)) @ (F2 @ X3))) @ (F11 @ (((minus_minus @ A) @ Y2) @ X3))))))) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((((has_derivative @ A) @ B) @ F2) @ F11) @ (((topolo174197925503356063within @ A) @ X3) @ S)))))))))). % 6.41/2.70 thf(fact_5702_has__derivative__iff__Ex, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),F11:(A > B),X3:A]: (((((((has_derivative @ A) @ B) @ F2) @ F11) @ (((topolo174197925503356063within @ A) @ X3) @ (top_top @ (set @ A)))) = ((((real_V3181309239436604168linear @ A) @ B) @ F11) & (? [E4:(A > B)]: (((! [H:A]: (((F2 @ (((plus_plus @ A) @ X3) @ H)) = (((plus_plus @ B) @ (((plus_plus @ B) @ (F2 @ X3)) @ (F11 @ H))) @ (E4 @ H))))) & (((((filterlim @ A) @ real) @ (^ [H:A]: ((((divide_divide @ real) @ ((real_V7770717601297561774m_norm @ B) @ (E4 @ H))) @ ((real_V7770717601297561774m_norm @ A) @ H))))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A))))))))))))))). % 6.41/2.70 thf(fact_5703_has__derivative__within, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),F11:(A > B),X3:A,S:(set @ A)]: (((((((has_derivative @ A) @ B) @ F2) @ F11) @ (((topolo174197925503356063within @ A) @ X3) @ S)) = ((((real_V3181309239436604168linear @ A) @ B) @ F11) & (((((filterlim @ A) @ B) @ (^ [Y2:A]: ((((real_V8093663219630862766scaleR @ B) @ (((divide_divide @ real) @ (one_one @ real)) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ Y2) @ X3)))) @ (((minus_minus @ B) @ (F2 @ Y2)) @ (((plus_plus @ B) @ (F2 @ X3)) @ (F11 @ (((minus_minus @ A) @ Y2) @ X3)))))))) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ (((topolo174197925503356063within @ A) @ X3) @ S)))))))))). % 6.41/2.70 thf(fact_5704_polyfun__extremal, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [C2:(nat > A),K:nat,N2:nat,B3:real]: ((((C2 @ K) != (zero_zero @ A)) => ((((ord_less_eq @ nat) @ (one_one @ nat)) @ K) => ((((ord_less_eq @ nat) @ K) @ N2) => (((eventually @ A) @ (^ [Z6:A]: ((((ord_less_eq @ real) @ B3) @ ((real_V7770717601297561774m_norm @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (C2 @ I3)) @ (((power_power @ A) @ Z6) @ I3))))) @ ((set_ord_atMost @ nat) @ N2))))))) @ (at_infinity @ A))))))))))). % 6.41/2.70 thf(fact_5705_summable__bounded__partials, axiom, (! [A:$tType]: ((((real_V8037385150606011577_space @ A) & (real_V822414075346904944vector @ A)) => (! [F2:(nat > A),G:(nat > real)]: (((((eventually @ nat) @ (^ [X02:nat]: ((! [A4:nat]: (((((ord_less_eq @ nat) @ X02) @ A4) => (! [B4:nat]: (((((ord_less @ nat) @ A4) @ B4) => (((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F2) @ (((set_or3652927894154168847AtMost @ nat) @ A4) @ B4)))) @ (G @ A4))))))))))) @ (at_top @ nat)) => ((((((filterlim @ nat) @ real) @ G) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (at_top @ nat)) => ((summable @ A) @ F2))))))))). % 6.41/2.70 thf(fact_5706_has__derivative__def, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (((has_derivative @ A) @ B) = (^ [F3:(A > B),F13:(A > B),F9:(filter @ A)]: (((((real_V3181309239436604168linear @ A) @ B) @ F13) & (((((filterlim @ A) @ B) @ (^ [Y2:A]: ((((real_V8093663219630862766scaleR @ B) @ ((inverse_inverse @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ Y2) @ ((((topolo3827282254853284352ce_Lim @ A) @ A) @ F9) @ (^ [X:A]: (X))))))) @ (((minus_minus @ B) @ (((minus_minus @ B) @ (F3 @ Y2)) @ (F3 @ ((((topolo3827282254853284352ce_Lim @ A) @ A) @ F9) @ (^ [X:A]: (X)))))) @ (F13 @ (((minus_minus @ A) @ Y2) @ ((((topolo3827282254853284352ce_Lim @ A) @ A) @ F9) @ (^ [X:A]: (X)))))))))) @ ((topolo7230453075368039082e_nhds @ B) @ (zero_zero @ B))) @ F9))))))))). % 6.41/2.70 thf(fact_5707_finite__greaterThanAtMost, axiom, (! [L:nat,U:nat]: (((finite_finite2 @ nat) @ (((set_or3652927894154168847AtMost @ nat) @ L) @ U))))). % 6.41/2.70 thf(fact_5708_greaterThanAtMost__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,L:A,U:A]: (((((member @ A) @ I) @ (((set_or3652927894154168847AtMost @ A) @ L) @ U)) = ((((ord_less @ A) @ L) @ I) & (((ord_less_eq @ A) @ I) @ U))))))))). % 6.41/2.70 thf(fact_5709_greaterThanAtMost__empty, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,K:A]: (((((ord_less_eq @ A) @ L) @ K) => ((((set_or3652927894154168847AtMost @ A) @ K) @ L) = (bot_bot @ (set @ A)))))))))). % 6.41/2.70 thf(fact_5710_greaterThanAtMost__empty__iff2, axiom, (! [A:$tType]: (((preorder @ A) => (! [K:A,L:A]: ((((bot_bot @ (set @ A)) = (((set_or3652927894154168847AtMost @ A) @ K) @ L)) = (~ ((((ord_less @ A) @ K) @ L)))))))))). % 6.41/2.70 thf(fact_5711_greaterThanAtMost__empty__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [K:A,L:A]: ((((((set_or3652927894154168847AtMost @ A) @ K) @ L) = (bot_bot @ (set @ A))) = (~ ((((ord_less @ A) @ K) @ L)))))))))). % 6.41/2.70 thf(fact_5712_infinite__Ioc__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A]: (((~ (((finite_finite2 @ A) @ (((set_or3652927894154168847AtMost @ A) @ A2) @ B2)))) = (((ord_less @ A) @ A2) @ B2)))))))). % 6.41/2.70 thf(fact_5713_image__add__greaterThanAtMost, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [C2:A,A2:A,B2:A]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ C2)) @ (((set_or3652927894154168847AtMost @ A) @ A2) @ B2)) = (((set_or3652927894154168847AtMost @ A) @ (((plus_plus @ A) @ C2) @ A2)) @ (((plus_plus @ A) @ C2) @ B2))))))))). % 6.41/2.70 thf(fact_5714_cSup__greaterThanAtMost, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [Y:A,X3:A]: (((((ord_less @ A) @ Y) @ X3) => (((complete_Sup_Sup @ A) @ (((set_or3652927894154168847AtMost @ A) @ Y) @ X3)) = X3)))))))). % 6.41/2.70 thf(fact_5715_Sup__greaterThanAtMost, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (((complete_Sup_Sup @ A) @ (((set_or3652927894154168847AtMost @ A) @ X3) @ Y)) = Y)))))))). % 6.41/2.70 thf(fact_5716_cInf__greaterThanAtMost, axiom, (! [A:$tType]: ((((condit6923001295902523014norder @ A) & (dense_linorder @ A)) => (! [Y:A,X3:A]: (((((ord_less @ A) @ Y) @ X3) => (((complete_Inf_Inf @ A) @ (((set_or3652927894154168847AtMost @ A) @ Y) @ X3)) = Y)))))))). % 6.41/2.70 thf(fact_5717_Inf__greaterThanAtMost, axiom, (! [A:$tType]: ((((comple6319245703460814977attice @ A) & (dense_linorder @ A)) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (((complete_Inf_Inf @ A) @ (((set_or3652927894154168847AtMost @ A) @ X3) @ Y)) = X3)))))))). % 6.41/2.70 thf(fact_5718_Ioc__inj, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: ((((((set_or3652927894154168847AtMost @ A) @ A2) @ B2) = (((set_or3652927894154168847AtMost @ A) @ C2) @ D2)) = (((((ord_less_eq @ A) @ B2) @ A2) & (((ord_less_eq @ A) @ D2) @ C2)) | ((A2 = C2) & (B2 = D2)))))))))). % 6.41/2.70 thf(fact_5719_atLeastSucAtMost__greaterThanAtMost, axiom, (! [L:nat,U:nat]: (((((set_or1337092689740270186AtMost @ nat) @ (suc @ L)) @ U) = (((set_or3652927894154168847AtMost @ nat) @ L) @ U))))). % 6.41/2.70 thf(fact_5720_Ioc__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ A2) @ B2)) @ (((set_or3652927894154168847AtMost @ A) @ C2) @ D2)) = ((((ord_less_eq @ A) @ B2) @ A2) | ((((ord_less_eq @ A) @ C2) @ A2) & (((ord_less_eq @ A) @ B2) @ D2)))))))))). % 6.41/2.70 thf(fact_5721_infinite__Ioc, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ (((finite_finite2 @ A) @ (((set_or3652927894154168847AtMost @ A) @ A2) @ B2))))))))))). % 6.41/2.70 thf(fact_5722_ivl__disj__int__two_I6_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M)) @ (((set_or3652927894154168847AtMost @ A) @ M) @ U)) = (bot_bot @ (set @ A))))))))). % 6.41/2.70 thf(fact_5723_Ioc__disjoint, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: ((((((inf_inf @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ A2) @ B2)) @ (((set_or3652927894154168847AtMost @ A) @ C2) @ D2)) = (bot_bot @ (set @ A))) = ((((ord_less_eq @ A) @ B2) @ A2) | ((((ord_less_eq @ A) @ D2) @ C2) | ((((ord_less_eq @ A) @ B2) @ C2) | (((ord_less_eq @ A) @ D2) @ A2))))))))))). % 6.41/2.70 thf(fact_5724_ivl__disj__int__two_I8_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M)) @ (((set_or3652927894154168847AtMost @ A) @ M) @ U)) = (bot_bot @ (set @ A))))))))). % 6.41/2.70 thf(fact_5725_ivl__disj__int__one_I3_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ ((set_ord_atMost @ A) @ L)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ U)) = (bot_bot @ (set @ A))))))))). % 6.41/2.70 thf(fact_5726_ivl__disj__int__two_I2_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M)) @ (((set_or5935395276787703475ssThan @ A) @ M) @ U)) = (bot_bot @ (set @ A))))))))). % 6.41/2.70 thf(fact_5727_Lim__ident__at, axiom, (! [A:$tType]: (((topological_t2_space @ A) => (! [X3:A,S:(set @ A)]: ((((((topolo174197925503356063within @ A) @ X3) @ S) != (bot_bot @ (filter @ A))) => (((((topolo3827282254853284352ce_Lim @ A) @ A) @ (((topolo174197925503356063within @ A) @ X3) @ S)) @ (^ [X:A]: (X))) = X3)))))))). % 6.41/2.70 thf(fact_5728_eventually__at__left__real, axiom, (! [B2:real,A2:real]: (((((ord_less @ real) @ B2) @ A2) => (((eventually @ real) @ (^ [X:real]: ((((member @ real) @ X) @ (((set_or5935395276787703475ssThan @ real) @ B2) @ A2))))) @ (((topolo174197925503356063within @ real) @ A2) @ ((set_ord_lessThan @ real) @ A2))))))). % 6.41/2.70 thf(fact_5729_sum_Ohead, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M:nat,N2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (((plus_plus @ A) @ (G @ M)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or3652927894154168847AtMost @ nat) @ M) @ N2))))))))))). % 6.41/2.70 thf(fact_5730_not__tendsto__and__filterlim__at__infinity, axiom, (! [A:$tType,B:$tType]: (((real_V822414075346904944vector @ A) => (! [F5:(filter @ B),F2:(B > A),C2:A]: (((F5 != (bot_bot @ (filter @ B))) => ((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ C2)) @ F5) => (~ ((((((filterlim @ B) @ A) @ F2) @ (at_infinity @ A)) @ F5))))))))))). % 6.41/2.70 thf(fact_5731_prod_Ohead, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M:nat,N2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M) @ N2)) = (((times_times @ A) @ (G @ M)) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or3652927894154168847AtMost @ nat) @ M) @ N2))))))))))). % 6.41/2.70 thf(fact_5732_greaterThanAtMost__subseteq__atLeastAtMost__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ A2) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D2)) = ((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ A2) & (((ord_less_eq @ A) @ B2) @ D2)))))))))). % 6.41/2.70 thf(fact_5733_tendsto__Lim, axiom, (! [A:$tType,B:$tType]: (((topological_t2_space @ B) => (! [Net:(filter @ A),F2:(A > B),L:B]: (((Net != (bot_bot @ (filter @ A))) => ((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L)) @ Net) => (((((topolo3827282254853284352ce_Lim @ A) @ B) @ Net) @ F2) = L))))))))). % 6.41/2.70 thf(fact_5734_greaterThanAtMost__subseteq__atLeastLessThan__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ A2) @ B2)) @ (((set_or7035219750837199246ssThan @ A) @ C2) @ D2)) = ((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ A2) & (((ord_less @ A) @ B2) @ D2)))))))))). % 6.41/2.70 thf(fact_5735_greaterThanLessThan__subseteq__greaterThanAtMost__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A2:A,B2:A,C2:A,D2:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ A2) @ B2)) @ (((set_or3652927894154168847AtMost @ A) @ C2) @ D2)) = ((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C2) @ A2) & (((ord_less_eq @ A) @ B2) @ D2)))))))))). % 6.41/2.70 thf(fact_5736_greaterThanAtMost__eq__atLeastAtMost__diff, axiom, (! [A:$tType]: (((order @ A) => ((set_or3652927894154168847AtMost @ A) = (^ [A4:A,B4:A]: ((((minus_minus @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A4) @ B4)) @ (((insert @ A) @ A4) @ (bot_bot @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5737_tendsto__add__filterlim__at__infinity_H, axiom, (! [B:$tType,A:$tType]: (((real_V822414075346904944vector @ B) => (! [F2:(A > B),F5:(filter @ A),G:(A > B),C2:B]: (((((((filterlim @ A) @ B) @ F2) @ (at_infinity @ B)) @ F5) => ((((((filterlim @ A) @ B) @ G) @ ((topolo7230453075368039082e_nhds @ B) @ C2)) @ F5) => (((((filterlim @ A) @ B) @ (^ [X:A]: ((((plus_plus @ B) @ (F2 @ X)) @ (G @ X))))) @ (at_infinity @ B)) @ F5))))))))). % 6.41/2.70 thf(fact_5738_tendsto__add__filterlim__at__infinity, axiom, (! [B:$tType,A:$tType]: (((real_V822414075346904944vector @ B) => (! [F2:(A > B),C2:B,F5:(filter @ A),G:(A > B)]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ C2)) @ F5) => ((((((filterlim @ A) @ B) @ G) @ (at_infinity @ B)) @ F5) => (((((filterlim @ A) @ B) @ (^ [X:A]: ((((plus_plus @ B) @ (F2 @ X)) @ (G @ X))))) @ (at_infinity @ B)) @ F5))))))))). % 6.41/2.70 thf(fact_5739_continuous__divide, axiom, (! [B:$tType,A:$tType]: ((((topological_t2_space @ A) & (real_V3459762299906320749_field @ B)) => (! [F5:(filter @ A),F2:(A > B),G:(A > B)]: ((((((topolo3448309680560233919inuous @ A) @ B) @ F5) @ F2) => (((((topolo3448309680560233919inuous @ A) @ B) @ F5) @ G) => (((G @ ((((topolo3827282254853284352ce_Lim @ A) @ A) @ F5) @ (^ [X:A]: (X)))) != (zero_zero @ B)) => ((((topolo3448309680560233919inuous @ A) @ B) @ F5) @ (^ [X:A]: ((((divide_divide @ B) @ (F2 @ X)) @ (G @ X)))))))))))))). % 6.41/2.70 thf(fact_5740_continuous__inverse, axiom, (! [B:$tType,A:$tType]: ((((topological_t2_space @ A) & (real_V8999393235501362500lgebra @ B)) => (! [F5:(filter @ A),F2:(A > B)]: ((((((topolo3448309680560233919inuous @ A) @ B) @ F5) @ F2) => (((F2 @ ((((topolo3827282254853284352ce_Lim @ A) @ A) @ F5) @ (^ [X:A]: (X)))) != (zero_zero @ B)) => ((((topolo3448309680560233919inuous @ A) @ B) @ F5) @ (^ [X:A]: (((inverse_inverse @ B) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5741_continuous__sgn, axiom, (! [B:$tType,A:$tType]: ((((topological_t2_space @ A) & (real_V822414075346904944vector @ B)) => (! [F5:(filter @ A),F2:(A > B)]: ((((((topolo3448309680560233919inuous @ A) @ B) @ F5) @ F2) => (((F2 @ ((((topolo3827282254853284352ce_Lim @ A) @ A) @ F5) @ (^ [X:A]: (X)))) != (zero_zero @ B)) => ((((topolo3448309680560233919inuous @ A) @ B) @ F5) @ (^ [X:A]: (((sgn_sgn @ B) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5742_tendsto__inverse__0, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (((((filterlim @ A) @ A) @ (inverse_inverse @ A)) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ (at_infinity @ A)))))). % 6.41/2.70 thf(fact_5743_filterlim__at__infinity__imp__filterlim__at__top, axiom, (! [A:$tType,F2:(A > real),F5:(filter @ A)]: (((((((filterlim @ A) @ real) @ F2) @ (at_infinity @ real)) @ F5) => ((((eventually @ A) @ (^ [X:A]: ((((ord_less @ real) @ (zero_zero @ real)) @ (F2 @ X))))) @ F5) => (((((filterlim @ A) @ real) @ F2) @ (at_top @ real)) @ F5)))))). % 6.41/2.70 thf(fact_5744_tendsto__mult__filterlim__at__infinity, axiom, (! [A:$tType,B:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(B > A),C2:A,F5:(filter @ B),G:(B > A)]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ C2)) @ F5) => ((C2 != (zero_zero @ A)) => ((((((filterlim @ B) @ A) @ G) @ (at_infinity @ A)) @ F5) => (((((filterlim @ B) @ A) @ (^ [X:B]: ((((times_times @ A) @ (F2 @ X)) @ (G @ X))))) @ (at_infinity @ A)) @ F5)))))))))). % 6.41/2.70 thf(fact_5745_tendsto__divide__0, axiom, (! [A:$tType,C:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [F2:(C > A),C2:A,F5:(filter @ C),G:(C > A)]: (((((((filterlim @ C) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ C2)) @ F5) => ((((((filterlim @ C) @ A) @ G) @ (at_infinity @ A)) @ F5) => (((((filterlim @ C) @ A) @ (^ [X:C]: ((((divide_divide @ A) @ (F2 @ X)) @ (G @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ (zero_zero @ A))) @ F5))))))))). % 6.41/2.70 thf(fact_5746_filterlim__power__at__infinity, axiom, (! [B:$tType,A:$tType]: (((real_V8999393235501362500lgebra @ B) => (! [F2:(A > B),F5:(filter @ A),N2:nat]: (((((((filterlim @ A) @ B) @ F2) @ (at_infinity @ B)) @ F5) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((((filterlim @ A) @ B) @ (^ [X:A]: ((((power_power @ B) @ (F2 @ X)) @ N2)))) @ (at_infinity @ B)) @ F5))))))))). % 6.41/2.70 thf(fact_5747_continuous__tan, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [F5:(filter @ A),F2:(A > A)]: ((((((topolo3448309680560233919inuous @ A) @ A) @ F5) @ F2) => ((((cos @ A) @ (F2 @ ((((topolo3827282254853284352ce_Lim @ A) @ A) @ F5) @ (^ [X:A]: (X))))) != (zero_zero @ A)) => ((((topolo3448309680560233919inuous @ A) @ A) @ F5) @ (^ [X:A]: (((tan @ A) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5748_continuous__cot, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [F5:(filter @ A),F2:(A > A)]: ((((((topolo3448309680560233919inuous @ A) @ A) @ F5) @ F2) => ((((sin @ A) @ (F2 @ ((((topolo3827282254853284352ce_Lim @ A) @ A) @ F5) @ (^ [X:A]: (X))))) != (zero_zero @ A)) => ((((topolo3448309680560233919inuous @ A) @ A) @ F5) @ (^ [X:A]: (((cot @ A) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5749_continuous__tanh, axiom, (! [A:$tType,C:$tType]: ((((topological_t2_space @ C) & ((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A))) => (! [F5:(filter @ C),F2:(C > A)]: ((((((topolo3448309680560233919inuous @ C) @ A) @ F5) @ F2) => ((((cosh @ A) @ (F2 @ ((((topolo3827282254853284352ce_Lim @ C) @ C) @ F5) @ (^ [X:C]: (X))))) != (zero_zero @ A)) => ((((topolo3448309680560233919inuous @ C) @ A) @ F5) @ (^ [X:C]: (((tanh @ A) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5750_continuous__arcosh, axiom, (! [A:$tType]: (((topological_t2_space @ A) => (! [F5:(filter @ A),F2:(A > real)]: ((((((topolo3448309680560233919inuous @ A) @ real) @ F5) @ F2) => ((((ord_less @ real) @ (one_one @ real)) @ (F2 @ ((((topolo3827282254853284352ce_Lim @ A) @ A) @ F5) @ (^ [X:A]: (X))))) => ((((topolo3448309680560233919inuous @ A) @ real) @ F5) @ (^ [X:A]: (((arcosh @ real) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5751_filterlim__inverse__at__infinity, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (((((filterlim @ A) @ A) @ (inverse_inverse @ A)) @ (at_infinity @ A)) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A)))))))). % 6.41/2.70 thf(fact_5752_eventually__at__infinity__pos, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [P4:(A > $o)]: (((((eventually @ A) @ P4) @ (at_infinity @ A)) = (? [B4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ B4) & (! [X:A]: (((((ord_less_eq @ real) @ B4) @ ((real_V7770717601297561774m_norm @ A) @ X)) => (P4 @ X)))))))))))))). % 6.41/2.70 thf(fact_5753_filterlim__inverse__at__iff, axiom, (! [B:$tType,A:$tType]: (((real_V8999393235501362500lgebra @ B) => (! [G:(A > B),F5:(filter @ A)]: (((((((filterlim @ A) @ B) @ (^ [X:A]: (((inverse_inverse @ B) @ (G @ X))))) @ (((topolo174197925503356063within @ B) @ (zero_zero @ B)) @ (top_top @ (set @ B)))) @ F5) = (((((filterlim @ A) @ B) @ G) @ (at_infinity @ B)) @ F5)))))))). % 6.41/2.70 thf(fact_5754_nth__sorted__list__of__set__greaterThanAtMost, axiom, (! [N2:nat,J:nat,I:nat]: (((((ord_less @ nat) @ N2) @ (((minus_minus @ nat) @ J) @ I)) => ((((nth @ nat) @ ((linord4507533701916653071of_set @ nat) @ (((set_or3652927894154168847AtMost @ nat) @ I) @ J))) @ N2) = (suc @ (((plus_plus @ nat) @ I) @ N2))))))). % 6.41/2.70 thf(fact_5755_continuous__log, axiom, (! [A:$tType]: (((topological_t2_space @ A) => (! [F5:(filter @ A),F2:(A > real),G:(A > real)]: ((((((topolo3448309680560233919inuous @ A) @ real) @ F5) @ F2) => (((((topolo3448309680560233919inuous @ A) @ real) @ F5) @ G) => ((((ord_less @ real) @ (zero_zero @ real)) @ (F2 @ ((((topolo3827282254853284352ce_Lim @ A) @ A) @ F5) @ (^ [X:A]: (X))))) => (((F2 @ ((((topolo3827282254853284352ce_Lim @ A) @ A) @ F5) @ (^ [X:A]: (X)))) != (one_one @ real)) => ((((ord_less @ real) @ (zero_zero @ real)) @ (G @ ((((topolo3827282254853284352ce_Lim @ A) @ A) @ F5) @ (^ [X:A]: (X))))) => ((((topolo3448309680560233919inuous @ A) @ real) @ F5) @ (^ [X:A]: (((log @ (F2 @ X)) @ (G @ X)))))))))))))))). % 6.41/2.70 thf(fact_5756_filterlim__divide__at__infinity, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),C2:A,F5:(filter @ A),G:(A > A)]: (((((((filterlim @ A) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ C2)) @ F5) => ((((((filterlim @ A) @ A) @ G) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A)))) @ F5) => ((C2 != (zero_zero @ A)) => (((((filterlim @ A) @ A) @ (^ [X:A]: ((((divide_divide @ A) @ (F2 @ X)) @ (G @ X))))) @ (at_infinity @ A)) @ F5)))))))))). % 6.41/2.70 thf(fact_5757_filterlim__at__infinity, axiom, (! [C:$tType,A:$tType]: (((real_V822414075346904944vector @ A) => (! [C2:real,F2:(C > A),F5:(filter @ C)]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ C2) => ((((((filterlim @ C) @ A) @ F2) @ (at_infinity @ A)) @ F5) = (! [R5:real]: (((((ord_less @ real) @ C2) @ R5) => (((eventually @ C) @ (^ [X:C]: ((((ord_less_eq @ real) @ R5) @ ((real_V7770717601297561774m_norm @ A) @ (F2 @ X)))))) @ F5)))))))))))). % 6.41/2.70 thf(fact_5758_filterlim__realpow__sequentially__gt1, axiom, (! [A:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [X3:A]: (((((ord_less @ real) @ (one_one @ real)) @ ((real_V7770717601297561774m_norm @ A) @ X3)) => (((((filterlim @ nat) @ A) @ ((power_power @ A) @ X3)) @ (at_infinity @ A)) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5759_lim__at__infinity__0, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),L:A]: (((((((filterlim @ A) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ (at_infinity @ A)) = (((((filterlim @ A) @ A) @ (((((comp @ A) @ A) @ A) @ F2) @ (inverse_inverse @ A))) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5760_lim__zero__infinity, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),L:A]: (((((((filterlim @ A) @ A) @ (^ [X:A]: ((F2 @ (((divide_divide @ A) @ (one_one @ A)) @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A)))) => (((((filterlim @ A) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ (at_infinity @ A))))))))). % 6.41/2.70 thf(fact_5761_has__derivative__at__within__iff__Ex, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [X3:A,S3:(set @ A),F2:(A > B),F11:(A > B)]: (((((member @ A) @ X3) @ S3) => (((topolo1002775350975398744n_open @ A) @ S3) => ((((((has_derivative @ A) @ B) @ F2) @ F11) @ (((topolo174197925503356063within @ A) @ X3) @ S3)) = ((((real_V3181309239436604168linear @ A) @ B) @ F11) & (? [E4:(A > B)]: (((! [H:A]: (((((member @ A) @ (((plus_plus @ A) @ X3) @ H)) @ S3) => ((F2 @ (((plus_plus @ A) @ X3) @ H)) = (((plus_plus @ B) @ (((plus_plus @ B) @ (F2 @ X3)) @ (F11 @ H))) @ (E4 @ H)))))) & (((((filterlim @ A) @ real) @ (^ [H:A]: ((((divide_divide @ real) @ ((real_V7770717601297561774m_norm @ B) @ (E4 @ H))) @ ((real_V7770717601297561774m_norm @ A) @ H))))) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A))))))))))))))))). % 6.41/2.70 thf(fact_5762_has__derivativeI__sandwich, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [E3:real,F11:(A > B),S:(set @ A),X3:A,F2:(A > B),H7:(A > real)]: (((((ord_less @ real) @ (zero_zero @ real)) @ E3) => ((((real_V3181309239436604168linear @ A) @ B) @ F11) => ((! [Y4:A]: (((((member @ A) @ Y4) @ S) => ((Y4 != X3) => ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ Y4) @ X3)) @ E3) => (((ord_less_eq @ real) @ (((divide_divide @ real) @ ((real_V7770717601297561774m_norm @ B) @ (((minus_minus @ B) @ (((minus_minus @ B) @ (F2 @ Y4)) @ (F2 @ X3))) @ (F11 @ (((minus_minus @ A) @ Y4) @ X3))))) @ ((real_V7770717601297561774m_norm @ A) @ (((minus_minus @ A) @ Y4) @ X3)))) @ (H7 @ Y4))))))) => ((((((filterlim @ A) @ real) @ H7) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((((has_derivative @ A) @ B) @ F2) @ F11) @ (((topolo174197925503356063within @ A) @ X3) @ S)))))))))))). % 6.41/2.70 thf(fact_5763_filterlim__pow__at__bot__even, axiom, (! [N2:nat,F2:(real > real),F5:(filter @ real)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((((filterlim @ real) @ real) @ F2) @ (at_bot @ real)) @ F5) => ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2) => (((((filterlim @ real) @ real) @ (^ [X:real]: ((((power_power @ real) @ (F2 @ X)) @ N2)))) @ (at_top @ real)) @ F5))))))). % 6.41/2.70 thf(fact_5764_finite__greaterThanAtMost__int, axiom, (! [L:int,U:int]: (((finite_finite2 @ int) @ (((set_or3652927894154168847AtMost @ int) @ L) @ U))))). % 6.41/2.70 thf(fact_5765_open__empty, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => ((topolo1002775350975398744n_open @ A) @ (bot_bot @ (set @ A))))))). % 6.41/2.70 thf(fact_5766_dist__add__cancel2, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [B2:A,A2:A,C2:A]: (((((real_V557655796197034286t_dist @ A) @ (((plus_plus @ A) @ B2) @ A2)) @ (((plus_plus @ A) @ C2) @ A2)) = (((real_V557655796197034286t_dist @ A) @ B2) @ C2)))))))). % 6.41/2.70 thf(fact_5767_dist__add__cancel, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [A2:A,B2:A,C2:A]: (((((real_V557655796197034286t_dist @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (((plus_plus @ A) @ A2) @ C2)) = (((real_V557655796197034286t_dist @ A) @ B2) @ C2)))))))). % 6.41/2.70 thf(fact_5768_dist__0__norm, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X3:A]: (((((real_V557655796197034286t_dist @ A) @ (zero_zero @ A)) @ X3) = ((real_V7770717601297561774m_norm @ A) @ X3)))))))). % 6.41/2.70 thf(fact_5769_zero__less__dist__iff, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X3:A,Y:A]: (((((ord_less @ real) @ (zero_zero @ real)) @ (((real_V557655796197034286t_dist @ A) @ X3) @ Y)) = (X3 != Y)))))))). % 6.41/2.70 thf(fact_5770_open__Inter, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [S3:(set @ (set @ A))]: ((((finite_finite2 @ (set @ A)) @ S3) => ((! [X5:(set @ A)]: (((((member @ (set @ A)) @ X5) @ S3) => ((topolo1002775350975398744n_open @ A) @ X5)))) => ((topolo1002775350975398744n_open @ A) @ ((complete_Inf_Inf @ (set @ A)) @ S3)))))))))). % 6.41/2.70 thf(fact_5771_open__INT, axiom, (! [A:$tType,B:$tType]: (((topolo4958980785337419405_space @ A) => (! [A3:(set @ B),B3:(B > (set @ A))]: ((((finite_finite2 @ B) @ A3) => ((! [X5:B]: (((((member @ B) @ X5) @ A3) => ((topolo1002775350975398744n_open @ A) @ (B3 @ X5))))) => ((topolo1002775350975398744n_open @ A) @ ((complete_Inf_Inf @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B3) @ A3))))))))))). % 6.41/2.70 thf(fact_5772_not__open__singleton, axiom, (! [A:$tType]: (((topolo8386298272705272623_space @ A) => (! [X3:A]: ((~ (((topolo1002775350975398744n_open @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))). % 6.41/2.70 thf(fact_5773_separation__t2, axiom, (! [A:$tType]: (((topological_t2_space @ A) => (! [X3:A,Y:A]: (((X3 != Y) = (? [U5:(set @ A),V6:(set @ A)]: ((((topolo1002775350975398744n_open @ A) @ U5) & (((topolo1002775350975398744n_open @ A) @ V6) & ((((member @ A) @ X3) @ U5) & ((((member @ A) @ Y) @ V6) & ((((inf_inf @ (set @ A)) @ U5) @ V6) = (bot_bot @ (set @ A)))))))))))))))). % 6.41/2.70 thf(fact_5774_hausdorff, axiom, (! [A:$tType]: (((topological_t2_space @ A) => (! [X3:A,Y:A]: (((X3 != Y) => (? [U6:(set @ A),V7:(set @ A)]: ((((topolo1002775350975398744n_open @ A) @ U6) & (((topolo1002775350975398744n_open @ A) @ V7) & ((((member @ A) @ X3) @ U6) & ((((member @ A) @ Y) @ V7) & ((((inf_inf @ (set @ A)) @ U6) @ V7) = (bot_bot @ (set @ A)))))))))))))))). % 6.41/2.70 thf(fact_5775_trivial__limit__at__bot__linorder, axiom, (! [A:$tType]: (((linorder @ A) => ((at_bot @ A) != (bot_bot @ (filter @ A))))))). % 6.41/2.70 thf(fact_5776_first__countable__basis, axiom, (! [A:$tType]: (((topolo3112930676232923870pology @ A) => (! [X3:A]: ((? [A8:(nat > (set @ A))]: (((! [I5:nat]: (((((member @ A) @ X3) @ (A8 @ I5)) & ((topolo1002775350975398744n_open @ A) @ (A8 @ I5))))) & (! [S10:(set @ A)]: (((((topolo1002775350975398744n_open @ A) @ S10) & (((member @ A) @ X3) @ S10)) => (? [I4:nat]: ((((ord_less_eq @ (set @ A)) @ (A8 @ I4)) @ S10))))))))))))))). % 6.41/2.70 thf(fact_5777_open__subopen, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => ((topolo1002775350975398744n_open @ A) = (^ [S7:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ S7) => (? [T8:(set @ A)]: ((((topolo1002775350975398744n_open @ A) @ T8) & ((((member @ A) @ X) @ T8) & (((ord_less_eq @ (set @ A)) @ T8) @ S7))))))))))))))). % 6.41/2.70 thf(fact_5778_openI, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [S3:(set @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ S3) => (? [T10:(set @ A)]: ((((topolo1002775350975398744n_open @ A) @ T10) & ((((member @ A) @ X5) @ T10) & (((ord_less_eq @ (set @ A)) @ T10) @ S3)))))))) => ((topolo1002775350975398744n_open @ A) @ S3)))))))). % 6.41/2.70 thf(fact_5779_norm__conv__dist, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => ((real_V7770717601297561774m_norm @ A) = (^ [X:A]: ((((real_V557655796197034286t_dist @ A) @ X) @ (zero_zero @ A))))))))). % 6.41/2.70 thf(fact_5780_open__Collect__ex, axiom, (! [B:$tType,A:$tType]: (((topolo4958980785337419405_space @ A) => (! [P:(B > (A > $o))]: (((! [I4:B]: (((topolo1002775350975398744n_open @ A) @ ((collect @ A) @ (P @ I4))))) => ((topolo1002775350975398744n_open @ A) @ ((collect @ A) @ (^ [X:A]: ((? [I3:B]: (((P @ I3) @ X)))))))))))))). % 6.41/2.70 thf(fact_5781_dist__triangle__lt, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X3:A,Z2:A,Y:A,E3:real]: (((((ord_less @ real) @ (((plus_plus @ real) @ (((real_V557655796197034286t_dist @ A) @ X3) @ Z2)) @ (((real_V557655796197034286t_dist @ A) @ Y) @ Z2))) @ E3) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X3) @ Y)) @ E3)))))))). % 6.41/2.70 thf(fact_5782_dist__triangle__less__add, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X1:A,Y:A,E1:real,X22:A,E22:real]: (((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X1) @ Y)) @ E1) => ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X22) @ Y)) @ E22) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X1) @ X22)) @ (((plus_plus @ real) @ E1) @ E22)))))))))). % 6.41/2.70 thf(fact_5783_open__ball, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X3:A,D2:real]: (((topolo1002775350975398744n_open @ A) @ ((collect @ A) @ (^ [Y2:A]: ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X3) @ Y2)) @ D2))))))))))). % 6.41/2.70 thf(fact_5784_dist__commute__lessI, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [Y:A,X3:A,E3:real]: (((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ Y) @ X3)) @ E3) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X3) @ Y)) @ E3)))))))). % 6.41/2.70 thf(fact_5785_open__dist, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => ((topolo1002775350975398744n_open @ A) = (^ [S7:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ S7) => (? [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) & (! [Y2:A]: (((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ Y2) @ X)) @ E4) => (((member @ A) @ Y2) @ S7))))))))))))))))). % 6.41/2.70 thf(fact_5786_Inf__notin__open, axiom, (! [A:$tType]: (((topolo8458572112393995274pology @ A) => (! [A3:(set @ A),X3:A]: ((((topolo1002775350975398744n_open @ A) @ A3) => ((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((ord_less @ A) @ X3) @ X5)))) => (~ ((((member @ A) @ ((complete_Inf_Inf @ A) @ A3)) @ A3))))))))))). % 6.41/2.70 thf(fact_5787_Sup__notin__open, axiom, (! [A:$tType]: (((topolo8458572112393995274pology @ A) => (! [A3:(set @ A),X3:A]: ((((topolo1002775350975398744n_open @ A) @ A3) => ((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((ord_less @ A) @ X5) @ X3)))) => (~ ((((member @ A) @ ((complete_Sup_Sup @ A) @ A3)) @ A3))))))))))). % 6.41/2.70 thf(fact_5788_dist__pos__lt, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X3:A,Y:A]: (((X3 != Y) => (((ord_less @ real) @ (zero_zero @ real)) @ (((real_V557655796197034286t_dist @ A) @ X3) @ Y))))))))). % 6.41/2.70 thf(fact_5789_dist__not__less__zero, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X3:A,Y:A]: ((~ ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X3) @ Y)) @ (zero_zero @ real)))))))))). % 6.41/2.70 thf(fact_5790_open__diagonal__complement, axiom, (! [A:$tType]: (((topological_t2_space @ A) => ((topolo1002775350975398744n_open @ ((product_prod @ A) @ A)) @ ((collect @ ((product_prod @ A) @ A)) @ (^ [Uu3:((product_prod @ A) @ A)]: ((? [X:A,Y2:A]: (((Uu3 = ((((product_Pair @ A) @ A) @ X) @ Y2)) & (X != Y2)))))))))))). % 6.41/2.70 thf(fact_5791_at__within__open__subset, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [A2:A,S3:(set @ A),T4:(set @ A)]: (((((member @ A) @ A2) @ S3) => (((topolo1002775350975398744n_open @ A) @ S3) => ((((ord_less_eq @ (set @ A)) @ S3) @ T4) => ((((topolo174197925503356063within @ A) @ A2) @ T4) = (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))))))))))))). % 6.41/2.70 thf(fact_5792_open__right, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [S3:(set @ A),X3:A,Y:A]: ((((topolo1002775350975398744n_open @ A) @ S3) => ((((member @ A) @ X3) @ S3) => ((((ord_less @ A) @ X3) @ Y) => (? [B5:A]: (((((ord_less @ A) @ X3) @ B5) & (((ord_less_eq @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ X3) @ B5)) @ S3))))))))))))). % 6.41/2.70 thf(fact_5793_open__left, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [S3:(set @ A),X3:A,Y:A]: ((((topolo1002775350975398744n_open @ A) @ S3) => ((((member @ A) @ X3) @ S3) => ((((ord_less @ A) @ Y) @ X3) => (? [B5:A]: (((((ord_less @ A) @ B5) @ X3) & (((ord_less_eq @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ B5) @ X3)) @ S3))))))))))))). % 6.41/2.70 thf(fact_5794_open__subdiagonal, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => ((topolo1002775350975398744n_open @ ((product_prod @ A) @ A)) @ ((collect @ ((product_prod @ A) @ A)) @ (^ [Uu3:((product_prod @ A) @ A)]: ((? [X:A,Y2:A]: (((Uu3 = ((((product_Pair @ A) @ A) @ X) @ Y2)) & (((ord_less @ A) @ X) @ Y2)))))))))))). % 6.41/2.70 thf(fact_5795_open__superdiagonal, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => ((topolo1002775350975398744n_open @ ((product_prod @ A) @ A)) @ ((collect @ ((product_prod @ A) @ A)) @ (^ [Uu3:((product_prod @ A) @ A)]: ((? [X:A,Y2:A]: (((Uu3 = ((((product_Pair @ A) @ A) @ X) @ Y2)) & (((ord_less @ A) @ Y2) @ X)))))))))))). % 6.41/2.70 thf(fact_5796_eventually__at__bot__linorder, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > $o)]: (((((eventually @ A) @ P) @ (at_bot @ A)) = (? [N5:A]: ((! [N:A]: (((((ord_less_eq @ A) @ N) @ N5) => (P @ N))))))))))))). % 6.41/2.70 thf(fact_5797_eventually__at__bot__dense, axiom, (! [A:$tType]: ((((linorder @ A) & (no_bot @ A)) => (! [P:(A > $o)]: (((((eventually @ A) @ P) @ (at_bot @ A)) = (? [N5:A]: ((! [N:A]: (((((ord_less @ A) @ N) @ N5) => (P @ N))))))))))))). % 6.41/2.70 thf(fact_5798_countable__basis__at__decseq, axiom, (! [A:$tType]: (((topolo3112930676232923870pology @ A) => (! [X3:A]: ((~ ((! [A8:(nat > (set @ A))]: (((! [I5:nat]: (((topolo1002775350975398744n_open @ A) @ (A8 @ I5)))) => ((! [I5:nat]: ((((member @ A) @ X3) @ (A8 @ I5)))) => (~ ((! [S10:(set @ A)]: ((((topolo1002775350975398744n_open @ A) @ S10) => ((((member @ A) @ X3) @ S10) => (((eventually @ nat) @ (^ [I3:nat]: ((((ord_less_eq @ (set @ A)) @ (A8 @ I3)) @ S10)))) @ (at_top @ nat)))))))))))))))))))). % 6.41/2.70 thf(fact_5799_eventually__at, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [P:(A > $o),A2:A,S3:(set @ A)]: (((((eventually @ A) @ P) @ (((topolo174197925503356063within @ A) @ A2) @ S3)) = (? [D5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D5) & (! [X:A]: (((((member @ A) @ X) @ S3) => (((X != A2) & (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X) @ A2)) @ D5)) => (P @ X))))))))))))))). % 6.41/2.70 thf(fact_5800_eventually__nhds__metric, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [P:(A > $o),A2:A]: (((((eventually @ A) @ P) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) = (? [D5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D5) & (! [X:A]: (((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X) @ A2)) @ D5) => (P @ X)))))))))))))). % 6.41/2.70 thf(fact_5801_has__field__derivative__transform__within, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),F11:A,A2:A,S3:(set @ A),D2:real,G:(A > A)]: ((((((has_field_derivative @ A) @ F2) @ F11) @ (((topolo174197925503356063within @ A) @ A2) @ S3)) => ((((ord_less @ real) @ (zero_zero @ real)) @ D2) => ((((member @ A) @ A2) @ S3) => ((! [X5:A]: (((((member @ A) @ X5) @ S3) => ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X5) @ A2)) @ D2) => ((F2 @ X5) = (G @ X5)))))) => ((((has_field_derivative @ A) @ G) @ F11) @ (((topolo174197925503356063within @ A) @ A2) @ S3)))))))))))). % 6.41/2.70 thf(fact_5802_has__derivative__transform__within, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),F11:(A > B),X3:A,S:(set @ A),D2:real,G:(A > B)]: (((((((has_derivative @ A) @ B) @ F2) @ F11) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => ((((ord_less @ real) @ (zero_zero @ real)) @ D2) => ((((member @ A) @ X3) @ S) => ((! [X17:A]: (((((member @ A) @ X17) @ S) => ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X17) @ X3)) @ D2) => ((F2 @ X17) = (G @ X17)))))) => (((((has_derivative @ A) @ B) @ G) @ F11) @ (((topolo174197925503356063within @ A) @ X3) @ S)))))))))))). % 6.41/2.70 thf(fact_5803_eventually__le__at__bot, axiom, (! [A:$tType]: (((linorder @ A) => (! [C2:A]: ((((eventually @ A) @ (^ [X:A]: ((((ord_less_eq @ A) @ X) @ C2)))) @ (at_bot @ A)))))))). % 6.41/2.70 thf(fact_5804_eventually__gt__at__bot, axiom, (! [A:$tType]: (((unboun7993243217541854897norder @ A) => (! [C2:A]: ((((eventually @ A) @ (^ [X:A]: ((((ord_less @ A) @ X) @ C2)))) @ (at_bot @ A)))))))). % 6.41/2.70 thf(fact_5805_Cauchy__def, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => ((topolo3814608138187158403Cauchy @ A) = (^ [X6:(nat > A)]: ((! [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) => (? [M9:nat]: ((! [M2:nat]: (((((ord_less_eq @ nat) @ M9) @ M2) => (! [N:nat]: (((((ord_less_eq @ nat) @ M9) @ N) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ (X6 @ M2)) @ (X6 @ N))) @ E4))))))))))))))))))). % 6.41/2.70 thf(fact_5806_Cauchy__altdef2, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => ((topolo3814608138187158403Cauchy @ A) = (^ [S8:(nat > A)]: ((! [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) => (? [N5:nat]: ((! [N:nat]: (((((ord_less_eq @ nat) @ N5) @ N) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ (S8 @ N)) @ (S8 @ N5))) @ E4)))))))))))))))). % 6.41/2.70 thf(fact_5807_metric__CauchyD, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X8:(nat > A),E3:real]: ((((topolo3814608138187158403Cauchy @ A) @ X8) => ((((ord_less @ real) @ (zero_zero @ real)) @ E3) => (? [M8:nat]: ((! [M6:nat]: (((((ord_less_eq @ nat) @ M8) @ M6) => (! [N6:nat]: (((((ord_less_eq @ nat) @ M8) @ N6) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ (X8 @ M6)) @ (X8 @ N6))) @ E3))))))))))))))))). % 6.41/2.70 thf(fact_5808_metric__CauchyI, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X8:(nat > A)]: (((! [E:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E) => (? [M10:nat]: ((! [M3:nat]: (((((ord_less_eq @ nat) @ M10) @ M3) => (! [N3:nat]: (((((ord_less_eq @ nat) @ M10) @ N3) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ (X8 @ M3)) @ (X8 @ N3))) @ E)))))))))))) => ((topolo3814608138187158403Cauchy @ A) @ X8)))))))). % 6.41/2.70 thf(fact_5809_lim__explicit, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [F2:(nat > A),F0:A]: (((((((filterlim @ nat) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ F0)) @ (at_top @ nat)) = (! [S7:(set @ A)]: ((((topolo1002775350975398744n_open @ A) @ S7) => ((((member @ A) @ F0) @ S7) => (? [N5:nat]: ((! [N:nat]: (((((ord_less_eq @ nat) @ N5) @ N) => (((member @ A) @ (F2 @ N)) @ S7))))))))))))))))). % 6.41/2.70 thf(fact_5810_Lim__transform__within, axiom, (! [B:$tType,A:$tType]: ((((real_V7819770556892013058_space @ A) & (topolo4958980785337419405_space @ B)) => (! [F2:(A > B),L:B,X3:A,S3:(set @ A),D2:real,G:(A > B)]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L)) @ (((topolo174197925503356063within @ A) @ X3) @ S3)) => ((((ord_less @ real) @ (zero_zero @ real)) @ D2) => ((! [X17:A]: (((((member @ A) @ X17) @ S3) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((real_V557655796197034286t_dist @ A) @ X17) @ X3)) => ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X17) @ X3)) @ D2) => ((F2 @ X17) = (G @ X17))))))) => (((((filterlim @ A) @ B) @ G) @ ((topolo7230453075368039082e_nhds @ B) @ L)) @ (((topolo174197925503356063within @ A) @ X3) @ S3))))))))))). % 6.41/2.70 thf(fact_5811_dist__triangle__half__r, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [Y:A,X1:A,E3:real,X22:A]: (((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ Y) @ X1)) @ (((divide_divide @ real) @ E3) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ Y) @ X22)) @ (((divide_divide @ real) @ E3) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X1) @ X22)) @ E3))))))))). % 6.41/2.70 thf(fact_5812_dist__triangle__half__l, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X1:A,Y:A,E3:real,X22:A]: (((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X1) @ Y)) @ (((divide_divide @ real) @ E3) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X22) @ Y)) @ (((divide_divide @ real) @ E3) @ ((numeral_numeral @ real) @ (bit0 @ one2)))) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X1) @ X22)) @ E3))))))))). % 6.41/2.70 thf(fact_5813_eventually__at__le, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [P:(A > $o),A2:A,S3:(set @ A)]: (((((eventually @ A) @ P) @ (((topolo174197925503356063within @ A) @ A2) @ S3)) = (? [D5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D5) & (! [X:A]: (((((member @ A) @ X) @ S3) => (((X != A2) & (((ord_less_eq @ real) @ (((real_V557655796197034286t_dist @ A) @ X) @ A2)) @ D5)) => (P @ X))))))))))))))). % 6.41/2.70 thf(fact_5814_dist__triangle__third, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X1:A,X22:A,E3:real,X32:A,X42:A]: (((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X1) @ X22)) @ (((divide_divide @ real) @ E3) @ ((numeral_numeral @ real) @ (bit1 @ one2)))) => ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X22) @ X32)) @ (((divide_divide @ real) @ E3) @ ((numeral_numeral @ real) @ (bit1 @ one2)))) => ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X32) @ X42)) @ (((divide_divide @ real) @ E3) @ ((numeral_numeral @ real) @ (bit1 @ one2)))) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X1) @ X42)) @ E3)))))))))). % 6.41/2.70 thf(fact_5815_at__within__nhd, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [X3:A,S3:(set @ A),T4:(set @ A),U3:(set @ A)]: (((((member @ A) @ X3) @ S3) => (((topolo1002775350975398744n_open @ A) @ S3) => (((((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ T4) @ S3)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ U3) @ S3)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) => ((((topolo174197925503356063within @ A) @ X3) @ T4) = (((topolo174197925503356063within @ A) @ X3) @ U3))))))))))). % 6.41/2.70 thf(fact_5816_filterlim__transform__within, axiom, (! [B:$tType,A:$tType]: (((real_V7819770556892013058_space @ A) => (! [G:(A > B),G6:(filter @ B),X3:A,S3:(set @ A),F5:(filter @ B),D2:real,F2:(A > B)]: (((((((filterlim @ A) @ B) @ G) @ G6) @ (((topolo174197925503356063within @ A) @ X3) @ S3)) => ((((ord_less_eq @ (filter @ B)) @ G6) @ F5) => ((((ord_less @ real) @ (zero_zero @ real)) @ D2) => ((! [X17:A]: (((((member @ A) @ X17) @ S3) => ((((ord_less @ real) @ (zero_zero @ real)) @ (((real_V557655796197034286t_dist @ A) @ X17) @ X3)) => ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X17) @ X3)) @ D2) => ((F2 @ X17) = (G @ X17))))))) => (((((filterlim @ A) @ B) @ F2) @ F5) @ (((topolo174197925503356063within @ A) @ X3) @ S3)))))))))))). % 6.41/2.70 thf(fact_5817_Cauchy__altdef, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => ((topolo3814608138187158403Cauchy @ A) = (^ [F3:(nat > A)]: ((! [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) => (? [M9:nat]: ((! [M2:nat]: (((((ord_less_eq @ nat) @ M9) @ M2) => (! [N:nat]: (((((ord_less @ nat) @ M2) @ N) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ (F3 @ M2)) @ (F3 @ N))) @ E4))))))))))))))))))). % 6.41/2.70 thf(fact_5818_CauchyI_H, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X8:(nat > A)]: (((! [E:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E) => (? [M10:nat]: ((! [M3:nat]: (((((ord_less_eq @ nat) @ M10) @ M3) => (! [N3:nat]: (((((ord_less @ nat) @ M3) @ N3) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ (X8 @ M3)) @ (X8 @ N3))) @ E)))))))))))) => ((topolo3814608138187158403Cauchy @ A) @ X8)))))))). % 6.41/2.70 thf(fact_5819_at__eq__bot__iff, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [A2:A]: ((((((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))) = (bot_bot @ (filter @ A))) = ((topolo1002775350975398744n_open @ A) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5820_filterlim__at__infinity__imp__filterlim__at__bot, axiom, (! [A:$tType,F2:(A > real),F5:(filter @ A)]: (((((((filterlim @ A) @ real) @ F2) @ (at_infinity @ real)) @ F5) => ((((eventually @ A) @ (^ [X:A]: ((((ord_less @ real) @ (F2 @ X)) @ (zero_zero @ real))))) @ F5) => (((((filterlim @ A) @ real) @ F2) @ (at_bot @ real)) @ F5)))))). % 6.41/2.70 thf(fact_5821_tendstoD, axiom, (! [A:$tType,B:$tType]: (((real_V7819770556892013058_space @ A) => (! [F2:(B > A),L:A,F5:(filter @ B),E3:real]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ F5) => ((((ord_less @ real) @ (zero_zero @ real)) @ E3) => (((eventually @ B) @ (^ [X:B]: ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ (F2 @ X)) @ L)) @ E3)))) @ F5))))))))). % 6.41/2.70 thf(fact_5822_tendstoI, axiom, (! [A:$tType,B:$tType]: (((real_V7819770556892013058_space @ A) => (! [F2:(B > A),L:A,F5:(filter @ B)]: (((! [E:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E) => (((eventually @ B) @ (^ [X:B]: ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ (F2 @ X)) @ L)) @ E)))) @ F5)))) => (((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ F5)))))))). % 6.41/2.70 thf(fact_5823_tendsto__iff, axiom, (! [B:$tType,A:$tType]: (((real_V7819770556892013058_space @ A) => (! [F2:(B > A),L:A,F5:(filter @ B)]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ F5) = (! [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) => (((eventually @ B) @ (^ [X:B]: ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ (F2 @ X)) @ L)) @ E4)))) @ F5))))))))))). % 6.41/2.70 thf(fact_5824_filterlim__at__bot__le, axiom, (! [A:$tType,B:$tType]: (((linorder @ B) => (! [F2:(A > B),F5:(filter @ A),C2:B]: (((((((filterlim @ A) @ B) @ F2) @ (at_bot @ B)) @ F5) = (! [Z9:B]: (((((ord_less_eq @ B) @ Z9) @ C2) => (((eventually @ A) @ (^ [X:A]: ((((ord_less_eq @ B) @ (F2 @ X)) @ Z9)))) @ F5))))))))))). % 6.41/2.70 thf(fact_5825_filterlim__at__bot, axiom, (! [A:$tType,B:$tType]: (((linorder @ B) => (! [F2:(A > B),F5:(filter @ A)]: (((((((filterlim @ A) @ B) @ F2) @ (at_bot @ B)) @ F5) = (! [Z9:B]: ((((eventually @ A) @ (^ [X:A]: ((((ord_less_eq @ B) @ (F2 @ X)) @ Z9)))) @ F5)))))))))). % 6.41/2.70 thf(fact_5826_filterlim__at__bot__dense, axiom, (! [A:$tType,B:$tType]: ((((dense_linorder @ B) & (no_bot @ B)) => (! [F2:(A > B),F5:(filter @ A)]: (((((((filterlim @ A) @ B) @ F2) @ (at_bot @ B)) @ F5) = (! [Z9:B]: ((((eventually @ A) @ (^ [X:A]: ((((ord_less @ B) @ (F2 @ X)) @ Z9)))) @ F5)))))))))). % 6.41/2.70 thf(fact_5827_metric__LIM__equal2, axiom, (! [B:$tType,A:$tType]: ((((real_V7819770556892013058_space @ A) & (topolo4958980785337419405_space @ B)) => (! [G:(A > B),L:B,A2:A,R:real,F2:(A > B)]: (((((((filterlim @ A) @ B) @ G) @ ((topolo7230453075368039082e_nhds @ B) @ L)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) => ((((ord_less @ real) @ (zero_zero @ real)) @ R) => ((! [X5:A]: (((X5 != A2) => ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X5) @ A2)) @ R) => ((F2 @ X5) = (G @ X5)))))) => (((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))))))))))))). % 6.41/2.70 thf(fact_5828_metric__LIM__I, axiom, (! [B:$tType,A:$tType]: ((((real_V7819770556892013058_space @ A) & (real_V7819770556892013058_space @ B)) => (! [A2:A,F2:(A > B),L5:B]: (((! [R4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R4) => (? [S9:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ S9) & (! [X5:A]: ((((X5 != A2) & (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X5) @ A2)) @ S9)) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ B) @ (F2 @ X5)) @ L5)) @ R4)))))))))) => (((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L5)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5829_metric__LIM__D, axiom, (! [B:$tType,A:$tType]: ((((real_V7819770556892013058_space @ A) & (real_V7819770556892013058_space @ B)) => (! [F2:(A > B),L5:B,A2:A,R2:real]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L5)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) => ((((ord_less @ real) @ (zero_zero @ real)) @ R2) => (? [S2:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ S2) & (! [X2:A]: ((((X2 != A2) & (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X2) @ A2)) @ S2)) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ B) @ (F2 @ X2)) @ L5)) @ R2))))))))))))))). % 6.41/2.70 thf(fact_5830_LIM__def, axiom, (! [B:$tType,A:$tType]: ((((real_V7819770556892013058_space @ A) & (real_V7819770556892013058_space @ B)) => (! [F2:(A > B),L5:B,A2:A]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L5)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) = (! [R5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R5) => (? [S8:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ S8) & (! [X:A]: ((((X != A2) & (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X) @ A2)) @ S8)) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ B) @ (F2 @ X)) @ L5)) @ R5))))))))))))))))). % 6.41/2.70 thf(fact_5831_metric__LIMSEQ__D, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X8:(nat > A),L5:A,R2:real]: (((((((filterlim @ nat) @ A) @ X8) @ ((topolo7230453075368039082e_nhds @ A) @ L5)) @ (at_top @ nat)) => ((((ord_less @ real) @ (zero_zero @ real)) @ R2) => (? [No:nat]: ((! [N6:nat]: (((((ord_less_eq @ nat) @ No) @ N6) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ (X8 @ N6)) @ L5)) @ R2)))))))))))))). % 6.41/2.70 thf(fact_5832_metric__LIMSEQ__I, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X8:(nat > A),L5:A]: (((! [R4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R4) => (? [No2:nat]: ((! [N3:nat]: (((((ord_less_eq @ nat) @ No2) @ N3) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ (X8 @ N3)) @ L5)) @ R4))))))))) => (((((filterlim @ nat) @ A) @ X8) @ ((topolo7230453075368039082e_nhds @ A) @ L5)) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5833_lim__sequentially, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X8:(nat > A),L5:A]: (((((((filterlim @ nat) @ A) @ X8) @ ((topolo7230453075368039082e_nhds @ A) @ L5)) @ (at_top @ nat)) = (! [R5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R5) => (? [No3:nat]: ((! [N:nat]: (((((ord_less_eq @ nat) @ No3) @ N) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ (X8 @ N)) @ L5)) @ R5)))))))))))))))). % 6.41/2.70 thf(fact_5834_metric__Cauchy__iff2, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => ((topolo3814608138187158403Cauchy @ A) = (^ [X6:(nat > A)]: ((! [J3:nat]: ((? [M9:nat]: ((! [M2:nat]: (((((ord_less_eq @ nat) @ M9) @ M2) => (! [N:nat]: (((((ord_less_eq @ nat) @ M9) @ N) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ (X6 @ M2)) @ (X6 @ N))) @ ((inverse_inverse @ real) @ ((semiring_1_of_nat @ real) @ (suc @ J3))))))))))))))))))))). % 6.41/2.70 thf(fact_5835_metric__LIM__compose2, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((real_V7819770556892013058_space @ A) & ((topolo4958980785337419405_space @ B) & (topolo4958980785337419405_space @ C))) => (! [F2:(A > B),B2:B,A2:A,G:(B > C),C2:C]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ B2)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) => ((((((filterlim @ B) @ C) @ G) @ ((topolo7230453075368039082e_nhds @ C) @ C2)) @ (((topolo174197925503356063within @ B) @ B2) @ (top_top @ (set @ B)))) => ((? [D4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D4) & (! [X5:A]: ((((X5 != A2) & (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X5) @ A2)) @ D4)) => ((F2 @ X5) != B2))))))) => (((((filterlim @ A) @ C) @ (^ [X:A]: ((G @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ C) @ C2)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))))))))))))). % 6.41/2.70 thf(fact_5836_filterlim__tendsto__pos__mult__at__bot, axiom, (! [A:$tType,F2:(A > real),C2:real,F5:(filter @ A),G:(A > real)]: (((((((filterlim @ A) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ C2)) @ F5) => ((((ord_less @ real) @ (zero_zero @ real)) @ C2) => ((((((filterlim @ A) @ real) @ G) @ (at_bot @ real)) @ F5) => (((((filterlim @ A) @ real) @ (^ [X:A]: ((((times_times @ real) @ (F2 @ X)) @ (G @ X))))) @ (at_bot @ real)) @ F5))))))). % 6.41/2.70 thf(fact_5837_filterlim__at__bot__lt, axiom, (! [A:$tType,B:$tType]: (((unboun7993243217541854897norder @ B) => (! [F2:(A > B),F5:(filter @ A),C2:B]: (((((((filterlim @ A) @ B) @ F2) @ (at_bot @ B)) @ F5) = (! [Z9:B]: (((((ord_less @ B) @ Z9) @ C2) => (((eventually @ A) @ (^ [X:A]: ((((ord_less_eq @ B) @ (F2 @ X)) @ Z9)))) @ F5))))))))))). % 6.41/2.70 thf(fact_5838_metric__isCont__LIM__compose2, axiom, (! [D:$tType,C:$tType,A:$tType]: ((((real_V7819770556892013058_space @ A) & ((topolo4958980785337419405_space @ C) & (topolo4958980785337419405_space @ D))) => (! [A2:A,F2:(A > C),G:(C > D),L:D]: ((((((topolo3448309680560233919inuous @ A) @ C) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ F2) => ((((((filterlim @ C) @ D) @ G) @ ((topolo7230453075368039082e_nhds @ D) @ L)) @ (((topolo174197925503356063within @ C) @ (F2 @ A2)) @ (top_top @ (set @ C)))) => ((? [D4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D4) & (! [X5:A]: ((((X5 != A2) & (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X5) @ A2)) @ D4)) => ((F2 @ X5) != (F2 @ A2)))))))) => (((((filterlim @ A) @ D) @ (^ [X:A]: ((G @ (F2 @ X))))) @ ((topolo7230453075368039082e_nhds @ D) @ L)) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))))))))))))). % 6.41/2.70 thf(fact_5839_tendsto__offset__zero__iff, axiom, (! [C:$tType,D:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & ((topolo4958980785337419405_space @ D) & (zero @ C))) => (! [A2:A,S3:(set @ A),F2:(A > D),L5:D]: ((((((nO_MATCH @ C) @ A) @ (zero_zero @ C)) @ A2) => ((((member @ A) @ A2) @ S3) => (((topolo1002775350975398744n_open @ A) @ S3) => ((((((filterlim @ A) @ D) @ F2) @ ((topolo7230453075368039082e_nhds @ D) @ L5)) @ (((topolo174197925503356063within @ A) @ A2) @ S3)) = (((((filterlim @ A) @ D) @ (^ [H:A]: ((F2 @ (((plus_plus @ A) @ A2) @ H))))) @ ((topolo7230453075368039082e_nhds @ D) @ L5)) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A)))))))))))))). % 6.41/2.70 thf(fact_5840_filterlim__inverse__at__bot, axiom, (! [A:$tType,F2:(A > real),F5:(filter @ A)]: (((((((filterlim @ A) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ (zero_zero @ real))) @ F5) => ((((eventually @ A) @ (^ [X:A]: ((((ord_less @ real) @ (F2 @ X)) @ (zero_zero @ real))))) @ F5) => (((((filterlim @ A) @ real) @ (^ [X:A]: (((inverse_inverse @ real) @ (F2 @ X))))) @ (at_bot @ real)) @ F5)))))). % 6.41/2.70 thf(fact_5841_filterlim__tendsto__neg__mult__at__bot, axiom, (! [A:$tType,F2:(A > real),C2:real,F5:(filter @ A),G:(A > real)]: (((((((filterlim @ A) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ C2)) @ F5) => ((((ord_less @ real) @ C2) @ (zero_zero @ real)) => ((((((filterlim @ A) @ real) @ G) @ (at_top @ real)) @ F5) => (((((filterlim @ A) @ real) @ (^ [X:A]: ((((times_times @ real) @ (F2 @ X)) @ (G @ X))))) @ (at_bot @ real)) @ F5))))))). % 6.41/2.70 thf(fact_5842_totally__bounded__metric, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => ((topolo6688025880775521714ounded @ A) = (^ [S7:(set @ A)]: ((! [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) => (? [K3:(set @ A)]: ((((finite_finite2 @ A) @ K3) & (((ord_less_eq @ (set @ A)) @ S7) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ A) @ (set @ A)) @ (^ [X:A]: (((collect @ A) @ (^ [Y2:A]: ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X) @ Y2)) @ E4))))))) @ K3)))))))))))))))). % 6.41/2.70 thf(fact_5843_LIMSEQ__iff__nz, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X8:(nat > A),L5:A]: (((((((filterlim @ nat) @ A) @ X8) @ ((topolo7230453075368039082e_nhds @ A) @ L5)) @ (at_top @ nat)) = (! [R5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R5) => (? [No3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ No3) & (! [N:nat]: (((((ord_less_eq @ nat) @ No3) @ N) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ (X8 @ N)) @ L5)) @ R5))))))))))))))))). % 6.41/2.70 thf(fact_5844_DERIV__pos__imp__increasing__at__bot, axiom, (! [B2:real,F2:(real > real),Flim:real]: (((! [X5:real]: (((((ord_less_eq @ real) @ X5) @ B2) => (? [Y5:real]: ((((((has_field_derivative @ real) @ F2) @ Y5) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))) & (((ord_less @ real) @ (zero_zero @ real)) @ Y5))))))) => ((((((filterlim @ real) @ real) @ F2) @ ((topolo7230453075368039082e_nhds @ real) @ Flim)) @ (at_bot @ real)) => (((ord_less @ real) @ Flim) @ (F2 @ B2))))))). % 6.41/2.70 thf(fact_5845_filterlim__pow__at__bot__odd, axiom, (! [N2:nat,F2:(real > real),F5:(filter @ real)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((((filterlim @ real) @ real) @ F2) @ (at_bot @ real)) @ F5) => ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N2))) => (((((filterlim @ real) @ real) @ (^ [X:real]: ((((power_power @ real) @ (F2 @ X)) @ N2)))) @ (at_bot @ real)) @ F5))))))). % 6.41/2.70 thf(fact_5846_Bfun__metric__def, axiom, (! [B:$tType,A:$tType]: (((real_V7819770556892013058_space @ B) => (((bfun @ A) @ B) = (^ [F3:(A > B),F9:(filter @ A)]: ((? [Y2:B,K6:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ K6) & (((eventually @ A) @ (^ [X:A]: ((((ord_less_eq @ real) @ (((real_V557655796197034286t_dist @ B) @ (F3 @ X)) @ Y2)) @ K6)))) @ F9))))))))))). % 6.41/2.70 thf(fact_5847_Bfun__def, axiom, (! [B:$tType,A:$tType]: (((real_V822414075346904944vector @ B) => (((bfun @ A) @ B) = (^ [F3:(A > B),F9:(filter @ A)]: ((? [K6:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ K6) & (((eventually @ A) @ (^ [X:A]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ B) @ (F3 @ X))) @ K6)))) @ F9))))))))))). % 6.41/2.70 thf(fact_5848_BfunE, axiom, (! [B:$tType,A:$tType]: (((real_V822414075346904944vector @ B) => (! [F2:(A > B),F5:(filter @ A)]: ((((((bfun @ A) @ B) @ F2) @ F5) => (~ ((! [B10:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ B10) => (~ ((((eventually @ A) @ (^ [X:A]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ B) @ (F2 @ X))) @ B10)))) @ F5))))))))))))))). % 6.41/2.70 thf(fact_5849_Bseq__add__iff, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),C2:A]: ((((((bfun @ nat) @ A) @ (^ [X:nat]: ((((plus_plus @ A) @ (F2 @ X)) @ C2)))) @ (at_top @ nat)) = ((((bfun @ nat) @ A) @ F2) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5850_Bseq__add, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),C2:A]: ((((((bfun @ nat) @ A) @ F2) @ (at_top @ nat)) => ((((bfun @ nat) @ A) @ (^ [X:nat]: ((((plus_plus @ A) @ (F2 @ X)) @ C2)))) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5851_Bseq__Suc__iff, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A)]: ((((((bfun @ nat) @ A) @ (^ [N:nat]: ((F2 @ (suc @ N))))) @ (at_top @ nat)) = ((((bfun @ nat) @ A) @ F2) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5852_Bseq__ignore__initial__segment, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X8:(nat > A),K:nat]: ((((((bfun @ nat) @ A) @ X8) @ (at_top @ nat)) => ((((bfun @ nat) @ A) @ (^ [N:nat]: ((X8 @ (((plus_plus @ nat) @ N) @ K))))) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5853_Bseq__offset, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => (! [X8:(nat > A),K:nat]: ((((((bfun @ nat) @ A) @ (^ [N:nat]: ((X8 @ (((plus_plus @ nat) @ N) @ K))))) @ (at_top @ nat)) => ((((bfun @ nat) @ A) @ X8) @ (at_top @ nat))))))))). % 6.41/2.70 thf(fact_5854_Bseq__cmult__iff, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [C2:A,F2:(nat > A)]: (((C2 != (zero_zero @ A)) => (((((bfun @ nat) @ A) @ (^ [X:nat]: ((((times_times @ A) @ C2) @ (F2 @ X))))) @ (at_top @ nat)) = ((((bfun @ nat) @ A) @ F2) @ (at_top @ nat)))))))))). % 6.41/2.70 thf(fact_5855_BseqD, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X8:(nat > A)]: ((((((bfun @ nat) @ A) @ X8) @ (at_top @ nat)) => (? [K8:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ K8) & (! [N6:nat]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (X8 @ N6))) @ K8))))))))))))). % 6.41/2.70 thf(fact_5856_BseqE, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X8:(nat > A)]: ((((((bfun @ nat) @ A) @ X8) @ (at_top @ nat)) => (~ ((! [K8:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ K8) => (~ ((! [N6:nat]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (X8 @ N6))) @ K8))))))))))))))))). % 6.41/2.70 thf(fact_5857_BseqI, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [K5:real,X8:(nat > A)]: (((((ord_less @ real) @ (zero_zero @ real)) @ K5) => ((! [N3:nat]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (X8 @ N3))) @ K5))) => ((((bfun @ nat) @ A) @ X8) @ (at_top @ nat)))))))))). % 6.41/2.70 thf(fact_5858_Bseq__def, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X8:(nat > A)]: ((((((bfun @ nat) @ A) @ X8) @ (at_top @ nat)) = (? [K6:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ K6) & (! [N:nat]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (X8 @ N))) @ K6))))))))))))). % 6.41/2.70 thf(fact_5859_Bseq__iff1a, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X8:(nat > A)]: ((((((bfun @ nat) @ A) @ X8) @ (at_top @ nat)) = (? [N5:nat]: ((! [N:nat]: ((((ord_less @ real) @ ((real_V7770717601297561774m_norm @ A) @ (X8 @ N))) @ ((semiring_1_of_nat @ real) @ (suc @ N5)))))))))))))). % 6.41/2.70 thf(fact_5860_Bseq__iff, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X8:(nat > A)]: ((((((bfun @ nat) @ A) @ X8) @ (at_top @ nat)) = (? [N5:nat]: ((! [N:nat]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (X8 @ N))) @ ((semiring_1_of_nat @ real) @ (suc @ N5)))))))))))))). % 6.41/2.70 thf(fact_5861_Bseq__iff2, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X8:(nat > A)]: ((((((bfun @ nat) @ A) @ X8) @ (at_top @ nat)) = (? [K3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ K3) & (? [X:A]: ((! [N:nat]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((plus_plus @ A) @ (X8 @ N)) @ ((uminus_uminus @ A) @ X)))) @ K3))))))))))))))). % 6.41/2.70 thf(fact_5862_Bseq__iff3, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [X8:(nat > A)]: ((((((bfun @ nat) @ A) @ X8) @ (at_top @ nat)) = (? [K3:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ K3) & (? [N5:nat]: ((! [N:nat]: ((((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (((plus_plus @ A) @ (X8 @ N)) @ ((uminus_uminus @ A) @ (X8 @ N5))))) @ K3))))))))))))))). % 6.41/2.70 thf(fact_5863_Bfun__inverse, axiom, (! [A:$tType,B:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [F2:(B > A),A2:A,F5:(filter @ B)]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ F5) => ((A2 != (zero_zero @ A)) => ((((bfun @ B) @ A) @ (^ [X:B]: (((inverse_inverse @ A) @ (F2 @ X))))) @ F5))))))))). % 6.41/2.70 thf(fact_5864_map__filter__on__comp, axiom, (! [A:$tType,C:$tType,B:$tType,G:(B > A),Y7:(set @ B),X8:(set @ A),F5:(filter @ B),F2:(A > C)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ G) @ Y7)) @ X8) => ((((eventually @ B) @ (^ [X:B]: ((((member @ B) @ X) @ Y7)))) @ F5) => ((((((map_filter_on @ A) @ C) @ X8) @ F2) @ (((((map_filter_on @ B) @ A) @ Y7) @ G) @ F5)) = (((((map_filter_on @ B) @ C) @ Y7) @ (((((comp @ A) @ C) @ B) @ F2) @ G)) @ F5))))))). % 6.41/2.70 thf(fact_5865_cauchy__filter__metric, axiom, (! [A:$tType]: ((((real_V768167426530841204y_dist @ A) & (topolo7287701948861334536_space @ A)) => ((topolo6773858410816713723filter @ A) = (^ [F9:(filter @ A)]: ((! [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) => (? [P3:(A > $o)]: (((((eventually @ A) @ P3) @ F9) & (! [X:A,Y2:A]: ((((P3 @ X) & (P3 @ Y2)) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X) @ Y2)) @ E4))))))))))))))))). % 6.41/2.70 thf(fact_5866_GMVT, axiom, (! [A2:real,B2:real,F2:(real > real),G:(real > real)]: (((((ord_less @ real) @ A2) @ B2) => ((! [X5:real]: ((((((ord_less_eq @ real) @ A2) @ X5) & (((ord_less_eq @ real) @ X5) @ B2)) => ((((topolo3448309680560233919inuous @ real) @ real) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))) @ F2)))) => ((! [X5:real]: ((((((ord_less @ real) @ A2) @ X5) & (((ord_less @ real) @ X5) @ B2)) => ((((differentiable @ real) @ real) @ F2) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real))))))) => ((! [X5:real]: ((((((ord_less_eq @ real) @ A2) @ X5) & (((ord_less_eq @ real) @ X5) @ B2)) => ((((topolo3448309680560233919inuous @ real) @ real) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))) @ G)))) => ((! [X5:real]: ((((((ord_less @ real) @ A2) @ X5) & (((ord_less @ real) @ X5) @ B2)) => ((((differentiable @ real) @ real) @ G) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real))))))) => (? [G_c:real,F_c:real,C3:real]: ((((((has_field_derivative @ real) @ G) @ G_c) @ (((topolo174197925503356063within @ real) @ C3) @ (top_top @ (set @ real)))) & (((((has_field_derivative @ real) @ F2) @ F_c) @ (((topolo174197925503356063within @ real) @ C3) @ (top_top @ (set @ real)))) & ((((ord_less @ real) @ A2) @ C3) & ((((ord_less @ real) @ C3) @ B2) & ((((times_times @ real) @ (((minus_minus @ real) @ (F2 @ B2)) @ (F2 @ A2))) @ G_c) = (((times_times @ real) @ (((minus_minus @ real) @ (G @ B2)) @ (G @ A2))) @ F_c)))))))))))))))). % 6.41/2.70 thf(fact_5867_differentiable__cmult__right__iff, axiom, (! [A:$tType,B:$tType]: ((((real_V822414075346904944vector @ B) & (real_V3459762299906320749_field @ A)) => (! [Q4:(B > A),C2:A,T2:B]: ((((((differentiable @ B) @ A) @ (^ [T3:B]: ((((times_times @ A) @ (Q4 @ T3)) @ C2)))) @ (((topolo174197925503356063within @ B) @ T2) @ (top_top @ (set @ B)))) = ((C2 = (zero_zero @ A)) | ((((differentiable @ B) @ A) @ Q4) @ (((topolo174197925503356063within @ B) @ T2) @ (top_top @ (set @ B)))))))))))). % 6.41/2.70 thf(fact_5868_differentiable__cmult__left__iff, axiom, (! [A:$tType,B:$tType]: ((((real_V822414075346904944vector @ B) & (real_V3459762299906320749_field @ A)) => (! [C2:A,Q4:(B > A),T2:B]: ((((((differentiable @ B) @ A) @ (^ [T3:B]: ((((times_times @ A) @ C2) @ (Q4 @ T3))))) @ (((topolo174197925503356063within @ B) @ T2) @ (top_top @ (set @ B)))) = ((C2 = (zero_zero @ A)) | ((((differentiable @ B) @ A) @ Q4) @ (((topolo174197925503356063within @ B) @ T2) @ (top_top @ (set @ B)))))))))))). % 6.41/2.70 thf(fact_5869_differentiable__add, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),F5:(filter @ A),G:(A > B)]: ((((((differentiable @ A) @ B) @ F2) @ F5) => (((((differentiable @ A) @ B) @ G) @ F5) => ((((differentiable @ A) @ B) @ (^ [X:A]: ((((plus_plus @ B) @ (F2 @ X)) @ (G @ X))))) @ F5))))))))). % 6.41/2.70 thf(fact_5870_differentiable__within__subset, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V822414075346904944vector @ B)) => (! [F2:(A > B),X3:A,S:(set @ A),T2:(set @ A)]: ((((((differentiable @ A) @ B) @ F2) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => ((((ord_less_eq @ (set @ A)) @ T2) @ S) => ((((differentiable @ A) @ B) @ F2) @ (((topolo174197925503356063within @ A) @ X3) @ T2)))))))))). % 6.41/2.70 thf(fact_5871_differentiable__sum, axiom, (! [C:$tType,A:$tType,B:$tType]: ((((real_V822414075346904944vector @ B) & (real_V822414075346904944vector @ C)) => (! [S:(set @ A),F2:(A > (B > C)),Net:(filter @ B)]: ((((finite_finite2 @ A) @ S) => ((! [X5:A]: (((((member @ A) @ X5) @ S) => ((((differentiable @ B) @ C) @ (F2 @ X5)) @ Net)))) => ((((differentiable @ B) @ C) @ (^ [X:B]: (((((groups7311177749621191930dd_sum @ A) @ C) @ (^ [A4:A]: (((F2 @ A4) @ X)))) @ S)))) @ Net))))))))). % 6.41/2.70 thf(fact_5872_differentiable__divide, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V3459762299906320749_field @ B)) => (! [F2:(A > B),X3:A,S:(set @ A),G:(A > B)]: ((((((differentiable @ A) @ B) @ F2) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((((differentiable @ A) @ B) @ G) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((G @ X3) != (zero_zero @ B)) => ((((differentiable @ A) @ B) @ (^ [X:A]: ((((divide_divide @ B) @ (F2 @ X)) @ (G @ X))))) @ (((topolo174197925503356063within @ A) @ X3) @ S))))))))))). % 6.41/2.70 thf(fact_5873_differentiable__inverse, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V3459762299906320749_field @ B)) => (! [F2:(A > B),X3:A,S:(set @ A)]: ((((((differentiable @ A) @ B) @ F2) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((F2 @ X3) != (zero_zero @ B)) => ((((differentiable @ A) @ B) @ (^ [X:A]: (((inverse_inverse @ B) @ (F2 @ X))))) @ (((topolo174197925503356063within @ A) @ X3) @ S)))))))))). % 6.41/2.70 thf(fact_5874_MVT, axiom, (! [A2:real,B2:real,F2:(real > real)]: (((((ord_less @ real) @ A2) @ B2) => (((((topolo81223032696312382ous_on @ real) @ real) @ (((set_or1337092689740270186AtMost @ real) @ A2) @ B2)) @ F2) => ((! [X5:real]: (((((ord_less @ real) @ A2) @ X5) => ((((ord_less @ real) @ X5) @ B2) => ((((differentiable @ real) @ real) @ F2) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))))))) => (? [L4:real,Z3:real]: (((((ord_less @ real) @ A2) @ Z3) & ((((ord_less @ real) @ Z3) @ B2) & (((((has_field_derivative @ real) @ F2) @ L4) @ (((topolo174197925503356063within @ real) @ Z3) @ (top_top @ (set @ real)))) & ((((minus_minus @ real) @ (F2 @ B2)) @ (F2 @ A2)) = (((times_times @ real) @ (((minus_minus @ real) @ B2) @ A2)) @ L4))))))))))))). % 6.41/2.70 thf(fact_5875_eventually__filtercomap__at__topological, axiom, (! [A:$tType,B:$tType]: (((topolo4958980785337419405_space @ B) => (! [P:(A > $o),F2:(A > B),A3:B,B3:(set @ B)]: (((((eventually @ A) @ P) @ ((((filtercomap @ A) @ B) @ F2) @ (((topolo174197925503356063within @ B) @ A3) @ B3))) = (? [S7:(set @ B)]: ((((topolo1002775350975398744n_open @ B) @ S7) & ((((member @ B) @ A3) @ S7) & (! [X:A]: (((((member @ B) @ (F2 @ X)) @ (((minus_minus @ (set @ B)) @ (((inf_inf @ (set @ B)) @ S7) @ B3)) @ (((insert @ B) @ A3) @ (bot_bot @ (set @ B))))) => (P @ X))))))))))))))). % 6.41/2.70 thf(fact_5876_at__within__eq, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => ((topolo174197925503356063within @ A) = (^ [X:A,S8:(set @ A)]: (((complete_Inf_Inf @ (filter @ A)) @ ((((image @ (set @ A)) @ (filter @ A)) @ (^ [S7:(set @ A)]: (((principal @ A) @ (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ S7) @ S8)) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))) @ ((collect @ (set @ A)) @ (^ [S7:(set @ A)]: ((((topolo1002775350975398744n_open @ A) @ S7) & (((member @ A) @ X) @ S7)))))))))))))). % 6.41/2.70 thf(fact_5877_filtercomap__bot, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: ((((((filtercomap @ A) @ B) @ F2) @ (bot_bot @ (filter @ B))) = (bot_bot @ (filter @ A)))))). % 6.41/2.70 thf(fact_5878_principal__le__iff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (filter @ A)) @ ((principal @ A) @ A3)) @ ((principal @ A) @ B3)) = (((ord_less_eq @ (set @ A)) @ A3) @ B3))))). % 6.41/2.70 thf(fact_5879_continuous__on__sgn, axiom, (! [B:$tType,A:$tType]: ((((topolo4958980785337419405_space @ A) & (real_V822414075346904944vector @ B)) => (! [S:(set @ A),F2:(A > B)]: ((((((topolo81223032696312382ous_on @ A) @ B) @ S) @ F2) => ((! [X5:A]: (((((member @ A) @ X5) @ S) => ((F2 @ X5) != (zero_zero @ B))))) => ((((topolo81223032696312382ous_on @ A) @ B) @ S) @ (^ [X:A]: (((sgn_sgn @ B) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5880_continuous__on__add, axiom, (! [B:$tType,D:$tType]: ((((topolo4958980785337419405_space @ D) & (topolo6943815403480290642id_add @ B)) => (! [S:(set @ D),F2:(D > B),G:(D > B)]: ((((((topolo81223032696312382ous_on @ D) @ B) @ S) @ F2) => (((((topolo81223032696312382ous_on @ D) @ B) @ S) @ G) => ((((topolo81223032696312382ous_on @ D) @ B) @ S) @ (^ [X:D]: ((((plus_plus @ B) @ (F2 @ X)) @ (G @ X))))))))))))). % 6.41/2.70 thf(fact_5881_continuous__on__Pair, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((topolo4958980785337419405_space @ A) & ((topolo4958980785337419405_space @ B) & (topolo4958980785337419405_space @ C))) => (! [S:(set @ A),F2:(A > B),G:(A > C)]: ((((((topolo81223032696312382ous_on @ A) @ B) @ S) @ F2) => (((((topolo81223032696312382ous_on @ A) @ C) @ S) @ G) => ((((topolo81223032696312382ous_on @ A) @ ((product_prod @ B) @ C)) @ S) @ (^ [X:A]: (((((product_Pair @ B) @ C) @ (F2 @ X)) @ (G @ X))))))))))))). % 6.41/2.70 thf(fact_5882_continuous__on__divide, axiom, (! [B:$tType,A:$tType]: ((((topolo4958980785337419405_space @ A) & (real_V3459762299906320749_field @ B)) => (! [S:(set @ A),F2:(A > B),G:(A > B)]: ((((((topolo81223032696312382ous_on @ A) @ B) @ S) @ F2) => (((((topolo81223032696312382ous_on @ A) @ B) @ S) @ G) => ((! [X5:A]: (((((member @ A) @ X5) @ S) => ((G @ X5) != (zero_zero @ B))))) => ((((topolo81223032696312382ous_on @ A) @ B) @ S) @ (^ [X:A]: ((((divide_divide @ B) @ (F2 @ X)) @ (G @ X)))))))))))))). % 6.41/2.70 thf(fact_5883_IVT2_H, axiom, (! [A:$tType,B:$tType]: ((((topolo1944317154257567458pology @ B) & (topolo8458572112393995274pology @ A)) => (! [F2:(A > B),B2:A,Y:B,A2:A]: (((((ord_less_eq @ B) @ (F2 @ B2)) @ Y) => ((((ord_less_eq @ B) @ Y) @ (F2 @ A2)) => ((((ord_less_eq @ A) @ A2) @ B2) => (((((topolo81223032696312382ous_on @ A) @ B) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) @ F2) => (? [X5:A]: (((((ord_less_eq @ A) @ A2) @ X5) & ((((ord_less_eq @ A) @ X5) @ B2) & ((F2 @ X5) = Y))))))))))))))). % 6.41/2.70 thf(fact_5884_IVT_H, axiom, (! [A:$tType,B:$tType]: ((((topolo1944317154257567458pology @ B) & (topolo8458572112393995274pology @ A)) => (! [F2:(A > B),A2:A,Y:B,B2:A]: (((((ord_less_eq @ B) @ (F2 @ A2)) @ Y) => ((((ord_less_eq @ B) @ Y) @ (F2 @ B2)) => ((((ord_less_eq @ A) @ A2) @ B2) => (((((topolo81223032696312382ous_on @ A) @ B) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) @ F2) => (? [X5:A]: (((((ord_less_eq @ A) @ A2) @ X5) & ((((ord_less_eq @ A) @ X5) @ B2) & ((F2 @ X5) = Y))))))))))))))). % 6.41/2.70 thf(fact_5885_continuous__on__subset, axiom, (! [B:$tType,A:$tType]: ((((topolo4958980785337419405_space @ A) & (topolo4958980785337419405_space @ B)) => (! [S:(set @ A),F2:(A > B),T2:(set @ A)]: ((((((topolo81223032696312382ous_on @ A) @ B) @ S) @ F2) => ((((ord_less_eq @ (set @ A)) @ T2) @ S) => ((((topolo81223032696312382ous_on @ A) @ B) @ T2) @ F2))))))))). % 6.41/2.70 thf(fact_5886_continuous__on__sing, axiom, (! [B:$tType,A:$tType]: ((((topolo4958980785337419405_space @ A) & (topolo4958980785337419405_space @ B)) => (! [X3:A,F2:(A > B)]: (((((topolo81223032696312382ous_on @ A) @ B) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) @ F2))))))). % 6.41/2.70 thf(fact_5887_continuous__on__empty, axiom, (! [B:$tType,A:$tType]: ((((topolo4958980785337419405_space @ A) & (topolo4958980785337419405_space @ B)) => (! [F2:(A > B)]: (((((topolo81223032696312382ous_on @ A) @ B) @ (bot_bot @ (set @ A))) @ F2))))))). % 6.41/2.70 thf(fact_5888_continuous__on__inverse, axiom, (! [B:$tType,A:$tType]: ((((topolo4958980785337419405_space @ A) & (real_V8999393235501362500lgebra @ B)) => (! [S:(set @ A),F2:(A > B)]: ((((((topolo81223032696312382ous_on @ A) @ B) @ S) @ F2) => ((! [X5:A]: (((((member @ A) @ X5) @ S) => ((F2 @ X5) != (zero_zero @ B))))) => ((((topolo81223032696312382ous_on @ A) @ B) @ S) @ (^ [X:A]: (((inverse_inverse @ B) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5889_continuous__on__compose2, axiom, (! [A:$tType,B:$tType,C:$tType]: ((((topolo4958980785337419405_space @ C) & ((topolo4958980785337419405_space @ B) & (topolo4958980785337419405_space @ A))) => (! [T2:(set @ A),G:(A > B),S:(set @ C),F2:(C > A)]: ((((((topolo81223032696312382ous_on @ A) @ B) @ T2) @ G) => (((((topolo81223032696312382ous_on @ C) @ A) @ S) @ F2) => ((((ord_less_eq @ (set @ A)) @ ((((image @ C) @ A) @ F2) @ S)) @ T2) => ((((topolo81223032696312382ous_on @ C) @ B) @ S) @ (^ [X:C]: ((G @ (F2 @ X)))))))))))))). % 6.41/2.70 thf(fact_5890_bot__eq__principal__empty, axiom, (! [A:$tType]: (((bot_bot @ (filter @ A)) = ((principal @ A) @ (bot_bot @ (set @ A))))))). % 6.41/2.70 thf(fact_5891_principal__eq__bot__iff, axiom, (! [A:$tType,X8:(set @ A)]: (((((principal @ A) @ X8) = (bot_bot @ (filter @ A))) = (X8 = (bot_bot @ (set @ A))))))). % 6.41/2.70 thf(fact_5892_filtercomap__neq__bot, axiom, (! [A:$tType,B:$tType,F5:(filter @ A),F2:(B > A)]: (((! [P8:(A > $o)]: (((((eventually @ A) @ P8) @ F5) => (? [X2:B]: ((P8 @ (F2 @ X2))))))) => (((((filtercomap @ B) @ A) @ F2) @ F5) != (bot_bot @ (filter @ B))))))). % 6.41/2.70 thf(fact_5893_continuous__onI__mono, axiom, (! [B:$tType,A:$tType]: ((((topolo1944317154257567458pology @ A) & ((dense_order @ B) & (topolo1944317154257567458pology @ B))) => (! [F2:(A > B),A3:(set @ A)]: ((((topolo1002775350975398744n_open @ B) @ ((((image @ A) @ B) @ F2) @ A3)) => ((! [X5:A,Y4:A]: (((((member @ A) @ X5) @ A3) => ((((member @ A) @ Y4) @ A3) => ((((ord_less_eq @ A) @ X5) @ Y4) => (((ord_less_eq @ B) @ (F2 @ X5)) @ (F2 @ Y4))))))) => ((((topolo81223032696312382ous_on @ A) @ B) @ A3) @ F2))))))))). % 6.41/2.70 thf(fact_5894_open__Collect__less, axiom, (! [B:$tType,A:$tType]: ((((topolo4958980785337419405_space @ A) & (topolo1944317154257567458pology @ B)) => (! [F2:(A > B),G:(A > B)]: ((((((topolo81223032696312382ous_on @ A) @ B) @ (top_top @ (set @ A))) @ F2) => (((((topolo81223032696312382ous_on @ A) @ B) @ (top_top @ (set @ A))) @ G) => ((topolo1002775350975398744n_open @ A) @ ((collect @ A) @ (^ [X:A]: ((((ord_less @ B) @ (F2 @ X)) @ (G @ X)))))))))))))). % 6.41/2.70 thf(fact_5895_nhds__discrete, axiom, (! [A:$tType]: (((topolo8865339358273720382pology @ A) => ((topolo7230453075368039082e_nhds @ A) = (^ [X:A]: (((principal @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))))). % 6.41/2.70 thf(fact_5896_continuous__on__tan, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [S:(set @ A),F2:(A > A)]: ((((((topolo81223032696312382ous_on @ A) @ A) @ S) @ F2) => ((! [X5:A]: (((((member @ A) @ X5) @ S) => (((cos @ A) @ (F2 @ X5)) != (zero_zero @ A))))) => ((((topolo81223032696312382ous_on @ A) @ A) @ S) @ (^ [X:A]: (((tan @ A) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5897_open__Collect__less__Int, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [S:(set @ A),F2:(A > real),G:(A > real)]: ((((((topolo81223032696312382ous_on @ A) @ real) @ S) @ F2) => (((((topolo81223032696312382ous_on @ A) @ real) @ S) @ G) => (? [A8:(set @ A)]: ((((topolo1002775350975398744n_open @ A) @ A8) & ((((inf_inf @ (set @ A)) @ A8) @ S) = ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ S) & (((ord_less @ real) @ (F2 @ X)) @ (G @ X)))))))))))))))))). % 6.41/2.70 thf(fact_5898_continuous__on__cot, axiom, (! [A:$tType]: ((((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A)) => (! [S:(set @ A),F2:(A > A)]: ((((((topolo81223032696312382ous_on @ A) @ A) @ S) @ F2) => ((! [X5:A]: (((((member @ A) @ X5) @ S) => (((sin @ A) @ (F2 @ X5)) != (zero_zero @ A))))) => ((((topolo81223032696312382ous_on @ A) @ A) @ S) @ (^ [X:A]: (((cot @ A) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5899_continuous__on__tanh, axiom, (! [A:$tType,C:$tType]: ((((topolo4958980785337419405_space @ C) & ((real_Vector_banach @ A) & (real_V3459762299906320749_field @ A))) => (! [A3:(set @ C),F2:(C > A)]: ((((((topolo81223032696312382ous_on @ C) @ A) @ A3) @ F2) => ((! [X5:C]: (((((member @ C) @ X5) @ A3) => (((cosh @ A) @ (F2 @ X5)) != (zero_zero @ A))))) => ((((topolo81223032696312382ous_on @ C) @ A) @ A3) @ (^ [X:C]: (((tanh @ A) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5900_eventually__filtercomap__at__top__linorder, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [P:(B > $o),F2:(B > A)]: (((((eventually @ B) @ P) @ ((((filtercomap @ B) @ A) @ F2) @ (at_top @ A))) = (? [N5:A]: ((! [X:B]: (((((ord_less_eq @ A) @ N5) @ (F2 @ X)) => (P @ X))))))))))))). % 6.41/2.70 thf(fact_5901_eventually__filtercomap__at__top__dense, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (no_top @ A)) => (! [P:(B > $o),F2:(B > A)]: (((((eventually @ B) @ P) @ ((((filtercomap @ B) @ A) @ F2) @ (at_top @ A))) = (? [N5:A]: ((! [X:B]: (((((ord_less @ A) @ N5) @ (F2 @ X)) => (P @ X))))))))))))). % 6.41/2.70 thf(fact_5902_filtercomap__neq__bot__surj, axiom, (! [A:$tType,B:$tType,F5:(filter @ A),F2:(B > A)]: (((F5 != (bot_bot @ (filter @ A))) => ((((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((((filtercomap @ B) @ A) @ F2) @ F5) != (bot_bot @ (filter @ B)))))))). % 6.41/2.70 thf(fact_5903_eventually__filtercomap__at__bot__linorder, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [P:(B > $o),F2:(B > A)]: (((((eventually @ B) @ P) @ ((((filtercomap @ B) @ A) @ F2) @ (at_bot @ A))) = (? [N5:A]: ((! [X:B]: (((((ord_less_eq @ A) @ (F2 @ X)) @ N5) => (P @ X))))))))))))). % 6.41/2.70 thf(fact_5904_eventually__filtercomap__at__bot__dense, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (no_bot @ A)) => (! [P:(B > $o),F2:(B > A)]: (((((eventually @ B) @ P) @ ((((filtercomap @ B) @ A) @ F2) @ (at_bot @ A))) = (? [N5:A]: ((! [X:B]: (((((ord_less @ A) @ (F2 @ X)) @ N5) => (P @ X))))))))))))). % 6.41/2.70 thf(fact_5905_open__Collect__positive, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [S:(set @ A),F2:(A > real)]: ((((((topolo81223032696312382ous_on @ A) @ real) @ S) @ F2) => (? [A8:(set @ A)]: ((((topolo1002775350975398744n_open @ A) @ A8) & ((((inf_inf @ (set @ A)) @ A8) @ S) = ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ S) & (((ord_less @ real) @ (zero_zero @ real)) @ (F2 @ X))))))))))))))))). % 6.41/2.70 thf(fact_5906_continuous__on__powr_H, axiom, (! [C:$tType]: (((topolo4958980785337419405_space @ C) => (! [S:(set @ C),F2:(C > real),G:(C > real)]: ((((((topolo81223032696312382ous_on @ C) @ real) @ S) @ F2) => (((((topolo81223032696312382ous_on @ C) @ real) @ S) @ G) => ((! [X5:C]: (((((member @ C) @ X5) @ S) => ((((ord_less_eq @ real) @ (zero_zero @ real)) @ (F2 @ X5)) & (((F2 @ X5) = (zero_zero @ real)) => (((ord_less @ real) @ (zero_zero @ real)) @ (G @ X5))))))) => ((((topolo81223032696312382ous_on @ C) @ real) @ S) @ (^ [X:C]: ((((powr @ real) @ (F2 @ X)) @ (G @ X)))))))))))))). % 6.41/2.70 thf(fact_5907_continuous__on__log, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [S:(set @ A),F2:(A > real),G:(A > real)]: ((((((topolo81223032696312382ous_on @ A) @ real) @ S) @ F2) => (((((topolo81223032696312382ous_on @ A) @ real) @ S) @ G) => ((! [X5:A]: (((((member @ A) @ X5) @ S) => (((ord_less @ real) @ (zero_zero @ real)) @ (F2 @ X5))))) => ((! [X5:A]: (((((member @ A) @ X5) @ S) => ((F2 @ X5) != (one_one @ real))))) => ((! [X5:A]: (((((member @ A) @ X5) @ S) => (((ord_less @ real) @ (zero_zero @ real)) @ (G @ X5))))) => ((((topolo81223032696312382ous_on @ A) @ real) @ S) @ (^ [X:A]: (((log @ (F2 @ X)) @ (G @ X)))))))))))))))). % 6.41/2.70 thf(fact_5908_tendsto__principal__singleton, axiom, (! [A:$tType,B:$tType]: (((topolo4958980785337419405_space @ A) => (! [F2:(B > A),X3:B]: ((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ (F2 @ X3))) @ ((principal @ B) @ (((insert @ B) @ X3) @ (bot_bot @ (set @ B))))))))))). % 6.41/2.70 thf(fact_5909_nhds__discrete__open, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [X3:A]: ((((topolo1002775350975398744n_open @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) => (((topolo7230453075368039082e_nhds @ A) @ X3) = ((principal @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))). % 6.41/2.70 thf(fact_5910_DERIV__atLeastAtMost__imp__continuous__on, axiom, (! [A:$tType]: ((((ord @ A) & (real_V3459762299906320749_field @ A)) => (! [A2:A,B2:A,F2:(A > A)]: (((! [X5:A]: (((((ord_less_eq @ A) @ A2) @ X5) => ((((ord_less_eq @ A) @ X5) @ B2) => (? [Y5:A]: (((((has_field_derivative @ A) @ F2) @ Y5) @ (((topolo174197925503356063within @ A) @ X5) @ (top_top @ (set @ A)))))))))) => ((((topolo81223032696312382ous_on @ A) @ A) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) @ F2)))))))). % 6.41/2.70 thf(fact_5911_Rolle__deriv, axiom, (! [A2:real,B2:real,F2:(real > real),F11:(real > (real > real))]: (((((ord_less @ real) @ A2) @ B2) => (((F2 @ A2) = (F2 @ B2)) => (((((topolo81223032696312382ous_on @ real) @ real) @ (((set_or1337092689740270186AtMost @ real) @ A2) @ B2)) @ F2) => ((! [X5:real]: (((((ord_less @ real) @ A2) @ X5) => ((((ord_less @ real) @ X5) @ B2) => (((((has_derivative @ real) @ real) @ F2) @ (F11 @ X5)) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))))))) => (? [Z3:real]: (((((ord_less @ real) @ A2) @ Z3) & ((((ord_less @ real) @ Z3) @ B2) & ((F11 @ Z3) = (^ [V5:real]: ((zero_zero @ real))))))))))))))). % 6.41/2.70 thf(fact_5912_mvt, axiom, (! [A2:real,B2:real,F2:(real > real),F11:(real > (real > real))]: (((((ord_less @ real) @ A2) @ B2) => (((((topolo81223032696312382ous_on @ real) @ real) @ (((set_or1337092689740270186AtMost @ real) @ A2) @ B2)) @ F2) => ((! [X5:real]: (((((ord_less @ real) @ A2) @ X5) => ((((ord_less @ real) @ X5) @ B2) => (((((has_derivative @ real) @ real) @ F2) @ (F11 @ X5)) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))))))) => (~ ((! [Xi:real]: (((((ord_less @ real) @ A2) @ Xi) => ((((ord_less @ real) @ Xi) @ B2) => ((((minus_minus @ real) @ (F2 @ B2)) @ (F2 @ A2)) != ((F11 @ Xi) @ (((minus_minus @ real) @ B2) @ A2))))))))))))))). % 6.41/2.70 thf(fact_5913_continuous__on__Icc__at__leftD, axiom, (! [B:$tType,A:$tType]: ((((topolo1944317154257567458pology @ A) & (topolo4958980785337419405_space @ B)) => (! [A2:A,B2:A,F2:(A > B)]: ((((((topolo81223032696312382ous_on @ A) @ B) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) @ F2) => ((((ord_less @ A) @ A2) @ B2) => (((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ (F2 @ B2))) @ (((topolo174197925503356063within @ A) @ B2) @ ((set_ord_lessThan @ A) @ B2))))))))))). % 6.41/2.70 thf(fact_5914_filterlim__base__iff, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,I6:(set @ A),F5:(A > (set @ B)),F2:(B > C),G6:(D > (set @ C)),J4:(set @ D)]: (((I6 != (bot_bot @ (set @ A))) => ((! [I4:A]: (((((member @ A) @ I4) @ I6) => (! [J2:A]: (((((member @ A) @ J2) @ I6) => ((((ord_less_eq @ (set @ B)) @ (F5 @ I4)) @ (F5 @ J2)) | (((ord_less_eq @ (set @ B)) @ (F5 @ J2)) @ (F5 @ I4))))))))) => ((((((filterlim @ B) @ C) @ F2) @ ((complete_Inf_Inf @ (filter @ C)) @ ((((image @ D) @ (filter @ C)) @ (^ [J3:D]: (((principal @ C) @ (G6 @ J3))))) @ J4))) @ ((complete_Inf_Inf @ (filter @ B)) @ ((((image @ A) @ (filter @ B)) @ (^ [I3:A]: (((principal @ B) @ (F5 @ I3))))) @ I6))) = (! [X:D]: (((((member @ D) @ X) @ J4) => (? [Y2:A]: (((((member @ A) @ Y2) @ I6) & (! [Z6:B]: (((((member @ B) @ Z6) @ (F5 @ Y2)) => (((member @ C) @ (F2 @ Z6)) @ (G6 @ X))))))))))))))))). % 6.41/2.70 thf(fact_5915_DERIV__pos__imp__increasing__open, axiom, (! [A2:real,B2:real,F2:(real > real)]: (((((ord_less @ real) @ A2) @ B2) => ((! [X5:real]: (((((ord_less @ real) @ A2) @ X5) => ((((ord_less @ real) @ X5) @ B2) => (? [Y5:real]: ((((((has_field_derivative @ real) @ F2) @ Y5) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))) & (((ord_less @ real) @ (zero_zero @ real)) @ Y5)))))))) => (((((topolo81223032696312382ous_on @ real) @ real) @ (((set_or1337092689740270186AtMost @ real) @ A2) @ B2)) @ F2) => (((ord_less @ real) @ (F2 @ A2)) @ (F2 @ B2)))))))). % 6.41/2.70 thf(fact_5916_DERIV__neg__imp__decreasing__open, axiom, (! [A2:real,B2:real,F2:(real > real)]: (((((ord_less @ real) @ A2) @ B2) => ((! [X5:real]: (((((ord_less @ real) @ A2) @ X5) => ((((ord_less @ real) @ X5) @ B2) => (? [Y5:real]: ((((((has_field_derivative @ real) @ F2) @ Y5) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))) & (((ord_less @ real) @ Y5) @ (zero_zero @ real))))))))) => (((((topolo81223032696312382ous_on @ real) @ real) @ (((set_or1337092689740270186AtMost @ real) @ A2) @ B2)) @ F2) => (((ord_less @ real) @ (F2 @ B2)) @ (F2 @ A2)))))))). % 6.41/2.70 thf(fact_5917_DERIV__isconst__end, axiom, (! [A2:real,B2:real,F2:(real > real)]: (((((ord_less @ real) @ A2) @ B2) => (((((topolo81223032696312382ous_on @ real) @ real) @ (((set_or1337092689740270186AtMost @ real) @ A2) @ B2)) @ F2) => ((! [X5:real]: (((((ord_less @ real) @ A2) @ X5) => ((((ord_less @ real) @ X5) @ B2) => ((((has_field_derivative @ real) @ F2) @ (zero_zero @ real)) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))))))) => ((F2 @ B2) = (F2 @ A2)))))))). % 6.41/2.70 thf(fact_5918_INF__principal__finite, axiom, (! [B:$tType,A:$tType,X8:(set @ A),F2:(A > (set @ B))]: ((((finite_finite2 @ A) @ X8) => (((complete_Inf_Inf @ (filter @ B)) @ ((((image @ A) @ (filter @ B)) @ (^ [X:A]: (((principal @ B) @ (F2 @ X))))) @ X8)) = ((principal @ B) @ ((complete_Inf_Inf @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ F2) @ X8)))))))). % 6.41/2.70 thf(fact_5919_at__within__def, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => ((topolo174197925503356063within @ A) = (^ [A4:A,S8:(set @ A)]: ((((inf_inf @ (filter @ A)) @ ((topolo7230453075368039082e_nhds @ A) @ A4)) @ ((principal @ A) @ (((minus_minus @ (set @ A)) @ S8) @ (((insert @ A) @ A4) @ (bot_bot @ (set @ A))))))))))))). % 6.41/2.70 thf(fact_5920_DERIV__isconst2, axiom, (! [A2:real,B2:real,F2:(real > real),X3:real]: (((((ord_less @ real) @ A2) @ B2) => (((((topolo81223032696312382ous_on @ real) @ real) @ (((set_or1337092689740270186AtMost @ real) @ A2) @ B2)) @ F2) => ((! [X5:real]: (((((ord_less @ real) @ A2) @ X5) => ((((ord_less @ real) @ X5) @ B2) => ((((has_field_derivative @ real) @ F2) @ (zero_zero @ real)) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))))))) => ((((ord_less_eq @ real) @ A2) @ X3) => ((((ord_less_eq @ real) @ X3) @ B2) => ((F2 @ X3) = (F2 @ A2)))))))))). % 6.41/2.70 thf(fact_5921_at__left__eq, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [Y:A,X3:A]: (((((ord_less @ A) @ Y) @ X3) => ((((topolo174197925503356063within @ A) @ X3) @ ((set_ord_lessThan @ A) @ X3)) = ((complete_Inf_Inf @ (filter @ A)) @ ((((image @ A) @ (filter @ A)) @ (^ [A4:A]: (((principal @ A) @ (((set_or5935395276787703475ssThan @ A) @ A4) @ X3))))) @ ((set_ord_lessThan @ A) @ X3))))))))))). % 6.41/2.70 thf(fact_5922_Rolle, axiom, (! [A2:real,B2:real,F2:(real > real)]: (((((ord_less @ real) @ A2) @ B2) => (((F2 @ A2) = (F2 @ B2)) => (((((topolo81223032696312382ous_on @ real) @ real) @ (((set_or1337092689740270186AtMost @ real) @ A2) @ B2)) @ F2) => ((! [X5:real]: (((((ord_less @ real) @ A2) @ X5) => ((((ord_less @ real) @ X5) @ B2) => ((((differentiable @ real) @ real) @ F2) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))))))) => (? [Z3:real]: (((((ord_less @ real) @ A2) @ Z3) & ((((ord_less @ real) @ Z3) @ B2) & ((((has_field_derivative @ real) @ F2) @ (zero_zero @ real)) @ (((topolo174197925503356063within @ real) @ Z3) @ (top_top @ (set @ real))))))))))))))). % 6.41/2.70 thf(fact_5923_complete__uniform, axiom, (! [A:$tType]: (((topolo7287701948861334536_space @ A) => ((topolo2479028161051973599mplete @ A) = (^ [S7:(set @ A)]: ((! [F9:(filter @ A)]: (((((ord_less_eq @ (filter @ A)) @ F9) @ ((principal @ A) @ S7)) => ((F9 != (bot_bot @ (filter @ A))) => (((topolo6773858410816713723filter @ A) @ F9) => (? [X:A]: (((((member @ A) @ X) @ S7) & (((ord_less_eq @ (filter @ A)) @ F9) @ ((topolo7230453075368039082e_nhds @ A) @ X))))))))))))))))). % 6.41/2.70 thf(fact_5924_at__within__order, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [X3:A,S:(set @ A)]: ((((top_top @ (set @ A)) != (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) => ((((topolo174197925503356063within @ A) @ X3) @ S) = (((inf_inf @ (filter @ A)) @ ((complete_Inf_Inf @ (filter @ A)) @ ((((image @ A) @ (filter @ A)) @ (^ [A4:A]: (((principal @ A) @ (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ ((set_ord_lessThan @ A) @ A4)) @ S)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))) @ ((set_ord_greaterThan @ A) @ X3)))) @ ((complete_Inf_Inf @ (filter @ A)) @ ((((image @ A) @ (filter @ A)) @ (^ [A4:A]: (((principal @ A) @ (((minus_minus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ ((set_ord_greaterThan @ A) @ A4)) @ S)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))) @ ((set_ord_lessThan @ A) @ X3)))))))))))). % 6.41/2.70 thf(fact_5925_continuous__on__IccI, axiom, (! [B:$tType,A:$tType]: ((((topolo1944317154257567458pology @ A) & (topolo4958980785337419405_space @ B)) => (! [F2:(A > B),A2:A,B2:A]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ (F2 @ A2))) @ (((topolo174197925503356063within @ A) @ A2) @ ((set_ord_greaterThan @ A) @ A2))) => ((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ (F2 @ B2))) @ (((topolo174197925503356063within @ A) @ B2) @ ((set_ord_lessThan @ A) @ B2))) => ((! [X5:A]: (((((ord_less @ A) @ A2) @ X5) => ((((ord_less @ A) @ X5) @ B2) => (((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ (F2 @ X5))) @ (((topolo174197925503356063within @ A) @ X5) @ (top_top @ (set @ A)))))))) => ((((ord_less @ A) @ A2) @ B2) => ((((topolo81223032696312382ous_on @ A) @ B) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) @ F2))))))))))). % 6.41/2.70 thf(fact_5926_greaterThan__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_greaterThan @ A) @ K)) = (((ord_less @ A) @ K) @ I)))))))). % 6.41/2.70 thf(fact_5927_greaterThan__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_greaterThan @ A) @ X3)) @ ((set_ord_greaterThan @ A) @ Y)) = (((ord_less_eq @ A) @ Y) @ X3)))))))). % 6.41/2.70 thf(fact_5928_Sup__greaterThanAtLeast, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X3:A]: (((((ord_less @ A) @ X3) @ (top_top @ A)) => (((complete_Sup_Sup @ A) @ ((set_ord_greaterThan @ A) @ X3)) = (top_top @ A))))))))). % 6.41/2.70 thf(fact_5929_greaterThan__non__empty, axiom, (! [A:$tType]: (((no_top @ A) => (! [X3:A]: ((((set_ord_greaterThan @ A) @ X3) != (bot_bot @ (set @ A))))))))). % 6.41/2.70 thf(fact_5930_infinite__Ioi, axiom, (! [A:$tType]: ((((linorder @ A) & (no_top @ A)) => (! [A2:A]: ((~ (((finite_finite2 @ A) @ ((set_ord_greaterThan @ A) @ A2)))))))))). % 6.41/2.70 thf(fact_5931_greaterThan__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_ord_greaterThan @ A) = (^ [L2:A]: (((collect @ A) @ ((ord_less @ A) @ L2))))))))). % 6.41/2.70 thf(fact_5932_trivial__limit__at__right__real, axiom, (! [A:$tType]: ((((dense_order @ A) & ((no_top @ A) & (topolo1944317154257567458pology @ A))) => (! [X3:A]: (((((topolo174197925503356063within @ A) @ X3) @ ((set_ord_greaterThan @ A) @ X3)) != (bot_bot @ (filter @ A))))))))). % 6.41/2.70 thf(fact_5933_INT__greaterThan__UNIV, axiom, (((complete_Inf_Inf @ (set @ nat)) @ ((((image @ nat) @ (set @ nat)) @ (set_ord_greaterThan @ nat)) @ (top_top @ (set @ nat)))) = (bot_bot @ (set @ nat)))). % 6.41/2.70 thf(fact_5934_eventually__at__right, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [X3:A,Y:A,P:(A > $o)]: (((((ord_less @ A) @ X3) @ Y) => ((((eventually @ A) @ P) @ (((topolo174197925503356063within @ A) @ X3) @ ((set_ord_greaterThan @ A) @ X3))) = (? [B4:A]: (((((ord_less @ A) @ X3) @ B4) & (! [Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => ((((ord_less @ A) @ Y2) @ B4) => (P @ Y2)))))))))))))))). % 6.41/2.70 thf(fact_5935_eventually__at__right__field, axiom, (! [A:$tType]: ((((linordered_field @ A) & (topolo1944317154257567458pology @ A)) => (! [P:(A > $o),X3:A]: (((((eventually @ A) @ P) @ (((topolo174197925503356063within @ A) @ X3) @ ((set_ord_greaterThan @ A) @ X3))) = (? [B4:A]: (((((ord_less @ A) @ X3) @ B4) & (! [Y2:A]: (((((ord_less @ A) @ X3) @ Y2) => ((((ord_less @ A) @ Y2) @ B4) => (P @ Y2))))))))))))))). % 6.41/2.70 thf(fact_5936_at__within__Icc__at__right, axiom, (! [A:$tType]: (((topolo2564578578187576103pology @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((topolo174197925503356063within @ A) @ A2) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) = (((topolo174197925503356063within @ A) @ A2) @ ((set_ord_greaterThan @ A) @ A2)))))))))). % 6.41/2.70 thf(fact_5937_ivl__disj__int__one_I7_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ U)) @ ((set_ord_greaterThan @ A) @ U)) = (bot_bot @ (set @ A))))))))). % 6.41/2.70 thf(fact_5938_trivial__limit__at__right__top, axiom, (! [A:$tType]: ((((order_top @ A) & (topolo1944317154257567458pology @ A)) => ((((topolo174197925503356063within @ A) @ (top_top @ A)) @ ((set_ord_greaterThan @ A) @ (top_top @ A))) = (bot_bot @ (filter @ A))))))). % 6.41/2.70 thf(fact_5939_ivl__disj__int__one_I5_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ U)) @ ((set_ord_greaterThan @ A) @ U)) = (bot_bot @ (set @ A))))))))). % 6.41/2.70 thf(fact_5940_eventually__at__right__less, axiom, (! [A:$tType]: ((((no_top @ A) & (topolo1944317154257567458pology @ A)) => (! [X3:A]: ((((eventually @ A) @ ((ord_less @ A) @ X3)) @ (((topolo174197925503356063within @ A) @ X3) @ ((set_ord_greaterThan @ A) @ X3))))))))). % 6.41/2.70 thf(fact_5941_less__separate, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => (? [A5:A,B5:A]: (((((member @ A) @ X3) @ ((set_ord_lessThan @ A) @ A5)) & ((((member @ A) @ Y) @ ((set_ord_greaterThan @ A) @ B5)) & ((((inf_inf @ (set @ A)) @ ((set_ord_lessThan @ A) @ A5)) @ ((set_ord_greaterThan @ A) @ B5)) = (bot_bot @ (set @ A)))))))))))))). % 6.41/2.70 thf(fact_5942_eventually__at__rightI, axiom, (! [A:$tType]: (((topolo2564578578187576103pology @ A) => (! [A2:A,B2:A,P:(A > $o)]: (((! [X5:A]: (((((member @ A) @ X5) @ (((set_or5935395276787703475ssThan @ A) @ A2) @ B2)) => (P @ X5)))) => ((((ord_less @ A) @ A2) @ B2) => (((eventually @ A) @ P) @ (((topolo174197925503356063within @ A) @ A2) @ ((set_ord_greaterThan @ A) @ A2))))))))))). % 6.41/2.70 thf(fact_5943_greaterThan__0, axiom, (((set_ord_greaterThan @ nat) @ (zero_zero @ nat)) = ((((image @ nat) @ nat) @ suc) @ (top_top @ (set @ nat))))). % 6.41/2.70 thf(fact_5944_eventually__at__right__real, axiom, (! [A2:real,B2:real]: (((((ord_less @ real) @ A2) @ B2) => (((eventually @ real) @ (^ [X:real]: ((((member @ real) @ X) @ (((set_or5935395276787703475ssThan @ real) @ A2) @ B2))))) @ (((topolo174197925503356063within @ real) @ A2) @ ((set_ord_greaterThan @ real) @ A2))))))). % 6.41/2.70 thf(fact_5945_greaterThan__Suc, axiom, (! [K:nat]: ((((set_ord_greaterThan @ nat) @ (suc @ K)) = (((minus_minus @ (set @ nat)) @ ((set_ord_greaterThan @ nat) @ K)) @ (((insert @ nat) @ (suc @ K)) @ (bot_bot @ (set @ nat)))))))). % 6.41/2.70 thf(fact_5946_filterlim__times__pos, axiom, (! [A:$tType,B:$tType]: ((((linordered_field @ A) & (topolo1944317154257567458pology @ A)) => (! [F2:(B > A),P4:A,F14:(filter @ B),C2:A,L:A]: (((((((filterlim @ B) @ A) @ F2) @ (((topolo174197925503356063within @ A) @ P4) @ ((set_ord_greaterThan @ A) @ P4))) @ F14) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((L = (((times_times @ A) @ C2) @ P4)) => (((((filterlim @ B) @ A) @ (^ [X:B]: ((((times_times @ A) @ C2) @ (F2 @ X))))) @ (((topolo174197925503356063within @ A) @ L) @ ((set_ord_greaterThan @ A) @ L))) @ F14)))))))))). % 6.41/2.70 thf(fact_5947_tendsto__imp__filterlim__at__right, axiom, (! [B:$tType,A:$tType]: (((topolo2564578578187576103pology @ B) => (! [F2:(A > B),L5:B,F5:(filter @ A)]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L5)) @ F5) => ((((eventually @ A) @ (^ [X:A]: ((((ord_less @ B) @ L5) @ (F2 @ X))))) @ F5) => (((((filterlim @ A) @ B) @ F2) @ (((topolo174197925503356063within @ B) @ L5) @ ((set_ord_greaterThan @ B) @ L5))) @ F5))))))))). % 6.41/2.70 thf(fact_5948_continuous__on__Icc__at__rightD, axiom, (! [B:$tType,A:$tType]: ((((topolo1944317154257567458pology @ A) & (topolo4958980785337419405_space @ B)) => (! [A2:A,B2:A,F2:(A > B)]: ((((((topolo81223032696312382ous_on @ A) @ B) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) @ F2) => ((((ord_less @ A) @ A2) @ B2) => (((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ (F2 @ A2))) @ (((topolo174197925503356063within @ A) @ A2) @ ((set_ord_greaterThan @ A) @ A2))))))))))). % 6.41/2.70 thf(fact_5949_filterlim__at__bot__at__right, axiom, (! [B:$tType,A:$tType]: ((((topolo1944317154257567458pology @ A) & (linorder @ B)) => (! [Q:(A > $o),F2:(A > B),P:(B > $o),G:(B > A),A2:A]: (((! [X5:A,Y4:A]: (((Q @ X5) => ((Q @ Y4) => ((((ord_less_eq @ A) @ X5) @ Y4) => (((ord_less_eq @ B) @ (F2 @ X5)) @ (F2 @ Y4))))))) => ((! [X5:B]: (((P @ X5) => ((F2 @ (G @ X5)) = X5)))) => ((! [X5:B]: (((P @ X5) => (Q @ (G @ X5))))) => ((((eventually @ A) @ Q) @ (((topolo174197925503356063within @ A) @ A2) @ ((set_ord_greaterThan @ A) @ A2))) => ((! [B5:A]: (((Q @ B5) => (((ord_less @ A) @ A2) @ B5)))) => ((((eventually @ B) @ P) @ (at_bot @ B)) => (((((filterlim @ A) @ B) @ F2) @ (at_bot @ B)) @ (((topolo174197925503356063within @ A) @ A2) @ ((set_ord_greaterThan @ A) @ A2))))))))))))))). % 6.41/2.70 thf(fact_5950_nhds__metric, axiom, (! [A:$tType]: (((real_V7819770556892013058_space @ A) => ((topolo7230453075368039082e_nhds @ A) = (^ [X:A]: (((complete_Inf_Inf @ (filter @ A)) @ ((((image @ real) @ (filter @ A)) @ (^ [E4:real]: (((principal @ A) @ ((collect @ A) @ (^ [Y2:A]: ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ Y2) @ X)) @ E4)))))))) @ ((set_ord_greaterThan @ real) @ (zero_zero @ real))))))))))). % 6.41/2.70 thf(fact_5951_at__right__eq, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [X3:A,Y:A]: (((((ord_less @ A) @ X3) @ Y) => ((((topolo174197925503356063within @ A) @ X3) @ ((set_ord_greaterThan @ A) @ X3)) = ((complete_Inf_Inf @ (filter @ A)) @ ((((image @ A) @ (filter @ A)) @ (^ [A4:A]: (((principal @ A) @ (((set_or5935395276787703475ssThan @ A) @ X3) @ A4))))) @ ((set_ord_greaterThan @ A) @ X3))))))))))). % 6.41/2.70 thf(fact_5952_isCont__If__ge, axiom, (! [B:$tType,A:$tType]: ((((topolo1944317154257567458pology @ A) & (topolo4958980785337419405_space @ B)) => (! [A2:A,G:(A > B),F2:(A > B)]: ((((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ ((set_ord_lessThan @ A) @ A2))) @ G) => ((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ (G @ A2))) @ (((topolo174197925503356063within @ A) @ A2) @ ((set_ord_greaterThan @ A) @ A2))) => ((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A)))) @ (^ [X:A]: (((((if @ B) @ (((ord_less_eq @ A) @ X) @ A2)) @ (G @ X)) @ (F2 @ X))))))))))))). % 6.41/2.70 thf(fact_5953_interval__cases, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [S3:(set @ A)]: (((! [A5:A,B5:A,X5:A]: (((((member @ A) @ A5) @ S3) => ((((member @ A) @ B5) @ S3) => ((((ord_less_eq @ A) @ A5) @ X5) => ((((ord_less_eq @ A) @ X5) @ B5) => (((member @ A) @ X5) @ S3))))))) => (? [A5:A,B5:A]: (((S3 = (bot_bot @ (set @ A))) | ((S3 = (top_top @ (set @ A))) | ((S3 = ((set_ord_lessThan @ A) @ B5)) | ((S3 = ((set_ord_atMost @ A) @ B5)) | ((S3 = ((set_ord_greaterThan @ A) @ A5)) | ((S3 = ((set_ord_atLeast @ A) @ A5)) | ((S3 = (((set_or5935395276787703475ssThan @ A) @ A5) @ B5)) | ((S3 = (((set_or3652927894154168847AtMost @ A) @ A5) @ B5)) | ((S3 = (((set_or7035219750837199246ssThan @ A) @ A5) @ B5)) | (S3 = (((set_or1337092689740270186AtMost @ A) @ A5) @ B5)))))))))))))))))))). % 6.41/2.70 thf(fact_5954_sequentially__imp__eventually__at__right, axiom, (! [A:$tType]: ((((topolo3112930676232923870pology @ A) & (topolo1944317154257567458pology @ A)) => (! [A2:A,B2:A,P:(A > $o)]: (((((ord_less @ A) @ A2) @ B2) => ((! [F4:(nat > A)]: (((! [N6:nat]: ((((ord_less @ A) @ A2) @ (F4 @ N6)))) => ((! [N6:nat]: ((((ord_less @ A) @ (F4 @ N6)) @ B2))) => ((((order_antimono @ nat) @ A) @ F4) => ((((((filterlim @ nat) @ A) @ F4) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ (at_top @ nat)) => (((eventually @ nat) @ (^ [N:nat]: ((P @ (F4 @ N))))) @ (at_top @ nat)))))))) => (((eventually @ A) @ P) @ (((topolo174197925503356063within @ A) @ A2) @ ((set_ord_greaterThan @ A) @ A2))))))))))). % 6.41/2.70 thf(fact_5955_tendsto__at__right__sequentially, axiom, (! [C:$tType,B:$tType]: ((((topolo3112930676232923870pology @ B) & ((topolo1944317154257567458pology @ B) & (topolo4958980785337419405_space @ C))) => (! [A2:B,B2:B,X8:(B > C),L5:C]: (((((ord_less @ B) @ A2) @ B2) => ((! [S5:(nat > B)]: (((! [N6:nat]: ((((ord_less @ B) @ A2) @ (S5 @ N6)))) => ((! [N6:nat]: ((((ord_less @ B) @ (S5 @ N6)) @ B2))) => ((((order_antimono @ nat) @ B) @ S5) => ((((((filterlim @ nat) @ B) @ S5) @ ((topolo7230453075368039082e_nhds @ B) @ A2)) @ (at_top @ nat)) => (((((filterlim @ nat) @ C) @ (^ [N:nat]: ((X8 @ (S5 @ N))))) @ ((topolo7230453075368039082e_nhds @ C) @ L5)) @ (at_top @ nat)))))))) => (((((filterlim @ B) @ C) @ X8) @ ((topolo7230453075368039082e_nhds @ C) @ L5)) @ (((topolo174197925503356063within @ B) @ A2) @ ((set_ord_greaterThan @ B) @ A2))))))))))). % 6.41/2.70 thf(fact_5956_atLeast__0, axiom, (((set_ord_atLeast @ nat) @ (zero_zero @ nat)) = (top_top @ (set @ nat)))). % 6.41/2.70 thf(fact_5957_atLeast__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_atLeast @ A) @ K)) = (((ord_less_eq @ A) @ K) @ I)))))))). % 6.41/2.70 thf(fact_5958_atLeast__empty__triv, axiom, (! [A:$tType]: ((((set_ord_atLeast @ (set @ A)) @ (bot_bot @ (set @ A))) = (top_top @ (set @ (set @ A))))))). % 6.41/2.70 thf(fact_5959_atLeast__subset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ X3)) @ ((set_ord_atLeast @ A) @ Y)) = (((ord_less_eq @ A) @ Y) @ X3)))))))). % 6.41/2.70 thf(fact_5960_image__add__atLeast, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [K:A,I:A]: ((((((image @ A) @ A) @ ((plus_plus @ A) @ K)) @ ((set_ord_atLeast @ A) @ I)) = ((set_ord_atLeast @ A) @ (((plus_plus @ A) @ K) @ I))))))))). % 6.41/2.70 thf(fact_5961_Icc__subset__Ici__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [L:A,H2:A,L3:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ H2)) @ ((set_ord_atLeast @ A) @ L3)) = ((~ ((((ord_less_eq @ A) @ L) @ H2))) | (((ord_less_eq @ A) @ L3) @ L))))))))). % 6.41/2.70 thf(fact_5962_atLeast__Suc__greaterThan, axiom, (! [K:nat]: ((((set_ord_atLeast @ nat) @ (suc @ K)) = ((set_ord_greaterThan @ nat) @ K))))). % 6.41/2.70 thf(fact_5963_atLeast__eq__UNIV__iff, axiom, (! [A:$tType]: (((order_bot @ A) => (! [X3:A]: (((((set_ord_atLeast @ A) @ X3) = (top_top @ (set @ A))) = (X3 = (bot_bot @ A))))))))). % 6.41/2.70 thf(fact_5964_infinite__Ici, axiom, (! [A:$tType]: ((((linorder @ A) & (no_top @ A)) => (! [A2:A]: ((~ (((finite_finite2 @ A) @ ((set_ord_atLeast @ A) @ A2)))))))))). % 6.41/2.70 thf(fact_5965_atLeast__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_ord_atLeast @ A) = (^ [L2:A]: (((collect @ A) @ ((ord_less_eq @ A) @ L2))))))))). % 6.41/2.70 thf(fact_5966_antimonoD, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F2:(A > B),X3:A,Y:A]: (((((order_antimono @ A) @ B) @ F2) => ((((ord_less_eq @ A) @ X3) @ Y) => (((ord_less_eq @ B) @ (F2 @ Y)) @ (F2 @ X3)))))))))). % 6.41/2.70 thf(fact_5967_antimonoE, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F2:(A > B),X3:A,Y:A]: (((((order_antimono @ A) @ B) @ F2) => ((((ord_less_eq @ A) @ X3) @ Y) => (((ord_less_eq @ B) @ (F2 @ Y)) @ (F2 @ X3)))))))))). % 6.41/2.70 thf(fact_5968_antimonoI, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F2:(A > B)]: (((! [X5:A,Y4:A]: (((((ord_less_eq @ A) @ X5) @ Y4) => (((ord_less_eq @ B) @ (F2 @ Y4)) @ (F2 @ X5))))) => (((order_antimono @ A) @ B) @ F2)))))))). % 6.41/2.70 thf(fact_5969_antimono__def, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (((order_antimono @ A) @ B) = (^ [F3:(A > B)]: ((! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => (((ord_less_eq @ B) @ (F3 @ Y2)) @ (F3 @ X)))))))))))). % 6.41/2.70 thf(fact_5970_not__empty__eq__Ici__eq__empty, axiom, (! [A:$tType]: (((preorder @ A) => (! [L:A]: (((bot_bot @ (set @ A)) != ((set_ord_atLeast @ A) @ L)))))))). % 6.41/2.70 thf(fact_5971_not__UNIV__le__Ici, axiom, (! [A:$tType]: (((no_bot @ A) => (! [L:A]: ((~ ((((ord_less_eq @ (set @ A)) @ (top_top @ (set @ A))) @ ((set_ord_atLeast @ A) @ L)))))))))). % 6.41/2.70 thf(fact_5972_not__Ici__le__Icc, axiom, (! [A:$tType]: (((no_top @ A) => (! [L:A,L3:A,H3:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ L)) @ (((set_or1337092689740270186AtMost @ A) @ L3) @ H3)))))))))). % 6.41/2.70 thf(fact_5973_not__Iic__le__Ici, axiom, (! [A:$tType]: (((no_bot @ A) => (! [H2:A,L3:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ H2)) @ ((set_ord_atLeast @ A) @ L3)))))))))). % 6.41/2.70 thf(fact_5974_not__Ici__le__Iic, axiom, (! [A:$tType]: (((no_top @ A) => (! [L:A,H3:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ L)) @ ((set_ord_atMost @ A) @ H3)))))))))). % 6.41/2.70 thf(fact_5975_Ioi__le__Ico, axiom, (! [A:$tType]: (((preorder @ A) => (! [A2:A]: ((((ord_less_eq @ (set @ A)) @ ((set_ord_greaterThan @ A) @ A2)) @ ((set_ord_atLeast @ A) @ A2)))))))). % 6.41/2.70 thf(fact_5976_decseq__Suc__iff, axiom, (! [A:$tType]: (((order @ A) => (((order_antimono @ nat) @ A) = (^ [F3:(nat > A)]: ((! [N:nat]: ((((ord_less_eq @ A) @ (F3 @ (suc @ N))) @ (F3 @ N))))))))))). % 6.41/2.70 thf(fact_5977_decseq__SucI, axiom, (! [A:$tType]: (((order @ A) => (! [X8:(nat > A)]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (X8 @ (suc @ N3))) @ (X8 @ N3)))) => (((order_antimono @ nat) @ A) @ X8)))))))). % 6.41/2.70 thf(fact_5978_decseq__SucD, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(nat > A),I:nat]: (((((order_antimono @ nat) @ A) @ A3) => (((ord_less_eq @ A) @ (A3 @ (suc @ I))) @ (A3 @ I))))))))). % 6.41/2.70 thf(fact_5979_decseqD, axiom, (! [A:$tType]: (((order @ A) => (! [F2:(nat > A),I:nat,J:nat]: (((((order_antimono @ nat) @ A) @ F2) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ A) @ (F2 @ J)) @ (F2 @ I)))))))))). % 6.41/2.70 thf(fact_5980_decseq__def, axiom, (! [A:$tType]: (((order @ A) => (((order_antimono @ nat) @ A) = (^ [X6:(nat > A)]: ((! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => (((ord_less_eq @ A) @ (X6 @ N)) @ (X6 @ M2)))))))))))). % 6.41/2.70 thf(fact_5981_Ici__subset__Ioi__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ A2)) @ ((set_ord_greaterThan @ A) @ B2)) = (((ord_less @ A) @ B2) @ A2)))))))). % 6.67/2.70 thf(fact_5982_ivl__disj__int__one_I8_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ U)) @ ((set_ord_atLeast @ A) @ U)) = (bot_bot @ (set @ A))))))))). % 6.67/2.70 thf(fact_5983_ivl__disj__int__one_I6_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) @ ((set_ord_atLeast @ A) @ U)) = (bot_bot @ (set @ A))))))))). % 6.67/2.70 thf(fact_5984_atMost__Int__atLeast, axiom, (! [A:$tType]: (((order @ A) => (! [N2:A]: (((((inf_inf @ (set @ A)) @ ((set_ord_atMost @ A) @ N2)) @ ((set_ord_atLeast @ A) @ N2)) = (((insert @ A) @ N2) @ (bot_bot @ (set @ A)))))))))). % 6.67/2.70 thf(fact_5985_decseq__ge, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [X8:(nat > A),L5:A,N2:nat]: (((((order_antimono @ nat) @ A) @ X8) => ((((((filterlim @ nat) @ A) @ X8) @ ((topolo7230453075368039082e_nhds @ A) @ L5)) @ (at_top @ nat)) => (((ord_less_eq @ A) @ L5) @ (X8 @ N2)))))))))). % 6.67/2.70 thf(fact_5986_atLeast__Suc, axiom, (! [K:nat]: ((((set_ord_atLeast @ nat) @ (suc @ K)) = (((minus_minus @ (set @ nat)) @ ((set_ord_atLeast @ nat) @ K)) @ (((insert @ nat) @ K) @ (bot_bot @ (set @ nat)))))))). % 6.67/2.70 thf(fact_5987_Gcd__eq__Max, axiom, (! [M7:(set @ nat)]: ((((finite_finite2 @ nat) @ M7) => ((M7 != (bot_bot @ (set @ nat))) => ((~ ((((member @ nat) @ (zero_zero @ nat)) @ M7))) => (((gcd_Gcd @ nat) @ M7) = ((lattic643756798349783984er_Max @ nat) @ ((complete_Inf_Inf @ (set @ nat)) @ ((((image @ nat) @ (set @ nat)) @ (^ [M2:nat]: (((collect @ nat) @ (^ [D5:nat]: ((((dvd_dvd @ nat) @ D5) @ M2))))))) @ M7)))))))))). % 6.67/2.70 thf(fact_5988_continuous__at__Sup__antimono, axiom, (! [B:$tType,A:$tType]: ((((condit6923001295902523014norder @ A) & ((topolo1944317154257567458pology @ A) & ((condit6923001295902523014norder @ B) & (topolo1944317154257567458pology @ B)))) => (! [F2:(A > B),S3:(set @ A)]: (((((order_antimono @ A) @ B) @ F2) => (((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ ((complete_Sup_Sup @ A) @ S3)) @ ((set_ord_lessThan @ A) @ ((complete_Sup_Sup @ A) @ S3)))) @ F2) => ((S3 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ S3) => ((F2 @ ((complete_Sup_Sup @ A) @ S3)) = ((complete_Inf_Inf @ B) @ ((((image @ A) @ B) @ F2) @ S3))))))))))))). % 6.67/2.70 thf(fact_5989_bdd__above_OI, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:(set @ A),M7:A]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((ord_less_eq @ A) @ X5) @ M7)))) => ((condit941137186595557371_above @ A) @ A3)))))))). % 6.67/2.70 thf(fact_5990_bdd__above__empty, axiom, (! [A:$tType]: (((preorder @ A) => ((condit941137186595557371_above @ A) @ (bot_bot @ (set @ A))))))). % 6.67/2.70 thf(fact_5991_Max__singleton, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A]: ((((lattic643756798349783984er_Max @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = X3))))))). % 6.67/2.70 thf(fact_5992_Max__divisors__self__nat, axiom, (! [N2:nat]: (((N2 != (zero_zero @ nat)) => (((lattic643756798349783984er_Max @ nat) @ ((collect @ nat) @ (^ [D5:nat]: ((((dvd_dvd @ nat) @ D5) @ N2))))) = N2))))). % 6.67/2.70 thf(fact_5993_Max_Obounded__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ ((lattic643756798349783984er_Max @ A) @ A3)) @ X3) = (! [X:A]: (((((member @ A) @ X) @ A3) => (((ord_less_eq @ A) @ X) @ X3))))))))))))). % 6.67/2.70 thf(fact_5994_Max__less__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ ((lattic643756798349783984er_Max @ A) @ A3)) @ X3) = (! [X:A]: (((((member @ A) @ X) @ A3) => (((ord_less @ A) @ X) @ X3))))))))))))). % 6.67/2.70 thf(fact_5995_Max__const, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [A3:(set @ B),C2:A]: ((((finite_finite2 @ B) @ A3) => ((A3 != (bot_bot @ (set @ B))) => (((lattic643756798349783984er_Max @ A) @ ((((image @ B) @ A) @ (^ [Uu3:B]: (C2))) @ A3)) = C2))))))))). % 6.67/2.70 thf(fact_5996_bdd__above__UN, axiom, (! [A:$tType,B:$tType]: (((lattice @ A) => (! [I6:(set @ B),A3:(B > (set @ A))]: ((((finite_finite2 @ B) @ I6) => (((condit941137186595557371_above @ A) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ A3) @ I6))) = (! [X:B]: (((((member @ B) @ X) @ I6) => ((condit941137186595557371_above @ A) @ (A3 @ X))))))))))))). % 6.67/2.70 thf(fact_5997_Max__insert, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ (((insert @ A) @ X3) @ A3)) = (((ord_max @ A) @ X3) @ ((lattic643756798349783984er_Max @ A) @ A3))))))))))). % 6.67/2.70 thf(fact_5998_bdd__above_OI2, axiom, (! [A:$tType,B:$tType]: (((preorder @ A) => (! [A3:(set @ B),F2:(B > A),M7:A]: (((! [X5:B]: (((((member @ B) @ X5) @ A3) => (((ord_less_eq @ A) @ (F2 @ X5)) @ M7)))) => ((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F2) @ A3))))))))). % 6.67/2.70 thf(fact_5999_Max_Oin__idem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => ((((ord_max @ A) @ X3) @ ((lattic643756798349783984er_Max @ A) @ A3)) = ((lattic643756798349783984er_Max @ A) @ A3)))))))))). % 6.67/2.70 thf(fact_6000_Max__in, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((member @ A) @ ((lattic643756798349783984er_Max @ A) @ A3)) @ A3))))))))). % 6.67/2.70 thf(fact_6001_bdd__above__mono, axiom, (! [A:$tType]: (((preorder @ A) => (! [B3:(set @ A),A3:(set @ A)]: ((((condit941137186595557371_above @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((condit941137186595557371_above @ A) @ A3))))))))). % 6.67/2.70 thf(fact_6002_bdd__above_OE, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:(set @ A)]: ((((condit941137186595557371_above @ A) @ A3) => (~ ((! [M8:A]: ((~ ((! [X2:A]: (((((member @ A) @ X2) @ A3) => (((ord_less_eq @ A) @ X2) @ M8))))))))))))))))). % 6.67/2.70 thf(fact_6003_bdd__above_Ounfold, axiom, (! [A:$tType]: (((preorder @ A) => ((condit941137186595557371_above @ A) = (^ [A6:(set @ A)]: ((? [M9:A]: ((! [X:A]: (((((member @ A) @ X) @ A6) => (((ord_less_eq @ A) @ X) @ M9))))))))))))). % 6.67/2.70 thf(fact_6004_cSup__upper, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X3:A,X8:(set @ A)]: (((((member @ A) @ X3) @ X8) => (((condit941137186595557371_above @ A) @ X8) => (((ord_less_eq @ A) @ X3) @ ((complete_Sup_Sup @ A) @ X8)))))))))). % 6.67/2.70 thf(fact_6005_cSup__upper2, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X3:A,X8:(set @ A),Y:A]: (((((member @ A) @ X3) @ X8) => ((((ord_less_eq @ A) @ Y) @ X3) => (((condit941137186595557371_above @ A) @ X8) => (((ord_less_eq @ A) @ Y) @ ((complete_Sup_Sup @ A) @ X8))))))))))). % 6.67/2.70 thf(fact_6006_Max__ge, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => (((ord_less_eq @ A) @ X3) @ ((lattic643756798349783984er_Max @ A) @ A3)))))))))). % 6.67/2.70 thf(fact_6007_Max__eqI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((! [Y4:A]: (((((member @ A) @ Y4) @ A3) => (((ord_less_eq @ A) @ Y4) @ X3)))) => ((((member @ A) @ X3) @ A3) => (((lattic643756798349783984er_Max @ A) @ A3) = X3)))))))))). % 6.67/2.70 thf(fact_6008_Max__eq__if, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ A) @ B3) => ((! [X5:A]: (((((member @ A) @ X5) @ A3) => (? [Xa:A]: (((((member @ A) @ Xa) @ B3) & (((ord_less_eq @ A) @ X5) @ Xa))))))) => ((! [X5:A]: (((((member @ A) @ X5) @ B3) => (? [Xa:A]: (((((member @ A) @ Xa) @ A3) & (((ord_less_eq @ A) @ X5) @ Xa))))))) => (((lattic643756798349783984er_Max @ A) @ A3) = ((lattic643756798349783984er_Max @ A) @ B3)))))))))))). % 6.67/2.70 thf(fact_6009_Max_OcoboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A2) @ A3) => (((ord_less_eq @ A) @ A2) @ ((lattic643756798349783984er_Max @ A) @ A3)))))))))). % 6.67/2.70 thf(fact_6010_bdd__above__finite, axiom, (! [A:$tType]: (((lattice @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((condit941137186595557371_above @ A) @ A3)))))))). % 6.67/2.70 thf(fact_6011_bdd__above__nat, axiom, ((condit941137186595557371_above @ nat) = (finite_finite2 @ nat))). % 6.67/2.70 thf(fact_6012_cSUP__upper2, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [F2:(B > A),A3:(set @ B),X3:B,U:A]: ((((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => ((((member @ B) @ X3) @ A3) => ((((ord_less_eq @ A) @ U) @ (F2 @ X3)) => (((ord_less_eq @ A) @ U) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3)))))))))))). % 6.67/2.70 thf(fact_6013_cSUP__upper, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [X3:B,A3:(set @ B),F2:(B > A)]: (((((member @ B) @ X3) @ A3) => (((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => (((ord_less_eq @ A) @ (F2 @ X3)) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))))))))))). % 6.67/2.70 thf(fact_6014_cSup__mono, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [B3:(set @ A),A3:(set @ A)]: (((B3 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ A3) => ((! [B5:A]: (((((member @ A) @ B5) @ B3) => (? [X2:A]: (((((member @ A) @ X2) @ A3) & (((ord_less_eq @ A) @ B5) @ X2))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ B3)) @ ((complete_Sup_Sup @ A) @ A3))))))))))). % 6.67/2.70 thf(fact_6015_cSup__le__iff, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [S3:(set @ A),A2:A]: (((S3 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ S3) => ((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ S3)) @ A2) = (! [X:A]: (((((member @ A) @ X) @ S3) => (((ord_less_eq @ A) @ X) @ A2))))))))))))). % 6.67/2.70 thf(fact_6016_less__cSup__iff, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),Y:A]: (((X8 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ X8) => ((((ord_less @ A) @ Y) @ ((complete_Sup_Sup @ A) @ X8)) = (? [X:A]: (((((member @ A) @ X) @ X8) & (((ord_less @ A) @ Y) @ X))))))))))))). % 6.67/2.70 thf(fact_6017_Max__eq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),M:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((lattic643756798349783984er_Max @ A) @ A3) = M) = ((((member @ A) @ M) @ A3) & (! [X:A]: (((((member @ A) @ X) @ A3) => (((ord_less_eq @ A) @ X) @ M)))))))))))))). % 6.67/2.70 thf(fact_6018_Max__ge__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ X3) @ ((lattic643756798349783984er_Max @ A) @ A3)) = (? [X:A]: (((((member @ A) @ X) @ A3) & (((ord_less_eq @ A) @ X3) @ X))))))))))))). % 6.67/2.70 thf(fact_6019_eq__Max__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),M:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((M = ((lattic643756798349783984er_Max @ A) @ A3)) = ((((member @ A) @ M) @ A3) & (! [X:A]: (((((member @ A) @ X) @ A3) => (((ord_less_eq @ A) @ X) @ M)))))))))))))). % 6.67/2.70 thf(fact_6020_Max_OboundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ ((lattic643756798349783984er_Max @ A) @ A3)) @ X3) => (! [A11:A]: (((((member @ A) @ A11) @ A3) => (((ord_less_eq @ A) @ A11) @ X3))))))))))))). % 6.67/2.70 thf(fact_6021_Max_OboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((! [A5:A]: (((((member @ A) @ A5) @ A3) => (((ord_less_eq @ A) @ A5) @ X3)))) => (((ord_less_eq @ A) @ ((lattic643756798349783984er_Max @ A) @ A3)) @ X3)))))))))). % 6.67/2.70 thf(fact_6022_Max__gr__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ X3) @ ((lattic643756798349783984er_Max @ A) @ A3)) = (? [X:A]: (((((member @ A) @ X) @ A3) & (((ord_less @ A) @ X3) @ X))))))))))))). % 6.67/2.70 thf(fact_6023_Max__insert2, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((! [B5:A]: (((((member @ A) @ B5) @ A3) => (((ord_less_eq @ A) @ B5) @ A2)))) => (((lattic643756798349783984er_Max @ A) @ (((insert @ A) @ A2) @ A3)) = A2))))))))). % 6.67/2.70 thf(fact_6024_Max__Sup, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ A3) = ((complete_Sup_Sup @ A) @ A3)))))))))). % 6.67/2.70 thf(fact_6025_cSup__eq__Max, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A)]: ((((finite_finite2 @ A) @ X8) => ((X8 != (bot_bot @ (set @ A))) => (((complete_Sup_Sup @ A) @ X8) = ((lattic643756798349783984er_Max @ A) @ X8)))))))))). % 6.67/2.70 thf(fact_6026_cSUP__lessD, axiom, (! [B:$tType,A:$tType]: (((condit1219197933456340205attice @ A) => (! [F2:(B > A),A3:(set @ B),Y:A,I:B]: ((((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => ((((ord_less @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ Y) => ((((member @ B) @ I) @ A3) => (((ord_less @ A) @ (F2 @ I)) @ Y)))))))))). % 6.67/2.70 thf(fact_6027_Max_Oinfinite, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: (((~ (((finite_finite2 @ A) @ A3))) => (((lattic643756798349783984er_Max @ A) @ A3) = ((the2 @ A) @ (none @ A)))))))))). % 6.67/2.70 thf(fact_6028_cSUP__mono, axiom, (! [B:$tType,A:$tType,C:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ B),G:(C > A),B3:(set @ C),F2:(B > A)]: (((A3 != (bot_bot @ (set @ B))) => (((condit941137186595557371_above @ A) @ ((((image @ C) @ A) @ G) @ B3)) => ((! [N3:B]: (((((member @ B) @ N3) @ A3) => (? [X2:C]: (((((member @ C) @ X2) @ B3) & (((ord_less_eq @ A) @ (F2 @ N3)) @ (G @ X2)))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ G) @ B3)))))))))))). % 6.67/2.70 thf(fact_6029_cSUP__le__iff, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ B),F2:(B > A),U:A]: (((A3 != (bot_bot @ (set @ B))) => (((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => ((((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ U) = (! [X:B]: (((((member @ B) @ X) @ A3) => (((ord_less_eq @ A) @ (F2 @ X)) @ U))))))))))))). % 6.67/2.70 thf(fact_6030_cSup__subset__mono, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ A),B3:(set @ A)]: (((A3 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A3)) @ ((complete_Sup_Sup @ A) @ B3))))))))))). % 6.67/2.70 thf(fact_6031_Max__mono, axiom, (! [A:$tType]: (((linorder @ A) => (! [M7:(set @ A),N7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ M7) @ N7) => ((M7 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ N7) => (((ord_less_eq @ A) @ ((lattic643756798349783984er_Max @ A) @ M7)) @ ((lattic643756798349783984er_Max @ A) @ N7))))))))))). % 6.67/2.70 thf(fact_6032_Max_Osubset__imp, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((A3 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => (((ord_less_eq @ A) @ ((lattic643756798349783984er_Max @ A) @ A3)) @ ((lattic643756798349783984er_Max @ A) @ B3))))))))))). % 6.67/2.70 thf(fact_6033_hom__Max__commute, axiom, (! [A:$tType]: (((linorder @ A) => (! [H2:(A > A),N7:(set @ A)]: (((! [X5:A,Y4:A]: (((H2 @ (((ord_max @ A) @ X5) @ Y4)) = (((ord_max @ A) @ (H2 @ X5)) @ (H2 @ Y4))))) => (((finite_finite2 @ A) @ N7) => ((N7 != (bot_bot @ (set @ A))) => ((H2 @ ((lattic643756798349783984er_Max @ A) @ N7)) = ((lattic643756798349783984er_Max @ A) @ ((((image @ A) @ A) @ H2) @ N7)))))))))))). % 6.67/2.70 thf(fact_6034_Max_Osubset, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((B3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => ((((ord_max @ A) @ ((lattic643756798349783984er_Max @ A) @ B3)) @ ((lattic643756798349783984er_Max @ A) @ A3)) = ((lattic643756798349783984er_Max @ A) @ A3))))))))))). % 6.67/2.70 thf(fact_6035_Max_Oinsert__not__elem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((~ ((((member @ A) @ X3) @ A3))) => ((A3 != (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ (((insert @ A) @ X3) @ A3)) = (((ord_max @ A) @ X3) @ ((lattic643756798349783984er_Max @ A) @ A3)))))))))))). % 6.67/2.70 thf(fact_6036_Max_Oclosed, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((! [X5:A,Y4:A]: ((((member @ A) @ (((ord_max @ A) @ X5) @ Y4)) @ (((insert @ A) @ X5) @ (((insert @ A) @ Y4) @ (bot_bot @ (set @ A))))))) => (((member @ A) @ ((lattic643756798349783984er_Max @ A) @ A3)) @ A3)))))))))). % 6.67/2.70 thf(fact_6037_less__cSUP__iff, axiom, (! [A:$tType,B:$tType]: (((condit6923001295902523014norder @ A) => (! [A3:(set @ B),F2:(B > A),A2:A]: (((A3 != (bot_bot @ (set @ B))) => (((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => ((((ord_less @ A) @ A2) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) = (? [X:B]: (((((member @ B) @ X) @ A3) & (((ord_less @ A) @ A2) @ (F2 @ X)))))))))))))). % 6.67/2.70 thf(fact_6038_card__le__Suc__Max, axiom, (! [S3:(set @ nat)]: ((((finite_finite2 @ nat) @ S3) => (((ord_less_eq @ nat) @ ((finite_card @ nat) @ S3)) @ (suc @ ((lattic643756798349783984er_Max @ nat) @ S3))))))). % 6.67/2.70 thf(fact_6039_Sup__nat__def, axiom, ((complete_Sup_Sup @ nat) = (^ [X6:(set @ nat)]: (((((if @ nat) @ (X6 = (bot_bot @ (set @ nat)))) @ (zero_zero @ nat)) @ ((lattic643756798349783984er_Max @ nat) @ X6)))))). % 6.67/2.70 thf(fact_6040_divide__nat__def, axiom, ((divide_divide @ nat) = (^ [M2:nat,N:nat]: (((((if @ nat) @ (N = (zero_zero @ nat))) @ (zero_zero @ nat)) @ ((lattic643756798349783984er_Max @ nat) @ ((collect @ nat) @ (^ [K3:nat]: ((((ord_less_eq @ nat) @ (((times_times @ nat) @ K3) @ N)) @ M2)))))))))). % 6.67/2.70 thf(fact_6041_Max__add__commute, axiom, (! [B:$tType,A:$tType]: (((linord4140545234300271783up_add @ A) => (! [S3:(set @ B),F2:(B > A),K:A]: ((((finite_finite2 @ B) @ S3) => ((S3 != (bot_bot @ (set @ B))) => (((lattic643756798349783984er_Max @ A) @ ((((image @ B) @ A) @ (^ [X:B]: ((((plus_plus @ A) @ (F2 @ X)) @ K)))) @ S3)) = (((plus_plus @ A) @ ((lattic643756798349783984er_Max @ A) @ ((((image @ B) @ A) @ F2) @ S3))) @ K)))))))))). % 6.67/2.70 thf(fact_6042_gcd__is__Max__divisors__nat, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((gcd_gcd @ nat) @ M) @ N2) = ((lattic643756798349783984er_Max @ nat) @ ((collect @ nat) @ (^ [D5:nat]: (((((dvd_dvd @ nat) @ D5) @ M) & (((dvd_dvd @ nat) @ D5) @ N2))))))))))). % 6.67/2.70 thf(fact_6043_cSUP__subset__mono, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ B),G:(B > A),B3:(set @ B),F2:(B > A)]: (((A3 != (bot_bot @ (set @ B))) => (((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ G) @ B3)) => ((((ord_less_eq @ (set @ B)) @ A3) @ B3) => ((! [X5:B]: (((((member @ B) @ X5) @ A3) => (((ord_less_eq @ A) @ (F2 @ X5)) @ (G @ X5))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ G) @ B3))))))))))))). % 6.67/2.70 thf(fact_6044_cSup__cInf, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [S3:(set @ A)]: (((S3 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ S3) => (((complete_Sup_Sup @ A) @ S3) = ((complete_Inf_Inf @ A) @ ((collect @ A) @ (^ [X:A]: ((! [Y2:A]: (((((member @ A) @ Y2) @ S3) => (((ord_less_eq @ A) @ Y2) @ X))))))))))))))))). % 6.67/2.70 thf(fact_6045_Max_Oremove, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => ((((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ A3) = X3)) & (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ A3) = (((ord_max @ A) @ X3) @ ((lattic643756798349783984er_Max @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))))))))))). % 6.67/2.70 thf(fact_6046_Max_Oinsert__remove, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ (((insert @ A) @ X3) @ A3)) = X3)) & (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ (((insert @ A) @ X3) @ A3)) = (((ord_max @ A) @ X3) @ ((lattic643756798349783984er_Max @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))))))). % 6.67/2.70 thf(fact_6047_cSUP__UNION, axiom, (! [B:$tType,D:$tType,C:$tType]: (((condit1219197933456340205attice @ B) => (! [A3:(set @ C),B3:(C > (set @ D)),F2:(D > B)]: (((A3 != (bot_bot @ (set @ C))) => ((! [X5:C]: (((((member @ C) @ X5) @ A3) => ((B3 @ X5) != (bot_bot @ (set @ D)))))) => (((condit941137186595557371_above @ B) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ C) @ (set @ B)) @ (^ [X:C]: (((((image @ D) @ B) @ F2) @ (B3 @ X))))) @ A3))) => (((complete_Sup_Sup @ B) @ ((((image @ D) @ B) @ F2) @ ((complete_Sup_Sup @ (set @ D)) @ ((((image @ C) @ (set @ D)) @ B3) @ A3)))) = ((complete_Sup_Sup @ B) @ ((((image @ C) @ B) @ (^ [X:C]: (((complete_Sup_Sup @ B) @ ((((image @ D) @ B) @ F2) @ (B3 @ X)))))) @ A3)))))))))))). % 6.67/2.70 thf(fact_6048_sum__le__card__Max, axiom, (! [A:$tType,A3:(set @ A),F2:(A > nat)]: ((((finite_finite2 @ A) @ A3) => (((ord_less_eq @ nat) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ A3)) @ (((times_times @ nat) @ ((finite_card @ A) @ A3)) @ ((lattic643756798349783984er_Max @ nat) @ ((((image @ A) @ nat) @ F2) @ A3)))))))). % 6.67/2.70 thf(fact_6049_continuous__at__Inf__antimono, axiom, (! [B:$tType,A:$tType]: ((((condit6923001295902523014norder @ A) & ((topolo1944317154257567458pology @ A) & ((condit6923001295902523014norder @ B) & (topolo1944317154257567458pology @ B)))) => (! [F2:(A > B),S3:(set @ A)]: (((((order_antimono @ A) @ B) @ F2) => (((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ ((complete_Inf_Inf @ A) @ S3)) @ ((set_ord_greaterThan @ A) @ ((complete_Inf_Inf @ A) @ S3)))) @ F2) => ((S3 != (bot_bot @ (set @ A))) => (((condit1013018076250108175_below @ A) @ S3) => ((F2 @ ((complete_Inf_Inf @ A) @ S3)) = ((complete_Sup_Sup @ B) @ ((((image @ A) @ B) @ F2) @ S3))))))))))))). % 6.67/2.70 thf(fact_6050_dual__Min, axiom, (! [A:$tType]: (((linorder @ A) => (((lattices_Min @ A) @ (^ [X:A,Y2:A]: ((((ord_less_eq @ A) @ Y2) @ X)))) = (lattic643756798349783984er_Max @ A)))))). % 6.67/2.70 thf(fact_6051_bdd__below_OI, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:(set @ A),M7:A]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((ord_less_eq @ A) @ M7) @ X5)))) => ((condit1013018076250108175_below @ A) @ A3)))))))). % 6.67/2.70 thf(fact_6052_bdd__belowI, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:(set @ A),M:A]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((ord_less_eq @ A) @ M) @ X5)))) => ((condit1013018076250108175_below @ A) @ A3)))))))). % 6.67/2.70 thf(fact_6053_bdd__below__empty, axiom, (! [A:$tType]: (((preorder @ A) => ((condit1013018076250108175_below @ A) @ (bot_bot @ (set @ A))))))). % 6.67/2.70 thf(fact_6054_Max__divisors__self__int, axiom, (! [N2:int]: (((N2 != (zero_zero @ int)) => (((lattic643756798349783984er_Max @ int) @ ((collect @ int) @ (^ [D5:int]: ((((dvd_dvd @ int) @ D5) @ N2))))) = ((abs_abs @ int) @ N2)))))). % 6.67/2.70 thf(fact_6055_bdd__below__UN, axiom, (! [A:$tType,B:$tType]: (((lattice @ A) => (! [I6:(set @ B),A3:(B > (set @ A))]: ((((finite_finite2 @ B) @ I6) => (((condit1013018076250108175_below @ A) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ A3) @ I6))) = (! [X:B]: (((((member @ B) @ X) @ I6) => ((condit1013018076250108175_below @ A) @ (A3 @ X))))))))))))). % 6.67/2.70 thf(fact_6056_linorder_OMin_Ocong, axiom, (! [A:$tType]: (((lattices_Min @ A) = (lattices_Min @ A))))). % 6.67/2.70 thf(fact_6057_bdd__below__finite, axiom, (! [A:$tType]: (((lattice @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((condit1013018076250108175_below @ A) @ A3)))))))). % 6.67/2.70 thf(fact_6058_bdd__below_Ounfold, axiom, (! [A:$tType]: (((preorder @ A) => ((condit1013018076250108175_below @ A) = (^ [A6:(set @ A)]: ((? [M9:A]: ((! [X:A]: (((((member @ A) @ X) @ A6) => (((ord_less_eq @ A) @ M9) @ X))))))))))))). % 6.67/2.70 thf(fact_6059_bdd__below_OE, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:(set @ A)]: ((((condit1013018076250108175_below @ A) @ A3) => (~ ((! [M8:A]: ((~ ((! [X2:A]: (((((member @ A) @ X2) @ A3) => (((ord_less_eq @ A) @ M8) @ X2))))))))))))))))). % 6.67/2.70 thf(fact_6060_bdd__below__mono, axiom, (! [A:$tType]: (((preorder @ A) => (! [B3:(set @ A),A3:(set @ A)]: ((((condit1013018076250108175_below @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((condit1013018076250108175_below @ A) @ A3))))))))). % 6.67/2.70 thf(fact_6061_bdd__below_OI2, axiom, (! [A:$tType,B:$tType]: (((preorder @ A) => (! [A3:(set @ B),M7:A,F2:(B > A)]: (((! [X5:B]: (((((member @ B) @ X5) @ A3) => (((ord_less_eq @ A) @ M7) @ (F2 @ X5))))) => ((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F2) @ A3))))))))). % 6.67/2.70 thf(fact_6062_bdd__belowI2, axiom, (! [A:$tType,B:$tType]: (((preorder @ A) => (! [A3:(set @ B),M:A,F2:(B > A)]: (((! [X5:B]: (((((member @ B) @ X5) @ A3) => (((ord_less_eq @ A) @ M) @ (F2 @ X5))))) => ((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F2) @ A3))))))))). % 6.67/2.70 thf(fact_6063_cInf__lower, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X3:A,X8:(set @ A)]: (((((member @ A) @ X3) @ X8) => (((condit1013018076250108175_below @ A) @ X8) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ X8)) @ X3))))))))). % 6.67/2.70 thf(fact_6064_cInf__lower2, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X3:A,X8:(set @ A),Y:A]: (((((member @ A) @ X3) @ X8) => ((((ord_less_eq @ A) @ X3) @ Y) => (((condit1013018076250108175_below @ A) @ X8) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ X8)) @ Y)))))))))). % 6.67/2.70 thf(fact_6065_cINF__lower, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [F2:(B > A),A3:(set @ B),X3:B]: ((((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => ((((member @ B) @ X3) @ A3) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ (F2 @ X3)))))))))). % 6.67/2.70 thf(fact_6066_cINF__lower2, axiom, (! [B:$tType,A:$tType]: (((condit1219197933456340205attice @ A) => (! [F2:(B > A),A3:(set @ B),X3:B,U:A]: ((((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => ((((member @ B) @ X3) @ A3) => ((((ord_less_eq @ A) @ (F2 @ X3)) @ U) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ U)))))))))). % 6.67/2.70 thf(fact_6067_le__cInf__iff, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [S3:(set @ A),A2:A]: (((S3 != (bot_bot @ (set @ A))) => (((condit1013018076250108175_below @ A) @ S3) => ((((ord_less_eq @ A) @ A2) @ ((complete_Inf_Inf @ A) @ S3)) = (! [X:A]: (((((member @ A) @ X) @ S3) => (((ord_less_eq @ A) @ A2) @ X))))))))))))). % 6.67/2.70 thf(fact_6068_cInf__mono, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [B3:(set @ A),A3:(set @ A)]: (((B3 != (bot_bot @ (set @ A))) => (((condit1013018076250108175_below @ A) @ A3) => ((! [B5:A]: (((((member @ A) @ B5) @ B3) => (? [X2:A]: (((((member @ A) @ X2) @ A3) & (((ord_less_eq @ A) @ X2) @ B5))))))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A3)) @ ((complete_Inf_Inf @ A) @ B3))))))))))). % 6.67/2.70 thf(fact_6069_cInf__less__iff, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A),Y:A]: (((X8 != (bot_bot @ (set @ A))) => (((condit1013018076250108175_below @ A) @ X8) => ((((ord_less @ A) @ ((complete_Inf_Inf @ A) @ X8)) @ Y) = (? [X:A]: (((((member @ A) @ X) @ X8) & (((ord_less @ A) @ X) @ Y))))))))))))). % 6.67/2.70 thf(fact_6070_less__cINF__D, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [F2:(B > A),A3:(set @ B),Y:A,I:B]: ((((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => ((((ord_less @ A) @ Y) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) => ((((member @ B) @ I) @ A3) => (((ord_less @ A) @ Y) @ (F2 @ I))))))))))). % 6.67/2.70 thf(fact_6071_gcd__is__Max__divisors__int, axiom, (! [N2:int,M:int]: (((N2 != (zero_zero @ int)) => ((((gcd_gcd @ int) @ M) @ N2) = ((lattic643756798349783984er_Max @ int) @ ((collect @ int) @ (^ [D5:int]: (((((dvd_dvd @ int) @ D5) @ M) & (((dvd_dvd @ int) @ D5) @ N2))))))))))). % 6.67/2.70 thf(fact_6072_le__cINF__iff, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ B),F2:(B > A),U:A]: (((A3 != (bot_bot @ (set @ B))) => (((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => ((((ord_less_eq @ A) @ U) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) = (! [X:B]: (((((member @ B) @ X) @ A3) => (((ord_less_eq @ A) @ U) @ (F2 @ X)))))))))))))). % 6.67/2.70 thf(fact_6073_cINF__mono, axiom, (! [C:$tType,A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [B3:(set @ B),F2:(C > A),A3:(set @ C),G:(B > A)]: (((B3 != (bot_bot @ (set @ B))) => (((condit1013018076250108175_below @ A) @ ((((image @ C) @ A) @ F2) @ A3)) => ((! [M3:B]: (((((member @ B) @ M3) @ B3) => (? [X2:C]: (((((member @ C) @ X2) @ A3) & (((ord_less_eq @ A) @ (F2 @ X2)) @ (G @ M3)))))))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ C) @ A) @ F2) @ A3))) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ G) @ B3)))))))))))). % 6.67/2.70 thf(fact_6074_cInf__superset__mono, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ A),B3:(set @ A)]: (((A3 != (bot_bot @ (set @ A))) => (((condit1013018076250108175_below @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ B3)) @ ((complete_Inf_Inf @ A) @ A3))))))))))). % 6.67/2.70 thf(fact_6075_cInf__insert, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),A2:A]: (((X8 != (bot_bot @ (set @ A))) => (((condit1013018076250108175_below @ A) @ X8) => (((complete_Inf_Inf @ A) @ (((insert @ A) @ A2) @ X8)) = (((inf_inf @ A) @ A2) @ ((complete_Inf_Inf @ A) @ X8))))))))))). % 6.67/2.70 thf(fact_6076_cInf__insert__If, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),A2:A]: ((((condit1013018076250108175_below @ A) @ X8) => (((X8 = (bot_bot @ (set @ A))) => (((complete_Inf_Inf @ A) @ (((insert @ A) @ A2) @ X8)) = A2)) & ((X8 != (bot_bot @ (set @ A))) => (((complete_Inf_Inf @ A) @ (((insert @ A) @ A2) @ X8)) = (((inf_inf @ A) @ A2) @ ((complete_Inf_Inf @ A) @ X8)))))))))))). % 6.67/2.70 thf(fact_6077_cINF__less__iff, axiom, (! [A:$tType,B:$tType]: (((condit6923001295902523014norder @ A) => (! [A3:(set @ B),F2:(B > A),A2:A]: (((A3 != (bot_bot @ (set @ B))) => (((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => ((((ord_less @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ A2) = (? [X:B]: (((((member @ B) @ X) @ A3) & (((ord_less @ A) @ (F2 @ X)) @ A2))))))))))))). % 6.67/2.70 thf(fact_6078_cINF__inf__distrib, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ B),F2:(B > A),G:(B > A)]: (((A3 != (bot_bot @ (set @ B))) => (((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => (((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ G) @ A3)) => ((((inf_inf @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ G) @ A3))) = ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ (^ [A4:B]: ((((inf_inf @ A) @ (F2 @ A4)) @ (G @ A4))))) @ A3)))))))))))). % 6.67/2.70 thf(fact_6079_cINF__superset__mono, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ B),G:(B > A),B3:(set @ B),F2:(B > A)]: (((A3 != (bot_bot @ (set @ B))) => (((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ G) @ B3)) => ((((ord_less_eq @ (set @ B)) @ A3) @ B3) => ((! [X5:B]: (((((member @ B) @ X5) @ B3) => (((ord_less_eq @ A) @ (G @ X5)) @ (F2 @ X5))))) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ G) @ B3))) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))))))))))))). % 6.67/2.70 thf(fact_6080_less__eq__cInf__inter, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ A),B3:(set @ A)]: ((((condit1013018076250108175_below @ A) @ A3) => (((condit1013018076250108175_below @ A) @ B3) => (((((inf_inf @ (set @ A)) @ A3) @ B3) != (bot_bot @ (set @ A))) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ ((complete_Inf_Inf @ A) @ A3)) @ ((complete_Inf_Inf @ A) @ B3))) @ ((complete_Inf_Inf @ A) @ (((inf_inf @ (set @ A)) @ A3) @ B3)))))))))))). % 6.67/2.70 thf(fact_6081_cINF__insert, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ B),F2:(B > A),A2:B]: (((A3 != (bot_bot @ (set @ B))) => (((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ (((insert @ B) @ A2) @ A3))) = (((inf_inf @ A) @ (F2 @ A2)) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3)))))))))))). % 6.67/2.70 thf(fact_6082_cInf__le__cSup, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ A)]: (((A3 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ A3) => (((condit1013018076250108175_below @ A) @ A3) => (((ord_less_eq @ A) @ ((complete_Inf_Inf @ A) @ A3)) @ ((complete_Sup_Sup @ A) @ A3))))))))))). % 6.67/2.70 thf(fact_6083_cInf__cSup, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [S3:(set @ A)]: (((S3 != (bot_bot @ (set @ A))) => (((condit1013018076250108175_below @ A) @ S3) => (((complete_Inf_Inf @ A) @ S3) = ((complete_Sup_Sup @ A) @ ((collect @ A) @ (^ [X:A]: ((! [Y2:A]: (((((member @ A) @ Y2) @ S3) => (((ord_less_eq @ A) @ X) @ Y2))))))))))))))))). % 6.67/2.70 thf(fact_6084_cINF__UNION, axiom, (! [B:$tType,D:$tType,C:$tType]: (((condit1219197933456340205attice @ B) => (! [A3:(set @ C),B3:(C > (set @ D)),F2:(D > B)]: (((A3 != (bot_bot @ (set @ C))) => ((! [X5:C]: (((((member @ C) @ X5) @ A3) => ((B3 @ X5) != (bot_bot @ (set @ D)))))) => (((condit1013018076250108175_below @ B) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ C) @ (set @ B)) @ (^ [X:C]: (((((image @ D) @ B) @ F2) @ (B3 @ X))))) @ A3))) => (((complete_Inf_Inf @ B) @ ((((image @ D) @ B) @ F2) @ ((complete_Sup_Sup @ (set @ D)) @ ((((image @ C) @ (set @ D)) @ B3) @ A3)))) = ((complete_Inf_Inf @ B) @ ((((image @ C) @ B) @ (^ [X:C]: (((complete_Inf_Inf @ B) @ ((((image @ D) @ B) @ F2) @ (B3 @ X)))))) @ A3)))))))))))). % 6.67/2.70 thf(fact_6085_uniformity__dist, axiom, (! [A:$tType]: (((real_V768167426530841204y_dist @ A) => ((topolo7806501430040627800ormity @ A) = ((complete_Inf_Inf @ (filter @ ((product_prod @ A) @ A))) @ ((((image @ real) @ (filter @ ((product_prod @ A) @ A))) @ (^ [E4:real]: (((principal @ ((product_prod @ A) @ A)) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X:A,Y2:A]: ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X) @ Y2)) @ E4))))))))) @ ((set_ord_greaterThan @ real) @ (zero_zero @ real))))))))). % 6.67/2.70 thf(fact_6086_compactE__image, axiom, (! [A:$tType,B:$tType]: (((topolo4958980785337419405_space @ A) => (! [S3:(set @ A),C5:(set @ B),F2:(B > (set @ A))]: ((((topolo2193935891317330818ompact @ A) @ S3) => ((! [T5:B]: (((((member @ B) @ T5) @ C5) => ((topolo1002775350975398744n_open @ A) @ (F2 @ T5))))) => ((((ord_less_eq @ (set @ A)) @ S3) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ F2) @ C5))) => (~ ((! [C9:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C9) @ C5) => (((finite_finite2 @ B) @ C9) => (~ ((((ord_less_eq @ (set @ A)) @ S3) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ F2) @ C9)))))))))))))))))))). % 6.67/2.70 thf(fact_6087_uniformity__bot, axiom, (! [A:$tType]: (((topolo7287701948861334536_space @ A) => ((topolo7806501430040627800ormity @ A) != (bot_bot @ (filter @ ((product_prod @ A) @ A)))))))). % 6.67/2.70 thf(fact_6088_compact__empty, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => ((topolo2193935891317330818ompact @ A) @ (bot_bot @ (set @ A))))))). % 6.67/2.70 thf(fact_6089_uniformity__transE, axiom, (! [A:$tType]: (((topolo7287701948861334536_space @ A) => (! [E5:(((product_prod @ A) @ A) > $o)]: (((((eventually @ ((product_prod @ A) @ A)) @ E5) @ (topolo7806501430040627800ormity @ A)) => (~ ((! [D7:(((product_prod @ A) @ A) > $o)]: (((((eventually @ ((product_prod @ A) @ A)) @ D7) @ (topolo7806501430040627800ormity @ A)) => (~ ((! [X2:A,Y5:A]: (((D7 @ ((((product_Pair @ A) @ A) @ X2) @ Y5)) => (! [Z4:A]: (((D7 @ ((((product_Pair @ A) @ A) @ Y5) @ Z4)) => (E5 @ ((((product_Pair @ A) @ A) @ X2) @ Z4)))))))))))))))))))))). % 6.67/2.70 thf(fact_6090_uniformity__trans, axiom, (! [A:$tType]: (((topolo7287701948861334536_space @ A) => (! [E5:(((product_prod @ A) @ A) > $o)]: (((((eventually @ ((product_prod @ A) @ A)) @ E5) @ (topolo7806501430040627800ormity @ A)) => (? [D7:(((product_prod @ A) @ A) > $o)]: (((((eventually @ ((product_prod @ A) @ A)) @ D7) @ (topolo7806501430040627800ormity @ A)) & (! [X2:A,Y5:A,Z4:A]: (((D7 @ ((((product_Pair @ A) @ A) @ X2) @ Y5)) => ((D7 @ ((((product_Pair @ A) @ A) @ Y5) @ Z4)) => (E5 @ ((((product_Pair @ A) @ A) @ X2) @ Z4)))))))))))))))). % 6.67/2.70 thf(fact_6091_uniformity__refl, axiom, (! [A:$tType]: (((topolo7287701948861334536_space @ A) => (! [E5:(((product_prod @ A) @ A) > $o),X3:A]: (((((eventually @ ((product_prod @ A) @ A)) @ E5) @ (topolo7806501430040627800ormity @ A)) => (E5 @ ((((product_Pair @ A) @ A) @ X3) @ X3))))))))). % 6.67/2.70 thf(fact_6092_compact__attains__inf, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [S3:(set @ A)]: ((((topolo2193935891317330818ompact @ A) @ S3) => ((S3 != (bot_bot @ (set @ A))) => (? [X5:A]: (((((member @ A) @ X5) @ S3) & (! [Xa:A]: (((((member @ A) @ Xa) @ S3) => (((ord_less_eq @ A) @ X5) @ Xa))))))))))))))). % 6.67/2.70 thf(fact_6093_compact__attains__sup, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [S3:(set @ A)]: ((((topolo2193935891317330818ompact @ A) @ S3) => ((S3 != (bot_bot @ (set @ A))) => (? [X5:A]: (((((member @ A) @ X5) @ S3) & (! [Xa:A]: (((((member @ A) @ Xa) @ S3) => (((ord_less_eq @ A) @ Xa) @ X5))))))))))))))). % 6.67/2.70 thf(fact_6094_uniformity__sym, axiom, (! [A:$tType]: (((topolo7287701948861334536_space @ A) => (! [E5:(((product_prod @ A) @ A) > $o)]: (((((eventually @ ((product_prod @ A) @ A)) @ E5) @ (topolo7806501430040627800ormity @ A)) => (((eventually @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X:A,Y2:A]: ((E5 @ ((((product_Pair @ A) @ A) @ Y2) @ X)))))) @ (topolo7806501430040627800ormity @ A))))))))). % 6.67/2.70 thf(fact_6095_continuous__attains__sup, axiom, (! [B:$tType,A:$tType]: ((((topolo4958980785337419405_space @ A) & (topolo1944317154257567458pology @ B)) => (! [S:(set @ A),F2:(A > B)]: ((((topolo2193935891317330818ompact @ A) @ S) => ((S != (bot_bot @ (set @ A))) => (((((topolo81223032696312382ous_on @ A) @ B) @ S) @ F2) => (? [X5:A]: (((((member @ A) @ X5) @ S) & (! [Xa:A]: (((((member @ A) @ Xa) @ S) => (((ord_less_eq @ B) @ (F2 @ Xa)) @ (F2 @ X5))))))))))))))))). % 6.67/2.70 thf(fact_6096_continuous__attains__inf, axiom, (! [B:$tType,A:$tType]: ((((topolo4958980785337419405_space @ A) & (topolo1944317154257567458pology @ B)) => (! [S:(set @ A),F2:(A > B)]: ((((topolo2193935891317330818ompact @ A) @ S) => ((S != (bot_bot @ (set @ A))) => (((((topolo81223032696312382ous_on @ A) @ B) @ S) @ F2) => (? [X5:A]: (((((member @ A) @ X5) @ S) & (! [Xa:A]: (((((member @ A) @ Xa) @ S) => (((ord_less_eq @ B) @ (F2 @ X5)) @ (F2 @ Xa))))))))))))))))). % 6.67/2.70 thf(fact_6097_Cauchy__uniform__iff, axiom, (! [A:$tType]: (((topolo7287701948861334536_space @ A) => ((topolo3814608138187158403Cauchy @ A) = (^ [X6:(nat > A)]: ((! [P3:(((product_prod @ A) @ A) > $o)]: (((((eventually @ ((product_prod @ A) @ A)) @ P3) @ (topolo7806501430040627800ormity @ A)) => (? [N5:nat]: ((! [N:nat]: (((((ord_less_eq @ nat) @ N5) @ N) => (! [M2:nat]: (((((ord_less_eq @ nat) @ N5) @ M2) => (P3 @ ((((product_Pair @ A) @ A) @ (X6 @ N)) @ (X6 @ M2))))))))))))))))))))). % 6.67/2.70 thf(fact_6098_uniformity__complex__def, axiom, ((topolo7806501430040627800ormity @ complex) = ((complete_Inf_Inf @ (filter @ ((product_prod @ complex) @ complex))) @ ((((image @ real) @ (filter @ ((product_prod @ complex) @ complex))) @ (^ [E4:real]: (((principal @ ((product_prod @ complex) @ complex)) @ ((collect @ ((product_prod @ complex) @ complex)) @ ((((product_case_prod @ complex) @ complex) @ $o) @ (^ [X:complex,Y2:complex]: ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ complex) @ X) @ Y2)) @ E4))))))))) @ ((set_ord_greaterThan @ real) @ (zero_zero @ real)))))). % 6.67/2.70 thf(fact_6099_uniformity__real__def, axiom, ((topolo7806501430040627800ormity @ real) = ((complete_Inf_Inf @ (filter @ ((product_prod @ real) @ real))) @ ((((image @ real) @ (filter @ ((product_prod @ real) @ real))) @ (^ [E4:real]: (((principal @ ((product_prod @ real) @ real)) @ ((collect @ ((product_prod @ real) @ real)) @ ((((product_case_prod @ real) @ real) @ $o) @ (^ [X:real,Y2:real]: ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ real) @ X) @ Y2)) @ E4))))))))) @ ((set_ord_greaterThan @ real) @ (zero_zero @ real)))))). % 6.67/2.70 thf(fact_6100_totally__bounded__def, axiom, (! [A:$tType]: (((topolo7287701948861334536_space @ A) => ((topolo6688025880775521714ounded @ A) = (^ [S7:(set @ A)]: ((! [E6:(((product_prod @ A) @ A) > $o)]: (((((eventually @ ((product_prod @ A) @ A)) @ E6) @ (topolo7806501430040627800ormity @ A)) => (? [X6:(set @ A)]: ((((finite_finite2 @ A) @ X6) & (! [X:A]: (((((member @ A) @ X) @ S7) => (? [Y2:A]: (((((member @ A) @ Y2) @ X6) & (E6 @ ((((product_Pair @ A) @ A) @ Y2) @ X))))))))))))))))))))). % 6.67/2.70 thf(fact_6101_eventually__uniformity__metric, axiom, (! [A:$tType]: (((real_V768167426530841204y_dist @ A) => (! [P:(((product_prod @ A) @ A) > $o)]: (((((eventually @ ((product_prod @ A) @ A)) @ P) @ (topolo7806501430040627800ormity @ A)) = (? [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) & (! [X:A,Y2:A]: (((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X) @ Y2)) @ E4) => (P @ ((((product_Pair @ A) @ A) @ X) @ Y2))))))))))))))). % 6.67/2.70 thf(fact_6102_tendsto__iff__uniformity, axiom, (! [A:$tType,B:$tType]: (((topolo7287701948861334536_space @ B) => (! [F2:(A > B),L:B,F5:(filter @ A)]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L)) @ F5) = (! [E6:(((product_prod @ B) @ B) > $o)]: (((((eventually @ ((product_prod @ B) @ B)) @ E6) @ (topolo7806501430040627800ormity @ B)) => (((eventually @ A) @ (^ [X:A]: ((E6 @ ((((product_Pair @ B) @ B) @ (F2 @ X)) @ L))))) @ F5))))))))))). % 6.67/2.70 thf(fact_6103_compactE, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [S3:(set @ A),T11:(set @ (set @ A))]: ((((topolo2193935891317330818ompact @ A) @ S3) => ((((ord_less_eq @ (set @ A)) @ S3) @ ((complete_Sup_Sup @ (set @ A)) @ T11)) => ((! [B10:(set @ A)]: (((((member @ (set @ A)) @ B10) @ T11) => ((topolo1002775350975398744n_open @ A) @ B10)))) => (~ ((! [T12:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ T12) @ T11) => (((finite_finite2 @ (set @ A)) @ T12) => (~ ((((ord_less_eq @ (set @ A)) @ S3) @ ((complete_Sup_Sup @ (set @ A)) @ T12))))))))))))))))))). % 6.67/2.70 thf(fact_6104_compactI, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [S:(set @ A)]: (((! [C8:(set @ (set @ A))]: (((! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ C8) => ((topolo1002775350975398744n_open @ A) @ X2)))) => ((((ord_less_eq @ (set @ A)) @ S) @ ((complete_Sup_Sup @ (set @ A)) @ C8)) => (? [C10:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ C10) @ C8) & (((finite_finite2 @ (set @ A)) @ C10) & (((ord_less_eq @ (set @ A)) @ S) @ ((complete_Sup_Sup @ (set @ A)) @ C10)))))))))) => ((topolo2193935891317330818ompact @ A) @ S)))))))). % 6.67/2.70 thf(fact_6105_compact__eq__Heine__Borel, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => ((topolo2193935891317330818ompact @ A) = (^ [S7:(set @ A)]: ((! [C7:(set @ (set @ A))]: ((((! [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ C7) => ((topolo1002775350975398744n_open @ A) @ X)))) & (((ord_less_eq @ (set @ A)) @ S7) @ ((complete_Sup_Sup @ (set @ A)) @ C7))) => (? [D8:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ D8) @ C7) & (((finite_finite2 @ (set @ A)) @ D8) & (((ord_less_eq @ (set @ A)) @ S7) @ ((complete_Sup_Sup @ (set @ A)) @ D8)))))))))))))))). % 6.67/2.70 thf(fact_6106_prod__filter__INF2, axiom, (! [B:$tType,C:$tType,A:$tType,J4:(set @ A),A3:(filter @ B),B3:(A > (filter @ C))]: (((J4 != (bot_bot @ (set @ A))) => (((((prod_filter @ B) @ C) @ A3) @ ((complete_Inf_Inf @ (filter @ C)) @ ((((image @ A) @ (filter @ C)) @ B3) @ J4))) = ((complete_Inf_Inf @ (filter @ ((product_prod @ B) @ C))) @ ((((image @ A) @ (filter @ ((product_prod @ B) @ C))) @ (^ [I3:A]: (((((prod_filter @ B) @ C) @ A3) @ (B3 @ I3))))) @ J4))))))). % 6.67/2.70 thf(fact_6107_prod__filter__INF1, axiom, (! [B:$tType,C:$tType,A:$tType,I6:(set @ A),A3:(A > (filter @ B)),B3:(filter @ C)]: (((I6 != (bot_bot @ (set @ A))) => (((((prod_filter @ B) @ C) @ ((complete_Inf_Inf @ (filter @ B)) @ ((((image @ A) @ (filter @ B)) @ A3) @ I6))) @ B3) = ((complete_Inf_Inf @ (filter @ ((product_prod @ B) @ C))) @ ((((image @ A) @ (filter @ ((product_prod @ B) @ C))) @ (^ [I3:A]: (((((prod_filter @ B) @ C) @ (A3 @ I3)) @ B3)))) @ I6))))))). % 6.67/2.70 thf(fact_6108_prod__filter__eq__bot, axiom, (! [A:$tType,B:$tType,A3:(filter @ A),B3:(filter @ B)]: (((((((prod_filter @ A) @ B) @ A3) @ B3) = (bot_bot @ (filter @ ((product_prod @ A) @ B)))) = ((A3 = (bot_bot @ (filter @ A))) | (B3 = (bot_bot @ (filter @ B)))))))). % 6.67/2.70 thf(fact_6109_eventually__prod__same, axiom, (! [A:$tType,P:(((product_prod @ A) @ A) > $o),F5:(filter @ A)]: (((((eventually @ ((product_prod @ A) @ A)) @ P) @ ((((prod_filter @ A) @ A) @ F5) @ F5)) = (? [Q7:(A > $o)]: (((((eventually @ A) @ Q7) @ F5) & (! [X:A,Y2:A]: (((Q7 @ X) => ((Q7 @ Y2) => (P @ ((((product_Pair @ A) @ A) @ X) @ Y2))))))))))))). % 6.67/2.70 thf(fact_6110_eventually__prod__filter, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),F5:(filter @ A),G6:(filter @ B)]: (((((eventually @ ((product_prod @ A) @ B)) @ P) @ ((((prod_filter @ A) @ B) @ F5) @ G6)) = (? [Pf:(A > $o),Pg:(B > $o)]: (((((eventually @ A) @ Pf) @ F5) & ((((eventually @ B) @ Pg) @ G6) & (! [X:A,Y2:B]: (((Pf @ X) => ((Pg @ Y2) => (P @ ((((product_Pair @ A) @ B) @ X) @ Y2)))))))))))))). % 6.67/2.70 thf(fact_6111_prod__filter__mono__iff, axiom, (! [A:$tType,B:$tType,A3:(filter @ A),B3:(filter @ B),C5:(filter @ A),D6:(filter @ B)]: (((A3 != (bot_bot @ (filter @ A))) => ((B3 != (bot_bot @ (filter @ B))) => ((((ord_less_eq @ (filter @ ((product_prod @ A) @ B))) @ ((((prod_filter @ A) @ B) @ A3) @ B3)) @ ((((prod_filter @ A) @ B) @ C5) @ D6)) = ((((ord_less_eq @ (filter @ A)) @ A3) @ C5) & (((ord_less_eq @ (filter @ B)) @ B3) @ D6)))))))). % 6.67/2.70 thf(fact_6112_eventually__prod__sequentially, axiom, (! [P:(((product_prod @ nat) @ nat) > $o)]: (((((eventually @ ((product_prod @ nat) @ nat)) @ P) @ ((((prod_filter @ nat) @ nat) @ (at_top @ nat)) @ (at_top @ nat))) = (? [N5:nat]: ((! [M2:nat]: (((((ord_less_eq @ nat) @ N5) @ M2) => (! [N:nat]: (((((ord_less_eq @ nat) @ N5) @ N) => (P @ ((((product_Pair @ nat) @ nat) @ N) @ M2)))))))))))))). % 6.67/2.70 thf(fact_6113_eventually__prod2, axiom, (! [A:$tType,B:$tType,A3:(filter @ A),P:(B > $o),B3:(filter @ B)]: (((A3 != (bot_bot @ (filter @ A))) => ((((eventually @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X:A]: (P)))) @ ((((prod_filter @ A) @ B) @ A3) @ B3)) = (((eventually @ B) @ P) @ B3)))))). % 6.67/2.70 thf(fact_6114_eventually__prod1, axiom, (! [A:$tType,B:$tType,B3:(filter @ A),P:(B > $o),A3:(filter @ B)]: (((B3 != (bot_bot @ (filter @ A))) => ((((eventually @ ((product_prod @ B) @ A)) @ ((((product_case_prod @ B) @ A) @ $o) @ (^ [X:B,Y2:A]: ((P @ X))))) @ ((((prod_filter @ B) @ A) @ A3) @ B3)) = (((eventually @ B) @ P) @ A3)))))). % 6.67/2.70 thf(fact_6115_nhds__prod, axiom, (! [A:$tType,B:$tType]: ((((topolo4958980785337419405_space @ B) & (topolo4958980785337419405_space @ A)) => (! [A2:A,B2:B]: ((((topolo7230453075368039082e_nhds @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((((prod_filter @ A) @ B) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ ((topolo7230453075368039082e_nhds @ B) @ B2))))))))). % 6.67/2.70 thf(fact_6116_filterlim__Pair, axiom, (! [C:$tType,B:$tType,A:$tType,F2:(A > B),G6:(filter @ B),F5:(filter @ A),G:(A > C),H7:(filter @ C)]: (((((((filterlim @ A) @ B) @ F2) @ G6) @ F5) => ((((((filterlim @ A) @ C) @ G) @ H7) @ F5) => (((((filterlim @ A) @ ((product_prod @ B) @ C)) @ (^ [X:A]: (((((product_Pair @ B) @ C) @ (F2 @ X)) @ (G @ X))))) @ ((((prod_filter @ B) @ C) @ G6) @ H7)) @ F5)))))). % 6.67/2.70 thf(fact_6117_tendsto__add__Pair, axiom, (! [A:$tType]: (((topolo6943815403480290642id_add @ A) => (! [A2:A,B2:A]: ((((((filterlim @ ((product_prod @ A) @ A)) @ A) @ (^ [X:((product_prod @ A) @ A)]: ((((plus_plus @ A) @ (((product_fst @ A) @ A) @ X)) @ (((product_snd @ A) @ A) @ X))))) @ ((topolo7230453075368039082e_nhds @ A) @ (((plus_plus @ A) @ A2) @ B2))) @ ((((prod_filter @ A) @ A) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ ((topolo7230453075368039082e_nhds @ A) @ B2))))))))). % 6.67/2.70 thf(fact_6118_uniformity__trans_H, axiom, (! [A:$tType]: (((topolo7287701948861334536_space @ A) => (! [E5:(((product_prod @ A) @ A) > $o)]: (((((eventually @ ((product_prod @ A) @ A)) @ E5) @ (topolo7806501430040627800ormity @ A)) => (((eventually @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A))) @ ((((product_case_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A)) @ $o) @ ((((product_case_prod @ A) @ A) @ (((product_prod @ A) @ A) > $o)) @ (^ [X:A,Y2:A]: (((((product_case_prod @ A) @ A) @ $o) @ (^ [Y6:A,Z6:A]: (((Y2 = Y6) => (E5 @ ((((product_Pair @ A) @ A) @ X) @ Z6))))))))))) @ ((((prod_filter @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A)) @ (topolo7806501430040627800ormity @ A)) @ (topolo7806501430040627800ormity @ A)))))))))). % 6.67/2.70 thf(fact_6119_prod__filter__INF, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,I6:(set @ A),J4:(set @ B),A3:(A > (filter @ C)),B3:(B > (filter @ D))]: (((I6 != (bot_bot @ (set @ A))) => ((J4 != (bot_bot @ (set @ B))) => (((((prod_filter @ C) @ D) @ ((complete_Inf_Inf @ (filter @ C)) @ ((((image @ A) @ (filter @ C)) @ A3) @ I6))) @ ((complete_Inf_Inf @ (filter @ D)) @ ((((image @ B) @ (filter @ D)) @ B3) @ J4))) = ((complete_Inf_Inf @ (filter @ ((product_prod @ C) @ D))) @ ((((image @ A) @ (filter @ ((product_prod @ C) @ D))) @ (^ [I3:A]: (((complete_Inf_Inf @ (filter @ ((product_prod @ C) @ D))) @ ((((image @ B) @ (filter @ ((product_prod @ C) @ D))) @ (^ [J3:B]: (((((prod_filter @ C) @ D) @ (A3 @ I3)) @ (B3 @ J3))))) @ J4))))) @ I6)))))))). % 6.67/2.70 thf(fact_6120_sequentially__imp__eventually__at__left, axiom, (! [A:$tType]: ((((topolo3112930676232923870pology @ A) & (topolo1944317154257567458pology @ A)) => (! [B2:A,A2:A,P:(A > $o)]: (((((ord_less @ A) @ B2) @ A2) => ((! [F4:(nat > A)]: (((! [N6:nat]: ((((ord_less @ A) @ B2) @ (F4 @ N6)))) => ((! [N6:nat]: ((((ord_less @ A) @ (F4 @ N6)) @ A2))) => ((((order_mono @ nat) @ A) @ F4) => ((((((filterlim @ nat) @ A) @ F4) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ (at_top @ nat)) => (((eventually @ nat) @ (^ [N:nat]: ((P @ (F4 @ N))))) @ (at_top @ nat)))))))) => (((eventually @ A) @ P) @ (((topolo174197925503356063within @ A) @ A2) @ ((set_ord_lessThan @ A) @ A2))))))))))). % 6.67/2.70 thf(fact_6121_relpow__finite__bounded1, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),K:nat]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ K) @ R)) @ ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ A))) @ ((((image @ nat) @ (set @ ((product_prod @ A) @ A))) @ (^ [N:nat]: ((((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R)))) @ ((collect @ nat) @ (^ [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) & (((ord_less_eq @ nat) @ N) @ ((finite_card @ ((product_prod @ A) @ A)) @ R)))))))))))))). % 6.67/2.70 thf(fact_6122_finite__relpow, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),N2:nat]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((finite_finite2 @ ((product_prod @ A) @ A)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N2) @ R))))))). % 6.67/2.70 thf(fact_6123_mono__funpow, axiom, (! [A:$tType]: ((((lattice @ A) & (order_bot @ A)) => (! [Q:(A > A)]: (((((order_mono @ A) @ A) @ Q) => (((order_mono @ nat) @ A) @ (^ [I3:nat]: (((((compow @ (A > A)) @ I3) @ Q) @ (bot_bot @ A)))))))))))). % 6.67/2.70 thf(fact_6124_relpow__Suc__I2, axiom, (! [A:$tType,X3:A,Y:A,R:(set @ ((product_prod @ A) @ A)),Z2:A,N2:nat]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N2) @ R)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N2)) @ R))))))). % 6.67/2.70 thf(fact_6125_relpow__Suc__E2, axiom, (! [A:$tType,X3:A,Z2:A,N2:nat,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N2)) @ R)) => (~ ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y4)) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N2) @ R))))))))))))). % 6.67/2.70 thf(fact_6126_relpow__Suc__D2, axiom, (! [A:$tType,X3:A,Z2:A,N2:nat,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N2)) @ R)) => (? [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y4)) @ R) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N2) @ R))))))))). % 6.67/2.70 thf(fact_6127_relpow__Suc__I, axiom, (! [A:$tType,X3:A,Y:A,N2:nat,R:(set @ ((product_prod @ A) @ A)),Z2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N2) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z2)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N2)) @ R))))))). % 6.67/2.70 thf(fact_6128_relpow__Suc__E, axiom, (! [A:$tType,X3:A,Z2:A,N2:nat,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N2)) @ R)) => (~ ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y4)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N2) @ R)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z2)) @ R)))))))))))). % 6.67/2.70 thf(fact_6129_relpow__0__E, axiom, (! [A:$tType,X3:A,Y:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (zero_zero @ nat)) @ R)) => (X3 = Y))))). % 6.67/2.70 thf(fact_6130_relpow__0__I, axiom, (! [A:$tType,X3:A,R:(set @ ((product_prod @ A) @ A))]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ X3)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (zero_zero @ nat)) @ R))))). % 6.67/2.70 thf(fact_6131_mono__invE, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F2:(A > B),X3:A,Y:A]: (((((order_mono @ A) @ B) @ F2) => ((((ord_less @ B) @ (F2 @ X3)) @ (F2 @ Y)) => (((ord_less_eq @ A) @ X3) @ Y))))))))). % 6.67/2.70 thf(fact_6132_funpow__mono, axiom, (! [A:$tType]: (((order @ A) => (! [F2:(A > A),A3:A,B3:A,N2:nat]: (((((order_mono @ A) @ A) @ F2) => ((((ord_less_eq @ A) @ A3) @ B3) => (((ord_less_eq @ A) @ ((((compow @ (A > A)) @ N2) @ F2) @ A3)) @ ((((compow @ (A > A)) @ N2) @ F2) @ B3)))))))))). % 6.67/2.70 thf(fact_6133_incseq__Suc__iff, axiom, (! [A:$tType]: (((order @ A) => (((order_mono @ nat) @ A) = (^ [F3:(nat > A)]: ((! [N:nat]: ((((ord_less_eq @ A) @ (F3 @ N)) @ (F3 @ (suc @ N)))))))))))). % 6.67/2.70 thf(fact_6134_incseq__SucI, axiom, (! [A:$tType]: (((order @ A) => (! [X8:(nat > A)]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (X8 @ N3)) @ (X8 @ (suc @ N3))))) => (((order_mono @ nat) @ A) @ X8)))))))). % 6.67/2.70 thf(fact_6135_incseq__SucD, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(nat > A),I:nat]: (((((order_mono @ nat) @ A) @ A3) => (((ord_less_eq @ A) @ (A3 @ I)) @ (A3 @ (suc @ I)))))))))). % 6.67/2.70 thf(fact_6136_monoD, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F2:(A > B),X3:A,Y:A]: (((((order_mono @ A) @ B) @ F2) => ((((ord_less_eq @ A) @ X3) @ Y) => (((ord_less_eq @ B) @ (F2 @ X3)) @ (F2 @ Y)))))))))). % 6.67/2.70 thf(fact_6137_monoE, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F2:(A > B),X3:A,Y:A]: (((((order_mono @ A) @ B) @ F2) => ((((ord_less_eq @ A) @ X3) @ Y) => (((ord_less_eq @ B) @ (F2 @ X3)) @ (F2 @ Y)))))))))). % 6.67/2.70 thf(fact_6138_monoI, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F2:(A > B)]: (((! [X5:A,Y4:A]: (((((ord_less_eq @ A) @ X5) @ Y4) => (((ord_less_eq @ B) @ (F2 @ X5)) @ (F2 @ Y4))))) => (((order_mono @ A) @ B) @ F2)))))))). % 6.67/2.70 thf(fact_6139_mono__def, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (((order_mono @ A) @ B) = (^ [F3:(A > B)]: ((! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => (((ord_less_eq @ B) @ (F3 @ X)) @ (F3 @ Y2)))))))))))). % 6.67/2.70 thf(fact_6140_incseqD, axiom, (! [A:$tType]: (((order @ A) => (! [F2:(nat > A),I:nat,J:nat]: (((((order_mono @ nat) @ A) @ F2) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ A) @ (F2 @ I)) @ (F2 @ J)))))))))). % 6.67/2.70 thf(fact_6141_incseq__def, axiom, (! [A:$tType]: (((order @ A) => (((order_mono @ nat) @ A) = (^ [X6:(nat > A)]: ((! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => (((ord_less_eq @ A) @ (X6 @ M2)) @ (X6 @ N)))))))))))). % 6.67/2.70 thf(fact_6142_mono__inf, axiom, (! [B:$tType,A:$tType]: ((((semilattice_inf @ A) & (semilattice_inf @ B)) => (! [F2:(A > B),A3:A,B3:A]: (((((order_mono @ A) @ B) @ F2) => (((ord_less_eq @ B) @ (F2 @ (((inf_inf @ A) @ A3) @ B3))) @ (((inf_inf @ B) @ (F2 @ A3)) @ (F2 @ B3)))))))))). % 6.67/2.70 thf(fact_6143_relpow__Suc__D2_H, axiom, (! [A:$tType,N2:nat,R:(set @ ((product_prod @ A) @ A)),X2:A,Y5:A,Z4:A]: ((((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y5)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N2) @ R)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ Z4)) @ R)) => (? [W:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ W)) @ R) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ W) @ Z4)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N2) @ R))))))))). % 6.67/2.70 thf(fact_6144_mono__add, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [A2:A]: ((((order_mono @ A) @ A) @ ((plus_plus @ A) @ A2)))))))). % 6.67/2.70 thf(fact_6145_mono__Suc, axiom, (((order_mono @ nat) @ nat) @ suc)). % 6.67/2.70 thf(fact_6146_mono__pow, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [F2:(A > A),N2:nat]: (((((order_mono @ A) @ A) @ F2) => (((order_mono @ A) @ A) @ (((compow @ (A > A)) @ N2) @ F2))))))))). % 6.67/2.70 thf(fact_6147_mono__strict__invE, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F2:(A > B),X3:A,Y:A]: (((((order_mono @ A) @ B) @ F2) => ((((ord_less @ B) @ (F2 @ X3)) @ (F2 @ Y)) => (((ord_less @ A) @ X3) @ Y))))))))). % 6.67/2.70 thf(fact_6148_cclfp__lowerbound, axiom, (! [A:$tType]: (((counta3822494911875563373attice @ A) => (! [F2:(A > A),A3:A]: (((((order_mono @ A) @ A) @ F2) => ((((ord_less_eq @ A) @ (F2 @ A3)) @ A3) => (((ord_less_eq @ A) @ ((order_532582986084564980_cclfp @ A) @ F2)) @ A3))))))))). % 6.67/2.70 thf(fact_6149_mono__times__nat, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((order_mono @ nat) @ nat) @ ((times_times @ nat) @ N2)))))). % 6.67/2.70 thf(fact_6150_mono__mult, axiom, (! [A:$tType]: (((ordered_semiring @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => (((order_mono @ A) @ A) @ ((times_times @ A) @ A2))))))))). % 6.67/2.70 thf(fact_6151_mono__image__least, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [F2:(A > B),M:A,N2:A,M5:B,N4:B]: (((((order_mono @ A) @ B) @ F2) => ((((((image @ A) @ B) @ F2) @ (((set_or7035219750837199246ssThan @ A) @ M) @ N2)) = (((set_or7035219750837199246ssThan @ B) @ M5) @ N4)) => ((((ord_less @ A) @ M) @ N2) => ((F2 @ M) = M5)))))))))). % 6.67/2.70 thf(fact_6152_Kleene__iter__gpfp, axiom, (! [A:$tType]: (((order_top @ A) => (! [F2:(A > A),P4:A,K:nat]: (((((order_mono @ A) @ A) @ F2) => ((((ord_less_eq @ A) @ P4) @ (F2 @ P4)) => (((ord_less_eq @ A) @ P4) @ ((((compow @ (A > A)) @ K) @ F2) @ (top_top @ A))))))))))). % 6.67/2.70 thf(fact_6153_Kleene__iter__lpfp, axiom, (! [A:$tType]: (((order_bot @ A) => (! [F2:(A > A),P4:A,K:nat]: (((((order_mono @ A) @ A) @ F2) => ((((ord_less_eq @ A) @ (F2 @ P4)) @ P4) => (((ord_less_eq @ A) @ ((((compow @ (A > A)) @ K) @ F2) @ (bot_bot @ A))) @ P4))))))))). % 6.67/2.70 thf(fact_6154_funpow__mono2, axiom, (! [A:$tType]: (((order @ A) => (! [F2:(A > A),I:nat,J:nat,X3:A,Y:A]: (((((order_mono @ A) @ A) @ F2) => ((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_less_eq @ A) @ X3) @ (F2 @ X3)) => (((ord_less_eq @ A) @ ((((compow @ (A > A)) @ I) @ F2) @ X3)) @ ((((compow @ (A > A)) @ J) @ F2) @ Y)))))))))))). % 6.67/2.70 thf(fact_6155_relpowp__relpow__eq, axiom, (! [A:$tType,N2:nat,R:(set @ ((product_prod @ A) @ A))]: (((((compow @ (A > (A > $o))) @ N2) @ (^ [X:A,Y2:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y2)) @ R)))) = (^ [X:A,Y2:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N2) @ R)))))))). % 6.67/2.70 thf(fact_6156_mono__Sup, axiom, (! [B:$tType,A:$tType]: ((((comple6319245703460814977attice @ A) & (comple6319245703460814977attice @ B)) => (! [F2:(A > B),A3:(set @ A)]: (((((order_mono @ A) @ B) @ F2) => (((ord_less_eq @ B) @ ((complete_Sup_Sup @ B) @ ((((image @ A) @ B) @ F2) @ A3))) @ (F2 @ ((complete_Sup_Sup @ A) @ A3)))))))))). % 6.67/2.70 thf(fact_6157_mono__SUP, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((comple6319245703460814977attice @ A) & (comple6319245703460814977attice @ B)) => (! [F2:(A > B),A3:(C > A),I6:(set @ C)]: (((((order_mono @ A) @ B) @ F2) => (((ord_less_eq @ B) @ ((complete_Sup_Sup @ B) @ ((((image @ C) @ B) @ (^ [X:C]: ((F2 @ (A3 @ X))))) @ I6))) @ (F2 @ ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ A3) @ I6))))))))))). % 6.67/2.70 thf(fact_6158_mono__Inf, axiom, (! [B:$tType,A:$tType]: ((((comple6319245703460814977attice @ A) & (comple6319245703460814977attice @ B)) => (! [F2:(A > B),A3:(set @ A)]: (((((order_mono @ A) @ B) @ F2) => (((ord_less_eq @ B) @ (F2 @ ((complete_Inf_Inf @ A) @ A3))) @ ((complete_Inf_Inf @ B) @ ((((image @ A) @ B) @ F2) @ A3)))))))))). % 6.67/2.70 thf(fact_6159_mono__INF, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((comple6319245703460814977attice @ A) & (comple6319245703460814977attice @ B)) => (! [F2:(A > B),A3:(C > A),I6:(set @ C)]: (((((order_mono @ A) @ B) @ F2) => (((ord_less_eq @ B) @ (F2 @ ((complete_Inf_Inf @ A) @ ((((image @ C) @ A) @ A3) @ I6)))) @ ((complete_Inf_Inf @ B) @ ((((image @ C) @ B) @ (^ [X:C]: ((F2 @ (A3 @ X))))) @ I6)))))))))). % 6.67/2.70 thf(fact_6160_relpow__E, axiom, (! [A:$tType,X3:A,Z2:A,N2:nat,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N2) @ R)) => (((N2 = (zero_zero @ nat)) => (X3 != Z2)) => (~ ((! [Y4:A,M3:nat]: (((N2 = (suc @ M3)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y4)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ M3) @ R)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z2)) @ R)))))))))))))). % 6.67/2.70 thf(fact_6161_relpow__E2, axiom, (! [A:$tType,X3:A,Z2:A,N2:nat,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N2) @ R)) => (((N2 = (zero_zero @ nat)) => (X3 != Z2)) => (~ ((! [Y4:A,M3:nat]: (((N2 = (suc @ M3)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y4)) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ M3) @ R))))))))))))))). % 6.67/2.70 thf(fact_6162_antimono__funpow, axiom, (! [A:$tType]: ((((lattice @ A) & (order_top @ A)) => (! [Q:(A > A)]: (((((order_mono @ A) @ A) @ Q) => (((order_antimono @ nat) @ A) @ (^ [I3:nat]: (((((compow @ (A > A)) @ I3) @ Q) @ (top_top @ A)))))))))))). % 6.67/2.70 thf(fact_6163_relpow__empty, axiom, (! [A:$tType,N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((compow @ (set @ ((product_prod @ A) @ A))) @ N2) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))) = (bot_bot @ (set @ ((product_prod @ A) @ A)))))))). % 6.67/2.70 thf(fact_6164_incseq__le, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => (! [X8:(nat > A),L5:A,N2:nat]: (((((order_mono @ nat) @ A) @ X8) => ((((((filterlim @ nat) @ A) @ X8) @ ((topolo7230453075368039082e_nhds @ A) @ L5)) @ (at_top @ nat)) => (((ord_less_eq @ A) @ (X8 @ N2)) @ L5))))))))). % 6.67/2.70 thf(fact_6165_funpow__increasing, axiom, (! [A:$tType]: ((((lattice @ A) & (order_top @ A)) => (! [M:nat,N2:nat,F2:(A > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((order_mono @ A) @ A) @ F2) => (((ord_less_eq @ A) @ ((((compow @ (A > A)) @ N2) @ F2) @ (top_top @ A))) @ ((((compow @ (A > A)) @ M) @ F2) @ (top_top @ A))))))))))). % 6.67/2.70 thf(fact_6166_funpow__decreasing, axiom, (! [A:$tType]: ((((lattice @ A) & (order_bot @ A)) => (! [M:nat,N2:nat,F2:(A > A)]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((order_mono @ A) @ A) @ F2) => (((ord_less_eq @ A) @ ((((compow @ (A > A)) @ M) @ F2) @ (bot_bot @ A))) @ ((((compow @ (A > A)) @ N2) @ F2) @ (bot_bot @ A))))))))))). % 6.67/2.70 thf(fact_6167_mono__Max__commute, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (linorder @ B)) => (! [F2:(A > B),A3:(set @ A)]: (((((order_mono @ A) @ B) @ F2) => (((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((F2 @ ((lattic643756798349783984er_Max @ A) @ A3)) = ((lattic643756798349783984er_Max @ B) @ ((((image @ A) @ B) @ F2) @ A3)))))))))))). % 6.67/2.70 thf(fact_6168_relpow__fun__conv, axiom, (! [A:$tType,A2:A,B2:A,N2:nat,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N2) @ R)) = (? [F3:(nat > A)]: ((((F3 @ (zero_zero @ nat)) = A2) & (((F3 @ N2) = B2) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ N2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (F3 @ I3)) @ (F3 @ (suc @ I3)))) @ R)))))))))))). % 6.67/2.70 thf(fact_6169_mono__cSup, axiom, (! [B:$tType,A:$tType]: ((((condit1219197933456340205attice @ A) & (condit1219197933456340205attice @ B)) => (! [F2:(A > B),A3:(set @ A)]: (((((order_mono @ A) @ B) @ F2) => (((condit941137186595557371_above @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((ord_less_eq @ B) @ ((complete_Sup_Sup @ B) @ ((((image @ A) @ B) @ F2) @ A3))) @ (F2 @ ((complete_Sup_Sup @ A) @ A3)))))))))))). % 6.67/2.70 thf(fact_6170_mono__cSUP, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((condit1219197933456340205attice @ A) & (condit1219197933456340205attice @ B)) => (! [F2:(A > B),A3:(C > A),I6:(set @ C)]: (((((order_mono @ A) @ B) @ F2) => (((condit941137186595557371_above @ A) @ ((((image @ C) @ A) @ A3) @ I6)) => ((I6 != (bot_bot @ (set @ C))) => (((ord_less_eq @ B) @ ((complete_Sup_Sup @ B) @ ((((image @ C) @ B) @ (^ [X:C]: ((F2 @ (A3 @ X))))) @ I6))) @ (F2 @ ((complete_Sup_Sup @ A) @ ((((image @ C) @ A) @ A3) @ I6))))))))))))). % 6.67/2.70 thf(fact_6171_mono__cInf, axiom, (! [B:$tType,A:$tType]: ((((condit1219197933456340205attice @ A) & (condit1219197933456340205attice @ B)) => (! [F2:(A > B),A3:(set @ A)]: (((((order_mono @ A) @ B) @ F2) => (((condit1013018076250108175_below @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((ord_less_eq @ B) @ (F2 @ ((complete_Inf_Inf @ A) @ A3))) @ ((complete_Inf_Inf @ B) @ ((((image @ A) @ B) @ F2) @ A3)))))))))))). % 6.67/2.70 thf(fact_6172_mono__cINF, axiom, (! [B:$tType,A:$tType,C:$tType]: ((((condit1219197933456340205attice @ A) & (condit1219197933456340205attice @ B)) => (! [F2:(A > B),A3:(C > A),I6:(set @ C)]: (((((order_mono @ A) @ B) @ F2) => (((condit1013018076250108175_below @ A) @ ((((image @ C) @ A) @ A3) @ I6)) => ((I6 != (bot_bot @ (set @ C))) => (((ord_less_eq @ B) @ (F2 @ ((complete_Inf_Inf @ A) @ ((((image @ C) @ A) @ A3) @ I6)))) @ ((complete_Inf_Inf @ B) @ ((((image @ C) @ B) @ (^ [X:C]: ((F2 @ (A3 @ X))))) @ I6)))))))))))). % 6.67/2.70 thf(fact_6173_relpow__finite__bounded, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),K:nat]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ K) @ R)) @ ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ A))) @ ((((image @ nat) @ (set @ ((product_prod @ A) @ A))) @ (^ [N:nat]: ((((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R)))) @ ((collect @ nat) @ (^ [N:nat]: ((((ord_less_eq @ nat) @ N) @ ((finite_card @ ((product_prod @ A) @ A)) @ R)))))))))))). % 6.67/2.70 thf(fact_6174_mono__ge2__power__minus__self, axiom, (! [K:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K) => (((order_mono @ nat) @ nat) @ (^ [M2:nat]: ((((minus_minus @ nat) @ (((power_power @ nat) @ K) @ M2)) @ M2)))))))). % 6.67/2.70 thf(fact_6175_finite__mono__remains__stable__implies__strict__prefix, axiom, (! [A:$tType]: (((order @ A) => (! [F2:(nat > A)]: ((((finite_finite2 @ A) @ ((((image @ nat) @ A) @ F2) @ (top_top @ (set @ nat)))) => ((((order_mono @ nat) @ A) @ F2) => ((! [N3:nat]: ((((F2 @ N3) = (F2 @ (suc @ N3))) => ((F2 @ (suc @ N3)) = (F2 @ (suc @ (suc @ N3))))))) => (? [N9:nat]: (((! [N6:nat]: (((((ord_less_eq @ nat) @ N6) @ N9) => (! [M6:nat]: (((((ord_less_eq @ nat) @ M6) @ N9) => ((((ord_less @ nat) @ M6) @ N6) => (((ord_less @ A) @ (F2 @ M6)) @ (F2 @ N6))))))))) & (! [N6:nat]: (((((ord_less_eq @ nat) @ N9) @ N6) => ((F2 @ N9) = (F2 @ N6))))))))))))))))). % 6.67/2.70 thf(fact_6176_tendsto__at__left__sequentially, axiom, (! [A:$tType,B:$tType]: ((((topolo3112930676232923870pology @ B) & ((topolo1944317154257567458pology @ B) & (topolo4958980785337419405_space @ A))) => (! [B2:B,A2:B,X8:(B > A),L5:A]: (((((ord_less @ B) @ B2) @ A2) => ((! [S5:(nat > B)]: (((! [N6:nat]: ((((ord_less @ B) @ (S5 @ N6)) @ A2))) => ((! [N6:nat]: ((((ord_less @ B) @ B2) @ (S5 @ N6)))) => ((((order_mono @ nat) @ B) @ S5) => ((((((filterlim @ nat) @ B) @ S5) @ ((topolo7230453075368039082e_nhds @ B) @ A2)) @ (at_top @ nat)) => (((((filterlim @ nat) @ A) @ (^ [N:nat]: ((X8 @ (S5 @ N))))) @ ((topolo7230453075368039082e_nhds @ A) @ L5)) @ (at_top @ nat)))))))) => (((((filterlim @ B) @ A) @ X8) @ ((topolo7230453075368039082e_nhds @ A) @ L5)) @ (((topolo174197925503356063within @ B) @ A2) @ ((set_ord_lessThan @ B) @ A2))))))))))). % 6.67/2.70 thf(fact_6177_continuous__at__Sup__mono, axiom, (! [B:$tType,A:$tType]: ((((condit6923001295902523014norder @ A) & ((topolo1944317154257567458pology @ A) & ((condit6923001295902523014norder @ B) & (topolo1944317154257567458pology @ B)))) => (! [F2:(A > B),S3:(set @ A)]: (((((order_mono @ A) @ B) @ F2) => (((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ ((complete_Sup_Sup @ A) @ S3)) @ ((set_ord_lessThan @ A) @ ((complete_Sup_Sup @ A) @ S3)))) @ F2) => ((S3 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ S3) => ((F2 @ ((complete_Sup_Sup @ A) @ S3)) = ((complete_Sup_Sup @ B) @ ((((image @ A) @ B) @ F2) @ S3))))))))))))). % 6.67/2.70 thf(fact_6178_continuous__at__Inf__mono, axiom, (! [B:$tType,A:$tType]: ((((condit6923001295902523014norder @ A) & ((topolo1944317154257567458pology @ A) & ((condit6923001295902523014norder @ B) & (topolo1944317154257567458pology @ B)))) => (! [F2:(A > B),S3:(set @ A)]: (((((order_mono @ A) @ B) @ F2) => (((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ ((complete_Inf_Inf @ A) @ S3)) @ ((set_ord_greaterThan @ A) @ ((complete_Inf_Inf @ A) @ S3)))) @ F2) => ((S3 != (bot_bot @ (set @ A))) => (((condit1013018076250108175_below @ A) @ S3) => ((F2 @ ((complete_Inf_Inf @ A) @ S3)) = ((complete_Inf_Inf @ B) @ ((((image @ A) @ B) @ F2) @ S3))))))))))))). % 6.67/2.70 thf(fact_6179_ntrancl__def, axiom, (! [A:$tType]: (((transitive_ntrancl @ A) = (^ [N:nat,R6:(set @ ((product_prod @ A) @ A))]: (((complete_Sup_Sup @ (set @ ((product_prod @ A) @ A))) @ ((((image @ nat) @ (set @ ((product_prod @ A) @ A))) @ (^ [I3:nat]: ((((compow @ (set @ ((product_prod @ A) @ A))) @ I3) @ R6)))) @ ((collect @ nat) @ (^ [I3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ I3) & (((ord_less_eq @ nat) @ I3) @ (suc @ N)))))))))))))). % 6.67/2.70 thf(fact_6180_trancl__finite__eq__relpow, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R) => (((transitive_trancl @ A) @ R) = ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ A))) @ ((((image @ nat) @ (set @ ((product_prod @ A) @ A))) @ (^ [N:nat]: ((((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R)))) @ ((collect @ nat) @ (^ [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) & (((ord_less_eq @ nat) @ N) @ ((finite_card @ ((product_prod @ A) @ A)) @ R))))))))))))). % 6.67/2.70 thf(fact_6181_trancl__empty, axiom, (! [A:$tType]: ((((transitive_trancl @ A) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))) = (bot_bot @ (set @ ((product_prod @ A) @ A))))))). % 6.67/2.70 thf(fact_6182_finite__trancl, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ ((transitive_trancl @ A) @ R2)) = ((finite_finite2 @ ((product_prod @ A) @ A)) @ R2))))). % 6.67/2.70 thf(fact_6183_ntrancl__Zero, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((transitive_ntrancl @ A) @ (zero_zero @ nat)) @ R) = R)))). % 6.67/2.70 thf(fact_6184_mono__Int, axiom, (! [B:$tType,A:$tType,F2:((set @ A) > (set @ B)),A3:(set @ A),B3:(set @ A)]: (((((order_mono @ (set @ A)) @ (set @ B)) @ F2) => (((ord_less_eq @ (set @ B)) @ (F2 @ (((inf_inf @ (set @ A)) @ A3) @ B3))) @ (((inf_inf @ (set @ B)) @ (F2 @ A3)) @ (F2 @ B3))))))). % 6.67/2.70 thf(fact_6185_rtranclp_Omono, axiom, (! [A:$tType,R2:(A > (A > $o))]: ((((order_mono @ (A > (A > $o))) @ (A > (A > $o))) @ (^ [P6:(A > (A > $o)),X15:A,X24:A]: (((? [A4:A]: (((X15 = A4) & (X24 = A4)))) | (? [A4:A,B4:A,C4:A]: (((X15 = A4) & ((X24 = C4) & (((P6 @ A4) @ B4) & ((R2 @ B4) @ C4))))))))))))). % 6.67/2.70 thf(fact_6186_tranclp_Omono, axiom, (! [A:$tType,R2:(A > (A > $o))]: ((((order_mono @ (A > (A > $o))) @ (A > (A > $o))) @ (^ [P6:(A > (A > $o)),X15:A,X24:A]: (((? [A4:A,B4:A]: (((X15 = A4) & ((X24 = B4) & ((R2 @ A4) @ B4))))) | (? [A4:A,B4:A,C4:A]: (((X15 = A4) & ((X24 = C4) & (((P6 @ A4) @ B4) & ((R2 @ B4) @ C4))))))))))))). % 6.67/2.70 thf(fact_6187_trancl__induct2, axiom, (! [A:$tType,B:$tType,Ax:A,Ay:B,Bx:A,By:B,R2:(set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))),P:(A > (B > $o))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ Bx) @ By))) @ ((transitive_trancl @ ((product_prod @ A) @ B)) @ R2)) => ((! [A5:A,B5:B]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ A5) @ B5))) @ R2) => ((P @ A5) @ B5)))) => ((! [A5:A,B5:B,Aa2:A,Ba:B]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ A5) @ B5))) @ ((transitive_trancl @ ((product_prod @ A) @ B)) @ R2)) => ((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ ((((product_Pair @ A) @ B) @ Aa2) @ Ba))) @ R2) => (((P @ A5) @ B5) => ((P @ Aa2) @ Ba)))))) => ((P @ Bx) @ By))))))). % 6.67/2.70 thf(fact_6188_converse__trancl__induct, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A)),P:(A > $o)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_trancl @ A) @ R2)) => ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ B2)) @ R2) => (P @ Y4)))) => ((! [Y4:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z3) @ B2)) @ ((transitive_trancl @ A) @ R2)) => ((P @ Z3) => (P @ Y4)))))) => (P @ A2))))))). % 6.67/2.70 thf(fact_6189_trancl__trans__induct, axiom, (! [A:$tType,X3:A,Y:A,R2:(set @ ((product_prod @ A) @ A)),P:(A > (A > $o))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ ((transitive_trancl @ A) @ R2)) => ((! [X5:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Y4)) @ R2) => ((P @ X5) @ Y4)))) => ((! [X5:A,Y4:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Y4)) @ ((transitive_trancl @ A) @ R2)) => (((P @ X5) @ Y4) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ ((transitive_trancl @ A) @ R2)) => (((P @ Y4) @ Z3) => ((P @ X5) @ Z3))))))) => ((P @ X3) @ Y))))))). % 6.67/2.70 thf(fact_6190_trancl__into__trancl2, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ ((transitive_trancl @ A) @ R2)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ C2)) @ ((transitive_trancl @ A) @ R2))))))). % 6.67/2.70 thf(fact_6191_Transitive__Closure_Otrancl__into__trancl, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_trancl @ A) @ R2)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ C2)) @ ((transitive_trancl @ A) @ R2))))))). % 6.67/2.70 thf(fact_6192_irrefl__trancl__rD, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),X3:A,Y:A]: (((! [X5:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ X5)) @ ((transitive_trancl @ A) @ R2)))))) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ R2) => (X3 != Y)))))). % 6.67/2.70 thf(fact_6193_converse__tranclE, axiom, (! [A:$tType,X3:A,Z2:A,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z2)) @ ((transitive_trancl @ A) @ R2)) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z2)) @ R2))) => (~ ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y4)) @ R2) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z2)) @ ((transitive_trancl @ A) @ R2)))))))))))))). % 6.67/2.70 thf(fact_6194_r__r__into__trancl, axiom, (! [A:$tType,A2:A,B2:A,R:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ C2)) @ ((transitive_trancl @ A) @ R))))))). % 6.67/2.70 thf(fact_6195_trancl__induct, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A)),P:(A > $o)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_trancl @ A) @ R2)) => ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ Y4)) @ R2) => (P @ Y4)))) => ((! [Y4:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ Y4)) @ ((transitive_trancl @ A) @ R2)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ R2) => ((P @ Y4) => (P @ Z3)))))) => (P @ B2))))))). % 6.67/2.70 thf(fact_6196_trancl__trans, axiom, (! [A:$tType,X3:A,Y:A,R2:(set @ ((product_prod @ A) @ A)),Z2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ ((transitive_trancl @ A) @ R2)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z2)) @ ((transitive_trancl @ A) @ R2)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z2)) @ ((transitive_trancl @ A) @ R2))))))). % 6.67/2.70 thf(fact_6197_tranclE, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_trancl @ A) @ R2)) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2))) => (~ ((! [C3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ C3)) @ ((transitive_trancl @ A) @ R2)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ C3) @ B2)) @ R2))))))))))))). % 6.67/2.70 thf(fact_6198_trancl_Or__into__trancl, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_trancl @ A) @ R2)))))). % 6.67/2.70 thf(fact_6199_trancl_Osimps, axiom, (! [A:$tType,A12:A,A23:A,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A12) @ A23)) @ ((transitive_trancl @ A) @ R2)) = ((? [A4:A,B4:A]: (((A12 = A4) & ((A23 = B4) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ B4)) @ R2))))) | (? [A4:A,B4:A,C4:A]: (((A12 = A4) & ((A23 = C4) & ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ B4)) @ ((transitive_trancl @ A) @ R2)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B4) @ C4)) @ R2))))))))))). % 6.67/2.70 thf(fact_6200_trancl_Ocases, axiom, (! [A:$tType,A12:A,A23:A,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A12) @ A23)) @ ((transitive_trancl @ A) @ R2)) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A12) @ A23)) @ R2))) => (~ ((! [B5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A12) @ B5)) @ ((transitive_trancl @ A) @ R2)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ A23)) @ R2))))))))))))). % 6.67/2.70 thf(fact_6201_finite__trancl__ntranl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R) => (((transitive_trancl @ A) @ R) = (((transitive_ntrancl @ A) @ (((minus_minus @ nat) @ ((finite_card @ ((product_prod @ A) @ A)) @ R)) @ (one_one @ nat))) @ R)))))). % 6.67/2.70 thf(fact_6202_trancl__set__ntrancl, axiom, (! [A:$tType,Xs:(list @ ((product_prod @ A) @ A))]: ((((transitive_trancl @ A) @ ((set2 @ ((product_prod @ A) @ A)) @ Xs)) = (((transitive_ntrancl @ A) @ (((minus_minus @ nat) @ ((finite_card @ ((product_prod @ A) @ A)) @ ((set2 @ ((product_prod @ A) @ A)) @ Xs))) @ (one_one @ nat))) @ ((set2 @ ((product_prod @ A) @ A)) @ Xs)))))). % 6.67/2.70 thf(fact_6203_trancl__power, axiom, (! [A:$tType,P4:((product_prod @ A) @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ P4) @ ((transitive_trancl @ A) @ R)) = (? [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) & (((member @ ((product_prod @ A) @ A)) @ P4) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R))))))))). % 6.67/2.70 thf(fact_6204_less__eq, axiom, (! [M:nat,N2:nat]: (((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ M) @ N2)) @ ((transitive_trancl @ nat) @ pred_nat)) = (((ord_less @ nat) @ M) @ N2))))). % 6.67/2.70 thf(fact_6205_finite_Omono, axiom, (! [A:$tType]: ((((order_mono @ ((set @ A) > $o)) @ ((set @ A) > $o)) @ (^ [P6:((set @ A) > $o),X:(set @ A)]: (((X = (bot_bot @ (set @ A))) | (? [A6:(set @ A),A4:A]: (((X = (((insert @ A) @ A4) @ A6)) & (P6 @ A6))))))))))). % 6.67/2.70 thf(fact_6206_remdups__adj__altdef, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((remdups_adj @ A) @ Xs) = Ys) = (? [F3:(nat > nat)]: (((((order_mono @ nat) @ nat) @ F3) & ((((((image @ nat) @ nat) @ F3) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs))) = (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Ys))) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ Xs) @ I3) = (((nth @ A) @ Ys) @ (F3 @ I3)))))) & (! [I3:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ I3) @ (one_one @ nat))) @ ((size_size @ (list @ A)) @ Xs)) => (((((nth @ A) @ Xs) @ I3) = (((nth @ A) @ Xs) @ (((plus_plus @ nat) @ I3) @ (one_one @ nat)))) = ((F3 @ I3) = (F3 @ (((plus_plus @ nat) @ I3) @ (one_one @ nat))))))))))))))))). % 6.67/2.70 thf(fact_6207_prod__filter__assoc, axiom, (! [A:$tType,B:$tType,C:$tType,F5:(filter @ A),G6:(filter @ B),H7:(filter @ C)]: ((((((prod_filter @ ((product_prod @ A) @ B)) @ C) @ ((((prod_filter @ A) @ B) @ F5) @ G6)) @ H7) = ((((filtermap @ ((product_prod @ A) @ ((product_prod @ B) @ C))) @ ((product_prod @ ((product_prod @ A) @ B)) @ C)) @ ((((product_case_prod @ A) @ ((product_prod @ B) @ C)) @ ((product_prod @ ((product_prod @ A) @ B)) @ C)) @ (^ [X:A]: (((((product_case_prod @ B) @ C) @ ((product_prod @ ((product_prod @ A) @ B)) @ C)) @ (^ [Y2:B]: ((((product_Pair @ ((product_prod @ A) @ B)) @ C) @ ((((product_Pair @ A) @ B) @ X) @ Y2))))))))) @ ((((prod_filter @ A) @ ((product_prod @ B) @ C)) @ F5) @ ((((prod_filter @ B) @ C) @ G6) @ H7))))))). % 6.67/2.70 thf(fact_6208_remdups__adj__Nil__iff, axiom, (! [A:$tType,Xs:(list @ A)]: (((((remdups_adj @ A) @ Xs) = (nil @ A)) = (Xs = (nil @ A)))))). % 6.67/2.70 thf(fact_6209_remdups__adj__set, axiom, (! [A:$tType,Xs:(list @ A)]: ((((set2 @ A) @ ((remdups_adj @ A) @ Xs)) = ((set2 @ A) @ Xs))))). % 6.67/2.70 thf(fact_6210_filtermap__bot, axiom, (! [B:$tType,A:$tType,F2:(B > A)]: ((((((filtermap @ B) @ A) @ F2) @ (bot_bot @ (filter @ B))) = (bot_bot @ (filter @ A)))))). % 6.67/2.70 thf(fact_6211_mono__compose, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType]: ((((order @ A) & (order @ C)) => (! [Q:(A > (B > C)),F2:(D > B)]: (((((order_mono @ A) @ (B > C)) @ Q) => (((order_mono @ A) @ (D > C)) @ (^ [I3:A,X:D]: (((Q @ I3) @ (F2 @ X)))))))))))). % 6.67/2.70 thf(fact_6212_filtermap__bot__iff, axiom, (! [A:$tType,B:$tType,F2:(B > A),F5:(filter @ B)]: (((((((filtermap @ B) @ A) @ F2) @ F5) = (bot_bot @ (filter @ A))) = (F5 = (bot_bot @ (filter @ B))))))). % 6.67/2.70 thf(fact_6213_remdups__adj_Osimps_I1_J, axiom, (! [A:$tType]: ((((remdups_adj @ A) @ (nil @ A)) = (nil @ A))))). % 6.67/2.70 thf(fact_6214_remdups__adj__distinct, axiom, (! [A:$tType,Xs:(list @ A)]: ((((distinct @ A) @ Xs) => (((remdups_adj @ A) @ Xs) = Xs))))). % 6.67/2.70 thf(fact_6215_remdups__adj__length, axiom, (! [A:$tType,Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ ((remdups_adj @ A) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))). % 6.67/2.70 thf(fact_6216_filtermap__Pair, axiom, (! [A:$tType,B:$tType,C:$tType,F2:(C > A),G:(C > B),F5:(filter @ C)]: ((((ord_less_eq @ (filter @ ((product_prod @ A) @ B))) @ ((((filtermap @ C) @ ((product_prod @ A) @ B)) @ (^ [X:C]: (((((product_Pair @ A) @ B) @ (F2 @ X)) @ (G @ X))))) @ F5)) @ ((((prod_filter @ A) @ B) @ ((((filtermap @ C) @ A) @ F2) @ F5)) @ ((((filtermap @ C) @ B) @ G) @ F5)))))). % 6.67/2.70 thf(fact_6217_filtermap__nhds__times, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [C2:A,A2:A]: (((C2 != (zero_zero @ A)) => (((((filtermap @ A) @ A) @ ((times_times @ A) @ C2)) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) = ((topolo7230453075368039082e_nhds @ A) @ (((times_times @ A) @ C2) @ A2)))))))))). % 6.67/2.70 thf(fact_6218_remdups__adj__adjacent, axiom, (! [A:$tType,I:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ (suc @ I)) @ ((size_size @ (list @ A)) @ ((remdups_adj @ A) @ Xs))) => ((((nth @ A) @ ((remdups_adj @ A) @ Xs)) @ I) != (((nth @ A) @ ((remdups_adj @ A) @ Xs)) @ (suc @ I))))))). % 6.67/2.70 thf(fact_6219_at__to__0, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [A2:A]: (((((topolo174197925503356063within @ A) @ A2) @ (top_top @ (set @ A))) = ((((filtermap @ A) @ A) @ (^ [X:A]: ((((plus_plus @ A) @ X) @ A2)))) @ (((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A))))))))))). % 6.67/2.70 thf(fact_6220_filtermap__times__pos__at__right, axiom, (! [A:$tType]: ((((linordered_field @ A) & (topolo1944317154257567458pology @ A)) => (! [C2:A,P4:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((((filtermap @ A) @ A) @ ((times_times @ A) @ C2)) @ (((topolo174197925503356063within @ A) @ P4) @ ((set_ord_greaterThan @ A) @ P4))) = (((topolo174197925503356063within @ A) @ (((times_times @ A) @ C2) @ P4)) @ ((set_ord_greaterThan @ A) @ (((times_times @ A) @ C2) @ P4))))))))))). % 6.67/2.70 thf(fact_6221_at__to__infinity, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => ((((topolo174197925503356063within @ A) @ (zero_zero @ A)) @ (top_top @ (set @ A))) = ((((filtermap @ A) @ A) @ (inverse_inverse @ A)) @ (at_infinity @ A))))))). % 6.67/2.70 thf(fact_6222_prod__filter__principal__singleton, axiom, (! [A:$tType,B:$tType,X3:A,F5:(filter @ B)]: ((((((prod_filter @ A) @ B) @ ((principal @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) @ F5) = ((((filtermap @ B) @ ((product_prod @ A) @ B)) @ (((product_Pair @ A) @ B) @ X3)) @ F5))))). % 6.67/2.70 thf(fact_6223_remdups__adj__length__ge1, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ ((size_size @ (list @ A)) @ ((remdups_adj @ A) @ Xs))))))). % 6.67/2.70 thf(fact_6224_prod__filter__principal__singleton2, axiom, (! [B:$tType,A:$tType,F5:(filter @ A),X3:B]: ((((((prod_filter @ A) @ B) @ F5) @ ((principal @ B) @ (((insert @ B) @ X3) @ (bot_bot @ (set @ B))))) = ((((filtermap @ A) @ ((product_prod @ A) @ B)) @ (^ [A4:A]: (((((product_Pair @ A) @ B) @ A4) @ X3)))) @ F5))))). % 6.67/2.70 thf(fact_6225_cauchy__filter__metric__filtermap, axiom, (! [A:$tType,B:$tType]: ((((real_V768167426530841204y_dist @ B) & (topolo7287701948861334536_space @ B)) => (! [F2:(A > B),F5:(filter @ A)]: ((((topolo6773858410816713723filter @ B) @ ((((filtermap @ A) @ B) @ F2) @ F5)) = (! [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) => (? [P3:(A > $o)]: (((((eventually @ A) @ P3) @ F5) & (! [X:A,Y2:A]: ((((P3 @ X) & (P3 @ Y2)) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ B) @ (F2 @ X)) @ (F2 @ Y2))) @ E4))))))))))))))))). % 6.67/2.70 thf(fact_6226_iteratesp_Omono, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [F2:(A > A)]: ((((order_mono @ (A > $o)) @ (A > $o)) @ (^ [P6:(A > $o),X:A]: (((? [Y2:A]: (((X = (F2 @ Y2)) & (P6 @ Y2)))) | (? [M9:(set @ A)]: (((X = ((complete_Sup_Sup @ A) @ M9)) & ((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ M9) & (! [Y2:A]: (((((member @ A) @ Y2) @ M9) => (P6 @ Y2)))))))))))))))))). % 6.67/2.70 thf(fact_6227_compact__imp__fip__image, axiom, (! [B:$tType,A:$tType]: (((topolo4958980785337419405_space @ A) => (! [S:(set @ A),I6:(set @ B),F2:(B > (set @ A))]: ((((topolo2193935891317330818ompact @ A) @ S) => ((! [I4:B]: (((((member @ B) @ I4) @ I6) => ((topolo7761053866217962861closed @ A) @ (F2 @ I4))))) => ((! [I9:(set @ B)]: ((((finite_finite2 @ B) @ I9) => ((((ord_less_eq @ (set @ B)) @ I9) @ I6) => ((((inf_inf @ (set @ A)) @ S) @ ((complete_Inf_Inf @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ F2) @ I9))) != (bot_bot @ (set @ A))))))) => ((((inf_inf @ (set @ A)) @ S) @ ((complete_Inf_Inf @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ F2) @ I6))) != (bot_bot @ (set @ A)))))))))))). % 6.67/2.70 thf(fact_6228_closed__empty, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => ((topolo7761053866217962861closed @ A) @ (bot_bot @ (set @ A))))))). % 6.67/2.70 thf(fact_6229_closed__singleton, axiom, (! [A:$tType]: (((topological_t1_space @ A) => (! [A2:A]: (((topolo7761053866217962861closed @ A) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))))))))). % 6.67/2.70 thf(fact_6230_closed__Union, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [S3:(set @ (set @ A))]: ((((finite_finite2 @ (set @ A)) @ S3) => ((! [X5:(set @ A)]: (((((member @ (set @ A)) @ X5) @ S3) => ((topolo7761053866217962861closed @ A) @ X5)))) => ((topolo7761053866217962861closed @ A) @ ((complete_Sup_Sup @ (set @ A)) @ S3)))))))))). % 6.67/2.70 thf(fact_6231_closed__UN, axiom, (! [A:$tType,B:$tType]: (((topolo4958980785337419405_space @ A) => (! [A3:(set @ B),B3:(B > (set @ A))]: ((((finite_finite2 @ B) @ A3) => ((! [X5:B]: (((((member @ B) @ X5) @ A3) => ((topolo7761053866217962861closed @ A) @ (B3 @ X5))))) => ((topolo7761053866217962861closed @ A) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B3) @ A3))))))))))). % 6.67/2.70 thf(fact_6232_filtermap__sequentually__ne__bot, axiom, (! [A:$tType,F2:(nat > A)]: ((((((filtermap @ nat) @ A) @ F2) @ (at_top @ nat)) != (bot_bot @ (filter @ A)))))). % 6.67/2.70 thf(fact_6233_closed__diagonal, axiom, (! [A:$tType]: (((topological_t2_space @ A) => ((topolo7761053866217962861closed @ ((product_prod @ A) @ A)) @ ((collect @ ((product_prod @ A) @ A)) @ (^ [Y2:((product_prod @ A) @ A)]: ((? [X:A]: ((Y2 = ((((product_Pair @ A) @ A) @ X) @ X)))))))))))). % 6.67/2.70 thf(fact_6234_chain__subset, axiom, (! [A:$tType,Ord:(A > (A > $o)),A3:(set @ A),B3:(set @ A)]: (((((comple1602240252501008431_chain @ A) @ Ord) @ A3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => (((comple1602240252501008431_chain @ A) @ Ord) @ B3)))))). % 6.67/2.70 thf(fact_6235_ccpo__Sup__upper, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [A3:(set @ A),X3:A]: (((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ A3) => ((((member @ A) @ X3) @ A3) => (((ord_less_eq @ A) @ X3) @ ((complete_Sup_Sup @ A) @ A3)))))))))). % 6.67/2.70 thf(fact_6236_ccpo__Sup__least, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [A3:(set @ A),Z2:A]: (((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ A3) => ((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((ord_less_eq @ A) @ X5) @ Z2)))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A3)) @ Z2))))))))). % 6.67/2.70 thf(fact_6237_finite__imp__closed, axiom, (! [A:$tType]: (((topological_t1_space @ A) => (! [S3:(set @ A)]: ((((finite_finite2 @ A) @ S3) => ((topolo7761053866217962861closed @ A) @ S3)))))))). % 6.67/2.70 thf(fact_6238_chain__empty, axiom, (! [A:$tType,Ord:(A > (A > $o))]: ((((comple1602240252501008431_chain @ A) @ Ord) @ (bot_bot @ (set @ A)))))). % 6.67/2.70 thf(fact_6239_closed__subdiagonal, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => ((topolo7761053866217962861closed @ ((product_prod @ A) @ A)) @ ((collect @ ((product_prod @ A) @ A)) @ (^ [Uu3:((product_prod @ A) @ A)]: ((? [X:A,Y2:A]: (((Uu3 = ((((product_Pair @ A) @ A) @ X) @ Y2)) & (((ord_less_eq @ A) @ X) @ Y2)))))))))))). % 6.67/2.70 thf(fact_6240_closed__superdiagonal, axiom, (! [A:$tType]: (((topolo1944317154257567458pology @ A) => ((topolo7761053866217962861closed @ ((product_prod @ A) @ A)) @ ((collect @ ((product_prod @ A) @ A)) @ (^ [Uu3:((product_prod @ A) @ A)]: ((? [X:A,Y2:A]: (((Uu3 = ((((product_Pair @ A) @ A) @ X) @ Y2)) & (((ord_less_eq @ A) @ Y2) @ X)))))))))))). % 6.67/2.70 thf(fact_6241_chain__singleton, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [X3:A]: ((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))). % 6.67/2.70 thf(fact_6242_closed__Collect__le, axiom, (! [B:$tType,A:$tType]: ((((topolo4958980785337419405_space @ A) & (topolo1944317154257567458pology @ B)) => (! [F2:(A > B),G:(A > B)]: ((((((topolo81223032696312382ous_on @ A) @ B) @ (top_top @ (set @ A))) @ F2) => (((((topolo81223032696312382ous_on @ A) @ B) @ (top_top @ (set @ A))) @ G) => ((topolo7761053866217962861closed @ A) @ ((collect @ A) @ (^ [X:A]: ((((ord_less_eq @ B) @ (F2 @ X)) @ (G @ X)))))))))))))). % 6.67/2.70 thf(fact_6243_t4__space, axiom, (! [A:$tType]: (((topological_t4_space @ A) => (! [S3:(set @ A),T4:(set @ A)]: ((((topolo7761053866217962861closed @ A) @ S3) => (((topolo7761053866217962861closed @ A) @ T4) => (((((inf_inf @ (set @ A)) @ S3) @ T4) = (bot_bot @ (set @ A))) => (? [U6:(set @ A),V7:(set @ A)]: ((((topolo1002775350975398744n_open @ A) @ U6) & (((topolo1002775350975398744n_open @ A) @ V7) & ((((ord_less_eq @ (set @ A)) @ S3) @ U6) & ((((ord_less_eq @ (set @ A)) @ T4) @ V7) & ((((inf_inf @ (set @ A)) @ U6) @ V7) = (bot_bot @ (set @ A)))))))))))))))))). % 6.67/2.70 thf(fact_6244_t3__space, axiom, (! [A:$tType]: (((topological_t3_space @ A) => (! [S3:(set @ A),Y:A]: ((((topolo7761053866217962861closed @ A) @ S3) => ((~ ((((member @ A) @ Y) @ S3))) => (? [U6:(set @ A),V7:(set @ A)]: ((((topolo1002775350975398744n_open @ A) @ U6) & (((topolo1002775350975398744n_open @ A) @ V7) & ((((member @ A) @ Y) @ U6) & ((((ord_less_eq @ (set @ A)) @ S3) @ V7) & ((((inf_inf @ (set @ A)) @ U6) @ V7) = (bot_bot @ (set @ A))))))))))))))))). % 6.67/2.70 thf(fact_6245_nhds__closed, axiom, (! [A:$tType]: (((topological_t3_space @ A) => (! [X3:A,A3:(set @ A)]: (((((member @ A) @ X3) @ A3) => (((topolo1002775350975398744n_open @ A) @ A3) => (? [A15:(set @ A)]: (((((member @ A) @ X3) @ A15) & (((topolo7761053866217962861closed @ A) @ A15) & ((((ord_less_eq @ (set @ A)) @ A15) @ A3) & (((eventually @ A) @ (^ [Y2:A]: ((((member @ A) @ Y2) @ A15)))) @ ((topolo7230453075368039082e_nhds @ A) @ X3))))))))))))))). % 6.67/2.70 thf(fact_6246_continuous__on__closed__Union, axiom, (! [A:$tType,C:$tType,B:$tType]: ((((topolo4958980785337419405_space @ B) & (topolo4958980785337419405_space @ C)) => (! [I6:(set @ A),U3:(A > (set @ B)),F2:(B > C)]: ((((finite_finite2 @ A) @ I6) => ((! [I4:A]: (((((member @ A) @ I4) @ I6) => ((topolo7761053866217962861closed @ B) @ (U3 @ I4))))) => ((! [I4:A]: (((((member @ A) @ I4) @ I6) => ((((topolo81223032696312382ous_on @ B) @ C) @ (U3 @ I4)) @ F2)))) => ((((topolo81223032696312382ous_on @ B) @ C) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ U3) @ I6))) @ F2)))))))))). % 6.67/2.70 thf(fact_6247_Lim__in__closed__set, axiom, (! [B:$tType,A:$tType]: (((topolo4958980785337419405_space @ A) => (! [S3:(set @ A),F2:(B > A),F5:(filter @ B),L:A]: ((((topolo7761053866217962861closed @ A) @ S3) => ((((eventually @ B) @ (^ [X:B]: ((((member @ A) @ (F2 @ X)) @ S3)))) @ F5) => ((F5 != (bot_bot @ (filter @ B))) => ((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ L)) @ F5) => (((member @ A) @ L) @ S3))))))))))). % 6.67/2.70 thf(fact_6248_compact__imp__fip, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => (! [S3:(set @ A),F5:(set @ (set @ A))]: ((((topolo2193935891317330818ompact @ A) @ S3) => ((! [T5:(set @ A)]: (((((member @ (set @ A)) @ T5) @ F5) => ((topolo7761053866217962861closed @ A) @ T5)))) => ((! [F15:(set @ (set @ A))]: ((((finite_finite2 @ (set @ A)) @ F15) => ((((ord_less_eq @ (set @ (set @ A))) @ F15) @ F5) => ((((inf_inf @ (set @ A)) @ S3) @ ((complete_Inf_Inf @ (set @ A)) @ F15)) != (bot_bot @ (set @ A))))))) => ((((inf_inf @ (set @ A)) @ S3) @ ((complete_Inf_Inf @ (set @ A)) @ F5)) != (bot_bot @ (set @ A)))))))))))). % 6.67/2.70 thf(fact_6249_compact__fip, axiom, (! [A:$tType]: (((topolo4958980785337419405_space @ A) => ((topolo2193935891317330818ompact @ A) = (^ [U5:(set @ A)]: ((! [A6:(set @ (set @ A))]: (((! [X:(set @ A)]: (((((member @ (set @ A)) @ X) @ A6) => ((topolo7761053866217962861closed @ A) @ X)))) => ((! [B6:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ B6) @ A6) => (((finite_finite2 @ (set @ A)) @ B6) => ((((inf_inf @ (set @ A)) @ U5) @ ((complete_Inf_Inf @ (set @ A)) @ B6)) != (bot_bot @ (set @ A))))))) => ((((inf_inf @ (set @ A)) @ U5) @ ((complete_Inf_Inf @ (set @ A)) @ A6)) != (bot_bot @ (set @ A)))))))))))))). % 6.67/2.70 thf(fact_6250_in__chain__finite, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [A3:(set @ A)]: (((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ A3) => (((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((member @ A) @ ((complete_Sup_Sup @ A) @ A3)) @ A3)))))))))). % 6.67/2.70 thf(fact_6251_inj__sgn__power, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((inj_on @ real) @ real) @ (^ [Y2:real]: ((((times_times @ real) @ ((sgn_sgn @ real) @ Y2)) @ (((power_power @ real) @ ((abs_abs @ real) @ Y2)) @ N2))))) @ (top_top @ (set @ real))))))). % 6.67/2.70 thf(fact_6252_inj__on__empty, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: (((((inj_on @ A) @ B) @ F2) @ (bot_bot @ (set @ A)))))). % 6.67/2.70 thf(fact_6253_inj__mult__left, axiom, (! [A:$tType]: (((idom @ A) => (! [A2:A]: ((((((inj_on @ A) @ A) @ ((times_times @ A) @ A2)) @ (top_top @ (set @ A))) = (A2 != (zero_zero @ A))))))))). % 6.67/2.70 thf(fact_6254_inj__divide__right, axiom, (! [A:$tType]: (((field @ A) => (! [A2:A]: ((((((inj_on @ A) @ A) @ (^ [B4:A]: ((((divide_divide @ A) @ B4) @ A2)))) @ (top_top @ (set @ A))) = (A2 != (zero_zero @ A))))))))). % 6.67/2.70 thf(fact_6255_inj__apfst, axiom, (! [B:$tType,C:$tType,A:$tType,F2:(A > C)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F2)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = ((((inj_on @ A) @ C) @ F2) @ (top_top @ (set @ A))))))). % 6.67/2.70 thf(fact_6256_inj__apsnd, axiom, (! [A:$tType,C:$tType,B:$tType,F2:(B > C)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F2)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = ((((inj_on @ B) @ C) @ F2) @ (top_top @ (set @ B))))))). % 6.67/2.70 thf(fact_6257_inj__on__insert, axiom, (! [B:$tType,A:$tType,F2:(A > B),A2:A,A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ (((insert @ A) @ A2) @ A3)) = (((((inj_on @ A) @ B) @ F2) @ A3) & (~ ((((member @ B) @ (F2 @ A2)) @ ((((image @ A) @ B) @ F2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))))))). % 6.67/2.70 thf(fact_6258_inj__on__Inter, axiom, (! [B:$tType,A:$tType,S3:(set @ (set @ A)),F2:(A > B)]: (((S3 != (bot_bot @ (set @ (set @ A)))) => ((! [A8:(set @ A)]: (((((member @ (set @ A)) @ A8) @ S3) => ((((inj_on @ A) @ B) @ F2) @ A8)))) => ((((inj_on @ A) @ B) @ F2) @ ((complete_Inf_Inf @ (set @ A)) @ S3))))))). % 6.67/2.70 thf(fact_6259_finite__inverse__image, axiom, (! [A:$tType,B:$tType,A3:(set @ A),F2:(B > A)]: ((((finite_finite2 @ A) @ A3) => (((((inj_on @ B) @ A) @ F2) @ (top_top @ (set @ B))) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [J3:B]: ((((member @ A) @ (F2 @ J3)) @ A3)))))))))). % 6.67/2.70 thf(fact_6260_finite__inverse__image__gen, axiom, (! [A:$tType,B:$tType,A3:(set @ A),F2:(B > A),D6:(set @ B)]: ((((finite_finite2 @ A) @ A3) => (((((inj_on @ B) @ A) @ F2) @ D6) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [J3:B]: (((((member @ B) @ J3) @ D6) & (((member @ A) @ (F2 @ J3)) @ A3))))))))))). % 6.67/2.70 thf(fact_6261_inj__on__mult, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A2:A,A3:(set @ A)]: (((A2 != (zero_zero @ A)) => ((((inj_on @ A) @ A) @ ((times_times @ A) @ A2)) @ A3)))))))). % 6.67/2.70 thf(fact_6262_inj__on__add_H, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,A3:(set @ A)]: (((((inj_on @ A) @ A) @ (^ [B4:A]: ((((plus_plus @ A) @ B4) @ A2)))) @ A3))))))). % 6.67/2.70 thf(fact_6263_inj__on__add, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,A3:(set @ A)]: (((((inj_on @ A) @ A) @ ((plus_plus @ A) @ A2)) @ A3))))))). % 6.67/2.70 thf(fact_6264_inj__add__left, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A]: (((((inj_on @ A) @ A) @ ((plus_plus @ A) @ A2)) @ (top_top @ (set @ A))))))))). % 6.67/2.70 thf(fact_6265_linorder__injI, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [F2:(A > B)]: (((! [X5:A,Y4:A]: (((((ord_less @ A) @ X5) @ Y4) => ((F2 @ X5) != (F2 @ Y4))))) => ((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A)))))))))). % 6.67/2.70 thf(fact_6266_sorted__list__of__set_Oinj__on, axiom, (! [A:$tType]: (((linorder @ A) => ((((inj_on @ A) @ A) @ (^ [X:A]: (X))) @ (top_top @ (set @ A))))))). % 6.67/2.70 thf(fact_6267_inj__fn, axiom, (! [A:$tType,F2:(A > A),N2:nat]: ((((((inj_on @ A) @ A) @ F2) @ (top_top @ (set @ A))) => ((((inj_on @ A) @ A) @ (((compow @ (A > A)) @ N2) @ F2)) @ (top_top @ (set @ A))))))). % 6.67/2.70 thf(fact_6268_inj__on__subset, axiom, (! [B:$tType,A:$tType,F2:(A > B),A3:(set @ A),B3:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ A3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => ((((inj_on @ A) @ B) @ F2) @ B3)))))). % 6.67/2.70 thf(fact_6269_subset__inj__on, axiom, (! [B:$tType,A:$tType,F2:(A > B),B3:(set @ A),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((inj_on @ A) @ B) @ F2) @ A3)))))). % 6.67/2.70 thf(fact_6270_linorder__inj__onI, axiom, (! [B:$tType,A:$tType]: (((order @ A) => (! [A3:(set @ A),F2:(A > B)]: (((! [X5:A,Y4:A]: (((((ord_less @ A) @ X5) @ Y4) => ((((member @ A) @ X5) @ A3) => ((((member @ A) @ Y4) @ A3) => ((F2 @ X5) != (F2 @ Y4))))))) => ((! [X5:A,Y4:A]: (((((member @ A) @ X5) @ A3) => ((((member @ A) @ Y4) @ A3) => ((((ord_less_eq @ A) @ X5) @ Y4) | (((ord_less_eq @ A) @ Y4) @ X5)))))) => ((((inj_on @ A) @ B) @ F2) @ A3))))))))). % 6.67/2.70 thf(fact_6271_subset__image__inj, axiom, (! [A:$tType,B:$tType,S3:(set @ A),F2:(B > A),T4:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ S3) @ ((((image @ B) @ A) @ F2) @ T4)) = (? [U5:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ U5) @ T4) & (((((inj_on @ B) @ A) @ F2) @ U5) & (S3 = ((((image @ B) @ A) @ F2) @ U5)))))))))). % 6.67/2.70 thf(fact_6272_inj__on__image__eq__iff, axiom, (! [B:$tType,A:$tType,F2:(A > B),C5:(set @ A),A3:(set @ A),B3:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ C5) => ((((ord_less_eq @ (set @ A)) @ A3) @ C5) => ((((ord_less_eq @ (set @ A)) @ B3) @ C5) => ((((((image @ A) @ B) @ F2) @ A3) = ((((image @ A) @ B) @ F2) @ B3)) = (A3 = B3)))))))). % 6.67/2.70 thf(fact_6273_inj__on__image__mem__iff, axiom, (! [B:$tType,A:$tType,F2:(A > B),B3:(set @ A),A2:A,A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ B3) => ((((member @ A) @ A2) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ B) @ (F2 @ A2)) @ ((((image @ A) @ B) @ F2) @ A3)) = (((member @ A) @ A2) @ A3)))))))). % 6.67/2.70 thf(fact_6274_finite__image__iff, axiom, (! [B:$tType,A:$tType,F2:(A > B),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ A3) => (((finite_finite2 @ B) @ ((((image @ A) @ B) @ F2) @ A3)) = ((finite_finite2 @ A) @ A3)))))). % 6.67/2.70 thf(fact_6275_finite__imageD, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B)]: ((((finite_finite2 @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => (((((inj_on @ B) @ A) @ F2) @ A3) => ((finite_finite2 @ B) @ A3)))))). % 6.67/2.70 thf(fact_6276_inj__on__strict__subset, axiom, (! [B:$tType,A:$tType,F2:(A > B),B3:(set @ A),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ B3) => ((((ord_less @ (set @ A)) @ A3) @ B3) => (((ord_less @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A3)) @ ((((image @ A) @ B) @ F2) @ B3))))))). % 6.67/2.70 thf(fact_6277_finite__UNIV__inj__surj, axiom, (! [A:$tType,F2:(A > A)]: ((((finite_finite2 @ A) @ (top_top @ (set @ A))) => (((((inj_on @ A) @ A) @ F2) @ (top_top @ (set @ A))) => (((((image @ A) @ A) @ F2) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))))). % 6.67/2.70 thf(fact_6278_finite__UNIV__surj__inj, axiom, (! [A:$tType,F2:(A > A)]: ((((finite_finite2 @ A) @ (top_top @ (set @ A))) => ((((((image @ A) @ A) @ F2) @ (top_top @ (set @ A))) = (top_top @ (set @ A))) => ((((inj_on @ A) @ A) @ F2) @ (top_top @ (set @ A)))))))). % 6.67/2.70 thf(fact_6279_inj__image__subset__iff, axiom, (! [B:$tType,A:$tType,F2:(A > B),A3:(set @ A),B3:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A3)) @ ((((image @ A) @ B) @ F2) @ B3)) = (((ord_less_eq @ (set @ A)) @ A3) @ B3)))))). % 6.67/2.70 thf(fact_6280_inj__on__iff__surj, axiom, (! [A:$tType,B:$tType,A3:(set @ A),A16:(set @ B)]: (((A3 != (bot_bot @ (set @ A))) => ((? [F3:(A > B)]: ((((((inj_on @ A) @ B) @ F3) @ A3) & (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F3) @ A3)) @ A16)))) = (? [G2:(B > A)]: ((((((image @ B) @ A) @ G2) @ A16) = A3)))))))). % 6.67/2.70 thf(fact_6281_finite__surj__inj, axiom, (! [A:$tType,A3:(set @ A),F2:(A > A)]: ((((finite_finite2 @ A) @ A3) => ((((ord_less_eq @ (set @ A)) @ A3) @ ((((image @ A) @ A) @ F2) @ A3)) => ((((inj_on @ A) @ A) @ F2) @ A3)))))). % 6.67/2.70 thf(fact_6282_inj__on__finite, axiom, (! [B:$tType,A:$tType,F2:(A > B),A3:(set @ A),B3:(set @ B)]: ((((((inj_on @ A) @ B) @ F2) @ A3) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A3)) @ B3) => (((finite_finite2 @ B) @ B3) => ((finite_finite2 @ A) @ A3))))))). % 6.67/2.70 thf(fact_6283_endo__inj__surj, axiom, (! [A:$tType,A3:(set @ A),F2:(A > A)]: ((((finite_finite2 @ A) @ A3) => ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ F2) @ A3)) @ A3) => (((((inj_on @ A) @ A) @ F2) @ A3) => (((((image @ A) @ A) @ F2) @ A3) = A3))))))). % 6.67/2.70 thf(fact_6284_inj__on__image__Int, axiom, (! [B:$tType,A:$tType,F2:(A > B),C5:(set @ A),A3:(set @ A),B3:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ C5) => ((((ord_less_eq @ (set @ A)) @ A3) @ C5) => ((((ord_less_eq @ (set @ A)) @ B3) @ C5) => (((((image @ A) @ B) @ F2) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) = (((inf_inf @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A3)) @ ((((image @ A) @ B) @ F2) @ B3))))))))). % 6.67/2.70 thf(fact_6285_inj__on__iff__eq__card, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F2:(A > B)]: ((((finite_finite2 @ A) @ A3) => (((((inj_on @ A) @ B) @ F2) @ A3) = (((finite_card @ B) @ ((((image @ A) @ B) @ F2) @ A3)) = ((finite_card @ A) @ A3))))))). % 6.67/2.70 thf(fact_6286_eq__card__imp__inj__on, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F2:(A > B)]: ((((finite_finite2 @ A) @ A3) => ((((finite_card @ B) @ ((((image @ A) @ B) @ F2) @ A3)) = ((finite_card @ A) @ A3)) => ((((inj_on @ A) @ B) @ F2) @ A3)))))). % 6.67/2.70 thf(fact_6287_inj__on__image__set__diff, axiom, (! [B:$tType,A:$tType,F2:(A > B),C5:(set @ A),A3:(set @ A),B3:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ C5) => ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ C5) => ((((ord_less_eq @ (set @ A)) @ B3) @ C5) => (((((image @ A) @ B) @ F2) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = (((minus_minus @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A3)) @ ((((image @ A) @ B) @ F2) @ B3))))))))). % 6.67/2.70 thf(fact_6288_pigeonhole, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B)]: (((((ord_less @ nat) @ ((finite_card @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ ((finite_card @ B) @ A3)) => (~ (((((inj_on @ B) @ A) @ F2) @ A3))))))). % 6.67/2.70 thf(fact_6289_continuous__inj__imp__mono, axiom, (! [B:$tType,A:$tType]: ((((topolo8458572112393995274pology @ A) & (topolo1944317154257567458pology @ B)) => (! [A2:A,X3:A,B2:A,F2:(A > B)]: (((((ord_less @ A) @ A2) @ X3) => ((((ord_less @ A) @ X3) @ B2) => (((((topolo81223032696312382ous_on @ A) @ B) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) @ F2) => (((((inj_on @ A) @ B) @ F2) @ (((set_or1337092689740270186AtMost @ A) @ A2) @ B2)) => (((((ord_less @ B) @ (F2 @ A2)) @ (F2 @ X3)) & (((ord_less @ B) @ (F2 @ X3)) @ (F2 @ B2))) | ((((ord_less @ B) @ (F2 @ B2)) @ (F2 @ X3)) & (((ord_less @ B) @ (F2 @ X3)) @ (F2 @ A2)))))))))))))). % 6.67/2.70 thf(fact_6290_the__inv__into__into, axiom, (! [B:$tType,A:$tType,F2:(A > B),A3:(set @ A),X3:B,B3:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ A3) => ((((member @ B) @ X3) @ ((((image @ A) @ B) @ F2) @ A3)) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((member @ A) @ (((((the_inv_into @ A) @ B) @ A3) @ F2) @ X3)) @ B3))))))). % 6.67/2.70 thf(fact_6291_inj__on__UNION__chain, axiom, (! [C:$tType,B:$tType,A:$tType,I6:(set @ A),A3:(A > (set @ B)),F2:(B > C)]: (((! [I4:A,J2:A]: (((((member @ A) @ I4) @ I6) => ((((member @ A) @ J2) @ I6) => ((((ord_less_eq @ (set @ B)) @ (A3 @ I4)) @ (A3 @ J2)) | (((ord_less_eq @ (set @ B)) @ (A3 @ J2)) @ (A3 @ I4))))))) => ((! [I4:A]: (((((member @ A) @ I4) @ I6) => ((((inj_on @ B) @ C) @ F2) @ (A3 @ I4))))) => ((((inj_on @ B) @ C) @ F2) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ A3) @ I6)))))))). % 6.67/2.70 thf(fact_6292_inj__on__INTER, axiom, (! [C:$tType,B:$tType,A:$tType,I6:(set @ A),F2:(B > C),A3:(A > (set @ B))]: (((I6 != (bot_bot @ (set @ A))) => ((! [I4:A]: (((((member @ A) @ I4) @ I6) => ((((inj_on @ B) @ C) @ F2) @ (A3 @ I4))))) => ((((inj_on @ B) @ C) @ F2) @ ((complete_Inf_Inf @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ A3) @ I6)))))))). % 6.67/2.70 thf(fact_6293_surjective__iff__injective__gen, axiom, (! [B:$tType,A:$tType,S3:(set @ A),T4:(set @ B),F2:(A > B)]: ((((finite_finite2 @ A) @ S3) => (((finite_finite2 @ B) @ T4) => ((((finite_card @ A) @ S3) = ((finite_card @ B) @ T4)) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ S3)) @ T4) => ((! [X:B]: (((((member @ B) @ X) @ T4) => (? [Y2:A]: (((((member @ A) @ Y2) @ S3) & ((F2 @ Y2) = X))))))) = ((((inj_on @ A) @ B) @ F2) @ S3))))))))). % 6.67/2.70 thf(fact_6294_card__bij__eq, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ A),B3:(set @ B),G:(B > A)]: ((((((inj_on @ A) @ B) @ F2) @ A3) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A3)) @ B3) => (((((inj_on @ B) @ A) @ G) @ B3) => ((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ G) @ B3)) @ A3) => (((finite_finite2 @ A) @ A3) => (((finite_finite2 @ B) @ B3) => (((finite_card @ A) @ A3) = ((finite_card @ B) @ B3))))))))))). % 6.67/2.70 thf(fact_6295_inj__image__Compl__subset, axiom, (! [B:$tType,A:$tType,F2:(A > B),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ ((uminus_uminus @ (set @ A)) @ A3))) @ ((uminus_uminus @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A3))))))). % 6.67/2.70 thf(fact_6296_image__INT, axiom, (! [B:$tType,A:$tType,C:$tType,F2:(A > B),C5:(set @ A),A3:(set @ C),B3:(C > (set @ A)),J:C]: ((((((inj_on @ A) @ B) @ F2) @ C5) => ((! [X5:C]: (((((member @ C) @ X5) @ A3) => (((ord_less_eq @ (set @ A)) @ (B3 @ X5)) @ C5)))) => ((((member @ C) @ J) @ A3) => (((((image @ A) @ B) @ F2) @ ((complete_Inf_Inf @ (set @ A)) @ ((((image @ C) @ (set @ A)) @ B3) @ A3))) = ((complete_Inf_Inf @ (set @ B)) @ ((((image @ C) @ (set @ B)) @ (^ [X:C]: (((((image @ A) @ B) @ F2) @ (B3 @ X))))) @ A3))))))))). % 6.67/2.70 thf(fact_6297_card__le__inj, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ B) @ B3) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ B) @ B3)) => (? [F4:(A > B)]: (((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F4) @ A3)) @ B3) & ((((inj_on @ A) @ B) @ F4) @ A3)))))))))). % 6.67/2.70 thf(fact_6298_card__inj__on__le, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ A),B3:(set @ B)]: ((((((inj_on @ A) @ B) @ F2) @ A3) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A3)) @ B3) => (((finite_finite2 @ B) @ B3) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ B) @ B3)))))))). % 6.67/2.70 thf(fact_6299_inj__on__iff__card__le, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B3:(set @ B)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ B) @ B3) => ((? [F3:(A > B)]: ((((((inj_on @ A) @ B) @ F3) @ A3) & (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F3) @ A3)) @ B3)))) = (((ord_less_eq @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ B) @ B3)))))))). % 6.67/2.70 thf(fact_6300_log__inj, axiom, (! [B2:real]: (((((ord_less @ real) @ (one_one @ real)) @ B2) => ((((inj_on @ real) @ real) @ (log @ B2)) @ ((set_ord_greaterThan @ real) @ (zero_zero @ real))))))). % 6.67/2.70 thf(fact_6301_funpow__inj__finite, axiom, (! [A:$tType,P4:(A > A),X3:A]: ((((((inj_on @ A) @ A) @ P4) @ (top_top @ (set @ A))) => (((finite_finite2 @ A) @ ((collect @ A) @ (^ [Y2:A]: ((? [N:nat]: ((Y2 = ((((compow @ (A > A)) @ N) @ P4) @ X3)))))))) => (~ ((! [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) => (((((compow @ (A > A)) @ N3) @ P4) @ X3) != X3))))))))))). % 6.67/2.70 thf(fact_6302_all__subset__image__inj, axiom, (! [A:$tType,B:$tType,F2:(B > A),S3:(set @ B),P:((set @ A) > $o)]: (((! [T8:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ T8) @ ((((image @ B) @ A) @ F2) @ S3)) => (P @ T8)))) = (! [T8:(set @ B)]: ((((((ord_less_eq @ (set @ B)) @ T8) @ S3) & ((((inj_on @ B) @ A) @ F2) @ T8)) => (P @ ((((image @ B) @ A) @ F2) @ T8))))))))). % 6.67/2.70 thf(fact_6303_ex__subset__image__inj, axiom, (! [A:$tType,B:$tType,F2:(B > A),S3:(set @ B),P:((set @ A) > $o)]: (((? [T8:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ T8) @ ((((image @ B) @ A) @ F2) @ S3)) & (P @ T8)))) = (? [T8:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ T8) @ S3) & (((((inj_on @ B) @ A) @ F2) @ T8) & (P @ ((((image @ B) @ A) @ F2) @ T8)))))))))). % 6.67/2.70 thf(fact_6304_inj__singleton, axiom, (! [A:$tType,A3:(set @ A)]: (((((inj_on @ A) @ (set @ A)) @ (^ [X:A]: ((((insert @ A) @ X) @ (bot_bot @ (set @ A)))))) @ A3)))). % 6.67/2.70 thf(fact_6305_inj__Some, axiom, (! [A:$tType,A3:(set @ A)]: (((((inj_on @ A) @ (option @ A)) @ (some @ A)) @ A3)))). % 6.67/2.70 thf(fact_6306_inj__Suc, axiom, (! [N7:(set @ nat)]: (((((inj_on @ nat) @ nat) @ suc) @ N7)))). % 6.67/2.70 thf(fact_6307_inj__of__nat, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => ((((inj_on @ nat) @ A) @ (semiring_1_of_nat @ A)) @ (top_top @ (set @ nat))))))). % 6.67/2.70 thf(fact_6308_inj__on__diff__nat, axiom, (! [N7:(set @ nat),K:nat]: (((! [N3:nat]: (((((member @ nat) @ N3) @ N7) => (((ord_less_eq @ nat) @ K) @ N3)))) => ((((inj_on @ nat) @ nat) @ (^ [N:nat]: ((((minus_minus @ nat) @ N) @ K)))) @ N7))))). % 6.67/2.70 thf(fact_6309_inj__on__set__encode, axiom, ((((inj_on @ (set @ nat)) @ nat) @ nat_set_encode) @ ((collect @ (set @ nat)) @ (finite_finite2 @ nat)))). % 6.67/2.70 thf(fact_6310_inj__on__convol__ident, axiom, (! [B:$tType,A:$tType,F2:(A > B),X8:(set @ A)]: (((((inj_on @ A) @ ((product_prod @ A) @ B)) @ (^ [X:A]: (((((product_Pair @ A) @ B) @ X) @ (F2 @ X))))) @ X8)))). % 6.67/2.70 thf(fact_6311_swap__inj__on, axiom, (! [B:$tType,A:$tType,A3:(set @ ((product_prod @ A) @ B))]: (((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ B) @ A)) @ ((((product_case_prod @ A) @ B) @ ((product_prod @ B) @ A)) @ (^ [I3:A,J3:B]: (((((product_Pair @ B) @ A) @ J3) @ I3))))) @ A3)))). % 6.67/2.70 thf(fact_6312_range__inj__infinite, axiom, (! [A:$tType,F2:(nat > A)]: ((((((inj_on @ nat) @ A) @ F2) @ (top_top @ (set @ nat))) => (~ (((finite_finite2 @ A) @ ((((image @ nat) @ A) @ F2) @ (top_top @ (set @ nat)))))))))). % 6.67/2.70 thf(fact_6313_finite__imp__nat__seg__image__inj__on, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (? [N3:nat,F4:(nat > A)]: (((A3 = ((((image @ nat) @ A) @ F4) @ ((collect @ nat) @ (^ [I3:nat]: ((((ord_less @ nat) @ I3) @ N3)))))) & ((((inj_on @ nat) @ A) @ F4) @ ((collect @ nat) @ (^ [I3:nat]: ((((ord_less @ nat) @ I3) @ N3)))))))))))). % 6.67/2.70 thf(fact_6314_finite__imp__inj__to__nat__seg, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (? [F4:(A > nat),N3:nat]: (((((((image @ A) @ nat) @ F4) @ A3) = ((collect @ nat) @ (^ [I3:nat]: ((((ord_less @ nat) @ I3) @ N3))))) & ((((inj_on @ A) @ nat) @ F4) @ A3)))))))). % 6.67/2.70 thf(fact_6315_inj__on__nth, axiom, (! [A:$tType,Xs:(list @ A),I6:(set @ nat)]: ((((distinct @ A) @ Xs) => ((! [X5:nat]: (((((member @ nat) @ X5) @ I6) => (((ord_less @ nat) @ X5) @ ((size_size @ (list @ A)) @ Xs))))) => ((((inj_on @ nat) @ A) @ ((nth @ A) @ Xs)) @ I6)))))). % 6.67/2.70 thf(fact_6316_infinite__iff__countable__subset, axiom, (! [A:$tType,S3:(set @ A)]: (((~ (((finite_finite2 @ A) @ S3))) = (? [F3:(nat > A)]: ((((((inj_on @ nat) @ A) @ F3) @ (top_top @ (set @ nat))) & (((ord_less_eq @ (set @ A)) @ ((((image @ nat) @ A) @ F3) @ (top_top @ (set @ nat)))) @ S3)))))))). % 6.67/2.70 thf(fact_6317_infinite__countable__subset, axiom, (! [A:$tType,S3:(set @ A)]: (((~ (((finite_finite2 @ A) @ S3))) => (? [F4:(nat > A)]: ((((((inj_on @ nat) @ A) @ F4) @ (top_top @ (set @ nat))) & (((ord_less_eq @ (set @ A)) @ ((((image @ nat) @ A) @ F4) @ (top_top @ (set @ nat)))) @ S3)))))))). % 6.67/2.70 thf(fact_6318_inj__on__funpow__least, axiom, (! [A:$tType,N2:nat,F2:(A > A),S:A]: (((((((compow @ (A > A)) @ N2) @ F2) @ S) = S) => ((! [M3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M3) => ((((ord_less @ nat) @ M3) @ N2) => (((((compow @ (A > A)) @ M3) @ F2) @ S) != S))))) => ((((inj_on @ nat) @ A) @ (^ [K3:nat]: (((((compow @ (A > A)) @ K3) @ F2) @ S)))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N2))))))). % 6.67/2.70 thf(fact_6319_flat__lub__def, axiom, (! [A:$tType]: (((partial_flat_lub @ A) = (^ [B4:A,A6:(set @ A)]: (((((if @ A) @ (((ord_less_eq @ (set @ A)) @ A6) @ (((insert @ A) @ B4) @ (bot_bot @ (set @ A))))) @ B4) @ ((the @ A) @ (^ [X:A]: ((((member @ A) @ X) @ (((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ B4) @ (bot_bot @ (set @ A))))))))))))))). % 6.67/2.70 thf(fact_6320_rtrancl__finite__eq__relpow, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R) => (((transitive_rtrancl @ A) @ R) = ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ A))) @ ((((image @ nat) @ (set @ ((product_prod @ A) @ A))) @ (^ [N:nat]: ((((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R)))) @ ((collect @ nat) @ (^ [N:nat]: ((((ord_less_eq @ nat) @ N) @ ((finite_card @ ((product_prod @ A) @ A)) @ R)))))))))))). % 6.67/2.70 thf(fact_6321_listrel__rtrancl__refl, axiom, (! [A:$tType,Xs:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Xs)) @ (((listrel @ A) @ A) @ ((transitive_rtrancl @ A) @ R2)))))). % 6.67/2.70 thf(fact_6322_listrel__rtrancl__eq__rtrancl__listrel1, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((((listrel @ A) @ A) @ ((transitive_rtrancl @ A) @ R2)) = ((transitive_rtrancl @ (list @ A)) @ ((listrel1 @ A) @ R2)))))). % 6.67/2.70 thf(fact_6323_listrel1__rtrancl__subset__rtrancl__listrel1, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((ord_less_eq @ (set @ ((product_prod @ (list @ A)) @ (list @ A)))) @ ((listrel1 @ A) @ ((transitive_rtrancl @ A) @ R2))) @ ((transitive_rtrancl @ (list @ A)) @ ((listrel1 @ A) @ R2)))))). % 6.67/2.70 thf(fact_6324_rtrancl__induct2, axiom, (! [A:$tType,B:$tType,Ax:A,Ay:B,Bx:A,By:B,R2:(set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))),P:(A > (B > $o))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ Bx) @ By))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R2)) => (((P @ Ax) @ Ay) => ((! [A5:A,B5:B,Aa2:A,Ba:B]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ A5) @ B5))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R2)) => ((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ ((((product_Pair @ A) @ B) @ Aa2) @ Ba))) @ R2) => (((P @ A5) @ B5) => ((P @ Aa2) @ Ba)))))) => ((P @ Bx) @ By))))))). % 6.67/2.70 thf(fact_6325_converse__rtranclE2, axiom, (! [B:$tType,A:$tType,Xa2:A,Xb:B,Za:A,Zb:B,R2:(set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Xa2) @ Xb)) @ ((((product_Pair @ A) @ B) @ Za) @ Zb))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R2)) => ((((((product_Pair @ A) @ B) @ Xa2) @ Xb) != ((((product_Pair @ A) @ B) @ Za) @ Zb)) => (~ ((! [A5:A,B5:B]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Xa2) @ Xb)) @ ((((product_Pair @ A) @ B) @ A5) @ B5))) @ R2) => (~ ((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ ((((product_Pair @ A) @ B) @ Za) @ Zb))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R2)))))))))))))). % 6.67/2.70 thf(fact_6326_converse__rtrancl__induct2, axiom, (! [A:$tType,B:$tType,Ax:A,Ay:B,Bx:A,By:B,R2:(set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))),P:(A > (B > $o))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ Bx) @ By))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R2)) => (((P @ Bx) @ By) => ((! [A5:A,B5:B,Aa2:A,Ba:B]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ ((((product_Pair @ A) @ B) @ Aa2) @ Ba))) @ R2) => ((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Aa2) @ Ba)) @ ((((product_Pair @ A) @ B) @ Bx) @ By))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R2)) => (((P @ Aa2) @ Ba) => ((P @ A5) @ B5)))))) => ((P @ Ax) @ Ay))))))). % 6.67/2.70 thf(fact_6327_converse__rtrancl__into__rtrancl, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ ((transitive_rtrancl @ A) @ R2)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ C2)) @ ((transitive_rtrancl @ A) @ R2))))))). % 6.67/2.70 thf(fact_6328_converse__rtrancl__induct, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A)),P:(A > $o)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_rtrancl @ A) @ R2)) => ((P @ B2) => ((! [Y4:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z3) @ B2)) @ ((transitive_rtrancl @ A) @ R2)) => ((P @ Z3) => (P @ Y4)))))) => (P @ A2))))))). % 6.67/2.70 thf(fact_6329_converse__rtranclE, axiom, (! [A:$tType,X3:A,Z2:A,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z2)) @ ((transitive_rtrancl @ A) @ R2)) => ((X3 != Z2) => (~ ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y4)) @ R2) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z2)) @ ((transitive_rtrancl @ A) @ R2)))))))))))))). % 6.67/2.70 thf(fact_6330_rtrancl__induct, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A)),P:(A > $o)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_rtrancl @ A) @ R2)) => ((P @ A2) => ((! [Y4:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ Y4)) @ ((transitive_rtrancl @ A) @ R2)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ R2) => ((P @ Y4) => (P @ Z3)))))) => (P @ B2))))))). % 6.67/2.70 thf(fact_6331_rtrancl__trans, axiom, (! [A:$tType,X3:A,Y:A,R2:(set @ ((product_prod @ A) @ A)),Z2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ ((transitive_rtrancl @ A) @ R2)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z2)) @ ((transitive_rtrancl @ A) @ R2)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z2)) @ ((transitive_rtrancl @ A) @ R2))))))). % 6.67/2.70 thf(fact_6332_rtranclE, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_rtrancl @ A) @ R2)) => ((A2 != B2) => (~ ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ Y4)) @ ((transitive_rtrancl @ A) @ R2)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ B2)) @ R2))))))))))))). % 6.67/2.70 thf(fact_6333_rtrancl_Ortrancl__into__rtrancl, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_rtrancl @ A) @ R2)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ C2)) @ ((transitive_rtrancl @ A) @ R2))))))). % 6.67/2.70 thf(fact_6334_rtrancl_Ortrancl__refl, axiom, (! [A:$tType,A2:A,R2:(set @ ((product_prod @ A) @ A))]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ ((transitive_rtrancl @ A) @ R2))))). % 6.67/2.70 thf(fact_6335_rtrancl_Osimps, axiom, (! [A:$tType,A12:A,A23:A,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A12) @ A23)) @ ((transitive_rtrancl @ A) @ R2)) = ((? [A4:A]: (((A12 = A4) & (A23 = A4)))) | (? [A4:A,B4:A,C4:A]: (((A12 = A4) & ((A23 = C4) & ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ B4)) @ ((transitive_rtrancl @ A) @ R2)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B4) @ C4)) @ R2))))))))))). % 6.67/2.70 thf(fact_6336_rtrancl_Ocases, axiom, (! [A:$tType,A12:A,A23:A,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A12) @ A23)) @ ((transitive_rtrancl @ A) @ R2)) => ((A23 != A12) => (~ ((! [B5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A12) @ B5)) @ ((transitive_rtrancl @ A) @ R2)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ A23)) @ R2))))))))))))). % 6.67/2.70 thf(fact_6337_listrel__rtrancl__trans, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A)),Zs2:(list @ A)]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ (((listrel @ A) @ A) @ ((transitive_rtrancl @ A) @ R2))) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ys) @ Zs2)) @ (((listrel @ A) @ A) @ ((transitive_rtrancl @ A) @ R2))) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Zs2)) @ (((listrel @ A) @ A) @ ((transitive_rtrancl @ A) @ R2)))))))). % 6.67/2.70 thf(fact_6338_tranclD, axiom, (! [A:$tType,X3:A,Y:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ ((transitive_trancl @ A) @ R)) => (? [Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z3)) @ R) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z3) @ Y)) @ ((transitive_rtrancl @ A) @ R))))))))). % 6.67/2.70 thf(fact_6339_rtranclD, axiom, (! [A:$tType,A2:A,B2:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_rtrancl @ A) @ R)) => ((A2 = B2) | ((A2 != B2) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_trancl @ A) @ R)))))))). % 6.67/2.70 thf(fact_6340_tranclD2, axiom, (! [A:$tType,X3:A,Y:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ ((transitive_trancl @ A) @ R)) => (? [Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z3)) @ ((transitive_rtrancl @ A) @ R)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z3) @ Y)) @ R)))))))). % 6.67/2.70 thf(fact_6341_trancl__into__rtrancl, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_trancl @ A) @ R2)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_rtrancl @ A) @ R2)))))). % 6.67/2.70 thf(fact_6342_rtrancl__eq__or__trancl, axiom, (! [A:$tType,X3:A,Y:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ ((transitive_rtrancl @ A) @ R)) = ((X3 = Y) | ((X3 != Y) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ ((transitive_trancl @ A) @ R)))))))). % 6.67/2.70 thf(fact_6343_rtrancl__into__trancl1, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_rtrancl @ A) @ R2)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ C2)) @ ((transitive_trancl @ A) @ R2))))))). % 6.67/2.70 thf(fact_6344_rtrancl__into__trancl2, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ ((transitive_rtrancl @ A) @ R2)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ C2)) @ ((transitive_trancl @ A) @ R2))))))). % 6.67/2.70 thf(fact_6345_rtrancl__trancl__trancl, axiom, (! [A:$tType,X3:A,Y:A,R2:(set @ ((product_prod @ A) @ A)),Z2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ ((transitive_rtrancl @ A) @ R2)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z2)) @ ((transitive_trancl @ A) @ R2)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z2)) @ ((transitive_trancl @ A) @ R2))))))). % 6.67/2.70 thf(fact_6346_trancl__rtrancl__trancl, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_trancl @ A) @ R2)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ ((transitive_rtrancl @ A) @ R2)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ C2)) @ ((transitive_trancl @ A) @ R2))))))). % 6.67/2.70 thf(fact_6347_rtrancl__listrel1__eq__len, axiom, (! [A:$tType,X3:(list @ A),Y:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X3) @ Y)) @ ((transitive_rtrancl @ (list @ A)) @ ((listrel1 @ A) @ R2))) => (((size_size @ (list @ A)) @ X3) = ((size_size @ (list @ A)) @ Y)))))). % 6.67/2.70 thf(fact_6348_listrel__reflcl__if__listrel1, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((listrel1 @ A) @ R2)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ (((listrel @ A) @ A) @ ((transitive_rtrancl @ A) @ R2))))))). % 6.67/2.70 thf(fact_6349_rtrancl__listrel1__if__listrel, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ (((listrel @ A) @ A) @ R2)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((transitive_rtrancl @ (list @ A)) @ ((listrel1 @ A) @ R2))))))). % 6.67/2.70 thf(fact_6350_pred__nat__trancl__eq__le, axiom, (! [M:nat,N2:nat]: (((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ M) @ N2)) @ ((transitive_rtrancl @ nat) @ pred_nat)) = (((ord_less_eq @ nat) @ M) @ N2))))). % 6.67/2.70 thf(fact_6351_listrel__subset__rtrancl__listrel1, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((ord_less_eq @ (set @ ((product_prod @ (list @ A)) @ (list @ A)))) @ (((listrel @ A) @ A) @ R2)) @ ((transitive_rtrancl @ (list @ A)) @ ((listrel1 @ A) @ R2)))))). % 6.67/2.70 thf(fact_6352_uniformly__continuous__onD, axiom, (! [B:$tType,A:$tType]: ((((topolo7287701948861334536_space @ A) & (topolo7287701948861334536_space @ B)) => (! [S:(set @ A),F2:(A > B),E5:(((product_prod @ B) @ B) > $o)]: ((((((topolo6026614971017936543ous_on @ A) @ B) @ S) @ F2) => ((((eventually @ ((product_prod @ B) @ B)) @ E5) @ (topolo7806501430040627800ormity @ B)) => (((eventually @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X:A,Y2:A]: (((((member @ A) @ X) @ S) => ((((member @ A) @ Y2) @ S) => (E5 @ ((((product_Pair @ B) @ B) @ (F2 @ X)) @ (F2 @ Y2))))))))) @ (topolo7806501430040627800ormity @ A)))))))))). % 6.67/2.70 thf(fact_6353_measure__function__int, axiom, ((fun_is_measure @ int) @ (((((comp @ int) @ nat) @ int) @ nat2) @ (abs_abs @ int)))). % 6.67/2.70 thf(fact_6354_measure__size, axiom, (! [A:$tType]: (((size @ A) => ((fun_is_measure @ A) @ (size_size @ A)))))). % 6.67/2.70 thf(fact_6355_uniformly__continuous__on__def, axiom, (! [B:$tType,A:$tType]: ((((real_V7819770556892013058_space @ A) & (real_V7819770556892013058_space @ B)) => (((topolo6026614971017936543ous_on @ A) @ B) = (^ [S8:(set @ A),F3:(A > B)]: ((! [E4:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ E4) => (? [D5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ D5) & (! [X:A]: (((((member @ A) @ X) @ S8) => (! [Y2:A]: (((((member @ A) @ Y2) @ S8) => ((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ Y2) @ X)) @ D5) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ B) @ (F3 @ Y2)) @ (F3 @ X))) @ E4))))))))))))))))))))). % 6.67/2.70 thf(fact_6356_isUCont__def, axiom, (! [B:$tType,A:$tType]: ((((real_V7819770556892013058_space @ A) & (real_V7819770556892013058_space @ B)) => (! [F2:(A > B)]: ((((((topolo6026614971017936543ous_on @ A) @ B) @ (top_top @ (set @ A))) @ F2) = (! [R5:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ R5) => (? [S8:real]: (((((ord_less @ real) @ (zero_zero @ real)) @ S8) & (! [X:A,Y2:A]: (((((ord_less @ real) @ (((real_V557655796197034286t_dist @ A) @ X) @ Y2)) @ S8) => (((ord_less @ real) @ (((real_V557655796197034286t_dist @ B) @ (F2 @ X)) @ (F2 @ Y2))) @ R5))))))))))))))))). % 6.67/2.70 thf(fact_6357_set__list__bind, axiom, (! [A:$tType,B:$tType,Xs:(list @ B),F2:(B > (list @ A))]: ((((set2 @ A) @ ((((bind @ B) @ A) @ Xs) @ F2)) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [X:B]: (((set2 @ A) @ (F2 @ X))))) @ ((set2 @ B) @ Xs))))))). % 6.67/2.70 thf(fact_6358_has__derivative__power__int, axiom, (! [A:$tType,C:$tType]: ((((real_V822414075346904944vector @ C) & (real_V3459762299906320749_field @ A)) => (! [F2:(C > A),X3:C,F11:(C > A),S3:(set @ C),N2:int]: ((((F2 @ X3) != (zero_zero @ A)) => ((((((has_derivative @ C) @ A) @ F2) @ F11) @ (((topolo174197925503356063within @ C) @ X3) @ S3)) => (((((has_derivative @ C) @ A) @ (^ [X:C]: ((((power_int @ A) @ (F2 @ X)) @ N2)))) @ (^ [H:C]: ((((times_times @ A) @ (F11 @ H)) @ (((times_times @ A) @ ((ring_1_of_int @ A) @ N2)) @ (((power_int @ A) @ (F2 @ X3)) @ (((minus_minus @ int) @ N2) @ (one_one @ int)))))))) @ (((topolo174197925503356063within @ C) @ X3) @ S3)))))))))). % 6.67/2.70 thf(fact_6359_power__int__1__left, axiom, (! [A:$tType]: (((division_ring @ A) => (! [N2:int]: (((((power_int @ A) @ (one_one @ A)) @ N2) = (one_one @ A)))))))). % 6.67/2.70 thf(fact_6360_power__int__mult__numeral, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,M:num,N2:num]: (((((power_int @ A) @ (((power_int @ A) @ X3) @ ((numeral_numeral @ int) @ M))) @ ((numeral_numeral @ int) @ N2)) = (((power_int @ A) @ X3) @ ((numeral_numeral @ int) @ (((times_times @ num) @ M) @ N2)))))))))). % 6.67/2.70 thf(fact_6361_power__int__1__right, axiom, (! [A:$tType]: ((((inverse @ A) & (monoid_mult @ A)) => (! [Y:A]: (((((power_int @ A) @ Y) @ (one_one @ int)) = Y))))))). % 6.67/2.70 thf(fact_6362_power__int__sgn, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,N2:int]: ((((sgn_sgn @ A) @ (((power_int @ A) @ A2) @ N2)) = (((power_int @ A) @ ((sgn_sgn @ A) @ A2)) @ N2)))))))). % 6.67/2.70 thf(fact_6363_bind__simps_I1_J, axiom, (! [B:$tType,A:$tType,F2:(B > (list @ A))]: ((((((bind @ B) @ A) @ (nil @ B)) @ F2) = (nil @ A))))). % 6.67/2.70 thf(fact_6364_power__int__mult__distrib__numeral1, axiom, (! [A:$tType]: (((field @ A) => (! [W2:num,Y:A,M:int]: (((((power_int @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ W2)) @ Y)) @ M) = (((times_times @ A) @ (((power_int @ A) @ ((numeral_numeral @ A) @ W2)) @ M)) @ (((power_int @ A) @ Y) @ M))))))))). % 6.67/2.70 thf(fact_6365_power__int__mult__distrib__numeral2, axiom, (! [A:$tType]: (((field @ A) => (! [X3:A,W2:num,M:int]: (((((power_int @ A) @ (((times_times @ A) @ X3) @ ((numeral_numeral @ A) @ W2))) @ M) = (((times_times @ A) @ (((power_int @ A) @ X3) @ M)) @ (((power_int @ A) @ ((numeral_numeral @ A) @ W2)) @ M))))))))). % 6.67/2.70 thf(fact_6366_power__int__0__left, axiom, (! [A:$tType]: (((division_ring @ A) => (! [M:int]: (((M != (zero_zero @ int)) => ((((power_int @ A) @ (zero_zero @ A)) @ M) = (zero_zero @ A))))))))). % 6.67/2.70 thf(fact_6367_power__int__eq__0__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,N2:int]: ((((((power_int @ A) @ X3) @ N2) = (zero_zero @ A)) = ((X3 = (zero_zero @ A)) & (N2 != (zero_zero @ int)))))))))). % 6.67/2.70 thf(fact_6368_power__int__0__right, axiom, (! [B:$tType]: ((((inverse @ B) & (power @ B)) => (! [X3:B]: (((((power_int @ B) @ X3) @ (zero_zero @ int)) = (one_one @ B)))))))). % 6.67/2.70 thf(fact_6369_abs__power__int__minus, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,N2:int]: ((((abs_abs @ A) @ (((power_int @ A) @ ((uminus_uminus @ A) @ A2)) @ N2)) = ((abs_abs @ A) @ (((power_int @ A) @ A2) @ N2))))))))). % 6.67/2.70 thf(fact_6370_power__int__of__nat, axiom, (! [A:$tType]: ((((inverse @ A) & (power @ A)) => (! [X3:A,N2:nat]: (((((power_int @ A) @ X3) @ ((semiring_1_of_nat @ int) @ N2)) = (((power_power @ A) @ X3) @ N2)))))))). % 6.67/2.70 thf(fact_6371_power__int__numeral, axiom, (! [A:$tType]: ((((inverse @ A) & (power @ A)) => (! [X3:A,N2:num]: (((((power_int @ A) @ X3) @ ((numeral_numeral @ int) @ N2)) = (((power_power @ A) @ X3) @ ((numeral_numeral @ nat) @ N2))))))))). % 6.67/2.70 thf(fact_6372_power__int__minus1__right, axiom, (! [A:$tType]: ((((inverse @ A) & (monoid_mult @ A)) => (! [Y:A]: (((((power_int @ A) @ Y) @ ((uminus_uminus @ int) @ (one_one @ int))) = ((inverse_inverse @ A) @ Y)))))))). % 6.67/2.70 thf(fact_6373_power__int__add__numeral, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,M:num,N2:num]: (((((times_times @ A) @ (((power_int @ A) @ X3) @ ((numeral_numeral @ int) @ M))) @ (((power_int @ A) @ X3) @ ((numeral_numeral @ int) @ N2))) = (((power_int @ A) @ X3) @ ((numeral_numeral @ int) @ (((plus_plus @ num) @ M) @ N2)))))))))). % 6.67/2.70 thf(fact_6374_power__int__add__numeral2, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,M:num,N2:num,B2:A]: (((((times_times @ A) @ (((power_int @ A) @ X3) @ ((numeral_numeral @ int) @ M))) @ (((times_times @ A) @ (((power_int @ A) @ X3) @ ((numeral_numeral @ int) @ N2))) @ B2)) = (((times_times @ A) @ (((power_int @ A) @ X3) @ ((numeral_numeral @ int) @ (((plus_plus @ num) @ M) @ N2)))) @ B2)))))))). % 6.67/2.70 thf(fact_6375_power__int__mono__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,N2:int]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ int) @ (zero_zero @ int)) @ N2) => ((((ord_less_eq @ A) @ (((power_int @ A) @ A2) @ N2)) @ (((power_int @ A) @ B2) @ N2)) = (((ord_less_eq @ A) @ A2) @ B2))))))))))). % 6.67/2.70 thf(fact_6376_list__bind__cong, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys:(list @ A),F2:(A > (list @ B)),G:(A > (list @ B))]: (((Xs = Ys) => ((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => ((F2 @ X5) = (G @ X5))))) => (((((bind @ A) @ B) @ Xs) @ F2) = ((((bind @ A) @ B) @ Ys) @ G))))))). % 6.67/2.70 thf(fact_6377_continuous__on__power__int, axiom, (! [B:$tType,A:$tType]: ((((topolo4958980785337419405_space @ A) & (real_V8999393235501362500lgebra @ B)) => (! [S:(set @ A),F2:(A > B),N2:int]: ((((((topolo81223032696312382ous_on @ A) @ B) @ S) @ F2) => ((! [X5:A]: (((((member @ A) @ X5) @ S) => ((F2 @ X5) != (zero_zero @ B))))) => ((((topolo81223032696312382ous_on @ A) @ B) @ S) @ (^ [X:A]: ((((power_int @ B) @ (F2 @ X)) @ N2)))))))))))). % 6.67/2.70 thf(fact_6378_power__int__abs, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,N2:int]: ((((abs_abs @ A) @ (((power_int @ A) @ A2) @ N2)) = (((power_int @ A) @ ((abs_abs @ A) @ A2)) @ N2)))))))). % 6.67/2.70 thf(fact_6379_power__int__mult__distrib, axiom, (! [A:$tType]: (((field @ A) => (! [X3:A,Y:A,M:int]: (((((power_int @ A) @ (((times_times @ A) @ X3) @ Y)) @ M) = (((times_times @ A) @ (((power_int @ A) @ X3) @ M)) @ (((power_int @ A) @ Y) @ M))))))))). % 6.67/2.70 thf(fact_6380_power__int__commutes, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,N2:int]: (((((times_times @ A) @ (((power_int @ A) @ X3) @ N2)) @ X3) = (((times_times @ A) @ X3) @ (((power_int @ A) @ X3) @ N2))))))))). % 6.67/2.70 thf(fact_6381_power__int__mult, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,M:int,N2:int]: (((((power_int @ A) @ X3) @ (((times_times @ int) @ M) @ N2)) = (((power_int @ A) @ (((power_int @ A) @ X3) @ M)) @ N2)))))))). % 6.67/2.70 thf(fact_6382_power__int__inverse, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,N2:int]: (((((power_int @ A) @ ((inverse_inverse @ A) @ X3)) @ N2) = ((inverse_inverse @ A) @ (((power_int @ A) @ X3) @ N2))))))))). % 6.67/2.70 thf(fact_6383_power__int__divide__distrib, axiom, (! [A:$tType]: (((field @ A) => (! [X3:A,Y:A,M:int]: (((((power_int @ A) @ (((divide_divide @ A) @ X3) @ Y)) @ M) = (((divide_divide @ A) @ (((power_int @ A) @ X3) @ M)) @ (((power_int @ A) @ Y) @ M))))))))). % 6.67/2.70 thf(fact_6384_power__int__not__zero, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,N2:int]: ((((X3 != (zero_zero @ A)) | (N2 = (zero_zero @ int))) => ((((power_int @ A) @ X3) @ N2) != (zero_zero @ A))))))))). % 6.67/2.70 thf(fact_6385_power__int__one__over, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,N2:int]: (((((power_int @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ X3)) @ N2) = (((divide_divide @ A) @ (one_one @ A)) @ (((power_int @ A) @ X3) @ N2))))))))). % 6.67/2.70 thf(fact_6386_zero__less__power__int, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,N2:int]: (((((ord_less @ A) @ (zero_zero @ A)) @ X3) => (((ord_less @ A) @ (zero_zero @ A)) @ (((power_int @ A) @ X3) @ N2))))))))). % 6.67/2.70 thf(fact_6387_power__int__minus, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,N2:int]: (((((power_int @ A) @ X3) @ ((uminus_uminus @ int) @ N2)) = ((inverse_inverse @ A) @ (((power_int @ A) @ X3) @ N2))))))))). % 6.67/2.70 thf(fact_6388_zero__le__power__int, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,N2:int]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((power_int @ A) @ X3) @ N2))))))))). % 6.67/2.70 thf(fact_6389_power__int__0__left__If, axiom, (! [A:$tType]: (((division_ring @ A) => (! [M:int]: ((((M = (zero_zero @ int)) => ((((power_int @ A) @ (zero_zero @ A)) @ M) = (one_one @ A))) & ((M != (zero_zero @ int)) => ((((power_int @ A) @ (zero_zero @ A)) @ M) = (zero_zero @ A)))))))))). % 6.67/2.70 thf(fact_6390_power__int__increasing, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [N2:int,N7:int,A2:A]: (((((ord_less_eq @ int) @ N2) @ N7) => ((((ord_less_eq @ A) @ (one_one @ A)) @ A2) => (((ord_less_eq @ A) @ (((power_int @ A) @ A2) @ N2)) @ (((power_int @ A) @ A2) @ N7)))))))))). % 6.67/2.70 thf(fact_6391_power__int__strict__increasing, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [N2:int,N7:int,A2:A]: (((((ord_less @ int) @ N2) @ N7) => ((((ord_less @ A) @ (one_one @ A)) @ A2) => (((ord_less @ A) @ (((power_int @ A) @ A2) @ N2)) @ (((power_int @ A) @ A2) @ N7)))))))))). % 6.67/2.70 thf(fact_6392_power__int__diff, axiom, (! [A:$tType]: (((field @ A) => (! [X3:A,M:int,N2:int]: ((((X3 != (zero_zero @ A)) | (M != N2)) => ((((power_int @ A) @ X3) @ (((minus_minus @ int) @ M) @ N2)) = (((divide_divide @ A) @ (((power_int @ A) @ X3) @ M)) @ (((power_int @ A) @ X3) @ N2)))))))))). % 6.67/2.70 thf(fact_6393_tendsto__power__int, axiom, (! [A:$tType,B:$tType]: (((real_V8999393235501362500lgebra @ A) => (! [F2:(B > A),A2:A,F5:(filter @ B),N2:int]: (((((((filterlim @ B) @ A) @ F2) @ ((topolo7230453075368039082e_nhds @ A) @ A2)) @ F5) => ((A2 != (zero_zero @ A)) => (((((filterlim @ B) @ A) @ (^ [X:B]: ((((power_int @ A) @ (F2 @ X)) @ N2)))) @ ((topolo7230453075368039082e_nhds @ A) @ (((power_int @ A) @ A2) @ N2))) @ F5))))))))). % 6.67/2.70 thf(fact_6394_continuous__at__within__power__int, axiom, (! [B:$tType,A:$tType]: ((((topological_t2_space @ A) & (real_V8999393235501362500lgebra @ B)) => (! [A2:A,S:(set @ A),F2:(A > B),N2:int]: ((((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ S)) @ F2) => (((F2 @ A2) != (zero_zero @ B)) => ((((topolo3448309680560233919inuous @ A) @ B) @ (((topolo174197925503356063within @ A) @ A2) @ S)) @ (^ [X:A]: ((((power_int @ B) @ (F2 @ X)) @ N2)))))))))))). % 6.67/2.70 thf(fact_6395_differentiable__power__int, axiom, (! [B:$tType,A:$tType]: ((((real_V822414075346904944vector @ A) & (real_V3459762299906320749_field @ B)) => (! [F2:(A > B),X3:A,S:(set @ A),N2:int]: ((((((differentiable @ A) @ B) @ F2) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((F2 @ X3) != (zero_zero @ B)) => ((((differentiable @ A) @ B) @ (^ [X:A]: ((((power_int @ B) @ (F2 @ X)) @ N2)))) @ (((topolo174197925503356063within @ A) @ X3) @ S)))))))))). % 6.67/2.70 thf(fact_6396_continuous__power__int, axiom, (! [B:$tType,A:$tType]: ((((topological_t2_space @ A) & (real_V8999393235501362500lgebra @ B)) => (! [F5:(filter @ A),F2:(A > B),N2:int]: ((((((topolo3448309680560233919inuous @ A) @ B) @ F5) @ F2) => (((F2 @ ((((topolo3827282254853284352ce_Lim @ A) @ A) @ F5) @ (^ [X:A]: (X)))) != (zero_zero @ B)) => ((((topolo3448309680560233919inuous @ A) @ B) @ F5) @ (^ [X:A]: ((((power_int @ B) @ (F2 @ X)) @ N2)))))))))))). % 6.67/2.70 thf(fact_6397_power__int__strict__decreasing, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [N2:int,N7:int,A2:A]: (((((ord_less @ int) @ N2) @ N7) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ A2) @ (one_one @ A)) => (((ord_less @ A) @ (((power_int @ A) @ A2) @ N7)) @ (((power_int @ A) @ A2) @ N2))))))))))). % 6.67/2.70 thf(fact_6398_power__int__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,Y:A,N2:int]: (((((ord_less_eq @ A) @ X3) @ Y) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => (((ord_less_eq @ A) @ (((power_int @ A) @ X3) @ N2)) @ (((power_int @ A) @ Y) @ N2))))))))))). % 6.67/2.70 thf(fact_6399_power__int__strict__antimono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,N2:int]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ int) @ N2) @ (zero_zero @ int)) => (((ord_less @ A) @ (((power_int @ A) @ B2) @ N2)) @ (((power_int @ A) @ A2) @ N2))))))))))). % 6.67/2.70 thf(fact_6400_one__le__power__int, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,N2:int]: (((((ord_less_eq @ A) @ (one_one @ A)) @ X3) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N2) => (((ord_less_eq @ A) @ (one_one @ A)) @ (((power_int @ A) @ X3) @ N2)))))))))). % 6.67/2.70 thf(fact_6401_one__less__power__int, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,N2:int]: (((((ord_less @ A) @ (one_one @ A)) @ A2) => ((((ord_less @ int) @ (zero_zero @ int)) @ N2) => (((ord_less @ A) @ (one_one @ A)) @ (((power_int @ A) @ A2) @ N2)))))))))). % 6.67/2.70 thf(fact_6402_power__int__add, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,M:int,N2:int]: ((((X3 != (zero_zero @ A)) | ((((plus_plus @ int) @ M) @ N2) != (zero_zero @ int))) => ((((power_int @ A) @ X3) @ (((plus_plus @ int) @ M) @ N2)) = (((times_times @ A) @ (((power_int @ A) @ X3) @ M)) @ (((power_int @ A) @ X3) @ N2)))))))))). % 6.67/2.70 thf(fact_6403_power__int__antimono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,N2:int]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ int) @ N2) @ (zero_zero @ int)) => (((ord_less_eq @ A) @ (((power_int @ A) @ B2) @ N2)) @ (((power_int @ A) @ A2) @ N2))))))))))). % 6.67/2.70 thf(fact_6404_power__int__strict__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A2:A,B2:A,N2:int]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ int) @ (zero_zero @ int)) @ N2) => (((ord_less @ A) @ (((power_int @ A) @ A2) @ N2)) @ (((power_int @ A) @ B2) @ N2))))))))))). % 6.67/2.70 thf(fact_6405_power__int__decreasing, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [N2:int,N7:int,A2:A]: (((((ord_less_eq @ int) @ N2) @ N7) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ A2) @ (one_one @ A)) => (((A2 != (zero_zero @ A)) | ((N7 != (zero_zero @ int)) | (N2 = (zero_zero @ int)))) => (((ord_less_eq @ A) @ (((power_int @ A) @ A2) @ N7)) @ (((power_int @ A) @ A2) @ N2)))))))))))). % 6.67/2.70 thf(fact_6406_power__int__le__one, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,N2:int]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N2) => ((((ord_less_eq @ A) @ X3) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((power_int @ A) @ X3) @ N2)) @ (one_one @ A))))))))))). % 6.67/2.70 thf(fact_6407_power__int__le__imp__le__exp, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,M:int,N2:int]: (((((ord_less @ A) @ (one_one @ A)) @ X3) => ((((ord_less_eq @ A) @ (((power_int @ A) @ X3) @ M)) @ (((power_int @ A) @ X3) @ N2)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N2) => (((ord_less_eq @ int) @ M) @ N2)))))))))). % 6.67/2.70 thf(fact_6408_power__int__le__imp__less__exp, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X3:A,M:int,N2:int]: (((((ord_less @ A) @ (one_one @ A)) @ X3) => ((((ord_less @ A) @ (((power_int @ A) @ X3) @ M)) @ (((power_int @ A) @ X3) @ N2)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N2) => (((ord_less @ int) @ M) @ N2)))))))))). % 6.67/2.70 thf(fact_6409_power__int__minus__mult, axiom, (! [A:$tType]: (((field @ A) => (! [X3:A,N2:int]: ((((X3 != (zero_zero @ A)) | (N2 != (zero_zero @ int))) => ((((times_times @ A) @ (((power_int @ A) @ X3) @ (((minus_minus @ int) @ N2) @ (one_one @ int)))) @ X3) = (((power_int @ A) @ X3) @ N2))))))))). % 6.67/2.70 thf(fact_6410_power__int__add__1, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,M:int]: ((((X3 != (zero_zero @ A)) | (M != ((uminus_uminus @ int) @ (one_one @ int)))) => ((((power_int @ A) @ X3) @ (((plus_plus @ int) @ M) @ (one_one @ int))) = (((times_times @ A) @ (((power_int @ A) @ X3) @ M)) @ X3))))))))). % 6.67/2.70 thf(fact_6411_power__int__add__1_H, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X3:A,M:int]: ((((X3 != (zero_zero @ A)) | (M != ((uminus_uminus @ int) @ (one_one @ int)))) => ((((power_int @ A) @ X3) @ (((plus_plus @ int) @ M) @ (one_one @ int))) = (((times_times @ A) @ X3) @ (((power_int @ A) @ X3) @ M)))))))))). % 6.67/2.70 thf(fact_6412_power__int__def, axiom, (! [A:$tType]: ((((inverse @ A) & (power @ A)) => ((power_int @ A) = (^ [X:A,N:int]: (((((if @ A) @ (((ord_less_eq @ int) @ (zero_zero @ int)) @ N)) @ (((power_power @ A) @ X) @ (nat2 @ N))) @ (((power_power @ A) @ ((inverse_inverse @ A) @ X)) @ (nat2 @ ((uminus_uminus @ int) @ N))))))))))). % 6.67/2.70 thf(fact_6413_powr__real__of__int_H, axiom, (! [X3:real,N2:int]: (((((ord_less_eq @ real) @ (zero_zero @ real)) @ X3) => (((X3 != (zero_zero @ real)) | (((ord_less @ int) @ (zero_zero @ int)) @ N2)) => ((((powr @ real) @ X3) @ ((ring_1_of_int @ real) @ N2)) = (((power_int @ real) @ X3) @ N2))))))). % 6.67/2.70 thf(fact_6414_DERIV__power__int, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [F2:(A > A),D2:A,X3:A,S:(set @ A),N2:int]: ((((((has_field_derivative @ A) @ F2) @ D2) @ (((topolo174197925503356063within @ A) @ X3) @ S)) => (((F2 @ X3) != (zero_zero @ A)) => ((((has_field_derivative @ A) @ (^ [X:A]: ((((power_int @ A) @ (F2 @ X)) @ N2)))) @ (((times_times @ A) @ (((times_times @ A) @ ((ring_1_of_int @ A) @ N2)) @ (((power_int @ A) @ (F2 @ X3)) @ (((minus_minus @ int) @ N2) @ (one_one @ int))))) @ D2)) @ (((topolo174197925503356063within @ A) @ X3) @ S)))))))))). % 6.67/2.70 thf(fact_6415_has__derivative__power__int_H, axiom, (! [A:$tType]: (((real_V3459762299906320749_field @ A) => (! [X3:A,N2:int,S3:(set @ A)]: (((X3 != (zero_zero @ A)) => (((((has_derivative @ A) @ A) @ (^ [X:A]: ((((power_int @ A) @ X) @ N2)))) @ (^ [Y2:A]: ((((times_times @ A) @ Y2) @ (((times_times @ A) @ ((ring_1_of_int @ A) @ N2)) @ (((power_int @ A) @ X3) @ (((minus_minus @ int) @ N2) @ (one_one @ int)))))))) @ (((topolo174197925503356063within @ A) @ X3) @ S3))))))))). % 6.67/2.70 thf(fact_6416_power__int__numeral__neg__numeral, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [M:num,N2:num]: (((((power_int @ A) @ ((numeral_numeral @ A) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N2))) = ((inverse_inverse @ A) @ ((numeral_numeral @ A) @ ((pow @ M) @ N2)))))))))). % 6.67/2.70 thf(fact_6417_eventually__less__ceiling, axiom, (! [B:$tType,A:$tType]: ((((archim2362893244070406136eiling @ B) & (topolo2564578578187576103pology @ B)) => (! [F2:(A > B),L:B,F5:(filter @ A)]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L)) @ F5) => ((~ ((((member @ B) @ L) @ (ring_1_Ints @ B)))) => (((eventually @ A) @ (^ [X:A]: ((((ord_less @ B) @ (F2 @ X)) @ ((ring_1_of_int @ B) @ ((archimedean_ceiling @ B) @ L)))))) @ F5))))))))). % 6.67/2.70 thf(fact_6418_Ints__sum, axiom, (! [A:$tType,B:$tType]: (((ring_1 @ B) => (! [A3:(set @ A),F2:(A > B)]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((member @ B) @ (F2 @ X5)) @ (ring_1_Ints @ B))))) => (((member @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F2) @ A3)) @ (ring_1_Ints @ B))))))))). % 6.67/2.70 thf(fact_6419_Ints__prod, axiom, (! [A:$tType,B:$tType]: ((((comm_monoid_mult @ B) & (ring_1 @ B)) => (! [A3:(set @ A),F2:(A > B)]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((member @ B) @ (F2 @ X5)) @ (ring_1_Ints @ B))))) => (((member @ B) @ ((((groups7121269368397514597t_prod @ A) @ B) @ F2) @ A3)) @ (ring_1_Ints @ B))))))))). % 6.67/2.70 thf(fact_6420_frac__eq__0__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((archimedean_frac @ A) @ X3) = (zero_zero @ A)) = (((member @ A) @ X3) @ (ring_1_Ints @ A))))))))). % 6.67/2.70 thf(fact_6421_floor__add2, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,Y:A]: ((((((member @ A) @ X3) @ (ring_1_Ints @ A)) | (((member @ A) @ Y) @ (ring_1_Ints @ A))) => (((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X3) @ Y)) = (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ ((archim6421214686448440834_floor @ A) @ Y)))))))))). % 6.67/2.70 thf(fact_6422_frac__gt__0__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((archimedean_frac @ A) @ X3)) = (~ ((((member @ A) @ X3) @ (ring_1_Ints @ A))))))))))). % 6.67/2.70 thf(fact_6423_Ints__double__eq__0__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [A2:A]: (((((member @ A) @ A2) @ (ring_1_Ints @ A)) => (((((plus_plus @ A) @ A2) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A)))))))))). % 6.67/2.70 thf(fact_6424_Ints__0, axiom, (! [A:$tType]: (((ring_1 @ A) => (((member @ A) @ (zero_zero @ A)) @ (ring_1_Ints @ A)))))). % 6.67/2.70 thf(fact_6425_Ints__minus, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A2:A]: (((((member @ A) @ A2) @ (ring_1_Ints @ A)) => (((member @ A) @ ((uminus_uminus @ A) @ A2)) @ (ring_1_Ints @ A))))))))). % 6.67/2.70 thf(fact_6426_minus__in__Ints__iff, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X3:A]: (((((member @ A) @ ((uminus_uminus @ A) @ X3)) @ (ring_1_Ints @ A)) = (((member @ A) @ X3) @ (ring_1_Ints @ A))))))))). % 6.67/2.70 thf(fact_6427_Ints__of__nat, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N2:nat]: ((((member @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ (ring_1_Ints @ A)))))))). % 6.67/2.70 thf(fact_6428_Ints__power, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A2:A,N2:nat]: (((((member @ A) @ A2) @ (ring_1_Ints @ A)) => (((member @ A) @ (((power_power @ A) @ A2) @ N2)) @ (ring_1_Ints @ A))))))))). % 6.67/2.70 thf(fact_6429_Ints__of__int, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z2:int]: ((((member @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (ring_1_Ints @ A)))))))). % 6.67/2.70 thf(fact_6430_Ints__induct, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Q4:A,P:(A > $o)]: (((((member @ A) @ Q4) @ (ring_1_Ints @ A)) => ((! [Z3:int]: ((P @ ((ring_1_of_int @ A) @ Z3)))) => (P @ Q4))))))))). % 6.67/2.70 thf(fact_6431_Ints__cases, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Q4:A]: (((((member @ A) @ Q4) @ (ring_1_Ints @ A)) => (~ ((! [Z3:int]: ((Q4 != ((ring_1_of_int @ A) @ Z3))))))))))))). % 6.67/2.70 thf(fact_6432_Ints__add, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A2:A,B2:A]: (((((member @ A) @ A2) @ (ring_1_Ints @ A)) => ((((member @ A) @ B2) @ (ring_1_Ints @ A)) => (((member @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ (ring_1_Ints @ A)))))))))). % 6.67/2.70 thf(fact_6433_Ints__1, axiom, (! [A:$tType]: (((ring_1 @ A) => (((member @ A) @ (one_one @ A)) @ (ring_1_Ints @ A)))))). % 6.67/2.70 thf(fact_6434_Ints__diff, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A2:A,B2:A]: (((((member @ A) @ A2) @ (ring_1_Ints @ A)) => ((((member @ A) @ B2) @ (ring_1_Ints @ A)) => (((member @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ (ring_1_Ints @ A)))))))))). % 6.67/2.70 thf(fact_6435_Ints__mult, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A2:A,B2:A]: (((((member @ A) @ A2) @ (ring_1_Ints @ A)) => ((((member @ A) @ B2) @ (ring_1_Ints @ A)) => (((member @ A) @ (((times_times @ A) @ A2) @ B2)) @ (ring_1_Ints @ A)))))))))). % 6.67/2.70 thf(fact_6436_Ints__numeral, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N2:num]: ((((member @ A) @ ((numeral_numeral @ A) @ N2)) @ (ring_1_Ints @ A)))))))). % 6.67/2.70 thf(fact_6437_Ints__abs, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((member @ A) @ A2) @ (ring_1_Ints @ A)) => (((member @ A) @ ((abs_abs @ A) @ A2)) @ (ring_1_Ints @ A))))))))). % 6.67/2.70 thf(fact_6438_Ints__subset__Rats, axiom, (! [A:$tType]: (((field_char_0 @ A) => (((ord_less_eq @ (set @ A)) @ (ring_1_Ints @ A)) @ (field_char_0_Rats @ A)))))). % 6.67/2.70 thf(fact_6439_Nats__subset__Ints, axiom, (! [A:$tType]: (((ring_1 @ A) => (((ord_less_eq @ (set @ A)) @ (semiring_1_Nats @ A)) @ (ring_1_Ints @ A)))))). % 6.67/2.70 thf(fact_6440_finite__int__segment, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [A2:A,B2:A]: (((finite_finite2 @ A) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ (ring_1_Ints @ A)) & ((((ord_less_eq @ A) @ A2) @ X) & (((ord_less_eq @ A) @ X) @ B2))))))))))))). % 6.67/2.70 thf(fact_6441_Ints__odd__nonzero, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [A2:A]: (((((member @ A) @ A2) @ (ring_1_Ints @ A)) => ((((plus_plus @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ A2)) @ A2) != (zero_zero @ A))))))))). % 6.67/2.70 thf(fact_6442_of__int__divide__in__Ints, axiom, (! [A:$tType]: (((idom_divide @ A) => (! [B2:int,A2:int]: (((((dvd_dvd @ int) @ B2) @ A2) => (((member @ A) @ (((divide_divide @ A) @ ((ring_1_of_int @ A) @ A2)) @ ((ring_1_of_int @ A) @ B2))) @ (ring_1_Ints @ A))))))))). % 6.67/2.70 thf(fact_6443_finite__abs__int__segment, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [A2:A]: (((finite_finite2 @ A) @ ((collect @ A) @ (^ [K3:A]: (((((member @ A) @ K3) @ (ring_1_Ints @ A)) & (((ord_less_eq @ A) @ ((abs_abs @ A) @ K3)) @ A2)))))))))))). % 6.67/2.70 thf(fact_6444_Nats__altdef2, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((semiring_1_Nats @ A) = ((collect @ A) @ (^ [N:A]: (((((member @ A) @ N) @ (ring_1_Ints @ A)) & (((ord_less_eq @ A) @ (zero_zero @ A)) @ N)))))))))). % 6.67/2.70 thf(fact_6445_Ints__def, axiom, (! [A:$tType]: (((ring_1 @ A) => ((ring_1_Ints @ A) = ((((image @ int) @ A) @ (ring_1_of_int @ A)) @ (top_top @ (set @ int)))))))). % 6.67/2.70 thf(fact_6446_Ints__odd__less__0, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A2:A]: (((((member @ A) @ A2) @ (ring_1_Ints @ A)) => ((((ord_less @ A) @ (((plus_plus @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ A2)) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A)))))))))). % 6.67/2.70 thf(fact_6447_Ints__nonzero__abs__ge1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A]: (((((member @ A) @ X3) @ (ring_1_Ints @ A)) => ((X3 != (zero_zero @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ ((abs_abs @ A) @ X3)))))))))). % 6.67/2.70 thf(fact_6448_Ints__nonzero__abs__less1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A]: (((((member @ A) @ X3) @ (ring_1_Ints @ A)) => ((((ord_less @ A) @ ((abs_abs @ A) @ X3)) @ (one_one @ A)) => (X3 = (zero_zero @ A)))))))))). % 6.67/2.70 thf(fact_6449_Ints__eq__abs__less1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X3:A,Y:A]: (((((member @ A) @ X3) @ (ring_1_Ints @ A)) => ((((member @ A) @ Y) @ (ring_1_Ints @ A)) => ((X3 = Y) = (((ord_less @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ X3) @ Y))) @ (one_one @ A))))))))))). % 6.67/2.70 thf(fact_6450_frac__neg, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A]: ((((((member @ A) @ X3) @ (ring_1_Ints @ A)) => (((archimedean_frac @ A) @ ((uminus_uminus @ A) @ X3)) = (zero_zero @ A))) & ((~ ((((member @ A) @ X3) @ (ring_1_Ints @ A)))) => (((archimedean_frac @ A) @ ((uminus_uminus @ A) @ X3)) = (((minus_minus @ A) @ (one_one @ A)) @ ((archimedean_frac @ A) @ X3))))))))))). % 6.67/2.70 thf(fact_6451_le__mult__floor__Ints, axiom, (! [A:$tType,B:$tType]: ((((archim2362893244070406136eiling @ B) & (linordered_idom @ A)) => (! [A2:B,B2:B]: (((((ord_less_eq @ B) @ (zero_zero @ B)) @ A2) => ((((member @ B) @ A2) @ (ring_1_Ints @ B)) => (((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ (((times_times @ int) @ ((archim6421214686448440834_floor @ B) @ A2)) @ ((archim6421214686448440834_floor @ B) @ B2)))) @ ((ring_1_of_int @ A) @ ((archim6421214686448440834_floor @ B) @ (((times_times @ B) @ A2) @ B2)))))))))))). % 6.67/2.70 thf(fact_6452_frac__unique__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X3:A,A2:A]: (((((archimedean_frac @ A) @ X3) = A2) = ((((member @ A) @ (((minus_minus @ A) @ X3) @ A2)) @ (ring_1_Ints @ A)) & ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ A2) @ (one_one @ A))))))))))). % 6.67/2.70 thf(fact_6453_mult__ceiling__le__Ints, axiom, (! [A:$tType,B:$tType]: ((((archim2362893244070406136eiling @ B) & (linordered_idom @ A)) => (! [A2:B,B2:B]: (((((ord_less_eq @ B) @ (zero_zero @ B)) @ A2) => ((((member @ B) @ A2) @ (ring_1_Ints @ B)) => (((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ B) @ (((times_times @ B) @ A2) @ B2)))) @ ((ring_1_of_int @ A) @ (((times_times @ int) @ ((archimedean_ceiling @ B) @ A2)) @ ((archimedean_ceiling @ B) @ B2)))))))))))). % 6.67/2.70 thf(fact_6454_eventually__floor__less, axiom, (! [B:$tType,A:$tType]: ((((archim2362893244070406136eiling @ B) & (topolo2564578578187576103pology @ B)) => (! [F2:(A > B),L:B,F5:(filter @ A)]: (((((((filterlim @ A) @ B) @ F2) @ ((topolo7230453075368039082e_nhds @ B) @ L)) @ F5) => ((~ ((((member @ B) @ L) @ (ring_1_Ints @ B)))) => (((eventually @ A) @ (^ [X:A]: ((((ord_less @ B) @ ((ring_1_of_int @ B) @ ((archim6421214686448440834_floor @ B) @ L))) @ (F2 @ X))))) @ F5))))))))). % 6.67/2.70 thf(fact_6455_Max_Oeq__fold_H, axiom, (! [A:$tType]: (((linorder @ A) => ((lattic643756798349783984er_Max @ A) = (^ [A6:(set @ A)]: (((the2 @ A) @ (((((finite_fold @ A) @ (option @ A)) @ (^ [X:A,Y2:(option @ A)]: (((some @ A) @ (((((case_option @ A) @ A) @ X) @ ((ord_max @ A) @ X)) @ Y2))))) @ (none @ A)) @ A6))))))))). % 6.67/2.70 thf(fact_6456_UN__le__eq__Un0, axiom, (! [A:$tType,M7:(nat > (set @ A)),N2:nat]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ M7) @ ((set_ord_atMost @ nat) @ N2))) = (((sup_sup @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ nat) @ (set @ A)) @ M7) @ (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ N2)))) @ (M7 @ (zero_zero @ nat))))))). % 6.67/2.70 thf(fact_6457_UnCI, axiom, (! [A:$tType,C2:A,B3:(set @ A),A3:(set @ A)]: ((((~ ((((member @ A) @ C2) @ B3))) => (((member @ A) @ C2) @ A3)) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)))))). % 6.67/2.70 thf(fact_6458_Un__iff, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = ((((member @ A) @ C2) @ A3) | (((member @ A) @ C2) @ B3)))))). % 6.67/2.70 thf(fact_6459_sup_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B2) @ C2)) @ A2) = ((((ord_less_eq @ A) @ B2) @ A2) & (((ord_less_eq @ A) @ C2) @ A2))))))))). % 6.67/2.70 thf(fact_6460_le__sup__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ X3) @ Y)) @ Z2) = ((((ord_less_eq @ A) @ X3) @ Z2) & (((ord_less_eq @ A) @ Y) @ Z2))))))))). % 6.67/2.70 thf(fact_6461_sup__bot_Oright__neutral, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ (bot_bot @ A)) = A2))))))). % 6.67/2.70 thf(fact_6462_sup__bot_Oneutr__eq__iff, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [A2:A,B2:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ A2) @ B2)) = ((A2 = (bot_bot @ A)) & (B2 = (bot_bot @ A)))))))))). % 6.67/2.70 thf(fact_6463_sup__bot_Oleft__neutral, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [A2:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ A2) = A2))))))). % 6.67/2.70 thf(fact_6464_sup__bot_Oeq__neutr__iff, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [A2:A,B2:A]: ((((((sup_sup @ A) @ A2) @ B2) = (bot_bot @ A)) = ((A2 = (bot_bot @ A)) & (B2 = (bot_bot @ A)))))))))). % 6.67/2.70 thf(fact_6465_sup__eq__bot__iff, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [X3:A,Y:A]: ((((((sup_sup @ A) @ X3) @ Y) = (bot_bot @ A)) = ((X3 = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))). % 6.67/2.70 thf(fact_6466_bot__eq__sup__iff, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [X3:A,Y:A]: ((((bot_bot @ A) = (((sup_sup @ A) @ X3) @ Y)) = ((X3 = (bot_bot @ A)) & (Y = (bot_bot @ A)))))))))). % 6.67/2.70 thf(fact_6467_sup__bot__right, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [X3:A]: (((((sup_sup @ A) @ X3) @ (bot_bot @ A)) = X3))))))). % 6.67/2.70 thf(fact_6468_sup__bot__left, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (! [X3:A]: (((((sup_sup @ A) @ (bot_bot @ A)) @ X3) = X3))))))). % 6.67/2.70 thf(fact_6469_Un__empty, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((((sup_sup @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))) = ((A3 = (bot_bot @ (set @ A))) & (B3 = (bot_bot @ (set @ A)))))))). % 6.67/2.70 thf(fact_6470_finite__Un, axiom, (! [A:$tType,F5:(set @ A),G6:(set @ A)]: ((((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ F5) @ G6)) = (((finite_finite2 @ A) @ F5) & ((finite_finite2 @ A) @ G6)))))). % 6.67/2.70 thf(fact_6471_Un__subset__iff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) @ C5) = ((((ord_less_eq @ (set @ A)) @ A3) @ C5) & (((ord_less_eq @ (set @ A)) @ B3) @ C5)))))). % 6.67/2.70 thf(fact_6472_Un__insert__right, axiom, (! [A:$tType,A3:(set @ A),A2:A,B3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B3)) = (((insert @ A) @ A2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)))))). % 6.67/2.70 thf(fact_6473_Un__insert__left, axiom, (! [A:$tType,A2:A,B3:(set @ A),C5:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((insert @ A) @ A2) @ B3)) @ C5) = (((insert @ A) @ A2) @ (((sup_sup @ (set @ A)) @ B3) @ C5)))))). % 6.67/2.70 thf(fact_6474_Int__Un__eq_I4_J, axiom, (! [A:$tType,T4:(set @ A),S3:(set @ A)]: (((((sup_sup @ (set @ A)) @ T4) @ (((inf_inf @ (set @ A)) @ S3) @ T4)) = T4)))). % 6.67/2.70 thf(fact_6475_Int__Un__eq_I3_J, axiom, (! [A:$tType,S3:(set @ A),T4:(set @ A)]: (((((sup_sup @ (set @ A)) @ S3) @ (((inf_inf @ (set @ A)) @ S3) @ T4)) = S3)))). % 6.67/2.70 thf(fact_6476_Int__Un__eq_I2_J, axiom, (! [A:$tType,S3:(set @ A),T4:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ S3) @ T4)) @ T4) = T4)))). % 6.67/2.70 thf(fact_6477_Int__Un__eq_I1_J, axiom, (! [A:$tType,S3:(set @ A),T4:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ S3) @ T4)) @ S3) = S3)))). % 6.67/2.70 thf(fact_6478_Un__Int__eq_I4_J, axiom, (! [A:$tType,T4:(set @ A),S3:(set @ A)]: (((((inf_inf @ (set @ A)) @ T4) @ (((sup_sup @ (set @ A)) @ S3) @ T4)) = T4)))). % 6.67/2.70 thf(fact_6479_Un__Int__eq_I3_J, axiom, (! [A:$tType,S3:(set @ A),T4:(set @ A)]: (((((inf_inf @ (set @ A)) @ S3) @ (((sup_sup @ (set @ A)) @ S3) @ T4)) = S3)))). % 6.67/2.70 thf(fact_6480_Un__Int__eq_I2_J, axiom, (! [A:$tType,S3:(set @ A),T4:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ S3) @ T4)) @ T4) = T4)))). % 6.67/2.70 thf(fact_6481_Un__Int__eq_I1_J, axiom, (! [A:$tType,S3:(set @ A),T4:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ S3) @ T4)) @ S3) = S3)))). % 6.67/2.70 thf(fact_6482_Un__Diff__cancel2, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((minus_minus @ (set @ A)) @ B3) @ A3)) @ A3) = (((sup_sup @ (set @ A)) @ B3) @ A3))))). % 6.67/2.70 thf(fact_6483_Un__Diff__cancel, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (((minus_minus @ (set @ A)) @ B3) @ A3)) = (((sup_sup @ (set @ A)) @ A3) @ B3))))). % 6.67/2.70 thf(fact_6484_fold__empty, axiom, (! [B:$tType,A:$tType,F2:(B > (A > A)),Z2:A]: (((((((finite_fold @ B) @ A) @ F2) @ Z2) @ (bot_bot @ (set @ B))) = Z2)))). % 6.67/2.70 thf(fact_6485_fold__infinite, axiom, (! [A:$tType,B:$tType,A3:(set @ A),F2:(A > (B > B)),Z2:B]: (((~ (((finite_finite2 @ A) @ A3))) => ((((((finite_fold @ A) @ B) @ F2) @ Z2) @ A3) = Z2))))). % 6.67/2.70 thf(fact_6486_Compl__Diff__eq, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = (((sup_sup @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A3)) @ B3))))). % 6.67/2.70 thf(fact_6487_if__image__distrib, axiom, (! [A:$tType,B:$tType,P:(B > $o),F2:(B > A),G:(B > A),S3:(set @ B)]: ((((((image @ B) @ A) @ (^ [X:B]: (((((if @ A) @ (P @ X)) @ (F2 @ X)) @ (G @ X))))) @ S3) = (((sup_sup @ (set @ A)) @ ((((image @ B) @ A) @ F2) @ (((inf_inf @ (set @ B)) @ S3) @ ((collect @ B) @ P)))) @ ((((image @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ S3) @ ((collect @ B) @ (^ [X:B]: ((~ ((P @ X))))))))))))). % 6.67/2.70 thf(fact_6488_set__union, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((set2 @ A) @ (((union @ A) @ Xs) @ Ys)) = (((sup_sup @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys)))))). % 6.67/2.70 thf(fact_6489_UN__simps_I2_J, axiom, (! [C:$tType,D:$tType,C5:(set @ C),A3:(C > (set @ D)),B3:(set @ D)]: ((((C5 = (bot_bot @ (set @ C))) => (((complete_Sup_Sup @ (set @ D)) @ ((((image @ C) @ (set @ D)) @ (^ [X:C]: ((((sup_sup @ (set @ D)) @ (A3 @ X)) @ B3)))) @ C5)) = (bot_bot @ (set @ D)))) & ((C5 != (bot_bot @ (set @ C))) => (((complete_Sup_Sup @ (set @ D)) @ ((((image @ C) @ (set @ D)) @ (^ [X:C]: ((((sup_sup @ (set @ D)) @ (A3 @ X)) @ B3)))) @ C5)) = (((sup_sup @ (set @ D)) @ ((complete_Sup_Sup @ (set @ D)) @ ((((image @ C) @ (set @ D)) @ A3) @ C5))) @ B3))))))). % 6.67/2.70 thf(fact_6490_UN__simps_I3_J, axiom, (! [E2:$tType,F:$tType,C5:(set @ F),A3:(set @ E2),B3:(F > (set @ E2))]: ((((C5 = (bot_bot @ (set @ F))) => (((complete_Sup_Sup @ (set @ E2)) @ ((((image @ F) @ (set @ E2)) @ (^ [X:F]: ((((sup_sup @ (set @ E2)) @ A3) @ (B3 @ X))))) @ C5)) = (bot_bot @ (set @ E2)))) & ((C5 != (bot_bot @ (set @ F))) => (((complete_Sup_Sup @ (set @ E2)) @ ((((image @ F) @ (set @ E2)) @ (^ [X:F]: ((((sup_sup @ (set @ E2)) @ A3) @ (B3 @ X))))) @ C5)) = (((sup_sup @ (set @ E2)) @ A3) @ ((complete_Sup_Sup @ (set @ E2)) @ ((((image @ F) @ (set @ E2)) @ B3) @ C5))))))))). % 6.67/2.70 thf(fact_6491_rtrancl__Un__separator__converseE, axiom, (! [A:$tType,A2:A,B2:A,P:(set @ ((product_prod @ A) @ A)),Q:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_rtrancl @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ P) @ Q))) => ((! [X5:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ B2)) @ ((transitive_rtrancl @ A) @ P)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X5)) @ Q) => (Y4 = X5))))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_rtrancl @ A) @ P))))))). % 6.67/2.70 thf(fact_6492_rtrancl__Un__separatorE, axiom, (! [A:$tType,A2:A,B2:A,P:(set @ ((product_prod @ A) @ A)),Q:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_rtrancl @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ P) @ Q))) => ((! [X5:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ X5)) @ ((transitive_rtrancl @ A) @ P)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Y4)) @ Q) => (X5 = Y4))))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_rtrancl @ A) @ P))))))). % 6.67/2.70 thf(fact_6493_mono__Un, axiom, (! [B:$tType,A:$tType,F2:((set @ A) > (set @ B)),A3:(set @ A),B3:(set @ A)]: (((((order_mono @ (set @ A)) @ (set @ B)) @ F2) => (((ord_less_eq @ (set @ B)) @ (((sup_sup @ (set @ B)) @ (F2 @ A3)) @ (F2 @ B3))) @ (F2 @ (((sup_sup @ (set @ A)) @ A3) @ B3))))))). % 6.67/2.70 thf(fact_6494_mono__sup, axiom, (! [B:$tType,A:$tType]: ((((semilattice_sup @ A) & (semilattice_sup @ B)) => (! [F2:(A > B),A3:A,B3:A]: (((((order_mono @ A) @ B) @ F2) => (((ord_less_eq @ B) @ (((sup_sup @ B) @ (F2 @ A3)) @ (F2 @ B3))) @ (F2 @ (((sup_sup @ A) @ A3) @ B3)))))))))). % 6.67/2.70 thf(fact_6495_insert__is__Un, axiom, (! [A:$tType]: (((insert @ A) = (^ [A4:A]: (((sup_sup @ (set @ A)) @ (((insert @ A) @ A4) @ (bot_bot @ (set @ A)))))))))). % 6.67/2.70 thf(fact_6496_Un__singleton__iff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),X3:A]: ((((((sup_sup @ (set @ A)) @ A3) @ B3) = (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = (((A3 = (bot_bot @ (set @ A))) & (B3 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) | (((A3 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) & (B3 = (bot_bot @ (set @ A)))) | ((A3 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) & (B3 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))))). % 6.67/2.70 thf(fact_6497_singleton__Un__iff, axiom, (! [A:$tType,X3:A,A3:(set @ A),B3:(set @ A)]: ((((((insert @ A) @ X3) @ (bot_bot @ (set @ A))) = (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((A3 = (bot_bot @ (set @ A))) & (B3 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) | (((A3 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) & (B3 = (bot_bot @ (set @ A)))) | ((A3 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) & (B3 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))))). % 6.67/2.70 thf(fact_6498_boolean__algebra_Odisj__zero__right, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A]: (((((sup_sup @ A) @ X3) @ (bot_bot @ A)) = X3))))))). % 6.67/2.70 thf(fact_6499_Un__empty__right, axiom, (! [A:$tType,A3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = A3)))). % 6.67/2.70 thf(fact_6500_Un__empty__left, axiom, (! [A:$tType,B3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (bot_bot @ (set @ A))) @ B3) = B3)))). % 6.67/2.70 thf(fact_6501_Sup__fold__sup, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (((complete_Sup_Sup @ A) @ A3) = (((((finite_fold @ A) @ A) @ (sup_sup @ A)) @ (bot_bot @ A)) @ A3))))))))). % 6.67/2.70 thf(fact_6502_cSup__union__distrib, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ A),B3:(set @ A)]: (((A3 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ A3) => ((B3 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ B3) => (((complete_Sup_Sup @ A) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((sup_sup @ A) @ ((complete_Sup_Sup @ A) @ A3)) @ ((complete_Sup_Sup @ A) @ B3))))))))))))). % 6.67/2.70 thf(fact_6503_Compl__partition2, axiom, (! [A:$tType,A3:(set @ A)]: (((((sup_sup @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A3)) @ A3) = (top_top @ (set @ A)))))). % 6.67/2.70 thf(fact_6504_Compl__partition, axiom, (! [A:$tType,A3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ ((uminus_uminus @ (set @ A)) @ A3)) = (top_top @ (set @ A)))))). % 6.67/2.70 thf(fact_6505_sup__Sup__fold__sup, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A),B3:A]: ((((finite_finite2 @ A) @ A3) => ((((sup_sup @ A) @ ((complete_Sup_Sup @ A) @ A3)) @ B3) = (((((finite_fold @ A) @ A) @ (sup_sup @ A)) @ B3) @ A3))))))))). % 6.67/2.70 thf(fact_6506_infinite__Un, axiom, (! [A:$tType,S3:(set @ A),T4:(set @ A)]: (((~ (((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ S3) @ T4)))) = ((~ (((finite_finite2 @ A) @ S3))) | (~ (((finite_finite2 @ A) @ T4)))))))). % 6.67/2.70 thf(fact_6507_Un__infinite, axiom, (! [A:$tType,S3:(set @ A),T4:(set @ A)]: (((~ (((finite_finite2 @ A) @ S3))) => (~ (((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ S3) @ T4)))))))). % 6.67/2.70 thf(fact_6508_finite__UnI, axiom, (! [A:$tType,F5:(set @ A),G6:(set @ A)]: ((((finite_finite2 @ A) @ F5) => (((finite_finite2 @ A) @ G6) => ((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ F5) @ G6))))))). % 6.67/2.70 thf(fact_6509_insert__def, axiom, (! [A:$tType]: (((insert @ A) = (^ [A4:A]: (((sup_sup @ (set @ A)) @ ((collect @ A) @ (^ [X:A]: ((X = A4))))))))))). % 6.67/2.70 thf(fact_6510_Un__Int__crazy, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) @ (((inf_inf @ (set @ A)) @ B3) @ C5))) @ (((inf_inf @ (set @ A)) @ C5) @ A3)) = (((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) @ (((sup_sup @ (set @ A)) @ B3) @ C5))) @ (((sup_sup @ (set @ A)) @ C5) @ A3)))))). % 6.67/2.70 thf(fact_6511_Int__Un__distrib, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ B3) @ C5)) = (((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) @ (((inf_inf @ (set @ A)) @ A3) @ C5)))))). % 6.67/2.70 thf(fact_6512_Un__Int__distrib, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (((inf_inf @ (set @ A)) @ B3) @ C5)) = (((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) @ (((sup_sup @ (set @ A)) @ A3) @ C5)))))). % 6.67/2.70 thf(fact_6513_Int__Un__distrib2, axiom, (! [A:$tType,B3:(set @ A),C5:(set @ A),A3:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ B3) @ C5)) @ A3) = (((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ B3) @ A3)) @ (((inf_inf @ (set @ A)) @ C5) @ A3)))))). % 6.67/2.70 thf(fact_6514_Un__Int__distrib2, axiom, (! [A:$tType,B3:(set @ A),C5:(set @ A),A3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ B3) @ C5)) @ A3) = (((inf_inf @ (set @ A)) @ (((sup_sup @ (set @ A)) @ B3) @ A3)) @ (((sup_sup @ (set @ A)) @ C5) @ A3)))))). % 6.67/2.70 thf(fact_6515_Un__Diff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) @ C5) = (((sup_sup @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ C5)) @ (((minus_minus @ (set @ A)) @ B3) @ C5)))))). % 6.67/2.70 thf(fact_6516_UnE, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) => ((~ ((((member @ A) @ C2) @ A3))) => (((member @ A) @ C2) @ B3)))))). % 6.67/2.70 thf(fact_6517_UnI1, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)))))). % 6.67/2.70 thf(fact_6518_UnI2, axiom, (! [A:$tType,C2:A,B3:(set @ A),A3:(set @ A)]: (((((member @ A) @ C2) @ B3) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)))))). % 6.67/2.70 thf(fact_6519_bex__Un, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),P:(A > $o)]: (((? [X:A]: (((((member @ A) @ X) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) & (P @ X)))) = ((? [X:A]: (((((member @ A) @ X) @ A3) & (P @ X)))) | (? [X:A]: (((((member @ A) @ X) @ B3) & (P @ X))))))))). % 6.67/2.70 thf(fact_6520_ball__Un, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),P:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) => (P @ X)))) = ((! [X:A]: (((((member @ A) @ X) @ A3) => (P @ X)))) & (! [X:A]: (((((member @ A) @ X) @ B3) => (P @ X))))))))). % 6.67/2.70 thf(fact_6521_Un__assoc, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) @ C5) = (((sup_sup @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ B3) @ C5)))))). % 6.67/2.70 thf(fact_6522_Un__absorb, axiom, (! [A:$tType,A3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ A3) = A3)))). % 6.67/2.70 thf(fact_6523_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: ((((sup_sup @ (set @ A)) @ B6) @ A6))))))). % 6.67/2.70 thf(fact_6524_Un__left__absorb, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((sup_sup @ (set @ A)) @ A3) @ B3))))). % 6.67/2.70 thf(fact_6525_Un__left__commute, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ B3) @ C5)) = (((sup_sup @ (set @ A)) @ B3) @ (((sup_sup @ (set @ A)) @ A3) @ C5)))))). % 6.67/2.70 thf(fact_6526_Un__def, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A6) | (((member @ A) @ X) @ B6))))))))))). % 6.67/2.70 thf(fact_6527_Collect__disj__eq, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((collect @ A) @ (^ [X:A]: (((P @ X) | (Q @ X))))) = (((sup_sup @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))). % 6.67/2.70 thf(fact_6528_Collect__imp__eq, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((collect @ A) @ (^ [X:A]: (((P @ X) => (Q @ X))))) = (((sup_sup @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ ((collect @ A) @ P))) @ ((collect @ A) @ Q)))))). % 6.67/2.70 thf(fact_6529_less__supI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,A2:A,B2:A]: (((((ord_less @ A) @ X3) @ A2) => (((ord_less @ A) @ X3) @ (((sup_sup @ A) @ A2) @ B2))))))))). % 6.67/2.70 thf(fact_6530_less__supI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,B2:A,A2:A]: (((((ord_less @ A) @ X3) @ B2) => (((ord_less @ A) @ X3) @ (((sup_sup @ A) @ A2) @ B2))))))))). % 6.67/2.70 thf(fact_6531_sup_Oabsorb3, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => ((((sup_sup @ A) @ A2) @ B2) = A2)))))))). % 6.67/2.70 thf(fact_6532_sup_Oabsorb4, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((sup_sup @ A) @ A2) @ B2) = B2)))))))). % 6.67/2.70 thf(fact_6533_sup_Ostrict__boundedE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less @ A) @ (((sup_sup @ A) @ B2) @ C2)) @ A2) => (~ (((((ord_less @ A) @ B2) @ A2) => (~ ((((ord_less @ A) @ C2) @ A2))))))))))))). % 6.67/2.70 thf(fact_6534_sup_Ostrict__order__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less @ A) = (^ [B4:A,A4:A]: (((A4 = (((sup_sup @ A) @ A4) @ B4)) & (A4 != B4))))))))). % 6.67/2.70 thf(fact_6535_sup_Ostrict__coboundedI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ C2) @ A2) => (((ord_less @ A) @ C2) @ (((sup_sup @ A) @ A2) @ B2))))))))). % 6.67/2.70 thf(fact_6536_sup_Ostrict__coboundedI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,B2:A,A2:A]: (((((ord_less @ A) @ C2) @ B2) => (((ord_less @ A) @ C2) @ (((sup_sup @ A) @ A2) @ B2))))))))). % 6.67/2.70 thf(fact_6537_Un__UNIV__right, axiom, (! [A:$tType,A3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))). % 6.67/2.70 thf(fact_6538_Un__UNIV__left, axiom, (! [A:$tType,B3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (top_top @ (set @ A))) @ B3) = (top_top @ (set @ A)))))). % 6.67/2.70 thf(fact_6539_Diff__Un, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ B3) @ C5)) = (((inf_inf @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ (((minus_minus @ (set @ A)) @ A3) @ C5)))))). % 6.67/2.70 thf(fact_6540_Diff__Int, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (((inf_inf @ (set @ A)) @ B3) @ C5)) = (((sup_sup @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ (((minus_minus @ (set @ A)) @ A3) @ C5)))))). % 6.67/2.70 thf(fact_6541_Int__Diff__Un, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = A3)))). % 6.67/2.70 thf(fact_6542_Un__Diff__Int, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) = A3)))). % 6.67/2.70 thf(fact_6543_Compl__Int, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) = (((sup_sup @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A3)) @ ((uminus_uminus @ (set @ A)) @ B3)))))). % 6.67/2.70 thf(fact_6544_Compl__Un, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((inf_inf @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A3)) @ ((uminus_uminus @ (set @ A)) @ B3)))))). % 6.67/2.70 thf(fact_6545_set__shuffles, axiom, (! [A:$tType,Zs2:(list @ A),Xs:(list @ A),Ys:(list @ A)]: (((((member @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ Xs) @ Ys)) => (((set2 @ A) @ Zs2) = (((sup_sup @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys))))))). % 6.67/2.70 thf(fact_6546_Diff__subset__conv, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ C5) = (((ord_less_eq @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ B3) @ C5)))))). % 6.67/2.70 thf(fact_6547_Diff__partition, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((sup_sup @ (set @ A)) @ A3) @ (((minus_minus @ (set @ A)) @ B3) @ A3)) = B3))))). % 6.67/2.70 thf(fact_6548_Un__Int__assoc__eq, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ A)]: ((((((sup_sup @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) @ C5) = (((inf_inf @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ B3) @ C5))) = (((ord_less_eq @ (set @ A)) @ C5) @ A3))))). % 6.67/2.70 thf(fact_6549_distrib__sup__le, axiom, (! [A:$tType]: (((lattice @ A) => (! [X3:A,Y:A,Z2:A]: ((((ord_less_eq @ A) @ (((sup_sup @ A) @ X3) @ (((inf_inf @ A) @ Y) @ Z2))) @ (((inf_inf @ A) @ (((sup_sup @ A) @ X3) @ Y)) @ (((sup_sup @ A) @ X3) @ Z2))))))))). % 6.67/2.70 thf(fact_6550_distrib__inf__le, axiom, (! [A:$tType]: (((lattice @ A) => (! [X3:A,Y:A,Z2:A]: ((((ord_less_eq @ A) @ (((sup_sup @ A) @ (((inf_inf @ A) @ X3) @ Y)) @ (((inf_inf @ A) @ X3) @ Z2))) @ (((inf_inf @ A) @ X3) @ (((sup_sup @ A) @ Y) @ Z2))))))))). % 6.67/2.70 thf(fact_6551_subset__Un__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((((sup_sup @ (set @ A)) @ A6) @ B6) = B6))))))). % 6.67/2.70 thf(fact_6552_subset__UnE, axiom, (! [A:$tType,C5:(set @ A),A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C5) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) => (~ ((! [A15:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A15) @ A3) => (! [B14:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B14) @ B3) => (C5 != (((sup_sup @ (set @ A)) @ A15) @ B14)))))))))))))). % 6.67/2.70 thf(fact_6553_Un__absorb2, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B3) @ A3) => ((((sup_sup @ (set @ A)) @ A3) @ B3) = A3))))). % 6.67/2.70 thf(fact_6554_Un__absorb1, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((sup_sup @ (set @ A)) @ A3) @ B3) = B3))))). % 6.67/2.70 thf(fact_6555_Un__upper2, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B3) @ (((sup_sup @ (set @ A)) @ A3) @ B3))))). % 6.67/2.70 thf(fact_6556_Un__upper1, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ A3) @ B3))))). % 6.67/2.70 thf(fact_6557_Un__least, axiom, (! [A:$tType,A3:(set @ A),C5:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ C5) => ((((ord_less_eq @ (set @ A)) @ B3) @ C5) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) @ C5)))))). % 6.67/2.70 thf(fact_6558_Un__mono, axiom, (! [A:$tType,A3:(set @ A),C5:(set @ A),B3:(set @ A),D6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ C5) => ((((ord_less_eq @ (set @ A)) @ B3) @ D6) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) @ (((sup_sup @ (set @ A)) @ C5) @ D6))))))). % 6.67/2.70 thf(fact_6559_Un__Pow__subset, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ (set @ (set @ A))) @ (((sup_sup @ (set @ (set @ A))) @ ((pow2 @ A) @ A3)) @ ((pow2 @ A) @ B3))) @ ((pow2 @ A) @ (((sup_sup @ (set @ A)) @ A3) @ B3)))))). % 6.67/2.70 thf(fact_6560_inf__sup__ord_I4_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [Y:A,X3:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X3) @ Y)))))))). % 6.67/2.70 thf(fact_6561_inf__sup__ord_I3_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X3:A,Y:A]: ((((ord_less_eq @ A) @ X3) @ (((sup_sup @ A) @ X3) @ Y)))))))). % 6.67/2.70 thf(fact_6562_le__supE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A,X3:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ X3) => (~ (((((ord_less_eq @ A) @ A2) @ X3) => (~ ((((ord_less_eq @ A) @ B2) @ X3))))))))))))). % 6.67/2.70 thf(fact_6563_le__supI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,X3:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ X3) => ((((ord_less_eq @ A) @ B2) @ X3) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ X3))))))))). % 6.67/2.70 thf(fact_6564_sup__ge1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,Y:A]: ((((ord_less_eq @ A) @ X3) @ (((sup_sup @ A) @ X3) @ Y)))))))). % 6.67/2.70 thf(fact_6565_sup__ge2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X3:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X3) @ Y)))))))). % 6.67/2.70 thf(fact_6566_le__supI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ X3) @ A2) => (((ord_less_eq @ A) @ X3) @ (((sup_sup @ A) @ A2) @ B2))))))))). % 6.67/2.70 thf(fact_6567_le__supI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,B2:A,A2:A]: (((((ord_less_eq @ A) @ X3) @ B2) => (((ord_less_eq @ A) @ X3) @ (((sup_sup @ A) @ A2) @ B2))))))))). % 6.67/2.70 thf(fact_6568_sup_Omono, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,A2:A,D2:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ A2) => ((((ord_less_eq @ A) @ D2) @ B2) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ C2) @ D2)) @ (((sup_sup @ A) @ A2) @ B2)))))))))). % 6.67/2.70 thf(fact_6569_sup__mono, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,C2:A,B2:A,D2:A]: (((((ord_less_eq @ A) @ A2) @ C2) => ((((ord_less_eq @ A) @ B2) @ D2) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ (((sup_sup @ A) @ C2) @ D2)))))))))). % 6.67/2.70 thf(fact_6570_sup__least, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X3:A,Z2:A]: (((((ord_less_eq @ A) @ Y) @ X3) => ((((ord_less_eq @ A) @ Z2) @ X3) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ Y) @ Z2)) @ X3))))))))). % 6.67/2.70 thf(fact_6571_le__iff__sup, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [X:A,Y2:A]: (((((sup_sup @ A) @ X) @ Y2) = Y2)))))))). % 6.67/2.70 thf(fact_6572_sup_OorderE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => (A2 = (((sup_sup @ A) @ A2) @ B2))))))))). % 6.67/2.70 thf(fact_6573_sup_OorderI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A]: (((A2 = (((sup_sup @ A) @ A2) @ B2)) => (((ord_less_eq @ A) @ B2) @ A2)))))))). % 6.67/2.70 thf(fact_6574_sup__unique, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [F2:(A > (A > A)),X3:A,Y:A]: (((! [X5:A,Y4:A]: ((((ord_less_eq @ A) @ X5) @ ((F2 @ X5) @ Y4)))) => ((! [X5:A,Y4:A]: ((((ord_less_eq @ A) @ Y4) @ ((F2 @ X5) @ Y4)))) => ((! [X5:A,Y4:A,Z3:A]: (((((ord_less_eq @ A) @ Y4) @ X5) => ((((ord_less_eq @ A) @ Z3) @ X5) => (((ord_less_eq @ A) @ ((F2 @ Y4) @ Z3)) @ X5))))) => ((((sup_sup @ A) @ X3) @ Y) = ((F2 @ X3) @ Y))))))))))). % 6.67/2.70 thf(fact_6575_sup_Oabsorb1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((sup_sup @ A) @ A2) @ B2) = A2)))))))). % 6.67/2.70 thf(fact_6576_sup_Oabsorb2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((sup_sup @ A) @ A2) @ B2) = B2)))))))). % 6.67/2.70 thf(fact_6577_sup__absorb1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X3:A]: (((((ord_less_eq @ A) @ Y) @ X3) => ((((sup_sup @ A) @ X3) @ Y) = X3)))))))). % 6.67/2.70 thf(fact_6578_sup__absorb2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) => ((((sup_sup @ A) @ X3) @ Y) = Y)))))))). % 6.67/2.70 thf(fact_6579_sup_OboundedE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B2) @ C2)) @ A2) => (~ (((((ord_less_eq @ A) @ B2) @ A2) => (~ ((((ord_less_eq @ A) @ C2) @ A2))))))))))))). % 6.67/2.70 thf(fact_6580_sup_OboundedI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ A2) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ B2) @ C2)) @ A2))))))))). % 6.67/2.70 thf(fact_6581_sup_Oorder__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: ((A4 = (((sup_sup @ A) @ A4) @ B4))))))))). % 6.67/2.70 thf(fact_6582_sup_Ocobounded1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ A2) @ (((sup_sup @ A) @ A2) @ B2)))))))). % 6.67/2.70 thf(fact_6583_sup_Ocobounded2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A2:A]: ((((ord_less_eq @ A) @ B2) @ (((sup_sup @ A) @ A2) @ B2)))))))). % 6.67/2.70 thf(fact_6584_sup_Oabsorb__iff1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: (((((sup_sup @ A) @ A4) @ B4) = A4)))))))). % 6.67/2.70 thf(fact_6585_sup_Oabsorb__iff2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: (((((sup_sup @ A) @ A4) @ B4) = B4)))))))). % 6.67/2.70 thf(fact_6586_sup_OcoboundedI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ A2) => (((ord_less_eq @ A) @ C2) @ (((sup_sup @ A) @ A2) @ B2))))))))). % 6.67/2.70 thf(fact_6587_sup_OcoboundedI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,B2:A,A2:A]: (((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ (((sup_sup @ A) @ A2) @ B2))))))))). % 6.67/2.70 thf(fact_6588_ivl__disj__un__two_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ M)) @ (((set_or7035219750837199246ssThan @ A) @ M) @ U)) = (((set_or7035219750837199246ssThan @ A) @ L) @ U)))))))))). % 6.67/2.70 thf(fact_6589_ivl__disj__un__two__touch_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M)) @ (((set_or1337092689740270186AtMost @ A) @ M) @ U)) = (((set_or1337092689740270186AtMost @ A) @ L) @ U)))))))))). % 6.67/2.70 thf(fact_6590_ivl__disj__un__two_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M)) @ (((set_or3652927894154168847AtMost @ A) @ M) @ U)) = (((set_or3652927894154168847AtMost @ A) @ L) @ U)))))))))). % 6.67/2.70 thf(fact_6591_sup__SUP__fold__sup, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),B3:A,F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((((sup_sup @ A) @ B3) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) = (((((finite_fold @ B) @ A) @ (((((comp @ A) @ (A > A)) @ B) @ (sup_sup @ A)) @ F2)) @ B3) @ A3))))))))). % 6.67/2.70 thf(fact_6592_image__Un, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B),B3:(set @ B)]: ((((((image @ B) @ A) @ F2) @ (((sup_sup @ (set @ B)) @ A3) @ B3)) = (((sup_sup @ (set @ A)) @ ((((image @ B) @ A) @ F2) @ A3)) @ ((((image @ B) @ A) @ F2) @ B3)))))). % 6.67/2.70 thf(fact_6593_cSUP__union, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ B),F2:(B > A),B3:(set @ B)]: (((A3 != (bot_bot @ (set @ B))) => (((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => ((B3 != (bot_bot @ (set @ B))) => (((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F2) @ B3)) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ (((sup_sup @ (set @ B)) @ A3) @ B3))) = (((sup_sup @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ B3)))))))))))))). % 6.67/2.70 thf(fact_6594_SUP__fold__sup, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3)) = (((((finite_fold @ B) @ A) @ (((((comp @ A) @ (A > A)) @ B) @ (sup_sup @ A)) @ F2)) @ (bot_bot @ A)) @ A3))))))))). % 6.67/2.70 thf(fact_6595_sup__shunt, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A,Y:A]: ((((((sup_sup @ A) @ X3) @ Y) = (top_top @ A)) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X3)) @ Y)))))))). % 6.67/2.70 thf(fact_6596_sup__neg__inf, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [P4:A,Q4:A,R2:A]: (((((ord_less_eq @ A) @ P4) @ (((sup_sup @ A) @ Q4) @ R2)) = (((ord_less_eq @ A) @ (((inf_inf @ A) @ P4) @ ((uminus_uminus @ A) @ Q4))) @ R2)))))))). % 6.67/2.70 thf(fact_6597_shunt2, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ (((inf_inf @ A) @ X3) @ ((uminus_uminus @ A) @ Y))) @ Z2) = (((ord_less_eq @ A) @ X3) @ (((sup_sup @ A) @ Y) @ Z2))))))))). % 6.67/2.70 thf(fact_6598_shunt1, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ (((inf_inf @ A) @ X3) @ Y)) @ Z2) = (((ord_less_eq @ A) @ X3) @ (((sup_sup @ A) @ ((uminus_uminus @ A) @ Y)) @ Z2))))))))). % 6.67/2.70 thf(fact_6599_boolean__algebra_Ocomplement__unique, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [A2:A,X3:A,Y:A]: ((((((inf_inf @ A) @ A2) @ X3) = (bot_bot @ A)) => (((((sup_sup @ A) @ A2) @ X3) = (top_top @ A)) => (((((inf_inf @ A) @ A2) @ Y) = (bot_bot @ A)) => (((((sup_sup @ A) @ A2) @ Y) = (top_top @ A)) => (X3 = Y))))))))))). % 6.67/2.70 thf(fact_6600_finite__Sup__in, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((! [X5:A,Y4:A]: (((((member @ A) @ X5) @ A3) => ((((member @ A) @ Y4) @ A3) => (((member @ A) @ (((sup_sup @ A) @ X5) @ Y4)) @ A3))))) => (((member @ A) @ ((complete_Sup_Sup @ A) @ A3)) @ A3)))))))))). % 6.67/2.70 thf(fact_6601_less__eq__Inf__inter, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ A) @ (((sup_sup @ A) @ ((complete_Inf_Inf @ A) @ A3)) @ ((complete_Inf_Inf @ A) @ B3))) @ ((complete_Inf_Inf @ A) @ (((inf_inf @ (set @ A)) @ A3) @ B3))))))))). % 6.67/2.70 thf(fact_6602_ivl__disj__un__two_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ M)) @ (((set_or1337092689740270186AtMost @ A) @ M) @ U)) = (((set_or1337092689740270186AtMost @ A) @ L) @ U)))))))))). % 6.67/2.70 thf(fact_6603_inf__Inf__fold__inf, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A),B3:A]: ((((finite_finite2 @ A) @ A3) => ((((inf_inf @ A) @ ((complete_Inf_Inf @ A) @ A3)) @ B3) = (((((finite_fold @ A) @ A) @ (inf_inf @ A)) @ B3) @ A3))))))))). % 6.67/2.70 thf(fact_6604_ivl__disj__un__one_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ ((set_ord_lessThan @ A) @ L)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ U)) = ((set_ord_lessThan @ A) @ U))))))))). % 6.67/2.70 thf(fact_6605_Union__image__empty, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F2:(B > (set @ A))]: (((((sup_sup @ (set @ A)) @ A3) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ F2) @ (bot_bot @ (set @ B))))) = A3)))). % 6.67/2.70 thf(fact_6606_card__Un__le, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ nat) @ ((finite_card @ A) @ (((sup_sup @ (set @ A)) @ A3) @ B3))) @ (((plus_plus @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3)))))). % 6.67/2.70 thf(fact_6607_atLeastLessThan__add__Un, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((set_or7035219750837199246ssThan @ nat) @ I) @ (((plus_plus @ nat) @ J) @ K)) = (((sup_sup @ (set @ nat)) @ (((set_or7035219750837199246ssThan @ nat) @ I) @ J)) @ (((set_or7035219750837199246ssThan @ nat) @ J) @ (((plus_plus @ nat) @ J) @ K)))))))). % 6.67/2.70 thf(fact_6608_ivl__disj__un__two_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M)) @ (((set_or3652927894154168847AtMost @ A) @ M) @ U)) = (((set_or1337092689740270186AtMost @ A) @ L) @ U)))))))))). % 6.67/2.70 thf(fact_6609_ivl__disj__un__one_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ U)) @ ((set_ord_atLeast @ A) @ U)) = ((set_ord_atLeast @ A) @ L))))))))). % 6.67/2.70 thf(fact_6610_ivl__disj__un__one_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ ((set_ord_atMost @ A) @ L)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ U)) = ((set_ord_atMost @ A) @ U))))))))). % 6.67/2.70 thf(fact_6611_ivl__disj__un__one_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ U)) @ ((set_ord_greaterThan @ A) @ U)) = ((set_ord_greaterThan @ A) @ L))))))))). % 6.67/2.70 thf(fact_6612_Pow__insert, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: ((((pow2 @ A) @ (((insert @ A) @ A2) @ A3)) = (((sup_sup @ (set @ (set @ A))) @ ((pow2 @ A) @ A3)) @ ((((image @ (set @ A)) @ (set @ A)) @ ((insert @ A) @ A2)) @ ((pow2 @ A) @ A3))))))). % 6.67/2.70 thf(fact_6613_UN__extend__simps_I2_J, axiom, (! [D:$tType,C:$tType,C5:(set @ C),A3:(C > (set @ D)),B3:(set @ D)]: ((((C5 = (bot_bot @ (set @ C))) => ((((sup_sup @ (set @ D)) @ ((complete_Sup_Sup @ (set @ D)) @ ((((image @ C) @ (set @ D)) @ A3) @ C5))) @ B3) = B3)) & ((C5 != (bot_bot @ (set @ C))) => ((((sup_sup @ (set @ D)) @ ((complete_Sup_Sup @ (set @ D)) @ ((((image @ C) @ (set @ D)) @ A3) @ C5))) @ B3) = ((complete_Sup_Sup @ (set @ D)) @ ((((image @ C) @ (set @ D)) @ (^ [X:C]: ((((sup_sup @ (set @ D)) @ (A3 @ X)) @ B3)))) @ C5)))))))). % 6.67/2.70 thf(fact_6614_UN__extend__simps_I3_J, axiom, (! [E2:$tType,F:$tType,C5:(set @ F),A3:(set @ E2),B3:(F > (set @ E2))]: ((((C5 = (bot_bot @ (set @ F))) => ((((sup_sup @ (set @ E2)) @ A3) @ ((complete_Sup_Sup @ (set @ E2)) @ ((((image @ F) @ (set @ E2)) @ B3) @ C5))) = A3)) & ((C5 != (bot_bot @ (set @ F))) => ((((sup_sup @ (set @ E2)) @ A3) @ ((complete_Sup_Sup @ (set @ E2)) @ ((((image @ F) @ (set @ E2)) @ B3) @ C5))) = ((complete_Sup_Sup @ (set @ E2)) @ ((((image @ F) @ (set @ E2)) @ (^ [X:F]: ((((sup_sup @ (set @ E2)) @ A3) @ (B3 @ X))))) @ C5)))))))). % 6.67/2.70 thf(fact_6615_Inter__Un__subset, axiom, (! [A:$tType,A3:(set @ (set @ A)),B3:(set @ (set @ A))]: ((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ ((complete_Inf_Inf @ (set @ A)) @ A3)) @ ((complete_Inf_Inf @ (set @ A)) @ B3))) @ ((complete_Inf_Inf @ (set @ A)) @ (((inf_inf @ (set @ (set @ A))) @ A3) @ B3)))))). % 6.67/2.70 thf(fact_6616_boolean__algebra__class_Oboolean__algebra_Ocompl__unique, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => (! [X3:A,Y:A]: ((((((inf_inf @ A) @ X3) @ Y) = (bot_bot @ A)) => (((((sup_sup @ A) @ X3) @ Y) = (top_top @ A)) => (((uminus_uminus @ A) @ X3) = Y))))))))). % 6.67/2.70 thf(fact_6617_cSup__insert__If, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),A2:A]: ((((condit941137186595557371_above @ A) @ X8) => (((X8 = (bot_bot @ (set @ A))) => (((complete_Sup_Sup @ A) @ (((insert @ A) @ A2) @ X8)) = A2)) & ((X8 != (bot_bot @ (set @ A))) => (((complete_Sup_Sup @ A) @ (((insert @ A) @ A2) @ X8)) = (((sup_sup @ A) @ A2) @ ((complete_Sup_Sup @ A) @ X8)))))))))))). % 6.67/2.70 thf(fact_6618_cSup__insert, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A),A2:A]: (((X8 != (bot_bot @ (set @ A))) => (((condit941137186595557371_above @ A) @ X8) => (((complete_Sup_Sup @ A) @ (((insert @ A) @ A2) @ X8)) = (((sup_sup @ A) @ A2) @ ((complete_Sup_Sup @ A) @ X8))))))))))). % 6.67/2.70 thf(fact_6619_ivl__disj__un__two__touch_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M)) @ (((set_or7035219750837199246ssThan @ A) @ M) @ U)) = (((set_or7035219750837199246ssThan @ A) @ L) @ U)))))))))). % 6.67/2.70 thf(fact_6620_sum_Ounion__inter, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),B3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((finite_finite2 @ B) @ B3) => ((((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A3) @ B3))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A3) @ B3))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ B3))))))))))). % 6.67/2.70 thf(fact_6621_prod_Ounion__inter, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),B3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((finite_finite2 @ B) @ B3) => ((((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A3) @ B3))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A3) @ B3))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A3)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ B3))))))))))). % 6.67/2.70 thf(fact_6622_Inf__fold__inf, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (((complete_Inf_Inf @ A) @ A3) = (((((finite_fold @ A) @ A) @ (inf_inf @ A)) @ (top_top @ A)) @ A3))))))))). % 6.67/2.70 thf(fact_6623_card__Un__Int, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ A) @ B3) => ((((plus_plus @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3)) = (((plus_plus @ nat) @ ((finite_card @ A) @ (((sup_sup @ (set @ A)) @ A3) @ B3))) @ ((finite_card @ A) @ (((inf_inf @ (set @ A)) @ A3) @ B3))))))))). % 6.67/2.70 thf(fact_6624_ivl__disj__un__two__touch_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M)) @ (((set_or1337092689740270186AtMost @ A) @ M) @ U)) = (((set_or3652927894154168847AtMost @ A) @ L) @ U)))))))))). % 6.67/2.70 thf(fact_6625_ivl__disj__un__two_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ M)) @ (((set_or7035219750837199246ssThan @ A) @ M) @ U)) = (((set_or5935395276787703475ssThan @ A) @ L) @ U)))))))))). % 6.67/2.70 thf(fact_6626_ivl__disj__un__one_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ ((set_ord_lessThan @ A) @ L)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ U)) = ((set_ord_atMost @ A) @ U))))))))). % 6.67/2.70 thf(fact_6627_ivl__disj__un__singleton_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [U:A]: (((((sup_sup @ (set @ A)) @ ((set_ord_lessThan @ A) @ U)) @ (((insert @ A) @ U) @ (bot_bot @ (set @ A)))) = ((set_ord_atMost @ A) @ U)))))))). % 6.67/2.70 thf(fact_6628_sum_Oeq__fold, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (((groups7311177749621191930dd_sum @ B) @ A) = (^ [G2:(B > A)]: (((((finite_fold @ B) @ A) @ (((((comp @ A) @ (A > A)) @ B) @ (plus_plus @ A)) @ G2)) @ (zero_zero @ A))))))))). % 6.67/2.70 thf(fact_6629_cInf__union__distrib, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ A),B3:(set @ A)]: (((A3 != (bot_bot @ (set @ A))) => (((condit1013018076250108175_below @ A) @ A3) => ((B3 != (bot_bot @ (set @ A))) => (((condit1013018076250108175_below @ A) @ B3) => (((complete_Inf_Inf @ A) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((inf_inf @ A) @ ((complete_Inf_Inf @ A) @ A3)) @ ((complete_Inf_Inf @ A) @ B3))))))))))))). % 6.67/2.70 thf(fact_6630_ivl__disj__un__two_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M)) @ (((set_or5935395276787703475ssThan @ A) @ M) @ U)) = (((set_or5935395276787703475ssThan @ A) @ L) @ U)))))))))). % 6.67/2.70 thf(fact_6631_Max_Ounion, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => ((B3 != (bot_bot @ (set @ A))) => (((lattic643756798349783984er_Max @ A) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((ord_max @ A) @ ((lattic643756798349783984er_Max @ A) @ A3)) @ ((lattic643756798349783984er_Max @ A) @ B3))))))))))))). % 6.67/2.70 thf(fact_6632_ivl__disj__un__one_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ ((set_ord_atMost @ A) @ L)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) = ((set_ord_lessThan @ A) @ U))))))))). % 6.67/2.70 thf(fact_6633_ivl__disj__un__one_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ U)) @ ((set_ord_greaterThan @ A) @ U)) = ((set_ord_atLeast @ A) @ L))))))))). % 6.67/2.70 thf(fact_6634_ivl__disj__un__singleton_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A]: (((((sup_sup @ (set @ A)) @ (((insert @ A) @ L) @ (bot_bot @ (set @ A)))) @ ((set_ord_greaterThan @ A) @ L)) = ((set_ord_atLeast @ A) @ L)))))))). % 6.67/2.70 thf(fact_6635_Max_Oeq__fold, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => (((lattic643756798349783984er_Max @ A) @ (((insert @ A) @ X3) @ A3)) = (((((finite_fold @ A) @ A) @ (ord_max @ A)) @ X3) @ A3))))))))). % 6.67/2.70 thf(fact_6636_ivl__disj__un__two__touch_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less @ A) @ L) @ M) => ((((ord_less @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M)) @ (((set_or7035219750837199246ssThan @ A) @ M) @ U)) = (((set_or5935395276787703475ssThan @ A) @ L) @ U)))))))))). % 6.67/2.70 thf(fact_6637_SUP__nat__binary, axiom, (! [A:$tType]: (((counta3822494911875563373attice @ A) => (! [A3:A,B3:A]: (((((sup_sup @ A) @ A3) @ ((complete_Sup_Sup @ A) @ ((((image @ nat) @ A) @ (^ [X:nat]: (B3))) @ ((collect @ nat) @ ((ord_less @ nat) @ (zero_zero @ nat)))))) = (((sup_sup @ A) @ A3) @ B3)))))))). % 6.67/2.70 thf(fact_6638_ivl__disj__un__one_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) @ ((set_ord_atLeast @ A) @ U)) = ((set_ord_greaterThan @ A) @ L))))))))). % 6.67/2.70 thf(fact_6639_conditionally__complete__lattice__class_OSUP__sup__distrib, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ B),F2:(B > A),G:(B > A)]: (((A3 != (bot_bot @ (set @ B))) => (((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => (((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ G) @ A3)) => ((((sup_sup @ A) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ G) @ A3))) = ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ (^ [A4:B]: ((((sup_sup @ A) @ (F2 @ A4)) @ (G @ A4))))) @ A3)))))))))))). % 6.67/2.70 thf(fact_6640_inj__on__disjoint__Un, axiom, (! [B:$tType,A:$tType,F2:(A > B),A3:(set @ A),G:(A > B),B3:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ A3) => (((((inj_on @ A) @ B) @ G) @ B3) => (((((inf_inf @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A3)) @ ((((image @ A) @ B) @ G) @ B3)) = (bot_bot @ (set @ B))) => ((((inj_on @ A) @ B) @ (^ [X:A]: (((((if @ B) @ (((member @ A) @ X) @ A3)) @ (F2 @ X)) @ (G @ X))))) @ (((sup_sup @ (set @ A)) @ A3) @ B3)))))))). % 6.67/2.70 thf(fact_6641_sup__bot_Osemilattice__neutr__order__axioms, axiom, (! [A:$tType]: (((bounde4967611905675639751up_bot @ A) => (((((semila1105856199041335345_order @ A) @ (sup_sup @ A)) @ (bot_bot @ A)) @ (^ [X:A,Y2:A]: ((((ord_less_eq @ A) @ Y2) @ X)))) @ (^ [X:A,Y2:A]: ((((ord_less @ A) @ Y2) @ X)))))))). % 6.67/2.70 thf(fact_6642_rtrancl__insert, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A))]: ((((transitive_rtrancl @ A) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2)) = (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ ((transitive_rtrancl @ A) @ R2)) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X:A,Y2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ A2)) @ ((transitive_rtrancl @ A) @ R2)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ Y2)) @ ((transitive_rtrancl @ A) @ R2)))))))))))). % 6.67/2.70 thf(fact_6643_trancl__insert2, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A))]: ((((transitive_trancl @ A) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2)) = (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ ((transitive_trancl @ A) @ R2)) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X:A,Y2:A]: ((((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ A2)) @ ((transitive_trancl @ A) @ R2)) | (X = A2)) & ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ Y2)) @ ((transitive_trancl @ A) @ R2)) | (Y2 = B2)))))))))))). % 6.67/2.70 thf(fact_6644_cSup__inter__less__eq, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ A),B3:(set @ A)]: ((((condit941137186595557371_above @ A) @ A3) => (((condit941137186595557371_above @ A) @ B3) => (((((inf_inf @ (set @ A)) @ A3) @ B3) != (bot_bot @ (set @ A))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ (((inf_inf @ (set @ A)) @ A3) @ B3))) @ (((sup_sup @ A) @ ((complete_Sup_Sup @ A) @ A3)) @ ((complete_Sup_Sup @ A) @ B3)))))))))))). % 6.67/2.70 thf(fact_6645_cSUP__insert, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ B),F2:(B > A),A2:B]: (((A3 != (bot_bot @ (set @ B))) => (((condit941137186595557371_above @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => (((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ (((insert @ B) @ A2) @ A3))) = (((sup_sup @ A) @ (F2 @ A2)) @ ((complete_Sup_Sup @ A) @ ((((image @ B) @ A) @ F2) @ A3)))))))))))). % 6.67/2.70 thf(fact_6646_sum_Ounion__inter__neutral, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),B3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((finite_finite2 @ B) @ B3) => ((! [X5:B]: (((((member @ B) @ X5) @ (((inf_inf @ (set @ B)) @ A3) @ B3)) => ((G @ X5) = (zero_zero @ A))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A3) @ B3)) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ B3)))))))))))). % 6.67/2.70 thf(fact_6647_sum__Un, axiom, (! [A:$tType,B:$tType]: (((ab_group_add @ A) => (! [A3:(set @ B),B3:(set @ B),F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((finite_finite2 @ B) @ B3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ (((sup_sup @ (set @ B)) @ A3) @ B3)) = (((minus_minus @ A) @ (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ A3)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ B3))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F2) @ (((inf_inf @ (set @ B)) @ A3) @ B3)))))))))))). % 6.67/2.70 thf(fact_6648_sum_Ounion__disjoint, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),B3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((finite_finite2 @ B) @ B3) => (((((inf_inf @ (set @ B)) @ A3) @ B3) = (bot_bot @ (set @ B))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A3) @ B3)) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A3)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ B3)))))))))))). % 6.67/2.70 thf(fact_6649_prod_Ounion__inter__neutral, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),B3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((finite_finite2 @ B) @ B3) => ((! [X5:B]: (((((member @ B) @ X5) @ (((inf_inf @ (set @ B)) @ A3) @ B3)) => ((G @ X5) = (one_one @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A3) @ B3)) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A3)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ B3)))))))))))). % 6.67/2.70 thf(fact_6650_prod_Ounion__disjoint, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),B3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((finite_finite2 @ B) @ B3) => (((((inf_inf @ (set @ B)) @ A3) @ B3) = (bot_bot @ (set @ B))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A3) @ B3)) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A3)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ B3)))))))))))). % 6.67/2.70 thf(fact_6651_ivl__disj__un__singleton_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ U)) @ (((insert @ A) @ U) @ (bot_bot @ (set @ A)))) = (((set_or1337092689740270186AtMost @ A) @ L) @ U))))))))). % 6.67/2.70 thf(fact_6652_sum__Un2, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ B) => (! [A3:(set @ A),B3:(set @ A),F2:(A > B)]: ((((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) => (((((groups7311177749621191930dd_sum @ A) @ B) @ F2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((plus_plus @ B) @ (((plus_plus @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F2) @ (((minus_minus @ (set @ A)) @ A3) @ B3))) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F2) @ (((minus_minus @ (set @ A)) @ B3) @ A3)))) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F2) @ (((inf_inf @ (set @ A)) @ A3) @ B3))))))))))). % 6.67/2.70 thf(fact_6653_sum_Ounion__diff2, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),B3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((finite_finite2 @ B) @ B3) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A3) @ B3)) = (((plus_plus @ A) @ (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A3) @ B3))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ B3) @ A3)))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A3) @ B3)))))))))))). % 6.67/2.70 thf(fact_6654_prod_Ounion__diff2, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),B3:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((finite_finite2 @ B) @ B3) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A3) @ B3)) = (((times_times @ A) @ (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A3) @ B3))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ B3) @ A3)))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A3) @ B3)))))))))))). % 6.67/2.70 thf(fact_6655_card__Un__disjoint, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ A) @ B3) => (((((inf_inf @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))) => (((finite_card @ A) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((plus_plus @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3))))))))). % 6.67/2.70 thf(fact_6656_inj__on__Un, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ A),B3:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((((inj_on @ A) @ B) @ F2) @ A3) & (((((inj_on @ A) @ B) @ F2) @ B3) & ((((inf_inf @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ (((minus_minus @ (set @ A)) @ A3) @ B3))) @ ((((image @ A) @ B) @ F2) @ (((minus_minus @ (set @ A)) @ B3) @ A3))) = (bot_bot @ (set @ B))))))))). % 6.67/2.70 thf(fact_6657_ivl__disj__un__singleton_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((insert @ A) @ L) @ (bot_bot @ (set @ A)))) @ (((set_or3652927894154168847AtMost @ A) @ L) @ U)) = (((set_or1337092689740270186AtMost @ A) @ L) @ U))))))))). % 6.67/2.70 thf(fact_6658_ivl__disj__un__two_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less_eq @ A) @ L) @ M) => ((((ord_less @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M)) @ (((set_or5935395276787703475ssThan @ A) @ M) @ U)) = (((set_or7035219750837199246ssThan @ A) @ L) @ U)))))))))). % 6.67/2.70 thf(fact_6659_cINF__union, axiom, (! [A:$tType,B:$tType]: (((condit1219197933456340205attice @ A) => (! [A3:(set @ B),F2:(B > A),B3:(set @ B)]: (((A3 != (bot_bot @ (set @ B))) => (((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => ((B3 != (bot_bot @ (set @ B))) => (((condit1013018076250108175_below @ A) @ ((((image @ B) @ A) @ F2) @ B3)) => (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ (((sup_sup @ (set @ B)) @ A3) @ B3))) = (((inf_inf @ A) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ B3)))))))))))))). % 6.67/2.70 thf(fact_6660_ivl__disj__un__singleton_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((insert @ A) @ L) @ (bot_bot @ (set @ A)))) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) = (((set_or7035219750837199246ssThan @ A) @ L) @ U))))))))). % 6.67/2.70 thf(fact_6661_inf__INF__fold__inf, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),B3:A,F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => ((((inf_inf @ A) @ B3) @ ((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3))) = (((((finite_fold @ B) @ A) @ (((((comp @ A) @ (A > A)) @ B) @ (inf_inf @ A)) @ F2)) @ B3) @ A3))))))))). % 6.67/2.70 thf(fact_6662_sum__Un__nat, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),F2:(A > nat)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ A) @ B3) => (((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ A3)) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ B3))) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ (((inf_inf @ (set @ A)) @ A3) @ B3))))))))). % 6.67/2.70 thf(fact_6663_ivl__disj__un__two_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M:A,U:A]: (((((ord_less @ A) @ L) @ M) => ((((ord_less_eq @ A) @ M) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ M)) @ (((set_or1337092689740270186AtMost @ A) @ M) @ U)) = (((set_or3652927894154168847AtMost @ A) @ L) @ U)))))))))). % 6.67/2.70 thf(fact_6664_ivl__disj__un__singleton_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) @ (((insert @ A) @ U) @ (bot_bot @ (set @ A)))) = (((set_or3652927894154168847AtMost @ A) @ L) @ U))))))))). % 6.67/2.70 thf(fact_6665_trancl__insert, axiom, (! [A:$tType,Y:A,X3:A,R2:(set @ ((product_prod @ A) @ A))]: ((((transitive_trancl @ A) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X3)) @ R2)) = (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ ((transitive_trancl @ A) @ R2)) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [A4:A,B4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ Y)) @ ((transitive_rtrancl @ A) @ R2)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ B4)) @ ((transitive_rtrancl @ A) @ R2)))))))))))). % 6.67/2.70 thf(fact_6666_prod__Un, axiom, (! [A:$tType,B:$tType]: (((field @ A) => (! [A3:(set @ B),B3:(set @ B),F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((finite_finite2 @ B) @ B3) => ((! [X5:B]: (((((member @ B) @ X5) @ (((inf_inf @ (set @ B)) @ A3) @ B3)) => ((F2 @ X5) != (zero_zero @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ (((sup_sup @ (set @ B)) @ A3) @ B3)) = (((divide_divide @ A) @ (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ A3)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ B3))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F2) @ (((inf_inf @ (set @ B)) @ A3) @ B3))))))))))))). % 6.67/2.70 thf(fact_6667_INF__fold__inf, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ B),F2:(B > A)]: ((((finite_finite2 @ B) @ A3) => (((complete_Inf_Inf @ A) @ ((((image @ B) @ A) @ F2) @ A3)) = (((((finite_fold @ B) @ A) @ (((((comp @ A) @ (A > A)) @ B) @ (inf_inf @ A)) @ F2)) @ (top_top @ A)) @ A3))))))))). % 6.67/2.70 thf(fact_6668_If__the__inv__into__in__Func, axiom, (! [B:$tType,A:$tType,G:(A > B),C5:(set @ A),B3:(set @ A),X3:A]: ((((((inj_on @ A) @ B) @ G) @ C5) => ((((ord_less_eq @ (set @ A)) @ C5) @ (((sup_sup @ (set @ A)) @ B3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) => (((member @ (B > A)) @ (^ [I3:B]: (((((if @ A) @ (((member @ B) @ I3) @ ((((image @ A) @ B) @ G) @ C5))) @ (((((the_inv_into @ A) @ B) @ C5) @ G) @ I3)) @ X3)))) @ ((((bNF_Wellorder_Func @ B) @ A) @ (top_top @ (set @ B))) @ (((sup_sup @ (set @ A)) @ B3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))))). % 6.67/2.70 thf(fact_6669_min__weak__def, axiom, (fun_min_weak = (((sup_sup @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))) @ ((min_ext @ ((product_prod @ nat) @ nat)) @ fun_pair_leq)) @ (((insert @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat)))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat))))) @ (bot_bot @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))))))). % 6.67/2.70 thf(fact_6670_Ball__fold, axiom, (! [A:$tType,A3:(set @ A),P:(A > $o)]: ((((finite_finite2 @ A) @ A3) => ((! [X:A]: (((((member @ A) @ X) @ A3) => (P @ X)))) = (((((finite_fold @ A) @ $o) @ (^ [K3:A,S8:$o]: ((S8 & (P @ K3))))) @ $true) @ A3)))))). % 6.67/2.70 thf(fact_6671_sup__int__def, axiom, ((sup_sup @ int) = (ord_max @ int))). % 6.67/2.70 thf(fact_6672_sup__set__def, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A6:(set @ A),B6:(set @ A)]: (((collect @ A) @ (((sup_sup @ (A > $o)) @ (^ [X:A]: ((((member @ A) @ X) @ A6)))) @ (^ [X:A]: ((((member @ A) @ X) @ B6))))))))))). % 6.67/2.70 thf(fact_6673_sup__nat__def, axiom, ((sup_sup @ nat) = (ord_max @ nat))). % 6.67/2.70 thf(fact_6674_sup__Un__eq2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: (((((sup_sup @ (A > (B > $o))) @ (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ R)))) @ (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ S3)))) = (^ [X:A,Y2:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ (((sup_sup @ (set @ ((product_prod @ A) @ B))) @ R) @ S3)))))))). % 6.67/2.70 thf(fact_6675_union__fold__insert, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((((sup_sup @ (set @ A)) @ A3) @ B3) = (((((finite_fold @ A) @ (set @ A)) @ (insert @ A)) @ B3) @ A3)))))). % 6.67/2.70 thf(fact_6676_card_Oeq__fold, axiom, (! [A:$tType]: (((finite_card @ A) = ((((finite_fold @ A) @ nat) @ (^ [Uu3:A]: (suc))) @ (zero_zero @ nat)))))). % 6.67/2.70 thf(fact_6677_sorted__list__of__set_Ofold__insort__key_Oeq__fold, axiom, (! [A:$tType]: (((linorder @ A) => ((linord4507533701916653071of_set @ A) = ((((finite_fold @ A) @ (list @ A)) @ (((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X)))) @ (nil @ A))))))). % 6.67/2.70 thf(fact_6678_image__fold__insert, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F2:(A > B)]: ((((finite_finite2 @ A) @ A3) => (((((image @ A) @ B) @ F2) @ A3) = (((((finite_fold @ A) @ (set @ B)) @ (^ [K3:A]: (((insert @ B) @ (F2 @ K3))))) @ (bot_bot @ (set @ B))) @ A3)))))). % 6.67/2.70 thf(fact_6679_Id__on__fold, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (((id_on @ A) @ A3) = (((((finite_fold @ A) @ (set @ ((product_prod @ A) @ A))) @ (^ [X:A]: (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ X))))) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))) @ A3)))))). % 6.67/2.70 thf(fact_6680_fold__union__pair, axiom, (! [B:$tType,A:$tType,B3:(set @ A),X3:B,A3:(set @ ((product_prod @ B) @ A))]: ((((finite_finite2 @ A) @ B3) => ((((sup_sup @ (set @ ((product_prod @ B) @ A))) @ ((complete_Sup_Sup @ (set @ ((product_prod @ B) @ A))) @ ((((image @ A) @ (set @ ((product_prod @ B) @ A))) @ (^ [Y2:A]: ((((insert @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X3) @ Y2)) @ (bot_bot @ (set @ ((product_prod @ B) @ A))))))) @ B3))) @ A3) = (((((finite_fold @ A) @ (set @ ((product_prod @ B) @ A))) @ (^ [Y2:A]: (((insert @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X3) @ Y2))))) @ A3) @ B3)))))). % 6.67/2.70 thf(fact_6681_Pow__fold, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (((pow2 @ A) @ A3) = (((((finite_fold @ A) @ (set @ (set @ A))) @ (^ [X:A,A6:(set @ (set @ A))]: ((((sup_sup @ (set @ (set @ A))) @ A6) @ ((((image @ (set @ A)) @ (set @ A)) @ ((insert @ A) @ X)) @ A6))))) @ (((insert @ (set @ A)) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ (set @ A))))) @ A3)))))). % 6.67/2.70 thf(fact_6682_max__weak__def, axiom, (fun_max_weak = (((sup_sup @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))) @ ((max_ext @ ((product_prod @ nat) @ nat)) @ fun_pair_leq)) @ (((insert @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat)))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat))))) @ (bot_bot @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))))))). % 6.67/2.70 thf(fact_6683_Func__map__surj, axiom, (! [C:$tType,A:$tType,D:$tType,B:$tType,F1:(B > A),A17:(set @ B),B1:(set @ A),F22:(C > D),B22:(set @ C),A26:(set @ D)]: (((((((image @ B) @ A) @ F1) @ A17) = B1) => (((((inj_on @ C) @ D) @ F22) @ B22) => ((((ord_less_eq @ (set @ D)) @ ((((image @ C) @ D) @ F22) @ B22)) @ A26) => (((B22 = (bot_bot @ (set @ C))) => (A26 = (bot_bot @ (set @ D)))) => (((((bNF_Wellorder_Func @ C) @ A) @ B22) @ B1) = ((((image @ (D > B)) @ (C > A)) @ (((((((bNF_We4925052301507509544nc_map @ C) @ B) @ A) @ D) @ B22) @ F1) @ F22)) @ ((((bNF_Wellorder_Func @ D) @ B) @ A26) @ A17)))))))))). % 6.67/2.70 thf(fact_6684_max__ext__additive, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),R:(set @ ((product_prod @ A) @ A)),C5:(set @ A),D6:(set @ A)]: (((((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ A3) @ B3)) @ ((max_ext @ A) @ R)) => ((((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ C5) @ D6)) @ ((max_ext @ A) @ R)) => (((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ C5)) @ (((sup_sup @ (set @ A)) @ B3) @ D6))) @ ((max_ext @ A) @ R))))))). % 6.67/2.70 thf(fact_6685_Func__map, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,G:(A > B),A26:(set @ A),A17:(set @ B),F1:(B > C),B1:(set @ C),F22:(D > A),B22:(set @ D)]: (((((member @ (A > B)) @ G) @ ((((bNF_Wellorder_Func @ A) @ B) @ A26) @ A17)) => ((((ord_less_eq @ (set @ C)) @ ((((image @ B) @ C) @ F1) @ A17)) @ B1) => ((((ord_less_eq @ (set @ A)) @ ((((image @ D) @ A) @ F22) @ B22)) @ A26) => (((member @ (D > C)) @ ((((((((bNF_We4925052301507509544nc_map @ D) @ B) @ C) @ A) @ B22) @ F1) @ F22) @ G)) @ ((((bNF_Wellorder_Func @ D) @ C) @ B22) @ B1)))))))). % 6.67/2.70 thf(fact_6686_max__ext_Ocases, axiom, (! [A:$tType,A12:(set @ A),A23:(set @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ A12) @ A23)) @ ((max_ext @ A) @ R)) => (~ ((((finite_finite2 @ A) @ A12) => (((finite_finite2 @ A) @ A23) => ((A23 != (bot_bot @ (set @ A))) => (~ ((! [X2:A]: (((((member @ A) @ X2) @ A12) => (? [Xa3:A]: (((((member @ A) @ Xa3) @ A23) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Xa3)) @ R)))))))))))))))))). % 6.67/2.70 thf(fact_6687_max__ext_Osimps, axiom, (! [A:$tType,A12:(set @ A),A23:(set @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ A12) @ A23)) @ ((max_ext @ A) @ R)) = (((finite_finite2 @ A) @ A12) & (((finite_finite2 @ A) @ A23) & ((A23 != (bot_bot @ (set @ A))) & (! [X:A]: (((((member @ A) @ X) @ A12) => (? [Y2:A]: (((((member @ A) @ Y2) @ A23) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y2)) @ R)))))))))))))). % 6.67/2.70 thf(fact_6688_max__ext_Omax__extI, axiom, (! [A:$tType,X8:(set @ A),Y7:(set @ A),R:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ A) @ X8) => (((finite_finite2 @ A) @ Y7) => ((Y7 != (bot_bot @ (set @ A))) => ((! [X5:A]: (((((member @ A) @ X5) @ X8) => (? [Xa:A]: (((((member @ A) @ Xa) @ Y7) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Xa)) @ R))))))) => (((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ X8) @ Y7)) @ ((max_ext @ A) @ R))))))))). % 6.67/2.70 thf(fact_6689_Func__non__emp, axiom, (! [A:$tType,B:$tType,B3:(set @ A),A3:(set @ B)]: (((B3 != (bot_bot @ (set @ A))) => (((((bNF_Wellorder_Func @ B) @ A) @ A3) @ B3) != (bot_bot @ (set @ (B > A)))))))). % 6.67/2.70 thf(fact_6690_Func__is__emp, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B3:(set @ B)]: (((((((bNF_Wellorder_Func @ A) @ B) @ A3) @ B3) = (bot_bot @ (set @ (A > B)))) = ((A3 != (bot_bot @ (set @ A))) & (B3 = (bot_bot @ (set @ B)))))))). % 6.67/2.70 thf(fact_6691_Set__filter__fold, axiom, (! [A:$tType,A3:(set @ A),P:(A > $o)]: ((((finite_finite2 @ A) @ A3) => ((((filter3 @ A) @ P) @ A3) = (((((finite_fold @ A) @ (set @ A)) @ (^ [X:A,A18:(set @ A)]: (((((if @ (set @ A)) @ (P @ X)) @ (((insert @ A) @ X) @ A18)) @ A18)))) @ (bot_bot @ (set @ A))) @ A3)))))). % 6.67/2.70 thf(fact_6692_Sup__fin_Oeq__fold_H, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((lattic5882676163264333800up_fin @ A) = (^ [A6:(set @ A)]: (((the2 @ A) @ (((((finite_fold @ A) @ (option @ A)) @ (^ [X:A,Y2:(option @ A)]: (((some @ A) @ (((((case_option @ A) @ A) @ X) @ ((sup_sup @ A) @ X)) @ Y2))))) @ (none @ A)) @ A6))))))))). % 6.67/2.70 thf(fact_6693_member__filter, axiom, (! [A:$tType,X3:A,P:(A > $o),A3:(set @ A)]: (((((member @ A) @ X3) @ (((filter3 @ A) @ P) @ A3)) = ((((member @ A) @ X3) @ A3) & (P @ X3)))))). % 6.67/2.70 thf(fact_6694_inf__Sup__absorb, axiom, (! [A:$tType]: (((lattice @ A) => (! [A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A2) @ A3) => ((((inf_inf @ A) @ A2) @ ((lattic5882676163264333800up_fin @ A) @ A3)) = A2))))))))). % 6.67/2.70 thf(fact_6695_Sup__fin_Osingleton, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A]: ((((lattic5882676163264333800up_fin @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = X3))))))). % 6.67/2.70 thf(fact_6696_Sup__fin_Oinsert, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ (((insert @ A) @ X3) @ A3)) = (((sup_sup @ A) @ X3) @ ((lattic5882676163264333800up_fin @ A) @ A3))))))))))). % 6.67/2.70 thf(fact_6697_Sup__fin_Oin__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => ((((sup_sup @ A) @ X3) @ ((lattic5882676163264333800up_fin @ A) @ A3)) = ((lattic5882676163264333800up_fin @ A) @ A3)))))))))). % 6.67/2.70 thf(fact_6698_Sup__fin_OcoboundedI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A2) @ A3) => (((ord_less_eq @ A) @ A2) @ ((lattic5882676163264333800up_fin @ A) @ A3)))))))))). % 6.67/2.70 thf(fact_6699_Sup__fin__Max, axiom, (! [A:$tType]: ((((semilattice_sup @ A) & (linorder @ A)) => ((lattic5882676163264333800up_fin @ A) = (lattic643756798349783984er_Max @ A)))))). % 6.67/2.70 thf(fact_6700_Set_Ofilter__def, axiom, (! [A:$tType]: (((filter3 @ A) = (^ [P3:(A > $o),A6:(set @ A)]: (((collect @ A) @ (^ [A4:A]: (((((member @ A) @ A4) @ A6) & (P3 @ A4))))))))))). % 6.67/2.70 thf(fact_6701_finite__filter, axiom, (! [A:$tType,S3:(set @ A),P:(A > $o)]: ((((finite_finite2 @ A) @ S3) => ((finite_finite2 @ A) @ (((filter3 @ A) @ P) @ S3)))))). % 6.67/2.70 thf(fact_6702_Sup__fin_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ ((lattic5882676163264333800up_fin @ A) @ A3)) @ X3) = (! [X:A]: (((((member @ A) @ X) @ A3) => (((ord_less_eq @ A) @ X) @ X3))))))))))))). % 6.67/2.70 thf(fact_6703_Sup__fin_OboundedI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((! [A5:A]: (((((member @ A) @ A5) @ A3) => (((ord_less_eq @ A) @ A5) @ X3)))) => (((ord_less_eq @ A) @ ((lattic5882676163264333800up_fin @ A) @ A3)) @ X3)))))))))). % 6.67/2.70 thf(fact_6704_Sup__fin_OboundedE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ ((lattic5882676163264333800up_fin @ A) @ A3)) @ X3) => (! [A11:A]: (((((member @ A) @ A11) @ A3) => (((ord_less_eq @ A) @ A11) @ X3))))))))))))). % 6.67/2.70 thf(fact_6705_cSup__eq__Sup__fin, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A)]: ((((finite_finite2 @ A) @ X8) => ((X8 != (bot_bot @ (set @ A))) => (((complete_Sup_Sup @ A) @ X8) = ((lattic5882676163264333800up_fin @ A) @ X8)))))))))). % 6.67/2.70 thf(fact_6706_Sup__fin__Sup, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ A3) = ((complete_Sup_Sup @ A) @ A3)))))))))). % 6.67/2.70 thf(fact_6707_Sup__fin_Oinfinite, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:(set @ A)]: (((~ (((finite_finite2 @ A) @ A3))) => (((lattic5882676163264333800up_fin @ A) @ A3) = ((the2 @ A) @ (none @ A)))))))))). % 6.67/2.70 thf(fact_6708_Sup__fin_Osubset__imp, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((A3 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => (((ord_less_eq @ A) @ ((lattic5882676163264333800up_fin @ A) @ A3)) @ ((lattic5882676163264333800up_fin @ A) @ B3))))))))))). % 6.67/2.70 thf(fact_6709_Sup__fin_Ohom__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [H2:(A > A),N7:(set @ A)]: (((! [X5:A,Y4:A]: (((H2 @ (((sup_sup @ A) @ X5) @ Y4)) = (((sup_sup @ A) @ (H2 @ X5)) @ (H2 @ Y4))))) => (((finite_finite2 @ A) @ N7) => ((N7 != (bot_bot @ (set @ A))) => ((H2 @ ((lattic5882676163264333800up_fin @ A) @ N7)) = ((lattic5882676163264333800up_fin @ A) @ ((((image @ A) @ A) @ H2) @ N7)))))))))))). % 6.67/2.70 thf(fact_6710_Sup__fin_Osubset, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((B3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => ((((sup_sup @ A) @ ((lattic5882676163264333800up_fin @ A) @ B3)) @ ((lattic5882676163264333800up_fin @ A) @ A3)) = ((lattic5882676163264333800up_fin @ A) @ A3))))))))))). % 6.67/2.70 thf(fact_6711_Sup__fin_Oinsert__not__elem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((~ ((((member @ A) @ X3) @ A3))) => ((A3 != (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ (((insert @ A) @ X3) @ A3)) = (((sup_sup @ A) @ X3) @ ((lattic5882676163264333800up_fin @ A) @ A3)))))))))))). % 6.67/2.70 thf(fact_6712_Sup__fin_Oclosed, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((! [X5:A,Y4:A]: ((((member @ A) @ (((sup_sup @ A) @ X5) @ Y4)) @ (((insert @ A) @ X5) @ (((insert @ A) @ Y4) @ (bot_bot @ (set @ A))))))) => (((member @ A) @ ((lattic5882676163264333800up_fin @ A) @ A3)) @ A3)))))))))). % 6.67/2.70 thf(fact_6713_Sup__fin_Ounion, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => ((B3 != (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((sup_sup @ A) @ ((lattic5882676163264333800up_fin @ A) @ A3)) @ ((lattic5882676163264333800up_fin @ A) @ B3))))))))))))). % 6.67/2.70 thf(fact_6714_Sup__fin_Oeq__fold, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => (((lattic5882676163264333800up_fin @ A) @ (((insert @ A) @ X3) @ A3)) = (((((finite_fold @ A) @ A) @ (sup_sup @ A)) @ X3) @ A3))))))))). % 6.67/2.70 thf(fact_6715_inter__Set__filter, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((inf_inf @ (set @ A)) @ A3) @ B3) = (((filter3 @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A3)))) @ B3)))))). % 6.67/2.70 thf(fact_6716_inf__Sup1__distrib, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((inf_inf @ A) @ X3) @ ((lattic5882676163264333800up_fin @ A) @ A3)) = ((lattic5882676163264333800up_fin @ A) @ ((collect @ A) @ (^ [Uu3:A]: ((? [A4:A]: (((Uu3 = (((inf_inf @ A) @ X3) @ A4)) & (((member @ A) @ A4) @ A3))))))))))))))))). % 6.67/2.70 thf(fact_6717_inf__Sup2__distrib, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => ((B3 != (bot_bot @ (set @ A))) => ((((inf_inf @ A) @ ((lattic5882676163264333800up_fin @ A) @ A3)) @ ((lattic5882676163264333800up_fin @ A) @ B3)) = ((lattic5882676163264333800up_fin @ A) @ ((collect @ A) @ (^ [Uu3:A]: ((? [A4:A,B4:A]: (((Uu3 = (((inf_inf @ A) @ A4) @ B4)) & ((((member @ A) @ A4) @ A3) & (((member @ A) @ B4) @ B3)))))))))))))))))))). % 6.67/2.70 thf(fact_6718_Sup__fin_Oinsert__remove, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ (((insert @ A) @ X3) @ A3)) = X3)) & (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ (((insert @ A) @ X3) @ A3)) = (((sup_sup @ A) @ X3) @ ((lattic5882676163264333800up_fin @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))))))). % 6.67/2.70 thf(fact_6719_Sup__fin_Oremove, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => ((((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ A3) = X3)) & (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic5882676163264333800up_fin @ A) @ A3) = (((sup_sup @ A) @ X3) @ ((lattic5882676163264333800up_fin @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))))))))))). % 6.67/2.70 thf(fact_6720_Inf__fin_Oeq__fold_H, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((lattic7752659483105999362nf_fin @ A) = (^ [A6:(set @ A)]: (((the2 @ A) @ (((((finite_fold @ A) @ (option @ A)) @ (^ [X:A,Y2:(option @ A)]: (((some @ A) @ (((((case_option @ A) @ A) @ X) @ ((inf_inf @ A) @ X)) @ Y2))))) @ (none @ A)) @ A6))))))))). % 6.67/2.70 thf(fact_6721_max__ext__def, axiom, (! [A:$tType]: (((max_ext @ A) = (^ [R6:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_case_prod @ (set @ A)) @ (set @ A)) @ $o) @ ((max_extp @ A) @ (^ [X:A,Y2:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y2)) @ R6)))))))))))). % 6.67/2.70 thf(fact_6722_Inf__fin_Osingleton, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X3:A]: ((((lattic7752659483105999362nf_fin @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = X3))))))). % 6.67/2.70 thf(fact_6723_sup__Inf__absorb, axiom, (! [A:$tType]: (((lattice @ A) => (! [A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A2) @ A3) => ((((sup_sup @ A) @ ((lattic7752659483105999362nf_fin @ A) @ A3)) @ A2) = A2))))))))). % 6.67/2.70 thf(fact_6724_Inf__fin_Oinsert, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ (((insert @ A) @ X3) @ A3)) = (((inf_inf @ A) @ X3) @ ((lattic7752659483105999362nf_fin @ A) @ A3))))))))))). % 6.67/2.70 thf(fact_6725_Inf__fin_OcoboundedI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A2) @ A3) => (((ord_less_eq @ A) @ ((lattic7752659483105999362nf_fin @ A) @ A3)) @ A2))))))))). % 6.67/2.70 thf(fact_6726_Inf__fin_Oin__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => ((((inf_inf @ A) @ X3) @ ((lattic7752659483105999362nf_fin @ A) @ A3)) = ((lattic7752659483105999362nf_fin @ A) @ A3)))))))))). % 6.67/2.70 thf(fact_6727_Inf__fin_OboundedE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ X3) @ ((lattic7752659483105999362nf_fin @ A) @ A3)) => (! [A11:A]: (((((member @ A) @ A11) @ A3) => (((ord_less_eq @ A) @ X3) @ A11))))))))))))). % 6.67/2.70 thf(fact_6728_Inf__fin_OboundedI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((! [A5:A]: (((((member @ A) @ A5) @ A3) => (((ord_less_eq @ A) @ X3) @ A5)))) => (((ord_less_eq @ A) @ X3) @ ((lattic7752659483105999362nf_fin @ A) @ A3))))))))))). % 6.67/2.70 thf(fact_6729_Inf__fin_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ X3) @ ((lattic7752659483105999362nf_fin @ A) @ A3)) = (! [X:A]: (((((member @ A) @ X) @ A3) => (((ord_less_eq @ A) @ X3) @ X))))))))))))). % 6.67/2.70 thf(fact_6730_Inf__fin__Inf, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ A3) = ((complete_Inf_Inf @ A) @ A3)))))))))). % 6.67/2.70 thf(fact_6731_cInf__eq__Inf__fin, axiom, (! [A:$tType]: (((condit1219197933456340205attice @ A) => (! [X8:(set @ A)]: ((((finite_finite2 @ A) @ X8) => ((X8 != (bot_bot @ (set @ A))) => (((complete_Inf_Inf @ A) @ X8) = ((lattic7752659483105999362nf_fin @ A) @ X8)))))))))). % 6.67/2.70 thf(fact_6732_Inf__fin_Oinfinite, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:(set @ A)]: (((~ (((finite_finite2 @ A) @ A3))) => (((lattic7752659483105999362nf_fin @ A) @ A3) = ((the2 @ A) @ (none @ A)))))))))). % 6.67/2.70 thf(fact_6733_Inf__fin_Osubset__imp, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((A3 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => (((ord_less_eq @ A) @ ((lattic7752659483105999362nf_fin @ A) @ B3)) @ ((lattic7752659483105999362nf_fin @ A) @ A3))))))))))). % 6.67/2.70 thf(fact_6734_Inf__fin_Ohom__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [H2:(A > A),N7:(set @ A)]: (((! [X5:A,Y4:A]: (((H2 @ (((inf_inf @ A) @ X5) @ Y4)) = (((inf_inf @ A) @ (H2 @ X5)) @ (H2 @ Y4))))) => (((finite_finite2 @ A) @ N7) => ((N7 != (bot_bot @ (set @ A))) => ((H2 @ ((lattic7752659483105999362nf_fin @ A) @ N7)) = ((lattic7752659483105999362nf_fin @ A) @ ((((image @ A) @ A) @ H2) @ N7)))))))))))). % 6.67/2.70 thf(fact_6735_Inf__fin_Osubset, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((B3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => ((((inf_inf @ A) @ ((lattic7752659483105999362nf_fin @ A) @ B3)) @ ((lattic7752659483105999362nf_fin @ A) @ A3)) = ((lattic7752659483105999362nf_fin @ A) @ A3))))))))))). % 6.67/2.70 thf(fact_6736_Inf__fin_Oclosed, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((! [X5:A,Y4:A]: ((((member @ A) @ (((inf_inf @ A) @ X5) @ Y4)) @ (((insert @ A) @ X5) @ (((insert @ A) @ Y4) @ (bot_bot @ (set @ A))))))) => (((member @ A) @ ((lattic7752659483105999362nf_fin @ A) @ A3)) @ A3)))))))))). % 6.67/2.70 thf(fact_6737_Inf__fin_Oinsert__not__elem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((~ ((((member @ A) @ X3) @ A3))) => ((A3 != (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ (((insert @ A) @ X3) @ A3)) = (((inf_inf @ A) @ X3) @ ((lattic7752659483105999362nf_fin @ A) @ A3)))))))))))). % 6.67/2.70 thf(fact_6738_Inf__fin_Ounion, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => ((B3 != (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((inf_inf @ A) @ ((lattic7752659483105999362nf_fin @ A) @ A3)) @ ((lattic7752659483105999362nf_fin @ A) @ B3))))))))))))). % 6.67/2.70 thf(fact_6739_Inf__fin__le__Sup__fin, axiom, (! [A:$tType]: (((lattice @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((ord_less_eq @ A) @ ((lattic7752659483105999362nf_fin @ A) @ A3)) @ ((lattic5882676163264333800up_fin @ A) @ A3)))))))))). % 6.67/2.70 thf(fact_6740_Inf__fin_Oeq__fold, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => (((lattic7752659483105999362nf_fin @ A) @ (((insert @ A) @ X3) @ A3)) = (((((finite_fold @ A) @ A) @ (inf_inf @ A)) @ X3) @ A3))))))))). % 6.67/2.70 thf(fact_6741_sup__Inf2__distrib, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => ((B3 != (bot_bot @ (set @ A))) => ((((sup_sup @ A) @ ((lattic7752659483105999362nf_fin @ A) @ A3)) @ ((lattic7752659483105999362nf_fin @ A) @ B3)) = ((lattic7752659483105999362nf_fin @ A) @ ((collect @ A) @ (^ [Uu3:A]: ((? [A4:A,B4:A]: (((Uu3 = (((sup_sup @ A) @ A4) @ B4)) & ((((member @ A) @ A4) @ A3) & (((member @ A) @ B4) @ B3)))))))))))))))))))). % 6.67/2.70 thf(fact_6742_sup__Inf1__distrib, axiom, (! [A:$tType]: (((distrib_lattice @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((sup_sup @ A) @ X3) @ ((lattic7752659483105999362nf_fin @ A) @ A3)) = ((lattic7752659483105999362nf_fin @ A) @ ((collect @ A) @ (^ [Uu3:A]: ((? [A4:A]: (((Uu3 = (((sup_sup @ A) @ X3) @ A4)) & (((member @ A) @ A4) @ A3))))))))))))))))). % 6.67/2.70 thf(fact_6743_Inf__fin_Oremove, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => ((((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ A3) = X3)) & (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ A3) = (((inf_inf @ A) @ X3) @ ((lattic7752659483105999362nf_fin @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))))))))))). % 6.67/2.70 thf(fact_6744_Inf__fin_Oinsert__remove, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ (((insert @ A) @ X3) @ A3)) = X3)) & (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic7752659483105999362nf_fin @ A) @ (((insert @ A) @ X3) @ A3)) = (((inf_inf @ A) @ X3) @ ((lattic7752659483105999362nf_fin @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))))))). % 6.67/2.70 thf(fact_6745_max__extp__max__ext__eq, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((max_extp @ A) @ (^ [X:A,Y2:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y2)) @ R)))) = (^ [X:(set @ A),Y2:(set @ A)]: ((((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ X) @ Y2)) @ ((max_ext @ A) @ R)))))))). % 6.67/2.70 thf(fact_6746_max__extp__eq, axiom, (! [A:$tType]: (((max_extp @ A) = (^ [R5:(A > (A > $o)),X:(set @ A),Y2:(set @ A)]: ((((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ X) @ Y2)) @ ((max_ext @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ R5)))))))))). % 6.67/2.70 thf(fact_6747_Min_Oeq__fold_H, axiom, (! [A:$tType]: (((linorder @ A) => ((lattic643756798350308766er_Min @ A) = (^ [A6:(set @ A)]: (((the2 @ A) @ (((((finite_fold @ A) @ (option @ A)) @ (^ [X:A,Y2:(option @ A)]: (((some @ A) @ (((((case_option @ A) @ A) @ X) @ ((ord_min @ A) @ X)) @ Y2))))) @ (none @ A)) @ A6))))))))). % 6.67/2.70 thf(fact_6748_insert__relcomp__union__fold, axiom, (! [C:$tType,B:$tType,A:$tType,S3:(set @ ((product_prod @ A) @ B)),X3:((product_prod @ C) @ A),X8:(set @ ((product_prod @ C) @ B))]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ S3) => ((((sup_sup @ (set @ ((product_prod @ C) @ B))) @ (((((relcomp @ C) @ A) @ B) @ (((insert @ ((product_prod @ C) @ A)) @ X3) @ (bot_bot @ (set @ ((product_prod @ C) @ A))))) @ S3)) @ X8) = (((((finite_fold @ ((product_prod @ A) @ B)) @ (set @ ((product_prod @ C) @ B))) @ ((((product_case_prod @ A) @ B) @ ((set @ ((product_prod @ C) @ B)) > (set @ ((product_prod @ C) @ B)))) @ (^ [W3:A,Z6:B,A18:(set @ ((product_prod @ C) @ B))]: (((((if @ (set @ ((product_prod @ C) @ B))) @ ((((product_snd @ C) @ A) @ X3) = W3)) @ (((insert @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ (((product_fst @ C) @ A) @ X3)) @ Z6)) @ A18)) @ A18))))) @ X8) @ S3)))))). % 6.67/2.70 thf(fact_6749_relcomp__empty1, axiom, (! [C:$tType,B:$tType,A:$tType,R:(set @ ((product_prod @ C) @ B))]: (((((((relcomp @ A) @ C) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ C)))) @ R) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))). % 6.67/2.70 thf(fact_6750_relcomp__empty2, axiom, (! [C:$tType,B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ C))]: (((((((relcomp @ A) @ C) @ B) @ R) @ (bot_bot @ (set @ ((product_prod @ C) @ B)))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))). % 6.67/2.70 thf(fact_6751_finite__relcomp, axiom, (! [A:$tType,C:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ B) @ C))]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ R) => (((finite_finite2 @ ((product_prod @ B) @ C)) @ S3) => ((finite_finite2 @ ((product_prod @ A) @ C)) @ (((((relcomp @ A) @ B) @ C) @ R) @ S3))))))). % 6.67/2.70 thf(fact_6752_Min__singleton, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A]: ((((lattic643756798350308766er_Min @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = X3))))))). % 6.67/2.70 thf(fact_6753_Min_Obounded__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ X3) @ ((lattic643756798350308766er_Min @ A) @ A3)) = (! [X:A]: (((((member @ A) @ X) @ A3) => (((ord_less_eq @ A) @ X3) @ X))))))))))))). % 6.67/2.70 thf(fact_6754_Min__gr__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ X3) @ ((lattic643756798350308766er_Min @ A) @ A3)) = (! [X:A]: (((((member @ A) @ X) @ A3) => (((ord_less @ A) @ X3) @ X))))))))))))). % 6.67/2.70 thf(fact_6755_Min__const, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [A3:(set @ B),C2:A]: ((((finite_finite2 @ B) @ A3) => ((A3 != (bot_bot @ (set @ B))) => (((lattic643756798350308766er_Min @ A) @ ((((image @ B) @ A) @ (^ [Uu3:B]: (C2))) @ A3)) = C2))))))))). % 6.67/2.70 thf(fact_6756_Min__insert, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ (((insert @ A) @ X3) @ A3)) = (((ord_min @ A) @ X3) @ ((lattic643756798350308766er_Min @ A) @ A3))))))))))). % 6.67/2.70 thf(fact_6757_minus__Max__eq__Min, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [S3:(set @ A)]: ((((finite_finite2 @ A) @ S3) => ((S3 != (bot_bot @ (set @ A))) => (((uminus_uminus @ A) @ ((lattic643756798349783984er_Max @ A) @ S3)) = ((lattic643756798350308766er_Min @ A) @ ((((image @ A) @ A) @ (uminus_uminus @ A)) @ S3))))))))))). % 6.67/2.70 thf(fact_6758_minus__Min__eq__Max, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [S3:(set @ A)]: ((((finite_finite2 @ A) @ S3) => ((S3 != (bot_bot @ (set @ A))) => (((uminus_uminus @ A) @ ((lattic643756798350308766er_Min @ A) @ S3)) = ((lattic643756798349783984er_Max @ A) @ ((((image @ A) @ A) @ (uminus_uminus @ A)) @ S3))))))))))). % 6.67/2.70 thf(fact_6759_Inf__fin__Min, axiom, (! [A:$tType]: ((((semilattice_inf @ A) & (linorder @ A)) => ((lattic7752659483105999362nf_fin @ A) = (lattic643756798350308766er_Min @ A)))))). % 6.67/2.70 thf(fact_6760_relcomp_Ocases, axiom, (! [A:$tType,C:$tType,B:$tType,A12:A,A23:C,R2:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A12) @ A23)) @ (((((relcomp @ A) @ B) @ C) @ R2) @ S)) => (~ ((! [B5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A12) @ B5)) @ R2) => (~ ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B5) @ A23)) @ S)))))))))))). % 6.67/2.70 thf(fact_6761_relcomp_Osimps, axiom, (! [B:$tType,C:$tType,A:$tType,A12:A,A23:C,R2:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A12) @ A23)) @ (((((relcomp @ A) @ B) @ C) @ R2) @ S)) = (? [A4:A,B4:B,C4:C]: (((A12 = A4) & ((A23 = C4) & ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ R2) & (((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B4) @ C4)) @ S)))))))))). % 6.67/2.70 thf(fact_6762_relcomp_OrelcompI, axiom, (! [A:$tType,C:$tType,B:$tType,A2:A,B2:B,R2:(set @ ((product_prod @ A) @ B)),C2:C,S:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R2) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B2) @ C2)) @ S) => (((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A2) @ C2)) @ (((((relcomp @ A) @ B) @ C) @ R2) @ S))))))). % 6.67/2.70 thf(fact_6763_relcompE, axiom, (! [A:$tType,B:$tType,C:$tType,Xz:((product_prod @ A) @ B),R2:(set @ ((product_prod @ A) @ C)),S:(set @ ((product_prod @ C) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ Xz) @ (((((relcomp @ A) @ C) @ B) @ R2) @ S)) => (~ ((! [X5:A,Y4:C,Z3:B]: (((Xz = ((((product_Pair @ A) @ B) @ X5) @ Z3)) => ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X5) @ Y4)) @ R2) => (~ ((((member @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ Y4) @ Z3)) @ S))))))))))))). % 6.67/2.70 thf(fact_6764_relcompEpair, axiom, (! [A:$tType,B:$tType,C:$tType,A2:A,C2:B,R2:(set @ ((product_prod @ A) @ C)),S:(set @ ((product_prod @ C) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ C2)) @ (((((relcomp @ A) @ C) @ B) @ R2) @ S)) => (~ ((! [B5:C]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A2) @ B5)) @ R2) => (~ ((((member @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ B5) @ C2)) @ S)))))))))))). % 6.67/2.70 thf(fact_6765_Min_OcoboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A2) @ A3) => (((ord_less_eq @ A) @ ((lattic643756798350308766er_Min @ A) @ A3)) @ A2))))))))). % 6.67/2.70 thf(fact_6766_Min__eqI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((! [Y4:A]: (((((member @ A) @ Y4) @ A3) => (((ord_less_eq @ A) @ X3) @ Y4)))) => ((((member @ A) @ X3) @ A3) => (((lattic643756798350308766er_Min @ A) @ A3) = X3)))))))))). % 6.67/2.70 thf(fact_6767_Min__le, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => (((ord_less_eq @ A) @ ((lattic643756798350308766er_Min @ A) @ A3)) @ X3))))))))). % 6.67/2.70 thf(fact_6768_relpow__add, axiom, (! [A:$tType,M:nat,N2:nat,R:(set @ ((product_prod @ A) @ A))]: (((((compow @ (set @ ((product_prod @ A) @ A))) @ (((plus_plus @ nat) @ M) @ N2)) @ R) = (((((relcomp @ A) @ A) @ A) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ M) @ R)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N2) @ R)))))). % 6.67/2.70 thf(fact_6769_relpow_Osimps_I2_J, axiom, (! [A:$tType,N2:nat,R:(set @ ((product_prod @ A) @ A))]: (((((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N2)) @ R) = (((((relcomp @ A) @ A) @ A) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N2) @ R)) @ R))))). % 6.67/2.70 thf(fact_6770_Min_Oin__idem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => ((((ord_min @ A) @ X3) @ ((lattic643756798350308766er_Min @ A) @ A3)) = ((lattic643756798350308766er_Min @ A) @ A3)))))))))). % 6.67/2.70 thf(fact_6771_Min__in, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((member @ A) @ ((lattic643756798350308766er_Min @ A) @ A3)) @ A3))))))))). % 6.67/2.70 thf(fact_6772_union__comp__emptyL, axiom, (! [A:$tType,A3:(set @ ((product_prod @ A) @ A)),C5:(set @ ((product_prod @ A) @ A)),B3:(set @ ((product_prod @ A) @ A))]: ((((((((relcomp @ A) @ A) @ A) @ A3) @ C5) = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (((((((relcomp @ A) @ A) @ A) @ B3) @ C5) = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => ((((((relcomp @ A) @ A) @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ A3) @ B3)) @ C5) = (bot_bot @ (set @ ((product_prod @ A) @ A))))))))). % 6.67/2.70 thf(fact_6773_union__comp__emptyR, axiom, (! [A:$tType,A3:(set @ ((product_prod @ A) @ A)),B3:(set @ ((product_prod @ A) @ A)),C5:(set @ ((product_prod @ A) @ A))]: ((((((((relcomp @ A) @ A) @ A) @ A3) @ B3) = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (((((((relcomp @ A) @ A) @ A) @ A3) @ C5) = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => ((((((relcomp @ A) @ A) @ A) @ A3) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ B3) @ C5)) = (bot_bot @ (set @ ((product_prod @ A) @ A))))))))). % 6.67/2.70 thf(fact_6774_Min_OboundedI, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((! [A5:A]: (((((member @ A) @ A5) @ A3) => (((ord_less_eq @ A) @ X3) @ A5)))) => (((ord_less_eq @ A) @ X3) @ ((lattic643756798350308766er_Min @ A) @ A3))))))))))). % 6.67/2.70 thf(fact_6775_Min_OboundedE, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ X3) @ ((lattic643756798350308766er_Min @ A) @ A3)) => (! [A11:A]: (((((member @ A) @ A11) @ A3) => (((ord_less_eq @ A) @ X3) @ A11))))))))))))). % 6.67/2.70 thf(fact_6776_eq__Min__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),M:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((M = ((lattic643756798350308766er_Min @ A) @ A3)) = ((((member @ A) @ M) @ A3) & (! [X:A]: (((((member @ A) @ X) @ A3) => (((ord_less_eq @ A) @ M) @ X)))))))))))))). % 6.67/2.70 thf(fact_6777_Min__le__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ A) @ ((lattic643756798350308766er_Min @ A) @ A3)) @ X3) = (? [X:A]: (((((member @ A) @ X) @ A3) & (((ord_less_eq @ A) @ X) @ X3))))))))))))). % 6.67/2.70 thf(fact_6778_Min__eq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),M:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((lattic643756798350308766er_Min @ A) @ A3) = M) = ((((member @ A) @ M) @ A3) & (! [X:A]: (((((member @ A) @ X) @ A3) => (((ord_less_eq @ A) @ M) @ X)))))))))))))). % 6.67/2.70 thf(fact_6779_Min__less__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((((ord_less @ A) @ ((lattic643756798350308766er_Min @ A) @ A3)) @ X3) = (? [X:A]: (((((member @ A) @ X) @ A3) & (((ord_less @ A) @ X) @ X3))))))))))))). % 6.67/2.70 thf(fact_6780_Min__insert2, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((! [B5:A]: (((((member @ A) @ B5) @ A3) => (((ord_less_eq @ A) @ A2) @ B5)))) => (((lattic643756798350308766er_Min @ A) @ (((insert @ A) @ A2) @ A3)) = A2))))))))). % 6.67/2.70 thf(fact_6781_cInf__eq__Min, axiom, (! [A:$tType]: (((condit6923001295902523014norder @ A) => (! [X8:(set @ A)]: ((((finite_finite2 @ A) @ X8) => ((X8 != (bot_bot @ (set @ A))) => (((complete_Inf_Inf @ A) @ X8) = ((lattic643756798350308766er_Min @ A) @ X8)))))))))). % 6.67/2.70 thf(fact_6782_Min__Inf, axiom, (! [A:$tType]: (((comple5582772986160207858norder @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ A3) = ((complete_Inf_Inf @ A) @ A3)))))))))). % 6.67/2.70 thf(fact_6783_relcomp__unfold, axiom, (! [B:$tType,C:$tType,A:$tType]: (((((relcomp @ A) @ C) @ B) = (^ [R5:(set @ ((product_prod @ A) @ C)),S8:(set @ ((product_prod @ C) @ B))]: (((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X:A,Z6:B]: ((? [Y2:C]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X) @ Y2)) @ R5) & (((member @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ Y2) @ Z6)) @ S8)))))))))))))). % 6.67/2.70 thf(fact_6784_Min_Oinfinite, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: (((~ (((finite_finite2 @ A) @ A3))) => (((lattic643756798350308766er_Min @ A) @ A3) = ((the2 @ A) @ (none @ A)))))))))). % 6.67/2.70 thf(fact_6785_Min_Osubset__imp, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((A3 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => (((ord_less_eq @ A) @ ((lattic643756798350308766er_Min @ A) @ B3)) @ ((lattic643756798350308766er_Min @ A) @ A3))))))))))). % 6.67/2.70 thf(fact_6786_Min__antimono, axiom, (! [A:$tType]: (((linorder @ A) => (! [M7:(set @ A),N7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ M7) @ N7) => ((M7 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ N7) => (((ord_less_eq @ A) @ ((lattic643756798350308766er_Min @ A) @ N7)) @ ((lattic643756798350308766er_Min @ A) @ M7))))))))))). % 6.67/2.70 thf(fact_6787_max__ext__compat, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((((relcomp @ A) @ A) @ A) @ R) @ S3)) @ R) => (((ord_less_eq @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (((((relcomp @ (set @ A)) @ (set @ A)) @ (set @ A)) @ ((max_ext @ A) @ R)) @ (((sup_sup @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ ((max_ext @ A) @ S3)) @ (((insert @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ A)))) @ (bot_bot @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))))))) @ ((max_ext @ A) @ R)))))). % 6.67/2.70 thf(fact_6788_hom__Min__commute, axiom, (! [A:$tType]: (((linorder @ A) => (! [H2:(A > A),N7:(set @ A)]: (((! [X5:A,Y4:A]: (((H2 @ (((ord_min @ A) @ X5) @ Y4)) = (((ord_min @ A) @ (H2 @ X5)) @ (H2 @ Y4))))) => (((finite_finite2 @ A) @ N7) => ((N7 != (bot_bot @ (set @ A))) => ((H2 @ ((lattic643756798350308766er_Min @ A) @ N7)) = ((lattic643756798350308766er_Min @ A) @ ((((image @ A) @ A) @ H2) @ N7)))))))))))). % 6.67/2.70 thf(fact_6789_min__ext__compat, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((((relcomp @ A) @ A) @ A) @ R) @ S3)) @ R) => (((ord_less_eq @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (((((relcomp @ (set @ A)) @ (set @ A)) @ (set @ A)) @ ((min_ext @ A) @ R)) @ (((sup_sup @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ ((min_ext @ A) @ S3)) @ (((insert @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ A)))) @ (bot_bot @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))))))) @ ((min_ext @ A) @ R)))))). % 6.67/2.70 thf(fact_6790_Min_Osubset, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((B3 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => ((((ord_min @ A) @ ((lattic643756798350308766er_Min @ A) @ B3)) @ ((lattic643756798350308766er_Min @ A) @ A3)) = ((lattic643756798350308766er_Min @ A) @ A3))))))))))). % 6.67/2.70 thf(fact_6791_Min_Oclosed, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((! [X5:A,Y4:A]: ((((member @ A) @ (((ord_min @ A) @ X5) @ Y4)) @ (((insert @ A) @ X5) @ (((insert @ A) @ Y4) @ (bot_bot @ (set @ A))))))) => (((member @ A) @ ((lattic643756798350308766er_Min @ A) @ A3)) @ A3)))))))))). % 6.67/2.70 thf(fact_6792_Min_Oinsert__not__elem, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((~ ((((member @ A) @ X3) @ A3))) => ((A3 != (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ (((insert @ A) @ X3) @ A3)) = (((ord_min @ A) @ X3) @ ((lattic643756798350308766er_Min @ A) @ A3)))))))))))). % 6.67/2.70 thf(fact_6793_mono__Min__commute, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (linorder @ B)) => (! [F2:(A > B),A3:(set @ A)]: (((((order_mono @ A) @ B) @ F2) => (((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => ((F2 @ ((lattic643756798350308766er_Min @ A) @ A3)) = ((lattic643756798350308766er_Min @ B) @ ((((image @ A) @ B) @ F2) @ A3)))))))))))). % 6.67/2.70 thf(fact_6794_Min_Ounion, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B3) => ((B3 != (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((ord_min @ A) @ ((lattic643756798350308766er_Min @ A) @ A3)) @ ((lattic643756798350308766er_Min @ A) @ B3))))))))))))). % 6.67/2.70 thf(fact_6795_Min_Oeq__fold, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => (((lattic643756798350308766er_Min @ A) @ (((insert @ A) @ X3) @ A3)) = (((((finite_fold @ A) @ A) @ (ord_min @ A)) @ X3) @ A3))))))))). % 6.67/2.70 thf(fact_6796_Min__add__commute, axiom, (! [B:$tType,A:$tType]: (((linord4140545234300271783up_add @ A) => (! [S3:(set @ B),F2:(B > A),K:A]: ((((finite_finite2 @ B) @ S3) => ((S3 != (bot_bot @ (set @ B))) => (((lattic643756798350308766er_Min @ A) @ ((((image @ B) @ A) @ (^ [X:B]: ((((plus_plus @ A) @ (F2 @ X)) @ K)))) @ S3)) = (((plus_plus @ A) @ ((lattic643756798350308766er_Min @ A) @ ((((image @ B) @ A) @ F2) @ S3))) @ K)))))))))). % 6.67/2.70 thf(fact_6797_relcomp__fold, axiom, (! [C:$tType,B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ B) @ C))]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ R) => (((finite_finite2 @ ((product_prod @ B) @ C)) @ S3) => ((((((relcomp @ A) @ B) @ C) @ R) @ S3) = (((((finite_fold @ ((product_prod @ A) @ B)) @ (set @ ((product_prod @ A) @ C))) @ ((((product_case_prod @ A) @ B) @ ((set @ ((product_prod @ A) @ C)) > (set @ ((product_prod @ A) @ C)))) @ (^ [X:A,Y2:B,A6:(set @ ((product_prod @ A) @ C))]: ((((((finite_fold @ ((product_prod @ B) @ C)) @ (set @ ((product_prod @ A) @ C))) @ ((((product_case_prod @ B) @ C) @ ((set @ ((product_prod @ A) @ C)) > (set @ ((product_prod @ A) @ C)))) @ (^ [W3:B,Z6:C,A18:(set @ ((product_prod @ A) @ C))]: (((((if @ (set @ ((product_prod @ A) @ C))) @ (Y2 = W3)) @ (((insert @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X) @ Z6)) @ A18)) @ A18))))) @ A6) @ S3))))) @ (bot_bot @ (set @ ((product_prod @ A) @ C)))) @ R))))))). % 6.67/2.70 thf(fact_6798_Min_Oinsert__remove, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ (((insert @ A) @ X3) @ A3)) = X3)) & (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ (((insert @ A) @ X3) @ A3)) = (((ord_min @ A) @ X3) @ ((lattic643756798350308766er_Min @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))))))). % 6.67/2.70 thf(fact_6799_Min_Oremove, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),X3:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => ((((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ A3) = X3)) & (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) != (bot_bot @ (set @ A))) => (((lattic643756798350308766er_Min @ A) @ A3) = (((ord_min @ A) @ X3) @ ((lattic643756798350308766er_Min @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))))))))))). % 6.67/2.70 thf(fact_6800_insert__relcomp__fold, axiom, (! [C:$tType,B:$tType,A:$tType,S3:(set @ ((product_prod @ A) @ B)),X3:((product_prod @ C) @ A),R:(set @ ((product_prod @ C) @ A))]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ S3) => ((((((relcomp @ C) @ A) @ B) @ (((insert @ ((product_prod @ C) @ A)) @ X3) @ R)) @ S3) = (((((finite_fold @ ((product_prod @ A) @ B)) @ (set @ ((product_prod @ C) @ B))) @ ((((product_case_prod @ A) @ B) @ ((set @ ((product_prod @ C) @ B)) > (set @ ((product_prod @ C) @ B)))) @ (^ [W3:A,Z6:B,A18:(set @ ((product_prod @ C) @ B))]: (((((if @ (set @ ((product_prod @ C) @ B))) @ ((((product_snd @ C) @ A) @ X3) = W3)) @ (((insert @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ (((product_fst @ C) @ A) @ X3)) @ Z6)) @ A18)) @ A18))))) @ (((((relcomp @ C) @ A) @ B) @ R) @ S3)) @ S3)))))). % 6.67/2.70 thf(fact_6801_card__Min__le__sum, axiom, (! [A:$tType,A3:(set @ A),F2:(A > nat)]: ((((finite_finite2 @ A) @ A3) => (((ord_less_eq @ nat) @ (((times_times @ nat) @ ((finite_card @ A) @ A3)) @ ((lattic643756798350308766er_Min @ nat) @ ((((image @ A) @ nat) @ F2) @ A3)))) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F2) @ A3)))))). % 6.67/2.70 thf(fact_6802_comp__fun__commute__relcomp__fold, axiom, (! [A:$tType,B:$tType,C:$tType,S3:(set @ ((product_prod @ A) @ B))]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ S3) => (((finite6289374366891150609ommute @ ((product_prod @ C) @ A)) @ (set @ ((product_prod @ C) @ B))) @ ((((product_case_prod @ C) @ A) @ ((set @ ((product_prod @ C) @ B)) > (set @ ((product_prod @ C) @ B)))) @ (^ [X:C,Y2:A,A6:(set @ ((product_prod @ C) @ B))]: ((((((finite_fold @ ((product_prod @ A) @ B)) @ (set @ ((product_prod @ C) @ B))) @ ((((product_case_prod @ A) @ B) @ ((set @ ((product_prod @ C) @ B)) > (set @ ((product_prod @ C) @ B)))) @ (^ [W3:A,Z6:B,A18:(set @ ((product_prod @ C) @ B))]: (((((if @ (set @ ((product_prod @ C) @ B))) @ (Y2 = W3)) @ (((insert @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ X) @ Z6)) @ A18)) @ A18))))) @ A6) @ S3))))))))). % 6.67/2.70 thf(fact_6803_dual__Max, axiom, (! [A:$tType]: (((linorder @ A) => (((lattices_Max @ A) @ (^ [X:A,Y2:A]: ((((ord_less_eq @ A) @ Y2) @ X)))) = (lattic643756798350308766er_Min @ A)))))). % 6.67/2.70 thf(fact_6804_linorder_OMax_Ocong, axiom, (! [A:$tType]: (((lattices_Max @ A) = (lattices_Max @ A))))). % 6.67/2.70 thf(fact_6805_comp__fun__commute__product__fold, axiom, (! [A:$tType,B:$tType,B3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => (((finite6289374366891150609ommute @ B) @ (set @ ((product_prod @ B) @ A))) @ (^ [X:B,Z6:(set @ ((product_prod @ B) @ A))]: ((((((finite_fold @ A) @ (set @ ((product_prod @ B) @ A))) @ (^ [Y2:A]: (((insert @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y2))))) @ Z6) @ B3)))))))). % 6.67/2.70 thf(fact_6806_f__arg__min__list__f, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [Xs:(list @ A),F2:(A > B)]: (((Xs != (nil @ A)) => ((F2 @ ((((arg_min_list @ A) @ B) @ F2) @ Xs)) = ((lattic643756798350308766er_Min @ B) @ ((((image @ A) @ B) @ F2) @ ((set2 @ A) @ Xs))))))))))). % 6.67/2.70 thf(fact_6807_UNION__fun__upd, axiom, (! [B:$tType,A:$tType,A3:(B > (set @ A)),I:B,B3:(set @ A),J4:(set @ B)]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (((((fun_upd @ B) @ (set @ A)) @ A3) @ I) @ B3)) @ J4)) = (((sup_sup @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ A3) @ (((minus_minus @ (set @ B)) @ J4) @ (((insert @ B) @ I) @ (bot_bot @ (set @ B))))))) @ ((((if @ (set @ A)) @ (((member @ B) @ I) @ J4)) @ B3) @ (bot_bot @ (set @ A)))))))). % 6.67/2.70 thf(fact_6808_finite__update__induct, axiom, (! [B:$tType,A:$tType,F2:(A > B),C2:B,P:((A > B) > $o)]: ((((finite_finite2 @ A) @ ((collect @ A) @ (^ [A4:A]: (((F2 @ A4) != C2))))) => ((P @ (^ [A4:A]: (C2))) => ((! [A5:A,B5:B,F4:(A > B)]: ((((finite_finite2 @ A) @ ((collect @ A) @ (^ [C4:A]: (((F4 @ C4) != C2))))) => (((F4 @ A5) = C2) => ((B5 != C2) => ((P @ F4) => (P @ (((((fun_upd @ A) @ B) @ F4) @ A5) @ B5)))))))) => (P @ F2))))))). % 6.67/2.70 thf(fact_6809_arg__min__list__in, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [Xs:(list @ A),F2:(A > B)]: (((Xs != (nil @ A)) => (((member @ A) @ ((((arg_min_list @ A) @ B) @ F2) @ Xs)) @ ((set2 @ A) @ Xs))))))))). % 6.67/2.70 thf(fact_6810_fun__upd__image, axiom, (! [A:$tType,B:$tType,X3:B,A3:(set @ B),F2:(B > A),Y:A]: ((((((member @ B) @ X3) @ A3) => (((((image @ B) @ A) @ (((((fun_upd @ B) @ A) @ F2) @ X3) @ Y)) @ A3) = (((insert @ A) @ Y) @ ((((image @ B) @ A) @ F2) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ X3) @ (bot_bot @ (set @ B)))))))) & ((~ ((((member @ B) @ X3) @ A3))) => (((((image @ B) @ A) @ (((((fun_upd @ B) @ A) @ F2) @ X3) @ Y)) @ A3) = ((((image @ B) @ A) @ F2) @ A3))))))). % 6.67/2.70 thf(fact_6811_comp__fun__commute__on_Ofold__set__union__disj, axiom, (! [B:$tType,A:$tType,S3:(set @ A),F2:(A > (B > B)),A3:(set @ A),B3:(set @ A),Z2:B]: ((((((finite4664212375090638736ute_on @ A) @ B) @ S3) @ F2) => ((((ord_less_eq @ (set @ A)) @ A3) @ S3) => ((((ord_less_eq @ (set @ A)) @ B3) @ S3) => (((finite_finite2 @ A) @ A3) => (((finite_finite2 @ A) @ B3) => (((((inf_inf @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))) => ((((((finite_fold @ A) @ B) @ F2) @ Z2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((((finite_fold @ A) @ B) @ F2) @ (((((finite_fold @ A) @ B) @ F2) @ Z2) @ A3)) @ B3))))))))))). % 6.67/2.70 thf(fact_6812_comp__fun__commute__on_Ofold__insert__remove, axiom, (! [B:$tType,A:$tType,S3:(set @ A),F2:(A > (B > B)),X3:A,A3:(set @ A),Z2:B]: ((((((finite4664212375090638736ute_on @ A) @ B) @ S3) @ F2) => ((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ S3) => (((finite_finite2 @ A) @ A3) => ((((((finite_fold @ A) @ B) @ F2) @ Z2) @ (((insert @ A) @ X3) @ A3)) = ((F2 @ X3) @ (((((finite_fold @ A) @ B) @ F2) @ Z2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))))))). % 6.67/2.70 thf(fact_6813_Finite__Set_Ofold__cong, axiom, (! [B:$tType,A:$tType,S3:(set @ A),F2:(A > (B > B)),G:(A > (B > B)),A3:(set @ A),S:B,T2:B,B3:(set @ A)]: ((((((finite4664212375090638736ute_on @ A) @ B) @ S3) @ F2) => (((((finite4664212375090638736ute_on @ A) @ B) @ S3) @ G) => ((((ord_less_eq @ (set @ A)) @ A3) @ S3) => (((finite_finite2 @ A) @ A3) => ((! [X5:A]: (((((member @ A) @ X5) @ A3) => ((F2 @ X5) = (G @ X5))))) => ((S = T2) => ((A3 = B3) => ((((((finite_fold @ A) @ B) @ F2) @ S) @ A3) = (((((finite_fold @ A) @ B) @ G) @ T2) @ B3)))))))))))). % 6.67/2.70 thf(fact_6814_comp__fun__commute__on_Ocomp__comp__fun__commute__on, axiom, (! [B:$tType,A:$tType,C:$tType,S3:(set @ A),F2:(A > (B > B)),G:(C > A),R:(set @ C)]: ((((((finite4664212375090638736ute_on @ A) @ B) @ S3) @ F2) => ((((ord_less_eq @ (set @ A)) @ ((((image @ C) @ A) @ G) @ (top_top @ (set @ C)))) @ S3) => ((((finite4664212375090638736ute_on @ C) @ B) @ R) @ (((((comp @ A) @ (B > B)) @ C) @ F2) @ G))))))). % 6.67/2.70 thf(fact_6815_comp__fun__commute__on_Ofold__insert, axiom, (! [B:$tType,A:$tType,S3:(set @ A),F2:(A > (B > B)),X3:A,A3:(set @ A),Z2:B]: ((((((finite4664212375090638736ute_on @ A) @ B) @ S3) @ F2) => ((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ S3) => (((finite_finite2 @ A) @ A3) => ((~ ((((member @ A) @ X3) @ A3))) => ((((((finite_fold @ A) @ B) @ F2) @ Z2) @ (((insert @ A) @ X3) @ A3)) = ((F2 @ X3) @ (((((finite_fold @ A) @ B) @ F2) @ Z2) @ A3)))))))))). % 6.67/2.70 thf(fact_6816_comp__fun__commute__on_Ofold__insert2, axiom, (! [B:$tType,A:$tType,S3:(set @ A),F2:(A > (B > B)),X3:A,A3:(set @ A),Z2:B]: ((((((finite4664212375090638736ute_on @ A) @ B) @ S3) @ F2) => ((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ S3) => (((finite_finite2 @ A) @ A3) => ((~ ((((member @ A) @ X3) @ A3))) => ((((((finite_fold @ A) @ B) @ F2) @ Z2) @ (((insert @ A) @ X3) @ A3)) = (((((finite_fold @ A) @ B) @ F2) @ ((F2 @ X3) @ Z2)) @ A3))))))))). % 6.67/2.70 thf(fact_6817_comp__fun__commute__on_Ofold__fun__left__comm, axiom, (! [B:$tType,A:$tType,S3:(set @ A),F2:(A > (B > B)),X3:A,A3:(set @ A),Z2:B]: ((((((finite4664212375090638736ute_on @ A) @ B) @ S3) @ F2) => ((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ S3) => (((finite_finite2 @ A) @ A3) => (((F2 @ X3) @ (((((finite_fold @ A) @ B) @ F2) @ Z2) @ A3)) = (((((finite_fold @ A) @ B) @ F2) @ ((F2 @ X3) @ Z2)) @ A3)))))))). % 6.67/2.70 thf(fact_6818_comp__fun__commute__on_Ofold__rec, axiom, (! [B:$tType,A:$tType,S3:(set @ A),F2:(A > (B > B)),A3:(set @ A),X3:A,Z2:B]: ((((((finite4664212375090638736ute_on @ A) @ B) @ S3) @ F2) => ((((ord_less_eq @ (set @ A)) @ A3) @ S3) => (((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => ((((((finite_fold @ A) @ B) @ F2) @ Z2) @ A3) = ((F2 @ X3) @ (((((finite_fold @ A) @ B) @ F2) @ Z2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))))). % 6.67/2.70 thf(fact_6819_image__map__upd, axiom, (! [B:$tType,A:$tType,X3:A,A3:(set @ A),M:(A > (option @ B)),Y:B]: (((~ ((((member @ A) @ X3) @ A3))) => (((((image @ A) @ (option @ B)) @ (((((fun_upd @ A) @ (option @ B)) @ M) @ X3) @ ((some @ B) @ Y))) @ A3) = ((((image @ A) @ (option @ B)) @ M) @ A3)))))). % 6.67/2.70 thf(fact_6820_finite__range__updI, axiom, (! [A:$tType,B:$tType,F2:(B > (option @ A)),A2:B,B2:A]: ((((finite_finite2 @ (option @ A)) @ ((((image @ B) @ (option @ A)) @ F2) @ (top_top @ (set @ B)))) => ((finite_finite2 @ (option @ A)) @ ((((image @ B) @ (option @ A)) @ (((((fun_upd @ B) @ (option @ A)) @ F2) @ A2) @ ((some @ A) @ B2))) @ (top_top @ (set @ B)))))))). % 6.67/2.70 thf(fact_6821_empty__upd__none, axiom, (! [B:$tType,A:$tType,X3:A]: (((((((fun_upd @ A) @ (option @ B)) @ (^ [X:A]: ((none @ B)))) @ X3) @ (none @ B)) = (^ [X:A]: ((none @ B))))))). % 6.67/2.70 thf(fact_6822_map__upd__Some__unfold, axiom, (! [B:$tType,A:$tType,M:(B > (option @ A)),A2:B,B2:A,X3:B,Y:A]: (((((((((fun_upd @ B) @ (option @ A)) @ M) @ A2) @ ((some @ A) @ B2)) @ X3) = ((some @ A) @ Y)) = (((X3 = A2) & (B2 = Y)) | ((X3 != A2) & ((M @ X3) = ((some @ A) @ Y)))))))). % 6.67/2.70 thf(fact_6823_map__upd__triv, axiom, (! [A:$tType,B:$tType,T2:(B > (option @ A)),K:B,X3:A]: ((((T2 @ K) = ((some @ A) @ X3)) => ((((((fun_upd @ B) @ (option @ A)) @ T2) @ K) @ ((some @ A) @ X3)) = T2))))). % 6.67/2.70 thf(fact_6824_map__upd__eqD1, axiom, (! [A:$tType,B:$tType,M:(A > (option @ B)),A2:A,X3:B,N2:(A > (option @ B)),Y:B]: ((((((((fun_upd @ A) @ (option @ B)) @ M) @ A2) @ ((some @ B) @ X3)) = (((((fun_upd @ A) @ (option @ B)) @ N2) @ A2) @ ((some @ B) @ Y))) => (X3 = Y))))). % 6.67/2.70 thf(fact_6825_map__upd__nonempty, axiom, (! [B:$tType,A:$tType,T2:(A > (option @ B)),K:A,X3:B]: (((((((fun_upd @ A) @ (option @ B)) @ T2) @ K) @ ((some @ B) @ X3)) != (^ [X:A]: ((none @ B))))))). % 6.67/2.70 thf(fact_6826_graph__map__upd, axiom, (! [A:$tType,B:$tType,M:(A > (option @ B)),K:A,V2:B]: (((((graph @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M) @ K) @ ((some @ B) @ V2))) = (((insert @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ V2)) @ (((graph @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M) @ K) @ (none @ B)))))))). % 6.67/2.70 thf(fact_6827_restrict__upd__same, axiom, (! [B:$tType,A:$tType,M:(A > (option @ B)),X3:A,Y:B]: ((((((restrict_map @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M) @ X3) @ ((some @ B) @ Y))) @ ((uminus_uminus @ (set @ A)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) = ((((restrict_map @ A) @ B) @ M) @ ((uminus_uminus @ (set @ A)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))). % 6.67/2.70 thf(fact_6828_restrict__out, axiom, (! [A:$tType,B:$tType,X3:A,A3:(set @ A),M:(A > (option @ B))]: (((~ ((((member @ A) @ X3) @ A3))) => ((((((restrict_map @ A) @ B) @ M) @ A3) @ X3) = (none @ B)))))). % 6.67/2.70 thf(fact_6829_restrict__map__empty, axiom, (! [B:$tType,A:$tType,D6:(set @ A)]: ((((((restrict_map @ A) @ B) @ (^ [X:A]: ((none @ B)))) @ D6) = (^ [X:A]: ((none @ B))))))). % 6.67/2.70 thf(fact_6830_restrict__map__to__empty, axiom, (! [B:$tType,A:$tType,M:(A > (option @ B))]: ((((((restrict_map @ A) @ B) @ M) @ (bot_bot @ (set @ A))) = (^ [X:A]: ((none @ B))))))). % 6.67/2.70 thf(fact_6831_graph__empty, axiom, (! [B:$tType,A:$tType]: (((((graph @ A) @ B) @ (^ [X:A]: ((none @ B)))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))). % 6.67/2.70 thf(fact_6832_fun__upd__restrict__conv, axiom, (! [A:$tType,B:$tType,X3:A,D6:(set @ A),M:(A > (option @ B)),Y:(option @ B)]: (((((member @ A) @ X3) @ D6) => ((((((fun_upd @ A) @ (option @ B)) @ ((((restrict_map @ A) @ B) @ M) @ D6)) @ X3) @ Y) = (((((fun_upd @ A) @ (option @ B)) @ ((((restrict_map @ A) @ B) @ M) @ (((minus_minus @ (set @ A)) @ D6) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))) @ X3) @ Y)))))). % 6.67/2.70 thf(fact_6833_restrict__fun__upd, axiom, (! [B:$tType,A:$tType,X3:A,D6:(set @ A),M:(A > (option @ B)),Y:(option @ B)]: ((((((member @ A) @ X3) @ D6) => (((((restrict_map @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M) @ X3) @ Y)) @ D6) = (((((fun_upd @ A) @ (option @ B)) @ ((((restrict_map @ A) @ B) @ M) @ (((minus_minus @ (set @ A)) @ D6) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))) @ X3) @ Y))) & ((~ ((((member @ A) @ X3) @ D6))) => (((((restrict_map @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M) @ X3) @ Y)) @ D6) = ((((restrict_map @ A) @ B) @ M) @ D6))))))). % 6.67/2.70 thf(fact_6834_fun__upd__None__restrict, axiom, (! [B:$tType,A:$tType,X3:A,D6:(set @ A),M:(A > (option @ B))]: ((((((member @ A) @ X3) @ D6) => ((((((fun_upd @ A) @ (option @ B)) @ ((((restrict_map @ A) @ B) @ M) @ D6)) @ X3) @ (none @ B)) = ((((restrict_map @ A) @ B) @ M) @ (((minus_minus @ (set @ A)) @ D6) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))) & ((~ ((((member @ A) @ X3) @ D6))) => ((((((fun_upd @ A) @ (option @ B)) @ ((((restrict_map @ A) @ B) @ M) @ D6)) @ X3) @ (none @ B)) = ((((restrict_map @ A) @ B) @ M) @ D6))))))). % 6.67/2.70 thf(fact_6835_graph__restrictD_I1_J, axiom, (! [B:$tType,A:$tType,K:A,V2:B,M:(A > (option @ B)),A3:(set @ A)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ V2)) @ (((graph @ A) @ B) @ ((((restrict_map @ A) @ B) @ M) @ A3))) => (((member @ A) @ K) @ A3))))). % 6.67/2.70 thf(fact_6836_graph__restrictD_I2_J, axiom, (! [A:$tType,B:$tType,K:A,V2:B,M:(A > (option @ B)),A3:(set @ A)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ V2)) @ (((graph @ A) @ B) @ ((((restrict_map @ A) @ B) @ M) @ A3))) => ((M @ K) = ((some @ B) @ V2)))))). % 6.67/2.70 thf(fact_6837_in__graphD, axiom, (! [A:$tType,B:$tType,K:A,V2:B,M:(A > (option @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ V2)) @ (((graph @ A) @ B) @ M)) => ((M @ K) = ((some @ B) @ V2)))))). % 6.67/2.70 thf(fact_6838_in__graphI, axiom, (! [A:$tType,B:$tType,M:(B > (option @ A)),K:B,V2:A]: ((((M @ K) = ((some @ A) @ V2)) => (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ K) @ V2)) @ (((graph @ B) @ A) @ M)))))). % 6.67/2.70 thf(fact_6839_restrict__map__def, axiom, (! [B:$tType,A:$tType]: ((((restrict_map @ A) @ B) = (^ [M2:(A > (option @ B)),A6:(set @ A),X:A]: (((((if @ (option @ B)) @ (((member @ A) @ X) @ A6)) @ (M2 @ X)) @ (none @ B)))))))). % 6.67/2.70 thf(fact_6840_graph__def, axiom, (! [B:$tType,A:$tType]: ((((graph @ A) @ B) = (^ [M2:(A > (option @ B))]: (((collect @ ((product_prod @ A) @ B)) @ (^ [Uu3:((product_prod @ A) @ B)]: ((? [A4:A,B4:B]: (((Uu3 = ((((product_Pair @ A) @ B) @ A4) @ B4)) & ((M2 @ A4) = ((some @ B) @ B4)))))))))))))). % 6.67/2.70 thf(fact_6841_graph__fun__upd__None, axiom, (! [B:$tType,A:$tType,M:(A > (option @ B)),K:A]: (((((graph @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M) @ K) @ (none @ B))) = ((collect @ ((product_prod @ A) @ B)) @ (^ [E4:((product_prod @ A) @ B)]: (((((member @ ((product_prod @ A) @ B)) @ E4) @ (((graph @ A) @ B) @ M)) & ((((product_fst @ A) @ B) @ E4) != K))))))))). % 6.67/2.70 thf(fact_6842_fun__upd__restrict, axiom, (! [A:$tType,B:$tType,M:(A > (option @ B)),D6:(set @ A),X3:A,Y:(option @ B)]: (((((((fun_upd @ A) @ (option @ B)) @ ((((restrict_map @ A) @ B) @ M) @ D6)) @ X3) @ Y) = (((((fun_upd @ A) @ (option @ B)) @ ((((restrict_map @ A) @ B) @ M) @ (((minus_minus @ (set @ A)) @ D6) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))) @ X3) @ Y))))). % 6.67/2.70 thf(fact_6843_restrict__complement__singleton__eq, axiom, (! [A:$tType,B:$tType,F2:(A > (option @ B)),X3:A]: ((((((restrict_map @ A) @ B) @ F2) @ ((uminus_uminus @ (set @ A)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))) = (((((fun_upd @ A) @ (option @ B)) @ F2) @ X3) @ (none @ B)))))). % 6.67/2.70 thf(fact_6844_restrict__map__upds, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B),D6:(set @ A),M:(A > (option @ B))]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => ((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ D6) => (((((restrict_map @ A) @ B) @ (((((map_upds @ A) @ B) @ M) @ Xs) @ Ys)) @ D6) = (((((map_upds @ A) @ B) @ ((((restrict_map @ A) @ B) @ M) @ (((minus_minus @ (set @ A)) @ D6) @ ((set2 @ A) @ Xs)))) @ Xs) @ Ys))))))). % 6.67/2.70 thf(fact_6845_ran__map__upd, axiom, (! [A:$tType,B:$tType,M:(B > (option @ A)),A2:B,B2:A]: ((((M @ A2) = (none @ A)) => ((((ran @ B) @ A) @ (((((fun_upd @ B) @ (option @ A)) @ M) @ A2) @ ((some @ A) @ B2))) = (((insert @ A) @ B2) @ (((ran @ B) @ A) @ M))))))). % 6.67/2.70 thf(fact_6846_map__upds__apply__nontin, axiom, (! [B:$tType,A:$tType,X3:A,Xs:(list @ A),F2:(A > (option @ B)),Ys:(list @ B)]: (((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)))) => (((((((map_upds @ A) @ B) @ F2) @ Xs) @ Ys) @ X3) = (F2 @ X3)))))). % 6.67/2.70 thf(fact_6847_ran__empty, axiom, (! [B:$tType,A:$tType]: (((((ran @ B) @ A) @ (^ [X:B]: ((none @ A)))) = (bot_bot @ (set @ A)))))). % 6.67/2.70 thf(fact_6848_map__upds__list__update2__drop, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),I:nat,M:(A > (option @ B)),Ys:(list @ B),Y:B]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ I) => ((((((map_upds @ A) @ B) @ M) @ Xs) @ ((((list_update @ B) @ Ys) @ I) @ Y)) = (((((map_upds @ A) @ B) @ M) @ Xs) @ Ys)))))). % 6.67/2.70 thf(fact_6849_map__upds__twist, axiom, (! [A:$tType,B:$tType,A2:A,As:(list @ A),M:(A > (option @ B)),B2:B,Bs:(list @ B)]: (((~ ((((member @ A) @ A2) @ ((set2 @ A) @ As)))) => ((((((map_upds @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M) @ A2) @ ((some @ B) @ B2))) @ As) @ Bs) = (((((fun_upd @ A) @ (option @ B)) @ (((((map_upds @ A) @ B) @ M) @ As) @ Bs)) @ A2) @ ((some @ B) @ B2))))))). % 6.67/2.70 thf(fact_6850_ranI, axiom, (! [A:$tType,B:$tType,M:(B > (option @ A)),A2:B,B2:A]: ((((M @ A2) = ((some @ A) @ B2)) => (((member @ A) @ B2) @ (((ran @ B) @ A) @ M)))))). % 6.67/2.70 thf(fact_6851_ran__restrictD, axiom, (! [B:$tType,A:$tType,Y:A,M:(B > (option @ A)),A3:(set @ B)]: (((((member @ A) @ Y) @ (((ran @ B) @ A) @ ((((restrict_map @ B) @ A) @ M) @ A3))) => (? [X5:B]: (((((member @ B) @ X5) @ A3) & ((M @ X5) = ((some @ A) @ Y))))))))). % 6.67/2.70 thf(fact_6852_ran__def, axiom, (! [B:$tType,A:$tType]: ((((ran @ A) @ B) = (^ [M2:(A > (option @ B))]: (((collect @ B) @ (^ [B4:B]: ((? [A4:A]: (((M2 @ A4) = ((some @ B) @ B4))))))))))))). % 6.67/2.70 thf(fact_6853_ran__map__upd__Some, axiom, (! [B:$tType,A:$tType,M:(B > (option @ A)),X3:B,Y:A,Z2:A]: ((((M @ X3) = ((some @ A) @ Y)) => (((((inj_on @ B) @ (option @ A)) @ M) @ (((dom @ B) @ A) @ M)) => ((~ ((((member @ A) @ Z2) @ (((ran @ B) @ A) @ M)))) => ((((ran @ B) @ A) @ (((((fun_upd @ B) @ (option @ A)) @ M) @ X3) @ ((some @ A) @ Z2))) = (((sup_sup @ (set @ A)) @ (((minus_minus @ (set @ A)) @ (((ran @ B) @ A) @ M)) @ (((insert @ A) @ Y) @ (bot_bot @ (set @ A))))) @ (((insert @ A) @ Z2) @ (bot_bot @ (set @ A))))))))))). % 6.67/2.70 thf(fact_6854_map__upds__fold__map__upd, axiom, (! [B:$tType,A:$tType]: ((((map_upds @ A) @ B) = (^ [M2:(A > (option @ B)),Ks:(list @ A),Vs:(list @ B)]: ((((((foldl @ (A > (option @ B))) @ ((product_prod @ A) @ B)) @ (^ [N:(A > (option @ B))]: (((((product_case_prod @ A) @ B) @ (A > (option @ B))) @ (^ [K3:A,V5:B]: ((((((fun_upd @ A) @ (option @ B)) @ N) @ K3) @ ((some @ B) @ V5)))))))) @ M2) @ ((((zip @ A) @ B) @ Ks) @ Vs)))))))). % 6.67/2.70 thf(fact_6855_dom__eq__empty__conv, axiom, (! [B:$tType,A:$tType,F2:(A > (option @ B))]: ((((((dom @ A) @ B) @ F2) = (bot_bot @ (set @ A))) = (F2 = (^ [X:A]: ((none @ B)))))))). % 6.67/2.70 thf(fact_6856_fun__upd__None__if__notin__dom, axiom, (! [B:$tType,A:$tType,K:A,M:(A > (option @ B))]: (((~ ((((member @ A) @ K) @ (((dom @ A) @ B) @ M)))) => ((((((fun_upd @ A) @ (option @ B)) @ M) @ K) @ (none @ B)) = M))))). % 6.67/2.70 thf(fact_6857_dom__const, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: (((((dom @ A) @ B) @ (^ [X:A]: (((some @ B) @ (F2 @ X))))) = (top_top @ (set @ A)))))). % 6.67/2.70 thf(fact_6858_dom__empty, axiom, (! [B:$tType,A:$tType]: (((((dom @ A) @ B) @ (^ [X:A]: ((none @ B)))) = (bot_bot @ (set @ A)))))). % 6.67/2.70 thf(fact_6859_finite__graph__iff__finite__dom, axiom, (! [B:$tType,A:$tType,M:(A > (option @ B))]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ (((graph @ A) @ B) @ M)) = ((finite_finite2 @ A) @ (((dom @ A) @ B) @ M)))))). % 6.67/2.70 thf(fact_6860_dom__fun__upd, axiom, (! [B:$tType,A:$tType,Y:(option @ B),F2:(A > (option @ B)),X3:A]: ((((Y = (none @ B)) => ((((dom @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ F2) @ X3) @ Y)) = (((minus_minus @ (set @ A)) @ (((dom @ A) @ B) @ F2)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))) & ((Y != (none @ B)) => ((((dom @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ F2) @ X3) @ Y)) = (((insert @ A) @ X3) @ (((dom @ A) @ B) @ F2)))))))). % 6.67/2.70 thf(fact_6861_finite__ran, axiom, (! [B:$tType,A:$tType,P4:(A > (option @ B))]: ((((finite_finite2 @ A) @ (((dom @ A) @ B) @ P4)) => ((finite_finite2 @ B) @ (((ran @ A) @ B) @ P4)))))). % 6.67/2.70 thf(fact_6862_insert__dom, axiom, (! [A:$tType,B:$tType,F2:(B > (option @ A)),X3:B,Y:A]: ((((F2 @ X3) = ((some @ A) @ Y)) => ((((insert @ B) @ X3) @ (((dom @ B) @ A) @ F2)) = (((dom @ B) @ A) @ F2)))))). % 6.67/2.70 thf(fact_6863_domD, axiom, (! [A:$tType,B:$tType,A2:A,M:(A > (option @ B))]: (((((member @ A) @ A2) @ (((dom @ A) @ B) @ M)) => (? [B5:B]: (((M @ A2) = ((some @ B) @ B5)))))))). % 6.67/2.70 thf(fact_6864_domI, axiom, (! [A:$tType,B:$tType,M:(B > (option @ A)),A2:B,B2:A]: ((((M @ A2) = ((some @ A) @ B2)) => (((member @ B) @ A2) @ (((dom @ B) @ A) @ M)))))). % 6.67/2.70 thf(fact_6865_domIff, axiom, (! [A:$tType,B:$tType,A2:A,M:(A > (option @ B))]: (((((member @ A) @ A2) @ (((dom @ A) @ B) @ M)) = ((M @ A2) != (none @ B)))))). % 6.67/2.70 thf(fact_6866_dom__def, axiom, (! [B:$tType,A:$tType]: ((((dom @ A) @ B) = (^ [M2:(A > (option @ B))]: (((collect @ A) @ (^ [A4:A]: (((M2 @ A4) != (none @ B))))))))))). % 6.67/2.70 thf(fact_6867_foldl__cong, axiom, (! [A:$tType,B:$tType,A2:A,B2:A,L:(list @ B),K:(list @ B),F2:(A > (B > A)),G:(A > (B > A))]: (((A2 = B2) => ((L = K) => ((! [A5:A,X5:B]: (((((member @ B) @ X5) @ ((set2 @ B) @ L)) => (((F2 @ A5) @ X5) = ((G @ A5) @ X5))))) => ((((((foldl @ A) @ B) @ F2) @ A2) @ L) = (((((foldl @ A) @ B) @ G) @ B2) @ K)))))))). % 6.67/2.70 thf(fact_6868_foldl__Nil, axiom, (! [A:$tType,B:$tType,F2:(B > (A > B)),A2:B]: (((((((foldl @ B) @ A) @ F2) @ A2) @ (nil @ A)) = A2)))). % 6.67/2.70 thf(fact_6869_finite__map__freshness, axiom, (! [A:$tType,B:$tType,F2:(A > (option @ B))]: ((((finite_finite2 @ A) @ (((dom @ A) @ B) @ F2)) => ((~ (((finite_finite2 @ A) @ (top_top @ (set @ A))))) => (? [X5:A]: (((F2 @ X5) = (none @ B))))))))). % 6.67/2.70 thf(fact_6870_dom__minus, axiom, (! [A:$tType,B:$tType,F2:(B > (option @ A)),X3:B,A3:(set @ B)]: ((((F2 @ X3) = (none @ A)) => ((((minus_minus @ (set @ B)) @ (((dom @ B) @ A) @ F2)) @ (((insert @ B) @ X3) @ A3)) = (((minus_minus @ (set @ B)) @ (((dom @ B) @ A) @ F2)) @ A3)))))). % 6.67/2.70 thf(fact_6871_finite__set__of__finite__maps, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ B) @ B3) => ((finite_finite2 @ (A > (option @ B))) @ ((collect @ (A > (option @ B))) @ (^ [M2:(A > (option @ B))]: ((((((dom @ A) @ B) @ M2) = A3) & (((ord_less_eq @ (set @ B)) @ (((ran @ A) @ B) @ M2)) @ B3))))))))))). % 6.67/2.70 thf(fact_6872_graph__eq__to__snd__dom, axiom, (! [B:$tType,A:$tType]: ((((graph @ A) @ B) = (^ [M2:(A > (option @ B))]: (((((image @ A) @ ((product_prod @ A) @ B)) @ (^ [X:A]: (((((product_Pair @ A) @ B) @ X) @ ((the2 @ B) @ (M2 @ X)))))) @ (((dom @ A) @ B) @ M2)))))))). % 6.67/2.70 thf(fact_6873_finite__Map__induct, axiom, (! [B:$tType,A:$tType,M:(A > (option @ B)),P:((A > (option @ B)) > $o)]: ((((finite_finite2 @ A) @ (((dom @ A) @ B) @ M)) => ((P @ (^ [X:A]: ((none @ B)))) => ((! [K2:A,V3:B,M3:(A > (option @ B))]: ((((finite_finite2 @ A) @ (((dom @ A) @ B) @ M3)) => ((~ ((((member @ A) @ K2) @ (((dom @ A) @ B) @ M3)))) => ((P @ M3) => (P @ (((((fun_upd @ A) @ (option @ B)) @ M3) @ K2) @ ((some @ B) @ V3)))))))) => (P @ M))))))). % 6.67/2.70 thf(fact_6874_dom__eq__singleton__conv, axiom, (! [A:$tType,B:$tType,F2:(A > (option @ B)),X3:A]: ((((((dom @ A) @ B) @ F2) = (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = (? [V5:B]: ((F2 = (((((fun_upd @ A) @ (option @ B)) @ (^ [X:A]: ((none @ B)))) @ X3) @ ((some @ B) @ V5))))))))). % 6.67/2.70 thf(fact_6875_min__list__Min, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((Xs != (nil @ A)) => (((min_list @ A) @ Xs) = ((lattic643756798350308766er_Min @ A) @ ((set2 @ A) @ Xs)))))))))). % 6.67/2.70 thf(fact_6876_minus__fold__remove, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((((minus_minus @ (set @ A)) @ B3) @ A3) = (((((finite_fold @ A) @ (set @ A)) @ (remove @ A)) @ B3) @ A3)))))). % 6.67/2.70 thf(fact_6877_member__remove, axiom, (! [A:$tType,X3:A,Y:A,A3:(set @ A)]: (((((member @ A) @ X3) @ (((remove @ A) @ Y) @ A3)) = ((((member @ A) @ X3) @ A3) & (X3 != Y)))))). % 6.67/2.70 thf(fact_6878_remove__code_I1_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((remove @ A) @ X3) @ ((set2 @ A) @ Xs)) = ((set2 @ A) @ (((removeAll @ A) @ X3) @ Xs)))))). % 6.67/2.70 thf(fact_6879_remove__def, axiom, (! [A:$tType]: (((remove @ A) = (^ [X:A,A6:(set @ A)]: ((((minus_minus @ (set @ A)) @ A6) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))). % 6.67/2.70 thf(fact_6880_finite__subsets__at__top__finite, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (((finite5375528669736107172at_top @ A) @ A3) = ((principal @ (set @ A)) @ (((insert @ (set @ A)) @ A3) @ (bot_bot @ (set @ (set @ A)))))))))). % 6.67/2.70 thf(fact_6881_pos__deriv__imp__strict__mono, axiom, (! [F2:(real > real),F11:(real > real)]: (((! [X5:real]: (((((has_field_derivative @ real) @ F2) @ (F11 @ X5)) @ (((topolo174197925503356063within @ real) @ X5) @ (top_top @ (set @ real)))))) => ((! [X5:real]: ((((ord_less @ real) @ (zero_zero @ real)) @ (F11 @ X5)))) => (((order_strict_mono @ real) @ real) @ F2)))))). % 6.67/2.70 thf(fact_6882_eventually__finite__subsets__at__top__weakI, axiom, (! [A:$tType,A3:(set @ A),P:((set @ A) > $o)]: (((! [X10:(set @ A)]: ((((finite_finite2 @ A) @ X10) => ((((ord_less_eq @ (set @ A)) @ X10) @ A3) => (P @ X10))))) => (((eventually @ (set @ A)) @ P) @ ((finite5375528669736107172at_top @ A) @ A3)))))). % 6.67/2.70 thf(fact_6883_eventually__finite__subsets__at__top__finite, axiom, (! [A:$tType,A3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A3) => ((((eventually @ (set @ A)) @ P) @ ((finite5375528669736107172at_top @ A) @ A3)) = (P @ A3)))))). % 6.67/2.70 thf(fact_6884_strict__mono__mono, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F2:(A > B)]: (((((order_strict_mono @ A) @ B) @ F2) => (((order_mono @ A) @ B) @ F2)))))))). % 6.67/2.70 thf(fact_6885_strict__mono__eq, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F2:(A > B),X3:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F2) => (((F2 @ X3) = (F2 @ Y)) = (X3 = Y))))))))). % 6.67/2.70 thf(fact_6886_strict__mono__less, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F2:(A > B),X3:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F2) => ((((ord_less @ B) @ (F2 @ X3)) @ (F2 @ Y)) = (((ord_less @ A) @ X3) @ Y))))))))). % 6.67/2.70 thf(fact_6887_strict__mono__def, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (((order_strict_mono @ A) @ B) = (^ [F3:(A > B)]: ((! [X:A,Y2:A]: (((((ord_less @ A) @ X) @ Y2) => (((ord_less @ B) @ (F3 @ X)) @ (F3 @ Y2)))))))))))). % 6.67/2.70 thf(fact_6888_strict__monoI, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F2:(A > B)]: (((! [X5:A,Y4:A]: (((((ord_less @ A) @ X5) @ Y4) => (((ord_less @ B) @ (F2 @ X5)) @ (F2 @ Y4))))) => (((order_strict_mono @ A) @ B) @ F2)))))))). % 6.67/2.70 thf(fact_6889_strict__monoD, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F2:(A > B),X3:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F2) => ((((ord_less @ A) @ X3) @ Y) => (((ord_less @ B) @ (F2 @ X3)) @ (F2 @ Y)))))))))). % 6.67/2.70 thf(fact_6890_strict__mono__add, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [K:A]: ((((order_strict_mono @ A) @ A) @ (^ [N:A]: ((((plus_plus @ A) @ N) @ K)))))))))). % 6.67/2.70 thf(fact_6891_strict__mono__less__eq, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F2:(A > B),X3:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F2) => ((((ord_less_eq @ B) @ (F2 @ X3)) @ (F2 @ Y)) = (((ord_less_eq @ A) @ X3) @ Y))))))))). % 6.67/2.70 thf(fact_6892_strict__mono__leD, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [R2:(A > B),M:A,N2:A]: (((((order_strict_mono @ A) @ B) @ R2) => ((((ord_less_eq @ A) @ M) @ N2) => (((ord_less_eq @ B) @ (R2 @ M)) @ (R2 @ N2)))))))))). % 6.67/2.70 thf(fact_6893_finite__subsets__at__top__neq__bot, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite5375528669736107172at_top @ A) @ A3) != (bot_bot @ (filter @ (set @ A))))))). % 6.67/2.70 thf(fact_6894_eventually__finite__subsets__at__top, axiom, (! [A:$tType,P:((set @ A) > $o),A3:(set @ A)]: (((((eventually @ (set @ A)) @ P) @ ((finite5375528669736107172at_top @ A) @ A3)) = (? [X6:(set @ A)]: ((((finite_finite2 @ A) @ X6) & ((((ord_less_eq @ (set @ A)) @ X6) @ A3) & (! [Y9:(set @ A)]: (((((finite_finite2 @ A) @ Y9) & ((((ord_less_eq @ (set @ A)) @ X6) @ Y9) & (((ord_less_eq @ (set @ A)) @ Y9) @ A3))) => (P @ Y9)))))))))))). % 6.67/2.70 thf(fact_6895_finite__subsets__at__top__def, axiom, (! [A:$tType]: (((finite5375528669736107172at_top @ A) = (^ [A6:(set @ A)]: (((complete_Inf_Inf @ (filter @ (set @ A))) @ ((((image @ (set @ A)) @ (filter @ (set @ A))) @ (^ [X6:(set @ A)]: (((principal @ (set @ A)) @ ((collect @ (set @ A)) @ (^ [Y9:(set @ A)]: ((((finite_finite2 @ A) @ Y9) & ((((ord_less_eq @ (set @ A)) @ X6) @ Y9) & (((ord_less_eq @ (set @ A)) @ Y9) @ A6)))))))))) @ ((collect @ (set @ A)) @ (^ [X6:(set @ A)]: ((((finite_finite2 @ A) @ X6) & (((ord_less_eq @ (set @ A)) @ X6) @ A6))))))))))))). % 6.67/2.70 thf(fact_6896_filterlim__finite__subsets__at__top, axiom, (! [A:$tType,B:$tType,F2:(A > (set @ B)),A3:(set @ B),F5:(filter @ A)]: (((((((filterlim @ A) @ (set @ B)) @ F2) @ ((finite5375528669736107172at_top @ B) @ A3)) @ F5) = (! [X6:(set @ B)]: (((((finite_finite2 @ B) @ X6) & (((ord_less_eq @ (set @ B)) @ X6) @ A3)) => (((eventually @ A) @ (^ [Y2:A]: ((((finite_finite2 @ B) @ (F2 @ Y2)) & ((((ord_less_eq @ (set @ B)) @ X6) @ (F2 @ Y2)) & (((ord_less_eq @ (set @ B)) @ (F2 @ Y2)) @ A3)))))) @ F5)))))))). % 6.67/2.70 thf(fact_6897_ring__1__class_Oof__int__def, axiom, (! [A:$tType]: (((ring_1 @ A) => ((ring_1_of_int @ A) = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ A) @ A) @ rep_Integ) @ (id @ A)) @ ((((product_case_prod @ nat) @ nat) @ A) @ (^ [I3:nat,J3:nat]: ((((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ I3)) @ ((semiring_1_of_nat @ A) @ J3))))))))))). % 6.67/2.70 thf(fact_6898_sorted__key__list__of__set__def, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (((linord144544945434240204of_set @ B) @ A) = (^ [F3:(B > A)]: (((((finite_folding_F @ B) @ (list @ B)) @ (((linorder_insort_key @ B) @ A) @ F3)) @ (nil @ B))))))))). % 6.67/2.70 thf(fact_6899_of__nat__eq__id, axiom, ((semiring_1_of_nat @ nat) = (id @ nat))). % 6.67/2.70 thf(fact_6900_case__prod__Pair, axiom, (! [B:$tType,A:$tType]: ((((((product_case_prod @ A) @ B) @ ((product_prod @ A) @ B)) @ ((product_Pair @ A) @ B)) = (id @ ((product_prod @ A) @ B)))))). % 6.67/2.70 thf(fact_6901_id__funpow, axiom, (! [A:$tType,N2:nat]: (((((compow @ (A > A)) @ N2) @ (id @ A)) = (id @ A))))). % 6.67/2.70 thf(fact_6902_apfst__id, axiom, (! [B:$tType,A:$tType]: ((((((product_apfst @ A) @ A) @ B) @ (id @ A)) = (id @ ((product_prod @ A) @ B)))))). % 6.67/2.70 thf(fact_6903_apsnd__id, axiom, (! [B:$tType,A:$tType]: ((((((product_apsnd @ B) @ B) @ A) @ (id @ B)) = (id @ ((product_prod @ A) @ B)))))). % 6.67/2.70 thf(fact_6904_push__bit__0__id, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (((bit_se4730199178511100633sh_bit @ A) @ (zero_zero @ nat)) = (id @ A)))))). % 6.67/2.70 thf(fact_6905_drop__bit__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (((bit_se4197421643247451524op_bit @ A) @ (zero_zero @ nat)) = (id @ A)))))). % 6.67/2.70 thf(fact_6906_comp__the__Some, axiom, (! [A:$tType]: (((((((comp @ (option @ A)) @ A) @ A) @ (the2 @ A)) @ (some @ A)) = (id @ A))))). % 6.67/2.70 thf(fact_6907_infinite__enumerate, axiom, (! [S3:(set @ nat)]: (((~ (((finite_finite2 @ nat) @ S3))) => (? [R4:(nat > nat)]: (((((order_strict_mono @ nat) @ nat) @ R4) & (! [N6:nat]: ((((member @ nat) @ (R4 @ N6)) @ S3)))))))))). % 6.67/2.70 thf(fact_6908_strict__mono__imp__increasing, axiom, (! [F2:(nat > nat),N2:nat]: (((((order_strict_mono @ nat) @ nat) @ F2) => (((ord_less_eq @ nat) @ N2) @ (F2 @ N2)))))). % 6.67/2.70 thf(fact_6909_strict__mono__Suc__iff, axiom, (! [A:$tType]: (((order @ A) => (((order_strict_mono @ nat) @ A) = (^ [F3:(nat > A)]: ((! [N:nat]: ((((ord_less @ A) @ (F3 @ N)) @ (F3 @ (suc @ N)))))))))))). % 6.67/2.70 thf(fact_6910_less__int__def, axiom, ((ord_less @ int) = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > $o)) @ (int > $o)) @ rep_Integ) @ ((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ $o) @ $o) @ rep_Integ) @ (id @ $o))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X:nat,Y2:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V5:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ X) @ V5)) @ (((plus_plus @ nat) @ U2) @ Y2))))))))))). % 6.67/2.70 thf(fact_6911_nat__def, axiom, (nat2 = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ nat) @ nat) @ rep_Integ) @ (id @ nat)) @ ((((product_case_prod @ nat) @ nat) @ nat) @ (minus_minus @ nat))))). % 6.67/2.70 thf(fact_6912_less__eq__int__def, axiom, ((ord_less_eq @ int) = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > $o)) @ (int > $o)) @ rep_Integ) @ ((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ $o) @ $o) @ rep_Integ) @ (id @ $o))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X:nat,Y2:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V5:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ X) @ V5)) @ (((plus_plus @ nat) @ U2) @ Y2))))))))))). % 6.67/2.70 thf(fact_6913_funpow__simps__right_I1_J, axiom, (! [A:$tType,F2:(A > A)]: (((((compow @ (A > A)) @ (zero_zero @ nat)) @ F2) = (id @ A))))). % 6.67/2.70 thf(fact_6914_fst__diag__id, axiom, (! [A:$tType,Z2:A]: ((((((((comp @ ((product_prod @ A) @ A)) @ A) @ A) @ ((product_fst @ A) @ A)) @ (^ [X:A]: (((((product_Pair @ A) @ A) @ X) @ X)))) @ Z2) = ((id @ A) @ Z2))))). % 6.67/2.70 thf(fact_6915_snd__diag__id, axiom, (! [A:$tType,Z2:A]: ((((((((comp @ ((product_prod @ A) @ A)) @ A) @ A) @ ((product_snd @ A) @ A)) @ (^ [X:A]: (((((product_Pair @ A) @ A) @ X) @ X)))) @ Z2) = ((id @ A) @ Z2))))). % 6.67/2.70 thf(fact_6916_summable__mono__reindex, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topolo4958980785337419405_space @ A)) => (! [G:(nat > nat),F2:(nat > A)]: (((((order_strict_mono @ nat) @ nat) @ G) => ((! [N3:nat]: (((~ ((((member @ nat) @ N3) @ ((((image @ nat) @ nat) @ G) @ (top_top @ (set @ nat)))))) => ((F2 @ N3) = (zero_zero @ A))))) => (((summable @ A) @ (^ [N:nat]: ((F2 @ (G @ N))))) = ((summable @ A) @ F2)))))))))). % 6.67/2.70 thf(fact_6917_sums__mono__reindex, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topolo4958980785337419405_space @ A)) => (! [G:(nat > nat),F2:(nat > A),C2:A]: (((((order_strict_mono @ nat) @ nat) @ G) => ((! [N3:nat]: (((~ ((((member @ nat) @ N3) @ ((((image @ nat) @ nat) @ G) @ (top_top @ (set @ nat)))))) => ((F2 @ N3) = (zero_zero @ A))))) => ((((sums @ A) @ (^ [N:nat]: ((F2 @ (G @ N))))) @ C2) = (((sums @ A) @ F2) @ C2)))))))))). % 6.67/2.70 thf(fact_6918_suminf__mono__reindex, axiom, (! [A:$tType]: ((((comm_monoid_add @ A) & (topological_t2_space @ A)) => (! [G:(nat > nat),F2:(nat > A)]: (((((order_strict_mono @ nat) @ nat) @ G) => ((! [N3:nat]: (((~ ((((member @ nat) @ N3) @ ((((image @ nat) @ nat) @ G) @ (top_top @ (set @ nat)))))) => ((F2 @ N3) = (zero_zero @ A))))) => (((suminf @ A) @ (^ [N:nat]: ((F2 @ (G @ N))))) = ((suminf @ A) @ F2)))))))))). % 6.67/2.70 thf(fact_6919_increasing__Bseq__subseq__iff, axiom, (! [A:$tType]: (((real_V822414075346904944vector @ A) => (! [F2:(nat > A),G:(nat > nat)]: (((! [X5:nat,Y4:nat]: (((((ord_less_eq @ nat) @ X5) @ Y4) => (((ord_less_eq @ real) @ ((real_V7770717601297561774m_norm @ A) @ (F2 @ X5))) @ ((real_V7770717601297561774m_norm @ A) @ (F2 @ Y4)))))) => ((((order_strict_mono @ nat) @ nat) @ G) => (((((bfun @ nat) @ A) @ (^ [X:nat]: ((F2 @ (G @ X))))) @ (at_top @ nat)) = ((((bfun @ nat) @ A) @ F2) @ (at_top @ nat))))))))))). % 6.67/2.70 thf(fact_6920_folding__on_Oinsert__remove, axiom, (! [B:$tType,A:$tType,S3:(set @ A),F2:(A > (B > B)),X3:A,A3:(set @ A),Z2:B]: ((((((finite_folding_on @ A) @ B) @ S3) @ F2) => ((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ S3) => (((finite_finite2 @ A) @ A3) => ((((((finite_folding_F @ A) @ B) @ F2) @ Z2) @ (((insert @ A) @ X3) @ A3)) = ((F2 @ X3) @ (((((finite_folding_F @ A) @ B) @ F2) @ Z2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))))))). % 6.67/2.70 thf(fact_6921_folding__on_Oremove, axiom, (! [B:$tType,A:$tType,S3:(set @ A),F2:(A > (B > B)),A3:(set @ A),X3:A,Z2:B]: ((((((finite_folding_on @ A) @ B) @ S3) @ F2) => ((((ord_less_eq @ (set @ A)) @ A3) @ S3) => (((finite_finite2 @ A) @ A3) => ((((member @ A) @ X3) @ A3) => ((((((finite_folding_F @ A) @ B) @ F2) @ Z2) @ A3) = ((F2 @ X3) @ (((((finite_folding_F @ A) @ B) @ F2) @ Z2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))))))). % 6.67/2.70 thf(fact_6922_card_Ofolding__on__axioms, axiom, (! [A:$tType]: (((((finite_folding_on @ A) @ nat) @ (top_top @ (set @ A))) @ (^ [Uu3:A]: (suc)))))). % 6.67/2.70 thf(fact_6923_folding__on_Oempty, axiom, (! [A:$tType,B:$tType,S3:(set @ A),F2:(A > (B > B)),Z2:B]: ((((((finite_folding_on @ A) @ B) @ S3) @ F2) => ((((((finite_folding_F @ A) @ B) @ F2) @ Z2) @ (bot_bot @ (set @ A))) = Z2))))). % 6.67/2.70 thf(fact_6924_folding__on_Oinfinite, axiom, (! [A:$tType,B:$tType,S3:(set @ A),F2:(A > (B > B)),A3:(set @ A),Z2:B]: ((((((finite_folding_on @ A) @ B) @ S3) @ F2) => ((~ (((finite_finite2 @ A) @ A3))) => ((((((finite_folding_F @ A) @ B) @ F2) @ Z2) @ A3) = Z2)))))). % 6.67/2.70 thf(fact_6925_sorted__list__of__set_Ofold__insort__key_Ofolding__on__axioms, axiom, (! [A:$tType]: (((linorder @ A) => ((((finite_folding_on @ A) @ (list @ A)) @ (top_top @ (set @ A))) @ (((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X)))))))). % 6.67/2.70 thf(fact_6926_card__def, axiom, (! [B:$tType]: (((finite_card @ B) = ((((finite_folding_F @ B) @ nat) @ (^ [Uu3:B]: (suc))) @ (zero_zero @ nat)))))). % 6.67/2.70 thf(fact_6927_folding__on_Oinsert, axiom, (! [B:$tType,A:$tType,S3:(set @ A),F2:(A > (B > B)),X3:A,A3:(set @ A),Z2:B]: ((((((finite_folding_on @ A) @ B) @ S3) @ F2) => ((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ S3) => (((finite_finite2 @ A) @ A3) => ((~ ((((member @ A) @ X3) @ A3))) => ((((((finite_folding_F @ A) @ B) @ F2) @ Z2) @ (((insert @ A) @ X3) @ A3)) = ((F2 @ X3) @ (((((finite_folding_F @ A) @ B) @ F2) @ Z2) @ A3)))))))))). % 6.67/2.70 thf(fact_6928_folding__idem__on_Oinsert__idem, axiom, (! [B:$tType,A:$tType,S3:(set @ A),F2:(A > (B > B)),X3:A,A3:(set @ A),Z2:B]: ((((((finite1890593828518410140dem_on @ A) @ B) @ S3) @ F2) => ((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ S3) => (((finite_finite2 @ A) @ A3) => ((((((finite_folding_F @ A) @ B) @ F2) @ Z2) @ (((insert @ A) @ X3) @ A3)) = ((F2 @ X3) @ (((((finite_folding_F @ A) @ B) @ F2) @ Z2) @ A3))))))))). % 6.67/2.70 thf(fact_6929_comp__fun__idem__on_Ofold__insert__idem, axiom, (! [B:$tType,A:$tType,S3:(set @ A),F2:(A > (B > B)),X3:A,A3:(set @ A),Z2:B]: ((((((finite673082921795544331dem_on @ A) @ B) @ S3) @ F2) => ((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ S3) => (((finite_finite2 @ A) @ A3) => ((((((finite_fold @ A) @ B) @ F2) @ Z2) @ (((insert @ A) @ X3) @ A3)) = ((F2 @ X3) @ (((((finite_fold @ A) @ B) @ F2) @ Z2) @ A3))))))))). % 6.67/2.70 thf(fact_6930_comp__fun__idem__on_Ocomp__comp__fun__idem__on, axiom, (! [B:$tType,A:$tType,C:$tType,S3:(set @ A),F2:(A > (B > B)),G:(C > A),R:(set @ C)]: ((((((finite673082921795544331dem_on @ A) @ B) @ S3) @ F2) => ((((ord_less_eq @ (set @ A)) @ ((((image @ C) @ A) @ G) @ (top_top @ (set @ C)))) @ S3) => ((((finite673082921795544331dem_on @ C) @ B) @ R) @ (((((comp @ A) @ (B > B)) @ C) @ F2) @ G))))))). % 6.67/2.70 thf(fact_6931_comp__fun__idem__on_Ofold__insert__idem2, axiom, (! [B:$tType,A:$tType,S3:(set @ A),F2:(A > (B > B)),X3:A,A3:(set @ A),Z2:B]: ((((((finite673082921795544331dem_on @ A) @ B) @ S3) @ F2) => ((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ S3) => (((finite_finite2 @ A) @ A3) => ((((((finite_fold @ A) @ B) @ F2) @ Z2) @ (((insert @ A) @ X3) @ A3)) = (((((finite_fold @ A) @ B) @ F2) @ ((F2 @ X3) @ Z2)) @ A3)))))))). % 6.67/2.70 thf(fact_6932_positive__rat, axiom, (! [A2:int,B2:int]: (((positive @ ((fract @ A2) @ B2)) = (((ord_less @ int) @ (zero_zero @ int)) @ (((times_times @ int) @ A2) @ B2)))))). % 6.67/2.70 thf(fact_6933_nth__image, axiom, (! [A:$tType,L:nat,Xs:(list @ A)]: (((((ord_less_eq @ nat) @ L) @ ((size_size @ (list @ A)) @ Xs)) => (((((image @ nat) @ A) @ ((nth @ A) @ Xs)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ L)) = ((set2 @ A) @ (((take @ A) @ L) @ Xs))))))). % 6.67/2.70 thf(fact_6934_take__take, axiom, (! [A:$tType,N2:nat,M:nat,Xs:(list @ A)]: (((((take @ A) @ N2) @ (((take @ A) @ M) @ Xs)) = (((take @ A) @ (((ord_min @ nat) @ N2) @ M)) @ Xs))))). % 6.67/2.70 thf(fact_6935_take__eq__Nil2, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((nil @ A) = (((take @ A) @ N2) @ Xs)) = ((N2 = (zero_zero @ nat)) | (Xs = (nil @ A))))))). % 6.67/2.70 thf(fact_6936_take__eq__Nil, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((((take @ A) @ N2) @ Xs) = (nil @ A)) = ((N2 = (zero_zero @ nat)) | (Xs = (nil @ A))))))). % 6.67/2.70 thf(fact_6937_take0, axiom, (! [A:$tType]: ((((take @ A) @ (zero_zero @ nat)) = (^ [Xs2:(list @ A)]: ((nil @ A))))))). % 6.67/2.70 thf(fact_6938_take__all__iff, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((((take @ A) @ N2) @ Xs) = Xs) = (((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N2))))). % 6.67/2.70 thf(fact_6939_take__all, axiom, (! [A:$tType,Xs:(list @ A),N2:nat]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N2) => ((((take @ A) @ N2) @ Xs) = Xs))))). % 6.67/2.70 thf(fact_6940_nth__take, axiom, (! [A:$tType,I:nat,N2:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ I) @ N2) => ((((nth @ A) @ (((take @ A) @ N2) @ Xs)) @ I) = (((nth @ A) @ Xs) @ I)))))). % 6.67/2.70 thf(fact_6941_take__update__cancel, axiom, (! [A:$tType,N2:nat,M:nat,Xs:(list @ A),Y:A]: (((((ord_less_eq @ nat) @ N2) @ M) => ((((take @ A) @ N2) @ ((((list_update @ A) @ Xs) @ M) @ Y)) = (((take @ A) @ N2) @ Xs)))))). % 6.67/2.70 thf(fact_6942_length__take, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ (((take @ A) @ N2) @ Xs)) = (((ord_min @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N2))))). % 6.67/2.70 thf(fact_6943_nths__upt__eq__take, axiom, (! [A:$tType,L:(list @ A),N2:nat]: (((((nths @ A) @ L) @ ((set_ord_lessThan @ nat) @ N2)) = (((take @ A) @ N2) @ L))))). % 6.67/2.70 thf(fact_6944_take__replicate, axiom, (! [A:$tType,I:nat,K:nat,X3:A]: (((((take @ A) @ I) @ (((replicate @ A) @ K) @ X3)) = (((replicate @ A) @ (((ord_min @ nat) @ I) @ K)) @ X3))))). % 6.67/2.70 thf(fact_6945_dom__map__upds, axiom, (! [B:$tType,A:$tType,M:(A > (option @ B)),Xs:(list @ A),Ys:(list @ B)]: (((((dom @ A) @ B) @ (((((map_upds @ A) @ B) @ M) @ Xs) @ Ys)) = (((sup_sup @ (set @ A)) @ ((set2 @ A) @ (((take @ A) @ ((size_size @ (list @ B)) @ Ys)) @ Xs))) @ (((dom @ A) @ B) @ M)))))). % 6.67/2.70 thf(fact_6946_in__set__takeD, axiom, (! [A:$tType,X3:A,N2:nat,Xs:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ (((take @ A) @ N2) @ Xs))) => (((member @ A) @ X3) @ ((set2 @ A) @ Xs)))))). % 6.67/2.70 thf(fact_6947_take__zip, axiom, (! [A:$tType,B:$tType,N2:nat,Xs:(list @ A),Ys:(list @ B)]: (((((take @ ((product_prod @ A) @ B)) @ N2) @ ((((zip @ A) @ B) @ Xs) @ Ys)) = ((((zip @ A) @ B) @ (((take @ A) @ N2) @ Xs)) @ (((take @ B) @ N2) @ Ys)))))). % 6.67/2.70 thf(fact_6948_take__Nil, axiom, (! [A:$tType,N2:nat]: (((((take @ A) @ N2) @ (nil @ A)) = (nil @ A))))). % 6.67/2.70 thf(fact_6949_take__update__swap, axiom, (! [A:$tType,M:nat,Xs:(list @ A),N2:nat,X3:A]: (((((take @ A) @ M) @ ((((list_update @ A) @ Xs) @ N2) @ X3)) = ((((list_update @ A) @ (((take @ A) @ M) @ Xs)) @ N2) @ X3))))). % 6.67/2.70 thf(fact_6950_take__equalityI, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((! [I4:nat]: (((((take @ A) @ I4) @ Xs) = (((take @ A) @ I4) @ Ys)))) => (Xs = Ys))))). % 6.67/2.70 thf(fact_6951_distinct__take, axiom, (! [A:$tType,Xs:(list @ A),I:nat]: ((((distinct @ A) @ Xs) => ((distinct @ A) @ (((take @ A) @ I) @ Xs)))))). % 6.67/2.70 thf(fact_6952_take__0, axiom, (! [A:$tType,Xs:(list @ A)]: (((((take @ A) @ (zero_zero @ nat)) @ Xs) = (nil @ A))))). % 6.67/2.70 thf(fact_6953_set__take__subset, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ (((take @ A) @ N2) @ Xs))) @ ((set2 @ A) @ Xs))))). % 6.67/2.70 thf(fact_6954_set__take__subset__set__take, axiom, (! [A:$tType,M:nat,N2:nat,Xs:(list @ A)]: (((((ord_less_eq @ nat) @ M) @ N2) => (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ (((take @ A) @ M) @ Xs))) @ ((set2 @ A) @ (((take @ A) @ N2) @ Xs))))))). % 6.67/2.70 thf(fact_6955_zip__obtain__same__length, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B),P:((list @ ((product_prod @ A) @ B)) > $o)]: (((! [Zs:(list @ A),Ws:(list @ B),N3:nat]: (((((size_size @ (list @ A)) @ Zs) = ((size_size @ (list @ B)) @ Ws)) => ((N3 = (((ord_min @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ B)) @ Ys))) => ((Zs = (((take @ A) @ N3) @ Xs)) => ((Ws = (((take @ B) @ N3) @ Ys)) => (P @ ((((zip @ A) @ B) @ Zs) @ Ws)))))))) => (P @ ((((zip @ A) @ B) @ Xs) @ Ys)))))). % 6.67/2.70 thf(fact_6956_nth__take__lemma, axiom, (! [A:$tType,K:nat,Xs:(list @ A),Ys:(list @ A)]: (((((ord_less_eq @ nat) @ K) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less_eq @ nat) @ K) @ ((size_size @ (list @ A)) @ Ys)) => ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ K) => ((((nth @ A) @ Xs) @ I4) = (((nth @ A) @ Ys) @ I4))))) => ((((take @ A) @ K) @ Xs) = (((take @ A) @ K) @ Ys)))))))). % 6.67/2.70 thf(fact_6957_less__rat__def, axiom, ((ord_less @ rat) = (^ [X:rat,Y2:rat]: ((positive @ (((minus_minus @ rat) @ Y2) @ X)))))). % 6.67/2.70 thf(fact_6958_map__upd__upds__conv__if, axiom, (! [A:$tType,B:$tType,X3:A,Ys:(list @ B),Xs:(list @ A),F2:(A > (option @ B)),Y:B]: ((((((member @ A) @ X3) @ ((set2 @ A) @ (((take @ A) @ ((size_size @ (list @ B)) @ Ys)) @ Xs))) => ((((((map_upds @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ F2) @ X3) @ ((some @ B) @ Y))) @ Xs) @ Ys) = (((((map_upds @ A) @ B) @ F2) @ Xs) @ Ys))) & ((~ ((((member @ A) @ X3) @ ((set2 @ A) @ (((take @ A) @ ((size_size @ (list @ B)) @ Ys)) @ Xs))))) => ((((((map_upds @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ F2) @ X3) @ ((some @ B) @ Y))) @ Xs) @ Ys) = (((((fun_upd @ A) @ (option @ B)) @ (((((map_upds @ A) @ B) @ F2) @ Xs) @ Ys)) @ X3) @ ((some @ B) @ Y)))))))). % 6.67/2.70 thf(fact_6959_Rat_Opositive_Orep__eq, axiom, (positive = (^ [X:rat]: ((((ord_less @ int) @ (zero_zero @ int)) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ (rep_Rat @ X))) @ (((product_snd @ int) @ int) @ (rep_Rat @ X)))))))). % 6.67/2.70 thf(fact_6960_lexord__take__index__conv, axiom, (! [A:$tType,X3:(list @ A),Y:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X3) @ Y)) @ ((lexord @ A) @ R2)) = (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ X3)) @ ((size_size @ (list @ A)) @ Y)) & ((((take @ A) @ ((size_size @ (list @ A)) @ X3)) @ Y) = X3)) | (? [I3:nat]: (((((ord_less @ nat) @ I3) @ (((ord_min @ nat) @ ((size_size @ (list @ A)) @ X3)) @ ((size_size @ (list @ A)) @ Y))) & (((((take @ A) @ I3) @ X3) = (((take @ A) @ I3) @ Y)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (((nth @ A) @ X3) @ I3)) @ (((nth @ A) @ Y) @ I3))) @ R2)))))))))). % 6.67/2.70 thf(fact_6961_lexord__linear, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),X3:(list @ A),Y:(list @ A)]: (((! [A5:A,B5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ B5)) @ R2) | ((A5 = B5) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ A5)) @ R2))))) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X3) @ Y)) @ ((lexord @ A) @ R2)) | ((X3 = Y) | (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Y) @ X3)) @ ((lexord @ A) @ R2)))))))). % 6.67/2.70 thf(fact_6962_lexord__irreflexive, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),Xs:(list @ A)]: (((! [X5:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ X5)) @ R2))))) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Xs)) @ ((lexord @ A) @ R2)))))))). % 6.67/2.70 thf(fact_6963_lexord__Nil__right, axiom, (! [A:$tType,X3:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: ((~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X3) @ (nil @ A))) @ ((lexord @ A) @ R2))))))). % 6.67/2.70 thf(fact_6964_lexord__partial__trans, axiom, (! [A:$tType,Xs:(list @ A),R2:(set @ ((product_prod @ A) @ A)),Ys:(list @ A),Zs2:(list @ A)]: (((! [X5:A,Y4:A,Z3:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Y4)) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ R2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Z3)) @ R2)))))) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((lexord @ A) @ R2)) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ys) @ Zs2)) @ ((lexord @ A) @ R2)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Zs2)) @ ((lexord @ A) @ R2)))))))). % 6.67/2.70 thf(fact_6965_List_Olexordp__def, axiom, (! [A:$tType]: (((lexordp @ A) = (^ [R5:(A > (A > $o)),Xs2:(list @ A),Ys3:(list @ A)]: ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs2) @ Ys3)) @ ((lexord @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ R5)))))))))). % 6.67/2.70 thf(fact_6966_Rat_Opositive__def, axiom, (positive = (((((((map_fun @ rat) @ ((product_prod @ int) @ int)) @ $o) @ $o) @ rep_Rat) @ (id @ $o)) @ (^ [X:((product_prod @ int) @ int)]: ((((ord_less @ int) @ (zero_zero @ int)) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X)) @ (((product_snd @ int) @ int) @ X)))))))). % 6.67/2.70 thf(fact_6967_lex__take__index, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((lex @ A) @ R2)) => (~ ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Ys)) => (((((take @ A) @ I4) @ Xs) = (((take @ A) @ I4) @ Ys)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (((nth @ A) @ Xs) @ I4)) @ (((nth @ A) @ Ys) @ I4))) @ R2)))))))))))))). % 6.67/2.70 thf(fact_6968_ntrancl__Suc, axiom, (! [A:$tType,N2:nat,R:(set @ ((product_prod @ A) @ A))]: (((((transitive_ntrancl @ A) @ (suc @ N2)) @ R) = (((((relcomp @ A) @ A) @ A) @ (((transitive_ntrancl @ A) @ N2) @ R)) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ (id2 @ A)) @ R)))))). % 6.67/2.70 thf(fact_6969_IdI, axiom, (! [A:$tType,A2:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ (id2 @ A))))). % 6.67/2.70 thf(fact_6970_pair__in__Id__conv, axiom, (! [A:$tType,A2:A,B2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ (id2 @ A)) = (A2 = B2))))). % 6.67/2.70 thf(fact_6971_rtrancl__empty, axiom, (! [A:$tType]: ((((transitive_rtrancl @ A) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))) = (id2 @ A))))). % 6.67/2.70 thf(fact_6972_Id__def, axiom, (! [A:$tType]: (((id2 @ A) = ((collect @ ((product_prod @ A) @ A)) @ (^ [P6:((product_prod @ A) @ A)]: ((? [X:A]: ((P6 = ((((product_Pair @ A) @ A) @ X) @ X))))))))))). % 6.67/2.70 thf(fact_6973_relpow_Osimps_I1_J, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((compow @ (set @ ((product_prod @ A) @ A))) @ (zero_zero @ nat)) @ R) = (id2 @ A))))). % 6.67/2.70 thf(fact_6974_IdD, axiom, (! [A:$tType,A2:A,B2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ (id2 @ A)) => (A2 = B2))))). % 6.67/2.70 thf(fact_6975_IdE, axiom, (! [A:$tType,P4:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ P4) @ (id2 @ A)) => (~ ((! [X5:A]: ((P4 != ((((product_Pair @ A) @ A) @ X5) @ X5)))))))))). % 6.67/2.70 thf(fact_6976_Nil__notin__lex, axiom, (! [A:$tType,Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: ((~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Ys)) @ ((lex @ A) @ R2))))))). % 6.67/2.70 thf(fact_6977_Nil2__notin__lex, axiom, (! [A:$tType,Xs:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: ((~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ (nil @ A))) @ ((lex @ A) @ R2))))))). % 6.67/2.70 thf(fact_6978_reflcl__set__eq, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((((sup_sup @ (A > (A > $o))) @ (^ [X:A,Y2:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y2)) @ R2)))) @ (^ [Y3:A,Z:A]: ((Y3 = Z)))) = (^ [X:A,Y2:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y2)) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ R2) @ (id2 @ A))))))))). % 6.67/2.70 thf(fact_6979_lexord__lex, axiom, (! [A:$tType,X3:(list @ A),Y:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X3) @ Y)) @ ((lex @ A) @ R2)) = ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X3) @ Y)) @ ((lexord @ A) @ R2)) & (((size_size @ (list @ A)) @ X3) = ((size_size @ (list @ A)) @ Y))))))). % 6.67/2.70 thf(fact_6980_lenlex__conv, axiom, (! [A:$tType]: (((lenlex @ A) = (^ [R5:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_case_prod @ (list @ A)) @ (list @ A)) @ $o) @ (^ [Xs2:(list @ A),Ys3:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xs2)) @ ((size_size @ (list @ A)) @ Ys3)) | ((((size_size @ (list @ A)) @ Xs2) = ((size_size @ (list @ A)) @ Ys3)) & (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs2) @ Ys3)) @ ((lex @ A) @ R5)))))))))))))). % 6.67/2.70 thf(fact_6981_map__of__zip__nth, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B),I:nat]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => (((distinct @ A) @ Xs) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ B)) @ Ys)) => (((((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Ys)) @ (((nth @ A) @ Xs) @ I)) = ((some @ B) @ (((nth @ B) @ Ys) @ I))))))))). % 6.67/2.70 thf(fact_6982_Nil__lenlex__iff1, axiom, (! [A:$tType,Ns:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Ns)) @ ((lenlex @ A) @ R2)) = (Ns != (nil @ A)))))). % 6.67/2.70 thf(fact_6983_map__of__eq__empty__iff, axiom, (! [B:$tType,A:$tType,Xys:(list @ ((product_prod @ A) @ B))]: ((((((map_of @ A) @ B) @ Xys) = (^ [X:A]: ((none @ B)))) = (Xys = (nil @ ((product_prod @ A) @ B))))))). % 6.67/2.70 thf(fact_6984_empty__eq__map__of__iff, axiom, (! [B:$tType,A:$tType,Xys:(list @ ((product_prod @ A) @ B))]: ((((^ [X:A]: ((none @ B))) = (((map_of @ A) @ B) @ Xys)) = (Xys = (nil @ ((product_prod @ A) @ B))))))). % 6.67/2.70 thf(fact_6985_map__of__zip__is__None, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys:(list @ B),X3:A]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => ((((((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Ys)) @ X3) = (none @ B)) = (~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs))))))))). % 6.67/2.70 thf(fact_6986_dom__map__of__zip, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys:(list @ B)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => ((((dom @ A) @ B) @ (((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Ys))) = ((set2 @ A) @ Xs)))))). % 6.67/2.70 thf(fact_6987_finite__dom__map__of, axiom, (! [B:$tType,A:$tType,L:(list @ ((product_prod @ A) @ B))]: (((finite_finite2 @ A) @ (((dom @ A) @ B) @ (((map_of @ A) @ B) @ L)))))). % 6.67/2.70 thf(fact_6988_finite__graph__map__of, axiom, (! [B:$tType,A:$tType,Al:(list @ ((product_prod @ A) @ B))]: (((finite_finite2 @ ((product_prod @ A) @ B)) @ (((graph @ A) @ B) @ (((map_of @ A) @ B) @ Al)))))). % 6.67/2.70 thf(fact_6989_map__of__Cons__code_I1_J, axiom, (! [B:$tType,A:$tType,K:B]: ((((((map_of @ B) @ A) @ (nil @ ((product_prod @ B) @ A))) @ K) = (none @ A))))). % 6.67/2.70 thf(fact_6990_map__of_Osimps_I1_J, axiom, (! [B:$tType,A:$tType]: (((((map_of @ A) @ B) @ (nil @ ((product_prod @ A) @ B))) = (^ [X:A]: ((none @ B))))))). % 6.67/2.70 thf(fact_6991_lenlex__irreflexive, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),Xs:(list @ A)]: (((! [X5:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ X5)) @ R2))))) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Xs)) @ ((lenlex @ A) @ R2)))))))). % 6.67/2.70 thf(fact_6992_Nil__lenlex__iff2, axiom, (! [A:$tType,Ns:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: ((~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ns) @ (nil @ A))) @ ((lenlex @ A) @ R2))))))). % 6.67/2.70 thf(fact_6993_weak__map__of__SomeI, axiom, (! [A:$tType,B:$tType,K:A,X3:B,L:(list @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ X3)) @ ((set2 @ ((product_prod @ A) @ B)) @ L)) => (? [X5:B]: ((((((map_of @ A) @ B) @ L) @ K) = ((some @ B) @ X5)))))))). % 6.67/2.70 thf(fact_6994_map__of__SomeD, axiom, (! [A:$tType,B:$tType,Xs:(list @ ((product_prod @ B) @ A)),K:B,Y:A]: (((((((map_of @ B) @ A) @ Xs) @ K) = ((some @ A) @ Y)) => (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ K) @ Y)) @ ((set2 @ ((product_prod @ B) @ A)) @ Xs)))))). % 6.67/2.70 thf(fact_6995_map__of__zip__inject, axiom, (! [B:$tType,A:$tType,Ys:(list @ A),Xs:(list @ B),Zs2:(list @ A)]: (((((size_size @ (list @ A)) @ Ys) = ((size_size @ (list @ B)) @ Xs)) => ((((size_size @ (list @ A)) @ Zs2) = ((size_size @ (list @ B)) @ Xs)) => (((distinct @ B) @ Xs) => (((((map_of @ B) @ A) @ ((((zip @ B) @ A) @ Xs) @ Ys)) = (((map_of @ B) @ A) @ ((((zip @ B) @ A) @ Xs) @ Zs2))) => (Ys = Zs2)))))))). % 6.67/2.70 thf(fact_6996_finite__range__map__of, axiom, (! [A:$tType,B:$tType,Xys:(list @ ((product_prod @ B) @ A))]: (((finite_finite2 @ (option @ A)) @ ((((image @ B) @ (option @ A)) @ (((map_of @ B) @ A) @ Xys)) @ (top_top @ (set @ B))))))). % 6.67/2.70 thf(fact_6997_map__of__zip__is__Some, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B),X3:A]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) = (? [Y2:B]: ((((((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Ys)) @ X3) = ((some @ B) @ Y2))))))))). % 6.67/2.70 thf(fact_6998_lenlex__length, axiom, (! [A:$tType,Ms:(list @ A),Ns:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ms) @ Ns)) @ ((lenlex @ A) @ R2)) => (((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Ms)) @ ((size_size @ (list @ A)) @ Ns)))))). % 6.67/2.70 thf(fact_6999_map__of__eq__None__iff, axiom, (! [A:$tType,B:$tType,Xys:(list @ ((product_prod @ B) @ A)),X3:B]: (((((((map_of @ B) @ A) @ Xys) @ X3) = (none @ A)) = (~ ((((member @ B) @ X3) @ ((((image @ ((product_prod @ B) @ A)) @ B) @ ((product_fst @ B) @ A)) @ ((set2 @ ((product_prod @ B) @ A)) @ Xys))))))))). % 6.67/2.70 thf(fact_7000_map__of__zip__upd, axiom, (! [A:$tType,B:$tType,Ys:(list @ B),Xs:(list @ A),Zs2:(list @ B),X3:A,Y:B,Z2:B]: (((((size_size @ (list @ B)) @ Ys) = ((size_size @ (list @ A)) @ Xs)) => ((((size_size @ (list @ B)) @ Zs2) = ((size_size @ (list @ A)) @ Xs)) => ((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)))) => (((((((fun_upd @ A) @ (option @ B)) @ (((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Ys))) @ X3) @ ((some @ B) @ Y)) = (((((fun_upd @ A) @ (option @ B)) @ (((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Zs2))) @ X3) @ ((some @ B) @ Z2))) => ((((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Ys)) = (((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Zs2)))))))))). % 6.67/2.70 thf(fact_7001_ran__map__of__zip, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => (((distinct @ A) @ Xs) => ((((ran @ A) @ B) @ (((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Ys))) = ((set2 @ B) @ Ys))))))). % 6.67/2.70 thf(fact_7002_map__of__is__SomeI, axiom, (! [A:$tType,B:$tType,Xys:(list @ ((product_prod @ A) @ B)),X3:A,Y:B]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xys)) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) @ ((set2 @ ((product_prod @ A) @ B)) @ Xys)) => (((((map_of @ A) @ B) @ Xys) @ X3) = ((some @ B) @ Y))))))). % 6.67/2.70 thf(fact_7003_Some__eq__map__of__iff, axiom, (! [B:$tType,A:$tType,Xys:(list @ ((product_prod @ A) @ B)),Y:B,X3:A]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xys)) => ((((some @ B) @ Y) = ((((map_of @ A) @ B) @ Xys) @ X3)) = (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) @ ((set2 @ ((product_prod @ A) @ B)) @ Xys))))))). % 6.67/2.70 thf(fact_7004_map__ident, axiom, (! [A:$tType]: (((((map @ A) @ A) @ (^ [X:A]: (X))) = (^ [Xs2:(list @ A)]: (Xs2)))))). % 6.67/2.70 thf(fact_7005_list_Omap__disc__iff, axiom, (! [B:$tType,A:$tType,F2:(A > B),A2:(list @ A)]: (((((((map @ A) @ B) @ F2) @ A2) = (nil @ B)) = (A2 = (nil @ A)))))). % 6.67/2.70 thf(fact_7006_Nil__is__map__conv, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B)]: ((((nil @ A) = ((((map @ B) @ A) @ F2) @ Xs)) = (Xs = (nil @ B)))))). % 6.67/2.70 thf(fact_7007_map__is__Nil__conv, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B)]: (((((((map @ B) @ A) @ F2) @ Xs) = (nil @ A)) = (Xs = (nil @ B)))))). % 6.67/2.70 thf(fact_7008_map__map, axiom, (! [B:$tType,A:$tType,C:$tType,F2:(B > A),G:(C > B),Xs:(list @ C)]: ((((((map @ B) @ A) @ F2) @ ((((map @ C) @ B) @ G) @ Xs)) = ((((map @ C) @ A) @ (((((comp @ B) @ A) @ C) @ F2) @ G)) @ Xs))))). % 6.67/2.70 thf(fact_7009_List_Omap_Ocompositionality, axiom, (! [B:$tType,C:$tType,A:$tType,F2:(B > C),G:(A > B),List:(list @ A)]: ((((((map @ B) @ C) @ F2) @ ((((map @ A) @ B) @ G) @ List)) = ((((map @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F2) @ G)) @ List))))). % 6.67/2.70 thf(fact_7010_list_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,G:(B > C),F2:(A > B),V2:(list @ A)]: ((((((map @ B) @ C) @ G) @ ((((map @ A) @ B) @ F2) @ V2)) = ((((map @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F2)) @ V2))))). % 6.67/2.70 thf(fact_7011_map__eq__conv, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B),G:(B > A)]: (((((((map @ B) @ A) @ F2) @ Xs) = ((((map @ B) @ A) @ G) @ Xs)) = (! [X:B]: (((((member @ B) @ X) @ ((set2 @ B) @ Xs)) => ((F2 @ X) = (G @ X))))))))). % 6.67/2.70 thf(fact_7012_length__map, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B)]: ((((size_size @ (list @ A)) @ ((((map @ B) @ A) @ F2) @ Xs)) = ((size_size @ (list @ B)) @ Xs))))). % 6.67/2.70 thf(fact_7013_list_Omap__id0, axiom, (! [A:$tType]: (((((map @ A) @ A) @ (id @ A)) = (id @ (list @ A)))))). % 6.67/2.70 thf(fact_7014_map__replicate, axiom, (! [A:$tType,B:$tType,F2:(B > A),N2:nat,X3:B]: ((((((map @ B) @ A) @ F2) @ (((replicate @ B) @ N2) @ X3)) = (((replicate @ A) @ N2) @ (F2 @ X3)))))). % 6.67/2.70 thf(fact_7015_list_Oset__map, axiom, (! [B:$tType,A:$tType,F2:(A > B),V2:(list @ A)]: ((((set2 @ B) @ ((((map @ A) @ B) @ F2) @ V2)) = ((((image @ A) @ B) @ F2) @ ((set2 @ A) @ V2)))))). % 6.67/2.70 thf(fact_7016_map__snd__enumerate, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((((map @ ((product_prod @ nat) @ A)) @ A) @ ((product_snd @ nat) @ A)) @ (((enumerate @ A) @ N2) @ Xs)) = Xs)))). % 6.67/2.70 thf(fact_7017_inj__map__eq__map, axiom, (! [B:$tType,A:$tType,F2:(A > B),Xs:(list @ A),Ys:(list @ A)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((((map @ A) @ B) @ F2) @ Xs) = ((((map @ A) @ B) @ F2) @ Ys)) = (Xs = Ys)))))). % 6.67/2.70 thf(fact_7018_map__fun__upd, axiom, (! [B:$tType,A:$tType,Y:A,Xs:(list @ A),F2:(A > B),V2:B]: (((~ ((((member @ A) @ Y) @ ((set2 @ A) @ Xs)))) => (((((map @ A) @ B) @ (((((fun_upd @ A) @ B) @ F2) @ Y) @ V2)) @ Xs) = ((((map @ A) @ B) @ F2) @ Xs)))))). % 6.67/2.70 thf(fact_7019_map__comp__map, axiom, (! [B:$tType,C:$tType,A:$tType,F2:(C > B),G:(A > C)]: (((((((comp @ (list @ C)) @ (list @ B)) @ (list @ A)) @ (((map @ C) @ B) @ F2)) @ (((map @ A) @ C) @ G)) = (((map @ A) @ B) @ (((((comp @ C) @ B) @ A) @ F2) @ G)))))). % 6.67/2.70 thf(fact_7020_List_Omap_Ocomp, axiom, (! [C:$tType,B:$tType,A:$tType,F2:(B > C),G:(A > B)]: (((((((comp @ (list @ B)) @ (list @ C)) @ (list @ A)) @ (((map @ B) @ C) @ F2)) @ (((map @ A) @ B) @ G)) = (((map @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F2) @ G)))))). % 6.67/2.70 thf(fact_7021_size__list__map, axiom, (! [A:$tType,B:$tType,F2:(A > nat),G:(B > A),Xs:(list @ B)]: (((((size_list @ A) @ F2) @ ((((map @ B) @ A) @ G) @ Xs)) = (((size_list @ B) @ (((((comp @ A) @ nat) @ B) @ F2) @ G)) @ Xs))))). % 6.67/2.70 thf(fact_7022_nth__map, axiom, (! [B:$tType,A:$tType,N2:nat,Xs:(list @ A),F2:(A > B)]: (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ B) @ ((((map @ A) @ B) @ F2) @ Xs)) @ N2) = (F2 @ (((nth @ A) @ Xs) @ N2))))))). % 6.67/2.70 thf(fact_7023_map__fst__zip, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys:(list @ B)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => (((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys)) = Xs))))). % 6.67/2.70 thf(fact_7024_map__snd__zip, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => (((((map @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys)) = Ys))))). % 6.67/2.70 thf(fact_7025_inj__map, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: ((((((inj_on @ (list @ A)) @ (list @ B)) @ (((map @ A) @ B) @ F2)) @ (top_top @ (set @ (list @ A)))) = ((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))))))). % 6.67/2.70 thf(fact_7026_inj__mapI, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((inj_on @ (list @ A)) @ (list @ B)) @ (((map @ A) @ B) @ F2)) @ (top_top @ (set @ (list @ A)))))))). % 6.67/2.70 thf(fact_7027_map__of__eq__Some__iff, axiom, (! [B:$tType,A:$tType,Xys:(list @ ((product_prod @ A) @ B)),X3:A,Y:B]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xys)) => ((((((map_of @ A) @ B) @ Xys) @ X3) = ((some @ B) @ Y)) = (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) @ ((set2 @ ((product_prod @ A) @ B)) @ Xys))))))). % 6.67/2.70 thf(fact_7028_List_Omap_Oidentity, axiom, (! [A:$tType]: (((((map @ A) @ A) @ (^ [X:A]: (X))) = (id @ (list @ A)))))). % 6.67/2.70 thf(fact_7029_list_Omap__id, axiom, (! [A:$tType,T2:(list @ A)]: ((((((map @ A) @ A) @ (id @ A)) @ T2) = T2)))). % 6.67/2.70 thf(fact_7030_list_Osize__gen__o__map, axiom, (! [B:$tType,A:$tType,F2:(B > nat),G:(A > B)]: (((((((comp @ (list @ B)) @ nat) @ (list @ A)) @ ((size_list @ B) @ F2)) @ (((map @ A) @ B) @ G)) = ((size_list @ A) @ (((((comp @ B) @ nat) @ A) @ F2) @ G)))))). % 6.67/2.70 thf(fact_7031_map2__map__map, axiom, (! [B:$tType,A:$tType,C:$tType,D:$tType,H2:(B > (C > A)),F2:(D > B),Xs:(list @ D),G:(D > C)]: ((((((map @ ((product_prod @ B) @ C)) @ A) @ ((((product_case_prod @ B) @ C) @ A) @ H2)) @ ((((zip @ B) @ C) @ ((((map @ D) @ B) @ F2) @ Xs)) @ ((((map @ D) @ C) @ G) @ Xs))) = ((((map @ D) @ A) @ (^ [X:D]: (((H2 @ (F2 @ X)) @ (G @ X))))) @ Xs))))). % 6.67/2.70 thf(fact_7032_zip__map__map, axiom, (! [B:$tType,A:$tType,C:$tType,D:$tType,F2:(C > A),Xs:(list @ C),G:(D > B),Ys:(list @ D)]: ((((((zip @ A) @ B) @ ((((map @ C) @ A) @ F2) @ Xs)) @ ((((map @ D) @ B) @ G) @ Ys)) = ((((map @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ C) @ D) @ ((product_prod @ A) @ B)) @ (^ [X:C,Y2:D]: (((((product_Pair @ A) @ B) @ (F2 @ X)) @ (G @ Y2)))))) @ ((((zip @ C) @ D) @ Xs) @ Ys)))))). % 6.67/2.70 thf(fact_7033_zip__map2, axiom, (! [B:$tType,A:$tType,C:$tType,Xs:(list @ A),F2:(C > B),Ys:(list @ C)]: ((((((zip @ A) @ B) @ Xs) @ ((((map @ C) @ B) @ F2) @ Ys)) = ((((map @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ C) @ ((product_prod @ A) @ B)) @ (^ [X:A,Y2:C]: (((((product_Pair @ A) @ B) @ X) @ (F2 @ Y2)))))) @ ((((zip @ A) @ C) @ Xs) @ Ys)))))). % 6.67/2.70 thf(fact_7034_zip__map1, axiom, (! [A:$tType,C:$tType,B:$tType,F2:(C > A),Xs:(list @ C),Ys:(list @ B)]: ((((((zip @ A) @ B) @ ((((map @ C) @ A) @ F2) @ Xs)) @ Ys) = ((((map @ ((product_prod @ C) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ C) @ B) @ ((product_prod @ A) @ B)) @ (^ [X:C]: ((((product_Pair @ A) @ B) @ (F2 @ X)))))) @ ((((zip @ C) @ B) @ Xs) @ Ys)))))). % 6.67/2.70 thf(fact_7035_map__zip__map2, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F2:(((product_prod @ B) @ C) > A),Xs:(list @ B),G:(D > C),Ys:(list @ D)]: ((((((map @ ((product_prod @ B) @ C)) @ A) @ F2) @ ((((zip @ B) @ C) @ Xs) @ ((((map @ D) @ C) @ G) @ Ys))) = ((((map @ ((product_prod @ B) @ D)) @ A) @ ((((product_case_prod @ B) @ D) @ A) @ (^ [X:B,Y2:D]: ((F2 @ ((((product_Pair @ B) @ C) @ X) @ (G @ Y2))))))) @ ((((zip @ B) @ D) @ Xs) @ Ys)))))). % 6.67/2.70 thf(fact_7036_map__zip__map, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F2:(((product_prod @ B) @ C) > A),G:(D > B),Xs:(list @ D),Ys:(list @ C)]: ((((((map @ ((product_prod @ B) @ C)) @ A) @ F2) @ ((((zip @ B) @ C) @ ((((map @ D) @ B) @ G) @ Xs)) @ Ys)) = ((((map @ ((product_prod @ D) @ C)) @ A) @ ((((product_case_prod @ D) @ C) @ A) @ (^ [X:D,Y2:C]: ((F2 @ ((((product_Pair @ B) @ C) @ (G @ X)) @ Y2)))))) @ ((((zip @ D) @ C) @ Xs) @ Ys)))))). % 6.67/2.70 thf(fact_7037_map__injective, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B),Ys:(list @ B)]: (((((((map @ B) @ A) @ F2) @ Xs) = ((((map @ B) @ A) @ F2) @ Ys)) => (((((inj_on @ B) @ A) @ F2) @ (top_top @ (set @ B))) => (Xs = Ys)))))). % 6.67/2.70 thf(fact_7038_remdups__map__remdups, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B)]: ((((remdups @ A) @ ((((map @ B) @ A) @ F2) @ ((remdups @ B) @ Xs))) = ((remdups @ A) @ ((((map @ B) @ A) @ F2) @ Xs)))))). % 6.67/2.70 thf(fact_7039_map__eq__imp__length__eq, axiom, (! [A:$tType,B:$tType,C:$tType,F2:(B > A),Xs:(list @ B),G:(C > A),Ys:(list @ C)]: (((((((map @ B) @ A) @ F2) @ Xs) = ((((map @ C) @ A) @ G) @ Ys)) => (((size_size @ (list @ B)) @ Xs) = ((size_size @ (list @ C)) @ Ys)))))). % 6.67/2.70 thf(fact_7040_rotate1__map, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B)]: ((((rotate1 @ A) @ ((((map @ B) @ A) @ F2) @ Xs)) = ((((map @ B) @ A) @ F2) @ ((rotate1 @ B) @ Xs)))))). % 6.67/2.70 thf(fact_7041_list_Omap__ident, axiom, (! [A:$tType,T2:(list @ A)]: ((((((map @ A) @ A) @ (^ [X:A]: (X))) @ T2) = T2)))). % 6.67/2.70 thf(fact_7042_map__concat, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ (list @ B))]: ((((((map @ B) @ A) @ F2) @ ((concat @ B) @ Xs)) = ((concat @ A) @ ((((map @ (list @ B)) @ (list @ A)) @ (((map @ B) @ A) @ F2)) @ Xs)))))). % 6.67/2.70 thf(fact_7043_map__update, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B),K:nat,Y:B]: ((((((map @ B) @ A) @ F2) @ ((((list_update @ B) @ Xs) @ K) @ Y)) = ((((list_update @ A) @ ((((map @ B) @ A) @ F2) @ Xs)) @ K) @ (F2 @ Y)))))). % 6.67/2.70 thf(fact_7044_list_Osimps_I8_J, axiom, (! [A:$tType,B:$tType,F2:(A > B)]: ((((((map @ A) @ B) @ F2) @ (nil @ A)) = (nil @ B))))). % 6.67/2.70 thf(fact_7045_list_Omap__cong, axiom, (! [B:$tType,A:$tType,X3:(list @ A),Ya:(list @ A),F2:(A > B),G:(A > B)]: (((X3 = Ya) => ((! [Z3:A]: (((((member @ A) @ Z3) @ ((set2 @ A) @ Ya)) => ((F2 @ Z3) = (G @ Z3))))) => (((((map @ A) @ B) @ F2) @ X3) = ((((map @ A) @ B) @ G) @ Ya))))))). % 6.67/2.70 thf(fact_7046_list_Omap__cong0, axiom, (! [B:$tType,A:$tType,X3:(list @ A),F2:(A > B),G:(A > B)]: (((! [Z3:A]: (((((member @ A) @ Z3) @ ((set2 @ A) @ X3)) => ((F2 @ Z3) = (G @ Z3))))) => (((((map @ A) @ B) @ F2) @ X3) = ((((map @ A) @ B) @ G) @ X3)))))). % 6.67/2.70 thf(fact_7047_list_Oinj__map__strong, axiom, (! [B:$tType,A:$tType,X3:(list @ A),Xa2:(list @ A),F2:(A > B),Fa:(A > B)]: (((! [Z3:A,Za2:A]: (((((member @ A) @ Z3) @ ((set2 @ A) @ X3)) => ((((member @ A) @ Za2) @ ((set2 @ A) @ Xa2)) => (((F2 @ Z3) = (Fa @ Za2)) => (Z3 = Za2)))))) => ((((((map @ A) @ B) @ F2) @ X3) = ((((map @ A) @ B) @ Fa) @ Xa2)) => (X3 = Xa2)))))). % 6.67/2.70 thf(fact_7048_map__ext, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),F2:(A > B),G:(A > B)]: (((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => ((F2 @ X5) = (G @ X5))))) => (((((map @ A) @ B) @ F2) @ Xs) = ((((map @ A) @ B) @ G) @ Xs)))))). % 6.67/2.70 thf(fact_7049_map__idI, axiom, (! [A:$tType,Xs:(list @ A),F2:(A > A)]: (((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => ((F2 @ X5) = X5)))) => (((((map @ A) @ A) @ F2) @ Xs) = Xs))))). % 6.67/2.70 thf(fact_7050_map__cong, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys:(list @ A),F2:(A > B),G:(A > B)]: (((Xs = Ys) => ((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Ys)) => ((F2 @ X5) = (G @ X5))))) => (((((map @ A) @ B) @ F2) @ Xs) = ((((map @ A) @ B) @ G) @ Ys))))))). % 6.67/2.70 thf(fact_7051_ex__map__conv, axiom, (! [A:$tType,B:$tType,Ys:(list @ B),F2:(A > B)]: (((? [Xs2:(list @ A)]: ((Ys = ((((map @ A) @ B) @ F2) @ Xs2)))) = (! [X:B]: (((((member @ B) @ X) @ ((set2 @ B) @ Ys)) => (? [Y2:A]: ((X = (F2 @ Y2))))))))))). % 6.67/2.70 thf(fact_7052_nths__map, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B),I6:(set @ nat)]: (((((nths @ A) @ ((((map @ B) @ A) @ F2) @ Xs)) @ I6) = ((((map @ B) @ A) @ F2) @ (((nths @ B) @ Xs) @ I6)))))). % 6.67/2.70 thf(fact_7053_pair__list__eqI, axiom, (! [B:$tType,A:$tType,Xs:(list @ ((product_prod @ A) @ B)),Ys:(list @ ((product_prod @ A) @ B))]: (((((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs) = ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Ys)) => ((((((map @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ Xs) = ((((map @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ Ys)) => (Xs = Ys)))))). % 6.67/2.70 thf(fact_7054_enumerate__Suc__eq, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((enumerate @ A) @ (suc @ N2)) @ Xs) = ((((map @ ((product_prod @ nat) @ A)) @ ((product_prod @ nat) @ A)) @ ((((product_apfst @ nat) @ nat) @ A) @ suc)) @ (((enumerate @ A) @ N2) @ Xs)))))). % 6.67/2.70 thf(fact_7055_map__replicate__const, axiom, (! [B:$tType,A:$tType,K:A,Lst:(list @ B)]: ((((((map @ B) @ A) @ (^ [X:B]: (K))) @ Lst) = (((replicate @ A) @ ((size_size @ (list @ B)) @ Lst)) @ K))))). % 6.67/2.70 thf(fact_7056_foldl__map, axiom, (! [A:$tType,B:$tType,C:$tType,G:(A > (B > A)),A2:A,F2:(C > B),Xs:(list @ C)]: (((((((foldl @ A) @ B) @ G) @ A2) @ ((((map @ C) @ B) @ F2) @ Xs)) = (((((foldl @ A) @ C) @ (^ [A4:A,X:C]: (((G @ A4) @ (F2 @ X))))) @ A2) @ Xs))))). % 6.67/2.70 thf(fact_7057_image__set, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B)]: ((((((image @ B) @ A) @ F2) @ ((set2 @ B) @ Xs)) = ((set2 @ A) @ ((((map @ B) @ A) @ F2) @ Xs)))))). % 6.67/2.70 thf(fact_7058_take__map, axiom, (! [A:$tType,B:$tType,N2:nat,F2:(B > A),Xs:(list @ B)]: (((((take @ A) @ N2) @ ((((map @ B) @ A) @ F2) @ Xs)) = ((((map @ B) @ A) @ F2) @ (((take @ B) @ N2) @ Xs)))))). % 6.67/2.70 thf(fact_7059_inj__on__map__eq__map, axiom, (! [B:$tType,A:$tType,F2:(A > B),Xs:(list @ A),Ys:(list @ A)]: ((((((inj_on @ A) @ B) @ F2) @ (((sup_sup @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys))) => ((((((map @ A) @ B) @ F2) @ Xs) = ((((map @ A) @ B) @ F2) @ Ys)) = (Xs = Ys)))))). % 6.67/2.70 thf(fact_7060_map__inj__on, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B),Ys:(list @ B)]: (((((((map @ B) @ A) @ F2) @ Xs) = ((((map @ B) @ A) @ F2) @ Ys)) => (((((inj_on @ B) @ A) @ F2) @ (((sup_sup @ (set @ B)) @ ((set2 @ B) @ Xs)) @ ((set2 @ B) @ Ys))) => (Xs = Ys)))))). % 6.67/2.70 thf(fact_7061_distinct__map, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B)]: ((((distinct @ A) @ ((((map @ B) @ A) @ F2) @ Xs)) = (((distinct @ B) @ Xs) & ((((inj_on @ B) @ A) @ F2) @ ((set2 @ B) @ Xs))))))). % 6.67/2.70 thf(fact_7062_remdups__adj__map__injective, axiom, (! [B:$tType,A:$tType,F2:(A > B),Xs:(list @ A)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => (((remdups_adj @ B) @ ((((map @ A) @ B) @ F2) @ Xs)) = ((((map @ A) @ B) @ F2) @ ((remdups_adj @ A) @ Xs))))))). % 6.67/2.70 thf(fact_7063_zip__map__fst__snd, axiom, (! [B:$tType,A:$tType,Zs2:(list @ ((product_prod @ A) @ B))]: ((((((zip @ A) @ B) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Zs2)) @ ((((map @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ Zs2)) = Zs2)))). % 6.67/2.70 thf(fact_7064_map__of__eqI, axiom, (! [B:$tType,A:$tType,Xs:(list @ ((product_prod @ A) @ B)),Ys:(list @ ((product_prod @ A) @ B))]: (((((set2 @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs)) = ((set2 @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Ys))) => ((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs))) => (((((map_of @ A) @ B) @ Xs) @ X5) = ((((map_of @ A) @ B) @ Ys) @ X5))))) => ((((map_of @ A) @ B) @ Xs) = (((map_of @ A) @ B) @ Ys))))))). % 6.67/2.70 thf(fact_7065_map__removeAll__inj, axiom, (! [B:$tType,A:$tType,F2:(A > B),X3:A,Xs:(list @ A)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => (((((map @ A) @ B) @ F2) @ (((removeAll @ A) @ X3) @ Xs)) = (((removeAll @ B) @ (F2 @ X3)) @ ((((map @ A) @ B) @ F2) @ Xs))))))). % 6.67/2.70 thf(fact_7066_inj__mapD, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: ((((((inj_on @ (list @ A)) @ (list @ B)) @ (((map @ A) @ B) @ F2)) @ (top_top @ (set @ (list @ A)))) => ((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))))))). % 6.67/2.70 thf(fact_7067_zip__assoc, axiom, (! [B:$tType,A:$tType,C:$tType,Xs:(list @ A),Ys:(list @ B),Zs2:(list @ C)]: ((((((zip @ A) @ ((product_prod @ B) @ C)) @ Xs) @ ((((zip @ B) @ C) @ Ys) @ Zs2)) = ((((map @ ((product_prod @ ((product_prod @ A) @ B)) @ C)) @ ((product_prod @ A) @ ((product_prod @ B) @ C))) @ ((((product_case_prod @ ((product_prod @ A) @ B)) @ C) @ ((product_prod @ A) @ ((product_prod @ B) @ C))) @ ((((product_case_prod @ A) @ B) @ (C > ((product_prod @ A) @ ((product_prod @ B) @ C)))) @ (^ [X:A,Y2:B,Z6:C]: (((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ X) @ ((((product_Pair @ B) @ C) @ Y2) @ Z6))))))) @ ((((zip @ ((product_prod @ A) @ B)) @ C) @ ((((zip @ A) @ B) @ Xs) @ Ys)) @ Zs2)))))). % 6.67/2.70 thf(fact_7068_zip__left__commute, axiom, (! [B:$tType,A:$tType,C:$tType,Xs:(list @ A),Ys:(list @ B),Zs2:(list @ C)]: ((((((zip @ A) @ ((product_prod @ B) @ C)) @ Xs) @ ((((zip @ B) @ C) @ Ys) @ Zs2)) = ((((map @ ((product_prod @ B) @ ((product_prod @ A) @ C))) @ ((product_prod @ A) @ ((product_prod @ B) @ C))) @ ((((product_case_prod @ B) @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ ((product_prod @ B) @ C))) @ (^ [Y2:B]: (((((product_case_prod @ A) @ C) @ ((product_prod @ A) @ ((product_prod @ B) @ C))) @ (^ [X:A,Z6:C]: (((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ X) @ ((((product_Pair @ B) @ C) @ Y2) @ Z6))))))))) @ ((((zip @ B) @ ((product_prod @ A) @ C)) @ Ys) @ ((((zip @ A) @ C) @ Xs) @ Zs2))))))). % 6.67/2.70 thf(fact_7069_zip__commute, axiom, (! [B:$tType,A:$tType]: ((((zip @ A) @ B) = (^ [Xs2:(list @ A),Ys3:(list @ B)]: (((((map @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ B) @ A) @ ((product_prod @ A) @ B)) @ (^ [X:B,Y2:A]: (((((product_Pair @ A) @ B) @ Y2) @ X))))) @ ((((zip @ B) @ A) @ Ys3) @ Xs2)))))))). % 6.67/2.70 thf(fact_7070_zip__eq__conv, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B),Zs2:(list @ ((product_prod @ A) @ B))]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => ((((((zip @ A) @ B) @ Xs) @ Ys) = Zs2) = ((((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Zs2) = Xs) & (((((map @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ Zs2) = Ys))))))). % 6.67/2.70 thf(fact_7071_distinct__insort__key, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),X3:B,Xs:(list @ B)]: ((((distinct @ A) @ ((((map @ B) @ A) @ F2) @ (((((linorder_insort_key @ B) @ A) @ F2) @ X3) @ Xs))) = ((~ ((((member @ A) @ (F2 @ X3)) @ ((((image @ B) @ A) @ F2) @ ((set2 @ B) @ Xs))))) & ((distinct @ A) @ ((((map @ B) @ A) @ F2) @ Xs)))))))))). % 6.67/2.70 thf(fact_7072_map__removeAll__inj__on, axiom, (! [B:$tType,A:$tType,F2:(A > B),X3:A,Xs:(list @ A)]: ((((((inj_on @ A) @ B) @ F2) @ (((insert @ A) @ X3) @ ((set2 @ A) @ Xs))) => (((((map @ A) @ B) @ F2) @ (((removeAll @ A) @ X3) @ Xs)) = (((removeAll @ B) @ (F2 @ X3)) @ ((((map @ A) @ B) @ F2) @ Xs))))))). % 6.67/2.70 thf(fact_7073_eq__key__imp__eq__value, axiom, (! [A:$tType,B:$tType,Xs:(list @ ((product_prod @ A) @ B)),K:A,V1:B,V22:B]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs)) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ V1)) @ ((set2 @ ((product_prod @ A) @ B)) @ Xs)) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ V22)) @ ((set2 @ ((product_prod @ A) @ B)) @ Xs)) => (V1 = V22))))))). % 6.67/2.70 thf(fact_7074_inj__on__mapI, axiom, (! [B:$tType,A:$tType,F2:(A > B),A3:(set @ (list @ A))]: ((((((inj_on @ A) @ B) @ F2) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ (list @ A)) @ (set @ A)) @ (set2 @ A)) @ A3))) => ((((inj_on @ (list @ A)) @ (list @ B)) @ (((map @ A) @ B) @ F2)) @ A3))))). % 6.67/2.70 thf(fact_7075_map__of__zip__map, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),F2:(A > B)]: (((((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ ((((map @ A) @ B) @ F2) @ Xs))) = (^ [X:A]: (((((if @ (option @ B)) @ (((member @ A) @ X) @ ((set2 @ A) @ Xs))) @ ((some @ B) @ (F2 @ X))) @ (none @ B)))))))). % 6.67/2.70 thf(fact_7076_map__fst__zip__take, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys:(list @ B)]: ((((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys)) = (((take @ A) @ (((ord_min @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ B)) @ Ys))) @ Xs))))). % 6.67/2.70 thf(fact_7077_map__snd__zip__take, axiom, (! [B:$tType,A:$tType,Xs:(list @ B),Ys:(list @ A)]: ((((((map @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ ((((zip @ B) @ A) @ Xs) @ Ys)) = (((take @ A) @ (((ord_min @ nat) @ ((size_size @ (list @ B)) @ Xs)) @ ((size_size @ (list @ A)) @ Ys))) @ Ys))))). % 6.67/2.70 thf(fact_7078_map__of__mapk__SomeI, axiom, (! [A:$tType,B:$tType,C:$tType,F2:(A > B),T2:(list @ ((product_prod @ A) @ C)),K:A,X3:C]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((((map_of @ A) @ C) @ T2) @ K) = ((some @ C) @ X3)) => (((((map_of @ B) @ C) @ ((((map @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ C)) @ ((((product_case_prod @ A) @ C) @ ((product_prod @ B) @ C)) @ (^ [K3:A]: ((((product_Pair @ B) @ C) @ (F2 @ K3)))))) @ T2)) @ (F2 @ K)) = ((some @ C) @ X3))))))). % 6.67/2.70 thf(fact_7079_set__map__of__compr, axiom, (! [B:$tType,A:$tType,Xs:(list @ ((product_prod @ A) @ B))]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs)) => (((set2 @ ((product_prod @ A) @ B)) @ Xs) = ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [K3:A,V5:B]: ((((((map_of @ A) @ B) @ Xs) @ K3) = ((some @ B) @ V5))))))))))). % 6.67/2.70 thf(fact_7080_set__relcomp, axiom, (! [B:$tType,C:$tType,A:$tType,Xys:(list @ ((product_prod @ A) @ C)),Yzs:(list @ ((product_prod @ C) @ B))]: (((((((relcomp @ A) @ C) @ B) @ ((set2 @ ((product_prod @ A) @ C)) @ Xys)) @ ((set2 @ ((product_prod @ C) @ B)) @ Yzs)) = ((set2 @ ((product_prod @ A) @ B)) @ ((concat @ ((product_prod @ A) @ B)) @ ((((map @ ((product_prod @ A) @ C)) @ (list @ ((product_prod @ A) @ B))) @ (^ [Xy:((product_prod @ A) @ C)]: (((concat @ ((product_prod @ A) @ B)) @ ((((map @ ((product_prod @ C) @ B)) @ (list @ ((product_prod @ A) @ B))) @ (^ [Yz:((product_prod @ C) @ B)]: (((((if @ (list @ ((product_prod @ A) @ B))) @ ((((product_snd @ A) @ C) @ Xy) = (((product_fst @ C) @ B) @ Yz))) @ (((cons @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ C) @ Xy)) @ (((product_snd @ C) @ B) @ Yz))) @ (nil @ ((product_prod @ A) @ B)))) @ (nil @ ((product_prod @ A) @ B)))))) @ Yzs))))) @ Xys))))))). % 6.67/2.70 thf(fact_7081_dual__max, axiom, (! [A:$tType]: (((linorder @ A) => (((max @ A) @ (^ [X:A,Y2:A]: ((((ord_less_eq @ A) @ Y2) @ X)))) = (ord_min @ A)))))). % 6.67/2.70 thf(fact_7082_list_Oinject, axiom, (! [A:$tType,X21:A,X222:(list @ A),Y21:A,Y222:(list @ A)]: ((((((cons @ A) @ X21) @ X222) = (((cons @ A) @ Y21) @ Y222)) = ((X21 = Y21) & (X222 = Y222)))))). % 6.67/2.70 thf(fact_7083_list_Osimps_I15_J, axiom, (! [A:$tType,X21:A,X222:(list @ A)]: ((((set2 @ A) @ (((cons @ A) @ X21) @ X222)) = (((insert @ A) @ X21) @ ((set2 @ A) @ X222)))))). % 6.67/2.70 thf(fact_7084_nth__Cons__Suc, axiom, (! [A:$tType,X3:A,Xs:(list @ A),N2:nat]: (((((nth @ A) @ (((cons @ A) @ X3) @ Xs)) @ (suc @ N2)) = (((nth @ A) @ Xs) @ N2))))). % 6.67/2.70 thf(fact_7085_nth__Cons__0, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((nth @ A) @ (((cons @ A) @ X3) @ Xs)) @ (zero_zero @ nat)) = X3)))). % 6.67/2.70 thf(fact_7086_take__Suc__Cons, axiom, (! [A:$tType,N2:nat,X3:A,Xs:(list @ A)]: (((((take @ A) @ (suc @ N2)) @ (((cons @ A) @ X3) @ Xs)) = (((cons @ A) @ X3) @ (((take @ A) @ N2) @ Xs)))))). % 6.67/2.70 thf(fact_7087_nths__singleton, axiom, (! [A:$tType,A3:(set @ nat),X3:A]: ((((((member @ nat) @ (zero_zero @ nat)) @ A3) => ((((nths @ A) @ (((cons @ A) @ X3) @ (nil @ A))) @ A3) = (((cons @ A) @ X3) @ (nil @ A)))) & ((~ ((((member @ nat) @ (zero_zero @ nat)) @ A3))) => ((((nths @ A) @ (((cons @ A) @ X3) @ (nil @ A))) @ A3) = (nil @ A))))))). % 6.67/2.70 thf(fact_7088_Cons__listrel1__Cons, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Y:A,Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ Y) @ Ys))) @ ((listrel1 @ A) @ R2)) = (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ R2) & (Xs = Ys)) | ((X3 = Y) & (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((listrel1 @ A) @ R2)))))))). % 6.67/2.70 thf(fact_7089_horner__sum__simps_I2_J, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_0 @ A) => (! [F2:(B > A),A2:A,X3:B,Xs:(list @ B)]: (((((((groups4207007520872428315er_sum @ B) @ A) @ F2) @ A2) @ (((cons @ B) @ X3) @ Xs)) = (((plus_plus @ A) @ (F2 @ X3)) @ (((times_times @ A) @ A2) @ (((((groups4207007520872428315er_sum @ B) @ A) @ F2) @ A2) @ Xs)))))))))). % 6.67/2.70 thf(fact_7090_lexord__cons__cons, axiom, (! [A:$tType,A2:A,X3:(list @ A),B2:A,Y:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ A2) @ X3)) @ (((cons @ A) @ B2) @ Y))) @ ((lexord @ A) @ R2)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2) | ((A2 = B2) & (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X3) @ Y)) @ ((lexord @ A) @ R2)))))))). % 6.67/2.70 thf(fact_7091_lexord__Nil__left, axiom, (! [A:$tType,Y:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Y)) @ ((lexord @ A) @ R2)) = (? [A4:A,X:(list @ A)]: ((Y = (((cons @ A) @ A4) @ X)))))))). % 6.67/2.70 thf(fact_7092_enumerate__simps_I2_J, axiom, (! [B:$tType,N2:nat,X3:B,Xs:(list @ B)]: (((((enumerate @ B) @ N2) @ (((cons @ B) @ X3) @ Xs)) = (((cons @ ((product_prod @ nat) @ B)) @ ((((product_Pair @ nat) @ B) @ N2) @ X3)) @ (((enumerate @ B) @ (suc @ N2)) @ Xs)))))). % 6.67/2.70 thf(fact_7093_map__upds__Cons, axiom, (! [A:$tType,B:$tType,M:(A > (option @ B)),A2:A,As:(list @ A),B2:B,Bs:(list @ B)]: (((((((map_upds @ A) @ B) @ M) @ (((cons @ A) @ A2) @ As)) @ (((cons @ B) @ B2) @ Bs)) = (((((map_upds @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M) @ A2) @ ((some @ B) @ B2))) @ As) @ Bs))))). % 6.67/2.70 thf(fact_7094_zip__Cons__Cons, axiom, (! [A:$tType,B:$tType,X3:A,Xs:(list @ A),Y:B,Ys:(list @ B)]: ((((((zip @ A) @ B) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ B) @ Y) @ Ys)) = (((cons @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) @ ((((zip @ A) @ B) @ Xs) @ Ys)))))). % 6.67/2.70 thf(fact_7095_nth__Cons__numeral, axiom, (! [A:$tType,X3:A,Xs:(list @ A),V2:num]: (((((nth @ A) @ (((cons @ A) @ X3) @ Xs)) @ ((numeral_numeral @ nat) @ V2)) = (((nth @ A) @ Xs) @ (((minus_minus @ nat) @ ((numeral_numeral @ nat) @ V2)) @ (one_one @ nat))))))). % 6.67/2.70 thf(fact_7096_take__Cons__numeral, axiom, (! [A:$tType,V2:num,X3:A,Xs:(list @ A)]: (((((take @ A) @ ((numeral_numeral @ nat) @ V2)) @ (((cons @ A) @ X3) @ Xs)) = (((cons @ A) @ X3) @ (((take @ A) @ (((minus_minus @ nat) @ ((numeral_numeral @ nat) @ V2)) @ (one_one @ nat))) @ Xs)))))). % 6.67/2.70 thf(fact_7097_Cons__in__lex, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Y:A,Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ Y) @ Ys))) @ ((lex @ A) @ R2)) = (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ R2) & (((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys))) | ((X3 = Y) & (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((lex @ A) @ R2)))))))). % 6.67/2.70 thf(fact_7098_concat__map__singleton, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B)]: ((((concat @ A) @ ((((map @ B) @ (list @ A)) @ (^ [X:B]: ((((cons @ A) @ (F2 @ X)) @ (nil @ A))))) @ Xs)) = ((((map @ B) @ A) @ F2) @ Xs))))). % 6.67/2.70 thf(fact_7099_nth__Cons__pos, axiom, (! [A:$tType,N2:nat,X3:A,Xs:(list @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((((nth @ A) @ (((cons @ A) @ X3) @ Xs)) @ N2) = (((nth @ A) @ Xs) @ (((minus_minus @ nat) @ N2) @ (one_one @ nat)))))))). % 6.67/2.70 thf(fact_7100_zip__same__conv__map, axiom, (! [A:$tType,Xs:(list @ A)]: ((((((zip @ A) @ A) @ Xs) @ Xs) = ((((map @ A) @ ((product_prod @ A) @ A)) @ (^ [X:A]: (((((product_Pair @ A) @ A) @ X) @ X)))) @ Xs))))). % 6.67/2.70 thf(fact_7101_product__concat__map, axiom, (! [B:$tType,A:$tType]: ((((product @ A) @ B) = (^ [Xs2:(list @ A),Ys3:(list @ B)]: (((concat @ ((product_prod @ A) @ B)) @ ((((map @ A) @ (list @ ((product_prod @ A) @ B))) @ (^ [X:A]: (((((map @ B) @ ((product_prod @ A) @ B)) @ (((product_Pair @ A) @ B) @ X)) @ Ys3)))) @ Xs2)))))))). % 6.67/2.70 thf(fact_7102_n__lists_Osimps_I2_J, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((n_lists @ A) @ (suc @ N2)) @ Xs) = ((concat @ (list @ A)) @ ((((map @ (list @ A)) @ (list @ (list @ A))) @ (^ [Ys3:(list @ A)]: (((((map @ A) @ (list @ A)) @ (^ [Y2:A]: ((((cons @ A) @ Y2) @ Ys3)))) @ Xs)))) @ (((n_lists @ A) @ N2) @ Xs))))))). % 6.67/2.70 thf(fact_7103_map__eq__Cons__conv, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B),Y:A,Ys:(list @ A)]: (((((((map @ B) @ A) @ F2) @ Xs) = (((cons @ A) @ Y) @ Ys)) = (? [Z6:B,Zs3:(list @ B)]: (((Xs = (((cons @ B) @ Z6) @ Zs3)) & (((F2 @ Z6) = Y) & (((((map @ B) @ A) @ F2) @ Zs3) = Ys))))))))). % 6.67/2.70 thf(fact_7104_Cons__eq__map__conv, axiom, (! [A:$tType,B:$tType,X3:A,Xs:(list @ A),F2:(B > A),Ys:(list @ B)]: ((((((cons @ A) @ X3) @ Xs) = ((((map @ B) @ A) @ F2) @ Ys)) = (? [Z6:B,Zs3:(list @ B)]: (((Ys = (((cons @ B) @ Z6) @ Zs3)) & ((X3 = (F2 @ Z6)) & (Xs = ((((map @ B) @ A) @ F2) @ Zs3)))))))))). % 6.67/2.70 thf(fact_7105_map__eq__Cons__D, axiom, (! [B:$tType,A:$tType,F2:(B > A),Xs:(list @ B),Y:A,Ys:(list @ A)]: (((((((map @ B) @ A) @ F2) @ Xs) = (((cons @ A) @ Y) @ Ys)) => (? [Z3:B,Zs:(list @ B)]: (((Xs = (((cons @ B) @ Z3) @ Zs)) & (((F2 @ Z3) = Y) & (((((map @ B) @ A) @ F2) @ Zs) = Ys))))))))). % 6.67/2.70 thf(fact_7106_Cons__eq__map__D, axiom, (! [A:$tType,B:$tType,X3:A,Xs:(list @ A),F2:(B > A),Ys:(list @ B)]: ((((((cons @ A) @ X3) @ Xs) = ((((map @ B) @ A) @ F2) @ Ys)) => (? [Z3:B,Zs:(list @ B)]: (((Ys = (((cons @ B) @ Z3) @ Zs)) & ((X3 = (F2 @ Z3)) & (Xs = ((((map @ B) @ A) @ F2) @ Zs)))))))))). % 6.67/2.70 thf(fact_7107_list_Osimps_I9_J, axiom, (! [B:$tType,A:$tType,F2:(A > B),X21:A,X222:(list @ A)]: ((((((map @ A) @ B) @ F2) @ (((cons @ A) @ X21) @ X222)) = (((cons @ B) @ (F2 @ X21)) @ ((((map @ A) @ B) @ F2) @ X222)))))). % 6.67/2.70 thf(fact_7108_product__lists_Osimps_I2_J, axiom, (! [A:$tType,Xs:(list @ A),Xss:(list @ (list @ A))]: ((((product_lists @ A) @ (((cons @ (list @ A)) @ Xs) @ Xss)) = ((concat @ (list @ A)) @ ((((map @ A) @ (list @ (list @ A))) @ (^ [X:A]: (((((map @ (list @ A)) @ (list @ A)) @ ((cons @ A) @ X)) @ ((product_lists @ A) @ Xss))))) @ Xs)))))). % 6.67/2.70 thf(fact_7109_List_Obind__def, axiom, (! [B:$tType,A:$tType]: ((((bind @ A) @ B) = (^ [Xs2:(list @ A),F3:(A > (list @ B))]: (((concat @ B) @ ((((map @ A) @ (list @ B)) @ F3) @ Xs2)))))))). % 6.67/2.70 thf(fact_7110_take__Cons, axiom, (! [A:$tType,N2:nat,X3:A,Xs:(list @ A)]: (((((take @ A) @ N2) @ (((cons @ A) @ X3) @ Xs)) = ((((case_nat @ (list @ A)) @ (nil @ A)) @ (^ [M2:nat]: ((((cons @ A) @ X3) @ (((take @ A) @ M2) @ Xs))))) @ N2))))). % 6.67/2.70 thf(fact_7111_shuffles_Osimps_I3_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Y:A,Ys:(list @ A)]: (((((shuffles @ A) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ Y) @ Ys)) = (((sup_sup @ (set @ (list @ A))) @ ((((image @ (list @ A)) @ (list @ A)) @ ((cons @ A) @ X3)) @ (((shuffles @ A) @ Xs) @ (((cons @ A) @ Y) @ Ys)))) @ ((((image @ (list @ A)) @ (list @ A)) @ ((cons @ A) @ Y)) @ (((shuffles @ A) @ (((cons @ A) @ X3) @ Xs)) @ Ys))))))). % 6.67/2.70 thf(fact_7112_Cons__shuffles__subset1, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Ys:(list @ A)]: ((((ord_less_eq @ (set @ (list @ A))) @ ((((image @ (list @ A)) @ (list @ A)) @ ((cons @ A) @ X3)) @ (((shuffles @ A) @ Xs) @ Ys))) @ (((shuffles @ A) @ (((cons @ A) @ X3) @ Xs)) @ Ys))))). % 6.67/2.70 thf(fact_7113_Cons__shuffles__subset2, axiom, (! [A:$tType,Y:A,Xs:(list @ A),Ys:(list @ A)]: ((((ord_less_eq @ (set @ (list @ A))) @ ((((image @ (list @ A)) @ (list @ A)) @ ((cons @ A) @ Y)) @ (((shuffles @ A) @ Xs) @ Ys))) @ (((shuffles @ A) @ Xs) @ (((cons @ A) @ Y) @ Ys)))))). % 6.67/2.70 thf(fact_7114_nth__Cons, axiom, (! [A:$tType,X3:A,Xs:(list @ A),N2:nat]: (((((nth @ A) @ (((cons @ A) @ X3) @ Xs)) @ N2) = ((((case_nat @ A) @ X3) @ ((nth @ A) @ Xs)) @ N2))))). % 6.67/2.70 thf(fact_7115_foldl__Cons, axiom, (! [B:$tType,A:$tType,F2:(B > (A > B)),A2:B,X3:A,Xs:(list @ A)]: (((((((foldl @ B) @ A) @ F2) @ A2) @ (((cons @ A) @ X3) @ Xs)) = (((((foldl @ B) @ A) @ F2) @ ((F2 @ A2) @ X3)) @ Xs))))). % 6.67/2.70 thf(fact_7116_gen__length__code_I2_J, axiom, (! [B:$tType,N2:nat,X3:B,Xs:(list @ B)]: (((((gen_length @ B) @ N2) @ (((cons @ B) @ X3) @ Xs)) = (((gen_length @ B) @ (suc @ N2)) @ Xs))))). % 6.67/2.70 thf(fact_7117_arg__min__list_Osimps_I1_J, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [F2:(A > B),X3:A]: ((((((arg_min_list @ A) @ B) @ F2) @ (((cons @ A) @ X3) @ (nil @ A))) = X3))))))). % 6.67/2.70 thf(fact_7118_find_Osimps_I2_J, axiom, (! [A:$tType,P:(A > $o),X3:A,Xs:(list @ A)]: ((((P @ X3) => ((((find @ A) @ P) @ (((cons @ A) @ X3) @ Xs)) = ((some @ A) @ X3))) & ((~ ((P @ X3))) => ((((find @ A) @ P) @ (((cons @ A) @ X3) @ Xs)) = (((find @ A) @ P) @ Xs))))))). % 6.67/2.70 thf(fact_7119_shufflesE, axiom, (! [A:$tType,Zs2:(list @ A),Xs:(list @ A),Ys:(list @ A)]: (((((member @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ Xs) @ Ys)) => (((Zs2 = Xs) => (Ys != (nil @ A))) => (((Zs2 = Ys) => (Xs != (nil @ A))) => ((! [X5:A,Xs4:(list @ A)]: (((Xs = (((cons @ A) @ X5) @ Xs4)) => (! [Z3:A,Zs4:(list @ A)]: (((Zs2 = (((cons @ A) @ Z3) @ Zs4)) => ((X5 = Z3) => (~ ((((member @ (list @ A)) @ Zs4) @ (((shuffles @ A) @ Xs4) @ Ys))))))))))) => (~ ((! [Y4:A,Ys5:(list @ A)]: (((Ys = (((cons @ A) @ Y4) @ Ys5)) => (! [Z3:A,Zs4:(list @ A)]: (((Zs2 = (((cons @ A) @ Z3) @ Zs4)) => ((Y4 = Z3) => (~ ((((member @ (list @ A)) @ Zs4) @ (((shuffles @ A) @ Xs) @ Ys5)))))))))))))))))))). % 6.67/2.70 thf(fact_7120_insort__key_Osimps_I1_J, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),X3:B]: (((((((linorder_insort_key @ B) @ A) @ F2) @ X3) @ (nil @ B)) = (((cons @ B) @ X3) @ (nil @ B))))))))). % 6.67/2.70 thf(fact_7121_list__update__code_I2_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Y:A]: ((((((list_update @ A) @ (((cons @ A) @ X3) @ Xs)) @ (zero_zero @ nat)) @ Y) = (((cons @ A) @ Y) @ Xs))))). % 6.67/2.70 thf(fact_7122_distinct__singleton, axiom, (! [A:$tType,X3:A]: (((distinct @ A) @ (((cons @ A) @ X3) @ (nil @ A)))))). % 6.67/2.70 thf(fact_7123_distinct_Osimps_I2_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((distinct @ A) @ (((cons @ A) @ X3) @ Xs)) = ((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)))) & ((distinct @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7124_length__Suc__conv, axiom, (! [A:$tType,Xs:(list @ A),N2:nat]: (((((size_size @ (list @ A)) @ Xs) = (suc @ N2)) = (? [Y2:A,Ys3:(list @ A)]: (((Xs = (((cons @ A) @ Y2) @ Ys3)) & (((size_size @ (list @ A)) @ Ys3) = N2)))))))). % 6.67/2.70 thf(fact_7125_Suc__length__conv, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((suc @ N2) = ((size_size @ (list @ A)) @ Xs)) = (? [Y2:A,Ys3:(list @ A)]: (((Xs = (((cons @ A) @ Y2) @ Ys3)) & (((size_size @ (list @ A)) @ Ys3) = N2)))))))). % 6.67/2.70 thf(fact_7126_length__Cons, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ (((cons @ A) @ X3) @ Xs)) = (suc @ ((size_size @ (list @ A)) @ Xs)))))). % 6.67/2.70 thf(fact_7127_list__induct4, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,Xs:(list @ A),Ys:(list @ B),Zs2:(list @ C),Ws2:(list @ D),P:((list @ A) > ((list @ B) > ((list @ C) > ((list @ D) > $o))))]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => ((((size_size @ (list @ B)) @ Ys) = ((size_size @ (list @ C)) @ Zs2)) => ((((size_size @ (list @ C)) @ Zs2) = ((size_size @ (list @ D)) @ Ws2)) => (((((P @ (nil @ A)) @ (nil @ B)) @ (nil @ C)) @ (nil @ D)) => ((! [X5:A,Xs3:(list @ A),Y4:B,Ys4:(list @ B),Z3:C,Zs:(list @ C),W:D,Ws:(list @ D)]: (((((size_size @ (list @ A)) @ Xs3) = ((size_size @ (list @ B)) @ Ys4)) => ((((size_size @ (list @ B)) @ Ys4) = ((size_size @ (list @ C)) @ Zs)) => ((((size_size @ (list @ C)) @ Zs) = ((size_size @ (list @ D)) @ Ws)) => (((((P @ Xs3) @ Ys4) @ Zs) @ Ws) => ((((P @ (((cons @ A) @ X5) @ Xs3)) @ (((cons @ B) @ Y4) @ Ys4)) @ (((cons @ C) @ Z3) @ Zs)) @ (((cons @ D) @ W) @ Ws)))))))) => ((((P @ Xs) @ Ys) @ Zs2) @ Ws2))))))))). % 6.67/2.70 thf(fact_7128_list__induct3, axiom, (! [B:$tType,A:$tType,C:$tType,Xs:(list @ A),Ys:(list @ B),Zs2:(list @ C),P:((list @ A) > ((list @ B) > ((list @ C) > $o)))]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => ((((size_size @ (list @ B)) @ Ys) = ((size_size @ (list @ C)) @ Zs2)) => ((((P @ (nil @ A)) @ (nil @ B)) @ (nil @ C)) => ((! [X5:A,Xs3:(list @ A),Y4:B,Ys4:(list @ B),Z3:C,Zs:(list @ C)]: (((((size_size @ (list @ A)) @ Xs3) = ((size_size @ (list @ B)) @ Ys4)) => ((((size_size @ (list @ B)) @ Ys4) = ((size_size @ (list @ C)) @ Zs)) => ((((P @ Xs3) @ Ys4) @ Zs) => (((P @ (((cons @ A) @ X5) @ Xs3)) @ (((cons @ B) @ Y4) @ Ys4)) @ (((cons @ C) @ Z3) @ Zs))))))) => (((P @ Xs) @ Ys) @ Zs2)))))))). % 6.67/2.70 thf(fact_7129_list__induct2, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B),P:((list @ A) > ((list @ B) > $o))]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => (((P @ (nil @ A)) @ (nil @ B)) => ((! [X5:A,Xs3:(list @ A),Y4:B,Ys4:(list @ B)]: (((((size_size @ (list @ A)) @ Xs3) = ((size_size @ (list @ B)) @ Ys4)) => (((P @ Xs3) @ Ys4) => ((P @ (((cons @ A) @ X5) @ Xs3)) @ (((cons @ B) @ Y4) @ Ys4)))))) => ((P @ Xs) @ Ys))))))). % 6.67/2.70 thf(fact_7130_listrelp_OCons, axiom, (! [A:$tType,B:$tType,R2:(A > (B > $o)),X3:A,Y:B,Xs:(list @ A),Ys:(list @ B)]: ((((R2 @ X3) @ Y) => ((((((listrelp @ A) @ B) @ R2) @ Xs) @ Ys) => (((((listrelp @ A) @ B) @ R2) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ B) @ Y) @ Ys))))))). % 6.67/2.70 thf(fact_7131_list__update_Osimps_I2_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A),I:nat,V2:A]: ((((((list_update @ A) @ (((cons @ A) @ X3) @ Xs)) @ I) @ V2) = ((((case_nat @ (list @ A)) @ (((cons @ A) @ V2) @ Xs)) @ (^ [J3:nat]: ((((cons @ A) @ X3) @ ((((list_update @ A) @ Xs) @ J3) @ V2))))) @ I))))). % 6.67/2.70 thf(fact_7132_set__ConsD, axiom, (! [A:$tType,Y:A,X3:A,Xs:(list @ A)]: (((((member @ A) @ Y) @ ((set2 @ A) @ (((cons @ A) @ X3) @ Xs))) => ((Y = X3) | (((member @ A) @ Y) @ ((set2 @ A) @ Xs))))))). % 6.67/2.70 thf(fact_7133_list_Oset__cases, axiom, (! [A:$tType,E3:A,A2:(list @ A)]: (((((member @ A) @ E3) @ ((set2 @ A) @ A2)) => ((! [Z23:(list @ A)]: ((A2 != (((cons @ A) @ E3) @ Z23)))) => (~ ((! [Z12:A,Z23:(list @ A)]: (((A2 = (((cons @ A) @ Z12) @ Z23)) => (~ ((((member @ A) @ E3) @ ((set2 @ A) @ Z23)))))))))))))). % 6.67/2.70 thf(fact_7134_list_Oset__intros_I1_J, axiom, (! [A:$tType,X21:A,X222:(list @ A)]: ((((member @ A) @ X21) @ ((set2 @ A) @ (((cons @ A) @ X21) @ X222)))))). % 6.67/2.70 thf(fact_7135_list_Oset__intros_I2_J, axiom, (! [A:$tType,Y:A,X222:(list @ A),X21:A]: (((((member @ A) @ Y) @ ((set2 @ A) @ X222)) => (((member @ A) @ Y) @ ((set2 @ A) @ (((cons @ A) @ X21) @ X222))))))). % 6.67/2.70 thf(fact_7136_list__nonempty__induct, axiom, (! [A:$tType,Xs:(list @ A),P:((list @ A) > $o)]: (((Xs != (nil @ A)) => ((! [X5:A]: ((P @ (((cons @ A) @ X5) @ (nil @ A))))) => ((! [X5:A,Xs3:(list @ A)]: (((Xs3 != (nil @ A)) => ((P @ Xs3) => (P @ (((cons @ A) @ X5) @ Xs3)))))) => (P @ Xs))))))). % 6.67/2.70 thf(fact_7137_list__induct2_H, axiom, (! [A:$tType,B:$tType,P:((list @ A) > ((list @ B) > $o)),Xs:(list @ A),Ys:(list @ B)]: ((((P @ (nil @ A)) @ (nil @ B)) => ((! [X5:A,Xs3:(list @ A)]: (((P @ (((cons @ A) @ X5) @ Xs3)) @ (nil @ B)))) => ((! [Y4:B,Ys4:(list @ B)]: (((P @ (nil @ A)) @ (((cons @ B) @ Y4) @ Ys4)))) => ((! [X5:A,Xs3:(list @ A),Y4:B,Ys4:(list @ B)]: ((((P @ Xs3) @ Ys4) => ((P @ (((cons @ A) @ X5) @ Xs3)) @ (((cons @ B) @ Y4) @ Ys4))))) => ((P @ Xs) @ Ys)))))))). % 6.67/2.70 thf(fact_7138_neq__Nil__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) = (? [Y2:A,Ys3:(list @ A)]: ((Xs = (((cons @ A) @ Y2) @ Ys3)))))))). % 6.67/2.70 thf(fact_7139_remdups__adj_Ocases, axiom, (! [A:$tType,X3:(list @ A)]: (((X3 != (nil @ A)) => ((! [X5:A]: ((X3 != (((cons @ A) @ X5) @ (nil @ A))))) => (~ ((! [X5:A,Y4:A,Xs3:(list @ A)]: ((X3 != (((cons @ A) @ X5) @ (((cons @ A) @ Y4) @ Xs3)))))))))))). % 6.67/2.70 thf(fact_7140_transpose_Ocases, axiom, (! [A:$tType,X3:(list @ (list @ A))]: (((X3 != (nil @ (list @ A))) => ((! [Xss2:(list @ (list @ A))]: ((X3 != (((cons @ (list @ A)) @ (nil @ A)) @ Xss2)))) => (~ ((! [X5:A,Xs3:(list @ A),Xss2:(list @ (list @ A))]: ((X3 != (((cons @ (list @ A)) @ (((cons @ A) @ X5) @ Xs3)) @ Xss2))))))))))). % 6.67/2.70 thf(fact_7141_min__list_Ocases, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:(list @ A)]: (((! [X5:A,Xs3:(list @ A)]: ((X3 != (((cons @ A) @ X5) @ Xs3)))) => (X3 = (nil @ A))))))))). % 6.67/2.70 thf(fact_7142_list_Oexhaust, axiom, (! [A:$tType,Y:(list @ A)]: (((Y != (nil @ A)) => (~ ((! [X212:A,X223:(list @ A)]: ((Y != (((cons @ A) @ X212) @ X223)))))))))). % 6.67/2.70 thf(fact_7143_list_OdiscI, axiom, (! [A:$tType,List:(list @ A),X21:A,X222:(list @ A)]: (((List = (((cons @ A) @ X21) @ X222)) => (List != (nil @ A)))))). % 6.67/2.70 thf(fact_7144_list_Odistinct_I1_J, axiom, (! [A:$tType,X21:A,X222:(list @ A)]: (((nil @ A) != (((cons @ A) @ X21) @ X222))))). % 6.67/2.70 thf(fact_7145_removeAll_Osimps_I2_J, axiom, (! [A:$tType,X3:A,Y:A,Xs:(list @ A)]: ((((X3 = Y) => ((((removeAll @ A) @ X3) @ (((cons @ A) @ Y) @ Xs)) = (((removeAll @ A) @ X3) @ Xs))) & ((X3 != Y) => ((((removeAll @ A) @ X3) @ (((cons @ A) @ Y) @ Xs)) = (((cons @ A) @ Y) @ (((removeAll @ A) @ X3) @ Xs)))))))). % 6.67/2.70 thf(fact_7146_remove1_Osimps_I2_J, axiom, (! [A:$tType,X3:A,Y:A,Xs:(list @ A)]: ((((X3 = Y) => ((((remove1 @ A) @ X3) @ (((cons @ A) @ Y) @ Xs)) = Xs)) & ((X3 != Y) => ((((remove1 @ A) @ X3) @ (((cons @ A) @ Y) @ Xs)) = (((cons @ A) @ Y) @ (((remove1 @ A) @ X3) @ Xs)))))))). % 6.67/2.70 thf(fact_7147_Cons__in__shuffles__rightI, axiom, (! [A:$tType,Zs2:(list @ A),Xs:(list @ A),Ys:(list @ A),Z2:A]: (((((member @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ Xs) @ Ys)) => (((member @ (list @ A)) @ (((cons @ A) @ Z2) @ Zs2)) @ (((shuffles @ A) @ Xs) @ (((cons @ A) @ Z2) @ Ys))))))). % 6.67/2.70 thf(fact_7148_Cons__in__shuffles__leftI, axiom, (! [A:$tType,Zs2:(list @ A),Xs:(list @ A),Ys:(list @ A),Z2:A]: (((((member @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ Xs) @ Ys)) => (((member @ (list @ A)) @ (((cons @ A) @ Z2) @ Zs2)) @ (((shuffles @ A) @ (((cons @ A) @ Z2) @ Xs)) @ Ys)))))). % 6.67/2.70 thf(fact_7149_not__Cons__self2, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((cons @ A) @ X3) @ Xs) != Xs)))). % 6.67/2.70 thf(fact_7150_ord_Omax__def, axiom, (! [A:$tType]: (((max @ A) = (^ [Less_eq:(A > (A > $o)),A4:A,B4:A]: (((((if @ A) @ ((Less_eq @ A4) @ B4)) @ B4) @ A4))))))). % 6.67/2.70 thf(fact_7151_ord_Omax_Ocong, axiom, (! [A:$tType]: (((max @ A) = (max @ A))))). % 6.67/2.70 thf(fact_7152_distinct__length__2__or__more, axiom, (! [A:$tType,A2:A,B2:A,Xs:(list @ A)]: ((((distinct @ A) @ (((cons @ A) @ A2) @ (((cons @ A) @ B2) @ Xs))) = ((A2 != B2) & (((distinct @ A) @ (((cons @ A) @ A2) @ Xs)) & ((distinct @ A) @ (((cons @ A) @ B2) @ Xs)))))))). % 6.67/2.70 thf(fact_7153_list__update__code_I3_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A),I:nat,Y:A]: ((((((list_update @ A) @ (((cons @ A) @ X3) @ Xs)) @ (suc @ I)) @ Y) = (((cons @ A) @ X3) @ ((((list_update @ A) @ Xs) @ I) @ Y)))))). % 6.67/2.70 thf(fact_7154_replicate__Suc, axiom, (! [A:$tType,N2:nat,X3:A]: (((((replicate @ A) @ (suc @ N2)) @ X3) = (((cons @ A) @ X3) @ (((replicate @ A) @ N2) @ X3)))))). % 6.67/2.70 thf(fact_7155_remdups_Osimps_I2_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (((remdups @ A) @ (((cons @ A) @ X3) @ Xs)) = ((remdups @ A) @ Xs))) & ((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)))) => (((remdups @ A) @ (((cons @ A) @ X3) @ Xs)) = (((cons @ A) @ X3) @ ((remdups @ A) @ Xs)))))))). % 6.67/2.70 thf(fact_7156_remdups__adj_Osimps_I3_J, axiom, (! [A:$tType,X3:A,Y:A,Xs:(list @ A)]: ((((X3 = Y) => (((remdups_adj @ A) @ (((cons @ A) @ X3) @ (((cons @ A) @ Y) @ Xs))) = ((remdups_adj @ A) @ (((cons @ A) @ X3) @ Xs)))) & ((X3 != Y) => (((remdups_adj @ A) @ (((cons @ A) @ X3) @ (((cons @ A) @ Y) @ Xs))) = (((cons @ A) @ X3) @ ((remdups_adj @ A) @ (((cons @ A) @ Y) @ Xs))))))))). % 6.67/2.70 thf(fact_7157_remdups__adj_Osimps_I2_J, axiom, (! [A:$tType,X3:A]: ((((remdups_adj @ A) @ (((cons @ A) @ X3) @ (nil @ A))) = (((cons @ A) @ X3) @ (nil @ A)))))). % 6.67/2.70 thf(fact_7158_remdups__adj_Oelims, axiom, (! [A:$tType,X3:(list @ A),Y:(list @ A)]: (((((remdups_adj @ A) @ X3) = Y) => (((X3 = (nil @ A)) => (Y != (nil @ A))) => ((! [X5:A]: (((X3 = (((cons @ A) @ X5) @ (nil @ A))) => (Y != (((cons @ A) @ X5) @ (nil @ A)))))) => (~ ((! [X5:A,Y4:A,Xs3:(list @ A)]: (((X3 = (((cons @ A) @ X5) @ (((cons @ A) @ Y4) @ Xs3))) => (~ ((((X5 = Y4) => (Y = ((remdups_adj @ A) @ (((cons @ A) @ X5) @ Xs3)))) & ((X5 != Y4) => (Y = (((cons @ A) @ X5) @ ((remdups_adj @ A) @ (((cons @ A) @ Y4) @ Xs3))))))))))))))))))). % 6.67/2.70 thf(fact_7159_zip__eq__ConsE, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B),Xy2:((product_prod @ A) @ B),Xys:(list @ ((product_prod @ A) @ B))]: (((((((zip @ A) @ B) @ Xs) @ Ys) = (((cons @ ((product_prod @ A) @ B)) @ Xy2) @ Xys)) => (~ ((! [X5:A,Xs4:(list @ A)]: (((Xs = (((cons @ A) @ X5) @ Xs4)) => (! [Y4:B,Ys5:(list @ B)]: (((Ys = (((cons @ B) @ Y4) @ Ys5)) => ((Xy2 = ((((product_Pair @ A) @ B) @ X5) @ Y4)) => (Xys != ((((zip @ A) @ B) @ Xs4) @ Ys5))))))))))))))). % 6.67/2.70 thf(fact_7160_successively_Ocases, axiom, (! [A:$tType,X3:((product_prod @ (A > (A > $o))) @ (list @ A))]: (((! [P8:(A > (A > $o))]: ((X3 != ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ P8) @ (nil @ A))))) => ((! [P8:(A > (A > $o)),X5:A]: ((X3 != ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ P8) @ (((cons @ A) @ X5) @ (nil @ A)))))) => (~ ((! [P8:(A > (A > $o)),X5:A,Y4:A,Xs3:(list @ A)]: ((X3 != ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ P8) @ (((cons @ A) @ X5) @ (((cons @ A) @ Y4) @ Xs3))))))))))))). % 6.67/2.70 thf(fact_7161_arg__min__list_Ocases, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [X3:((product_prod @ (A > B)) @ (list @ A))]: (((! [F4:(A > B),X5:A]: ((X3 != ((((product_Pair @ (A > B)) @ (list @ A)) @ F4) @ (((cons @ A) @ X5) @ (nil @ A)))))) => ((! [F4:(A > B),X5:A,Y4:A,Zs:(list @ A)]: ((X3 != ((((product_Pair @ (A > B)) @ (list @ A)) @ F4) @ (((cons @ A) @ X5) @ (((cons @ A) @ Y4) @ Zs)))))) => (~ ((! [A5:(A > B)]: ((X3 != ((((product_Pair @ (A > B)) @ (list @ A)) @ A5) @ (nil @ A))))))))))))))). % 6.67/2.70 thf(fact_7162_sorted__wrt_Ocases, axiom, (! [A:$tType,X3:((product_prod @ (A > (A > $o))) @ (list @ A))]: (((! [P8:(A > (A > $o))]: ((X3 != ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ P8) @ (nil @ A))))) => (~ ((! [P8:(A > (A > $o)),X5:A,Ys4:(list @ A)]: ((X3 != ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ P8) @ (((cons @ A) @ X5) @ Ys4))))))))))). % 6.67/2.70 thf(fact_7163_splice_Ocases, axiom, (! [A:$tType,X3:((product_prod @ (list @ A)) @ (list @ A))]: (((! [Ys4:(list @ A)]: ((X3 != ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Ys4)))) => (~ ((! [X5:A,Xs3:(list @ A),Ys4:(list @ A)]: ((X3 != ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X5) @ Xs3)) @ Ys4)))))))))). % 6.67/2.70 thf(fact_7164_shuffles_Ocases, axiom, (! [A:$tType,X3:((product_prod @ (list @ A)) @ (list @ A))]: (((! [Ys4:(list @ A)]: ((X3 != ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Ys4)))) => ((! [Xs3:(list @ A)]: ((X3 != ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs3) @ (nil @ A))))) => (~ ((! [X5:A,Xs3:(list @ A),Y4:A,Ys4:(list @ A)]: ((X3 != ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X5) @ Xs3)) @ (((cons @ A) @ Y4) @ Ys4)))))))))))). % 6.67/2.70 thf(fact_7165_map__tailrec__rev_Ocases, axiom, (! [A:$tType,B:$tType,X3:((product_prod @ (A > B)) @ ((product_prod @ (list @ A)) @ (list @ B)))]: (((! [F4:(A > B),Bs2:(list @ B)]: ((X3 != ((((product_Pair @ (A > B)) @ ((product_prod @ (list @ A)) @ (list @ B))) @ F4) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ Bs2))))) => (~ ((! [F4:(A > B),A5:A,As2:(list @ A),Bs2:(list @ B)]: ((X3 != ((((product_Pair @ (A > B)) @ ((product_prod @ (list @ A)) @ (list @ B))) @ F4) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ A5) @ As2)) @ Bs2))))))))))). % 6.67/2.70 thf(fact_7166_listrel1I2, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A)),X3:A]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((listrel1 @ A) @ R2)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ X3) @ Ys))) @ ((listrel1 @ A) @ R2)))))). % 6.67/2.70 thf(fact_7167_inj__on__Cons1, axiom, (! [A:$tType,X3:A,A3:(set @ (list @ A))]: (((((inj_on @ (list @ A)) @ (list @ A)) @ ((cons @ A) @ X3)) @ A3)))). % 6.67/2.70 thf(fact_7168_listrelp_Ocases, axiom, (! [A:$tType,B:$tType,R2:(A > (B > $o)),A12:(list @ A),A23:(list @ B)]: (((((((listrelp @ A) @ B) @ R2) @ A12) @ A23) => (((A12 = (nil @ A)) => (A23 != (nil @ B))) => (~ ((! [X5:A,Y4:B,Xs3:(list @ A)]: (((A12 = (((cons @ A) @ X5) @ Xs3)) => (! [Ys4:(list @ B)]: (((A23 = (((cons @ B) @ Y4) @ Ys4)) => (((R2 @ X5) @ Y4) => (~ ((((((listrelp @ A) @ B) @ R2) @ Xs3) @ Ys4))))))))))))))))). % 6.67/2.70 thf(fact_7169_listrelp_Osimps, axiom, (! [B:$tType,A:$tType]: ((((listrelp @ A) @ B) = (^ [R5:(A > (B > $o)),A1:(list @ A),A22:(list @ B)]: ((((A1 = (nil @ A)) & (A22 = (nil @ B))) | (? [X:A,Y2:B,Xs2:(list @ A),Ys3:(list @ B)]: (((A1 = (((cons @ A) @ X) @ Xs2)) & ((A22 = (((cons @ B) @ Y2) @ Ys3)) & (((R5 @ X) @ Y2) & (((((listrelp @ A) @ B) @ R5) @ Xs2) @ Ys3))))))))))))). % 6.67/2.70 thf(fact_7170_set__subset__Cons, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ (((cons @ A) @ X3) @ Xs)))))). % 6.67/2.70 thf(fact_7171_impossible__Cons, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),X3:A]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ A)) @ Ys)) => (Xs != (((cons @ A) @ X3) @ Ys)))))). % 6.67/2.70 thf(fact_7172_insort__key_Osimps_I2_J, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),X3:B,Y:B,Ys:(list @ B)]: ((((((ord_less_eq @ A) @ (F2 @ X3)) @ (F2 @ Y)) => ((((((linorder_insort_key @ B) @ A) @ F2) @ X3) @ (((cons @ B) @ Y) @ Ys)) = (((cons @ B) @ X3) @ (((cons @ B) @ Y) @ Ys)))) & ((~ ((((ord_less_eq @ A) @ (F2 @ X3)) @ (F2 @ Y)))) => ((((((linorder_insort_key @ B) @ A) @ F2) @ X3) @ (((cons @ B) @ Y) @ Ys)) = (((cons @ B) @ Y) @ (((((linorder_insort_key @ B) @ A) @ F2) @ X3) @ Ys))))))))))). % 6.67/2.70 thf(fact_7173_Cons__in__subseqsD, axiom, (! [A:$tType,Y:A,Ys:(list @ A),Xs:(list @ A)]: (((((member @ (list @ A)) @ (((cons @ A) @ Y) @ Ys)) @ ((set2 @ (list @ A)) @ ((subseqs @ A) @ Xs))) => (((member @ (list @ A)) @ Ys) @ ((set2 @ (list @ A)) @ ((subseqs @ A) @ Xs))))))). % 6.67/2.70 thf(fact_7174_inj__split__Cons, axiom, (! [A:$tType,X8:(set @ ((product_prod @ (list @ A)) @ A))]: (((((inj_on @ ((product_prod @ (list @ A)) @ A)) @ (list @ A)) @ ((((product_case_prod @ (list @ A)) @ A) @ (list @ A)) @ (^ [Xs2:(list @ A),N:A]: ((((cons @ A) @ N) @ Xs2))))) @ X8)))). % 6.67/2.70 thf(fact_7175_ord_Olexordp_Omono, axiom, (! [A:$tType,Less:(A > (A > $o))]: ((((order_mono @ ((list @ A) > ((list @ A) > $o))) @ ((list @ A) > ((list @ A) > $o))) @ (^ [P6:((list @ A) > ((list @ A) > $o)),X15:(list @ A),X24:(list @ A)]: (((? [Y2:A,Ys3:(list @ A)]: (((X15 = (nil @ A)) & (X24 = (((cons @ A) @ Y2) @ Ys3))))) | ((? [X:A,Y2:A,Xs2:(list @ A),Ys3:(list @ A)]: (((X15 = (((cons @ A) @ X) @ Xs2)) & ((X24 = (((cons @ A) @ Y2) @ Ys3)) & ((Less @ X) @ Y2))))) | (? [X:A,Y2:A,Xs2:(list @ A),Ys3:(list @ A)]: (((X15 = (((cons @ A) @ X) @ Xs2)) & ((X24 = (((cons @ A) @ Y2) @ Ys3)) & ((~ (((Less @ X) @ Y2))) & ((~ (((Less @ Y2) @ X))) & ((P6 @ Xs2) @ Ys3))))))))))))))). % 6.67/2.70 thf(fact_7176_arg__min__list_Osimps_I2_J, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [F2:(A > B),X3:A,Y:A,Zs2:(list @ A)]: ((((((arg_min_list @ A) @ B) @ F2) @ (((cons @ A) @ X3) @ (((cons @ A) @ Y) @ Zs2))) = ((((if @ A) @ (((ord_less_eq @ B) @ (F2 @ X3)) @ (F2 @ ((((arg_min_list @ A) @ B) @ F2) @ (((cons @ A) @ Y) @ Zs2))))) @ X3) @ ((((arg_min_list @ A) @ B) @ F2) @ (((cons @ A) @ Y) @ Zs2)))))))))). % 6.67/2.70 thf(fact_7177_Suc__le__length__iff, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((ord_less_eq @ nat) @ (suc @ N2)) @ ((size_size @ (list @ A)) @ Xs)) = (? [X:A,Ys3:(list @ A)]: (((Xs = (((cons @ A) @ X) @ Ys3)) & (((ord_less_eq @ nat) @ N2) @ ((size_size @ (list @ A)) @ Ys3))))))))). % 6.67/2.70 thf(fact_7178_distinct__set__subseqs, axiom, (! [A:$tType,Xs:(list @ A)]: ((((distinct @ A) @ Xs) => ((distinct @ (set @ A)) @ ((((map @ (list @ A)) @ (set @ A)) @ (set2 @ A)) @ ((subseqs @ A) @ Xs))))))). % 6.67/2.70 thf(fact_7179_insort__is__Cons, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [Xs:(list @ B),F2:(B > A),A2:B]: (((! [X5:B]: (((((member @ B) @ X5) @ ((set2 @ B) @ Xs)) => (((ord_less_eq @ A) @ (F2 @ A2)) @ (F2 @ X5))))) => ((((((linorder_insort_key @ B) @ A) @ F2) @ A2) @ Xs) = (((cons @ B) @ A2) @ Xs))))))))). % 6.67/2.70 thf(fact_7180_map__of__Cons__code_I2_J, axiom, (! [C:$tType,B:$tType,L:B,K:B,V2:C,Ps:(list @ ((product_prod @ B) @ C))]: ((((L = K) => (((((map_of @ B) @ C) @ (((cons @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ L) @ V2)) @ Ps)) @ K) = ((some @ C) @ V2))) & ((L != K) => (((((map_of @ B) @ C) @ (((cons @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ L) @ V2)) @ Ps)) @ K) = ((((map_of @ B) @ C) @ Ps) @ K))))))). % 6.67/2.70 thf(fact_7181_listrel1I1, axiom, (! [A:$tType,X3:A,Y:A,R2:(set @ ((product_prod @ A) @ A)),Xs:(list @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ R2) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ Y) @ Xs))) @ ((listrel1 @ A) @ R2)))))). % 6.67/2.70 thf(fact_7182_Cons__listrel1E1, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X3) @ Xs)) @ Ys)) @ ((listrel1 @ A) @ R2)) => ((! [Y4:A]: (((Ys = (((cons @ A) @ Y4) @ Xs)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y4)) @ R2)))))) => (~ ((! [Zs:(list @ A)]: (((Ys = (((cons @ A) @ X3) @ Zs)) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Zs)) @ ((listrel1 @ A) @ R2)))))))))))))). % 6.67/2.70 thf(fact_7183_Cons__listrel1E2, axiom, (! [A:$tType,Xs:(list @ A),Y:A,Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ (((cons @ A) @ Y) @ Ys))) @ ((listrel1 @ A) @ R2)) => ((! [X5:A]: (((Xs = (((cons @ A) @ X5) @ Ys)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Y)) @ R2)))))) => (~ ((! [Zs:(list @ A)]: (((Xs = (((cons @ A) @ Y) @ Zs)) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Zs) @ Ys)) @ ((listrel1 @ A) @ R2)))))))))))))). % 6.67/2.70 thf(fact_7184_listrel_OCons, axiom, (! [B:$tType,A:$tType,X3:A,Y:B,R2:(set @ ((product_prod @ A) @ B)),Xs:(list @ A),Ys:(list @ B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) @ R2) => ((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs) @ Ys)) @ (((listrel @ A) @ B) @ R2)) => (((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ B) @ Y) @ Ys))) @ (((listrel @ A) @ B) @ R2))))))). % 6.67/2.70 thf(fact_7185_listrel__Cons1, axiom, (! [B:$tType,A:$tType,Y:A,Ys:(list @ A),Xs:(list @ B),R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ Y) @ Ys)) @ Xs)) @ (((listrel @ A) @ B) @ R2)) => (~ ((! [Y4:B,Ys4:(list @ B)]: (((Xs = (((cons @ B) @ Y4) @ Ys4)) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y) @ Y4)) @ R2) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Ys) @ Ys4)) @ (((listrel @ A) @ B) @ R2)))))))))))))). % 6.67/2.70 thf(fact_7186_listrel__Cons2, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Y:B,Ys:(list @ B),R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs) @ (((cons @ B) @ Y) @ Ys))) @ (((listrel @ A) @ B) @ R2)) => (~ ((! [X5:A,Xs3:(list @ A)]: (((Xs = (((cons @ A) @ X5) @ Xs3)) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y)) @ R2) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs3) @ Ys)) @ (((listrel @ A) @ B) @ R2)))))))))))))). % 6.67/2.70 thf(fact_7187_count__list_Osimps_I2_J, axiom, (! [A:$tType,X3:A,Y:A,Xs:(list @ A)]: ((((X3 = Y) => ((((count_list @ A) @ (((cons @ A) @ X3) @ Xs)) @ Y) = (((plus_plus @ nat) @ (((count_list @ A) @ Xs) @ Y)) @ (one_one @ nat)))) & ((X3 != Y) => ((((count_list @ A) @ (((cons @ A) @ X3) @ Xs)) @ Y) = (((count_list @ A) @ Xs) @ Y))))))). % 6.67/2.70 thf(fact_7188_the__elem__set, axiom, (! [A:$tType,X3:A]: ((((the_elem @ A) @ ((set2 @ A) @ (((cons @ A) @ X3) @ (nil @ A)))) = X3)))). % 6.67/2.70 thf(fact_7189_rtrancl__listrel1__ConsI1, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A)),X3:A]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((transitive_rtrancl @ (list @ A)) @ ((listrel1 @ A) @ R2))) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ X3) @ Ys))) @ ((transitive_rtrancl @ (list @ A)) @ ((listrel1 @ A) @ R2))))))). % 6.67/2.70 thf(fact_7190_lexordp_Omono, axiom, (! [A:$tType]: (((ord @ A) => (((order_mono @ ((list @ A) > ((list @ A) > $o))) @ ((list @ A) > ((list @ A) > $o))) @ (^ [P6:((list @ A) > ((list @ A) > $o)),X15:(list @ A),X24:(list @ A)]: (((? [Y2:A,Ys3:(list @ A)]: (((X15 = (nil @ A)) & (X24 = (((cons @ A) @ Y2) @ Ys3))))) | ((? [X:A,Y2:A,Xs2:(list @ A),Ys3:(list @ A)]: (((X15 = (((cons @ A) @ X) @ Xs2)) & ((X24 = (((cons @ A) @ Y2) @ Ys3)) & (((ord_less @ A) @ X) @ Y2))))) | (? [X:A,Y2:A,Xs2:(list @ A),Ys3:(list @ A)]: (((X15 = (((cons @ A) @ X) @ Xs2)) & ((X24 = (((cons @ A) @ Y2) @ Ys3)) & ((~ ((((ord_less @ A) @ X) @ Y2))) & ((~ ((((ord_less @ A) @ Y2) @ X))) & ((P6 @ Xs2) @ Ys3)))))))))))))))). % 6.67/2.70 thf(fact_7191_zip__replicate1, axiom, (! [A:$tType,B:$tType,N2:nat,X3:A,Ys:(list @ B)]: ((((((zip @ A) @ B) @ (((replicate @ A) @ N2) @ X3)) @ Ys) = ((((map @ B) @ ((product_prod @ A) @ B)) @ (((product_Pair @ A) @ B) @ X3)) @ (((take @ B) @ N2) @ Ys)))))). % 6.67/2.70 thf(fact_7192_list_Osize_I4_J, axiom, (! [A:$tType,X21:A,X222:(list @ A)]: ((((size_size @ (list @ A)) @ (((cons @ A) @ X21) @ X222)) = (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ X222)) @ (suc @ (zero_zero @ nat))))))). % 6.67/2.70 thf(fact_7193_nth__Cons_H, axiom, (! [A:$tType,N2:nat,X3:A,Xs:(list @ A)]: ((((N2 = (zero_zero @ nat)) => ((((nth @ A) @ (((cons @ A) @ X3) @ Xs)) @ N2) = X3)) & ((N2 != (zero_zero @ nat)) => ((((nth @ A) @ (((cons @ A) @ X3) @ Xs)) @ N2) = (((nth @ A) @ Xs) @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))))))))). % 6.67/2.70 thf(fact_7194_zip__replicate2, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),N2:nat,Y:B]: ((((((zip @ A) @ B) @ Xs) @ (((replicate @ B) @ N2) @ Y)) = ((((map @ A) @ ((product_prod @ A) @ B)) @ (^ [X:A]: (((((product_Pair @ A) @ B) @ X) @ Y)))) @ (((take @ A) @ N2) @ Xs)))))). % 6.67/2.70 thf(fact_7195_listrel_Ocases, axiom, (! [B:$tType,A:$tType,A12:(list @ A),A23:(list @ B),R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ A12) @ A23)) @ (((listrel @ A) @ B) @ R2)) => (((A12 = (nil @ A)) => (A23 != (nil @ B))) => (~ ((! [X5:A,Y4:B,Xs3:(list @ A)]: (((A12 = (((cons @ A) @ X5) @ Xs3)) => (! [Ys4:(list @ B)]: (((A23 = (((cons @ B) @ Y4) @ Ys4)) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X5) @ Y4)) @ R2) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs3) @ Ys4)) @ (((listrel @ A) @ B) @ R2)))))))))))))))))). % 6.67/2.70 thf(fact_7196_listrel_Osimps, axiom, (! [B:$tType,A:$tType,A12:(list @ A),A23:(list @ B),R2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ A12) @ A23)) @ (((listrel @ A) @ B) @ R2)) = (((A12 = (nil @ A)) & (A23 = (nil @ B))) | (? [X:A,Y2:B,Xs2:(list @ A),Ys3:(list @ B)]: (((A12 = (((cons @ A) @ X) @ Xs2)) & ((A23 = (((cons @ B) @ Y2) @ Ys3)) & ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ R2) & (((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs2) @ Ys3)) @ (((listrel @ A) @ B) @ R2)))))))))))). % 6.67/2.70 thf(fact_7197_remdups__adj__replicate, axiom, (! [A:$tType,N2:nat,X3:A]: ((((N2 = (zero_zero @ nat)) => (((remdups_adj @ A) @ (((replicate @ A) @ N2) @ X3)) = (nil @ A))) & ((N2 != (zero_zero @ nat)) => (((remdups_adj @ A) @ (((replicate @ A) @ N2) @ X3)) = (((cons @ A) @ X3) @ (nil @ A)))))))). % 6.67/2.70 thf(fact_7198_remdups__adj__singleton, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: (((((remdups_adj @ A) @ Xs) = (((cons @ A) @ X3) @ (nil @ A))) => (Xs = (((replicate @ A) @ ((size_size @ (list @ A)) @ Xs)) @ X3)))))). % 6.67/2.70 thf(fact_7199_Id__on__set, axiom, (! [A:$tType,Xs:(list @ A)]: ((((id_on @ A) @ ((set2 @ A) @ Xs)) = ((set2 @ ((product_prod @ A) @ A)) @ ((((map @ A) @ ((product_prod @ A) @ A)) @ (^ [X:A]: (((((product_Pair @ A) @ A) @ X) @ X)))) @ Xs)))))). % 6.67/2.70 thf(fact_7200_list_Osize__gen_I2_J, axiom, (! [A:$tType,X3:(A > nat),X21:A,X222:(list @ A)]: (((((size_list @ A) @ X3) @ (((cons @ A) @ X21) @ X222)) = (((plus_plus @ nat) @ (((plus_plus @ nat) @ (X3 @ X21)) @ (((size_list @ A) @ X3) @ X222))) @ (suc @ (zero_zero @ nat))))))). % 6.67/2.70 thf(fact_7201_shuffles_Oelims, axiom, (! [A:$tType,X3:(list @ A),Xa2:(list @ A),Y:(set @ (list @ A))]: ((((((shuffles @ A) @ X3) @ Xa2) = Y) => (((X3 = (nil @ A)) => (Y != (((insert @ (list @ A)) @ Xa2) @ (bot_bot @ (set @ (list @ A)))))) => (((Xa2 = (nil @ A)) => (Y != (((insert @ (list @ A)) @ X3) @ (bot_bot @ (set @ (list @ A)))))) => (~ ((! [X5:A,Xs3:(list @ A)]: (((X3 = (((cons @ A) @ X5) @ Xs3)) => (! [Y4:A,Ys4:(list @ A)]: (((Xa2 = (((cons @ A) @ Y4) @ Ys4)) => (Y != (((sup_sup @ (set @ (list @ A))) @ ((((image @ (list @ A)) @ (list @ A)) @ ((cons @ A) @ X5)) @ (((shuffles @ A) @ Xs3) @ (((cons @ A) @ Y4) @ Ys4)))) @ ((((image @ (list @ A)) @ (list @ A)) @ ((cons @ A) @ Y4)) @ (((shuffles @ A) @ (((cons @ A) @ X5) @ Xs3)) @ Ys4)))))))))))))))))). % 6.67/2.70 thf(fact_7202_nth__equal__first__eq, axiom, (! [A:$tType,X3:A,Xs:(list @ A),N2:nat]: (((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)))) => ((((ord_less_eq @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => (((((nth @ A) @ (((cons @ A) @ X3) @ Xs)) @ N2) = X3) = (N2 = (zero_zero @ nat)))))))). % 6.67/2.70 thf(fact_7203_nth__non__equal__first__eq, axiom, (! [A:$tType,X3:A,Y:A,Xs:(list @ A),N2:nat]: (((X3 != Y) => (((((nth @ A) @ (((cons @ A) @ X3) @ Xs)) @ N2) = Y) = (((((nth @ A) @ Xs) @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))) = Y) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N2))))))). % 6.67/2.70 thf(fact_7204_take__Cons_H, axiom, (! [A:$tType,N2:nat,X3:A,Xs:(list @ A)]: ((((N2 = (zero_zero @ nat)) => ((((take @ A) @ N2) @ (((cons @ A) @ X3) @ Xs)) = (nil @ A))) & ((N2 != (zero_zero @ nat)) => ((((take @ A) @ N2) @ (((cons @ A) @ X3) @ Xs)) = (((cons @ A) @ X3) @ (((take @ A) @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))) @ Xs)))))))). % 6.67/2.70 thf(fact_7205_Cons__replicate__eq, axiom, (! [A:$tType,X3:A,Xs:(list @ A),N2:nat,Y:A]: ((((((cons @ A) @ X3) @ Xs) = (((replicate @ A) @ N2) @ Y)) = ((X3 = Y) & ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) & (Xs = (((replicate @ A) @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))) @ X3)))))))). % 6.67/2.70 thf(fact_7206_map__of__map__restrict, axiom, (! [B:$tType,A:$tType,F2:(A > B),Ks2:(list @ A)]: (((((map_of @ A) @ B) @ ((((map @ A) @ ((product_prod @ A) @ B)) @ (^ [K3:A]: (((((product_Pair @ A) @ B) @ K3) @ (F2 @ K3))))) @ Ks2)) = ((((restrict_map @ A) @ B) @ (((((comp @ B) @ (option @ B)) @ A) @ (some @ B)) @ F2)) @ ((set2 @ A) @ Ks2)))))). % 6.67/2.70 thf(fact_7207_map__of__map__keys, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),M:(A > (option @ B))]: (((((set2 @ A) @ Xs) = (((dom @ A) @ B) @ M)) => ((((map_of @ A) @ B) @ ((((map @ A) @ ((product_prod @ A) @ B)) @ (^ [K3:A]: (((((product_Pair @ A) @ B) @ K3) @ ((the2 @ B) @ (M @ K3)))))) @ Xs)) = M))))). % 6.67/2.70 thf(fact_7208_Cons__lenlex__iff, axiom, (! [A:$tType,M:A,Ms:(list @ A),N2:A,Ns:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ M) @ Ms)) @ (((cons @ A) @ N2) @ Ns))) @ ((lenlex @ A) @ R2)) = ((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ms)) @ ((size_size @ (list @ A)) @ Ns)) | (((((size_size @ (list @ A)) @ Ms) = ((size_size @ (list @ A)) @ Ns)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ M) @ N2)) @ R2)) | ((M = N2) & (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ms) @ Ns)) @ ((lenlex @ A) @ R2))))))))). % 6.67/2.70 thf(fact_7209_map__of_Osimps_I2_J, axiom, (! [B:$tType,A:$tType,P4:((product_prod @ A) @ B),Ps:(list @ ((product_prod @ A) @ B))]: (((((map_of @ A) @ B) @ (((cons @ ((product_prod @ A) @ B)) @ P4) @ Ps)) = (((((fun_upd @ A) @ (option @ B)) @ (((map_of @ A) @ B) @ Ps)) @ (((product_fst @ A) @ B) @ P4)) @ ((some @ B) @ (((product_snd @ A) @ B) @ P4))))))). % 6.67/2.70 thf(fact_7210_rtrancl__listrel1__ConsI2, axiom, (! [A:$tType,X3:A,Y:A,R2:(set @ ((product_prod @ A) @ A)),Xs:(list @ A),Ys:(list @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ ((transitive_rtrancl @ A) @ R2)) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((transitive_rtrancl @ (list @ A)) @ ((listrel1 @ A) @ R2))) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X3) @ Xs)) @ (((cons @ A) @ Y) @ Ys))) @ ((transitive_rtrancl @ (list @ A)) @ ((listrel1 @ A) @ R2)))))))). % 6.67/2.70 thf(fact_7211_Pow__set_I2_J, axiom, (! [B:$tType,X3:B,Xs:(list @ B)]: ((((pow2 @ B) @ ((set2 @ B) @ (((cons @ B) @ X3) @ Xs))) = (((sup_sup @ (set @ (set @ B))) @ ((pow2 @ B) @ ((set2 @ B) @ Xs))) @ ((((image @ (set @ B)) @ (set @ B)) @ ((insert @ B) @ X3)) @ ((pow2 @ B) @ ((set2 @ B) @ Xs)))))))). % 6.67/2.70 thf(fact_7212_sorted__list__of__set__nonempty, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((A3 != (bot_bot @ (set @ A))) => (((linord4507533701916653071of_set @ A) @ A3) = (((cons @ A) @ ((lattic643756798350308766er_Min @ A) @ A3)) @ ((linord4507533701916653071of_set @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ ((lattic643756798350308766er_Min @ A) @ A3)) @ (bot_bot @ (set @ A))))))))))))))). % 6.67/2.70 thf(fact_7213_product__code, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys:(list @ B)]: ((((((product_product @ A) @ B) @ ((set2 @ A) @ Xs)) @ ((set2 @ B) @ Ys)) = ((set2 @ ((product_prod @ A) @ B)) @ ((concat @ ((product_prod @ A) @ B)) @ ((((map @ A) @ (list @ ((product_prod @ A) @ B))) @ (^ [X:A]: (((((map @ B) @ ((product_prod @ A) @ B)) @ (((product_Pair @ A) @ B) @ X)) @ Ys)))) @ Xs))))))). % 6.67/2.70 thf(fact_7214_set__Cons__sing__Nil, axiom, (! [A:$tType,A3:(set @ A)]: (((((set_Cons @ A) @ A3) @ (((insert @ (list @ A)) @ (nil @ A)) @ (bot_bot @ (set @ (list @ A))))) = ((((image @ A) @ (list @ A)) @ (^ [X:A]: ((((cons @ A) @ X) @ (nil @ A))))) @ A3))))). % 6.67/2.70 thf(fact_7215_n__lists__Nil, axiom, (! [A:$tType,N2:nat]: ((((N2 = (zero_zero @ nat)) => ((((n_lists @ A) @ N2) @ (nil @ A)) = (((cons @ (list @ A)) @ (nil @ A)) @ (nil @ (list @ A))))) & ((N2 != (zero_zero @ nat)) => ((((n_lists @ A) @ N2) @ (nil @ A)) = (nil @ (list @ A)))))))). % 6.67/2.70 thf(fact_7216_listset_Osimps_I2_J, axiom, (! [A:$tType,A3:(set @ A),As3:(list @ (set @ A))]: ((((listset @ A) @ (((cons @ (set @ A)) @ A3) @ As3)) = (((set_Cons @ A) @ A3) @ ((listset @ A) @ As3)))))). % 6.67/2.70 thf(fact_7217_subseqs_Osimps_I1_J, axiom, (! [A:$tType]: ((((subseqs @ A) @ (nil @ A)) = (((cons @ (list @ A)) @ (nil @ A)) @ (nil @ (list @ A))))))). % 6.67/2.70 thf(fact_7218_product__lists_Osimps_I1_J, axiom, (! [A:$tType]: ((((product_lists @ A) @ (nil @ (list @ A))) = (((cons @ (list @ A)) @ (nil @ A)) @ (nil @ (list @ A))))))). % 6.67/2.70 thf(fact_7219_n__lists_Osimps_I1_J, axiom, (! [A:$tType,Xs:(list @ A)]: (((((n_lists @ A) @ (zero_zero @ nat)) @ Xs) = (((cons @ (list @ A)) @ (nil @ A)) @ (nil @ (list @ A))))))). % 6.67/2.70 thf(fact_7220_sorted__list__of__set__greaterThanAtMost, axiom, (! [I:nat,J:nat]: (((((ord_less_eq @ nat) @ (suc @ I)) @ J) => (((linord4507533701916653071of_set @ nat) @ (((set_or3652927894154168847AtMost @ nat) @ I) @ J)) = (((cons @ nat) @ (suc @ I)) @ ((linord4507533701916653071of_set @ nat) @ (((set_or3652927894154168847AtMost @ nat) @ (suc @ I)) @ J)))))))). % 6.67/2.70 thf(fact_7221_sorted__list__of__set__greaterThanLessThan, axiom, (! [I:nat,J:nat]: (((((ord_less @ nat) @ (suc @ I)) @ J) => (((linord4507533701916653071of_set @ nat) @ (((set_or5935395276787703475ssThan @ nat) @ I) @ J)) = (((cons @ nat) @ (suc @ I)) @ ((linord4507533701916653071of_set @ nat) @ (((set_or5935395276787703475ssThan @ nat) @ (suc @ I)) @ J)))))))). % 6.67/2.70 thf(fact_7222_set__Cons__def, axiom, (! [A:$tType]: (((set_Cons @ A) = (^ [A6:(set @ A),XS2:(set @ (list @ A))]: (((collect @ (list @ A)) @ (^ [Z6:(list @ A)]: ((? [X:A,Xs2:(list @ A)]: (((Z6 = (((cons @ A) @ X) @ Xs2)) & ((((member @ A) @ X) @ A6) & (((member @ (list @ A)) @ Xs2) @ XS2)))))))))))))). % 6.67/2.70 thf(fact_7223_concat__inth, axiom, (! [A:$tType,Xs:(list @ A),X3:A,Ys:(list @ A)]: (((((nth @ A) @ (((append @ A) @ Xs) @ (((append @ A) @ (((cons @ A) @ X3) @ (nil @ A))) @ Ys))) @ ((size_size @ (list @ A)) @ Xs)) = X3)))). % 6.67/2.70 thf(fact_7224_map__upds__append1, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys:(list @ B),M:(A > (option @ B)),X3:A]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ B)) @ Ys)) => ((((((map_upds @ A) @ B) @ M) @ (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (nil @ A)))) @ Ys) = (((((fun_upd @ A) @ (option @ B)) @ (((((map_upds @ A) @ B) @ M) @ Xs) @ Ys)) @ X3) @ ((some @ B) @ (((nth @ B) @ Ys) @ ((size_size @ (list @ A)) @ Xs))))))))). % 6.67/2.70 thf(fact_7225_append_Oassoc, axiom, (! [A:$tType,A2:(list @ A),B2:(list @ A),C2:(list @ A)]: (((((append @ A) @ (((append @ A) @ A2) @ B2)) @ C2) = (((append @ A) @ A2) @ (((append @ A) @ B2) @ C2)))))). % 6.67/2.70 thf(fact_7226_append__assoc, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A)]: (((((append @ A) @ (((append @ A) @ Xs) @ Ys)) @ Zs2) = (((append @ A) @ Xs) @ (((append @ A) @ Ys) @ Zs2)))))). % 6.67/2.70 thf(fact_7227_append__same__eq, axiom, (! [A:$tType,Ys:(list @ A),Xs:(list @ A),Zs2:(list @ A)]: ((((((append @ A) @ Ys) @ Xs) = (((append @ A) @ Zs2) @ Xs)) = (Ys = Zs2))))). % 6.67/2.70 thf(fact_7228_same__append__eq, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = (((append @ A) @ Xs) @ Zs2)) = (Ys = Zs2))))). % 6.67/2.70 thf(fact_7229_append__is__Nil__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = (nil @ A)) = ((Xs = (nil @ A)) & (Ys = (nil @ A))))))). % 6.67/2.70 thf(fact_7230_Nil__is__append__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((nil @ A) = (((append @ A) @ Xs) @ Ys)) = ((Xs = (nil @ A)) & (Ys = (nil @ A))))))). % 6.67/2.70 thf(fact_7231_self__append__conv2, axiom, (! [A:$tType,Y:(list @ A),Xs:(list @ A)]: (((Y = (((append @ A) @ Xs) @ Y)) = (Xs = (nil @ A)))))). % 6.67/2.70 thf(fact_7232_append__self__conv2, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = Ys) = (Xs = (nil @ A)))))). % 6.67/2.70 thf(fact_7233_self__append__conv, axiom, (! [A:$tType,Y:(list @ A),Ys:(list @ A)]: (((Y = (((append @ A) @ Y) @ Ys)) = (Ys = (nil @ A)))))). % 6.67/2.70 thf(fact_7234_append__self__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = Xs) = (Ys = (nil @ A)))))). % 6.67/2.70 thf(fact_7235_append__Nil2, axiom, (! [A:$tType,Xs:(list @ A)]: (((((append @ A) @ Xs) @ (nil @ A)) = Xs)))). % 6.67/2.70 thf(fact_7236_append_Oright__neutral, axiom, (! [A:$tType,A2:(list @ A)]: (((((append @ A) @ A2) @ (nil @ A)) = A2)))). % 6.67/2.70 thf(fact_7237_append__eq__append__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Us:(list @ A),Vs2:(list @ A)]: ((((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys)) | (((size_size @ (list @ A)) @ Us) = ((size_size @ (list @ A)) @ Vs2))) => (((((append @ A) @ Xs) @ Us) = (((append @ A) @ Ys) @ Vs2)) = ((Xs = Ys) & (Us = Vs2))))))). % 6.67/2.70 thf(fact_7238_map__append, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B),Ys:(list @ B)]: ((((((map @ B) @ A) @ F2) @ (((append @ B) @ Xs) @ Ys)) = (((append @ A) @ ((((map @ B) @ A) @ F2) @ Xs)) @ ((((map @ B) @ A) @ F2) @ Ys)))))). % 6.67/2.70 thf(fact_7239_concat__append, axiom, (! [A:$tType,Xs:(list @ (list @ A)),Ys:(list @ (list @ A))]: ((((concat @ A) @ (((append @ (list @ A)) @ Xs) @ Ys)) = (((append @ A) @ ((concat @ A) @ Xs)) @ ((concat @ A) @ Ys)))))). % 6.67/2.70 thf(fact_7240_removeAll__append, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Ys:(list @ A)]: (((((removeAll @ A) @ X3) @ (((append @ A) @ Xs) @ Ys)) = (((append @ A) @ (((removeAll @ A) @ X3) @ Xs)) @ (((removeAll @ A) @ X3) @ Ys)))))). % 6.67/2.70 thf(fact_7241_foldl__append, axiom, (! [A:$tType,B:$tType,F2:(A > (B > A)),A2:A,Xs:(list @ B),Ys:(list @ B)]: (((((((foldl @ A) @ B) @ F2) @ A2) @ (((append @ B) @ Xs) @ Ys)) = (((((foldl @ A) @ B) @ F2) @ (((((foldl @ A) @ B) @ F2) @ A2) @ Xs)) @ Ys))))). % 6.67/2.70 thf(fact_7242_append1__eq__conv, axiom, (! [A:$tType,Xs:(list @ A),X3:A,Ys:(list @ A),Y:A]: ((((((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (nil @ A))) = (((append @ A) @ Ys) @ (((cons @ A) @ Y) @ (nil @ A)))) = ((Xs = Ys) & (X3 = Y)))))). % 6.67/2.70 thf(fact_7243_length__append, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((size_size @ (list @ A)) @ (((append @ A) @ Xs) @ Ys)) = (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ A)) @ Ys)))))). % 6.67/2.70 thf(fact_7244_set__append, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((set2 @ A) @ (((append @ A) @ Xs) @ Ys)) = (((sup_sup @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys)))))). % 6.67/2.70 thf(fact_7245_zip__append, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Us:(list @ B),Ys:(list @ A),Vs2:(list @ B)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Us)) => (((((zip @ A) @ B) @ (((append @ A) @ Xs) @ Ys)) @ (((append @ B) @ Us) @ Vs2)) = (((append @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Us)) @ ((((zip @ A) @ B) @ Ys) @ Vs2))))))). % 6.67/2.70 thf(fact_7246_fun__upds__append__drop, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B),M:(A > (option @ B)),Zs2:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => ((((((map_upds @ A) @ B) @ M) @ (((append @ A) @ Xs) @ Zs2)) @ Ys) = (((((map_upds @ A) @ B) @ M) @ Xs) @ Ys)))))). % 6.67/2.70 thf(fact_7247_fun__upds__append2__drop, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B),M:(A > (option @ B)),Zs2:(list @ B)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => ((((((map_upds @ A) @ B) @ M) @ Xs) @ (((append @ B) @ Ys) @ Zs2)) = (((((map_upds @ A) @ B) @ M) @ Xs) @ Ys)))))). % 6.67/2.70 thf(fact_7248_size__list__append, axiom, (! [A:$tType,F2:(A > nat),Xs:(list @ A),Ys:(list @ A)]: (((((size_list @ A) @ F2) @ (((append @ A) @ Xs) @ Ys)) = (((plus_plus @ nat) @ (((size_list @ A) @ F2) @ Xs)) @ (((size_list @ A) @ F2) @ Ys)))))). % 6.67/2.70 thf(fact_7249_bind__simps_I2_J, axiom, (! [A:$tType,B:$tType,X3:B,Xs:(list @ B),F2:(B > (list @ A))]: ((((((bind @ B) @ A) @ (((cons @ B) @ X3) @ Xs)) @ F2) = (((append @ A) @ (F2 @ X3)) @ ((((bind @ B) @ A) @ Xs) @ F2)))))). % 6.67/2.70 thf(fact_7250_nth__append__length, axiom, (! [A:$tType,Xs:(list @ A),X3:A,Ys:(list @ A)]: (((((nth @ A) @ (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ Ys))) @ ((size_size @ (list @ A)) @ Xs)) = X3)))). % 6.67/2.70 thf(fact_7251_nth__append__length__plus, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),N2:nat]: (((((nth @ A) @ (((append @ A) @ Xs) @ Ys)) @ (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N2)) = (((nth @ A) @ Ys) @ N2))))). % 6.67/2.70 thf(fact_7252_take__append, axiom, (! [A:$tType,N2:nat,Xs:(list @ A),Ys:(list @ A)]: (((((take @ A) @ N2) @ (((append @ A) @ Xs) @ Ys)) = (((append @ A) @ (((take @ A) @ N2) @ Xs)) @ (((take @ A) @ (((minus_minus @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs))) @ Ys)))))). % 6.67/2.70 thf(fact_7253_list__update__length, axiom, (! [A:$tType,Xs:(list @ A),X3:A,Ys:(list @ A),Y:A]: ((((((list_update @ A) @ (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ Ys))) @ ((size_size @ (list @ A)) @ Xs)) @ Y) = (((append @ A) @ Xs) @ (((cons @ A) @ Y) @ Ys)))))). % 6.67/2.70 thf(fact_7254_sorted__list__of__set__lessThan__Suc, axiom, (! [K:nat]: ((((linord4507533701916653071of_set @ nat) @ ((set_ord_lessThan @ nat) @ (suc @ K))) = (((append @ nat) @ ((linord4507533701916653071of_set @ nat) @ ((set_ord_lessThan @ nat) @ K))) @ (((cons @ nat) @ K) @ (nil @ nat))))))). % 6.67/2.70 thf(fact_7255_sorted__list__of__set__atMost__Suc, axiom, (! [K:nat]: ((((linord4507533701916653071of_set @ nat) @ ((set_ord_atMost @ nat) @ (suc @ K))) = (((append @ nat) @ ((linord4507533701916653071of_set @ nat) @ ((set_ord_atMost @ nat) @ K))) @ (((cons @ nat) @ (suc @ K)) @ (nil @ nat))))))). % 6.67/2.70 thf(fact_7256_distinct__append, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((distinct @ A) @ (((append @ A) @ Xs) @ Ys)) = (((distinct @ A) @ Xs) & (((distinct @ A) @ Ys) & ((((inf_inf @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys)) = (bot_bot @ (set @ A))))))))). % 6.67/2.70 thf(fact_7257_map__eq__append__conv, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B),Ys:(list @ A),Zs2:(list @ A)]: (((((((map @ B) @ A) @ F2) @ Xs) = (((append @ A) @ Ys) @ Zs2)) = (? [Us2:(list @ B),Vs:(list @ B)]: (((Xs = (((append @ B) @ Us2) @ Vs)) & ((Ys = ((((map @ B) @ A) @ F2) @ Us2)) & (Zs2 = ((((map @ B) @ A) @ F2) @ Vs)))))))))). % 6.67/2.70 thf(fact_7258_append__eq__map__conv, axiom, (! [A:$tType,B:$tType,Ys:(list @ A),Zs2:(list @ A),F2:(B > A),Xs:(list @ B)]: ((((((append @ A) @ Ys) @ Zs2) = ((((map @ B) @ A) @ F2) @ Xs)) = (? [Us2:(list @ B),Vs:(list @ B)]: (((Xs = (((append @ B) @ Us2) @ Vs)) & ((Ys = ((((map @ B) @ A) @ F2) @ Us2)) & (Zs2 = ((((map @ B) @ A) @ F2) @ Vs)))))))))). % 6.67/2.70 thf(fact_7259_split__list, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (? [Ys4:(list @ A),Zs:(list @ A)]: ((Xs = (((append @ A) @ Ys4) @ (((cons @ A) @ X3) @ Zs))))))))). % 6.67/2.70 thf(fact_7260_split__list__last, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (? [Ys4:(list @ A),Zs:(list @ A)]: (((Xs = (((append @ A) @ Ys4) @ (((cons @ A) @ X3) @ Zs))) & (~ ((((member @ A) @ X3) @ ((set2 @ A) @ Zs))))))))))). % 6.67/2.70 thf(fact_7261_split__list__prop, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) & (P @ X2)))) => (? [Ys4:(list @ A),X5:A]: (((? [Zs:(list @ A)]: ((Xs = (((append @ A) @ Ys4) @ (((cons @ A) @ X5) @ Zs))))) & (P @ X5)))))))). % 6.67/2.70 thf(fact_7262_split__list__first, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (? [Ys4:(list @ A),Zs:(list @ A)]: (((Xs = (((append @ A) @ Ys4) @ (((cons @ A) @ X3) @ Zs))) & (~ ((((member @ A) @ X3) @ ((set2 @ A) @ Ys4))))))))))). % 6.67/2.70 thf(fact_7263_split__list__propE, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) & (P @ X2)))) => (~ ((! [Ys4:(list @ A),X5:A]: (((? [Zs:(list @ A)]: ((Xs = (((append @ A) @ Ys4) @ (((cons @ A) @ X5) @ Zs))))) => (~ ((P @ X5)))))))))))). % 6.67/2.70 thf(fact_7264_append__Cons__eq__iff, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Ys:(list @ A),Xs5:(list @ A),Ys6:(list @ A)]: (((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)))) => ((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Ys)))) => (((((append @ A) @ Xs) @ (((cons @ A) @ X3) @ Ys)) = (((append @ A) @ Xs5) @ (((cons @ A) @ X3) @ Ys6))) = ((Xs = Xs5) & (Ys = Ys6)))))))). % 6.67/2.70 thf(fact_7265_in__set__conv__decomp, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) = (? [Ys3:(list @ A),Zs3:(list @ A)]: ((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ X3) @ Zs3))))))))). % 6.67/2.70 thf(fact_7266_split__list__last__prop, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) & (P @ X2)))) => (? [Ys4:(list @ A),X5:A,Zs:(list @ A)]: (((Xs = (((append @ A) @ Ys4) @ (((cons @ A) @ X5) @ Zs))) & ((P @ X5) & (! [Xa:A]: (((((member @ A) @ Xa) @ ((set2 @ A) @ Zs)) => (~ ((P @ Xa)))))))))))))). % 6.67/2.70 thf(fact_7267_split__list__first__prop, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) & (P @ X2)))) => (? [Ys4:(list @ A),X5:A]: (((? [Zs:(list @ A)]: ((Xs = (((append @ A) @ Ys4) @ (((cons @ A) @ X5) @ Zs))))) & ((P @ X5) & (! [Xa:A]: (((((member @ A) @ Xa) @ ((set2 @ A) @ Ys4)) => (~ ((P @ Xa)))))))))))))). % 6.67/2.70 thf(fact_7268_split__list__last__propE, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) & (P @ X2)))) => (~ ((! [Ys4:(list @ A),X5:A,Zs:(list @ A)]: (((Xs = (((append @ A) @ Ys4) @ (((cons @ A) @ X5) @ Zs))) => ((P @ X5) => (~ ((! [Xa:A]: (((((member @ A) @ Xa) @ ((set2 @ A) @ Zs)) => (~ ((P @ Xa)))))))))))))))))). % 6.67/2.70 thf(fact_7269_split__list__first__propE, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) & (P @ X2)))) => (~ ((! [Ys4:(list @ A),X5:A]: (((? [Zs:(list @ A)]: ((Xs = (((append @ A) @ Ys4) @ (((cons @ A) @ X5) @ Zs))))) => ((P @ X5) => (~ ((! [Xa:A]: (((((member @ A) @ Xa) @ ((set2 @ A) @ Ys4)) => (~ ((P @ Xa)))))))))))))))))). % 6.67/2.70 thf(fact_7270_in__set__conv__decomp__last, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) = (? [Ys3:(list @ A),Zs3:(list @ A)]: (((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ X3) @ Zs3))) & (~ ((((member @ A) @ X3) @ ((set2 @ A) @ Zs3))))))))))). % 6.67/2.70 thf(fact_7271_in__set__conv__decomp__first, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) = (? [Ys3:(list @ A),Zs3:(list @ A)]: (((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ X3) @ Zs3))) & (~ ((((member @ A) @ X3) @ ((set2 @ A) @ Ys3))))))))))). % 6.67/2.70 thf(fact_7272_split__list__last__prop__iff, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) & (P @ X)))) = (? [Ys3:(list @ A),X:A,Zs3:(list @ A)]: (((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ X) @ Zs3))) & ((P @ X) & (! [Y2:A]: (((((member @ A) @ Y2) @ ((set2 @ A) @ Zs3)) => (~ ((P @ Y2)))))))))))))). % 6.67/2.70 thf(fact_7273_split__list__first__prop__iff, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((? [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) & (P @ X)))) = (? [Ys3:(list @ A),X:A]: (((? [Zs3:(list @ A)]: ((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ X) @ Zs3))))) & ((P @ X) & (! [Y2:A]: (((((member @ A) @ Y2) @ ((set2 @ A) @ Ys3)) => (~ ((P @ Y2)))))))))))))). % 6.67/2.70 thf(fact_7274_concat__eq__appendD, axiom, (! [A:$tType,Xss:(list @ (list @ A)),Ys:(list @ A),Zs2:(list @ A)]: (((((concat @ A) @ Xss) = (((append @ A) @ Ys) @ Zs2)) => ((Xss != (nil @ (list @ A))) => (? [Xss1:(list @ (list @ A)),Xs3:(list @ A),Xs4:(list @ A),Xss22:(list @ (list @ A))]: (((Xss = (((append @ (list @ A)) @ Xss1) @ (((cons @ (list @ A)) @ (((append @ A) @ Xs3) @ Xs4)) @ Xss22))) & ((Ys = (((append @ A) @ ((concat @ A) @ Xss1)) @ Xs3)) & (Zs2 = (((append @ A) @ Xs4) @ ((concat @ A) @ Xss22)))))))))))). % 6.67/2.70 thf(fact_7275_concat__eq__append__conv, axiom, (! [A:$tType,Xss:(list @ (list @ A)),Ys:(list @ A),Zs2:(list @ A)]: (((((concat @ A) @ Xss) = (((append @ A) @ Ys) @ Zs2)) = (((Xss = (nil @ (list @ A))) => ((Ys = (nil @ A)) & (Zs2 = (nil @ A)))) & ((Xss != (nil @ (list @ A))) => (? [Xss12:(list @ (list @ A)),Xs2:(list @ A),Xs6:(list @ A),Xss23:(list @ (list @ A))]: (((Xss = (((append @ (list @ A)) @ Xss12) @ (((cons @ (list @ A)) @ (((append @ A) @ Xs2) @ Xs6)) @ Xss23))) & ((Ys = (((append @ A) @ ((concat @ A) @ Xss12)) @ Xs2)) & (Zs2 = (((append @ A) @ Xs6) @ ((concat @ A) @ Xss23))))))))))))). % 6.67/2.70 thf(fact_7276_rev__nonempty__induct, axiom, (! [A:$tType,Xs:(list @ A),P:((list @ A) > $o)]: (((Xs != (nil @ A)) => ((! [X5:A]: ((P @ (((cons @ A) @ X5) @ (nil @ A))))) => ((! [X5:A,Xs3:(list @ A)]: (((Xs3 != (nil @ A)) => ((P @ Xs3) => (P @ (((append @ A) @ Xs3) @ (((cons @ A) @ X5) @ (nil @ A)))))))) => (P @ Xs))))))). % 6.67/2.70 thf(fact_7277_append__eq__Cons__conv, axiom, (! [A:$tType,Ys:(list @ A),Zs2:(list @ A),X3:A,Xs:(list @ A)]: ((((((append @ A) @ Ys) @ Zs2) = (((cons @ A) @ X3) @ Xs)) = (((Ys = (nil @ A)) & (Zs2 = (((cons @ A) @ X3) @ Xs))) | (? [Ys7:(list @ A)]: (((Ys = (((cons @ A) @ X3) @ Ys7)) & ((((append @ A) @ Ys7) @ Zs2) = Xs))))))))). % 6.67/2.70 thf(fact_7278_Cons__eq__append__conv, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A)]: ((((((cons @ A) @ X3) @ Xs) = (((append @ A) @ Ys) @ Zs2)) = (((Ys = (nil @ A)) & ((((cons @ A) @ X3) @ Xs) = Zs2)) | (? [Ys7:(list @ A)]: ((((((cons @ A) @ X3) @ Ys7) = Ys) & (Xs = (((append @ A) @ Ys7) @ Zs2)))))))))). % 6.67/2.70 thf(fact_7279_rev__exhaust, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (~ ((! [Ys4:(list @ A),Y4:A]: ((Xs != (((append @ A) @ Ys4) @ (((cons @ A) @ Y4) @ (nil @ A)))))))))))). % 6.67/2.70 thf(fact_7280_rev__induct, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((P @ (nil @ A)) => ((! [X5:A,Xs3:(list @ A)]: (((P @ Xs3) => (P @ (((append @ A) @ Xs3) @ (((cons @ A) @ X5) @ (nil @ A))))))) => (P @ Xs)))))). % 6.67/2.70 thf(fact_7281_append__Cons, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Ys:(list @ A)]: (((((append @ A) @ (((cons @ A) @ X3) @ Xs)) @ Ys) = (((cons @ A) @ X3) @ (((append @ A) @ Xs) @ Ys)))))). % 6.67/2.70 thf(fact_7282_Cons__eq__appendI, axiom, (! [A:$tType,X3:A,Xs1:(list @ A),Ys:(list @ A),Xs:(list @ A),Zs2:(list @ A)]: ((((((cons @ A) @ X3) @ Xs1) = Ys) => ((Xs = (((append @ A) @ Xs1) @ Zs2)) => ((((cons @ A) @ X3) @ Xs) = (((append @ A) @ Ys) @ Zs2))))))). % 6.67/2.70 thf(fact_7283_concat_Osimps_I2_J, axiom, (! [A:$tType,X3:(list @ A),Xs:(list @ (list @ A))]: ((((concat @ A) @ (((cons @ (list @ A)) @ X3) @ Xs)) = (((append @ A) @ X3) @ ((concat @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7284_replicate__app__Cons__same, axiom, (! [A:$tType,N2:nat,X3:A,Xs:(list @ A)]: (((((append @ A) @ (((replicate @ A) @ N2) @ X3)) @ (((cons @ A) @ X3) @ Xs)) = (((cons @ A) @ X3) @ (((append @ A) @ (((replicate @ A) @ N2) @ X3)) @ Xs)))))). % 6.67/2.70 thf(fact_7285_append__listrel1I, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A)),Us:(list @ A),Vs2:(list @ A)]: (((((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((listrel1 @ A) @ R2)) & (Us = Vs2)) | ((Xs = Ys) & (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Us) @ Vs2)) @ ((listrel1 @ A) @ R2)))) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Us)) @ (((append @ A) @ Ys) @ Vs2))) @ ((listrel1 @ A) @ R2)))))). % 6.67/2.70 thf(fact_7286_lexord__append__leftI, axiom, (! [A:$tType,U:(list @ A),V2:(list @ A),R2:(set @ ((product_prod @ A) @ A)),X3:(list @ A)]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ U) @ V2)) @ ((lexord @ A) @ R2)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ X3) @ U)) @ (((append @ A) @ X3) @ V2))) @ ((lexord @ A) @ R2)))))). % 6.67/2.70 thf(fact_7287_lex__append__leftI, axiom, (! [A:$tType,Ys:(list @ A),Zs2:(list @ A),R2:(set @ ((product_prod @ A) @ A)),Xs:(list @ A)]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ys) @ Zs2)) @ ((lex @ A) @ R2)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Ys)) @ (((append @ A) @ Xs) @ Zs2))) @ ((lex @ A) @ R2)))))). % 6.67/2.70 thf(fact_7288_comm__append__are__replicate, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = (((append @ A) @ Ys) @ Xs)) => (? [M3:nat,N3:nat,Zs:(list @ A)]: (((((concat @ A) @ (((replicate @ (list @ A)) @ M3) @ Zs)) = Xs) & (((concat @ A) @ (((replicate @ (list @ A)) @ N3) @ Zs)) = Ys)))))))). % 6.67/2.70 thf(fact_7289_replicate__add, axiom, (! [A:$tType,N2:nat,M:nat,X3:A]: (((((replicate @ A) @ (((plus_plus @ nat) @ N2) @ M)) @ X3) = (((append @ A) @ (((replicate @ A) @ N2) @ X3)) @ (((replicate @ A) @ M) @ X3)))))). % 6.67/2.70 thf(fact_7290_eq__Nil__appendI, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((Xs = Ys) => (Xs = (((append @ A) @ (nil @ A)) @ Ys)))))). % 6.67/2.70 thf(fact_7291_append_Oleft__neutral, axiom, (! [A:$tType,A2:(list @ A)]: (((((append @ A) @ (nil @ A)) @ A2) = A2)))). % 6.67/2.70 thf(fact_7292_append__Nil, axiom, (! [A:$tType,Ys:(list @ A)]: (((((append @ A) @ (nil @ A)) @ Ys) = Ys)))). % 6.67/2.70 thf(fact_7293_append__eq__appendI, axiom, (! [A:$tType,Xs:(list @ A),Xs1:(list @ A),Zs2:(list @ A),Ys:(list @ A),Us:(list @ A)]: ((((((append @ A) @ Xs) @ Xs1) = Zs2) => ((Ys = (((append @ A) @ Xs1) @ Us)) => ((((append @ A) @ Xs) @ Ys) = (((append @ A) @ Zs2) @ Us))))))). % 6.67/2.70 thf(fact_7294_append__eq__append__conv2, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A),Ts:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = (((append @ A) @ Zs2) @ Ts)) = (? [Us2:(list @ A)]: ((((Xs = (((append @ A) @ Zs2) @ Us2)) & ((((append @ A) @ Us2) @ Ys) = Ts)) | (((((append @ A) @ Xs) @ Us2) = Zs2) & (Ys = (((append @ A) @ Us2) @ Ts)))))))))). % 6.67/2.70 thf(fact_7295_append__replicate__commute, axiom, (! [A:$tType,N2:nat,X3:A,K:nat]: (((((append @ A) @ (((replicate @ A) @ N2) @ X3)) @ (((replicate @ A) @ K) @ X3)) = (((append @ A) @ (((replicate @ A) @ K) @ X3)) @ (((replicate @ A) @ N2) @ X3)))))). % 6.67/2.70 thf(fact_7296_enumerate__append__eq, axiom, (! [A:$tType,N2:nat,Xs:(list @ A),Ys:(list @ A)]: (((((enumerate @ A) @ N2) @ (((append @ A) @ Xs) @ Ys)) = (((append @ ((product_prod @ nat) @ A)) @ (((enumerate @ A) @ N2) @ Xs)) @ (((enumerate @ A) @ (((plus_plus @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs))) @ Ys)))))). % 6.67/2.70 thf(fact_7297_remdups__append2, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((remdups @ A) @ (((append @ A) @ Xs) @ ((remdups @ A) @ Ys))) = ((remdups @ A) @ (((append @ A) @ Xs) @ Ys)))))). % 6.67/2.70 thf(fact_7298_remove1__append, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Ys:(list @ A)]: ((((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => ((((remove1 @ A) @ X3) @ (((append @ A) @ Xs) @ Ys)) = (((append @ A) @ (((remove1 @ A) @ X3) @ Xs)) @ Ys))) & ((~ ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)))) => ((((remove1 @ A) @ X3) @ (((append @ A) @ Xs) @ Ys)) = (((append @ A) @ Xs) @ (((remove1 @ A) @ X3) @ Ys)))))))). % 6.67/2.70 thf(fact_7299_same__length__different, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((Xs != Ys) => ((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys)) => (? [Pre:(list @ A),X5:A,Xs4:(list @ A),Y4:A,Ys5:(list @ A)]: (((X5 != Y4) & ((Xs = (((append @ A) @ Pre) @ (((append @ A) @ (((cons @ A) @ X5) @ (nil @ A))) @ Xs4))) & (Ys = (((append @ A) @ Pre) @ (((append @ A) @ (((cons @ A) @ Y4) @ (nil @ A))) @ Ys5)))))))))))). % 6.67/2.70 thf(fact_7300_not__distinct__decomp, axiom, (! [A:$tType,Ws2:(list @ A)]: (((~ (((distinct @ A) @ Ws2))) => (? [Xs3:(list @ A),Ys4:(list @ A),Zs:(list @ A),Y4:A]: ((Ws2 = (((append @ A) @ Xs3) @ (((append @ A) @ (((cons @ A) @ Y4) @ (nil @ A))) @ (((append @ A) @ Ys4) @ (((append @ A) @ (((cons @ A) @ Y4) @ (nil @ A))) @ Zs))))))))))). % 6.67/2.70 thf(fact_7301_not__distinct__conv__prefix, axiom, (! [A:$tType,As:(list @ A)]: (((~ (((distinct @ A) @ As))) = (? [Xs2:(list @ A),Y2:A,Ys3:(list @ A)]: (((((member @ A) @ Y2) @ ((set2 @ A) @ Xs2)) & (((distinct @ A) @ Xs2) & (As = (((append @ A) @ Xs2) @ (((cons @ A) @ Y2) @ Ys3))))))))))). % 6.67/2.70 thf(fact_7302_replicate__append__same, axiom, (! [A:$tType,I:nat,X3:A]: (((((append @ A) @ (((replicate @ A) @ I) @ X3)) @ (((cons @ A) @ X3) @ (nil @ A))) = (((cons @ A) @ X3) @ (((replicate @ A) @ I) @ X3)))))). % 6.67/2.70 thf(fact_7303_list__update__append1, axiom, (! [A:$tType,I:nat,Xs:(list @ A),Ys:(list @ A),X3:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => (((((list_update @ A) @ (((append @ A) @ Xs) @ Ys)) @ I) @ X3) = (((append @ A) @ ((((list_update @ A) @ Xs) @ I) @ X3)) @ Ys)))))). % 6.67/2.70 thf(fact_7304_remdups__adj__append__two, axiom, (! [A:$tType,Xs:(list @ A),X3:A,Y:A]: ((((remdups_adj @ A) @ (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (((cons @ A) @ Y) @ (nil @ A))))) = (((append @ A) @ ((remdups_adj @ A) @ (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (nil @ A))))) @ ((((if @ (list @ A)) @ (X3 = Y)) @ (nil @ A)) @ (((cons @ A) @ Y) @ (nil @ A)))))))). % 6.67/2.70 thf(fact_7305_remove1__split, axiom, (! [A:$tType,A2:A,Xs:(list @ A),Ys:(list @ A)]: (((((member @ A) @ A2) @ ((set2 @ A) @ Xs)) => (((((remove1 @ A) @ A2) @ Xs) = Ys) = (? [Ls:(list @ A),Rs:(list @ A)]: (((Xs = (((append @ A) @ Ls) @ (((cons @ A) @ A2) @ Rs))) & ((~ ((((member @ A) @ A2) @ ((set2 @ A) @ Ls)))) & (Ys = (((append @ A) @ Ls) @ Rs))))))))))). % 6.67/2.70 thf(fact_7306_lexord__append__leftD, axiom, (! [A:$tType,X3:(list @ A),U:(list @ A),V2:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ X3) @ U)) @ (((append @ A) @ X3) @ V2))) @ ((lexord @ A) @ R2)) => ((! [A5:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ A5)) @ R2))))) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ U) @ V2)) @ ((lexord @ A) @ R2))))))). % 6.67/2.70 thf(fact_7307_lexord__append__rightI, axiom, (! [A:$tType,Y:(list @ A),X3:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((? [B11:A,Z4:(list @ A)]: ((Y = (((cons @ A) @ B11) @ Z4)))) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X3) @ (((append @ A) @ X3) @ Y))) @ ((lexord @ A) @ R2)))))). % 6.67/2.70 thf(fact_7308_lexord__sufE, axiom, (! [A:$tType,Xs:(list @ A),Zs2:(list @ A),Ys:(list @ A),Qs:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Zs2)) @ (((append @ A) @ Ys) @ Qs))) @ ((lexord @ A) @ R2)) => ((Xs != Ys) => ((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys)) => ((((size_size @ (list @ A)) @ Zs2) = ((size_size @ (list @ A)) @ Qs)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((lexord @ A) @ R2))))))))). % 6.67/2.70 thf(fact_7309_lex__append__leftD, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A)]: (((! [X5:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ X5)) @ R2))))) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Ys)) @ (((append @ A) @ Xs) @ Zs2))) @ ((lex @ A) @ R2)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ys) @ Zs2)) @ ((lex @ A) @ R2))))))). % 6.67/2.70 thf(fact_7310_lex__append__left__iff, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A)]: (((! [X5:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ X5)) @ R2))))) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Ys)) @ (((append @ A) @ Xs) @ Zs2))) @ ((lex @ A) @ R2)) = (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ys) @ Zs2)) @ ((lex @ A) @ R2))))))). % 6.67/2.70 thf(fact_7311_rotate1_Osimps_I2_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((rotate1 @ A) @ (((cons @ A) @ X3) @ Xs)) = (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (nil @ A))))))). % 6.67/2.70 thf(fact_7312_lex__append__rightI, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A)),Vs2:(list @ A),Us:(list @ A)]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((lex @ A) @ R2)) => ((((size_size @ (list @ A)) @ Vs2) = ((size_size @ (list @ A)) @ Us)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Us)) @ (((append @ A) @ Ys) @ Vs2))) @ ((lex @ A) @ R2))))))). % 6.67/2.70 thf(fact_7313_lenlex__append1, axiom, (! [A:$tType,Us:(list @ A),Xs:(list @ A),R:(set @ ((product_prod @ A) @ A)),Vs2:(list @ A),Ys:(list @ A)]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Us) @ Xs)) @ ((lenlex @ A) @ R)) => ((((size_size @ (list @ A)) @ Vs2) = ((size_size @ (list @ A)) @ Ys)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Us) @ Vs2)) @ (((append @ A) @ Xs) @ Ys))) @ ((lenlex @ A) @ R))))))). % 6.67/2.70 thf(fact_7314_nths__append, axiom, (! [A:$tType,L:(list @ A),L3:(list @ A),A3:(set @ nat)]: (((((nths @ A) @ (((append @ A) @ L) @ L3)) @ A3) = (((append @ A) @ (((nths @ A) @ L) @ A3)) @ (((nths @ A) @ L3) @ ((collect @ nat) @ (^ [J3:nat]: ((((member @ nat) @ (((plus_plus @ nat) @ J3) @ ((size_size @ (list @ A)) @ L))) @ A3)))))))))). % 6.67/2.70 thf(fact_7315_length__Suc__conv__rev, axiom, (! [A:$tType,Xs:(list @ A),N2:nat]: (((((size_size @ (list @ A)) @ Xs) = (suc @ N2)) = (? [Y2:A,Ys3:(list @ A)]: (((Xs = (((append @ A) @ Ys3) @ (((cons @ A) @ Y2) @ (nil @ A)))) & (((size_size @ (list @ A)) @ Ys3) = N2)))))))). % 6.67/2.70 thf(fact_7316_length__append__singleton, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((size_size @ (list @ A)) @ (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (nil @ A)))) = (suc @ ((size_size @ (list @ A)) @ Xs)))))). % 6.67/2.70 thf(fact_7317_subseqs_Osimps_I2_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((subseqs @ A) @ (((cons @ A) @ X3) @ Xs)) = (((append @ (list @ A)) @ ((((map @ (list @ A)) @ (list @ A)) @ ((cons @ A) @ X3)) @ ((subseqs @ A) @ Xs))) @ ((subseqs @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7318_nth__append, axiom, (! [A:$tType,N2:nat,Xs:(list @ A),Ys:(list @ A)]: ((((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ (((append @ A) @ Xs) @ Ys)) @ N2) = (((nth @ A) @ Xs) @ N2))) & ((~ ((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)))) => ((((nth @ A) @ (((append @ A) @ Xs) @ Ys)) @ N2) = (((nth @ A) @ Ys) @ (((minus_minus @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs))))))))). % 6.67/2.70 thf(fact_7319_list__update__append, axiom, (! [A:$tType,N2:nat,Xs:(list @ A),Ys:(list @ A),X3:A]: ((((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => (((((list_update @ A) @ (((append @ A) @ Xs) @ Ys)) @ N2) @ X3) = (((append @ A) @ ((((list_update @ A) @ Xs) @ N2) @ X3)) @ Ys))) & ((~ ((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)))) => (((((list_update @ A) @ (((append @ A) @ Xs) @ Ys)) @ N2) @ X3) = (((append @ A) @ Xs) @ ((((list_update @ A) @ Ys) @ (((minus_minus @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs))) @ X3)))))))). % 6.67/2.70 thf(fact_7320_listrel1E, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((listrel1 @ A) @ R2)) => (~ ((! [X5:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Y4)) @ R2) => (! [Us3:(list @ A),Vs3:(list @ A)]: (((Xs = (((append @ A) @ Us3) @ (((cons @ A) @ X5) @ Vs3))) => (Ys != (((append @ A) @ Us3) @ (((cons @ A) @ Y4) @ Vs3))))))))))))))). % 6.67/2.70 thf(fact_7321_listrel1I, axiom, (! [A:$tType,X3:A,Y:A,R2:(set @ ((product_prod @ A) @ A)),Xs:(list @ A),Us:(list @ A),Vs2:(list @ A),Ys:(list @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ R2) => ((Xs = (((append @ A) @ Us) @ (((cons @ A) @ X3) @ Vs2))) => ((Ys = (((append @ A) @ Us) @ (((cons @ A) @ Y) @ Vs2))) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((listrel1 @ A) @ R2)))))))). % 6.67/2.70 thf(fact_7322_lexord__append__left__rightI, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A)),U:(list @ A),X3:(list @ A),Y:(list @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ U) @ (((cons @ A) @ A2) @ X3))) @ (((append @ A) @ U) @ (((cons @ A) @ B2) @ Y)))) @ ((lexord @ A) @ R2)))))). % 6.67/2.70 thf(fact_7323_lexord__same__pref__iff, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Ys)) @ (((append @ A) @ Xs) @ Zs2))) @ ((lexord @ A) @ R2)) = ((? [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ X)) @ R2)))) | (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ys) @ Zs2)) @ ((lexord @ A) @ R2))))))). % 6.67/2.70 thf(fact_7324_lexord__sufI, axiom, (! [A:$tType,U:(list @ A),W2:(list @ A),R2:(set @ ((product_prod @ A) @ A)),V2:(list @ A),Z2:(list @ A)]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ U) @ W2)) @ ((lexord @ A) @ R2)) => ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ W2)) @ ((size_size @ (list @ A)) @ U)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ U) @ V2)) @ (((append @ A) @ W2) @ Z2))) @ ((lexord @ A) @ R2))))))). % 6.67/2.70 thf(fact_7325_product_Osimps_I2_J, axiom, (! [A:$tType,B:$tType,X3:A,Xs:(list @ A),Ys:(list @ B)]: ((((((product @ A) @ B) @ (((cons @ A) @ X3) @ Xs)) @ Ys) = (((append @ ((product_prod @ A) @ B)) @ ((((map @ B) @ ((product_prod @ A) @ B)) @ (((product_Pair @ A) @ B) @ X3)) @ Ys)) @ ((((product @ A) @ B) @ Xs) @ Ys)))))). % 6.67/2.70 thf(fact_7326_snoc__listrel1__snoc__iff, axiom, (! [A:$tType,Xs:(list @ A),X3:A,Ys:(list @ A),Y:A,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ (nil @ A)))) @ (((append @ A) @ Ys) @ (((cons @ A) @ Y) @ (nil @ A))))) @ ((listrel1 @ A) @ R2)) = (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((listrel1 @ A) @ R2)) & (X3 = Y)) | ((Xs = Ys) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ R2))))))). % 6.67/2.70 thf(fact_7327_horner__sum__append, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_1 @ A) => (! [F2:(B > A),A2:A,Xs:(list @ B),Ys:(list @ B)]: (((((((groups4207007520872428315er_sum @ B) @ A) @ F2) @ A2) @ (((append @ B) @ Xs) @ Ys)) = (((plus_plus @ A) @ (((((groups4207007520872428315er_sum @ B) @ A) @ F2) @ A2) @ Xs)) @ (((times_times @ A) @ (((power_power @ A) @ A2) @ ((size_size @ (list @ B)) @ Xs))) @ (((((groups4207007520872428315er_sum @ B) @ A) @ F2) @ A2) @ Ys)))))))))). % 6.67/2.70 thf(fact_7328_nths__Cons, axiom, (! [A:$tType,X3:A,L:(list @ A),A3:(set @ nat)]: (((((nths @ A) @ (((cons @ A) @ X3) @ L)) @ A3) = (((append @ A) @ ((((if @ (list @ A)) @ (((member @ nat) @ (zero_zero @ nat)) @ A3)) @ (((cons @ A) @ X3) @ (nil @ A))) @ (nil @ A))) @ (((nths @ A) @ L) @ ((collect @ nat) @ (^ [J3:nat]: ((((member @ nat) @ (suc @ J3)) @ A3)))))))))). % 6.67/2.70 thf(fact_7329_comm__append__is__replicate, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((Xs != (nil @ A)) => ((Ys != (nil @ A)) => (((((append @ A) @ Xs) @ Ys) = (((append @ A) @ Ys) @ Xs)) => (? [N3:nat,Zs:(list @ A)]: (((((ord_less @ nat) @ (one_one @ nat)) @ N3) & (((concat @ A) @ (((replicate @ (list @ A)) @ N3) @ Zs)) = (((append @ A) @ Xs) @ Ys))))))))))). % 6.67/2.70 thf(fact_7330_listrel1__def, axiom, (! [A:$tType]: (((listrel1 @ A) = (^ [R5:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_case_prod @ (list @ A)) @ (list @ A)) @ $o) @ (^ [Xs2:(list @ A),Ys3:(list @ A)]: ((? [Us2:(list @ A),Z6:A,Z7:A,Vs:(list @ A)]: (((Xs2 = (((append @ A) @ Us2) @ (((cons @ A) @ Z6) @ Vs))) & ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z6) @ Z7)) @ R5) & (Ys3 = (((append @ A) @ Us2) @ (((cons @ A) @ Z7) @ Vs))))))))))))))))). % 6.67/2.70 thf(fact_7331_lexord__def, axiom, (! [A:$tType]: (((lexord @ A) = (^ [R5:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_case_prod @ (list @ A)) @ (list @ A)) @ $o) @ (^ [X:(list @ A),Y2:(list @ A)]: ((? [A4:A,V5:(list @ A)]: (((Y2 = (((append @ A) @ X) @ (((cons @ A) @ A4) @ V5))) | (? [U2:(list @ A),B4:A,C4:A,W3:(list @ A),Z6:(list @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B4) @ C4)) @ R5) & ((X = (((append @ A) @ U2) @ (((cons @ A) @ B4) @ W3))) & (Y2 = (((append @ A) @ U2) @ (((cons @ A) @ C4) @ Z6)))))))))))))))))))). % 6.67/2.70 thf(fact_7332_take__Suc__conv__app__nth, axiom, (! [A:$tType,I:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((take @ A) @ (suc @ I)) @ Xs) = (((append @ A) @ (((take @ A) @ I) @ Xs)) @ (((cons @ A) @ (((nth @ A) @ Xs) @ I)) @ (nil @ A)))))))). % 6.67/2.70 thf(fact_7333_lex__conv, axiom, (! [A:$tType]: (((lex @ A) = (^ [R5:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_case_prod @ (list @ A)) @ (list @ A)) @ $o) @ (^ [Xs2:(list @ A),Ys3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) = ((size_size @ (list @ A)) @ Ys3)) & (? [Xys2:(list @ A),X:A,Y2:A,Xs6:(list @ A),Ys7:(list @ A)]: (((Xs2 = (((append @ A) @ Xys2) @ (((cons @ A) @ X) @ Xs6))) & ((Ys3 = (((append @ A) @ Xys2) @ (((cons @ A) @ Y2) @ Ys7))) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y2)) @ R5)))))))))))))))). % 6.67/2.70 thf(fact_7334_nth__repl, axiom, (! [A:$tType,M:nat,Xs:(list @ A),N2:nat,X3:A]: (((((ord_less @ nat) @ M) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => ((M != N2) => ((((nth @ A) @ (((append @ A) @ (((take @ A) @ N2) @ Xs)) @ (((append @ A) @ (((cons @ A) @ X3) @ (nil @ A))) @ (((drop @ A) @ (((plus_plus @ nat) @ N2) @ (one_one @ nat))) @ Xs)))) @ M) = (((nth @ A) @ Xs) @ M)))))))). % 6.67/2.70 thf(fact_7335_pos__n__replace, axiom, (! [A:$tType,N2:nat,Xs:(list @ A),Y:A]: (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => (((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ (((append @ A) @ (((take @ A) @ N2) @ Xs)) @ (((append @ A) @ (((cons @ A) @ Y) @ (nil @ A))) @ (((drop @ A) @ (suc @ N2)) @ Xs))))))))). % 6.67/2.70 thf(fact_7336_drop0, axiom, (! [A:$tType]: ((((drop @ A) @ (zero_zero @ nat)) = (^ [X:(list @ A)]: (X)))))). % 6.67/2.70 thf(fact_7337_drop__drop, axiom, (! [A:$tType,N2:nat,M:nat,Xs:(list @ A)]: (((((drop @ A) @ N2) @ (((drop @ A) @ M) @ Xs)) = (((drop @ A) @ (((plus_plus @ nat) @ N2) @ M)) @ Xs))))). % 6.67/2.70 thf(fact_7338_drop__Suc__Cons, axiom, (! [A:$tType,N2:nat,X3:A,Xs:(list @ A)]: (((((drop @ A) @ (suc @ N2)) @ (((cons @ A) @ X3) @ Xs)) = (((drop @ A) @ N2) @ Xs))))). % 6.67/2.70 thf(fact_7339_length__drop, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ (((drop @ A) @ N2) @ Xs)) = (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N2))))). % 6.67/2.70 thf(fact_7340_append__take__drop__id, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((append @ A) @ (((take @ A) @ N2) @ Xs)) @ (((drop @ A) @ N2) @ Xs)) = Xs)))). % 6.67/2.70 thf(fact_7341_drop__update__cancel, axiom, (! [A:$tType,N2:nat,M:nat,Xs:(list @ A),X3:A]: (((((ord_less @ nat) @ N2) @ M) => ((((drop @ A) @ M) @ ((((list_update @ A) @ Xs) @ N2) @ X3)) = (((drop @ A) @ M) @ Xs)))))). % 6.67/2.70 thf(fact_7342_drop__replicate, axiom, (! [A:$tType,I:nat,K:nat,X3:A]: (((((drop @ A) @ I) @ (((replicate @ A) @ K) @ X3)) = (((replicate @ A) @ (((minus_minus @ nat) @ K) @ I)) @ X3))))). % 6.67/2.70 thf(fact_7343_drop__eq__Nil2, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((nil @ A) = (((drop @ A) @ N2) @ Xs)) = (((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N2))))). % 6.67/2.70 thf(fact_7344_drop__eq__Nil, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((((drop @ A) @ N2) @ Xs) = (nil @ A)) = (((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N2))))). % 6.67/2.70 thf(fact_7345_drop__all, axiom, (! [A:$tType,Xs:(list @ A),N2:nat]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N2) => ((((drop @ A) @ N2) @ Xs) = (nil @ A)))))). % 6.67/2.70 thf(fact_7346_drop__append, axiom, (! [A:$tType,N2:nat,Xs:(list @ A),Ys:(list @ A)]: (((((drop @ A) @ N2) @ (((append @ A) @ Xs) @ Ys)) = (((append @ A) @ (((drop @ A) @ N2) @ Xs)) @ (((drop @ A) @ (((minus_minus @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs))) @ Ys)))))). % 6.67/2.70 thf(fact_7347_drop__Cons__numeral, axiom, (! [A:$tType,V2:num,X3:A,Xs:(list @ A)]: (((((drop @ A) @ ((numeral_numeral @ nat) @ V2)) @ (((cons @ A) @ X3) @ Xs)) = (((drop @ A) @ (((minus_minus @ nat) @ ((numeral_numeral @ nat) @ V2)) @ (one_one @ nat))) @ Xs))))). % 6.67/2.70 thf(fact_7348_nth__drop, axiom, (! [A:$tType,N2:nat,Xs:(list @ A),I:nat]: (((((ord_less_eq @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ (((drop @ A) @ N2) @ Xs)) @ I) = (((nth @ A) @ Xs) @ (((plus_plus @ nat) @ N2) @ I))))))). % 6.67/2.70 thf(fact_7349_nth__via__drop, axiom, (! [A:$tType,N2:nat,Xs:(list @ A),Y:A,Ys:(list @ A)]: ((((((drop @ A) @ N2) @ Xs) = (((cons @ A) @ Y) @ Ys)) => ((((nth @ A) @ Xs) @ N2) = Y))))). % 6.67/2.70 thf(fact_7350_set__drop__subset__set__drop, axiom, (! [A:$tType,N2:nat,M:nat,Xs:(list @ A)]: (((((ord_less_eq @ nat) @ N2) @ M) => (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ (((drop @ A) @ M) @ Xs))) @ ((set2 @ A) @ (((drop @ A) @ N2) @ Xs))))))). % 6.67/2.70 thf(fact_7351_drop__zip, axiom, (! [A:$tType,B:$tType,N2:nat,Xs:(list @ A),Ys:(list @ B)]: (((((drop @ ((product_prod @ A) @ B)) @ N2) @ ((((zip @ A) @ B) @ Xs) @ Ys)) = ((((zip @ A) @ B) @ (((drop @ A) @ N2) @ Xs)) @ (((drop @ B) @ N2) @ Ys)))))). % 6.67/2.70 thf(fact_7352_drop__0, axiom, (! [A:$tType,Xs:(list @ A)]: (((((drop @ A) @ (zero_zero @ nat)) @ Xs) = Xs)))). % 6.67/2.70 thf(fact_7353_drop__Nil, axiom, (! [A:$tType,N2:nat]: (((((drop @ A) @ N2) @ (nil @ A)) = (nil @ A))))). % 6.67/2.70 thf(fact_7354_in__set__dropD, axiom, (! [A:$tType,X3:A,N2:nat,Xs:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ (((drop @ A) @ N2) @ Xs))) => (((member @ A) @ X3) @ ((set2 @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7355_take__drop, axiom, (! [A:$tType,N2:nat,M:nat,Xs:(list @ A)]: (((((take @ A) @ N2) @ (((drop @ A) @ M) @ Xs)) = (((drop @ A) @ M) @ (((take @ A) @ (((plus_plus @ nat) @ N2) @ M)) @ Xs)))))). % 6.67/2.70 thf(fact_7356_drop__take, axiom, (! [A:$tType,N2:nat,M:nat,Xs:(list @ A)]: (((((drop @ A) @ N2) @ (((take @ A) @ M) @ Xs)) = (((take @ A) @ (((minus_minus @ nat) @ M) @ N2)) @ (((drop @ A) @ N2) @ Xs)))))). % 6.67/2.70 thf(fact_7357_distinct__drop, axiom, (! [A:$tType,Xs:(list @ A),I:nat]: ((((distinct @ A) @ Xs) => ((distinct @ A) @ (((drop @ A) @ I) @ Xs)))))). % 6.67/2.70 thf(fact_7358_set__drop__subset, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ (((drop @ A) @ N2) @ Xs))) @ ((set2 @ A) @ Xs))))). % 6.67/2.70 thf(fact_7359_drop__eq__nths, axiom, (! [A:$tType]: (((drop @ A) = (^ [N:nat,Xs2:(list @ A)]: ((((nths @ A) @ Xs2) @ ((collect @ nat) @ ((ord_less_eq @ nat) @ N))))))))). % 6.67/2.70 thf(fact_7360_drop__map, axiom, (! [A:$tType,B:$tType,N2:nat,F2:(B > A),Xs:(list @ B)]: (((((drop @ A) @ N2) @ ((((map @ B) @ A) @ F2) @ Xs)) = ((((map @ B) @ A) @ F2) @ (((drop @ B) @ N2) @ Xs)))))). % 6.67/2.70 thf(fact_7361_take__add, axiom, (! [A:$tType,I:nat,J:nat,Xs:(list @ A)]: (((((take @ A) @ (((plus_plus @ nat) @ I) @ J)) @ Xs) = (((append @ A) @ (((take @ A) @ I) @ Xs)) @ (((take @ A) @ J) @ (((drop @ A) @ I) @ Xs))))))). % 6.67/2.70 thf(fact_7362_append__eq__conv__conj, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A)]: ((((((append @ A) @ Xs) @ Ys) = Zs2) = ((Xs = (((take @ A) @ ((size_size @ (list @ A)) @ Xs)) @ Zs2)) & (Ys = (((drop @ A) @ ((size_size @ (list @ A)) @ Xs)) @ Zs2))))))). % 6.67/2.70 thf(fact_7363_drop__update__swap, axiom, (! [A:$tType,M:nat,N2:nat,Xs:(list @ A),X3:A]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((drop @ A) @ M) @ ((((list_update @ A) @ Xs) @ N2) @ X3)) = ((((list_update @ A) @ (((drop @ A) @ M) @ Xs)) @ (((minus_minus @ nat) @ N2) @ M)) @ X3)))))). % 6.67/2.70 thf(fact_7364_drop__Cons, axiom, (! [A:$tType,N2:nat,X3:A,Xs:(list @ A)]: (((((drop @ A) @ N2) @ (((cons @ A) @ X3) @ Xs)) = ((((case_nat @ (list @ A)) @ (((cons @ A) @ X3) @ Xs)) @ (^ [M2:nat]: ((((drop @ A) @ M2) @ Xs)))) @ N2))))). % 6.67/2.70 thf(fact_7365_nths__drop, axiom, (! [A:$tType,N2:nat,Xs:(list @ A),I6:(set @ nat)]: (((((nths @ A) @ (((drop @ A) @ N2) @ Xs)) @ I6) = (((nths @ A) @ Xs) @ ((((image @ nat) @ nat) @ ((plus_plus @ nat) @ N2)) @ I6)))))). % 6.67/2.70 thf(fact_7366_drop__Cons_H, axiom, (! [A:$tType,N2:nat,X3:A,Xs:(list @ A)]: ((((N2 = (zero_zero @ nat)) => ((((drop @ A) @ N2) @ (((cons @ A) @ X3) @ Xs)) = (((cons @ A) @ X3) @ Xs))) & ((N2 != (zero_zero @ nat)) => ((((drop @ A) @ N2) @ (((cons @ A) @ X3) @ Xs)) = (((drop @ A) @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))) @ Xs))))))). % 6.67/2.70 thf(fact_7367_append__eq__append__conv__if, axiom, (! [A:$tType,Xs_1:(list @ A),Xs_2:(list @ A),Ys_1:(list @ A),Ys_2:(list @ A)]: ((((((append @ A) @ Xs_1) @ Xs_2) = (((append @ A) @ Ys_1) @ Ys_2)) = (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs_1)) @ ((size_size @ (list @ A)) @ Ys_1)) => ((Xs_1 = (((take @ A) @ ((size_size @ (list @ A)) @ Xs_1)) @ Ys_1)) & (Xs_2 = (((append @ A) @ (((drop @ A) @ ((size_size @ (list @ A)) @ Xs_1)) @ Ys_1)) @ Ys_2)))) & ((~ ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs_1)) @ ((size_size @ (list @ A)) @ Ys_1)))) => (((((take @ A) @ ((size_size @ (list @ A)) @ Ys_1)) @ Xs_1) = Ys_1) & ((((append @ A) @ (((drop @ A) @ ((size_size @ (list @ A)) @ Ys_1)) @ Xs_1)) @ Xs_2) = Ys_2)))))))). % 6.67/2.70 thf(fact_7368_zip__append2, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B),Zs2:(list @ B)]: ((((((zip @ A) @ B) @ Xs) @ (((append @ B) @ Ys) @ Zs2)) = (((append @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ (((take @ A) @ ((size_size @ (list @ B)) @ Ys)) @ Xs)) @ Ys)) @ ((((zip @ A) @ B) @ (((drop @ A) @ ((size_size @ (list @ B)) @ Ys)) @ Xs)) @ Zs2)))))). % 6.67/2.70 thf(fact_7369_zip__append1, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ B)]: ((((((zip @ A) @ B) @ (((append @ A) @ Xs) @ Ys)) @ Zs2) = (((append @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ (((take @ B) @ ((size_size @ (list @ A)) @ Xs)) @ Zs2))) @ ((((zip @ A) @ B) @ Ys) @ (((drop @ B) @ ((size_size @ (list @ A)) @ Xs)) @ Zs2))))))). % 6.67/2.70 thf(fact_7370_Cons__nth__drop__Suc, axiom, (! [A:$tType,I:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((cons @ A) @ (((nth @ A) @ Xs) @ I)) @ (((drop @ A) @ (suc @ I)) @ Xs)) = (((drop @ A) @ I) @ Xs)))))). % 6.67/2.70 thf(fact_7371_set__take__disj__set__drop__if__distinct, axiom, (! [A:$tType,Vs2:(list @ A),I:nat,J:nat]: ((((distinct @ A) @ Vs2) => ((((ord_less_eq @ nat) @ I) @ J) => ((((inf_inf @ (set @ A)) @ ((set2 @ A) @ (((take @ A) @ I) @ Vs2))) @ ((set2 @ A) @ (((drop @ A) @ J) @ Vs2))) = (bot_bot @ (set @ A)))))))). % 6.67/2.70 thf(fact_7372_id__take__nth__drop, axiom, (! [A:$tType,I:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => (Xs = (((append @ A) @ (((take @ A) @ I) @ Xs)) @ (((cons @ A) @ (((nth @ A) @ Xs) @ I)) @ (((drop @ A) @ (suc @ I)) @ Xs)))))))). % 6.67/2.70 thf(fact_7373_upd__conv__take__nth__drop, axiom, (! [A:$tType,I:nat,Xs:(list @ A),A2:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => (((((list_update @ A) @ Xs) @ I) @ A2) = (((append @ A) @ (((take @ A) @ I) @ Xs)) @ (((cons @ A) @ A2) @ (((drop @ A) @ (suc @ I)) @ Xs)))))))). % 6.67/2.70 thf(fact_7374_lexn__conv, axiom, (! [A:$tType]: (((lexn @ A) = (^ [R5:(set @ ((product_prod @ A) @ A)),N:nat]: (((collect @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_case_prod @ (list @ A)) @ (list @ A)) @ $o) @ (^ [Xs2:(list @ A),Ys3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs2) = N) & ((((size_size @ (list @ A)) @ Ys3) = N) & (? [Xys2:(list @ A),X:A,Y2:A,Xs6:(list @ A),Ys7:(list @ A)]: (((Xs2 = (((append @ A) @ Xys2) @ (((cons @ A) @ X) @ Xs6))) & ((Ys3 = (((append @ A) @ Xys2) @ (((cons @ A) @ Y2) @ Ys7))) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y2)) @ R5))))))))))))))))). % 6.67/2.70 thf(fact_7375_list__encode_Oelims, axiom, (! [X3:(list @ nat),Y:nat]: ((((nat_list_encode @ X3) = Y) => (((X3 = (nil @ nat)) => (Y != (zero_zero @ nat))) => (~ ((! [X5:nat,Xs3:(list @ nat)]: (((X3 = (((cons @ nat) @ X5) @ Xs3)) => (Y != (suc @ (nat_prod_encode @ ((((product_Pair @ nat) @ nat) @ X5) @ (nat_list_encode @ Xs3))))))))))))))). % 6.67/2.70 thf(fact_7376_lexn_Osimps_I1_J, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((((lexn @ A) @ R2) @ (zero_zero @ nat)) = (bot_bot @ (set @ ((product_prod @ (list @ A)) @ (list @ A)))))))). % 6.67/2.70 thf(fact_7377_list__encode_Osimps_I1_J, axiom, ((nat_list_encode @ (nil @ nat)) = (zero_zero @ nat))). % 6.67/2.70 thf(fact_7378_lexn__length, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),R2:(set @ ((product_prod @ A) @ A)),N2:nat]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ (((lexn @ A) @ R2) @ N2)) => ((((size_size @ (list @ A)) @ Xs) = N2) & (((size_size @ (list @ A)) @ Ys) = N2)))))). % 6.67/2.70 thf(fact_7379_lex__def, axiom, (! [A:$tType]: (((lex @ A) = (^ [R5:(set @ ((product_prod @ A) @ A))]: (((complete_Sup_Sup @ (set @ ((product_prod @ (list @ A)) @ (list @ A)))) @ ((((image @ nat) @ (set @ ((product_prod @ (list @ A)) @ (list @ A)))) @ ((lexn @ A) @ R5)) @ (top_top @ (set @ nat)))))))))). % 6.67/2.70 thf(fact_7380_list__encode_Osimps_I2_J, axiom, (! [X3:nat,Xs:(list @ nat)]: (((nat_list_encode @ (((cons @ nat) @ X3) @ Xs)) = (suc @ (nat_prod_encode @ ((((product_Pair @ nat) @ nat) @ X3) @ (nat_list_encode @ Xs)))))))). % 6.67/2.70 thf(fact_7381_upto__aux__rec, axiom, (upto_aux = (^ [I3:int,J3:int,Js:(list @ int)]: (((((if @ (list @ int)) @ (((ord_less @ int) @ J3) @ I3)) @ Js) @ (((upto_aux @ I3) @ (((minus_minus @ int) @ J3) @ (one_one @ int))) @ (((cons @ int) @ J3) @ Js))))))). % 6.67/2.70 thf(fact_7382_list_Osimps_I5_J, axiom, (! [B:$tType,A:$tType,F1:B,F22:(A > ((list @ A) > B)),X21:A,X222:(list @ A)]: (((((((case_list @ B) @ A) @ F1) @ F22) @ (((cons @ A) @ X21) @ X222)) = ((F22 @ X21) @ X222))))). % 6.67/2.70 thf(fact_7383_list_Ocase__distrib, axiom, (! [B:$tType,C:$tType,A:$tType,H2:(B > C),F1:B,F22:(A > ((list @ A) > B)),List:(list @ A)]: (((H2 @ (((((case_list @ B) @ A) @ F1) @ F22) @ List)) = (((((case_list @ C) @ A) @ (H2 @ F1)) @ (^ [X15:A,X24:(list @ A)]: ((H2 @ ((F22 @ X15) @ X24))))) @ List))))). % 6.67/2.70 thf(fact_7384_list_Osimps_I4_J, axiom, (! [A:$tType,B:$tType,F1:B,F22:(A > ((list @ A) > B))]: (((((((case_list @ B) @ A) @ F1) @ F22) @ (nil @ A)) = F1)))). % 6.67/2.70 thf(fact_7385_remdups__adj__Cons, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((remdups_adj @ A) @ (((cons @ A) @ X3) @ Xs)) = (((((case_list @ (list @ A)) @ A) @ (((cons @ A) @ X3) @ (nil @ A))) @ (^ [Y2:A,Xs2:(list @ A)]: (((((if @ (list @ A)) @ (X3 = Y2)) @ (((cons @ A) @ Y2) @ Xs2)) @ (((cons @ A) @ X3) @ (((cons @ A) @ Y2) @ Xs2)))))) @ ((remdups_adj @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7386_min__list_Osimps, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:A,Xs:(list @ A)]: ((((min_list @ A) @ (((cons @ A) @ X3) @ Xs)) = (((((case_list @ A) @ A) @ X3) @ (^ [A4:A,List2:(list @ A)]: ((((ord_min @ A) @ X3) @ ((min_list @ A) @ Xs))))) @ Xs)))))))). % 6.67/2.70 thf(fact_7387_transpose_Oelims, axiom, (! [A:$tType,X3:(list @ (list @ A)),Y:(list @ (list @ A))]: (((((transpose @ A) @ X3) = Y) => (((X3 = (nil @ (list @ A))) => (Y != (nil @ (list @ A)))) => ((! [Xss2:(list @ (list @ A))]: (((X3 = (((cons @ (list @ A)) @ (nil @ A)) @ Xss2)) => (Y != ((transpose @ A) @ Xss2))))) => (~ ((! [X5:A,Xs3:(list @ A),Xss2:(list @ (list @ A))]: (((X3 = (((cons @ (list @ A)) @ (((cons @ A) @ X5) @ Xs3)) @ Xss2)) => (Y != (((cons @ (list @ A)) @ (((cons @ A) @ X5) @ ((concat @ A) @ ((((map @ (list @ A)) @ (list @ A)) @ ((((case_list @ (list @ A)) @ A) @ (nil @ A)) @ (^ [H:A,T3:(list @ A)]: ((((cons @ A) @ H) @ (nil @ A)))))) @ Xss2)))) @ ((transpose @ A) @ (((cons @ (list @ A)) @ Xs3) @ ((concat @ (list @ A)) @ ((((map @ (list @ A)) @ (list @ (list @ A))) @ ((((case_list @ (list @ (list @ A))) @ A) @ (nil @ (list @ A))) @ (^ [H:A,T3:(list @ A)]: ((((cons @ (list @ A)) @ T3) @ (nil @ (list @ A))))))) @ Xss2))))))))))))))))). % 6.67/2.70 thf(fact_7388_transpose_Osimps_I3_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Xss:(list @ (list @ A))]: ((((transpose @ A) @ (((cons @ (list @ A)) @ (((cons @ A) @ X3) @ Xs)) @ Xss)) = (((cons @ (list @ A)) @ (((cons @ A) @ X3) @ ((concat @ A) @ ((((map @ (list @ A)) @ (list @ A)) @ ((((case_list @ (list @ A)) @ A) @ (nil @ A)) @ (^ [H:A,T3:(list @ A)]: ((((cons @ A) @ H) @ (nil @ A)))))) @ Xss)))) @ ((transpose @ A) @ (((cons @ (list @ A)) @ Xs) @ ((concat @ (list @ A)) @ ((((map @ (list @ A)) @ (list @ (list @ A))) @ ((((case_list @ (list @ (list @ A))) @ A) @ (nil @ (list @ A))) @ (^ [H:A,T3:(list @ A)]: ((((cons @ (list @ A)) @ T3) @ (nil @ (list @ A))))))) @ Xss))))))))). % 6.67/2.70 thf(fact_7389_list_Odisc__eq__case_I1_J, axiom, (! [A:$tType,List:(list @ A)]: (((List = (nil @ A)) = (((((case_list @ $o) @ A) @ $true) @ (^ [Uu3:A,Uv3:(list @ A)]: ($false))) @ List))))). % 6.67/2.70 thf(fact_7390_list_Odisc__eq__case_I2_J, axiom, (! [A:$tType,List:(list @ A)]: (((List != (nil @ A)) = (((((case_list @ $o) @ A) @ $false) @ (^ [Uu3:A,Uv3:(list @ A)]: ($true))) @ List))))). % 6.67/2.70 thf(fact_7391_transpose_Osimps_I2_J, axiom, (! [A:$tType,Xss:(list @ (list @ A))]: ((((transpose @ A) @ (((cons @ (list @ A)) @ (nil @ A)) @ Xss)) = ((transpose @ A) @ Xss))))). % 6.67/2.70 thf(fact_7392_transpose_Osimps_I1_J, axiom, (! [A:$tType]: ((((transpose @ A) @ (nil @ (list @ A))) = (nil @ (list @ A)))))). % 6.67/2.70 thf(fact_7393_transpose__map__map, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ (list @ B))]: ((((transpose @ A) @ ((((map @ (list @ B)) @ (list @ A)) @ (((map @ B) @ A) @ F2)) @ Xs)) = ((((map @ (list @ B)) @ (list @ A)) @ (((map @ B) @ A) @ F2)) @ ((transpose @ B) @ Xs)))))). % 6.67/2.70 thf(fact_7394_transpose__empty, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: (((((transpose @ A) @ Xs) = (nil @ (list @ A))) = (! [X:(list @ A)]: (((((member @ (list @ A)) @ X) @ ((set2 @ (list @ A)) @ Xs)) => (X = (nil @ A))))))))). % 6.67/2.70 thf(fact_7395_zip__Cons, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Y:B,Ys:(list @ B)]: ((((((zip @ A) @ B) @ Xs) @ (((cons @ B) @ Y) @ Ys)) = (((((case_list @ (list @ ((product_prod @ A) @ B))) @ A) @ (nil @ ((product_prod @ A) @ B))) @ (^ [Z6:A,Zs3:(list @ A)]: ((((cons @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Z6) @ Y)) @ ((((zip @ A) @ B) @ Zs3) @ Ys))))) @ Xs))))). % 6.67/2.70 thf(fact_7396_zip__Cons1, axiom, (! [A:$tType,B:$tType,X3:A,Xs:(list @ A),Ys:(list @ B)]: ((((((zip @ A) @ B) @ (((cons @ A) @ X3) @ Xs)) @ Ys) = (((((case_list @ (list @ ((product_prod @ A) @ B))) @ B) @ (nil @ ((product_prod @ A) @ B))) @ (^ [Y2:B,Ys3:(list @ B)]: ((((cons @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y2)) @ ((((zip @ A) @ B) @ Xs) @ Ys3))))) @ Ys))))). % 6.67/2.70 thf(fact_7397_upto_Opsimps, axiom, (! [I:int,J:int]: (((((accp @ ((product_prod @ int) @ int)) @ upto_rel) @ ((((product_Pair @ int) @ int) @ I) @ J)) => (((((ord_less_eq @ int) @ I) @ J) => (((upto @ I) @ J) = (((cons @ int) @ I) @ ((upto @ (((plus_plus @ int) @ I) @ (one_one @ int))) @ J)))) & ((~ ((((ord_less_eq @ int) @ I) @ J))) => (((upto @ I) @ J) = (nil @ int)))))))). % 6.67/2.70 thf(fact_7398_upto_Opelims, axiom, (! [X3:int,Xa2:int,Y:(list @ int)]: (((((upto @ X3) @ Xa2) = Y) => ((((accp @ ((product_prod @ int) @ int)) @ upto_rel) @ ((((product_Pair @ int) @ int) @ X3) @ Xa2)) => (~ (((((((ord_less_eq @ int) @ X3) @ Xa2) => (Y = (((cons @ int) @ X3) @ ((upto @ (((plus_plus @ int) @ X3) @ (one_one @ int))) @ Xa2)))) & ((~ ((((ord_less_eq @ int) @ X3) @ Xa2))) => (Y = (nil @ int)))) => (~ ((((accp @ ((product_prod @ int) @ int)) @ upto_rel) @ ((((product_Pair @ int) @ int) @ X3) @ Xa2)))))))))))). % 6.67/2.70 thf(fact_7399_upto__empty, axiom, (! [J:int,I:int]: (((((ord_less @ int) @ J) @ I) => (((upto @ I) @ J) = (nil @ int)))))). % 6.67/2.70 thf(fact_7400_upto__Nil2, axiom, (! [I:int,J:int]: ((((nil @ int) = ((upto @ I) @ J)) = (((ord_less @ int) @ J) @ I))))). % 6.67/2.70 thf(fact_7401_upto__Nil, axiom, (! [I:int,J:int]: (((((upto @ I) @ J) = (nil @ int)) = (((ord_less @ int) @ J) @ I))))). % 6.67/2.70 thf(fact_7402_upto__single, axiom, (! [I:int]: ((((upto @ I) @ I) = (((cons @ int) @ I) @ (nil @ int)))))). % 6.67/2.70 thf(fact_7403_nth__upto, axiom, (! [I:int,K:nat,J:int]: (((((ord_less_eq @ int) @ (((plus_plus @ int) @ I) @ ((semiring_1_of_nat @ int) @ K))) @ J) => ((((nth @ int) @ ((upto @ I) @ J)) @ K) = (((plus_plus @ int) @ I) @ ((semiring_1_of_nat @ int) @ K))))))). % 6.67/2.70 thf(fact_7404_length__upto, axiom, (! [I:int,J:int]: ((((size_size @ (list @ int)) @ ((upto @ I) @ J)) = (nat2 @ (((plus_plus @ int) @ (((minus_minus @ int) @ J) @ I)) @ (one_one @ int))))))). % 6.67/2.70 thf(fact_7405_upto__rec__numeral_I1_J, axiom, (! [M:num,N2:num]: ((((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ M)) @ ((numeral_numeral @ int) @ N2)) => (((upto @ ((numeral_numeral @ int) @ M)) @ ((numeral_numeral @ int) @ N2)) = (((cons @ int) @ ((numeral_numeral @ int) @ M)) @ ((upto @ (((plus_plus @ int) @ ((numeral_numeral @ int) @ M)) @ (one_one @ int))) @ ((numeral_numeral @ int) @ N2))))) & ((~ ((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ M)) @ ((numeral_numeral @ int) @ N2)))) => (((upto @ ((numeral_numeral @ int) @ M)) @ ((numeral_numeral @ int) @ N2)) = (nil @ int))))))). % 6.67/2.70 thf(fact_7406_upto__rec__numeral_I2_J, axiom, (! [M:num,N2:num]: ((((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N2))) => (((upto @ ((numeral_numeral @ int) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N2))) = (((cons @ int) @ ((numeral_numeral @ int) @ M)) @ ((upto @ (((plus_plus @ int) @ ((numeral_numeral @ int) @ M)) @ (one_one @ int))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N2)))))) & ((~ ((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N2))))) => (((upto @ ((numeral_numeral @ int) @ M)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N2))) = (nil @ int))))))). % 6.67/2.70 thf(fact_7407_upto__rec__numeral_I3_J, axiom, (! [M:num,N2:num]: ((((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((numeral_numeral @ int) @ N2)) => (((upto @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((numeral_numeral @ int) @ N2)) = (((cons @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((upto @ (((plus_plus @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ (one_one @ int))) @ ((numeral_numeral @ int) @ N2))))) & ((~ ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((numeral_numeral @ int) @ N2)))) => (((upto @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((numeral_numeral @ int) @ N2)) = (nil @ int))))))). % 6.67/2.70 thf(fact_7408_upto__rec__numeral_I4_J, axiom, (! [M:num,N2:num]: ((((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N2))) => (((upto @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N2))) = (((cons @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((upto @ (((plus_plus @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ (one_one @ int))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N2)))))) & ((~ ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N2))))) => (((upto @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N2))) = (nil @ int))))))). % 6.67/2.70 thf(fact_7409_atLeastAtMost__upto, axiom, ((set_or1337092689740270186AtMost @ int) = (^ [I3:int,J3:int]: (((set2 @ int) @ ((upto @ I3) @ J3)))))). % 6.67/2.70 thf(fact_7410_distinct__upto, axiom, (! [I:int,J:int]: (((distinct @ int) @ ((upto @ I) @ J))))). % 6.67/2.70 thf(fact_7411_upto__aux__def, axiom, (upto_aux = (^ [I3:int,J3:int]: (((append @ int) @ ((upto @ I3) @ J3)))))). % 6.67/2.70 thf(fact_7412_upto__code, axiom, (upto = (^ [I3:int,J3:int]: ((((upto_aux @ I3) @ J3) @ (nil @ int)))))). % 6.67/2.70 thf(fact_7413_upto__split2, axiom, (! [I:int,J:int,K:int]: (((((ord_less_eq @ int) @ I) @ J) => ((((ord_less_eq @ int) @ J) @ K) => (((upto @ I) @ K) = (((append @ int) @ ((upto @ I) @ J)) @ ((upto @ (((plus_plus @ int) @ J) @ (one_one @ int))) @ K)))))))). % 6.67/2.70 thf(fact_7414_upto__split1, axiom, (! [I:int,J:int,K:int]: (((((ord_less_eq @ int) @ I) @ J) => ((((ord_less_eq @ int) @ J) @ K) => (((upto @ I) @ K) = (((append @ int) @ ((upto @ I) @ (((minus_minus @ int) @ J) @ (one_one @ int)))) @ ((upto @ J) @ K)))))))). % 6.67/2.70 thf(fact_7415_atLeastLessThan__upto, axiom, ((set_or7035219750837199246ssThan @ int) = (^ [I3:int,J3:int]: (((set2 @ int) @ ((upto @ I3) @ (((minus_minus @ int) @ J3) @ (one_one @ int)))))))). % 6.67/2.70 thf(fact_7416_greaterThanAtMost__upto, axiom, ((set_or3652927894154168847AtMost @ int) = (^ [I3:int,J3:int]: (((set2 @ int) @ ((upto @ (((plus_plus @ int) @ I3) @ (one_one @ int))) @ J3)))))). % 6.67/2.70 thf(fact_7417_upto__rec1, axiom, (! [I:int,J:int]: (((((ord_less_eq @ int) @ I) @ J) => (((upto @ I) @ J) = (((cons @ int) @ I) @ ((upto @ (((plus_plus @ int) @ I) @ (one_one @ int))) @ J))))))). % 6.67/2.70 thf(fact_7418_upto_Osimps, axiom, (upto = (^ [I3:int,J3:int]: (((((if @ (list @ int)) @ (((ord_less_eq @ int) @ I3) @ J3)) @ (((cons @ int) @ I3) @ ((upto @ (((plus_plus @ int) @ I3) @ (one_one @ int))) @ J3))) @ (nil @ int)))))). % 6.67/2.70 thf(fact_7419_upto_Oelims, axiom, (! [X3:int,Xa2:int,Y:(list @ int)]: (((((upto @ X3) @ Xa2) = Y) => (((((ord_less_eq @ int) @ X3) @ Xa2) => (Y = (((cons @ int) @ X3) @ ((upto @ (((plus_plus @ int) @ X3) @ (one_one @ int))) @ Xa2)))) & ((~ ((((ord_less_eq @ int) @ X3) @ Xa2))) => (Y = (nil @ int)))))))). % 6.67/2.70 thf(fact_7420_upto__rec2, axiom, (! [I:int,J:int]: (((((ord_less_eq @ int) @ I) @ J) => (((upto @ I) @ J) = (((append @ int) @ ((upto @ I) @ (((minus_minus @ int) @ J) @ (one_one @ int)))) @ (((cons @ int) @ J) @ (nil @ int)))))))). % 6.67/2.70 thf(fact_7421_greaterThanLessThan__upto, axiom, ((set_or5935395276787703475ssThan @ int) = (^ [I3:int,J3:int]: (((set2 @ int) @ ((upto @ (((plus_plus @ int) @ I3) @ (one_one @ int))) @ (((minus_minus @ int) @ J3) @ (one_one @ int)))))))). % 6.67/2.70 thf(fact_7422_upto__split3, axiom, (! [I:int,J:int,K:int]: (((((ord_less_eq @ int) @ I) @ J) => ((((ord_less_eq @ int) @ J) @ K) => (((upto @ I) @ K) = (((append @ int) @ ((upto @ I) @ (((minus_minus @ int) @ J) @ (one_one @ int)))) @ (((cons @ int) @ J) @ ((upto @ (((plus_plus @ int) @ J) @ (one_one @ int))) @ K))))))))). % 6.67/2.70 thf(fact_7423_transpose_Opelims, axiom, (! [A:$tType,X3:(list @ (list @ A)),Y:(list @ (list @ A))]: (((((transpose @ A) @ X3) = Y) => ((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ X3) => (((X3 = (nil @ (list @ A))) => ((Y = (nil @ (list @ A))) => (~ ((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (nil @ (list @ A))))))) => ((! [Xss2:(list @ (list @ A))]: (((X3 = (((cons @ (list @ A)) @ (nil @ A)) @ Xss2)) => ((Y = ((transpose @ A) @ Xss2)) => (~ ((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (((cons @ (list @ A)) @ (nil @ A)) @ Xss2)))))))) => (~ ((! [X5:A,Xs3:(list @ A),Xss2:(list @ (list @ A))]: (((X3 = (((cons @ (list @ A)) @ (((cons @ A) @ X5) @ Xs3)) @ Xss2)) => ((Y = (((cons @ (list @ A)) @ (((cons @ A) @ X5) @ ((concat @ A) @ ((((map @ (list @ A)) @ (list @ A)) @ ((((case_list @ (list @ A)) @ A) @ (nil @ A)) @ (^ [H:A,T3:(list @ A)]: ((((cons @ A) @ H) @ (nil @ A)))))) @ Xss2)))) @ ((transpose @ A) @ (((cons @ (list @ A)) @ Xs3) @ ((concat @ (list @ A)) @ ((((map @ (list @ A)) @ (list @ (list @ A))) @ ((((case_list @ (list @ (list @ A))) @ A) @ (nil @ (list @ A))) @ (^ [H:A,T3:(list @ A)]: ((((cons @ (list @ A)) @ T3) @ (nil @ (list @ A))))))) @ Xss2)))))) => (~ ((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (((cons @ (list @ A)) @ (((cons @ A) @ X5) @ Xs3)) @ Xss2))))))))))))))))). % 6.67/2.70 thf(fact_7424_transpose_Opsimps_I3_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A),Xss:(list @ (list @ A))]: (((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (((cons @ (list @ A)) @ (((cons @ A) @ X3) @ Xs)) @ Xss)) => (((transpose @ A) @ (((cons @ (list @ A)) @ (((cons @ A) @ X3) @ Xs)) @ Xss)) = (((cons @ (list @ A)) @ (((cons @ A) @ X3) @ ((concat @ A) @ ((((map @ (list @ A)) @ (list @ A)) @ ((((case_list @ (list @ A)) @ A) @ (nil @ A)) @ (^ [H:A,T3:(list @ A)]: ((((cons @ A) @ H) @ (nil @ A)))))) @ Xss)))) @ ((transpose @ A) @ (((cons @ (list @ A)) @ Xs) @ ((concat @ (list @ A)) @ ((((map @ (list @ A)) @ (list @ (list @ A))) @ ((((case_list @ (list @ (list @ A))) @ A) @ (nil @ (list @ A))) @ (^ [H:A,T3:(list @ A)]: ((((cons @ (list @ A)) @ T3) @ (nil @ (list @ A))))))) @ Xss)))))))))). % 6.67/2.70 thf(fact_7425_transpose_Opsimps_I2_J, axiom, (! [A:$tType,Xss:(list @ (list @ A))]: (((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (((cons @ (list @ A)) @ (nil @ A)) @ Xss)) => (((transpose @ A) @ (((cons @ (list @ A)) @ (nil @ A)) @ Xss)) = ((transpose @ A) @ Xss)))))). % 6.67/2.70 thf(fact_7426_transpose_Opsimps_I1_J, axiom, (! [A:$tType]: (((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (nil @ (list @ A))) => (((transpose @ A) @ (nil @ (list @ A))) = (nil @ (list @ A))))))). % 6.67/2.70 thf(fact_7427_transpose_Opinduct, axiom, (! [A:$tType,A0:(list @ (list @ A)),P:((list @ (list @ A)) > $o)]: (((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ A0) => (((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (nil @ (list @ A))) => (P @ (nil @ (list @ A)))) => ((! [Xss2:(list @ (list @ A))]: (((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (((cons @ (list @ A)) @ (nil @ A)) @ Xss2)) => ((P @ Xss2) => (P @ (((cons @ (list @ A)) @ (nil @ A)) @ Xss2)))))) => ((! [X5:A,Xs3:(list @ A),Xss2:(list @ (list @ A))]: (((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (((cons @ (list @ A)) @ (((cons @ A) @ X5) @ Xs3)) @ Xss2)) => ((P @ (((cons @ (list @ A)) @ Xs3) @ ((concat @ (list @ A)) @ ((((map @ (list @ A)) @ (list @ (list @ A))) @ ((((case_list @ (list @ (list @ A))) @ A) @ (nil @ (list @ A))) @ (^ [H:A,T3:(list @ A)]: ((((cons @ (list @ A)) @ T3) @ (nil @ (list @ A))))))) @ Xss2)))) => (P @ (((cons @ (list @ A)) @ (((cons @ A) @ X5) @ Xs3)) @ Xss2)))))) => (P @ A0)))))))). % 6.67/2.70 thf(fact_7428_remdups__adj_Opelims, axiom, (! [A:$tType,X3:(list @ A),Y:(list @ A)]: (((((remdups_adj @ A) @ X3) = Y) => ((((accp @ (list @ A)) @ (remdups_adj_rel @ A)) @ X3) => (((X3 = (nil @ A)) => ((Y = (nil @ A)) => (~ ((((accp @ (list @ A)) @ (remdups_adj_rel @ A)) @ (nil @ A)))))) => ((! [X5:A]: (((X3 = (((cons @ A) @ X5) @ (nil @ A))) => ((Y = (((cons @ A) @ X5) @ (nil @ A))) => (~ ((((accp @ (list @ A)) @ (remdups_adj_rel @ A)) @ (((cons @ A) @ X5) @ (nil @ A))))))))) => (~ ((! [X5:A,Y4:A,Xs3:(list @ A)]: (((X3 = (((cons @ A) @ X5) @ (((cons @ A) @ Y4) @ Xs3))) => ((((X5 = Y4) => (Y = ((remdups_adj @ A) @ (((cons @ A) @ X5) @ Xs3)))) & ((X5 != Y4) => (Y = (((cons @ A) @ X5) @ ((remdups_adj @ A) @ (((cons @ A) @ Y4) @ Xs3)))))) => (~ ((((accp @ (list @ A)) @ (remdups_adj_rel @ A)) @ (((cons @ A) @ X5) @ (((cons @ A) @ Y4) @ Xs3)))))))))))))))))). % 6.67/2.70 thf(fact_7429_take__hd__drop, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => ((((append @ A) @ (((take @ A) @ N2) @ Xs)) @ (((cons @ A) @ ((hd @ A) @ (((drop @ A) @ N2) @ Xs))) @ (nil @ A))) = (((take @ A) @ (suc @ N2)) @ Xs)))))). % 6.67/2.70 thf(fact_7430_hd__remdups__adj, axiom, (! [A:$tType,Xs:(list @ A)]: ((((hd @ A) @ ((remdups_adj @ A) @ Xs)) = ((hd @ A) @ Xs))))). % 6.67/2.70 thf(fact_7431_hd__append2, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((Xs != (nil @ A)) => (((hd @ A) @ (((append @ A) @ Xs) @ Ys)) = ((hd @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7432_hd__replicate, axiom, (! [A:$tType,N2:nat,X3:A]: (((N2 != (zero_zero @ nat)) => (((hd @ A) @ (((replicate @ A) @ N2) @ X3)) = X3))))). % 6.67/2.70 thf(fact_7433_hd__take, axiom, (! [A:$tType,J:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ J) => (((hd @ A) @ (((take @ A) @ J) @ Xs)) = ((hd @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7434_hd__in__set, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (((member @ A) @ ((hd @ A) @ Xs)) @ ((set2 @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7435_list_Oset__sel_I1_J, axiom, (! [A:$tType,A2:(list @ A)]: (((A2 != (nil @ A)) => (((member @ A) @ ((hd @ A) @ A2)) @ ((set2 @ A) @ A2)))))). % 6.67/2.70 thf(fact_7436_hd__append, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((Xs = (nil @ A)) => (((hd @ A) @ (((append @ A) @ Xs) @ Ys)) = ((hd @ A) @ Ys))) & ((Xs != (nil @ A)) => (((hd @ A) @ (((append @ A) @ Xs) @ Ys)) = ((hd @ A) @ Xs))))))). % 6.67/2.70 thf(fact_7437_longest__common__prefix, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((? [Ps2:(list @ A),Xs4:(list @ A),Ys5:(list @ A)]: (((Xs = (((append @ A) @ Ps2) @ Xs4)) & ((Ys = (((append @ A) @ Ps2) @ Ys5)) & ((Xs4 = (nil @ A)) | ((Ys5 = (nil @ A)) | (((hd @ A) @ Xs4) != ((hd @ A) @ Ys5))))))))))). % 6.67/2.70 thf(fact_7438_list_Osel_I1_J, axiom, (! [A:$tType,X21:A,X222:(list @ A)]: ((((hd @ A) @ (((cons @ A) @ X21) @ X222)) = X21)))). % 6.67/2.70 thf(fact_7439_hd__zip, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B)]: (((Xs != (nil @ A)) => ((Ys != (nil @ B)) => (((hd @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys)) = ((((product_Pair @ A) @ B) @ ((hd @ A) @ Xs)) @ ((hd @ B) @ Ys)))))))). % 6.67/2.70 thf(fact_7440_hd__concat, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: (((Xs != (nil @ (list @ A))) => ((((hd @ (list @ A)) @ Xs) != (nil @ A)) => (((hd @ A) @ ((concat @ A) @ Xs)) = ((hd @ A) @ ((hd @ (list @ A)) @ Xs)))))))). % 6.67/2.70 thf(fact_7441_hd__map, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),F2:(A > B)]: (((Xs != (nil @ A)) => (((hd @ B) @ ((((map @ A) @ B) @ F2) @ Xs)) = (F2 @ ((hd @ A) @ Xs))))))). % 6.67/2.70 thf(fact_7442_list_Omap__sel_I1_J, axiom, (! [B:$tType,A:$tType,A2:(list @ A),F2:(A > B)]: (((A2 != (nil @ A)) => (((hd @ B) @ ((((map @ A) @ B) @ F2) @ A2)) = (F2 @ ((hd @ A) @ A2))))))). % 6.67/2.70 thf(fact_7443_hd__conv__nth, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (((hd @ A) @ Xs) = (((nth @ A) @ Xs) @ (zero_zero @ nat))))))). % 6.67/2.70 thf(fact_7444_hd__drop__conv__nth, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => (((hd @ A) @ (((drop @ A) @ N2) @ Xs)) = (((nth @ A) @ Xs) @ N2)))))). % 6.67/2.70 thf(fact_7445_remdups__adj__singleton__iff, axiom, (! [A:$tType,Xs:(list @ A)]: (((((size_size @ (list @ A)) @ ((remdups_adj @ A) @ Xs)) = (suc @ (zero_zero @ nat))) = ((Xs != (nil @ A)) & (Xs = (((replicate @ A) @ ((size_size @ (list @ A)) @ Xs)) @ ((hd @ A) @ Xs)))))))). % 6.67/2.70 thf(fact_7446_list__encode_Opelims, axiom, (! [X3:(list @ nat),Y:nat]: ((((nat_list_encode @ X3) = Y) => ((((accp @ (list @ nat)) @ nat_list_encode_rel) @ X3) => (((X3 = (nil @ nat)) => ((Y = (zero_zero @ nat)) => (~ ((((accp @ (list @ nat)) @ nat_list_encode_rel) @ (nil @ nat)))))) => (~ ((! [X5:nat,Xs3:(list @ nat)]: (((X3 = (((cons @ nat) @ X5) @ Xs3)) => ((Y = (suc @ (nat_prod_encode @ ((((product_Pair @ nat) @ nat) @ X5) @ (nat_list_encode @ Xs3))))) => (~ ((((accp @ (list @ nat)) @ nat_list_encode_rel) @ (((cons @ nat) @ X5) @ Xs3)))))))))))))))). % 6.67/2.70 thf(fact_7447_horner__sum__bit__eq__take__bit, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A2:A,N2:nat]: (((((((groups4207007520872428315er_sum @ $o) @ A) @ (zero_neq_one_of_bool @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((((map @ nat) @ $o) @ ((bit_se5641148757651400278ts_bit @ A) @ A2)) @ ((upt @ (zero_zero @ nat)) @ N2))) = (((bit_se2584673776208193580ke_bit @ A) @ N2) @ A2)))))))). % 6.67/2.70 thf(fact_7448_remdups__upt, axiom, (! [M:nat,N2:nat]: ((((remdups @ nat) @ ((upt @ M) @ N2)) = ((upt @ M) @ N2))))). % 6.67/2.70 thf(fact_7449_hd__upt, axiom, (! [I:nat,J:nat]: (((((ord_less @ nat) @ I) @ J) => (((hd @ nat) @ ((upt @ I) @ J)) = I))))). % 6.67/2.70 thf(fact_7450_drop__upt, axiom, (! [M:nat,I:nat,J:nat]: (((((drop @ nat) @ M) @ ((upt @ I) @ J)) = ((upt @ (((plus_plus @ nat) @ I) @ M)) @ J))))). % 6.67/2.70 thf(fact_7451_length__upt, axiom, (! [I:nat,J:nat]: ((((size_size @ (list @ nat)) @ ((upt @ I) @ J)) = (((minus_minus @ nat) @ J) @ I))))). % 6.67/2.70 thf(fact_7452_take__upt, axiom, (! [I:nat,M:nat,N2:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ M)) @ N2) => ((((take @ nat) @ M) @ ((upt @ I) @ N2)) = ((upt @ I) @ (((plus_plus @ nat) @ I) @ M))))))). % 6.67/2.70 thf(fact_7453_upt__conv__Nil, axiom, (! [J:nat,I:nat]: (((((ord_less_eq @ nat) @ J) @ I) => (((upt @ I) @ J) = (nil @ nat)))))). % 6.67/2.70 thf(fact_7454_sorted__list__of__set__range, axiom, (! [M:nat,N2:nat]: ((((linord4507533701916653071of_set @ nat) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2)) = ((upt @ M) @ N2))))). % 6.67/2.70 thf(fact_7455_upt__eq__Nil__conv, axiom, (! [I:nat,J:nat]: (((((upt @ I) @ J) = (nil @ nat)) = ((J = (zero_zero @ nat)) | (((ord_less_eq @ nat) @ J) @ I)))))). % 6.67/2.70 thf(fact_7456_nth__upt, axiom, (! [I:nat,K:nat,J:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ J) => ((((nth @ nat) @ ((upt @ I) @ J)) @ K) = (((plus_plus @ nat) @ I) @ K)))))). % 6.67/2.70 thf(fact_7457_map__fst__enumerate, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((((map @ ((product_prod @ nat) @ A)) @ nat) @ ((product_fst @ nat) @ A)) @ (((enumerate @ A) @ N2) @ Xs)) = ((upt @ N2) @ (((plus_plus @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs))))))). % 6.67/2.70 thf(fact_7458_upt__rec__numeral, axiom, (! [M:num,N2:num]: ((((((ord_less @ nat) @ ((numeral_numeral @ nat) @ M)) @ ((numeral_numeral @ nat) @ N2)) => (((upt @ ((numeral_numeral @ nat) @ M)) @ ((numeral_numeral @ nat) @ N2)) = (((cons @ nat) @ ((numeral_numeral @ nat) @ M)) @ ((upt @ (suc @ ((numeral_numeral @ nat) @ M))) @ ((numeral_numeral @ nat) @ N2))))) & ((~ ((((ord_less @ nat) @ ((numeral_numeral @ nat) @ M)) @ ((numeral_numeral @ nat) @ N2)))) => (((upt @ ((numeral_numeral @ nat) @ M)) @ ((numeral_numeral @ nat) @ N2)) = (nil @ nat))))))). % 6.67/2.70 thf(fact_7459_upt__0, axiom, (! [I:nat]: ((((upt @ I) @ (zero_zero @ nat)) = (nil @ nat))))). % 6.67/2.70 thf(fact_7460_upt__conv__Cons__Cons, axiom, (! [M:nat,N2:nat,Ns:(list @ nat),Q4:nat]: ((((((cons @ nat) @ M) @ (((cons @ nat) @ N2) @ Ns)) = ((upt @ M) @ Q4)) = ((((cons @ nat) @ N2) @ Ns) = ((upt @ (suc @ M)) @ Q4)))))). % 6.67/2.70 thf(fact_7461_enumerate__map__upt, axiom, (! [A:$tType,N2:nat,F2:(nat > A),M:nat]: (((((enumerate @ A) @ N2) @ ((((map @ nat) @ A) @ F2) @ ((upt @ N2) @ M))) = ((((map @ nat) @ ((product_prod @ nat) @ A)) @ (^ [K3:nat]: (((((product_Pair @ nat) @ A) @ K3) @ (F2 @ K3))))) @ ((upt @ N2) @ M)))))). % 6.67/2.70 thf(fact_7462_map__replicate__trivial, axiom, (! [A:$tType,X3:A,I:nat]: ((((((map @ nat) @ A) @ (^ [I3:nat]: (X3))) @ ((upt @ (zero_zero @ nat)) @ I)) = (((replicate @ A) @ I) @ X3))))). % 6.67/2.70 thf(fact_7463_map__add__upt, axiom, (! [N2:nat,M:nat]: ((((((map @ nat) @ nat) @ (^ [I3:nat]: ((((plus_plus @ nat) @ I3) @ N2)))) @ ((upt @ (zero_zero @ nat)) @ M)) = ((upt @ N2) @ (((plus_plus @ nat) @ M) @ N2)))))). % 6.67/2.70 thf(fact_7464_map__Suc__upt, axiom, (! [M:nat,N2:nat]: ((((((map @ nat) @ nat) @ suc) @ ((upt @ M) @ N2)) = ((upt @ (suc @ M)) @ (suc @ N2)))))). % 6.67/2.70 thf(fact_7465_atMost__upto, axiom, ((set_ord_atMost @ nat) = (^ [N:nat]: (((set2 @ nat) @ ((upt @ (zero_zero @ nat)) @ (suc @ N))))))). % 6.67/2.70 thf(fact_7466_greaterThanAtMost__upt, axiom, ((set_or3652927894154168847AtMost @ nat) = (^ [N:nat,M2:nat]: (((set2 @ nat) @ ((upt @ (suc @ N)) @ (suc @ M2))))))). % 6.67/2.70 thf(fact_7467_atLeast__upt, axiom, ((set_ord_lessThan @ nat) = (^ [N:nat]: (((set2 @ nat) @ ((upt @ (zero_zero @ nat)) @ N)))))). % 6.67/2.70 thf(fact_7468_atLeastAtMost__upt, axiom, ((set_or1337092689740270186AtMost @ nat) = (^ [N:nat,M2:nat]: (((set2 @ nat) @ ((upt @ N) @ (suc @ M2))))))). % 6.67/2.70 thf(fact_7469_atLeastLessThan__upt, axiom, ((set_or7035219750837199246ssThan @ nat) = (^ [I3:nat,J3:nat]: (((set2 @ nat) @ ((upt @ I3) @ J3)))))). % 6.67/2.70 thf(fact_7470_greaterThanLessThan__upt, axiom, ((set_or5935395276787703475ssThan @ nat) = (^ [N:nat,M2:nat]: (((set2 @ nat) @ ((upt @ (suc @ N)) @ M2)))))). % 6.67/2.70 thf(fact_7471_distinct__upt, axiom, (! [I:nat,J:nat]: (((distinct @ nat) @ ((upt @ I) @ J))))). % 6.67/2.70 thf(fact_7472_upt__conv__Cons, axiom, (! [I:nat,J:nat]: (((((ord_less @ nat) @ I) @ J) => (((upt @ I) @ J) = (((cons @ nat) @ I) @ ((upt @ (suc @ I)) @ J))))))). % 6.67/2.70 thf(fact_7473_enumerate__eq__zip, axiom, (! [A:$tType]: (((enumerate @ A) = (^ [N:nat,Xs2:(list @ A)]: (((((zip @ nat) @ A) @ ((upt @ N) @ (((plus_plus @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs2)))) @ Xs2))))))). % 6.67/2.70 thf(fact_7474_map__upt__Suc, axiom, (! [A:$tType,F2:(nat > A),N2:nat]: ((((((map @ nat) @ A) @ F2) @ ((upt @ (zero_zero @ nat)) @ (suc @ N2))) = (((cons @ A) @ (F2 @ (zero_zero @ nat))) @ ((((map @ nat) @ A) @ (^ [I3:nat]: ((F2 @ (suc @ I3))))) @ ((upt @ (zero_zero @ nat)) @ N2))))))). % 6.67/2.70 thf(fact_7475_map__decr__upt, axiom, (! [M:nat,N2:nat]: ((((((map @ nat) @ nat) @ (^ [N:nat]: ((((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat)))))) @ ((upt @ (suc @ M)) @ (suc @ N2))) = ((upt @ M) @ N2))))). % 6.67/2.70 thf(fact_7476_map__nth, axiom, (! [A:$tType,Xs:(list @ A)]: ((((((map @ nat) @ A) @ ((nth @ A) @ Xs)) @ ((upt @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs))) = Xs)))). % 6.67/2.70 thf(fact_7477_upt__add__eq__append, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((upt @ I) @ (((plus_plus @ nat) @ J) @ K)) = (((append @ nat) @ ((upt @ I) @ J)) @ ((upt @ J) @ (((plus_plus @ nat) @ J) @ K)))))))). % 6.67/2.70 thf(fact_7478_nth__map__upt, axiom, (! [A:$tType,I:nat,N2:nat,M:nat,F2:(nat > A)]: (((((ord_less @ nat) @ I) @ (((minus_minus @ nat) @ N2) @ M)) => ((((nth @ A) @ ((((map @ nat) @ A) @ F2) @ ((upt @ M) @ N2))) @ I) = (F2 @ (((plus_plus @ nat) @ M) @ I))))))). % 6.67/2.70 thf(fact_7479_upt__eq__Cons__conv, axiom, (! [I:nat,J:nat,X3:nat,Xs:(list @ nat)]: (((((upt @ I) @ J) = (((cons @ nat) @ X3) @ Xs)) = ((((ord_less @ nat) @ I) @ J) & ((I = X3) & (((upt @ (((plus_plus @ nat) @ I) @ (one_one @ nat))) @ J) = Xs))))))). % 6.67/2.70 thf(fact_7480_upt__rec, axiom, (upt = (^ [I3:nat,J3:nat]: (((((if @ (list @ nat)) @ (((ord_less @ nat) @ I3) @ J3)) @ (((cons @ nat) @ I3) @ ((upt @ (suc @ I3)) @ J3))) @ (nil @ nat)))))). % 6.67/2.70 thf(fact_7481_enumerate__replicate__eq, axiom, (! [A:$tType,N2:nat,M:nat,A2:A]: (((((enumerate @ A) @ N2) @ (((replicate @ A) @ M) @ A2)) = ((((map @ nat) @ ((product_prod @ nat) @ A)) @ (^ [Q5:nat]: (((((product_Pair @ nat) @ A) @ Q5) @ A2)))) @ ((upt @ N2) @ (((plus_plus @ nat) @ N2) @ M))))))). % 6.67/2.70 thf(fact_7482_map__upt__eqI, axiom, (! [A:$tType,Xs:(list @ A),N2:nat,M:nat,F2:(nat > A)]: (((((size_size @ (list @ A)) @ Xs) = (((minus_minus @ nat) @ N2) @ M)) => ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ Xs) @ I4) = (F2 @ (((plus_plus @ nat) @ M) @ I4)))))) => (((((map @ nat) @ A) @ F2) @ ((upt @ M) @ N2)) = Xs)))))). % 6.67/2.70 thf(fact_7483_upt__Suc, axiom, (! [I:nat,J:nat]: ((((((ord_less_eq @ nat) @ I) @ J) => (((upt @ I) @ (suc @ J)) = (((append @ nat) @ ((upt @ I) @ J)) @ (((cons @ nat) @ J) @ (nil @ nat))))) & ((~ ((((ord_less_eq @ nat) @ I) @ J))) => (((upt @ I) @ (suc @ J)) = (nil @ nat))))))). % 6.67/2.70 thf(fact_7484_upt__Suc__append, axiom, (! [I:nat,J:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((upt @ I) @ (suc @ J)) = (((append @ nat) @ ((upt @ I) @ J)) @ (((cons @ nat) @ J) @ (nil @ nat)))))))). % 6.67/2.70 thf(fact_7485_transpose__rectangle, axiom, (! [A:$tType,Xs:(list @ (list @ A)),N2:nat]: ((((Xs = (nil @ (list @ A))) => (N2 = (zero_zero @ nat))) => ((! [I4:nat]: (((((ord_less @ nat) @ I4) @ ((size_size @ (list @ (list @ A))) @ Xs)) => (((size_size @ (list @ A)) @ (((nth @ (list @ A)) @ Xs) @ I4)) = N2)))) => (((transpose @ A) @ Xs) = ((((map @ nat) @ (list @ A)) @ (^ [I3:nat]: (((((map @ nat) @ A) @ (^ [J3:nat]: ((((nth @ A) @ (((nth @ (list @ A)) @ Xs) @ J3)) @ I3)))) @ ((upt @ (zero_zero @ nat)) @ ((size_size @ (list @ (list @ A))) @ Xs)))))) @ ((upt @ (zero_zero @ nat)) @ N2)))))))). % 6.67/2.70 thf(fact_7486_extract__SomeE, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),Ys:(list @ A),Y:A,Zs2:(list @ A)]: ((((((extract @ A) @ P) @ Xs) = ((some @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ A) @ (list @ A))) @ Ys) @ ((((product_Pair @ A) @ (list @ A)) @ Y) @ Zs2)))) => ((Xs = (((append @ A) @ Ys) @ (((cons @ A) @ Y) @ Zs2))) & ((P @ Y) & (~ ((? [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Ys)) & (P @ X2)))))))))))). % 6.67/2.70 thf(fact_7487_extract__Some__iff, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),Ys:(list @ A),Y:A,Zs2:(list @ A)]: ((((((extract @ A) @ P) @ Xs) = ((some @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ A) @ (list @ A))) @ Ys) @ ((((product_Pair @ A) @ (list @ A)) @ Y) @ Zs2)))) = ((Xs = (((append @ A) @ Ys) @ (((cons @ A) @ Y) @ Zs2))) & ((P @ Y) & (~ ((? [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Ys)) & (P @ X)))))))))))). % 6.67/2.70 thf(fact_7488_extract__None__iff, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((((extract @ A) @ P) @ Xs) = (none @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))))) = (~ ((? [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) & (P @ X)))))))))). % 6.67/2.70 thf(fact_7489_extract__Nil__code, axiom, (! [A:$tType,P:(A > $o)]: (((((extract @ A) @ P) @ (nil @ A)) = (none @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))))))). % 6.67/2.70 thf(fact_7490_extract__Cons__code, axiom, (! [A:$tType,P:(A > $o),X3:A,Xs:(list @ A)]: ((((P @ X3) => ((((extract @ A) @ P) @ (((cons @ A) @ X3) @ Xs)) = ((some @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ A) @ (list @ A))) @ (nil @ A)) @ ((((product_Pair @ A) @ (list @ A)) @ X3) @ Xs))))) & ((~ ((P @ X3))) => ((((extract @ A) @ P) @ (((cons @ A) @ X3) @ Xs)) = (((((case_option @ (option @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))))) @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))) @ (none @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))))) @ ((((product_case_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))) @ (option @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))))) @ (^ [Ys3:(list @ A)]: (((((product_case_prod @ A) @ (list @ A)) @ (option @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))))) @ (^ [Y2:A,Zs3:(list @ A)]: (((some @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ A) @ (list @ A))) @ (((cons @ A) @ X3) @ Ys3)) @ ((((product_Pair @ A) @ (list @ A)) @ Y2) @ Zs3)))))))))) @ (((extract @ A) @ P) @ Xs)))))))). % 6.67/2.70 thf(fact_7491_sum__list__map__eq__sum__count2, axiom, (! [A:$tType,Xs:(list @ A),X8:(set @ A),F2:(A > nat)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ X8) => (((finite_finite2 @ A) @ X8) => (((groups8242544230860333062m_list @ nat) @ ((((map @ A) @ nat) @ F2) @ Xs)) = ((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [X:A]: ((((times_times @ nat) @ (((count_list @ A) @ Xs) @ X)) @ (F2 @ X))))) @ X8))))))). % 6.67/2.70 thf(fact_7492_nth__transpose, axiom, (! [A:$tType,I:nat,Xs:(list @ (list @ A))]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ (list @ A))) @ ((transpose @ A) @ Xs))) => ((((nth @ (list @ A)) @ ((transpose @ A) @ Xs)) @ I) = ((((map @ (list @ A)) @ A) @ (^ [Xs2:(list @ A)]: ((((nth @ A) @ Xs2) @ I)))) @ (((filter2 @ (list @ A)) @ (^ [Ys3:(list @ A)]: ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Ys3))))) @ Xs))))))). % 6.67/2.70 thf(fact_7493_filter__filter, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o),Xs:(list @ A)]: (((((filter2 @ A) @ P) @ (((filter2 @ A) @ Q) @ Xs)) = (((filter2 @ A) @ (^ [X:A]: (((Q @ X) & (P @ X))))) @ Xs))))). % 6.67/2.70 thf(fact_7494_filter__True, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (P @ X5)))) => ((((filter2 @ A) @ P) @ Xs) = Xs))))). % 6.67/2.70 thf(fact_7495_filter__append, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),Ys:(list @ A)]: (((((filter2 @ A) @ P) @ (((append @ A) @ Xs) @ Ys)) = (((append @ A) @ (((filter2 @ A) @ P) @ Xs)) @ (((filter2 @ A) @ P) @ Ys)))))). % 6.67/2.70 thf(fact_7496_remove1__filter__not, axiom, (! [A:$tType,P:(A > $o),X3:A,Xs:(list @ A)]: (((~ ((P @ X3))) => ((((remove1 @ A) @ X3) @ (((filter2 @ A) @ P) @ Xs)) = (((filter2 @ A) @ P) @ Xs)))))). % 6.67/2.70 thf(fact_7497_removeAll__filter__not, axiom, (! [A:$tType,P:(A > $o),X3:A,Xs:(list @ A)]: (((~ ((P @ X3))) => ((((removeAll @ A) @ X3) @ (((filter2 @ A) @ P) @ Xs)) = (((filter2 @ A) @ P) @ Xs)))))). % 6.67/2.70 thf(fact_7498_set__filter, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((set2 @ A) @ (((filter2 @ A) @ P) @ Xs)) = ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) & (P @ X))))))))). % 6.67/2.70 thf(fact_7499_sum__list_ONil, axiom, (! [A:$tType]: (((monoid_add @ A) => (((groups8242544230860333062m_list @ A) @ (nil @ A)) = (zero_zero @ A)))))). % 6.67/2.70 thf(fact_7500_sum__list__eq__0__iff, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [Ns:(list @ A)]: (((((groups8242544230860333062m_list @ A) @ Ns) = (zero_zero @ A)) = (! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Ns)) => (X = (zero_zero @ A)))))))))))). % 6.67/2.70 thf(fact_7501_sum__list_OCons, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [X3:A,Xs:(list @ A)]: ((((groups8242544230860333062m_list @ A) @ (((cons @ A) @ X3) @ Xs)) = (((plus_plus @ A) @ X3) @ ((groups8242544230860333062m_list @ A) @ Xs))))))))). % 6.67/2.70 thf(fact_7502_filter__False, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (~ ((P @ X5)))))) => ((((filter2 @ A) @ P) @ Xs) = (nil @ A)))))). % 6.67/2.70 thf(fact_7503_sum__list__append, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [Xs:(list @ A),Ys:(list @ A)]: ((((groups8242544230860333062m_list @ A) @ (((append @ A) @ Xs) @ Ys)) = (((plus_plus @ A) @ ((groups8242544230860333062m_list @ A) @ Xs)) @ ((groups8242544230860333062m_list @ A) @ Ys))))))))). % 6.67/2.70 thf(fact_7504_sum__list__0, axiom, (! [B:$tType,A:$tType]: (((monoid_add @ A) => (! [Xs:(list @ B)]: ((((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ (^ [X:B]: ((zero_zero @ A)))) @ Xs)) = (zero_zero @ A)))))))). % 6.67/2.70 thf(fact_7505_length__filter__map, axiom, (! [A:$tType,B:$tType,P:(A > $o),F2:(B > A),Xs:(list @ B)]: ((((size_size @ (list @ A)) @ (((filter2 @ A) @ P) @ ((((map @ B) @ A) @ F2) @ Xs))) = ((size_size @ (list @ B)) @ (((filter2 @ B) @ (((((comp @ A) @ $o) @ B) @ P) @ F2)) @ Xs)))))). % 6.67/2.70 thf(fact_7506_sum__list__upt, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => (((groups8242544230860333062m_list @ nat) @ ((upt @ M) @ N2)) = ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X:nat]: (X))) @ (((set_or7035219750837199246ssThan @ nat) @ M) @ N2))))))). % 6.67/2.70 thf(fact_7507_filter__map, axiom, (! [A:$tType,B:$tType,P:(A > $o),F2:(B > A),Xs:(list @ B)]: (((((filter2 @ A) @ P) @ ((((map @ B) @ A) @ F2) @ Xs)) = ((((map @ B) @ A) @ F2) @ (((filter2 @ B) @ (((((comp @ A) @ $o) @ B) @ P) @ F2)) @ Xs)))))). % 6.67/2.70 thf(fact_7508_sum__list__filter__le__nat, axiom, (! [A:$tType,F2:(A > nat),P:(A > $o),Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((groups8242544230860333062m_list @ nat) @ ((((map @ A) @ nat) @ F2) @ (((filter2 @ A) @ P) @ Xs)))) @ ((groups8242544230860333062m_list @ nat) @ ((((map @ A) @ nat) @ F2) @ Xs)))))). % 6.67/2.70 thf(fact_7509_sum__list__addf, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [F2:(B > A),G:(B > A),Xs:(list @ B)]: ((((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ (^ [X:B]: ((((plus_plus @ A) @ (F2 @ X)) @ (G @ X))))) @ Xs)) = (((plus_plus @ A) @ ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ F2) @ Xs))) @ ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ G) @ Xs)))))))))). % 6.67/2.70 thf(fact_7510_sum__list__map__filter_H, axiom, (! [A:$tType,B:$tType]: (((monoid_add @ A) => (! [F2:(B > A),P:(B > $o),Xs:(list @ B)]: ((((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ F2) @ (((filter2 @ B) @ P) @ Xs))) = ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ (^ [X:B]: (((((if @ A) @ (P @ X)) @ (F2 @ X)) @ (zero_zero @ A))))) @ Xs))))))))). % 6.67/2.70 thf(fact_7511_filter__concat, axiom, (! [A:$tType,P4:(A > $o),Xs:(list @ (list @ A))]: (((((filter2 @ A) @ P4) @ ((concat @ A) @ Xs)) = ((concat @ A) @ ((((map @ (list @ A)) @ (list @ A)) @ ((filter2 @ A) @ P4)) @ Xs)))))). % 6.67/2.70 thf(fact_7512_sum__list__map__filter, axiom, (! [A:$tType,B:$tType]: (((monoid_add @ A) => (! [Xs:(list @ B),P:(B > $o),F2:(B > A)]: (((! [X5:B]: (((((member @ B) @ X5) @ ((set2 @ B) @ Xs)) => ((~ ((P @ X5))) => ((F2 @ X5) = (zero_zero @ A)))))) => (((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ F2) @ (((filter2 @ B) @ P) @ Xs))) = ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ F2) @ Xs)))))))))). % 6.67/2.70 thf(fact_7513_distinct__map__filter, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B),P:(B > $o)]: ((((distinct @ A) @ ((((map @ B) @ A) @ F2) @ Xs)) => ((distinct @ A) @ ((((map @ B) @ A) @ F2) @ (((filter2 @ B) @ P) @ Xs))))))). % 6.67/2.70 thf(fact_7514_length__concat, axiom, (! [B:$tType,Xss:(list @ (list @ B))]: ((((size_size @ (list @ B)) @ ((concat @ B) @ Xss)) = ((groups8242544230860333062m_list @ nat) @ ((((map @ (list @ B)) @ nat) @ (size_size @ (list @ B))) @ Xss)))))). % 6.67/2.70 thf(fact_7515_sum__list__nonpos, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [Xs:(list @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ A) @ X5) @ (zero_zero @ A))))) => (((ord_less_eq @ A) @ ((groups8242544230860333062m_list @ A) @ Xs)) @ (zero_zero @ A))))))))). % 6.67/2.70 thf(fact_7516_sum__list__nonneg__eq__0__iff, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [Xs:(list @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ X5)))) => ((((groups8242544230860333062m_list @ A) @ Xs) = (zero_zero @ A)) = (! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (X = (zero_zero @ A))))))))))))). % 6.67/2.70 thf(fact_7517_Groups__List_Osum__list__nonneg, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [Xs:(list @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ X5)))) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ ((groups8242544230860333062m_list @ A) @ Xs))))))))). % 6.67/2.70 thf(fact_7518_empty__filter__conv, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((nil @ A) = (((filter2 @ A) @ P) @ Xs)) = (! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (~ ((P @ X)))))))))). % 6.67/2.70 thf(fact_7519_filter__empty__conv, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((((filter2 @ A) @ P) @ Xs) = (nil @ A)) = (! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (~ ((P @ X)))))))))). % 6.67/2.70 thf(fact_7520_Cons__eq__filterD, axiom, (! [A:$tType,X3:A,Xs:(list @ A),P:(A > $o),Ys:(list @ A)]: ((((((cons @ A) @ X3) @ Xs) = (((filter2 @ A) @ P) @ Ys)) => (? [Us3:(list @ A),Vs3:(list @ A)]: (((Ys = (((append @ A) @ Us3) @ (((cons @ A) @ X3) @ Vs3))) & ((! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Us3)) => (~ ((P @ X2)))))) & ((P @ X3) & (Xs = (((filter2 @ A) @ P) @ Vs3))))))))))). % 6.67/2.70 thf(fact_7521_filter__eq__ConsD, axiom, (! [A:$tType,P:(A > $o),Ys:(list @ A),X3:A,Xs:(list @ A)]: ((((((filter2 @ A) @ P) @ Ys) = (((cons @ A) @ X3) @ Xs)) => (? [Us3:(list @ A),Vs3:(list @ A)]: (((Ys = (((append @ A) @ Us3) @ (((cons @ A) @ X3) @ Vs3))) & ((! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Us3)) => (~ ((P @ X2)))))) & ((P @ X3) & (Xs = (((filter2 @ A) @ P) @ Vs3))))))))))). % 6.67/2.70 thf(fact_7522_Cons__eq__filter__iff, axiom, (! [A:$tType,X3:A,Xs:(list @ A),P:(A > $o),Ys:(list @ A)]: ((((((cons @ A) @ X3) @ Xs) = (((filter2 @ A) @ P) @ Ys)) = (? [Us2:(list @ A),Vs:(list @ A)]: (((Ys = (((append @ A) @ Us2) @ (((cons @ A) @ X3) @ Vs))) & ((! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Us2)) => (~ ((P @ X)))))) & ((P @ X3) & (Xs = (((filter2 @ A) @ P) @ Vs))))))))))). % 6.67/2.70 thf(fact_7523_filter__eq__Cons__iff, axiom, (! [A:$tType,P:(A > $o),Ys:(list @ A),X3:A,Xs:(list @ A)]: ((((((filter2 @ A) @ P) @ Ys) = (((cons @ A) @ X3) @ Xs)) = (? [Us2:(list @ A),Vs:(list @ A)]: (((Ys = (((append @ A) @ Us2) @ (((cons @ A) @ X3) @ Vs))) & ((! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Us2)) => (~ ((P @ X)))))) & ((P @ X3) & (Xs = (((filter2 @ A) @ P) @ Vs))))))))))). % 6.67/2.70 thf(fact_7524_inj__on__filter__key__eq, axiom, (! [B:$tType,A:$tType,F2:(A > B),Y:A,Xs:(list @ A)]: ((((((inj_on @ A) @ B) @ F2) @ (((insert @ A) @ Y) @ ((set2 @ A) @ Xs))) => ((((filter2 @ A) @ (^ [X:A]: (((F2 @ Y) = (F2 @ X))))) @ Xs) = (((filter2 @ A) @ ((^ [Y3:A,Z:A]: ((Y3 = Z))) @ Y)) @ Xs)))))). % 6.67/2.70 thf(fact_7525_member__le__sum__list, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [X3:A,Xs:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ A) @ X3) @ ((groups8242544230860333062m_list @ A) @ Xs))))))))). % 6.67/2.70 thf(fact_7526_length__filter__le, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ (((filter2 @ A) @ P) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))). % 6.67/2.70 thf(fact_7527_filter__is__subset, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ (((filter2 @ A) @ P) @ Xs))) @ ((set2 @ A) @ Xs))))). % 6.67/2.70 thf(fact_7528_distinct__filter, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: ((((distinct @ A) @ Xs) => ((distinct @ A) @ (((filter2 @ A) @ P) @ Xs)))))). % 6.67/2.70 thf(fact_7529_length__filter__less, axiom, (! [A:$tType,X3:A,Xs:(list @ A),P:(A > $o)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => ((~ ((P @ X3))) => (((ord_less @ nat) @ ((size_size @ (list @ A)) @ (((filter2 @ A) @ P) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))))). % 6.67/2.70 thf(fact_7530_filter__set, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: (((((filter3 @ A) @ P) @ ((set2 @ A) @ Xs)) = ((set2 @ A) @ (((filter2 @ A) @ P) @ Xs)))))). % 6.67/2.70 thf(fact_7531_replicate__length__filter, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((replicate @ A) @ ((size_size @ (list @ A)) @ (((filter2 @ A) @ ((^ [Y3:A,Z:A]: ((Y3 = Z))) @ X3)) @ Xs))) @ X3) = (((filter2 @ A) @ ((^ [Y3:A,Z:A]: ((Y3 = Z))) @ X3)) @ Xs))))). % 6.67/2.70 thf(fact_7532_filter__id__conv, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((((filter2 @ A) @ P) @ Xs) = Xs) = (! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (P @ X)))))))). % 6.67/2.70 thf(fact_7533_filter__cong, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),P:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Ys)) => ((P @ X5) = (Q @ X5))))) => ((((filter2 @ A) @ P) @ Xs) = (((filter2 @ A) @ Q) @ Ys))))))). % 6.67/2.70 thf(fact_7534_inter__set__filter, axiom, (! [A:$tType,A3:(set @ A),Xs:(list @ A)]: (((((inf_inf @ (set @ A)) @ A3) @ ((set2 @ A) @ Xs)) = ((set2 @ A) @ (((filter2 @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A3)))) @ Xs)))))). % 6.67/2.70 thf(fact_7535_sum__length__filter__compl, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: (((((plus_plus @ nat) @ ((size_size @ (list @ A)) @ (((filter2 @ A) @ P) @ Xs))) @ ((size_size @ (list @ A)) @ (((filter2 @ A) @ (^ [X:A]: ((~ ((P @ X)))))) @ Xs))) = ((size_size @ (list @ A)) @ Xs))))). % 6.67/2.70 thf(fact_7536_remdups__filter, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((remdups @ A) @ (((filter2 @ A) @ P) @ Xs)) = (((filter2 @ A) @ P) @ ((remdups @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7537_filter__replicate, axiom, (! [A:$tType,P:(A > $o),X3:A,N2:nat]: ((((P @ X3) => ((((filter2 @ A) @ P) @ (((replicate @ A) @ N2) @ X3)) = (((replicate @ A) @ N2) @ X3))) & ((~ ((P @ X3))) => ((((filter2 @ A) @ P) @ (((replicate @ A) @ N2) @ X3)) = (nil @ A))))))). % 6.67/2.70 thf(fact_7538_filter_Osimps_I1_J, axiom, (! [A:$tType,P:(A > $o)]: (((((filter2 @ A) @ P) @ (nil @ A)) = (nil @ A))))). % 6.67/2.70 thf(fact_7539_filter_Osimps_I2_J, axiom, (! [A:$tType,P:(A > $o),X3:A,Xs:(list @ A)]: ((((P @ X3) => ((((filter2 @ A) @ P) @ (((cons @ A) @ X3) @ Xs)) = (((cons @ A) @ X3) @ (((filter2 @ A) @ P) @ Xs)))) & ((~ ((P @ X3))) => ((((filter2 @ A) @ P) @ (((cons @ A) @ X3) @ Xs)) = (((filter2 @ A) @ P) @ Xs))))))). % 6.67/2.70 thf(fact_7540_filter__shuffles, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),Ys:(list @ A)]: ((((((image @ (list @ A)) @ (list @ A)) @ ((filter2 @ A) @ P)) @ (((shuffles @ A) @ Xs) @ Ys)) = (((shuffles @ A) @ (((filter2 @ A) @ P) @ Xs)) @ (((filter2 @ A) @ P) @ Ys)))))). % 6.67/2.70 thf(fact_7541_removeAll__filter__not__eq, axiom, (! [A:$tType]: (((removeAll @ A) = (^ [X:A]: (((filter2 @ A) @ (^ [Y2:A]: ((X != Y2)))))))))). % 6.67/2.70 thf(fact_7542_filter__remove1, axiom, (! [A:$tType,Q:(A > $o),X3:A,Xs:(list @ A)]: (((((filter2 @ A) @ Q) @ (((remove1 @ A) @ X3) @ Xs)) = (((remove1 @ A) @ X3) @ (((filter2 @ A) @ Q) @ Xs)))))). % 6.67/2.70 thf(fact_7543_partition__in__shuffles, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: ((((member @ (list @ A)) @ Xs) @ (((shuffles @ A) @ (((filter2 @ A) @ P) @ Xs)) @ (((filter2 @ A) @ (^ [X:A]: ((~ ((P @ X)))))) @ Xs)))))). % 6.67/2.70 thf(fact_7544_filter__insort__triv, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [P:(B > $o),X3:B,F2:(B > A),Xs:(list @ B)]: (((~ ((P @ X3))) => ((((filter2 @ B) @ P) @ (((((linorder_insort_key @ B) @ A) @ F2) @ X3) @ Xs)) = (((filter2 @ B) @ P) @ Xs))))))))). % 6.67/2.70 thf(fact_7545_sum__list__abs, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [Xs:(list @ A)]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ ((groups8242544230860333062m_list @ A) @ Xs))) @ ((groups8242544230860333062m_list @ A) @ ((((map @ A) @ A) @ (abs_abs @ A)) @ Xs))))))))). % 6.67/2.70 thf(fact_7546_filter__in__nths, axiom, (! [A:$tType,Xs:(list @ A),S:(set @ nat)]: ((((distinct @ A) @ Xs) => ((((filter2 @ A) @ (^ [X:A]: ((((member @ A) @ X) @ ((set2 @ A) @ (((nths @ A) @ Xs) @ S)))))) @ Xs) = (((nths @ A) @ Xs) @ S)))))). % 6.67/2.70 thf(fact_7547_sum__list__replicate, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N2:nat,C2:A]: ((((groups8242544230860333062m_list @ A) @ (((replicate @ A) @ N2) @ C2)) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ C2)))))))). % 6.67/2.70 thf(fact_7548_sum__list__mono, axiom, (! [B:$tType,A:$tType]: ((((monoid_add @ B) & (ordere6658533253407199908up_add @ B)) => (! [Xs:(list @ A),F2:(A > B),G:(A > B)]: (((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ B) @ (F2 @ X5)) @ (G @ X5))))) => (((ord_less_eq @ B) @ ((groups8242544230860333062m_list @ B) @ ((((map @ A) @ B) @ F2) @ Xs))) @ ((groups8242544230860333062m_list @ B) @ ((((map @ A) @ B) @ G) @ Xs)))))))))). % 6.67/2.70 thf(fact_7549_distinct__sum__list__conv__Sum, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [Xs:(list @ A)]: ((((distinct @ A) @ Xs) => (((groups8242544230860333062m_list @ A) @ Xs) = ((((groups7311177749621191930dd_sum @ A) @ A) @ (^ [X:A]: (X))) @ ((set2 @ A) @ Xs)))))))))). % 6.67/2.70 thf(fact_7550_set__minus__filter__out, axiom, (! [A:$tType,Xs:(list @ A),Y:A]: (((((minus_minus @ (set @ A)) @ ((set2 @ A) @ Xs)) @ (((insert @ A) @ Y) @ (bot_bot @ (set @ A)))) = ((set2 @ A) @ (((filter2 @ A) @ (^ [X:A]: ((X != Y)))) @ Xs)))))). % 6.67/2.70 thf(fact_7551_filter__shuffles__disjoint1_I2_J, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A)]: ((((((inf_inf @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys)) = (bot_bot @ (set @ A))) => ((((member @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ Xs) @ Ys)) => ((((filter2 @ A) @ (^ [X:A]: ((~ ((((member @ A) @ X) @ ((set2 @ A) @ Xs))))))) @ Zs2) = Ys)))))). % 6.67/2.70 thf(fact_7552_filter__shuffles__disjoint1_I1_J, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A)]: ((((((inf_inf @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys)) = (bot_bot @ (set @ A))) => ((((member @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ Xs) @ Ys)) => ((((filter2 @ A) @ (^ [X:A]: ((((member @ A) @ X) @ ((set2 @ A) @ Xs))))) @ Zs2) = Xs)))))). % 6.67/2.70 thf(fact_7553_filter__shuffles__disjoint2_I2_J, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A)]: ((((((inf_inf @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys)) = (bot_bot @ (set @ A))) => ((((member @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ Xs) @ Ys)) => ((((filter2 @ A) @ (^ [X:A]: ((~ ((((member @ A) @ X) @ ((set2 @ A) @ Ys))))))) @ Zs2) = Xs)))))). % 6.67/2.70 thf(fact_7554_filter__shuffles__disjoint2_I1_J, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A)]: ((((((inf_inf @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys)) = (bot_bot @ (set @ A))) => ((((member @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ Xs) @ Ys)) => ((((filter2 @ A) @ (^ [X:A]: ((((member @ A) @ X) @ ((set2 @ A) @ Ys))))) @ Zs2) = Ys)))))). % 6.67/2.70 thf(fact_7555_filter__eq__nths, axiom, (! [A:$tType]: (((filter2 @ A) = (^ [P3:(A > $o),Xs2:(list @ A)]: ((((nths @ A) @ Xs2) @ ((collect @ nat) @ (^ [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs2)) & (P3 @ (((nth @ A) @ Xs2) @ I3))))))))))))). % 6.67/2.70 thf(fact_7556_length__filter__conv__card, axiom, (! [A:$tType,P4:(A > $o),Xs:(list @ A)]: ((((size_size @ (list @ A)) @ (((filter2 @ A) @ P4) @ Xs)) = ((finite_card @ nat) @ ((collect @ nat) @ (^ [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) & (P4 @ (((nth @ A) @ Xs) @ I3))))))))))). % 6.67/2.70 thf(fact_7557_sum__list__strict__mono, axiom, (! [B:$tType,A:$tType]: ((((monoid_add @ B) & (strict9044650504122735259up_add @ B)) => (! [Xs:(list @ A),F2:(A > B),G:(A > B)]: (((Xs != (nil @ A)) => ((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (((ord_less @ B) @ (F2 @ X5)) @ (G @ X5))))) => (((ord_less @ B) @ ((groups8242544230860333062m_list @ B) @ ((((map @ A) @ B) @ F2) @ Xs))) @ ((groups8242544230860333062m_list @ B) @ ((((map @ A) @ B) @ G) @ Xs))))))))))). % 6.67/2.70 thf(fact_7558_elem__le__sum__list, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [K:nat,Ns:(list @ A)]: (((((ord_less @ nat) @ K) @ ((size_size @ (list @ A)) @ Ns)) => (((ord_less_eq @ A) @ (((nth @ A) @ Ns) @ K)) @ ((groups8242544230860333062m_list @ A) @ Ns))))))))). % 6.67/2.70 thf(fact_7559_sum__list__distinct__conv__sum__set, axiom, (! [C:$tType,B:$tType]: (((comm_monoid_add @ C) => (! [Xs:(list @ B),F2:(B > C)]: ((((distinct @ B) @ Xs) => (((groups8242544230860333062m_list @ C) @ ((((map @ B) @ C) @ F2) @ Xs)) = ((((groups7311177749621191930dd_sum @ B) @ C) @ F2) @ ((set2 @ B) @ Xs)))))))))). % 6.67/2.70 thf(fact_7560_sum_Odistinct__set__conv__list, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [Xs:(list @ B),G:(B > A)]: ((((distinct @ B) @ Xs) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ ((set2 @ B) @ Xs)) = ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ G) @ Xs)))))))))). % 6.67/2.70 thf(fact_7561_sum__list__map__remove1, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [X3:B,Xs:(list @ B),F2:(B > A)]: (((((member @ B) @ X3) @ ((set2 @ B) @ Xs)) => (((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ F2) @ Xs)) = (((plus_plus @ A) @ (F2 @ X3)) @ ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ F2) @ (((remove1 @ B) @ X3) @ Xs)))))))))))). % 6.67/2.70 thf(fact_7562_sum__code, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [G:(B > A),Xs:(list @ B)]: ((((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ ((set2 @ B) @ Xs)) = ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ G) @ ((remdups @ B) @ Xs)))))))))). % 6.67/2.70 thf(fact_7563_size__list__conv__sum__list, axiom, (! [B:$tType]: (((size_list @ B) = (^ [F3:(B > nat),Xs2:(list @ B)]: ((((plus_plus @ nat) @ ((groups8242544230860333062m_list @ nat) @ ((((map @ B) @ nat) @ F3) @ Xs2))) @ ((size_size @ (list @ B)) @ Xs2)))))))). % 6.67/2.70 thf(fact_7564_sum__list__triv, axiom, (! [C:$tType,B:$tType]: (((semiring_1 @ B) => (! [R2:B,Xs:(list @ C)]: ((((groups8242544230860333062m_list @ B) @ ((((map @ C) @ B) @ (^ [X:C]: (R2))) @ Xs)) = (((times_times @ B) @ ((semiring_1_of_nat @ B) @ ((size_size @ (list @ C)) @ Xs))) @ R2)))))))). % 6.67/2.70 thf(fact_7565_sum__list__Suc, axiom, (! [A:$tType,F2:(A > nat),Xs:(list @ A)]: ((((groups8242544230860333062m_list @ nat) @ ((((map @ A) @ nat) @ (^ [X:A]: ((suc @ (F2 @ X))))) @ Xs)) = (((plus_plus @ nat) @ ((groups8242544230860333062m_list @ nat) @ ((((map @ A) @ nat) @ F2) @ Xs))) @ ((size_size @ (list @ A)) @ Xs)))))). % 6.67/2.70 thf(fact_7566_distinct__length__filter, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: ((((distinct @ A) @ Xs) => (((size_size @ (list @ A)) @ (((filter2 @ A) @ P) @ Xs)) = ((finite_card @ A) @ (((inf_inf @ (set @ A)) @ ((collect @ A) @ P)) @ ((set2 @ A) @ Xs)))))))). % 6.67/2.70 thf(fact_7567_sum__list__sum__nth, axiom, (! [B:$tType]: (((comm_monoid_add @ B) => ((groups8242544230860333062m_list @ B) = (^ [Xs2:(list @ B)]: (((((groups7311177749621191930dd_sum @ nat) @ B) @ ((nth @ B) @ Xs2)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ B)) @ Xs2)))))))))). % 6.67/2.70 thf(fact_7568_card__length__sum__list__rec, axiom, (! [M:nat,N7:nat]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ M) => (((finite_card @ (list @ nat)) @ ((collect @ (list @ nat)) @ (^ [L2:(list @ nat)]: (((((size_size @ (list @ nat)) @ L2) = M) & (((groups8242544230860333062m_list @ nat) @ L2) = N7)))))) = (((plus_plus @ nat) @ ((finite_card @ (list @ nat)) @ ((collect @ (list @ nat)) @ (^ [L2:(list @ nat)]: (((((size_size @ (list @ nat)) @ L2) = (((minus_minus @ nat) @ M) @ (one_one @ nat))) & (((groups8242544230860333062m_list @ nat) @ L2) = N7))))))) @ ((finite_card @ (list @ nat)) @ ((collect @ (list @ nat)) @ (^ [L2:(list @ nat)]: (((((size_size @ (list @ nat)) @ L2) = M) & ((((plus_plus @ nat) @ ((groups8242544230860333062m_list @ nat) @ L2)) @ (one_one @ nat)) = N7)))))))))))). % 6.67/2.70 thf(fact_7569_card__length__sum__list, axiom, (! [M:nat,N7:nat]: ((((finite_card @ (list @ nat)) @ ((collect @ (list @ nat)) @ (^ [L2:(list @ nat)]: (((((size_size @ (list @ nat)) @ L2) = M) & (((groups8242544230860333062m_list @ nat) @ L2) = N7)))))) = ((binomial @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ N7) @ M)) @ (one_one @ nat))) @ N7))))). % 6.67/2.70 thf(fact_7570_sum__list__map__eq__sum__count, axiom, (! [A:$tType,F2:(A > nat),Xs:(list @ A)]: ((((groups8242544230860333062m_list @ nat) @ ((((map @ A) @ nat) @ F2) @ Xs)) = ((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [X:A]: ((((times_times @ nat) @ (((count_list @ A) @ Xs) @ X)) @ (F2 @ X))))) @ ((set2 @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7571_sum__list__update, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [K:nat,Xs:(list @ A),X3:A]: (((((ord_less @ nat) @ K) @ ((size_size @ (list @ A)) @ Xs)) => (((groups8242544230860333062m_list @ A) @ ((((list_update @ A) @ Xs) @ K) @ X3)) = (((minus_minus @ A) @ (((plus_plus @ A) @ ((groups8242544230860333062m_list @ A) @ Xs)) @ X3)) @ (((nth @ A) @ Xs) @ K)))))))))). % 6.67/2.70 thf(fact_7572_transpose__aux__filter__head, axiom, (! [A:$tType,Xss:(list @ (list @ A))]: ((((concat @ A) @ ((((map @ (list @ A)) @ (list @ A)) @ ((((case_list @ (list @ A)) @ A) @ (nil @ A)) @ (^ [H:A,T3:(list @ A)]: ((((cons @ A) @ H) @ (nil @ A)))))) @ Xss)) = ((((map @ (list @ A)) @ A) @ (hd @ A)) @ (((filter2 @ (list @ A)) @ (^ [Ys3:(list @ A)]: ((Ys3 != (nil @ A))))) @ Xss)))))). % 6.67/2.70 thf(fact_7573_map__filter__def, axiom, (! [B:$tType,A:$tType]: ((((map_filter @ A) @ B) = (^ [F3:(A > (option @ B)),Xs2:(list @ A)]: (((((map @ A) @ B) @ (((((comp @ (option @ B)) @ B) @ A) @ (the2 @ B)) @ F3)) @ (((filter2 @ A) @ (^ [X:A]: (((F3 @ X) != (none @ B))))) @ Xs2)))))))). % 6.67/2.70 thf(fact_7574_transpose__aux__filter__tail, axiom, (! [A:$tType,Xss:(list @ (list @ A))]: ((((concat @ (list @ A)) @ ((((map @ (list @ A)) @ (list @ (list @ A))) @ ((((case_list @ (list @ (list @ A))) @ A) @ (nil @ (list @ A))) @ (^ [H:A,T3:(list @ A)]: ((((cons @ (list @ A)) @ T3) @ (nil @ (list @ A))))))) @ Xss)) = ((((map @ (list @ A)) @ (list @ A)) @ (tl @ A)) @ (((filter2 @ (list @ A)) @ (^ [Ys3:(list @ A)]: ((Ys3 != (nil @ A))))) @ Xss)))))). % 6.67/2.70 thf(fact_7575_tl__upt, axiom, (! [M:nat,N2:nat]: ((((tl @ nat) @ ((upt @ M) @ N2)) = ((upt @ (suc @ M)) @ N2))))). % 6.67/2.70 thf(fact_7576_tl__append2, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((Xs != (nil @ A)) => (((tl @ A) @ (((append @ A) @ Xs) @ Ys)) = (((append @ A) @ ((tl @ A) @ Xs)) @ Ys)))))). % 6.67/2.70 thf(fact_7577_remdups__adj__Cons__alt, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: (((((cons @ A) @ X3) @ ((tl @ A) @ ((remdups_adj @ A) @ (((cons @ A) @ X3) @ Xs)))) = ((remdups_adj @ A) @ (((cons @ A) @ X3) @ Xs)))))). % 6.67/2.70 thf(fact_7578_length__tl, axiom, (! [A:$tType,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ ((tl @ A) @ Xs)) = (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)))))). % 6.67/2.70 thf(fact_7579_list_Ocollapse, axiom, (! [A:$tType,List:(list @ A)]: (((List != (nil @ A)) => ((((cons @ A) @ ((hd @ A) @ List)) @ ((tl @ A) @ List)) = List))))). % 6.67/2.70 thf(fact_7580_hd__Cons__tl, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => ((((cons @ A) @ ((hd @ A) @ Xs)) @ ((tl @ A) @ Xs)) = Xs))))). % 6.67/2.70 thf(fact_7581_tl__replicate, axiom, (! [A:$tType,N2:nat,X3:A]: ((((tl @ A) @ (((replicate @ A) @ N2) @ X3)) = (((replicate @ A) @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))) @ X3))))). % 6.67/2.70 thf(fact_7582_list_Omap__sel_I2_J, axiom, (! [B:$tType,A:$tType,A2:(list @ A),F2:(A > B)]: (((A2 != (nil @ A)) => (((tl @ B) @ ((((map @ A) @ B) @ F2) @ A2)) = ((((map @ A) @ B) @ F2) @ ((tl @ A) @ A2))))))). % 6.67/2.70 thf(fact_7583_map__tl, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B)]: ((((((map @ B) @ A) @ F2) @ ((tl @ B) @ Xs)) = ((tl @ A) @ ((((map @ B) @ A) @ F2) @ Xs)))))). % 6.67/2.70 thf(fact_7584_map__of__filter__in, axiom, (! [B:$tType,A:$tType,Xs:(list @ ((product_prod @ B) @ A)),K:B,Z2:A,P:(B > (A > $o))]: (((((((map_of @ B) @ A) @ Xs) @ K) = ((some @ A) @ Z2)) => (((P @ K) @ Z2) => (((((map_of @ B) @ A) @ (((filter2 @ ((product_prod @ B) @ A)) @ ((((product_case_prod @ B) @ A) @ $o) @ P)) @ Xs)) @ K) = ((some @ A) @ Z2))))))). % 6.67/2.70 thf(fact_7585_tl__append, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((tl @ A) @ (((append @ A) @ Xs) @ Ys)) = (((((case_list @ (list @ A)) @ A) @ ((tl @ A) @ Ys)) @ (^ [Z6:A,Zs3:(list @ A)]: ((((append @ A) @ Zs3) @ Ys)))) @ Xs))))). % 6.67/2.70 thf(fact_7586_list_Osel_I3_J, axiom, (! [A:$tType,X21:A,X222:(list @ A)]: ((((tl @ A) @ (((cons @ A) @ X21) @ X222)) = X222)))). % 6.67/2.70 thf(fact_7587_Nil__tl, axiom, (! [A:$tType,Xs:(list @ A)]: ((((nil @ A) = ((tl @ A) @ Xs)) = ((Xs = (nil @ A)) | (? [X:A]: ((Xs = (((cons @ A) @ X) @ (nil @ A)))))))))). % 6.67/2.70 thf(fact_7588_tl__Nil, axiom, (! [A:$tType,Xs:(list @ A)]: (((((tl @ A) @ Xs) = (nil @ A)) = ((Xs = (nil @ A)) | (? [X:A]: ((Xs = (((cons @ A) @ X) @ (nil @ A)))))))))). % 6.67/2.70 thf(fact_7589_list_Osel_I2_J, axiom, (! [A:$tType]: ((((tl @ A) @ (nil @ A)) = (nil @ A))))). % 6.67/2.70 thf(fact_7590_map__filter__simps_I2_J, axiom, (! [B:$tType,A:$tType,F2:(B > (option @ A))]: ((((((map_filter @ B) @ A) @ F2) @ (nil @ B)) = (nil @ A))))). % 6.67/2.70 thf(fact_7591_tl__def, axiom, (! [A:$tType]: (((tl @ A) = ((((case_list @ (list @ A)) @ A) @ (nil @ A)) @ (^ [X213:A,X224:(list @ A)]: (X224))))))). % 6.67/2.70 thf(fact_7592_take__tl, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((take @ A) @ N2) @ ((tl @ A) @ Xs)) = ((tl @ A) @ (((take @ A) @ (suc @ N2)) @ Xs)))))). % 6.67/2.70 thf(fact_7593_drop__Suc, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((drop @ A) @ (suc @ N2)) @ Xs) = (((drop @ A) @ N2) @ ((tl @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7594_tl__drop, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((tl @ A) @ (((drop @ A) @ N2) @ Xs)) = (((drop @ A) @ N2) @ ((tl @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7595_distinct__tl, axiom, (! [A:$tType,Xs:(list @ A)]: ((((distinct @ A) @ Xs) => ((distinct @ A) @ ((tl @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7596_list_Oset__sel_I2_J, axiom, (! [A:$tType,A2:(list @ A),X3:A]: (((A2 != (nil @ A)) => ((((member @ A) @ X3) @ ((set2 @ A) @ ((tl @ A) @ A2))) => (((member @ A) @ X3) @ ((set2 @ A) @ A2))))))). % 6.67/2.70 thf(fact_7597_list_Oexpand, axiom, (! [A:$tType,List:(list @ A),List3:(list @ A)]: ((((List = (nil @ A)) = (List3 = (nil @ A))) => (((List != (nil @ A)) => ((List3 != (nil @ A)) => ((((hd @ A) @ List) = ((hd @ A) @ List3)) & (((tl @ A) @ List) = ((tl @ A) @ List3))))) => (List = List3)))))). % 6.67/2.70 thf(fact_7598_list_Oexhaust__sel, axiom, (! [A:$tType,List:(list @ A)]: (((List != (nil @ A)) => (List = (((cons @ A) @ ((hd @ A) @ List)) @ ((tl @ A) @ List))))))). % 6.67/2.70 thf(fact_7599_map__filter__simps_I1_J, axiom, (! [A:$tType,B:$tType,F2:(B > (option @ A)),X3:B,Xs:(list @ B)]: ((((((map_filter @ B) @ A) @ F2) @ (((cons @ B) @ X3) @ Xs)) = (((((case_option @ (list @ A)) @ A) @ ((((map_filter @ B) @ A) @ F2) @ Xs)) @ (^ [Y2:A]: ((((cons @ A) @ Y2) @ ((((map_filter @ B) @ A) @ F2) @ Xs))))) @ (F2 @ X3)))))). % 6.67/2.70 thf(fact_7600_tl__take, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((tl @ A) @ (((take @ A) @ N2) @ Xs)) = (((take @ A) @ (((minus_minus @ nat) @ N2) @ (one_one @ nat))) @ ((tl @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7601_list_Ocase__eq__if, axiom, (! [A:$tType,B:$tType]: ((((case_list @ B) @ A) = (^ [F12:B,F23:(A > ((list @ A) > B)),List2:(list @ A)]: (((((if @ B) @ (List2 = (nil @ A))) @ F12) @ ((F23 @ ((hd @ A) @ List2)) @ ((tl @ A) @ List2))))))))). % 6.67/2.70 thf(fact_7602_Nitpick_Osize__list__simp_I2_J, axiom, (! [A:$tType]: (((size_size @ (list @ A)) = (^ [Xs2:(list @ A)]: (((((if @ nat) @ (Xs2 = (nil @ A))) @ (zero_zero @ nat)) @ (suc @ ((size_size @ (list @ A)) @ ((tl @ A) @ Xs2)))))))))). % 6.67/2.70 thf(fact_7603_nth__tl, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ ((tl @ A) @ Xs))) => ((((nth @ A) @ ((tl @ A) @ Xs)) @ N2) = (((nth @ A) @ Xs) @ (suc @ N2))))))). % 6.67/2.70 thf(fact_7604_remdups__adj__append, axiom, (! [A:$tType,Xs_1:(list @ A),X3:A,Xs_2:(list @ A)]: ((((remdups_adj @ A) @ (((append @ A) @ Xs_1) @ (((cons @ A) @ X3) @ Xs_2))) = (((append @ A) @ ((remdups_adj @ A) @ (((append @ A) @ Xs_1) @ (((cons @ A) @ X3) @ (nil @ A))))) @ ((tl @ A) @ ((remdups_adj @ A) @ (((cons @ A) @ X3) @ Xs_2)))))))). % 6.67/2.70 thf(fact_7605_Cons__in__shuffles__iff, axiom, (! [A:$tType,Z2:A,Zs2:(list @ A),Xs:(list @ A),Ys:(list @ A)]: (((((member @ (list @ A)) @ (((cons @ A) @ Z2) @ Zs2)) @ (((shuffles @ A) @ Xs) @ Ys)) = (((Xs != (nil @ A)) & ((((hd @ A) @ Xs) = Z2) & (((member @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ ((tl @ A) @ Xs)) @ Ys)))) | ((Ys != (nil @ A)) & ((((hd @ A) @ Ys) = Z2) & (((member @ (list @ A)) @ Zs2) @ (((shuffles @ A) @ Xs) @ ((tl @ A) @ Ys)))))))))). % 6.67/2.70 thf(fact_7606_list_Osplit__sel__asm, axiom, (! [B:$tType,A:$tType,P:(B > $o),F1:B,F22:(A > ((list @ A) > B)),List:(list @ A)]: (((P @ (((((case_list @ B) @ A) @ F1) @ F22) @ List)) = (~ ((((List = (nil @ A)) & (~ ((P @ F1)))) | ((List = (((cons @ A) @ ((hd @ A) @ List)) @ ((tl @ A) @ List))) & (~ ((P @ ((F22 @ ((hd @ A) @ List)) @ ((tl @ A) @ List))))))))))))). % 6.67/2.70 thf(fact_7607_list_Osplit__sel, axiom, (! [B:$tType,A:$tType,P:(B > $o),F1:B,F22:(A > ((list @ A) > B)),List:(list @ A)]: (((P @ (((((case_list @ B) @ A) @ F1) @ F22) @ List)) = (((List = (nil @ A)) => (P @ F1)) & ((List = (((cons @ A) @ ((hd @ A) @ List)) @ ((tl @ A) @ List))) => (P @ ((F22 @ ((hd @ A) @ List)) @ ((tl @ A) @ List))))))))). % 6.67/2.70 thf(fact_7608_nths__shift__lemma__Suc, axiom, (! [A:$tType,P:(nat > $o),Xs:(list @ A),Is:(list @ nat)]: ((((((map @ ((product_prod @ A) @ nat)) @ A) @ ((product_fst @ A) @ nat)) @ (((filter2 @ ((product_prod @ A) @ nat)) @ (^ [P6:((product_prod @ A) @ nat)]: ((P @ (suc @ (((product_snd @ A) @ nat) @ P6)))))) @ ((((zip @ A) @ nat) @ Xs) @ Is))) = ((((map @ ((product_prod @ A) @ nat)) @ A) @ ((product_fst @ A) @ nat)) @ (((filter2 @ ((product_prod @ A) @ nat)) @ (^ [P6:((product_prod @ A) @ nat)]: ((P @ (((product_snd @ A) @ nat) @ P6))))) @ ((((zip @ A) @ nat) @ Xs) @ ((((map @ nat) @ nat) @ suc) @ Is)))))))). % 6.67/2.70 thf(fact_7609_take__Suc, axiom, (! [A:$tType,Xs:(list @ A),N2:nat]: (((Xs != (nil @ A)) => ((((take @ A) @ (suc @ N2)) @ Xs) = (((cons @ A) @ ((hd @ A) @ Xs)) @ (((take @ A) @ N2) @ ((tl @ A) @ Xs)))))))). % 6.67/2.70 thf(fact_7610_rotate1__hd__tl, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (((rotate1 @ A) @ Xs) = (((append @ A) @ ((tl @ A) @ Xs)) @ (((cons @ A) @ ((hd @ A) @ Xs)) @ (nil @ A)))))))). % 6.67/2.70 thf(fact_7611_nths__shift__lemma, axiom, (! [A:$tType,A3:(set @ nat),Xs:(list @ A),I:nat]: ((((((map @ ((product_prod @ A) @ nat)) @ A) @ ((product_fst @ A) @ nat)) @ (((filter2 @ ((product_prod @ A) @ nat)) @ (^ [P6:((product_prod @ A) @ nat)]: ((((member @ nat) @ (((product_snd @ A) @ nat) @ P6)) @ A3)))) @ ((((zip @ A) @ nat) @ Xs) @ ((upt @ I) @ (((plus_plus @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)))))) = ((((map @ ((product_prod @ A) @ nat)) @ A) @ ((product_fst @ A) @ nat)) @ (((filter2 @ ((product_prod @ A) @ nat)) @ (^ [P6:((product_prod @ A) @ nat)]: ((((member @ nat) @ (((plus_plus @ nat) @ (((product_snd @ A) @ nat) @ P6)) @ I)) @ A3)))) @ ((((zip @ A) @ nat) @ Xs) @ ((upt @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs))))))))). % 6.67/2.70 thf(fact_7612_Nitpick_Osize__list__simp_I1_J, axiom, (! [A:$tType]: (((size_list @ A) = (^ [F3:(A > nat),Xs2:(list @ A)]: (((((if @ nat) @ (Xs2 = (nil @ A))) @ (zero_zero @ nat)) @ (suc @ (((plus_plus @ nat) @ (F3 @ ((hd @ A) @ Xs2))) @ (((size_list @ A) @ F3) @ ((tl @ A) @ Xs2))))))))))). % 6.67/2.70 thf(fact_7613_nths__def, axiom, (! [A:$tType]: (((nths @ A) = (^ [Xs2:(list @ A),A6:(set @ nat)]: (((((map @ ((product_prod @ A) @ nat)) @ A) @ ((product_fst @ A) @ nat)) @ (((filter2 @ ((product_prod @ A) @ nat)) @ (^ [P6:((product_prod @ A) @ nat)]: ((((member @ nat) @ (((product_snd @ A) @ nat) @ P6)) @ A6)))) @ ((((zip @ A) @ nat) @ Xs2) @ ((upt @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs2))))))))))). % 6.67/2.70 thf(fact_7614_map__filter__map__filter, axiom, (! [A:$tType,B:$tType,F2:(B > A),P:(B > $o),Xs:(list @ B)]: ((((((map @ B) @ A) @ F2) @ (((filter2 @ B) @ P) @ Xs)) = ((((map_filter @ B) @ A) @ (^ [X:B]: (((((if @ (option @ A)) @ (P @ X)) @ ((some @ A) @ (F2 @ X))) @ (none @ A))))) @ Xs))))). % 6.67/2.70 thf(fact_7615_transpose__max__length, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: (((((((foldr @ (list @ A)) @ nat) @ (^ [Xs2:(list @ A)]: (((ord_max @ nat) @ ((size_size @ (list @ A)) @ Xs2))))) @ ((transpose @ A) @ Xs)) @ (zero_zero @ nat)) = ((size_size @ (list @ (list @ A))) @ (((filter2 @ (list @ A)) @ (^ [X:(list @ A)]: ((X != (nil @ A))))) @ Xs)))))). % 6.67/2.70 thf(fact_7616_transpose__aux__max, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Xss:(list @ (list @ B))]: (((((ord_max @ nat) @ (suc @ ((size_size @ (list @ A)) @ Xs))) @ (((((foldr @ (list @ B)) @ nat) @ (^ [Xs2:(list @ B)]: (((ord_max @ nat) @ ((size_size @ (list @ B)) @ Xs2))))) @ Xss) @ (zero_zero @ nat))) = (suc @ (((ord_max @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (((((foldr @ (list @ B)) @ nat) @ (^ [X:(list @ B)]: (((ord_max @ nat) @ (((minus_minus @ nat) @ ((size_size @ (list @ B)) @ X)) @ (suc @ (zero_zero @ nat))))))) @ (((filter2 @ (list @ B)) @ (^ [Ys3:(list @ B)]: ((Ys3 != (nil @ B))))) @ Xss)) @ (zero_zero @ nat)))))))). % 6.67/2.70 thf(fact_7617_foldr__append, axiom, (! [B:$tType,A:$tType,F2:(B > (A > A)),Xs:(list @ B),Ys:(list @ B),A2:A]: (((((((foldr @ B) @ A) @ F2) @ (((append @ B) @ Xs) @ Ys)) @ A2) = (((((foldr @ B) @ A) @ F2) @ Xs) @ (((((foldr @ B) @ A) @ F2) @ Ys) @ A2)))))). % 6.67/2.70 thf(fact_7618_foldr__replicate, axiom, (! [A:$tType,B:$tType,F2:(B > (A > A)),N2:nat,X3:B]: ((((((foldr @ B) @ A) @ F2) @ (((replicate @ B) @ N2) @ X3)) = (((compow @ (A > A)) @ N2) @ (F2 @ X3)))))). % 6.67/2.70 thf(fact_7619_foldr__map, axiom, (! [C:$tType,B:$tType,A:$tType,G:(B > (A > A)),F2:(C > B),Xs:(list @ C),A2:A]: (((((((foldr @ B) @ A) @ G) @ ((((map @ C) @ B) @ F2) @ Xs)) @ A2) = (((((foldr @ C) @ A) @ (((((comp @ B) @ (A > A)) @ C) @ G) @ F2)) @ Xs) @ A2))))). % 6.67/2.70 thf(fact_7620_foldr__Cons, axiom, (! [B:$tType,A:$tType,F2:(A > (B > B)),X3:A,Xs:(list @ A)]: ((((((foldr @ A) @ B) @ F2) @ (((cons @ A) @ X3) @ Xs)) = (((((comp @ B) @ B) @ B) @ (F2 @ X3)) @ ((((foldr @ A) @ B) @ F2) @ Xs)))))). % 6.67/2.70 thf(fact_7621_foldr__cong, axiom, (! [B:$tType,A:$tType,A2:A,B2:A,L:(list @ B),K:(list @ B),F2:(B > (A > A)),G:(B > (A > A))]: (((A2 = B2) => ((L = K) => ((! [A5:A,X5:B]: (((((member @ B) @ X5) @ ((set2 @ B) @ L)) => (((F2 @ X5) @ A5) = ((G @ X5) @ A5))))) => ((((((foldr @ B) @ A) @ F2) @ L) @ A2) = (((((foldr @ B) @ A) @ G) @ K) @ B2)))))))). % 6.67/2.70 thf(fact_7622_foldr__Nil, axiom, (! [A:$tType,B:$tType,F2:(A > (B > B))]: ((((((foldr @ A) @ B) @ F2) @ (nil @ A)) = (id @ B))))). % 6.67/2.70 thf(fact_7623_foldr__filter, axiom, (! [A:$tType,B:$tType,F2:(B > (A > A)),P:(B > $o),Xs:(list @ B)]: ((((((foldr @ B) @ A) @ F2) @ (((filter2 @ B) @ P) @ Xs)) = ((((foldr @ B) @ A) @ (^ [X:B]: (((((if @ (A > A)) @ (P @ X)) @ (F2 @ X)) @ (id @ A))))) @ Xs))))). % 6.67/2.70 thf(fact_7624_sum__list_Oeq__foldr, axiom, (! [A:$tType]: (((monoid_add @ A) => ((groups8242544230860333062m_list @ A) = (^ [Xs2:(list @ A)]: ((((((foldr @ A) @ A) @ (plus_plus @ A)) @ Xs2) @ (zero_zero @ A))))))))). % 6.67/2.70 thf(fact_7625_horner__sum__foldr, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_0 @ A) => (((groups4207007520872428315er_sum @ B) @ A) = (^ [F3:(B > A),A4:A,Xs2:(list @ B)]: ((((((foldr @ B) @ A) @ (^ [X:B,B4:A]: ((((plus_plus @ A) @ (F3 @ X)) @ (((times_times @ A) @ A4) @ B4))))) @ Xs2) @ (zero_zero @ A))))))))). % 6.67/2.70 thf(fact_7626_length__transpose, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: ((((size_size @ (list @ (list @ A))) @ ((transpose @ A) @ Xs)) = (((((foldr @ (list @ A)) @ nat) @ (^ [Xs2:(list @ A)]: (((ord_max @ nat) @ ((size_size @ (list @ A)) @ Xs2))))) @ Xs) @ (zero_zero @ nat)))))). % 6.67/2.70 thf(fact_7627_sorted__wrt__less__sum__mono__lowerbound, axiom, (! [B:$tType]: (((ordere6911136660526730532id_add @ B) => (! [F2:(nat > B),Ns:(list @ nat)]: (((! [X5:nat,Y4:nat]: (((((ord_less_eq @ nat) @ X5) @ Y4) => (((ord_less_eq @ B) @ (F2 @ X5)) @ (F2 @ Y4))))) => ((((sorted_wrt @ nat) @ (ord_less @ nat)) @ Ns) => (((ord_less_eq @ B) @ ((((groups7311177749621191930dd_sum @ nat) @ B) @ F2) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ nat)) @ Ns)))) @ ((groups8242544230860333062m_list @ B) @ ((((map @ nat) @ B) @ F2) @ Ns))))))))))). % 6.67/2.70 thf(fact_7628_length__product__lists, axiom, (! [B:$tType,Xss:(list @ (list @ B))]: ((((size_size @ (list @ (list @ B))) @ ((product_lists @ B) @ Xss)) = (((((foldr @ nat) @ nat) @ (times_times @ nat)) @ ((((map @ (list @ B)) @ nat) @ (size_size @ (list @ B))) @ Xss)) @ (one_one @ nat)))))). % 6.67/2.70 thf(fact_7629_sorted__map__same, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),G:((list @ B) > A),Xs:(list @ B)]: ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F2) @ (((filter2 @ B) @ (^ [X:B]: (((F2 @ X) = (G @ Xs))))) @ Xs))))))))). % 6.67/2.70 thf(fact_7630_sorted__map, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),Xs:(list @ B)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F2) @ Xs)) = (((sorted_wrt @ B) @ (^ [X:B,Y2:B]: ((((ord_less_eq @ A) @ (F2 @ X)) @ (F2 @ Y2))))) @ Xs)))))))). % 6.67/2.70 thf(fact_7631_sorted__filter, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),Xs:(list @ B),P:(B > $o)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F2) @ Xs)) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F2) @ (((filter2 @ B) @ P) @ Xs)))))))))). % 6.67/2.70 thf(fact_7632_sorted__wrt__map, axiom, (! [A:$tType,B:$tType,R:(A > (A > $o)),F2:(B > A),Xs:(list @ B)]: (((((sorted_wrt @ A) @ R) @ ((((map @ B) @ A) @ F2) @ Xs)) = (((sorted_wrt @ B) @ (^ [X:B,Y2:B]: (((R @ (F2 @ X)) @ (F2 @ Y2))))) @ Xs))))). % 6.67/2.70 thf(fact_7633_sorted__insort__key, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),X3:B,Xs:(list @ B)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F2) @ (((((linorder_insort_key @ B) @ A) @ F2) @ X3) @ Xs))) = (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F2) @ Xs))))))))). % 6.67/2.70 thf(fact_7634_sorted__map__remove1, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),Xs:(list @ B),X3:B]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F2) @ Xs)) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F2) @ (((remove1 @ B) @ X3) @ Xs)))))))))). % 6.67/2.70 thf(fact_7635_sorted__same, axiom, (! [A:$tType]: (((linorder @ A) => (! [G:((list @ A) > A),Xs:(list @ A)]: ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((filter2 @ A) @ (^ [X:A]: ((X = (G @ Xs))))) @ Xs)))))))). % 6.67/2.70 thf(fact_7636_sorted__wrt__filter, axiom, (! [A:$tType,F2:(A > (A > $o)),Xs:(list @ A),P:(A > $o)]: (((((sorted_wrt @ A) @ F2) @ Xs) => (((sorted_wrt @ A) @ F2) @ (((filter2 @ A) @ P) @ Xs)))))). % 6.67/2.70 thf(fact_7637_sorted__tl, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((tl @ A) @ Xs))))))))). % 6.67/2.70 thf(fact_7638_sorted__upt, axiom, (! [M:nat,N2:nat]: ((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((upt @ M) @ N2))))). % 6.67/2.70 thf(fact_7639_sorted__wrt__upt, axiom, (! [M:nat,N2:nat]: ((((sorted_wrt @ nat) @ (ord_less @ nat)) @ ((upt @ M) @ N2))))). % 6.67/2.70 thf(fact_7640_sorted__wrt__true, axiom, (! [A:$tType,Xs:(list @ A)]: ((((sorted_wrt @ A) @ (^ [Uu3:A,Uv3:A]: ($true))) @ Xs)))). % 6.67/2.70 thf(fact_7641_sorted__list__of__set_Ostrict__sorted__key__list__of__set, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: ((((sorted_wrt @ A) @ (ord_less @ A)) @ ((linord4507533701916653071of_set @ A) @ A3)))))))). % 6.67/2.70 thf(fact_7642_sorted__remdups__adj, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((remdups_adj @ A) @ Xs))))))))). % 6.67/2.70 thf(fact_7643_sorted__insort, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X))) @ X3) @ Xs)) = (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs)))))))). % 6.67/2.70 thf(fact_7644_strict__sorted__imp__sorted, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs)))))))). % 6.67/2.70 thf(fact_7645_sorted__replicate, axiom, (! [A:$tType]: (((linorder @ A) => (! [N2:nat,X3:A]: ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((replicate @ A) @ N2) @ X3)))))))). % 6.67/2.70 thf(fact_7646_sorted__nths, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),I6:(set @ nat)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((nths @ A) @ Xs) @ I6))))))))). % 6.67/2.70 thf(fact_7647_sorted__remove1, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),A2:A]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((remove1 @ A) @ A2) @ Xs))))))))). % 6.67/2.70 thf(fact_7648_sorted__list__of__set_Osorted__sorted__key__list__of__set, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((linord4507533701916653071of_set @ A) @ A3)))))))). % 6.67/2.70 thf(fact_7649_sorted2, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y:A,Zs2:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((cons @ A) @ X3) @ (((cons @ A) @ Y) @ Zs2))) = ((((ord_less_eq @ A) @ X3) @ Y) & (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((cons @ A) @ Y) @ Zs2)))))))))). % 6.67/2.70 thf(fact_7650_sorted1, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A]: ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((cons @ A) @ X3) @ (nil @ A))))))))). % 6.67/2.70 thf(fact_7651_sorted__wrt1, axiom, (! [A:$tType,P:(A > (A > $o)),X3:A]: ((((sorted_wrt @ A) @ P) @ (((cons @ A) @ X3) @ (nil @ A)))))). % 6.67/2.70 thf(fact_7652_strict__sorted__simps_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (((sorted_wrt @ A) @ (ord_less @ A)) @ (nil @ A)))))). % 6.67/2.70 thf(fact_7653_sorted__wrt_Osimps_I1_J, axiom, (! [A:$tType,P:(A > (A > $o))]: ((((sorted_wrt @ A) @ P) @ (nil @ A))))). % 6.67/2.70 thf(fact_7654_sorted0, axiom, (! [A:$tType]: (((linorder @ A) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (nil @ A)))))). % 6.67/2.70 thf(fact_7655_sorted__remdups, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((remdups @ A) @ Xs))))))))). % 6.67/2.70 thf(fact_7656_sorted__take, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),N2:nat]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((take @ A) @ N2) @ Xs))))))))). % 6.67/2.70 thf(fact_7657_sorted__wrt__take, axiom, (! [A:$tType,F2:(A > (A > $o)),Xs:(list @ A),N2:nat]: (((((sorted_wrt @ A) @ F2) @ Xs) => (((sorted_wrt @ A) @ F2) @ (((take @ A) @ N2) @ Xs)))))). % 6.67/2.70 thf(fact_7658_sorted__drop, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),N2:nat]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((drop @ A) @ N2) @ Xs))))))))). % 6.67/2.70 thf(fact_7659_sorted__wrt__drop, axiom, (! [A:$tType,F2:(A > (A > $o)),Xs:(list @ A),N2:nat]: (((((sorted_wrt @ A) @ F2) @ Xs) => (((sorted_wrt @ A) @ F2) @ (((drop @ A) @ N2) @ Xs)))))). % 6.67/2.70 thf(fact_7660_strict__sorted__equal, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Ys:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less @ A)) @ Xs) => ((((sorted_wrt @ A) @ (ord_less @ A)) @ Ys) => ((((set2 @ A) @ Ys) = ((set2 @ A) @ Xs)) => (Ys = Xs)))))))))). % 6.67/2.70 thf(fact_7661_sorted__wrt__mono__rel, axiom, (! [A:$tType,Xs:(list @ A),P:(A > (A > $o)),Q:(A > (A > $o))]: (((! [X5:A,Y4:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => ((((member @ A) @ Y4) @ ((set2 @ A) @ Xs)) => (((P @ X5) @ Y4) => ((Q @ X5) @ Y4)))))) => ((((sorted_wrt @ A) @ P) @ Xs) => (((sorted_wrt @ A) @ Q) @ Xs)))))). % 6.67/2.70 thf(fact_7662_sorted__wrt__iff__nth__less, axiom, (! [A:$tType]: (((sorted_wrt @ A) = (^ [P3:(A > (A > $o)),Xs2:(list @ A)]: ((! [I3:nat,J3:nat]: (((((ord_less @ nat) @ I3) @ J3) => ((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs2)) => ((P3 @ (((nth @ A) @ Xs2) @ I3)) @ (((nth @ A) @ Xs2) @ J3)))))))))))). % 6.67/2.70 thf(fact_7663_sorted__wrt__nth__less, axiom, (! [A:$tType,P:(A > (A > $o)),Xs:(list @ A),I:nat,J:nat]: (((((sorted_wrt @ A) @ P) @ Xs) => ((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => ((P @ (((nth @ A) @ Xs) @ I)) @ (((nth @ A) @ Xs) @ J)))))))). % 6.67/2.70 thf(fact_7664_sorted__wrt01, axiom, (! [A:$tType,Xs:(list @ A),P:(A > (A > $o))]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)) => (((sorted_wrt @ A) @ P) @ Xs))))). % 6.67/2.70 thf(fact_7665_sorted__distinct__set__unique, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Ys:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((distinct @ A) @ Xs) => ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Ys) => (((distinct @ A) @ Ys) => ((((set2 @ A) @ Xs) = ((set2 @ A) @ Ys)) => (Xs = Ys)))))))))))). % 6.67/2.70 thf(fact_7666_strict__sorted__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less @ A)) @ L) = ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ L) & ((distinct @ A) @ L))))))))). % 6.67/2.70 thf(fact_7667_sorted__wrt__append, axiom, (! [A:$tType,P:(A > (A > $o)),Xs:(list @ A),Ys:(list @ A)]: (((((sorted_wrt @ A) @ P) @ (((append @ A) @ Xs) @ Ys)) = ((((sorted_wrt @ A) @ P) @ Xs) & ((((sorted_wrt @ A) @ P) @ Ys) & (! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (! [Y2:A]: (((((member @ A) @ Y2) @ ((set2 @ A) @ Ys)) => ((P @ X) @ Y2))))))))))))). % 6.67/2.70 thf(fact_7668_sorted__append, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Ys:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((append @ A) @ Xs) @ Ys)) = ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) & ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Ys) & (! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (! [Y2:A]: (((((member @ A) @ Y2) @ ((set2 @ A) @ Ys)) => (((ord_less_eq @ A) @ X) @ Y2)))))))))))))))). % 6.67/2.70 thf(fact_7669_sorted__simps_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Ys:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((cons @ A) @ X3) @ Ys)) = ((! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Ys)) => (((ord_less_eq @ A) @ X3) @ X)))) & (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Ys))))))))). % 6.67/2.70 thf(fact_7670_strict__sorted__simps_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Ys:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less @ A)) @ (((cons @ A) @ X3) @ Ys)) = ((! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Ys)) => (((ord_less @ A) @ X3) @ X)))) & (((sorted_wrt @ A) @ (ord_less @ A)) @ Ys))))))))). % 6.67/2.70 thf(fact_7671_sorted__wrt_Oelims_I3_J, axiom, (! [A:$tType,X3:(A > (A > $o)),Xa2:(list @ A)]: (((~ ((((sorted_wrt @ A) @ X3) @ Xa2))) => (~ ((! [X5:A,Ys4:(list @ A)]: (((Xa2 = (((cons @ A) @ X5) @ Ys4)) => ((! [Xa3:A]: (((((member @ A) @ Xa3) @ ((set2 @ A) @ Ys4)) => ((X3 @ X5) @ Xa3)))) & (((sorted_wrt @ A) @ X3) @ Ys4))))))))))). % 6.67/2.70 thf(fact_7672_sorted__wrt_Osimps_I2_J, axiom, (! [A:$tType,P:(A > (A > $o)),X3:A,Ys:(list @ A)]: (((((sorted_wrt @ A) @ P) @ (((cons @ A) @ X3) @ Ys)) = ((! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Ys)) => ((P @ X3) @ X)))) & (((sorted_wrt @ A) @ P) @ Ys)))))). % 6.67/2.70 thf(fact_7673_sorted01, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs)))))))). % 6.67/2.70 thf(fact_7674_sorted__iff__nth__mono__less, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) = (! [I3:nat,J3:nat]: (((((ord_less @ nat) @ I3) @ J3) => ((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ I3)) @ (((nth @ A) @ Xs) @ J3))))))))))))). % 6.67/2.70 thf(fact_7675_sorted__wrt_Oelims_I1_J, axiom, (! [A:$tType,X3:(A > (A > $o)),Xa2:(list @ A),Y:$o]: ((((((sorted_wrt @ A) @ X3) @ Xa2) = Y) => (((Xa2 = (nil @ A)) => (~ (Y))) => (~ ((! [X5:A,Ys4:(list @ A)]: (((Xa2 = (((cons @ A) @ X5) @ Ys4)) => (Y = (~ (((! [Y2:A]: (((((member @ A) @ Y2) @ ((set2 @ A) @ Ys4)) => ((X3 @ X5) @ Y2)))) & (((sorted_wrt @ A) @ X3) @ Ys4))))))))))))))). % 6.67/2.70 thf(fact_7676_sorted__wrt_Oelims_I2_J, axiom, (! [A:$tType,X3:(A > (A > $o)),Xa2:(list @ A)]: (((((sorted_wrt @ A) @ X3) @ Xa2) => ((Xa2 != (nil @ A)) => (~ ((! [X5:A,Ys4:(list @ A)]: (((Xa2 = (((cons @ A) @ X5) @ Ys4)) => (~ (((! [Xa:A]: (((((member @ A) @ Xa) @ ((set2 @ A) @ Ys4)) => ((X3 @ X5) @ Xa)))) & (((sorted_wrt @ A) @ X3) @ Ys4)))))))))))))). % 6.67/2.70 thf(fact_7677_finite__sorted__distinct__unique, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (? [X5:(list @ A)]: (((((set2 @ A) @ X5) = A3) & ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ X5) & (((distinct @ A) @ X5) & (! [Y5:(list @ A)]: ((((((set2 @ A) @ Y5) = A3) & ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Y5) & ((distinct @ A) @ Y5))) => (Y5 = X5)))))))))))))))). % 6.67/2.70 thf(fact_7678_sorted__list__of__set_Oidem__if__sorted__distinct, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((distinct @ A) @ Xs) => (((linord4507533701916653071of_set @ A) @ ((set2 @ A) @ Xs)) = Xs))))))))). % 6.67/2.70 thf(fact_7679_filter__insort, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),Xs:(list @ B),P:(B > $o),X3:B]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F2) @ Xs)) => ((P @ X3) => ((((filter2 @ B) @ P) @ (((((linorder_insort_key @ B) @ A) @ F2) @ X3) @ Xs)) = (((((linorder_insort_key @ B) @ A) @ F2) @ X3) @ (((filter2 @ B) @ P) @ Xs))))))))))). % 6.67/2.70 thf(fact_7680_concat__conv__foldr, axiom, (! [A:$tType]: (((concat @ A) = (^ [Xss3:(list @ (list @ A))]: ((((((foldr @ (list @ A)) @ (list @ A)) @ (append @ A)) @ Xss3) @ (nil @ A)))))))). % 6.67/2.70 thf(fact_7681_insort__remove1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A2:A,Xs:(list @ A)]: (((((member @ A) @ A2) @ ((set2 @ A) @ Xs)) => ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => ((((((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X))) @ A2) @ (((remove1 @ A) @ A2) @ Xs)) = Xs))))))))). % 6.67/2.70 thf(fact_7682_sorted__iff__nth__Suc, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) = (! [I3:nat]: (((((ord_less @ nat) @ (suc @ I3)) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ I3)) @ (((nth @ A) @ Xs) @ (suc @ I3))))))))))))). % 6.67/2.70 thf(fact_7683_sorted__list__of__set_Ofinite__set__strict__sorted, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (~ ((! [L4:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less @ A)) @ L4) => ((((set2 @ A) @ L4) = A3) => (((size_size @ (list @ A)) @ L4) != ((finite_card @ A) @ A3))))))))))))))). % 6.67/2.70 thf(fact_7684_sorted__nth__mono, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),I:nat,J:nat]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => ((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ I)) @ (((nth @ A) @ Xs) @ J))))))))))). % 6.67/2.70 thf(fact_7685_sorted__iff__nth__mono, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) = (! [I3:nat,J3:nat]: (((((ord_less_eq @ nat) @ I3) @ J3) => ((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ I3)) @ (((nth @ A) @ Xs) @ J3))))))))))))). % 6.67/2.70 thf(fact_7686_sorted__wrt__less__idx, axiom, (! [Ns:(list @ nat),I:nat]: (((((sorted_wrt @ nat) @ (ord_less @ nat)) @ Ns) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ nat)) @ Ns)) => (((ord_less_eq @ nat) @ I) @ (((nth @ nat) @ Ns) @ I))))))). % 6.67/2.70 thf(fact_7687_sorted__find__Min, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),P:(A > $o)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => ((? [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) & (P @ X2)))) => ((((find @ A) @ P) @ Xs) = ((some @ A) @ ((lattic643756798350308766er_Min @ A) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) & (P @ X)))))))))))))))). % 6.67/2.70 thf(fact_7688_sorted__enumerate, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((((map @ ((product_prod @ nat) @ A)) @ nat) @ ((product_fst @ nat) @ A)) @ (((enumerate @ A) @ N2) @ Xs)))))). % 6.67/2.70 thf(fact_7689_map__sorted__distinct__set__unique, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),Xs:(list @ B),Ys:(list @ B)]: ((((((inj_on @ B) @ A) @ F2) @ (((sup_sup @ (set @ B)) @ ((set2 @ B) @ Xs)) @ ((set2 @ B) @ Ys))) => ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F2) @ Xs)) => (((distinct @ A) @ ((((map @ B) @ A) @ F2) @ Xs)) => ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F2) @ Ys)) => (((distinct @ A) @ ((((map @ B) @ A) @ F2) @ Ys)) => ((((set2 @ B) @ Xs) = ((set2 @ B) @ Ys)) => (Xs = Ys))))))))))))). % 6.67/2.70 thf(fact_7690_sorted__list__of__set_Osorted__key__list__of__set__unique, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:(set @ A),L:(list @ A)]: ((((finite_finite2 @ A) @ A3) => (((((sorted_wrt @ A) @ (ord_less @ A)) @ L) & ((((set2 @ A) @ L) = A3) & (((size_size @ (list @ A)) @ L) = ((finite_card @ A) @ A3)))) = (((linord4507533701916653071of_set @ A) @ A3) = L))))))))). % 6.67/2.70 thf(fact_7691_sorted__insort__is__snoc, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),A2:A]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => ((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ A) @ X5) @ A2)))) => ((((((linorder_insort_key @ A) @ A) @ (^ [X:A]: (X))) @ A2) @ Xs) = (((append @ A) @ Xs) @ (((cons @ A) @ A2) @ (nil @ A)))))))))))). % 6.67/2.70 thf(fact_7692_insort__key__remove1, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [A2:B,Xs:(list @ B),F2:(B > A)]: (((((member @ B) @ A2) @ ((set2 @ B) @ Xs)) => ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ B) @ A) @ F2) @ Xs)) => ((((hd @ B) @ (((filter2 @ B) @ (^ [X:B]: (((F2 @ A2) = (F2 @ X))))) @ Xs)) = A2) => ((((((linorder_insort_key @ B) @ A) @ F2) @ A2) @ (((remove1 @ B) @ A2) @ Xs)) = Xs)))))))))). % 6.67/2.70 thf(fact_7693_nth__nth__transpose__sorted, axiom, (! [A:$tType,Xs:(list @ (list @ A)),I:nat,J:nat]: (((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((rev @ nat) @ ((((map @ (list @ A)) @ nat) @ (size_size @ (list @ A))) @ Xs))) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ (list @ A))) @ ((transpose @ A) @ Xs))) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ (list @ A))) @ (((filter2 @ (list @ A)) @ (^ [Ys3:(list @ A)]: ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Ys3))))) @ Xs))) => ((((nth @ A) @ (((nth @ (list @ A)) @ ((transpose @ A) @ Xs)) @ I)) @ J) = (((nth @ A) @ (((nth @ (list @ A)) @ Xs) @ J)) @ I)))))))). % 6.67/2.70 thf(fact_7694_transpose__column, axiom, (! [A:$tType,Xs:(list @ (list @ A)),I:nat]: (((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((rev @ nat) @ ((((map @ (list @ A)) @ nat) @ (size_size @ (list @ A))) @ Xs))) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ (list @ A))) @ Xs)) => (((((map @ (list @ A)) @ A) @ (^ [Ys3:(list @ A)]: ((((nth @ A) @ Ys3) @ I)))) @ (((filter2 @ (list @ A)) @ (^ [Ys3:(list @ A)]: ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Ys3))))) @ ((transpose @ A) @ Xs))) = (((nth @ (list @ A)) @ Xs) @ I))))))). % 6.67/2.70 thf(fact_7695_rev__rev__ident, axiom, (! [A:$tType,Xs:(list @ A)]: ((((rev @ A) @ ((rev @ A) @ Xs)) = Xs)))). % 6.67/2.70 thf(fact_7696_rev__is__rev__conv, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((rev @ A) @ Xs) = ((rev @ A) @ Ys)) = (Xs = Ys))))). % 6.67/2.70 thf(fact_7697_rev__is__Nil__conv, axiom, (! [A:$tType,Xs:(list @ A)]: (((((rev @ A) @ Xs) = (nil @ A)) = (Xs = (nil @ A)))))). % 6.67/2.70 thf(fact_7698_Nil__is__rev__conv, axiom, (! [A:$tType,Xs:(list @ A)]: ((((nil @ A) = ((rev @ A) @ Xs)) = (Xs = (nil @ A)))))). % 6.67/2.70 thf(fact_7699_set__rev, axiom, (! [A:$tType,Xs:(list @ A)]: ((((set2 @ A) @ ((rev @ A) @ Xs)) = ((set2 @ A) @ Xs))))). % 6.67/2.70 thf(fact_7700_length__rev, axiom, (! [A:$tType,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ ((rev @ A) @ Xs)) = ((size_size @ (list @ A)) @ Xs))))). % 6.67/2.70 thf(fact_7701_rev__append, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: ((((rev @ A) @ (((append @ A) @ Xs) @ Ys)) = (((append @ A) @ ((rev @ A) @ Ys)) @ ((rev @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7702_distinct__rev, axiom, (! [A:$tType,Xs:(list @ A)]: ((((distinct @ A) @ ((rev @ A) @ Xs)) = ((distinct @ A) @ Xs))))). % 6.67/2.70 thf(fact_7703_rev__replicate, axiom, (! [A:$tType,N2:nat,X3:A]: ((((rev @ A) @ (((replicate @ A) @ N2) @ X3)) = (((replicate @ A) @ N2) @ X3))))). % 6.67/2.70 thf(fact_7704_remdups__adj__rev, axiom, (! [A:$tType,Xs:(list @ A)]: ((((remdups_adj @ A) @ ((rev @ A) @ Xs)) = ((rev @ A) @ ((remdups_adj @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7705_inj__on__rev, axiom, (! [A:$tType,A3:(set @ (list @ A))]: (((((inj_on @ (list @ A)) @ (list @ A)) @ (rev @ A)) @ A3)))). % 6.67/2.70 thf(fact_7706_singleton__rev__conv, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((((cons @ A) @ X3) @ (nil @ A)) = ((rev @ A) @ Xs)) = ((((cons @ A) @ X3) @ (nil @ A)) = Xs))))). % 6.67/2.70 thf(fact_7707_rev__singleton__conv, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: (((((rev @ A) @ Xs) = (((cons @ A) @ X3) @ (nil @ A))) = (Xs = (((cons @ A) @ X3) @ (nil @ A))))))). % 6.67/2.70 thf(fact_7708_rev__eq__Cons__iff, axiom, (! [A:$tType,Xs:(list @ A),Y:A,Ys:(list @ A)]: (((((rev @ A) @ Xs) = (((cons @ A) @ Y) @ Ys)) = (Xs = (((append @ A) @ ((rev @ A) @ Ys)) @ (((cons @ A) @ Y) @ (nil @ A)))))))). % 6.67/2.70 thf(fact_7709_foldl__conv__foldr, axiom, (! [B:$tType,A:$tType]: ((((foldl @ A) @ B) = (^ [F3:(A > (B > A)),A4:A,Xs2:(list @ B)]: ((((((foldr @ B) @ A) @ (^ [X:B,Y2:A]: (((F3 @ Y2) @ X)))) @ ((rev @ B) @ Xs2)) @ A4))))))). % 6.67/2.70 thf(fact_7710_foldr__conv__foldl, axiom, (! [A:$tType,B:$tType]: ((((foldr @ B) @ A) = (^ [F3:(B > (A > A)),Xs2:(list @ B),A4:A]: ((((((foldl @ A) @ B) @ (^ [X:A,Y2:B]: (((F3 @ Y2) @ X)))) @ A4) @ ((rev @ B) @ Xs2)))))))). % 6.67/2.70 thf(fact_7711_sorted__wrt__rev, axiom, (! [A:$tType,P:(A > (A > $o)),Xs:(list @ A)]: (((((sorted_wrt @ A) @ P) @ ((rev @ A) @ Xs)) = (((sorted_wrt @ A) @ (^ [X:A,Y2:A]: (((P @ Y2) @ X)))) @ Xs))))). % 6.67/2.70 thf(fact_7712_sorted__upto, axiom, (! [M:int,N2:int]: ((((sorted_wrt @ int) @ (ord_less_eq @ int)) @ ((upto @ M) @ N2))))). % 6.67/2.70 thf(fact_7713_sorted__wrt__upto, axiom, (! [I:int,J:int]: ((((sorted_wrt @ int) @ (ord_less @ int)) @ ((upto @ I) @ J))))). % 6.67/2.70 thf(fact_7714_rev_Osimps_I1_J, axiom, (! [A:$tType]: ((((rev @ A) @ (nil @ A)) = (nil @ A))))). % 6.67/2.70 thf(fact_7715_rev__swap, axiom, (! [A:$tType,Xs:(list @ A),Ys:(list @ A)]: (((((rev @ A) @ Xs) = Ys) = (Xs = ((rev @ A) @ Ys)))))). % 6.67/2.70 thf(fact_7716_rev_Osimps_I2_J, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((rev @ A) @ (((cons @ A) @ X3) @ Xs)) = (((append @ A) @ ((rev @ A) @ Xs)) @ (((cons @ A) @ X3) @ (nil @ A))))))). % 6.67/2.70 thf(fact_7717_zip__rev, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys:(list @ B)]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys)) => (((((zip @ A) @ B) @ ((rev @ A) @ Xs)) @ ((rev @ B) @ Ys)) = ((rev @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys))))))). % 6.67/2.70 thf(fact_7718_rev__filter, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((rev @ A) @ (((filter2 @ A) @ P) @ Xs)) = (((filter2 @ A) @ P) @ ((rev @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7719_rev__map, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xs:(list @ B)]: ((((rev @ A) @ ((((map @ B) @ A) @ F2) @ Xs)) = ((((map @ B) @ A) @ F2) @ ((rev @ B) @ Xs)))))). % 6.67/2.70 thf(fact_7720_rev__concat, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: ((((rev @ A) @ ((concat @ A) @ Xs)) = ((concat @ A) @ ((((map @ (list @ A)) @ (list @ A)) @ (rev @ A)) @ ((rev @ (list @ A)) @ Xs))))))). % 6.67/2.70 thf(fact_7721_take__rev, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((take @ A) @ N2) @ ((rev @ A) @ Xs)) = ((rev @ A) @ (((drop @ A) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N2)) @ Xs)))))). % 6.67/2.70 thf(fact_7722_rev__take, axiom, (! [A:$tType,I:nat,Xs:(list @ A)]: ((((rev @ A) @ (((take @ A) @ I) @ Xs)) = (((drop @ A) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ I)) @ ((rev @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7723_rev__drop, axiom, (! [A:$tType,I:nat,Xs:(list @ A)]: ((((rev @ A) @ (((drop @ A) @ I) @ Xs)) = (((take @ A) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ I)) @ ((rev @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7724_drop__rev, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((drop @ A) @ N2) @ ((rev @ A) @ Xs)) = ((rev @ A) @ (((take @ A) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N2)) @ Xs)))))). % 6.67/2.70 thf(fact_7725_rev__nth, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ ((rev @ A) @ Xs)) @ N2) = (((nth @ A) @ Xs) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (suc @ N2)))))))). % 6.67/2.70 thf(fact_7726_rev__update, axiom, (! [A:$tType,K:nat,Xs:(list @ A),Y:A]: (((((ord_less @ nat) @ K) @ ((size_size @ (list @ A)) @ Xs)) => (((rev @ A) @ ((((list_update @ A) @ Xs) @ K) @ Y)) = ((((list_update @ A) @ ((rev @ A) @ Xs)) @ (((minus_minus @ nat) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ K)) @ (one_one @ nat))) @ Y)))))). % 6.67/2.70 thf(fact_7727_sorted__transpose, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: ((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((rev @ nat) @ ((((map @ (list @ A)) @ nat) @ (size_size @ (list @ A))) @ ((transpose @ A) @ Xs))))))). % 6.67/2.70 thf(fact_7728_sorted__rev__iff__nth__Suc, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((rev @ A) @ Xs)) = (! [I3:nat]: (((((ord_less @ nat) @ (suc @ I3)) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ (suc @ I3))) @ (((nth @ A) @ Xs) @ I3)))))))))))). % 6.67/2.70 thf(fact_7729_sorted__rev__nth__mono, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),I:nat,J:nat]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((rev @ A) @ Xs)) => ((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ J)) @ (((nth @ A) @ Xs) @ I))))))))))). % 6.67/2.70 thf(fact_7730_sorted__rev__iff__nth__mono, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((rev @ A) @ Xs)) = (! [I3:nat,J3:nat]: (((((ord_less_eq @ nat) @ I3) @ J3) => ((((ord_less @ nat) @ J3) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ A) @ (((nth @ A) @ Xs) @ J3)) @ (((nth @ A) @ Xs) @ I3))))))))))))). % 6.67/2.70 thf(fact_7731_foldr__max__sorted, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Y:A]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((rev @ A) @ Xs)) => (((Xs = (nil @ A)) => ((((((foldr @ A) @ A) @ (ord_max @ A)) @ Xs) @ Y) = Y)) & ((Xs != (nil @ A)) => ((((((foldr @ A) @ A) @ (ord_max @ A)) @ Xs) @ Y) = (((ord_max @ A) @ (((nth @ A) @ Xs) @ (zero_zero @ nat))) @ Y))))))))))). % 6.67/2.70 thf(fact_7732_length__transpose__sorted, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: (((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((rev @ nat) @ ((((map @ (list @ A)) @ nat) @ (size_size @ (list @ A))) @ Xs))) => (((Xs = (nil @ (list @ A))) => (((size_size @ (list @ (list @ A))) @ ((transpose @ A) @ Xs)) = (zero_zero @ nat))) & ((Xs != (nil @ (list @ A))) => (((size_size @ (list @ (list @ A))) @ ((transpose @ A) @ Xs)) = ((size_size @ (list @ A)) @ (((nth @ (list @ A)) @ Xs) @ (zero_zero @ nat)))))))))). % 6.67/2.70 thf(fact_7733_transpose__column__length, axiom, (! [A:$tType,Xs:(list @ (list @ A)),I:nat]: (((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((rev @ nat) @ ((((map @ (list @ A)) @ nat) @ (size_size @ (list @ A))) @ Xs))) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ (list @ A))) @ Xs)) => (((size_size @ (list @ (list @ A))) @ (((filter2 @ (list @ A)) @ (^ [Ys3:(list @ A)]: ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Ys3))))) @ ((transpose @ A) @ Xs))) = ((size_size @ (list @ A)) @ (((nth @ (list @ A)) @ Xs) @ I)))))))). % 6.67/2.70 thf(fact_7734_folding__insort__key_Ofinite__set__strict__sorted, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A),A3:(set @ B)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((ord_less_eq @ (set @ B)) @ A3) @ S3) => (((finite_finite2 @ B) @ A3) => (~ ((! [L4:(list @ B)]: (((((sorted_wrt @ A) @ Less) @ ((((map @ B) @ A) @ F2) @ L4)) => ((((set2 @ B) @ L4) = A3) => (((size_size @ (list @ B)) @ L4) != ((finite_card @ B) @ A3)))))))))))))). % 6.67/2.70 thf(fact_7735_transpose__transpose, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: (((((sorted_wrt @ nat) @ (ord_less_eq @ nat)) @ ((rev @ nat) @ ((((map @ (list @ A)) @ nat) @ (size_size @ (list @ A))) @ Xs))) => (((transpose @ A) @ ((transpose @ A) @ Xs)) = (((takeWhile @ (list @ A)) @ (^ [X:(list @ A)]: ((X != (nil @ A))))) @ Xs)))))). % 6.67/2.70 thf(fact_7736_takeWhile__idem, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: (((((takeWhile @ A) @ P) @ (((takeWhile @ A) @ P) @ Xs)) = (((takeWhile @ A) @ P) @ Xs))))). % 6.67/2.70 thf(fact_7737_takeWhile__eq__all__conv, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((((takeWhile @ A) @ P) @ Xs) = Xs) = (! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (P @ X)))))))). % 6.67/2.70 thf(fact_7738_takeWhile__append1, axiom, (! [A:$tType,X3:A,Xs:(list @ A),P:(A > $o),Ys:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => ((~ ((P @ X3))) => ((((takeWhile @ A) @ P) @ (((append @ A) @ Xs) @ Ys)) = (((takeWhile @ A) @ P) @ Xs))))))). % 6.67/2.70 thf(fact_7739_takeWhile__append2, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o),Ys:(list @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (P @ X5)))) => ((((takeWhile @ A) @ P) @ (((append @ A) @ Xs) @ Ys)) = (((append @ A) @ Xs) @ (((takeWhile @ A) @ P) @ Ys))))))). % 6.67/2.70 thf(fact_7740_takeWhile__replicate, axiom, (! [A:$tType,P:(A > $o),X3:A,N2:nat]: ((((P @ X3) => ((((takeWhile @ A) @ P) @ (((replicate @ A) @ N2) @ X3)) = (((replicate @ A) @ N2) @ X3))) & ((~ ((P @ X3))) => ((((takeWhile @ A) @ P) @ (((replicate @ A) @ N2) @ X3)) = (nil @ A))))))). % 6.67/2.70 thf(fact_7741_length__concat__rev, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: ((((size_size @ (list @ A)) @ ((concat @ A) @ ((rev @ (list @ A)) @ Xs))) = ((size_size @ (list @ A)) @ ((concat @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7742_sorted__takeWhile, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),P:(A > $o)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((takeWhile @ A) @ P) @ Xs))))))))). % 6.67/2.70 thf(fact_7743_nth__length__takeWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ Xs))) @ ((size_size @ (list @ A)) @ Xs)) => (~ ((P @ (((nth @ A) @ Xs) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ Xs)))))))))). % 6.67/2.70 thf(fact_7744_takeWhile__nth, axiom, (! [A:$tType,J:nat,P:(A > $o),Xs:(list @ A)]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ Xs))) => ((((nth @ A) @ (((takeWhile @ A) @ P) @ Xs)) @ J) = (((nth @ A) @ Xs) @ J)))))). % 6.67/2.70 thf(fact_7745_takeWhile__map, axiom, (! [A:$tType,B:$tType,P:(A > $o),F2:(B > A),Xs:(list @ B)]: (((((takeWhile @ A) @ P) @ ((((map @ B) @ A) @ F2) @ Xs)) = ((((map @ B) @ A) @ F2) @ (((takeWhile @ B) @ (((((comp @ A) @ $o) @ B) @ P) @ F2)) @ Xs)))))). % 6.67/2.70 thf(fact_7746_folding__insort__key_Odistinct__if__distinct__map, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A),Xs:(list @ B)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => (((distinct @ A) @ ((((map @ B) @ A) @ F2) @ Xs)) => ((distinct @ B) @ Xs)))))). % 6.67/2.70 thf(fact_7747_zip__takeWhile__snd, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),P:(B > $o),Ys:(list @ B)]: ((((((zip @ A) @ B) @ Xs) @ (((takeWhile @ B) @ P) @ Ys)) = (((takeWhile @ ((product_prod @ A) @ B)) @ (((((comp @ B) @ $o) @ ((product_prod @ A) @ B)) @ P) @ ((product_snd @ A) @ B))) @ ((((zip @ A) @ B) @ Xs) @ Ys)))))). % 6.67/2.70 thf(fact_7748_zip__takeWhile__fst, axiom, (! [A:$tType,B:$tType,P:(A > $o),Xs:(list @ A),Ys:(list @ B)]: ((((((zip @ A) @ B) @ (((takeWhile @ A) @ P) @ Xs)) @ Ys) = (((takeWhile @ ((product_prod @ A) @ B)) @ (((((comp @ A) @ $o) @ ((product_prod @ A) @ B)) @ P) @ ((product_fst @ A) @ B))) @ ((((zip @ A) @ B) @ Xs) @ Ys)))))). % 6.67/2.70 thf(fact_7749_takeWhile__tail, axiom, (! [A:$tType,P:(A > $o),X3:A,Xs:(list @ A),L:(list @ A)]: (((~ ((P @ X3))) => ((((takeWhile @ A) @ P) @ (((append @ A) @ Xs) @ (((cons @ A) @ X3) @ L))) = (((takeWhile @ A) @ P) @ Xs)))))). % 6.67/2.70 thf(fact_7750_folding__insort__key_Oinj__on, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((inj_on @ B) @ A) @ F2) @ S3))))). % 6.67/2.70 thf(fact_7751_takeWhile_Osimps_I2_J, axiom, (! [A:$tType,P:(A > $o),X3:A,Xs:(list @ A)]: ((((P @ X3) => ((((takeWhile @ A) @ P) @ (((cons @ A) @ X3) @ Xs)) = (((cons @ A) @ X3) @ (((takeWhile @ A) @ P) @ Xs)))) & ((~ ((P @ X3))) => ((((takeWhile @ A) @ P) @ (((cons @ A) @ X3) @ Xs)) = (nil @ A))))))). % 6.67/2.70 thf(fact_7752_takeWhile_Osimps_I1_J, axiom, (! [A:$tType,P:(A > $o)]: (((((takeWhile @ A) @ P) @ (nil @ A)) = (nil @ A))))). % 6.67/2.70 thf(fact_7753_set__takeWhileD, axiom, (! [A:$tType,X3:A,P:(A > $o),Xs:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ (((takeWhile @ A) @ P) @ Xs))) => ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) & (P @ X3)))))). % 6.67/2.70 thf(fact_7754_takeWhile__cong, axiom, (! [A:$tType,L:(list @ A),K:(list @ A),P:(A > $o),Q:(A > $o)]: (((L = K) => ((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ L)) => ((P @ X5) = (Q @ X5))))) => ((((takeWhile @ A) @ P) @ L) = (((takeWhile @ A) @ Q) @ K))))))). % 6.67/2.70 thf(fact_7755_takeWhile__eq__take, axiom, (! [A:$tType]: (((takeWhile @ A) = (^ [P3:(A > $o),Xs2:(list @ A)]: ((((take @ A) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P3) @ Xs2))) @ Xs2))))))). % 6.67/2.70 thf(fact_7756_distinct__takeWhile, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: ((((distinct @ A) @ Xs) => ((distinct @ A) @ (((takeWhile @ A) @ P) @ Xs)))))). % 6.67/2.70 thf(fact_7757_length__takeWhile__le, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))). % 6.67/2.70 thf(fact_7758_takeWhile__eq__Nil__iff, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((((takeWhile @ A) @ P) @ Xs) = (nil @ A)) = ((Xs = (nil @ A)) | (~ ((P @ ((hd @ A) @ Xs))))))))). % 6.67/2.70 thf(fact_7759_takeWhile__append, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o),Ys:(list @ A)]: ((((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (P @ X5)))) => ((((takeWhile @ A) @ P) @ (((append @ A) @ Xs) @ Ys)) = (((append @ A) @ Xs) @ (((takeWhile @ A) @ P) @ Ys)))) & ((~ ((! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => (P @ X2)))))) => ((((takeWhile @ A) @ P) @ (((append @ A) @ Xs) @ Ys)) = (((takeWhile @ A) @ P) @ Xs))))))). % 6.67/2.70 thf(fact_7760_length__takeWhile__less__P__nth, axiom, (! [A:$tType,J:nat,P:(A > $o),Xs:(list @ A)]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ J) => (P @ (((nth @ A) @ Xs) @ I4))))) => ((((ord_less_eq @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((ord_less_eq @ nat) @ J) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ Xs)))))))). % 6.67/2.70 thf(fact_7761_takeWhile__eq__take__P__nth, axiom, (! [A:$tType,N2:nat,Xs:(list @ A),P:(A > $o)]: (((! [I4:nat]: (((((ord_less @ nat) @ I4) @ N2) => ((((ord_less @ nat) @ I4) @ ((size_size @ (list @ A)) @ Xs)) => (P @ (((nth @ A) @ Xs) @ I4)))))) => (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => (~ ((P @ (((nth @ A) @ Xs) @ N2))))) => ((((takeWhile @ A) @ P) @ Xs) = (((take @ A) @ N2) @ Xs))))))). % 6.67/2.70 thf(fact_7762_sorted__list__of__set_Ofolding__insort__key__axioms, axiom, (! [A:$tType]: (((linorder @ A) => ((((((folding_insort_key @ A) @ A) @ (ord_less_eq @ A)) @ (ord_less @ A)) @ (top_top @ (set @ A))) @ (^ [X:A]: (X))))))). % 6.67/2.70 thf(fact_7763_filter__equals__takeWhile__sorted__rev, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F2:(B > A),Xs:(list @ B),T2:A]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((rev @ A) @ ((((map @ B) @ A) @ F2) @ Xs))) => ((((filter2 @ B) @ (^ [X:B]: ((((ord_less @ A) @ T2) @ (F2 @ X))))) @ Xs) = (((takeWhile @ B) @ (^ [X:B]: ((((ord_less @ A) @ T2) @ (F2 @ X))))) @ Xs))))))))). % 6.67/2.70 thf(fact_7764_folding__insort__key_Osorted__key__list__of__set__unique, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A),A3:(set @ B),L:(list @ B)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((ord_less_eq @ (set @ B)) @ A3) @ S3) => (((finite_finite2 @ B) @ A3) => (((((sorted_wrt @ A) @ Less) @ ((((map @ B) @ A) @ F2) @ L)) & ((((set2 @ B) @ L) = A3) & (((size_size @ (list @ B)) @ L) = ((finite_card @ B) @ A3)))) = ((((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ A3) = L)))))))). % 6.67/2.70 thf(fact_7765_folding__insort__key_Osorted__key__list__of__set__remove, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A),X3:B,A3:(set @ B)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((ord_less_eq @ (set @ B)) @ (((insert @ B) @ X3) @ A3)) @ S3) => (((finite_finite2 @ B) @ A3) => ((((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ X3) @ (bot_bot @ (set @ B))))) = (((remove1 @ B) @ X3) @ (((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ A3))))))))). % 6.67/2.70 thf(fact_7766_linorder_Osorted__key__list__of__set_Ocong, axiom, (! [B:$tType,A:$tType]: ((((sorted8670434370408473282of_set @ A) @ B) = ((sorted8670434370408473282of_set @ A) @ B))))). % 6.67/2.70 thf(fact_7767_folding__insort__key_Osorted__key__list__of__set__inject, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A),A3:(set @ B),B3:(set @ B)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((ord_less_eq @ (set @ B)) @ A3) @ S3) => ((((ord_less_eq @ (set @ B)) @ B3) @ S3) => (((((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ A3) = (((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ B3)) => (((finite_finite2 @ B) @ A3) => (((finite_finite2 @ B) @ B3) => (A3 = B3)))))))))). % 6.67/2.70 thf(fact_7768_folding__insort__key_Osorted__key__list__of__set__empty, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ (bot_bot @ (set @ B))) = (nil @ B)))))). % 6.67/2.70 thf(fact_7769_folding__insort__key_Oset__sorted__key__list__of__set, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A),A3:(set @ B)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((ord_less_eq @ (set @ B)) @ A3) @ S3) => (((finite_finite2 @ B) @ A3) => (((set2 @ B) @ (((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ A3)) = A3))))))). % 6.67/2.70 thf(fact_7770_folding__insort__key_Olength__sorted__key__list__of__set, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A),A3:(set @ B)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((ord_less_eq @ (set @ B)) @ A3) @ S3) => (((size_size @ (list @ B)) @ (((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ A3)) = ((finite_card @ B) @ A3))))))). % 6.67/2.70 thf(fact_7771_folding__insort__key_Odistinct__sorted__key__list__of__set, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A),A3:(set @ B)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((ord_less_eq @ (set @ B)) @ A3) @ S3) => ((distinct @ A) @ ((((map @ B) @ A) @ F2) @ (((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ A3)))))))). % 6.67/2.70 thf(fact_7772_folding__insort__key_Ostrict__sorted__key__list__of__set, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A),A3:(set @ B)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((ord_less_eq @ (set @ B)) @ A3) @ S3) => (((sorted_wrt @ A) @ Less) @ ((((map @ B) @ A) @ F2) @ (((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ A3)))))))). % 6.67/2.70 thf(fact_7773_folding__insort__key_Osorted__sorted__key__list__of__set, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A),A3:(set @ B)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((ord_less_eq @ (set @ B)) @ A3) @ S3) => (((sorted_wrt @ A) @ Less_eq2) @ ((((map @ B) @ A) @ F2) @ (((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ A3)))))))). % 6.67/2.70 thf(fact_7774_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A),A3:(set @ B)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((ord_less_eq @ (set @ B)) @ A3) @ S3) => (((finite_finite2 @ B) @ A3) => (((((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ A3) = (nil @ B)) = (A3 = (bot_bot @ (set @ B)))))))))). % 6.67/2.70 thf(fact_7775_folding__insort__key_Oidem__if__sorted__distinct, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A),Xs:(list @ B)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((ord_less_eq @ (set @ B)) @ ((set2 @ B) @ Xs)) @ S3) => ((((sorted_wrt @ A) @ Less_eq2) @ ((((map @ B) @ A) @ F2) @ Xs)) => (((distinct @ B) @ Xs) => ((((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ ((set2 @ B) @ Xs)) = Xs)))))))). % 6.67/2.70 thf(fact_7776_folding__insort__key_Osorted__key__list__of__set__insert__remove, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A),X3:B,A3:(set @ B)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((ord_less_eq @ (set @ B)) @ (((insert @ B) @ X3) @ A3)) @ S3) => (((finite_finite2 @ B) @ A3) => ((((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ (((insert @ B) @ X3) @ A3)) = ((((((insort_key @ A) @ B) @ Less_eq2) @ F2) @ X3) @ (((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ (((minus_minus @ (set @ B)) @ A3) @ (((insert @ B) @ X3) @ (bot_bot @ (set @ B))))))))))))). % 6.67/2.70 thf(fact_7777_folding__insort__key_Osorted__key__list__of__set__insert, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A),X3:B,A3:(set @ B)]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((ord_less_eq @ (set @ B)) @ (((insert @ B) @ X3) @ A3)) @ S3) => (((finite_finite2 @ B) @ A3) => ((~ ((((member @ B) @ X3) @ A3))) => ((((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ (((insert @ B) @ X3) @ A3)) = ((((((insort_key @ A) @ B) @ Less_eq2) @ F2) @ X3) @ (((((sorted8670434370408473282of_set @ A) @ B) @ Less_eq2) @ F2) @ A3)))))))))). % 6.67/2.70 thf(fact_7778_linorder_Oinsort__key_Ocong, axiom, (! [B:$tType,A:$tType]: ((((insort_key @ A) @ B) = ((insort_key @ A) @ B))))). % 6.67/2.70 thf(fact_7779_folding__insort__key_Oinsort__key__commute, axiom, (! [A:$tType,B:$tType,Less_eq2:(A > (A > $o)),Less:(A > (A > $o)),S3:(set @ B),F2:(B > A),X3:B,Y:B]: ((((((((folding_insort_key @ A) @ B) @ Less_eq2) @ Less) @ S3) @ F2) => ((((member @ B) @ X3) @ S3) => ((((member @ B) @ Y) @ S3) => ((((((comp @ (list @ B)) @ (list @ B)) @ (list @ B)) @ (((((insort_key @ A) @ B) @ Less_eq2) @ F2) @ Y)) @ (((((insort_key @ A) @ B) @ Less_eq2) @ F2) @ X3)) = (((((comp @ (list @ B)) @ (list @ B)) @ (list @ B)) @ (((((insort_key @ A) @ B) @ Less_eq2) @ F2) @ X3)) @ (((((insort_key @ A) @ B) @ Less_eq2) @ F2) @ Y))))))))). % 6.67/2.70 thf(fact_7780_extract__def, axiom, (! [A:$tType]: (((extract @ A) = (^ [P3:(A > $o),Xs2:(list @ A)]: ((((((case_list @ (option @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))))) @ A) @ (none @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))))) @ (^ [Y2:A,Ys3:(list @ A)]: (((some @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ A) @ (list @ A))) @ (((takeWhile @ A) @ (((((comp @ $o) @ $o) @ A) @ (~)) @ P3)) @ Xs2)) @ ((((product_Pair @ A) @ (list @ A)) @ Y2) @ Ys3)))))) @ (((dropWhile @ A) @ (((((comp @ $o) @ $o) @ A) @ (~)) @ P3)) @ Xs2)))))))). % 6.67/2.70 thf(fact_7781_lenlex__append2, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),Us:(list @ A),Xs:(list @ A),Ys:(list @ A)]: ((((irrefl @ A) @ R) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Us) @ Xs)) @ (((append @ A) @ Us) @ Ys))) @ ((lenlex @ A) @ R)) = (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ ((lenlex @ A) @ R))))))). % 6.67/2.70 thf(fact_7782_dropWhile__idem, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: (((((dropWhile @ A) @ P) @ (((dropWhile @ A) @ P) @ Xs)) = (((dropWhile @ A) @ P) @ Xs))))). % 6.67/2.70 thf(fact_7783_dropWhile__eq__Nil__conv, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((((dropWhile @ A) @ P) @ Xs) = (nil @ A)) = (! [X:A]: (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (P @ X)))))))). % 6.67/2.70 thf(fact_7784_dropWhile__append2, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o),Ys:(list @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (P @ X5)))) => ((((dropWhile @ A) @ P) @ (((append @ A) @ Xs) @ Ys)) = (((dropWhile @ A) @ P) @ Ys)))))). % 6.67/2.70 thf(fact_7785_dropWhile__append1, axiom, (! [A:$tType,X3:A,Xs:(list @ A),P:(A > $o),Ys:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => ((~ ((P @ X3))) => ((((dropWhile @ A) @ P) @ (((append @ A) @ Xs) @ Ys)) = (((append @ A) @ (((dropWhile @ A) @ P) @ Xs)) @ Ys))))))). % 6.67/2.70 thf(fact_7786_dropWhile__replicate, axiom, (! [A:$tType,P:(A > $o),X3:A,N2:nat]: ((((P @ X3) => ((((dropWhile @ A) @ P) @ (((replicate @ A) @ N2) @ X3)) = (nil @ A))) & ((~ ((P @ X3))) => ((((dropWhile @ A) @ P) @ (((replicate @ A) @ N2) @ X3)) = (((replicate @ A) @ N2) @ X3))))))). % 6.67/2.70 thf(fact_7787_takeWhile__dropWhile__id, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: (((((append @ A) @ (((takeWhile @ A) @ P) @ Xs)) @ (((dropWhile @ A) @ P) @ Xs)) = Xs)))). % 6.67/2.70 thf(fact_7788_lexord__same__pref__if__irrefl, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),Xs:(list @ A),Ys:(list @ A),Zs2:(list @ A)]: ((((irrefl @ A) @ R2) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Ys)) @ (((append @ A) @ Xs) @ Zs2))) @ ((lexord @ A) @ R2)) = (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ys) @ Zs2)) @ ((lexord @ A) @ R2))))))). % 6.67/2.70 thf(fact_7789_sorted__dropWhile, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),P:(A > $o)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((dropWhile @ A) @ P) @ Xs))))))))). % 6.67/2.70 thf(fact_7790_hd__dropWhile, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((((dropWhile @ A) @ P) @ Xs) != (nil @ A)) => (~ ((P @ ((hd @ A) @ (((dropWhile @ A) @ P) @ Xs))))))))). % 6.67/2.70 thf(fact_7791_dropWhile__eq__self__iff, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((((dropWhile @ A) @ P) @ Xs) = Xs) = ((Xs = (nil @ A)) | (~ ((P @ ((hd @ A) @ Xs))))))))). % 6.67/2.70 thf(fact_7792_length__dropWhile__le, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ (((dropWhile @ A) @ P) @ Xs))) @ ((size_size @ (list @ A)) @ Xs))))). % 6.67/2.70 thf(fact_7793_distinct__dropWhile, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: ((((distinct @ A) @ Xs) => ((distinct @ A) @ (((dropWhile @ A) @ P) @ Xs)))))). % 6.67/2.70 thf(fact_7794_set__dropWhileD, axiom, (! [A:$tType,X3:A,P:(A > $o),Xs:(list @ A)]: (((((member @ A) @ X3) @ ((set2 @ A) @ (((dropWhile @ A) @ P) @ Xs))) => (((member @ A) @ X3) @ ((set2 @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7795_dropWhile__cong, axiom, (! [A:$tType,L:(list @ A),K:(list @ A),P:(A > $o),Q:(A > $o)]: (((L = K) => ((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ L)) => ((P @ X5) = (Q @ X5))))) => ((((dropWhile @ A) @ P) @ L) = (((dropWhile @ A) @ Q) @ K))))))). % 6.67/2.70 thf(fact_7796_irreflI, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((! [A5:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ A5)) @ R))))) => ((irrefl @ A) @ R))))). % 6.67/2.70 thf(fact_7797_irrefl__def, axiom, (! [A:$tType]: (((irrefl @ A) = (^ [R5:(set @ ((product_prod @ A) @ A))]: ((! [A4:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ A4)) @ R5))))))))))). % 6.67/2.70 thf(fact_7798_dropWhile_Osimps_I1_J, axiom, (! [A:$tType,P:(A > $o)]: (((((dropWhile @ A) @ P) @ (nil @ A)) = (nil @ A))))). % 6.67/2.70 thf(fact_7799_dropWhile_Osimps_I2_J, axiom, (! [A:$tType,P:(A > $o),X3:A,Xs:(list @ A)]: ((((P @ X3) => ((((dropWhile @ A) @ P) @ (((cons @ A) @ X3) @ Xs)) = (((dropWhile @ A) @ P) @ Xs))) & ((~ ((P @ X3))) => ((((dropWhile @ A) @ P) @ (((cons @ A) @ X3) @ Xs)) = (((cons @ A) @ X3) @ Xs))))))). % 6.67/2.70 thf(fact_7800_remdups__adj__Cons_H, axiom, (! [A:$tType,X3:A,Xs:(list @ A)]: ((((remdups_adj @ A) @ (((cons @ A) @ X3) @ Xs)) = (((cons @ A) @ X3) @ ((remdups_adj @ A) @ (((dropWhile @ A) @ (^ [Y2:A]: ((Y2 = X3)))) @ Xs))))))). % 6.67/2.70 thf(fact_7801_irrefl__lex, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((irrefl @ A) @ R2) => ((irrefl @ (list @ A)) @ ((lex @ A) @ R2)))))). % 6.67/2.70 thf(fact_7802_lexord__irrefl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((irrefl @ A) @ R) => ((irrefl @ (list @ A)) @ ((lexord @ A) @ R)))))). % 6.67/2.70 thf(fact_7803_dropWhile__append3, axiom, (! [A:$tType,P:(A > $o),Y:A,Xs:(list @ A),Ys:(list @ A)]: (((~ ((P @ Y))) => ((((dropWhile @ A) @ P) @ (((append @ A) @ Xs) @ (((cons @ A) @ Y) @ Ys))) = (((append @ A) @ (((dropWhile @ A) @ P) @ Xs)) @ (((cons @ A) @ Y) @ Ys))))))). % 6.67/2.70 thf(fact_7804_dropWhile__map, axiom, (! [A:$tType,B:$tType,P:(A > $o),F2:(B > A),Xs:(list @ B)]: (((((dropWhile @ A) @ P) @ ((((map @ B) @ A) @ F2) @ Xs)) = ((((map @ B) @ A) @ F2) @ (((dropWhile @ B) @ (((((comp @ A) @ $o) @ B) @ P) @ F2)) @ Xs)))))). % 6.67/2.70 thf(fact_7805_dropWhile__eq__Cons__conv, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),Y:A,Ys:(list @ A)]: ((((((dropWhile @ A) @ P) @ Xs) = (((cons @ A) @ Y) @ Ys)) = ((Xs = (((append @ A) @ (((takeWhile @ A) @ P) @ Xs)) @ (((cons @ A) @ Y) @ Ys))) & (~ ((P @ Y)))))))). % 6.67/2.70 thf(fact_7806_takeWhile__eq__filter, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ (((dropWhile @ A) @ P) @ Xs))) => (~ ((P @ X5)))))) => ((((takeWhile @ A) @ P) @ Xs) = (((filter2 @ A) @ P) @ Xs)))))). % 6.67/2.70 thf(fact_7807_dropWhile__eq__drop, axiom, (! [A:$tType]: (((dropWhile @ A) = (^ [P3:(A > $o),Xs2:(list @ A)]: ((((drop @ A) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P3) @ Xs2))) @ Xs2))))))). % 6.67/2.70 thf(fact_7808_dropWhile__append, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o),Ys:(list @ A)]: ((((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (P @ X5)))) => ((((dropWhile @ A) @ P) @ (((append @ A) @ Xs) @ Ys)) = (((dropWhile @ A) @ P) @ Ys))) & ((~ ((! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => (P @ X2)))))) => ((((dropWhile @ A) @ P) @ (((append @ A) @ Xs) @ Ys)) = (((append @ A) @ (((dropWhile @ A) @ P) @ Xs)) @ Ys))))))). % 6.67/2.70 thf(fact_7809_lexl__not__refl, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),X3:(list @ A)]: ((((irrefl @ A) @ R2) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X3) @ X3)) @ ((lex @ A) @ R2)))))))). % 6.67/2.70 thf(fact_7810_find__dropWhile, axiom, (! [A:$tType]: (((find @ A) = (^ [P3:(A > $o),Xs2:(list @ A)]: ((((((case_list @ (option @ A)) @ A) @ (none @ A)) @ (^ [X:A,Xa4:(list @ A)]: (((some @ A) @ X)))) @ (((dropWhile @ A) @ (((((comp @ $o) @ $o) @ A) @ (~)) @ P3)) @ Xs2)))))))). % 6.67/2.70 thf(fact_7811_remdups__adj__append__dropWhile, axiom, (! [A:$tType,Xs:(list @ A),Y:A,Ys:(list @ A)]: ((((remdups_adj @ A) @ (((append @ A) @ Xs) @ (((cons @ A) @ Y) @ Ys))) = (((append @ A) @ ((remdups_adj @ A) @ (((append @ A) @ Xs) @ (((cons @ A) @ Y) @ (nil @ A))))) @ ((remdups_adj @ A) @ (((dropWhile @ A) @ (^ [X:A]: ((X = Y)))) @ Ys))))))). % 6.67/2.70 thf(fact_7812_tl__remdups__adj, axiom, (! [A:$tType,Ys:(list @ A)]: (((Ys != (nil @ A)) => (((tl @ A) @ ((remdups_adj @ A) @ Ys)) = ((remdups_adj @ A) @ (((dropWhile @ A) @ (^ [X:A]: ((X = ((hd @ A) @ Ys))))) @ ((tl @ A) @ Ys)))))))). % 6.67/2.70 thf(fact_7813_dropWhile__nth, axiom, (! [A:$tType,J:nat,P:(A > $o),Xs:(list @ A)]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ (((dropWhile @ A) @ P) @ Xs))) => ((((nth @ A) @ (((dropWhile @ A) @ P) @ Xs)) @ J) = (((nth @ A) @ Xs) @ (((plus_plus @ nat) @ J) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ Xs))))))))). % 6.67/2.70 thf(fact_7814_dropWhile__neq__rev, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((distinct @ A) @ Xs) => ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => ((((dropWhile @ A) @ (^ [Y2:A]: ((Y2 != X3)))) @ ((rev @ A) @ Xs)) = (((cons @ A) @ X3) @ ((rev @ A) @ (((takeWhile @ A) @ (^ [Y2:A]: ((Y2 != X3)))) @ Xs))))))))). % 6.67/2.70 thf(fact_7815_takeWhile__neq__rev, axiom, (! [A:$tType,Xs:(list @ A),X3:A]: ((((distinct @ A) @ Xs) => ((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => ((((takeWhile @ A) @ (^ [Y2:A]: ((Y2 != X3)))) @ ((rev @ A) @ Xs)) = ((rev @ A) @ ((tl @ A) @ (((dropWhile @ A) @ (^ [Y2:A]: ((Y2 != X3)))) @ Xs))))))))). % 6.67/2.70 thf(fact_7816_partition__filter__conv, axiom, (! [A:$tType]: (((partition @ A) = (^ [F3:(A > $o),Xs2:(list @ A)]: (((((product_Pair @ (list @ A)) @ (list @ A)) @ (((filter2 @ A) @ F3) @ Xs2)) @ (((filter2 @ A) @ (((((comp @ $o) @ $o) @ A) @ (~)) @ F3)) @ Xs2)))))))). % 6.67/2.70 thf(fact_7817_min__list_Oelims, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:(list @ A),Y:A]: (((((min_list @ A) @ X3) = Y) => ((! [X5:A,Xs3:(list @ A)]: (((X3 = (((cons @ A) @ X5) @ Xs3)) => (Y != (((((case_list @ A) @ A) @ X5) @ (^ [A4:A,List2:(list @ A)]: ((((ord_min @ A) @ X5) @ ((min_list @ A) @ Xs3))))) @ Xs3))))) => (~ (((X3 = (nil @ A)) => (Y != (undefined @ A))))))))))))). % 6.67/2.70 thf(fact_7818_hd__def, axiom, (! [A:$tType]: (((hd @ A) = ((((case_list @ A) @ A) @ (undefined @ A)) @ (^ [X213:A,X224:(list @ A)]: (X213))))))). % 6.67/2.70 thf(fact_7819_option_Othe__def, axiom, (! [A:$tType]: (((the2 @ A) = ((((case_option @ A) @ A) @ (undefined @ A)) @ (^ [X24:A]: (X24))))))). % 6.67/2.70 thf(fact_7820_partition__filter1, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: (((((product_fst @ (list @ A)) @ (list @ A)) @ (((partition @ A) @ P) @ Xs)) = (((filter2 @ A) @ P) @ Xs))))). % 6.67/2.70 thf(fact_7821_partition_Osimps_I1_J, axiom, (! [A:$tType,P:(A > $o)]: (((((partition @ A) @ P) @ (nil @ A)) = ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A)))))). % 6.67/2.70 thf(fact_7822_partition__P, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),Yes:(list @ A),No4:(list @ A)]: ((((((partition @ A) @ P) @ Xs) = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes) @ No4)) => ((! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Yes)) => (P @ X2)))) & (! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ No4)) => (~ ((P @ X2))))))))))). % 6.67/2.70 thf(fact_7823_partition_Osimps_I2_J, axiom, (! [A:$tType,P:(A > $o),X3:A,Xs:(list @ A)]: (((((partition @ A) @ P) @ (((cons @ A) @ X3) @ Xs)) = (((((product_case_prod @ (list @ A)) @ (list @ A)) @ ((product_prod @ (list @ A)) @ (list @ A))) @ (^ [Yes2:(list @ A),No3:(list @ A)]: (((((if @ ((product_prod @ (list @ A)) @ (list @ A))) @ (P @ X3)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X3) @ Yes2)) @ No3)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes2) @ (((cons @ A) @ X3) @ No3)))))) @ (((partition @ A) @ P) @ Xs)))))). % 6.67/2.70 thf(fact_7824_partition__filter2, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: (((((product_snd @ (list @ A)) @ (list @ A)) @ (((partition @ A) @ P) @ Xs)) = (((filter2 @ A) @ (((((comp @ $o) @ $o) @ A) @ (~)) @ P)) @ Xs))))). % 6.67/2.70 thf(fact_7825_Func__empty, axiom, (! [B:$tType,A:$tType,B3:(set @ B)]: ((((((bNF_Wellorder_Func @ A) @ B) @ (bot_bot @ (set @ A))) @ B3) = (((insert @ (A > B)) @ (^ [X:A]: ((undefined @ B)))) @ (bot_bot @ (set @ (A > B)))))))). % 6.67/2.70 thf(fact_7826_partition__set, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),Yes:(list @ A),No4:(list @ A)]: ((((((partition @ A) @ P) @ Xs) = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes) @ No4)) => ((((sup_sup @ (set @ A)) @ ((set2 @ A) @ Yes)) @ ((set2 @ A) @ No4)) = ((set2 @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7827_arg__min__list_Oelims, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [X3:(A > B),Xa2:(list @ A),Y:A]: (((((((arg_min_list @ A) @ B) @ X3) @ Xa2) = Y) => ((! [X5:A]: (((Xa2 = (((cons @ A) @ X5) @ (nil @ A))) => (Y != X5)))) => ((! [X5:A,Y4:A,Zs:(list @ A)]: (((Xa2 = (((cons @ A) @ X5) @ (((cons @ A) @ Y4) @ Zs))) => (Y != ((((if @ A) @ (((ord_less_eq @ B) @ (X3 @ X5)) @ (X3 @ ((((arg_min_list @ A) @ B) @ X3) @ (((cons @ A) @ Y4) @ Zs))))) @ X5) @ ((((arg_min_list @ A) @ B) @ X3) @ (((cons @ A) @ Y4) @ Zs))))))) => (~ (((Xa2 = (nil @ A)) => (Y != (undefined @ A)))))))))))))). % 6.67/2.70 thf(fact_7828_min__list_Opelims, axiom, (! [A:$tType]: (((ord @ A) => (! [X3:(list @ A),Y:A]: (((((min_list @ A) @ X3) = Y) => ((((accp @ (list @ A)) @ (min_list_rel @ A)) @ X3) => ((! [X5:A,Xs3:(list @ A)]: (((X3 = (((cons @ A) @ X5) @ Xs3)) => ((Y = (((((case_list @ A) @ A) @ X5) @ (^ [A4:A,List2:(list @ A)]: ((((ord_min @ A) @ X5) @ ((min_list @ A) @ Xs3))))) @ Xs3)) => (~ ((((accp @ (list @ A)) @ (min_list_rel @ A)) @ (((cons @ A) @ X5) @ Xs3)))))))) => (~ (((X3 = (nil @ A)) => ((Y = (undefined @ A)) => (~ ((((accp @ (list @ A)) @ (min_list_rel @ A)) @ (nil @ A))))))))))))))))). % 6.67/2.70 thf(fact_7829_lists__length__Suc__eq, axiom, (! [A:$tType,A3:(set @ A),N2:nat]: ((((collect @ (list @ A)) @ (^ [Xs2:(list @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs2)) @ A3) & (((size_size @ (list @ A)) @ Xs2) = (suc @ N2)))))) = ((((image @ ((product_prod @ (list @ A)) @ A)) @ (list @ A)) @ ((((product_case_prod @ (list @ A)) @ A) @ (list @ A)) @ (^ [Xs2:(list @ A),N:A]: ((((cons @ A) @ N) @ Xs2))))) @ ((((product_Sigma @ (list @ A)) @ A) @ ((collect @ (list @ A)) @ (^ [Xs2:(list @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs2)) @ A3) & (((size_size @ (list @ A)) @ Xs2) = N2)))))) @ (^ [Uu3:(list @ A)]: (A3)))))))). % 6.67/2.70 thf(fact_7830_mem__Sigma__iff, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,A3:(set @ A),B3:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) = ((((member @ A) @ A2) @ A3) & (((member @ B) @ B2) @ (B3 @ A2))))))). % 6.67/2.70 thf(fact_7831_SigmaI, axiom, (! [B:$tType,A:$tType,A2:A,A3:(set @ A),B2:B,B3:(A > (set @ B))]: (((((member @ A) @ A2) @ A3) => ((((member @ B) @ B2) @ (B3 @ A2)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3))))))). % 6.67/2.70 thf(fact_7832_Collect__case__prod, axiom, (! [B:$tType,A:$tType,P:(A > $o),Q:(B > $o)]: ((((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A4:A,B4:B]: (((P @ A4) & (Q @ B4)))))) = ((((product_Sigma @ A) @ B) @ ((collect @ A) @ P)) @ (^ [Uu3:A]: (((collect @ B) @ Q)))))))). % 6.67/2.70 thf(fact_7833_Sigma__empty1, axiom, (! [B:$tType,A:$tType,B3:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ (bot_bot @ (set @ A))) @ B3) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))). % 6.67/2.70 thf(fact_7834_Compl__Times__UNIV2, axiom, (! [B:$tType,A:$tType,A3:(set @ A)]: ((((uminus_uminus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: ((top_top @ (set @ B)))))) = ((((product_Sigma @ A) @ B) @ ((uminus_uminus @ (set @ A)) @ A3)) @ (^ [Uu3:A]: ((top_top @ (set @ B))))))))). % 6.67/2.70 thf(fact_7835_Compl__Times__UNIV1, axiom, (! [B:$tType,A:$tType,A3:(set @ B)]: ((((uminus_uminus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu3:A]: (A3)))) = ((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu3:A]: (((uminus_uminus @ (set @ B)) @ A3)))))))). % 6.67/2.70 thf(fact_7836_Times__empty, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B3:(set @ B)]: (((((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3))) = (bot_bot @ (set @ ((product_prod @ A) @ B)))) = ((A3 = (bot_bot @ (set @ A))) | (B3 = (bot_bot @ (set @ B)))))))). % 6.67/2.70 thf(fact_7837_Sigma__empty2, axiom, (! [B:$tType,A:$tType,A3:(set @ A)]: ((((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: ((bot_bot @ (set @ B))))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))). % 6.67/2.70 thf(fact_7838_finite__SigmaI, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(A > (set @ B))]: ((((finite_finite2 @ A) @ A3) => ((! [A5:A]: (((((member @ A) @ A5) @ A3) => ((finite_finite2 @ B) @ (B3 @ A5))))) => ((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3))))))). % 6.67/2.70 thf(fact_7839_UNIV__Times__UNIV, axiom, (! [B:$tType,A:$tType]: ((((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu3:A]: ((top_top @ (set @ B))))) = (top_top @ (set @ ((product_prod @ A) @ B))))))). % 6.67/2.70 thf(fact_7840_fst__image__times, axiom, (! [B:$tType,A:$tType,B3:(set @ B),A3:(set @ A)]: ((((B3 = (bot_bot @ (set @ B))) => (((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3)))) = (bot_bot @ (set @ A)))) & ((B3 != (bot_bot @ (set @ B))) => (((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3)))) = A3)))))). % 6.67/2.70 thf(fact_7841_snd__image__times, axiom, (! [B:$tType,A:$tType,A3:(set @ B),B3:(set @ A)]: ((((A3 = (bot_bot @ (set @ B))) => (((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu3:B]: (B3)))) = (bot_bot @ (set @ A)))) & ((A3 != (bot_bot @ (set @ B))) => (((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu3:B]: (B3)))) = B3)))))). % 6.67/2.70 thf(fact_7842_set__product, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys:(list @ B)]: ((((set2 @ ((product_prod @ A) @ B)) @ ((((product @ A) @ B) @ Xs) @ Ys)) = ((((product_Sigma @ A) @ B) @ ((set2 @ A) @ Xs)) @ (^ [Uu3:A]: (((set2 @ B) @ Ys)))))))). % 6.67/2.70 thf(fact_7843_insert__Times__insert, axiom, (! [B:$tType,A:$tType,A2:A,A3:(set @ A),B2:B,B3:(set @ B)]: ((((((product_Sigma @ A) @ B) @ (((insert @ A) @ A2) @ A3)) @ (^ [Uu3:A]: ((((insert @ B) @ B2) @ B3)))) = (((insert @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ (((sup_sup @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: ((((insert @ B) @ B2) @ B3))))) @ ((((product_Sigma @ A) @ B) @ (((insert @ A) @ A2) @ A3)) @ (^ [Uu3:A]: (B3))))))))). % 6.67/2.70 thf(fact_7844_card__SigmaI, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(A > (set @ B))]: ((((finite_finite2 @ A) @ A3) => ((! [X5:A]: (((((member @ A) @ X5) @ A3) => ((finite_finite2 @ B) @ (B3 @ X5))))) => (((finite_card @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) = ((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [A4:A]: (((finite_card @ B) @ (B3 @ A4))))) @ A3))))))). % 6.67/2.70 thf(fact_7845_inj__on__apfst, axiom, (! [B:$tType,C:$tType,A:$tType,F2:(A > C),A3:(set @ A)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F2)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: ((top_top @ (set @ B)))))) = ((((inj_on @ A) @ C) @ F2) @ A3))))). % 6.67/2.70 thf(fact_7846_inj__on__apsnd, axiom, (! [A:$tType,C:$tType,B:$tType,F2:(B > C),A3:(set @ B)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F2)) @ ((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu3:A]: (A3)))) = ((((inj_on @ B) @ C) @ F2) @ A3))))). % 6.67/2.70 thf(fact_7847_fst__image__Sigma, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(A > (set @ B))]: ((((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) = ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A3) & ((B3 @ X) != (bot_bot @ (set @ B))))))))))). % 6.67/2.70 thf(fact_7848_trancl__subset__Sigma__aux, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A)),A3:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ ((transitive_rtrancl @ A) @ R2)) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R2) @ ((((product_Sigma @ A) @ A) @ A3) @ (^ [Uu3:A]: (A3)))) => ((A2 = B2) | (((member @ A) @ A2) @ A3))))))). % 6.67/2.70 thf(fact_7849_SigmaE2, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,A3:(set @ A),B3:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) => (~ (((((member @ A) @ A2) @ A3) => (~ ((((member @ B) @ B2) @ (B3 @ A2))))))))))). % 6.67/2.70 thf(fact_7850_SigmaD2, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,A3:(set @ A),B3:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) => (((member @ B) @ B2) @ (B3 @ A2)))))). % 6.67/2.70 thf(fact_7851_SigmaD1, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,A3:(set @ A),B3:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) => (((member @ A) @ A2) @ A3))))). % 6.67/2.70 thf(fact_7852_SigmaE, axiom, (! [A:$tType,B:$tType,C2:((product_prod @ A) @ B),A3:(set @ A),B3:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ C2) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) => (~ ((! [X5:A]: (((((member @ A) @ X5) @ A3) => (! [Y4:B]: (((((member @ B) @ Y4) @ (B3 @ X5)) => (C2 != ((((product_Pair @ A) @ B) @ X5) @ Y4)))))))))))))). % 6.67/2.70 thf(fact_7853_times__eq__iff, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B3:(set @ B),C5:(set @ A),D6:(set @ B)]: (((((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3))) = ((((product_Sigma @ A) @ B) @ C5) @ (^ [Uu3:A]: (D6)))) = (((A3 = C5) & (B3 = D6)) | (((A3 = (bot_bot @ (set @ A))) | (B3 = (bot_bot @ (set @ B)))) & ((C5 = (bot_bot @ (set @ A))) | (D6 = (bot_bot @ (set @ B)))))))))). % 6.67/2.70 thf(fact_7854_Sigma__empty__iff, axiom, (! [B:$tType,A:$tType,I6:(set @ A),X8:(A > (set @ B))]: (((((((product_Sigma @ A) @ B) @ I6) @ X8) = (bot_bot @ (set @ ((product_prod @ A) @ B)))) = (! [X:A]: (((((member @ A) @ X) @ I6) => ((X8 @ X) = (bot_bot @ (set @ B)))))))))). % 6.67/2.70 thf(fact_7855_Sigma__Union, axiom, (! [B:$tType,A:$tType,X8:(set @ (set @ A)),B3:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ ((complete_Sup_Sup @ (set @ A)) @ X8)) @ B3) = ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ ((((image @ (set @ A)) @ (set @ ((product_prod @ A) @ B))) @ (^ [A6:(set @ A)]: (((((product_Sigma @ A) @ B) @ A6) @ B3)))) @ X8)))))). % 6.67/2.70 thf(fact_7856_Times__subset__cancel2, axiom, (! [A:$tType,B:$tType,X3:A,C5:(set @ A),A3:(set @ B),B3:(set @ B)]: (((((member @ A) @ X3) @ C5) => ((((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ ((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu3:B]: (C5)))) @ ((((product_Sigma @ B) @ A) @ B3) @ (^ [Uu3:B]: (C5)))) = (((ord_less_eq @ (set @ B)) @ A3) @ B3)))))). % 6.67/2.70 thf(fact_7857_Sigma__mono, axiom, (! [B:$tType,A:$tType,A3:(set @ A),C5:(set @ A),B3:(A > (set @ B)),D6:(A > (set @ B))]: (((((ord_less_eq @ (set @ A)) @ A3) @ C5) => ((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((ord_less_eq @ (set @ B)) @ (B3 @ X5)) @ (D6 @ X5))))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) @ ((((product_Sigma @ A) @ B) @ C5) @ D6))))))). % 6.67/2.70 thf(fact_7858_Restr__subset, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((inf_inf @ (set @ ((product_prod @ A) @ A))) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R2) @ ((((product_Sigma @ A) @ A) @ B3) @ (^ [Uu3:A]: (B3))))) @ ((((product_Sigma @ A) @ A) @ A3) @ (^ [Uu3:A]: (A3)))) = (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R2) @ ((((product_Sigma @ A) @ A) @ A3) @ (^ [Uu3:A]: (A3))))))))). % 6.67/2.70 thf(fact_7859_Times__Un__distrib1, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ B)]: ((((((product_Sigma @ A) @ B) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) @ (^ [Uu3:A]: (C5))) = (((sup_sup @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (C5)))) @ ((((product_Sigma @ A) @ B) @ B3) @ (^ [Uu3:A]: (C5)))))))). % 6.67/2.70 thf(fact_7860_Sigma__Un__distrib2, axiom, (! [B:$tType,A:$tType,I6:(set @ A),A3:(A > (set @ B)),B3:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ I6) @ (^ [I3:A]: ((((sup_sup @ (set @ B)) @ (A3 @ I3)) @ (B3 @ I3))))) = (((sup_sup @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I6) @ A3)) @ ((((product_Sigma @ A) @ B) @ I6) @ B3)))))). % 6.67/2.70 thf(fact_7861_Sigma__Un__distrib1, axiom, (! [B:$tType,A:$tType,I6:(set @ A),J4:(set @ A),C5:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ (((sup_sup @ (set @ A)) @ I6) @ J4)) @ C5) = (((sup_sup @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I6) @ C5)) @ ((((product_Sigma @ A) @ B) @ J4) @ C5)))))). % 6.67/2.70 thf(fact_7862_member__product, axiom, (! [B:$tType,A:$tType,X3:((product_prod @ A) @ B),A3:(set @ A),B3:(set @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X3) @ ((((product_product @ A) @ B) @ A3) @ B3)) = (((member @ ((product_prod @ A) @ B)) @ X3) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3)))))))). % 6.67/2.70 thf(fact_7863_Product__Type_Oproduct__def, axiom, (! [B:$tType,A:$tType]: ((((product_product @ A) @ B) = (^ [A6:(set @ A),B6:(set @ B)]: (((((product_Sigma @ A) @ B) @ A6) @ (^ [Uu3:A]: (B6))))))))). % 6.67/2.70 thf(fact_7864_finite__cartesian__product, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ B) @ B3) => ((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3))))))))). % 6.67/2.70 thf(fact_7865_Times__Int__distrib1, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ B)]: ((((((product_Sigma @ A) @ B) @ (((inf_inf @ (set @ A)) @ A3) @ B3)) @ (^ [Uu3:A]: (C5))) = (((inf_inf @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (C5)))) @ ((((product_Sigma @ A) @ B) @ B3) @ (^ [Uu3:A]: (C5)))))))). % 6.67/2.70 thf(fact_7866_Sigma__Int__distrib2, axiom, (! [B:$tType,A:$tType,I6:(set @ A),A3:(A > (set @ B)),B3:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ I6) @ (^ [I3:A]: ((((inf_inf @ (set @ B)) @ (A3 @ I3)) @ (B3 @ I3))))) = (((inf_inf @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I6) @ A3)) @ ((((product_Sigma @ A) @ B) @ I6) @ B3)))))). % 6.67/2.70 thf(fact_7867_Times__Int__Times, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B),C5:(set @ A),D6:(set @ B)]: (((((inf_inf @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3)))) @ ((((product_Sigma @ A) @ B) @ C5) @ (^ [Uu3:A]: (D6)))) = ((((product_Sigma @ A) @ B) @ (((inf_inf @ (set @ A)) @ A3) @ C5)) @ (^ [Uu3:A]: ((((inf_inf @ (set @ B)) @ B3) @ D6)))))))). % 6.67/2.70 thf(fact_7868_Sigma__Int__distrib1, axiom, (! [B:$tType,A:$tType,I6:(set @ A),J4:(set @ A),C5:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ (((inf_inf @ (set @ A)) @ I6) @ J4)) @ C5) = (((inf_inf @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I6) @ C5)) @ ((((product_Sigma @ A) @ B) @ J4) @ C5)))))). % 6.67/2.70 thf(fact_7869_Sigma__Diff__distrib1, axiom, (! [B:$tType,A:$tType,I6:(set @ A),J4:(set @ A),C5:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ (((minus_minus @ (set @ A)) @ I6) @ J4)) @ C5) = (((minus_minus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I6) @ C5)) @ ((((product_Sigma @ A) @ B) @ J4) @ C5)))))). % 6.67/2.70 thf(fact_7870_Sigma__cong, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ A),C5:(A > (set @ B)),D6:(A > (set @ B))]: (((A3 = B3) => ((! [X5:A]: (((((member @ A) @ X5) @ B3) => ((C5 @ X5) = (D6 @ X5))))) => (((((product_Sigma @ A) @ B) @ A3) @ C5) = ((((product_Sigma @ A) @ B) @ B3) @ D6))))))). % 6.67/2.70 thf(fact_7871_Times__eq__cancel2, axiom, (! [A:$tType,B:$tType,X3:A,C5:(set @ A),A3:(set @ B),B3:(set @ B)]: (((((member @ A) @ X3) @ C5) => ((((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu3:B]: (C5))) = ((((product_Sigma @ B) @ A) @ B3) @ (^ [Uu3:B]: (C5)))) = (A3 = B3)))))). % 6.67/2.70 thf(fact_7872_Times__Diff__distrib1, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ A),C5:(set @ B)]: ((((((product_Sigma @ A) @ B) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ (^ [Uu3:A]: (C5))) = (((minus_minus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (C5)))) @ ((((product_Sigma @ A) @ B) @ B3) @ (^ [Uu3:A]: (C5)))))))). % 6.67/2.70 thf(fact_7873_Sigma__Diff__distrib2, axiom, (! [B:$tType,A:$tType,I6:(set @ A),A3:(A > (set @ B)),B3:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ I6) @ (^ [I3:A]: ((((minus_minus @ (set @ B)) @ (A3 @ I3)) @ (B3 @ I3))))) = (((minus_minus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I6) @ A3)) @ ((((product_Sigma @ A) @ B) @ I6) @ B3)))))). % 6.67/2.70 thf(fact_7874_Collect__case__prod__Sigma, axiom, (! [B:$tType,A:$tType,P:(A > $o),Q:(A > (B > $o))]: ((((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X:A,Y2:B]: (((P @ X) & ((Q @ X) @ Y2)))))) = ((((product_Sigma @ A) @ B) @ ((collect @ A) @ P)) @ (^ [X:A]: (((collect @ B) @ (Q @ X))))))))). % 6.67/2.70 thf(fact_7875_swap__product, axiom, (! [B:$tType,A:$tType,A3:(set @ B),B3:(set @ A)]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ B) @ A) @ ((product_prod @ A) @ B)) @ (^ [I3:B,J3:A]: (((((product_Pair @ A) @ B) @ J3) @ I3))))) @ ((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu3:B]: (B3)))) = ((((product_Sigma @ A) @ B) @ B3) @ (^ [Uu3:A]: (A3))))))). % 6.67/2.70 thf(fact_7876_times__subset__iff, axiom, (! [A:$tType,B:$tType,A3:(set @ A),C5:(set @ B),B3:(set @ A),D6:(set @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (C5)))) @ ((((product_Sigma @ A) @ B) @ B3) @ (^ [Uu3:A]: (D6)))) = ((A3 = (bot_bot @ (set @ A))) | ((C5 = (bot_bot @ (set @ B))) | ((((ord_less_eq @ (set @ A)) @ A3) @ B3) & (((ord_less_eq @ (set @ B)) @ C5) @ D6)))))))). % 6.67/2.70 thf(fact_7877_mem__Times__iff, axiom, (! [A:$tType,B:$tType,X3:((product_prod @ A) @ B),A3:(set @ A),B3:(set @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X3) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3)))) = ((((member @ A) @ (((product_fst @ A) @ B) @ X3)) @ A3) & (((member @ B) @ (((product_snd @ A) @ B) @ X3)) @ B3)))))). % 6.67/2.70 thf(fact_7878_image__paired__Times, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,F2:(C > A),G:(D > B),A3:(set @ C),B3:(set @ D)]: ((((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ C) @ D) @ ((product_prod @ A) @ B)) @ (^ [X:C,Y2:D]: (((((product_Pair @ A) @ B) @ (F2 @ X)) @ (G @ Y2)))))) @ ((((product_Sigma @ C) @ D) @ A3) @ (^ [Uu3:C]: (B3)))) = ((((product_Sigma @ A) @ B) @ ((((image @ C) @ A) @ F2) @ A3)) @ (^ [Uu3:A]: (((((image @ D) @ B) @ G) @ B3)))))))). % 6.67/2.70 thf(fact_7879_finite__SigmaI2, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(A > (set @ B))]: ((((finite_finite2 @ A) @ ((collect @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A3) & ((B3 @ X) != (bot_bot @ (set @ B)))))))) => ((! [A5:A]: (((((member @ A) @ A5) @ A3) => ((finite_finite2 @ B) @ (B3 @ A5))))) => ((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3))))))). % 6.67/2.70 thf(fact_7880_finite__cartesian__productD1, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B)]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3)))) => ((B3 != (bot_bot @ (set @ B))) => ((finite_finite2 @ A) @ A3)))))). % 6.67/2.70 thf(fact_7881_finite__cartesian__productD2, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B3:(set @ B)]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3)))) => ((A3 != (bot_bot @ (set @ A))) => ((finite_finite2 @ B) @ B3)))))). % 6.67/2.70 thf(fact_7882_finite__cartesian__product__iff, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B3:(set @ B)]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3)))) = ((A3 = (bot_bot @ (set @ A))) | ((B3 = (bot_bot @ (set @ B))) | (((finite_finite2 @ A) @ A3) & ((finite_finite2 @ B) @ B3)))))))). % 6.67/2.70 thf(fact_7883_UN__Times__distrib, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,E5:(C > (set @ A)),F5:(D > (set @ B)),A3:(set @ C),B3:(set @ D)]: ((((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ ((((image @ ((product_prod @ C) @ D)) @ (set @ ((product_prod @ A) @ B))) @ ((((product_case_prod @ C) @ D) @ (set @ ((product_prod @ A) @ B))) @ (^ [A4:C,B4:D]: (((((product_Sigma @ A) @ B) @ (E5 @ A4)) @ (^ [Uu3:A]: ((F5 @ B4)))))))) @ ((((product_Sigma @ C) @ D) @ A3) @ (^ [Uu3:C]: (B3))))) = ((((product_Sigma @ A) @ B) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image @ C) @ (set @ A)) @ E5) @ A3))) @ (^ [Uu3:A]: (((complete_Sup_Sup @ (set @ B)) @ ((((image @ D) @ (set @ B)) @ F5) @ B3))))))))). % 6.67/2.70 thf(fact_7884_snd__image__Sigma, axiom, (! [A:$tType,B:$tType,A3:(set @ B),B3:(B > (set @ A))]: ((((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A3) @ B3)) = ((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ B3) @ A3)))))). % 6.67/2.70 thf(fact_7885_subset__fst__imageI, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B),S3:(set @ ((product_prod @ A) @ B)),Y:B]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3)))) @ S3) => ((((member @ B) @ Y) @ B3) => (((ord_less_eq @ (set @ A)) @ A3) @ ((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ S3))))))). % 6.67/2.70 thf(fact_7886_subset__snd__imageI, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B),S3:(set @ ((product_prod @ A) @ B)),X3:A]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3)))) @ S3) => ((((member @ A) @ X3) @ A3) => (((ord_less_eq @ (set @ B)) @ B3) @ ((((image @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ S3))))))). % 6.67/2.70 thf(fact_7887_subset__fst__snd, axiom, (! [B:$tType,A:$tType,A3:(set @ ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A3) @ ((((product_Sigma @ A) @ B) @ ((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ A3)) @ (^ [Uu3:A]: (((((image @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ A3)))))))). % 6.67/2.70 thf(fact_7888_card__cartesian__product__singleton, axiom, (! [A:$tType,B:$tType,X3:A,A3:(set @ B)]: ((((finite_card @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) @ (^ [Uu3:A]: (A3)))) = ((finite_card @ B) @ A3))))). % 6.67/2.70 thf(fact_7889_Sigma__interval__disjoint, axiom, (! [A:$tType,B:$tType]: (((order @ A) => (! [A3:(set @ B),V2:(B > A),W2:A]: (((((inf_inf @ (set @ ((product_prod @ B) @ A))) @ ((((product_Sigma @ B) @ A) @ A3) @ (^ [I3:B]: (((set_ord_atMost @ A) @ (V2 @ I3)))))) @ ((((product_Sigma @ B) @ A) @ A3) @ (^ [I3:B]: ((((set_or3652927894154168847AtMost @ A) @ (V2 @ I3)) @ W2))))) = (bot_bot @ (set @ ((product_prod @ B) @ A)))))))))). % 6.67/2.70 thf(fact_7890_sum_OSigma, axiom, (! [A:$tType,C:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A3:(set @ B),B3:(B > (set @ C)),G:(B > (C > A))]: ((((finite_finite2 @ B) @ A3) => ((! [X5:B]: (((((member @ B) @ X5) @ A3) => ((finite_finite2 @ C) @ (B3 @ X5))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X:B]: (((((groups7311177749621191930dd_sum @ C) @ A) @ (G @ X)) @ (B3 @ X))))) @ A3) = ((((groups7311177749621191930dd_sum @ ((product_prod @ B) @ C)) @ A) @ ((((product_case_prod @ B) @ C) @ A) @ G)) @ ((((product_Sigma @ B) @ C) @ A3) @ B3))))))))))). % 6.67/2.70 thf(fact_7891_prod_OSigma, axiom, (! [A:$tType,C:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A3:(set @ B),B3:(B > (set @ C)),G:(B > (C > A))]: ((((finite_finite2 @ B) @ A3) => ((! [X5:B]: (((((member @ B) @ X5) @ A3) => ((finite_finite2 @ C) @ (B3 @ X5))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X:B]: (((((groups7121269368397514597t_prod @ C) @ A) @ (G @ X)) @ (B3 @ X))))) @ A3) = ((((groups7121269368397514597t_prod @ ((product_prod @ B) @ C)) @ A) @ ((((product_case_prod @ B) @ C) @ A) @ G)) @ ((((product_Sigma @ B) @ C) @ A3) @ B3))))))))))). % 6.67/2.70 thf(fact_7892_Sigma__def, axiom, (! [B:$tType,A:$tType]: ((((product_Sigma @ A) @ B) = (^ [A6:(set @ A),B6:(A > (set @ B))]: (((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ ((((image @ A) @ (set @ ((product_prod @ A) @ B))) @ (^ [X:A]: (((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ ((((image @ B) @ (set @ ((product_prod @ A) @ B))) @ (^ [Y2:B]: ((((insert @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ (bot_bot @ (set @ ((product_prod @ A) @ B))))))) @ (B6 @ X)))))) @ A6)))))))). % 6.67/2.70 thf(fact_7893_product__fold, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ B) @ B3) => (((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3))) = (((((finite_fold @ A) @ (set @ ((product_prod @ A) @ B))) @ (^ [X:A,Z6:(set @ ((product_prod @ A) @ B))]: ((((((finite_fold @ B) @ (set @ ((product_prod @ A) @ B))) @ (^ [Y2:B]: (((insert @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2))))) @ Z6) @ B3)))) @ (bot_bot @ (set @ ((product_prod @ A) @ B)))) @ A3))))))). % 6.67/2.70 thf(fact_7894_uniformly__continuous__on__uniformity, axiom, (! [B:$tType,A:$tType]: ((((topolo7287701948861334536_space @ A) & (topolo7287701948861334536_space @ B)) => (((topolo6026614971017936543ous_on @ A) @ B) = (^ [S8:(set @ A),F3:(A > B)]: ((((((filterlim @ ((product_prod @ A) @ A)) @ ((product_prod @ B) @ B)) @ ((((product_case_prod @ A) @ A) @ ((product_prod @ B) @ B)) @ (^ [X:A,Y2:A]: (((((product_Pair @ B) @ B) @ (F3 @ X)) @ (F3 @ Y2)))))) @ (topolo7806501430040627800ormity @ B)) @ (((inf_inf @ (filter @ ((product_prod @ A) @ A))) @ (topolo7806501430040627800ormity @ A)) @ ((principal @ ((product_prod @ A) @ A)) @ ((((product_Sigma @ A) @ A) @ S8) @ (^ [Uu3:A]: (S8))))))))))))). % 6.67/2.70 thf(fact_7895_sum__mult__sum__if__inj, axiom, (! [A:$tType,C:$tType,B:$tType]: (((semiring_0 @ B) => (! [F2:(A > B),G:(C > B),A3:(set @ A),B3:(set @ C)]: ((((((inj_on @ ((product_prod @ A) @ C)) @ B) @ ((((product_case_prod @ A) @ C) @ B) @ (^ [A4:A,B4:C]: ((((times_times @ B) @ (F2 @ A4)) @ (G @ B4)))))) @ ((((product_Sigma @ A) @ C) @ A3) @ (^ [Uu3:A]: (B3)))) => ((((times_times @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F2) @ A3)) @ ((((groups7311177749621191930dd_sum @ C) @ B) @ G) @ B3)) = ((((groups7311177749621191930dd_sum @ B) @ B) @ (id @ B)) @ ((collect @ B) @ (^ [Uu3:B]: ((? [A4:A,B4:C]: (((Uu3 = (((times_times @ B) @ (F2 @ A4)) @ (G @ B4))) & ((((member @ A) @ A4) @ A3) & (((member @ C) @ B4) @ B3))))))))))))))))). % 6.67/2.70 thf(fact_7896_infinite__cartesian__product, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B)]: (((~ (((finite_finite2 @ A) @ A3))) => ((~ (((finite_finite2 @ B) @ B3))) => (~ (((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3))))))))))). % 6.67/2.70 thf(fact_7897_Gr__incl, axiom, (! [A:$tType,B:$tType,A3:(set @ A),F2:(A > B),B3:(set @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((bNF_Gr @ A) @ B) @ A3) @ F2)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: (B3)))) = (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A3)) @ B3))))). % 6.67/2.70 thf(fact_7898_GrD2, axiom, (! [A:$tType,B:$tType,X3:A,Fx:B,A3:(set @ A),F2:(A > B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Fx)) @ ((((bNF_Gr @ A) @ B) @ A3) @ F2)) => ((F2 @ X3) = Fx))))). % 6.67/2.70 thf(fact_7899_GrD1, axiom, (! [B:$tType,A:$tType,X3:A,Fx:B,A3:(set @ A),F2:(A > B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Fx)) @ ((((bNF_Gr @ A) @ B) @ A3) @ F2)) => (((member @ A) @ X3) @ A3))))). % 6.67/2.70 thf(fact_7900_Gr__def, axiom, (! [B:$tType,A:$tType]: ((((bNF_Gr @ A) @ B) = (^ [A6:(set @ A),F3:(A > B)]: (((collect @ ((product_prod @ A) @ B)) @ (^ [Uu3:((product_prod @ A) @ B)]: ((? [A4:A]: (((Uu3 = ((((product_Pair @ A) @ B) @ A4) @ (F3 @ A4))) & (((member @ A) @ A4) @ A6))))))))))))). % 6.67/2.70 thf(fact_7901_pairs__le__eq__Sigma, axiom, (! [M:nat]: ((((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [I3:nat,J3:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I3) @ J3)) @ M))))) = ((((product_Sigma @ nat) @ nat) @ ((set_ord_atMost @ nat) @ M)) @ (^ [R5:nat]: (((set_ord_atMost @ nat) @ (((minus_minus @ nat) @ M) @ R5))))))))). % 6.67/2.70 thf(fact_7902_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca3754400796208372196lChain @ A) @ B) = (^ [R5:(set @ ((product_prod @ A) @ A)),As4:(A > B)]: ((! [I3:A,J3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I3) @ J3)) @ R5) => (((ord_less_eq @ B) @ (As4 @ I3)) @ (As4 @ J3)))))))))))). % 6.67/2.70 thf(fact_7903_natLess__def, axiom, (bNF_Ca8459412986667044542atLess = ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (ord_less @ nat))))). % 6.67/2.70 thf(fact_7904_rotate__drop__take, axiom, (! [A:$tType]: (((rotate @ A) = (^ [N:nat,Xs2:(list @ A)]: ((((append @ A) @ (((drop @ A) @ (((modulo_modulo @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs2))) @ Xs2)) @ (((take @ A) @ (((modulo_modulo @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs2))) @ Xs2)))))))). % 6.67/2.70 thf(fact_7905_image__split__eq__Sigma, axiom, (! [C:$tType,B:$tType,A:$tType,F2:(C > A),G:(C > B),A3:(set @ C)]: ((((((image @ C) @ ((product_prod @ A) @ B)) @ (^ [X:C]: (((((product_Pair @ A) @ B) @ (F2 @ X)) @ (G @ X))))) @ A3) = ((((product_Sigma @ A) @ B) @ ((((image @ C) @ A) @ F2) @ A3)) @ (^ [X:A]: (((((image @ C) @ B) @ G) @ (((inf_inf @ (set @ C)) @ ((((vimage @ C) @ A) @ F2) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) @ A3))))))))). % 6.67/2.70 thf(fact_7906_vimage__eq, axiom, (! [A:$tType,B:$tType,A2:A,F2:(A > B),B3:(set @ B)]: (((((member @ A) @ A2) @ ((((vimage @ A) @ B) @ F2) @ B3)) = (((member @ B) @ (F2 @ A2)) @ B3))))). % 6.67/2.70 thf(fact_7907_vimageI, axiom, (! [B:$tType,A:$tType,F2:(B > A),A2:B,B2:A,B3:(set @ A)]: ((((F2 @ A2) = B2) => ((((member @ A) @ B2) @ B3) => (((member @ B) @ A2) @ ((((vimage @ B) @ A) @ F2) @ B3))))))). % 6.67/2.70 thf(fact_7908_vimage__Collect__eq, axiom, (! [B:$tType,A:$tType,F2:(A > B),P:(B > $o)]: ((((((vimage @ A) @ B) @ F2) @ ((collect @ B) @ P)) = ((collect @ A) @ (^ [Y2:A]: ((P @ (F2 @ Y2))))))))). % 6.67/2.70 thf(fact_7909_vimage__ident, axiom, (! [A:$tType,Y7:(set @ A)]: ((((((vimage @ A) @ A) @ (^ [X:A]: (X))) @ Y7) = Y7)))). % 6.67/2.70 thf(fact_7910_vimage__UNIV, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: ((((((vimage @ A) @ B) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A)))))). % 6.67/2.70 thf(fact_7911_vimage__empty, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: ((((((vimage @ A) @ B) @ F2) @ (bot_bot @ (set @ B))) = (bot_bot @ (set @ A)))))). % 6.67/2.70 thf(fact_7912_vimage__Int, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ B),B3:(set @ B)]: ((((((vimage @ A) @ B) @ F2) @ (((inf_inf @ (set @ B)) @ A3) @ B3)) = (((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ F2) @ A3)) @ ((((vimage @ A) @ B) @ F2) @ B3)))))). % 6.67/2.70 thf(fact_7913_vimage__Un, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ B),B3:(set @ B)]: ((((((vimage @ A) @ B) @ F2) @ (((sup_sup @ (set @ B)) @ A3) @ B3)) = (((sup_sup @ (set @ A)) @ ((((vimage @ A) @ B) @ F2) @ A3)) @ ((((vimage @ A) @ B) @ F2) @ B3)))))). % 6.67/2.70 thf(fact_7914_rotate__is__Nil__conv, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((((rotate @ A) @ N2) @ Xs) = (nil @ A)) = (Xs = (nil @ A)))))). % 6.67/2.70 thf(fact_7915_set__rotate, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((set2 @ A) @ (((rotate @ A) @ N2) @ Xs)) = ((set2 @ A) @ Xs))))). % 6.67/2.70 thf(fact_7916_length__rotate, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ (((rotate @ A) @ N2) @ Xs)) = ((size_size @ (list @ A)) @ Xs))))). % 6.67/2.70 thf(fact_7917_distinct__rotate, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((distinct @ A) @ (((rotate @ A) @ N2) @ Xs)) = ((distinct @ A) @ Xs))))). % 6.67/2.70 thf(fact_7918_vimage__const, axiom, (! [B:$tType,A:$tType,C2:B,A3:(set @ B)]: ((((((member @ B) @ C2) @ A3) => (((((vimage @ A) @ B) @ (^ [X:A]: (C2))) @ A3) = (top_top @ (set @ A)))) & ((~ ((((member @ B) @ C2) @ A3))) => (((((vimage @ A) @ B) @ (^ [X:A]: (C2))) @ A3) = (bot_bot @ (set @ A)))))))). % 6.67/2.70 thf(fact_7919_rotate__Suc, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((rotate @ A) @ (suc @ N2)) @ Xs) = ((rotate1 @ A) @ (((rotate @ A) @ N2) @ Xs)))))). % 6.67/2.70 thf(fact_7920_rotate0, axiom, (! [A:$tType]: ((((rotate @ A) @ (zero_zero @ nat)) = (id @ (list @ A)))))). % 6.67/2.70 thf(fact_7921_image__vimage__eq, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ A)]: ((((((image @ B) @ A) @ F2) @ ((((vimage @ B) @ A) @ F2) @ A3)) = (((inf_inf @ (set @ A)) @ A3) @ ((((image @ B) @ A) @ F2) @ (top_top @ (set @ B)))))))). % 6.67/2.70 thf(fact_7922_rotate__length01, axiom, (! [A:$tType,Xs:(list @ A),N2:nat]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)) => ((((rotate @ A) @ N2) @ Xs) = Xs))))). % 6.67/2.70 thf(fact_7923_rotate__id, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((((modulo_modulo @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) = (zero_zero @ nat)) => ((((rotate @ A) @ N2) @ Xs) = Xs))))). % 6.67/2.70 thf(fact_7924_vimage__if, axiom, (! [B:$tType,A:$tType,C2:B,A3:(set @ B),D2:B,B3:(set @ A)]: ((((((member @ B) @ C2) @ A3) => (((((member @ B) @ D2) @ A3) => (((((vimage @ A) @ B) @ (^ [X:A]: (((((if @ B) @ (((member @ A) @ X) @ B3)) @ C2) @ D2)))) @ A3) = (top_top @ (set @ A)))) & ((~ ((((member @ B) @ D2) @ A3))) => (((((vimage @ A) @ B) @ (^ [X:A]: (((((if @ B) @ (((member @ A) @ X) @ B3)) @ C2) @ D2)))) @ A3) = B3)))) & ((~ ((((member @ B) @ C2) @ A3))) => (((((member @ B) @ D2) @ A3) => (((((vimage @ A) @ B) @ (^ [X:A]: (((((if @ B) @ (((member @ A) @ X) @ B3)) @ C2) @ D2)))) @ A3) = ((uminus_uminus @ (set @ A)) @ B3))) & ((~ ((((member @ B) @ D2) @ A3))) => (((((vimage @ A) @ B) @ (^ [X:A]: (((((if @ B) @ (((member @ A) @ X) @ B3)) @ C2) @ D2)))) @ A3) = (bot_bot @ (set @ A)))))))))). % 6.67/2.70 thf(fact_7925_Pair__vimage__Sigma, axiom, (! [B:$tType,A:$tType,X3:B,A3:(set @ B),F2:(B > (set @ A))]: ((((((member @ B) @ X3) @ A3) => (((((vimage @ A) @ ((product_prod @ B) @ A)) @ (((product_Pair @ B) @ A) @ X3)) @ ((((product_Sigma @ B) @ A) @ A3) @ F2)) = (F2 @ X3))) & ((~ ((((member @ B) @ X3) @ A3))) => (((((vimage @ A) @ ((product_prod @ B) @ A)) @ (((product_Pair @ B) @ A) @ X3)) @ ((((product_Sigma @ B) @ A) @ A3) @ F2)) = (bot_bot @ (set @ A)))))))). % 6.67/2.70 thf(fact_7926_vimage__snd, axiom, (! [B:$tType,A:$tType,A3:(set @ B)]: ((((((vimage @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ A3) = ((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu3:A]: (A3))))))). % 6.67/2.70 thf(fact_7927_vimage__fst, axiom, (! [B:$tType,A:$tType,A3:(set @ A)]: ((((((vimage @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ A3) = ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu3:A]: ((top_top @ (set @ B))))))))). % 6.67/2.70 thf(fact_7928_vimage__Times, axiom, (! [A:$tType,B:$tType,C:$tType,F2:(A > ((product_prod @ B) @ C)),A3:(set @ B),B3:(set @ C)]: ((((((vimage @ A) @ ((product_prod @ B) @ C)) @ F2) @ ((((product_Sigma @ B) @ C) @ A3) @ (^ [Uu3:B]: (B3)))) = (((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ (((((comp @ ((product_prod @ B) @ C)) @ B) @ A) @ ((product_fst @ B) @ C)) @ F2)) @ A3)) @ ((((vimage @ A) @ C) @ (((((comp @ ((product_prod @ B) @ C)) @ C) @ A) @ ((product_snd @ B) @ C)) @ F2)) @ B3)))))). % 6.67/2.70 thf(fact_7929_vimage__singleton__eq, axiom, (! [A:$tType,B:$tType,A2:A,F2:(A > B),B2:B]: (((((member @ A) @ A2) @ ((((vimage @ A) @ B) @ F2) @ (((insert @ B) @ B2) @ (bot_bot @ (set @ B))))) = ((F2 @ A2) = B2))))). % 6.67/2.70 thf(fact_7930_vimage__insert, axiom, (! [A:$tType,B:$tType,F2:(A > B),A2:B,B3:(set @ B)]: ((((((vimage @ A) @ B) @ F2) @ (((insert @ B) @ A2) @ B3)) = (((sup_sup @ (set @ A)) @ ((((vimage @ A) @ B) @ F2) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B))))) @ ((((vimage @ A) @ B) @ F2) @ B3)))))). % 6.67/2.70 thf(fact_7931_vimage__Suc__insert__Suc, axiom, (! [N2:nat,A3:(set @ nat)]: ((((((vimage @ nat) @ nat) @ suc) @ (((insert @ nat) @ (suc @ N2)) @ A3)) = (((insert @ nat) @ N2) @ ((((vimage @ nat) @ nat) @ suc) @ A3)))))). % 6.67/2.70 thf(fact_7932_vimage__Suc__insert__0, axiom, (! [A3:(set @ nat)]: ((((((vimage @ nat) @ nat) @ suc) @ (((insert @ nat) @ (zero_zero @ nat)) @ A3)) = ((((vimage @ nat) @ nat) @ suc) @ A3))))). % 6.67/2.70 thf(fact_7933_rotate__add, axiom, (! [A:$tType,M:nat,N2:nat]: ((((rotate @ A) @ (((plus_plus @ nat) @ M) @ N2)) = (((((comp @ (list @ A)) @ (list @ A)) @ (list @ A)) @ ((rotate @ A) @ M)) @ ((rotate @ A) @ N2)))))). % 6.67/2.70 thf(fact_7934_rotate__map, axiom, (! [A:$tType,B:$tType,N2:nat,F2:(B > A),Xs:(list @ B)]: (((((rotate @ A) @ N2) @ ((((map @ B) @ A) @ F2) @ Xs)) = ((((map @ B) @ A) @ F2) @ (((rotate @ B) @ N2) @ Xs)))))). % 6.67/2.70 thf(fact_7935_rotate__def, axiom, (! [A:$tType]: (((rotate @ A) = (^ [N:nat]: ((((compow @ ((list @ A) > (list @ A))) @ N) @ (rotate1 @ A)))))))). % 6.67/2.70 thf(fact_7936_rotate__rotate, axiom, (! [A:$tType,M:nat,N2:nat,Xs:(list @ A)]: (((((rotate @ A) @ M) @ (((rotate @ A) @ N2) @ Xs)) = (((rotate @ A) @ (((plus_plus @ nat) @ M) @ N2)) @ Xs))))). % 6.67/2.70 thf(fact_7937_vimage__def, axiom, (! [B:$tType,A:$tType]: ((((vimage @ A) @ B) = (^ [F3:(A > B),B6:(set @ B)]: (((collect @ A) @ (^ [X:A]: ((((member @ B) @ (F3 @ X)) @ B6)))))))))). % 6.67/2.70 thf(fact_7938_vimage__Collect, axiom, (! [B:$tType,A:$tType,P:(B > $o),F2:(A > B),Q:(A > $o)]: (((! [X5:A]: (((P @ (F2 @ X5)) = (Q @ X5)))) => (((((vimage @ A) @ B) @ F2) @ ((collect @ B) @ P)) = ((collect @ A) @ Q)))))). % 6.67/2.70 thf(fact_7939_vimageI2, axiom, (! [B:$tType,A:$tType,F2:(B > A),A2:B,A3:(set @ A)]: (((((member @ A) @ (F2 @ A2)) @ A3) => (((member @ B) @ A2) @ ((((vimage @ B) @ A) @ F2) @ A3)))))). % 6.67/2.70 thf(fact_7940_vimageE, axiom, (! [A:$tType,B:$tType,A2:A,F2:(A > B),B3:(set @ B)]: (((((member @ A) @ A2) @ ((((vimage @ A) @ B) @ F2) @ B3)) => (((member @ B) @ (F2 @ A2)) @ B3))))). % 6.67/2.70 thf(fact_7941_vimageD, axiom, (! [A:$tType,B:$tType,A2:A,F2:(A > B),A3:(set @ B)]: (((((member @ A) @ A2) @ ((((vimage @ A) @ B) @ F2) @ A3)) => (((member @ B) @ (F2 @ A2)) @ A3))))). % 6.67/2.70 thf(fact_7942_rotate1__rotate__swap, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: ((((rotate1 @ A) @ (((rotate @ A) @ N2) @ Xs)) = (((rotate @ A) @ N2) @ ((rotate1 @ A) @ Xs)))))). % 6.67/2.70 thf(fact_7943_finite__vimage__Suc__iff, axiom, (! [F5:(set @ nat)]: ((((finite_finite2 @ nat) @ ((((vimage @ nat) @ nat) @ suc) @ F5)) = ((finite_finite2 @ nat) @ F5))))). % 6.67/2.70 thf(fact_7944_continuous__imp__open__vimage, axiom, (! [A:$tType,B:$tType]: ((((topolo4958980785337419405_space @ B) & (topolo4958980785337419405_space @ A)) => (! [S:(set @ A),F2:(A > B),B3:(set @ B)]: ((((((topolo81223032696312382ous_on @ A) @ B) @ S) @ F2) => (((topolo1002775350975398744n_open @ A) @ S) => (((topolo1002775350975398744n_open @ B) @ B3) => ((((ord_less_eq @ (set @ A)) @ ((((vimage @ A) @ B) @ F2) @ B3)) @ S) => ((topolo1002775350975398744n_open @ A) @ ((((vimage @ A) @ B) @ F2) @ B3)))))))))))). % 6.67/2.70 thf(fact_7945_vimage__mono, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ A),F2:(B > A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ (set @ B)) @ ((((vimage @ B) @ A) @ F2) @ A3)) @ ((((vimage @ B) @ A) @ F2) @ B3)))))). % 6.67/2.70 thf(fact_7946_subset__vimage__iff, axiom, (! [B:$tType,A:$tType,A3:(set @ A),F2:(A > B),B3:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ A3) @ ((((vimage @ A) @ B) @ F2) @ B3)) = (! [X:A]: (((((member @ A) @ X) @ A3) => (((member @ B) @ (F2 @ X)) @ B3)))))))). % 6.67/2.70 thf(fact_7947_rotate__conv__mod, axiom, (! [A:$tType]: (((rotate @ A) = (^ [N:nat,Xs2:(list @ A)]: ((((rotate @ A) @ (((modulo_modulo @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs2))) @ Xs2))))))). % 6.67/2.70 thf(fact_7948_rotate__append, axiom, (! [A:$tType,L:(list @ A),Q4:(list @ A)]: (((((rotate @ A) @ ((size_size @ (list @ A)) @ L)) @ (((append @ A) @ L) @ Q4)) = (((append @ A) @ Q4) @ L))))). % 6.67/2.70 thf(fact_7949_finite__vimage__IntI, axiom, (! [A:$tType,B:$tType,F5:(set @ A),H2:(B > A),A3:(set @ B)]: ((((finite_finite2 @ A) @ F5) => (((((inj_on @ B) @ A) @ H2) @ A3) => ((finite_finite2 @ B) @ (((inf_inf @ (set @ B)) @ ((((vimage @ B) @ A) @ H2) @ F5)) @ A3))))))). % 6.67/2.70 thf(fact_7950_finite__vimageI, axiom, (! [B:$tType,A:$tType,F5:(set @ A),H2:(B > A)]: ((((finite_finite2 @ A) @ F5) => (((((inj_on @ B) @ A) @ H2) @ (top_top @ (set @ B))) => ((finite_finite2 @ B) @ ((((vimage @ B) @ A) @ H2) @ F5))))))). % 6.67/2.70 thf(fact_7951_vimage__inter__cong, axiom, (! [B:$tType,A:$tType,S3:(set @ A),F2:(A > B),G:(A > B),Y:(set @ B)]: (((! [W:A]: (((((member @ A) @ W) @ S3) => ((F2 @ W) = (G @ W))))) => ((((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ F2) @ Y)) @ S3) = (((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ G) @ Y)) @ S3)))))). % 6.67/2.70 thf(fact_7952_vimage__Diff, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ B),B3:(set @ B)]: ((((((vimage @ A) @ B) @ F2) @ (((minus_minus @ (set @ B)) @ A3) @ B3)) = (((minus_minus @ (set @ A)) @ ((((vimage @ A) @ B) @ F2) @ A3)) @ ((((vimage @ A) @ B) @ F2) @ B3)))))). % 6.67/2.70 thf(fact_7953_vimage__Compl, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ B)]: ((((((vimage @ A) @ B) @ F2) @ ((uminus_uminus @ (set @ B)) @ A3)) = ((uminus_uminus @ (set @ A)) @ ((((vimage @ A) @ B) @ F2) @ A3)))))). % 6.67/2.70 thf(fact_7954_surj__vimage__empty, axiom, (! [B:$tType,A:$tType,F2:(B > A),A3:(set @ A)]: (((((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((((((vimage @ B) @ A) @ F2) @ A3) = (bot_bot @ (set @ B))) = (A3 = (bot_bot @ (set @ A)))))))). % 6.67/2.70 thf(fact_7955_image__subset__iff__subset__vimage, axiom, (! [B:$tType,A:$tType,F2:(B > A),A3:(set @ B),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F2) @ A3)) @ B3) = (((ord_less_eq @ (set @ B)) @ A3) @ ((((vimage @ B) @ A) @ F2) @ B3)))))). % 6.67/2.70 thf(fact_7956_image__vimage__subset, axiom, (! [B:$tType,A:$tType,F2:(B > A),A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F2) @ ((((vimage @ B) @ A) @ F2) @ A3))) @ A3)))). % 6.67/2.70 thf(fact_7957_vimage__subsetD, axiom, (! [A:$tType,B:$tType,F2:(B > A),B3:(set @ A),A3:(set @ B)]: (((((((image @ B) @ A) @ F2) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((((ord_less_eq @ (set @ B)) @ ((((vimage @ B) @ A) @ F2) @ B3)) @ A3) => (((ord_less_eq @ (set @ A)) @ B3) @ ((((image @ B) @ A) @ F2) @ A3))))))). % 6.67/2.70 thf(fact_7958_finite__vimageD, axiom, (! [A:$tType,B:$tType,H2:(A > B),F5:(set @ B)]: ((((finite_finite2 @ A) @ ((((vimage @ A) @ B) @ H2) @ F5)) => ((((((image @ A) @ B) @ H2) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((finite_finite2 @ B) @ F5)))))). % 6.67/2.70 thf(fact_7959_finite__vimageD_H, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ B)]: ((((finite_finite2 @ A) @ ((((vimage @ A) @ B) @ F2) @ A3)) => ((((ord_less_eq @ (set @ B)) @ A3) @ ((((image @ A) @ B) @ F2) @ (top_top @ (set @ A)))) => ((finite_finite2 @ B) @ A3)))))). % 6.67/2.70 thf(fact_7960_inf__img__fin__domE, axiom, (! [B:$tType,A:$tType,F2:(B > A),A3:(set @ B)]: ((((finite_finite2 @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => ((~ (((finite_finite2 @ B) @ A3))) => (~ ((! [Y4:A]: (((((member @ A) @ Y4) @ ((((image @ B) @ A) @ F2) @ A3)) => ((finite_finite2 @ B) @ ((((vimage @ B) @ A) @ F2) @ (((insert @ A) @ Y4) @ (bot_bot @ (set @ A))))))))))))))). % 6.67/2.70 thf(fact_7961_inf__img__fin__dom, axiom, (! [B:$tType,A:$tType,F2:(B > A),A3:(set @ B)]: ((((finite_finite2 @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => ((~ (((finite_finite2 @ B) @ A3))) => (? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F2) @ A3)) & (~ (((finite_finite2 @ B) @ ((((vimage @ B) @ A) @ F2) @ (((insert @ A) @ X5) @ (bot_bot @ (set @ A))))))))))))))). % 6.67/2.70 thf(fact_7962_vimage__subsetI, axiom, (! [B:$tType,A:$tType,F2:(A > B),B3:(set @ B),A3:(set @ A)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((ord_less_eq @ (set @ B)) @ B3) @ ((((image @ A) @ B) @ F2) @ A3)) => (((ord_less_eq @ (set @ A)) @ ((((vimage @ A) @ B) @ F2) @ B3)) @ A3)))))). % 6.67/2.70 thf(fact_7963_finite__finite__vimage__IntI, axiom, (! [A:$tType,B:$tType,F5:(set @ A),H2:(B > A),A3:(set @ B)]: ((((finite_finite2 @ A) @ F5) => ((! [Y4:A]: (((((member @ A) @ Y4) @ F5) => ((finite_finite2 @ B) @ (((inf_inf @ (set @ B)) @ ((((vimage @ B) @ A) @ H2) @ (((insert @ A) @ Y4) @ (bot_bot @ (set @ A))))) @ A3))))) => ((finite_finite2 @ B) @ (((inf_inf @ (set @ B)) @ ((((vimage @ B) @ A) @ H2) @ F5)) @ A3))))))). % 6.67/2.70 thf(fact_7964_vimage__eq__UN, axiom, (! [B:$tType,A:$tType]: ((((vimage @ A) @ B) = (^ [F3:(A > B),B6:(set @ B)]: (((complete_Sup_Sup @ (set @ A)) @ ((((image @ B) @ (set @ A)) @ (^ [Y2:B]: (((((vimage @ A) @ B) @ F3) @ (((insert @ B) @ Y2) @ (bot_bot @ (set @ B))))))) @ B6)))))))). % 6.67/2.70 thf(fact_7965_inf__img__fin__dom_H, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B)]: ((((finite_finite2 @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => ((~ (((finite_finite2 @ B) @ A3))) => (? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F2) @ A3)) & (~ (((finite_finite2 @ B) @ (((inf_inf @ (set @ B)) @ ((((vimage @ B) @ A) @ F2) @ (((insert @ A) @ X5) @ (bot_bot @ (set @ A))))) @ A3)))))))))))). % 6.67/2.70 thf(fact_7966_inf__img__fin__domE_H, axiom, (! [A:$tType,B:$tType,F2:(B > A),A3:(set @ B)]: ((((finite_finite2 @ A) @ ((((image @ B) @ A) @ F2) @ A3)) => ((~ (((finite_finite2 @ B) @ A3))) => (~ ((! [Y4:A]: (((((member @ A) @ Y4) @ ((((image @ B) @ A) @ F2) @ A3)) => ((finite_finite2 @ B) @ (((inf_inf @ (set @ B)) @ ((((vimage @ B) @ A) @ F2) @ (((insert @ A) @ Y4) @ (bot_bot @ (set @ A))))) @ A3)))))))))))). % 6.67/2.70 thf(fact_7967_card__vimage__inj, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ B)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((ord_less_eq @ (set @ B)) @ A3) @ ((((image @ A) @ B) @ F2) @ (top_top @ (set @ A)))) => (((finite_card @ A) @ ((((vimage @ A) @ B) @ F2) @ A3)) = ((finite_card @ B) @ A3))))))). % 6.67/2.70 thf(fact_7968_card__vimage__inj__on__le, axiom, (! [A:$tType,B:$tType,F2:(A > B),D6:(set @ A),A3:(set @ B)]: ((((((inj_on @ A) @ B) @ F2) @ D6) => (((finite_finite2 @ B) @ A3) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ (((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ F2) @ A3)) @ D6))) @ ((finite_card @ B) @ A3))))))). % 6.67/2.70 thf(fact_7969_rotate__rev, axiom, (! [A:$tType,N2:nat,Xs:(list @ A)]: (((((rotate @ A) @ N2) @ ((rev @ A) @ Xs)) = ((rev @ A) @ (((rotate @ A) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (((modulo_modulo @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)))) @ Xs)))))). % 6.67/2.70 thf(fact_7970_set__decode__div__2, axiom, (! [X3:nat]: (((nat_set_decode @ (((divide_divide @ nat) @ X3) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = ((((vimage @ nat) @ nat) @ suc) @ (nat_set_decode @ X3)))))). % 6.67/2.70 thf(fact_7971_set__encode__vimage__Suc, axiom, (! [A3:(set @ nat)]: (((nat_set_encode @ ((((vimage @ nat) @ nat) @ suc) @ A3)) = (((divide_divide @ nat) @ (nat_set_encode @ A3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))). % 6.67/2.70 thf(fact_7972_inj__vimage__singleton, axiom, (! [B:$tType,A:$tType,F2:(A > B),A2:B]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => (((ord_less_eq @ (set @ A)) @ ((((vimage @ A) @ B) @ F2) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B))))) @ (((insert @ A) @ ((the @ A) @ (^ [X:A]: (((F2 @ X) = A2))))) @ (bot_bot @ (set @ A)))))))). % 6.67/2.70 thf(fact_7973_inj__on__vimage__singleton, axiom, (! [B:$tType,A:$tType,F2:(A > B),A3:(set @ A),A2:B]: ((((((inj_on @ A) @ B) @ F2) @ A3) => (((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ F2) @ (((insert @ B) @ A2) @ (bot_bot @ (set @ B))))) @ A3)) @ (((insert @ A) @ ((the @ A) @ (^ [X:A]: (((((member @ A) @ X) @ A3) & ((F2 @ X) = A2)))))) @ (bot_bot @ (set @ A)))))))). % 6.67/2.70 thf(fact_7974_inv__image__partition, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o),Ys:(list @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ ((set2 @ A) @ Xs)) => (P @ X5)))) => ((! [Y4:A]: (((((member @ A) @ Y4) @ ((set2 @ A) @ Ys)) => (~ ((P @ Y4)))))) => (((((vimage @ (list @ A)) @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((partition @ A) @ P)) @ (((insert @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys)) @ (bot_bot @ (set @ ((product_prod @ (list @ A)) @ (list @ A)))))) = (((shuffles @ A) @ Xs) @ Ys))))))). % 6.67/2.70 thf(fact_7975_nth__rotate, axiom, (! [A:$tType,N2:nat,Xs:(list @ A),M:nat]: (((((ord_less @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ (((rotate @ A) @ M) @ Xs)) @ N2) = (((nth @ A) @ Xs) @ (((modulo_modulo @ nat) @ (((plus_plus @ nat) @ M) @ N2)) @ ((size_size @ (list @ A)) @ Xs)))))))). % 6.67/2.70 thf(fact_7976_hd__rotate__conv__nth, axiom, (! [A:$tType,Xs:(list @ A),N2:nat]: (((Xs != (nil @ A)) => (((hd @ A) @ (((rotate @ A) @ N2) @ Xs)) = (((nth @ A) @ Xs) @ (((modulo_modulo @ nat) @ N2) @ ((size_size @ (list @ A)) @ Xs)))))))). % 6.67/2.70 thf(fact_7977_dual__min, axiom, (! [A:$tType]: (((linorder @ A) => (((min @ A) @ (^ [X:A,Y2:A]: ((((ord_less_eq @ A) @ Y2) @ X)))) = (ord_max @ A)))))). % 6.67/2.70 thf(fact_7978_listrel1__subset__listrel, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),R3:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R2) @ R3) => ((((refl_on @ A) @ (top_top @ (set @ A))) @ R3) => (((ord_less_eq @ (set @ ((product_prod @ (list @ A)) @ (list @ A)))) @ ((listrel1 @ A) @ R2)) @ (((listrel @ A) @ A) @ R3))))))). % 6.67/2.70 thf(fact_7979_refl__onD2, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X3:A,Y:A]: (((((refl_on @ A) @ A3) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ R2) => (((member @ A) @ Y) @ A3)))))). % 6.67/2.70 thf(fact_7980_refl__onD1, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),X3:A,Y:A]: (((((refl_on @ A) @ A3) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ R2) => (((member @ A) @ X3) @ A3)))))). % 6.67/2.70 thf(fact_7981_refl__onD, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),A2:A]: (((((refl_on @ A) @ A3) @ R2) => ((((member @ A) @ A2) @ A3) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ R2)))))). % 6.67/2.70 thf(fact_7982_refl__on__empty, axiom, (! [A:$tType]: ((((refl_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))). % 6.67/2.70 thf(fact_7983_ord_Omin__def, axiom, (! [A:$tType]: (((min @ A) = (^ [Less_eq:(A > (A > $o)),A4:A,B4:A]: (((((if @ A) @ ((Less_eq @ A4) @ B4)) @ A4) @ B4))))))). % 6.67/2.70 thf(fact_7984_ord_Omin_Ocong, axiom, (! [A:$tType]: (((min @ A) = (min @ A))))). % 6.67/2.70 thf(fact_7985_refl__on__def, axiom, (! [A:$tType]: (((refl_on @ A) = (^ [A6:(set @ A),R5:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R5) @ ((((product_Sigma @ A) @ A) @ A6) @ (^ [Uu3:A]: (A6)))) & (! [X:A]: (((((member @ A) @ X) @ A6) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ X)) @ R5))))))))))). % 6.67/2.70 thf(fact_7986_refl__onI, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A3:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R2) @ ((((product_Sigma @ A) @ A) @ A3) @ (^ [Uu3:A]: (A3)))) => ((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ X5)) @ R2)))) => (((refl_on @ A) @ A3) @ R2)))))). % 6.67/2.70 thf(fact_7987_refl__on__def_H, axiom, (! [A:$tType]: (((refl_on @ A) = (^ [A6:(set @ A),R5:(set @ ((product_prod @ A) @ A))]: (((! [X:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ X) @ R5) => (((((product_case_prod @ A) @ A) @ $o) @ (^ [Y2:A,Z6:A]: (((((member @ A) @ Y2) @ A6) & (((member @ A) @ Z6) @ A6))))) @ X)))) & (! [X:A]: (((((member @ A) @ X) @ A6) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ X)) @ R5))))))))))). % 6.67/2.70 thf(fact_7988_refl__on__singleton, axiom, (! [A:$tType,X3:A]: ((((refl_on @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ X3)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))). % 6.67/2.70 thf(fact_7989_refl__on__domain, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A)),A2:A,B2:A]: (((((refl_on @ A) @ A3) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2) => ((((member @ A) @ A2) @ A3) & (((member @ A) @ B2) @ A3))))))). % 6.67/2.70 thf(fact_7990_Restr__natLeq, axiom, (! [N2:nat]: (((((inf_inf @ (set @ ((product_prod @ nat) @ nat))) @ bNF_Ca8665028551170535155natLeq) @ ((((product_Sigma @ nat) @ nat) @ ((collect @ nat) @ (^ [X:nat]: ((((ord_less @ nat) @ X) @ N2))))) @ (^ [Uu3:nat]: (((collect @ nat) @ (^ [X:nat]: ((((ord_less @ nat) @ X) @ N2)))))))) = ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [X:nat,Y2:nat]: (((((ord_less @ nat) @ X) @ N2) & ((((ord_less @ nat) @ Y2) @ N2) & (((ord_less_eq @ nat) @ X) @ Y2))))))))))). % 6.67/2.70 thf(fact_7991_natLeq__def, axiom, (bNF_Ca8665028551170535155natLeq = ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (ord_less_eq @ nat))))). % 6.67/2.70 thf(fact_7992_Restr__natLeq2, axiom, (! [N2:nat]: (((((inf_inf @ (set @ ((product_prod @ nat) @ nat))) @ bNF_Ca8665028551170535155natLeq) @ ((((product_Sigma @ nat) @ nat) @ (((order_underS @ nat) @ bNF_Ca8665028551170535155natLeq) @ N2)) @ (^ [Uu3:nat]: ((((order_underS @ nat) @ bNF_Ca8665028551170535155natLeq) @ N2))))) = ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [X:nat,Y2:nat]: (((((ord_less @ nat) @ X) @ N2) & ((((ord_less @ nat) @ Y2) @ N2) & (((ord_less_eq @ nat) @ X) @ Y2))))))))))). % 6.67/2.70 thf(fact_7993_linear__order__on__singleton, axiom, (! [A:$tType,X3:A]: ((((order_679001287576687338der_on @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ X3)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))). % 6.67/2.70 thf(fact_7994_lnear__order__on__empty, axiom, (! [A:$tType]: ((((order_679001287576687338der_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))). % 6.67/2.70 thf(fact_7995_underS__def, axiom, (! [A:$tType]: (((order_underS @ A) = (^ [R5:(set @ ((product_prod @ A) @ A)),A4:A]: (((collect @ A) @ (^ [B4:A]: (((B4 != A4) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B4) @ A4)) @ R5))))))))))). % 6.67/2.70 thf(fact_7996_natLeq__underS__less, axiom, (! [N2:nat]: (((((order_underS @ nat) @ bNF_Ca8665028551170535155natLeq) @ N2) = ((collect @ nat) @ (^ [X:nat]: ((((ord_less @ nat) @ X) @ N2)))))))). % 6.67/2.70 thf(fact_7997_underS__I, axiom, (! [A:$tType,I:A,J:A,R:(set @ ((product_prod @ A) @ A))]: (((I != J) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R) => (((member @ A) @ I) @ (((order_underS @ A) @ R) @ J))))))). % 6.67/2.70 thf(fact_7998_underS__E, axiom, (! [A:$tType,I:A,R:(set @ ((product_prod @ A) @ A)),J:A]: (((((member @ A) @ I) @ (((order_underS @ A) @ R) @ J)) => ((I != J) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R)))))). % 6.67/2.70 thf(fact_7999_total__on__singleton, axiom, (! [A:$tType,X3:A]: ((((total_on @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ X3)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))). % 6.67/2.70 thf(fact_8000_Arg__bounded, axiom, (! [Z2:complex]: (((((ord_less @ real) @ ((uminus_uminus @ real) @ pi)) @ (arg @ Z2)) & (((ord_less_eq @ real) @ (arg @ Z2)) @ pi))))). % 6.67/2.70 thf(fact_8001_total__lenlex, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((((total_on @ A) @ (top_top @ (set @ A))) @ R2) => (((total_on @ (list @ A)) @ (top_top @ (set @ (list @ A)))) @ ((lenlex @ A) @ R2)))))). % 6.67/2.70 thf(fact_8002_total__lexord, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((((total_on @ A) @ (top_top @ (set @ A))) @ R2) => (((total_on @ (list @ A)) @ (top_top @ (set @ (list @ A)))) @ ((lexord @ A) @ R2)))))). % 6.67/2.70 thf(fact_8003_total__on__empty, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((total_on @ A) @ (bot_bot @ (set @ A))) @ R2)))). % 6.67/2.70 thf(fact_8004_total__onI, axiom, (! [A:$tType,A3:(set @ A),R2:(set @ ((product_prod @ A) @ A))]: (((! [X5:A,Y4:A]: (((((member @ A) @ X5) @ A3) => ((((member @ A) @ Y4) @ A3) => ((X5 != Y4) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Y4)) @ R2) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X5)) @ R2))))))) => (((total_on @ A) @ A3) @ R2))))). % 6.67/2.70 thf(fact_8005_total__on__def, axiom, (! [A:$tType]: (((total_on @ A) = (^ [A6:(set @ A),R5:(set @ ((product_prod @ A) @ A))]: ((! [X:A]: (((((member @ A) @ X) @ A6) => (! [Y2:A]: (((((member @ A) @ Y2) @ A6) => ((X != Y2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y2)) @ R5) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ X)) @ R5))))))))))))))). % 6.67/2.70 thf(fact_8006_Arg__correct, axiom, (! [Z2:complex]: (((Z2 != (zero_zero @ complex)) => ((((sgn_sgn @ complex) @ Z2) = (cis @ (arg @ Z2))) & ((((ord_less @ real) @ ((uminus_uminus @ real) @ pi)) @ (arg @ Z2)) & (((ord_less_eq @ real) @ (arg @ Z2)) @ pi))))))). % 6.67/2.70 thf(fact_8007_Total__subset__Id, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((((total_on @ A) @ ((field2 @ A) @ R2)) @ R2) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R2) @ (id2 @ A)) => ((R2 = (bot_bot @ (set @ ((product_prod @ A) @ A)))) | (? [A5:A]: ((R2 = (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ A5)) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))))))))). % 6.67/2.70 thf(fact_8008_Field__empty, axiom, (! [A:$tType]: ((((field2 @ A) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))) = (bot_bot @ (set @ A)))))). % 6.67/2.70 thf(fact_8009_Field__insert, axiom, (! [A:$tType,A2:A,B2:A,R2:(set @ ((product_prod @ A) @ A))]: ((((field2 @ A) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2)) = (((sup_sup @ (set @ A)) @ (((insert @ A) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))) @ ((field2 @ A) @ R2)))))). % 6.67/2.70 thf(fact_8010_underS__Field2, axiom, (! [A:$tType,A2:A,R2:(set @ ((product_prod @ A) @ A))]: (((((member @ A) @ A2) @ ((field2 @ A) @ R2)) => (((ord_less @ (set @ A)) @ (((order_underS @ A) @ R2) @ A2)) @ ((field2 @ A) @ R2)))))). % 6.67/2.70 thf(fact_8011_Order__Relation_OunderS__Field, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A2:A]: ((((ord_less_eq @ (set @ A)) @ (((order_underS @ A) @ R2) @ A2)) @ ((field2 @ A) @ R2))))). % 6.67/2.70 thf(fact_8012_underS__empty, axiom, (! [A:$tType,A2:A,R2:(set @ ((product_prod @ A) @ A))]: (((~ ((((member @ A) @ A2) @ ((field2 @ A) @ R2)))) => ((((order_underS @ A) @ R2) @ A2) = (bot_bot @ (set @ A))))))). % 6.67/2.70 thf(fact_8013_underS__Field3, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A2:A]: (((((field2 @ A) @ R2) != (bot_bot @ (set @ A))) => (((ord_less @ (set @ A)) @ (((order_underS @ A) @ R2) @ A2)) @ ((field2 @ A) @ R2)))))). % 6.67/2.70 thf(fact_8014_FieldI2, axiom, (! [A:$tType,I:A,J:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R) => (((member @ A) @ J) @ ((field2 @ A) @ R)))))). % 6.67/2.70 thf(fact_8015_FieldI1, axiom, (! [A:$tType,I:A,J:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R) => (((member @ A) @ I) @ ((field2 @ A) @ R)))))). % 6.67/2.70 thf(fact_8016_mono__Field, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R2) @ S) => (((ord_less_eq @ (set @ A)) @ ((field2 @ A) @ R2)) @ ((field2 @ A) @ S)))))). % 6.67/2.70 thf(fact_8017_finite__Field, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R2) => ((finite_finite2 @ A) @ ((field2 @ A) @ R2)))))). % 6.67/2.70 thf(fact_8018_Field__Restr__subset, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ ((field2 @ A) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R2) @ ((((product_Sigma @ A) @ A) @ A3) @ (^ [Uu3:A]: (A3)))))) @ A3)))). % 6.67/2.70 thf(fact_8019_Field__natLeq__on, axiom, (! [N2:nat]: ((((field2 @ nat) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [X:nat,Y2:nat]: (((((ord_less @ nat) @ X) @ N2) & ((((ord_less @ nat) @ Y2) @ N2) & (((ord_less_eq @ nat) @ X) @ Y2)))))))) = ((collect @ nat) @ (^ [X:nat]: ((((ord_less @ nat) @ X) @ N2)))))))). % 6.67/2.70 thf(fact_8020_underS__incl__iff, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A2:A,B2:A]: (((((order_679001287576687338der_on @ A) @ ((field2 @ A) @ R2)) @ R2) => ((((member @ A) @ A2) @ ((field2 @ A) @ R2)) => ((((member @ A) @ B2) @ ((field2 @ A) @ R2)) => ((((ord_less_eq @ (set @ A)) @ (((order_underS @ A) @ R2) @ A2)) @ (((order_underS @ A) @ R2) @ B2)) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2)))))))). % 6.67/2.70 thf(fact_8021_Linear__order__in__diff__Id, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A2:A,B2:A]: (((((order_679001287576687338der_on @ A) @ ((field2 @ A) @ R2)) @ R2) => ((((member @ A) @ A2) @ ((field2 @ A) @ R2)) => ((((member @ A) @ B2) @ ((field2 @ A) @ R2)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ B2)) @ R2) = (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ A2)) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R2) @ (id2 @ A)))))))))))). % 6.67/2.70 thf(fact_8022_Refl__Field__Restr2, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A3:(set @ A)]: (((((refl_on @ A) @ ((field2 @ A) @ R2)) @ R2) => ((((ord_less_eq @ (set @ A)) @ A3) @ ((field2 @ A) @ R2)) => (((field2 @ A) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R2) @ ((((product_Sigma @ A) @ A) @ A3) @ (^ [Uu3:A]: (A3))))) = A3)))))). % 6.67/2.70 thf(fact_8023_cis__Arg__unique, axiom, (! [Z2:complex,X3:real]: (((((sgn_sgn @ complex) @ Z2) = (cis @ X3)) => ((((ord_less @ real) @ ((uminus_uminus @ real) @ pi)) @ X3) => ((((ord_less_eq @ real) @ X3) @ pi) => ((arg @ Z2) = X3))))))). % 6.67/2.70 thf(fact_8024_Under__def, axiom, (! [A:$tType]: (((order_Under @ A) = (^ [R5:(set @ ((product_prod @ A) @ A)),A6:(set @ A)]: (((collect @ A) @ (^ [B4:A]: (((((member @ A) @ B4) @ ((field2 @ A) @ R5)) & (! [X:A]: (((((member @ A) @ X) @ A6) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B4) @ X)) @ R5)))))))))))))). % 6.67/2.70 thf(fact_8025_UnderS__def, axiom, (! [A:$tType]: (((order_UnderS @ A) = (^ [R5:(set @ ((product_prod @ A) @ A)),A6:(set @ A)]: (((collect @ A) @ (^ [B4:A]: (((((member @ A) @ B4) @ ((field2 @ A) @ R5)) & (! [X:A]: (((((member @ A) @ X) @ A6) => ((B4 != X) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B4) @ X)) @ R5))))))))))))))). % 6.67/2.70 thf(fact_8026_Above__def, axiom, (! [A:$tType]: (((order_Above @ A) = (^ [R5:(set @ ((product_prod @ A) @ A)),A6:(set @ A)]: (((collect @ A) @ (^ [B4:A]: (((((member @ A) @ B4) @ ((field2 @ A) @ R5)) & (! [X:A]: (((((member @ A) @ X) @ A6) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ B4)) @ R5)))))))))))))). % 6.67/2.70 thf(fact_8027_bij__betw__roots__unity, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((((bij_betw @ nat) @ complex) @ (^ [K3:nat]: ((cis @ (((divide_divide @ real) @ (((times_times @ real) @ (((times_times @ real) @ ((numeral_numeral @ real) @ (bit0 @ one2))) @ pi)) @ ((semiring_1_of_nat @ real) @ K3))) @ ((semiring_1_of_nat @ real) @ N2)))))) @ ((set_ord_lessThan @ nat) @ N2)) @ ((collect @ complex) @ (^ [Z6:complex]: (((((power_power @ complex) @ Z6) @ N2) = (one_one @ complex)))))))))). % 6.67/2.70 thf(fact_8028_bij__betw__add, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,A3:(set @ A),B3:(set @ A)]: (((((((bij_betw @ A) @ A) @ ((plus_plus @ A) @ A2)) @ A3) @ B3) = (((((image @ A) @ A) @ ((plus_plus @ A) @ A2)) @ A3) = B3)))))))). % 6.67/2.70 thf(fact_8029_bij__betw__combine, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ A),B3:(set @ B),C5:(set @ A),D6:(set @ B)]: (((((((bij_betw @ A) @ B) @ F2) @ A3) @ B3) => ((((((bij_betw @ A) @ B) @ F2) @ C5) @ D6) => (((((inf_inf @ (set @ B)) @ B3) @ D6) = (bot_bot @ (set @ B))) => (((((bij_betw @ A) @ B) @ F2) @ (((sup_sup @ (set @ A)) @ A3) @ C5)) @ (((sup_sup @ (set @ B)) @ B3) @ D6)))))))). % 6.67/2.70 thf(fact_8030_bij__betw__partition, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ A),C5:(set @ A),B3:(set @ B),D6:(set @ B)]: (((((((bij_betw @ A) @ B) @ F2) @ (((sup_sup @ (set @ A)) @ A3) @ C5)) @ (((sup_sup @ (set @ B)) @ B3) @ D6)) => ((((((bij_betw @ A) @ B) @ F2) @ C5) @ D6) => (((((inf_inf @ (set @ A)) @ A3) @ C5) = (bot_bot @ (set @ A))) => (((((inf_inf @ (set @ B)) @ B3) @ D6) = (bot_bot @ (set @ B))) => (((((bij_betw @ A) @ B) @ F2) @ A3) @ B3)))))))). % 6.67/2.70 thf(fact_8031_bij__betw__disjoint__Un, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ A),C5:(set @ B),G:(A > B),B3:(set @ A),D6:(set @ B)]: (((((((bij_betw @ A) @ B) @ F2) @ A3) @ C5) => ((((((bij_betw @ A) @ B) @ G) @ B3) @ D6) => (((((inf_inf @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))) => (((((inf_inf @ (set @ B)) @ C5) @ D6) = (bot_bot @ (set @ B))) => (((((bij_betw @ A) @ B) @ (^ [X:A]: (((((if @ B) @ (((member @ A) @ X) @ A3)) @ (F2 @ X)) @ (G @ X))))) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) @ (((sup_sup @ (set @ B)) @ C5) @ D6))))))))). % 6.67/2.70 thf(fact_8032_bij__betw__subset, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ A),A16:(set @ B),B3:(set @ A),B15:(set @ B)]: (((((((bij_betw @ A) @ B) @ F2) @ A3) @ A16) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => ((((((image @ A) @ B) @ F2) @ B3) = B15) => (((((bij_betw @ A) @ B) @ F2) @ B3) @ B15))))))). % 6.67/2.70 thf(fact_8033_bij__betw__byWitness, axiom, (! [A:$tType,B:$tType,A3:(set @ A),F11:(B > A),F2:(A > B),A16:(set @ B)]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => ((F11 @ (F2 @ X5)) = X5)))) => ((! [X5:B]: (((((member @ B) @ X5) @ A16) => ((F2 @ (F11 @ X5)) = X5)))) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A3)) @ A16) => ((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F11) @ A16)) @ A3) => (((((bij_betw @ A) @ B) @ F2) @ A3) @ A16)))))))). % 6.67/2.70 thf(fact_8034_bij__betw__UNION__chain, axiom, (! [B:$tType,C:$tType,A:$tType,I6:(set @ A),A3:(A > (set @ B)),F2:(B > C),A16:(A > (set @ C))]: (((! [I4:A,J2:A]: (((((member @ A) @ I4) @ I6) => ((((member @ A) @ J2) @ I6) => ((((ord_less_eq @ (set @ B)) @ (A3 @ I4)) @ (A3 @ J2)) | (((ord_less_eq @ (set @ B)) @ (A3 @ J2)) @ (A3 @ I4))))))) => ((! [I4:A]: (((((member @ A) @ I4) @ I6) => (((((bij_betw @ B) @ C) @ F2) @ (A3 @ I4)) @ (A16 @ I4))))) => (((((bij_betw @ B) @ C) @ F2) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image @ A) @ (set @ B)) @ A3) @ I6))) @ ((complete_Sup_Sup @ (set @ C)) @ ((((image @ A) @ (set @ C)) @ A16) @ I6)))))))). % 6.67/2.70 thf(fact_8035_bij__betw__comp__iff2, axiom, (! [C:$tType,A:$tType,B:$tType,F11:(A > B),A16:(set @ A),A19:(set @ B),F2:(C > A),A3:(set @ C)]: (((((((bij_betw @ A) @ B) @ F11) @ A16) @ A19) => ((((ord_less_eq @ (set @ A)) @ ((((image @ C) @ A) @ F2) @ A3)) @ A16) => ((((((bij_betw @ C) @ A) @ F2) @ A3) @ A16) = (((((bij_betw @ C) @ B) @ (((((comp @ A) @ B) @ C) @ F11) @ F2)) @ A3) @ A19))))))). % 6.67/2.70 thf(fact_8036_finite__same__card__bij, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B3:(set @ B)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ B) @ B3) => ((((finite_card @ A) @ A3) = ((finite_card @ B) @ B3)) => (? [H6:(A > B)]: ((((((bij_betw @ A) @ B) @ H6) @ A3) @ B3))))))))). % 6.67/2.70 thf(fact_8037_bij__betw__iff__card, axiom, (! [A:$tType,B:$tType,A3:(set @ A),B3:(set @ B)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ B) @ B3) => ((? [F3:(A > B)]: ((((((bij_betw @ A) @ B) @ F3) @ A3) @ B3))) = (((finite_card @ A) @ A3) = ((finite_card @ B) @ B3)))))))). % 6.67/2.70 thf(fact_8038_bij__betw__empty2, axiom, (! [B:$tType,A:$tType,F2:(A > B),A3:(set @ A)]: (((((((bij_betw @ A) @ B) @ F2) @ A3) @ (bot_bot @ (set @ B))) => (A3 = (bot_bot @ (set @ A))))))). % 6.67/2.70 thf(fact_8039_bij__betw__empty1, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ B)]: (((((((bij_betw @ A) @ B) @ F2) @ (bot_bot @ (set @ A))) @ A3) => (A3 = (bot_bot @ (set @ B))))))). % 6.67/2.70 thf(fact_8040_bij__betw__finite, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ A),B3:(set @ B)]: (((((((bij_betw @ A) @ B) @ F2) @ A3) @ B3) => (((finite_finite2 @ A) @ A3) = ((finite_finite2 @ B) @ B3)))))). % 6.67/2.70 thf(fact_8041_notIn__Un__bij__betw, axiom, (! [A:$tType,B:$tType,B2:A,A3:(set @ A),F2:(A > B),A16:(set @ B)]: (((~ ((((member @ A) @ B2) @ A3))) => ((~ ((((member @ B) @ (F2 @ B2)) @ A16))) => ((((((bij_betw @ A) @ B) @ F2) @ A3) @ A16) => (((((bij_betw @ A) @ B) @ F2) @ (((sup_sup @ (set @ A)) @ A3) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))) @ (((sup_sup @ (set @ B)) @ A16) @ (((insert @ B) @ (F2 @ B2)) @ (bot_bot @ (set @ B))))))))))). % 6.67/2.70 thf(fact_8042_notIn__Un__bij__betw3, axiom, (! [A:$tType,B:$tType,B2:A,A3:(set @ A),F2:(A > B),A16:(set @ B)]: (((~ ((((member @ A) @ B2) @ A3))) => ((~ ((((member @ B) @ (F2 @ B2)) @ A16))) => ((((((bij_betw @ A) @ B) @ F2) @ A3) @ A16) = (((((bij_betw @ A) @ B) @ F2) @ (((sup_sup @ (set @ A)) @ A3) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))) @ (((sup_sup @ (set @ B)) @ A16) @ (((insert @ B) @ (F2 @ B2)) @ (bot_bot @ (set @ B))))))))))). % 6.67/2.70 thf(fact_8043_bij__betw__funpow, axiom, (! [A:$tType,F2:(A > A),S3:(set @ A),N2:nat]: (((((((bij_betw @ A) @ A) @ F2) @ S3) @ S3) => (((((bij_betw @ A) @ A) @ (((compow @ (A > A)) @ N2) @ F2)) @ S3) @ S3))))). % 6.67/2.70 thf(fact_8044_bij__fn, axiom, (! [A:$tType,F2:(A > A),N2:nat]: (((((((bij_betw @ A) @ A) @ F2) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))) => (((((bij_betw @ A) @ A) @ (((compow @ (A > A)) @ N2) @ F2)) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))))))). % 6.67/2.70 thf(fact_8045_finite__vimage__iff, axiom, (! [A:$tType,B:$tType,H2:(A > B),F5:(set @ B)]: (((((((bij_betw @ A) @ B) @ H2) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((finite_finite2 @ A) @ ((((vimage @ A) @ B) @ H2) @ F5)) = ((finite_finite2 @ B) @ F5)))))). % 6.67/2.70 thf(fact_8046_infinite__imp__bij__betw2, axiom, (! [A:$tType,A3:(set @ A),A2:A]: (((~ (((finite_finite2 @ A) @ A3))) => (? [H6:(A > A)]: ((((((bij_betw @ A) @ A) @ H6) @ A3) @ (((sup_sup @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))))). % 6.67/2.70 thf(fact_8047_infinite__imp__bij__betw, axiom, (! [A:$tType,A3:(set @ A),A2:A]: (((~ (((finite_finite2 @ A) @ A3))) => (? [H6:(A > A)]: ((((((bij_betw @ A) @ A) @ H6) @ A3) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))))). % 6.67/2.70 thf(fact_8048_vimage__subset__eq, axiom, (! [B:$tType,A:$tType,F2:(A > B),B3:(set @ B),A3:(set @ A)]: (((((((bij_betw @ A) @ B) @ F2) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => ((((ord_less_eq @ (set @ A)) @ ((((vimage @ A) @ B) @ F2) @ B3)) @ A3) = (((ord_less_eq @ (set @ B)) @ B3) @ ((((image @ A) @ B) @ F2) @ A3))))))). % 6.67/2.70 thf(fact_8049_ex__bij__betw__nat__finite, axiom, (! [A:$tType,M7:(set @ A)]: ((((finite_finite2 @ A) @ M7) => (? [H6:(nat > A)]: ((((((bij_betw @ nat) @ A) @ H6) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ M7))) @ M7))))))). % 6.67/2.70 thf(fact_8050_ex__bij__betw__nat__finite__1, axiom, (! [A:$tType,M7:(set @ A)]: ((((finite_finite2 @ A) @ M7) => (? [H6:(nat > A)]: ((((((bij_betw @ nat) @ A) @ H6) @ (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ ((finite_card @ A) @ M7))) @ M7))))))). % 6.67/2.70 thf(fact_8051_sum_Oreindex__bij__betw__not__neutral, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comm_monoid_add @ A) => (! [S6:(set @ B),T6:(set @ C),H2:(B > C),S3:(set @ B),T4:(set @ C),G:(C > A)]: ((((finite_finite2 @ B) @ S6) => (((finite_finite2 @ C) @ T6) => ((((((bij_betw @ B) @ C) @ H2) @ (((minus_minus @ (set @ B)) @ S3) @ S6)) @ (((minus_minus @ (set @ C)) @ T4) @ T6)) => ((! [A5:B]: (((((member @ B) @ A5) @ S6) => ((G @ (H2 @ A5)) = (zero_zero @ A))))) => ((! [B5:C]: (((((member @ C) @ B5) @ T6) => ((G @ B5) = (zero_zero @ A))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X:B]: ((G @ (H2 @ X))))) @ S3) = ((((groups7311177749621191930dd_sum @ C) @ A) @ G) @ T4))))))))))))). % 6.67/2.70 thf(fact_8052_prod_Oreindex__bij__betw__not__neutral, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comm_monoid_mult @ A) => (! [S6:(set @ B),T6:(set @ C),H2:(B > C),S3:(set @ B),T4:(set @ C),G:(C > A)]: ((((finite_finite2 @ B) @ S6) => (((finite_finite2 @ C) @ T6) => ((((((bij_betw @ B) @ C) @ H2) @ (((minus_minus @ (set @ B)) @ S3) @ S6)) @ (((minus_minus @ (set @ C)) @ T4) @ T6)) => ((! [A5:B]: (((((member @ B) @ A5) @ S6) => ((G @ (H2 @ A5)) = (one_one @ A))))) => ((! [B5:C]: (((((member @ C) @ B5) @ T6) => ((G @ B5) = (one_one @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X:B]: ((G @ (H2 @ X))))) @ S3) = ((((groups7121269368397514597t_prod @ C) @ A) @ G) @ T4))))))))))))). % 6.67/2.70 thf(fact_8053_bij__betw__nth, axiom, (! [A:$tType,Xs:(list @ A),A3:(set @ nat),B3:(set @ A)]: ((((distinct @ A) @ Xs) => ((A3 = ((set_ord_lessThan @ nat) @ ((size_size @ (list @ A)) @ Xs))) => ((B3 = ((set2 @ A) @ Xs)) => (((((bij_betw @ nat) @ A) @ ((nth @ A) @ Xs)) @ A3) @ B3))))))). % 6.67/2.70 thf(fact_8054_ex__bij__betw__strict__mono__card, axiom, (! [A:$tType]: (((wellorder @ A) => (! [M7:(set @ A)]: ((((finite_finite2 @ A) @ M7) => (~ ((! [H6:(nat > A)]: (((((((bij_betw @ nat) @ A) @ H6) @ ((set_ord_lessThan @ nat) @ ((finite_card @ A) @ M7))) @ M7) => (~ (((((strict_mono_on @ nat) @ A) @ H6) @ ((set_ord_lessThan @ nat) @ ((finite_card @ A) @ M7))))))))))))))))). % 6.67/2.70 thf(fact_8055_Schroeder__Bernstein, axiom, (! [A:$tType,B:$tType,F2:(A > B),A3:(set @ A),B3:(set @ B),G:(B > A)]: ((((((inj_on @ A) @ B) @ F2) @ A3) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F2) @ A3)) @ B3) => (((((inj_on @ B) @ A) @ G) @ B3) => ((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ G) @ B3)) @ A3) => (? [H6:(A > B)]: ((((((bij_betw @ A) @ B) @ H6) @ A3) @ B3)))))))))). % 6.67/2.70 thf(fact_8056_bij__betw__nth__root__unity, axiom, (! [C2:complex,N2:nat]: (((C2 != (zero_zero @ complex)) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => (((((bij_betw @ complex) @ complex) @ ((times_times @ complex) @ (((times_times @ complex) @ ((real_Vector_of_real @ complex) @ ((root @ N2) @ ((real_V7770717601297561774m_norm @ complex) @ C2)))) @ (cis @ (((divide_divide @ real) @ (arg @ C2)) @ ((semiring_1_of_nat @ real) @ N2)))))) @ ((collect @ complex) @ (^ [Z6:complex]: (((((power_power @ complex) @ Z6) @ N2) = (one_one @ complex)))))) @ ((collect @ complex) @ (^ [Z6:complex]: (((((power_power @ complex) @ Z6) @ N2) = C2)))))))))). % 6.67/2.70 thf(fact_8057_bij__betw__Suc, axiom, (! [M7:(set @ nat),N7:(set @ nat)]: (((((((bij_betw @ nat) @ nat) @ suc) @ M7) @ N7) = (((((image @ nat) @ nat) @ suc) @ M7) = N7))))). % 6.67/2.70 thf(fact_8058_le__rel__bool__arg__iff, axiom, (! [A:$tType]: (((ord @ A) => ((ord_less_eq @ ($o > A)) = (^ [X6:($o > A),Y9:($o > A)]: (((((ord_less_eq @ A) @ (X6 @ $false)) @ (Y9 @ $false)) & (((ord_less_eq @ A) @ (X6 @ $true)) @ (Y9 @ $true)))))))))). % 6.67/2.70 thf(fact_8059_ex__bij__betw__finite__nat, axiom, (! [A:$tType,M7:(set @ A)]: ((((finite_finite2 @ A) @ M7) => (? [H6:(A > nat)]: ((((((bij_betw @ A) @ nat) @ H6) @ M7) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ M7))))))))). % 6.67/2.70 thf(fact_8060_finite__Linear__order__induct, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),X3:A,P:(A > $o)]: (((((order_679001287576687338der_on @ A) @ ((field2 @ A) @ R2)) @ R2) => ((((member @ A) @ X3) @ ((field2 @ A) @ R2)) => (((finite_finite2 @ ((product_prod @ A) @ A)) @ R2) => ((! [X5:A]: (((((member @ A) @ X5) @ ((field2 @ A) @ R2)) => ((! [Y5:A]: (((((member @ A) @ Y5) @ (((order_aboveS @ A) @ R2) @ X5)) => (P @ Y5)))) => (P @ X5))))) => (P @ X3)))))))). % 6.67/2.70 thf(fact_8061_Arg__def, axiom, (arg = (^ [Z6:complex]: (((((if @ real) @ (Z6 = (zero_zero @ complex))) @ (zero_zero @ real)) @ ((fChoice @ real) @ (^ [A4:real]: (((((sgn_sgn @ complex) @ Z6) = (cis @ A4)) & ((((ord_less @ real) @ ((uminus_uminus @ real) @ pi)) @ A4) & (((ord_less_eq @ real) @ A4) @ pi))))))))))). % 6.67/2.70 thf(fact_8062_some__in__eq, axiom, (! [A:$tType,A3:(set @ A)]: (((((member @ A) @ ((fChoice @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A3))))) @ A3) = (A3 != (bot_bot @ (set @ A))))))). % 6.67/2.70 thf(fact_8063_exE__some, axiom, (! [A:$tType,P:(A > $o),C2:A]: (((? [X_12:A]: ((P @ X_12))) => ((C2 = ((fChoice @ A) @ P)) => (P @ C2)))))). % 6.67/2.70 thf(fact_8064_aboveS__def, axiom, (! [A:$tType]: (((order_aboveS @ A) = (^ [R5:(set @ ((product_prod @ A) @ A)),A4:A]: (((collect @ A) @ (^ [B4:A]: (((B4 != A4) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ B4)) @ R5))))))))))). % 6.67/2.70 thf(fact_8065_arg__min__SOME__Min, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [S3:(set @ A),F2:(A > B)]: ((((finite_finite2 @ A) @ S3) => (((((lattic7623131987881927897min_on @ A) @ B) @ F2) @ S3) = ((fChoice @ A) @ (^ [Y2:A]: (((((member @ A) @ Y2) @ S3) & ((F2 @ Y2) = ((lattic643756798350308766er_Min @ B) @ ((((image @ A) @ B) @ F2) @ S3))))))))))))))). % 6.67/2.70 thf(fact_8066_cofinal__def, axiom, (! [A:$tType]: (((bNF_Ca7293521722713021262ofinal @ A) = (^ [A6:(set @ A),R5:(set @ ((product_prod @ A) @ A))]: ((! [X:A]: (((((member @ A) @ X) @ ((field2 @ A) @ R5)) => (? [Y2:A]: (((((member @ A) @ Y2) @ A6) & ((X != Y2) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y2)) @ R5)))))))))))))). % 6.67/2.70 thf(fact_8067_bsqr__def, axiom, (! [A:$tType]: (((bNF_Wellorder_bsqr @ A) = (^ [R5:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A))) @ ((((product_case_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A)) @ $o) @ ((((product_case_prod @ A) @ A) @ (((product_prod @ A) @ A) > $o)) @ (^ [A1:A,A22:A]: (((((product_case_prod @ A) @ A) @ $o) @ (^ [B16:A,B23:A]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A1) @ (((insert @ A) @ A22) @ (((insert @ A) @ B16) @ (((insert @ A) @ B23) @ (bot_bot @ (set @ A))))))) @ ((field2 @ A) @ R5)) & (((A1 = B16) & (A22 = B23)) | ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ ((((bNF_We1388413361240627857o_max2 @ A) @ R5) @ A1) @ A22)) @ ((((bNF_We1388413361240627857o_max2 @ A) @ R5) @ B16) @ B23))) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R5) @ (id2 @ A))) | (((((((bNF_We1388413361240627857o_max2 @ A) @ R5) @ A1) @ A22) = ((((bNF_We1388413361240627857o_max2 @ A) @ R5) @ B16) @ B23)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A1) @ B16)) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R5) @ (id2 @ A)))) | ((((((bNF_We1388413361240627857o_max2 @ A) @ R5) @ A1) @ A22) = ((((bNF_We1388413361240627857o_max2 @ A) @ R5) @ B16) @ B23)) & ((A1 = B16) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A22) @ B23)) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R5) @ (id2 @ A))))))))))))))))))))))). % 6.67/2.70 thf(fact_8068_Eps__case__prod__eq, axiom, (! [A:$tType,B:$tType,X3:A,Y:B]: ((((fChoice @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X9:A,Y6:B]: (((X3 = X9) & (Y = Y6)))))) = ((((product_Pair @ A) @ B) @ X3) @ Y))))). % 6.67/2.70 thf(fact_8069_split__paired__Eps, axiom, (! [B:$tType,A:$tType]: (((fChoice @ ((product_prod @ A) @ B)) = (^ [P3:(((product_prod @ A) @ B) > $o)]: (((fChoice @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A4:A,B4:B]: ((P3 @ ((((product_Pair @ A) @ B) @ A4) @ B4)))))))))))). % 6.67/2.70 thf(fact_8070_to__nat__on__def, axiom, (! [A:$tType]: (((countable_to_nat_on @ A) = (^ [S7:(set @ A)]: (((fChoice @ (A > nat)) @ (^ [F3:(A > nat)]: (((((finite_finite2 @ A) @ S7) => (((((bij_betw @ A) @ nat) @ F3) @ S7) @ ((set_ord_lessThan @ nat) @ ((finite_card @ A) @ S7)))) & ((~ (((finite_finite2 @ A) @ S7))) => (((((bij_betw @ A) @ nat) @ F3) @ S7) @ (top_top @ (set @ nat)))))))))))))). % 6.67/2.70 thf(fact_8071_Refl__under__underS, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A2:A]: (((((refl_on @ A) @ ((field2 @ A) @ R2)) @ R2) => ((((member @ A) @ A2) @ ((field2 @ A) @ R2)) => ((((order_under @ A) @ R2) @ A2) = (((sup_sup @ (set @ A)) @ (((order_underS @ A) @ R2) @ A2)) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))))))))). % 6.67/2.70 thf(fact_8072_under__def, axiom, (! [A:$tType]: (((order_under @ A) = (^ [R5:(set @ ((product_prod @ A) @ A)),A4:A]: (((collect @ A) @ (^ [B4:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B4) @ A4)) @ R5)))))))))). % 6.67/2.70 thf(fact_8073_underS__subset__under, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A2:A]: ((((ord_less_eq @ (set @ A)) @ (((order_underS @ A) @ R2) @ A2)) @ (((order_under @ A) @ R2) @ A2))))). % 6.67/2.70 thf(fact_8074_under__Field, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A2:A]: ((((ord_less_eq @ (set @ A)) @ (((order_under @ A) @ R2) @ A2)) @ ((field2 @ A) @ R2))))). % 6.67/2.70 thf(fact_8075_to__nat__on__finite, axiom, (! [A:$tType,S3:(set @ A)]: ((((finite_finite2 @ A) @ S3) => (((((bij_betw @ A) @ nat) @ ((countable_to_nat_on @ A) @ S3)) @ S3) @ ((set_ord_lessThan @ nat) @ ((finite_card @ A) @ S3))))))). % 6.67/2.70 thf(fact_8076_Linear__order__wf__diff__Id, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((((order_679001287576687338der_on @ A) @ ((field2 @ A) @ R2)) @ R2) => (((wf @ A) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R2) @ (id2 @ A))) = (! [A6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ ((field2 @ A) @ R2)) => ((A6 != (bot_bot @ (set @ A))) => (? [X:A]: (((((member @ A) @ X) @ A6) & (! [Y2:A]: (((((member @ A) @ Y2) @ A6) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y2)) @ R2)))))))))))))))). % 6.67/2.70 thf(fact_8077_slice__eq__mask, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N2:nat,M:nat,A2:A]: (((((bit_se4730199178511100633sh_bit @ A) @ N2) @ (((bit_se2584673776208193580ke_bit @ A) @ M) @ (((bit_se4197421643247451524op_bit @ A) @ N2) @ A2))) = (((bit_se5824344872417868541ns_and @ A) @ A2) @ (((bit_se5824344872417868541ns_and @ A) @ ((bit_se2239418461657761734s_mask @ A) @ (((plus_plus @ nat) @ M) @ N2))) @ ((bit_ri4277139882892585799ns_not @ A) @ ((bit_se2239418461657761734s_mask @ A) @ N2))))))))))). % 6.67/2.70 thf(fact_8078_wf__empty, axiom, (! [A:$tType]: (((wf @ A) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))). % 6.67/2.70 thf(fact_8079_wf__listrel1__iff, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((wf @ (list @ A)) @ ((listrel1 @ A) @ R2)) = ((wf @ A) @ R2))))). % 6.67/2.70 thf(fact_8080_wf__lex, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((wf @ A) @ R2) => ((wf @ (list @ A)) @ ((lex @ A) @ R2)))))). % 6.67/2.70 thf(fact_8081_wf__lenlex, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: ((((wf @ A) @ R2) => ((wf @ (list @ A)) @ ((lenlex @ A) @ R2)))))). % 6.67/2.70 thf(fact_8082_not__negative__int__iff, axiom, (! [K:int]: (((((ord_less @ int) @ ((bit_ri4277139882892585799ns_not @ int) @ K)) @ (zero_zero @ int)) = (((ord_less_eq @ int) @ (zero_zero @ int)) @ K))))). % 6.67/2.70 thf(fact_8083_not__nonnegative__int__iff, axiom, (! [K:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((bit_ri4277139882892585799ns_not @ int) @ K)) = (((ord_less @ int) @ K) @ (zero_zero @ int)))))). % 6.67/2.70 thf(fact_8084_bit_Oconj__cancel__left, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X3:A]: (((((bit_se5824344872417868541ns_and @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ X3)) @ X3) = (zero_zero @ A)))))))). % 6.67/2.70 thf(fact_8085_bit_Oconj__cancel__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X3:A]: (((((bit_se5824344872417868541ns_and @ A) @ X3) @ ((bit_ri4277139882892585799ns_not @ A) @ X3)) = (zero_zero @ A)))))))). % 6.67/2.70 thf(fact_8086_bit_Ocompl__one, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((bit_ri4277139882892585799ns_not @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) = (zero_zero @ A)))))). % 6.67/2.70 thf(fact_8087_bit_Ocompl__zero, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((bit_ri4277139882892585799ns_not @ A) @ (zero_zero @ A)) = ((uminus_uminus @ A) @ (one_one @ A))))))). % 6.67/2.70 thf(fact_8088_wf__insert, axiom, (! [A:$tType,Y:A,X3:A,R2:(set @ ((product_prod @ A) @ A))]: ((((wf @ A) @ (((insert @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X3)) @ R2)) = (((wf @ A) @ R2) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y)) @ ((transitive_rtrancl @ A) @ R2))))))))). % 6.67/2.70 thf(fact_8089_reduction__pairI, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ A) @ A))]: ((((wf @ A) @ R) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((((relcomp @ A) @ A) @ A) @ R) @ S3)) @ R) => ((fun_reduction_pair @ A) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ S3))))))). % 6.67/2.70 thf(fact_8090_wf__bounded__measure, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),Ub:(A > nat),F2:(A > nat)]: (((! [A5:A,B5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ A5)) @ R2) => ((((ord_less_eq @ nat) @ (Ub @ B5)) @ (Ub @ A5)) & ((((ord_less_eq @ nat) @ (F2 @ B5)) @ (Ub @ A5)) & (((ord_less @ nat) @ (F2 @ A5)) @ (F2 @ B5))))))) => ((wf @ A) @ R2))))). % 6.67/2.70 thf(fact_8091_wfI, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R2) @ ((((product_Sigma @ A) @ A) @ A3) @ (^ [Uu3:A]: (B3)))) => ((! [X5:A,P8:(A > $o)]: (((! [Xa:A]: (((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Xa)) @ R2) => (P8 @ Y4)))) => (P8 @ Xa)))) => ((((member @ A) @ X5) @ A3) => ((((member @ A) @ X5) @ B3) => (P8 @ X5)))))) => ((wf @ A) @ R2)))))). % 6.67/2.70 thf(fact_8092_wf__lexn, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),N2:nat]: ((((wf @ A) @ R2) => ((wf @ (list @ A)) @ (((lexn @ A) @ R2) @ N2)))))). % 6.67/2.70 thf(fact_8093_wf__int__ge__less__than2, axiom, (! [D2:int]: (((wf @ int) @ (int_ge_less_than2 @ D2))))). % 6.67/2.70 thf(fact_8094_wf__int__ge__less__than, axiom, (! [D2:int]: (((wf @ int) @ (int_ge_less_than @ D2))))). % 6.67/2.70 thf(fact_8095_wf__no__loop, axiom, (! [B:$tType,R:(set @ ((product_prod @ B) @ B))]: ((((((((relcomp @ B) @ B) @ B) @ R) @ R) = (bot_bot @ (set @ ((product_prod @ B) @ B)))) => ((wf @ B) @ R))))). % 6.67/2.70 thf(fact_8096_wfE__min_H, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),Q:(set @ A)]: ((((wf @ A) @ R) => ((Q != (bot_bot @ (set @ A))) => (~ ((! [Z3:A]: (((((member @ A) @ Z3) @ Q) => (~ ((! [Y5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ Z3)) @ R) => (~ ((((member @ A) @ Y5) @ Q)))))))))))))))))). % 6.67/2.70 thf(fact_8097_wf__linord__ex__has__least, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ A)),P:(B > $o),K:B,M:(B > A)]: ((((wf @ A) @ R2) => ((! [X5:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X5) @ Y4)) @ ((transitive_trancl @ A) @ R2)) = (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X5)) @ ((transitive_rtrancl @ A) @ R2))))))) => ((P @ K) => (? [X5:B]: (((P @ X5) & (! [Y5:B]: (((P @ Y5) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (M @ X5)) @ (M @ Y5))) @ ((transitive_rtrancl @ A) @ R2)))))))))))))). % 6.67/2.70 thf(fact_8098_wf__def, axiom, (! [A:$tType]: (((wf @ A) = (^ [R5:(set @ ((product_prod @ A) @ A))]: ((! [P3:(A > $o)]: (((! [X:A]: (((! [Y2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ X)) @ R5) => (P3 @ Y2)))) => (P3 @ X)))) => (! [X6:A]: ((P3 @ X6)))))))))))). % 6.67/2.70 thf(fact_8099_wfE__min, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),X3:A,Q:(set @ A)]: ((((wf @ A) @ R) => ((((member @ A) @ X3) @ Q) => (~ ((! [Z3:A]: (((((member @ A) @ Z3) @ Q) => (~ ((! [Y5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ Z3)) @ R) => (~ ((((member @ A) @ Y5) @ Q)))))))))))))))))). % 6.67/2.70 thf(fact_8100_wfI__min, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((! [X5:A,Q8:(set @ A)]: (((((member @ A) @ X5) @ Q8) => (? [Xa:A]: (((((member @ A) @ Xa) @ Q8) & (! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Xa)) @ R) => (~ ((((member @ A) @ Y4) @ Q8)))))))))))) => ((wf @ A) @ R))))). % 6.67/2.70 thf(fact_8101_wfUNIVI, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A))]: (((! [P8:(A > $o),X5:A]: (((! [Xa:A]: (((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Xa)) @ R2) => (P8 @ Y4)))) => (P8 @ Xa)))) => (P8 @ X5)))) => ((wf @ A) @ R2))))). % 6.67/2.70 thf(fact_8102_wf__asym, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A2:A,X3:A]: ((((wf @ A) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ X3)) @ R2) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ A2)) @ R2)))))))). % 6.67/2.70 thf(fact_8103_wf__induct, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),P:(A > $o),A2:A]: ((((wf @ A) @ R2) => ((! [X5:A]: (((! [Y5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ X5)) @ R2) => (P @ Y5)))) => (P @ X5)))) => (P @ A2)))))). % 6.67/2.70 thf(fact_8104_wf__irrefl, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A2:A]: ((((wf @ A) @ R2) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ R2))))))). % 6.67/2.70 thf(fact_8105_wf__not__sym, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A2:A,X3:A]: ((((wf @ A) @ R2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ X3)) @ R2) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ A2)) @ R2)))))))). % 6.67/2.70 thf(fact_8106_wf__not__refl, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A2:A]: ((((wf @ A) @ R2) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A2) @ A2)) @ R2))))))). % 6.67/2.70 thf(fact_8107_wf__eq__minimal, axiom, (! [A:$tType]: (((wf @ A) = (^ [R5:(set @ ((product_prod @ A) @ A))]: ((! [Q7:(set @ A)]: (((? [X:A]: ((((member @ A) @ X) @ Q7))) => (? [X:A]: (((((member @ A) @ X) @ Q7) & (! [Y2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ X)) @ R5) => (~ ((((member @ A) @ Y2) @ Q7)))))))))))))))))). % 6.67/2.70 thf(fact_8108_wf__induct__rule, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),P:(A > $o),A2:A]: ((((wf @ A) @ R2) => ((! [X5:A]: (((! [Y5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ X5)) @ R2) => (P @ Y5)))) => (P @ X5)))) => (P @ A2)))))). % 6.67/2.70 thf(fact_8109_wf__iff__no__infinite__down__chain, axiom, (! [A:$tType]: (((wf @ A) = (^ [R5:(set @ ((product_prod @ A) @ A))]: ((~ ((? [F3:(nat > A)]: ((! [I3:nat]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (F3 @ (suc @ I3))) @ (F3 @ I3))) @ R5))))))))))))). % 6.67/2.70 thf(fact_8110_wf__no__infinite__down__chainE, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),F2:(nat > A)]: ((((wf @ A) @ R2) => (~ ((! [K2:nat]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (F2 @ (suc @ K2))) @ (F2 @ K2))) @ R2))))))))). % 6.67/2.70 thf(fact_8111_minus__eq__not__plus__1, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((uminus_uminus @ A) = (^ [A4:A]: ((((plus_plus @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ A4)) @ (one_one @ A))))))))). % 6.67/2.70 thf(fact_8112_not__diff__distrib, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A2:A,B2:A]: ((((bit_ri4277139882892585799ns_not @ A) @ (((minus_minus @ A) @ A2) @ B2)) = (((plus_plus @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ A2)) @ B2)))))))). % 6.67/2.70 thf(fact_8113_not__add__distrib, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A2:A,B2:A]: ((((bit_ri4277139882892585799ns_not @ A) @ (((plus_plus @ A) @ A2) @ B2)) = (((minus_minus @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ A2)) @ B2)))))))). % 6.67/2.70 thf(fact_8114_wf, axiom, (! [A:$tType]: (((wellorder @ A) => ((wf @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (ord_less @ A)))))))). % 6.67/2.70 thf(fact_8115_wf__if__measure, axiom, (! [A:$tType,P:(A > $o),F2:(A > nat),G:(A > A)]: (((! [X5:A]: (((P @ X5) => (((ord_less @ nat) @ (F2 @ (G @ X5))) @ (F2 @ X5))))) => ((wf @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [Y2:A,X:A]: (((P @ X) & (Y2 = (G @ X)))))))))))). % 6.67/2.70 thf(fact_8116_wf__less, axiom, ((wf @ nat) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (ord_less @ nat))))). % 6.67/2.70 thf(fact_8117_wf__eq__minimal2, axiom, (! [A:$tType]: (((wf @ A) = (^ [R5:(set @ ((product_prod @ A) @ A))]: ((! [A6:(set @ A)]: ((((((ord_less_eq @ (set @ A)) @ A6) @ ((field2 @ A) @ R5)) & (A6 != (bot_bot @ (set @ A)))) => (? [X:A]: (((((member @ A) @ X) @ A6) & (! [Y2:A]: (((((member @ A) @ Y2) @ A6) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ X)) @ R5)))))))))))))))))). % 6.67/2.70 thf(fact_8118_wf__bounded__set, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ A)),Ub:(A > (set @ B)),F2:(A > (set @ B))]: (((! [A5:A,B5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ A5)) @ R2) => (((finite_finite2 @ B) @ (Ub @ A5)) & ((((ord_less_eq @ (set @ B)) @ (Ub @ B5)) @ (Ub @ A5)) & ((((ord_less_eq @ (set @ B)) @ (F2 @ B5)) @ (Ub @ A5)) & (((ord_less @ (set @ B)) @ (F2 @ A5)) @ (F2 @ B5)))))))) => ((wf @ A) @ R2))))). % 6.67/2.70 thf(fact_8119_take__bit__not__mask__eq__0, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((bit_se2584673776208193580ke_bit @ A) @ M) @ ((bit_ri4277139882892585799ns_not @ A) @ ((bit_se2239418461657761734s_mask @ A) @ N2))) = (zero_zero @ A))))))))). % 6.67/2.70 thf(fact_8120_push__bit__mask__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [M:nat,N2:nat]: (((((bit_se4730199178511100633sh_bit @ A) @ M) @ ((bit_se2239418461657761734s_mask @ A) @ N2)) = (((bit_se5824344872417868541ns_and @ A) @ ((bit_se2239418461657761734s_mask @ A) @ (((plus_plus @ nat) @ N2) @ M))) @ ((bit_ri4277139882892585799ns_not @ A) @ ((bit_se2239418461657761734s_mask @ A) @ M)))))))))). % 6.67/2.70 thf(fact_8121_finite__subset__wf, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((wf @ (set @ A)) @ ((collect @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_case_prod @ (set @ A)) @ (set @ A)) @ $o) @ (^ [X6:(set @ A),Y9:(set @ A)]: (((((ord_less @ (set @ A)) @ X6) @ Y9) & (((ord_less_eq @ (set @ A)) @ Y9) @ A3))))))))))). % 6.67/2.70 thf(fact_8122_bit_Ocompl__unique, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X3:A,Y:A]: ((((((bit_se5824344872417868541ns_and @ A) @ X3) @ Y) = (zero_zero @ A)) => (((((bit_se1065995026697491101ons_or @ A) @ X3) @ Y) = ((uminus_uminus @ A) @ (one_one @ A))) => (((bit_ri4277139882892585799ns_not @ A) @ X3) = Y))))))))). % 6.67/2.70 thf(fact_8123_bit__not__iff__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A2:A,N2:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ A2)) @ N2) = (((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N2) != (zero_zero @ A)) & (~ ((((bit_se5641148757651400278ts_bit @ A) @ A2) @ N2))))))))))). % 6.67/2.70 thf(fact_8124_bit_Oabstract__boolean__algebra__sym__diff__axioms, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((((((boolea3799213064322606851m_diff @ A) @ (bit_se5824344872417868541ns_and @ A)) @ (bit_se1065995026697491101ons_or @ A)) @ (bit_ri4277139882892585799ns_not @ A)) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (bit_se5824344971392196577ns_xor @ A)))))). % 6.67/2.70 thf(fact_8125_dependent__wf__choice, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),P:((A > B) > (A > (B > $o)))]: ((((wf @ A) @ R) => ((! [F4:(A > B),G4:(A > B),X5:A,R4:B]: (((! [Z4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z4) @ X5)) @ R) => ((F4 @ Z4) = (G4 @ Z4))))) => ((((P @ F4) @ X5) @ R4) = (((P @ G4) @ X5) @ R4))))) => ((! [X5:A,F4:(A > B)]: (((! [Y5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y5) @ X5)) @ R) => (((P @ F4) @ Y5) @ (F4 @ Y5))))) => (? [X_12:B]: ((((P @ F4) @ X5) @ X_12)))))) => (? [F4:(A > B)]: ((! [X2:A]: ((((P @ F4) @ X2) @ (F4 @ X2)))))))))))). % 6.67/2.70 thf(fact_8126_dependent__wellorder__choice, axiom, (! [B:$tType,A:$tType]: (((wellorder @ A) => (! [P:((A > B) > (A > (B > $o)))]: (((! [R4:B,F4:(A > B),G4:(A > B),X5:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X5) => ((F4 @ Y5) = (G4 @ Y5))))) => ((((P @ F4) @ X5) @ R4) = (((P @ G4) @ X5) @ R4))))) => ((! [X5:A,F4:(A > B)]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X5) => (((P @ F4) @ Y5) @ (F4 @ Y5))))) => (? [X_12:B]: ((((P @ F4) @ X5) @ X_12)))))) => (? [F4:(A > B)]: ((! [X2:A]: ((((P @ F4) @ X2) @ (F4 @ X2)))))))))))))). % 6.67/2.70 thf(fact_8127_chains__extend, axiom, (! [A:$tType,C2:(set @ (set @ A)),S3:(set @ (set @ A)),Z2:(set @ A)]: (((((member @ (set @ (set @ A))) @ C2) @ ((chains @ A) @ S3)) => ((((member @ (set @ A)) @ Z2) @ S3) => ((! [X5:(set @ A)]: (((((member @ (set @ A)) @ X5) @ C2) => (((ord_less_eq @ (set @ A)) @ X5) @ Z2)))) => (((member @ (set @ (set @ A))) @ (((sup_sup @ (set @ (set @ A))) @ (((insert @ (set @ A)) @ Z2) @ (bot_bot @ (set @ (set @ A))))) @ C2)) @ ((chains @ A) @ S3)))))))). % 6.67/2.70 thf(fact_8128_bit_Oabstract__boolean__algebra__axioms, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((((((boolea2506097494486148201lgebra @ A) @ (bit_se5824344872417868541ns_and @ A)) @ (bit_se1065995026697491101ons_or @ A)) @ (bit_ri4277139882892585799ns_not @ A)) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))). % 6.67/2.70 thf(fact_8129_chainsD, axiom, (! [A:$tType,C2:(set @ (set @ A)),S3:(set @ (set @ A)),X3:(set @ A),Y:(set @ A)]: (((((member @ (set @ (set @ A))) @ C2) @ ((chains @ A) @ S3)) => ((((member @ (set @ A)) @ X3) @ C2) => ((((member @ (set @ A)) @ Y) @ C2) => ((((ord_less_eq @ (set @ A)) @ X3) @ Y) | (((ord_less_eq @ (set @ A)) @ Y) @ X3)))))))). % 6.67/2.70 thf(fact_8130_Zorn__Lemma2, axiom, (! [A:$tType,A3:(set @ (set @ A))]: (((! [X5:(set @ (set @ A))]: (((((member @ (set @ (set @ A))) @ X5) @ ((chains @ A) @ A3)) => (? [Xa:(set @ A)]: (((((member @ (set @ A)) @ Xa) @ A3) & (! [Xb2:(set @ A)]: (((((member @ (set @ A)) @ Xb2) @ X5) => (((ord_less_eq @ (set @ A)) @ Xb2) @ Xa)))))))))) => (? [X5:(set @ A)]: (((((member @ (set @ A)) @ X5) @ A3) & (! [Xa:(set @ A)]: (((((member @ (set @ A)) @ Xa) @ A3) => ((((ord_less_eq @ (set @ A)) @ X5) @ Xa) => (Xa = X5)))))))))))). % 6.67/2.70 thf(fact_8131_Zorn__Lemma, axiom, (! [A:$tType,A3:(set @ (set @ A))]: (((! [X5:(set @ (set @ A))]: (((((member @ (set @ (set @ A))) @ X5) @ ((chains @ A) @ A3)) => (((member @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ X5)) @ A3)))) => (? [X5:(set @ A)]: (((((member @ (set @ A)) @ X5) @ A3) & (! [Xa:(set @ A)]: (((((member @ (set @ A)) @ Xa) @ A3) => ((((ord_less_eq @ (set @ A)) @ X5) @ Xa) => (Xa = X5)))))))))))). % 6.67/2.70 thf(fact_8132_boolean__algebra_Oabstract__boolean__algebra__axioms, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => ((((((boolea2506097494486148201lgebra @ A) @ (inf_inf @ A)) @ (sup_sup @ A)) @ (uminus_uminus @ A)) @ (bot_bot @ A)) @ (top_top @ A)))))). % 6.67/2.70 thf(fact_8133_and__not__num__eq__None__iff, axiom, (! [M:num,N2:num]: (((((bit_and_not_num @ M) @ N2) = (none @ num)) = ((((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ M)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N2))) = (zero_zero @ int)))))). % 6.67/2.70 thf(fact_8134_partial__order__on__well__order__on, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A3:(set @ A)]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R2) => ((((order_7125193373082350890der_on @ A) @ A3) @ R2) => ((wf @ A) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R2) @ (id2 @ A)))))))). % 6.67/2.70 thf(fact_8135_partial__order__on__empty, axiom, (! [A:$tType]: ((((order_7125193373082350890der_on @ A) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))). % 6.67/2.70 thf(fact_8136_and__not__num_Osimps_I1_J, axiom, (((bit_and_not_num @ one2) @ one2) = (none @ num))). % 6.67/2.70 thf(fact_8137_and__not__num_Osimps_I3_J, axiom, (! [N2:num]: ((((bit_and_not_num @ one2) @ (bit1 @ N2)) = (none @ num))))). % 6.67/2.70 thf(fact_8138_finite__Partial__order__induct, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),X3:A,P:(A > $o)]: (((((order_7125193373082350890der_on @ A) @ ((field2 @ A) @ R2)) @ R2) => ((((member @ A) @ X3) @ ((field2 @ A) @ R2)) => (((finite_finite2 @ ((product_prod @ A) @ A)) @ R2) => ((! [X5:A]: (((((member @ A) @ X5) @ ((field2 @ A) @ R2)) => ((! [Y5:A]: (((((member @ A) @ Y5) @ (((order_aboveS @ A) @ R2) @ X5)) => (P @ Y5)))) => (P @ X5))))) => (P @ X3)))))))). % 6.67/2.70 thf(fact_8139_and__not__num_Oelims, axiom, (! [X3:num,Xa2:num,Y:(option @ num)]: (((((bit_and_not_num @ X3) @ Xa2) = Y) => (((X3 = one2) => ((Xa2 = one2) => (Y != (none @ num)))) => (((X3 = one2) => ((? [N3:num]: ((Xa2 = (bit0 @ N3)))) => (Y != ((some @ num) @ one2)))) => (((X3 = one2) => ((? [N3:num]: ((Xa2 = (bit1 @ N3)))) => (Y != (none @ num)))) => ((! [M3:num]: (((X3 = (bit0 @ M3)) => ((Xa2 = one2) => (Y != ((some @ num) @ (bit0 @ M3))))))) => ((! [M3:num]: (((X3 = (bit0 @ M3)) => (! [N3:num]: (((Xa2 = (bit0 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_and_not_num @ M3) @ N3))))))))) => ((! [M3:num]: (((X3 = (bit0 @ M3)) => (! [N3:num]: (((Xa2 = (bit1 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_and_not_num @ M3) @ N3))))))))) => ((! [M3:num]: (((X3 = (bit1 @ M3)) => ((Xa2 = one2) => (Y != ((some @ num) @ (bit0 @ M3))))))) => ((! [M3:num]: (((X3 = (bit1 @ M3)) => (! [N3:num]: (((Xa2 = (bit0 @ N3)) => (Y != (((((case_option @ (option @ num)) @ num) @ ((some @ num) @ one2)) @ (^ [N10:num]: (((some @ num) @ (bit1 @ N10))))) @ ((bit_and_not_num @ M3) @ N3))))))))) => (~ ((! [M3:num]: (((X3 = (bit1 @ M3)) => (! [N3:num]: (((Xa2 = (bit1 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_and_not_num @ M3) @ N3))))))))))))))))))))))). % 6.67/2.70 thf(fact_8140_max__ext__eq, axiom, (! [A:$tType]: (((max_ext @ A) = (^ [R6:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_case_prod @ (set @ A)) @ (set @ A)) @ $o) @ (^ [X6:(set @ A),Y9:(set @ A)]: ((((finite_finite2 @ A) @ X6) & (((finite_finite2 @ A) @ Y9) & ((Y9 != (bot_bot @ (set @ A))) & (! [X:A]: (((((member @ A) @ X) @ X6) => (? [Y2:A]: (((((member @ A) @ Y2) @ Y9) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y2)) @ R6)))))))))))))))))))). % 6.67/2.70 thf(fact_8141_bex__empty, axiom, (! [A:$tType,P:(A > $o)]: ((~ ((? [X2:A]: (((((member @ A) @ X2) @ (bot_bot @ (set @ A))) & (P @ X2))))))))). % 6.67/2.70 thf(fact_8142_map__option__eq__Some, axiom, (! [A:$tType,B:$tType,F2:(B > A),Xo:(option @ B),Y:A]: (((((((map_option @ B) @ A) @ F2) @ Xo) = ((some @ A) @ Y)) = (? [Z6:B]: (((Xo = ((some @ B) @ Z6)) & ((F2 @ Z6) = Y)))))))). % 6.67/2.70 thf(fact_8143_option_Omap__disc__iff, axiom, (! [B:$tType,A:$tType,F2:(A > B),A2:(option @ A)]: (((((((map_option @ A) @ B) @ F2) @ A2) = (none @ B)) = (A2 = (none @ A)))))). % 6.67/2.70 thf(fact_8144_map__option__is__None, axiom, (! [A:$tType,B:$tType,F2:(B > A),Opt:(option @ B)]: (((((((map_option @ B) @ A) @ F2) @ Opt) = (none @ A)) = (Opt = (none @ B)))))). % 6.67/2.70 thf(fact_8145_None__eq__map__option__iff, axiom, (! [A:$tType,B:$tType,F2:(B > A),X3:(option @ B)]: ((((none @ A) = ((((map_option @ B) @ A) @ F2) @ X3)) = (X3 = (none @ B)))))). % 6.67/2.70 thf(fact_8146_finite__Collect__bex, axiom, (! [B:$tType,A:$tType,A3:(set @ A),Q:(B > (A > $o))]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [X:B]: ((? [Y2:A]: (((((member @ A) @ Y2) @ A3) & ((Q @ X) @ Y2)))))))) = (! [X:A]: (((((member @ A) @ X) @ A3) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [Y2:B]: (((Q @ Y2) @ X))))))))))))). % 6.67/2.70 thf(fact_8147_map__option__o__empty, axiom, (! [C:$tType,B:$tType,A:$tType,F2:(C > B)]: (((((((comp @ (option @ C)) @ (option @ B)) @ A) @ (((map_option @ C) @ B) @ F2)) @ (^ [X:A]: ((none @ C)))) = (^ [X:A]: ((none @ B))))))). % 6.67/2.70 thf(fact_8148_bex__UNIV, axiom, (! [A:$tType,P:(A > $o)]: (((? [X:A]: (((((member @ A) @ X) @ (top_top @ (set @ A))) & (P @ X)))) = (? [X6:A]: ((P @ X6))))))). % 6.67/2.70 thf(fact_8149_case__map__option, axiom, (! [B:$tType,A:$tType,C:$tType,G:A,H2:(B > A),F2:(C > B),X3:(option @ C)]: (((((((case_option @ A) @ B) @ G) @ H2) @ ((((map_option @ C) @ B) @ F2) @ X3)) = (((((case_option @ A) @ C) @ G) @ (((((comp @ B) @ A) @ C) @ H2) @ F2)) @ X3))))). % 6.67/2.70 thf(fact_8150_map__option__o__map__upd, axiom, (! [A:$tType,B:$tType,C:$tType,F2:(C > B),M:(A > (option @ C)),A2:A,B2:C]: (((((((comp @ (option @ C)) @ (option @ B)) @ A) @ (((map_option @ C) @ B) @ F2)) @ (((((fun_upd @ A) @ (option @ C)) @ M) @ A2) @ ((some @ C) @ B2))) = (((((fun_upd @ A) @ (option @ B)) @ (((((comp @ (option @ C)) @ (option @ B)) @ A) @ (((map_option @ C) @ B) @ F2)) @ M)) @ A2) @ ((some @ B) @ (F2 @ B2))))))). % 6.67/2.70 thf(fact_8151_nths__nths, axiom, (! [A:$tType,Xs:(list @ A),A3:(set @ nat),B3:(set @ nat)]: (((((nths @ A) @ (((nths @ A) @ Xs) @ A3)) @ B3) = (((nths @ A) @ Xs) @ ((collect @ nat) @ (^ [I3:nat]: (((((member @ nat) @ I3) @ A3) & (((member @ nat) @ ((finite_card @ nat) @ ((collect @ nat) @ (^ [I10:nat]: (((((member @ nat) @ I10) @ A3) & (((ord_less @ nat) @ I10) @ I3))))))) @ B3)))))))))). % 6.67/2.70 thf(fact_8152_vimage__image__eq, axiom, (! [B:$tType,A:$tType,F2:(A > B),A3:(set @ A)]: ((((((vimage @ A) @ B) @ F2) @ ((((image @ A) @ B) @ F2) @ A3)) = ((collect @ A) @ (^ [Y2:A]: ((? [X:A]: (((((member @ A) @ X) @ A3) & ((F2 @ X) = (F2 @ Y2)))))))))))). % 6.67/2.70 thf(fact_8153_option_Osize__gen__o__map, axiom, (! [B:$tType,A:$tType,F2:(B > nat),G:(A > B)]: (((((((comp @ (option @ B)) @ nat) @ (option @ A)) @ ((size_option @ B) @ F2)) @ (((map_option @ A) @ B) @ G)) = ((size_option @ A) @ (((((comp @ B) @ nat) @ A) @ F2) @ G)))))). % 6.67/2.70 thf(fact_8154_map__option_Ocomp, axiom, (! [C:$tType,B:$tType,A:$tType,F2:(B > C),G:(A > B)]: (((((((comp @ (option @ B)) @ (option @ C)) @ (option @ A)) @ (((map_option @ B) @ C) @ F2)) @ (((map_option @ A) @ B) @ G)) = (((map_option @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F2) @ G)))))). % 6.67/2.70 thf(fact_8155_option_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,G:(B > C),F2:(A > B),V2:(option @ A)]: ((((((map_option @ B) @ C) @ G) @ ((((map_option @ A) @ B) @ F2) @ V2)) = ((((map_option @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F2)) @ V2))))). % 6.67/2.70 thf(fact_8156_map__option_Ocompositionality, axiom, (! [B:$tType,C:$tType,A:$tType,F2:(B > C),G:(A > B),Option:(option @ A)]: ((((((map_option @ B) @ C) @ F2) @ ((((map_option @ A) @ B) @ G) @ Option)) = ((((map_option @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F2) @ G)) @ Option))))). % 6.67/2.70 thf(fact_8157_option_Osimps_I8_J, axiom, (! [A:$tType,B:$tType,F2:(A > B)]: ((((((map_option @ A) @ B) @ F2) @ (none @ A)) = (none @ B))))). % 6.67/2.70 thf(fact_8158_option_Omap__sel, axiom, (! [B:$tType,A:$tType,A2:(option @ A),F2:(A > B)]: (((A2 != (none @ A)) => (((the2 @ B) @ ((((map_option @ A) @ B) @ F2) @ A2)) = (F2 @ ((the2 @ A) @ A2))))))). % 6.67/2.70 thf(fact_8159_option_Omap__id, axiom, (! [A:$tType,T2:(option @ A)]: ((((((map_option @ A) @ A) @ (id @ A)) @ T2) = T2)))). % 6.67/2.70 thf(fact_8160_option_Omap__id0, axiom, (! [A:$tType]: (((((map_option @ A) @ A) @ (id @ A)) = (id @ (option @ A)))))). % 6.67/2.70 thf(fact_8161_map__option_Oidentity, axiom, (! [A:$tType]: (((((map_option @ A) @ A) @ (^ [X:A]: (X))) = (id @ (option @ A)))))). % 6.67/2.70 thf(fact_8162_option_Omap__ident, axiom, (! [A:$tType,T2:(option @ A)]: ((((((map_option @ A) @ A) @ (^ [X:A]: (X))) @ T2) = T2)))). % 6.67/2.70 thf(fact_8163_Bex__def, axiom, (! [A:$tType]: (((bex @ A) = (^ [A6:(set @ A),P3:(A > $o)]: ((? [X:A]: (((((member @ A) @ X) @ A6) & (P3 @ X)))))))))). % 6.67/2.70 thf(fact_8164_Bex__def__raw, axiom, (! [A:$tType]: (((bex @ A) = (^ [A6:(set @ A),P3:(A > $o)]: ((? [X:A]: (((((member @ A) @ X) @ A6) & (P3 @ X)))))))))). % 6.67/2.70 thf(fact_8165_map__option__cong, axiom, (! [B:$tType,A:$tType,X3:(option @ A),Y:(option @ A),F2:(A > B),G:(A > B)]: (((X3 = Y) => ((! [A5:A]: (((Y = ((some @ A) @ A5)) => ((F2 @ A5) = (G @ A5))))) => (((((map_option @ A) @ B) @ F2) @ X3) = ((((map_option @ A) @ B) @ G) @ Y))))))). % 6.67/2.70 thf(fact_8166_option_Osimps_I9_J, axiom, (! [B:$tType,A:$tType,F2:(A > B),X22:A]: ((((((map_option @ A) @ B) @ F2) @ ((some @ A) @ X22)) = ((some @ B) @ (F2 @ X22)))))). % 6.67/2.70 thf(fact_8167_Bex__fold, axiom, (! [A:$tType,A3:(set @ A),P:(A > $o)]: ((((finite_finite2 @ A) @ A3) => ((? [X:A]: (((((member @ A) @ X) @ A3) & (P @ X)))) = (((((finite_fold @ A) @ $o) @ (^ [K3:A,S8:$o]: ((S8 | (P @ K3))))) @ $false) @ A3)))))). % 6.67/2.70 thf(fact_8168_option_Oinj__map, axiom, (! [B:$tType,A:$tType,F2:(A > B)]: ((((((inj_on @ A) @ B) @ F2) @ (top_top @ (set @ A))) => ((((inj_on @ (option @ A)) @ (option @ B)) @ (((map_option @ A) @ B) @ F2)) @ (top_top @ (set @ (option @ A)))))))). % 6.67/2.70 thf(fact_8169_image__def, axiom, (! [B:$tType,A:$tType]: ((((image @ A) @ B) = (^ [F3:(A > B),A6:(set @ A)]: (((collect @ B) @ (^ [Y2:B]: ((? [X:A]: (((((member @ A) @ X) @ A6) & (Y2 = (F3 @ X)))))))))))))). % 6.67/2.70 thf(fact_8170_map__option__case, axiom, (! [A:$tType,B:$tType]: ((((map_option @ B) @ A) = (^ [F3:(B > A)]: (((((case_option @ (option @ A)) @ B) @ (none @ A)) @ (^ [X:B]: (((some @ A) @ (F3 @ X))))))))))). % 6.67/2.70 thf(fact_8171_max__extp_Ocases, axiom, (! [A:$tType,R:(A > (A > $o)),A12:(set @ A),A23:(set @ A)]: ((((((max_extp @ A) @ R) @ A12) @ A23) => (~ ((((finite_finite2 @ A) @ A12) => (((finite_finite2 @ A) @ A23) => ((A23 != ((collect @ A) @ (bot_bot @ (A > $o)))) => (~ ((! [X2:A]: (((((member @ A) @ X2) @ A12) => (? [Xa3:A]: (((((member @ A) @ Xa3) @ A23) & ((R @ X2) @ Xa3)))))))))))))))))). % 6.67/2.70 thf(fact_8172_max__extp_Osimps, axiom, (! [A:$tType]: (((max_extp @ A) = (^ [R6:(A > (A > $o)),A1:(set @ A),A22:(set @ A)]: ((((finite_finite2 @ A) @ A1) & (((finite_finite2 @ A) @ A22) & ((A22 != ((collect @ A) @ (bot_bot @ (A > $o)))) & (! [X:A]: (((((member @ A) @ X) @ A1) => (? [Y2:A]: (((((member @ A) @ Y2) @ A22) & ((R6 @ X) @ Y2)))))))))))))))). % 6.67/2.70 thf(fact_8173_max__extp_Omax__extI, axiom, (! [A:$tType,X8:(set @ A),Y7:(set @ A),R:(A > (A > $o))]: ((((finite_finite2 @ A) @ X8) => (((finite_finite2 @ A) @ Y7) => ((Y7 != ((collect @ A) @ (bot_bot @ (A > $o)))) => ((! [X5:A]: (((((member @ A) @ X5) @ X8) => (? [Xa:A]: (((((member @ A) @ Xa) @ Y7) & ((R @ X5) @ Xa))))))) => ((((max_extp @ A) @ R) @ X8) @ Y7)))))))). % 6.67/2.70 thf(fact_8174_map__of__map, axiom, (! [B:$tType,C:$tType,A:$tType,F2:(C > B),Xs:(list @ ((product_prod @ A) @ C))]: (((((map_of @ A) @ B) @ ((((map @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ C) @ ((product_prod @ A) @ B)) @ (^ [K3:A,V5:C]: (((((product_Pair @ A) @ B) @ K3) @ (F2 @ V5)))))) @ Xs)) = (((((comp @ (option @ C)) @ (option @ B)) @ A) @ (((map_option @ C) @ B) @ F2)) @ (((map_of @ A) @ C) @ Xs)))))). % 6.67/2.70 thf(fact_8175_min__ext__def, axiom, (! [A:$tType]: (((min_ext @ A) = (^ [R5:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (set @ A)) @ (set @ A))) @ (^ [Uu3:((product_prod @ (set @ A)) @ (set @ A))]: ((? [X6:(set @ A),Y9:(set @ A)]: (((Uu3 = ((((product_Pair @ (set @ A)) @ (set @ A)) @ X6) @ Y9)) & ((X6 != (bot_bot @ (set @ A))) & (! [X:A]: (((((member @ A) @ X) @ Y9) => (? [Y2:A]: (((((member @ A) @ Y2) @ X6) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ X)) @ R5)))))))))))))))))))). % 6.67/2.70 thf(fact_8176_map__project__def, axiom, (! [B:$tType,A:$tType]: ((((map_project @ A) @ B) = (^ [F3:(A > (option @ B)),A6:(set @ A)]: (((collect @ B) @ (^ [B4:B]: ((? [X:A]: (((((member @ A) @ X) @ A6) & ((F3 @ X) = ((some @ B) @ B4)))))))))))))). % 6.67/2.70 thf(fact_8177_and__num_Oelims, axiom, (! [X3:num,Xa2:num,Y:(option @ num)]: (((((bit_un7362597486090784418nd_num @ X3) @ Xa2) = Y) => (((X3 = one2) => ((Xa2 = one2) => (Y != ((some @ num) @ one2)))) => (((X3 = one2) => ((? [N3:num]: ((Xa2 = (bit0 @ N3)))) => (Y != (none @ num)))) => (((X3 = one2) => ((? [N3:num]: ((Xa2 = (bit1 @ N3)))) => (Y != ((some @ num) @ one2)))) => (((? [M3:num]: ((X3 = (bit0 @ M3)))) => ((Xa2 = one2) => (Y != (none @ num)))) => ((! [M3:num]: (((X3 = (bit0 @ M3)) => (! [N3:num]: (((Xa2 = (bit0 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_un7362597486090784418nd_num @ M3) @ N3))))))))) => ((! [M3:num]: (((X3 = (bit0 @ M3)) => (! [N3:num]: (((Xa2 = (bit1 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_un7362597486090784418nd_num @ M3) @ N3))))))))) => (((? [M3:num]: ((X3 = (bit1 @ M3)))) => ((Xa2 = one2) => (Y != ((some @ num) @ one2)))) => ((! [M3:num]: (((X3 = (bit1 @ M3)) => (! [N3:num]: (((Xa2 = (bit0 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_un7362597486090784418nd_num @ M3) @ N3))))))))) => (~ ((! [M3:num]: (((X3 = (bit1 @ M3)) => (! [N3:num]: (((Xa2 = (bit1 @ N3)) => (Y != (((((case_option @ (option @ num)) @ num) @ ((some @ num) @ one2)) @ (^ [N10:num]: (((some @ num) @ (bit1 @ N10))))) @ ((bit_un7362597486090784418nd_num @ M3) @ N3))))))))))))))))))))))). % 6.67/2.70 thf(fact_8178_and__num_Osimps_I4_J, axiom, (! [M:num]: ((((bit_un7362597486090784418nd_num @ (bit0 @ M)) @ one2) = (none @ num))))). % 6.67/2.70 thf(fact_8179_and__num_Osimps_I2_J, axiom, (! [N2:num]: ((((bit_un7362597486090784418nd_num @ one2) @ (bit0 @ N2)) = (none @ num))))). % 6.67/2.70 thf(fact_8180_and__num__eq__None__iff, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M:num,N2:num]: (((((bit_un7362597486090784418nd_num @ M) @ N2) = (none @ num)) = ((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N2)) = (zero_zero @ A))))))))). % 6.67/2.70 thf(fact_8181_numeral__and__num, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M:num,N2:num]: (((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N2)) = (((((case_option @ A) @ num) @ (zero_zero @ A)) @ (numeral_numeral @ A)) @ ((bit_un7362597486090784418nd_num @ M) @ N2))))))))). % 6.67/2.70 thf(fact_8182_xor__num_Oelims, axiom, (! [X3:num,Xa2:num,Y:(option @ num)]: (((((bit_un2480387367778600638or_num @ X3) @ Xa2) = Y) => (((X3 = one2) => ((Xa2 = one2) => (Y != (none @ num)))) => (((X3 = one2) => (! [N3:num]: (((Xa2 = (bit0 @ N3)) => (Y != ((some @ num) @ (bit1 @ N3))))))) => (((X3 = one2) => (! [N3:num]: (((Xa2 = (bit1 @ N3)) => (Y != ((some @ num) @ (bit0 @ N3))))))) => ((! [M3:num]: (((X3 = (bit0 @ M3)) => ((Xa2 = one2) => (Y != ((some @ num) @ (bit1 @ M3))))))) => ((! [M3:num]: (((X3 = (bit0 @ M3)) => (! [N3:num]: (((Xa2 = (bit0 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_un2480387367778600638or_num @ M3) @ N3))))))))) => ((! [M3:num]: (((X3 = (bit0 @ M3)) => (! [N3:num]: (((Xa2 = (bit1 @ N3)) => (Y != ((some @ num) @ (((((case_option @ num) @ num) @ one2) @ bit1) @ ((bit_un2480387367778600638or_num @ M3) @ N3)))))))))) => ((! [M3:num]: (((X3 = (bit1 @ M3)) => ((Xa2 = one2) => (Y != ((some @ num) @ (bit0 @ M3))))))) => ((! [M3:num]: (((X3 = (bit1 @ M3)) => (! [N3:num]: (((Xa2 = (bit0 @ N3)) => (Y != ((some @ num) @ (((((case_option @ num) @ num) @ one2) @ bit1) @ ((bit_un2480387367778600638or_num @ M3) @ N3)))))))))) => (~ ((! [M3:num]: (((X3 = (bit1 @ M3)) => (! [N3:num]: (((Xa2 = (bit1 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_un2480387367778600638or_num @ M3) @ N3))))))))))))))))))))))). % 6.67/2.70 thf(fact_8183_option_Orec__o__map, axiom, (! [B:$tType,C:$tType,A:$tType,G:C,Ga:(B > C),F2:(A > B)]: (((((((comp @ (option @ B)) @ C) @ (option @ A)) @ ((((rec_option @ C) @ B) @ G) @ Ga)) @ (((map_option @ A) @ B) @ F2)) = ((((rec_option @ C) @ A) @ G) @ (^ [X:A]: ((Ga @ (F2 @ X))))))))). % 6.67/2.70 thf(fact_8184_option_Osimps_I6_J, axiom, (! [A:$tType,C:$tType,F1:C,F22:(A > C)]: (((((((rec_option @ C) @ A) @ F1) @ F22) @ (none @ A)) = F1)))). % 6.67/2.70 thf(fact_8185_xor__num_Osimps_I1_J, axiom, (((bit_un2480387367778600638or_num @ one2) @ one2) = (none @ num))). % 6.67/2.70 thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice, axiom, (! [A10:$tType,A20:$tType]: (((comple6319245703460814977attice @ A20) => (condit1219197933456340205attice @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice, axiom, (! [A10:$tType,A20:$tType]: (((counta3822494911875563373attice @ A20) => (counta3822494911875563373attice @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice, axiom, (! [A10:$tType,A20:$tType]: (((comple592849572758109894attice @ A20) => (comple592849572758109894attice @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot, axiom, (! [A10:$tType,A20:$tType]: (((bounded_lattice @ A20) => (bounde4967611905675639751up_bot @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Lattices_Obounded__semilattice__inf__top, axiom, (! [A10:$tType,A20:$tType]: (((bounded_lattice @ A20) => (bounde4346867609351753570nf_top @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A10:$tType,A20:$tType]: (((comple6319245703460814977attice @ A20) => (comple6319245703460814977attice @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Boolean__Algebras_Oboolean__algebra, axiom, (! [A10:$tType,A20:$tType]: (((boolea8198339166811842893lgebra @ A20) => (boolea8198339166811842893lgebra @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Lattices_Obounded__lattice__bot, axiom, (! [A10:$tType,A20:$tType]: (((bounded_lattice @ A20) => (bounded_lattice_bot @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Complete__Partial__Order_Occpo, axiom, (! [A10:$tType,A20:$tType]: (((comple6319245703460814977attice @ A20) => (comple9053668089753744459l_ccpo @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A10:$tType,A20:$tType]: (((semilattice_sup @ A20) => (semilattice_sup @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Lattices_Osemilattice__inf, axiom, (! [A10:$tType,A20:$tType]: (((semilattice_inf @ A20) => (semilattice_inf @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Lattices_Odistrib__lattice, axiom, (! [A10:$tType,A20:$tType]: (((distrib_lattice @ A20) => (distrib_lattice @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Lattices_Obounded__lattice, axiom, (! [A10:$tType,A20:$tType]: (((bounded_lattice @ A20) => (bounded_lattice @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A10:$tType,A20:$tType]: (((order_top @ A20) => (order_top @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A10:$tType,A20:$tType]: (((order_bot @ A20) => (order_bot @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A10:$tType,A20:$tType]: (((preorder @ A20) => (preorder @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A10:$tType,A20:$tType]: ((((finite_finite @ A10) & (finite_finite @ A20)) => (finite_finite @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Lattices_Olattice, axiom, (! [A10:$tType,A20:$tType]: (((lattice @ A20) => (lattice @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Orderings_Oorder, axiom, (! [A10:$tType,A20:$tType]: (((order @ A20) => (order @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Orderings_Otop, axiom, (! [A10:$tType,A20:$tType]: (((top @ A20) => (top @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Orderings_Oord, axiom, (! [A10:$tType,A20:$tType]: (((ord @ A20) => (ord @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Orderings_Obot, axiom, (! [A10:$tType,A20:$tType]: (((bot @ A20) => (bot @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_fun___Groups_Ouminus, axiom, (! [A10:$tType,A20:$tType]: (((uminus @ A20) => (uminus @ (A10 > A20)))))). % 6.67/2.70 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, (condit6923001295902523014norder @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1, axiom, (condit1219197933456340205attice @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations, axiom, (bit_un5681908812861735899ations @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, axiom, (semiri1453513574482234551roduct @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat, axiom, (euclid5411537665997757685th_nat @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le, axiom, (ordere1937475149494474687imp_le @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring, axiom, (euclid3128863361964157862miring @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel, axiom, (euclid4440199948858584721cancel @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral, axiom, (unique1627219031080169319umeral @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel, axiom, (euclid8851590272496341667cancel @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel, axiom, (semiri6575147826004484403cancel @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add, axiom, (strict9044650504122735259up_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add, axiom, (ordere580206878836729694up_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (ordere2412721322843649153imp_le @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations, axiom, (bit_se359711467146920520ations @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict, axiom, (linord2810124833399127020strict @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add, axiom, (strict7427464778891057005id_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add, axiom, (ordere8940638589300402666id_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring, axiom, (euclid3725896446679973847miring @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Topological__Spaces_Otopological__space, axiom, (topolo4958980785337419405_space @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology, axiom, (topolo1944317154257567458pology @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Topological__Spaces_Odiscrete__topology, axiom, (topolo8865339358273720382pology @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict, axiom, (linord715952674999750819strict @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__add, axiom, (topolo5987344860129210374id_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add, axiom, (linord4140545234300271783up_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations, axiom, (bit_ri3973907225187159222ations @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology, axiom, (topolo2564578578187576103pology @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors, axiom, (semiri2026040879449505780visors @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring, axiom, (linord181362715937106298miring @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Limits_Otopological__semigroup__mult, axiom, (topolo4211221413907600880p_mult @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict, axiom, (linord8928482502909563296strict @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors, axiom, (semiri3467727345109120633visors @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add, axiom, (ordere6658533253407199908up_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs, axiom, (ordere166539214618696060dd_abs @ int)). % 6.67/2.70 thf(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize, axiom, (semiri6843258321239162965malize @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add, axiom, (ordere6911136660526730532id_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add, axiom, (linord5086331880401160121up_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add, axiom, (cancel2418104881723323429up_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors, axiom, (ring_15535105094025558882visors @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Limits_Otopological__monoid__add, axiom, (topolo6943815403480290642id_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add, axiom, (cancel1802427076303600483id_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Olinordered__ring__strict, axiom, (linord4710134922213307826strict @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel, axiom, (comm_s4317794764714335236cancel @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits, axiom, (bit_semiring_bits @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Topological__Spaces_Ot2__space, axiom, (topological_t2_space @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Topological__Spaces_Ot1__space, axiom, (topological_t1_space @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Oordered__comm__semiring, axiom, (ordere2520102378445227354miring @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Olinordered__semiring__1, axiom, (linord6961819062388156250ring_1 @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Olinordered__semiring, axiom, (linordered_semiring @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Oordered__semiring__0, axiom, (ordered_semiring_0 @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Olinordered__semidom, axiom, (linordered_semidom @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_2, axiom, (semilattice_sup @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Lattices_Osemilattice__inf_3, axiom, (semilattice_inf @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Lattices_Odistrib__lattice_4, axiom, (distrib_lattice @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Oab__semigroup__mult, axiom, (ab_semigroup_mult @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Osemiring__1__cancel, axiom, (semiring_1_cancel @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Oalgebraic__semidom, axiom, (algebraic_semidom @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult, axiom, (comm_monoid_mult @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Oab__semigroup__add, axiom, (ab_semigroup_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Oordered__semiring, axiom, (ordered_semiring @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Oordered__ring__abs, axiom, (ordered_ring_abs @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Parity_Osemiring__parity, axiom, (semiring_parity @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Osemiring__modulo, axiom, (semiring_modulo @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Olinordered__ring, axiom, (linordered_ring @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, (linordered_idom @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1, axiom, (comm_semiring_1 @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Ocomm__semiring__0, axiom, (comm_semiring_0 @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Osemigroup__mult, axiom, (semigroup_mult @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Osemidom__modulo, axiom, (semidom_modulo @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Osemidom__divide, axiom, (semidom_divide @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Num_Osemiring__numeral, axiom, (semiring_numeral @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Osemigroup__add, axiom, (semigroup_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Ozero__less__one, axiom, (zero_less_one @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Ocomm__semiring, axiom, (comm_semiring @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Nat_Osemiring__char__0, axiom, (semiring_char_0 @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Oab__group__add, axiom, (ab_group_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Ozero__neq__one, axiom, (zero_neq_one @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Oordered__ring, axiom, (ordered_ring @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Oidom__abs__sgn, axiom, (idom_abs_sgn @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Parity_Oring__parity, axiom, (ring_parity @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Orderings_Opreorder_5, axiom, (preorder @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, (linorder @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Omonoid__mult, axiom, (monoid_mult @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Oidom__modulo, axiom, (idom_modulo @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Oidom__divide, axiom, (idom_divide @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Ocomm__ring__1, axiom, (comm_ring_1 @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Omonoid__add, axiom, (monoid_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Osemiring__1, axiom, (semiring_1 @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Osemiring__0, axiom, (semiring_0 @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Orderings_Ono__top, axiom, (no_top @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Orderings_Ono__bot, axiom, (no_bot @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Lattices_Olattice_6, axiom, (lattice @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Ogroup__add, axiom, (group_add @ int)). % 6.67/2.70 thf(tcon_Int_Oint___GCD_Osemiring__gcd, axiom, (semiring_gcd @ int)). % 6.67/2.70 thf(tcon_Int_Oint___GCD_Osemiring__Gcd, axiom, (semiring_Gcd @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Omult__zero, axiom, (mult_zero @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Ocomm__ring, axiom, (comm_ring @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Orderings_Oorder_7, axiom, (order @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Num_Oneg__numeral, axiom, (neg_numeral @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Nat_Oring__char__0, axiom, (ring_char_0 @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Osemiring, axiom, (semiring @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Osemidom, axiom, (semidom @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Orderings_Oord_8, axiom, (ord @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Ouminus_9, axiom, (uminus @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Oring__1, axiom, (ring_1 @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Oabs__if, axiom, (abs_if @ int)). % 6.67/2.70 thf(tcon_Int_Oint___GCD_Oring__gcd, axiom, (ring_gcd @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Power_Opower, axiom, (power @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Num_Onumeral, axiom, (numeral @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Ozero, axiom, (zero @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Oplus, axiom, (plus @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Oring, axiom, (ring @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Oidom, axiom, (idom @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Groups_Oone, axiom, (one @ int)). % 6.67/2.70 thf(tcon_Int_Oint___Rings_Odvd, axiom, (dvd @ int)). % 6.67/2.70 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_10, axiom, (condit6923001295902523014norder @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11, axiom, (condit1219197933456340205attice @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_12, axiom, (bit_un5681908812861735899ations @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_13, axiom, (semiri1453513574482234551roduct @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_14, axiom, (euclid5411537665997757685th_nat @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_15, axiom, (ordere1937475149494474687imp_le @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_16, axiom, (euclid3128863361964157862miring @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_17, axiom, (euclid4440199948858584721cancel @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_18, axiom, (unique1627219031080169319umeral @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_19, axiom, (semiri6575147826004484403cancel @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_20, axiom, (strict9044650504122735259up_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff, axiom, (ordere1170586879665033532d_diff @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_21, axiom, (ordere580206878836729694up_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_22, axiom, (ordere2412721322843649153imp_le @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_23, axiom, (bit_se359711467146920520ations @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_24, axiom, (linord2810124833399127020strict @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_25, axiom, (strict7427464778891057005id_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_26, axiom, (ordere8940638589300402666id_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni5634975068530333245id_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_27, axiom, (euclid3725896446679973847miring @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_28, axiom, (topolo4958980785337419405_space @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_29, axiom, (topolo1944317154257567458pology @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_30, axiom, (topolo8865339358273720382pology @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_31, axiom, (topolo5987344860129210374id_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_32, axiom, (linord4140545234300271783up_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_33, axiom, (topolo2564578578187576103pology @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_34, axiom, (semiri2026040879449505780visors @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_35, axiom, (linord181362715937106298miring @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_36, axiom, (topolo4211221413907600880p_mult @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_37, axiom, (linord8928482502909563296strict @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_38, axiom, (semiri3467727345109120633visors @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_39, axiom, (ordere6658533253407199908up_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_40, axiom, (semiri6843258321239162965malize @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_41, axiom, (ordere6911136660526730532id_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_42, axiom, (cancel2418104881723323429up_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_43, axiom, (topolo6943815403480290642id_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_44, axiom, (cancel1802427076303600483id_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_45, axiom, (comm_s4317794764714335236cancel @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_46, axiom, (bit_semiring_bits @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_47, axiom, (topological_t2_space @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space_48, axiom, (topological_t1_space @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_49, axiom, (ordere2520102378445227354miring @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_50, axiom, (cancel_semigroup_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Olinordered__semiring_51, axiom, (linordered_semiring @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_52, axiom, (ordered_semiring_0 @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_53, axiom, (linordered_semidom @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_54, axiom, (semilattice_sup @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_55, axiom, (semilattice_inf @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_56, axiom, (distrib_lattice @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_57, axiom, (ab_semigroup_mult @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_58, axiom, (semiring_1_cancel @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_59, axiom, (algebraic_semidom @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_60, axiom, (comm_monoid_mult @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff, axiom, (comm_monoid_diff @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_61, axiom, (ab_semigroup_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Oordered__semiring_62, axiom, (ordered_semiring @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Parity_Osemiring__parity_63, axiom, (semiring_parity @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_64, axiom, (comm_monoid_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Osemiring__modulo_65, axiom, (semiring_modulo @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_66, axiom, (comm_semiring_1 @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_67, axiom, (comm_semiring_0 @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Osemigroup__mult_68, axiom, (semigroup_mult @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Osemidom__modulo_69, axiom, (semidom_modulo @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Osemidom__divide_70, axiom, (semidom_divide @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Num_Osemiring__numeral_71, axiom, (semiring_numeral @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Osemigroup__add_72, axiom, (semigroup_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Ozero__less__one_73, axiom, (zero_less_one @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Ocomm__semiring_74, axiom, (comm_semiring @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Orderings_Oorder__bot_75, axiom, (order_bot @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_76, axiom, (semiring_char_0 @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Ozero__neq__one_77, axiom, (zero_neq_one @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Orderings_Opreorder_78, axiom, (preorder @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Orderings_Olinorder_79, axiom, (linorder @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Omonoid__mult_80, axiom, (monoid_mult @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Omonoid__add_81, axiom, (monoid_add @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Osemiring__1_82, axiom, (semiring_1 @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Osemiring__0_83, axiom, (semiring_0 @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Orderings_Ono__top_84, axiom, (no_top @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Lattices_Olattice_85, axiom, (lattice @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___GCD_Osemiring__gcd_86, axiom, (semiring_gcd @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_87, axiom, (semiring_Gcd @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Omult__zero_88, axiom, (mult_zero @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Orderings_Oorder_89, axiom, (order @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Osemiring_90, axiom, (semiring @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Osemidom_91, axiom, (semidom @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Orderings_Oord_92, axiom, (ord @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Orderings_Obot_93, axiom, (bot @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Power_Opower_94, axiom, (power @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Num_Onumeral_95, axiom, (numeral @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Ozero_96, axiom, (zero @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Oplus_97, axiom, (plus @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Groups_Oone_98, axiom, (one @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Rings_Odvd_99, axiom, (dvd @ nat)). % 6.67/2.70 thf(tcon_Nat_Onat___Nat_Osize, axiom, (size @ nat)). % 6.67/2.70 thf(tcon_Num_Onum___Orderings_Opreorder_100, axiom, (preorder @ num)). % 6.67/2.70 thf(tcon_Num_Onum___Orderings_Olinorder_101, axiom, (linorder @ num)). % 6.67/2.70 thf(tcon_Num_Onum___Orderings_Oorder_102, axiom, (order @ num)). % 6.67/2.70 thf(tcon_Num_Onum___Orderings_Oord_103, axiom, (ord @ num)). % 6.67/2.70 thf(tcon_Num_Onum___Groups_Oplus_104, axiom, (plus @ num)). % 6.67/2.70 thf(tcon_Num_Onum___Nat_Osize_105, axiom, (size @ num)). % 6.67/2.70 thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_106, axiom, (semiri1453513574482234551roduct @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_107, axiom, (ordere1937475149494474687imp_le @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_108, axiom, (semiri6575147826004484403cancel @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_109, axiom, (strict9044650504122735259up_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_110, axiom, (ordere580206878836729694up_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_111, axiom, (ordere2412721322843649153imp_le @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_112, axiom, (linord2810124833399127020strict @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_113, axiom, (strict7427464778891057005id_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_114, axiom, (ordere8940638589300402666id_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field, axiom, (archim462609752435547400_field @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_115, axiom, (linord715952674999750819strict @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder, axiom, (unboun7993243217541854897norder @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_116, axiom, (linord4140545234300271783up_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_117, axiom, (semiri2026040879449505780visors @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_118, axiom, (linord181362715937106298miring @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_119, axiom, (linord8928482502909563296strict @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_120, axiom, (semiri3467727345109120633visors @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_121, axiom, (ordere6658533253407199908up_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_122, axiom, (ordere166539214618696060dd_abs @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling, axiom, (archim2362893244070406136eiling @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_123, axiom, (ordere6911136660526730532id_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_124, axiom, (linord5086331880401160121up_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_125, axiom, (cancel2418104881723323429up_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_126, axiom, (ring_15535105094025558882visors @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_127, axiom, (cancel1802427076303600483id_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_128, axiom, (linord4710134922213307826strict @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_129, axiom, (comm_s4317794764714335236cancel @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_130, axiom, (ordere2520102378445227354miring @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_131, axiom, (linord6961819062388156250ring_1 @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_132, axiom, (ordered_ab_group_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_133, axiom, (cancel_semigroup_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Olinordered__semiring_134, axiom, (linordered_semiring @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_135, axiom, (ordered_semiring_0 @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Olinordered__semidom_136, axiom, (linordered_semidom @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Orderings_Odense__linorder, axiom, (dense_linorder @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_137, axiom, (semilattice_sup @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_138, axiom, (semilattice_inf @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_139, axiom, (distrib_lattice @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_140, axiom, (ab_semigroup_mult @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_141, axiom, (semiring_1_cancel @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_142, axiom, (comm_monoid_mult @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_143, axiom, (ab_semigroup_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Fields_Olinordered__field, axiom, (linordered_field @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Oordered__semiring_144, axiom, (ordered_semiring @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_145, axiom, (ordered_ring_abs @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_146, axiom, (comm_monoid_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Olinordered__ring_147, axiom, (linordered_ring @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Olinordered__idom_148, axiom, (linordered_idom @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_149, axiom, (comm_semiring_1 @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_150, axiom, (comm_semiring_0 @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Orderings_Odense__order, axiom, (dense_order @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Osemigroup__mult_151, axiom, (semigroup_mult @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Osemidom__divide_152, axiom, (semidom_divide @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Num_Osemiring__numeral_153, axiom, (semiring_numeral @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Osemigroup__add_154, axiom, (semigroup_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Fields_Ofield__abs__sgn, axiom, (field_abs_sgn @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Fields_Odivision__ring, axiom, (division_ring @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Ozero__less__one_155, axiom, (zero_less_one @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Ocomm__semiring_156, axiom, (comm_semiring @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Nat_Osemiring__char__0_157, axiom, (semiring_char_0 @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Oab__group__add_158, axiom, (ab_group_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Fields_Ofield__char__0, axiom, (field_char_0 @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Ozero__neq__one_159, axiom, (zero_neq_one @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Oordered__ring_160, axiom, (ordered_ring @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_161, axiom, (idom_abs_sgn @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Orderings_Opreorder_162, axiom, (preorder @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Orderings_Olinorder_163, axiom, (linorder @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Omonoid__mult_164, axiom, (monoid_mult @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Oidom__divide_165, axiom, (idom_divide @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_166, axiom, (comm_ring_1 @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Omonoid__add_167, axiom, (monoid_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Osemiring__1_168, axiom, (semiring_1 @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Osemiring__0_169, axiom, (semiring_0 @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Orderings_Ono__top_170, axiom, (no_top @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Orderings_Ono__bot_171, axiom, (no_bot @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Lattices_Olattice_172, axiom, (lattice @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Ogroup__add_173, axiom, (group_add @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Omult__zero_174, axiom, (mult_zero @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Ocomm__ring_175, axiom, (comm_ring @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Orderings_Oorder_176, axiom, (order @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Num_Oneg__numeral_177, axiom, (neg_numeral @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Nat_Oring__char__0_178, axiom, (ring_char_0 @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Osemiring_179, axiom, (semiring @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Fields_Oinverse, axiom, (inverse @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Osemidom_180, axiom, (semidom @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Orderings_Oord_181, axiom, (ord @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Ouminus_182, axiom, (uminus @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Oring__1_183, axiom, (ring_1 @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Oabs__if_184, axiom, (abs_if @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Fields_Ofield, axiom, (field @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Power_Opower_185, axiom, (power @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Num_Onumeral_186, axiom, (numeral @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Ozero_187, axiom, (zero @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Oplus_188, axiom, (plus @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Oring_189, axiom, (ring @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Oidom_190, axiom, (idom @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Groups_Oone_191, axiom, (one @ rat)). % 6.67/2.70 thf(tcon_Rat_Orat___Rings_Odvd_192, axiom, (dvd @ rat)). % 6.67/2.70 thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_193, axiom, (! [A10:$tType]: ((condit1219197933456340205attice @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_194, axiom, (! [A10:$tType]: ((counta3822494911875563373attice @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_195, axiom, (! [A10:$tType]: ((comple592849572758109894attice @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_196, axiom, (! [A10:$tType]: ((bounde4967611905675639751up_bot @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_197, axiom, (! [A10:$tType]: ((bounde4346867609351753570nf_top @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_198, axiom, (! [A10:$tType]: ((comple6319245703460814977attice @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_199, axiom, (! [A10:$tType]: ((boolea8198339166811842893lgebra @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_200, axiom, (! [A10:$tType]: ((bounded_lattice_bot @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_201, axiom, (! [A10:$tType]: ((comple9053668089753744459l_ccpo @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_202, axiom, (! [A10:$tType]: ((semilattice_sup @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_203, axiom, (! [A10:$tType]: ((semilattice_inf @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Lattices_Odistrib__lattice_204, axiom, (! [A10:$tType]: ((distrib_lattice @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Lattices_Obounded__lattice_205, axiom, (! [A10:$tType]: ((bounded_lattice @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Orderings_Oorder__top_206, axiom, (! [A10:$tType]: ((order_top @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Orderings_Oorder__bot_207, axiom, (! [A10:$tType]: ((order_bot @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Orderings_Opreorder_208, axiom, (! [A10:$tType]: ((preorder @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Finite__Set_Ofinite_209, axiom, (! [A10:$tType]: (((finite_finite @ A10) => (finite_finite @ (set @ A10)))))). % 6.67/2.70 thf(tcon_Set_Oset___Lattices_Olattice_210, axiom, (! [A10:$tType]: ((lattice @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Orderings_Oorder_211, axiom, (! [A10:$tType]: ((order @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Orderings_Otop_212, axiom, (! [A10:$tType]: ((top @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Orderings_Oord_213, axiom, (! [A10:$tType]: ((ord @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Orderings_Obot_214, axiom, (! [A10:$tType]: ((bot @ (set @ A10))))). % 6.67/2.70 thf(tcon_Set_Oset___Groups_Ouminus_215, axiom, (! [A10:$tType]: ((uminus @ (set @ A10))))). % 6.67/2.70 thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_216, axiom, (condit1219197933456340205attice @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_217, axiom, (counta3822494911875563373attice @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_218, axiom, (comple592849572758109894attice @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_219, axiom, (topolo4958980785337419405_space @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_220, axiom, (topolo1944317154257567458pology @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_221, axiom, (topolo8865339358273720382pology @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_222, axiom, (bounde4967611905675639751up_bot @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_223, axiom, (bounde4346867609351753570nf_top @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_224, axiom, (comple6319245703460814977attice @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_225, axiom, (topolo2564578578187576103pology @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_226, axiom, (boolea8198339166811842893lgebra @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_227, axiom, (bounded_lattice_bot @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_228, axiom, (topological_t2_space @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_229, axiom, (topological_t1_space @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_230, axiom, (comple9053668089753744459l_ccpo @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_231, axiom, (semilattice_sup @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_232, axiom, (semilattice_inf @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_233, axiom, (distrib_lattice @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_234, axiom, (bounded_lattice @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Orderings_Oorder__top_235, axiom, (order_top @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Orderings_Oorder__bot_236, axiom, (order_bot @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Orderings_Opreorder_237, axiom, (preorder @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Orderings_Olinorder_238, axiom, (linorder @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Finite__Set_Ofinite_239, axiom, (finite_finite @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Lattices_Olattice_240, axiom, (lattice @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Orderings_Oorder_241, axiom, (order @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Orderings_Otop_242, axiom, (top @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Orderings_Oord_243, axiom, (ord @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Orderings_Obot_244, axiom, (bot @ $o)). % 6.67/2.70 thf(tcon_HOL_Obool___Groups_Ouminus_245, axiom, (uminus @ $o)). % 6.67/2.70 thf(tcon_List_Olist___Nat_Osize_246, axiom, (! [A10:$tType]: ((size @ (list @ A10))))). % 6.67/2.70 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_247, axiom, (condit6923001295902523014norder @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_248, axiom, (condit1219197933456340205attice @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_249, axiom, (semiri1453513574482234551roduct @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum, axiom, (condit5016429287641298734tinuum @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_250, axiom, (ordere1937475149494474687imp_le @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Topological__Spaces_Olinear__continuum__topology, axiom, (topolo8458572112393995274pology @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Topological__Spaces_Ofirst__countable__topology, axiom, (topolo3112930676232923870pology @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__div__algebra, axiom, (real_V8999393235501362500lgebra @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra__1, axiom, (real_V2822296259951069270ebra_1 @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_251, axiom, (semiri6575147826004484403cancel @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra, axiom, (real_V4412858255891104859lgebra @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oordered__real__vector, axiom, (real_V5355595471888546746vector @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add_252, axiom, (strict9044650504122735259up_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_253, axiom, (ordere580206878836729694up_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_254, axiom, (ordere2412721322843649153imp_le @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_255, axiom, (linord2810124833399127020strict @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector, axiom, (real_V822414075346904944vector @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Ostrict__ordered__comm__monoid__add_256, axiom, (strict7427464778891057005id_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_257, axiom, (ordere8940638589300402666id_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_258, axiom, (topolo4958980785337419405_space @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_259, axiom, (topolo1944317154257567458pology @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field, axiom, (real_V3459762299906320749_field @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra, axiom, (real_V5047593784448816457lgebra @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_260, axiom, (archim462609752435547400_field @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_261, axiom, (linord715952674999750819strict @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ouniformity__dist, axiom, (real_V768167426530841204y_dist @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_262, axiom, (unboun7993243217541854897norder @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_263, axiom, (topolo5987344860129210374id_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_264, axiom, (linord4140545234300271783up_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_265, axiom, (topolo2564578578187576103pology @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_266, axiom, (semiri2026040879449505780visors @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_267, axiom, (linord181362715937106298miring @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1, axiom, (real_V2191834092415804123ebra_1 @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space, axiom, (real_V8037385150606011577_space @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Limits_Otopological__semigroup__mult_268, axiom, (topolo4211221413907600880p_mult @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Topological__Spaces_Ouniform__space, axiom, (topolo7287701948861334536_space @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Topological__Spaces_Operfect__space, axiom, (topolo8386298272705272623_space @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_269, axiom, (linord8928482502909563296strict @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_270, axiom, (semiri3467727345109120633visors @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space, axiom, (real_V7819770556892013058_space @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Limits_Otopological__ab__group__add, axiom, (topolo1287966508704411220up_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_271, axiom, (ordere6658533253407199908up_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_272, axiom, (ordere166539214618696060dd_abs @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_273, axiom, (archim2362893244070406136eiling @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector, axiom, (real_V4867850818363320053vector @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add_274, axiom, (ordere6911136660526730532id_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_275, axiom, (linord5086331880401160121up_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_276, axiom, (cancel2418104881723323429up_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_277, axiom, (ring_15535105094025558882visors @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__field, axiom, (real_V7773925162809079976_field @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Limits_Otopological__monoid__add_278, axiom, (topolo6943815403480290642id_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_279, axiom, (cancel1802427076303600483id_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_280, axiom, (linord4710134922213307826strict @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_281, axiom, (comm_s4317794764714335236cancel @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Limits_Otopological__group__add, axiom, (topolo1633459387980952147up_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_282, axiom, (topological_t2_space @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Topological__Spaces_Ot1__space_283, axiom, (topological_t1_space @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_284, axiom, (ordere2520102378445227354miring @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_285, axiom, (linord6961819062388156250ring_1 @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_286, axiom, (ordered_ab_group_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_287, axiom, (cancel_semigroup_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Olinordered__semiring_288, axiom, (linordered_semiring @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Real__Vector__Spaces_Obanach, axiom, (real_Vector_banach @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Oordered__semiring__0_289, axiom, (ordered_semiring_0 @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Olinordered__semidom_290, axiom, (linordered_semidom @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Orderings_Odense__linorder_291, axiom, (dense_linorder @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_292, axiom, (semilattice_sup @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Lattices_Osemilattice__inf_293, axiom, (semilattice_inf @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Lattices_Odistrib__lattice_294, axiom, (distrib_lattice @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_295, axiom, (ab_semigroup_mult @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_296, axiom, (semiring_1_cancel @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_297, axiom, (comm_monoid_mult @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_298, axiom, (ab_semigroup_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Fields_Olinordered__field_299, axiom, (linordered_field @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Oordered__semiring_300, axiom, (ordered_semiring @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_301, axiom, (ordered_ring_abs @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_302, axiom, (comm_monoid_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Olinordered__ring_303, axiom, (linordered_ring @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Olinordered__idom_304, axiom, (linordered_idom @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_305, axiom, (comm_semiring_1 @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_306, axiom, (comm_semiring_0 @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Orderings_Odense__order_307, axiom, (dense_order @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Osemigroup__mult_308, axiom, (semigroup_mult @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Osemidom__divide_309, axiom, (semidom_divide @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Num_Osemiring__numeral_310, axiom, (semiring_numeral @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Osemigroup__add_311, axiom, (semigroup_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Fields_Ofield__abs__sgn_312, axiom, (field_abs_sgn @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Fields_Odivision__ring_313, axiom, (division_ring @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Ozero__less__one_314, axiom, (zero_less_one @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Ocomm__semiring_315, axiom, (comm_semiring @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Nat_Osemiring__char__0_316, axiom, (semiring_char_0 @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Oab__group__add_317, axiom, (ab_group_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Fields_Ofield__char__0_318, axiom, (field_char_0 @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Ozero__neq__one_319, axiom, (zero_neq_one @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Oordered__ring_320, axiom, (ordered_ring @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_321, axiom, (idom_abs_sgn @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Orderings_Opreorder_322, axiom, (preorder @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Orderings_Olinorder_323, axiom, (linorder @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Omonoid__mult_324, axiom, (monoid_mult @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Transcendental_Oln, axiom, (ln @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Oidom__divide_325, axiom, (idom_divide @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_326, axiom, (comm_ring_1 @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Omonoid__add_327, axiom, (monoid_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Osemiring__1_328, axiom, (semiring_1 @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Osemiring__0_329, axiom, (semiring_0 @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Orderings_Ono__top_330, axiom, (no_top @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Orderings_Ono__bot_331, axiom, (no_bot @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Lattices_Olattice_332, axiom, (lattice @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Ogroup__add_333, axiom, (group_add @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Omult__zero_334, axiom, (mult_zero @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Ocomm__ring_335, axiom, (comm_ring @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Orderings_Oorder_336, axiom, (order @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Num_Oneg__numeral_337, axiom, (neg_numeral @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Nat_Oring__char__0_338, axiom, (ring_char_0 @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Osemiring_339, axiom, (semiring @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Fields_Oinverse_340, axiom, (inverse @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Osemidom_341, axiom, (semidom @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Orderings_Oord_342, axiom, (ord @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Ouminus_343, axiom, (uminus @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Oring__1_344, axiom, (ring_1 @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Oabs__if_345, axiom, (abs_if @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Fields_Ofield_346, axiom, (field @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Power_Opower_347, axiom, (power @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Num_Onumeral_348, axiom, (numeral @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Ozero_349, axiom, (zero @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Oplus_350, axiom, (plus @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Oring_351, axiom, (ring @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Oidom_352, axiom, (idom @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Groups_Oone_353, axiom, (one @ real)). % 6.67/2.70 thf(tcon_Real_Oreal___Rings_Odvd_354, axiom, (dvd @ real)). % 6.67/2.70 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_355, axiom, (! [A10:$tType,A20:$tType]: ((((finite_finite @ A10) & (finite_finite @ A20)) => (finite_finite @ ((sum_sum @ A10) @ A20)))))). % 6.67/2.70 thf(tcon_Sum__Type_Osum___Nat_Osize_356, axiom, (! [A10:$tType,A20:$tType]: ((size @ ((sum_sum @ A10) @ A20))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_357, axiom, (! [A10:$tType]: ((condit1219197933456340205attice @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_358, axiom, (! [A10:$tType]: ((counta3822494911875563373attice @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_359, axiom, (! [A10:$tType]: ((bounde4967611905675639751up_bot @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_360, axiom, (! [A10:$tType]: ((bounde4346867609351753570nf_top @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_361, axiom, (! [A10:$tType]: ((comple6319245703460814977attice @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_362, axiom, (! [A10:$tType]: ((bounded_lattice_bot @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_363, axiom, (! [A10:$tType]: ((comple9053668089753744459l_ccpo @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_364, axiom, (! [A10:$tType]: ((semilattice_sup @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_365, axiom, (! [A10:$tType]: ((semilattice_inf @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_366, axiom, (! [A10:$tType]: ((distrib_lattice @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_367, axiom, (! [A10:$tType]: ((bounded_lattice @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Orderings_Oorder__top_368, axiom, (! [A10:$tType]: ((order_top @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_369, axiom, (! [A10:$tType]: ((order_bot @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Orderings_Opreorder_370, axiom, (! [A10:$tType]: ((preorder @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Lattices_Olattice_371, axiom, (! [A10:$tType]: ((lattice @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Orderings_Oorder_372, axiom, (! [A10:$tType]: ((order @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Orderings_Otop_373, axiom, (! [A10:$tType]: ((top @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Orderings_Oord_374, axiom, (! [A10:$tType]: ((ord @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Filter_Ofilter___Orderings_Obot_375, axiom, (! [A10:$tType]: ((bot @ (filter @ A10))))). % 6.67/2.70 thf(tcon_Option_Ooption___Finite__Set_Ofinite_376, axiom, (! [A10:$tType]: (((finite_finite @ A10) => (finite_finite @ (option @ A10)))))). % 6.67/2.70 thf(tcon_Option_Ooption___Nat_Osize_377, axiom, (! [A10:$tType]: ((size @ (option @ A10))))). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_378, axiom, (semiri1453513574482234551roduct @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_379, axiom, (topolo3112930676232923870pology @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_380, axiom, (real_V8999393235501362500lgebra @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_381, axiom, (real_V2822296259951069270ebra_1 @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_382, axiom, (semiri6575147826004484403cancel @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_383, axiom, (real_V4412858255891104859lgebra @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_384, axiom, (real_V822414075346904944vector @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_385, axiom, (topolo4958980785337419405_space @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_386, axiom, (real_V3459762299906320749_field @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_387, axiom, (real_V5047593784448816457lgebra @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_388, axiom, (real_V768167426530841204y_dist @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_389, axiom, (topolo5987344860129210374id_add @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_390, axiom, (semiri2026040879449505780visors @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_391, axiom, (real_V2191834092415804123ebra_1 @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_392, axiom, (real_V8037385150606011577_space @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_393, axiom, (topolo4211221413907600880p_mult @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_394, axiom, (topolo7287701948861334536_space @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_395, axiom, (topolo8386298272705272623_space @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_396, axiom, (semiri3467727345109120633visors @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_397, axiom, (real_V7819770556892013058_space @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_398, axiom, (topolo1287966508704411220up_add @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_399, axiom, (real_V4867850818363320053vector @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_400, axiom, (cancel2418104881723323429up_add @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_401, axiom, (ring_15535105094025558882visors @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_402, axiom, (real_V7773925162809079976_field @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_403, axiom, (topolo6943815403480290642id_add @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_404, axiom, (cancel1802427076303600483id_add @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_405, axiom, (comm_s4317794764714335236cancel @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Limits_Otopological__group__add_406, axiom, (topolo1633459387980952147up_add @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_407, axiom, (topological_t2_space @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_408, axiom, (topological_t1_space @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_409, axiom, (cancel_semigroup_add @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_410, axiom, (real_Vector_banach @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_411, axiom, (ab_semigroup_mult @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_412, axiom, (semiring_1_cancel @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_413, axiom, (comm_monoid_mult @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_414, axiom, (ab_semigroup_add @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_415, axiom, (comm_monoid_add @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_416, axiom, (comm_semiring_1 @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_417, axiom, (comm_semiring_0 @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_418, axiom, (semigroup_mult @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_419, axiom, (semidom_divide @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_420, axiom, (semiring_numeral @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_421, axiom, (semigroup_add @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_422, axiom, (field_abs_sgn @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_423, axiom, (division_ring @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_424, axiom, (comm_semiring @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_425, axiom, (semiring_char_0 @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_426, axiom, (ab_group_add @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_427, axiom, (field_char_0 @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_428, axiom, (zero_neq_one @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_429, axiom, (idom_abs_sgn @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_430, axiom, (monoid_mult @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_431, axiom, (idom_divide @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_432, axiom, (comm_ring_1 @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_433, axiom, (monoid_add @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_434, axiom, (semiring_1 @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_435, axiom, (semiring_0 @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_436, axiom, (group_add @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Omult__zero_437, axiom, (mult_zero @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_438, axiom, (comm_ring @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_439, axiom, (neg_numeral @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_440, axiom, (ring_char_0 @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Osemiring_441, axiom, (semiring @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Fields_Oinverse_442, axiom, (inverse @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Osemidom_443, axiom, (semidom @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Ouminus_444, axiom, (uminus @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Oring__1_445, axiom, (ring_1 @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Fields_Ofield_446, axiom, (field @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Power_Opower_447, axiom, (power @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Num_Onumeral_448, axiom, (numeral @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Ozero_449, axiom, (zero @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Oplus_450, axiom, (plus @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Oring_451, axiom, (ring @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Oidom_452, axiom, (idom @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Groups_Oone_453, axiom, (one @ complex)). % 6.67/2.70 thf(tcon_Complex_Ocomplex___Rings_Odvd_454, axiom, (dvd @ complex)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_455, axiom, (condit6923001295902523014norder @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_456, axiom, (condit1219197933456340205attice @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_457, axiom, (counta3822494911875563373attice @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_458, axiom, (comple592849572758109894attice @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_459, axiom, (strict9044650504122735259up_add @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_460, axiom, (strict7427464778891057005id_add @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_461, axiom, (canoni5634975068530333245id_add @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_462, axiom, (bounde4967611905675639751up_bot @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_463, axiom, (bounde4346867609351753570nf_top @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder, axiom, (comple5582772986160207858norder @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_464, axiom, (linord4140545234300271783up_add @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_465, axiom, (comple6319245703460814977attice @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_466, axiom, (linord181362715937106298miring @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_467, axiom, (semiri3467727345109120633visors @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_468, axiom, (ordere6658533253407199908up_add @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_469, axiom, (ordere6911136660526730532id_add @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_470, axiom, (bounded_lattice_bot @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_471, axiom, (ordere2520102378445227354miring @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_472, axiom, (comple9053668089753744459l_ccpo @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_473, axiom, (semilattice_sup @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_474, axiom, (semilattice_inf @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_475, axiom, (distrib_lattice @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_476, axiom, (bounded_lattice @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_477, axiom, (ab_semigroup_mult @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_478, axiom, (comm_monoid_mult @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_479, axiom, (ab_semigroup_add @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_480, axiom, (ordered_semiring @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_481, axiom, (comm_monoid_add @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_482, axiom, (comm_semiring_1 @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_483, axiom, (comm_semiring_0 @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_484, axiom, (semigroup_mult @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_485, axiom, (semiring_numeral @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_486, axiom, (semigroup_add @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_487, axiom, (zero_less_one @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_488, axiom, (comm_semiring @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_489, axiom, (wellorder @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_490, axiom, (order_top @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_491, axiom, (order_bot @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_492, axiom, (semiring_char_0 @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_493, axiom, (zero_neq_one @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_494, axiom, (preorder @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_495, axiom, (linorder @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_496, axiom, (monoid_mult @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_497, axiom, (monoid_add @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_498, axiom, (semiring_1 @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_499, axiom, (semiring_0 @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_500, axiom, (lattice @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_501, axiom, (mult_zero @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_502, axiom, (order @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_503, axiom, (semiring @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Orderings_Otop_504, axiom, (top @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_505, axiom, (ord @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Orderings_Obot_506, axiom, (bot @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Power_Opower_507, axiom, (power @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Num_Onumeral_508, axiom, (numeral @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_509, axiom, (zero @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Oplus_510, axiom, (plus @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Groups_Oone_511, axiom, (one @ extended_enat)). % 6.67/2.70 thf(tcon_Extended__Nat_Oenat___Rings_Odvd_512, axiom, (dvd @ extended_enat)). % 6.67/2.70 thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_513, axiom, (! [A10:$tType,A20:$tType]: ((((topolo4958980785337419405_space @ A10) & (topolo4958980785337419405_space @ A20)) => (topolo4958980785337419405_space @ ((product_prod @ A10) @ A20)))))). % 6.67/2.70 thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_514, axiom, (! [A10:$tType,A20:$tType]: ((((topological_t2_space @ A10) & (topological_t2_space @ A20)) => (topological_t2_space @ ((product_prod @ A10) @ A20)))))). % 6.67/2.70 thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_515, axiom, (! [A10:$tType,A20:$tType]: ((((topological_t1_space @ A10) & (topological_t1_space @ A20)) => (topological_t1_space @ ((product_prod @ A10) @ A20)))))). % 6.67/2.70 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_516, axiom, (! [A10:$tType,A20:$tType]: ((((finite_finite @ A10) & (finite_finite @ A20)) => (finite_finite @ ((product_prod @ A10) @ A20)))))). % 6.67/2.70 thf(tcon_Product__Type_Oprod___Nat_Osize_517, axiom, (! [A10:$tType,A20:$tType]: ((size @ ((product_prod @ A10) @ A20))))). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_518, axiom, (condit6923001295902523014norder @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_519, axiom, (condit1219197933456340205attice @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_520, axiom, (counta3822494911875563373attice @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_521, axiom, (comple592849572758109894attice @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_522, axiom, (bounde4967611905675639751up_bot @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_523, axiom, (bounde4346867609351753570nf_top @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_524, axiom, (comple5582772986160207858norder @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_525, axiom, (comple6319245703460814977attice @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_526, axiom, (boolea8198339166811842893lgebra @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_527, axiom, (bounded_lattice_bot @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_528, axiom, (comple9053668089753744459l_ccpo @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_529, axiom, (semilattice_sup @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_530, axiom, (semilattice_inf @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_531, axiom, (distrib_lattice @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_532, axiom, (bounded_lattice @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Orderings_Owellorder_533, axiom, (wellorder @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_534, axiom, (order_top @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_535, axiom, (order_bot @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_536, axiom, (preorder @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_537, axiom, (linorder @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_538, axiom, (finite_finite @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Lattices_Olattice_539, axiom, (lattice @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Orderings_Oorder_540, axiom, (order @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Orderings_Otop_541, axiom, (top @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Orderings_Oord_542, axiom, (ord @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Orderings_Obot_543, axiom, (bot @ product_unit)). % 6.67/2.70 thf(tcon_Product__Type_Ounit___Groups_Ouminus_544, axiom, (uminus @ product_unit)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_545, axiom, (bit_un5681908812861735899ations @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_546, axiom, (semiri1453513574482234551roduct @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_547, axiom, (euclid5411537665997757685th_nat @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_548, axiom, (ordere1937475149494474687imp_le @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_549, axiom, (euclid3128863361964157862miring @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_550, axiom, (euclid4440199948858584721cancel @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_551, axiom, (unique1627219031080169319umeral @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_552, axiom, (euclid8851590272496341667cancel @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_553, axiom, (semiri6575147826004484403cancel @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_554, axiom, (strict9044650504122735259up_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_555, axiom, (ordere580206878836729694up_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_556, axiom, (ordere2412721322843649153imp_le @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_557, axiom, (bit_se359711467146920520ations @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_558, axiom, (linord2810124833399127020strict @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_559, axiom, (strict7427464778891057005id_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_560, axiom, (ordere8940638589300402666id_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_561, axiom, (euclid3725896446679973847miring @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_562, axiom, (linord715952674999750819strict @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_563, axiom, (linord4140545234300271783up_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_564, axiom, (bit_ri3973907225187159222ations @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_565, axiom, (semiri2026040879449505780visors @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_566, axiom, (linord181362715937106298miring @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_567, axiom, (linord8928482502909563296strict @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_568, axiom, (semiri3467727345109120633visors @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_569, axiom, (ordere6658533253407199908up_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_570, axiom, (ordere166539214618696060dd_abs @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_571, axiom, (ordere6911136660526730532id_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_572, axiom, (linord5086331880401160121up_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_573, axiom, (cancel2418104881723323429up_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_574, axiom, (ring_15535105094025558882visors @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_575, axiom, (cancel1802427076303600483id_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_576, axiom, (linord4710134922213307826strict @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_577, axiom, (comm_s4317794764714335236cancel @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_578, axiom, (bit_semiring_bits @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_579, axiom, (ordere2520102378445227354miring @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_580, axiom, (linord6961819062388156250ring_1 @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_581, axiom, (ordered_ab_group_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_582, axiom, (cancel_semigroup_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_583, axiom, (linordered_semiring @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_584, axiom, (ordered_semiring_0 @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_585, axiom, (linordered_semidom @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_586, axiom, (ab_semigroup_mult @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_587, axiom, (semiring_1_cancel @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_588, axiom, (algebraic_semidom @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_589, axiom, (comm_monoid_mult @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_590, axiom, (ab_semigroup_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_591, axiom, (ordered_semiring @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_592, axiom, (ordered_ring_abs @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_593, axiom, (semiring_parity @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_594, axiom, (comm_monoid_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_595, axiom, (semiring_modulo @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_596, axiom, (linordered_ring @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_597, axiom, (linordered_idom @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_598, axiom, (comm_semiring_1 @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_599, axiom, (comm_semiring_0 @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_600, axiom, (semigroup_mult @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_601, axiom, (semidom_modulo @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_602, axiom, (semidom_divide @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_603, axiom, (semiring_numeral @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_604, axiom, (semigroup_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_605, axiom, (zero_less_one @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_606, axiom, (comm_semiring @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_607, axiom, (semiring_char_0 @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_608, axiom, (ab_group_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_609, axiom, (zero_neq_one @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_610, axiom, (ordered_ring @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_611, axiom, (idom_abs_sgn @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_612, axiom, (ring_parity @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_613, axiom, (preorder @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_614, axiom, (linorder @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_615, axiom, (monoid_mult @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_616, axiom, (idom_modulo @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_617, axiom, (idom_divide @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_618, axiom, (comm_ring_1 @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_619, axiom, (monoid_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_620, axiom, (semiring_1 @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_621, axiom, (semiring_0 @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_622, axiom, (group_add @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_623, axiom, (mult_zero @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_624, axiom, (comm_ring @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_625, axiom, (order @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_626, axiom, (neg_numeral @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_627, axiom, (ring_char_0 @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_628, axiom, (semiring @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_629, axiom, (semidom @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_630, axiom, (ord @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_631, axiom, (uminus @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_632, axiom, (ring_1 @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_633, axiom, (abs_if @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Power_Opower_634, axiom, (power @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_635, axiom, (numeral @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_636, axiom, (zero @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_637, axiom, (plus @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Oring_638, axiom, (ring @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_639, axiom, (idom @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Groups_Oone_640, axiom, (one @ code_integer)). % 6.67/2.70 thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_641, axiom, (dvd @ code_integer)). % 6.67/2.70 thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_642, axiom, (size @ vEBT_VEBT)). % 6.67/2.70 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))). % 6.67/2.70 thf(help_If_2_1_T, axiom, (! [A:$tType,X3:A,Y:A]: ((((((if @ A) @ $false) @ X3) @ Y) = Y)))). % 6.67/2.70 thf(help_If_1_1_T, axiom, (! [A:$tType,X3:A,Y:A]: ((((((if @ A) @ $true) @ X3) @ Y) = X3)))). % 6.67/2.70 thf(help_fChoice_1_1_T, axiom, (! [A:$tType,P:(A > $o)]: (((P @ ((fChoice @ A) @ P)) = (? [X6:A]: ((P @ X6))))))). % 6.67/2.70 thf(conj_0, hypothesis, (! [X2:nat]: ((((vEBT_V8194947554948674370ptions @ (((nth @ vEBT_VEBT) @ treeList) @ maxs)) @ X2) => thesis)))). % 6.67/2.70 thf(conj_1, conjecture, thesis). % 6.67/2.70 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.0zb77TbOdX/DTF2DTF3756.p % 6.67/2.70 ------------------------ % 8.42/4.47 ---- Cleaned THF --- % 8.42/4.47 thf(ty_t_VEBT__Definitions_OVEBT,type, % 8.42/4.47 vEBT_VEBT: $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_Code__Numeral_Ointeger,type, % 8.42/4.47 code_integer: $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_Code__Evaluation_Oterm,type, % 8.42/4.47 code_term: $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_Product__Type_Ounit,type, % 8.42/4.47 product_unit: $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_Product__Type_Oprod,type, % 8.42/4.47 product_prod: $tType > $tType > $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_Extended__Nat_Oenat,type, % 8.42/4.47 extended_enat: $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_Complex_Ocomplex,type, % 8.42/4.47 complex: $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_Sum__Type_Osum,type, % 8.42/4.47 sum_sum: $tType > $tType > $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_Option_Ooption,type, % 8.42/4.47 option: $tType > $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_Filter_Ofilter,type, % 8.42/4.47 filter: $tType > $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_Real_Oreal,type, % 8.42/4.47 real: $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_List_Olist,type, % 8.42/4.47 list: $tType > $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_Set_Oset,type, % 8.42/4.47 set: $tType > $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_Rat_Orat,type, % 8.42/4.47 rat: $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_Num_Onum,type, % 8.42/4.47 num: $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_Nat_Onat,type, % 8.42/4.47 nat: $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_Int_Oint,type, % 8.42/4.47 int: $tType ). % 8.42/4.47 % 8.42/4.47 thf(ty_t_itself,type, % 8.42/4.47 itself: $tType > $tType ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_HOL_Otype,type, % 8.42/4.47 type: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Nat_Osize,type, % 8.42/4.47 size: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Odvd,type, % 8.42/4.47 dvd: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Oone,type, % 8.42/4.47 one: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Oidom,type, % 8.42/4.47 idom: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Oring,type, % 8.42/4.47 ring: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Oplus,type, % 8.42/4.47 plus: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Ozero,type, % 8.42/4.47 zero: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Num_Onumeral,type, % 8.42/4.47 numeral: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Power_Opower,type, % 8.42/4.47 power: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Fields_Ofield,type, % 8.42/4.47 field: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_GCD_Oring__gcd,type, % 8.42/4.47 ring_gcd: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Oabs__if,type, % 8.42/4.47 abs_if: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Oring__1,type, % 8.42/4.47 ring_1: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Ouminus,type, % 8.42/4.47 uminus: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Orderings_Obot,type, % 8.42/4.47 bot: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Orderings_Oord,type, % 8.42/4.47 ord: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Orderings_Otop,type, % 8.42/4.47 top: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Osemidom,type, % 8.42/4.47 semidom: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Fields_Oinverse,type, % 8.42/4.47 inverse: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Osemiring,type, % 8.42/4.47 semiring: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Nat_Oring__char__0,type, % 8.42/4.47 ring_char_0: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Num_Oneg__numeral,type, % 8.42/4.47 neg_numeral: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Orderings_Oorder,type, % 8.42/4.47 order: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Ocomm__ring,type, % 8.42/4.47 comm_ring: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Omult__zero,type, % 8.42/4.47 mult_zero: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_GCD_Osemiring__Gcd,type, % 8.42/4.47 semiring_Gcd: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_GCD_Osemiring__gcd,type, % 8.42/4.47 semiring_gcd: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Ogroup__add,type, % 8.42/4.47 group_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Lattices_Olattice,type, % 8.42/4.47 lattice: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Orderings_Ono__bot,type, % 8.42/4.47 no_bot: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Orderings_Ono__top,type, % 8.42/4.47 no_top: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Osemiring__0,type, % 8.42/4.47 semiring_0: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Osemiring__1,type, % 8.42/4.47 semiring_1: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Finite__Set_Ofinite,type, % 8.42/4.47 finite_finite: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Omonoid__add,type, % 8.42/4.47 monoid_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Ocomm__ring__1,type, % 8.42/4.47 comm_ring_1: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Oidom__divide,type, % 8.42/4.47 idom_divide: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Oidom__modulo,type, % 8.42/4.47 idom_modulo: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Transcendental_Oln,type, % 8.42/4.47 ln: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Omonoid__mult,type, % 8.42/4.47 monoid_mult: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Orderings_Olinorder,type, % 8.42/4.47 linorder: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Orderings_Opreorder,type, % 8.42/4.47 preorder: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Parity_Oring__parity,type, % 8.42/4.47 ring_parity: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Oidom__abs__sgn,type, % 8.42/4.47 idom_abs_sgn: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Oordered__ring,type, % 8.42/4.47 ordered_ring: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Ozero__neq__one,type, % 8.42/4.47 zero_neq_one: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Enum_Ofinite__lattice,type, % 8.42/4.47 finite_lattice: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Fields_Ofield__char__0,type, % 8.42/4.47 field_char_0: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Oab__group__add,type, % 8.42/4.47 ab_group_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Nat_Osemiring__char__0,type, % 8.42/4.47 semiring_char_0: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Orderings_Oorder__bot,type, % 8.42/4.47 order_bot: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Orderings_Oorder__top,type, % 8.42/4.47 order_top: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Orderings_Owellorder,type, % 8.42/4.47 wellorder: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Ocomm__semiring,type, % 8.42/4.47 comm_semiring: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Ozero__less__one,type, % 8.42/4.47 zero_less_one: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Fields_Odivision__ring,type, % 8.42/4.47 division_ring: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Fields_Ofield__abs__sgn,type, % 8.42/4.47 field_abs_sgn: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Osemigroup__add,type, % 8.42/4.47 semigroup_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Num_Osemiring__numeral,type, % 8.42/4.47 semiring_numeral: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Osemidom__divide,type, % 8.42/4.47 semidom_divide: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Osemidom__modulo,type, % 8.42/4.47 semidom_modulo: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Osemigroup__mult,type, % 8.42/4.47 semigroup_mult: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Orderings_Odense__order,type, % 8.42/4.47 dense_order: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Ocomm__semiring__0,type, % 8.42/4.47 comm_semiring_0: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Ocomm__semiring__1,type, % 8.42/4.47 comm_semiring_1: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Olinordered__idom,type, % 8.42/4.47 linordered_idom: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Olinordered__ring,type, % 8.42/4.47 linordered_ring: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Osemiring__modulo,type, % 8.42/4.47 semiring_modulo: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Ocomm__monoid__add,type, % 8.42/4.47 comm_monoid_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Parity_Osemiring__parity,type, % 8.42/4.47 semiring_parity: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Oordered__ring__abs,type, % 8.42/4.47 ordered_ring_abs: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Oordered__semiring,type, % 8.42/4.47 ordered_semiring: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Fields_Olinordered__field,type, % 8.42/4.47 linordered_field: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Oab__semigroup__add,type, % 8.42/4.47 ab_semigroup_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Ocomm__monoid__diff,type, % 8.42/4.47 comm_monoid_diff: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Ocomm__monoid__mult,type, % 8.42/4.47 comm_monoid_mult: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Oalgebraic__semidom,type, % 8.42/4.47 algebraic_semidom: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Osemiring__1__cancel,type, % 8.42/4.47 semiring_1_cancel: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Oab__semigroup__mult,type, % 8.42/4.47 ab_semigroup_mult: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Lattices_Obounded__lattice,type, % 8.42/4.47 bounded_lattice: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Lattices_Odistrib__lattice,type, % 8.42/4.47 distrib_lattice: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Lattices_Osemilattice__inf,type, % 8.42/4.47 semilattice_inf: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Lattices_Osemilattice__sup,type, % 8.42/4.47 semilattice_sup: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Orderings_Odense__linorder,type, % 8.42/4.47 dense_linorder: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Olinordered__semidom,type, % 8.42/4.47 linordered_semidom: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Oordered__semiring__0,type, % 8.42/4.47 ordered_semiring_0: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Real__Vector__Spaces_Obanach,type, % 8.42/4.47 real_Vector_banach: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Olinordered__semiring,type, % 8.42/4.47 linordered_semiring: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Complete__Partial__Order_Occpo,type, % 8.42/4.47 comple9053668089753744459l_ccpo: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Enum_Ofinite__distrib__lattice,type, % 8.42/4.47 finite8700451911770168679attice: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Ocancel__semigroup__add,type, % 8.42/4.47 cancel_semigroup_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Oordered__ab__group__add,type, % 8.42/4.47 ordered_ab_group_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Olinordered__semiring__1,type, % 8.42/4.47 linord6961819062388156250ring_1: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Oordered__comm__semiring,type, % 8.42/4.47 ordere2520102378445227354miring: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Topological__Spaces_Ot1__space,type, % 8.42/4.47 topological_t1_space: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Topological__Spaces_Ot2__space,type, % 8.42/4.47 topological_t2_space: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Topological__Spaces_Ot3__space,type, % 8.42/4.47 topological_t3_space: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Topological__Spaces_Ot4__space,type, % 8.42/4.47 topological_t4_space: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Bit__Operations_Osemiring__bits,type, % 8.42/4.47 bit_semiring_bits: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Lattices_Obounded__lattice__bot,type, % 8.42/4.47 bounded_lattice_bot: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Limits_Otopological__group__add,type, % 8.42/4.47 topolo1633459387980952147up_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type, % 8.42/4.47 comm_s4317794764714335236cancel: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Olinordered__ring__strict,type, % 8.42/4.47 linord4710134922213307826strict: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type, % 8.42/4.47 cancel1802427076303600483id_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Limits_Otopological__monoid__add,type, % 8.42/4.47 topolo6943815403480290642id_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Real__Vector__Spaces_Oreal__field,type, % 8.42/4.47 real_V7773925162809079976_field: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Oring__1__no__zero__divisors,type, % 8.42/4.47 ring_15535105094025558882visors: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type, % 8.42/4.47 cancel2418104881723323429up_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Olinordered__ab__group__add,type, % 8.42/4.47 linord5086331880401160121up_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Oordered__comm__monoid__add,type, % 8.42/4.47 ordere6911136660526730532id_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type, % 8.42/4.47 real_V4867850818363320053vector: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type, % 8.42/4.47 archim2362893244070406136eiling: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_GCD_Osemiring__gcd__mult__normalize,type, % 8.42/4.47 semiri6843258321239162965malize: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Oordered__ab__group__add__abs,type, % 8.42/4.47 ordere166539214618696060dd_abs: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type, % 8.42/4.47 ordere6658533253407199908up_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Limits_Otopological__ab__group__add,type, % 8.42/4.47 topolo1287966508704411220up_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Real__Vector__Spaces_Ometric__space,type, % 8.42/4.47 real_V7819770556892013058_space: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type, % 8.42/4.47 semiri3467727345109120633visors: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type, % 8.42/4.47 boolea8198339166811842893lgebra: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Olinordered__semiring__strict,type, % 8.42/4.47 linord8928482502909563296strict: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Topological__Spaces_Operfect__space,type, % 8.42/4.47 topolo8386298272705272623_space: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Topological__Spaces_Ouniform__space,type, % 8.42/4.47 topolo7287701948861334536_space: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Limits_Otopological__semigroup__mult,type, % 8.42/4.47 topolo4211221413907600880p_mult: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Real__Vector__Spaces_Ocomplete__space,type, % 8.42/4.47 real_V8037385150606011577_space: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Real__Vector__Spaces_Oreal__algebra__1,type, % 8.42/4.47 real_V2191834092415804123ebra_1: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type, % 8.42/4.47 linord181362715937106298miring: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type, % 8.42/4.47 semiri2026040879449505780visors: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Topological__Spaces_Oorder__topology,type, % 8.42/4.47 topolo2564578578187576103pology: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Bit__Operations_Oring__bit__operations,type, % 8.42/4.47 bit_ri3973907225187159222ations: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type, % 8.42/4.47 comple6319245703460814977attice: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type, % 8.42/4.47 linord4140545234300271783up_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Limits_Otopological__comm__monoid__add,type, % 8.42/4.47 topolo5987344860129210374id_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Orderings_Ounbounded__dense__linorder,type, % 8.42/4.47 unboun7993243217541854897norder: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Real__Vector__Spaces_Ouniformity__dist,type, % 8.42/4.47 real_V768167426530841204y_dist: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Olinordered__semiring__1__strict,type, % 8.42/4.47 linord715952674999750819strict: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type, % 8.42/4.47 archim462609752435547400_field: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type, % 8.42/4.47 comple5582772986160207858norder: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Real__Vector__Spaces_Oreal__div__algebra,type, % 8.42/4.47 real_V5047593784448816457lgebra: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type, % 8.42/4.47 bounde4346867609351753570nf_top: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type, % 8.42/4.47 bounde4967611905675639751up_bot: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__field,type, % 8.42/4.47 real_V3459762299906320749_field: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Topological__Spaces_Odiscrete__topology,type, % 8.42/4.47 topolo8865339358273720382pology: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Topological__Spaces_Olinorder__topology,type, % 8.42/4.47 topolo1944317154257567458pology: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Topological__Spaces_Otopological__space,type, % 8.42/4.47 topolo4958980785337419405_space: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type, % 8.42/4.47 euclid3725896446679973847miring: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type, % 8.42/4.47 canoni5634975068530333245id_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type, % 8.42/4.47 ordere8940638589300402666id_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type, % 8.42/4.47 strict7427464778891057005id_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type, % 8.42/4.47 real_V822414075346904944vector: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type, % 8.42/4.47 linord2810124833399127020strict: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type, % 8.42/4.47 bit_se359711467146920520ations: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type, % 8.42/4.47 ordere2412721322843649153imp_le: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type, % 8.42/4.47 ordere580206878836729694up_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type, % 8.42/4.47 ordere1170586879665033532d_diff: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type, % 8.42/4.47 strict9044650504122735259up_add: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Real__Vector__Spaces_Oordered__real__vector,type, % 8.42/4.47 real_V5355595471888546746vector: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra,type, % 8.42/4.47 real_V4412858255891104859lgebra: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type, % 8.42/4.47 semiri6575147826004484403cancel: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type, % 8.42/4.47 euclid8851590272496341667cancel: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra__1,type, % 8.42/4.47 real_V2822296259951069270ebra_1: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type, % 8.42/4.47 unique1627219031080169319umeral: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type, % 8.42/4.47 comple592849572758109894attice: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Real__Vector__Spaces_Oreal__normed__div__algebra,type, % 8.42/4.47 real_V8999393235501362500lgebra: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type, % 8.42/4.47 topolo3112930676232923870pology: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type, % 8.42/4.47 euclid4440199948858584721cancel: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type, % 8.42/4.47 euclid3128863361964157862miring: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Topological__Spaces_Olinear__continuum__topology,type, % 8.42/4.47 topolo8458572112393995274pology: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type, % 8.42/4.47 ordere1937475149494474687imp_le: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type, % 8.42/4.47 condit5016429287641298734tinuum: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type, % 8.42/4.47 euclid5411537665997757685th_nat: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Countable__Complete__Lattices_Ocountable__complete__lattice,type, % 8.42/4.47 counta3822494911875563373attice: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type, % 8.42/4.47 semiri1453513574482234551roduct: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type, % 8.42/4.47 bit_un5681908812861735899ations: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type, % 8.42/4.47 condit1219197933456340205attice: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type, % 8.42/4.47 condit6923001295902523014norder: % 8.42/4.47 !>[A: $tType] : $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Archimedean__Field_Oceiling,type, % 8.42/4.47 archimedean_ceiling: % 8.42/4.47 !>[A: $tType] : ( A > int ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type, % 8.42/4.47 archim6421214686448440834_floor: % 8.42/4.47 !>[A: $tType] : ( A > int ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Archimedean__Field_Ofrac,type, % 8.42/4.47 archimedean_frac: % 8.42/4.47 !>[A: $tType] : ( A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Archimedean__Field_Oround,type, % 8.42/4.47 archimedean_round: % 8.42/4.47 !>[A: $tType] : ( A > int ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal,type, % 8.42/4.47 bNF_Ca7293521722713021262ofinal: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type, % 8.42/4.47 bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type, % 8.42/4.47 bNF_Ca8459412986667044542atLess: set @ ( product_prod @ nat @ nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type, % 8.42/4.47 bNF_Ca3754400796208372196lChain: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_BNF__Def_OGr,type, % 8.42/4.47 bNF_Gr: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type, % 8.42/4.47 bNF_Greatest_image2: % 8.42/4.47 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_BNF__Wellorder__Constructions_OFunc,type, % 8.42/4.47 bNF_Wellorder_Func: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( A > B ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_BNF__Wellorder__Constructions_OFunc__map,type, % 8.42/4.47 bNF_We4925052301507509544nc_map: % 8.42/4.47 !>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set @ B ) > ( C > A ) > ( B > D ) > ( D > C ) > B > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_BNF__Wellorder__Constructions_Obsqr,type, % 8.42/4.47 bNF_Wellorder_bsqr: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type, % 8.42/4.47 bNF_We1388413361240627857o_max2: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type, % 8.42/4.47 basic_BNF_size_prod: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Binomial_Obinomial,type, % 8.42/4.47 binomial: nat > nat > nat ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Binomial_Ogbinomial,type, % 8.42/4.47 gbinomial: % 8.42/4.47 !>[A: $tType] : ( A > nat > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Oand__int__rel,type, % 8.42/4.47 bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Oand__not__num,type, % 8.42/4.47 bit_and_not_num: num > num > ( option @ num ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Oconcat__bit,type, % 8.42/4.47 bit_concat_bit: nat > int > int > int ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type, % 8.42/4.47 bit_ri4277139882892585799ns_not: % 8.42/4.47 !>[A: $tType] : ( A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type, % 8.42/4.47 bit_ri4674362597316999326ke_bit: % 8.42/4.47 !>[A: $tType] : ( nat > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type, % 8.42/4.47 bit_se5824344872417868541ns_and: % 8.42/4.47 !>[A: $tType] : ( A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type, % 8.42/4.47 bit_se4197421643247451524op_bit: % 8.42/4.47 !>[A: $tType] : ( nat > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type, % 8.42/4.47 bit_se8732182000553998342ip_bit: % 8.42/4.47 !>[A: $tType] : ( nat > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type, % 8.42/4.47 bit_se2239418461657761734s_mask: % 8.42/4.47 !>[A: $tType] : ( nat > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type, % 8.42/4.47 bit_se1065995026697491101ons_or: % 8.42/4.47 !>[A: $tType] : ( A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type, % 8.42/4.47 bit_se4730199178511100633sh_bit: % 8.42/4.47 !>[A: $tType] : ( nat > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type, % 8.42/4.47 bit_se5668285175392031749et_bit: % 8.42/4.47 !>[A: $tType] : ( nat > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type, % 8.42/4.47 bit_se2584673776208193580ke_bit: % 8.42/4.47 !>[A: $tType] : ( nat > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type, % 8.42/4.47 bit_se2638667681897837118et_bit: % 8.42/4.47 !>[A: $tType] : ( nat > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type, % 8.42/4.47 bit_se5824344971392196577ns_xor: % 8.42/4.47 !>[A: $tType] : ( A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type, % 8.42/4.47 bit_se5641148757651400278ts_bit: % 8.42/4.47 !>[A: $tType] : ( A > nat > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Otake__bit__num,type, % 8.42/4.47 bit_take_bit_num: nat > num > ( option @ num ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type, % 8.42/4.47 bit_un7362597486090784418nd_num: num > num > ( option @ num ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type, % 8.42/4.47 bit_un2480387367778600638or_num: num > num > ( option @ num ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type, % 8.42/4.47 boolea2506097494486148201lgebra: % 8.42/4.47 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type, % 8.42/4.47 boolea3799213064322606851m_diff: % 8.42/4.47 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Code__Numeral_Obit__cut__integer,type, % 8.42/4.47 code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Code__Numeral_Odivmod__abs,type, % 8.42/4.47 code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Code__Numeral_Odivmod__integer,type, % 8.42/4.47 code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type, % 8.42/4.47 code_int_of_integer: code_integer > int ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type, % 8.42/4.47 code_integer_of_int: int > code_integer ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Code__Numeral_Onat__of__integer,type, % 8.42/4.47 code_nat_of_integer: code_integer > nat ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Complete__Lattices_OInf__class_OInf,type, % 8.42/4.47 complete_Inf_Inf: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Complete__Lattices_OSup__class_OSup,type, % 8.42/4.47 complete_Sup_Sup: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Complete__Partial__Order_Ochain,type, % 8.42/4.47 comple1602240252501008431_chain: % 8.42/4.47 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Complex_OArg,type, % 8.42/4.47 arg: complex > real ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Complex_Ocis,type, % 8.42/4.47 cis: real > complex ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Complex_Ocomplex_OComplex,type, % 8.42/4.47 complex2: real > real > complex ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type, % 8.42/4.47 condit941137186595557371_above: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type, % 8.42/4.47 condit1013018076250108175_below: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Countable__Set_Oto__nat__on,type, % 8.42/4.47 countable_to_nat_on: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > A > nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Deriv_Odifferentiable,type, % 8.42/4.47 differentiable: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Deriv_Ohas__derivative,type, % 8.42/4.47 has_derivative: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( filter @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Deriv_Ohas__field__derivative,type, % 8.42/4.47 has_field_derivative: % 8.42/4.47 !>[A: $tType] : ( ( A > A ) > A > ( filter @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Divides_Oadjust__div,type, % 8.42/4.47 adjust_div: ( product_prod @ int @ int ) > int ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Divides_Odivmod__nat,type, % 8.42/4.47 divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Divides_Oeucl__rel__int,type, % 8.42/4.47 eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type, % 8.42/4.47 unique5940410009612947441es_aux: % 8.42/4.47 !>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type, % 8.42/4.47 unique8689654367752047608divmod: % 8.42/4.47 !>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type, % 8.42/4.47 unique1321980374590559556d_step: % 8.42/4.47 !>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type, % 8.42/4.47 comm_s3205402744901411588hammer: % 8.42/4.47 !>[A: $tType] : ( A > nat > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type, % 8.42/4.47 semiring_char_0_fact: % 8.42/4.47 !>[A: $tType] : ( nat > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fields_Oinverse__class_Oinverse,type, % 8.42/4.47 inverse_inverse: % 8.42/4.47 !>[A: $tType] : ( A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Filter_Oat__bot,type, % 8.42/4.47 at_bot: % 8.42/4.47 !>[A: $tType] : ( filter @ A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Filter_Oat__top,type, % 8.42/4.47 at_top: % 8.42/4.47 !>[A: $tType] : ( filter @ A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Filter_Oeventually,type, % 8.42/4.47 eventually: % 8.42/4.47 !>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Filter_Ofiltercomap,type, % 8.42/4.47 filtercomap: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Filter_Ofilterlim,type, % 8.42/4.47 filterlim: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Filter_Ofiltermap,type, % 8.42/4.47 filtermap: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Filter_Ofinite__subsets__at__top,type, % 8.42/4.47 finite5375528669736107172at_top: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > ( filter @ ( set @ A ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Filter_Omap__filter__on,type, % 8.42/4.47 map_filter_on: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Filter_Oprincipal,type, % 8.42/4.47 principal: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Filter_Oprod__filter,type, % 8.42/4.47 prod_filter: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( filter @ A ) > ( filter @ B ) > ( filter @ ( product_prod @ A @ B ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Finite__Set_OFpow,type, % 8.42/4.47 finite_Fpow: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Finite__Set_Ocard,type, % 8.42/4.47 finite_card: % 8.42/4.47 !>[B: $tType] : ( ( set @ B ) > nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Finite__Set_Ocomp__fun__commute,type, % 8.42/4.47 finite6289374366891150609ommute: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Finite__Set_Ocomp__fun__commute__on,type, % 8.42/4.47 finite4664212375090638736ute_on: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Finite__Set_Ocomp__fun__idem__on,type, % 8.42/4.47 finite673082921795544331dem_on: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Finite__Set_Ofinite,type, % 8.42/4.47 finite_finite2: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Finite__Set_Ofold,type, % 8.42/4.47 finite_fold: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Finite__Set_Ofolding__idem__on,type, % 8.42/4.47 finite1890593828518410140dem_on: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Finite__Set_Ofolding__on,type, % 8.42/4.47 finite_folding_on: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Finite__Set_Ofolding__on_OF,type, % 8.42/4.47 finite_folding_F: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun_Obij__betw,type, % 8.42/4.47 bij_betw: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun_Ocomp,type, % 8.42/4.47 comp: % 8.42/4.47 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun_Ofun__upd,type, % 8.42/4.47 fun_upd: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun_Oid,type, % 8.42/4.47 id: % 8.42/4.47 !>[A: $tType] : ( A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun_Oinj__on,type, % 8.42/4.47 inj_on: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun_Omap__fun,type, % 8.42/4.47 map_fun: % 8.42/4.47 !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun_Ostrict__mono__on,type, % 8.42/4.47 strict_mono_on: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun_Othe__inv__into,type, % 8.42/4.47 the_inv_into: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun__Def_Ois__measure,type, % 8.42/4.47 fun_is_measure: % 8.42/4.47 !>[A: $tType] : ( ( A > nat ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun__Def_Omax__strict,type, % 8.42/4.47 fun_max_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun__Def_Omax__weak,type, % 8.42/4.47 fun_max_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun__Def_Omin__strict,type, % 8.42/4.47 fun_min_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun__Def_Omin__weak,type, % 8.42/4.47 fun_min_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun__Def_Opair__leq,type, % 8.42/4.47 fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun__Def_Opair__less,type, % 8.42/4.47 fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Fun__Def_Oreduction__pair,type, % 8.42/4.47 fun_reduction_pair: % 8.42/4.47 !>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_GCD_OGcd__class_OGcd,type, % 8.42/4.47 gcd_Gcd: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_GCD_Obezw,type, % 8.42/4.47 bezw: nat > nat > ( product_prod @ int @ int ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_GCD_Obezw__rel,type, % 8.42/4.47 bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_GCD_Ogcd__class_Ogcd,type, % 8.42/4.47 gcd_gcd: % 8.42/4.47 !>[A: $tType] : ( A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_GCD_Ogcd__nat__rel,type, % 8.42/4.47 gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_GCD_Osemiring__1__class_Osemiring__char,type, % 8.42/4.47 semiri4206861660011772517g_char: % 8.42/4.47 !>[A: $tType] : ( ( itself @ A ) > nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Groups_Oabs__class_Oabs,type, % 8.42/4.47 abs_abs: % 8.42/4.47 !>[A: $tType] : ( A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Groups_Ominus__class_Ominus,type, % 8.42/4.47 minus_minus: % 8.42/4.47 !>[A: $tType] : ( A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Groups_Oone__class_Oone,type, % 8.42/4.47 one_one: % 8.42/4.47 !>[A: $tType] : A ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Groups_Oplus__class_Oplus,type, % 8.42/4.47 plus_plus: % 8.42/4.47 !>[A: $tType] : ( A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Groups_Osgn__class_Osgn,type, % 8.42/4.47 sgn_sgn: % 8.42/4.47 !>[A: $tType] : ( A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Groups_Otimes__class_Otimes,type, % 8.42/4.47 times_times: % 8.42/4.47 !>[A: $tType] : ( A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Groups_Ouminus__class_Ouminus,type, % 8.42/4.47 uminus_uminus: % 8.42/4.47 !>[A: $tType] : ( A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Groups_Ozero__class_Ozero,type, % 8.42/4.47 zero_zero: % 8.42/4.47 !>[A: $tType] : A ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type, % 8.42/4.47 groups7311177749621191930dd_sum: % 8.42/4.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type, % 8.42/4.47 groups1027152243600224163dd_sum: % 8.42/4.47 !>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type, % 8.42/4.47 groups7121269368397514597t_prod: % 8.42/4.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type, % 8.42/4.47 groups1962203154675924110t_prod: % 8.42/4.47 !>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type, % 8.42/4.47 groups4207007520872428315er_sum: % 8.42/4.47 !>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type, % 8.42/4.47 groups8242544230860333062m_list: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_HOL_ONO__MATCH,type, % 8.42/4.47 nO_MATCH: % 8.42/4.47 !>[A: $tType,B: $tType] : ( A > B > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_HOL_OThe,type, % 8.42/4.47 the: % 8.42/4.47 !>[A: $tType] : ( ( A > $o ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_HOL_Oundefined,type, % 8.42/4.47 undefined: % 8.42/4.47 !>[A: $tType] : A ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_If,type, % 8.42/4.47 if: % 8.42/4.47 !>[A: $tType] : ( $o > A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Int_OAbs__Integ,type, % 8.42/4.47 abs_Integ: ( product_prod @ nat @ nat ) > int ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Int_ORep__Integ,type, % 8.42/4.47 rep_Integ: int > ( product_prod @ nat @ nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Int_Oint__ge__less__than,type, % 8.42/4.47 int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Int_Oint__ge__less__than2,type, % 8.42/4.47 int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Int_Onat,type, % 8.42/4.47 nat2: int > nat ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Int_Opower__int,type, % 8.42/4.47 power_int: % 8.42/4.47 !>[A: $tType] : ( A > int > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Int_Oring__1__class_OInts,type, % 8.42/4.47 ring_1_Ints: % 8.42/4.47 !>[A: $tType] : ( set @ A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Int_Oring__1__class_Oof__int,type, % 8.42/4.47 ring_1_of_int: % 8.42/4.47 !>[A: $tType] : ( int > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Lattices_Oinf__class_Oinf,type, % 8.42/4.47 inf_inf: % 8.42/4.47 !>[A: $tType] : ( A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Lattices_Osemilattice__neutr__order,type, % 8.42/4.47 semila1105856199041335345_order: % 8.42/4.47 !>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Lattices_Osup__class_Osup,type, % 8.42/4.47 sup_sup: % 8.42/4.47 !>[A: $tType] : ( A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Lattices__Big_Olinorder_OMax,type, % 8.42/4.47 lattices_Max: % 8.42/4.47 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Lattices__Big_Olinorder_OMin,type, % 8.42/4.47 lattices_Min: % 8.42/4.47 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Lattices__Big_Olinorder__class_OMax,type, % 8.42/4.47 lattic643756798349783984er_Max: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Lattices__Big_Olinorder__class_OMin,type, % 8.42/4.47 lattic643756798350308766er_Min: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type, % 8.42/4.47 lattic7623131987881927897min_on: % 8.42/4.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type, % 8.42/4.47 lattic7752659483105999362nf_fin: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type, % 8.42/4.47 lattic5882676163264333800up_fin: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Limits_OBfun,type, % 8.42/4.47 bfun: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Limits_Oat__infinity,type, % 8.42/4.47 at_infinity: % 8.42/4.47 !>[A: $tType] : ( filter @ A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oappend,type, % 8.42/4.47 append: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oarg__min__list,type, % 8.42/4.47 arg_min_list: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Obind,type, % 8.42/4.47 bind: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oconcat,type, % 8.42/4.47 concat: % 8.42/4.47 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Ocount__list,type, % 8.42/4.47 count_list: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > A > nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Odistinct,type, % 8.42/4.47 distinct: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Odrop,type, % 8.42/4.47 drop: % 8.42/4.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_OdropWhile,type, % 8.42/4.47 dropWhile: % 8.42/4.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oenumerate,type, % 8.42/4.47 enumerate: % 8.42/4.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oextract,type, % 8.42/4.47 extract: % 8.42/4.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Ofilter,type, % 8.42/4.47 filter2: % 8.42/4.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Ofind,type, % 8.42/4.47 find: % 8.42/4.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Ofolding__insort__key,type, % 8.42/4.47 folding_insort_key: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Ofoldl,type, % 8.42/4.47 foldl: % 8.42/4.47 !>[B: $tType,A: $tType] : ( ( B > A > B ) > B > ( list @ A ) > B ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Ofoldr,type, % 8.42/4.47 foldr: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Ogen__length,type, % 8.42/4.47 gen_length: % 8.42/4.47 !>[A: $tType] : ( nat > ( list @ A ) > nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olenlex,type, % 8.42/4.47 lenlex: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olex,type, % 8.42/4.47 lex: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olexn,type, % 8.42/4.47 lexn: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olexord,type, % 8.42/4.47 lexord: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olexordp,type, % 8.42/4.47 lexordp: % 8.42/4.47 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olinorder_Oinsort__key,type, % 8.42/4.47 insort_key: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olinorder_Osorted__key__list__of__set,type, % 8.42/4.47 sorted8670434370408473282of_set: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( set @ B ) > ( list @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olinorder__class_Oinsort__key,type, % 8.42/4.47 linorder_insort_key: % 8.42/4.47 !>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olinorder__class_Osorted__key__list__of__set,type, % 8.42/4.47 linord144544945434240204of_set: % 8.42/4.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > ( list @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type, % 8.42/4.47 linord4507533701916653071of_set: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olist_OCons,type, % 8.42/4.47 cons: % 8.42/4.47 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olist_ONil,type, % 8.42/4.47 nil: % 8.42/4.47 !>[A: $tType] : ( list @ A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olist_Ocase__list,type, % 8.42/4.47 case_list: % 8.42/4.47 !>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olist_Ohd,type, % 8.42/4.47 hd: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olist_Omap,type, % 8.42/4.47 map: % 8.42/4.47 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olist_Oset,type, % 8.42/4.47 set2: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olist_Osize__list,type, % 8.42/4.47 size_list: % 8.42/4.47 !>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olist_Otl,type, % 8.42/4.47 tl: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olist__update,type, % 8.42/4.47 list_update: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olistrel,type, % 8.42/4.47 listrel: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olistrel1,type, % 8.42/4.47 listrel1: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olistrel1p,type, % 8.42/4.47 listrel1p: % 8.42/4.47 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olistrelp,type, % 8.42/4.47 listrelp: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Olistset,type, % 8.42/4.47 listset: % 8.42/4.47 !>[A: $tType] : ( ( list @ ( set @ A ) ) > ( set @ ( list @ A ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Omap__filter,type, % 8.42/4.47 map_filter: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Omap__project,type, % 8.42/4.47 map_project: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > ( set @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Omin__list,type, % 8.42/4.47 min_list: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Omin__list__rel,type, % 8.42/4.47 min_list_rel: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_On__lists,type, % 8.42/4.47 n_lists: % 8.42/4.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Onth,type, % 8.42/4.47 nth: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > nat > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Onths,type, % 8.42/4.47 nths: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Opartition,type, % 8.42/4.47 partition: % 8.42/4.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oproduct,type, % 8.42/4.47 product: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oproduct__lists,type, % 8.42/4.47 product_lists: % 8.42/4.47 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oremdups,type, % 8.42/4.47 remdups: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oremdups__adj,type, % 8.42/4.47 remdups_adj: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oremdups__adj__rel,type, % 8.42/4.47 remdups_adj_rel: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oremove1,type, % 8.42/4.47 remove1: % 8.42/4.47 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_OremoveAll,type, % 8.42/4.47 removeAll: % 8.42/4.47 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oreplicate,type, % 8.42/4.47 replicate: % 8.42/4.47 !>[A: $tType] : ( nat > A > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Orev,type, % 8.42/4.47 rev: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Orotate,type, % 8.42/4.47 rotate: % 8.42/4.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Orotate1,type, % 8.42/4.47 rotate1: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oset__Cons,type, % 8.42/4.47 set_Cons: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) > ( set @ ( list @ A ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oshuffles,type, % 8.42/4.47 shuffles: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Osorted__wrt,type, % 8.42/4.47 sorted_wrt: % 8.42/4.47 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Osubseqs,type, % 8.42/4.47 subseqs: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Otake,type, % 8.42/4.47 take: % 8.42/4.47 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_OtakeWhile,type, % 8.42/4.47 takeWhile: % 8.42/4.47 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Otranspose,type, % 8.42/4.47 transpose: % 8.42/4.47 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Otranspose__rel,type, % 8.42/4.47 transpose_rel: % 8.42/4.47 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Ounion,type, % 8.42/4.47 union: % 8.42/4.47 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oupt,type, % 8.42/4.47 upt: nat > nat > ( list @ nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oupto,type, % 8.42/4.47 upto: int > int > ( list @ int ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oupto__aux,type, % 8.42/4.47 upto_aux: int > int > ( list @ int ) > ( list @ int ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Oupto__rel,type, % 8.42/4.47 upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_List_Ozip,type, % 8.42/4.47 zip: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Map_Odom,type, % 8.42/4.47 dom: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Map_Ograph,type, % 8.42/4.47 graph: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Map_Omap__of,type, % 8.42/4.47 map_of: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Map_Omap__upds,type, % 8.42/4.47 map_upds: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Map_Oran,type, % 8.42/4.47 ran: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Map_Orestrict__map,type, % 8.42/4.47 restrict_map: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat_OSuc,type, % 8.42/4.47 suc: nat > nat ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat_Ocompow,type, % 8.42/4.47 compow: % 8.42/4.47 !>[A: $tType] : ( nat > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat_Ofunpow,type, % 8.42/4.47 funpow: % 8.42/4.47 !>[A: $tType] : ( nat > ( A > A ) > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat_Onat_Ocase__nat,type, % 8.42/4.47 case_nat: % 8.42/4.47 !>[A: $tType] : ( A > ( nat > A ) > nat > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat_Onat_Opred,type, % 8.42/4.47 pred: nat > nat ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat_Oold_Onat_Orec__nat,type, % 8.42/4.47 rec_nat: % 8.42/4.47 !>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat_Oold_Onat_Orec__set__nat,type, % 8.42/4.47 rec_set_nat: % 8.42/4.47 !>[T: $tType] : ( T > ( nat > T > T ) > nat > T > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat_Osemiring__1__class_ONats,type, % 8.42/4.47 semiring_1_Nats: % 8.42/4.47 !>[A: $tType] : ( set @ A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type, % 8.42/4.47 semiring_1_of_nat: % 8.42/4.47 !>[A: $tType] : ( nat > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type, % 8.42/4.47 semiri8178284476397505188at_aux: % 8.42/4.47 !>[A: $tType] : ( ( A > A ) > nat > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat_Osize__class_Osize,type, % 8.42/4.47 size_size: % 8.42/4.47 !>[A: $tType] : ( A > nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat__Bijection_Olist__encode,type, % 8.42/4.47 nat_list_encode: ( list @ nat ) > nat ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat__Bijection_Olist__encode__rel,type, % 8.42/4.47 nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type, % 8.42/4.47 nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type, % 8.42/4.47 nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat__Bijection_Oprod__encode,type, % 8.42/4.47 nat_prod_encode: ( product_prod @ nat @ nat ) > nat ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat__Bijection_Oset__decode,type, % 8.42/4.47 nat_set_decode: nat > ( set @ nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat__Bijection_Oset__encode,type, % 8.42/4.47 nat_set_encode: ( set @ nat ) > nat ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Nat__Bijection_Otriangle,type, % 8.42/4.47 nat_triangle: nat > nat ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_NthRoot_Oroot,type, % 8.42/4.47 root: nat > real > real ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_NthRoot_Osqrt,type, % 8.42/4.47 sqrt: real > real ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Num_OBitM,type, % 8.42/4.47 bitM: num > num ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Num_Oinc,type, % 8.42/4.47 inc: num > num ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type, % 8.42/4.47 neg_numeral_dbl_dec: % 8.42/4.47 !>[A: $tType] : ( A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Num_Oneg__numeral__class_Osub,type, % 8.42/4.47 neg_numeral_sub: % 8.42/4.47 !>[A: $tType] : ( num > num > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Num_Onum_OBit0,type, % 8.42/4.47 bit0: num > num ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Num_Onum_OBit1,type, % 8.42/4.47 bit1: num > num ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Num_Onum_OOne,type, % 8.42/4.47 one2: num ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Num_Onum_Ocase__num,type, % 8.42/4.47 case_num: % 8.42/4.47 !>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Num_Onum_Osize__num,type, % 8.42/4.47 size_num: num > nat ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Num_Onum__of__nat,type, % 8.42/4.47 num_of_nat: nat > num ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Num_Onumeral__class_Onumeral,type, % 8.42/4.47 numeral_numeral: % 8.42/4.47 !>[A: $tType] : ( num > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Num_Opow,type, % 8.42/4.47 pow: num > num > num ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Num_Opred__numeral,type, % 8.42/4.47 pred_numeral: num > nat ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Option_Ooption_ONone,type, % 8.42/4.47 none: % 8.42/4.47 !>[A: $tType] : ( option @ A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Option_Ooption_OSome,type, % 8.42/4.47 some: % 8.42/4.47 !>[A: $tType] : ( A > ( option @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Option_Ooption_Ocase__option,type, % 8.42/4.47 case_option: % 8.42/4.47 !>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Option_Ooption_Omap__option,type, % 8.42/4.47 map_option: % 8.42/4.47 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( option @ A ) > ( option @ Aa ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Option_Ooption_Orec__option,type, % 8.42/4.47 rec_option: % 8.42/4.47 !>[C: $tType,A: $tType] : ( C > ( A > C ) > ( option @ A ) > C ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Option_Ooption_Osize__option,type, % 8.42/4.47 size_option: % 8.42/4.47 !>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Option_Ooption_Othe,type, % 8.42/4.47 the2: % 8.42/4.47 !>[A: $tType] : ( ( option @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Option_Othese,type, % 8.42/4.47 these: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Order__Continuity_Ocountable__complete__lattice__class_Occlfp,type, % 8.42/4.47 order_532582986084564980_cclfp: % 8.42/4.47 !>[A: $tType] : ( ( A > A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Order__Relation_OAbove,type, % 8.42/4.47 order_Above: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Order__Relation_OUnder,type, % 8.42/4.47 order_Under: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Order__Relation_OUnderS,type, % 8.42/4.47 order_UnderS: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Order__Relation_OaboveS,type, % 8.42/4.47 order_aboveS: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Order__Relation_Olinear__order__on,type, % 8.42/4.47 order_679001287576687338der_on: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Order__Relation_Opartial__order__on,type, % 8.42/4.47 order_7125193373082350890der_on: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Order__Relation_Ounder,type, % 8.42/4.47 order_under: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Order__Relation_OunderS,type, % 8.42/4.47 order_underS: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Orderings_Obot__class_Obot,type, % 8.42/4.47 bot_bot: % 8.42/4.47 !>[A: $tType] : A ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Orderings_Oord_OLeast,type, % 8.42/4.47 least: % 8.42/4.47 !>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Orderings_Oord_Omax,type, % 8.42/4.47 max: % 8.42/4.47 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Orderings_Oord_Omin,type, % 8.42/4.47 min: % 8.42/4.47 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Orderings_Oord__class_Oless,type, % 8.42/4.47 ord_less: % 8.42/4.47 !>[A: $tType] : ( A > A > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type, % 8.42/4.47 ord_less_eq: % 8.42/4.47 !>[A: $tType] : ( A > A > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Orderings_Oord__class_Omax,type, % 8.42/4.47 ord_max: % 8.42/4.47 !>[A: $tType] : ( A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Orderings_Oord__class_Omin,type, % 8.42/4.47 ord_min: % 8.42/4.47 !>[A: $tType] : ( A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Orderings_Oorder__class_OGreatest,type, % 8.42/4.47 order_Greatest: % 8.42/4.47 !>[A: $tType] : ( ( A > $o ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Orderings_Oorder__class_Oantimono,type, % 8.42/4.47 order_antimono: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Orderings_Oorder__class_Omono,type, % 8.42/4.47 order_mono: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type, % 8.42/4.47 order_strict_mono: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Orderings_Otop__class_Otop,type, % 8.42/4.47 top_top: % 8.42/4.47 !>[A: $tType] : A ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Partial__Function_Oflat__lub,type, % 8.42/4.47 partial_flat_lub: % 8.42/4.47 !>[A: $tType] : ( A > ( set @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Power_Opower__class_Opower,type, % 8.42/4.47 power_power: % 8.42/4.47 !>[A: $tType] : ( A > nat > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Product__Type_OPair,type, % 8.42/4.47 product_Pair: % 8.42/4.47 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Product__Type_OSigma,type, % 8.42/4.47 product_Sigma: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Product__Type_Oapfst,type, % 8.42/4.47 product_apfst: % 8.42/4.47 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Product__Type_Oapsnd,type, % 8.42/4.47 product_apsnd: % 8.42/4.47 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type, % 8.42/4.47 product_rec_prod: % 8.42/4.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type, % 8.42/4.47 product_rec_set_prod: % 8.42/4.47 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Product__Type_Oprod_Ocase__prod,type, % 8.42/4.47 product_case_prod: % 8.42/4.47 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Product__Type_Oprod_Ofst,type, % 8.42/4.47 product_fst: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Product__Type_Oprod_Osnd,type, % 8.42/4.47 product_snd: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Product__Type_Oproduct,type, % 8.42/4.47 product_product: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Rat_OFract,type, % 8.42/4.47 fract: int > int > rat ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Rat_OFrct,type, % 8.42/4.47 frct: ( product_prod @ int @ int ) > rat ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Rat_ORep__Rat,type, % 8.42/4.47 rep_Rat: rat > ( product_prod @ int @ int ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Rat_Ofield__char__0__class_ORats,type, % 8.42/4.47 field_char_0_Rats: % 8.42/4.47 !>[A: $tType] : ( set @ A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Rat_Onormalize,type, % 8.42/4.47 normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Rat_Oof__int,type, % 8.42/4.47 of_int: int > rat ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Rat_Opositive,type, % 8.42/4.47 positive: rat > $o ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Rat_Oquotient__of,type, % 8.42/4.47 quotient_of: rat > ( product_prod @ int @ int ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Real__Vector__Spaces_Obounded__linear,type, % 8.42/4.47 real_V3181309239436604168linear: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type, % 8.42/4.47 real_V557655796197034286t_dist: % 8.42/4.47 !>[A: $tType] : ( A > A > real ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type, % 8.42/4.47 real_V7770717601297561774m_norm: % 8.42/4.47 !>[A: $tType] : ( A > real ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Real__Vector__Spaces_Oof__real,type, % 8.42/4.47 real_Vector_of_real: % 8.42/4.47 !>[A: $tType] : ( real > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type, % 8.42/4.47 real_V8093663219630862766scaleR: % 8.42/4.47 !>[A: $tType] : ( real > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Relation_OField,type, % 8.42/4.47 field2: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Relation_OId,type, % 8.42/4.47 id2: % 8.42/4.47 !>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Relation_OId__on,type, % 8.42/4.47 id_on: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Relation_Oirrefl,type, % 8.42/4.47 irrefl: % 8.42/4.47 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Relation_Orefl__on,type, % 8.42/4.47 refl_on: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Relation_Orelcomp,type, % 8.42/4.47 relcomp: % 8.42/4.47 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Relation_Ototal__on,type, % 8.42/4.47 total_on: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Rings_Odivide__class_Odivide,type, % 8.42/4.47 divide_divide: % 8.42/4.47 !>[A: $tType] : ( A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Rings_Odvd__class_Odvd,type, % 8.42/4.47 dvd_dvd: % 8.42/4.47 !>[A: $tType] : ( A > A > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Rings_Omodulo__class_Omodulo,type, % 8.42/4.47 modulo_modulo: % 8.42/4.47 !>[A: $tType] : ( A > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type, % 8.42/4.47 zero_neq_one_of_bool: % 8.42/4.47 !>[A: $tType] : ( $o > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Series_Osuminf,type, % 8.42/4.47 suminf: % 8.42/4.47 !>[A: $tType] : ( ( nat > A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Series_Osummable,type, % 8.42/4.47 summable: % 8.42/4.47 !>[A: $tType] : ( ( nat > A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Series_Osums,type, % 8.42/4.47 sums: % 8.42/4.47 !>[A: $tType] : ( ( nat > A ) > A > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set_OBall,type, % 8.42/4.47 ball: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set_OBex,type, % 8.42/4.47 bex: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set_OCollect,type, % 8.42/4.47 collect: % 8.42/4.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set_OPow,type, % 8.42/4.47 pow2: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set_Ofilter,type, % 8.42/4.47 filter3: % 8.42/4.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set_Oimage,type, % 8.42/4.47 image: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set_Oinsert,type, % 8.42/4.47 insert: % 8.42/4.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set_Ois__singleton,type, % 8.42/4.47 is_singleton: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set_Oremove,type, % 8.42/4.47 remove: % 8.42/4.47 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set_Othe__elem,type, % 8.42/4.47 the_elem: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set_Ovimage,type, % 8.42/4.47 vimage: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type, % 8.42/4.47 set_fo6178422350223883121st_nat: % 8.42/4.47 !>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set__Interval_Oord__class_OatLeast,type, % 8.42/4.47 set_ord_atLeast: % 8.42/4.47 !>[A: $tType] : ( A > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type, % 8.42/4.47 set_or1337092689740270186AtMost: % 8.42/4.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type, % 8.42/4.47 set_or7035219750837199246ssThan: % 8.42/4.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set__Interval_Oord__class_OatMost,type, % 8.42/4.47 set_ord_atMost: % 8.42/4.47 !>[A: $tType] : ( A > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type, % 8.42/4.47 set_ord_greaterThan: % 8.42/4.47 !>[A: $tType] : ( A > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type, % 8.42/4.47 set_or3652927894154168847AtMost: % 8.42/4.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type, % 8.42/4.47 set_or5935395276787703475ssThan: % 8.42/4.47 !>[A: $tType] : ( A > A > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Set__Interval_Oord__class_OlessThan,type, % 8.42/4.47 set_ord_lessThan: % 8.42/4.47 !>[A: $tType] : ( A > ( set @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Topological__Spaces_Ocontinuous,type, % 8.42/4.47 topolo3448309680560233919inuous: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( filter @ A ) > ( A > B ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Topological__Spaces_Ocontinuous__on,type, % 8.42/4.47 topolo81223032696312382ous_on: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Topological__Spaces_Omonoseq,type, % 8.42/4.47 topological_monoseq: % 8.42/4.47 !>[A: $tType] : ( ( nat > A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Topological__Spaces_Oopen__class_Oopen,type, % 8.42/4.47 topolo1002775350975398744n_open: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Topological__Spaces_Ot2__space__class_OLim,type, % 8.42/4.47 topolo3827282254853284352ce_Lim: % 8.42/4.47 !>[F: $tType,A: $tType] : ( ( filter @ F ) > ( F > A ) > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type, % 8.42/4.47 topolo174197925503356063within: % 8.42/4.47 !>[A: $tType] : ( A > ( set @ A ) > ( filter @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Topological__Spaces_Otopological__space__class_Oclosed,type, % 8.42/4.47 topolo7761053866217962861closed: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Topological__Spaces_Otopological__space__class_Ocompact,type, % 8.42/4.47 topolo2193935891317330818ompact: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type, % 8.42/4.47 topolo7230453075368039082e_nhds: % 8.42/4.47 !>[A: $tType] : ( A > ( filter @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type, % 8.42/4.47 topolo3814608138187158403Cauchy: % 8.42/4.47 !>[A: $tType] : ( ( nat > A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type, % 8.42/4.47 topolo6773858410816713723filter: % 8.42/4.47 !>[A: $tType] : ( ( filter @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocomplete,type, % 8.42/4.47 topolo2479028161051973599mplete: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type, % 8.42/4.47 topolo6688025880775521714ounded: % 8.42/4.47 !>[A: $tType] : ( ( set @ A ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity,type, % 8.42/4.47 topolo7806501430040627800ormity: % 8.42/4.47 !>[A: $tType] : ( filter @ ( product_prod @ A @ A ) ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Topological__Spaces_Ouniformly__continuous__on,type, % 8.42/4.47 topolo6026614971017936543ous_on: % 8.42/4.47 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $o ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Transcendental_Oarccos,type, % 8.42/4.47 arccos: real > real ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Transcendental_Oarcosh,type, % 8.42/4.47 arcosh: % 8.42/4.47 !>[A: $tType] : ( A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Transcendental_Oarcsin,type, % 8.42/4.47 arcsin: real > real ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Transcendental_Oarctan,type, % 8.42/4.47 arctan: real > real ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Transcendental_Oarsinh,type, % 8.42/4.47 arsinh: % 8.42/4.47 !>[A: $tType] : ( A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Transcendental_Oartanh,type, % 8.42/4.47 artanh: % 8.42/4.47 !>[A: $tType] : ( A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Transcendental_Ocos,type, % 8.42/4.47 cos: % 8.42/4.47 !>[A: $tType] : ( A > A ) ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Transcendental_Ocos__coeff,type, % 8.42/4.47 cos_coeff: nat > real ). % 8.42/4.47 % 8.42/4.47 thf(sy_c_Transcendental_Ocosh,type, % 8.42/4.47 cosh: % 8.42/4.47 !>[A: $tType] : ( A > A ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Transcendental_Ocot,type, % 8.42/4.48 cot: % 8.42/4.48 !>[A: $tType] : ( A > A ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Transcendental_Odiffs,type, % 8.42/4.48 diffs: % 8.42/4.48 !>[A: $tType] : ( ( nat > A ) > nat > A ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Transcendental_Oexp,type, % 8.42/4.48 exp: % 8.42/4.48 !>[A: $tType] : ( A > A ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Transcendental_Oln__class_Oln,type, % 8.42/4.48 ln_ln: % 8.42/4.48 !>[A: $tType] : ( A > A ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Transcendental_Olog,type, % 8.42/4.48 log: real > real > real ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Transcendental_Opi,type, % 8.42/4.48 pi: real ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Transcendental_Opowr,type, % 8.42/4.48 powr: % 8.42/4.48 !>[A: $tType] : ( A > A > A ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Transcendental_Osin,type, % 8.42/4.48 sin: % 8.42/4.48 !>[A: $tType] : ( A > A ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Transcendental_Osin__coeff,type, % 8.42/4.48 sin_coeff: nat > real ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Transcendental_Osinh,type, % 8.42/4.48 sinh: % 8.42/4.48 !>[A: $tType] : ( A > A ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Transcendental_Otan,type, % 8.42/4.48 tan: % 8.42/4.48 !>[A: $tType] : ( A > A ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Transcendental_Otanh,type, % 8.42/4.48 tanh: % 8.42/4.48 !>[A: $tType] : ( A > A ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Transitive__Closure_Ontrancl,type, % 8.42/4.48 transitive_ntrancl: % 8.42/4.48 !>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Transitive__Closure_Ortrancl,type, % 8.42/4.48 transitive_rtrancl: % 8.42/4.48 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Transitive__Closure_Otrancl,type, % 8.42/4.48 transitive_trancl: % 8.42/4.48 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type, % 8.42/4.48 vEBT_Leaf: $o > $o > vEBT_VEBT ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_OVEBT_ONode,type, % 8.42/4.48 vEBT_Node: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > vEBT_VEBT ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type, % 8.42/4.48 vEBT_size_VEBT: vEBT_VEBT > nat ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type, % 8.42/4.48 vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type, % 8.42/4.48 vEBT_VEBT_high: nat > nat > nat ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type, % 8.42/4.48 vEBT_V5917875025757280293ildren: nat > ( list @ vEBT_VEBT ) > nat > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type, % 8.42/4.48 vEBT_VEBT_low: nat > nat > nat ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type, % 8.42/4.48 vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type, % 8.42/4.48 vEBT_V4351362008482014158ma_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type, % 8.42/4.48 vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type, % 8.42/4.48 vEBT_V5765760719290551771er_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type, % 8.42/4.48 vEBT_VEBT_valid: vEBT_VEBT > nat > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type, % 8.42/4.48 vEBT_VEBT_valid_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type, % 8.42/4.48 vEBT_invar_vebt: vEBT_VEBT > nat > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_Oset__vebt,type, % 8.42/4.48 vEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_Ovebt__buildup,type, % 8.42/4.48 vEBT_vebt_buildup: nat > vEBT_VEBT ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type, % 8.42/4.48 vEBT_v4011308405150292612up_rel: nat > nat > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type, % 8.42/4.48 vEBT_VEBT_bit_concat: nat > nat > nat > nat ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type, % 8.42/4.48 vEBT_VEBT_minNull: vEBT_VEBT > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type, % 8.42/4.48 vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type, % 8.42/4.48 vEBT_VEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Member_Ovebt__member,type, % 8.42/4.48 vEBT_vebt_member: vEBT_VEBT > nat > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__Member_Ovebt__member__rel,type, % 8.42/4.48 vEBT_vebt_member_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type, % 8.42/4.48 vEBT_VEBT_add: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type, % 8.42/4.48 vEBT_VEBT_greater: ( option @ nat ) > ( option @ nat ) > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type, % 8.42/4.48 vEBT_VEBT_less: ( option @ nat ) > ( option @ nat ) > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type, % 8.42/4.48 vEBT_VEBT_lesseq: ( option @ nat ) > ( option @ nat ) > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type, % 8.42/4.48 vEBT_VEBT_max_in_set: ( set @ nat ) > nat > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type, % 8.42/4.48 vEBT_VEBT_min_in_set: ( set @ nat ) > nat > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type, % 8.42/4.48 vEBT_VEBT_mul: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift,type, % 8.42/4.48 vEBT_V2048590022279873568_shift: % 8.42/4.48 !>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type, % 8.42/4.48 vEBT_VEBT_power: ( option @ nat ) > ( option @ nat ) > ( option @ nat ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__MinMax_Ovebt__maxt,type, % 8.42/4.48 vEBT_vebt_maxt: vEBT_VEBT > ( option @ nat ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type, % 8.42/4.48 vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__MinMax_Ovebt__mint,type, % 8.42/4.48 vEBT_vebt_mint: vEBT_VEBT > ( option @ nat ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type, % 8.42/4.48 vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Wellfounded_Oaccp,type, % 8.42/4.48 accp: % 8.42/4.48 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Wellfounded_Ofinite__psubset,type, % 8.42/4.48 finite_psubset: % 8.42/4.48 !>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Wellfounded_Olex__prod,type, % 8.42/4.48 lex_prod: % 8.42/4.48 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Wellfounded_Omax__ext,type, % 8.42/4.48 max_ext: % 8.42/4.48 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Wellfounded_Omax__extp,type, % 8.42/4.48 max_extp: % 8.42/4.48 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ A ) > $o ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Wellfounded_Omeasure,type, % 8.42/4.48 measure: % 8.42/4.48 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Wellfounded_Omin__ext,type, % 8.42/4.48 min_ext: % 8.42/4.48 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Wellfounded_Omlex__prod,type, % 8.42/4.48 mlex_prod: % 8.42/4.48 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Wellfounded_Opred__nat,type, % 8.42/4.48 pred_nat: set @ ( product_prod @ nat @ nat ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Wellfounded_Owf,type, % 8.42/4.48 wf: % 8.42/4.48 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Wfrec_Osame__fst,type, % 8.42/4.48 same_fst: % 8.42/4.48 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_Zorn_Ochains,type, % 8.42/4.48 chains: % 8.42/4.48 !>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ ( set @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_fChoice,type, % 8.42/4.48 fChoice: % 8.42/4.48 !>[A: $tType] : ( ( A > $o ) > A ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_c_member,type, % 8.42/4.48 member: % 8.42/4.48 !>[A: $tType] : ( A > ( set @ A ) > $o ) ). % 8.42/4.48 % 8.42/4.48 thf(sy_v_deg,type, % 8.42/4.48 deg: nat ). % 8.42/4.48 % 8.42/4.48 thf(sy_v_m____,type, % 8.42/4.48 m: nat ). % 8.42/4.48 % 8.42/4.48 thf(sy_v_ma,type, % 8.42/4.48 ma: nat ). % 8.42/4.48 % 8.42/4.48 thf(sy_v_maxs____,type, % 8.42/4.48 maxs: nat ). % 8.42/4.48 % 8.42/4.48 thf(sy_v_mi,type, % 8.42/4.48 mi: nat ). % 8.42/4.48 % 8.42/4.48 thf(sy_v_n____,type, % 8.42/4.48 n: nat ). % 8.42/4.48 % 8.42/4.48 thf(sy_v_summary,type, % 8.42/4.48 summary: vEBT_VEBT ). % 8.42/4.48 % 8.42/4.48 thf(sy_v_thesis____,type, % 8.42/4.48 thesis: $o ). % 8.42/4.48 % 8.42/4.48 thf(sy_v_treeList,type, % 8.42/4.48 treeList: list @ vEBT_VEBT ). % 8.42/4.48 % 8.42/4.48 thf(fact_0__092_060open_062_092_060exists_062x_O_Aboth__member__options_A_ItreeList_A_B_Amaxs_J_Ax_092_060close_062,axiom, % 8.42/4.48 ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ maxs ) @ X_1 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_1_not__min__Null__member,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT] : % 8.42/4.48 ( ~ ( vEBT_VEBT_minNull @ T2 ) % 8.42/4.48 => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_1 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_2__092_060open_062Some_Amaxs_A_061_Avebt__maxt_Asummary_092_060close_062,axiom, % 8.42/4.48 ( ( some @ nat @ maxs ) % 8.42/4.48 = ( vEBT_vebt_maxt @ summary ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_3_both__member__options__def,axiom, % 8.42/4.48 ( vEBT_V8194947554948674370ptions % 8.42/4.48 = ( ^ [T3: vEBT_VEBT,X: nat] : % 8.42/4.48 ( ( vEBT_V5719532721284313246member @ T3 @ X ) % 8.42/4.48 | ( vEBT_VEBT_membermima @ T3 @ X ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_4__C5_C_I7_J,axiom, % 8.42/4.48 ( ( mi = ma ) % 8.42/4.48 => ! [X2: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ treeList ) ) % 8.42/4.48 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_5__C5_C_I1_J,axiom, % 8.42/4.48 ! [X2: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ treeList ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ X2 @ n ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_6_in__children__def,axiom, % 8.42/4.48 ( vEBT_V5917875025757280293ildren % 8.42/4.48 = ( ^ [N: nat,TreeList: list @ vEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ N ) ) @ ( vEBT_VEBT_low @ X @ N ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_7_both__member__options__equiv__member,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat,X3: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ( vEBT_V8194947554948674370ptions @ T2 @ X3 ) % 8.42/4.48 = ( vEBT_vebt_member @ T2 @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_8_valid__member__both__member__options,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat,X3: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ( vEBT_V8194947554948674370ptions @ T2 @ X3 ) % 8.42/4.48 => ( vEBT_vebt_member @ T2 @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_9_intind,axiom, % 8.42/4.48 ! [A: $tType,I: nat,N2: nat,P: A > $o,X3: A] : % 8.42/4.48 ( ( ord_less @ nat @ I @ N2 ) % 8.42/4.48 => ( ( P @ X3 ) % 8.42/4.48 => ( P @ ( nth @ A @ ( replicate @ A @ N2 @ X3 ) @ I ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_10_assms_I2_J,axiom, % 8.42/4.48 mi != ma ). % 8.42/4.48 % 8.42/4.48 thf(fact_11_list__update__id,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,I: nat] : % 8.42/4.48 ( ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ I ) ) % 8.42/4.48 = Xs ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_12_nth__list__update__neq,axiom, % 8.42/4.48 ! [A: $tType,I: nat,J: nat,Xs: list @ A,X3: A] : % 8.42/4.48 ( ( I != J ) % 8.42/4.48 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ J ) % 8.42/4.48 = ( nth @ A @ Xs @ J ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_13_min__Null__member,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,X3: nat] : % 8.42/4.48 ( ( vEBT_VEBT_minNull @ T2 ) % 8.42/4.48 => ~ ( vEBT_vebt_member @ T2 @ X3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_14__C5_C_I2_J,axiom, % 8.42/4.48 vEBT_invar_vebt @ summary @ m ). % 8.42/4.48 % 8.42/4.48 thf(fact_15_bit__split__inv,axiom, % 8.42/4.48 ! [X3: nat,D2: nat] : % 8.42/4.48 ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X3 @ D2 ) @ ( vEBT_VEBT_low @ X3 @ D2 ) @ D2 ) % 8.42/4.48 = X3 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_16__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxs_O_ASome_Amaxs_A_061_Avebt__maxt_Asummary_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom, % 8.42/4.48 ~ ! [Maxs: nat] : % 8.42/4.48 ( ( some @ nat @ Maxs ) % 8.42/4.48 != ( vEBT_vebt_maxt @ summary ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_17_member__valid__both__member__options,axiom, % 8.42/4.48 ! [Tree: vEBT_VEBT,N2: nat,X3: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ Tree @ N2 ) % 8.42/4.48 => ( ( vEBT_vebt_member @ Tree @ X3 ) % 8.42/4.48 => ( ( vEBT_V5719532721284313246member @ Tree @ X3 ) % 8.42/4.48 | ( vEBT_VEBT_membermima @ Tree @ X3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_18_maxt__member,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat,Maxi: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ( ( vEBT_vebt_maxt @ T2 ) % 8.42/4.48 = ( some @ nat @ Maxi ) ) % 8.42/4.48 => ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_19_list__update__overwrite,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,I: nat,X3: A,Y: A] : % 8.42/4.48 ( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ I @ Y ) % 8.42/4.48 = ( list_update @ A @ Xs @ I @ Y ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_20_member__correct,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat,X3: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ( vEBT_vebt_member @ T2 @ X3 ) % 8.42/4.48 = ( member @ nat @ X3 @ ( vEBT_set_vebt @ T2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_21__092_060open_062both__member__options_Asummary_A_Ihigh_Ama_An_J_092_060close_062,axiom, % 8.42/4.48 vEBT_V8194947554948674370ptions @ summary @ ( vEBT_VEBT_high @ ma @ n ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_22_nth__replicate,axiom, % 8.42/4.48 ! [A: $tType,I: nat,N2: nat,X3: A] : % 8.42/4.48 ( ( ord_less @ nat @ I @ N2 ) % 8.42/4.48 => ( ( nth @ A @ ( replicate @ A @ N2 @ X3 ) @ I ) % 8.42/4.48 = X3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_23__C5_C_I8_J,axiom, % 8.42/4.48 ord_less_eq @ nat @ mi @ ma ). % 8.42/4.48 % 8.42/4.48 thf(fact_24_greater__shift,axiom, % 8.42/4.48 ( ( ord_less @ nat ) % 8.42/4.48 = ( ^ [Y2: nat,X: nat] : ( vEBT_VEBT_greater @ ( some @ nat @ X ) @ ( some @ nat @ Y2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_25_less__shift,axiom, % 8.42/4.48 ( ( ord_less @ nat ) % 8.42/4.48 = ( ^ [X: nat,Y2: nat] : ( vEBT_VEBT_less @ ( some @ nat @ X ) @ ( some @ nat @ Y2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_26__092_060open_062high_Ama_An_A_060_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062,axiom, % 8.42/4.48 ord_less @ nat @ ( vEBT_VEBT_high @ ma @ n ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ summary ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_27_mint__member,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat,Maxi: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ( ( vEBT_vebt_mint @ T2 ) % 8.42/4.48 = ( some @ nat @ Maxi ) ) % 8.42/4.48 => ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_28_list__update__swap,axiom, % 8.42/4.48 ! [A: $tType,I: nat,I2: nat,Xs: list @ A,X3: A,X4: A] : % 8.42/4.48 ( ( I != I2 ) % 8.42/4.48 => ( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ I2 @ X4 ) % 8.42/4.48 = ( list_update @ A @ ( list_update @ A @ Xs @ I2 @ X4 ) @ I @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_29_maxt__corr__help,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat,Maxi: nat,X3: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ( ( vEBT_vebt_maxt @ T2 ) % 8.42/4.48 = ( some @ nat @ Maxi ) ) % 8.42/4.48 => ( ( vEBT_vebt_member @ T2 @ X3 ) % 8.42/4.48 => ( ord_less_eq @ nat @ X3 @ Maxi ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_30_maxt__corr,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat,X3: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ( ( vEBT_vebt_maxt @ T2 ) % 8.42/4.48 = ( some @ nat @ X3 ) ) % 8.42/4.48 => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_31_maxt__sound,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat,X3: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 ) % 8.42/4.48 => ( ( vEBT_vebt_maxt @ T2 ) % 8.42/4.48 = ( some @ nat @ X3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_32_buildup__nothing__in__leaf,axiom, % 8.42/4.48 ! [N2: nat,X3: nat] : % 8.42/4.48 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X3 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_33_buildup__nothing__in__min__max,axiom, % 8.42/4.48 ! [N2: nat,X3: nat] : % 8.42/4.48 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X3 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_34_option_Oinject,axiom, % 8.42/4.48 ! [A: $tType,X22: A,Y22: A] : % 8.42/4.48 ( ( ( some @ A @ X22 ) % 8.42/4.48 = ( some @ A @ Y22 ) ) % 8.42/4.48 = ( X22 = Y22 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_35_valid__eq,axiom, % 8.42/4.48 vEBT_VEBT_valid = vEBT_invar_vebt ). % 8.42/4.48 % 8.42/4.48 thf(fact_36_valid__eq1,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,D2: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ D2 ) % 8.42/4.48 => ( vEBT_VEBT_valid @ T2 @ D2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_37_valid__eq2,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,D2: nat] : % 8.42/4.48 ( ( vEBT_VEBT_valid @ T2 @ D2 ) % 8.42/4.48 => ( vEBT_invar_vebt @ T2 @ D2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_38_inthall,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,P: A > $o,N2: nat] : % 8.42/4.48 ( ! [X5: A] : % 8.42/4.48 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 8.42/4.48 => ( P @ X5 ) ) % 8.42/4.48 => ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 => ( P @ ( nth @ A @ Xs @ N2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_39_deg__not__0,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_40__092_060open_062high_Ama_An_A_092_060le_062_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062,axiom, % 8.42/4.48 ord_less_eq @ nat @ ( vEBT_VEBT_high @ ma @ n ) @ ( the2 @ nat @ ( vEBT_vebt_maxt @ summary ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_41_valid__tree__deg__neq__0,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT] : % 8.42/4.48 ~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_42_valid__0__not,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT] : % 8.42/4.48 ~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_43_min__in__set__def,axiom, % 8.42/4.48 ( vEBT_VEBT_min_in_set % 8.42/4.48 = ( ^ [Xs2: set @ nat,X: nat] : % 8.42/4.48 ( ( member @ nat @ X @ Xs2 ) % 8.42/4.48 & ! [Y2: nat] : % 8.42/4.48 ( ( member @ nat @ Y2 @ Xs2 ) % 8.42/4.48 => ( ord_less_eq @ nat @ X @ Y2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_44_max__in__set__def,axiom, % 8.42/4.48 ( vEBT_VEBT_max_in_set % 8.42/4.48 = ( ^ [Xs2: set @ nat,X: nat] : % 8.42/4.48 ( ( member @ nat @ X @ Xs2 ) % 8.42/4.48 & ! [Y2: nat] : % 8.42/4.48 ( ( member @ nat @ Y2 @ Xs2 ) % 8.42/4.48 => ( ord_less_eq @ nat @ Y2 @ X ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_45_mem__Collect__eq,axiom, % 8.42/4.48 ! [A: $tType,A2: A,P: A > $o] : % 8.42/4.48 ( ( member @ A @ A2 @ ( collect @ A @ P ) ) % 8.42/4.48 = ( P @ A2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_46_Collect__mem__eq,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A] : % 8.42/4.48 ( ( collect @ A % 8.42/4.48 @ ^ [X: A] : ( member @ A @ X @ A3 ) ) % 8.42/4.48 = A3 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_47_Collect__cong,axiom, % 8.42/4.48 ! [A: $tType,P: A > $o,Q: A > $o] : % 8.42/4.48 ( ! [X5: A] : % 8.42/4.48 ( ( P @ X5 ) % 8.42/4.48 = ( Q @ X5 ) ) % 8.42/4.48 => ( ( collect @ A @ P ) % 8.42/4.48 = ( collect @ A @ Q ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_48_ext,axiom, % 8.42/4.48 ! [B: $tType,A: $tType,F2: A > B,G: A > B] : % 8.42/4.48 ( ! [X5: A] : % 8.42/4.48 ( ( F2 @ X5 ) % 8.42/4.48 = ( G @ X5 ) ) % 8.42/4.48 => ( F2 = G ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_49_buildup__gives__valid,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.48 => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N2 ) @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_50_set__vebt__set__vebt_H__valid,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ( vEBT_set_vebt @ T2 ) % 8.42/4.48 = ( vEBT_VEBT_set_vebt @ T2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_51_mint__corr__help,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat,Mini: nat,X3: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ( ( vEBT_vebt_mint @ T2 ) % 8.42/4.48 = ( some @ nat @ Mini ) ) % 8.42/4.48 => ( ( vEBT_vebt_member @ T2 @ X3 ) % 8.42/4.48 => ( ord_less_eq @ nat @ Mini @ X3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_52_replicate__eq__replicate,axiom, % 8.42/4.48 ! [A: $tType,M: nat,X3: A,N2: nat,Y: A] : % 8.42/4.48 ( ( ( replicate @ A @ M @ X3 ) % 8.42/4.48 = ( replicate @ A @ N2 @ Y ) ) % 8.42/4.48 = ( ( M = N2 ) % 8.42/4.48 & ( ( M % 8.42/4.48 != ( zero_zero @ nat ) ) % 8.42/4.48 => ( X3 = Y ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_53_length__replicate,axiom, % 8.42/4.48 ! [A: $tType,N2: nat,X3: A] : % 8.42/4.48 ( ( size_size @ ( list @ A ) @ ( replicate @ A @ N2 @ X3 ) ) % 8.42/4.48 = N2 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_54_length__list__update,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,I: nat,X3: A] : % 8.42/4.48 ( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I @ X3 ) ) % 8.42/4.48 = ( size_size @ ( list @ A ) @ Xs ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_55_mint__sound,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat,X3: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 ) % 8.42/4.48 => ( ( vEBT_vebt_mint @ T2 ) % 8.42/4.48 = ( some @ nat @ X3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_56_mint__corr,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat,X3: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ( ( vEBT_vebt_mint @ T2 ) % 8.42/4.48 = ( some @ nat @ X3 ) ) % 8.42/4.48 => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_57_Ball__set__replicate,axiom, % 8.42/4.48 ! [A: $tType,N2: nat,A2: A,P: A > $o] : % 8.42/4.48 ( ( ! [X: A] : % 8.42/4.48 ( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N2 @ A2 ) ) ) % 8.42/4.48 => ( P @ X ) ) ) % 8.42/4.48 = ( ( P @ A2 ) % 8.42/4.48 | ( N2 % 8.42/4.48 = ( zero_zero @ nat ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_58_Bex__set__replicate,axiom, % 8.42/4.48 ! [A: $tType,N2: nat,A2: A,P: A > $o] : % 8.42/4.48 ( ( ? [X: A] : % 8.42/4.48 ( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N2 @ A2 ) ) ) % 8.42/4.48 & ( P @ X ) ) ) % 8.42/4.48 = ( ( P @ A2 ) % 8.42/4.48 & ( N2 % 8.42/4.48 != ( zero_zero @ nat ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_59_in__set__replicate,axiom, % 8.42/4.48 ! [A: $tType,X3: A,N2: nat,Y: A] : % 8.42/4.48 ( ( member @ A @ X3 @ ( set2 @ A @ ( replicate @ A @ N2 @ Y ) ) ) % 8.42/4.48 = ( ( X3 = Y ) % 8.42/4.48 & ( N2 % 8.42/4.48 != ( zero_zero @ nat ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_60_list__update__beyond,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,I: nat,X3: A] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I ) % 8.42/4.48 => ( ( list_update @ A @ Xs @ I @ X3 ) % 8.42/4.48 = Xs ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_61__C5_C_I4_J,axiom, % 8.42/4.48 ( m % 8.42/4.48 = ( suc @ n ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_62_nth__list__update__eq,axiom, % 8.42/4.48 ! [A: $tType,I: nat,Xs: list @ A,X3: A] : % 8.42/4.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ I ) % 8.42/4.48 = X3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_63_lesseq__shift,axiom, % 8.42/4.48 ( ( ord_less_eq @ nat ) % 8.42/4.48 = ( ^ [X: nat,Y2: nat] : ( vEBT_VEBT_lesseq @ ( some @ nat @ X ) @ ( some @ nat @ Y2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_64_set__swap,axiom, % 8.42/4.48 ! [A: $tType,I: nat,Xs: list @ A,J: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 => ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) ) % 8.42/4.48 = ( set2 @ A @ Xs ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_65_Ex__list__of__length,axiom, % 8.42/4.48 ! [A: $tType,N2: nat] : % 8.42/4.48 ? [Xs3: list @ A] : % 8.42/4.48 ( ( size_size @ ( list @ A ) @ Xs3 ) % 8.42/4.48 = N2 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_66_neq__if__length__neq,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 8.42/4.48 ( ( ( size_size @ ( list @ A ) @ Xs ) % 8.42/4.48 != ( size_size @ ( list @ A ) @ Ys ) ) % 8.42/4.48 => ( Xs != Ys ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_67_option_Osel,axiom, % 8.42/4.48 ! [A: $tType,X22: A] : % 8.42/4.48 ( ( the2 @ A @ ( some @ A @ X22 ) ) % 8.42/4.48 = X22 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_68_length__pos__if__in__set,axiom, % 8.42/4.48 ! [A: $tType,X3: A,Xs: list @ A] : % 8.42/4.48 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.42/4.48 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_69_length__induct,axiom, % 8.42/4.48 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] : % 8.42/4.48 ( ! [Xs3: list @ A] : % 8.42/4.48 ( ! [Ys2: list @ A] : % 8.42/4.48 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) % 8.42/4.48 => ( P @ Ys2 ) ) % 8.42/4.48 => ( P @ Xs3 ) ) % 8.42/4.48 => ( P @ Xs ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_70_list__eq__iff__nth__eq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ^ [Y3: list @ A,Z: list @ A] : ( Y3 = Z ) ) % 8.42/4.48 = ( ^ [Xs2: list @ A,Ys3: list @ A] : % 8.42/4.48 ( ( ( size_size @ ( list @ A ) @ Xs2 ) % 8.42/4.48 = ( size_size @ ( list @ A ) @ Ys3 ) ) % 8.42/4.48 & ! [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) % 8.42/4.48 => ( ( nth @ A @ Xs2 @ I3 ) % 8.42/4.48 = ( nth @ A @ Ys3 @ I3 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_71_Skolem__list__nth,axiom, % 8.42/4.48 ! [A: $tType,K: nat,P: nat > A > $o] : % 8.42/4.48 ( ( ! [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ K ) % 8.42/4.48 => ? [X6: A] : ( P @ I3 @ X6 ) ) ) % 8.42/4.48 = ( ? [Xs2: list @ A] : % 8.42/4.48 ( ( ( size_size @ ( list @ A ) @ Xs2 ) % 8.42/4.48 = K ) % 8.42/4.48 & ! [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ K ) % 8.42/4.48 => ( P @ I3 @ ( nth @ A @ Xs2 @ I3 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_72_nth__equalityI,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 8.42/4.48 ( ( ( size_size @ ( list @ A ) @ Xs ) % 8.42/4.48 = ( size_size @ ( list @ A ) @ Ys ) ) % 8.42/4.48 => ( ! [I4: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 => ( ( nth @ A @ Xs @ I4 ) % 8.42/4.48 = ( nth @ A @ Ys @ I4 ) ) ) % 8.42/4.48 => ( Xs = Ys ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_73_replicate__length__same,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,X3: A] : % 8.42/4.48 ( ! [X5: A] : % 8.42/4.48 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 8.42/4.48 => ( X5 = X3 ) ) % 8.42/4.48 => ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X3 ) % 8.42/4.48 = Xs ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_74_replicate__eqI,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,N2: nat,X3: A] : % 8.42/4.48 ( ( ( size_size @ ( list @ A ) @ Xs ) % 8.42/4.48 = N2 ) % 8.42/4.48 => ( ! [Y4: A] : % 8.42/4.48 ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) ) % 8.42/4.48 => ( Y4 = X3 ) ) % 8.42/4.48 => ( Xs % 8.42/4.48 = ( replicate @ A @ N2 @ X3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_75_all__set__conv__all__nth,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,P: A > $o] : % 8.42/4.48 ( ( ! [X: A] : % 8.42/4.48 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 8.42/4.48 => ( P @ X ) ) ) % 8.42/4.48 = ( ! [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 => ( P @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_76_all__nth__imp__all__set,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,P: A > $o,X3: A] : % 8.42/4.48 ( ! [I4: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 => ( P @ ( nth @ A @ Xs @ I4 ) ) ) % 8.42/4.48 => ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.42/4.48 => ( P @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_77_in__set__conv__nth,axiom, % 8.42/4.48 ! [A: $tType,X3: A,Xs: list @ A] : % 8.42/4.48 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.42/4.48 = ( ? [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 & ( ( nth @ A @ Xs @ I3 ) % 8.42/4.48 = X3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_78_list__ball__nth,axiom, % 8.42/4.48 ! [A: $tType,N2: nat,Xs: list @ A,P: A > $o] : % 8.42/4.48 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 => ( ! [X5: A] : % 8.42/4.48 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 8.42/4.48 => ( P @ X5 ) ) % 8.42/4.48 => ( P @ ( nth @ A @ Xs @ N2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_79_nth__mem,axiom, % 8.42/4.48 ! [A: $tType,N2: nat,Xs: list @ A] : % 8.42/4.48 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 => ( member @ A @ ( nth @ A @ Xs @ N2 ) @ ( set2 @ A @ Xs ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_80_set__update__memI,axiom, % 8.42/4.48 ! [A: $tType,N2: nat,Xs: list @ A,X3: A] : % 8.42/4.48 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 => ( member @ A @ X3 @ ( set2 @ A @ ( list_update @ A @ Xs @ N2 @ X3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_81_list__update__same__conv,axiom, % 8.42/4.48 ! [A: $tType,I: nat,Xs: list @ A,X3: A] : % 8.42/4.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 => ( ( ( list_update @ A @ Xs @ I @ X3 ) % 8.42/4.48 = Xs ) % 8.42/4.48 = ( ( nth @ A @ Xs @ I ) % 8.42/4.48 = X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_82_nth__list__update,axiom, % 8.42/4.48 ! [A: $tType,I: nat,Xs: list @ A,J: nat,X3: A] : % 8.42/4.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 => ( ( ( I = J ) % 8.42/4.48 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ J ) % 8.42/4.48 = X3 ) ) % 8.42/4.48 & ( ( I != J ) % 8.42/4.48 => ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ J ) % 8.42/4.48 = ( nth @ A @ Xs @ J ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_83_le0,axiom, % 8.42/4.48 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_84_bot__nat__0_Oextremum,axiom, % 8.42/4.48 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_85_less__nat__zero__code,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_86_neq0__conv,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( N2 % 8.42/4.48 != ( zero_zero @ nat ) ) % 8.42/4.48 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_87_bot__nat__0_Onot__eq__extremum,axiom, % 8.42/4.48 ! [A2: nat] : % 8.42/4.48 ( ( A2 % 8.42/4.48 != ( zero_zero @ nat ) ) % 8.42/4.48 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_88_not__gr__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( canoni5634975068530333245id_add @ A ) % 8.42/4.48 => ! [N2: A] : % 8.42/4.48 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) % 8.42/4.48 = ( N2 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_89_le__zero__eq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( canoni5634975068530333245id_add @ A ) % 8.42/4.48 => ! [N2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) ) % 8.42/4.48 = ( N2 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_90__C5_C_I5_J,axiom, % 8.42/4.48 ( deg % 8.42/4.48 = ( plus_plus @ nat @ n @ m ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_91_ex__least__nat__le,axiom, % 8.42/4.48 ! [P: nat > $o,N2: nat] : % 8.42/4.48 ( ( P @ N2 ) % 8.42/4.48 => ( ~ ( P @ ( zero_zero @ nat ) ) % 8.42/4.48 => ? [K2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ K2 @ N2 ) % 8.42/4.48 & ! [I5: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I5 @ K2 ) % 8.42/4.48 => ~ ( P @ I5 ) ) % 8.42/4.48 & ( P @ K2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_92_buildup__gives__empty,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) ) % 8.42/4.48 = ( bot_bot @ ( set @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_93_dual__order_Orefl,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_94_even__odd__cases,axiom, % 8.42/4.48 ! [X3: nat] : % 8.42/4.48 ( ! [N3: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( plus_plus @ nat @ N3 @ N3 ) ) % 8.42/4.48 => ~ ! [N3: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_95_add__shift,axiom, % 8.42/4.48 ! [X3: nat,Y: nat,Z2: nat] : % 8.42/4.48 ( ( ( plus_plus @ nat @ X3 @ Y ) % 8.42/4.48 = Z2 ) % 8.42/4.48 = ( ( vEBT_VEBT_add @ ( some @ nat @ X3 ) @ ( some @ nat @ Y ) ) % 8.42/4.48 = ( some @ nat @ Z2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_96_order__refl,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_97_add__left__cancel,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( cancel_semigroup_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ( plus_plus @ A @ A2 @ B2 ) % 8.42/4.48 = ( plus_plus @ A @ A2 @ C2 ) ) % 8.42/4.48 = ( B2 = C2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_98_add__right__cancel,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( cancel_semigroup_add @ A ) % 8.42/4.48 => ! [B2: A,A2: A,C2: A] : % 8.42/4.48 ( ( ( plus_plus @ A @ B2 @ A2 ) % 8.42/4.48 = ( plus_plus @ A @ C2 @ A2 ) ) % 8.42/4.48 = ( B2 = C2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_99_bot__apply,axiom, % 8.42/4.48 ! [C: $tType,D: $tType] : % 8.42/4.48 ( ( bot @ C ) % 8.42/4.48 => ( ( bot_bot @ ( D > C ) ) % 8.42/4.48 = ( ^ [X: D] : ( bot_bot @ C ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_100_nat_Oinject,axiom, % 8.42/4.48 ! [X22: nat,Y22: nat] : % 8.42/4.48 ( ( ( suc @ X22 ) % 8.42/4.48 = ( suc @ Y22 ) ) % 8.42/4.48 = ( X22 = Y22 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_101_old_Onat_Oinject,axiom, % 8.42/4.48 ! [Nat: nat,Nat2: nat] : % 8.42/4.48 ( ( ( suc @ Nat ) % 8.42/4.48 = ( suc @ Nat2 ) ) % 8.42/4.48 = ( Nat = Nat2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_102_add__le__cancel__left,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere2412721322843649153imp_le @ A ) % 8.42/4.48 => ! [C2: A,A2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) % 8.42/4.48 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_103_add__le__cancel__right,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere2412721322843649153imp_le @ A ) % 8.42/4.48 => ! [A2: A,C2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) % 8.42/4.48 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_104_add_Oright__neutral,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( monoid_add @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 = A2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_105_double__zero__sym,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord5086331880401160121up_add @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ( zero_zero @ A ) % 8.42/4.48 = ( plus_plus @ A @ A2 @ A2 ) ) % 8.42/4.48 = ( A2 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_106_add__cancel__left__left,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( cancel1802427076303600483id_add @ A ) % 8.42/4.48 => ! [B2: A,A2: A] : % 8.42/4.48 ( ( ( plus_plus @ A @ B2 @ A2 ) % 8.42/4.48 = A2 ) % 8.42/4.48 = ( B2 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_107_add__cancel__left__right,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( cancel1802427076303600483id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ( plus_plus @ A @ A2 @ B2 ) % 8.42/4.48 = A2 ) % 8.42/4.48 = ( B2 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_108_add__cancel__right__left,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( cancel1802427076303600483id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( A2 % 8.42/4.48 = ( plus_plus @ A @ B2 @ A2 ) ) % 8.42/4.48 = ( B2 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_109_add__cancel__right__right,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( cancel1802427076303600483id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( A2 % 8.42/4.48 = ( plus_plus @ A @ A2 @ B2 ) ) % 8.42/4.48 = ( B2 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_110_add__eq__0__iff__both__eq__0,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( canoni5634975068530333245id_add @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ( plus_plus @ A @ X3 @ Y ) % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 = ( ( X3 % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 & ( Y % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_111_zero__eq__add__iff__both__eq__0,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( canoni5634975068530333245id_add @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ( zero_zero @ A ) % 8.42/4.48 = ( plus_plus @ A @ X3 @ Y ) ) % 8.42/4.48 = ( ( X3 % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 & ( Y % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_112_add__0,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( monoid_add @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 = A2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_113_add__less__cancel__left,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere2412721322843649153imp_le @ A ) % 8.42/4.48 => ! [C2: A,A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) % 8.42/4.48 = ( ord_less @ A @ A2 @ B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_114_add__less__cancel__right,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere2412721322843649153imp_le @ A ) % 8.42/4.48 => ! [A2: A,C2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) % 8.42/4.48 = ( ord_less @ A @ A2 @ B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_115_lessI,axiom, % 8.42/4.48 ! [N2: nat] : ( ord_less @ nat @ N2 @ ( suc @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_116_Suc__mono,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.48 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_117_Suc__less__eq,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) % 8.42/4.48 = ( ord_less @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_118_Suc__le__mono,axiom, % 8.42/4.48 ! [N2: nat,M: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( suc @ M ) ) % 8.42/4.48 = ( ord_less_eq @ nat @ N2 @ M ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_119_add__Suc__right,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( plus_plus @ nat @ M @ ( suc @ N2 ) ) % 8.42/4.48 = ( suc @ ( plus_plus @ nat @ M @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_120_add__is__0,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ( plus_plus @ nat @ M @ N2 ) % 8.42/4.48 = ( zero_zero @ nat ) ) % 8.42/4.48 = ( ( M % 8.42/4.48 = ( zero_zero @ nat ) ) % 8.42/4.48 & ( N2 % 8.42/4.48 = ( zero_zero @ nat ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_121_Nat_Oadd__0__right,axiom, % 8.42/4.48 ! [M: nat] : % 8.42/4.48 ( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) ) % 8.42/4.48 = M ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_122_nat__add__left__cancel__less,axiom, % 8.42/4.48 ! [K: nat,M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) ) % 8.42/4.48 = ( ord_less @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_123_nat__add__left__cancel__le,axiom, % 8.42/4.48 ! [K: nat,M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) ) % 8.42/4.48 = ( ord_less_eq @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_124_add__le__same__cancel1,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere1937475149494474687imp_le @ A ) % 8.42/4.48 => ! [B2: A,A2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 ) % 8.42/4.48 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_125_add__le__same__cancel2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere1937475149494474687imp_le @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 ) % 8.42/4.48 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_126_le__add__same__cancel1,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere1937475149494474687imp_le @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) ) % 8.42/4.48 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_127_le__add__same__cancel2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere1937475149494474687imp_le @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) ) % 8.42/4.48 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_128_double__add__le__zero__iff__single__add__le__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord5086331880401160121up_add @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) ) % 8.42/4.48 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_129_zero__le__double__add__iff__zero__le__single__add,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord5086331880401160121up_add @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) ) % 8.42/4.48 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_130_zero__less__double__add__iff__zero__less__single__add,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord5086331880401160121up_add @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) ) % 8.42/4.48 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_131_double__add__less__zero__iff__single__add__less__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord5086331880401160121up_add @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) ) % 8.42/4.48 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_132_less__add__same__cancel2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere1937475149494474687imp_le @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) ) % 8.42/4.48 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_133_less__add__same__cancel1,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere1937475149494474687imp_le @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) ) % 8.42/4.48 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_134_add__less__same__cancel2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere1937475149494474687imp_le @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 ) % 8.42/4.48 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_135_add__less__same__cancel1,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere1937475149494474687imp_le @ A ) % 8.42/4.48 => ! [B2: A,A2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 ) % 8.42/4.48 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_136_zero__less__Suc,axiom, % 8.42/4.48 ! [N2: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_137_less__Suc0,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.48 = ( N2 % 8.42/4.48 = ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_138_add__gr__0,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N2 ) ) % 8.42/4.48 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.42/4.48 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_139_one__is__add,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ( suc @ ( zero_zero @ nat ) ) % 8.42/4.48 = ( plus_plus @ nat @ M @ N2 ) ) % 8.42/4.48 = ( ( ( M % 8.42/4.48 = ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.48 & ( N2 % 8.42/4.48 = ( zero_zero @ nat ) ) ) % 8.42/4.48 | ( ( M % 8.42/4.48 = ( zero_zero @ nat ) ) % 8.42/4.48 & ( N2 % 8.42/4.48 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_140_add__is__1,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ( plus_plus @ nat @ M @ N2 ) % 8.42/4.48 = ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.48 = ( ( ( M % 8.42/4.48 = ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.48 & ( N2 % 8.42/4.48 = ( zero_zero @ nat ) ) ) % 8.42/4.48 | ( ( M % 8.42/4.48 = ( zero_zero @ nat ) ) % 8.42/4.48 & ( N2 % 8.42/4.48 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_141_ab__semigroup__add__class_Oadd__ac_I1_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ab_semigroup_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.48 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_142_add__mono__thms__linordered__semiring_I4_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6658533253407199908up_add @ A ) % 8.42/4.48 => ! [I: A,J: A,K: A,L: A] : % 8.42/4.48 ( ( ( I = J ) % 8.42/4.48 & ( K = L ) ) % 8.42/4.48 => ( ( plus_plus @ A @ I @ K ) % 8.42/4.48 = ( plus_plus @ A @ J @ L ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_143_nat__arith_Osuc1,axiom, % 8.42/4.48 ! [A3: nat,K: nat,A2: nat] : % 8.42/4.48 ( ( A3 % 8.42/4.48 = ( plus_plus @ nat @ K @ A2 ) ) % 8.42/4.48 => ( ( suc @ A3 ) % 8.42/4.48 = ( plus_plus @ nat @ K @ ( suc @ A2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_144_add__Suc,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( plus_plus @ nat @ ( suc @ M ) @ N2 ) % 8.42/4.48 = ( suc @ ( plus_plus @ nat @ M @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_145_group__cancel_Oadd1,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( comm_monoid_add @ A ) % 8.42/4.48 => ! [A3: A,K: A,A2: A,B2: A] : % 8.42/4.48 ( ( A3 % 8.42/4.48 = ( plus_plus @ A @ K @ A2 ) ) % 8.42/4.48 => ( ( plus_plus @ A @ A3 @ B2 ) % 8.42/4.48 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_146_group__cancel_Oadd2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( comm_monoid_add @ A ) % 8.42/4.48 => ! [B3: A,K: A,B2: A,A2: A] : % 8.42/4.48 ( ( B3 % 8.42/4.48 = ( plus_plus @ A @ K @ B2 ) ) % 8.42/4.48 => ( ( plus_plus @ A @ A2 @ B3 ) % 8.42/4.48 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_147_add_Oassoc,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( semigroup_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.48 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_148_add_Oleft__cancel,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( group_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ( plus_plus @ A @ A2 @ B2 ) % 8.42/4.48 = ( plus_plus @ A @ A2 @ C2 ) ) % 8.42/4.48 = ( B2 = C2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_149_add_Oright__cancel,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( group_add @ A ) % 8.42/4.48 => ! [B2: A,A2: A,C2: A] : % 8.42/4.48 ( ( ( plus_plus @ A @ B2 @ A2 ) % 8.42/4.48 = ( plus_plus @ A @ C2 @ A2 ) ) % 8.42/4.48 = ( B2 = C2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_150_add_Ocommute,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ab_semigroup_add @ A ) % 8.42/4.48 => ( ( plus_plus @ A ) % 8.42/4.48 = ( ^ [A4: A,B4: A] : ( plus_plus @ A @ B4 @ A4 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_151_add_Oleft__commute,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ab_semigroup_add @ A ) % 8.42/4.48 => ! [B2: A,A2: A,C2: A] : % 8.42/4.48 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) % 8.42/4.48 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_152_add__left__imp__eq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( cancel_semigroup_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ( plus_plus @ A @ A2 @ B2 ) % 8.42/4.48 = ( plus_plus @ A @ A2 @ C2 ) ) % 8.42/4.48 => ( B2 = C2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_153_add__right__imp__eq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( cancel_semigroup_add @ A ) % 8.42/4.48 => ! [B2: A,A2: A,C2: A] : % 8.42/4.48 ( ( ( plus_plus @ A @ B2 @ A2 ) % 8.42/4.48 = ( plus_plus @ A @ C2 @ A2 ) ) % 8.42/4.48 => ( B2 = C2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_154_Suc__inject,axiom, % 8.42/4.48 ! [X3: nat,Y: nat] : % 8.42/4.48 ( ( ( suc @ X3 ) % 8.42/4.48 = ( suc @ Y ) ) % 8.42/4.48 => ( X3 = Y ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_155_n__not__Suc__n,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( N2 % 8.42/4.48 != ( suc @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_156_add__Suc__shift,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( plus_plus @ nat @ ( suc @ M ) @ N2 ) % 8.42/4.48 = ( plus_plus @ nat @ M @ ( suc @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_157_bot__fun__def,axiom, % 8.42/4.48 ! [B: $tType,A: $tType] : % 8.42/4.48 ( ( bot @ B ) % 8.42/4.48 => ( ( bot_bot @ ( A > B ) ) % 8.42/4.48 = ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_158_less__imp__Suc__add,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.48 => ? [K2: nat] : % 8.42/4.48 ( N2 % 8.42/4.48 = ( suc @ ( plus_plus @ nat @ M @ K2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_159_less__iff__Suc__add,axiom, % 8.42/4.48 ( ( ord_less @ nat ) % 8.42/4.48 = ( ^ [M2: nat,N: nat] : % 8.42/4.48 ? [K3: nat] : % 8.42/4.48 ( N % 8.42/4.48 = ( suc @ ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_160_less__add__Suc2,axiom, % 8.42/4.48 ! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_161_less__add__Suc1,axiom, % 8.42/4.48 ! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_162_less__natE,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.48 => ~ ! [Q2: nat] : % 8.42/4.48 ( N2 % 8.42/4.48 != ( suc @ ( plus_plus @ nat @ M @ Q2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_163_bot_Oextremum,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order_bot @ A ) % 8.42/4.48 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_164_bot_Oextremum__unique,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order_bot @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) ) % 8.42/4.48 = ( A2 % 8.42/4.48 = ( bot_bot @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_165_bot_Oextremum__uniqueI,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order_bot @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) ) % 8.42/4.48 => ( A2 % 8.42/4.48 = ( bot_bot @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_166_bot_Oextremum__strict,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order_bot @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_167_bot_Onot__eq__extremum,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order_bot @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( A2 % 8.42/4.48 != ( bot_bot @ A ) ) % 8.42/4.48 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_168_add__mono__thms__linordered__semiring_I3_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6658533253407199908up_add @ A ) % 8.42/4.48 => ! [I: A,J: A,K: A,L: A] : % 8.42/4.48 ( ( ( ord_less_eq @ A @ I @ J ) % 8.42/4.48 & ( K = L ) ) % 8.42/4.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_169_add__mono__thms__linordered__semiring_I2_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6658533253407199908up_add @ A ) % 8.42/4.48 => ! [I: A,J: A,K: A,L: A] : % 8.42/4.48 ( ( ( I = J ) % 8.42/4.48 & ( ord_less_eq @ A @ K @ L ) ) % 8.42/4.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_170_add__mono__thms__linordered__semiring_I1_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6658533253407199908up_add @ A ) % 8.42/4.48 => ! [I: A,J: A,K: A,L: A] : % 8.42/4.48 ( ( ( ord_less_eq @ A @ I @ J ) % 8.42/4.48 & ( ord_less_eq @ A @ K @ L ) ) % 8.42/4.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_171_add__mono,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6658533253407199908up_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ C2 @ D2 ) % 8.42/4.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_172_add__left__mono,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6658533253407199908up_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_173_less__eqE,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( canoni5634975068530333245id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ~ ! [C3: A] : % 8.42/4.48 ( B2 % 8.42/4.48 != ( plus_plus @ A @ A2 @ C3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_174_add__right__mono,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6658533253407199908up_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_175_le__iff__add,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( canoni5634975068530333245id_add @ A ) % 8.42/4.48 => ( ( ord_less_eq @ A ) % 8.42/4.48 = ( ^ [A4: A,B4: A] : % 8.42/4.48 ? [C4: A] : % 8.42/4.48 ( B4 % 8.42/4.48 = ( plus_plus @ A @ A4 @ C4 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_176_add__le__imp__le__left,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere2412721322843649153imp_le @ A ) % 8.42/4.48 => ! [C2: A,A2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) % 8.42/4.48 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_177_add__le__imp__le__right,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere2412721322843649153imp_le @ A ) % 8.42/4.48 => ! [A2: A,C2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) % 8.42/4.48 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_178_comm__monoid__add__class_Oadd__0,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( comm_monoid_add @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 = A2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_179_add_Ocomm__neutral,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( comm_monoid_add @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 = A2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_180_add_Ogroup__left__neutral,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( group_add @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 = A2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_181_add__mono__thms__linordered__field_I5_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere580206878836729694up_add @ A ) % 8.42/4.48 => ! [I: A,J: A,K: A,L: A] : % 8.42/4.48 ( ( ( ord_less @ A @ I @ J ) % 8.42/4.48 & ( ord_less @ A @ K @ L ) ) % 8.42/4.48 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_182_add__mono__thms__linordered__field_I2_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere580206878836729694up_add @ A ) % 8.42/4.48 => ! [I: A,J: A,K: A,L: A] : % 8.42/4.48 ( ( ( I = J ) % 8.42/4.48 & ( ord_less @ A @ K @ L ) ) % 8.42/4.48 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_183_add__mono__thms__linordered__field_I1_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere580206878836729694up_add @ A ) % 8.42/4.48 => ! [I: A,J: A,K: A,L: A] : % 8.42/4.48 ( ( ( ord_less @ A @ I @ J ) % 8.42/4.48 & ( K = L ) ) % 8.42/4.48 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_184_add__strict__mono,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( strict9044650504122735259up_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less @ A @ C2 @ D2 ) % 8.42/4.48 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_185_add__strict__left__mono,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere580206878836729694up_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_186_add__strict__right__mono,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere580206878836729694up_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_187_add__less__imp__less__left,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere2412721322843649153imp_le @ A ) % 8.42/4.48 => ! [C2: A,A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) % 8.42/4.48 => ( ord_less @ A @ A2 @ B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_188_add__less__imp__less__right,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere2412721322843649153imp_le @ A ) % 8.42/4.48 => ! [A2: A,C2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) % 8.42/4.48 => ( ord_less @ A @ A2 @ B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_189_nat_Odistinct_I1_J,axiom, % 8.42/4.48 ! [X22: nat] : % 8.42/4.48 ( ( zero_zero @ nat ) % 8.42/4.48 != ( suc @ X22 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_190_old_Onat_Odistinct_I2_J,axiom, % 8.42/4.48 ! [Nat3: nat] : % 8.42/4.48 ( ( suc @ Nat3 ) % 8.42/4.48 != ( zero_zero @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_191_old_Onat_Odistinct_I1_J,axiom, % 8.42/4.48 ! [Nat2: nat] : % 8.42/4.48 ( ( zero_zero @ nat ) % 8.42/4.48 != ( suc @ Nat2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_192_nat_OdiscI,axiom, % 8.42/4.48 ! [Nat: nat,X22: nat] : % 8.42/4.48 ( ( Nat % 8.42/4.48 = ( suc @ X22 ) ) % 8.42/4.48 => ( Nat % 8.42/4.48 != ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_193_old_Onat_Oexhaust,axiom, % 8.42/4.48 ! [Y: nat] : % 8.42/4.48 ( ( Y % 8.42/4.48 != ( zero_zero @ nat ) ) % 8.42/4.48 => ~ ! [Nat4: nat] : % 8.42/4.48 ( Y % 8.42/4.48 != ( suc @ Nat4 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_194_nat__induct,axiom, % 8.42/4.48 ! [P: nat > $o,N2: nat] : % 8.42/4.48 ( ( P @ ( zero_zero @ nat ) ) % 8.42/4.48 => ( ! [N3: nat] : % 8.42/4.48 ( ( P @ N3 ) % 8.42/4.48 => ( P @ ( suc @ N3 ) ) ) % 8.42/4.48 => ( P @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_195_diff__induct,axiom, % 8.42/4.48 ! [P: nat > nat > $o,M: nat,N2: nat] : % 8.42/4.48 ( ! [X5: nat] : ( P @ X5 @ ( zero_zero @ nat ) ) % 8.42/4.48 => ( ! [Y4: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y4 ) ) % 8.42/4.48 => ( ! [X5: nat,Y4: nat] : % 8.42/4.48 ( ( P @ X5 @ Y4 ) % 8.42/4.48 => ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) ) % 8.42/4.48 => ( P @ M @ N2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_196_zero__induct,axiom, % 8.42/4.48 ! [P: nat > $o,K: nat] : % 8.42/4.48 ( ( P @ K ) % 8.42/4.48 => ( ! [N3: nat] : % 8.42/4.48 ( ( P @ ( suc @ N3 ) ) % 8.42/4.48 => ( P @ N3 ) ) % 8.42/4.48 => ( P @ ( zero_zero @ nat ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_197_Suc__neq__Zero,axiom, % 8.42/4.48 ! [M: nat] : % 8.42/4.48 ( ( suc @ M ) % 8.42/4.48 != ( zero_zero @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_198_Zero__neq__Suc,axiom, % 8.42/4.48 ! [M: nat] : % 8.42/4.48 ( ( zero_zero @ nat ) % 8.42/4.48 != ( suc @ M ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_199_Zero__not__Suc,axiom, % 8.42/4.48 ! [M: nat] : % 8.42/4.48 ( ( zero_zero @ nat ) % 8.42/4.48 != ( suc @ M ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_200_not0__implies__Suc,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( N2 % 8.42/4.48 != ( zero_zero @ nat ) ) % 8.42/4.48 => ? [M3: nat] : % 8.42/4.48 ( N2 % 8.42/4.48 = ( suc @ M3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_201_plus__nat_Oadd__0,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.48 = N2 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_202_add__eq__self__zero,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ( plus_plus @ nat @ M @ N2 ) % 8.42/4.48 = M ) % 8.42/4.48 => ( N2 % 8.42/4.48 = ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_203_Nat_OlessE,axiom, % 8.42/4.48 ! [I: nat,K: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I @ K ) % 8.42/4.48 => ( ( K % 8.42/4.48 != ( suc @ I ) ) % 8.42/4.48 => ~ ! [J2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I @ J2 ) % 8.42/4.48 => ( K % 8.42/4.48 != ( suc @ J2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_204_Suc__lessD,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( suc @ M ) @ N2 ) % 8.42/4.48 => ( ord_less @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_205_Suc__lessE,axiom, % 8.42/4.48 ! [I: nat,K: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( suc @ I ) @ K ) % 8.42/4.48 => ~ ! [J2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I @ J2 ) % 8.42/4.48 => ( K % 8.42/4.48 != ( suc @ J2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_206_Suc__lessI,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.48 => ( ( ( suc @ M ) % 8.42/4.48 != N2 ) % 8.42/4.48 => ( ord_less @ nat @ ( suc @ M ) @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_207_less__SucE,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M @ ( suc @ N2 ) ) % 8.42/4.48 => ( ~ ( ord_less @ nat @ M @ N2 ) % 8.42/4.48 => ( M = N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_208_less__SucI,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.48 => ( ord_less @ nat @ M @ ( suc @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_209_Ex__less__Suc,axiom, % 8.42/4.48 ! [N2: nat,P: nat > $o] : % 8.42/4.48 ( ( ? [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ ( suc @ N2 ) ) % 8.42/4.48 & ( P @ I3 ) ) ) % 8.42/4.48 = ( ( P @ N2 ) % 8.42/4.48 | ? [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ N2 ) % 8.42/4.48 & ( P @ I3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_210_less__Suc__eq,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M @ ( suc @ N2 ) ) % 8.42/4.48 = ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.48 | ( M = N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_211_not__less__eq,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ~ ( ord_less @ nat @ M @ N2 ) ) % 8.42/4.48 = ( ord_less @ nat @ N2 @ ( suc @ M ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_212_All__less__Suc,axiom, % 8.42/4.48 ! [N2: nat,P: nat > $o] : % 8.42/4.48 ( ( ! [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ ( suc @ N2 ) ) % 8.42/4.48 => ( P @ I3 ) ) ) % 8.42/4.48 = ( ( P @ N2 ) % 8.42/4.48 & ! [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ N2 ) % 8.42/4.48 => ( P @ I3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_213_Suc__less__eq2,axiom, % 8.42/4.48 ! [N2: nat,M: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( suc @ N2 ) @ M ) % 8.42/4.48 = ( ? [M4: nat] : % 8.42/4.48 ( ( M % 8.42/4.48 = ( suc @ M4 ) ) % 8.42/4.48 & ( ord_less @ nat @ N2 @ M4 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_214_less__antisym,axiom, % 8.42/4.48 ! [N2: nat,M: nat] : % 8.42/4.48 ( ~ ( ord_less @ nat @ N2 @ M ) % 8.42/4.48 => ( ( ord_less @ nat @ N2 @ ( suc @ M ) ) % 8.42/4.48 => ( M = N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_215_Suc__less__SucD,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) % 8.42/4.48 => ( ord_less @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_216_less__trans__Suc,axiom, % 8.42/4.48 ! [I: nat,J: nat,K: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I @ J ) % 8.42/4.48 => ( ( ord_less @ nat @ J @ K ) % 8.42/4.48 => ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_217_less__Suc__induct,axiom, % 8.42/4.48 ! [I: nat,J: nat,P: nat > nat > $o] : % 8.42/4.48 ( ( ord_less @ nat @ I @ J ) % 8.42/4.48 => ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) ) % 8.42/4.48 => ( ! [I4: nat,J2: nat,K2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I4 @ J2 ) % 8.42/4.48 => ( ( ord_less @ nat @ J2 @ K2 ) % 8.42/4.48 => ( ( P @ I4 @ J2 ) % 8.42/4.48 => ( ( P @ J2 @ K2 ) % 8.42/4.48 => ( P @ I4 @ K2 ) ) ) ) ) % 8.42/4.48 => ( P @ I @ J ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_218_strict__inc__induct,axiom, % 8.42/4.48 ! [I: nat,J: nat,P: nat > $o] : % 8.42/4.48 ( ( ord_less @ nat @ I @ J ) % 8.42/4.48 => ( ! [I4: nat] : % 8.42/4.48 ( ( J % 8.42/4.48 = ( suc @ I4 ) ) % 8.42/4.48 => ( P @ I4 ) ) % 8.42/4.48 => ( ! [I4: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I4 @ J ) % 8.42/4.48 => ( ( P @ ( suc @ I4 ) ) % 8.42/4.48 => ( P @ I4 ) ) ) % 8.42/4.48 => ( P @ I ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_219_not__less__less__Suc__eq,axiom, % 8.42/4.48 ! [N2: nat,M: nat] : % 8.42/4.48 ( ~ ( ord_less @ nat @ N2 @ M ) % 8.42/4.48 => ( ( ord_less @ nat @ N2 @ ( suc @ M ) ) % 8.42/4.48 = ( N2 = M ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_220_add__lessD1,axiom, % 8.42/4.48 ! [I: nat,J: nat,K: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) % 8.42/4.48 => ( ord_less @ nat @ I @ K ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_221_add__less__mono,axiom, % 8.42/4.48 ! [I: nat,J: nat,K: nat,L: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I @ J ) % 8.42/4.48 => ( ( ord_less @ nat @ K @ L ) % 8.42/4.48 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_222_not__add__less1,axiom, % 8.42/4.48 ! [I: nat,J: nat] : % 8.42/4.48 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_223_not__add__less2,axiom, % 8.42/4.48 ! [J: nat,I: nat] : % 8.42/4.48 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_224_add__less__mono1,axiom, % 8.42/4.48 ! [I: nat,J: nat,K: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I @ J ) % 8.42/4.48 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_225_trans__less__add1,axiom, % 8.42/4.48 ! [I: nat,J: nat,M: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I @ J ) % 8.42/4.48 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_226_trans__less__add2,axiom, % 8.42/4.48 ! [I: nat,J: nat,M: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I @ J ) % 8.42/4.48 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_227_less__add__eq__less,axiom, % 8.42/4.48 ! [K: nat,L: nat,M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ K @ L ) % 8.42/4.48 => ( ( ( plus_plus @ nat @ M @ L ) % 8.42/4.48 = ( plus_plus @ nat @ K @ N2 ) ) % 8.42/4.48 => ( ord_less @ nat @ M @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_228_Suc__leD,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 ) % 8.42/4.48 => ( ord_less_eq @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_229_le__SucE,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) % 8.42/4.48 => ( ~ ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.48 => ( M % 8.42/4.48 = ( suc @ N2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_230_le__SucI,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.48 => ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_231_Suc__le__D,axiom, % 8.42/4.48 ! [N2: nat,M5: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( suc @ N2 ) @ M5 ) % 8.42/4.48 => ? [M3: nat] : % 8.42/4.48 ( M5 % 8.42/4.48 = ( suc @ M3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_232_le__Suc__eq,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) % 8.42/4.48 = ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.48 | ( M % 8.42/4.48 = ( suc @ N2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_233_Suc__n__not__le__n,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ~ ( ord_less_eq @ nat @ ( suc @ N2 ) @ N2 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_234_not__less__eq__eq,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ~ ( ord_less_eq @ nat @ M @ N2 ) ) % 8.42/4.48 = ( ord_less_eq @ nat @ ( suc @ N2 ) @ M ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_235_full__nat__induct,axiom, % 8.42/4.48 ! [P: nat > $o,N2: nat] : % 8.42/4.48 ( ! [N3: nat] : % 8.42/4.48 ( ! [M6: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( suc @ M6 ) @ N3 ) % 8.42/4.48 => ( P @ M6 ) ) % 8.42/4.48 => ( P @ N3 ) ) % 8.42/4.48 => ( P @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_236_nat__induct__at__least,axiom, % 8.42/4.48 ! [M: nat,N2: nat,P: nat > $o] : % 8.42/4.48 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.48 => ( ( P @ M ) % 8.42/4.48 => ( ! [N3: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ M @ N3 ) % 8.42/4.48 => ( ( P @ N3 ) % 8.42/4.48 => ( P @ ( suc @ N3 ) ) ) ) % 8.42/4.48 => ( P @ N2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_237_transitive__stepwise__le,axiom, % 8.42/4.48 ! [M: nat,N2: nat,R: nat > nat > $o] : % 8.42/4.48 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.48 => ( ! [X5: nat] : ( R @ X5 @ X5 ) % 8.42/4.48 => ( ! [X5: nat,Y4: nat,Z3: nat] : % 8.42/4.48 ( ( R @ X5 @ Y4 ) % 8.42/4.48 => ( ( R @ Y4 @ Z3 ) % 8.42/4.48 => ( R @ X5 @ Z3 ) ) ) % 8.42/4.48 => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) ) % 8.42/4.48 => ( R @ M @ N2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_238_add__leE,axiom, % 8.42/4.48 ! [M: nat,K: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N2 ) % 8.42/4.48 => ~ ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.48 => ~ ( ord_less_eq @ nat @ K @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_239_le__add1,axiom, % 8.42/4.48 ! [N2: nat,M: nat] : ( ord_less_eq @ nat @ N2 @ ( plus_plus @ nat @ N2 @ M ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_240_le__add2,axiom, % 8.42/4.48 ! [N2: nat,M: nat] : ( ord_less_eq @ nat @ N2 @ ( plus_plus @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_241_add__leD1,axiom, % 8.42/4.48 ! [M: nat,K: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N2 ) % 8.42/4.48 => ( ord_less_eq @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_242_add__leD2,axiom, % 8.42/4.48 ! [M: nat,K: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N2 ) % 8.42/4.48 => ( ord_less_eq @ nat @ K @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_243_le__Suc__ex,axiom, % 8.42/4.48 ! [K: nat,L: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ K @ L ) % 8.42/4.48 => ? [N3: nat] : % 8.42/4.48 ( L % 8.42/4.48 = ( plus_plus @ nat @ K @ N3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_244_add__le__mono,axiom, % 8.42/4.48 ! [I: nat,J: nat,K: nat,L: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.48 => ( ( ord_less_eq @ nat @ K @ L ) % 8.42/4.48 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_245_add__le__mono1,axiom, % 8.42/4.48 ! [I: nat,J: nat,K: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.48 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_246_trans__le__add1,axiom, % 8.42/4.48 ! [I: nat,J: nat,M: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.48 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_247_trans__le__add2,axiom, % 8.42/4.48 ! [I: nat,J: nat,M: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.48 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_248_nat__le__iff__add,axiom, % 8.42/4.48 ( ( ord_less_eq @ nat ) % 8.42/4.48 = ( ^ [M2: nat,N: nat] : % 8.42/4.48 ? [K3: nat] : % 8.42/4.48 ( N % 8.42/4.48 = ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_249_subset__code_I1_J,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,B3: set @ A] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B3 ) % 8.42/4.48 = ( ! [X: A] : % 8.42/4.48 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 8.42/4.48 => ( member @ A @ X @ B3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_250_add__decreasing,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6911136660526730532id_add @ A ) % 8.42/4.48 => ! [A2: A,C2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ C2 @ B2 ) % 8.42/4.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_251_add__increasing,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6911136660526730532id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ B2 @ C2 ) % 8.42/4.48 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_252_add__decreasing2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6911136660526730532id_add @ A ) % 8.42/4.48 => ! [C2: A,A2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_253_add__increasing2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6911136660526730532id_add @ A ) % 8.42/4.48 => ! [C2: A,B2: A,A2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.42/4.48 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_254_add__nonneg__nonneg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6911136660526730532id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_255_add__nonpos__nonpos,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6911136660526730532id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_256_add__nonneg__eq__0__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6911136660526730532id_add @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.48 => ( ( ( plus_plus @ A @ X3 @ Y ) % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 = ( ( X3 % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 & ( Y % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_257_add__nonpos__eq__0__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6911136660526730532id_add @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ( plus_plus @ A @ X3 @ Y ) % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 = ( ( X3 % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 & ( Y % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_258_add__less__le__mono,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere580206878836729694up_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ C2 @ D2 ) % 8.42/4.48 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_259_add__le__less__mono,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere580206878836729694up_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less @ A @ C2 @ D2 ) % 8.42/4.48 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_260_add__mono__thms__linordered__field_I3_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere580206878836729694up_add @ A ) % 8.42/4.48 => ! [I: A,J: A,K: A,L: A] : % 8.42/4.48 ( ( ( ord_less @ A @ I @ J ) % 8.42/4.48 & ( ord_less_eq @ A @ K @ L ) ) % 8.42/4.48 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_261_add__mono__thms__linordered__field_I4_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere580206878836729694up_add @ A ) % 8.42/4.48 => ! [I: A,J: A,K: A,L: A] : % 8.42/4.48 ( ( ( ord_less_eq @ A @ I @ J ) % 8.42/4.48 & ( ord_less @ A @ K @ L ) ) % 8.42/4.48 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_262_pos__add__strict,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( strict7427464778891057005id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less @ A @ B2 @ C2 ) % 8.42/4.48 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_263_canonically__ordered__monoid__add__class_OlessE,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( canoni5634975068530333245id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ~ ! [C3: A] : % 8.42/4.48 ( ( B2 % 8.42/4.48 = ( plus_plus @ A @ A2 @ C3 ) ) % 8.42/4.48 => ( C3 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_264_add__pos__pos,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6911136660526730532id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.48 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_265_add__neg__neg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6911136660526730532id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_266_lift__Suc__mono__less,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [F2: nat > A,N2: nat,N4: nat] : % 8.42/4.48 ( ! [N3: nat] : ( ord_less @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) ) % 8.42/4.48 => ( ( ord_less @ nat @ N2 @ N4 ) % 8.42/4.48 => ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ N4 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_267_lift__Suc__mono__less__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [F2: nat > A,N2: nat,M: nat] : % 8.42/4.48 ( ! [N3: nat] : ( ord_less @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) ) % 8.42/4.48 => ( ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ M ) ) % 8.42/4.48 = ( ord_less @ nat @ N2 @ M ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_268_lift__Suc__mono__le,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [F2: nat > A,N2: nat,N4: nat] : % 8.42/4.48 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) ) % 8.42/4.48 => ( ( ord_less_eq @ nat @ N2 @ N4 ) % 8.42/4.48 => ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( F2 @ N4 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_269_lift__Suc__antimono__le,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [F2: nat > A,N2: nat,N4: nat] : % 8.42/4.48 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) ) % 8.42/4.48 => ( ( ord_less_eq @ nat @ N2 @ N4 ) % 8.42/4.48 => ( ord_less_eq @ A @ ( F2 @ N4 ) @ ( F2 @ N2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_270_less__Suc__eq__0__disj,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M @ ( suc @ N2 ) ) % 8.42/4.48 = ( ( M % 8.42/4.48 = ( zero_zero @ nat ) ) % 8.42/4.48 | ? [J3: nat] : % 8.42/4.48 ( ( M % 8.42/4.48 = ( suc @ J3 ) ) % 8.42/4.48 & ( ord_less @ nat @ J3 @ N2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_271_gr0__implies__Suc,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.48 => ? [M3: nat] : % 8.42/4.48 ( N2 % 8.42/4.48 = ( suc @ M3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_272_All__less__Suc2,axiom, % 8.42/4.48 ! [N2: nat,P: nat > $o] : % 8.42/4.48 ( ( ! [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ ( suc @ N2 ) ) % 8.42/4.48 => ( P @ I3 ) ) ) % 8.42/4.48 = ( ( P @ ( zero_zero @ nat ) ) % 8.42/4.48 & ! [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ N2 ) % 8.42/4.48 => ( P @ ( suc @ I3 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_273_gr0__conv__Suc,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.48 = ( ? [M2: nat] : % 8.42/4.48 ( N2 % 8.42/4.48 = ( suc @ M2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_274_Ex__less__Suc2,axiom, % 8.42/4.48 ! [N2: nat,P: nat > $o] : % 8.42/4.48 ( ( ? [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ ( suc @ N2 ) ) % 8.42/4.48 & ( P @ I3 ) ) ) % 8.42/4.48 = ( ( P @ ( zero_zero @ nat ) ) % 8.42/4.48 | ? [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ N2 ) % 8.42/4.48 & ( P @ ( suc @ I3 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_275_less__imp__add__positive,axiom, % 8.42/4.48 ! [I: nat,J: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I @ J ) % 8.42/4.48 => ? [K2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 ) % 8.42/4.48 & ( ( plus_plus @ nat @ I @ K2 ) % 8.42/4.48 = J ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_276_le__imp__less__Suc,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.48 => ( ord_less @ nat @ M @ ( suc @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_277_less__eq__Suc__le,axiom, % 8.42/4.48 ( ( ord_less @ nat ) % 8.42/4.48 = ( ^ [N: nat] : ( ord_less_eq @ nat @ ( suc @ N ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_278_less__Suc__eq__le,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M @ ( suc @ N2 ) ) % 8.42/4.48 = ( ord_less_eq @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_279_le__less__Suc__eq,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.48 => ( ( ord_less @ nat @ N2 @ ( suc @ M ) ) % 8.42/4.48 = ( N2 = M ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_280_Suc__le__lessD,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 ) % 8.42/4.48 => ( ord_less @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_281_inc__induct,axiom, % 8.42/4.48 ! [I: nat,J: nat,P: nat > $o] : % 8.42/4.48 ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.48 => ( ( P @ J ) % 8.42/4.48 => ( ! [N3: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ I @ N3 ) % 8.42/4.48 => ( ( ord_less @ nat @ N3 @ J ) % 8.42/4.48 => ( ( P @ ( suc @ N3 ) ) % 8.42/4.48 => ( P @ N3 ) ) ) ) % 8.42/4.48 => ( P @ I ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_282_dec__induct,axiom, % 8.42/4.48 ! [I: nat,J: nat,P: nat > $o] : % 8.42/4.48 ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.48 => ( ( P @ I ) % 8.42/4.48 => ( ! [N3: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ I @ N3 ) % 8.42/4.48 => ( ( ord_less @ nat @ N3 @ J ) % 8.42/4.48 => ( ( P @ N3 ) % 8.42/4.48 => ( P @ ( suc @ N3 ) ) ) ) ) % 8.42/4.48 => ( P @ J ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_283_Suc__le__eq,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 ) % 8.42/4.48 = ( ord_less @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_284_Suc__leI,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.48 => ( ord_less_eq @ nat @ ( suc @ M ) @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_285_mono__nat__linear__lb,axiom, % 8.42/4.48 ! [F2: nat > nat,M: nat,K: nat] : % 8.42/4.48 ( ! [M3: nat,N3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M3 @ N3 ) % 8.42/4.48 => ( ord_less @ nat @ ( F2 @ M3 ) @ ( F2 @ N3 ) ) ) % 8.42/4.48 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus @ nat @ M @ K ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_286_add__strict__increasing2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere8940638589300402666id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less @ A @ B2 @ C2 ) % 8.42/4.48 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_287_add__strict__increasing,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere8940638589300402666id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ B2 @ C2 ) % 8.42/4.48 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_288_add__pos__nonneg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6911136660526730532id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.48 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_289_add__nonpos__neg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6911136660526730532id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_290_add__nonneg__pos,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6911136660526730532id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.48 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_291_add__neg__nonpos,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ordere6911136660526730532id_add @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_292_ex__least__nat__less,axiom, % 8.42/4.48 ! [P: nat > $o,N2: nat] : % 8.42/4.48 ( ( P @ N2 ) % 8.42/4.48 => ( ~ ( P @ ( zero_zero @ nat ) ) % 8.42/4.48 => ? [K2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ K2 @ N2 ) % 8.42/4.48 & ! [I5: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ I5 @ K2 ) % 8.42/4.48 => ~ ( P @ I5 ) ) % 8.42/4.48 & ( P @ ( suc @ K2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_293_set__update__subsetI,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,A3: set @ A,X3: A,I: nat] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 ) % 8.42/4.48 => ( ( member @ A @ X3 @ A3 ) % 8.42/4.48 => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X3 ) ) @ A3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_294_nle__le,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) % 8.42/4.48 = ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.42/4.48 & ( B2 != A2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_295_le__cases3,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A,Z2: A] : % 8.42/4.48 ( ( ( ord_less_eq @ A @ X3 @ Y ) % 8.42/4.48 => ~ ( ord_less_eq @ A @ Y @ Z2 ) ) % 8.42/4.48 => ( ( ( ord_less_eq @ A @ Y @ X3 ) % 8.42/4.48 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) ) % 8.42/4.48 => ( ( ( ord_less_eq @ A @ X3 @ Z2 ) % 8.42/4.48 => ~ ( ord_less_eq @ A @ Z2 @ Y ) ) % 8.42/4.48 => ( ( ( ord_less_eq @ A @ Z2 @ Y ) % 8.42/4.48 => ~ ( ord_less_eq @ A @ Y @ X3 ) ) % 8.42/4.48 => ( ( ( ord_less_eq @ A @ Y @ Z2 ) % 8.42/4.48 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) ) % 8.42/4.48 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 ) % 8.42/4.48 => ~ ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_296_order__class_Oorder__eq__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) ) % 8.42/4.48 = ( ^ [X: A,Y2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X @ Y2 ) % 8.42/4.48 & ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_297_ord__eq__le__trans,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ord @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( A2 = B2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ B2 @ C2 ) % 8.42/4.48 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_298_ord__le__eq__trans,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ord @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( B2 = C2 ) % 8.42/4.48 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_299_order__antisym,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.42/4.48 => ( ( ord_less_eq @ A @ Y @ X3 ) % 8.42/4.48 => ( X3 = Y ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_300_order_Otrans,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ B2 @ C2 ) % 8.42/4.48 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_301_order__trans,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A,Z2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.42/4.48 => ( ( ord_less_eq @ A @ Y @ Z2 ) % 8.42/4.48 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_302_linorder__wlog,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [P: A > A > $o,A2: A,B2: A] : % 8.42/4.48 ( ! [A5: A,B5: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A5 @ B5 ) % 8.42/4.48 => ( P @ A5 @ B5 ) ) % 8.42/4.48 => ( ! [A5: A,B5: A] : % 8.42/4.48 ( ( P @ B5 @ A5 ) % 8.42/4.48 => ( P @ A5 @ B5 ) ) % 8.42/4.48 => ( P @ A2 @ B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_303_dual__order_Oeq__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) ) % 8.42/4.48 = ( ^ [A4: A,B4: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ B4 @ A4 ) % 8.42/4.48 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_304_dual__order_Oantisym,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [B2: A,A2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( A2 = B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_305_dual__order_Otrans,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [B2: A,A2: A,C2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ C2 @ B2 ) % 8.42/4.48 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_306_antisym,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.42/4.48 => ( A2 = B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_307_le__funD,axiom, % 8.42/4.48 ! [B: $tType,A: $tType] : % 8.42/4.48 ( ( ord @ B ) % 8.42/4.48 => ! [F2: A > B,G: A > B,X3: A] : % 8.42/4.48 ( ( ord_less_eq @ ( A > B ) @ F2 @ G ) % 8.42/4.48 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_308_le__funE,axiom, % 8.42/4.48 ! [B: $tType,A: $tType] : % 8.42/4.48 ( ( ord @ B ) % 8.42/4.48 => ! [F2: A > B,G: A > B,X3: A] : % 8.42/4.48 ( ( ord_less_eq @ ( A > B ) @ F2 @ G ) % 8.42/4.48 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_309_le__funI,axiom, % 8.42/4.48 ! [B: $tType,A: $tType] : % 8.42/4.48 ( ( ord @ B ) % 8.42/4.48 => ! [F2: A > B,G: A > B] : % 8.42/4.48 ( ! [X5: A] : ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G @ X5 ) ) % 8.42/4.48 => ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_310_le__fun__def,axiom, % 8.42/4.48 ! [B: $tType,A: $tType] : % 8.42/4.48 ( ( ord @ B ) % 8.42/4.48 => ( ( ord_less_eq @ ( A > B ) ) % 8.42/4.48 = ( ^ [F3: A > B,G2: A > B] : % 8.42/4.48 ! [X: A] : ( ord_less_eq @ B @ ( F3 @ X ) @ ( G2 @ X ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_311_Orderings_Oorder__eq__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) ) % 8.42/4.48 = ( ^ [A4: A,B4: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A4 @ B4 ) % 8.42/4.48 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_312_order__subst1,axiom, % 8.42/4.48 ! [A: $tType,B: $tType] : % 8.42/4.48 ( ( ( order @ B ) % 8.42/4.48 & ( order @ A ) ) % 8.42/4.48 => ! [A2: A,F2: B > A,B2: B,C2: B] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) ) % 8.42/4.48 => ( ( ord_less_eq @ B @ B2 @ C2 ) % 8.42/4.48 => ( ! [X5: B,Y4: B] : % 8.42/4.48 ( ( ord_less_eq @ B @ X5 @ Y4 ) % 8.42/4.48 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 8.42/4.48 => ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_313_order__subst2,axiom, % 8.42/4.48 ! [A: $tType,C: $tType] : % 8.42/4.48 ( ( ( order @ C ) % 8.42/4.48 & ( order @ A ) ) % 8.42/4.48 => ! [A2: A,B2: A,F2: A > C,C2: C] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 ) % 8.42/4.48 => ( ! [X5: A,Y4: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X5 @ Y4 ) % 8.42/4.48 => ( ord_less_eq @ C @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 8.42/4.48 => ( ord_less_eq @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_314_order__eq__refl,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( X3 = Y ) % 8.42/4.48 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_315_linorder__linear,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.42/4.48 | ( ord_less_eq @ A @ Y @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_316_ord__eq__le__subst,axiom, % 8.42/4.48 ! [A: $tType,B: $tType] : % 8.42/4.48 ( ( ( ord @ B ) % 8.42/4.48 & ( ord @ A ) ) % 8.42/4.48 => ! [A2: A,F2: B > A,B2: B,C2: B] : % 8.42/4.48 ( ( A2 % 8.42/4.48 = ( F2 @ B2 ) ) % 8.42/4.48 => ( ( ord_less_eq @ B @ B2 @ C2 ) % 8.42/4.48 => ( ! [X5: B,Y4: B] : % 8.42/4.48 ( ( ord_less_eq @ B @ X5 @ Y4 ) % 8.42/4.48 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 8.42/4.48 => ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_317_ord__le__eq__subst,axiom, % 8.42/4.48 ! [A: $tType,B: $tType] : % 8.42/4.48 ( ( ( ord @ B ) % 8.42/4.48 & ( ord @ A ) ) % 8.42/4.48 => ! [A2: A,B2: A,F2: A > B,C2: B] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ( F2 @ B2 ) % 8.42/4.48 = C2 ) % 8.42/4.48 => ( ! [X5: A,Y4: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X5 @ Y4 ) % 8.42/4.48 => ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 8.42/4.48 => ( ord_less_eq @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_318_linorder__le__cases,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ~ ( ord_less_eq @ A @ X3 @ Y ) % 8.42/4.48 => ( ord_less_eq @ A @ Y @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_319_order__antisym__conv,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [Y: A,X3: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ Y @ X3 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ X3 @ Y ) % 8.42/4.48 = ( X3 = Y ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_320_zero__reorient,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( zero @ A ) % 8.42/4.48 => ! [X3: A] : % 8.42/4.48 ( ( ( zero_zero @ A ) % 8.42/4.48 = X3 ) % 8.42/4.48 = ( X3 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_321_lt__ex,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( no_bot @ A ) % 8.42/4.48 => ! [X3: A] : % 8.42/4.48 ? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_322_gt__ex,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( no_top @ A ) % 8.42/4.48 => ! [X3: A] : % 8.42/4.48 ? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_323_dense,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( dense_order @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ? [Z3: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ Z3 ) % 8.42/4.48 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_324_less__imp__neq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ( X3 != Y ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_325_order_Oasym,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_326_ord__eq__less__trans,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ord @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( A2 = B2 ) % 8.42/4.48 => ( ( ord_less @ A @ B2 @ C2 ) % 8.42/4.48 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_327_ord__less__eq__trans,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ord @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( B2 = C2 ) % 8.42/4.48 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_328_less__induct,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( wellorder @ A ) % 8.42/4.48 => ! [P: A > $o,A2: A] : % 8.42/4.48 ( ! [X5: A] : % 8.42/4.48 ( ! [Y5: A] : % 8.42/4.48 ( ( ord_less @ A @ Y5 @ X5 ) % 8.42/4.48 => ( P @ Y5 ) ) % 8.42/4.48 => ( P @ X5 ) ) % 8.42/4.48 => ( P @ A2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_329_antisym__conv3,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [Y: A,X3: A] : % 8.42/4.48 ( ~ ( ord_less @ A @ Y @ X3 ) % 8.42/4.48 => ( ( ~ ( ord_less @ A @ X3 @ Y ) ) % 8.42/4.48 = ( X3 = Y ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_330_linorder__cases,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ~ ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ( ( X3 != Y ) % 8.42/4.48 => ( ord_less @ A @ Y @ X3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_331_dual__order_Oasym,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [B2: A,A2: A] : % 8.42/4.48 ( ( ord_less @ A @ B2 @ A2 ) % 8.42/4.48 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_332_dual__order_Oirrefl,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ~ ( ord_less @ A @ A2 @ A2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_333_exists__least__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( wellorder @ A ) % 8.42/4.48 => ( ( ^ [P2: A > $o] : % 8.42/4.48 ? [X7: A] : ( P2 @ X7 ) ) % 8.42/4.48 = ( ^ [P3: A > $o] : % 8.42/4.48 ? [N: A] : % 8.42/4.48 ( ( P3 @ N ) % 8.42/4.48 & ! [M2: A] : % 8.42/4.48 ( ( ord_less @ A @ M2 @ N ) % 8.42/4.48 => ~ ( P3 @ M2 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_334_linorder__less__wlog,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [P: A > A > $o,A2: A,B2: A] : % 8.42/4.48 ( ! [A5: A,B5: A] : % 8.42/4.48 ( ( ord_less @ A @ A5 @ B5 ) % 8.42/4.48 => ( P @ A5 @ B5 ) ) % 8.42/4.48 => ( ! [A5: A] : ( P @ A5 @ A5 ) % 8.42/4.48 => ( ! [A5: A,B5: A] : % 8.42/4.48 ( ( P @ B5 @ A5 ) % 8.42/4.48 => ( P @ A5 @ B5 ) ) % 8.42/4.48 => ( P @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_335_order_Ostrict__trans,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less @ A @ B2 @ C2 ) % 8.42/4.48 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_336_not__less__iff__gr__or__eq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ~ ( ord_less @ A @ X3 @ Y ) ) % 8.42/4.48 = ( ( ord_less @ A @ Y @ X3 ) % 8.42/4.48 | ( X3 = Y ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_337_dual__order_Ostrict__trans,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [B2: A,A2: A,C2: A] : % 8.42/4.48 ( ( ord_less @ A @ B2 @ A2 ) % 8.42/4.48 => ( ( ord_less @ A @ C2 @ B2 ) % 8.42/4.48 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_338_order_Ostrict__implies__not__eq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ( A2 != B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_339_dual__order_Ostrict__implies__not__eq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [B2: A,A2: A] : % 8.42/4.48 ( ( ord_less @ A @ B2 @ A2 ) % 8.42/4.48 => ( A2 != B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_340_measure__induct,axiom, % 8.42/4.48 ! [B: $tType,A: $tType] : % 8.42/4.48 ( ( wellorder @ B ) % 8.42/4.48 => ! [F2: A > B,P: A > $o,A2: A] : % 8.42/4.48 ( ! [X5: A] : % 8.42/4.48 ( ! [Y5: A] : % 8.42/4.48 ( ( ord_less @ B @ ( F2 @ Y5 ) @ ( F2 @ X5 ) ) % 8.42/4.48 => ( P @ Y5 ) ) % 8.42/4.48 => ( P @ X5 ) ) % 8.42/4.48 => ( P @ A2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_341_measure__induct__rule,axiom, % 8.42/4.48 ! [B: $tType,A: $tType] : % 8.42/4.48 ( ( wellorder @ B ) % 8.42/4.48 => ! [F2: A > B,P: A > $o,A2: A] : % 8.42/4.48 ( ! [X5: A] : % 8.42/4.48 ( ! [Y5: A] : % 8.42/4.48 ( ( ord_less @ B @ ( F2 @ Y5 ) @ ( F2 @ X5 ) ) % 8.42/4.48 => ( P @ Y5 ) ) % 8.42/4.48 => ( P @ X5 ) ) % 8.42/4.48 => ( P @ A2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_342_linorder__neqE,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( X3 != Y ) % 8.42/4.48 => ( ~ ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ( ord_less @ A @ Y @ X3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_343_order__less__asym,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ~ ( ord_less @ A @ Y @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_344_linorder__neq__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( X3 != Y ) % 8.42/4.48 = ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 | ( ord_less @ A @ Y @ X3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_345_order__less__asym_H,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_346_order__less__trans,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A,Z2: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ( ( ord_less @ A @ Y @ Z2 ) % 8.42/4.48 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_347_ord__eq__less__subst,axiom, % 8.42/4.48 ! [A: $tType,B: $tType] : % 8.42/4.48 ( ( ( ord @ B ) % 8.42/4.48 & ( ord @ A ) ) % 8.42/4.48 => ! [A2: A,F2: B > A,B2: B,C2: B] : % 8.42/4.48 ( ( A2 % 8.42/4.48 = ( F2 @ B2 ) ) % 8.42/4.48 => ( ( ord_less @ B @ B2 @ C2 ) % 8.42/4.48 => ( ! [X5: B,Y4: B] : % 8.42/4.48 ( ( ord_less @ B @ X5 @ Y4 ) % 8.42/4.48 => ( ord_less @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 8.42/4.48 => ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_348_ord__less__eq__subst,axiom, % 8.42/4.48 ! [A: $tType,B: $tType] : % 8.42/4.48 ( ( ( ord @ B ) % 8.42/4.48 & ( ord @ A ) ) % 8.42/4.48 => ! [A2: A,B2: A,F2: A > B,C2: B] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ( F2 @ B2 ) % 8.42/4.48 = C2 ) % 8.42/4.48 => ( ! [X5: A,Y4: A] : % 8.42/4.48 ( ( ord_less @ A @ X5 @ Y4 ) % 8.42/4.48 => ( ord_less @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 8.42/4.48 => ( ord_less @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_349_order__less__irrefl,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [X3: A] : % 8.42/4.48 ~ ( ord_less @ A @ X3 @ X3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_350_order__less__subst1,axiom, % 8.42/4.48 ! [A: $tType,B: $tType] : % 8.42/4.48 ( ( ( order @ B ) % 8.42/4.48 & ( order @ A ) ) % 8.42/4.48 => ! [A2: A,F2: B > A,B2: B,C2: B] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ ( F2 @ B2 ) ) % 8.42/4.48 => ( ( ord_less @ B @ B2 @ C2 ) % 8.42/4.48 => ( ! [X5: B,Y4: B] : % 8.42/4.48 ( ( ord_less @ B @ X5 @ Y4 ) % 8.42/4.48 => ( ord_less @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 8.42/4.48 => ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_351_order__less__subst2,axiom, % 8.42/4.48 ! [A: $tType,C: $tType] : % 8.42/4.48 ( ( ( order @ C ) % 8.42/4.48 & ( order @ A ) ) % 8.42/4.48 => ! [A2: A,B2: A,F2: A > C,C2: C] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less @ C @ ( F2 @ B2 ) @ C2 ) % 8.42/4.48 => ( ! [X5: A,Y4: A] : % 8.42/4.48 ( ( ord_less @ A @ X5 @ Y4 ) % 8.42/4.48 => ( ord_less @ C @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 8.42/4.48 => ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_352_order__less__not__sym,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ~ ( ord_less @ A @ Y @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_353_order__less__imp__triv,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A,P: $o] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ( ( ord_less @ A @ Y @ X3 ) % 8.42/4.48 => P ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_354_linorder__less__linear,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 | ( X3 = Y ) % 8.42/4.48 | ( ord_less @ A @ Y @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_355_order__less__imp__not__eq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ( X3 != Y ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_356_order__less__imp__not__eq2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ( Y != X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_357_order__less__imp__not__less,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ~ ( ord_less @ A @ Y @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_358_nat__neq__iff,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( M != N2 ) % 8.42/4.48 = ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.48 | ( ord_less @ nat @ N2 @ M ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_359_less__not__refl,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ~ ( ord_less @ nat @ N2 @ N2 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_360_less__not__refl2,axiom, % 8.42/4.48 ! [N2: nat,M: nat] : % 8.42/4.48 ( ( ord_less @ nat @ N2 @ M ) % 8.42/4.48 => ( M != N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_361_less__not__refl3,axiom, % 8.42/4.48 ! [S: nat,T2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ S @ T2 ) % 8.42/4.48 => ( S != T2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_362_less__irrefl__nat,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ~ ( ord_less @ nat @ N2 @ N2 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_363_nat__less__induct,axiom, % 8.42/4.48 ! [P: nat > $o,N2: nat] : % 8.42/4.48 ( ! [N3: nat] : % 8.42/4.48 ( ! [M6: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M6 @ N3 ) % 8.42/4.48 => ( P @ M6 ) ) % 8.42/4.48 => ( P @ N3 ) ) % 8.42/4.48 => ( P @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_364_infinite__descent,axiom, % 8.42/4.48 ! [P: nat > $o,N2: nat] : % 8.42/4.48 ( ! [N3: nat] : % 8.42/4.48 ( ~ ( P @ N3 ) % 8.42/4.48 => ? [M6: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M6 @ N3 ) % 8.42/4.48 & ~ ( P @ M6 ) ) ) % 8.42/4.48 => ( P @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_365_linorder__neqE__nat,axiom, % 8.42/4.48 ! [X3: nat,Y: nat] : % 8.42/4.48 ( ( X3 != Y ) % 8.42/4.48 => ( ~ ( ord_less @ nat @ X3 @ Y ) % 8.42/4.48 => ( ord_less @ nat @ Y @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_366_infinite__descent__measure,axiom, % 8.42/4.48 ! [A: $tType,P: A > $o,V: A > nat,X3: A] : % 8.42/4.48 ( ! [X5: A] : % 8.42/4.48 ( ~ ( P @ X5 ) % 8.42/4.48 => ? [Y5: A] : % 8.42/4.48 ( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X5 ) ) % 8.42/4.48 & ~ ( P @ Y5 ) ) ) % 8.42/4.48 => ( P @ X3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_367_le__refl,axiom, % 8.42/4.48 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_368_le__trans,axiom, % 8.42/4.48 ! [I: nat,J: nat,K: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.48 => ( ( ord_less_eq @ nat @ J @ K ) % 8.42/4.48 => ( ord_less_eq @ nat @ I @ K ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_369_eq__imp__le,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( M = N2 ) % 8.42/4.48 => ( ord_less_eq @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_370_le__antisym,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.48 => ( ( ord_less_eq @ nat @ N2 @ M ) % 8.42/4.48 => ( M = N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_371_nat__le__linear,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.48 | ( ord_less_eq @ nat @ N2 @ M ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_372_Nat_Oex__has__greatest__nat,axiom, % 8.42/4.48 ! [P: nat > $o,K: nat,B2: nat] : % 8.42/4.48 ( ( P @ K ) % 8.42/4.48 => ( ! [Y4: nat] : % 8.42/4.48 ( ( P @ Y4 ) % 8.42/4.48 => ( ord_less_eq @ nat @ Y4 @ B2 ) ) % 8.42/4.48 => ? [X5: nat] : % 8.42/4.48 ( ( P @ X5 ) % 8.42/4.48 & ! [Y5: nat] : % 8.42/4.48 ( ( P @ Y5 ) % 8.42/4.48 => ( ord_less_eq @ nat @ Y5 @ X5 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_373_size__neq__size__imp__neq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( size @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ( size_size @ A @ X3 ) % 8.42/4.48 != ( size_size @ A @ Y ) ) % 8.42/4.48 => ( X3 != Y ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_374_zero__le,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( canoni5634975068530333245id_add @ A ) % 8.42/4.48 => ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_375_leD,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [Y: A,X3: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ Y @ X3 ) % 8.42/4.48 => ~ ( ord_less @ A @ X3 @ Y ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_376_leI,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ~ ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ( ord_less_eq @ A @ Y @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_377_nless__le,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ~ ( ord_less @ A @ A2 @ B2 ) ) % 8.42/4.48 = ( ~ ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 | ( A2 = B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_378_antisym__conv1,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ~ ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ( ( ord_less_eq @ A @ X3 @ Y ) % 8.42/4.48 = ( X3 = Y ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_379_antisym__conv2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.42/4.48 => ( ( ~ ( ord_less @ A @ X3 @ Y ) ) % 8.42/4.48 = ( X3 = Y ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_380_dense__ge,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( dense_linorder @ A ) % 8.42/4.48 => ! [Z2: A,Y: A] : % 8.42/4.48 ( ! [X5: A] : % 8.42/4.48 ( ( ord_less @ A @ Z2 @ X5 ) % 8.42/4.48 => ( ord_less_eq @ A @ Y @ X5 ) ) % 8.42/4.48 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_381_dense__le,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( dense_linorder @ A ) % 8.42/4.48 => ! [Y: A,Z2: A] : % 8.42/4.48 ( ! [X5: A] : % 8.42/4.48 ( ( ord_less @ A @ X5 @ Y ) % 8.42/4.48 => ( ord_less_eq @ A @ X5 @ Z2 ) ) % 8.42/4.48 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_382_less__le__not__le,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ( ( ord_less @ A ) % 8.42/4.48 = ( ^ [X: A,Y2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X @ Y2 ) % 8.42/4.48 & ~ ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_383_not__le__imp__less,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [Y: A,X3: A] : % 8.42/4.48 ( ~ ( ord_less_eq @ A @ Y @ X3 ) % 8.42/4.48 => ( ord_less @ A @ X3 @ Y ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_384_order_Oorder__iff__strict,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ( ( ord_less_eq @ A ) % 8.42/4.48 = ( ^ [A4: A,B4: A] : % 8.42/4.48 ( ( ord_less @ A @ A4 @ B4 ) % 8.42/4.48 | ( A4 = B4 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_385_order_Ostrict__iff__order,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ( ( ord_less @ A ) % 8.42/4.48 = ( ^ [A4: A,B4: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A4 @ B4 ) % 8.42/4.48 & ( A4 != B4 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_386_order_Ostrict__trans1,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less @ A @ B2 @ C2 ) % 8.42/4.48 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_387_order_Ostrict__trans2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ B2 @ C2 ) % 8.42/4.48 => ( ord_less @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_388_order_Ostrict__iff__not,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ( ( ord_less @ A ) % 8.42/4.48 = ( ^ [A4: A,B4: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A4 @ B4 ) % 8.42/4.48 & ~ ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_389_dense__ge__bounded,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( dense_linorder @ A ) % 8.42/4.48 => ! [Z2: A,X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ Z2 @ X3 ) % 8.42/4.48 => ( ! [W: A] : % 8.42/4.48 ( ( ord_less @ A @ Z2 @ W ) % 8.42/4.48 => ( ( ord_less @ A @ W @ X3 ) % 8.42/4.48 => ( ord_less_eq @ A @ Y @ W ) ) ) % 8.42/4.48 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_390_dense__le__bounded,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( dense_linorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A,Z2: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ( ! [W: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ W ) % 8.42/4.48 => ( ( ord_less @ A @ W @ Y ) % 8.42/4.48 => ( ord_less_eq @ A @ W @ Z2 ) ) ) % 8.42/4.48 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_391_dual__order_Oorder__iff__strict,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ( ( ord_less_eq @ A ) % 8.42/4.48 = ( ^ [B4: A,A4: A] : % 8.42/4.48 ( ( ord_less @ A @ B4 @ A4 ) % 8.42/4.48 | ( A4 = B4 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_392_dual__order_Ostrict__iff__order,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ( ( ord_less @ A ) % 8.42/4.48 = ( ^ [B4: A,A4: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ B4 @ A4 ) % 8.42/4.48 & ( A4 != B4 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_393_dual__order_Ostrict__trans1,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [B2: A,A2: A,C2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.42/4.48 => ( ( ord_less @ A @ C2 @ B2 ) % 8.42/4.48 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_394_dual__order_Ostrict__trans2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [B2: A,A2: A,C2: A] : % 8.42/4.48 ( ( ord_less @ A @ B2 @ A2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ C2 @ B2 ) % 8.42/4.48 => ( ord_less @ A @ C2 @ A2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_395_dual__order_Ostrict__iff__not,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ( ( ord_less @ A ) % 8.42/4.48 = ( ^ [B4: A,A4: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ B4 @ A4 ) % 8.42/4.48 & ~ ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_396_order_Ostrict__implies__order,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_397_dual__order_Ostrict__implies__order,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [B2: A,A2: A] : % 8.42/4.48 ( ( ord_less @ A @ B2 @ A2 ) % 8.42/4.48 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_398_order__le__less,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ( ( ord_less_eq @ A ) % 8.42/4.48 = ( ^ [X: A,Y2: A] : % 8.42/4.48 ( ( ord_less @ A @ X @ Y2 ) % 8.42/4.48 | ( X = Y2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_399_order__less__le,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ( ( ord_less @ A ) % 8.42/4.48 = ( ^ [X: A,Y2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X @ Y2 ) % 8.42/4.48 & ( X != Y2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_400_linorder__not__le,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ~ ( ord_less_eq @ A @ X3 @ Y ) ) % 8.42/4.48 = ( ord_less @ A @ Y @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_401_linorder__not__less,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ~ ( ord_less @ A @ X3 @ Y ) ) % 8.42/4.48 = ( ord_less_eq @ A @ Y @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_402_order__less__imp__le,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_403_order__le__neq__trans,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( A2 != B2 ) % 8.42/4.48 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_404_order__neq__le__trans,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( A2 != B2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_405_order__le__less__trans,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A,Z2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.42/4.48 => ( ( ord_less @ A @ Y @ Z2 ) % 8.42/4.48 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_406_order__less__le__trans,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( preorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A,Z2: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ( ( ord_less_eq @ A @ Y @ Z2 ) % 8.42/4.48 => ( ord_less @ A @ X3 @ Z2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_407_order__le__less__subst1,axiom, % 8.42/4.48 ! [A: $tType,B: $tType] : % 8.42/4.48 ( ( ( order @ B ) % 8.42/4.48 & ( order @ A ) ) % 8.42/4.48 => ! [A2: A,F2: B > A,B2: B,C2: B] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) ) % 8.42/4.48 => ( ( ord_less @ B @ B2 @ C2 ) % 8.42/4.48 => ( ! [X5: B,Y4: B] : % 8.42/4.48 ( ( ord_less @ B @ X5 @ Y4 ) % 8.42/4.48 => ( ord_less @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 8.42/4.48 => ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_408_order__le__less__subst2,axiom, % 8.42/4.48 ! [A: $tType,C: $tType] : % 8.42/4.48 ( ( ( order @ C ) % 8.42/4.48 & ( order @ A ) ) % 8.42/4.48 => ! [A2: A,B2: A,F2: A > C,C2: C] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less @ C @ ( F2 @ B2 ) @ C2 ) % 8.42/4.48 => ( ! [X5: A,Y4: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X5 @ Y4 ) % 8.42/4.48 => ( ord_less_eq @ C @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 8.42/4.48 => ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_409_order__less__le__subst1,axiom, % 8.42/4.48 ! [A: $tType,B: $tType] : % 8.42/4.48 ( ( ( order @ B ) % 8.42/4.48 & ( order @ A ) ) % 8.42/4.48 => ! [A2: A,F2: B > A,B2: B,C2: B] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ ( F2 @ B2 ) ) % 8.42/4.48 => ( ( ord_less_eq @ B @ B2 @ C2 ) % 8.42/4.48 => ( ! [X5: B,Y4: B] : % 8.42/4.48 ( ( ord_less_eq @ B @ X5 @ Y4 ) % 8.42/4.48 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 8.42/4.48 => ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_410_order__less__le__subst2,axiom, % 8.42/4.48 ! [A: $tType,C: $tType] : % 8.42/4.48 ( ( ( order @ C ) % 8.42/4.48 & ( order @ A ) ) % 8.42/4.48 => ! [A2: A,B2: A,F2: A > C,C2: C] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 ) % 8.42/4.48 => ( ! [X5: A,Y4: A] : % 8.42/4.48 ( ( ord_less @ A @ X5 @ Y4 ) % 8.42/4.48 => ( ord_less @ C @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) ) % 8.42/4.48 => ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_411_linorder__le__less__linear,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.42/4.48 | ( ord_less @ A @ Y @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_412_order__le__imp__less__or__eq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.42/4.48 => ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 | ( X3 = Y ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_413_gr__zeroI,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( canoni5634975068530333245id_add @ A ) % 8.42/4.48 => ! [N2: A] : % 8.42/4.48 ( ( N2 % 8.42/4.48 != ( zero_zero @ A ) ) % 8.42/4.48 => ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_414_not__less__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( canoni5634975068530333245id_add @ A ) % 8.42/4.48 => ! [N2: A] : % 8.42/4.48 ~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_415_gr__implies__not__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( canoni5634975068530333245id_add @ A ) % 8.42/4.48 => ! [M: A,N2: A] : % 8.42/4.48 ( ( ord_less @ A @ M @ N2 ) % 8.42/4.48 => ( N2 % 8.42/4.48 != ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_416_zero__less__iff__neq__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( canoni5634975068530333245id_add @ A ) % 8.42/4.48 => ! [N2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) % 8.42/4.48 = ( N2 % 8.42/4.48 != ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_417_bot__nat__0_Oextremum__strict,axiom, % 8.42/4.48 ! [A2: nat] : % 8.42/4.48 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_418_gr0I,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( N2 % 8.42/4.48 != ( zero_zero @ nat ) ) % 8.42/4.48 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_419_not__gr0,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.42/4.48 = ( N2 % 8.42/4.48 = ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_420_not__less0,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_421_less__zeroE,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_422_gr__implies__not0,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.48 => ( N2 % 8.42/4.48 != ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_423_infinite__descent0,axiom, % 8.42/4.48 ! [P: nat > $o,N2: nat] : % 8.42/4.48 ( ( P @ ( zero_zero @ nat ) ) % 8.42/4.48 => ( ! [N3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 8.42/4.48 => ( ~ ( P @ N3 ) % 8.42/4.48 => ? [M6: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M6 @ N3 ) % 8.42/4.48 & ~ ( P @ M6 ) ) ) ) % 8.42/4.48 => ( P @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_424_infinite__descent0__measure,axiom, % 8.42/4.48 ! [A: $tType,V: A > nat,P: A > $o,X3: A] : % 8.42/4.48 ( ! [X5: A] : % 8.42/4.48 ( ( ( V @ X5 ) % 8.42/4.48 = ( zero_zero @ nat ) ) % 8.42/4.48 => ( P @ X5 ) ) % 8.42/4.48 => ( ! [X5: A] : % 8.42/4.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X5 ) ) % 8.42/4.48 => ( ~ ( P @ X5 ) % 8.42/4.48 => ? [Y5: A] : % 8.42/4.48 ( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X5 ) ) % 8.42/4.48 & ~ ( P @ Y5 ) ) ) ) % 8.42/4.48 => ( P @ X3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_425_less__eq__nat_Osimps_I1_J,axiom, % 8.42/4.48 ! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_426_bot__nat__0_Oextremum__unique,axiom, % 8.42/4.48 ! [A2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) ) % 8.42/4.48 = ( A2 % 8.42/4.48 = ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_427_bot__nat__0_Oextremum__uniqueI,axiom, % 8.42/4.48 ! [A2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) ) % 8.42/4.48 => ( A2 % 8.42/4.48 = ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_428_le__0__eq,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) ) % 8.42/4.48 = ( N2 % 8.42/4.48 = ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_429_nat__less__le,axiom, % 8.42/4.48 ( ( ord_less @ nat ) % 8.42/4.48 = ( ^ [M2: nat,N: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ M2 @ N ) % 8.42/4.48 & ( M2 != N ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_430_less__imp__le__nat,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.48 => ( ord_less_eq @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_431_le__eq__less__or__eq,axiom, % 8.42/4.48 ( ( ord_less_eq @ nat ) % 8.42/4.48 = ( ^ [M2: nat,N: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M2 @ N ) % 8.42/4.48 | ( M2 = N ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_432_less__or__eq__imp__le,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.48 | ( M = N2 ) ) % 8.42/4.48 => ( ord_less_eq @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_433_le__neq__implies__less,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.48 => ( ( M != N2 ) % 8.42/4.48 => ( ord_less @ nat @ M @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_434_less__mono__imp__le__mono,axiom, % 8.42/4.48 ! [F2: nat > nat,I: nat,J: nat] : % 8.42/4.48 ( ! [I4: nat,J2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I4 @ J2 ) % 8.42/4.48 => ( ord_less @ nat @ ( F2 @ I4 ) @ ( F2 @ J2 ) ) ) % 8.42/4.48 => ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.48 => ( ord_less_eq @ nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_435_double__eq__0__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord5086331880401160121up_add @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ( plus_plus @ A @ A2 @ A2 ) % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 = ( A2 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_436_assms_I1_J,axiom, % 8.42/4.48 vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ). % 8.42/4.48 % 8.42/4.48 thf(fact_437_mint__corr__help__empty,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ( ( vEBT_vebt_mint @ T2 ) % 8.42/4.48 = ( none @ nat ) ) % 8.42/4.48 => ( ( vEBT_VEBT_set_vebt @ T2 ) % 8.42/4.48 = ( bot_bot @ ( set @ nat ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_438_maxt__corr__help__empty,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ( ( vEBT_vebt_maxt @ T2 ) % 8.42/4.48 = ( none @ nat ) ) % 8.42/4.48 => ( ( vEBT_VEBT_set_vebt @ T2 ) % 8.42/4.48 = ( bot_bot @ ( set @ nat ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_439_field__le__epsilon,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ! [E: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ E ) % 8.42/4.48 => ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ Y @ E ) ) ) % 8.42/4.48 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_440_ex__has__greatest__nat__lemma,axiom, % 8.42/4.48 ! [A: $tType,P: A > $o,K: A,F2: A > nat,N2: nat] : % 8.42/4.48 ( ( P @ K ) % 8.42/4.48 => ( ! [X5: A] : % 8.42/4.48 ( ( P @ X5 ) % 8.42/4.48 => ? [Y5: A] : % 8.42/4.48 ( ( P @ Y5 ) % 8.42/4.48 & ~ ( ord_less_eq @ nat @ ( F2 @ Y5 ) @ ( F2 @ X5 ) ) ) ) % 8.42/4.48 => ? [Y4: A] : % 8.42/4.48 ( ( P @ Y4 ) % 8.42/4.48 & ~ ( ord_less @ nat @ ( F2 @ Y4 ) @ ( plus_plus @ nat @ ( F2 @ K ) @ N2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_441_option_Osize__gen_I2_J,axiom, % 8.42/4.48 ! [A: $tType,X3: A > nat,X22: A] : % 8.42/4.48 ( ( size_option @ A @ X3 @ ( some @ A @ X22 ) ) % 8.42/4.48 = ( plus_plus @ nat @ ( X3 @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_442_add__less__zeroD,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_idom @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less @ A @ X3 @ ( zero_zero @ A ) ) % 8.42/4.48 | ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_443_empty__subsetI,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_444_subset__empty,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) % 8.42/4.48 = ( A3 % 8.42/4.48 = ( bot_bot @ ( set @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_445_empty__iff,axiom, % 8.42/4.48 ! [A: $tType,C2: A] : % 8.42/4.48 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_446_all__not__in__conv,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A] : % 8.42/4.48 ( ( ! [X: A] : % 8.42/4.48 ~ ( member @ A @ X @ A3 ) ) % 8.42/4.48 = ( A3 % 8.42/4.48 = ( bot_bot @ ( set @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_447_deg__deg__n,axiom, % 8.42/4.48 ! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 ) % 8.42/4.48 => ( Deg = N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_448_deg__SUcn__Node,axiom, % 8.42/4.48 ! [Tree: vEBT_VEBT,N2: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) ) % 8.42/4.48 => ? [Info2: option @ ( product_prod @ nat @ nat ),TreeList3: list @ vEBT_VEBT,S2: vEBT_VEBT] : % 8.42/4.48 ( Tree % 8.42/4.48 = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N2 ) ) @ TreeList3 @ S2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_449_empty__Collect__eq,axiom, % 8.42/4.48 ! [A: $tType,P: A > $o] : % 8.42/4.48 ( ( ( bot_bot @ ( set @ A ) ) % 8.42/4.48 = ( collect @ A @ P ) ) % 8.42/4.48 = ( ! [X: A] : % 8.42/4.48 ~ ( P @ X ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_450_Collect__empty__eq,axiom, % 8.42/4.48 ! [A: $tType,P: A > $o] : % 8.42/4.48 ( ( ( collect @ A @ P ) % 8.42/4.48 = ( bot_bot @ ( set @ A ) ) ) % 8.42/4.48 = ( ! [X: A] : % 8.42/4.48 ~ ( P @ X ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_451_psubsetI,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.48 => ( ( A3 != B3 ) % 8.42/4.48 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_452_subset__antisym,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.48 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 8.42/4.48 => ( A3 = B3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_453_subsetI,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.42/4.48 ( ! [X5: A] : % 8.42/4.48 ( ( member @ A @ X5 @ A3 ) % 8.42/4.48 => ( member @ A @ X5 @ B3 ) ) % 8.42/4.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_454_mi__eq__ma__no__ch,axiom, % 8.42/4.48 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT] : % 8.42/4.48 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) % 8.42/4.48 => ( ( Mi = Ma ) % 8.42/4.48 => ( ! [X2: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 8.42/4.48 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) % 8.42/4.48 & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_455_not__Some__eq,axiom, % 8.42/4.48 ! [A: $tType,X3: option @ A] : % 8.42/4.48 ( ( ! [Y2: A] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( some @ A @ Y2 ) ) ) % 8.42/4.48 = ( X3 % 8.42/4.48 = ( none @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_456_not__None__eq,axiom, % 8.42/4.48 ! [A: $tType,X3: option @ A] : % 8.42/4.48 ( ( X3 % 8.42/4.48 != ( none @ A ) ) % 8.42/4.48 = ( ? [Y2: A] : % 8.42/4.48 ( X3 % 8.42/4.48 = ( some @ A @ Y2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_457_option_Ocollapse,axiom, % 8.42/4.48 ! [A: $tType,Option: option @ A] : % 8.42/4.48 ( ( Option % 8.42/4.48 != ( none @ A ) ) % 8.42/4.48 => ( ( some @ A @ ( the2 @ A @ Option ) ) % 8.42/4.48 = Option ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_458_add__def,axiom, % 8.42/4.48 ( vEBT_VEBT_add % 8.42/4.48 = ( vEBT_V2048590022279873568_shift @ nat @ ( plus_plus @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_459_psubsetE,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.42/4.48 ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.48 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.48 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_460_psubset__eq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ord_less @ ( set @ A ) ) % 8.42/4.48 = ( ^ [A6: set @ A,B6: set @ A] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) % 8.42/4.48 & ( A6 != B6 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_461_less__fun__def,axiom, % 8.42/4.48 ! [B: $tType,A: $tType] : % 8.42/4.48 ( ( ord @ B ) % 8.42/4.48 => ( ( ord_less @ ( A > B ) ) % 8.42/4.48 = ( ^ [F3: A > B,G2: A > B] : % 8.42/4.48 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 ) % 8.42/4.48 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_462_psubset__imp__subset,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.42/4.48 ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_463_psubset__subset__trans,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 8.42/4.48 ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.48 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 ) % 8.42/4.48 => ( ord_less @ ( set @ A ) @ A3 @ C5 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_464_subset__not__subset__eq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ord_less @ ( set @ A ) ) % 8.42/4.48 = ( ^ [A6: set @ A,B6: set @ A] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) % 8.42/4.48 & ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_465_subset__psubset__trans,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.48 => ( ( ord_less @ ( set @ A ) @ B3 @ C5 ) % 8.42/4.48 => ( ord_less @ ( set @ A ) @ A3 @ C5 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_466_subset__iff__psubset__eq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) ) % 8.42/4.48 = ( ^ [A6: set @ A,B6: set @ A] : % 8.42/4.48 ( ( ord_less @ ( set @ A ) @ A6 @ B6 ) % 8.42/4.48 | ( A6 = B6 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_467_Collect__mono__iff,axiom, % 8.42/4.48 ! [A: $tType,P: A > $o,Q: A > $o] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) % 8.42/4.48 = ( ! [X: A] : % 8.42/4.48 ( ( P @ X ) % 8.42/4.48 => ( Q @ X ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_468_set__eq__subset,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ^ [Y3: set @ A,Z: set @ A] : ( Y3 = Z ) ) % 8.42/4.48 = ( ^ [A6: set @ A,B6: set @ A] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) % 8.42/4.48 & ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_469_subset__trans,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.48 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 ) % 8.42/4.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_470_Collect__mono,axiom, % 8.42/4.48 ! [A: $tType,P: A > $o,Q: A > $o] : % 8.42/4.48 ( ! [X5: A] : % 8.42/4.48 ( ( P @ X5 ) % 8.42/4.48 => ( Q @ X5 ) ) % 8.42/4.48 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_471_subset__refl,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_472_subset__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) ) % 8.42/4.48 = ( ^ [A6: set @ A,B6: set @ A] : % 8.42/4.48 ! [T3: A] : % 8.42/4.48 ( ( member @ A @ T3 @ A6 ) % 8.42/4.48 => ( member @ A @ T3 @ B6 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_473_equalityD2,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.42/4.48 ( ( A3 = B3 ) % 8.42/4.48 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_474_equalityD1,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.42/4.48 ( ( A3 = B3 ) % 8.42/4.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_475_subset__eq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) ) % 8.42/4.48 = ( ^ [A6: set @ A,B6: set @ A] : % 8.42/4.48 ! [X: A] : % 8.42/4.48 ( ( member @ A @ X @ A6 ) % 8.42/4.48 => ( member @ A @ X @ B6 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_476_equalityE,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.42/4.48 ( ( A3 = B3 ) % 8.42/4.48 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.48 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_477_subsetD,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.48 => ( ( member @ A @ C2 @ A3 ) % 8.42/4.48 => ( member @ A @ C2 @ B3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_478_in__mono,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A,X3: A] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.48 => ( ( member @ A @ X3 @ A3 ) % 8.42/4.48 => ( member @ A @ X3 @ B3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_479_bot__set__def,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( bot_bot @ ( set @ A ) ) % 8.42/4.48 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_480_bot__nat__def,axiom, % 8.42/4.48 ( ( bot_bot @ nat ) % 8.42/4.48 = ( zero_zero @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_481_combine__options__cases,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,X3: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] : % 8.42/4.48 ( ( ( X3 % 8.42/4.48 = ( none @ A ) ) % 8.42/4.48 => ( P @ X3 @ Y ) ) % 8.42/4.48 => ( ( ( Y % 8.42/4.48 = ( none @ B ) ) % 8.42/4.48 => ( P @ X3 @ Y ) ) % 8.42/4.48 => ( ! [A5: A,B5: B] : % 8.42/4.48 ( ( X3 % 8.42/4.48 = ( some @ A @ A5 ) ) % 8.42/4.48 => ( ( Y % 8.42/4.48 = ( some @ B @ B5 ) ) % 8.42/4.48 => ( P @ X3 @ Y ) ) ) % 8.42/4.48 => ( P @ X3 @ Y ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_482_split__option__all,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ^ [P2: ( option @ A ) > $o] : % 8.42/4.48 ! [X7: option @ A] : ( P2 @ X7 ) ) % 8.42/4.48 = ( ^ [P3: ( option @ A ) > $o] : % 8.42/4.48 ( ( P3 @ ( none @ A ) ) % 8.42/4.48 & ! [X: A] : ( P3 @ ( some @ A @ X ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_483_split__option__ex,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( ^ [P2: ( option @ A ) > $o] : % 8.42/4.48 ? [X7: option @ A] : ( P2 @ X7 ) ) % 8.42/4.48 = ( ^ [P3: ( option @ A ) > $o] : % 8.42/4.48 ( ( P3 @ ( none @ A ) ) % 8.42/4.48 | ? [X: A] : ( P3 @ ( some @ A @ X ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_484_option_Oexhaust,axiom, % 8.42/4.48 ! [A: $tType,Y: option @ A] : % 8.42/4.48 ( ( Y % 8.42/4.48 != ( none @ A ) ) % 8.42/4.48 => ~ ! [X23: A] : % 8.42/4.48 ( Y % 8.42/4.48 != ( some @ A @ X23 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_485_option_OdiscI,axiom, % 8.42/4.48 ! [A: $tType,Option: option @ A,X22: A] : % 8.42/4.48 ( ( Option % 8.42/4.48 = ( some @ A @ X22 ) ) % 8.42/4.48 => ( Option % 8.42/4.48 != ( none @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_486_option_Odistinct_I1_J,axiom, % 8.42/4.48 ! [A: $tType,X22: A] : % 8.42/4.48 ( ( none @ A ) % 8.42/4.48 != ( some @ A @ X22 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_487_vebt__maxt_Osimps_I3_J,axiom, % 8.42/4.48 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] : % 8.42/4.48 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) ) % 8.42/4.48 = ( some @ nat @ Ma ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_488_vebt__mint_Osimps_I3_J,axiom, % 8.42/4.48 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT] : % 8.42/4.48 ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) ) % 8.42/4.48 = ( some @ nat @ Mi ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_489_option_Oexpand,axiom, % 8.42/4.48 ! [A: $tType,Option: option @ A,Option2: option @ A] : % 8.42/4.48 ( ( ( Option % 8.42/4.48 = ( none @ A ) ) % 8.42/4.48 = ( Option2 % 8.42/4.48 = ( none @ A ) ) ) % 8.42/4.48 => ( ( ( Option % 8.42/4.48 != ( none @ A ) ) % 8.42/4.48 => ( ( Option2 % 8.42/4.48 != ( none @ A ) ) % 8.42/4.48 => ( ( the2 @ A @ Option ) % 8.42/4.48 = ( the2 @ A @ Option2 ) ) ) ) % 8.42/4.48 => ( Option = Option2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_490_option_Osize_I3_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( size_size @ ( option @ A ) @ ( none @ A ) ) % 8.42/4.48 = ( suc @ ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_491_option_Osize__gen_I1_J,axiom, % 8.42/4.48 ! [A: $tType,X3: A > nat] : % 8.42/4.48 ( ( size_option @ A @ X3 @ ( none @ A ) ) % 8.42/4.48 = ( suc @ ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_492_option_Oexhaust__sel,axiom, % 8.42/4.48 ! [A: $tType,Option: option @ A] : % 8.42/4.48 ( ( Option % 8.42/4.48 != ( none @ A ) ) % 8.42/4.48 => ( Option % 8.42/4.48 = ( some @ A @ ( the2 @ A @ Option ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_493_option_Osize__neq,axiom, % 8.42/4.48 ! [A: $tType,X3: option @ A] : % 8.42/4.48 ( ( size_size @ ( option @ A ) @ X3 ) % 8.42/4.48 != ( zero_zero @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_494_linorder__neqE__linordered__idom,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_idom @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( X3 != Y ) % 8.42/4.48 => ( ~ ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ( ord_less @ A @ Y @ X3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_495_linordered__field__no__ub,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X2: A] : % 8.42/4.48 ? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_496_linordered__field__no__lb,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X2: A] : % 8.42/4.48 ? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_497_not__psubset__empty,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A] : % 8.42/4.48 ~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_498_ex__in__conv,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A] : % 8.42/4.48 ( ( ? [X: A] : ( member @ A @ X @ A3 ) ) % 8.42/4.48 = ( A3 % 8.42/4.48 != ( bot_bot @ ( set @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_499_equals0I,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A] : % 8.42/4.48 ( ! [Y4: A] : % 8.42/4.48 ~ ( member @ A @ Y4 @ A3 ) % 8.42/4.48 => ( A3 % 8.42/4.48 = ( bot_bot @ ( set @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_500_equals0D,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,A2: A] : % 8.42/4.48 ( ( A3 % 8.42/4.48 = ( bot_bot @ ( set @ A ) ) ) % 8.42/4.48 => ~ ( member @ A @ A2 @ A3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_501_emptyE,axiom, % 8.42/4.48 ! [A: $tType,A2: A] : % 8.42/4.48 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_502_ex__has__least__nat,axiom, % 8.42/4.48 ! [A: $tType,P: A > $o,K: A,M: A > nat] : % 8.42/4.48 ( ( P @ K ) % 8.42/4.48 => ? [X5: A] : % 8.42/4.48 ( ( P @ X5 ) % 8.42/4.48 & ! [Y5: A] : % 8.42/4.48 ( ( P @ Y5 ) % 8.42/4.48 => ( ord_less_eq @ nat @ ( M @ X5 ) @ ( M @ Y5 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_503_Lattices__Big_Oex__has__greatest__nat,axiom, % 8.42/4.48 ! [A: $tType,P: A > $o,K: A,F2: A > nat,B2: nat] : % 8.42/4.48 ( ( P @ K ) % 8.42/4.48 => ( ! [Y4: A] : % 8.42/4.48 ( ( P @ Y4 ) % 8.42/4.48 => ( ord_less @ nat @ ( F2 @ Y4 ) @ B2 ) ) % 8.42/4.48 => ? [X5: A] : % 8.42/4.48 ( ( P @ X5 ) % 8.42/4.48 & ! [Y5: A] : % 8.42/4.48 ( ( P @ Y5 ) % 8.42/4.48 => ( ord_less_eq @ nat @ ( F2 @ Y5 ) @ ( F2 @ X5 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_504_option_Osize_I4_J,axiom, % 8.42/4.48 ! [A: $tType,X22: A] : % 8.42/4.48 ( ( size_size @ ( option @ A ) @ ( some @ A @ X22 ) ) % 8.42/4.48 = ( suc @ ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_505_VEBT__internal_Omembermima_Osimps_I3_J,axiom, % 8.42/4.48 ! [Mi: nat,Ma: nat,Va: list @ vEBT_VEBT,Vb: vEBT_VEBT,X3: nat] : % 8.42/4.48 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ Va @ Vb ) @ X3 ) % 8.42/4.48 = ( ( X3 = Mi ) % 8.42/4.48 | ( X3 = Ma ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_506_vebt__member_Osimps_I4_J,axiom, % 8.42/4.48 ! [V2: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] : % 8.42/4.48 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X3 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_507_vebt__member_Osimps_I3_J,axiom, % 8.42/4.48 ! [V2: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X3: nat] : % 8.42/4.48 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X3 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_508_VEBT__internal_OminNull_Osimps_I5_J,axiom, % 8.42/4.48 ! [Uz: product_prod @ nat @ nat,Va: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] : % 8.42/4.48 ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va @ Vb @ Vc ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_509_VEBT__internal_Onaive__member_Osimps_I2_J,axiom, % 8.42/4.48 ! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : % 8.42/4.48 ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Ux ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_510_VEBT_Oinject_I1_J,axiom, % 8.42/4.48 ! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,Y11: option @ ( product_prod @ nat @ nat ),Y12: nat,Y13: list @ vEBT_VEBT,Y14: vEBT_VEBT] : % 8.42/4.48 ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) % 8.42/4.48 = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) ) % 8.42/4.48 = ( ( X11 = Y11 ) % 8.42/4.48 & ( X12 = Y12 ) % 8.42/4.48 & ( X13 = Y13 ) % 8.42/4.48 & ( X14 = Y14 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_511_old_Oprod_Oinject,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] : % 8.42/4.48 ( ( ( product_Pair @ A @ B @ A2 @ B2 ) % 8.42/4.48 = ( product_Pair @ A @ B @ A7 @ B7 ) ) % 8.42/4.48 = ( ( A2 = A7 ) % 8.42/4.48 & ( B2 = B7 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_512_prod_Oinject,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] : % 8.42/4.48 ( ( ( product_Pair @ A @ B @ X1 @ X22 ) % 8.42/4.48 = ( product_Pair @ A @ B @ Y1 @ Y22 ) ) % 8.42/4.48 = ( ( X1 = Y1 ) % 8.42/4.48 & ( X22 = Y22 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_513_nth__enumerate__eq,axiom, % 8.42/4.48 ! [A: $tType,M: nat,Xs: list @ A,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 => ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) @ M ) % 8.42/4.48 = ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N2 @ M ) @ ( nth @ A @ Xs @ M ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_514_set__vebt__finite,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( finite_finite2 @ nat @ ( vEBT_VEBT_set_vebt @ T2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_515_Euclid__induct,axiom, % 8.42/4.48 ! [P: nat > nat > $o,A2: nat,B2: nat] : % 8.42/4.48 ( ! [A5: nat,B5: nat] : % 8.42/4.48 ( ( P @ A5 @ B5 ) % 8.42/4.48 = ( P @ B5 @ A5 ) ) % 8.42/4.48 => ( ! [A5: nat] : ( P @ A5 @ ( zero_zero @ nat ) ) % 8.42/4.48 => ( ! [A5: nat,B5: nat] : % 8.42/4.48 ( ( P @ A5 @ B5 ) % 8.42/4.48 => ( P @ A5 @ ( plus_plus @ nat @ A5 @ B5 ) ) ) % 8.42/4.48 => ( P @ A2 @ B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_516_List_Ofinite__set,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_517_length__enumerate,axiom, % 8.42/4.48 ! [A: $tType,N2: nat,Xs: list @ A] : % 8.42/4.48 ( ( size_size @ ( list @ ( product_prod @ nat @ A ) ) @ ( enumerate @ A @ N2 @ Xs ) ) % 8.42/4.48 = ( size_size @ ( list @ A ) @ Xs ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_518_psubset__trans,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 8.42/4.48 ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.48 => ( ( ord_less @ ( set @ A ) @ B3 @ C5 ) % 8.42/4.48 => ( ord_less @ ( set @ A ) @ A3 @ C5 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_519_psubsetD,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] : % 8.42/4.48 ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.48 => ( ( member @ A @ C2 @ A3 ) % 8.42/4.48 => ( member @ A @ C2 @ B3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_520_VEBT__internal_Ooption__shift_Ocases,axiom, % 8.42/4.48 ! [A: $tType,X3: product_prod @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] : % 8.42/4.48 ( ! [Uu2: A > A > A,Uv2: option @ A] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) ) % 8.42/4.48 => ( ! [Uw2: A > A > A,V3: A] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) ) % 8.42/4.48 => ~ ! [F4: A > A > A,A5: A,B5: A] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ ( A > A > A ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F4 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ A5 ) @ ( some @ A @ B5 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_521_VEBT__internal_Ooption__comp__shift_Ocases,axiom, % 8.42/4.48 ! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) )] : % 8.42/4.48 ( ! [Uu2: A > A > $o,Uv2: option @ A] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uu2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( none @ A ) @ Uv2 ) ) ) % 8.42/4.48 => ( ! [Uw2: A > A > $o,V3: A] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ Uw2 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ V3 ) @ ( none @ A ) ) ) ) % 8.42/4.48 => ~ ! [F4: A > A > $o,X5: A,Y4: A] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( option @ A ) @ ( option @ A ) ) @ F4 @ ( product_Pair @ ( option @ A ) @ ( option @ A ) @ ( some @ A @ X5 ) @ ( some @ A @ Y4 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_522_finite__list,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A] : % 8.42/4.48 ( ( finite_finite2 @ A @ A3 ) % 8.42/4.48 => ? [Xs3: list @ A] : % 8.42/4.48 ( ( set2 @ A @ Xs3 ) % 8.42/4.48 = A3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_523_vebt__member_Osimps_I2_J,axiom, % 8.42/4.48 ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X3: nat] : % 8.42/4.48 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X3 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_524_VEBT__internal_OminNull_Osimps_I4_J,axiom, % 8.42/4.48 ! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_525_ex__min__if__finite,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [S3: set @ A] : % 8.42/4.48 ( ( finite_finite2 @ A @ S3 ) % 8.42/4.48 => ( ( S3 % 8.42/4.48 != ( bot_bot @ ( set @ A ) ) ) % 8.42/4.48 => ? [X5: A] : % 8.42/4.48 ( ( member @ A @ X5 @ S3 ) % 8.42/4.48 & ~ ? [Xa: A] : % 8.42/4.48 ( ( member @ A @ Xa @ S3 ) % 8.42/4.48 & ( ord_less @ A @ Xa @ X5 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_526_infinite__growing,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [X8: set @ A] : % 8.42/4.48 ( ( X8 % 8.42/4.48 != ( bot_bot @ ( set @ A ) ) ) % 8.42/4.48 => ( ! [X5: A] : % 8.42/4.48 ( ( member @ A @ X5 @ X8 ) % 8.42/4.48 => ? [Xa: A] : % 8.42/4.48 ( ( member @ A @ Xa @ X8 ) % 8.42/4.48 & ( ord_less @ A @ X5 @ Xa ) ) ) % 8.42/4.48 => ~ ( finite_finite2 @ A @ X8 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_527_VEBT__internal_Omembermima_Osimps_I2_J,axiom, % 8.42/4.48 ! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : % 8.42/4.48 ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Uz ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_528_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom, % 8.42/4.48 ! [A: $tType,F2: A > A > A,A2: A,B2: A] : % 8.42/4.48 ( ( vEBT_V2048590022279873568_shift @ A @ F2 @ ( some @ A @ A2 ) @ ( some @ A @ B2 ) ) % 8.42/4.48 = ( some @ A @ ( F2 @ A2 @ B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_529_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom, % 8.42/4.48 ! [A: $tType,Uu: A > A > A,Uv: option @ A] : % 8.42/4.48 ( ( vEBT_V2048590022279873568_shift @ A @ Uu @ ( none @ A ) @ Uv ) % 8.42/4.48 = ( none @ A ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_530_old_Oprod_Oexhaust,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] : % 8.42/4.48 ~ ! [A5: A,B5: B] : % 8.42/4.48 ( Y % 8.42/4.48 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_531_surj__pair,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,P4: product_prod @ A @ B] : % 8.42/4.48 ? [X5: A,Y4: B] : % 8.42/4.48 ( P4 % 8.42/4.48 = ( product_Pair @ A @ B @ X5 @ Y4 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_532_prod__cases,axiom, % 8.42/4.48 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] : % 8.42/4.48 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) ) % 8.42/4.48 => ( P @ P4 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_533_Pair__inject,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] : % 8.42/4.48 ( ( ( product_Pair @ A @ B @ A2 @ B2 ) % 8.42/4.48 = ( product_Pair @ A @ B @ A7 @ B7 ) ) % 8.42/4.48 => ~ ( ( A2 = A7 ) % 8.42/4.48 => ( B2 != B7 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_534_prod__cases3,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] : % 8.42/4.48 ~ ! [A5: A,B5: B,C3: C] : % 8.42/4.48 ( Y % 8.42/4.48 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_535_prod__cases4,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] : % 8.42/4.48 ~ ! [A5: A,B5: B,C3: C,D3: D] : % 8.42/4.48 ( Y % 8.42/4.48 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_536_prod__cases5,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] : % 8.42/4.48 ~ ! [A5: A,B5: B,C3: C,D3: D,E: E2] : % 8.42/4.48 ( Y % 8.42/4.48 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D3 @ E ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_537_prod__cases6,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) )] : % 8.42/4.48 ~ ! [A5: A,B5: B,C3: C,D3: D,E: E2,F4: F] : % 8.42/4.48 ( Y % 8.42/4.48 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F ) @ D3 @ ( product_Pair @ E2 @ F @ E @ F4 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_538_prod__cases7,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) )] : % 8.42/4.48 ~ ! [A5: A,B5: B,C3: C,D3: D,E: E2,F4: F,G4: G3] : % 8.42/4.48 ( Y % 8.42/4.48 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F @ G3 ) @ E @ ( product_Pair @ F @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_539_prod__induct3,axiom, % 8.42/4.48 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] : % 8.42/4.48 ( ! [A5: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) % 8.42/4.48 => ( P @ X3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_540_prod__induct4,axiom, % 8.42/4.48 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] : % 8.42/4.48 ( ! [A5: A,B5: B,C3: C,D3: 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 @ C3 @ D3 ) ) ) ) % 8.42/4.48 => ( P @ X3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_541_prod__induct5,axiom, % 8.42/4.48 ! [E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] : % 8.42/4.48 ( ! [A5: A,B5: B,C3: C,D3: D,E: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D3 @ E ) ) ) ) ) % 8.42/4.48 => ( P @ X3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_542_prod__induct6,axiom, % 8.42/4.48 ! [F: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) )] : % 8.42/4.48 ( ! [A5: A,B5: B,C3: C,D3: D,E: E2,F4: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F ) @ D3 @ ( product_Pair @ E2 @ F @ E @ F4 ) ) ) ) ) ) % 8.42/4.48 => ( P @ X3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_543_prod__induct7,axiom, % 8.42/4.48 ! [G3: $tType,F: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) )] : % 8.42/4.48 ( ! [A5: A,B5: B,C3: C,D3: D,E: E2,F4: F,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F @ G3 ) @ E @ ( product_Pair @ F @ G3 @ F4 @ G4 ) ) ) ) ) ) ) % 8.42/4.48 => ( P @ X3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_544_vebt__maxt_Osimps_I2_J,axiom, % 8.42/4.48 ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] : % 8.42/4.48 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) % 8.42/4.48 = ( none @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_545_vebt__mint_Osimps_I2_J,axiom, % 8.42/4.48 ! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT] : % 8.42/4.48 ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) ) % 8.42/4.48 = ( none @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_546_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom, % 8.42/4.48 ! [A: $tType,Uw: A > A > A,V2: A] : % 8.42/4.48 ( ( vEBT_V2048590022279873568_shift @ A @ Uw @ ( some @ A @ V2 ) @ ( none @ A ) ) % 8.42/4.48 = ( none @ A ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_547_VEBT__internal_Ooption__shift_Oelims,axiom, % 8.42/4.48 ! [A: $tType,X3: A > A > A,Xa2: option @ A,Xb: option @ A,Y: option @ A] : % 8.42/4.48 ( ( ( vEBT_V2048590022279873568_shift @ A @ X3 @ Xa2 @ Xb ) % 8.42/4.48 = Y ) % 8.42/4.48 => ( ( ( Xa2 % 8.42/4.48 = ( none @ A ) ) % 8.42/4.48 => ( Y % 8.42/4.48 != ( none @ A ) ) ) % 8.42/4.48 => ( ( ? [V3: A] : % 8.42/4.48 ( Xa2 % 8.42/4.48 = ( some @ A @ V3 ) ) % 8.42/4.48 => ( ( Xb % 8.42/4.48 = ( none @ A ) ) % 8.42/4.48 => ( Y % 8.42/4.48 != ( none @ A ) ) ) ) % 8.42/4.48 => ~ ! [A5: A] : % 8.42/4.48 ( ( Xa2 % 8.42/4.48 = ( some @ A @ A5 ) ) % 8.42/4.48 => ! [B5: A] : % 8.42/4.48 ( ( Xb % 8.42/4.48 = ( some @ A @ B5 ) ) % 8.42/4.48 => ( Y % 8.42/4.48 != ( some @ A @ ( X3 @ A5 @ B5 ) ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_548_vebt__buildup_Ocases,axiom, % 8.42/4.48 ! [X3: nat] : % 8.42/4.48 ( ( X3 % 8.42/4.48 != ( zero_zero @ nat ) ) % 8.42/4.48 => ( ( X3 % 8.42/4.48 != ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.48 => ~ ! [Va2: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( suc @ ( suc @ Va2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_549_old_Oprod_Orec,axiom, % 8.42/4.48 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] : % 8.42/4.48 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) % 8.42/4.48 = ( F1 @ A2 @ B2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_550_finite__has__minimal,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [A3: set @ A] : % 8.42/4.48 ( ( finite_finite2 @ A @ A3 ) % 8.42/4.48 => ( ( A3 % 8.42/4.48 != ( bot_bot @ ( set @ A ) ) ) % 8.42/4.48 => ? [X5: A] : % 8.42/4.48 ( ( member @ A @ X5 @ A3 ) % 8.42/4.48 & ! [Xa: A] : % 8.42/4.48 ( ( member @ A @ Xa @ A3 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ Xa @ X5 ) % 8.42/4.48 => ( X5 = Xa ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_551_finite__has__maximal,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [A3: set @ A] : % 8.42/4.48 ( ( finite_finite2 @ A @ A3 ) % 8.42/4.48 => ( ( A3 % 8.42/4.48 != ( bot_bot @ ( set @ A ) ) ) % 8.42/4.48 => ? [X5: A] : % 8.42/4.48 ( ( member @ A @ X5 @ A3 ) % 8.42/4.48 & ! [Xa: A] : % 8.42/4.48 ( ( member @ A @ Xa @ A3 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ X5 @ Xa ) % 8.42/4.48 => ( X5 = Xa ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_552_finite__code,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( finite_finite @ A ) % 8.42/4.48 => ( ( finite_finite2 @ A ) % 8.42/4.48 = ( ^ [A6: set @ A] : $true ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_553_bot__empty__eq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( bot_bot @ ( A > $o ) ) % 8.42/4.48 = ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_554_Collect__empty__eq__bot,axiom, % 8.42/4.48 ! [A: $tType,P: A > $o] : % 8.42/4.48 ( ( ( collect @ A @ P ) % 8.42/4.48 = ( bot_bot @ ( set @ A ) ) ) % 8.42/4.48 = ( P % 8.42/4.48 = ( bot_bot @ ( A > $o ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_555_infinite__nat__iff__unbounded__le,axiom, % 8.42/4.48 ! [S3: set @ nat] : % 8.42/4.48 ( ( ~ ( finite_finite2 @ nat @ S3 ) ) % 8.42/4.48 = ( ! [M2: nat] : % 8.42/4.48 ? [N: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ M2 @ N ) % 8.42/4.48 & ( member @ nat @ N @ S3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_556_finite__nat__set__iff__bounded__le,axiom, % 8.42/4.48 ( ( finite_finite2 @ nat ) % 8.42/4.48 = ( ^ [N5: set @ nat] : % 8.42/4.48 ? [M2: nat] : % 8.42/4.48 ! [X: nat] : % 8.42/4.48 ( ( member @ nat @ X @ N5 ) % 8.42/4.48 => ( ord_less_eq @ nat @ X @ M2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_557_unbounded__k__infinite,axiom, % 8.42/4.48 ! [K: nat,S3: set @ nat] : % 8.42/4.48 ( ! [M3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ K @ M3 ) % 8.42/4.48 => ? [N6: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M3 @ N6 ) % 8.42/4.48 & ( member @ nat @ N6 @ S3 ) ) ) % 8.42/4.48 => ~ ( finite_finite2 @ nat @ S3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_558_bounded__nat__set__is__finite,axiom, % 8.42/4.48 ! [N7: set @ nat,N2: nat] : % 8.42/4.48 ( ! [X5: nat] : % 8.42/4.48 ( ( member @ nat @ X5 @ N7 ) % 8.42/4.48 => ( ord_less @ nat @ X5 @ N2 ) ) % 8.42/4.48 => ( finite_finite2 @ nat @ N7 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_559_finite__maxlen,axiom, % 8.42/4.48 ! [A: $tType,M7: set @ ( list @ A )] : % 8.42/4.48 ( ( finite_finite2 @ ( list @ A ) @ M7 ) % 8.42/4.48 => ? [N3: nat] : % 8.42/4.48 ! [X2: list @ A] : % 8.42/4.48 ( ( member @ ( list @ A ) @ X2 @ M7 ) % 8.42/4.48 => ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X2 ) @ N3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_560_subrelI,axiom, % 8.42/4.48 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] : % 8.42/4.48 ( ! [X5: A,Y4: B] : % 8.42/4.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ R2 ) % 8.42/4.48 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ S ) ) % 8.42/4.48 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_561_finite__set__choice,axiom, % 8.42/4.48 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] : % 8.42/4.48 ( ( finite_finite2 @ A @ A3 ) % 8.42/4.48 => ( ! [X5: A] : % 8.42/4.48 ( ( member @ A @ X5 @ A3 ) % 8.42/4.48 => ? [X_12: B] : ( P @ X5 @ X_12 ) ) % 8.42/4.48 => ? [F4: A > B] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( member @ A @ X2 @ A3 ) % 8.42/4.48 => ( P @ X2 @ ( F4 @ X2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_562_finite,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( finite_finite @ A ) % 8.42/4.48 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_563_bounded__Max__nat,axiom, % 8.42/4.48 ! [P: nat > $o,X3: nat,M7: nat] : % 8.42/4.48 ( ( P @ X3 ) % 8.42/4.48 => ( ! [X5: nat] : % 8.42/4.48 ( ( P @ X5 ) % 8.42/4.48 => ( ord_less_eq @ nat @ X5 @ M7 ) ) % 8.42/4.48 => ~ ! [M3: nat] : % 8.42/4.48 ( ( P @ M3 ) % 8.42/4.48 => ~ ! [X2: nat] : % 8.42/4.48 ( ( P @ X2 ) % 8.42/4.48 => ( ord_less_eq @ nat @ X2 @ M3 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_564_fold__atLeastAtMost__nat_Ocases,axiom, % 8.42/4.48 ! [A: $tType,X3: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] : % 8.42/4.48 ~ ! [F4: nat > A > A,A5: nat,B5: nat,Acc: A] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F4 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A5 @ ( product_Pair @ nat @ A @ B5 @ Acc ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_565_finite__has__minimal2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [A3: set @ A,A2: A] : % 8.42/4.48 ( ( finite_finite2 @ A @ A3 ) % 8.42/4.48 => ( ( member @ A @ A2 @ A3 ) % 8.42/4.48 => ? [X5: A] : % 8.42/4.48 ( ( member @ A @ X5 @ A3 ) % 8.42/4.48 & ( ord_less_eq @ A @ X5 @ A2 ) % 8.42/4.48 & ! [Xa: A] : % 8.42/4.48 ( ( member @ A @ Xa @ A3 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ Xa @ X5 ) % 8.42/4.48 => ( X5 = Xa ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_566_finite__has__maximal2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [A3: set @ A,A2: A] : % 8.42/4.48 ( ( finite_finite2 @ A @ A3 ) % 8.42/4.48 => ( ( member @ A @ A2 @ A3 ) % 8.42/4.48 => ? [X5: A] : % 8.42/4.48 ( ( member @ A @ X5 @ A3 ) % 8.42/4.48 & ( ord_less_eq @ A @ A2 @ X5 ) % 8.42/4.48 & ! [Xa: A] : % 8.42/4.48 ( ( member @ A @ Xa @ A3 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ X5 @ Xa ) % 8.42/4.48 => ( X5 = Xa ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_567_finite__transitivity__chain,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,R: A > A > $o] : % 8.42/4.48 ( ( finite_finite2 @ A @ A3 ) % 8.42/4.48 => ( ! [X5: A] : % 8.42/4.48 ~ ( R @ X5 @ X5 ) % 8.42/4.48 => ( ! [X5: A,Y4: A,Z3: A] : % 8.42/4.48 ( ( R @ X5 @ Y4 ) % 8.42/4.48 => ( ( R @ Y4 @ Z3 ) % 8.42/4.48 => ( R @ X5 @ Z3 ) ) ) % 8.42/4.48 => ( ! [X5: A] : % 8.42/4.48 ( ( member @ A @ X5 @ A3 ) % 8.42/4.48 => ? [Y5: A] : % 8.42/4.48 ( ( member @ A @ Y5 @ A3 ) % 8.42/4.48 & ( R @ X5 @ Y5 ) ) ) % 8.42/4.48 => ( A3 % 8.42/4.48 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_568_infinite__imp__nonempty,axiom, % 8.42/4.48 ! [A: $tType,S3: set @ A] : % 8.42/4.48 ( ~ ( finite_finite2 @ A @ S3 ) % 8.42/4.48 => ( S3 % 8.42/4.48 != ( bot_bot @ ( set @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_569_finite_OemptyI,axiom, % 8.42/4.48 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_570_rev__finite__subset,axiom, % 8.42/4.48 ! [A: $tType,B3: set @ A,A3: set @ A] : % 8.42/4.48 ( ( finite_finite2 @ A @ B3 ) % 8.42/4.48 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.48 => ( finite_finite2 @ A @ A3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_571_infinite__super,axiom, % 8.42/4.48 ! [A: $tType,S3: set @ A,T4: set @ A] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) @ S3 @ T4 ) % 8.42/4.48 => ( ~ ( finite_finite2 @ A @ S3 ) % 8.42/4.48 => ~ ( finite_finite2 @ A @ T4 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_572_finite__subset,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.48 => ( ( finite_finite2 @ A @ B3 ) % 8.42/4.48 => ( finite_finite2 @ A @ A3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_573_finite__psubset__induct,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] : % 8.42/4.48 ( ( finite_finite2 @ A @ A3 ) % 8.42/4.48 => ( ! [A8: set @ A] : % 8.42/4.48 ( ( finite_finite2 @ A @ A8 ) % 8.42/4.48 => ( ! [B8: set @ A] : % 8.42/4.48 ( ( ord_less @ ( set @ A ) @ B8 @ A8 ) % 8.42/4.48 => ( P @ B8 ) ) % 8.42/4.48 => ( P @ A8 ) ) ) % 8.42/4.48 => ( P @ A3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_574_finite__nat__set__iff__bounded,axiom, % 8.42/4.48 ( ( finite_finite2 @ nat ) % 8.42/4.48 = ( ^ [N5: set @ nat] : % 8.42/4.48 ? [M2: nat] : % 8.42/4.48 ! [X: nat] : % 8.42/4.48 ( ( member @ nat @ X @ N5 ) % 8.42/4.48 => ( ord_less @ nat @ X @ M2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_575_infinite__nat__iff__unbounded,axiom, % 8.42/4.48 ! [S3: set @ nat] : % 8.42/4.48 ( ( ~ ( finite_finite2 @ nat @ S3 ) ) % 8.42/4.48 = ( ! [M2: nat] : % 8.42/4.48 ? [N: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M2 @ N ) % 8.42/4.48 & ( member @ nat @ N @ S3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_576_arg__min__if__finite_I2_J,axiom, % 8.42/4.48 ! [B: $tType,A: $tType] : % 8.42/4.48 ( ( order @ B ) % 8.42/4.48 => ! [S3: set @ A,F2: A > B] : % 8.42/4.48 ( ( finite_finite2 @ A @ S3 ) % 8.42/4.48 => ( ( S3 % 8.42/4.48 != ( bot_bot @ ( set @ A ) ) ) % 8.42/4.48 => ~ ? [X2: A] : % 8.42/4.48 ( ( member @ A @ X2 @ S3 ) % 8.42/4.48 & ( ord_less @ B @ ( F2 @ X2 ) @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S3 ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_577_arg__min__least,axiom, % 8.42/4.48 ! [B: $tType,A: $tType] : % 8.42/4.48 ( ( linorder @ B ) % 8.42/4.48 => ! [S3: set @ A,Y: A,F2: A > B] : % 8.42/4.48 ( ( finite_finite2 @ A @ S3 ) % 8.42/4.48 => ( ( S3 % 8.42/4.48 != ( bot_bot @ ( set @ A ) ) ) % 8.42/4.48 => ( ( member @ A @ Y @ S3 ) % 8.42/4.48 => ( ord_less_eq @ B @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S3 ) ) @ ( F2 @ Y ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_578_nat__descend__induct,axiom, % 8.42/4.48 ! [N2: nat,P: nat > $o,M: nat] : % 8.42/4.48 ( ! [K2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ N2 @ K2 ) % 8.42/4.48 => ( P @ K2 ) ) % 8.42/4.48 => ( ! [K2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ K2 @ N2 ) % 8.42/4.48 => ( ! [I5: nat] : % 8.42/4.48 ( ( ord_less @ nat @ K2 @ I5 ) % 8.42/4.48 => ( P @ I5 ) ) % 8.42/4.48 => ( P @ K2 ) ) ) % 8.42/4.48 => ( P @ M ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_579_subset__emptyI,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ A] : % 8.42/4.48 ( ! [X5: A] : % 8.42/4.48 ~ ( member @ A @ X5 @ A3 ) % 8.42/4.48 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_580_exists__least__lemma,axiom, % 8.42/4.48 ! [P: nat > $o] : % 8.42/4.48 ( ~ ( P @ ( zero_zero @ nat ) ) % 8.42/4.48 => ( ? [X_12: nat] : ( P @ X_12 ) % 8.42/4.48 => ? [N3: nat] : % 8.42/4.48 ( ~ ( P @ N3 ) % 8.42/4.48 & ( P @ ( suc @ N3 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_581_dependent__nat__choice,axiom, % 8.42/4.48 ! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] : % 8.42/4.48 ( ? [X_12: A] : ( P @ ( zero_zero @ nat ) @ X_12 ) % 8.42/4.48 => ( ! [X5: A,N3: nat] : % 8.42/4.48 ( ( P @ N3 @ X5 ) % 8.42/4.48 => ? [Y5: A] : % 8.42/4.48 ( ( P @ ( suc @ N3 ) @ Y5 ) % 8.42/4.48 & ( Q @ N3 @ X5 @ Y5 ) ) ) % 8.42/4.48 => ? [F4: nat > A] : % 8.42/4.48 ! [N6: nat] : % 8.42/4.48 ( ( P @ N6 @ ( F4 @ N6 ) ) % 8.42/4.48 & ( Q @ N6 @ ( F4 @ N6 ) @ ( F4 @ ( suc @ N6 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_582_list__decode_Ocases,axiom, % 8.42/4.48 ! [X3: nat] : % 8.42/4.48 ( ( X3 % 8.42/4.48 != ( zero_zero @ nat ) ) % 8.42/4.48 => ~ ! [N3: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( suc @ N3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_583_verit__sum__simplify,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( cancel1802427076303600483id_add @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 = A2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_584_add__0__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( semiri1453513574482234551roduct @ A ) % 8.42/4.48 => ! [B2: A,A2: A] : % 8.42/4.48 ( ( B2 % 8.42/4.48 = ( plus_plus @ A @ B2 @ A2 ) ) % 8.42/4.48 = ( A2 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_585_field__lbound__gt__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [D1: A,D22: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ D1 ) % 8.42/4.48 => ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 ) % 8.42/4.48 => ? [E: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ E ) % 8.42/4.48 & ( ord_less @ A @ E @ D1 ) % 8.42/4.48 & ( ord_less @ A @ E @ D22 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_586_less__by__empty,axiom, % 8.42/4.48 ! [A: $tType,A3: set @ ( product_prod @ A @ A ),B3: set @ ( product_prod @ A @ A )] : % 8.42/4.48 ( ( A3 % 8.42/4.48 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) % 8.42/4.48 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_587_verit__comp__simplify1_I2_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_588_verit__la__disequality,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( A2 = B2 ) % 8.42/4.48 | ~ ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 | ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_589_verit__comp__simplify1_I1_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( order @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ~ ( ord_less @ A @ A2 @ A2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_590_ssubst__Pair__rhs,axiom, % 8.42/4.48 ! [B: $tType,A: $tType,R2: A,S: B,R: set @ ( product_prod @ A @ B ),S4: B] : % 8.42/4.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S ) @ R ) % 8.42/4.48 => ( ( S4 = S ) % 8.42/4.48 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S4 ) @ R ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_591_prod__decode__aux_Ocases,axiom, % 8.42/4.48 ! [X3: product_prod @ nat @ nat] : % 8.42/4.48 ~ ! [K2: nat,M3: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ nat @ nat @ K2 @ M3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_592_arg__min__if__finite_I1_J,axiom, % 8.42/4.48 ! [B: $tType,A: $tType] : % 8.42/4.48 ( ( order @ B ) % 8.42/4.48 => ! [S3: set @ A,F2: A > B] : % 8.42/4.48 ( ( finite_finite2 @ A @ S3 ) % 8.42/4.48 => ( ( S3 % 8.42/4.48 != ( bot_bot @ ( set @ A ) ) ) % 8.42/4.48 => ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S3 ) @ S3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_593_verit__comp__simplify1_I3_J,axiom, % 8.42/4.48 ! [B: $tType] : % 8.42/4.48 ( ( linorder @ B ) % 8.42/4.48 => ! [B7: B,A7: B] : % 8.42/4.48 ( ( ~ ( ord_less_eq @ B @ B7 @ A7 ) ) % 8.42/4.48 = ( ord_less @ B @ A7 @ B7 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_594_triangle__Suc,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( nat_triangle @ ( suc @ N2 ) ) % 8.42/4.48 = ( plus_plus @ nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_595_less__numeral__extra_I3_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord181362715937106298miring @ A ) % 8.42/4.48 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_596_minf_I8_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [T2: A] : % 8.42/4.48 ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ X2 @ Z3 ) % 8.42/4.48 => ~ ( ord_less_eq @ A @ T2 @ X2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_597_minf_I6_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [T2: A] : % 8.42/4.48 ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ X2 @ Z3 ) % 8.42/4.48 => ( ord_less_eq @ A @ X2 @ T2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_598_pinf_I8_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [T2: A] : % 8.42/4.48 ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ Z3 @ X2 ) % 8.42/4.48 => ( ord_less_eq @ A @ T2 @ X2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_599_pinf_I6_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [T2: A] : % 8.42/4.48 ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ Z3 @ X2 ) % 8.42/4.48 => ~ ( ord_less_eq @ A @ X2 @ T2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_600_complete__interval,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( condit6923001295902523014norder @ A ) % 8.42/4.48 => ! [A2: A,B2: A,P: A > $o] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( P @ A2 ) % 8.42/4.48 => ( ~ ( P @ B2 ) % 8.42/4.48 => ? [C3: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ C3 ) % 8.42/4.48 & ( ord_less_eq @ A @ C3 @ B2 ) % 8.42/4.48 & ! [X2: A] : % 8.42/4.48 ( ( ( ord_less_eq @ A @ A2 @ X2 ) % 8.42/4.48 & ( ord_less @ A @ X2 @ C3 ) ) % 8.42/4.48 => ( P @ X2 ) ) % 8.42/4.48 & ! [D4: A] : % 8.42/4.48 ( ! [X5: A] : % 8.42/4.48 ( ( ( ord_less_eq @ A @ A2 @ X5 ) % 8.42/4.48 & ( ord_less @ A @ X5 @ D4 ) ) % 8.42/4.48 => ( P @ X5 ) ) % 8.42/4.48 => ( ord_less_eq @ A @ D4 @ C3 ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_601_le__numeral__extra_I3_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord181362715937106298miring @ A ) % 8.42/4.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_602_find__Some__iff,axiom, % 8.42/4.48 ! [A: $tType,P: A > $o,Xs: list @ A,X3: A] : % 8.42/4.48 ( ( ( find @ A @ P @ Xs ) % 8.42/4.48 = ( some @ A @ X3 ) ) % 8.42/4.48 = ( ? [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 & ( P @ ( nth @ A @ Xs @ I3 ) ) % 8.42/4.48 & ( X3 % 8.42/4.48 = ( nth @ A @ Xs @ I3 ) ) % 8.42/4.48 & ! [J3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ J3 @ I3 ) % 8.42/4.48 => ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_603_find__Some__iff2,axiom, % 8.42/4.48 ! [A: $tType,X3: A,P: A > $o,Xs: list @ A] : % 8.42/4.48 ( ( ( some @ A @ X3 ) % 8.42/4.48 = ( find @ A @ P @ Xs ) ) % 8.42/4.48 = ( ? [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 & ( P @ ( nth @ A @ Xs @ I3 ) ) % 8.42/4.48 & ( X3 % 8.42/4.48 = ( nth @ A @ Xs @ I3 ) ) % 8.42/4.48 & ! [J3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ J3 @ I3 ) % 8.42/4.48 => ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_604_triangle__0,axiom, % 8.42/4.48 ( ( nat_triangle @ ( zero_zero @ nat ) ) % 8.42/4.48 = ( zero_zero @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_605_find__cong,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] : % 8.42/4.48 ( ( Xs = Ys ) % 8.42/4.48 => ( ! [X5: A] : % 8.42/4.48 ( ( member @ A @ X5 @ ( set2 @ A @ Ys ) ) % 8.42/4.48 => ( ( P @ X5 ) % 8.42/4.48 = ( Q @ X5 ) ) ) % 8.42/4.48 => ( ( find @ A @ P @ Xs ) % 8.42/4.48 = ( find @ A @ Q @ Ys ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_606_ex__gt__or__lt,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( condit5016429287641298734tinuum @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ? [B5: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B5 ) % 8.42/4.48 | ( ord_less @ A @ B5 @ A2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_607_pinf_I1_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] : % 8.42/4.48 ( ? [Z4: A] : % 8.42/4.48 ! [X5: A] : % 8.42/4.48 ( ( ord_less @ A @ Z4 @ X5 ) % 8.42/4.48 => ( ( P @ X5 ) % 8.42/4.48 = ( P5 @ X5 ) ) ) % 8.42/4.48 => ( ? [Z4: A] : % 8.42/4.48 ! [X5: A] : % 8.42/4.48 ( ( ord_less @ A @ Z4 @ X5 ) % 8.42/4.48 => ( ( Q @ X5 ) % 8.42/4.48 = ( Q3 @ X5 ) ) ) % 8.42/4.48 => ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ Z3 @ X2 ) % 8.42/4.48 => ( ( ( P @ X2 ) % 8.42/4.48 & ( Q @ X2 ) ) % 8.42/4.48 = ( ( P5 @ X2 ) % 8.42/4.48 & ( Q3 @ X2 ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_608_pinf_I2_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] : % 8.42/4.48 ( ? [Z4: A] : % 8.42/4.48 ! [X5: A] : % 8.42/4.48 ( ( ord_less @ A @ Z4 @ X5 ) % 8.42/4.48 => ( ( P @ X5 ) % 8.42/4.48 = ( P5 @ X5 ) ) ) % 8.42/4.48 => ( ? [Z4: A] : % 8.42/4.48 ! [X5: A] : % 8.42/4.48 ( ( ord_less @ A @ Z4 @ X5 ) % 8.42/4.48 => ( ( Q @ X5 ) % 8.42/4.48 = ( Q3 @ X5 ) ) ) % 8.42/4.48 => ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ Z3 @ X2 ) % 8.42/4.48 => ( ( ( P @ X2 ) % 8.42/4.48 | ( Q @ X2 ) ) % 8.42/4.48 = ( ( P5 @ X2 ) % 8.42/4.48 | ( Q3 @ X2 ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_609_pinf_I3_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [T2: A] : % 8.42/4.48 ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ Z3 @ X2 ) % 8.42/4.48 => ( X2 != T2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_610_pinf_I4_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [T2: A] : % 8.42/4.48 ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ Z3 @ X2 ) % 8.42/4.48 => ( X2 != T2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_611_pinf_I5_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [T2: A] : % 8.42/4.48 ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ Z3 @ X2 ) % 8.42/4.48 => ~ ( ord_less @ A @ X2 @ T2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_612_pinf_I7_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [T2: A] : % 8.42/4.48 ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ Z3 @ X2 ) % 8.42/4.48 => ( ord_less @ A @ T2 @ X2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_613_pinf_I11_J,axiom, % 8.42/4.48 ! [C: $tType,D: $tType] : % 8.42/4.48 ( ( ord @ C ) % 8.42/4.48 => ! [F5: D] : % 8.42/4.48 ? [Z3: C] : % 8.42/4.48 ! [X2: C] : % 8.42/4.48 ( ( ord_less @ C @ Z3 @ X2 ) % 8.42/4.48 => ( F5 = F5 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_614_minf_I1_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] : % 8.42/4.48 ( ? [Z4: A] : % 8.42/4.48 ! [X5: A] : % 8.42/4.48 ( ( ord_less @ A @ X5 @ Z4 ) % 8.42/4.48 => ( ( P @ X5 ) % 8.42/4.48 = ( P5 @ X5 ) ) ) % 8.42/4.48 => ( ? [Z4: A] : % 8.42/4.48 ! [X5: A] : % 8.42/4.48 ( ( ord_less @ A @ X5 @ Z4 ) % 8.42/4.48 => ( ( Q @ X5 ) % 8.42/4.48 = ( Q3 @ X5 ) ) ) % 8.42/4.48 => ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ X2 @ Z3 ) % 8.42/4.48 => ( ( ( P @ X2 ) % 8.42/4.48 & ( Q @ X2 ) ) % 8.42/4.48 = ( ( P5 @ X2 ) % 8.42/4.48 & ( Q3 @ X2 ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_615_minf_I2_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] : % 8.42/4.48 ( ? [Z4: A] : % 8.42/4.48 ! [X5: A] : % 8.42/4.48 ( ( ord_less @ A @ X5 @ Z4 ) % 8.42/4.48 => ( ( P @ X5 ) % 8.42/4.48 = ( P5 @ X5 ) ) ) % 8.42/4.48 => ( ? [Z4: A] : % 8.42/4.48 ! [X5: A] : % 8.42/4.48 ( ( ord_less @ A @ X5 @ Z4 ) % 8.42/4.48 => ( ( Q @ X5 ) % 8.42/4.48 = ( Q3 @ X5 ) ) ) % 8.42/4.48 => ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ X2 @ Z3 ) % 8.42/4.48 => ( ( ( P @ X2 ) % 8.42/4.48 | ( Q @ X2 ) ) % 8.42/4.48 = ( ( P5 @ X2 ) % 8.42/4.48 | ( Q3 @ X2 ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_616_minf_I3_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [T2: A] : % 8.42/4.48 ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ X2 @ Z3 ) % 8.42/4.48 => ( X2 != T2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_617_minf_I4_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [T2: A] : % 8.42/4.48 ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ X2 @ Z3 ) % 8.42/4.48 => ( X2 != T2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_618_minf_I5_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [T2: A] : % 8.42/4.48 ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ X2 @ Z3 ) % 8.42/4.48 => ( ord_less @ A @ X2 @ T2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_619_minf_I7_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linorder @ A ) % 8.42/4.48 => ! [T2: A] : % 8.42/4.48 ? [Z3: A] : % 8.42/4.48 ! [X2: A] : % 8.42/4.48 ( ( ord_less @ A @ X2 @ Z3 ) % 8.42/4.48 => ~ ( ord_less @ A @ T2 @ X2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_620_minf_I11_J,axiom, % 8.42/4.48 ! [C: $tType,D: $tType] : % 8.42/4.48 ( ( ord @ C ) % 8.42/4.48 => ! [F5: D] : % 8.42/4.48 ? [Z3: C] : % 8.42/4.48 ! [X2: C] : % 8.42/4.48 ( ( ord_less @ C @ X2 @ Z3 ) % 8.42/4.48 => ( F5 = F5 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_621_is__num__normalize_I1_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( neg_numeral @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.48 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_622_find__None__iff2,axiom, % 8.42/4.48 ! [A: $tType,P: A > $o,Xs: list @ A] : % 8.42/4.48 ( ( ( none @ A ) % 8.42/4.48 = ( find @ A @ P @ Xs ) ) % 8.42/4.48 = ( ~ ? [X: A] : % 8.42/4.48 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 8.42/4.48 & ( P @ X ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_623_find__None__iff,axiom, % 8.42/4.48 ! [A: $tType,P: A > $o,Xs: list @ A] : % 8.42/4.48 ( ( ( find @ A @ P @ Xs ) % 8.42/4.48 = ( none @ A ) ) % 8.42/4.48 = ( ~ ? [X: A] : % 8.42/4.48 ( ( member @ A @ X @ ( set2 @ A @ Xs ) ) % 8.42/4.48 & ( P @ X ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_624_nth__zip,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys: list @ B] : % 8.42/4.48 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) ) % 8.42/4.48 => ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I ) % 8.42/4.48 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys @ I ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_625__C5_C_I10_J,axiom, % 8.42/4.48 ( ( mi != ma ) % 8.42/4.48 => ! [I5: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) ) % 8.42/4.48 => ( ( ( ( vEBT_VEBT_high @ ma @ n ) % 8.42/4.48 = I5 ) % 8.42/4.48 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I5 ) @ ( vEBT_VEBT_low @ ma @ n ) ) ) % 8.42/4.48 & ! [X2: nat] : % 8.42/4.48 ( ( ( ( vEBT_VEBT_high @ X2 @ n ) % 8.42/4.48 = I5 ) % 8.42/4.48 & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I5 ) @ ( vEBT_VEBT_low @ X2 @ n ) ) ) % 8.42/4.48 => ( ( ord_less @ nat @ mi @ X2 ) % 8.42/4.48 & ( ord_less_eq @ nat @ X2 @ ma ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_626_vebt__member_Ocases,axiom, % 8.42/4.48 ! [X3: product_prod @ vEBT_VEBT @ nat] : % 8.42/4.48 ( ! [A5: $o,B5: $o,X5: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X5 ) ) % 8.42/4.48 => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X5 ) ) % 8.42/4.48 => ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ X5 ) ) % 8.42/4.48 => ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ X5 ) ) % 8.42/4.48 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_627_VEBT__internal_Omembermima_Ocases,axiom, % 8.42/4.48 ! [X3: product_prod @ vEBT_VEBT @ nat] : % 8.42/4.48 ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) ) % 8.42/4.48 => ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Uz2 ) ) % 8.42/4.48 => ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ X5 ) ) % 8.42/4.48 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ X5 ) ) % 8.42/4.48 => ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT,X5: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ X5 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_628_count__notin,axiom, % 8.42/4.48 ! [A: $tType,X3: A,Xs: list @ A] : % 8.42/4.48 ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.42/4.48 => ( ( count_list @ A @ Xs @ X3 ) % 8.42/4.48 = ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_629_listrel1__iff__update,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] : % 8.42/4.48 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) % 8.42/4.48 = ( ? [Y2: A,N: nat] : % 8.42/4.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N ) @ Y2 ) @ R2 ) % 8.42/4.48 & ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.42/4.48 & ( Ys % 8.42/4.48 = ( list_update @ A @ Xs @ N @ Y2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_630_pair__lessI2,axiom, % 8.42/4.48 ! [A2: nat,B2: nat,S: nat,T2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less @ nat @ S @ T2 ) % 8.42/4.48 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_631_pair__less__iff1,axiom, % 8.42/4.48 ! [X3: nat,Y: nat,Z2: nat] : % 8.42/4.48 ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X3 @ Y ) @ ( product_Pair @ nat @ nat @ X3 @ Z2 ) ) @ fun_pair_less ) % 8.42/4.48 = ( ord_less @ nat @ Y @ Z2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_632_gen__length__def,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( gen_length @ A ) % 8.42/4.48 = ( ^ [N: nat,Xs2: list @ A] : ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_633_set__encode__empty,axiom, % 8.42/4.48 ( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) ) % 8.42/4.48 = ( zero_zero @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_634_Leaf__0__not,axiom, % 8.42/4.48 ! [A2: $o,B2: $o] : % 8.42/4.48 ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_635_power__shift,axiom, % 8.42/4.48 ! [X3: nat,Y: nat,Z2: nat] : % 8.42/4.48 ( ( ( power_power @ nat @ X3 @ Y ) % 8.42/4.48 = Z2 ) % 8.42/4.48 = ( ( vEBT_VEBT_power @ ( some @ nat @ X3 ) @ ( some @ nat @ Y ) ) % 8.42/4.48 = ( some @ nat @ Z2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_636__C5_C_I3_J,axiom, % 8.42/4.48 ( ( size_size @ ( list @ vEBT_VEBT ) @ treeList ) % 8.42/4.48 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_637__C5_C_I9_J,axiom, % 8.42/4.48 ord_less @ nat @ ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_638_VEBT_Oinject_I2_J,axiom, % 8.42/4.48 ! [X21: $o,X222: $o,Y21: $o,Y222: $o] : % 8.42/4.48 ( ( ( vEBT_Leaf @ X21 @ X222 ) % 8.42/4.48 = ( vEBT_Leaf @ Y21 @ Y222 ) ) % 8.42/4.48 = ( ( X21 = Y21 ) % 8.42/4.48 & ( X222 = Y222 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_639_local_Opower__def,axiom, % 8.42/4.48 ( vEBT_VEBT_power % 8.42/4.48 = ( vEBT_V2048590022279873568_shift @ nat @ ( power_power @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_640_high__bound__aux,axiom, % 8.42/4.48 ! [Ma: nat,N2: nat,M: nat] : % 8.42/4.48 ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) ) % 8.42/4.48 => ( ord_less @ nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_641_member__bound,axiom, % 8.42/4.48 ! [Tree: vEBT_VEBT,X3: nat,N2: nat] : % 8.42/4.48 ( ( vEBT_vebt_member @ Tree @ X3 ) % 8.42/4.48 => ( ( vEBT_invar_vebt @ Tree @ N2 ) % 8.42/4.48 => ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_642__C5_C_I6_J,axiom, % 8.42/4.48 ! [I5: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) ) % 8.42/4.48 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I5 ) @ X6 ) ) % 8.42/4.48 = ( vEBT_V8194947554948674370ptions @ summary @ I5 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_643_numeral__le__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord181362715937106298miring @ A ) % 8.42/4.48 => ! [M: num,N2: num] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) % 8.42/4.48 = ( ord_less_eq @ num @ M @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_644_numeral__less__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord181362715937106298miring @ A ) % 8.42/4.48 => ! [M: num,N2: num] : % 8.42/4.48 ( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) % 8.42/4.48 = ( ord_less @ num @ M @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_645_numeral__plus__numeral,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( numeral @ A ) % 8.42/4.48 => ! [M: num,N2: num] : % 8.42/4.48 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) % 8.42/4.48 = ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_646_add__numeral__left,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( numeral @ A ) % 8.42/4.48 => ! [V2: num,W2: num,Z2: A] : % 8.42/4.48 ( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Z2 ) ) % 8.42/4.48 = ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W2 ) ) @ Z2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_647_mi__ma__2__deg,axiom, % 8.42/4.48 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 ) % 8.42/4.48 => ( ( ord_less_eq @ nat @ Mi @ Ma ) % 8.42/4.48 & ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_648_Suc__numeral,axiom, % 8.42/4.48 ! [N2: num] : % 8.42/4.48 ( ( suc @ ( numeral_numeral @ nat @ N2 ) ) % 8.42/4.48 = ( numeral_numeral @ nat @ ( plus_plus @ num @ N2 @ one2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_649_add__2__eq__Suc_H,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.48 = ( suc @ ( suc @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_650_add__2__eq__Suc,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.48 = ( suc @ ( suc @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_651_numeral__Bit0,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( numeral @ A ) % 8.42/4.48 => ! [N2: num] : % 8.42/4.48 ( ( numeral_numeral @ A @ ( bit0 @ N2 ) ) % 8.42/4.48 = ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_652_numeral__2__eq__2,axiom, % 8.42/4.48 ( ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) % 8.42/4.48 = ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_653_less__2__cases,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.48 => ( ( N2 % 8.42/4.48 = ( zero_zero @ nat ) ) % 8.42/4.48 | ( N2 % 8.42/4.48 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_654_less__2__cases__iff,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.48 = ( ( N2 % 8.42/4.48 = ( zero_zero @ nat ) ) % 8.42/4.48 | ( N2 % 8.42/4.48 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_655_VEBT__internal_Oexp__split__high__low_I1_J,axiom, % 8.42/4.48 ! [X3: nat,N2: nat,M: nat] : % 8.42/4.48 ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) ) % 8.42/4.48 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.48 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.42/4.48 => ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_656_VEBT__internal_Oexp__split__high__low_I2_J,axiom, % 8.42/4.48 ! [X3: nat,N2: nat,M: nat] : % 8.42/4.48 ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) ) % 8.42/4.48 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.48 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.42/4.48 => ( ord_less @ nat @ ( vEBT_VEBT_low @ X3 @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_657_numeral__1__eq__Suc__0,axiom, % 8.42/4.48 ( ( numeral_numeral @ nat @ one2 ) % 8.42/4.48 = ( suc @ ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_658_Suc__nat__number__of__add,axiom, % 8.42/4.48 ! [V2: num,N2: nat] : % 8.42/4.48 ( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N2 ) ) % 8.42/4.48 = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V2 @ one2 ) ) @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_659_VEBT_Osize_I4_J,axiom, % 8.42/4.48 ! [X21: $o,X222: $o] : % 8.42/4.48 ( ( size_size @ vEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) ) % 8.42/4.48 = ( zero_zero @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_660_VEBT__internal_Ovalid_H_Ocases,axiom, % 8.42/4.48 ! [X3: product_prod @ vEBT_VEBT @ nat] : % 8.42/4.48 ( ! [Uu2: $o,Uv2: $o,D3: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D3 ) ) % 8.42/4.48 => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_661_VEBT_Odistinct_I1_J,axiom, % 8.42/4.48 ! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] : % 8.42/4.48 ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) % 8.42/4.48 != ( vEBT_Leaf @ X21 @ X222 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_662_VEBT_Oexhaust,axiom, % 8.42/4.48 ! [Y: vEBT_VEBT] : % 8.42/4.48 ( ! [X112: option @ ( product_prod @ nat @ nat ),X122: nat,X132: list @ vEBT_VEBT,X142: vEBT_VEBT] : % 8.42/4.48 ( Y % 8.42/4.48 != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) ) % 8.42/4.48 => ~ ! [X212: $o,X223: $o] : % 8.42/4.48 ( Y % 8.42/4.48 != ( vEBT_Leaf @ X212 @ X223 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_663_zero__neq__numeral,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( semiring_char_0 @ A ) % 8.42/4.48 => ! [N2: num] : % 8.42/4.48 ( ( zero_zero @ A ) % 8.42/4.48 != ( numeral_numeral @ A @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_664_VEBT__internal_Omembermima_Osimps_I1_J,axiom, % 8.42/4.48 ! [Uu: $o,Uv: $o,Uw: nat] : % 8.42/4.48 ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_665_num_Osize_I4_J,axiom, % 8.42/4.48 ( ( size_size @ num @ one2 ) % 8.42/4.48 = ( zero_zero @ nat ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_666_listrel1__eq__len,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] : % 8.42/4.48 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) % 8.42/4.48 => ( ( size_size @ ( list @ A ) @ Xs ) % 8.42/4.48 = ( size_size @ ( list @ A ) @ Ys ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_667_listrel1__mono,axiom, % 8.42/4.48 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] : % 8.42/4.48 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) % 8.42/4.48 => ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R2 ) @ ( listrel1 @ A @ S ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_668_VEBT__internal_OminNull_Osimps_I3_J,axiom, % 8.42/4.48 ! [Uu: $o] : % 8.42/4.48 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_669_VEBT__internal_OminNull_Osimps_I2_J,axiom, % 8.42/4.48 ! [Uv: $o] : % 8.42/4.48 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_670_VEBT__internal_OminNull_Osimps_I1_J,axiom, % 8.42/4.48 vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_671_invar__vebt_Ointros_I2_J,axiom, % 8.42/4.48 ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] : % 8.42/4.48 ( ! [X5: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ X5 @ N2 ) ) % 8.42/4.48 => ( ( vEBT_invar_vebt @ Summary @ M ) % 8.42/4.48 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) % 8.42/4.48 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 8.42/4.48 => ( ( M = N2 ) % 8.42/4.48 => ( ( Deg % 8.42/4.48 = ( plus_plus @ nat @ N2 @ M ) ) % 8.42/4.48 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) % 8.42/4.48 => ( ! [X5: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 8.42/4.48 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_672_invar__vebt_Ointros_I3_J,axiom, % 8.42/4.48 ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] : % 8.42/4.48 ( ! [X5: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ X5 @ N2 ) ) % 8.42/4.48 => ( ( vEBT_invar_vebt @ Summary @ M ) % 8.42/4.48 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) % 8.42/4.48 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 8.42/4.48 => ( ( M % 8.42/4.48 = ( suc @ N2 ) ) % 8.42/4.48 => ( ( Deg % 8.42/4.48 = ( plus_plus @ nat @ N2 @ M ) ) % 8.42/4.48 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) % 8.42/4.48 => ( ! [X5: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 8.42/4.48 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_673_zip__same,axiom, % 8.42/4.48 ! [A: $tType,A2: A,B2: A,Xs: list @ A] : % 8.42/4.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Xs ) ) ) % 8.42/4.48 = ( ( member @ A @ A2 @ ( set2 @ A @ Xs ) ) % 8.42/4.48 & ( A2 = B2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_674_in__set__zipE,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,X3: A,Y: B,Xs: list @ A,Ys: list @ B] : % 8.42/4.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) % 8.42/4.48 => ~ ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.42/4.48 => ~ ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_675_set__zip__leftD,axiom, % 8.42/4.48 ! [B: $tType,A: $tType,X3: A,Y: B,Xs: list @ A,Ys: list @ B] : % 8.42/4.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) % 8.42/4.48 => ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_676_set__zip__rightD,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,X3: A,Y: B,Xs: list @ A,Ys: list @ B] : % 8.42/4.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) % 8.42/4.48 => ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_677_zip__update,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,Xs: list @ A,I: nat,X3: A,Ys: list @ B,Y: B] : % 8.42/4.48 ( ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ X3 ) @ ( list_update @ B @ Ys @ I @ Y ) ) % 8.42/4.48 = ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I @ ( product_Pair @ A @ B @ X3 @ Y ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_678_zero__le__numeral,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord181362715937106298miring @ A ) % 8.42/4.48 => ! [N2: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_679_not__numeral__le__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord181362715937106298miring @ A ) % 8.42/4.48 => ! [N2: num] : % 8.42/4.48 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( zero_zero @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_680_not__numeral__less__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord181362715937106298miring @ A ) % 8.42/4.48 => ! [N2: num] : % 8.42/4.48 ~ ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( zero_zero @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_681_zero__less__numeral,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord181362715937106298miring @ A ) % 8.42/4.48 => ! [N2: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_682_set__encode__eq,axiom, % 8.42/4.48 ! [A3: set @ nat,B3: set @ nat] : % 8.42/4.48 ( ( finite_finite2 @ nat @ A3 ) % 8.42/4.48 => ( ( finite_finite2 @ nat @ B3 ) % 8.42/4.48 => ( ( ( nat_set_encode @ A3 ) % 8.42/4.48 = ( nat_set_encode @ B3 ) ) % 8.42/4.48 = ( A3 = B3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_683_vebt__buildup_Osimps_I1_J,axiom, % 8.42/4.48 ( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) ) % 8.42/4.48 = ( vEBT_Leaf @ $false @ $false ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_684_in__set__impl__in__set__zip2,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Y: B] : % 8.42/4.48 ( ( ( size_size @ ( list @ A ) @ Xs ) % 8.42/4.48 = ( size_size @ ( list @ B ) @ Ys ) ) % 8.42/4.48 => ( ( member @ B @ Y @ ( set2 @ B @ Ys ) ) % 8.42/4.48 => ~ ! [X5: A] : % 8.42/4.48 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_685_in__set__impl__in__set__zip1,axiom, % 8.42/4.48 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X3: A] : % 8.42/4.48 ( ( ( size_size @ ( list @ A ) @ Xs ) % 8.42/4.48 = ( size_size @ ( list @ B ) @ Ys ) ) % 8.42/4.48 => ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.42/4.48 => ~ ! [Y4: B] : % 8.42/4.48 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_686_VEBT__internal_Onaive__member_Ocases,axiom, % 8.42/4.48 ! [X3: product_prod @ vEBT_VEBT @ nat] : % 8.42/4.48 ( ! [A5: $o,B5: $o,X5: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X5 ) ) % 8.42/4.48 => ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Ux2 ) ) % 8.42/4.48 => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT,S2: vEBT_VEBT,X5: nat] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) @ X5 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_687_invar__vebt_Ointros_I1_J,axiom, % 8.42/4.48 ! [A2: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_688_vebt__buildup_Osimps_I2_J,axiom, % 8.42/4.48 ( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.48 = ( vEBT_Leaf @ $false @ $false ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_689_invar__vebt_Osimps,axiom, % 8.42/4.48 ( vEBT_invar_vebt % 8.42/4.48 = ( ^ [A1: vEBT_VEBT,A22: nat] : % 8.42/4.48 ( ( ? [A4: $o,B4: $o] : % 8.42/4.48 ( A1 % 8.42/4.48 = ( vEBT_Leaf @ A4 @ B4 ) ) % 8.42/4.48 & ( A22 % 8.42/4.48 = ( suc @ ( zero_zero @ nat ) ) ) ) % 8.42/4.48 | ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT] : % 8.42/4.48 ( ( A1 % 8.42/4.48 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A22 @ TreeList @ Summary3 ) ) % 8.42/4.48 & ! [X: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ X @ N ) ) % 8.42/4.48 & ( vEBT_invar_vebt @ Summary3 @ N ) % 8.42/4.48 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) % 8.42/4.48 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) % 8.42/4.48 & ( A22 % 8.42/4.48 = ( plus_plus @ nat @ N @ N ) ) % 8.42/4.48 & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 ) % 8.42/4.48 & ! [X: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 8.42/4.48 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.42/4.48 | ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT] : % 8.42/4.48 ( ( A1 % 8.42/4.48 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A22 @ TreeList @ Summary3 ) ) % 8.42/4.48 & ! [X: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ X @ N ) ) % 8.42/4.48 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) ) % 8.42/4.48 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) % 8.42/4.48 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) % 8.42/4.48 & ( A22 % 8.42/4.48 = ( plus_plus @ nat @ N @ ( suc @ N ) ) ) % 8.42/4.48 & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 ) % 8.42/4.48 & ! [X: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 8.42/4.48 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.42/4.48 | ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] : % 8.42/4.48 ( ( A1 % 8.42/4.48 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) ) % 8.42/4.48 & ! [X: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ X @ N ) ) % 8.42/4.48 & ( vEBT_invar_vebt @ Summary3 @ N ) % 8.42/4.48 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) % 8.42/4.48 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) % 8.42/4.48 & ( A22 % 8.42/4.48 = ( plus_plus @ nat @ N @ N ) ) % 8.42/4.48 & ! [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) % 8.42/4.48 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X6 ) ) % 8.42/4.48 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) ) % 8.42/4.48 & ( ( Mi3 = Ma3 ) % 8.42/4.48 => ! [X: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 8.42/4.48 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.42/4.48 & ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 8.42/4.48 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A22 ) ) % 8.42/4.48 & ( ( Mi3 != Ma3 ) % 8.42/4.48 => ! [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) % 8.42/4.48 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N ) % 8.42/4.48 = I3 ) % 8.42/4.48 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) ) % 8.42/4.48 & ! [X: nat] : % 8.42/4.48 ( ( ( ( vEBT_VEBT_high @ X @ N ) % 8.42/4.48 = I3 ) % 8.42/4.48 & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X @ N ) ) ) % 8.42/4.48 => ( ( ord_less @ nat @ Mi3 @ X ) % 8.42/4.48 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) % 8.42/4.48 | ? [TreeList: list @ vEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] : % 8.42/4.48 ( ( A1 % 8.42/4.48 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) ) % 8.42/4.48 & ! [X: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ X @ N ) ) % 8.42/4.48 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) ) % 8.42/4.48 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) % 8.42/4.48 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) % 8.42/4.48 & ( A22 % 8.42/4.48 = ( plus_plus @ nat @ N @ ( suc @ N ) ) ) % 8.42/4.48 & ! [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) % 8.42/4.48 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X6 ) ) % 8.42/4.48 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) ) % 8.42/4.48 & ( ( Mi3 = Ma3 ) % 8.42/4.48 => ! [X: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList ) ) % 8.42/4.48 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.42/4.48 & ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 8.42/4.48 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A22 ) ) % 8.42/4.48 & ( ( Mi3 != Ma3 ) % 8.42/4.48 => ! [I3: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) % 8.42/4.48 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N ) % 8.42/4.48 = I3 ) % 8.42/4.48 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) ) % 8.42/4.48 & ! [X: nat] : % 8.42/4.48 ( ( ( ( vEBT_VEBT_high @ X @ N ) % 8.42/4.48 = I3 ) % 8.42/4.48 & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X @ N ) ) ) % 8.42/4.48 => ( ( ord_less @ nat @ Mi3 @ X ) % 8.42/4.48 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_690_invar__vebt_Ocases,axiom, % 8.42/4.48 ! [A12: vEBT_VEBT,A23: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ A12 @ A23 ) % 8.42/4.48 => ( ( ? [A5: $o,B5: $o] : % 8.42/4.48 ( A12 % 8.42/4.48 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.42/4.48 => ( A23 % 8.42/4.48 != ( suc @ ( zero_zero @ nat ) ) ) ) % 8.42/4.48 => ( ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] : % 8.42/4.48 ( ( A12 % 8.42/4.48 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) ) % 8.42/4.48 => ( ( A23 = Deg2 ) % 8.42/4.48 => ( ! [X2: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ X2 @ N3 ) ) % 8.42/4.48 => ( ( vEBT_invar_vebt @ Summary2 @ M3 ) % 8.42/4.48 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 8.42/4.48 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 8.42/4.48 => ( ( M3 = N3 ) % 8.42/4.48 => ( ( Deg2 % 8.42/4.48 = ( plus_plus @ nat @ N3 @ M3 ) ) % 8.42/4.48 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 ) % 8.42/4.48 => ~ ! [X2: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.42/4.48 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) ) % 8.42/4.48 => ( ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] : % 8.42/4.48 ( ( A12 % 8.42/4.48 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList3 @ Summary2 ) ) % 8.42/4.48 => ( ( A23 = Deg2 ) % 8.42/4.48 => ( ! [X2: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ X2 @ N3 ) ) % 8.42/4.48 => ( ( vEBT_invar_vebt @ Summary2 @ M3 ) % 8.42/4.48 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 8.42/4.48 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 8.42/4.48 => ( ( M3 % 8.42/4.48 = ( suc @ N3 ) ) % 8.42/4.48 => ( ( Deg2 % 8.42/4.48 = ( plus_plus @ nat @ N3 @ M3 ) ) % 8.42/4.48 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 ) % 8.42/4.48 => ~ ! [X2: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.42/4.48 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) ) % 8.42/4.48 => ( ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] : % 8.42/4.48 ( ( A12 % 8.42/4.48 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) ) % 8.42/4.48 => ( ( A23 = Deg2 ) % 8.42/4.48 => ( ! [X2: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ X2 @ N3 ) ) % 8.42/4.48 => ( ( vEBT_invar_vebt @ Summary2 @ M3 ) % 8.42/4.48 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 8.42/4.48 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 8.42/4.48 => ( ( M3 = N3 ) % 8.42/4.48 => ( ( Deg2 % 8.42/4.48 = ( plus_plus @ nat @ N3 @ M3 ) ) % 8.42/4.48 => ( ! [I5: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 8.42/4.48 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ X6 ) ) % 8.42/4.48 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) ) % 8.42/4.48 => ( ( ( Mi2 = Ma2 ) % 8.42/4.48 => ! [X2: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.42/4.48 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) % 8.42/4.48 => ( ( ord_less_eq @ nat @ Mi2 @ Ma2 ) % 8.42/4.48 => ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 8.42/4.48 => ~ ( ( Mi2 != Ma2 ) % 8.42/4.48 => ! [I5: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 8.42/4.48 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 ) % 8.42/4.48 = I5 ) % 8.42/4.48 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) ) % 8.42/4.48 & ! [X2: nat] : % 8.42/4.48 ( ( ( ( vEBT_VEBT_high @ X2 @ N3 ) % 8.42/4.48 = I5 ) % 8.42/4.48 & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) ) % 8.42/4.48 => ( ( ord_less @ nat @ Mi2 @ X2 ) % 8.42/4.48 & ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) % 8.42/4.48 => ~ ! [TreeList3: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] : % 8.42/4.48 ( ( A12 % 8.42/4.48 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) ) % 8.42/4.48 => ( ( A23 = Deg2 ) % 8.42/4.48 => ( ! [X2: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ X2 @ N3 ) ) % 8.42/4.48 => ( ( vEBT_invar_vebt @ Summary2 @ M3 ) % 8.42/4.48 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 8.42/4.48 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 8.42/4.48 => ( ( M3 % 8.42/4.48 = ( suc @ N3 ) ) % 8.42/4.48 => ( ( Deg2 % 8.42/4.48 = ( plus_plus @ nat @ N3 @ M3 ) ) % 8.42/4.48 => ( ! [I5: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 8.42/4.48 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ X6 ) ) % 8.42/4.48 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) ) % 8.42/4.48 => ( ( ( Mi2 = Ma2 ) % 8.42/4.48 => ! [X2: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.42/4.48 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) % 8.42/4.48 => ( ( ord_less_eq @ nat @ Mi2 @ Ma2 ) % 8.42/4.48 => ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 8.42/4.48 => ~ ( ( Mi2 != Ma2 ) % 8.42/4.48 => ! [I5: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) ) % 8.42/4.48 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 ) % 8.42/4.48 = I5 ) % 8.42/4.48 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) ) % 8.42/4.48 & ! [X2: nat] : % 8.42/4.48 ( ( ( ( vEBT_VEBT_high @ X2 @ N3 ) % 8.42/4.48 = I5 ) % 8.42/4.48 & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I5 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) ) % 8.42/4.48 => ( ( ord_less @ nat @ Mi2 @ X2 ) % 8.42/4.48 & ( ord_less_eq @ nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_691_VEBT__internal_OminNull_Ocases,axiom, % 8.42/4.48 ! [X3: vEBT_VEBT] : % 8.42/4.48 ( ( X3 % 8.42/4.48 != ( vEBT_Leaf @ $false @ $false ) ) % 8.42/4.48 => ( ! [Uv2: $o] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( vEBT_Leaf @ $true @ Uv2 ) ) % 8.42/4.48 => ( ! [Uu2: $o] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( vEBT_Leaf @ Uu2 @ $true ) ) % 8.42/4.48 => ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) % 8.42/4.48 => ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_692_num_Osize_I5_J,axiom, % 8.42/4.48 ! [X22: num] : % 8.42/4.48 ( ( size_size @ num @ ( bit0 @ X22 ) ) % 8.42/4.48 = ( plus_plus @ nat @ ( size_size @ num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_693_invar__vebt_Ointros_I4_J,axiom, % 8.42/4.48 ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] : % 8.42/4.48 ( ! [X5: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ X5 @ N2 ) ) % 8.42/4.48 => ( ( vEBT_invar_vebt @ Summary @ M ) % 8.42/4.48 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) % 8.42/4.48 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 8.42/4.48 => ( ( M = N2 ) % 8.42/4.48 => ( ( Deg % 8.42/4.48 = ( plus_plus @ nat @ N2 @ M ) ) % 8.42/4.48 => ( ! [I4: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 8.42/4.48 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) ) % 8.42/4.48 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) ) % 8.42/4.48 => ( ( ( Mi = Ma ) % 8.42/4.48 => ! [X5: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 8.42/4.48 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) % 8.42/4.48 => ( ( ord_less_eq @ nat @ Mi @ Ma ) % 8.42/4.48 => ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) % 8.42/4.48 => ( ( ( Mi != Ma ) % 8.42/4.48 => ! [I4: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 8.42/4.48 => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 ) % 8.42/4.48 = I4 ) % 8.42/4.48 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) ) % 8.42/4.48 & ! [X5: nat] : % 8.42/4.48 ( ( ( ( vEBT_VEBT_high @ X5 @ N2 ) % 8.42/4.48 = I4 ) % 8.42/4.48 & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) ) % 8.42/4.48 => ( ( ord_less @ nat @ Mi @ X5 ) % 8.42/4.48 & ( ord_less_eq @ nat @ X5 @ Ma ) ) ) ) ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_694_VEBT__internal_OminNull_Oelims_I3_J,axiom, % 8.42/4.48 ! [X3: vEBT_VEBT] : % 8.42/4.48 ( ~ ( vEBT_VEBT_minNull @ X3 ) % 8.42/4.48 => ( ! [Uv2: $o] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( vEBT_Leaf @ $true @ Uv2 ) ) % 8.42/4.48 => ( ! [Uu2: $o] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( vEBT_Leaf @ Uu2 @ $true ) ) % 8.42/4.48 => ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_695_VEBT__internal_OminNull_Oelims_I2_J,axiom, % 8.42/4.48 ! [X3: vEBT_VEBT] : % 8.42/4.48 ( ( vEBT_VEBT_minNull @ X3 ) % 8.42/4.48 => ( ( X3 % 8.42/4.48 != ( vEBT_Leaf @ $false @ $false ) ) % 8.42/4.48 => ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_696_invar__vebt_Ointros_I5_J,axiom, % 8.42/4.48 ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] : % 8.42/4.48 ( ! [X5: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ X5 @ N2 ) ) % 8.42/4.48 => ( ( vEBT_invar_vebt @ Summary @ M ) % 8.42/4.48 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) % 8.42/4.48 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 8.42/4.48 => ( ( M % 8.42/4.48 = ( suc @ N2 ) ) % 8.42/4.48 => ( ( Deg % 8.42/4.48 = ( plus_plus @ nat @ N2 @ M ) ) % 8.42/4.48 => ( ! [I4: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 8.42/4.48 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) ) % 8.42/4.48 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) ) % 8.42/4.48 => ( ( ( Mi = Ma ) % 8.42/4.48 => ! [X5: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 8.42/4.48 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) % 8.42/4.48 => ( ( ord_less_eq @ nat @ Mi @ Ma ) % 8.42/4.48 => ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) % 8.42/4.48 => ( ( ( Mi != Ma ) % 8.42/4.48 => ! [I4: nat] : % 8.42/4.48 ( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 8.42/4.48 => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 ) % 8.42/4.48 = I4 ) % 8.42/4.48 => ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) ) % 8.42/4.48 & ! [X5: nat] : % 8.42/4.48 ( ( ( ( vEBT_VEBT_high @ X5 @ N2 ) % 8.42/4.48 = I4 ) % 8.42/4.48 & ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) ) % 8.42/4.48 => ( ( ord_less @ nat @ Mi @ X5 ) % 8.42/4.48 & ( ord_less_eq @ nat @ X5 @ Ma ) ) ) ) ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_697_set__encode__inf,axiom, % 8.42/4.48 ! [A3: set @ nat] : % 8.42/4.48 ( ~ ( finite_finite2 @ nat @ A3 ) % 8.42/4.48 => ( ( nat_set_encode @ A3 ) % 8.42/4.48 = ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_698_pair__lessI1,axiom, % 8.42/4.48 ! [A2: nat,B2: nat,S: nat,T2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ A2 @ B2 ) % 8.42/4.48 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_699_count__le__length,axiom, % 8.42/4.48 ! [A: $tType,Xs: list @ A,X3: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X3 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_700_vebt__mint_Ocases,axiom, % 8.42/4.48 ! [X3: vEBT_VEBT] : % 8.42/4.48 ( ! [A5: $o,B5: $o] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( vEBT_Leaf @ A5 @ B5 ) ) % 8.42/4.48 => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 8.42/4.48 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 8.42/4.48 ( X3 % 8.42/4.48 != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_701_VEBT__internal_OminNull_Oelims_I1_J,axiom, % 8.42/4.48 ! [X3: vEBT_VEBT,Y: $o] : % 8.42/4.48 ( ( ( vEBT_VEBT_minNull @ X3 ) % 8.42/4.48 = Y ) % 8.42/4.48 => ( ( ( X3 % 8.42/4.48 = ( vEBT_Leaf @ $false @ $false ) ) % 8.42/4.48 => ~ Y ) % 8.42/4.48 => ( ( ? [Uv2: $o] : % 8.42/4.48 ( X3 % 8.42/4.48 = ( vEBT_Leaf @ $true @ Uv2 ) ) % 8.42/4.48 => Y ) % 8.42/4.48 => ( ( ? [Uu2: $o] : % 8.42/4.48 ( X3 % 8.42/4.48 = ( vEBT_Leaf @ Uu2 @ $true ) ) % 8.42/4.48 => Y ) % 8.42/4.48 => ( ( ? [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 8.42/4.48 ( X3 % 8.42/4.48 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) % 8.42/4.48 => ~ Y ) % 8.42/4.48 => ~ ( ? [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 8.42/4.48 ( X3 % 8.42/4.48 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) % 8.42/4.48 => Y ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_702_length__code,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( size_size @ ( list @ A ) ) % 8.42/4.48 = ( gen_length @ A @ ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_703_sum__power2__eq__zero__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_idom @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 = ( ( X3 % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 & ( Y % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_704_zero__less__power2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_idom @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 8.42/4.48 = ( A2 % 8.42/4.48 != ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_705_power2__less__eq__zero__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_idom @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) % 8.42/4.48 = ( A2 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_706_power2__eq__iff__nonneg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.48 => ( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.48 = ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 8.42/4.48 = ( X3 = Y ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_707_zero__eq__power2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( semiri2026040879449505780visors @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 = ( A2 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_708_power__mono__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [A2: A,B2: A,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.48 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) % 8.42/4.48 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_709_power__eq__0__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( semiri2026040879449505780visors @ A ) % 8.42/4.48 => ! [A2: A,N2: nat] : % 8.42/4.48 ( ( ( power_power @ A @ A2 @ N2 ) % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 = ( ( A2 % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_710_member__inv,axiom, % 8.42/4.48 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : % 8.42/4.48 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 ) % 8.42/4.48 => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) % 8.42/4.48 & ( ( X3 = Mi ) % 8.42/4.48 | ( X3 = Ma ) % 8.42/4.48 | ( ( ord_less @ nat @ X3 @ Ma ) % 8.42/4.48 & ( ord_less @ nat @ Mi @ X3 ) % 8.42/4.48 & ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) % 8.42/4.48 & ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_711_both__member__options__ding,axiom, % 8.42/4.48 ! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X3: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 ) % 8.42/4.48 => ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) % 8.42/4.48 => ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.42/4.48 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_712_nat__zero__less__power__iff,axiom, % 8.42/4.48 ! [X3: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X3 @ N2 ) ) % 8.42/4.48 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 ) % 8.42/4.48 | ( N2 % 8.42/4.48 = ( zero_zero @ nat ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_713_pow__sum,axiom, % 8.42/4.48 ! [A2: nat,B2: nat] : % 8.42/4.48 ( ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) ) % 8.42/4.48 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_714_high__def,axiom, % 8.42/4.48 ( vEBT_VEBT_high % 8.42/4.48 = ( ^ [X: nat,N: nat] : ( divide_divide @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_715_division__ring__divide__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( division_ring @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 = ( zero_zero @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_716_divide__cancel__right,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( field @ A ) % 8.42/4.48 => ! [A2: A,C2: A,B2: A] : % 8.42/4.48 ( ( ( divide_divide @ A @ A2 @ C2 ) % 8.42/4.48 = ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.48 = ( ( C2 % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 | ( A2 = B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_717_divide__cancel__left,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( field @ A ) % 8.42/4.48 => ! [C2: A,A2: A,B2: A] : % 8.42/4.48 ( ( ( divide_divide @ A @ C2 @ A2 ) % 8.42/4.48 = ( divide_divide @ A @ C2 @ B2 ) ) % 8.42/4.48 = ( ( C2 % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 | ( A2 = B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_718_div__by__0,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( semidom_divide @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 = ( zero_zero @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_719_divide__eq__0__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( field @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ( divide_divide @ A @ A2 @ B2 ) % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 = ( ( A2 % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 | ( B2 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_720_div__0,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( semidom_divide @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 = ( zero_zero @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_721_power__0__Suc,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( semiring_1 @ A ) % 8.42/4.48 => ! [N2: nat] : % 8.42/4.48 ( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N2 ) ) % 8.42/4.48 = ( zero_zero @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_722_power__zero__numeral,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( semiring_1 @ A ) % 8.42/4.48 => ! [K: num] : % 8.42/4.48 ( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ K ) ) % 8.42/4.48 = ( zero_zero @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_723_power__Suc0__right,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( monoid_mult @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.48 = A2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_724_power__Suc__0,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 8.42/4.48 = ( suc @ ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_725_nat__power__eq__Suc__0__iff,axiom, % 8.42/4.48 ! [X3: nat,M: nat] : % 8.42/4.48 ( ( ( power_power @ nat @ X3 @ M ) % 8.42/4.48 = ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.48 = ( ( M % 8.42/4.48 = ( zero_zero @ nat ) ) % 8.42/4.48 | ( X3 % 8.42/4.48 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_726_add__divide__distrib,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( division_ring @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.48 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_727_divide__right__mono__neg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_728_divide__nonpos__nonpos,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_729_divide__nonpos__nonneg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.48 => ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_730_divide__nonneg__nonpos,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_731_divide__nonneg__nonneg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_732_zero__le__divide__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) ) % 8.42/4.48 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) % 8.42/4.48 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_733_divide__right__mono,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.48 => ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_734_divide__le__0__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) % 8.42/4.48 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) % 8.42/4.48 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_735_divide__neg__neg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_736_divide__neg__pos,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.48 => ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_737_divide__pos__neg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 8.42/4.48 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_738_divide__pos__pos,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 8.42/4.48 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.48 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_739_divide__less__0__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) % 8.42/4.48 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) % 8.42/4.48 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_740_divide__less__cancel,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,C2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.48 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.48 => ( ord_less @ A @ A2 @ B2 ) ) % 8.42/4.48 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ord_less @ A @ B2 @ A2 ) ) % 8.42/4.48 & ( C2 % 8.42/4.48 != ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_741_zero__less__divide__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) ) % 8.42/4.48 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) % 8.42/4.48 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_742_divide__strict__right__mono,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,B2: A,C2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.48 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_743_divide__strict__right__mono__neg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [B2: A,A2: A,C2: A] : % 8.42/4.48 ( ( ord_less @ A @ B2 @ A2 ) % 8.42/4.48 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_744_frac__le,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [Y: A,X3: A,W2: A,Z2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.48 => ( ( ord_less_eq @ A @ X3 @ Y ) % 8.42/4.48 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ W2 @ Z2 ) % 8.42/4.48 => ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_745_frac__less,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A,W2: A,Z2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.42/4.48 => ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ W2 @ Z2 ) % 8.42/4.48 => ( ord_less @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_746_frac__less2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A,W2: A,Z2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ X3 @ Y ) % 8.42/4.48 => ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 ) % 8.42/4.48 => ( ( ord_less @ A @ W2 @ Z2 ) % 8.42/4.48 => ( ord_less @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_747_divide__le__cancel,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,C2: A,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.48 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.48 => ( ord_less_eq @ A @ A2 @ B2 ) ) % 8.42/4.48 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_748_divide__nonneg__neg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.42/4.48 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_749_divide__nonneg__pos,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.42/4.48 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_750_divide__nonpos__neg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less @ A @ Y @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_751_divide__nonpos__pos,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.48 => ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_752_field__sum__of__halves,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A] : % 8.42/4.48 ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) % 8.42/4.48 = X3 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_753_half__gt__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_754_half__gt__zero__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) % 8.42/4.48 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_755_field__less__half__sum,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ X3 @ Y ) % 8.42/4.48 => ( ord_less @ A @ X3 @ ( divide_divide @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_756_power__not__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( semiri2026040879449505780visors @ A ) % 8.42/4.48 => ! [A2: A,N2: nat] : % 8.42/4.48 ( ( A2 % 8.42/4.48 != ( zero_zero @ A ) ) % 8.42/4.48 => ( ( power_power @ A @ A2 @ N2 ) % 8.42/4.48 != ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_757_zero__le__power,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [A2: A,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_758_power__mono,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [A2: A,B2: A,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_759_zero__less__power,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [A2: A,N2: nat] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_760_nat__power__less__imp__less,axiom, % 8.42/4.48 ! [I: nat,M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ I ) % 8.42/4.48 => ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N2 ) ) % 8.42/4.48 => ( ord_less @ nat @ M @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_761_power__less__imp__less__base,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [A2: A,N2: nat,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.48 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_762_power__le__imp__le__base,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [A2: A,N2: nat,B2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N2 ) ) @ ( power_power @ A @ B2 @ ( suc @ N2 ) ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.48 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_763_power__inject__base,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [A2: A,N2: nat,B2: A] : % 8.42/4.48 ( ( ( power_power @ A @ A2 @ ( suc @ N2 ) ) % 8.42/4.48 = ( power_power @ A @ B2 @ ( suc @ N2 ) ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.48 => ( A2 = B2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_764_zero__power,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( semiring_1 @ A ) % 8.42/4.48 => ! [N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.48 => ( ( power_power @ A @ ( zero_zero @ A ) @ N2 ) % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_765_power__gt__expt,axiom, % 8.42/4.48 ! [N2: nat,K: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 8.42/4.48 => ( ord_less @ nat @ K @ ( power_power @ nat @ N2 @ K ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_766_nat__one__le__power,axiom, % 8.42/4.48 ! [I: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I ) % 8.42/4.48 => ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_767_power__eq__iff__eq__base,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [N2: nat,A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.48 => ( ( ( power_power @ A @ A2 @ N2 ) % 8.42/4.48 = ( power_power @ A @ B2 @ N2 ) ) % 8.42/4.48 = ( A2 = B2 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_768_power__eq__imp__eq__base,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [A2: A,N2: nat,B2: A] : % 8.42/4.48 ( ( ( power_power @ A @ A2 @ N2 ) % 8.42/4.48 = ( power_power @ A @ B2 @ N2 ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.48 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.48 => ( A2 = B2 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_769_zero__power2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( semiring_1 @ A ) % 8.42/4.48 => ( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.48 = ( zero_zero @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_770_less__exp,axiom, % 8.42/4.48 ! [N2: nat] : ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_771_power2__nat__le__imp__le,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N2 ) % 8.42/4.48 => ( ord_less_eq @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_772_power2__nat__le__eq__le,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 8.42/4.48 = ( ord_less_eq @ nat @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_773_self__le__ge2__pow,axiom, % 8.42/4.48 ! [K: nat,M: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) % 8.42/4.48 => ( ord_less_eq @ nat @ M @ ( power_power @ nat @ K @ M ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_774_power2__le__imp__le,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.48 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_775_power2__eq__imp__eq,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.48 = ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.48 => ( X3 = Y ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_776_zero__le__power2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_idom @ A ) % 8.42/4.48 => ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_777_power__strict__mono,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [A2: A,B2: A,N2: nat] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.48 => ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_778_power2__less__0,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_idom @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ~ ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_779_power2__less__imp__less,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.48 => ( ord_less @ A @ X3 @ Y ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_780_sum__power2__le__zero__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_idom @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) % 8.42/4.48 = ( ( X3 % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 & ( Y % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_781_sum__power2__ge__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_idom @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_782_not__sum__power2__lt__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_idom @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_783_sum__power2__gt__zero__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_idom @ A ) % 8.42/4.48 => ! [X3: A,Y: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.42/4.48 = ( ( X3 % 8.42/4.48 != ( zero_zero @ A ) ) % 8.42/4.48 | ( Y % 8.42/4.48 != ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_784_both__member__options__from__chilf__to__complete__tree,axiom, % 8.42/4.48 ! [X3: nat,Deg: nat,TreeList2: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] : % 8.42/4.48 ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) % 8.42/4.48 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg ) % 8.42/4.48 => ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.42/4.48 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_785_add__self__div__2,axiom, % 8.42/4.48 ! [M: nat] : % 8.42/4.48 ( ( divide_divide @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.48 = M ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_786_div2__Suc__Suc,axiom, % 8.42/4.48 ! [M: nat] : % 8.42/4.48 ( ( divide_divide @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.48 = ( suc @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_787_both__member__options__from__complete__tree__to__child,axiom, % 8.42/4.48 ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : % 8.42/4.48 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg ) % 8.42/4.48 => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 ) % 8.42/4.48 => ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.42/4.48 | ( X3 = Mi ) % 8.42/4.48 | ( X3 = Ma ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_788_semiring__norm_I76_J,axiom, % 8.42/4.48 ! [N2: num] : ( ord_less @ num @ one2 @ ( bit0 @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_789_div__less,axiom, % 8.42/4.48 ! [M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.48 => ( ( divide_divide @ nat @ M @ N2 ) % 8.42/4.48 = ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_790_div__by__Suc__0,axiom, % 8.42/4.48 ! [M: nat] : % 8.42/4.48 ( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.48 = M ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_791_set__n__deg__not__0,axiom, % 8.42/4.48 ! [TreeList2: list @ vEBT_VEBT,N2: nat,M: nat] : % 8.42/4.48 ( ! [X5: vEBT_VEBT] : % 8.42/4.48 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 8.42/4.48 => ( vEBT_invar_vebt @ X5 @ N2 ) ) % 8.42/4.48 => ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) % 8.42/4.48 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 8.42/4.48 => ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_792_Suc__n__div__2__gt__zero,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.48 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_793_div__2__gt__zero,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 8.42/4.48 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_794_deg1Leaf,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) ) % 8.42/4.48 = ( ? [A4: $o,B4: $o] : % 8.42/4.48 ( T2 % 8.42/4.48 = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_795_deg__1__Leaf,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) ) % 8.42/4.48 => ? [A5: $o,B5: $o] : % 8.42/4.48 ( T2 % 8.42/4.48 = ( vEBT_Leaf @ A5 @ B5 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_796_deg__1__Leafy,axiom, % 8.42/4.48 ! [T2: vEBT_VEBT,N2: nat] : % 8.42/4.48 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.48 => ( ( N2 % 8.42/4.48 = ( one_one @ nat ) ) % 8.42/4.48 => ? [A5: $o,B5: $o] : % 8.42/4.48 ( T2 % 8.42/4.48 = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_797_div__by__1,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( semidom_divide @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( divide_divide @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.48 = A2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_798_semiring__norm_I78_J,axiom, % 8.42/4.48 ! [M: num,N2: num] : % 8.42/4.48 ( ( ord_less @ num @ ( bit0 @ M ) @ ( bit0 @ N2 ) ) % 8.42/4.48 = ( ord_less @ num @ M @ N2 ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_799_semiring__norm_I75_J,axiom, % 8.42/4.48 ! [M: num] : % 8.42/4.48 ~ ( ord_less @ num @ M @ one2 ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_800_divide__eq__1__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( field @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ( divide_divide @ A @ A2 @ B2 ) % 8.42/4.48 = ( one_one @ A ) ) % 8.42/4.48 = ( ( B2 % 8.42/4.48 != ( zero_zero @ A ) ) % 8.42/4.48 & ( A2 = B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_801_div__self,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( semidom_divide @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( A2 % 8.42/4.48 != ( zero_zero @ A ) ) % 8.42/4.48 => ( ( divide_divide @ A @ A2 @ A2 ) % 8.42/4.48 = ( one_one @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_802_one__eq__divide__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( field @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ( one_one @ A ) % 8.42/4.48 = ( divide_divide @ A @ A2 @ B2 ) ) % 8.42/4.48 = ( ( B2 % 8.42/4.48 != ( zero_zero @ A ) ) % 8.42/4.48 & ( A2 = B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_803_divide__self,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( division_ring @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( A2 % 8.42/4.48 != ( zero_zero @ A ) ) % 8.42/4.48 => ( ( divide_divide @ A @ A2 @ A2 ) % 8.42/4.48 = ( one_one @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_804_divide__self__if,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( division_ring @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ( A2 % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 => ( ( divide_divide @ A @ A2 @ A2 ) % 8.42/4.48 = ( zero_zero @ A ) ) ) % 8.42/4.48 & ( ( A2 % 8.42/4.48 != ( zero_zero @ A ) ) % 8.42/4.48 => ( ( divide_divide @ A @ A2 @ A2 ) % 8.42/4.48 = ( one_one @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_805_divide__eq__eq__1,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [B2: A,A2: A] : % 8.42/4.48 ( ( ( divide_divide @ A @ B2 @ A2 ) % 8.42/4.48 = ( one_one @ A ) ) % 8.42/4.48 = ( ( A2 % 8.42/4.48 != ( zero_zero @ A ) ) % 8.42/4.48 & ( A2 = B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_806_eq__divide__eq__1,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [B2: A,A2: A] : % 8.42/4.48 ( ( ( one_one @ A ) % 8.42/4.48 = ( divide_divide @ A @ B2 @ A2 ) ) % 8.42/4.48 = ( ( A2 % 8.42/4.48 != ( zero_zero @ A ) ) % 8.42/4.48 & ( A2 = B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_807_one__divide__eq__0__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 ) % 8.42/4.48 = ( zero_zero @ A ) ) % 8.42/4.48 = ( A2 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_808_zero__eq__1__divide__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ( zero_zero @ A ) % 8.42/4.48 = ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) % 8.42/4.48 = ( A2 % 8.42/4.48 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_809_power__inject__exp,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [A2: A,M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 8.42/4.48 => ( ( ( power_power @ A @ A2 @ M ) % 8.42/4.48 = ( power_power @ A @ A2 @ N2 ) ) % 8.42/4.48 = ( M = N2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_810_less__one,axiom, % 8.42/4.48 ! [N2: nat] : % 8.42/4.48 ( ( ord_less @ nat @ N2 @ ( one_one @ nat ) ) % 8.42/4.48 = ( N2 % 8.42/4.48 = ( zero_zero @ nat ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_811_divide__le__0__1__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) ) % 8.42/4.48 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_812_zero__le__divide__1__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) % 8.42/4.48 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_813_zero__less__divide__1__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) % 8.42/4.48 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_814_less__divide__eq__1__pos,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) ) % 8.42/4.48 = ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_815_less__divide__eq__1__neg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) ) % 8.42/4.48 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_816_divide__less__eq__1__pos,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) ) % 8.42/4.48 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_817_divide__less__eq__1__neg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) ) % 8.42/4.48 = ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_818_divide__less__0__1__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) ) % 8.42/4.48 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_819_power__strict__increasing__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [B2: A,X3: nat,Y: nat] : % 8.42/4.48 ( ( ord_less @ A @ ( one_one @ A ) @ B2 ) % 8.42/4.48 => ( ( ord_less @ A @ ( power_power @ A @ B2 @ X3 ) @ ( power_power @ A @ B2 @ Y ) ) % 8.42/4.48 = ( ord_less @ nat @ X3 @ Y ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_820_le__divide__eq__1__pos,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) ) % 8.42/4.48 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_821_le__divide__eq__1__neg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) ) % 8.42/4.48 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_822_divide__le__eq__1__pos,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) ) % 8.42/4.48 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_823_divide__le__eq__1__neg,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_field @ A ) % 8.42/4.48 => ! [A2: A,B2: A] : % 8.42/4.48 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) ) % 8.42/4.48 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_824_one__add__one,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( numeral @ A ) % 8.42/4.48 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) % 8.42/4.48 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_825_power__strict__decreasing__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [B2: A,M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.48 => ( ( ord_less @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.48 => ( ( ord_less @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N2 ) ) % 8.42/4.48 = ( ord_less @ nat @ N2 @ M ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_826_power__increasing__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [B2: A,X3: nat,Y: nat] : % 8.42/4.48 ( ( ord_less @ A @ ( one_one @ A ) @ B2 ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X3 ) @ ( power_power @ A @ B2 @ Y ) ) % 8.42/4.48 = ( ord_less_eq @ nat @ X3 @ Y ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_827_Suc__1,axiom, % 8.42/4.48 ( ( suc @ ( one_one @ nat ) ) % 8.42/4.48 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_828_one__plus__numeral,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( numeral @ A ) % 8.42/4.48 => ! [N2: num] : % 8.42/4.48 ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) ) % 8.42/4.48 = ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_829_numeral__plus__one,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( numeral @ A ) % 8.42/4.48 => ! [N2: num] : % 8.42/4.48 ( ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) ) % 8.42/4.48 = ( numeral_numeral @ A @ ( plus_plus @ num @ N2 @ one2 ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_830_numeral__le__one__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord181362715937106298miring @ A ) % 8.42/4.48 => ! [N2: num] : % 8.42/4.48 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) ) % 8.42/4.48 = ( ord_less_eq @ num @ N2 @ one2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_831_one__less__numeral__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord181362715937106298miring @ A ) % 8.42/4.48 => ! [N2: num] : % 8.42/4.48 ( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) ) % 8.42/4.48 = ( ord_less @ num @ one2 @ N2 ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_832_one__div__two__eq__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( euclid5411537665997757685th_nat @ A ) % 8.42/4.48 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.48 = ( zero_zero @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_833_power__decreasing__iff,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linordered_semidom @ A ) % 8.42/4.48 => ! [B2: A,M: nat,N2: nat] : % 8.42/4.48 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.48 => ( ( ord_less @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.48 => ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N2 ) ) % 8.42/4.48 = ( ord_less_eq @ nat @ N2 @ M ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_834_one__reorient,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( one @ A ) % 8.42/4.48 => ! [X3: A] : % 8.42/4.48 ( ( ( one_one @ A ) % 8.42/4.48 = X3 ) % 8.42/4.48 = ( X3 % 8.42/4.48 = ( one_one @ A ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_835_le__numeral__extra_I4_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord181362715937106298miring @ A ) % 8.42/4.48 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_836_zero__neq__one,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( zero_neq_one @ A ) % 8.42/4.48 => ( ( zero_zero @ A ) % 8.42/4.48 != ( one_one @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_837_less__numeral__extra_I4_J,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord181362715937106298miring @ A ) % 8.42/4.48 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_838_div__add__self2,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.48 => ! [B2: A,A2: A] : % 8.42/4.48 ( ( B2 % 8.42/4.48 != ( zero_zero @ A ) ) % 8.42/4.48 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 ) % 8.42/4.48 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_839_div__add__self1,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.48 => ! [B2: A,A2: A] : % 8.42/4.48 ( ( B2 % 8.42/4.48 != ( zero_zero @ A ) ) % 8.42/4.48 => ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 ) % 8.42/4.48 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_840_zero__less__one__class_Ozero__le__one,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( zero_less_one @ A ) % 8.42/4.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_841_linordered__nonzero__semiring__class_Ozero__le__one,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.48 ( ( linord181362715937106298miring @ A ) % 8.42/4.48 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ). % 8.42/4.48 % 8.42/4.48 thf(fact_842_not__one__le__zero,axiom, % 8.42/4.48 ! [A: $tType] : % 8.42/4.49 ( ( linord181362715937106298miring @ A ) % 8.42/4.49 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_843_not__one__less__zero,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord181362715937106298miring @ A ) % 8.42/4.49 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_844_zero__less__one,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( zero_less_one @ A ) % 8.42/4.49 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_845_less__numeral__extra_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord181362715937106298miring @ A ) % 8.42/4.49 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_846_one__le__numeral,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord181362715937106298miring @ A ) % 8.42/4.49 => ! [N2: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_847_not__numeral__less__one,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord181362715937106298miring @ A ) % 8.42/4.49 => ! [N2: num] : % 8.42/4.49 ~ ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_848_add__mono1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord181362715937106298miring @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_849_less__add__one,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_850_right__inverse__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( B2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ( divide_divide @ A @ A2 @ B2 ) % 8.42/4.49 = ( one_one @ A ) ) % 8.42/4.49 = ( A2 = B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_851_one__plus__numeral__commute,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( numeral @ A ) % 8.42/4.49 => ! [X3: num] : % 8.42/4.49 ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X3 ) ) % 8.42/4.49 = ( plus_plus @ A @ ( numeral_numeral @ A @ X3 ) @ ( one_one @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_852_one__le__power,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_853_power__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( power @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( power_power @ A @ A2 @ ( zero_zero @ nat ) ) % 8.42/4.49 = ( one_one @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_854_div__eq__dividend__iff,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.42/4.49 => ( ( ( divide_divide @ nat @ M @ N2 ) % 8.42/4.49 = M ) % 8.42/4.49 = ( N2 % 8.42/4.49 = ( one_one @ nat ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_855_div__less__dividend,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( one_one @ nat ) @ N2 ) % 8.42/4.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.42/4.49 => ( ord_less @ nat @ ( divide_divide @ nat @ M @ N2 ) @ M ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_856_One__nat__def,axiom, % 8.42/4.49 ( ( one_one @ nat ) % 8.42/4.49 = ( suc @ ( zero_zero @ nat ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_857_Suc__eq__plus1,axiom, % 8.42/4.49 ( suc % 8.42/4.49 = ( ^ [N: nat] : ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_858_plus__1__eq__Suc,axiom, % 8.42/4.49 ( ( plus_plus @ nat @ ( one_one @ nat ) ) % 8.42/4.49 = suc ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_859_Suc__eq__plus1__left,axiom, % 8.42/4.49 ( suc % 8.42/4.49 = ( plus_plus @ nat @ ( one_one @ nat ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_860_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom, % 8.42/4.49 ! [Uu: $o,Uv: $o,D2: nat] : % 8.42/4.49 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D2 ) % 8.42/4.49 = ( D2 % 8.42/4.49 = ( one_one @ nat ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_861_zero__less__two,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord181362715937106298miring @ A ) % 8.42/4.49 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_862_power__le__one,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( one_one @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_863_less__divide__eq__1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 & ( ord_less @ A @ A2 @ B2 ) ) % 8.42/4.49 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 & ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_864_divide__less__eq__1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 & ( ord_less @ A @ B2 @ A2 ) ) % 8.42/4.49 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 & ( ord_less @ A @ A2 @ B2 ) ) % 8.42/4.49 | ( A2 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_865_less__half__sum,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 => ( ord_less @ A @ A2 @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_866_gt__half__sum,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 => ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_867_power__0__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_1 @ A ) % 8.42/4.49 => ! [N2: nat] : % 8.42/4.49 ( ( ( N2 % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 => ( ( power_power @ A @ ( zero_zero @ A ) @ N2 ) % 8.42/4.49 = ( one_one @ A ) ) ) % 8.42/4.49 & ( ( N2 % 8.42/4.49 != ( zero_zero @ nat ) ) % 8.42/4.49 => ( ( power_power @ A @ ( zero_zero @ A ) @ N2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_868_power__gt1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 8.42/4.49 => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( suc @ N2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_869_power__less__imp__less__exp,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [A2: A,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) % 8.42/4.49 => ( ord_less @ nat @ M @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_870_power__strict__increasing,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [N2: nat,N7: nat,A2: A] : % 8.42/4.49 ( ( ord_less @ nat @ N2 @ N7 ) % 8.42/4.49 => ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 8.42/4.49 => ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ A2 @ N7 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_871_power__increasing,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [N2: nat,N7: nat,A2: A] : % 8.42/4.49 ( ( ord_less_eq @ nat @ N2 @ N7 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ A2 @ N7 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_872_nat__induct__non__zero,axiom, % 8.42/4.49 ! [N2: nat,P: nat > $o] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ( P @ ( one_one @ nat ) ) % 8.42/4.49 => ( ! [N3: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 8.42/4.49 => ( ( P @ N3 ) % 8.42/4.49 => ( P @ ( suc @ N3 ) ) ) ) % 8.42/4.49 => ( P @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_873_vebt__member_Osimps_I1_J,axiom, % 8.42/4.49 ! [A2: $o,B2: $o,X3: nat] : % 8.42/4.49 ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A2 @ B2 ) @ X3 ) % 8.42/4.49 = ( ( ( X3 % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 => A2 ) % 8.42/4.49 & ( ( X3 % 8.42/4.49 != ( zero_zero @ nat ) ) % 8.42/4.49 => ( ( ( X3 % 8.42/4.49 = ( one_one @ nat ) ) % 8.42/4.49 => B2 ) % 8.42/4.49 & ( X3 % 8.42/4.49 = ( one_one @ nat ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_874_VEBT__internal_Onaive__member_Osimps_I1_J,axiom, % 8.42/4.49 ! [A2: $o,B2: $o,X3: nat] : % 8.42/4.49 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A2 @ B2 ) @ X3 ) % 8.42/4.49 = ( ( ( X3 % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 => A2 ) % 8.42/4.49 & ( ( X3 % 8.42/4.49 != ( zero_zero @ nat ) ) % 8.42/4.49 => ( ( ( X3 % 8.42/4.49 = ( one_one @ nat ) ) % 8.42/4.49 => B2 ) % 8.42/4.49 & ( X3 % 8.42/4.49 = ( one_one @ nat ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_875_le__divide__eq__1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 & ( ord_less_eq @ A @ A2 @ B2 ) ) % 8.42/4.49 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 & ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_876_divide__le__eq__1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 & ( ord_less_eq @ A @ B2 @ A2 ) ) % 8.42/4.49 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 & ( ord_less_eq @ A @ A2 @ B2 ) ) % 8.42/4.49 | ( A2 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_877_power__Suc__le__self,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_878_power__Suc__less__one,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ N2 ) ) @ ( one_one @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_879_power__strict__decreasing,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [N2: nat,N7: nat,A2: A] : % 8.42/4.49 ( ( ord_less @ nat @ N2 @ N7 ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( power_power @ A @ A2 @ N7 ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_880_power__decreasing,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [N2: nat,N7: nat,A2: A] : % 8.42/4.49 ( ( ord_less_eq @ nat @ N2 @ N7 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N7 ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_881_power__le__imp__le__exp,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [A2: A,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) % 8.42/4.49 => ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_882_self__le__power,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ord_less_eq @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_883_one__less__power,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_884_nat__1__add__1,axiom, % 8.42/4.49 ( ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) ) % 8.42/4.49 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_885_vebt__maxt_Osimps_I1_J,axiom, % 8.42/4.49 ! [B2: $o,A2: $o] : % 8.42/4.49 ( ( B2 % 8.42/4.49 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) ) % 8.42/4.49 = ( some @ nat @ ( one_one @ nat ) ) ) ) % 8.42/4.49 & ( ~ B2 % 8.42/4.49 => ( ( A2 % 8.42/4.49 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) ) % 8.42/4.49 = ( some @ nat @ ( zero_zero @ nat ) ) ) ) % 8.42/4.49 & ( ~ A2 % 8.42/4.49 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) ) % 8.42/4.49 = ( none @ nat ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_886_vebt__mint_Osimps_I1_J,axiom, % 8.42/4.49 ! [A2: $o,B2: $o] : % 8.42/4.49 ( ( A2 % 8.42/4.49 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) ) % 8.42/4.49 = ( some @ nat @ ( zero_zero @ nat ) ) ) ) % 8.42/4.49 & ( ~ A2 % 8.42/4.49 => ( ( B2 % 8.42/4.49 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) ) % 8.42/4.49 = ( some @ nat @ ( one_one @ nat ) ) ) ) % 8.42/4.49 & ( ~ B2 % 8.42/4.49 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) ) % 8.42/4.49 = ( none @ nat ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_887_div__le__mono,axiom, % 8.42/4.49 ! [M: nat,N2: nat,K: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.49 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N2 @ K ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_888_div__le__dividend,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N2 ) @ M ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_889_ex__power__ivl2,axiom, % 8.42/4.49 ! [B2: nat,K: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) % 8.42/4.49 => ? [N3: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K ) % 8.42/4.49 & ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_890_ex__power__ivl1,axiom, % 8.42/4.49 ! [B2: nat,K: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K ) % 8.42/4.49 => ? [N3: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K ) % 8.42/4.49 & ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_891_Euclidean__Division_Odiv__eq__0__iff,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ( divide_divide @ nat @ M @ N2 ) % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 = ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.49 | ( N2 % 8.42/4.49 = ( zero_zero @ nat ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_892_Suc__div__le__mono,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N2 ) @ ( divide_divide @ nat @ ( suc @ M ) @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_893_vebt__maxt_Oelims,axiom, % 8.42/4.49 ! [X3: vEBT_VEBT,Y: option @ nat] : % 8.42/4.49 ( ( ( vEBT_vebt_maxt @ X3 ) % 8.42/4.49 = Y ) % 8.42/4.49 => ( ! [A5: $o,B5: $o] : % 8.42/4.49 ( ( X3 % 8.42/4.49 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.42/4.49 => ~ ( ( B5 % 8.42/4.49 => ( Y % 8.42/4.49 = ( some @ nat @ ( one_one @ nat ) ) ) ) % 8.42/4.49 & ( ~ B5 % 8.42/4.49 => ( ( A5 % 8.42/4.49 => ( Y % 8.42/4.49 = ( some @ nat @ ( zero_zero @ nat ) ) ) ) % 8.42/4.49 & ( ~ A5 % 8.42/4.49 => ( Y % 8.42/4.49 = ( none @ nat ) ) ) ) ) ) ) % 8.42/4.49 => ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 8.42/4.49 ( X3 % 8.42/4.49 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 8.42/4.49 => ( Y % 8.42/4.49 != ( none @ nat ) ) ) % 8.42/4.49 => ~ ! [Mi2: nat,Ma2: nat] : % 8.42/4.49 ( ? [Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 8.42/4.49 ( X3 % 8.42/4.49 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) % 8.42/4.49 => ( Y % 8.42/4.49 != ( some @ nat @ Ma2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_894_vebt__mint_Oelims,axiom, % 8.42/4.49 ! [X3: vEBT_VEBT,Y: option @ nat] : % 8.42/4.49 ( ( ( vEBT_vebt_mint @ X3 ) % 8.42/4.49 = Y ) % 8.42/4.49 => ( ! [A5: $o,B5: $o] : % 8.42/4.49 ( ( X3 % 8.42/4.49 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.42/4.49 => ~ ( ( A5 % 8.42/4.49 => ( Y % 8.42/4.49 = ( some @ nat @ ( zero_zero @ nat ) ) ) ) % 8.42/4.49 & ( ~ A5 % 8.42/4.49 => ( ( B5 % 8.42/4.49 => ( Y % 8.42/4.49 = ( some @ nat @ ( one_one @ nat ) ) ) ) % 8.42/4.49 & ( ~ B5 % 8.42/4.49 => ( Y % 8.42/4.49 = ( none @ nat ) ) ) ) ) ) ) % 8.42/4.49 => ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 8.42/4.49 ( X3 % 8.42/4.49 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 8.42/4.49 => ( Y % 8.42/4.49 != ( none @ nat ) ) ) % 8.42/4.49 => ~ ! [Mi2: nat] : % 8.42/4.49 ( ? [Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 8.42/4.49 ( X3 % 8.42/4.49 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) % 8.42/4.49 => ( Y % 8.42/4.49 != ( some @ nat @ Mi2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_895_div__greater__zero__iff,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N2 ) ) % 8.42/4.49 = ( ( ord_less_eq @ nat @ N2 @ M ) % 8.42/4.49 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_896_div__le__mono2,axiom, % 8.42/4.49 ! [M: nat,N2: nat,K: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.49 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N2 ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_897_bits__1__div__2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_898_nat__induct2,axiom, % 8.42/4.49 ! [P: nat > $o,N2: nat] : % 8.42/4.49 ( ( P @ ( zero_zero @ nat ) ) % 8.42/4.49 => ( ( P @ ( one_one @ nat ) ) % 8.42/4.49 => ( ! [N3: nat] : % 8.42/4.49 ( ( P @ N3 ) % 8.42/4.49 => ( P @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.42/4.49 => ( P @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_899_div__exp__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ! [A2: A,M: nat,N2: nat] : % 8.42/4.49 ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_900_exp__add__not__zero__imp__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) ) % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 != ( zero_zero @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_901_exp__add__not__zero__imp__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) ) % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) % 8.42/4.49 != ( zero_zero @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_902_bit__concat__def,axiom, % 8.42/4.49 ( vEBT_VEBT_bit_concat % 8.42/4.49 = ( ^ [H: nat,L2: nat,D5: nat] : ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ D5 ) ) @ L2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_903_low__inv,axiom, % 8.42/4.49 ! [X3: nat,N2: nat,Y: nat] : % 8.42/4.49 ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.42/4.49 => ( ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ X3 ) @ N2 ) % 8.42/4.49 = X3 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_904_high__inv,axiom, % 8.42/4.49 ! [X3: nat,N2: nat,Y: nat] : % 8.42/4.49 ( ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.42/4.49 => ( ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ X3 ) @ N2 ) % 8.42/4.49 = Y ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_905_enat__ord__number_I1_J,axiom, % 8.42/4.49 ! [M: num,N2: num] : % 8.42/4.49 ( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N2 ) ) % 8.42/4.49 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_906_enat__ord__number_I2_J,axiom, % 8.42/4.49 ! [M: num,N2: num] : % 8.42/4.49 ( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N2 ) ) % 8.42/4.49 = ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_907_mul__shift,axiom, % 8.42/4.49 ! [X3: nat,Y: nat,Z2: nat] : % 8.42/4.49 ( ( ( times_times @ nat @ X3 @ Y ) % 8.42/4.49 = Z2 ) % 8.42/4.49 = ( ( vEBT_VEBT_mul @ ( some @ nat @ X3 ) @ ( some @ nat @ Y ) ) % 8.42/4.49 = ( some @ nat @ Z2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_908_mul__def,axiom, % 8.42/4.49 ( vEBT_VEBT_mul % 8.42/4.49 = ( vEBT_V2048590022279873568_shift @ nat @ ( times_times @ nat ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_909_i0__less,axiom, % 8.42/4.49 ! [N2: extended_enat] : % 8.42/4.49 ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 ) % 8.42/4.49 = ( N2 % 8.42/4.49 != ( zero_zero @ extended_enat ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_910_mult__cancel__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiri6575147826004484403cancel @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( ( times_times @ A @ A2 @ C2 ) % 8.42/4.49 = ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ( C2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 | ( A2 = B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_911_mult__cancel__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiri6575147826004484403cancel @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ( times_times @ A @ C2 @ A2 ) % 8.42/4.49 = ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ( C2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 | ( A2 = B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_912_mult__eq__0__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiri3467727345109120633visors @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ( times_times @ A @ A2 @ B2 ) % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 = ( ( A2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 | ( B2 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_913_mult__zero__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( mult_zero @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( times_times @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_914_mult__zero__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( mult_zero @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( times_times @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_915_bits__div__by__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_916_bits__div__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_917_mult_Oright__neutral,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( monoid_mult @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( times_times @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_918_mult__1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( monoid_mult @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( times_times @ A @ ( one_one @ A ) @ A2 ) % 8.42/4.49 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_919_half__negative__int__iff,axiom, % 8.42/4.49 ! [K: int] : % 8.42/4.49 ( ( ord_less @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ int ) ) % 8.42/4.49 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_920_times__divide__eq__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [B2: A,C2: A,A2: A] : % 8.42/4.49 ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 8.42/4.49 = ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_921_divide__divide__eq__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_922_divide__divide__eq__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_923_times__divide__eq__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_924_mult__cancel2,axiom, % 8.42/4.49 ! [M: nat,K: nat,N2: nat] : % 8.42/4.49 ( ( ( times_times @ nat @ M @ K ) % 8.42/4.49 = ( times_times @ nat @ N2 @ K ) ) % 8.42/4.49 = ( ( M = N2 ) % 8.42/4.49 | ( K % 8.42/4.49 = ( zero_zero @ nat ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_925_mult__cancel1,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ( times_times @ nat @ K @ M ) % 8.42/4.49 = ( times_times @ nat @ K @ N2 ) ) % 8.42/4.49 = ( ( M = N2 ) % 8.42/4.49 | ( K % 8.42/4.49 = ( zero_zero @ nat ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_926_mult__0__right,axiom, % 8.42/4.49 ! [M: nat] : % 8.42/4.49 ( ( times_times @ nat @ M @ ( zero_zero @ nat ) ) % 8.42/4.49 = ( zero_zero @ nat ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_927_mult__is__0,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ( times_times @ nat @ M @ N2 ) % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 = ( ( M % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 | ( N2 % 8.42/4.49 = ( zero_zero @ nat ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_928_nat__1__eq__mult__iff,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ( one_one @ nat ) % 8.42/4.49 = ( times_times @ nat @ M @ N2 ) ) % 8.42/4.49 = ( ( M % 8.42/4.49 = ( one_one @ nat ) ) % 8.42/4.49 & ( N2 % 8.42/4.49 = ( one_one @ nat ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_929_nat__mult__eq__1__iff,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ( times_times @ nat @ M @ N2 ) % 8.42/4.49 = ( one_one @ nat ) ) % 8.42/4.49 = ( ( M % 8.42/4.49 = ( one_one @ nat ) ) % 8.42/4.49 & ( N2 % 8.42/4.49 = ( one_one @ nat ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_930_mult__cancel__right2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ring_15535105094025558882visors @ A ) % 8.42/4.49 => ! [A2: A,C2: A] : % 8.42/4.49 ( ( ( times_times @ A @ A2 @ C2 ) % 8.42/4.49 = C2 ) % 8.42/4.49 = ( ( C2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 | ( A2 % 8.42/4.49 = ( one_one @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_931_mult__cancel__right1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ring_15535105094025558882visors @ A ) % 8.42/4.49 => ! [C2: A,B2: A] : % 8.42/4.49 ( ( C2 % 8.42/4.49 = ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ( C2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 | ( B2 % 8.42/4.49 = ( one_one @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_932_mult__cancel__left2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ring_15535105094025558882visors @ A ) % 8.42/4.49 => ! [C2: A,A2: A] : % 8.42/4.49 ( ( ( times_times @ A @ C2 @ A2 ) % 8.42/4.49 = C2 ) % 8.42/4.49 = ( ( C2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 | ( A2 % 8.42/4.49 = ( one_one @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_933_mult__cancel__left1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ring_15535105094025558882visors @ A ) % 8.42/4.49 => ! [C2: A,B2: A] : % 8.42/4.49 ( ( C2 % 8.42/4.49 = ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ( C2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 | ( B2 % 8.42/4.49 = ( one_one @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_934_sum__squares__eq__zero__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [X3: A,Y: A] : % 8.42/4.49 ( ( ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 = ( ( X3 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 & ( Y % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_935_nonzero__mult__divide__mult__cancel__right2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_936_nonzero__mult__div__cancel__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_divide @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( B2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 ) % 8.42/4.49 = A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_937_nonzero__mult__divide__mult__cancel__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_938_nonzero__mult__divide__mult__cancel__left2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_939_nonzero__mult__div__cancel__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_divide @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( A2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ A2 ) % 8.42/4.49 = B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_940_nonzero__mult__divide__mult__cancel__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_941_mult__divide__mult__cancel__left__if,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ( C2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) % 8.42/4.49 & ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_942_div__mult__mult1__if,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ( C2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) % 8.42/4.49 & ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_943_div__mult__mult2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_944_div__mult__mult1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_945_distrib__right__numeral,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ( numeral @ A ) % 8.42/4.49 & ( semiring @ A ) ) % 8.42/4.49 => ! [A2: A,B2: A,V2: num] : % 8.42/4.49 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( numeral_numeral @ A @ V2 ) ) % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_946_distrib__left__numeral,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ( numeral @ A ) % 8.42/4.49 & ( semiring @ A ) ) % 8.42/4.49 => ! [V2: num,B2: A,C2: A] : % 8.42/4.49 ( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_947_mult__eq__1__iff,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ( times_times @ nat @ M @ N2 ) % 8.42/4.49 = ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.49 = ( ( M % 8.42/4.49 = ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.49 & ( N2 % 8.42/4.49 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_948_one__eq__mult__iff,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ( suc @ ( zero_zero @ nat ) ) % 8.42/4.49 = ( times_times @ nat @ M @ N2 ) ) % 8.42/4.49 = ( ( M % 8.42/4.49 = ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.49 & ( N2 % 8.42/4.49 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_949_mult__less__cancel2,axiom, % 8.42/4.49 ! [M: nat,K: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) % 8.42/4.49 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.42/4.49 & ( ord_less @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_950_nat__0__less__mult__iff,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N2 ) ) % 8.42/4.49 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.42/4.49 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_951_nat__mult__less__cancel__disj,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 8.42/4.49 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.42/4.49 & ( ord_less @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_952_mult__Suc__right,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( times_times @ nat @ M @ ( suc @ N2 ) ) % 8.42/4.49 = ( plus_plus @ nat @ M @ ( times_times @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_953_nat__mult__div__cancel__disj,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ( K % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 => ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 8.42/4.49 = ( zero_zero @ nat ) ) ) % 8.42/4.49 & ( ( K % 8.42/4.49 != ( zero_zero @ nat ) ) % 8.42/4.49 => ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 8.42/4.49 = ( divide_divide @ nat @ M @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_954_divide__le__eq__numeral1_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [B2: A,W2: num,A2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A2 ) % 8.42/4.49 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_955_le__divide__eq__numeral1_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [A2: A,B2: A,W2: num] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) ) % 8.42/4.49 = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_956_divide__eq__eq__numeral1_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [B2: A,W2: num,A2: A] : % 8.42/4.49 ( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) % 8.42/4.49 = A2 ) % 8.42/4.49 = ( ( ( ( numeral_numeral @ A @ W2 ) % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( B2 % 8.42/4.49 = ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) ) % 8.42/4.49 & ( ( ( numeral_numeral @ A @ W2 ) % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 => ( A2 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_957_eq__divide__eq__numeral1_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,W2: num] : % 8.42/4.49 ( ( A2 % 8.42/4.49 = ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) ) % 8.42/4.49 = ( ( ( ( numeral_numeral @ A @ W2 ) % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) % 8.42/4.49 = B2 ) ) % 8.42/4.49 & ( ( ( numeral_numeral @ A @ W2 ) % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 => ( A2 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_958_less__divide__eq__numeral1_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [A2: A,B2: A,W2: num] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) ) % 8.42/4.49 = ( ord_less @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_959_divide__less__eq__numeral1_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [B2: A,W2: num,A2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A2 ) % 8.42/4.49 = ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_960_nonzero__divide__mult__cancel__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( A2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_961_nonzero__divide__mult__cancel__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( B2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A2 @ B2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_962_div__mult__self1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( B2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 ) % 8.42/4.49 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_963_div__mult__self2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( B2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 ) % 8.42/4.49 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_964_div__mult__self3,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [B2: A,C2: A,A2: A] : % 8.42/4.49 ( ( B2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 ) % 8.42/4.49 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_965_div__mult__self4,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [B2: A,C2: A,A2: A] : % 8.42/4.49 ( ( B2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 ) % 8.42/4.49 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_966_one__le__mult__iff,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N2 ) ) % 8.42/4.49 = ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M ) % 8.42/4.49 & ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_967_mult__le__cancel2,axiom, % 8.42/4.49 ! [M: nat,K: nat,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) % 8.42/4.49 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.42/4.49 => ( ord_less_eq @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_968_nat__mult__le__cancel__disj,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 8.42/4.49 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.42/4.49 => ( ord_less_eq @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_969_div__mult__self__is__m,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ( divide_divide @ nat @ ( times_times @ nat @ M @ N2 ) @ N2 ) % 8.42/4.49 = M ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_970_div__mult__self1__is__m,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ( divide_divide @ nat @ ( times_times @ nat @ N2 @ M ) @ N2 ) % 8.42/4.49 = M ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_971_ab__semigroup__mult__class_Omult__ac_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ab_semigroup_mult @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_972_mult_Oassoc,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semigroup_mult @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_973_mult_Ocommute,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ab_semigroup_mult @ A ) % 8.42/4.49 => ( ( times_times @ A ) % 8.42/4.49 = ( ^ [A4: A,B4: A] : ( times_times @ A @ B4 @ A4 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_974_mult_Oleft__commute,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ab_semigroup_mult @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) % 8.42/4.49 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_975_not__iless0,axiom, % 8.42/4.49 ! [N2: extended_enat] : % 8.42/4.49 ~ ( ord_less @ extended_enat @ N2 @ ( zero_zero @ extended_enat ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_976_enat__less__induct,axiom, % 8.42/4.49 ! [P: extended_enat > $o,N2: extended_enat] : % 8.42/4.49 ( ! [N3: extended_enat] : % 8.42/4.49 ( ! [M6: extended_enat] : % 8.42/4.49 ( ( ord_less @ extended_enat @ M6 @ N3 ) % 8.42/4.49 => ( P @ M6 ) ) % 8.42/4.49 => ( P @ N3 ) ) % 8.42/4.49 => ( P @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_977_mult__right__cancel,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiri6575147826004484403cancel @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ( times_times @ A @ A2 @ C2 ) % 8.42/4.49 = ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( A2 = B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_978_mult__left__cancel,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiri6575147826004484403cancel @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ( times_times @ A @ C2 @ A2 ) % 8.42/4.49 = ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( A2 = B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_979_no__zero__divisors,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiri3467727345109120633visors @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( A2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( B2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( times_times @ A @ A2 @ B2 ) % 8.42/4.49 != ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_980_divisors__zero,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiri3467727345109120633visors @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ( times_times @ A @ A2 @ B2 ) % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 => ( ( A2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 | ( B2 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_981_mult__not__zero,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( mult_zero @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ( times_times @ A @ A2 @ B2 ) % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( A2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 & ( B2 % 8.42/4.49 != ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_982_comm__monoid__mult__class_Omult__1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_monoid_mult @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( times_times @ A @ ( one_one @ A ) @ A2 ) % 8.42/4.49 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_983_mult_Ocomm__neutral,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_monoid_mult @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( times_times @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_984_combine__common__factor,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring @ A ) % 8.42/4.49 => ! [A2: A,E3: A,B2: A,C2: A] : % 8.42/4.49 ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ C2 ) ) % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ E3 ) @ C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_985_distrib__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_986_distrib__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_987_comm__semiring__class_Odistrib,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_semiring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_988_ring__class_Oring__distribs_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_989_ring__class_Oring__distribs_I2_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_990_crossproduct__noteq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiri1453513574482234551roduct @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.42/4.49 ( ( ( A2 != B2 ) % 8.42/4.49 & ( C2 != D2 ) ) % 8.42/4.49 = ( ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) % 8.42/4.49 != ( plus_plus @ A @ ( times_times @ A @ A2 @ D2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_991_crossproduct__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiri1453513574482234551roduct @ A ) % 8.42/4.49 => ! [W2: A,Y: A,X3: A,Z2: A] : % 8.42/4.49 ( ( ( plus_plus @ A @ ( times_times @ A @ W2 @ Y ) @ ( times_times @ A @ X3 @ Z2 ) ) % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ W2 @ Z2 ) @ ( times_times @ A @ X3 @ Y ) ) ) % 8.42/4.49 = ( ( W2 = X3 ) % 8.42/4.49 | ( Y = Z2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_992_times__divide__times__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [X3: A,Y: A,Z2: A,W2: A] : % 8.42/4.49 ( ( times_times @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ Z2 @ W2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ Y @ W2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_993_divide__divide__times__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [X3: A,Y: A,Z2: A,W2: A] : % 8.42/4.49 ( ( divide_divide @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ Z2 @ W2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( times_times @ A @ X3 @ W2 ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_994_divide__divide__eq__left_H,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_995_Suc__mult__cancel1,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ( times_times @ nat @ ( suc @ K ) @ M ) % 8.42/4.49 = ( times_times @ nat @ ( suc @ K ) @ N2 ) ) % 8.42/4.49 = ( M = N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_996_mult__0,axiom, % 8.42/4.49 ! [N2: nat] : % 8.42/4.49 ( ( times_times @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 = ( zero_zero @ nat ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_997_nat__mult__eq__cancel__disj,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ( times_times @ nat @ K @ M ) % 8.42/4.49 = ( times_times @ nat @ K @ N2 ) ) % 8.42/4.49 = ( ( K % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 | ( M = N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_998_mult__le__mono2,axiom, % 8.42/4.49 ! [I: nat,J: nat,K: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.49 => ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_999_mult__le__mono1,axiom, % 8.42/4.49 ! [I: nat,J: nat,K: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.49 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1000_mult__le__mono,axiom, % 8.42/4.49 ! [I: nat,J: nat,K: nat,L: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ K @ L ) % 8.42/4.49 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1001_le__square,axiom, % 8.42/4.49 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1002_le__cube,axiom, % 8.42/4.49 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1003_add__mult__distrib2,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N2 ) ) % 8.42/4.49 = ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1004_add__mult__distrib,axiom, % 8.42/4.49 ! [M: nat,N2: nat,K: nat] : % 8.42/4.49 ( ( times_times @ nat @ ( plus_plus @ nat @ M @ N2 ) @ K ) % 8.42/4.49 = ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1005_left__add__mult__distrib,axiom, % 8.42/4.49 ! [I: nat,U: nat,J: nat,K: nat] : % 8.42/4.49 ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) ) % 8.42/4.49 = ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1006_nat__mult__1,axiom, % 8.42/4.49 ! [N2: nat] : % 8.42/4.49 ( ( times_times @ nat @ ( one_one @ nat ) @ N2 ) % 8.42/4.49 = N2 ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1007_nat__mult__1__right,axiom, % 8.42/4.49 ! [N2: nat] : % 8.42/4.49 ( ( times_times @ nat @ N2 @ ( one_one @ nat ) ) % 8.42/4.49 = N2 ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1008_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordere2520102378445227354miring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1009_zero__le__mult__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 8.42/4.49 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) % 8.42/4.49 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1010_mult__nonneg__nonpos2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_semiring_0 @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1011_mult__nonpos__nonneg,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_semiring_0 @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1012_mult__nonneg__nonpos,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_semiring_0 @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1013_mult__nonneg__nonneg,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_semiring_0 @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1014_split__mult__neg__le,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_semiring_0 @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) % 8.42/4.49 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1015_mult__le__0__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) % 8.42/4.49 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) % 8.42/4.49 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1016_mult__right__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_semiring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1017_mult__right__mono__neg,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ring @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1018_mult__left__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_semiring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1019_mult__nonpos__nonpos,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ring @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1020_mult__left__mono__neg,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ring @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1021_split__mult__pos__le,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ring @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) % 8.42/4.49 | ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1022_zero__le__square,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_ring @ A ) % 8.42/4.49 => ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1023_mult__mono_H,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_semiring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ C2 @ D2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1024_mult__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_semiring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ C2 @ D2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1025_mult__neg__neg,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1026_not__square__less__zero,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_ring @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ~ ( ord_less @ A @ ( times_times @ A @ A2 @ A2 ) @ ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1027_mult__less__0__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) % 8.42/4.49 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1028_mult__neg__pos,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord8928482502909563296strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1029_mult__pos__neg,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord8928482502909563296strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1030_mult__pos__pos,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord8928482502909563296strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.49 => ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1031_mult__pos__neg2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord8928482502909563296strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1032_zero__less__mult__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) % 8.42/4.49 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1033_zero__less__mult__pos,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord8928482502909563296strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1034_zero__less__mult__pos2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord8928482502909563296strict @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A2 ) ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1035_mult__less__cancel__left__neg,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1036_mult__less__cancel__left__pos,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1037_mult__strict__left__mono__neg,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( ord_less @ A @ B2 @ A2 ) % 8.42/4.49 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1038_mult__strict__left__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord8928482502909563296strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1039_mult__less__cancel__left__disj,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 & ( ord_less @ A @ A2 @ B2 ) ) % 8.42/4.49 | ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 & ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1040_mult__strict__right__mono__neg,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( ord_less @ A @ B2 @ A2 ) % 8.42/4.49 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1041_mult__strict__right__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord8928482502909563296strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1042_mult__less__cancel__right__disj,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 & ( ord_less @ A @ A2 @ B2 ) ) % 8.42/4.49 | ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 & ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1043_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord2810124833399127020strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1044_add__scale__eq__noteq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiri1453513574482234551roduct @ A ) % 8.42/4.49 => ! [R2: A,A2: A,B2: A,C2: A,D2: A] : % 8.42/4.49 ( ( R2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ( A2 = B2 ) % 8.42/4.49 & ( C2 != D2 ) ) % 8.42/4.49 => ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R2 @ C2 ) ) % 8.42/4.49 != ( plus_plus @ A @ B2 @ ( times_times @ A @ R2 @ D2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1045_less__1__mult,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [M: A,N2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( one_one @ A ) @ M ) % 8.42/4.49 => ( ( ord_less @ A @ ( one_one @ A ) @ N2 ) % 8.42/4.49 => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1046_frac__eq__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [Y: A,Z2: A,X3: A,W2: A] : % 8.42/4.49 ( ( Y % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( Z2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ( divide_divide @ A @ X3 @ Y ) % 8.42/4.49 = ( divide_divide @ A @ W2 @ Z2 ) ) % 8.42/4.49 = ( ( times_times @ A @ X3 @ Z2 ) % 8.42/4.49 = ( times_times @ A @ W2 @ Y ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1047_divide__eq__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [B2: A,C2: A,A2: A] : % 8.42/4.49 ( ( ( divide_divide @ A @ B2 @ C2 ) % 8.42/4.49 = A2 ) % 8.42/4.49 = ( ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( B2 % 8.42/4.49 = ( times_times @ A @ A2 @ C2 ) ) ) % 8.42/4.49 & ( ( C2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 => ( A2 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1048_eq__divide__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( A2 % 8.42/4.49 = ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( times_times @ A @ A2 @ C2 ) % 8.42/4.49 = B2 ) ) % 8.42/4.49 & ( ( C2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 => ( A2 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1049_divide__eq__imp,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [C2: A,B2: A,A2: A] : % 8.42/4.49 ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( B2 % 8.42/4.49 = ( times_times @ A @ A2 @ C2 ) ) % 8.42/4.49 => ( ( divide_divide @ A @ B2 @ C2 ) % 8.42/4.49 = A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1050_eq__divide__imp,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ( times_times @ A @ A2 @ C2 ) % 8.42/4.49 = B2 ) % 8.42/4.49 => ( A2 % 8.42/4.49 = ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1051_nonzero__divide__eq__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [C2: A,B2: A,A2: A] : % 8.42/4.49 ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ( divide_divide @ A @ B2 @ C2 ) % 8.42/4.49 = A2 ) % 8.42/4.49 = ( B2 % 8.42/4.49 = ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1052_nonzero__eq__divide__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( A2 % 8.42/4.49 = ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ( times_times @ A @ A2 @ C2 ) % 8.42/4.49 = B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1053_power__Suc,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( power @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( power_power @ A @ A2 @ ( suc @ N2 ) ) % 8.42/4.49 = ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1054_power__Suc2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( monoid_mult @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( power_power @ A @ A2 @ ( suc @ N2 ) ) % 8.42/4.49 = ( times_times @ A @ ( power_power @ A @ A2 @ N2 ) @ A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1055_Suc__mult__less__cancel1,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N2 ) ) % 8.42/4.49 = ( ord_less @ nat @ M @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1056_power__add,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( monoid_mult @ A ) % 8.42/4.49 => ! [A2: A,M: nat,N2: nat] : % 8.42/4.49 ( ( power_power @ A @ A2 @ ( plus_plus @ nat @ M @ N2 ) ) % 8.42/4.49 = ( times_times @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1057_mult__less__mono1,axiom, % 8.42/4.49 ! [I: nat,J: nat,K: nat] : % 8.42/4.49 ( ( ord_less @ nat @ I @ J ) % 8.42/4.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.42/4.49 => ( ord_less @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1058_mult__less__mono2,axiom, % 8.42/4.49 ! [I: nat,J: nat,K: nat] : % 8.42/4.49 ( ( ord_less @ nat @ I @ J ) % 8.42/4.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.42/4.49 => ( ord_less @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1059_nat__mult__eq__cancel1,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.42/4.49 => ( ( ( times_times @ nat @ K @ M ) % 8.42/4.49 = ( times_times @ nat @ K @ N2 ) ) % 8.42/4.49 = ( M = N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1060_nat__mult__less__cancel1,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.42/4.49 => ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 8.42/4.49 = ( ord_less @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1061_Suc__mult__le__cancel1,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N2 ) ) % 8.42/4.49 = ( ord_less_eq @ nat @ M @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1062_mult__Suc,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( times_times @ nat @ ( suc @ M ) @ N2 ) % 8.42/4.49 = ( plus_plus @ nat @ N2 @ ( times_times @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1063_mult__eq__self__implies__10,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( M % 8.42/4.49 = ( times_times @ nat @ M @ N2 ) ) % 8.42/4.49 => ( ( N2 % 8.42/4.49 = ( one_one @ nat ) ) % 8.42/4.49 | ( M % 8.42/4.49 = ( zero_zero @ nat ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1064_less__mult__imp__div__less,axiom, % 8.42/4.49 ! [M: nat,I: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ M @ ( times_times @ nat @ I @ N2 ) ) % 8.42/4.49 => ( ord_less @ nat @ ( divide_divide @ nat @ M @ N2 ) @ I ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1065_times__div__less__eq__dividend,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ ( divide_divide @ nat @ M @ N2 ) ) @ M ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1066_div__times__less__eq__dividend,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N2 ) @ N2 ) @ M ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1067_power__odd__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( monoid_mult @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 8.42/4.49 = ( times_times @ A @ A2 @ ( power_power @ A @ ( power_power @ A @ A2 @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1068_Suc__double__not__eq__double,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) % 8.42/4.49 != ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1069_double__not__eq__Suc__double,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) % 8.42/4.49 != ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1070_mult__le__cancel__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ A2 @ B2 ) ) % 8.42/4.49 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1071_mult__le__cancel__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ A2 @ B2 ) ) % 8.42/4.49 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1072_mult__left__less__imp__less,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semiring @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1073_mult__strict__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord8928482502909563296strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less @ A @ C2 @ D2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1074_mult__less__cancel__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ A2 @ B2 ) ) % 8.42/4.49 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1075_mult__right__less__imp__less,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semiring @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1076_mult__strict__mono_H,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord8928482502909563296strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less @ A @ C2 @ D2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1077_mult__less__cancel__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ A2 @ B2 ) ) % 8.42/4.49 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1078_mult__le__cancel__left__neg,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1079_mult__le__cancel__left__pos,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1080_mult__left__le__imp__le,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord8928482502909563296strict @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1081_mult__right__le__imp__le,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord8928482502909563296strict @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1082_mult__le__less__imp__less,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord8928482502909563296strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less @ A @ C2 @ D2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1083_mult__less__le__imp__less,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord8928482502909563296strict @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ C2 @ D2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1084_mult__left__le,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord181362715937106298miring @ A ) % 8.42/4.49 => ! [C2: A,A2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1085_mult__le__one,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord181362715937106298miring @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1086_mult__right__le__one__le,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [X3: A,Y: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.49 => ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ X3 @ Y ) @ X3 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1087_mult__left__le__one__le,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [X3: A,Y: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.49 => ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ Y @ X3 ) @ X3 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1088_sum__squares__le__zero__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [X3: A,Y: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) % 8.42/4.49 = ( ( X3 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 & ( Y % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1089_sum__squares__ge__zero,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_ring @ A ) % 8.42/4.49 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1090_sum__squares__gt__zero__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord4710134922213307826strict @ A ) % 8.42/4.49 => ! [X3: A,Y: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) ) % 8.42/4.49 = ( ( X3 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 | ( Y % 8.42/4.49 != ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1091_not__sum__squares__lt__zero,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_ring @ A ) % 8.42/4.49 => ! [X3: A,Y: A] : % 8.42/4.49 ~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1092_divide__less__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [B2: A,C2: A,A2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1093_less__divide__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1094_neg__divide__less__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [C2: A,B2: A,A2: A] : % 8.42/4.49 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 8.42/4.49 = ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1095_neg__less__divide__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1096_pos__divide__less__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [C2: A,B2: A,A2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 8.42/4.49 = ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1097_pos__less__divide__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1098_mult__imp__div__pos__less,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [Y: A,X3: A,Z2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.49 => ( ( ord_less @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) ) % 8.42/4.49 => ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ Z2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1099_mult__imp__less__div__pos,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [Y: A,Z2: A,X3: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.49 => ( ( ord_less @ A @ ( times_times @ A @ Z2 @ Y ) @ X3 ) % 8.42/4.49 => ( ord_less @ A @ Z2 @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1100_divide__strict__left__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( ord_less @ A @ B2 @ A2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 8.42/4.49 => ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1101_divide__strict__left__mono__neg,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 8.42/4.49 => ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1102_divide__eq__eq__numeral_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [B2: A,C2: A,W2: num] : % 8.42/4.49 ( ( ( divide_divide @ A @ B2 @ C2 ) % 8.42/4.49 = ( numeral_numeral @ A @ W2 ) ) % 8.42/4.49 = ( ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( B2 % 8.42/4.49 = ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) ) % 8.42/4.49 & ( ( C2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 => ( ( numeral_numeral @ A @ W2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1103_eq__divide__eq__numeral_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [W2: num,B2: A,C2: A] : % 8.42/4.49 ( ( ( numeral_numeral @ A @ W2 ) % 8.42/4.49 = ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) % 8.42/4.49 = B2 ) ) % 8.42/4.49 & ( ( C2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 => ( ( numeral_numeral @ A @ W2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1104_add__divide__eq__if__simps_I2_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [Z2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ( Z2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 => ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 ) % 8.42/4.49 = B2 ) ) % 8.42/4.49 & ( ( Z2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 ) % 8.42/4.49 = ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1105_add__divide__eq__if__simps_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [Z2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ( Z2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 => ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) ) % 8.42/4.49 = A2 ) ) % 8.42/4.49 & ( ( Z2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1106_add__frac__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [Y: A,Z2: A,X3: A,W2: A] : % 8.42/4.49 ( ( Y % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( Z2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1107_add__frac__num,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [Y: A,X3: A,Z2: A] : % 8.42/4.49 ( ( Y % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Y ) @ Z2 ) % 8.42/4.49 = ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1108_add__num__frac,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ! [Y: A,Z2: A,X3: A] : % 8.42/4.49 ( ( Y % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ X3 @ Y ) ) % 8.42/4.49 = ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1109_add__divide__eq__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [Z2: A,X3: A,Y: A] : % 8.42/4.49 ( ( Z2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( plus_plus @ A @ X3 @ ( divide_divide @ A @ Y @ Z2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1110_divide__add__eq__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [Z2: A,X3: A,Y: A] : % 8.42/4.49 ( ( Z2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ Y ) % 8.42/4.49 = ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1111_power__gt1__lemma,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 8.42/4.49 => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1112_power__less__power__Suc,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less @ A @ ( one_one @ A ) @ A2 ) % 8.42/4.49 => ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1113_one__less__mult,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 8.42/4.49 => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M ) % 8.42/4.49 => ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1114_n__less__m__mult__n,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M ) % 8.42/4.49 => ( ord_less @ nat @ N2 @ ( times_times @ nat @ M @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1115_n__less__n__mult__m,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M ) % 8.42/4.49 => ( ord_less @ nat @ N2 @ ( times_times @ nat @ N2 @ M ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1116_nat__mult__le__cancel1,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 8.42/4.49 = ( ord_less_eq @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1117_nat__mult__div__cancel1,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.42/4.49 => ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 8.42/4.49 = ( divide_divide @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1118_div__less__iff__less__mult,axiom, % 8.42/4.49 ! [Q4: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 ) % 8.42/4.49 => ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q4 ) @ N2 ) % 8.42/4.49 = ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ Q4 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1119_mult__le__cancel__left1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [C2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) ) % 8.42/4.49 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1120_mult__le__cancel__left2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [C2: A,A2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) ) % 8.42/4.49 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1121_mult__le__cancel__right1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [C2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) ) % 8.42/4.49 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1122_mult__le__cancel__right2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [A2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) ) % 8.42/4.49 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1123_mult__less__cancel__left1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [C2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ ( one_one @ A ) @ B2 ) ) % 8.42/4.49 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1124_mult__less__cancel__left2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [C2: A,A2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 ) % 8.42/4.49 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ A2 @ ( one_one @ A ) ) ) % 8.42/4.49 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1125_mult__less__cancel__right1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [C2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ ( one_one @ A ) @ B2 ) ) % 8.42/4.49 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1126_mult__less__cancel__right2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [A2: A,C2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 ) % 8.42/4.49 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ A2 @ ( one_one @ A ) ) ) % 8.42/4.49 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1127_field__le__mult__one__interval,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [X3: A,Y: A] : % 8.42/4.49 ( ! [Z3: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 ) % 8.42/4.49 => ( ( ord_less @ A @ Z3 @ ( one_one @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X3 ) @ Y ) ) ) % 8.42/4.49 => ( ord_less_eq @ A @ X3 @ Y ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1128_divide__le__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [B2: A,C2: A,A2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1129_le__divide__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1130_divide__left__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1131_neg__divide__le__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [C2: A,B2: A,A2: A] : % 8.42/4.49 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 8.42/4.49 = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1132_neg__le__divide__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1133_pos__divide__le__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [C2: A,B2: A,A2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 8.42/4.49 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1134_pos__le__divide__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1135_mult__imp__div__pos__le,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [Y: A,X3: A,Z2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.49 => ( ( ord_less_eq @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ Z2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1136_mult__imp__le__div__pos,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [Y: A,Z2: A,X3: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ Y ) @ X3 ) % 8.42/4.49 => ( ord_less_eq @ A @ Z2 @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1137_divide__left__mono__neg,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1138_convex__bound__le,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord6961819062388156250ring_1 @ A ) % 8.42/4.49 => ! [X3: A,A2: A,Y: A,U: A,V2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ X3 @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ Y @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 ) % 8.42/4.49 => ( ( ( plus_plus @ A @ U @ V2 ) % 8.42/4.49 = ( one_one @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X3 ) @ ( times_times @ A @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1139_divide__less__eq__numeral_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [B2: A,C2: A,W2: num] : % 8.42/4.49 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1140_less__divide__eq__numeral_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [W2: num,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1141_power__Suc__less,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N2 ) ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1142_mult__2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_numeral @ A ) % 8.42/4.49 => ! [Z2: A] : % 8.42/4.49 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) % 8.42/4.49 = ( plus_plus @ A @ Z2 @ Z2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1143_mult__2__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_numeral @ A ) % 8.42/4.49 => ! [Z2: A] : % 8.42/4.49 ( ( times_times @ A @ Z2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( plus_plus @ A @ Z2 @ Z2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1144_left__add__twice,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_numeral @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) ) % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1145_div__nat__eqI,axiom, % 8.42/4.49 ! [N2: nat,Q4: nat,M: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ Q4 ) @ M ) % 8.42/4.49 => ( ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ ( suc @ Q4 ) ) ) % 8.42/4.49 => ( ( divide_divide @ nat @ M @ N2 ) % 8.42/4.49 = Q4 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1146_less__eq__div__iff__mult__less__eq,axiom, % 8.42/4.49 ! [Q4: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N2 @ Q4 ) ) % 8.42/4.49 = ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q4 ) @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1147_split__div,axiom, % 8.42/4.49 ! [P: nat > $o,M: nat,N2: nat] : % 8.42/4.49 ( ( P @ ( divide_divide @ nat @ M @ N2 ) ) % 8.42/4.49 = ( ( ( N2 % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 => ( P @ ( zero_zero @ nat ) ) ) % 8.42/4.49 & ( ( N2 % 8.42/4.49 != ( zero_zero @ nat ) ) % 8.42/4.49 => ! [I3: nat,J3: nat] : % 8.42/4.49 ( ( ord_less @ nat @ J3 @ N2 ) % 8.42/4.49 => ( ( M % 8.42/4.49 = ( plus_plus @ nat @ ( times_times @ nat @ N2 @ I3 ) @ J3 ) ) % 8.42/4.49 => ( P @ I3 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1148_dividend__less__div__times,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ord_less @ nat @ M @ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N2 ) @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1149_dividend__less__times__div,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ord_less @ nat @ M @ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ N2 @ ( divide_divide @ nat @ M @ N2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1150_nat__bit__induct,axiom, % 8.42/4.49 ! [P: nat > $o,N2: nat] : % 8.42/4.49 ( ( P @ ( zero_zero @ nat ) ) % 8.42/4.49 => ( ! [N3: nat] : % 8.42/4.49 ( ( P @ N3 ) % 8.42/4.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 8.42/4.49 => ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) % 8.42/4.49 => ( ! [N3: nat] : % 8.42/4.49 ( ( P @ N3 ) % 8.42/4.49 => ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) % 8.42/4.49 => ( P @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1151_convex__bound__lt,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord715952674999750819strict @ A ) % 8.42/4.49 => ! [X3: A,A2: A,Y: A,U: A,V2: A] : % 8.42/4.49 ( ( ord_less @ A @ X3 @ A2 ) % 8.42/4.49 => ( ( ord_less @ A @ Y @ A2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 ) % 8.42/4.49 => ( ( ( plus_plus @ A @ U @ V2 ) % 8.42/4.49 = ( one_one @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X3 ) @ ( times_times @ A @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1152_divide__le__eq__numeral_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [B2: A,C2: A,W2: num] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1153_le__divide__eq__numeral_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [W2: num,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) ) % 8.42/4.49 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1154_split__div_H,axiom, % 8.42/4.49 ! [P: nat > $o,M: nat,N2: nat] : % 8.42/4.49 ( ( P @ ( divide_divide @ nat @ M @ N2 ) ) % 8.42/4.49 = ( ( ( N2 % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 & ( P @ ( zero_zero @ nat ) ) ) % 8.42/4.49 | ? [Q5: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ Q5 ) @ M ) % 8.42/4.49 & ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ ( suc @ Q5 ) ) ) % 8.42/4.49 & ( P @ Q5 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1155_power2__sum,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_semiring_1 @ A ) % 8.42/4.49 => ! [X3: A,Y: A] : % 8.42/4.49 ( ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) @ Y ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1156_zero__le__even__power_H,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1157_triangle__def,axiom, % 8.42/4.49 ( nat_triangle % 8.42/4.49 = ( ^ [N: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1158_odd__0__le__power__imp__0__le,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1159_odd__power__less__zero,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 => ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( zero_zero @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1160_arith__geo__mean,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [U: A,X3: A,Y: A] : % 8.42/4.49 ( ( ( power_power @ A @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( times_times @ A @ X3 @ Y ) ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 8.42/4.49 => ( ord_less_eq @ A @ U @ ( divide_divide @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1161_sum__squares__bound,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_field @ A ) % 8.42/4.49 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) @ Y ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1162_set__bit__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_se359711467146920520ations @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A2 ) % 8.42/4.49 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1163_pos2,axiom, % 8.42/4.49 ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1164_unset__bit__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_se359711467146920520ations @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A2 ) % 8.42/4.49 = ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1165_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1166_discrete,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ( ( ord_less @ A ) % 8.42/4.49 = ( ^ [A4: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1167_unique__euclidean__semiring__numeral__class_Odiv__less,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( divide_divide @ A @ A2 @ B2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1168_div__positive,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.42/4.49 => ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1169_zle__add1__eq__le,axiom, % 8.42/4.49 ! [W2: int,Z2: int] : % 8.42/4.49 ( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) ) % 8.42/4.49 = ( ord_less_eq @ int @ W2 @ Z2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1170_set__bit__negative__int__iff,axiom, % 8.42/4.49 ! [N2: nat,K: int] : % 8.42/4.49 ( ( ord_less @ int @ ( bit_se5668285175392031749et_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) % 8.42/4.49 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1171_unset__bit__negative__int__iff,axiom, % 8.42/4.49 ! [N2: nat,K: int] : % 8.42/4.49 ( ( ord_less @ int @ ( bit_se2638667681897837118et_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) % 8.42/4.49 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1172_div__pos__pos__trivial,axiom, % 8.42/4.49 ! [K: int,L: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 8.42/4.49 => ( ( ord_less @ int @ K @ L ) % 8.42/4.49 => ( ( divide_divide @ int @ K @ L ) % 8.42/4.49 = ( zero_zero @ int ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1173_div__neg__neg__trivial,axiom, % 8.42/4.49 ! [K: int,L: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) ) % 8.42/4.49 => ( ( ord_less @ int @ L @ K ) % 8.42/4.49 => ( ( divide_divide @ int @ K @ L ) % 8.42/4.49 = ( zero_zero @ int ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1174_nonneg1__imp__zdiv__pos__iff,axiom, % 8.42/4.49 ! [A2: int,B2: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) ) % 8.42/4.49 = ( ( ord_less_eq @ int @ B2 @ A2 ) % 8.42/4.49 & ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1175_pos__imp__zdiv__nonneg__iff,axiom, % 8.42/4.49 ! [B2: int,A2: int] : % 8.42/4.49 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) ) % 8.42/4.49 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1176_neg__imp__zdiv__nonneg__iff,axiom, % 8.42/4.49 ! [B2: int,A2: int] : % 8.42/4.49 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) ) % 8.42/4.49 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) ) % 8.42/4.49 = ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1177_int__one__le__iff__zero__less,axiom, % 8.42/4.49 ! [Z2: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 ) % 8.42/4.49 = ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1178_pos__imp__zdiv__pos__iff,axiom, % 8.42/4.49 ! [K: int,I: int] : % 8.42/4.49 ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 8.42/4.49 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I @ K ) ) % 8.42/4.49 = ( ord_less_eq @ int @ K @ I ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1179_div__nonpos__pos__le0,axiom, % 8.42/4.49 ! [A2: int,B2: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) % 8.42/4.49 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.42/4.49 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1180_div__nonneg__neg__le0,axiom, % 8.42/4.49 ! [A2: int,B2: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) ) % 8.42/4.49 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1181_div__positive__int,axiom, % 8.42/4.49 ! [L: int,K: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ L @ K ) % 8.42/4.49 => ( ( ord_less @ int @ ( zero_zero @ int ) @ L ) % 8.42/4.49 => ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1182_div__int__pos__iff,axiom, % 8.42/4.49 ! [K: int,L: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) ) % 8.42/4.49 = ( ( K % 8.42/4.49 = ( zero_zero @ int ) ) % 8.42/4.49 | ( L % 8.42/4.49 = ( zero_zero @ int ) ) % 8.42/4.49 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 8.42/4.49 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) % 8.42/4.49 | ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 8.42/4.49 & ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1183_zless__imp__add1__zle,axiom, % 8.42/4.49 ! [W2: int,Z2: int] : % 8.42/4.49 ( ( ord_less @ int @ W2 @ Z2 ) % 8.42/4.49 => ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1184_zdiv__mono2__neg,axiom, % 8.42/4.49 ! [A2: int,B7: int,B2: int] : % 8.42/4.49 ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) % 8.42/4.49 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 ) % 8.42/4.49 => ( ( ord_less_eq @ int @ B7 @ B2 ) % 8.42/4.49 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B7 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1185_zdiv__mono1__neg,axiom, % 8.42/4.49 ! [A2: int,A7: int,B2: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ A2 @ A7 ) % 8.42/4.49 => ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) ) % 8.42/4.49 => ( ord_less_eq @ int @ ( divide_divide @ int @ A7 @ B2 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1186_int__div__pos__eq,axiom, % 8.42/4.49 ! [A2: int,B2: int,Q4: int,R2: int] : % 8.42/4.49 ( ( A2 % 8.42/4.49 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) ) % 8.42/4.49 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 ) % 8.42/4.49 => ( ( ord_less @ int @ R2 @ B2 ) % 8.42/4.49 => ( ( divide_divide @ int @ A2 @ B2 ) % 8.42/4.49 = Q4 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1187_int__div__neg__eq,axiom, % 8.42/4.49 ! [A2: int,B2: int,Q4: int,R2: int] : % 8.42/4.49 ( ( A2 % 8.42/4.49 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) ) % 8.42/4.49 => ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) ) % 8.42/4.49 => ( ( ord_less @ int @ B2 @ R2 ) % 8.42/4.49 => ( ( divide_divide @ int @ A2 @ B2 ) % 8.42/4.49 = Q4 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1188_zdiv__eq__0__iff,axiom, % 8.42/4.49 ! [I: int,K: int] : % 8.42/4.49 ( ( ( divide_divide @ int @ I @ K ) % 8.42/4.49 = ( zero_zero @ int ) ) % 8.42/4.49 = ( ( K % 8.42/4.49 = ( zero_zero @ int ) ) % 8.42/4.49 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I ) % 8.42/4.49 & ( ord_less @ int @ I @ K ) ) % 8.42/4.49 | ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) % 8.42/4.49 & ( ord_less @ int @ K @ I ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1189_zdiv__mono2,axiom, % 8.42/4.49 ! [A2: int,B7: int,B2: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 ) % 8.42/4.49 => ( ( ord_less_eq @ int @ B7 @ B2 ) % 8.42/4.49 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B7 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1190_zdiv__mono1,axiom, % 8.42/4.49 ! [A2: int,A7: int,B2: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ A2 @ A7 ) % 8.42/4.49 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.42/4.49 => ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A7 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1191_split__zdiv,axiom, % 8.42/4.49 ! [P: int > $o,N2: int,K: int] : % 8.42/4.49 ( ( P @ ( divide_divide @ int @ N2 @ K ) ) % 8.42/4.49 = ( ( ( K % 8.42/4.49 = ( zero_zero @ int ) ) % 8.42/4.49 => ( P @ ( zero_zero @ int ) ) ) % 8.42/4.49 & ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 8.42/4.49 => ! [I3: int,J3: int] : % 8.42/4.49 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 ) % 8.42/4.49 & ( ord_less @ int @ J3 @ K ) % 8.42/4.49 & ( N2 % 8.42/4.49 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) ) % 8.42/4.49 => ( P @ I3 ) ) ) % 8.42/4.49 & ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 8.42/4.49 => ! [I3: int,J3: int] : % 8.42/4.49 ( ( ( ord_less @ int @ K @ J3 ) % 8.42/4.49 & ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) ) % 8.42/4.49 & ( N2 % 8.42/4.49 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) ) % 8.42/4.49 => ( P @ I3 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1192_le__imp__0__less,axiom, % 8.42/4.49 ! [Z2: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 8.42/4.49 => ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1193_add1__zle__eq,axiom, % 8.42/4.49 ! [W2: int,Z2: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z2 ) % 8.42/4.49 = ( ord_less @ int @ W2 @ Z2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1194_odd__less__0__iff,axiom, % 8.42/4.49 ! [Z2: int] : % 8.42/4.49 ( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 ) @ ( zero_zero @ int ) ) % 8.42/4.49 = ( ord_less @ int @ Z2 @ ( zero_zero @ int ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1195_pos__zmult__eq__1__iff,axiom, % 8.42/4.49 ! [M: int,N2: int] : % 8.42/4.49 ( ( ord_less @ int @ ( zero_zero @ int ) @ M ) % 8.42/4.49 => ( ( ( times_times @ int @ M @ N2 ) % 8.42/4.49 = ( one_one @ int ) ) % 8.42/4.49 = ( ( M % 8.42/4.49 = ( one_one @ int ) ) % 8.42/4.49 & ( N2 % 8.42/4.49 = ( one_one @ int ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1196_zless__add1__eq,axiom, % 8.42/4.49 ! [W2: int,Z2: int] : % 8.42/4.49 ( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) ) % 8.42/4.49 = ( ( ord_less @ int @ W2 @ Z2 ) % 8.42/4.49 | ( W2 = Z2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1197_int__gr__induct,axiom, % 8.42/4.49 ! [K: int,I: int,P: int > $o] : % 8.42/4.49 ( ( ord_less @ int @ K @ I ) % 8.42/4.49 => ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) % 8.42/4.49 => ( ! [I4: int] : % 8.42/4.49 ( ( ord_less @ int @ K @ I4 ) % 8.42/4.49 => ( ( P @ I4 ) % 8.42/4.49 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) ) % 8.42/4.49 => ( P @ I ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1198_pos__imp__zdiv__neg__iff,axiom, % 8.42/4.49 ! [B2: int,A2: int] : % 8.42/4.49 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.42/4.49 => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) % 8.42/4.49 = ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1199_neg__imp__zdiv__neg__iff,axiom, % 8.42/4.49 ! [B2: int,A2: int] : % 8.42/4.49 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) ) % 8.42/4.49 => ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) % 8.42/4.49 = ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1200_int__div__less__self,axiom, % 8.42/4.49 ! [X3: int,K: int] : % 8.42/4.49 ( ( ord_less @ int @ ( zero_zero @ int ) @ X3 ) % 8.42/4.49 => ( ( ord_less @ int @ ( one_one @ int ) @ K ) % 8.42/4.49 => ( ord_less @ int @ ( divide_divide @ int @ X3 @ K ) @ X3 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1201_div__neg__pos__less0,axiom, % 8.42/4.49 ! [A2: int,B2: int] : % 8.42/4.49 ( ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) % 8.42/4.49 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.42/4.49 => ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1202_odd__nonzero,axiom, % 8.42/4.49 ! [Z2: int] : % 8.42/4.49 ( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 ) % 8.42/4.49 != ( zero_zero @ int ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1203_times__int__code_I2_J,axiom, % 8.42/4.49 ! [L: int] : % 8.42/4.49 ( ( times_times @ int @ ( zero_zero @ int ) @ L ) % 8.42/4.49 = ( zero_zero @ int ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1204_times__int__code_I1_J,axiom, % 8.42/4.49 ! [K: int] : % 8.42/4.49 ( ( times_times @ int @ K @ ( zero_zero @ int ) ) % 8.42/4.49 = ( zero_zero @ int ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1205_bot__enat__def,axiom, % 8.42/4.49 ( ( bot_bot @ extended_enat ) % 8.42/4.49 = ( zero_zero @ extended_enat ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1206_incr__mult__lemma,axiom, % 8.42/4.49 ! [D2: int,P: int > $o,K: int] : % 8.42/4.49 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 ) % 8.42/4.49 => ( ! [X5: int] : % 8.42/4.49 ( ( P @ X5 ) % 8.42/4.49 => ( P @ ( plus_plus @ int @ X5 @ D2 ) ) ) % 8.42/4.49 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 8.42/4.49 => ! [X2: int] : % 8.42/4.49 ( ( P @ X2 ) % 8.42/4.49 => ( P @ ( plus_plus @ int @ X2 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1207_zmult__zless__mono2,axiom, % 8.42/4.49 ! [I: int,J: int,K: int] : % 8.42/4.49 ( ( ord_less @ int @ I @ J ) % 8.42/4.49 => ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 8.42/4.49 => ( ord_less @ int @ ( times_times @ int @ K @ I ) @ ( times_times @ int @ K @ J ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1208_less__int__code_I1_J,axiom, % 8.42/4.49 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1209_q__pos__lemma,axiom, % 8.42/4.49 ! [B7: int,Q6: int,R3: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B7 @ Q6 ) @ R3 ) ) % 8.42/4.49 => ( ( ord_less @ int @ R3 @ B7 ) % 8.42/4.49 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 ) % 8.42/4.49 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q6 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1210_zdiv__mono2__lemma,axiom, % 8.42/4.49 ! [B2: int,Q4: int,R2: int,B7: int,Q6: int,R3: int] : % 8.42/4.49 ( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) % 8.42/4.49 = ( plus_plus @ int @ ( times_times @ int @ B7 @ Q6 ) @ R3 ) ) % 8.42/4.49 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B7 @ Q6 ) @ R3 ) ) % 8.42/4.49 => ( ( ord_less @ int @ R3 @ B7 ) % 8.42/4.49 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 ) % 8.42/4.49 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 ) % 8.42/4.49 => ( ( ord_less_eq @ int @ B7 @ B2 ) % 8.42/4.49 => ( ord_less_eq @ int @ Q4 @ Q6 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1211_zdiv__mono2__neg__lemma,axiom, % 8.42/4.49 ! [B2: int,Q4: int,R2: int,B7: int,Q6: int,R3: int] : % 8.42/4.49 ( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) % 8.42/4.49 = ( plus_plus @ int @ ( times_times @ int @ B7 @ Q6 ) @ R3 ) ) % 8.42/4.49 => ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B7 @ Q6 ) @ R3 ) @ ( zero_zero @ int ) ) % 8.42/4.49 => ( ( ord_less @ int @ R2 @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 ) % 8.42/4.49 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B7 ) % 8.42/4.49 => ( ( ord_less_eq @ int @ B7 @ B2 ) % 8.42/4.49 => ( ord_less_eq @ int @ Q6 @ Q4 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1212_unique__quotient__lemma,axiom, % 8.42/4.49 ! [B2: int,Q6: int,R3: int,Q4: int,R2: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R3 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) ) % 8.42/4.49 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 ) % 8.42/4.49 => ( ( ord_less @ int @ R3 @ B2 ) % 8.42/4.49 => ( ( ord_less @ int @ R2 @ B2 ) % 8.42/4.49 => ( ord_less_eq @ int @ Q6 @ Q4 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1213_unique__quotient__lemma__neg,axiom, % 8.42/4.49 ! [B2: int,Q6: int,R3: int,Q4: int,R2: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R3 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) ) % 8.42/4.49 => ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) ) % 8.42/4.49 => ( ( ord_less @ int @ B2 @ R2 ) % 8.42/4.49 => ( ( ord_less @ int @ B2 @ R3 ) % 8.42/4.49 => ( ord_less_eq @ int @ Q4 @ Q6 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1214_enat__0__less__mult__iff,axiom, % 8.42/4.49 ! [M: extended_enat,N2: extended_enat] : % 8.42/4.49 ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ ( times_times @ extended_enat @ M @ N2 ) ) % 8.42/4.49 = ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ M ) % 8.42/4.49 & ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1215_realpow__pos__nth2,axiom, % 8.42/4.49 ! [A2: real,N2: nat] : % 8.42/4.49 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.42/4.49 => ? [R4: real] : % 8.42/4.49 ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 ) % 8.42/4.49 & ( ( power_power @ real @ R4 @ ( suc @ N2 ) ) % 8.42/4.49 = A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1216_realpow__pos__nth__unique,axiom, % 8.42/4.49 ! [N2: nat,A2: real] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.42/4.49 => ? [X5: real] : % 8.42/4.49 ( ( ord_less @ real @ ( zero_zero @ real ) @ X5 ) % 8.42/4.49 & ( ( power_power @ real @ X5 @ N2 ) % 8.42/4.49 = A2 ) % 8.42/4.49 & ! [Y5: real] : % 8.42/4.49 ( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) % 8.42/4.49 & ( ( power_power @ real @ Y5 @ N2 ) % 8.42/4.49 = A2 ) ) % 8.42/4.49 => ( Y5 = X5 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1217_realpow__pos__nth,axiom, % 8.42/4.49 ! [N2: nat,A2: real] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.42/4.49 => ? [R4: real] : % 8.42/4.49 ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 ) % 8.42/4.49 & ( ( power_power @ real @ R4 @ N2 ) % 8.42/4.49 = A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1218_mult__le__cancel__iff2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [Z2: A,X3: A,Y: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ X3 ) @ ( times_times @ A @ Z2 @ Y ) ) % 8.42/4.49 = ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1219_mult__le__cancel__iff1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [Z2: A,X3: A,Y: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ Y @ Z2 ) ) % 8.42/4.49 = ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1220_divides__aux__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ! [Q4: A,R2: A] : % 8.42/4.49 ( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q4 @ R2 ) ) % 8.42/4.49 = ( R2 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1221_low__def,axiom, % 8.42/4.49 ( vEBT_VEBT_low % 8.42/4.49 = ( ^ [X: nat,N: nat] : ( modulo_modulo @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1222_even__succ__div__exp,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1223_set__decode__Suc,axiom, % 8.42/4.49 ! [N2: nat,X3: nat] : % 8.42/4.49 ( ( member @ nat @ ( suc @ N2 ) @ ( nat_set_decode @ X3 ) ) % 8.42/4.49 = ( member @ nat @ N2 @ ( nat_set_decode @ ( divide_divide @ nat @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1224_length__product,axiom, % 8.42/4.49 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 8.42/4.49 ( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs @ Ys ) ) % 8.42/4.49 = ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1225_mult__less__iff1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [Z2: A,X3: A,Y: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ Y @ Z2 ) ) % 8.42/4.49 = ( ord_less @ A @ X3 @ Y ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1226_num_Osize__gen_I2_J,axiom, % 8.42/4.49 ! [X22: num] : % 8.42/4.49 ( ( size_num @ ( bit0 @ X22 ) ) % 8.42/4.49 = ( plus_plus @ nat @ ( size_num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1227_nat__dvd__1__iff__1,axiom, % 8.42/4.49 ! [M: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ M @ ( one_one @ nat ) ) % 8.42/4.49 = ( M % 8.42/4.49 = ( one_one @ nat ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1228_dvd__0__left__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_semiring_1 @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 = ( A2 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1229_dvd__0__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_semiring_1 @ A ) % 8.42/4.49 => ! [A2: A] : ( dvd_dvd @ A @ A2 @ ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1230_dvd__add__triv__right__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_s4317794764714335236cancel @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) ) % 8.42/4.49 = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1231_dvd__add__triv__left__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_s4317794764714335236cancel @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) ) % 8.42/4.49 = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1232_bits__mod__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1233_mod__self,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_modulo @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ A2 @ A2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1234_mod__by__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_modulo @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1235_mod__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_modulo @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1236_dvd__1__left,axiom, % 8.42/4.49 ! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1237_dvd__1__iff__1,axiom, % 8.42/4.49 ! [M: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.49 = ( M % 8.42/4.49 = ( suc @ ( zero_zero @ nat ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1238_div__dvd__div,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ A2 @ C2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( divide_divide @ A @ C2 @ A2 ) ) % 8.42/4.49 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1239_not__real__square__gt__zero,axiom, % 8.42/4.49 ! [X3: real] : % 8.42/4.49 ( ( ~ ( ord_less @ real @ ( zero_zero @ real ) @ ( times_times @ real @ X3 @ X3 ) ) ) % 8.42/4.49 = ( X3 % 8.42/4.49 = ( zero_zero @ real ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1240_mod__add__self2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 ) % 8.42/4.49 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1241_mod__add__self1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 ) % 8.42/4.49 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1242_nat__mult__dvd__cancel__disj,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 8.42/4.49 = ( ( K % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 | ( dvd_dvd @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1243_mod__less,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.49 => ( ( modulo_modulo @ nat @ M @ N2 ) % 8.42/4.49 = M ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1244_dvd__mult__cancel__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( idom @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ( C2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 | ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1245_dvd__mult__cancel__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( idom @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( ( C2 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 | ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1246_dvd__times__left__cancel__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( A2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) % 8.42/4.49 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1247_dvd__times__right__cancel__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( A2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) ) % 8.42/4.49 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1248_unit__prod,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1249_dvd__add__times__triv__right__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_s4317794764714335236cancel @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) ) ) % 8.42/4.49 = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1250_dvd__add__times__triv__left__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_s4317794764714335236cancel @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A2 ) @ B2 ) ) % 8.42/4.49 = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1251_mod__mult__self1__is__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A2 ) @ B2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1252_mod__mult__self2__is__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1253_bits__mod__by__1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1254_mod__by__1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_modulo @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1255_dvd__div__mult__self,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A2 ) @ A2 ) % 8.42/4.49 = B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1256_dvd__mult__div__cancel,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ A2 ) ) % 8.42/4.49 = B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1257_unit__div,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1258_unit__div__1__unit,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( one_one @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1259_unit__div__1__div__1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) % 8.42/4.49 = A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1260_div__add,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ C2 @ A2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ C2 @ B2 ) % 8.42/4.49 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1261_bits__mod__div__trivial,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1262_mod__div__trivial,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1263_mod__mult__self4,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [B2: A,C2: A,A2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 ) % 8.42/4.49 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1264_mod__mult__self3,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [C2: A,B2: A,A2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 ) % 8.42/4.49 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1265_mod__mult__self2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 ) % 8.42/4.49 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1266_mod__mult__self1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 ) % 8.42/4.49 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1267_dvd__imp__mod__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_modulo @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( modulo_modulo @ A @ B2 @ A2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1268_mod__by__Suc__0,axiom, % 8.42/4.49 ! [M: nat] : % 8.42/4.49 ( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.49 = ( zero_zero @ nat ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1269_set__decode__zero,axiom, % 8.42/4.49 ( ( nat_set_decode @ ( zero_zero @ nat ) ) % 8.42/4.49 = ( bot_bot @ ( set @ nat ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1270_set__encode__inverse,axiom, % 8.42/4.49 ! [A3: set @ nat] : % 8.42/4.49 ( ( finite_finite2 @ nat @ A3 ) % 8.42/4.49 => ( ( nat_set_decode @ ( nat_set_encode @ A3 ) ) % 8.42/4.49 = A3 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1271_unit__div__mult__self,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A2 ) @ A2 ) % 8.42/4.49 = B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1272_unit__mult__div__div,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) % 8.42/4.49 = ( divide_divide @ A @ B2 @ A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1273_even__Suc,axiom, % 8.42/4.49 ! [N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) % 8.42/4.49 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1274_even__Suc__Suc__iff,axiom, % 8.42/4.49 ! [N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ N2 ) ) ) % 8.42/4.49 = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1275_pow__divides__pow__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_gcd @ A ) % 8.42/4.49 => ! [N2: nat,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) % 8.42/4.49 = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1276_Suc__mod__mult__self4,axiom, % 8.42/4.49 ! [N2: nat,K: nat,M: nat] : % 8.42/4.49 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ K ) @ M ) ) @ N2 ) % 8.42/4.49 = ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1277_Suc__mod__mult__self3,axiom, % 8.42/4.49 ! [K: nat,N2: nat,M: nat] : % 8.42/4.49 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N2 ) @ M ) ) @ N2 ) % 8.42/4.49 = ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1278_Suc__mod__mult__self2,axiom, % 8.42/4.49 ! [M: nat,N2: nat,K: nat] : % 8.42/4.49 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N2 @ K ) ) ) @ N2 ) % 8.42/4.49 = ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1279_Suc__mod__mult__self1,axiom, % 8.42/4.49 ! [M: nat,K: nat,N2: nat] : % 8.42/4.49 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ K @ N2 ) ) ) @ N2 ) % 8.42/4.49 = ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1280_even__add,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_parity @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) % 8.42/4.49 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1281_odd__add,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_parity @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) % 8.42/4.49 = ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) % 8.42/4.49 != ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1282_odd__Suc__div__two,axiom, % 8.42/4.49 ! [N2: nat] : % 8.42/4.49 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 => ( ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1283_even__Suc__div__two,axiom, % 8.42/4.49 ! [N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 => ( ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1284_mod2__Suc__Suc,axiom, % 8.42/4.49 ! [M: nat] : % 8.42/4.49 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1285_Suc__times__numeral__mod__eq,axiom, % 8.42/4.49 ! [K: num,N2: nat] : % 8.42/4.49 ( ( ( numeral_numeral @ nat @ K ) % 8.42/4.49 != ( one_one @ nat ) ) % 8.42/4.49 => ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N2 ) ) @ ( numeral_numeral @ nat @ K ) ) % 8.42/4.49 = ( one_one @ nat ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1286_set__decode__0,axiom, % 8.42/4.49 ! [X3: nat] : % 8.42/4.49 ( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X3 ) ) % 8.42/4.49 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X3 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1287_zero__le__power__eq__numeral,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [A2: A,W2: num] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) ) % 8.42/4.49 = ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) ) % 8.42/4.49 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) ) % 8.42/4.49 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1288_power__less__zero__eq__numeral,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [A2: A,W2: num] : % 8.42/4.49 ( ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) ) % 8.42/4.49 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) ) % 8.42/4.49 & ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1289_power__less__zero__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less @ A @ ( power_power @ A @ A2 @ N2 ) @ ( zero_zero @ A ) ) % 8.42/4.49 = ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 & ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1290_even__plus__one__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_parity @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) % 8.42/4.49 = ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1291_not__mod__2__eq__1__eq__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_parity @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 != ( one_one @ A ) ) % 8.42/4.49 = ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1292_not__mod__2__eq__0__eq__1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_parity @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 = ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( one_one @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1293_not__mod2__eq__Suc__0__eq__0,axiom, % 8.42/4.49 ! [N2: nat] : % 8.42/4.49 ( ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.49 != ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.49 = ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( zero_zero @ nat ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1294_add__self__mod__2,axiom, % 8.42/4.49 ! [M: nat] : % 8.42/4.49 ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( zero_zero @ nat ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1295_zero__less__power__eq__numeral,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [A2: A,W2: num] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) ) % 8.42/4.49 = ( ( ( numeral_numeral @ nat @ W2 ) % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) ) % 8.42/4.49 & ( A2 % 8.42/4.49 != ( zero_zero @ A ) ) ) % 8.42/4.49 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) ) % 8.42/4.49 & ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1296_even__succ__div__two,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid5411537665997757685th_nat @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1297_odd__succ__div__two,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid5411537665997757685th_nat @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1298_even__succ__div__2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1299_even__power,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_parity @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A2 @ N2 ) ) % 8.42/4.49 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1300_mod2__gr__0,axiom, % 8.42/4.49 ! [M: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 8.42/4.49 = ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( one_one @ nat ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1301_odd__two__times__div__two__succ,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid5411537665997757685th_nat @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 => ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) ) % 8.42/4.49 = A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1302_power__le__zero__eq__numeral,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [A2: A,W2: num] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) ) % 8.42/4.49 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W2 ) ) % 8.42/4.49 & ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) ) % 8.42/4.49 & ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) % 8.42/4.49 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) ) % 8.42/4.49 & ( A2 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1303_even__succ__mod__exp,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.42/4.49 = ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1304_real__arch__pow,axiom, % 8.42/4.49 ! [X3: real,Y: real] : % 8.42/4.49 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 8.42/4.49 => ? [N3: nat] : ( ord_less @ real @ Y @ ( power_power @ real @ X3 @ N3 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1305_real__arch__pow__inv,axiom, % 8.42/4.49 ! [Y: real,X3: real] : % 8.42/4.49 ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 8.42/4.49 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 8.42/4.49 => ? [N3: nat] : ( ord_less @ real @ ( power_power @ real @ X3 @ N3 ) @ Y ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1306_int__distrib_I2_J,axiom, % 8.42/4.49 ! [W2: int,Z1: int,Z22: int] : % 8.42/4.49 ( ( times_times @ int @ W2 @ ( plus_plus @ int @ Z1 @ Z22 ) ) % 8.42/4.49 = ( plus_plus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1307_int__distrib_I1_J,axiom, % 8.42/4.49 ! [Z1: int,Z22: int,W2: int] : % 8.42/4.49 ( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W2 ) % 8.42/4.49 = ( plus_plus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1308_zdvd__period,axiom, % 8.42/4.49 ! [A2: int,D2: int,X3: int,T2: int,C2: int] : % 8.42/4.49 ( ( dvd_dvd @ int @ A2 @ D2 ) % 8.42/4.49 => ( ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ X3 @ T2 ) ) % 8.42/4.49 = ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ ( plus_plus @ int @ X3 @ ( times_times @ int @ C2 @ D2 ) ) @ T2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1309_zdvd__reduce,axiom, % 8.42/4.49 ! [K: int,N2: int,M: int] : % 8.42/4.49 ( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N2 @ ( times_times @ int @ K @ M ) ) ) % 8.42/4.49 = ( dvd_dvd @ int @ K @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1310_less__eq__real__def,axiom, % 8.42/4.49 ( ( ord_less_eq @ real ) % 8.42/4.49 = ( ^ [X: real,Y2: real] : % 8.42/4.49 ( ( ord_less @ real @ X @ Y2 ) % 8.42/4.49 | ( X = Y2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1311_mod__eq__0__iff__dvd,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_modulo @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ( modulo_modulo @ A @ A2 @ B2 ) % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 = ( dvd_dvd @ A @ B2 @ A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1312_dvd__eq__mod__eq__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_modulo @ A ) % 8.42/4.49 => ( ( dvd_dvd @ A ) % 8.42/4.49 = ( ^ [A4: A,B4: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ B4 @ A4 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1313_mod__0__imp__dvd,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_modulo @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ( modulo_modulo @ A @ A2 @ B2 ) % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 => ( dvd_dvd @ A @ B2 @ A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1314_dvd__refl,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_monoid_mult @ A ) % 8.42/4.49 => ! [A2: A] : ( dvd_dvd @ A @ A2 @ A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1315_dvd__trans,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_monoid_mult @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ B2 @ C2 ) % 8.42/4.49 => ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1316_dvd__mod__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_modulo @ A ) % 8.42/4.49 => ! [C2: A,B2: A,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ C2 @ B2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) % 8.42/4.49 = ( dvd_dvd @ A @ C2 @ A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1317_dvd__mod__imp__dvd,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_modulo @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ C2 @ B2 ) % 8.42/4.49 => ( dvd_dvd @ A @ C2 @ A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1318_int__ge__induct,axiom, % 8.42/4.49 ! [K: int,I: int,P: int > $o] : % 8.42/4.49 ( ( ord_less_eq @ int @ K @ I ) % 8.42/4.49 => ( ( P @ K ) % 8.42/4.49 => ( ! [I4: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ K @ I4 ) % 8.42/4.49 => ( ( P @ I4 ) % 8.42/4.49 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) ) % 8.42/4.49 => ( P @ I ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1319_unit__imp__mod__eq__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid3725896446679973847miring @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( modulo_modulo @ A @ A2 @ B2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1320_mod__greater__zero__iff__not__dvd,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ N2 ) ) % 8.42/4.49 = ( ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1321_mod__add__right__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) % 8.42/4.49 = ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1322_mod__add__left__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 ) % 8.42/4.49 = ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1323_mod__add__cong,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [A2: A,C2: A,A7: A,B2: A,B7: A] : % 8.42/4.49 ( ( ( modulo_modulo @ A @ A2 @ C2 ) % 8.42/4.49 = ( modulo_modulo @ A @ A7 @ C2 ) ) % 8.42/4.49 => ( ( ( modulo_modulo @ A @ B2 @ C2 ) % 8.42/4.49 = ( modulo_modulo @ A @ B7 @ C2 ) ) % 8.42/4.49 => ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( modulo_modulo @ A @ ( plus_plus @ A @ A7 @ B7 ) @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1324_mod__add__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) % 8.42/4.49 = ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1325_dvd__0__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_semiring_1 @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( A2 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1326_dvd__field__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( field @ A ) % 8.42/4.49 => ( ( dvd_dvd @ A ) % 8.42/4.49 = ( ^ [A4: A,B4: A] : % 8.42/4.49 ( ( A4 % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 => ( B4 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1327_mod__Suc__Suc__eq,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) ) @ N2 ) % 8.42/4.49 = ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1328_mod__Suc__eq,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( modulo_modulo @ nat @ ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) @ N2 ) % 8.42/4.49 = ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1329_dvd__productE,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_gcd @ A ) % 8.42/4.49 => ! [P4: A,A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ P4 @ ( times_times @ A @ A2 @ B2 ) ) % 8.42/4.49 => ~ ! [X5: A,Y4: A] : % 8.42/4.49 ( ( P4 % 8.42/4.49 = ( times_times @ A @ X5 @ Y4 ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ X5 @ A2 ) % 8.42/4.49 => ~ ( dvd_dvd @ A @ Y4 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1330_division__decomp,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_gcd @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 => ? [B9: A,C6: A] : % 8.42/4.49 ( ( A2 % 8.42/4.49 = ( times_times @ A @ B9 @ C6 ) ) % 8.42/4.49 & ( dvd_dvd @ A @ B9 @ B2 ) % 8.42/4.49 & ( dvd_dvd @ A @ C6 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1331_dvdE,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( dvd @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ B2 @ A2 ) % 8.42/4.49 => ~ ! [K2: A] : % 8.42/4.49 ( A2 % 8.42/4.49 != ( times_times @ A @ B2 @ K2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1332_dvdI,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( dvd @ A ) % 8.42/4.49 => ! [A2: A,B2: A,K: A] : % 8.42/4.49 ( ( A2 % 8.42/4.49 = ( times_times @ A @ B2 @ K ) ) % 8.42/4.49 => ( dvd_dvd @ A @ B2 @ A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1333_dvd__def,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( dvd @ A ) % 8.42/4.49 => ( ( dvd_dvd @ A ) % 8.42/4.49 = ( ^ [B4: A,A4: A] : % 8.42/4.49 ? [K3: A] : % 8.42/4.49 ( A4 % 8.42/4.49 = ( times_times @ A @ B4 @ K3 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1334_dvd__mult,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_monoid_mult @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ C2 ) % 8.42/4.49 => ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1335_dvd__mult2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_monoid_mult @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ B2 ) % 8.42/4.49 => ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1336_dvd__mult__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_monoid_mult @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 => ( dvd_dvd @ A @ A2 @ C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1337_dvd__triv__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_monoid_mult @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1338_mult__dvd__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_monoid_mult @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ C2 @ D2 ) % 8.42/4.49 => ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1339_dvd__mult__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_monoid_mult @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 => ( dvd_dvd @ A @ B2 @ C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1340_dvd__triv__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_monoid_mult @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1341_one__dvd,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_monoid_mult @ A ) % 8.42/4.49 => ! [A2: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1342_unit__imp__dvd,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( dvd_dvd @ A @ B2 @ A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1343_dvd__unit__imp__unit,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1344_dvd__add__right__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_s4317794764714335236cancel @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1345_dvd__add__left__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_s4317794764714335236cancel @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ C2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1346_dvd__add,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_semiring_1 @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ A2 @ C2 ) % 8.42/4.49 => ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1347_dvd__div__eq__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_divide @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ C2 @ A2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ C2 @ B2 ) % 8.42/4.49 => ( ( ( divide_divide @ A @ A2 @ C2 ) % 8.42/4.49 = ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( A2 = B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1348_dvd__div__eq__cancel,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_divide @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( ( divide_divide @ A @ A2 @ C2 ) % 8.42/4.49 = ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ C2 @ A2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ C2 @ B2 ) % 8.42/4.49 => ( A2 = B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1349_div__div__div__same,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [D2: A,B2: A,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ D2 @ B2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ B2 @ A2 ) % 8.42/4.49 => ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1350_mod__less__eq__dividend,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N2 ) @ M ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1351_gcd__nat_Oextremum__uniqueI,axiom, % 8.42/4.49 ! [A2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 ) % 8.42/4.49 => ( A2 % 8.42/4.49 = ( zero_zero @ nat ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1352_gcd__nat_Onot__eq__extremum,axiom, % 8.42/4.49 ! [A2: nat] : % 8.42/4.49 ( ( A2 % 8.42/4.49 != ( zero_zero @ nat ) ) % 8.42/4.49 = ( ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) ) % 8.42/4.49 & ( A2 % 8.42/4.49 != ( zero_zero @ nat ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1353_gcd__nat_Oextremum__unique,axiom, % 8.42/4.49 ! [A2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 ) % 8.42/4.49 = ( A2 % 8.42/4.49 = ( zero_zero @ nat ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1354_gcd__nat_Oextremum__strict,axiom, % 8.42/4.49 ! [A2: nat] : % 8.42/4.49 ~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 ) % 8.42/4.49 & ( ( zero_zero @ nat ) % 8.42/4.49 != A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1355_gcd__nat_Oextremum,axiom, % 8.42/4.49 ! [A2: nat] : ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1356_zdvd__antisym__nonneg,axiom, % 8.42/4.49 ! [M: int,N2: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M ) % 8.42/4.49 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) % 8.42/4.49 => ( ( dvd_dvd @ int @ M @ N2 ) % 8.42/4.49 => ( ( dvd_dvd @ int @ N2 @ M ) % 8.42/4.49 => ( M = N2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1357_zdvd__mult__cancel,axiom, % 8.42/4.49 ! [K: int,M: int,N2: int] : % 8.42/4.49 ( ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N2 ) ) % 8.42/4.49 => ( ( K % 8.42/4.49 != ( zero_zero @ int ) ) % 8.42/4.49 => ( dvd_dvd @ int @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1358_even__iff__mod__2__eq__zero,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_parity @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 = ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1359_subset__decode__imp__le,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) ) % 8.42/4.49 => ( ord_less_eq @ nat @ M @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1360_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1361_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.49 => ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1362_mod__eq__self__iff__div__eq__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid3725896446679973847miring @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ( modulo_modulo @ A @ A2 @ B2 ) % 8.42/4.49 = A2 ) % 8.42/4.49 = ( ( divide_divide @ A @ A2 @ B2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1363_mod__eqE,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid8851590272496341667cancel @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( ( modulo_modulo @ A @ A2 @ C2 ) % 8.42/4.49 = ( modulo_modulo @ A @ B2 @ C2 ) ) % 8.42/4.49 => ~ ! [D3: A] : % 8.42/4.49 ( B2 % 8.42/4.49 != ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ D3 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1364_div__add1__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid3128863361964157862miring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1365_not__is__unit__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ~ ( dvd_dvd @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1366_minf_I10_J,axiom, % 8.42/4.49 ! [B: $tType] : % 8.42/4.49 ( ( ( plus @ B ) % 8.42/4.49 & ( linorder @ B ) % 8.42/4.49 & ( dvd @ B ) ) % 8.42/4.49 => ! [D2: B,S: B] : % 8.42/4.49 ? [Z3: B] : % 8.42/4.49 ! [X2: B] : % 8.42/4.49 ( ( ord_less @ B @ X2 @ Z3 ) % 8.42/4.49 => ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) % 8.42/4.49 = ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1367_minf_I9_J,axiom, % 8.42/4.49 ! [B: $tType] : % 8.42/4.49 ( ( ( plus @ B ) % 8.42/4.49 & ( linorder @ B ) % 8.42/4.49 & ( dvd @ B ) ) % 8.42/4.49 => ! [D2: B,S: B] : % 8.42/4.49 ? [Z3: B] : % 8.42/4.49 ! [X2: B] : % 8.42/4.49 ( ( ord_less @ B @ X2 @ Z3 ) % 8.42/4.49 => ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) % 8.42/4.49 = ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1368_pinf_I10_J,axiom, % 8.42/4.49 ! [B: $tType] : % 8.42/4.49 ( ( ( plus @ B ) % 8.42/4.49 & ( linorder @ B ) % 8.42/4.49 & ( dvd @ B ) ) % 8.42/4.49 => ! [D2: B,S: B] : % 8.42/4.49 ? [Z3: B] : % 8.42/4.49 ! [X2: B] : % 8.42/4.49 ( ( ord_less @ B @ Z3 @ X2 ) % 8.42/4.49 => ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) % 8.42/4.49 = ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1369_pinf_I9_J,axiom, % 8.42/4.49 ! [B: $tType] : % 8.42/4.49 ( ( ( plus @ B ) % 8.42/4.49 & ( linorder @ B ) % 8.42/4.49 & ( dvd @ B ) ) % 8.42/4.49 => ! [D2: B,S: B] : % 8.42/4.49 ? [Z3: B] : % 8.42/4.49 ! [X2: B] : % 8.42/4.49 ( ( ord_less @ B @ Z3 @ X2 ) % 8.42/4.49 => ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) % 8.42/4.49 = ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1370_dvd__div__eq__0__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_divide @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ B2 @ A2 ) % 8.42/4.49 => ( ( ( divide_divide @ A @ A2 @ B2 ) % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 = ( A2 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1371_unit__mult__right__cancel,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( ( times_times @ A @ B2 @ A2 ) % 8.42/4.49 = ( times_times @ A @ C2 @ A2 ) ) % 8.42/4.49 = ( B2 = C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1372_unit__mult__left__cancel,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( ( times_times @ A @ A2 @ B2 ) % 8.42/4.49 = ( times_times @ A @ A2 @ C2 ) ) % 8.42/4.49 = ( B2 = C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1373_mult__unit__dvd__iff_H,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1374_dvd__mult__unit__iff_H,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1375_mult__unit__dvd__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1376_dvd__mult__unit__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1377_is__unit__mult__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) % 8.42/4.49 = ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 & ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1378_mod__Suc,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ( ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) % 8.42/4.49 = N2 ) % 8.42/4.49 => ( ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) % 8.42/4.49 = ( zero_zero @ nat ) ) ) % 8.42/4.49 & ( ( ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) % 8.42/4.49 != N2 ) % 8.42/4.49 => ( ( modulo_modulo @ nat @ ( suc @ M ) @ N2 ) % 8.42/4.49 = ( suc @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1379_finite__set__decode,axiom, % 8.42/4.49 ! [N2: nat] : ( finite_finite2 @ nat @ ( nat_set_decode @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1380_mod__induct,axiom, % 8.42/4.49 ! [P: nat > $o,N2: nat,P4: nat,M: nat] : % 8.42/4.49 ( ( P @ N2 ) % 8.42/4.49 => ( ( ord_less @ nat @ N2 @ P4 ) % 8.42/4.49 => ( ( ord_less @ nat @ M @ P4 ) % 8.42/4.49 => ( ! [N3: nat] : % 8.42/4.49 ( ( ord_less @ nat @ N3 @ P4 ) % 8.42/4.49 => ( ( P @ N3 ) % 8.42/4.49 => ( P @ ( modulo_modulo @ nat @ ( suc @ N3 ) @ P4 ) ) ) ) % 8.42/4.49 => ( P @ M ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1381_dvd__div__mult,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [C2: A,B2: A,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ C2 @ B2 ) % 8.42/4.49 => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 ) % 8.42/4.49 = ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1382_div__mult__swap,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [C2: A,B2: A,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ C2 @ B2 ) % 8.42/4.49 => ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1383_div__div__eq__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [C2: A,B2: A,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ C2 @ B2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ B2 @ A2 ) % 8.42/4.49 => ( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1384_dvd__div__mult2__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [B2: A,C2: A,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) % 8.42/4.49 => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1385_dvd__mult__imp__div,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) % 8.42/4.49 => ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1386_div__mult__div__if__dvd,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [B2: A,A2: A,D2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ B2 @ A2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ D2 @ C2 ) % 8.42/4.49 => ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( divide_divide @ A @ C2 @ D2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1387_gcd__nat__induct,axiom, % 8.42/4.49 ! [P: nat > nat > $o,M: nat,N2: nat] : % 8.42/4.49 ( ! [M3: nat] : ( P @ M3 @ ( zero_zero @ nat ) ) % 8.42/4.49 => ( ! [M3: nat,N3: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 8.42/4.49 => ( ( P @ N3 @ ( modulo_modulo @ nat @ M3 @ N3 ) ) % 8.42/4.49 => ( P @ M3 @ N3 ) ) ) % 8.42/4.49 => ( P @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1388_mod__less__divisor,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ord_less @ nat @ ( modulo_modulo @ nat @ M @ N2 ) @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1389_dvd__div__unit__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1390_div__unit__dvd__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1391_unit__div__cancel,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( ( divide_divide @ A @ B2 @ A2 ) % 8.42/4.49 = ( divide_divide @ A @ C2 @ A2 ) ) % 8.42/4.49 = ( B2 = C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1392_div__plus__div__distrib__dvd__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [C2: A,B2: A,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ C2 @ B2 ) % 8.42/4.49 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1393_div__plus__div__distrib__dvd__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ C2 @ A2 ) % 8.42/4.49 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1394_mod__Suc__le__divisor,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ ( suc @ N2 ) ) @ N2 ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1395_mod__eq__0D,axiom, % 8.42/4.49 ! [M: nat,D2: nat] : % 8.42/4.49 ( ( ( modulo_modulo @ nat @ M @ D2 ) % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 => ? [Q2: nat] : % 8.42/4.49 ( M % 8.42/4.49 = ( times_times @ nat @ D2 @ Q2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1396_le__imp__power__dvd,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_semiring_1 @ A ) % 8.42/4.49 => ! [M: nat,N2: nat,A2: A] : % 8.42/4.49 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.49 => ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1397_power__le__dvd,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_semiring_1 @ A ) % 8.42/4.49 => ! [A2: A,N2: nat,B2: A,M: nat] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N2 ) @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.49 => ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1398_dvd__power__le,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_semiring_1 @ A ) % 8.42/4.49 => ! [X3: A,Y: A,N2: nat,M: nat] : % 8.42/4.49 ( ( dvd_dvd @ A @ X3 @ Y ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ N2 @ M ) % 8.42/4.49 => ( dvd_dvd @ A @ ( power_power @ A @ X3 @ N2 ) @ ( power_power @ A @ Y @ M ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1399_nat__mod__eq__iff,axiom, % 8.42/4.49 ! [X3: nat,N2: nat,Y: nat] : % 8.42/4.49 ( ( ( modulo_modulo @ nat @ X3 @ N2 ) % 8.42/4.49 = ( modulo_modulo @ nat @ Y @ N2 ) ) % 8.42/4.49 = ( ? [Q1: nat,Q22: nat] : % 8.42/4.49 ( ( plus_plus @ nat @ X3 @ ( times_times @ nat @ N2 @ Q1 ) ) % 8.42/4.49 = ( plus_plus @ nat @ Y @ ( times_times @ nat @ N2 @ Q22 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1400_nat__dvd__not__less,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.42/4.49 => ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.49 => ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1401_dvd__pos__nat,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ M @ N2 ) % 8.42/4.49 => ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1402_zdvd__imp__le,axiom, % 8.42/4.49 ! [Z2: int,N2: int] : % 8.42/4.49 ( ( dvd_dvd @ int @ Z2 @ N2 ) % 8.42/4.49 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 ) % 8.42/4.49 => ( ord_less_eq @ int @ Z2 @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1403_zdvd__not__zless,axiom, % 8.42/4.49 ! [M: int,N2: int] : % 8.42/4.49 ( ( ord_less @ int @ ( zero_zero @ int ) @ M ) % 8.42/4.49 => ( ( ord_less @ int @ M @ N2 ) % 8.42/4.49 => ~ ( dvd_dvd @ int @ N2 @ M ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1404_bezout__add__nat,axiom, % 8.42/4.49 ! [A2: nat,B2: nat] : % 8.42/4.49 ? [D3: nat,X5: nat,Y4: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ D3 @ A2 ) % 8.42/4.49 & ( dvd_dvd @ nat @ D3 @ B2 ) % 8.42/4.49 & ( ( ( times_times @ nat @ A2 @ X5 ) % 8.42/4.49 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D3 ) ) % 8.42/4.49 | ( ( times_times @ nat @ B2 @ X5 ) % 8.42/4.49 = ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y4 ) @ D3 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1405_bezout__lemma__nat,axiom, % 8.42/4.49 ! [D2: nat,A2: nat,B2: nat,X3: nat,Y: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ D2 @ A2 ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ D2 @ B2 ) % 8.42/4.49 => ( ( ( ( times_times @ nat @ A2 @ X3 ) % 8.42/4.49 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y ) @ D2 ) ) % 8.42/4.49 | ( ( times_times @ nat @ B2 @ X3 ) % 8.42/4.49 = ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y ) @ D2 ) ) ) % 8.42/4.49 => ? [X5: nat,Y4: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ D2 @ A2 ) % 8.42/4.49 & ( dvd_dvd @ nat @ D2 @ ( plus_plus @ nat @ A2 @ B2 ) ) % 8.42/4.49 & ( ( ( times_times @ nat @ A2 @ X5 ) % 8.42/4.49 = ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ Y4 ) @ D2 ) ) % 8.42/4.49 | ( ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ X5 ) % 8.42/4.49 = ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y4 ) @ D2 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1406_parity__cases,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_parity @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 => ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 != ( zero_zero @ A ) ) ) % 8.42/4.49 => ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 => ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 != ( one_one @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1407_mod2__eq__if,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_parity @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 => ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) % 8.42/4.49 & ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 => ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( one_one @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1408_unique__euclidean__semiring__numeral__class_Omod__less,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( modulo_modulo @ A @ A2 @ B2 ) % 8.42/4.49 = A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1409_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1410_cong__exp__iff__simps_I2_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ! [N2: num,Q4: num] : % 8.42/4.49 ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ Q4 ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1411_cong__exp__iff__simps_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ! [N2: num] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ one2 ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1412_mult__div__mod__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_modulo @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) % 8.42/4.49 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1413_mod__mult__div__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_modulo @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) % 8.42/4.49 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1414_mod__div__mult__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_modulo @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) ) % 8.42/4.49 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1415_div__mult__mod__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_modulo @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) % 8.42/4.49 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1416_mod__div__decomp,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_modulo @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( A2 % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1417_cancel__div__mod__rules_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_modulo @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 ) % 8.42/4.49 = ( plus_plus @ A @ A2 @ C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1418_cancel__div__mod__rules_I2_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_modulo @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 ) % 8.42/4.49 = ( plus_plus @ A @ A2 @ C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1419_div__mult1__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid3128863361964157862miring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1420_unit__dvdE,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ~ ( ( A2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ! [C3: A] : % 8.42/4.49 ( B2 % 8.42/4.49 != ( times_times @ A @ A2 @ C3 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1421_unity__coeff__ex,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ( dvd @ A ) % 8.42/4.49 & ( semiring_0 @ A ) ) % 8.42/4.49 => ! [P: A > $o,L: A] : % 8.42/4.49 ( ( ? [X: A] : ( P @ ( times_times @ A @ L @ X ) ) ) % 8.42/4.49 = ( ? [X: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X @ ( zero_zero @ A ) ) ) % 8.42/4.49 & ( P @ X ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1422_dvd__div__eq__mult,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( A2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ( divide_divide @ A @ B2 @ A2 ) % 8.42/4.49 = C2 ) % 8.42/4.49 = ( B2 % 8.42/4.49 = ( times_times @ A @ C2 @ A2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1423_div__dvd__iff__mult,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( B2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ B2 @ A2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1424_dvd__div__iff__mult,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [C2: A,B2: A,A2: A] : % 8.42/4.49 ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ C2 @ B2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1425_dvd__div__div__eq__mult,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A,D2: A] : % 8.42/4.49 ( ( A2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( C2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ C2 @ D2 ) % 8.42/4.49 => ( ( ( divide_divide @ A @ B2 @ A2 ) % 8.42/4.49 = ( divide_divide @ A @ D2 @ C2 ) ) % 8.42/4.49 = ( ( times_times @ A @ B2 @ C2 ) % 8.42/4.49 = ( times_times @ A @ A2 @ D2 ) ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1426_unit__div__eq__0__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( ( divide_divide @ A @ A2 @ B2 ) % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 = ( A2 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1427_is__unit__div__mult2__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [B2: A,C2: A,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1428_unit__div__mult__swap,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1429_unit__div__commute,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1430_div__mult__unit2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [C2: A,B2: A,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ B2 @ A2 ) % 8.42/4.49 => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1431_unit__eq__div2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( A2 % 8.42/4.49 = ( divide_divide @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ( times_times @ A @ A2 @ B2 ) % 8.42/4.49 = C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1432_unit__eq__div1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( ( divide_divide @ A @ A2 @ B2 ) % 8.42/4.49 = C2 ) % 8.42/4.49 = ( A2 % 8.42/4.49 = ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1433_mod__le__divisor,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N2 ) @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1434_is__unit__power__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N2 ) @ ( one_one @ A ) ) % 8.42/4.49 = ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 | ( N2 % 8.42/4.49 = ( zero_zero @ nat ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1435_div__less__mono,axiom, % 8.42/4.49 ! [A3: nat,B3: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ A3 @ B3 ) % 8.42/4.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ( ( modulo_modulo @ nat @ A3 @ N2 ) % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 => ( ( ( modulo_modulo @ nat @ B3 @ N2 ) % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 => ( ord_less @ nat @ ( divide_divide @ nat @ A3 @ N2 ) @ ( divide_divide @ nat @ B3 @ N2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1436_mod__eq__nat1E,axiom, % 8.42/4.49 ! [M: nat,Q4: nat,N2: nat] : % 8.42/4.49 ( ( ( modulo_modulo @ nat @ M @ Q4 ) % 8.42/4.49 = ( modulo_modulo @ nat @ N2 @ Q4 ) ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ N2 @ M ) % 8.42/4.49 => ~ ! [S2: nat] : % 8.42/4.49 ( M % 8.42/4.49 != ( plus_plus @ nat @ N2 @ ( times_times @ nat @ Q4 @ S2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1437_mod__eq__nat2E,axiom, % 8.42/4.49 ! [M: nat,Q4: nat,N2: nat] : % 8.42/4.49 ( ( ( modulo_modulo @ nat @ M @ Q4 ) % 8.42/4.49 = ( modulo_modulo @ nat @ N2 @ Q4 ) ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.49 => ~ ! [S2: nat] : % 8.42/4.49 ( N2 % 8.42/4.49 != ( plus_plus @ nat @ M @ ( times_times @ nat @ Q4 @ S2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1438_nat__mod__eq__lemma,axiom, % 8.42/4.49 ! [X3: nat,N2: nat,Y: nat] : % 8.42/4.49 ( ( ( modulo_modulo @ nat @ X3 @ N2 ) % 8.42/4.49 = ( modulo_modulo @ nat @ Y @ N2 ) ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ Y @ X3 ) % 8.42/4.49 => ? [Q2: nat] : % 8.42/4.49 ( X3 % 8.42/4.49 = ( plus_plus @ nat @ Y @ ( times_times @ nat @ N2 @ Q2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1439_dvd__imp__le,axiom, % 8.42/4.49 ! [K: nat,N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ K @ N2 ) % 8.42/4.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ord_less_eq @ nat @ K @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1440_div__mod__decomp,axiom, % 8.42/4.49 ! [A3: nat,N2: nat] : % 8.42/4.49 ( A3 % 8.42/4.49 = ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A3 @ N2 ) @ N2 ) @ ( modulo_modulo @ nat @ A3 @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1441_mod__mult2__eq,axiom, % 8.42/4.49 ! [M: nat,N2: nat,Q4: nat] : % 8.42/4.49 ( ( modulo_modulo @ nat @ M @ ( times_times @ nat @ N2 @ Q4 ) ) % 8.42/4.49 = ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M @ N2 ) @ Q4 ) ) @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1442_dvd__mult__cancel,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 8.42/4.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.42/4.49 => ( dvd_dvd @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1443_nat__mult__dvd__cancel1,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) % 8.42/4.49 = ( dvd_dvd @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1444_bezout__add__strong__nat,axiom, % 8.42/4.49 ! [A2: nat,B2: nat] : % 8.42/4.49 ( ( A2 % 8.42/4.49 != ( zero_zero @ nat ) ) % 8.42/4.49 => ? [D3: nat,X5: nat,Y4: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ D3 @ A2 ) % 8.42/4.49 & ( dvd_dvd @ nat @ D3 @ B2 ) % 8.42/4.49 & ( ( times_times @ nat @ A2 @ X5 ) % 8.42/4.49 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D3 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1445_even__zero,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_parity @ A ) % 8.42/4.49 => ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1446_is__unitE,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,C2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.42/4.49 => ~ ( ( A2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ! [B5: A] : % 8.42/4.49 ( ( B5 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ B5 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 ) % 8.42/4.49 = B5 ) % 8.42/4.49 => ( ( ( divide_divide @ A @ ( one_one @ A ) @ B5 ) % 8.42/4.49 = A2 ) % 8.42/4.49 => ( ( ( times_times @ A @ A2 @ B5 ) % 8.42/4.49 = ( one_one @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ C2 @ A2 ) % 8.42/4.49 != ( times_times @ A @ C2 @ B5 ) ) ) ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1447_is__unit__div__mult__cancel__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( A2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1448_is__unit__div__mult__cancel__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( A2 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) % 8.42/4.49 => ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) ) % 8.42/4.49 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1449_odd__even__add,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_parity @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 => ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) % 8.42/4.49 => ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1450_dvd__power__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( algebraic_semidom @ A ) % 8.42/4.49 => ! [X3: A,M: nat,N2: nat] : % 8.42/4.49 ( ( X3 % 8.42/4.49 != ( zero_zero @ A ) ) % 8.42/4.49 => ( ( dvd_dvd @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ X3 @ N2 ) ) % 8.42/4.49 = ( ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) % 8.42/4.49 | ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1451_dvd__power,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_semiring_1 @ A ) % 8.42/4.49 => ! [N2: nat,X3: A] : % 8.42/4.49 ( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 | ( X3 % 8.42/4.49 = ( one_one @ A ) ) ) % 8.42/4.49 => ( dvd_dvd @ A @ X3 @ ( power_power @ A @ X3 @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1452_split__mod,axiom, % 8.42/4.49 ! [P: nat > $o,M: nat,N2: nat] : % 8.42/4.49 ( ( P @ ( modulo_modulo @ nat @ M @ N2 ) ) % 8.42/4.49 = ( ( ( N2 % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 => ( P @ M ) ) % 8.42/4.49 & ( ( N2 % 8.42/4.49 != ( zero_zero @ nat ) ) % 8.42/4.49 => ! [I3: nat,J3: nat] : % 8.42/4.49 ( ( ord_less @ nat @ J3 @ N2 ) % 8.42/4.49 => ( ( M % 8.42/4.49 = ( plus_plus @ nat @ ( times_times @ nat @ N2 @ I3 ) @ J3 ) ) % 8.42/4.49 => ( P @ J3 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1453_dvd__mult__cancel1,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ M @ N2 ) @ M ) % 8.42/4.49 = ( N2 % 8.42/4.49 = ( one_one @ nat ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1454_dvd__mult__cancel2,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ N2 @ M ) @ M ) % 8.42/4.49 = ( N2 % 8.42/4.49 = ( one_one @ nat ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1455_power__dvd__imp__le,axiom, % 8.42/4.49 ! [I: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N2 ) ) % 8.42/4.49 => ( ( ord_less @ nat @ ( one_one @ nat ) @ I ) % 8.42/4.49 => ( ord_less_eq @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1456_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.42/4.49 => ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1457_product__nth,axiom, % 8.42/4.49 ! [A: $tType,B: $tType,N2: nat,Xs: list @ A,Ys: list @ B] : % 8.42/4.49 ( ( ord_less @ nat @ N2 @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) % 8.42/4.49 => ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) @ N2 ) % 8.42/4.49 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1458_power__mono__odd,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [N2: nat,A2: A,B2: A] : % 8.42/4.49 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1459_Suc__times__mod__eq,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M ) % 8.42/4.49 => ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M @ N2 ) ) @ M ) % 8.42/4.49 = ( one_one @ nat ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1460_odd__pos,axiom, % 8.42/4.49 ! [N2: nat] : % 8.42/4.49 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1461_dvd__power__iff__le,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N2 ) ) % 8.42/4.49 = ( ord_less_eq @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1462_even__unset__bit__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_se359711467146920520ations @ A ) % 8.42/4.49 => ! [M: nat,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) ) % 8.42/4.49 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 | ( M % 8.42/4.49 = ( zero_zero @ nat ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1463_even__set__bit__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_se359711467146920520ations @ A ) % 8.42/4.49 => ! [M: nat,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) ) % 8.42/4.49 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 & ( M % 8.42/4.49 != ( zero_zero @ nat ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1464_num_Osize__gen_I1_J,axiom, % 8.42/4.49 ( ( size_num @ one2 ) % 8.42/4.49 = ( zero_zero @ nat ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1465_divmod__digit__0_I2_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 ) % 8.42/4.49 => ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) % 8.42/4.49 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1466_bits__stable__imp__add__self,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = A2 ) % 8.42/4.49 => ( ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1467_div__exp__mod__exp__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ! [A2: A,N2: nat,M: nat] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) % 8.42/4.49 = ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N2 @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1468_oddE,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_parity @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 => ~ ! [B5: A] : % 8.42/4.49 ( A2 % 8.42/4.49 != ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) @ ( one_one @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1469_zero__le__even__power,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [N2: nat,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1470_zero__le__odd__power,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [N2: nat,A2: A] : % 8.42/4.49 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) % 8.42/4.49 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1471_zero__le__power__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) % 8.42/4.49 = ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1472_verit__le__mono__div,axiom, % 8.42/4.49 ! [A3: nat,B3: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ A3 @ B3 ) % 8.42/4.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ord_less_eq @ nat % 8.42/4.49 @ ( plus_plus @ nat @ ( divide_divide @ nat @ A3 @ N2 ) % 8.42/4.49 @ ( if @ nat % 8.42/4.49 @ ( ( modulo_modulo @ nat @ B3 @ N2 ) % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 @ ( one_one @ nat ) % 8.42/4.49 @ ( zero_zero @ nat ) ) ) % 8.42/4.49 @ ( divide_divide @ nat @ B3 @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1473_even__set__encode__iff,axiom, % 8.42/4.49 ! [A3: set @ nat] : % 8.42/4.49 ( ( finite_finite2 @ nat @ A3 ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A3 ) ) % 8.42/4.49 = ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1474_divmod__digit__0_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 ) % 8.42/4.49 => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1475_zero__less__power__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N2 ) ) % 8.42/4.49 = ( ( N2 % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 & ( A2 % 8.42/4.49 != ( zero_zero @ A ) ) ) % 8.42/4.49 | ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 & ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1476_mod__double__modulus,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ! [M: A,X3: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ M ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.42/4.49 => ( ( ( modulo_modulo @ A @ X3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) % 8.42/4.49 = ( modulo_modulo @ A @ X3 @ M ) ) % 8.42/4.49 | ( ( modulo_modulo @ A @ X3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) % 8.42/4.49 = ( plus_plus @ A @ ( modulo_modulo @ A @ X3 @ M ) @ M ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1477_unset__bit__Suc,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_se359711467146920520ations @ A ) % 8.42/4.49 => ! [N2: nat,A2: A] : % 8.42/4.49 ( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N2 ) @ A2 ) % 8.42/4.49 = ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1478_set__bit__Suc,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_se359711467146920520ations @ A ) % 8.42/4.49 => ! [N2: nat,A2: A] : % 8.42/4.49 ( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N2 ) @ A2 ) % 8.42/4.49 = ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1479_power__le__zero__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_idom @ A ) % 8.42/4.49 => ! [A2: A,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( zero_zero @ A ) ) % 8.42/4.49 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 & ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 & ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) % 8.42/4.49 | ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 & ( A2 % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1480_divmod__digit__1_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) % 8.42/4.49 => ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) ) % 8.42/4.49 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1481_flip__bit__Suc,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_se359711467146920520ations @ A ) % 8.42/4.49 => ! [N2: nat,A2: A] : % 8.42/4.49 ( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N2 ) @ A2 ) % 8.42/4.49 = ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1482_pos__eucl__rel__int__mult__2,axiom, % 8.42/4.49 ! [B2: int,A2: int,Q4: int,R2: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 ) % 8.42/4.49 => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 8.42/4.49 => ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q4 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1483_signed__take__bit__Suc,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_ri3973907225187159222ations @ A ) % 8.42/4.49 => ! [N2: nat,A2: A] : % 8.42/4.49 ( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N2 ) @ A2 ) % 8.42/4.49 = ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1484_even__mult__exp__div__exp__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ! [A2: A,M: nat,N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 8.42/4.49 = ( ( ord_less @ nat @ N2 @ M ) % 8.42/4.49 | ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 | ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.49 & ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1485_even__flip__bit__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_se359711467146920520ations @ A ) % 8.42/4.49 => ! [M: nat,A2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) ) % 8.42/4.49 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.42/4.49 != ( M % 8.42/4.49 = ( zero_zero @ nat ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1486_one__mod__2__pow__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid5411537665997757685th_nat @ A ) % 8.42/4.49 => ! [N2: nat] : % 8.42/4.49 ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.42/4.49 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1487_set__encode__insert,axiom, % 8.42/4.49 ! [A3: set @ nat,N2: nat] : % 8.42/4.49 ( ( finite_finite2 @ nat @ A3 ) % 8.42/4.49 => ( ~ ( member @ nat @ N2 @ A3 ) % 8.42/4.49 => ( ( nat_set_encode @ ( insert @ nat @ N2 @ A3 ) ) % 8.42/4.49 = ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( nat_set_encode @ A3 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1488_insert__absorb2,axiom, % 8.42/4.49 ! [A: $tType,X3: A,A3: set @ A] : % 8.42/4.49 ( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A3 ) ) % 8.42/4.49 = ( insert @ A @ X3 @ A3 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1489_insert__iff,axiom, % 8.42/4.49 ! [A: $tType,A2: A,B2: A,A3: set @ A] : % 8.42/4.49 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) ) % 8.42/4.49 = ( ( A2 = B2 ) % 8.42/4.49 | ( member @ A @ A2 @ A3 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1490_insertCI,axiom, % 8.42/4.49 ! [A: $tType,A2: A,B3: set @ A,B2: A] : % 8.42/4.49 ( ( ~ ( member @ A @ A2 @ B3 ) % 8.42/4.49 => ( A2 = B2 ) ) % 8.42/4.49 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1491_diff__self,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( group_add @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( minus_minus @ A @ A2 @ A2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1492_diff__0__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( group_add @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1493_zero__diff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_monoid_diff @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1494_diff__zero,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( cancel1802427076303600483id_add @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) ) % 8.42/4.49 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1495_cancel__comm__monoid__add__class_Odiff__cancel,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( cancel1802427076303600483id_add @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( minus_minus @ A @ A2 @ A2 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1496_add__diff__cancel,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 ) % 8.42/4.49 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1497_diff__add__cancel,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 ) % 8.42/4.49 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1498_add__diff__cancel__left,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( cancel2418104881723323429up_add @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( minus_minus @ A @ A2 @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1499_add__diff__cancel__left_H,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( cancel2418104881723323429up_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 ) % 8.42/4.49 = B2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1500_add__diff__cancel__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( cancel2418104881723323429up_add @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( minus_minus @ A @ A2 @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1501_add__diff__cancel__right_H,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( cancel2418104881723323429up_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 ) % 8.42/4.49 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1502_singletonI,axiom, % 8.42/4.49 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1503_finite__insert,axiom, % 8.42/4.49 ! [A: $tType,A2: A,A3: set @ A] : % 8.42/4.49 ( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) ) % 8.42/4.49 = ( finite_finite2 @ A @ A3 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1504_insert__subset,axiom, % 8.42/4.49 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] : % 8.42/4.49 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 ) % 8.42/4.49 = ( ( member @ A @ X3 @ B3 ) % 8.42/4.49 & ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1505_diff__Suc__Suc,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) % 8.42/4.49 = ( minus_minus @ nat @ M @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1506_Suc__diff__diff,axiom, % 8.42/4.49 ! [M: nat,N2: nat,K: nat] : % 8.42/4.49 ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) ) % 8.42/4.49 = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1507_diff__0__eq__0,axiom, % 8.42/4.49 ! [N2: nat] : % 8.42/4.49 ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 = ( zero_zero @ nat ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1508_diff__self__eq__0,axiom, % 8.42/4.49 ! [M: nat] : % 8.42/4.49 ( ( minus_minus @ nat @ M @ M ) % 8.42/4.49 = ( zero_zero @ nat ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1509_diff__diff__cancel,axiom, % 8.42/4.49 ! [I: nat,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ I @ N2 ) % 8.42/4.49 => ( ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ N2 @ I ) ) % 8.42/4.49 = I ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1510_diff__diff__left,axiom, % 8.42/4.49 ! [I: nat,J: nat,K: nat] : % 8.42/4.49 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K ) % 8.42/4.49 = ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1511_flip__bit__negative__int__iff,axiom, % 8.42/4.49 ! [N2: nat,K: int] : % 8.42/4.49 ( ( ord_less @ int @ ( bit_se8732182000553998342ip_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) % 8.42/4.49 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1512_of__bool__less__eq__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord181362715937106298miring @ A ) % 8.42/4.49 => ! [P: $o,Q: $o] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) % 8.42/4.49 = ( P % 8.42/4.49 => Q ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1513_of__bool__eq_I1_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( zero_neq_one @ A ) % 8.42/4.49 => ( ( zero_neq_one_of_bool @ A @ $false ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1514_of__bool__eq__0__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( zero_neq_one @ A ) % 8.42/4.49 => ! [P: $o] : % 8.42/4.49 ( ( ( zero_neq_one_of_bool @ A @ P ) % 8.42/4.49 = ( zero_zero @ A ) ) % 8.42/4.49 = ~ P ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1515_of__bool__less__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linord181362715937106298miring @ A ) % 8.42/4.49 => ! [P: $o,Q: $o] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) % 8.42/4.49 = ( ~ P % 8.42/4.49 & Q ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1516_of__bool__eq__1__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( zero_neq_one @ A ) % 8.42/4.49 => ! [P: $o] : % 8.42/4.49 ( ( ( zero_neq_one_of_bool @ A @ P ) % 8.42/4.49 = ( one_one @ A ) ) % 8.42/4.49 = P ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1517_of__bool__eq_I2_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( zero_neq_one @ A ) % 8.42/4.49 => ( ( zero_neq_one_of_bool @ A @ $true ) % 8.42/4.49 = ( one_one @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1518_signed__take__bit__of__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_ri3973907225187159222ations @ A ) % 8.42/4.49 => ! [N2: nat] : % 8.42/4.49 ( ( bit_ri4674362597316999326ke_bit @ A @ N2 @ ( zero_zero @ A ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1519_diff__ge__0__iff__ge,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) ) % 8.42/4.49 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1520_diff__gt__0__iff__gt,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) ) % 8.42/4.49 = ( ord_less @ A @ B2 @ A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1521_le__add__diff__inverse2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.42/4.49 => ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 ) % 8.42/4.49 = A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1522_le__add__diff__inverse,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.42/4.49 => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) ) % 8.42/4.49 = A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1523_diff__numeral__special_I9_J,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( neg_numeral @ A ) % 8.42/4.49 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1524_diff__add__zero,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_monoid_diff @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1525_div__diff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( idom_modulo @ A ) % 8.42/4.49 => ! [C2: A,A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ C2 @ A2 ) % 8.42/4.49 => ( ( dvd_dvd @ A @ C2 @ B2 ) % 8.42/4.49 => ( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1526_singleton__insert__inj__eq_H,axiom, % 8.42/4.49 ! [A: $tType,A2: A,A3: set @ A,B2: A] : % 8.42/4.49 ( ( ( insert @ A @ A2 @ A3 ) % 8.42/4.49 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.42/4.49 = ( ( A2 = B2 ) % 8.42/4.49 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1527_singleton__insert__inj__eq,axiom, % 8.42/4.49 ! [A: $tType,B2: A,A2: A,A3: set @ A] : % 8.42/4.49 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) % 8.42/4.49 = ( insert @ A @ A2 @ A3 ) ) % 8.42/4.49 = ( ( A2 = B2 ) % 8.42/4.49 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1528_zero__less__of__bool__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [P: $o] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) % 8.42/4.49 = P ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1529_zero__less__diff,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) % 8.42/4.49 = ( ord_less @ nat @ M @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1530_diff__is__0__eq,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ( minus_minus @ nat @ M @ N2 ) % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 = ( ord_less_eq @ nat @ M @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1531_diff__is__0__eq_H,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.49 => ( ( minus_minus @ nat @ M @ N2 ) % 8.42/4.49 = ( zero_zero @ nat ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1532_of__bool__less__one__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [P: $o] : % 8.42/4.49 ( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) ) % 8.42/4.49 = ~ P ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1533_of__bool__not__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ring_1 @ A ) % 8.42/4.49 => ! [P: $o] : % 8.42/4.49 ( ( zero_neq_one_of_bool @ A @ ~ P ) % 8.42/4.49 = ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1534_Nat_Odiff__diff__right,axiom, % 8.42/4.49 ! [K: nat,J: nat,I: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ K @ J ) % 8.42/4.49 => ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) % 8.42/4.49 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1535_Nat_Oadd__diff__assoc2,axiom, % 8.42/4.49 ! [K: nat,J: nat,I: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ K @ J ) % 8.42/4.49 => ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) % 8.42/4.49 = ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1536_Nat_Oadd__diff__assoc,axiom, % 8.42/4.49 ! [K: nat,J: nat,I: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ K @ J ) % 8.42/4.49 => ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) % 8.42/4.49 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1537_diff__Suc__1,axiom, % 8.42/4.49 ! [N2: nat] : % 8.42/4.49 ( ( minus_minus @ nat @ ( suc @ N2 ) @ ( one_one @ nat ) ) % 8.42/4.49 = N2 ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1538_Suc__0__mod__eq,axiom, % 8.42/4.49 ! [N2: nat] : % 8.42/4.49 ( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 8.42/4.49 = ( zero_neq_one_of_bool @ nat % 8.42/4.49 @ ( N2 % 8.42/4.49 != ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1539_mod__pos__pos__trivial,axiom, % 8.42/4.49 ! [K: int,L: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 8.42/4.49 => ( ( ord_less @ int @ K @ L ) % 8.42/4.49 => ( ( modulo_modulo @ int @ K @ L ) % 8.42/4.49 = K ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1540_mod__neg__neg__trivial,axiom, % 8.42/4.49 ! [K: int,L: int] : % 8.42/4.49 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) ) % 8.42/4.49 => ( ( ord_less @ int @ L @ K ) % 8.42/4.49 => ( ( modulo_modulo @ int @ K @ L ) % 8.42/4.49 = K ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1541_signed__take__bit__Suc__1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_ri3973907225187159222ations @ A ) % 8.42/4.49 => ! [N2: nat] : % 8.42/4.49 ( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N2 ) @ ( one_one @ A ) ) % 8.42/4.49 = ( one_one @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1542_Suc__pred,axiom, % 8.42/4.49 ! [N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ( suc @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) % 8.42/4.49 = N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1543_diff__Suc__diff__eq2,axiom, % 8.42/4.49 ! [K: nat,J: nat,I: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ K @ J ) % 8.42/4.49 => ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J @ K ) ) @ I ) % 8.42/4.49 = ( minus_minus @ nat @ ( suc @ J ) @ ( plus_plus @ nat @ K @ I ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1544_diff__Suc__diff__eq1,axiom, % 8.42/4.49 ! [K: nat,J: nat,I: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ K @ J ) % 8.42/4.49 => ( ( minus_minus @ nat @ I @ ( suc @ ( minus_minus @ nat @ J @ K ) ) ) % 8.42/4.49 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ ( suc @ J ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1545_signed__take__bit__Suc__bit0,axiom, % 8.42/4.49 ! [N2: nat,K: num] : % 8.42/4.49 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) % 8.42/4.49 = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1546_Suc__diff__1,axiom, % 8.42/4.49 ! [N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) % 8.42/4.49 = N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1547_set__replicate,axiom, % 8.42/4.49 ! [A: $tType,N2: nat,X3: A] : % 8.42/4.49 ( ( N2 % 8.42/4.49 != ( zero_zero @ nat ) ) % 8.42/4.49 => ( ( set2 @ A @ ( replicate @ A @ N2 @ X3 ) ) % 8.42/4.49 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1548_even__diff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ring_parity @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A2 @ B2 ) ) % 8.42/4.49 = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1549_of__bool__half__eq__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid5411537665997757685th_nat @ A ) % 8.42/4.49 => ! [B2: $o] : % 8.42/4.49 ( ( divide_divide @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1550_odd__Suc__minus__one,axiom, % 8.42/4.49 ! [N2: nat] : % 8.42/4.49 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.49 => ( ( suc @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) % 8.42/4.49 = N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1551_even__diff__nat,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N2 ) ) % 8.42/4.49 = ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.49 | ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1552_semiring__parity__class_Oeven__mask__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_parity @ A ) % 8.42/4.49 => ! [N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) ) % 8.42/4.49 = ( N2 % 8.42/4.49 = ( zero_zero @ nat ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1553_one__div__2__pow__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( euclid5411537665997757685th_nat @ A ) % 8.42/4.49 => ! [N2: nat] : % 8.42/4.49 ( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.42/4.49 = ( zero_neq_one_of_bool @ A % 8.42/4.49 @ ( N2 % 8.42/4.49 = ( zero_zero @ nat ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1554_bits__1__div__exp,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_semiring_bits @ A ) % 8.42/4.49 => ! [N2: nat] : % 8.42/4.49 ( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.42/4.49 = ( zero_neq_one_of_bool @ A % 8.42/4.49 @ ( N2 % 8.42/4.49 = ( zero_zero @ nat ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1555_flip__bit__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( bit_se359711467146920520ations @ A ) % 8.42/4.49 => ! [A2: A] : % 8.42/4.49 ( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A2 ) % 8.42/4.49 = ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1556_mod__int__unique,axiom, % 8.42/4.49 ! [K: int,L: int,Q4: int,R2: int] : % 8.42/4.49 ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 8.42/4.49 => ( ( modulo_modulo @ int @ K @ L ) % 8.42/4.49 = R2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1557_gcd__nat_Oasym,axiom, % 8.42/4.49 ! [A2: nat,B2: nat] : % 8.42/4.49 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.42/4.49 & ( A2 != B2 ) ) % 8.42/4.49 => ~ ( ( dvd_dvd @ nat @ B2 @ A2 ) % 8.42/4.49 & ( B2 != A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1558_gcd__nat_Orefl,axiom, % 8.42/4.49 ! [A2: nat] : ( dvd_dvd @ nat @ A2 @ A2 ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1559_gcd__nat_Otrans,axiom, % 8.42/4.49 ! [A2: nat,B2: nat,C2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ B2 @ C2 ) % 8.42/4.49 => ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1560_gcd__nat_Oeq__iff,axiom, % 8.42/4.49 ( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) ) % 8.42/4.49 = ( ^ [A4: nat,B4: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ A4 @ B4 ) % 8.42/4.49 & ( dvd_dvd @ nat @ B4 @ A4 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1561_gcd__nat_Oirrefl,axiom, % 8.42/4.49 ! [A2: nat] : % 8.42/4.49 ~ ( ( dvd_dvd @ nat @ A2 @ A2 ) % 8.42/4.49 & ( A2 != A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1562_gcd__nat_Oantisym,axiom, % 8.42/4.49 ! [A2: nat,B2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ B2 @ A2 ) % 8.42/4.49 => ( A2 = B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1563_gcd__nat_Ostrict__trans,axiom, % 8.42/4.49 ! [A2: nat,B2: nat,C2: nat] : % 8.42/4.49 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.42/4.49 & ( A2 != B2 ) ) % 8.42/4.49 => ( ( ( dvd_dvd @ nat @ B2 @ C2 ) % 8.42/4.49 & ( B2 != C2 ) ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ A2 @ C2 ) % 8.42/4.49 & ( A2 != C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1564_gcd__nat_Ostrict__trans1,axiom, % 8.42/4.49 ! [A2: nat,B2: nat,C2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.42/4.49 => ( ( ( dvd_dvd @ nat @ B2 @ C2 ) % 8.42/4.49 & ( B2 != C2 ) ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ A2 @ C2 ) % 8.42/4.49 & ( A2 != C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1565_gcd__nat_Ostrict__trans2,axiom, % 8.42/4.49 ! [A2: nat,B2: nat,C2: nat] : % 8.42/4.49 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.42/4.49 & ( A2 != B2 ) ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ B2 @ C2 ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ A2 @ C2 ) % 8.42/4.49 & ( A2 != C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1566_gcd__nat_Ostrict__iff__not,axiom, % 8.42/4.49 ! [A2: nat,B2: nat] : % 8.42/4.49 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.42/4.49 & ( A2 != B2 ) ) % 8.42/4.49 = ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.42/4.49 & ~ ( dvd_dvd @ nat @ B2 @ A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1567_gcd__nat_Oorder__iff__strict,axiom, % 8.42/4.49 ( ( dvd_dvd @ nat ) % 8.42/4.49 = ( ^ [A4: nat,B4: nat] : % 8.42/4.49 ( ( ( dvd_dvd @ nat @ A4 @ B4 ) % 8.42/4.49 & ( A4 != B4 ) ) % 8.42/4.49 | ( A4 = B4 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1568_gcd__nat_Ostrict__iff__order,axiom, % 8.42/4.49 ! [A2: nat,B2: nat] : % 8.42/4.49 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.42/4.49 & ( A2 != B2 ) ) % 8.42/4.49 = ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.42/4.49 & ( A2 != B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1569_gcd__nat_Ostrict__implies__order,axiom, % 8.42/4.49 ! [A2: nat,B2: nat] : % 8.42/4.49 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.42/4.49 & ( A2 != B2 ) ) % 8.42/4.49 => ( dvd_dvd @ nat @ A2 @ B2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1570_gcd__nat_Ostrict__implies__not__eq,axiom, % 8.42/4.49 ! [A2: nat,B2: nat] : % 8.42/4.49 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.42/4.49 & ( A2 != B2 ) ) % 8.42/4.49 => ( A2 != B2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1571_gcd__nat_Onot__eq__order__implies__strict,axiom, % 8.42/4.49 ! [A2: nat,B2: nat] : % 8.42/4.49 ( ( A2 != B2 ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.42/4.49 & ( A2 != B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1572_dvd__antisym,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ M @ N2 ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ N2 @ M ) % 8.42/4.49 => ( M = N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1573_dvd__diff__nat,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ K @ M ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ K @ N2 ) % 8.42/4.49 => ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1574_eucl__rel__int,axiom, % 8.42/4.49 ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K @ L ) @ ( modulo_modulo @ int @ K @ L ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1575_mk__disjoint__insert,axiom, % 8.42/4.49 ! [A: $tType,A2: A,A3: set @ A] : % 8.42/4.49 ( ( member @ A @ A2 @ A3 ) % 8.42/4.49 => ? [B10: set @ A] : % 8.42/4.49 ( ( A3 % 8.42/4.49 = ( insert @ A @ A2 @ B10 ) ) % 8.42/4.49 & ~ ( member @ A @ A2 @ B10 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1576_insert__commute,axiom, % 8.42/4.49 ! [A: $tType,X3: A,Y: A,A3: set @ A] : % 8.42/4.49 ( ( insert @ A @ X3 @ ( insert @ A @ Y @ A3 ) ) % 8.42/4.49 = ( insert @ A @ Y @ ( insert @ A @ X3 @ A3 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1577_insert__eq__iff,axiom, % 8.42/4.49 ! [A: $tType,A2: A,A3: set @ A,B2: A,B3: set @ A] : % 8.42/4.49 ( ~ ( member @ A @ A2 @ A3 ) % 8.42/4.49 => ( ~ ( member @ A @ B2 @ B3 ) % 8.42/4.49 => ( ( ( insert @ A @ A2 @ A3 ) % 8.42/4.49 = ( insert @ A @ B2 @ B3 ) ) % 8.42/4.49 = ( ( ( A2 = B2 ) % 8.42/4.49 => ( A3 = B3 ) ) % 8.42/4.49 & ( ( A2 != B2 ) % 8.42/4.49 => ? [C7: set @ A] : % 8.42/4.49 ( ( A3 % 8.42/4.49 = ( insert @ A @ B2 @ C7 ) ) % 8.42/4.49 & ~ ( member @ A @ B2 @ C7 ) % 8.42/4.49 & ( B3 % 8.42/4.49 = ( insert @ A @ A2 @ C7 ) ) % 8.42/4.49 & ~ ( member @ A @ A2 @ C7 ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1578_insert__absorb,axiom, % 8.42/4.49 ! [A: $tType,A2: A,A3: set @ A] : % 8.42/4.49 ( ( member @ A @ A2 @ A3 ) % 8.42/4.49 => ( ( insert @ A @ A2 @ A3 ) % 8.42/4.49 = A3 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1579_insert__ident,axiom, % 8.42/4.49 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] : % 8.42/4.49 ( ~ ( member @ A @ X3 @ A3 ) % 8.42/4.49 => ( ~ ( member @ A @ X3 @ B3 ) % 8.42/4.49 => ( ( ( insert @ A @ X3 @ A3 ) % 8.42/4.49 = ( insert @ A @ X3 @ B3 ) ) % 8.42/4.49 = ( A3 = B3 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1580_Set_Oset__insert,axiom, % 8.42/4.49 ! [A: $tType,X3: A,A3: set @ A] : % 8.42/4.49 ( ( member @ A @ X3 @ A3 ) % 8.42/4.49 => ~ ! [B10: set @ A] : % 8.42/4.49 ( ( A3 % 8.42/4.49 = ( insert @ A @ X3 @ B10 ) ) % 8.42/4.49 => ( member @ A @ X3 @ B10 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1581_insertI2,axiom, % 8.42/4.49 ! [A: $tType,A2: A,B3: set @ A,B2: A] : % 8.42/4.49 ( ( member @ A @ A2 @ B3 ) % 8.42/4.49 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1582_insertI1,axiom, % 8.42/4.49 ! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1583_insertE,axiom, % 8.42/4.49 ! [A: $tType,A2: A,B2: A,A3: set @ A] : % 8.42/4.49 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) ) % 8.42/4.49 => ( ( A2 != B2 ) % 8.42/4.49 => ( member @ A @ A2 @ A3 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1584_of__bool__eq__iff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( zero_neq_one @ A ) % 8.42/4.49 => ! [P4: $o,Q4: $o] : % 8.42/4.49 ( ( ( zero_neq_one_of_bool @ A @ P4 ) % 8.42/4.49 = ( zero_neq_one_of_bool @ A @ Q4 ) ) % 8.42/4.49 = ( P4 = Q4 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1585_diff__commute,axiom, % 8.42/4.49 ! [I: nat,J: nat,K: nat] : % 8.42/4.49 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K ) % 8.42/4.49 = ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1586_diff__right__commute,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( cancel2418104881723323429up_add @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) % 8.42/4.49 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1587_diff__eq__diff__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.42/4.49 ( ( ( minus_minus @ A @ A2 @ B2 ) % 8.42/4.49 = ( minus_minus @ A @ C2 @ D2 ) ) % 8.42/4.49 => ( ( A2 = B2 ) % 8.42/4.49 = ( C2 = D2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1588_full__exhaustive__int_H_Ocases,axiom, % 8.42/4.49 ! [X3: product_prod @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] : % 8.42/4.49 ~ ! [F4: ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: int,I4: int] : % 8.42/4.49 ( X3 % 8.42/4.49 != ( product_Pair @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D3 @ I4 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1589_exhaustive__int_H_Ocases,axiom, % 8.42/4.49 ! [X3: product_prod @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] : % 8.42/4.49 ~ ! [F4: int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: int,I4: int] : % 8.42/4.49 ( X3 % 8.42/4.49 != ( product_Pair @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D3 @ I4 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1590_small__lazy_H_Ocases,axiom, % 8.42/4.49 ! [X3: product_prod @ int @ int] : % 8.42/4.49 ~ ! [D3: int,I4: int] : % 8.42/4.49 ( X3 % 8.42/4.49 != ( product_Pair @ int @ int @ D3 @ I4 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1591_unique__quotient,axiom, % 8.42/4.49 ! [A2: int,B2: int,Q4: int,R2: int,Q6: int,R3: int] : % 8.42/4.49 ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 8.42/4.49 => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q6 @ R3 ) ) % 8.42/4.49 => ( Q4 = Q6 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1592_unique__remainder,axiom, % 8.42/4.49 ! [A2: int,B2: int,Q4: int,R2: int,Q6: int,R3: int] : % 8.42/4.49 ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 8.42/4.49 => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q6 @ R3 ) ) % 8.42/4.49 => ( R2 = R3 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1593_of__bool__conj,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semiring_1 @ A ) % 8.42/4.49 => ! [P: $o,Q: $o] : % 8.42/4.49 ( ( zero_neq_one_of_bool @ A % 8.42/4.49 @ ( P % 8.42/4.49 & Q ) ) % 8.42/4.49 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1594_diff__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A,D2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ D2 @ C2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1595_diff__left__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1596_diff__right__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1597_diff__eq__diff__less__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.42/4.49 ( ( ( minus_minus @ A @ A2 @ B2 ) % 8.42/4.49 = ( minus_minus @ A @ C2 @ D2 ) ) % 8.42/4.49 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 = ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1598_eq__iff__diff__eq__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( group_add @ A ) % 8.42/4.49 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) ) % 8.42/4.49 = ( ^ [A4: A,B4: A] : % 8.42/4.49 ( ( minus_minus @ A @ A4 @ B4 ) % 8.42/4.49 = ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1599_diff__strict__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A,D2: A,C2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less @ A @ D2 @ C2 ) % 8.42/4.49 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1600_diff__eq__diff__less,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.42/4.49 ( ( ( minus_minus @ A @ A2 @ B2 ) % 8.42/4.49 = ( minus_minus @ A @ C2 @ D2 ) ) % 8.42/4.49 => ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 = ( ord_less @ A @ C2 @ D2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1601_diff__strict__left__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ! [B2: A,A2: A,C2: A] : % 8.42/4.49 ( ( ord_less @ A @ B2 @ A2 ) % 8.42/4.49 => ( ord_less @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1602_diff__strict__right__mono,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1603_right__diff__distrib_H,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_s4317794764714335236cancel @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1604_left__diff__distrib_H,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_s4317794764714335236cancel @ A ) % 8.42/4.49 => ! [B2: A,C2: A,A2: A] : % 8.42/4.49 ( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A2 ) % 8.42/4.49 = ( minus_minus @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1605_right__diff__distrib,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1606_left__diff__distrib,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1607_add__diff__add,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ab_group_add @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A,D2: A] : % 8.42/4.49 ( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) % 8.42/4.49 = ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( minus_minus @ A @ C2 @ D2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1608_group__cancel_Osub1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ab_group_add @ A ) % 8.42/4.49 => ! [A3: A,K: A,A2: A,B2: A] : % 8.42/4.49 ( ( A3 % 8.42/4.49 = ( plus_plus @ A @ K @ A2 ) ) % 8.42/4.49 => ( ( minus_minus @ A @ A3 @ B2 ) % 8.42/4.49 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1609_diff__eq__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ( minus_minus @ A @ A2 @ B2 ) % 8.42/4.49 = C2 ) % 8.42/4.49 = ( A2 % 8.42/4.49 = ( plus_plus @ A @ C2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1610_eq__diff__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( group_add @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( A2 % 8.42/4.49 = ( minus_minus @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ( plus_plus @ A @ A2 @ B2 ) % 8.42/4.49 = C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1611_add__diff__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1612_diff__diff__eq2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1613_diff__add__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ab_group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1614_diff__add__eq__diff__diff__swap,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) % 8.42/4.49 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1615_add__implies__diff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( cancel1802427076303600483id_add @ A ) % 8.42/4.49 => ! [C2: A,B2: A,A2: A] : % 8.42/4.49 ( ( ( plus_plus @ A @ C2 @ B2 ) % 8.42/4.49 = A2 ) % 8.42/4.49 => ( C2 % 8.42/4.49 = ( minus_minus @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1616_diff__diff__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( cancel2418104881723323429up_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1617_diff__divide__distrib,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( division_ring @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1618_singletonD,axiom, % 8.42/4.49 ! [A: $tType,B2: A,A2: A] : % 8.42/4.49 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.42/4.49 => ( B2 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1619_singleton__iff,axiom, % 8.42/4.49 ! [A: $tType,B2: A,A2: A] : % 8.42/4.49 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.42/4.49 = ( B2 = A2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1620_doubleton__eq__iff,axiom, % 8.42/4.49 ! [A: $tType,A2: A,B2: A,C2: A,D2: A] : % 8.42/4.49 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.42/4.49 = ( insert @ A @ C2 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.42/4.49 = ( ( ( A2 = C2 ) % 8.42/4.49 & ( B2 = D2 ) ) % 8.42/4.49 | ( ( A2 = D2 ) % 8.42/4.49 & ( B2 = C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1621_insert__not__empty,axiom, % 8.42/4.49 ! [A: $tType,A2: A,A3: set @ A] : % 8.42/4.49 ( ( insert @ A @ A2 @ A3 ) % 8.42/4.49 != ( bot_bot @ ( set @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1622_singleton__inject,axiom, % 8.42/4.49 ! [A: $tType,A2: A,B2: A] : % 8.42/4.49 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) % 8.42/4.49 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.42/4.49 => ( A2 = B2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1623_finite_OinsertI,axiom, % 8.42/4.49 ! [A: $tType,A3: set @ A,A2: A] : % 8.42/4.49 ( ( finite_finite2 @ A @ A3 ) % 8.42/4.49 => ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1624_insert__subsetI,axiom, % 8.42/4.49 ! [A: $tType,X3: A,A3: set @ A,X8: set @ A] : % 8.42/4.49 ( ( member @ A @ X3 @ A3 ) % 8.42/4.49 => ( ( ord_less_eq @ ( set @ A ) @ X8 @ A3 ) % 8.42/4.49 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ X8 ) @ A3 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1625_insert__mono,axiom, % 8.42/4.49 ! [A: $tType,C5: set @ A,D6: set @ A,A2: A] : % 8.42/4.49 ( ( ord_less_eq @ ( set @ A ) @ C5 @ D6 ) % 8.42/4.49 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C5 ) @ ( insert @ A @ A2 @ D6 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1626_subset__insert,axiom, % 8.42/4.49 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] : % 8.42/4.49 ( ~ ( member @ A @ X3 @ A3 ) % 8.42/4.49 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) ) % 8.42/4.49 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1627_subset__insertI,axiom, % 8.42/4.49 ! [A: $tType,B3: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( insert @ A @ A2 @ B3 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1628_subset__insertI2,axiom, % 8.42/4.49 ! [A: $tType,A3: set @ A,B3: set @ A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.49 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B3 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1629_dvd__diff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_ring_1 @ A ) % 8.42/4.49 => ! [X3: A,Y: A,Z2: A] : % 8.42/4.49 ( ( dvd_dvd @ A @ X3 @ Y ) % 8.42/4.49 => ( ( dvd_dvd @ A @ X3 @ Z2 ) % 8.42/4.49 => ( dvd_dvd @ A @ X3 @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1630_zero__induct__lemma,axiom, % 8.42/4.49 ! [P: nat > $o,K: nat,I: nat] : % 8.42/4.49 ( ( P @ K ) % 8.42/4.49 => ( ! [N3: nat] : % 8.42/4.49 ( ( P @ ( suc @ N3 ) ) % 8.42/4.49 => ( P @ N3 ) ) % 8.42/4.49 => ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1631_minus__nat_Odiff__0,axiom, % 8.42/4.49 ! [M: nat] : % 8.42/4.49 ( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) ) % 8.42/4.49 = M ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1632_diffs0__imp__equal,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ( minus_minus @ nat @ M @ N2 ) % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 => ( ( ( minus_minus @ nat @ N2 @ M ) % 8.42/4.49 = ( zero_zero @ nat ) ) % 8.42/4.49 => ( M = N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1633_diff__less__mono2,axiom, % 8.42/4.49 ! [M: nat,N2: nat,L: nat] : % 8.42/4.49 ( ( ord_less @ nat @ M @ N2 ) % 8.42/4.49 => ( ( ord_less @ nat @ M @ L ) % 8.42/4.49 => ( ord_less @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1634_less__imp__diff__less,axiom, % 8.42/4.49 ! [J: nat,K: nat,N2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ J @ K ) % 8.42/4.49 => ( ord_less @ nat @ ( minus_minus @ nat @ J @ N2 ) @ K ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1635_dvd__minus__self,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ M ) ) % 8.42/4.49 = ( ( ord_less @ nat @ N2 @ M ) % 8.42/4.49 | ( dvd_dvd @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1636_eq__diff__iff,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ K @ M ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ K @ N2 ) % 8.42/4.49 => ( ( ( minus_minus @ nat @ M @ K ) % 8.42/4.49 = ( minus_minus @ nat @ N2 @ K ) ) % 8.42/4.49 = ( M = N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1637_le__diff__iff,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ K @ M ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ K @ N2 ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) ) % 8.42/4.49 = ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1638_Nat_Odiff__diff__eq,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ K @ M ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ K @ N2 ) % 8.42/4.49 => ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) ) % 8.42/4.49 = ( minus_minus @ nat @ M @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1639_diff__le__mono,axiom, % 8.42/4.49 ! [M: nat,N2: nat,L: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.49 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N2 @ L ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1640_diff__le__self,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1641_le__diff__iff_H,axiom, % 8.42/4.49 ! [A2: nat,C2: nat,B2: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ A2 @ C2 ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ B2 @ C2 ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A2 ) @ ( minus_minus @ nat @ C2 @ B2 ) ) % 8.42/4.49 = ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1642_diff__le__mono2,axiom, % 8.42/4.49 ! [M: nat,N2: nat,L: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.49 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1643_dvd__diffD,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ K @ N2 ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ N2 @ M ) % 8.42/4.49 => ( dvd_dvd @ nat @ K @ M ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1644_dvd__diffD1,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ K @ M ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ N2 @ M ) % 8.42/4.49 => ( dvd_dvd @ nat @ K @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1645_less__eq__dvd__minus,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.49 => ( ( dvd_dvd @ nat @ M @ N2 ) % 8.42/4.49 = ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1646_Nat_Odiff__cancel,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) ) % 8.42/4.49 = ( minus_minus @ nat @ M @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1647_diff__cancel2,axiom, % 8.42/4.49 ! [M: nat,K: nat,N2: nat] : % 8.42/4.49 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) % 8.42/4.49 = ( minus_minus @ nat @ M @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1648_diff__add__inverse,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ N2 ) % 8.42/4.49 = M ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1649_diff__add__inverse2,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ N2 ) % 8.42/4.49 = M ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1650_diff__mult__distrib2,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) ) % 8.42/4.49 = ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1651_diff__mult__distrib,axiom, % 8.42/4.49 ! [M: nat,N2: nat,K: nat] : % 8.42/4.49 ( ( times_times @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K ) % 8.42/4.49 = ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1652_bezout1__nat,axiom, % 8.42/4.49 ! [A2: nat,B2: nat] : % 8.42/4.49 ? [D3: nat,X5: nat,Y4: nat] : % 8.42/4.49 ( ( dvd_dvd @ nat @ D3 @ A2 ) % 8.42/4.49 & ( dvd_dvd @ nat @ D3 @ B2 ) % 8.42/4.49 & ( ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X5 ) @ ( times_times @ nat @ B2 @ Y4 ) ) % 8.42/4.49 = D3 ) % 8.42/4.49 | ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X5 ) @ ( times_times @ nat @ A2 @ Y4 ) ) % 8.42/4.49 = D3 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1653_eucl__rel__int__by0,axiom, % 8.42/4.49 ! [K: int] : ( eucl_rel_int @ K @ ( zero_zero @ int ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1654_div__int__unique,axiom, % 8.42/4.49 ! [K: int,L: int,Q4: int,R2: int] : % 8.42/4.49 ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 8.42/4.49 => ( ( divide_divide @ int @ K @ L ) % 8.42/4.49 = Q4 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1655_zero__less__eq__of__bool,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [P: $o] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1656_of__bool__less__eq__one,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [P: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1657_split__of__bool__asm,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( zero_neq_one @ A ) % 8.42/4.49 => ! [P: A > $o,P4: $o] : % 8.42/4.49 ( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) ) % 8.42/4.49 = ( ~ ( ( P4 % 8.42/4.49 & ~ ( P @ ( one_one @ A ) ) ) % 8.42/4.49 | ( ~ P4 % 8.42/4.49 & ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1658_split__of__bool,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( zero_neq_one @ A ) % 8.42/4.49 => ! [P: A > $o,P4: $o] : % 8.42/4.49 ( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) ) % 8.42/4.49 = ( ( P4 % 8.42/4.49 => ( P @ ( one_one @ A ) ) ) % 8.42/4.49 & ( ~ P4 % 8.42/4.49 => ( P @ ( zero_zero @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1659_of__bool__def,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( zero_neq_one @ A ) % 8.42/4.49 => ( ( zero_neq_one_of_bool @ A ) % 8.42/4.49 = ( ^ [P6: $o] : ( if @ A @ P6 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1660_le__iff__diff__le__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ( ( ord_less_eq @ A ) % 8.42/4.49 = ( ^ [A4: A,B4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1661_less__iff__diff__less__0,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ( ( ord_less @ A ) % 8.42/4.49 = ( ^ [A4: A,B4: A] : ( ord_less @ A @ ( minus_minus @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1662_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordere1170586879665033532d_diff @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ( minus_minus @ A @ B2 @ A2 ) % 8.42/4.49 = C2 ) % 8.42/4.49 = ( B2 % 8.42/4.49 = ( plus_plus @ A @ C2 @ A2 ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1663_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordere1170586879665033532d_diff @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ A2 ) ) % 8.42/4.49 = B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1664_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordere1170586879665033532d_diff @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) % 8.42/4.49 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1665_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordere1170586879665033532d_diff @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) % 8.42/4.49 = ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1666_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordere1170586879665033532d_diff @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 ) % 8.42/4.49 = ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1667_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordere1170586879665033532d_diff @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 ) % 8.42/4.49 = ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1668_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordere1170586879665033532d_diff @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) % 8.42/4.49 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1669_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordere1170586879665033532d_diff @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) % 8.42/4.49 = ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1670_le__add__diff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordere1170586879665033532d_diff @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1671_diff__add,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordere1170586879665033532d_diff @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ A2 ) % 8.42/4.49 = B2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1672_le__diff__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1673_diff__le__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1674_add__le__add__imp__diff__le,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [I: A,K: A,N2: A,J: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ N2 @ ( plus_plus @ A @ J @ K ) ) % 8.42/4.49 => ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N2 ) % 8.42/4.49 => ( ( ord_less_eq @ A @ N2 @ ( plus_plus @ A @ J @ K ) ) % 8.42/4.49 => ( ord_less_eq @ A @ ( minus_minus @ A @ N2 @ K ) @ J ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1675_add__le__imp__le__diff,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [I: A,K: A,N2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N2 ) % 8.42/4.49 => ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N2 @ K ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1676_less__diff__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ! [A2: A,C2: A,B2: A] : % 8.42/4.49 ( ( ord_less @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) ) % 8.42/4.49 = ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1677_diff__less__eq,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ab_group_add @ A ) % 8.42/4.49 => ! [A2: A,B2: A,C2: A] : % 8.42/4.49 ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 8.42/4.49 = ( ord_less @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1678_linordered__semidom__class_Oadd__diff__inverse,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( linordered_semidom @ A ) % 8.42/4.49 => ! [A2: A,B2: A] : % 8.42/4.49 ( ~ ( ord_less @ A @ A2 @ B2 ) % 8.42/4.49 => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) ) % 8.42/4.49 = A2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1679_mult__diff__mult,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ring @ A ) % 8.42/4.49 => ! [X3: A,Y: A,A2: A,B2: A] : % 8.42/4.49 ( ( minus_minus @ A @ ( times_times @ A @ X3 @ Y ) @ ( times_times @ A @ A2 @ B2 ) ) % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ X3 @ ( minus_minus @ A @ Y @ B2 ) ) @ ( times_times @ A @ ( minus_minus @ A @ X3 @ A2 ) @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1680_square__diff__square__factored,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( comm_ring @ A ) % 8.42/4.49 => ! [X3: A,Y: A] : % 8.42/4.49 ( ( minus_minus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) % 8.42/4.49 = ( times_times @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( minus_minus @ A @ X3 @ Y ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1681_eq__add__iff2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ring @ A ) % 8.42/4.49 => ! [A2: A,E3: A,C2: A,B2: A,D2: A] : % 8.42/4.49 ( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) ) % 8.42/4.49 = ( C2 % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1682_eq__add__iff1,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ring @ A ) % 8.42/4.49 => ! [A2: A,E3: A,C2: A,B2: A,D2: A] : % 8.42/4.49 ( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) % 8.42/4.49 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) ) % 8.42/4.49 = ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C2 ) % 8.42/4.49 = D2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1683_finite_Ocases,axiom, % 8.42/4.49 ! [A: $tType,A2: set @ A] : % 8.42/4.49 ( ( finite_finite2 @ A @ A2 ) % 8.42/4.49 => ( ( A2 % 8.42/4.49 != ( bot_bot @ ( set @ A ) ) ) % 8.42/4.49 => ~ ! [A8: set @ A] : % 8.42/4.49 ( ? [A5: A] : % 8.42/4.49 ( A2 % 8.42/4.49 = ( insert @ A @ A5 @ A8 ) ) % 8.42/4.49 => ~ ( finite_finite2 @ A @ A8 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1684_finite_Osimps,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( finite_finite2 @ A ) % 8.42/4.49 = ( ^ [A4: set @ A] : % 8.42/4.49 ( ( A4 % 8.42/4.49 = ( bot_bot @ ( set @ A ) ) ) % 8.42/4.49 | ? [A6: set @ A,B4: A] : % 8.42/4.49 ( ( A4 % 8.42/4.49 = ( insert @ A @ B4 @ A6 ) ) % 8.42/4.49 & ( finite_finite2 @ A @ A6 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1685_finite__induct,axiom, % 8.42/4.49 ! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] : % 8.42/4.49 ( ( finite_finite2 @ A @ F5 ) % 8.42/4.49 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 8.42/4.49 => ( ! [X5: A,F6: set @ A] : % 8.42/4.49 ( ( finite_finite2 @ A @ F6 ) % 8.42/4.49 => ( ~ ( member @ A @ X5 @ F6 ) % 8.42/4.49 => ( ( P @ F6 ) % 8.42/4.49 => ( P @ ( insert @ A @ X5 @ F6 ) ) ) ) ) % 8.42/4.49 => ( P @ F5 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1686_finite__ne__induct,axiom, % 8.42/4.49 ! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] : % 8.42/4.49 ( ( finite_finite2 @ A @ F5 ) % 8.42/4.49 => ( ( F5 % 8.42/4.49 != ( bot_bot @ ( set @ A ) ) ) % 8.42/4.49 => ( ! [X5: A] : ( P @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.42/4.49 => ( ! [X5: A,F6: set @ A] : % 8.42/4.49 ( ( finite_finite2 @ A @ F6 ) % 8.42/4.49 => ( ( F6 % 8.42/4.49 != ( bot_bot @ ( set @ A ) ) ) % 8.42/4.49 => ( ~ ( member @ A @ X5 @ F6 ) % 8.42/4.49 => ( ( P @ F6 ) % 8.42/4.49 => ( P @ ( insert @ A @ X5 @ F6 ) ) ) ) ) ) % 8.42/4.49 => ( P @ F5 ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1687_infinite__finite__induct,axiom, % 8.42/4.49 ! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] : % 8.42/4.49 ( ! [A8: set @ A] : % 8.42/4.49 ( ~ ( finite_finite2 @ A @ A8 ) % 8.42/4.49 => ( P @ A8 ) ) % 8.42/4.49 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 8.42/4.49 => ( ! [X5: A,F6: set @ A] : % 8.42/4.49 ( ( finite_finite2 @ A @ F6 ) % 8.42/4.49 => ( ~ ( member @ A @ X5 @ F6 ) % 8.42/4.49 => ( ( P @ F6 ) % 8.42/4.49 => ( P @ ( insert @ A @ X5 @ F6 ) ) ) ) ) % 8.42/4.49 => ( P @ A3 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1688_Euclidean__Division_Opos__mod__bound,axiom, % 8.42/4.49 ! [L: int,K: int] : % 8.42/4.49 ( ( ord_less @ int @ ( zero_zero @ int ) @ L ) % 8.42/4.49 => ( ord_less @ int @ ( modulo_modulo @ int @ K @ L ) @ L ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1689_neg__mod__bound,axiom, % 8.42/4.49 ! [L: int,K: int] : % 8.42/4.49 ( ( ord_less @ int @ L @ ( zero_zero @ int ) ) % 8.42/4.49 => ( ord_less @ int @ L @ ( modulo_modulo @ int @ K @ L ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1690_subset__singleton__iff,axiom, % 8.42/4.49 ! [A: $tType,X8: set @ A,A2: A] : % 8.42/4.49 ( ( ord_less_eq @ ( set @ A ) @ X8 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.42/4.49 = ( ( X8 % 8.42/4.49 = ( bot_bot @ ( set @ A ) ) ) % 8.42/4.49 | ( X8 % 8.42/4.49 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1691_subset__singletonD,axiom, % 8.42/4.49 ! [A: $tType,A3: set @ A,X3: A] : % 8.42/4.49 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.42/4.49 => ( ( A3 % 8.42/4.49 = ( bot_bot @ ( set @ A ) ) ) % 8.42/4.49 | ( A3 % 8.42/4.49 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1692_dvd__minus__mod,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( semidom_modulo @ A ) % 8.42/4.49 => ! [B2: A,A2: A] : ( dvd_dvd @ A @ B2 @ ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1693_diff__less__Suc,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N2 ) @ ( suc @ M ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1694_Suc__diff__Suc,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less @ nat @ N2 @ M ) % 8.42/4.49 => ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N2 ) ) ) % 8.42/4.49 = ( minus_minus @ nat @ M @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1695_diff__less,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.49 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.42/4.49 => ( ord_less @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1696_Suc__diff__le,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ N2 @ M ) % 8.42/4.49 => ( ( minus_minus @ nat @ ( suc @ M ) @ N2 ) % 8.42/4.49 = ( suc @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1697_diff__less__mono,axiom, % 8.42/4.49 ! [A2: nat,B2: nat,C2: nat] : % 8.42/4.49 ( ( ord_less @ nat @ A2 @ B2 ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ C2 @ A2 ) % 8.42/4.49 => ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1698_less__diff__iff,axiom, % 8.42/4.49 ! [K: nat,M: nat,N2: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ K @ M ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ K @ N2 ) % 8.42/4.49 => ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) ) % 8.42/4.49 = ( ord_less @ nat @ M @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1699_diff__add__0,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( minus_minus @ nat @ N2 @ ( plus_plus @ nat @ N2 @ M ) ) % 8.42/4.49 = ( zero_zero @ nat ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1700_add__diff__inverse__nat,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ~ ( ord_less @ nat @ M @ N2 ) % 8.42/4.49 => ( ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ M @ N2 ) ) % 8.42/4.49 = M ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1701_less__diff__conv,axiom, % 8.42/4.49 ! [I: nat,J: nat,K: nat] : % 8.42/4.49 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) % 8.42/4.49 = ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1702_Nat_Ole__imp__diff__is__add,axiom, % 8.42/4.49 ! [I: nat,J: nat,K: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.49 => ( ( ( minus_minus @ nat @ J @ I ) % 8.42/4.49 = K ) % 8.42/4.49 = ( J % 8.42/4.49 = ( plus_plus @ nat @ K @ I ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1703_Nat_Odiff__add__assoc2,axiom, % 8.42/4.49 ! [K: nat,J: nat,I: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ K @ J ) % 8.42/4.49 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) % 8.42/4.49 = ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1704_Nat_Odiff__add__assoc,axiom, % 8.42/4.49 ! [K: nat,J: nat,I: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ K @ J ) % 8.42/4.49 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) % 8.42/4.49 = ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1705_Nat_Ole__diff__conv2,axiom, % 8.42/4.49 ! [K: nat,J: nat,I: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ K @ J ) % 8.42/4.49 => ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) % 8.42/4.49 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1706_le__diff__conv,axiom, % 8.42/4.49 ! [J: nat,K: nat,I: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) % 8.42/4.49 = ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1707_diff__Suc__eq__diff__pred,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ( minus_minus @ nat @ M @ ( suc @ N2 ) ) % 8.42/4.49 = ( minus_minus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N2 ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1708_mod__geq,axiom, % 8.42/4.49 ! [M: nat,N2: nat] : % 8.42/4.49 ( ~ ( ord_less @ nat @ M @ N2 ) % 8.42/4.49 => ( ( modulo_modulo @ nat @ M @ N2 ) % 8.42/4.49 = ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1709_mod__if,axiom, % 8.42/4.49 ( ( modulo_modulo @ nat ) % 8.42/4.49 = ( ^ [M2: nat,N: nat] : ( if @ nat @ ( ord_less @ nat @ M2 @ N ) @ M2 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1710_le__mod__geq,axiom, % 8.42/4.49 ! [N2: nat,M: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ N2 @ M ) % 8.42/4.49 => ( ( modulo_modulo @ nat @ M @ N2 ) % 8.42/4.49 = ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1711_mod__eq__dvd__iff__nat,axiom, % 8.42/4.49 ! [N2: nat,M: nat,Q4: nat] : % 8.42/4.49 ( ( ord_less_eq @ nat @ N2 @ M ) % 8.42/4.49 => ( ( ( modulo_modulo @ nat @ M @ Q4 ) % 8.42/4.49 = ( modulo_modulo @ nat @ N2 @ Q4 ) ) % 8.42/4.49 = ( dvd_dvd @ nat @ Q4 @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ). % 8.42/4.49 % 8.42/4.49 thf(fact_1712_ordered__ring__class_Ole__add__iff2,axiom, % 8.42/4.49 ! [A: $tType] : % 8.42/4.49 ( ( ordered_ring @ A ) % 8.42/4.49 => ! [A2: A,E3: A,C2: A,B2: A,D2: A] : % 8.42/4.49 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) ) % 8.42/4.50 = ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1713_ordered__ring__class_Ole__add__iff1,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( ordered_ring @ A ) % 8.42/4.50 => ! [A2: A,E3: A,C2: A,B2: A,D2: A] : % 8.42/4.50 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) ) % 8.42/4.50 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C2 ) @ D2 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1714_less__add__iff2,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( ordered_ring @ A ) % 8.42/4.50 => ! [A2: A,E3: A,C2: A,B2: A,D2: A] : % 8.42/4.50 ( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) ) % 8.42/4.50 = ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1715_less__add__iff1,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( ordered_ring @ A ) % 8.42/4.50 => ! [A2: A,E3: A,C2: A,B2: A,D2: A] : % 8.42/4.50 ( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D2 ) ) % 8.42/4.50 = ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C2 ) @ D2 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1716_divide__diff__eq__iff,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( division_ring @ A ) % 8.42/4.50 => ! [Z2: A,X3: A,Y: A] : % 8.42/4.50 ( ( Z2 % 8.42/4.50 != ( zero_zero @ A ) ) % 8.42/4.50 => ( ( minus_minus @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ Y ) % 8.42/4.50 = ( divide_divide @ A @ ( minus_minus @ A @ X3 @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1717_diff__divide__eq__iff,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( division_ring @ A ) % 8.42/4.50 => ! [Z2: A,X3: A,Y: A] : % 8.42/4.50 ( ( Z2 % 8.42/4.50 != ( zero_zero @ A ) ) % 8.42/4.50 => ( ( minus_minus @ A @ X3 @ ( divide_divide @ A @ Y @ Z2 ) ) % 8.42/4.50 = ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1718_diff__frac__eq,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( field @ A ) % 8.42/4.50 => ! [Y: A,Z2: A,X3: A,W2: A] : % 8.42/4.50 ( ( Y % 8.42/4.50 != ( zero_zero @ A ) ) % 8.42/4.50 => ( ( Z2 % 8.42/4.50 != ( zero_zero @ A ) ) % 8.42/4.50 => ( ( minus_minus @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) ) % 8.42/4.50 = ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1719_add__divide__eq__if__simps_I4_J,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( division_ring @ A ) % 8.42/4.50 => ! [Z2: A,A2: A,B2: A] : % 8.42/4.50 ( ( ( Z2 % 8.42/4.50 = ( zero_zero @ A ) ) % 8.42/4.50 => ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) ) % 8.42/4.50 = A2 ) ) % 8.42/4.50 & ( ( Z2 % 8.42/4.50 != ( zero_zero @ A ) ) % 8.42/4.50 => ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) ) % 8.42/4.50 = ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A2 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1720_square__diff__one__factored,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( ring_1 @ A ) % 8.42/4.50 => ! [X3: A] : % 8.42/4.50 ( ( minus_minus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( one_one @ A ) ) % 8.42/4.50 = ( times_times @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) @ ( minus_minus @ A @ X3 @ ( one_one @ A ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1721_finite__ranking__induct,axiom, % 8.42/4.50 ! [A: $tType,B: $tType] : % 8.42/4.50 ( ( linorder @ A ) % 8.42/4.50 => ! [S3: set @ B,P: ( set @ B ) > $o,F2: B > A] : % 8.42/4.50 ( ( finite_finite2 @ B @ S3 ) % 8.42/4.50 => ( ( P @ ( bot_bot @ ( set @ B ) ) ) % 8.42/4.50 => ( ! [X5: B,S5: set @ B] : % 8.42/4.50 ( ( finite_finite2 @ B @ S5 ) % 8.42/4.50 => ( ! [Y5: B] : % 8.42/4.50 ( ( member @ B @ Y5 @ S5 ) % 8.42/4.50 => ( ord_less_eq @ A @ ( F2 @ Y5 ) @ ( F2 @ X5 ) ) ) % 8.42/4.50 => ( ( P @ S5 ) % 8.42/4.50 => ( P @ ( insert @ B @ X5 @ S5 ) ) ) ) ) % 8.42/4.50 => ( P @ S3 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1722_finite__linorder__min__induct,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( linorder @ A ) % 8.42/4.50 => ! [A3: set @ A,P: ( set @ A ) > $o] : % 8.42/4.50 ( ( finite_finite2 @ A @ A3 ) % 8.42/4.50 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 8.42/4.50 => ( ! [B5: A,A8: set @ A] : % 8.42/4.50 ( ( finite_finite2 @ A @ A8 ) % 8.42/4.50 => ( ! [X2: A] : % 8.42/4.50 ( ( member @ A @ X2 @ A8 ) % 8.42/4.50 => ( ord_less @ A @ B5 @ X2 ) ) % 8.42/4.50 => ( ( P @ A8 ) % 8.42/4.50 => ( P @ ( insert @ A @ B5 @ A8 ) ) ) ) ) % 8.42/4.50 => ( P @ A3 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1723_finite__linorder__max__induct,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( linorder @ A ) % 8.42/4.50 => ! [A3: set @ A,P: ( set @ A ) > $o] : % 8.42/4.50 ( ( finite_finite2 @ A @ A3 ) % 8.42/4.50 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 8.42/4.50 => ( ! [B5: A,A8: set @ A] : % 8.42/4.50 ( ( finite_finite2 @ A @ A8 ) % 8.42/4.50 => ( ! [X2: A] : % 8.42/4.50 ( ( member @ A @ X2 @ A8 ) % 8.42/4.50 => ( ord_less @ A @ X2 @ B5 ) ) % 8.42/4.50 => ( ( P @ A8 ) % 8.42/4.50 => ( P @ ( insert @ A @ B5 @ A8 ) ) ) ) ) % 8.42/4.50 => ( P @ A3 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1724_inf__period_I4_J,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( ( comm_ring @ A ) % 8.42/4.50 & ( dvd @ A ) ) % 8.42/4.50 => ! [D2: A,D6: A,T2: A] : % 8.42/4.50 ( ( dvd_dvd @ A @ D2 @ D6 ) % 8.42/4.50 => ! [X2: A,K4: A] : % 8.42/4.50 ( ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X2 @ T2 ) ) ) % 8.42/4.50 = ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K4 @ D6 ) ) @ T2 ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1725_inf__period_I3_J,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( ( comm_ring @ A ) % 8.42/4.50 & ( dvd @ A ) ) % 8.42/4.50 => ! [D2: A,D6: A,T2: A] : % 8.42/4.50 ( ( dvd_dvd @ A @ D2 @ D6 ) % 8.42/4.50 => ! [X2: A,K4: A] : % 8.42/4.50 ( ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X2 @ T2 ) ) % 8.42/4.50 = ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K4 @ D6 ) ) @ T2 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1726_finite__subset__induct,axiom, % 8.42/4.50 ! [A: $tType,F5: set @ A,A3: set @ A,P: ( set @ A ) > $o] : % 8.42/4.50 ( ( finite_finite2 @ A @ F5 ) % 8.42/4.50 => ( ( ord_less_eq @ ( set @ A ) @ F5 @ A3 ) % 8.42/4.50 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 8.42/4.50 => ( ! [A5: A,F6: set @ A] : % 8.42/4.50 ( ( finite_finite2 @ A @ F6 ) % 8.42/4.50 => ( ( member @ A @ A5 @ A3 ) % 8.42/4.50 => ( ~ ( member @ A @ A5 @ F6 ) % 8.42/4.50 => ( ( P @ F6 ) % 8.42/4.50 => ( P @ ( insert @ A @ A5 @ F6 ) ) ) ) ) ) % 8.42/4.50 => ( P @ F5 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1727_finite__subset__induct_H,axiom, % 8.42/4.50 ! [A: $tType,F5: set @ A,A3: set @ A,P: ( set @ A ) > $o] : % 8.42/4.50 ( ( finite_finite2 @ A @ F5 ) % 8.42/4.50 => ( ( ord_less_eq @ ( set @ A ) @ F5 @ A3 ) % 8.42/4.50 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 8.42/4.50 => ( ! [A5: A,F6: set @ A] : % 8.42/4.50 ( ( finite_finite2 @ A @ F6 ) % 8.42/4.50 => ( ( member @ A @ A5 @ A3 ) % 8.42/4.50 => ( ( ord_less_eq @ ( set @ A ) @ F6 @ A3 ) % 8.42/4.50 => ( ~ ( member @ A @ A5 @ F6 ) % 8.42/4.50 => ( ( P @ F6 ) % 8.42/4.50 => ( P @ ( insert @ A @ A5 @ F6 ) ) ) ) ) ) ) % 8.42/4.50 => ( P @ F5 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1728_minus__mult__div__eq__mod,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( semiring_modulo @ A ) % 8.42/4.50 => ! [A2: A,B2: A] : % 8.42/4.50 ( ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) % 8.42/4.50 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1729_minus__mod__eq__mult__div,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( semiring_modulo @ A ) % 8.42/4.50 => ! [A2: A,B2: A] : % 8.42/4.50 ( ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) % 8.42/4.50 = ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1730_minus__mod__eq__div__mult,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( semiring_modulo @ A ) % 8.42/4.50 => ! [A2: A,B2: A] : % 8.42/4.50 ( ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) % 8.42/4.50 = ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1731_minus__div__mult__eq__mod,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( semiring_modulo @ A ) % 8.42/4.50 => ! [A2: A,B2: A] : % 8.42/4.50 ( ( minus_minus @ A @ A2 @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) ) % 8.42/4.50 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1732_eucl__rel__int__dividesI,axiom, % 8.42/4.50 ! [L: int,K: int,Q4: int] : % 8.42/4.50 ( ( L % 8.42/4.50 != ( zero_zero @ int ) ) % 8.42/4.50 => ( ( K % 8.42/4.50 = ( times_times @ int @ Q4 @ L ) ) % 8.42/4.50 => ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1733_Euclidean__Division_Opos__mod__sign,axiom, % 8.42/4.50 ! [L: int,K: int] : % 8.42/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ L ) % 8.42/4.50 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1734_neg__mod__sign,axiom, % 8.42/4.50 ! [L: int,K: int] : % 8.42/4.50 ( ( ord_less @ int @ L @ ( zero_zero @ int ) ) % 8.42/4.50 => ( ord_less_eq @ int @ ( modulo_modulo @ int @ K @ L ) @ ( zero_zero @ int ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1735_neg__mod__conj,axiom, % 8.42/4.50 ! [B2: int,A2: int] : % 8.42/4.50 ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) ) % 8.42/4.50 => ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) % 8.42/4.50 & ( ord_less @ int @ B2 @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1736_pos__mod__conj,axiom, % 8.42/4.50 ! [B2: int,A2: int] : % 8.42/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.42/4.50 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A2 @ B2 ) ) % 8.42/4.50 & ( ord_less @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1737_zmod__trivial__iff,axiom, % 8.42/4.50 ! [I: int,K: int] : % 8.42/4.50 ( ( ( modulo_modulo @ int @ I @ K ) % 8.42/4.50 = I ) % 8.42/4.50 = ( ( K % 8.42/4.50 = ( zero_zero @ int ) ) % 8.42/4.50 | ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I ) % 8.42/4.50 & ( ord_less @ int @ I @ K ) ) % 8.42/4.50 | ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) % 8.42/4.50 & ( ord_less @ int @ K @ I ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1738_mod__int__pos__iff,axiom, % 8.42/4.50 ! [K: int,L: int] : % 8.42/4.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) ) % 8.42/4.50 = ( ( dvd_dvd @ int @ L @ K ) % 8.42/4.50 | ( ( L % 8.42/4.50 = ( zero_zero @ int ) ) % 8.42/4.50 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) % 8.42/4.50 | ( ord_less @ int @ ( zero_zero @ int ) @ L ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1739_diff__Suc__less,axiom, % 8.42/4.50 ! [N2: nat,I: nat] : % 8.42/4.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.50 => ( ord_less @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1740_zdiv__mono__strict,axiom, % 8.42/4.50 ! [A3: int,B3: int,N2: int] : % 8.42/4.50 ( ( ord_less @ int @ A3 @ B3 ) % 8.42/4.50 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 ) % 8.42/4.50 => ( ( ( modulo_modulo @ int @ A3 @ N2 ) % 8.42/4.50 = ( zero_zero @ int ) ) % 8.42/4.50 => ( ( ( modulo_modulo @ int @ B3 @ N2 ) % 8.42/4.50 = ( zero_zero @ int ) ) % 8.42/4.50 => ( ord_less @ int @ ( divide_divide @ int @ A3 @ N2 ) @ ( divide_divide @ int @ B3 @ N2 ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1741_nat__diff__split__asm,axiom, % 8.42/4.50 ! [P: nat > $o,A2: nat,B2: nat] : % 8.42/4.50 ( ( P @ ( minus_minus @ nat @ A2 @ B2 ) ) % 8.42/4.50 = ( ~ ( ( ( ord_less @ nat @ A2 @ B2 ) % 8.42/4.50 & ~ ( P @ ( zero_zero @ nat ) ) ) % 8.42/4.50 | ? [D5: nat] : % 8.42/4.50 ( ( A2 % 8.42/4.50 = ( plus_plus @ nat @ B2 @ D5 ) ) % 8.42/4.50 & ~ ( P @ D5 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1742_nat__diff__split,axiom, % 8.42/4.50 ! [P: nat > $o,A2: nat,B2: nat] : % 8.42/4.50 ( ( P @ ( minus_minus @ nat @ A2 @ B2 ) ) % 8.42/4.50 = ( ( ( ord_less @ nat @ A2 @ B2 ) % 8.42/4.50 => ( P @ ( zero_zero @ nat ) ) ) % 8.42/4.50 & ! [D5: nat] : % 8.42/4.50 ( ( A2 % 8.42/4.50 = ( plus_plus @ nat @ B2 @ D5 ) ) % 8.42/4.50 => ( P @ D5 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1743_less__diff__conv2,axiom, % 8.42/4.50 ! [K: nat,J: nat,I: nat] : % 8.42/4.50 ( ( ord_less_eq @ nat @ K @ J ) % 8.42/4.50 => ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) % 8.42/4.50 = ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1744_nat__diff__add__eq2,axiom, % 8.42/4.50 ! [I: nat,J: nat,U: nat,M: nat,N2: nat] : % 8.42/4.50 ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.50 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 8.42/4.50 = ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1745_nat__diff__add__eq1,axiom, % 8.42/4.50 ! [J: nat,I: nat,U: nat,M: nat,N2: nat] : % 8.42/4.50 ( ( ord_less_eq @ nat @ J @ I ) % 8.42/4.50 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 8.42/4.50 = ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1746_nat__le__add__iff2,axiom, % 8.42/4.50 ! [I: nat,J: nat,U: nat,M: nat,N2: nat] : % 8.42/4.50 ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.50 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 8.42/4.50 = ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1747_nat__le__add__iff1,axiom, % 8.42/4.50 ! [J: nat,I: nat,U: nat,M: nat,N2: nat] : % 8.42/4.50 ( ( ord_less_eq @ nat @ J @ I ) % 8.42/4.50 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 8.42/4.50 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1748_nat__eq__add__iff2,axiom, % 8.42/4.50 ! [I: nat,J: nat,U: nat,M: nat,N2: nat] : % 8.42/4.50 ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.50 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) % 8.42/4.50 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 8.42/4.50 = ( M % 8.42/4.50 = ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1749_nat__eq__add__iff1,axiom, % 8.42/4.50 ! [J: nat,I: nat,U: nat,M: nat,N2: nat] : % 8.42/4.50 ( ( ord_less_eq @ nat @ J @ I ) % 8.42/4.50 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) % 8.42/4.50 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 8.42/4.50 = ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) % 8.42/4.50 = N2 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1750_dvd__minus__add,axiom, % 8.42/4.50 ! [Q4: nat,N2: nat,R2: nat,M: nat] : % 8.42/4.50 ( ( ord_less_eq @ nat @ Q4 @ N2 ) % 8.42/4.50 => ( ( ord_less_eq @ nat @ Q4 @ ( times_times @ nat @ R2 @ M ) ) % 8.42/4.50 => ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ Q4 ) ) % 8.42/4.50 = ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ ( times_times @ nat @ R2 @ M ) @ Q4 ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1751_set__update__subset__insert,axiom, % 8.42/4.50 ! [A: $tType,Xs: list @ A,I: nat,X3: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X3 ) ) @ ( insert @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1752_mod__nat__eqI,axiom, % 8.42/4.50 ! [R2: nat,N2: nat,M: nat] : % 8.42/4.50 ( ( ord_less @ nat @ R2 @ N2 ) % 8.42/4.50 => ( ( ord_less_eq @ nat @ R2 @ M ) % 8.42/4.50 => ( ( dvd_dvd @ nat @ N2 @ ( minus_minus @ nat @ M @ R2 ) ) % 8.42/4.50 => ( ( modulo_modulo @ nat @ M @ N2 ) % 8.42/4.50 = R2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1753_exp__div__exp__eq,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( bit_semiring_bits @ A ) % 8.42/4.50 => ! [M: nat,N2: nat] : % 8.42/4.50 ( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.42/4.50 = ( times_times @ A % 8.42/4.50 @ ( zero_neq_one_of_bool @ A % 8.42/4.50 @ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) % 8.42/4.50 != ( zero_zero @ A ) ) % 8.42/4.50 & ( ord_less_eq @ nat @ N2 @ M ) ) ) % 8.42/4.50 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1754_frac__le__eq,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( linordered_field @ A ) % 8.42/4.50 => ! [Y: A,Z2: A,X3: A,W2: A] : % 8.42/4.50 ( ( Y % 8.42/4.50 != ( zero_zero @ A ) ) % 8.42/4.50 => ( ( Z2 % 8.42/4.50 != ( zero_zero @ A ) ) % 8.42/4.50 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) ) % 8.42/4.50 = ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1755_frac__less__eq,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( linordered_field @ A ) % 8.42/4.50 => ! [Y: A,Z2: A,X3: A,W2: A] : % 8.42/4.50 ( ( Y % 8.42/4.50 != ( zero_zero @ A ) ) % 8.42/4.50 => ( ( Z2 % 8.42/4.50 != ( zero_zero @ A ) ) % 8.42/4.50 => ( ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) ) % 8.42/4.50 = ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1756_signed__take__bit__int__less__exp,axiom, % 8.42/4.50 ! [N2: nat,K: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1757_power__diff,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( semidom_divide @ A ) % 8.42/4.50 => ! [A2: A,N2: nat,M: nat] : % 8.42/4.50 ( ( A2 % 8.42/4.50 != ( zero_zero @ A ) ) % 8.42/4.50 => ( ( ord_less_eq @ nat @ N2 @ M ) % 8.42/4.50 => ( ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N2 ) ) % 8.42/4.50 = ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1758_mod__pos__neg__trivial,axiom, % 8.42/4.50 ! [K: int,L: int] : % 8.42/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 8.42/4.50 => ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) ) % 8.42/4.50 => ( ( modulo_modulo @ int @ K @ L ) % 8.42/4.50 = ( plus_plus @ int @ K @ L ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1759_Suc__diff__eq__diff__pred,axiom, % 8.42/4.50 ! [N2: nat,M: nat] : % 8.42/4.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.50 => ( ( minus_minus @ nat @ ( suc @ M ) @ N2 ) % 8.42/4.50 = ( minus_minus @ nat @ M @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1760_Suc__pred_H,axiom, % 8.42/4.50 ! [N2: nat] : % 8.42/4.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.50 => ( N2 % 8.42/4.50 = ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1761_div__geq,axiom, % 8.42/4.50 ! [N2: nat,M: nat] : % 8.42/4.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.50 => ( ~ ( ord_less @ nat @ M @ N2 ) % 8.42/4.50 => ( ( divide_divide @ nat @ M @ N2 ) % 8.42/4.50 = ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1762_div__if,axiom, % 8.42/4.50 ( ( divide_divide @ nat ) % 8.42/4.50 = ( ^ [M2: nat,N: nat] : % 8.42/4.50 ( if @ nat % 8.42/4.50 @ ( ( ord_less @ nat @ M2 @ N ) % 8.42/4.50 | ( N % 8.42/4.50 = ( zero_zero @ nat ) ) ) % 8.42/4.50 @ ( zero_zero @ nat ) % 8.42/4.50 @ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1763_set__replicate__Suc,axiom, % 8.42/4.50 ! [A: $tType,N2: nat,X3: A] : % 8.42/4.50 ( ( set2 @ A @ ( replicate @ A @ ( suc @ N2 ) @ X3 ) ) % 8.42/4.50 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1764_add__eq__if,axiom, % 8.42/4.50 ( ( plus_plus @ nat ) % 8.42/4.50 = ( ^ [M2: nat,N: nat] : % 8.42/4.50 ( if @ nat % 8.42/4.50 @ ( M2 % 8.42/4.50 = ( zero_zero @ nat ) ) % 8.42/4.50 @ N % 8.42/4.50 @ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1765_set__replicate__conv__if,axiom, % 8.42/4.50 ! [A: $tType,N2: nat,X3: A] : % 8.42/4.50 ( ( ( N2 % 8.42/4.50 = ( zero_zero @ nat ) ) % 8.42/4.50 => ( ( set2 @ A @ ( replicate @ A @ N2 @ X3 ) ) % 8.42/4.50 = ( bot_bot @ ( set @ A ) ) ) ) % 8.42/4.50 & ( ( N2 % 8.42/4.50 != ( zero_zero @ nat ) ) % 8.42/4.50 => ( ( set2 @ A @ ( replicate @ A @ N2 @ X3 ) ) % 8.42/4.50 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1766_nat__less__add__iff2,axiom, % 8.42/4.50 ! [I: nat,J: nat,U: nat,M: nat,N2: nat] : % 8.42/4.50 ( ( ord_less_eq @ nat @ I @ J ) % 8.42/4.50 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 8.42/4.50 = ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1767_nat__less__add__iff1,axiom, % 8.42/4.50 ! [J: nat,I: nat,U: nat,M: nat,N2: nat] : % 8.42/4.50 ( ( ord_less_eq @ nat @ J @ I ) % 8.42/4.50 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) ) % 8.42/4.50 = ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1768_mult__eq__if,axiom, % 8.42/4.50 ( ( times_times @ nat ) % 8.42/4.50 = ( ^ [M2: nat,N: nat] : % 8.42/4.50 ( if @ nat % 8.42/4.50 @ ( M2 % 8.42/4.50 = ( zero_zero @ nat ) ) % 8.42/4.50 @ ( zero_zero @ nat ) % 8.42/4.50 @ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1769_scaling__mono,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( linordered_field @ A ) % 8.42/4.50 => ! [U: A,V2: A,R2: A,S: A] : % 8.42/4.50 ( ( ord_less_eq @ A @ U @ V2 ) % 8.42/4.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 ) % 8.42/4.50 => ( ( ord_less_eq @ A @ R2 @ S ) % 8.42/4.50 => ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R2 @ ( minus_minus @ A @ V2 @ U ) ) @ S ) ) @ V2 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1770_exp__not__zero__imp__exp__diff__not__zero,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( bit_semiring_bits @ A ) % 8.42/4.50 => ! [N2: nat,M: nat] : % 8.42/4.50 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.50 != ( zero_zero @ A ) ) % 8.42/4.50 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) % 8.42/4.50 != ( zero_zero @ A ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1771_signed__take__bit__int__greater__eq__self__iff,axiom, % 8.42/4.50 ! [K: int,N2: nat] : % 8.42/4.50 ( ( ord_less_eq @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) ) % 8.42/4.50 = ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1772_signed__take__bit__int__less__self__iff,axiom, % 8.42/4.50 ! [N2: nat,K: int] : % 8.42/4.50 ( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ K ) % 8.42/4.50 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1773_power__diff__power__eq,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( euclid4440199948858584721cancel @ A ) % 8.42/4.50 => ! [A2: A,N2: nat,M: nat] : % 8.42/4.50 ( ( A2 % 8.42/4.50 != ( zero_zero @ A ) ) % 8.42/4.50 => ( ( ( ord_less_eq @ nat @ N2 @ M ) % 8.42/4.50 => ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) % 8.42/4.50 = ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N2 ) ) ) ) % 8.42/4.50 & ( ~ ( ord_less_eq @ nat @ N2 @ M ) % 8.42/4.50 => ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N2 ) ) % 8.42/4.50 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1774_split__zmod,axiom, % 8.42/4.50 ! [P: int > $o,N2: int,K: int] : % 8.42/4.50 ( ( P @ ( modulo_modulo @ int @ N2 @ K ) ) % 8.42/4.50 = ( ( ( K % 8.42/4.50 = ( zero_zero @ int ) ) % 8.42/4.50 => ( P @ N2 ) ) % 8.42/4.50 & ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 8.42/4.50 => ! [I3: int,J3: int] : % 8.42/4.50 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 ) % 8.42/4.50 & ( ord_less @ int @ J3 @ K ) % 8.42/4.50 & ( N2 % 8.42/4.50 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) ) % 8.42/4.50 => ( P @ J3 ) ) ) % 8.42/4.50 & ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 8.42/4.50 => ! [I3: int,J3: int] : % 8.42/4.50 ( ( ( ord_less @ int @ K @ J3 ) % 8.42/4.50 & ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) ) % 8.42/4.50 & ( N2 % 8.42/4.50 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) ) % 8.42/4.50 => ( P @ J3 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1775_int__mod__neg__eq,axiom, % 8.42/4.50 ! [A2: int,B2: int,Q4: int,R2: int] : % 8.42/4.50 ( ( A2 % 8.42/4.50 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) ) % 8.42/4.50 => ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) ) % 8.42/4.50 => ( ( ord_less @ int @ B2 @ R2 ) % 8.42/4.50 => ( ( modulo_modulo @ int @ A2 @ B2 ) % 8.42/4.50 = R2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1776_int__mod__pos__eq,axiom, % 8.42/4.50 ! [A2: int,B2: int,Q4: int,R2: int] : % 8.42/4.50 ( ( A2 % 8.42/4.50 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) ) % 8.42/4.50 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 ) % 8.42/4.50 => ( ( ord_less @ int @ R2 @ B2 ) % 8.42/4.50 => ( ( modulo_modulo @ int @ A2 @ B2 ) % 8.42/4.50 = R2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1777_power__eq__if,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( power @ A ) % 8.42/4.50 => ( ( power_power @ A ) % 8.42/4.50 = ( ^ [P6: A,M2: nat] : % 8.42/4.50 ( if @ A % 8.42/4.50 @ ( M2 % 8.42/4.50 = ( zero_zero @ nat ) ) % 8.42/4.50 @ ( one_one @ A ) % 8.42/4.50 @ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1778_power__minus__mult,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( monoid_mult @ A ) % 8.42/4.50 => ! [N2: nat,A2: A] : % 8.42/4.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.50 => ( ( times_times @ A @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ A2 ) % 8.42/4.50 = ( power_power @ A @ A2 @ N2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1779_diff__le__diff__pow,axiom, % 8.42/4.50 ! [K: nat,M: nat,N2: nat] : % 8.42/4.50 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) % 8.42/4.50 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N2 ) @ ( minus_minus @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1780_le__div__geq,axiom, % 8.42/4.50 ! [N2: nat,M: nat] : % 8.42/4.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.42/4.50 => ( ( ord_less_eq @ nat @ N2 @ M ) % 8.42/4.50 => ( ( divide_divide @ nat @ M @ N2 ) % 8.42/4.50 = ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1781_bits__induct,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( bit_semiring_bits @ A ) % 8.42/4.50 => ! [P: A > $o,A2: A] : % 8.42/4.50 ( ! [A5: A] : % 8.42/4.50 ( ( ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.50 = A5 ) % 8.42/4.50 => ( P @ A5 ) ) % 8.42/4.50 => ( ! [A5: A,B5: $o] : % 8.42/4.50 ( ( P @ A5 ) % 8.42/4.50 => ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.42/4.50 = A5 ) % 8.42/4.50 => ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) ) ) ) % 8.42/4.50 => ( P @ A2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1782_verit__le__mono__div__int,axiom, % 8.42/4.50 ! [A3: int,B3: int,N2: int] : % 8.42/4.50 ( ( ord_less @ int @ A3 @ B3 ) % 8.42/4.50 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 ) % 8.42/4.50 => ( ord_less_eq @ int % 8.42/4.50 @ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ N2 ) % 8.42/4.50 @ ( if @ int % 8.42/4.50 @ ( ( modulo_modulo @ int @ B3 @ N2 ) % 8.42/4.50 = ( zero_zero @ int ) ) % 8.42/4.50 @ ( one_one @ int ) % 8.42/4.50 @ ( zero_zero @ int ) ) ) % 8.42/4.50 @ ( divide_divide @ int @ B3 @ N2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1783_split__pos__lemma,axiom, % 8.42/4.50 ! [K: int,P: int > int > $o,N2: int] : % 8.42/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 8.42/4.50 => ( ( P @ ( divide_divide @ int @ N2 @ K ) @ ( modulo_modulo @ int @ N2 @ K ) ) % 8.42/4.50 = ( ! [I3: int,J3: int] : % 8.42/4.50 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 ) % 8.42/4.50 & ( ord_less @ int @ J3 @ K ) % 8.42/4.50 & ( N2 % 8.42/4.50 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) ) % 8.42/4.50 => ( P @ I3 @ J3 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1784_split__neg__lemma,axiom, % 8.42/4.50 ! [K: int,P: int > int > $o,N2: int] : % 8.42/4.50 ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 8.42/4.50 => ( ( P @ ( divide_divide @ int @ N2 @ K ) @ ( modulo_modulo @ int @ N2 @ K ) ) % 8.42/4.50 = ( ! [I3: int,J3: int] : % 8.42/4.50 ( ( ( ord_less @ int @ K @ J3 ) % 8.42/4.50 & ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) ) % 8.42/4.50 & ( N2 % 8.42/4.50 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J3 ) ) ) % 8.42/4.50 => ( P @ I3 @ J3 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1785_int__power__div__base,axiom, % 8.42/4.50 ! [M: nat,K: int] : % 8.42/4.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.42/4.50 => ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 8.42/4.50 => ( ( divide_divide @ int @ ( power_power @ int @ K @ M ) @ K ) % 8.42/4.50 = ( power_power @ int @ K @ ( minus_minus @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1786_set__decode__plus__power__2,axiom, % 8.42/4.50 ! [N2: nat,Z2: nat] : % 8.42/4.50 ( ~ ( member @ nat @ N2 @ ( nat_set_decode @ Z2 ) ) % 8.42/4.50 => ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ Z2 ) ) % 8.42/4.50 = ( insert @ nat @ N2 @ ( nat_set_decode @ Z2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1787_exp__mod__exp,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( euclid5411537665997757685th_nat @ A ) % 8.42/4.50 => ! [M: nat,N2: nat] : % 8.42/4.50 ( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.42/4.50 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M @ N2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1788_power2__diff,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( comm_ring_1 @ A ) % 8.42/4.50 => ! [X3: A,Y: A] : % 8.42/4.50 ( ( power_power @ A @ ( minus_minus @ A @ X3 @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.42/4.50 = ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) @ Y ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1789_mult__exp__mod__exp__eq,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( bit_semiring_bits @ A ) % 8.42/4.50 => ! [M: nat,N2: nat,A2: A] : % 8.42/4.50 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.42/4.50 => ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.42/4.50 = ( times_times @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1790_even__mod__4__div__2,axiom, % 8.42/4.50 ! [N2: nat] : % 8.42/4.50 ( ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) % 8.42/4.50 = ( suc @ ( zero_zero @ nat ) ) ) % 8.42/4.50 => ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1791_eucl__rel__int__iff,axiom, % 8.42/4.50 ! [K: int,L: int,Q4: int,R2: int] : % 8.42/4.50 ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 8.42/4.50 = ( ( K % 8.42/4.50 = ( plus_plus @ int @ ( times_times @ int @ L @ Q4 ) @ R2 ) ) % 8.42/4.50 & ( ( ord_less @ int @ ( zero_zero @ int ) @ L ) % 8.42/4.50 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 ) % 8.42/4.50 & ( ord_less @ int @ R2 @ L ) ) ) % 8.42/4.50 & ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L ) % 8.42/4.50 => ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) ) % 8.42/4.50 => ( ( ord_less @ int @ L @ R2 ) % 8.42/4.50 & ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) ) ) ) % 8.42/4.50 & ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) ) % 8.42/4.50 => ( Q4 % 8.42/4.50 = ( zero_zero @ int ) ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1792_divmod__digit__1_I2_J,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.50 => ! [A2: A,B2: A] : % 8.42/4.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.42/4.50 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.42/4.50 => ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) % 8.42/4.50 => ( ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 ) % 8.42/4.50 = ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1793_even__mask__div__iff_H,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( euclid5411537665997757685th_nat @ A ) % 8.42/4.50 => ! [M: nat,N2: nat] : % 8.42/4.50 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 8.42/4.50 = ( ord_less_eq @ nat @ M @ N2 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1794_even__mask__div__iff,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( bit_semiring_bits @ A ) % 8.42/4.50 => ! [M: nat,N2: nat] : % 8.42/4.50 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 8.42/4.50 = ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) % 8.42/4.50 = ( zero_zero @ A ) ) % 8.42/4.50 | ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1795_divmod__step__eq,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( unique1627219031080169319umeral @ A ) % 8.42/4.50 => ! [L: num,R2: A,Q4: A] : % 8.42/4.50 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 ) % 8.42/4.50 => ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R2 ) ) % 8.42/4.50 = ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R2 @ ( numeral_numeral @ A @ L ) ) ) ) ) % 8.42/4.50 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 ) % 8.42/4.50 => ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R2 ) ) % 8.42/4.50 = ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ R2 ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1796_inrange,axiom, % 8.42/4.50 ! [T2: vEBT_VEBT,N2: nat] : % 8.42/4.50 ( ( vEBT_invar_vebt @ T2 @ N2 ) % 8.42/4.50 => ( ord_less_eq @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1797_Suc__if__eq,axiom, % 8.42/4.50 ! [A: $tType,F2: nat > A,H2: nat > A,G: A,N2: nat] : % 8.42/4.50 ( ! [N3: nat] : % 8.42/4.50 ( ( F2 @ ( suc @ N3 ) ) % 8.42/4.50 = ( H2 @ N3 ) ) % 8.42/4.50 => ( ( ( F2 @ ( zero_zero @ nat ) ) % 8.42/4.50 = G ) % 8.42/4.50 => ( ( ( N2 % 8.42/4.50 = ( zero_zero @ nat ) ) % 8.42/4.50 => ( ( F2 @ N2 ) % 8.42/4.50 = G ) ) % 8.42/4.50 & ( ( N2 % 8.42/4.50 != ( zero_zero @ nat ) ) % 8.42/4.50 => ( ( F2 @ N2 ) % 8.42/4.50 = ( H2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1798_Divides_Oadjust__div__eq,axiom, % 8.42/4.50 ! [Q4: int,R2: int] : % 8.42/4.50 ( ( adjust_div @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 8.42/4.50 = ( plus_plus @ int @ Q4 % 8.42/4.50 @ ( zero_neq_one_of_bool @ int % 8.42/4.50 @ ( R2 % 8.42/4.50 != ( zero_zero @ int ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1799_signed__take__bit__rec,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( bit_ri3973907225187159222ations @ A ) % 8.42/4.50 => ( ( bit_ri4674362597316999326ke_bit @ A ) % 8.42/4.50 = ( ^ [N: nat,A4: A] : % 8.42/4.50 ( if @ A % 8.42/4.50 @ ( N % 8.42/4.50 = ( zero_zero @ nat ) ) % 8.42/4.50 @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) % 8.42/4.50 @ ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1800_neg__eucl__rel__int__mult__2,axiom, % 8.42/4.50 ! [B2: int,A2: int,Q4: int,R2: int] : % 8.42/4.50 ( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) ) % 8.42/4.50 => ( ( eucl_rel_int @ ( plus_plus @ int @ A2 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 8.42/4.50 => ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q4 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) @ ( one_one @ int ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1801_diff__shunt__var,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( boolea8198339166811842893lgebra @ A ) % 8.42/4.50 => ! [X3: A,Y: A] : % 8.42/4.50 ( ( ( minus_minus @ A @ X3 @ Y ) % 8.42/4.50 = ( bot_bot @ A ) ) % 8.42/4.50 = ( ord_less_eq @ A @ X3 @ Y ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1802_take__bit__rec,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.42/4.50 => ( ( bit_se2584673776208193580ke_bit @ A ) % 8.42/4.50 = ( ^ [N: nat,A4: A] : % 8.42/4.50 ( if @ A % 8.42/4.50 @ ( N % 8.42/4.50 = ( zero_zero @ nat ) ) % 8.42/4.50 @ ( zero_zero @ A ) % 8.42/4.50 @ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1803_add_Oinverse__inverse,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( group_add @ A ) % 8.42/4.50 => ! [A2: A] : % 8.42/4.50 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) ) % 8.42/4.50 = A2 ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1804_neg__equal__iff__equal,axiom, % 8.42/4.50 ! [A: $tType] : % 8.42/4.50 ( ( group_add @ A ) % 8.42/4.50 => ! [A2: A,B2: A] : % 8.42/4.50 ( ( ( uminus_uminus @ A @ A2 ) % 8.42/4.50 = ( uminus_uminus @ A @ B2 ) ) % 8.42/4.50 = ( A2 = B2 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1805_Diff__cancel,axiom, % 8.42/4.50 ! [A: $tType,A3: set @ A] : % 8.42/4.50 ( ( minus_minus @ ( set @ A ) @ A3 @ A3 ) % 8.42/4.50 = ( bot_bot @ ( set @ A ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1806_empty__Diff,axiom, % 8.42/4.50 ! [A: $tType,A3: set @ A] : % 8.42/4.50 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) % 8.42/4.50 = ( bot_bot @ ( set @ A ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1807_Diff__empty,axiom, % 8.42/4.50 ! [A: $tType,A3: set @ A] : % 8.42/4.50 ( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) % 8.42/4.50 = A3 ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1808_finite__Diff2,axiom, % 8.42/4.50 ! [A: $tType,B3: set @ A,A3: set @ A] : % 8.42/4.50 ( ( finite_finite2 @ A @ B3 ) % 8.42/4.50 => ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 8.42/4.50 = ( finite_finite2 @ A @ A3 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1809_finite__Diff,axiom, % 8.42/4.50 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.42/4.50 ( ( finite_finite2 @ A @ A3 ) % 8.42/4.50 => ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1810_Compl__subset__Compl__iff,axiom, % 8.42/4.50 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.42/4.50 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) % 8.42/4.50 = ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1811_Compl__anti__mono,axiom, % 8.42/4.50 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.42/4.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.42/4.50 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ). % 8.42/4.50 % 8.42/4.50 thf(fact_1812_Diff__insert0,axiom, % 8.42/4.50 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] : % 8.42/4.50 ( ~ ( member @ A @ X3 @ A3 ) % 8.42/4.50 => ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) ) % 8.42/4.50 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1813_insert__Diff1,axiom, % 8.50/4.50 ! [A: $tType,X3: A,B3: set @ A,A3: set @ A] : % 8.50/4.50 ( ( member @ A @ X3 @ B3 ) % 8.50/4.50 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 ) % 8.50/4.50 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1814_compl__le__compl__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.50 => ! [X3: A,Y: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) ) % 8.50/4.50 = ( ord_less_eq @ A @ Y @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1815_neg__le__iff__le,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ordered_ab_group_add @ A ) % 8.50/4.50 => ! [B2: A,A2: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) % 8.50/4.50 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1816_neg__equal__zero,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linord5086331880401160121up_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( ( uminus_uminus @ A @ A2 ) % 8.50/4.50 = A2 ) % 8.50/4.50 = ( A2 % 8.50/4.50 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1817_equal__neg__zero,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linord5086331880401160121up_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( A2 % 8.50/4.50 = ( uminus_uminus @ A @ A2 ) ) % 8.50/4.50 = ( A2 % 8.50/4.50 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1818_neg__equal__0__iff__equal,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( ( uminus_uminus @ A @ A2 ) % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 = ( A2 % 8.50/4.50 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1819_neg__0__equal__iff__equal,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( ( zero_zero @ A ) % 8.50/4.50 = ( uminus_uminus @ A @ A2 ) ) % 8.50/4.50 = ( ( zero_zero @ A ) % 8.50/4.50 = A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1820_add_Oinverse__neutral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1821_compl__less__compl__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.50 => ! [X3: A,Y: A] : % 8.50/4.50 ( ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) ) % 8.50/4.50 = ( ord_less @ A @ Y @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1822_neg__less__iff__less,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ordered_ab_group_add @ A ) % 8.50/4.50 => ! [B2: A,A2: A] : % 8.50/4.50 ( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) % 8.50/4.50 = ( ord_less @ A @ A2 @ B2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1823_mult__minus__left,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 8.50/4.50 = ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1824_minus__mult__minus,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) % 8.50/4.50 = ( times_times @ A @ A2 @ B2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1825_mult__minus__right,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) % 8.50/4.50 = ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1826_minus__add__distrib,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ab_group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) ) % 8.50/4.50 = ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1827_minus__add__cancel,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) ) % 8.50/4.50 = B2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1828_add__minus__cancel,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) % 8.50/4.50 = B2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1829_minus__diff__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) ) % 8.50/4.50 = ( minus_minus @ A @ B2 @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1830_dvd__minus__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( comm_ring_1 @ A ) % 8.50/4.50 => ! [X3: A,Y: A] : % 8.50/4.50 ( ( dvd_dvd @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) % 8.50/4.50 = ( dvd_dvd @ A @ X3 @ Y ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1831_minus__dvd__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( comm_ring_1 @ A ) % 8.50/4.50 => ! [X3: A,Y: A] : % 8.50/4.50 ( ( dvd_dvd @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) % 8.50/4.50 = ( dvd_dvd @ A @ X3 @ Y ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1832_Diff__eq__empty__iff,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.50 ( ( ( minus_minus @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.50 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.50 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1833_atLeastAtMost__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ord @ A ) % 8.50/4.50 => ! [I: A,L: A,U: A] : % 8.50/4.50 ( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L @ U ) ) % 8.50/4.50 = ( ( ord_less_eq @ A @ L @ I ) % 8.50/4.50 & ( ord_less_eq @ A @ I @ U ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1834_Icc__eq__Icc,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( order @ A ) % 8.50/4.50 => ! [L: A,H2: A,L3: A,H3: A] : % 8.50/4.50 ( ( ( set_or1337092689740270186AtMost @ A @ L @ H2 ) % 8.50/4.50 = ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) % 8.50/4.50 = ( ( ( L = L3 ) % 8.50/4.50 & ( H2 = H3 ) ) % 8.50/4.50 | ( ~ ( ord_less_eq @ A @ L @ H2 ) % 8.50/4.50 & ~ ( ord_less_eq @ A @ L3 @ H3 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1835_insert__Diff__single,axiom, % 8.50/4.50 ! [A: $tType,A2: A,A3: set @ A] : % 8.50/4.50 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.50 = ( insert @ A @ A2 @ A3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1836_finite__Diff__insert,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] : % 8.50/4.50 ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) ) % 8.50/4.50 = ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1837_take__bit__of__0,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [N2: nat] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( zero_zero @ A ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1838_finite__atLeastAtMost,axiom, % 8.50/4.50 ! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1839_neg__less__eq__nonneg,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linord5086331880401160121up_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 ) % 8.50/4.50 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1840_less__eq__neg__nonpos,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linord5086331880401160121up_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) ) % 8.50/4.50 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1841_neg__le__0__iff__le,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ordered_ab_group_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) ) % 8.50/4.50 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1842_neg__0__le__iff__le,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ordered_ab_group_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) ) % 8.50/4.50 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1843_less__neg__neg,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linord5086331880401160121up_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) ) % 8.50/4.50 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1844_neg__less__pos,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linord5086331880401160121up_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 ) % 8.50/4.50 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1845_neg__0__less__iff__less,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ordered_ab_group_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) ) % 8.50/4.50 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1846_neg__less__0__iff__less,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ordered_ab_group_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) ) % 8.50/4.50 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1847_add_Oright__inverse,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( plus_plus @ A @ A2 @ ( uminus_uminus @ A @ A2 ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1848_ab__left__minus,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1849_verit__minus__simplify_I3_J,axiom, % 8.50/4.50 ! [B: $tType] : % 8.50/4.50 ( ( group_add @ B ) % 8.50/4.50 => ! [B2: B] : % 8.50/4.50 ( ( minus_minus @ B @ ( zero_zero @ B ) @ B2 ) % 8.50/4.50 = ( uminus_uminus @ B @ B2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1850_diff__0,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.50 = ( uminus_uminus @ A @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1851_add__neg__numeral__simps_I3_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( neg_numeral @ A ) % 8.50/4.50 => ! [M: num,N2: num] : % 8.50/4.50 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) % 8.50/4.50 = ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1852_uminus__add__conv__diff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ab_group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 8.50/4.50 = ( minus_minus @ A @ B2 @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1853_diff__minus__eq__add,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( minus_minus @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) % 8.50/4.50 = ( plus_plus @ A @ A2 @ B2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1854_divide__minus1,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( field @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( divide_divide @ A @ X3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 8.50/4.50 = ( uminus_uminus @ A @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1855_atLeastatMost__empty__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( preorder @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 8.50/4.50 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.50 = ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1856_atLeastatMost__empty__iff2,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( preorder @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ( bot_bot @ ( set @ A ) ) % 8.50/4.50 = ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.50 = ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1857_atLeastatMost__subset__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( preorder @ A ) % 8.50/4.50 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.50/4.50 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) ) % 8.50/4.50 = ( ~ ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.50 | ( ( ord_less_eq @ A @ C2 @ A2 ) % 8.50/4.50 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1858_atLeastatMost__empty,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( order @ A ) % 8.50/4.50 => ! [B2: A,A2: A] : % 8.50/4.50 ( ( ord_less @ A @ B2 @ A2 ) % 8.50/4.50 => ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 8.50/4.50 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1859_infinite__Icc__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( dense_linorder @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) ) % 8.50/4.50 = ( ord_less @ A @ A2 @ B2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1860_subset__Compl__singleton,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,B2: A] : % 8.50/4.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.50 = ( ~ ( member @ A @ B2 @ A3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1861_take__bit__0,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ A @ ( zero_zero @ nat ) @ A2 ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1862_atLeastAtMost__singleton,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( order @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( set_or1337092689740270186AtMost @ A @ A2 @ A2 ) % 8.50/4.50 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1863_atLeastAtMost__singleton__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( order @ A ) % 8.50/4.50 => ! [A2: A,B2: A,C2: A] : % 8.50/4.50 ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 8.50/4.50 = ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.50 = ( ( A2 = B2 ) % 8.50/4.50 & ( B2 = C2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1864_take__bit__Suc__1,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [N2: nat] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( one_one @ A ) ) % 8.50/4.50 = ( one_one @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1865_add__neg__numeral__special_I8_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( neg_numeral @ A ) % 8.50/4.50 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1866_add__neg__numeral__special_I7_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( neg_numeral @ A ) % 8.50/4.50 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1867_diff__numeral__special_I12_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( neg_numeral @ A ) % 8.50/4.50 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1868_mod__minus1__right,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( euclid8851590272496341667cancel @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( modulo_modulo @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1869_take__bit__of__1__eq__0__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [N2: nat] : % 8.50/4.50 ( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( one_one @ A ) ) % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 = ( N2 % 8.50/4.50 = ( zero_zero @ nat ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1870_semiring__norm_I168_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( neg_numeral @ A ) % 8.50/4.50 => ! [V2: num,W2: num,Y: A] : % 8.50/4.50 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) ) % 8.50/4.50 = ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W2 ) ) ) @ Y ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1871_neg__numeral__le__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num,N2: num] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) % 8.50/4.50 = ( ord_less_eq @ num @ N2 @ M ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1872_neg__numeral__less__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num,N2: num] : % 8.50/4.50 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) % 8.50/4.50 = ( ord_less @ num @ N2 @ M ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1873_zle__diff1__eq,axiom, % 8.50/4.50 ! [W2: int,Z2: int] : % 8.50/4.50 ( ( ord_less_eq @ int @ W2 @ ( minus_minus @ int @ Z2 @ ( one_one @ int ) ) ) % 8.50/4.50 = ( ord_less @ int @ W2 @ Z2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1874_take__bit__of__Suc__0,axiom, % 8.50/4.50 ! [N2: nat] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.50 = ( zero_neq_one_of_bool @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1875_not__neg__one__le__neg__numeral__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num] : % 8.50/4.50 ( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) % 8.50/4.50 = ( M != one2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1876_divide__le__eq__numeral1_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [B2: A,W2: num,A2: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A2 ) % 8.50/4.50 = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1877_le__divide__eq__numeral1_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [A2: A,B2: A,W2: num] : % 8.50/4.50 ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) % 8.50/4.50 = ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1878_divide__eq__eq__numeral1_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [B2: A,W2: num,A2: A] : % 8.50/4.50 ( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) % 8.50/4.50 = A2 ) % 8.50/4.50 = ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 => ( B2 % 8.50/4.50 = ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) % 8.50/4.50 & ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 => ( A2 % 8.50/4.50 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1879_eq__divide__eq__numeral1_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [A2: A,B2: A,W2: num] : % 8.50/4.50 ( ( A2 % 8.50/4.50 = ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) % 8.50/4.50 = ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 => ( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) % 8.50/4.50 = B2 ) ) % 8.50/4.50 & ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 => ( A2 % 8.50/4.50 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1880_neg__numeral__less__neg__one__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num] : % 8.50/4.50 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 8.50/4.50 = ( M != one2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1881_less__divide__eq__numeral1_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [A2: A,B2: A,W2: num] : % 8.50/4.50 ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) % 8.50/4.50 = ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1882_divide__less__eq__numeral1_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [B2: A,W2: num,A2: A] : % 8.50/4.50 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A2 ) % 8.50/4.50 = ( ord_less @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1883_take__bit__of__1,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [N2: nat] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( one_one @ A ) ) % 8.50/4.50 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1884_add__neg__numeral__special_I9_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( neg_numeral @ A ) % 8.50/4.50 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 8.50/4.50 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1885_even__take__bit__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [N2: nat,A2: A] : % 8.50/4.50 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) % 8.50/4.50 = ( ( N2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 | ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1886_signed__take__bit__Suc__minus__bit0,axiom, % 8.50/4.50 ! [N2: nat,K: num] : % 8.50/4.50 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) ) % 8.50/4.50 = ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1887_take__bit__Suc__0,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ ( zero_zero @ nat ) ) @ A2 ) % 8.50/4.50 = ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1888_signed__take__bit__0,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_ri3973907225187159222ations @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A2 ) % 8.50/4.50 = ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1889_take__bit__of__exp,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [M: nat,N2: nat] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.50/4.50 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N2 @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1890_take__bit__of__2,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [N2: nat] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.50/4.50 = ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1891_minus__int__code_I2_J,axiom, % 8.50/4.50 ! [L: int] : % 8.50/4.50 ( ( minus_minus @ int @ ( zero_zero @ int ) @ L ) % 8.50/4.50 = ( uminus_uminus @ int @ L ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1892_compl__mono,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.50 => ! [X3: A,Y: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.50 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1893_compl__le__swap1,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.50 => ! [Y: A,X3: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X3 ) ) % 8.50/4.50 => ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1894_compl__le__swap2,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.50 => ! [Y: A,X3: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X3 ) % 8.50/4.50 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1895_equation__minus__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( A2 % 8.50/4.50 = ( uminus_uminus @ A @ B2 ) ) % 8.50/4.50 = ( B2 % 8.50/4.50 = ( uminus_uminus @ A @ A2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1896_minus__equation__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ( uminus_uminus @ A @ A2 ) % 8.50/4.50 = B2 ) % 8.50/4.50 = ( ( uminus_uminus @ A @ B2 ) % 8.50/4.50 = A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1897_compl__less__swap1,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.50 => ! [Y: A,X3: A] : % 8.50/4.50 ( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X3 ) ) % 8.50/4.50 => ( ord_less @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1898_compl__less__swap2,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.50 => ! [Y: A,X3: A] : % 8.50/4.50 ( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X3 ) % 8.50/4.50 => ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1899_Compl__insert,axiom, % 8.50/4.50 ! [A: $tType,X3: A,A3: set @ A] : % 8.50/4.50 ( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) ) % 8.50/4.50 = ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1900_take__bit__add,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [N2: nat,A2: A,B2: A] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ B2 ) ) ) % 8.50/4.50 = ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1901_take__bit__tightened,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [N2: nat,A2: A,B2: A,M: nat] : % 8.50/4.50 ( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) % 8.50/4.50 = ( bit_se2584673776208193580ke_bit @ A @ N2 @ B2 ) ) % 8.50/4.50 => ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.50 => ( ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) % 8.50/4.50 = ( bit_se2584673776208193580ke_bit @ A @ M @ B2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1902_take__bit__nat__less__eq__self,axiom, % 8.50/4.50 ! [N2: nat,M: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) @ M ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1903_take__bit__tightened__less__eq__nat,axiom, % 8.50/4.50 ! [M: nat,N2: nat,Q4: nat] : % 8.50/4.50 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.50 => ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M @ Q4 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ Q4 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1904_le__imp__neg__le,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ordered_ab_group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.50 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1905_minus__le__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ordered_ab_group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 8.50/4.50 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1906_le__minus__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ordered_ab_group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) % 8.50/4.50 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1907_minus__less__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ordered_ab_group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 8.50/4.50 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1908_less__minus__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ordered_ab_group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) % 8.50/4.50 = ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1909_verit__negate__coefficient_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ordered_ab_group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.50 => ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1910_square__eq__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( idom @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ( times_times @ A @ A2 @ A2 ) % 8.50/4.50 = ( times_times @ A @ B2 @ B2 ) ) % 8.50/4.50 = ( ( A2 = B2 ) % 8.50/4.50 | ( A2 % 8.50/4.50 = ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1911_minus__mult__commute,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 8.50/4.50 = ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1912_is__num__normalize_I8_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( neg_numeral @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) ) % 8.50/4.50 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1913_add_Oinverse__distrib__swap,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) ) % 8.50/4.50 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1914_group__cancel_Oneg1,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ab_group_add @ A ) % 8.50/4.50 => ! [A3: A,K: A,A2: A] : % 8.50/4.50 ( ( A3 % 8.50/4.50 = ( plus_plus @ A @ K @ A2 ) ) % 8.50/4.50 => ( ( uminus_uminus @ A @ A3 ) % 8.50/4.50 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1915_minus__diff__commute,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ab_group_add @ A ) % 8.50/4.50 => ! [B2: A,A2: A] : % 8.50/4.50 ( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) % 8.50/4.50 = ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1916_minus__divide__right,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( field @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 8.50/4.50 = ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1917_minus__divide__divide,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( field @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) % 8.50/4.50 = ( divide_divide @ A @ A2 @ B2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1918_minus__divide__left,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 8.50/4.50 = ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1919_Diff__infinite__finite,axiom, % 8.50/4.50 ! [A: $tType,T4: set @ A,S3: set @ A] : % 8.50/4.50 ( ( finite_finite2 @ A @ T4 ) % 8.50/4.50 => ( ~ ( finite_finite2 @ A @ S3 ) % 8.50/4.50 => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S3 @ T4 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1920_double__diff,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 8.50/4.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.50 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C5 ) % 8.50/4.50 => ( ( minus_minus @ ( set @ A ) @ B3 @ ( minus_minus @ ( set @ A ) @ C5 @ A3 ) ) % 8.50/4.50 = A3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1921_Diff__subset,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ A3 ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1922_Diff__mono,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,C5: set @ A,D6: set @ A,B3: set @ A] : % 8.50/4.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) % 8.50/4.50 => ( ( ord_less_eq @ ( set @ A ) @ D6 @ B3 ) % 8.50/4.50 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ C5 @ D6 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1923_insert__Diff__if,axiom, % 8.50/4.50 ! [A: $tType,X3: A,B3: set @ A,A3: set @ A] : % 8.50/4.50 ( ( ( member @ A @ X3 @ B3 ) % 8.50/4.50 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 ) % 8.50/4.50 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) % 8.50/4.50 & ( ~ ( member @ A @ X3 @ B3 ) % 8.50/4.50 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 ) % 8.50/4.50 = ( insert @ A @ X3 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1924_minus__int__code_I1_J,axiom, % 8.50/4.50 ! [K: int] : % 8.50/4.50 ( ( minus_minus @ int @ K @ ( zero_zero @ int ) ) % 8.50/4.50 = K ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1925_uminus__int__code_I1_J,axiom, % 8.50/4.50 ( ( uminus_uminus @ int @ ( zero_zero @ int ) ) % 8.50/4.50 = ( zero_zero @ int ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1926_int__distrib_I4_J,axiom, % 8.50/4.50 ! [W2: int,Z1: int,Z22: int] : % 8.50/4.50 ( ( times_times @ int @ W2 @ ( minus_minus @ int @ Z1 @ Z22 ) ) % 8.50/4.50 = ( minus_minus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1927_int__distrib_I3_J,axiom, % 8.50/4.50 ! [Z1: int,Z22: int,W2: int] : % 8.50/4.50 ( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W2 ) % 8.50/4.50 = ( minus_minus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1928_psubset__imp__ex__mem,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.50 ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.50 => ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1929_zdvd__zdiffD,axiom, % 8.50/4.50 ! [K: int,M: int,N2: int] : % 8.50/4.50 ( ( dvd_dvd @ int @ K @ ( minus_minus @ int @ M @ N2 ) ) % 8.50/4.50 => ( ( dvd_dvd @ int @ K @ N2 ) % 8.50/4.50 => ( dvd_dvd @ int @ K @ M ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1930_zmod__minus1,axiom, % 8.50/4.50 ! [B2: int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.50/4.50 => ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 ) % 8.50/4.50 = ( minus_minus @ int @ B2 @ ( one_one @ int ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1931_take__bit__minus__small__eq,axiom, % 8.50/4.50 ! [K: int,N2: nat] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 8.50/4.50 => ( ( ord_less_eq @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.50/4.50 => ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ K ) ) % 8.50/4.50 = ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1932_infinite__Icc,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( dense_linorder @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.50 => ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1933_atLeastAtMost__singleton_H,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( order @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( A2 = B2 ) % 8.50/4.50 => ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 8.50/4.50 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1934_take__bit__tightened__less__eq__int,axiom, % 8.50/4.50 ! [M: nat,N2: nat,K: int] : % 8.50/4.50 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.50 => ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1935_take__bit__int__greater__self__iff,axiom, % 8.50/4.50 ! [K: int,N2: nat] : % 8.50/4.50 ( ( ord_less @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) % 8.50/4.50 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1936_not__take__bit__negative,axiom, % 8.50/4.50 ! [N2: nat,K: int] : % 8.50/4.50 ~ ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1937_signed__take__bit__eq__iff__take__bit__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_ri3973907225187159222ations @ A ) % 8.50/4.50 => ! [N2: nat,A2: A,B2: A] : % 8.50/4.50 ( ( ( bit_ri4674362597316999326ke_bit @ A @ N2 @ A2 ) % 8.50/4.50 = ( bit_ri4674362597316999326ke_bit @ A @ N2 @ B2 ) ) % 8.50/4.50 = ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A2 ) % 8.50/4.50 = ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1938_signed__take__bit__take__bit,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_ri3973907225187159222ations @ A ) % 8.50/4.50 => ! [M: nat,N2: nat,A2: A] : % 8.50/4.50 ( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) % 8.50/4.50 = ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N2 @ M ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 ) @ ( bit_ri4674362597316999326ke_bit @ A @ M ) @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1939_ex__nat__less,axiom, % 8.50/4.50 ! [N2: nat,P: nat > $o] : % 8.50/4.50 ( ( ? [M2: nat] : % 8.50/4.50 ( ( ord_less_eq @ nat @ M2 @ N2 ) % 8.50/4.50 & ( P @ M2 ) ) ) % 8.50/4.50 = ( ? [X: nat] : % 8.50/4.50 ( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.50/4.50 & ( P @ X ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1940_all__nat__less,axiom, % 8.50/4.50 ! [N2: nat,P: nat > $o] : % 8.50/4.50 ( ( ! [M2: nat] : % 8.50/4.50 ( ( ord_less_eq @ nat @ M2 @ N2 ) % 8.50/4.50 => ( P @ M2 ) ) ) % 8.50/4.50 = ( ! [X: nat] : % 8.50/4.50 ( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.50/4.50 => ( P @ X ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1941_neg__numeral__le__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num,N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1942_not__numeral__le__neg__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num,N2: num] : % 8.50/4.50 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1943_zero__neq__neg__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_char_0 @ A ) % 8.50/4.50 => ! [N2: num] : % 8.50/4.50 ( ( zero_zero @ A ) % 8.50/4.50 != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1944_not__numeral__less__neg__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num,N2: num] : % 8.50/4.50 ~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1945_neg__numeral__less__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num,N2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1946_le__minus__one__simps_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1947_le__minus__one__simps_I4_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1948_zero__neq__neg__one,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_char_0 @ A ) % 8.50/4.50 => ( ( zero_zero @ A ) % 8.50/4.50 != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1949_add__eq__0__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ( plus_plus @ A @ A2 @ B2 ) % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 = ( B2 % 8.50/4.50 = ( uminus_uminus @ A @ A2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1950_ab__group__add__class_Oab__left__minus,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ab_group_add @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1951_add_Oinverse__unique,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ( plus_plus @ A @ A2 @ B2 ) % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 => ( ( uminus_uminus @ A @ A2 ) % 8.50/4.50 = B2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1952_eq__neg__iff__add__eq__0,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( A2 % 8.50/4.50 = ( uminus_uminus @ A @ B2 ) ) % 8.50/4.50 = ( ( plus_plus @ A @ A2 @ B2 ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1953_neg__eq__iff__add__eq__0,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ( uminus_uminus @ A @ A2 ) % 8.50/4.50 = B2 ) % 8.50/4.50 = ( ( plus_plus @ A @ A2 @ B2 ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1954_less__minus__one__simps_I4_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1955_less__minus__one__simps_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1956_nonzero__minus__divide__right,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [B2: A,A2: A] : % 8.50/4.50 ( ( B2 % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 => ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 8.50/4.50 = ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1957_nonzero__minus__divide__divide,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [B2: A,A2: A] : % 8.50/4.50 ( ( B2 % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 => ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) % 8.50/4.50 = ( divide_divide @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1958_square__eq__1__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_15535105094025558882visors @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( ( times_times @ A @ X3 @ X3 ) % 8.50/4.50 = ( one_one @ A ) ) % 8.50/4.50 = ( ( X3 % 8.50/4.50 = ( one_one @ A ) ) % 8.50/4.50 | ( X3 % 8.50/4.50 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1959_group__cancel_Osub2,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ab_group_add @ A ) % 8.50/4.50 => ! [B3: A,K: A,B2: A,A2: A] : % 8.50/4.50 ( ( B3 % 8.50/4.50 = ( plus_plus @ A @ K @ B2 ) ) % 8.50/4.50 => ( ( minus_minus @ A @ A2 @ B3 ) % 8.50/4.50 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1960_diff__conv__add__uminus,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( group_add @ A ) % 8.50/4.50 => ( ( minus_minus @ A ) % 8.50/4.50 = ( ^ [A4: A,B4: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1961_ab__group__add__class_Oab__diff__conv__add__uminus,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ab_group_add @ A ) % 8.50/4.50 => ( ( minus_minus @ A ) % 8.50/4.50 = ( ^ [A4: A,B4: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1962_take__bit__unset__bit__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [N2: nat,M: nat,A2: A] : % 8.50/4.50 ( ( ( ord_less_eq @ nat @ N2 @ M ) % 8.50/4.50 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) ) % 8.50/4.50 = ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) % 8.50/4.50 & ( ~ ( ord_less_eq @ nat @ N2 @ M ) % 8.50/4.50 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) ) % 8.50/4.50 = ( bit_se2638667681897837118et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1963_take__bit__set__bit__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [N2: nat,M: nat,A2: A] : % 8.50/4.50 ( ( ( ord_less_eq @ nat @ N2 @ M ) % 8.50/4.50 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) ) % 8.50/4.50 = ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) % 8.50/4.50 & ( ~ ( ord_less_eq @ nat @ N2 @ M ) % 8.50/4.50 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) ) % 8.50/4.50 = ( bit_se5668285175392031749et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1964_take__bit__flip__bit__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [N2: nat,M: nat,A2: A] : % 8.50/4.50 ( ( ( ord_less_eq @ nat @ N2 @ M ) % 8.50/4.50 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) ) % 8.50/4.50 = ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) % 8.50/4.50 & ( ~ ( ord_less_eq @ nat @ N2 @ M ) % 8.50/4.50 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) ) % 8.50/4.50 = ( bit_se8732182000553998342ip_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1965_dvd__neg__div,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( idom_divide @ A ) % 8.50/4.50 => ! [B2: A,A2: A] : % 8.50/4.50 ( ( dvd_dvd @ A @ B2 @ A2 ) % 8.50/4.50 => ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 8.50/4.50 = ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1966_dvd__div__neg,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( idom_divide @ A ) % 8.50/4.50 => ! [B2: A,A2: A] : % 8.50/4.50 ( ( dvd_dvd @ A @ B2 @ A2 ) % 8.50/4.50 => ( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) % 8.50/4.50 = ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1967_subset__Compl__self__eq,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A] : % 8.50/4.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) % 8.50/4.50 = ( A3 % 8.50/4.50 = ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1968_Diff__insert__absorb,axiom, % 8.50/4.50 ! [A: $tType,X3: A,A3: set @ A] : % 8.50/4.50 ( ~ ( member @ A @ X3 @ A3 ) % 8.50/4.50 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.50 = A3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1969_Diff__insert2,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] : % 8.50/4.50 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) % 8.50/4.50 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1970_insert__Diff,axiom, % 8.50/4.50 ! [A: $tType,A2: A,A3: set @ A] : % 8.50/4.50 ( ( member @ A @ A2 @ A3 ) % 8.50/4.50 => ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.50 = A3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1971_Diff__insert,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] : % 8.50/4.50 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) % 8.50/4.50 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1972_subset__Diff__insert,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,B3: set @ A,X3: A,C5: set @ A] : % 8.50/4.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ ( insert @ A @ X3 @ C5 ) ) ) % 8.50/4.50 = ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ C5 ) ) % 8.50/4.50 & ~ ( member @ A @ X3 @ A3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1973_real__0__less__add__iff,axiom, % 8.50/4.50 ! [X3: real,Y: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X3 @ Y ) ) % 8.50/4.50 = ( ord_less @ real @ ( uminus_uminus @ real @ X3 ) @ Y ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1974_real__add__less__0__iff,axiom, % 8.50/4.50 ! [X3: real,Y: real] : % 8.50/4.50 ( ( ord_less @ real @ ( plus_plus @ real @ X3 @ Y ) @ ( zero_zero @ real ) ) % 8.50/4.50 = ( ord_less @ real @ Y @ ( uminus_uminus @ real @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1975_zminus1__lemma,axiom, % 8.50/4.50 ! [A2: int,B2: int,Q4: int,R2: int] : % 8.50/4.50 ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) % 8.50/4.50 => ( ( B2 % 8.50/4.50 != ( zero_zero @ int ) ) % 8.50/4.50 => ( eucl_rel_int @ ( uminus_uminus @ int @ A2 ) @ B2 % 8.50/4.50 @ ( product_Pair @ int @ int % 8.50/4.50 @ ( if @ int % 8.50/4.50 @ ( R2 % 8.50/4.50 = ( zero_zero @ int ) ) % 8.50/4.50 @ ( uminus_uminus @ int @ Q4 ) % 8.50/4.50 @ ( minus_minus @ int @ ( uminus_uminus @ int @ Q4 ) @ ( one_one @ int ) ) ) % 8.50/4.50 @ ( if @ int % 8.50/4.50 @ ( R2 % 8.50/4.50 = ( zero_zero @ int ) ) % 8.50/4.50 @ ( zero_zero @ int ) % 8.50/4.50 @ ( minus_minus @ int @ B2 @ R2 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1976_int__le__induct,axiom, % 8.50/4.50 ! [I: int,K: int,P: int > $o] : % 8.50/4.50 ( ( ord_less_eq @ int @ I @ K ) % 8.50/4.50 => ( ( P @ K ) % 8.50/4.50 => ( ! [I4: int] : % 8.50/4.50 ( ( ord_less_eq @ int @ I4 @ K ) % 8.50/4.50 => ( ( P @ I4 ) % 8.50/4.50 => ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) ) % 8.50/4.50 => ( P @ I ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1977_int__less__induct,axiom, % 8.50/4.50 ! [I: int,K: int,P: int > $o] : % 8.50/4.50 ( ( ord_less @ int @ I @ K ) % 8.50/4.50 => ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) ) % 8.50/4.50 => ( ! [I4: int] : % 8.50/4.50 ( ( ord_less @ int @ I4 @ K ) % 8.50/4.50 => ( ( P @ I4 ) % 8.50/4.50 => ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) ) % 8.50/4.50 => ( P @ I ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1978_pos__zmult__eq__1__iff__lemma,axiom, % 8.50/4.50 ! [M: int,N2: int] : % 8.50/4.50 ( ( ( times_times @ int @ M @ N2 ) % 8.50/4.50 = ( one_one @ int ) ) % 8.50/4.50 => ( ( M % 8.50/4.50 = ( one_one @ int ) ) % 8.50/4.50 | ( M % 8.50/4.50 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1979_zmult__eq__1__iff,axiom, % 8.50/4.50 ! [M: int,N2: int] : % 8.50/4.50 ( ( ( times_times @ int @ M @ N2 ) % 8.50/4.50 = ( one_one @ int ) ) % 8.50/4.50 = ( ( ( M % 8.50/4.50 = ( one_one @ int ) ) % 8.50/4.50 & ( N2 % 8.50/4.50 = ( one_one @ int ) ) ) % 8.50/4.50 | ( ( M % 8.50/4.50 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) % 8.50/4.50 & ( N2 % 8.50/4.50 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1980_signed__take__bit__eq__take__bit__shift,axiom, % 8.50/4.50 ( ( bit_ri4674362597316999326ke_bit @ int ) % 8.50/4.50 = ( ^ [N: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( plus_plus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1981_take__bit__Suc__minus__bit0,axiom, % 8.50/4.50 ! [N2: nat,K: num] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) ) % 8.50/4.50 = ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1982_take__bit__int__less__eq,axiom, % 8.50/4.50 ! [N2: nat,K: int] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) % 8.50/4.50 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.50 => ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1983_atLeastatMost__psubset__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( preorder @ A ) % 8.50/4.50 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.50/4.50 ( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) ) % 8.50/4.50 = ( ( ~ ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.50 | ( ( ord_less_eq @ A @ C2 @ A2 ) % 8.50/4.50 & ( ord_less_eq @ A @ B2 @ D2 ) % 8.50/4.50 & ( ( ord_less @ A @ C2 @ A2 ) % 8.50/4.50 | ( ord_less @ A @ B2 @ D2 ) ) ) ) % 8.50/4.50 & ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1984_take__bit__signed__take__bit,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_ri3973907225187159222ations @ A ) % 8.50/4.50 => ! [M: nat,N2: nat,A2: A] : % 8.50/4.50 ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) % 8.50/4.50 => ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N2 @ A2 ) ) % 8.50/4.50 = ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1985_not__zero__le__neg__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [N2: num] : % 8.50/4.50 ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1986_neg__numeral__le__zero,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1987_neg__numeral__less__zero,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [N2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1988_not__zero__less__neg__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [N2: num] : % 8.50/4.50 ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1989_le__minus__one__simps_I1_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1990_le__minus__one__simps_I3_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1991_less__minus__one__simps_I3_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1992_less__minus__one__simps_I1_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1993_neg__numeral__le__one,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1994_neg__one__le__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1995_neg__numeral__le__neg__one,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1996_not__numeral__le__neg__one,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num] : % 8.50/4.50 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1997_not__one__le__neg__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num] : % 8.50/4.50 ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1998_not__neg__one__less__neg__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num] : % 8.50/4.50 ~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_1999_not__one__less__neg__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num] : % 8.50/4.50 ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2000_not__numeral__less__neg__one,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num] : % 8.50/4.50 ~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2001_neg__one__less__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2002_neg__numeral__less__one,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2003_eq__minus__divide__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [A2: A,B2: A,C2: A] : % 8.50/4.50 ( ( A2 % 8.50/4.50 = ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) ) % 8.50/4.50 = ( ( ( C2 % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 => ( ( times_times @ A @ A2 @ C2 ) % 8.50/4.50 = ( uminus_uminus @ A @ B2 ) ) ) % 8.50/4.50 & ( ( C2 % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 => ( A2 % 8.50/4.50 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2004_minus__divide__eq__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [B2: A,C2: A,A2: A] : % 8.50/4.50 ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.50/4.50 = A2 ) % 8.50/4.50 = ( ( ( C2 % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 => ( ( uminus_uminus @ A @ B2 ) % 8.50/4.50 = ( times_times @ A @ A2 @ C2 ) ) ) % 8.50/4.50 & ( ( C2 % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 => ( A2 % 8.50/4.50 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2005_nonzero__neg__divide__eq__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [B2: A,A2: A,C2: A] : % 8.50/4.50 ( ( B2 % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 => ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 8.50/4.50 = C2 ) % 8.50/4.50 = ( ( uminus_uminus @ A @ A2 ) % 8.50/4.50 = ( times_times @ A @ C2 @ B2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2006_nonzero__neg__divide__eq__eq2,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [B2: A,C2: A,A2: A] : % 8.50/4.50 ( ( B2 % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 => ( ( C2 % 8.50/4.50 = ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) % 8.50/4.50 = ( ( times_times @ A @ C2 @ B2 ) % 8.50/4.50 = ( uminus_uminus @ A @ A2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2007_divide__eq__minus__1__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( field @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ( divide_divide @ A @ A2 @ B2 ) % 8.50/4.50 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 8.50/4.50 = ( ( B2 % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 & ( A2 % 8.50/4.50 = ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2008_finite__empty__induct,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] : % 8.50/4.50 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.50 => ( ( P @ A3 ) % 8.50/4.50 => ( ! [A5: A,A8: set @ A] : % 8.50/4.50 ( ( finite_finite2 @ A @ A8 ) % 8.50/4.50 => ( ( member @ A @ A5 @ A8 ) % 8.50/4.50 => ( ( P @ A8 ) % 8.50/4.50 => ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) % 8.50/4.50 => ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2009_infinite__coinduct,axiom, % 8.50/4.50 ! [A: $tType,X8: ( set @ A ) > $o,A3: set @ A] : % 8.50/4.50 ( ( X8 @ A3 ) % 8.50/4.50 => ( ! [A8: set @ A] : % 8.50/4.50 ( ( X8 @ A8 ) % 8.50/4.50 => ? [X2: A] : % 8.50/4.50 ( ( member @ A @ X2 @ A8 ) % 8.50/4.50 & ( ( X8 @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.50 | ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) % 8.50/4.50 => ~ ( finite_finite2 @ A @ A3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2010_infinite__remove,axiom, % 8.50/4.50 ! [A: $tType,S3: set @ A,A2: A] : % 8.50/4.50 ( ~ ( finite_finite2 @ A @ S3 ) % 8.50/4.50 => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2011_Diff__single__insert,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,X3: A,B3: set @ A] : % 8.50/4.50 ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) % 8.50/4.50 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2012_subset__insert__iff,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,X3: A,B3: set @ A] : % 8.50/4.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) ) % 8.50/4.50 = ( ( ( member @ A @ X3 @ A3 ) % 8.50/4.50 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) % 8.50/4.50 & ( ~ ( member @ A @ X3 @ A3 ) % 8.50/4.50 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2013_take__bit__Suc__minus__1__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_ri3973907225187159222ations @ A ) % 8.50/4.50 => ! [N2: nat] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 8.50/4.50 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2014_atLeast0__atMost__Suc,axiom, % 8.50/4.50 ! [N2: nat] : % 8.50/4.50 ( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) % 8.50/4.50 = ( insert @ nat @ ( suc @ N2 ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2015_mod__pos__geq,axiom, % 8.50/4.50 ! [L: int,K: int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ L ) % 8.50/4.50 => ( ( ord_less_eq @ int @ L @ K ) % 8.50/4.50 => ( ( modulo_modulo @ int @ K @ L ) % 8.50/4.50 = ( modulo_modulo @ int @ ( minus_minus @ int @ K @ L ) @ L ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2016_Icc__eq__insert__lb__nat,axiom, % 8.50/4.50 ! [M: nat,N2: nat] : % 8.50/4.50 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.50 => ( ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) % 8.50/4.50 = ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2017_atLeastAtMostSuc__conv,axiom, % 8.50/4.50 ! [M: nat,N2: nat] : % 8.50/4.50 ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) % 8.50/4.50 => ( ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) % 8.50/4.50 = ( insert @ nat @ ( suc @ N2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2018_atLeastAtMost__insertL,axiom, % 8.50/4.50 ! [M: nat,N2: nat] : % 8.50/4.50 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.50 => ( ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) % 8.50/4.50 = ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2019_plusinfinity,axiom, % 8.50/4.50 ! [D2: int,P5: int > $o,P: int > $o] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 ) % 8.50/4.50 => ( ! [X5: int,K2: int] : % 8.50/4.50 ( ( P5 @ X5 ) % 8.50/4.50 = ( P5 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D2 ) ) ) ) % 8.50/4.50 => ( ? [Z4: int] : % 8.50/4.50 ! [X5: int] : % 8.50/4.50 ( ( ord_less @ int @ Z4 @ X5 ) % 8.50/4.50 => ( ( P @ X5 ) % 8.50/4.50 = ( P5 @ X5 ) ) ) % 8.50/4.50 => ( ? [X_12: int] : ( P5 @ X_12 ) % 8.50/4.50 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2020_minusinfinity,axiom, % 8.50/4.50 ! [D2: int,P1: int > $o,P: int > $o] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 ) % 8.50/4.50 => ( ! [X5: int,K2: int] : % 8.50/4.50 ( ( P1 @ X5 ) % 8.50/4.50 = ( P1 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D2 ) ) ) ) % 8.50/4.50 => ( ? [Z4: int] : % 8.50/4.50 ! [X5: int] : % 8.50/4.50 ( ( ord_less @ int @ X5 @ Z4 ) % 8.50/4.50 => ( ( P @ X5 ) % 8.50/4.50 = ( P1 @ X5 ) ) ) % 8.50/4.50 => ( ? [X_12: int] : ( P1 @ X_12 ) % 8.50/4.50 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2021_int__induct,axiom, % 8.50/4.50 ! [P: int > $o,K: int,I: int] : % 8.50/4.50 ( ( P @ K ) % 8.50/4.50 => ( ! [I4: int] : % 8.50/4.50 ( ( ord_less_eq @ int @ K @ I4 ) % 8.50/4.50 => ( ( P @ I4 ) % 8.50/4.50 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) ) % 8.50/4.50 => ( ! [I4: int] : % 8.50/4.50 ( ( ord_less_eq @ int @ I4 @ K ) % 8.50/4.50 => ( ( P @ I4 ) % 8.50/4.50 => ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) ) % 8.50/4.50 => ( P @ I ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2022_subset__eq__atLeast0__atMost__finite,axiom, % 8.50/4.50 ! [N7: set @ nat,N2: nat] : % 8.50/4.50 ( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.50/4.50 => ( finite_finite2 @ nat @ N7 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2023_take__bit__int__less__exp,axiom, % 8.50/4.50 ! [N2: nat,K: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2024_pos__minus__divide__less__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [C2: A,B2: A,A2: A] : % 8.50/4.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 ) % 8.50/4.50 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2025_pos__less__minus__divide__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [C2: A,A2: A,B2: A] : % 8.50/4.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) ) % 8.50/4.50 = ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2026_neg__minus__divide__less__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [C2: A,B2: A,A2: A] : % 8.50/4.50 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 ) % 8.50/4.50 = ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2027_neg__less__minus__divide__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [C2: A,A2: A,B2: A] : % 8.50/4.50 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) ) % 8.50/4.50 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2028_minus__divide__less__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [B2: A,C2: A,A2: A] : % 8.50/4.50 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 ) % 8.50/4.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2029_less__minus__divide__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [A2: A,B2: A,C2: A] : % 8.50/4.50 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) ) % 8.50/4.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2030_divide__eq__eq__numeral_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [B2: A,C2: A,W2: num] : % 8.50/4.50 ( ( ( divide_divide @ A @ B2 @ C2 ) % 8.50/4.50 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) % 8.50/4.50 = ( ( ( C2 % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 => ( B2 % 8.50/4.50 = ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) ) % 8.50/4.50 & ( ( C2 % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 => ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2031_eq__divide__eq__numeral_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [W2: num,B2: A,C2: A] : % 8.50/4.50 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) % 8.50/4.50 = ( divide_divide @ A @ B2 @ C2 ) ) % 8.50/4.50 = ( ( ( C2 % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 => ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) % 8.50/4.50 = B2 ) ) % 8.50/4.50 & ( ( C2 % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 => ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2032_minus__divide__add__eq__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [Z2: A,X3: A,Y: A] : % 8.50/4.50 ( ( Z2 % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X3 @ Z2 ) ) @ Y ) % 8.50/4.50 = ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X3 ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2033_add__divide__eq__if__simps_I3_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [Z2: A,A2: A,B2: A] : % 8.50/4.50 ( ( ( Z2 % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 ) % 8.50/4.50 = B2 ) ) % 8.50/4.50 & ( ( Z2 % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 ) % 8.50/4.50 = ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2034_add__divide__eq__if__simps_I6_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [Z2: A,A2: A,B2: A] : % 8.50/4.50 ( ( ( Z2 % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 ) % 8.50/4.50 = ( uminus_uminus @ A @ B2 ) ) ) % 8.50/4.50 & ( ( Z2 % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 ) % 8.50/4.50 = ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2035_add__divide__eq__if__simps_I5_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [Z2: A,A2: A,B2: A] : % 8.50/4.50 ( ( ( Z2 % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 => ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 ) % 8.50/4.50 = ( uminus_uminus @ A @ B2 ) ) ) % 8.50/4.50 & ( ( Z2 % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 => ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 ) % 8.50/4.50 = ( divide_divide @ A @ ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2036_minus__divide__diff__eq__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( division_ring @ A ) % 8.50/4.50 => ! [Z2: A,X3: A,Y: A] : % 8.50/4.50 ( ( Z2 % 8.50/4.50 != ( zero_zero @ A ) ) % 8.50/4.50 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X3 @ Z2 ) ) @ Y ) % 8.50/4.50 = ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X3 ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2037_remove__induct,axiom, % 8.50/4.50 ! [A: $tType,P: ( set @ A ) > $o,B3: set @ A] : % 8.50/4.50 ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.50 => ( ( ~ ( finite_finite2 @ A @ B3 ) % 8.50/4.50 => ( P @ B3 ) ) % 8.50/4.50 => ( ! [A8: set @ A] : % 8.50/4.50 ( ( finite_finite2 @ A @ A8 ) % 8.50/4.50 => ( ( A8 % 8.50/4.50 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.50 => ( ( ord_less_eq @ ( set @ A ) @ A8 @ B3 ) % 8.50/4.50 => ( ! [X2: A] : % 8.50/4.50 ( ( member @ A @ X2 @ A8 ) % 8.50/4.50 => ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) % 8.50/4.50 => ( P @ A8 ) ) ) ) ) % 8.50/4.50 => ( P @ B3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2038_finite__remove__induct,axiom, % 8.50/4.50 ! [A: $tType,B3: set @ A,P: ( set @ A ) > $o] : % 8.50/4.50 ( ( finite_finite2 @ A @ B3 ) % 8.50/4.50 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.50 => ( ! [A8: set @ A] : % 8.50/4.50 ( ( finite_finite2 @ A @ A8 ) % 8.50/4.50 => ( ( A8 % 8.50/4.50 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.50 => ( ( ord_less_eq @ ( set @ A ) @ A8 @ B3 ) % 8.50/4.50 => ( ! [X2: A] : % 8.50/4.50 ( ( member @ A @ X2 @ A8 ) % 8.50/4.50 => ( P @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) % 8.50/4.50 => ( P @ A8 ) ) ) ) ) % 8.50/4.50 => ( P @ B3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2039_signed__take__bit__int__greater__self__iff,axiom, % 8.50/4.50 ! [K: int,N2: nat] : % 8.50/4.50 ( ( ord_less @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) ) % 8.50/4.50 = ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2040_finite__induct__select,axiom, % 8.50/4.50 ! [A: $tType,S3: set @ A,P: ( set @ A ) > $o] : % 8.50/4.50 ( ( finite_finite2 @ A @ S3 ) % 8.50/4.50 => ( ( P @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.50 => ( ! [T5: set @ A] : % 8.50/4.50 ( ( ord_less @ ( set @ A ) @ T5 @ S3 ) % 8.50/4.50 => ( ( P @ T5 ) % 8.50/4.50 => ? [X2: A] : % 8.50/4.50 ( ( member @ A @ X2 @ ( minus_minus @ ( set @ A ) @ S3 @ T5 ) ) % 8.50/4.50 & ( P @ ( insert @ A @ X2 @ T5 ) ) ) ) ) % 8.50/4.50 => ( P @ S3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2041_psubset__insert__iff,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,X3: A,B3: set @ A] : % 8.50/4.50 ( ( ord_less @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) ) % 8.50/4.50 = ( ( ( member @ A @ X3 @ B3 ) % 8.50/4.50 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.50 & ( ~ ( member @ A @ X3 @ B3 ) % 8.50/4.50 => ( ( ( member @ A @ X3 @ A3 ) % 8.50/4.50 => ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) % 8.50/4.50 & ( ~ ( member @ A @ X3 @ A3 ) % 8.50/4.50 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2042_decr__mult__lemma,axiom, % 8.50/4.50 ! [D2: int,P: int > $o,K: int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 ) % 8.50/4.50 => ( ! [X5: int] : % 8.50/4.50 ( ( P @ X5 ) % 8.50/4.50 => ( P @ ( minus_minus @ int @ X5 @ D2 ) ) ) % 8.50/4.50 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 8.50/4.50 => ! [X2: int] : % 8.50/4.50 ( ( P @ X2 ) % 8.50/4.50 => ( P @ ( minus_minus @ int @ X2 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2043_verit__less__mono__div__int2,axiom, % 8.50/4.50 ! [A3: int,B3: int,N2: int] : % 8.50/4.50 ( ( ord_less_eq @ int @ A3 @ B3 ) % 8.50/4.50 => ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N2 ) ) % 8.50/4.50 => ( ord_less_eq @ int @ ( divide_divide @ int @ B3 @ N2 ) @ ( divide_divide @ int @ A3 @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2044_div__eq__minus1,axiom, % 8.50/4.50 ! [B2: int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.50/4.50 => ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 ) % 8.50/4.50 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2045_take__bit__Suc__bit0,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [N2: nat,K: num] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) % 8.50/4.50 = ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2046_take__bit__nat__eq__self,axiom, % 8.50/4.50 ! [M: nat,N2: nat] : % 8.50/4.50 ( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.50/4.50 => ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) % 8.50/4.50 = M ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2047_take__bit__nat__less__exp,axiom, % 8.50/4.50 ! [N2: nat,M: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2048_take__bit__nat__eq__self__iff,axiom, % 8.50/4.50 ! [N2: nat,M: nat] : % 8.50/4.50 ( ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) % 8.50/4.50 = M ) % 8.50/4.50 = ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2049_take__bit__int__less__self__iff,axiom, % 8.50/4.50 ! [N2: nat,K: int] : % 8.50/4.50 ( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ K ) % 8.50/4.50 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2050_take__bit__int__greater__eq__self__iff,axiom, % 8.50/4.50 ! [K: int,N2: nat] : % 8.50/4.50 ( ( ord_less_eq @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) % 8.50/4.50 = ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2051_le__minus__divide__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [A2: A,B2: A,C2: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) ) % 8.50/4.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2052_minus__divide__le__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [B2: A,C2: A,A2: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 ) % 8.50/4.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2053_neg__le__minus__divide__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [C2: A,A2: A,B2: A] : % 8.50/4.50 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) ) % 8.50/4.50 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2054_neg__minus__divide__le__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [C2: A,B2: A,A2: A] : % 8.50/4.50 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 ) % 8.50/4.50 = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2055_pos__le__minus__divide__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [C2: A,A2: A,B2: A] : % 8.50/4.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) ) % 8.50/4.50 = ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2056_pos__minus__divide__le__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [C2: A,B2: A,A2: A] : % 8.50/4.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 ) % 8.50/4.50 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2057_divide__less__eq__numeral_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [B2: A,C2: A,W2: num] : % 8.50/4.50 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) % 8.50/4.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2058_less__divide__eq__numeral_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [W2: num,B2: A,C2: A] : % 8.50/4.50 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.50/4.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2059_signed__take__bit__int__less__eq,axiom, % 8.50/4.50 ! [N2: nat,K: int] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) % 8.50/4.50 => ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2060_neg__one__power__add__eq__neg__one__power__diff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_1 @ A ) % 8.50/4.50 => ! [K: nat,N2: nat] : % 8.50/4.50 ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.50 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N2 @ K ) ) % 8.50/4.50 = ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2061_signed__take__bit__int__eq__self,axiom, % 8.50/4.50 ! [N2: nat,K: int] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ K ) % 8.50/4.50 => ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.50/4.50 => ( ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) % 8.50/4.50 = K ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2062_signed__take__bit__int__eq__self__iff,axiom, % 8.50/4.50 ! [N2: nat,K: int] : % 8.50/4.50 ( ( ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) % 8.50/4.50 = K ) % 8.50/4.50 = ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ K ) % 8.50/4.50 & ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2063_take__bit__eq__0__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [N2: nat,A2: A] : % 8.50/4.50 ( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 = ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2064_take__bit__nat__less__self__iff,axiom, % 8.50/4.50 ! [N2: nat,M: nat] : % 8.50/4.50 ( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) @ M ) % 8.50/4.50 = ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2065_take__bit__int__eq__self,axiom, % 8.50/4.50 ! [K: int,N2: nat] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 8.50/4.50 => ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.50/4.50 => ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) % 8.50/4.50 = K ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2066_take__bit__int__eq__self__iff,axiom, % 8.50/4.50 ! [N2: nat,K: int] : % 8.50/4.50 ( ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) % 8.50/4.50 = K ) % 8.50/4.50 = ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 8.50/4.50 & ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2067_divide__le__eq__numeral_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [B2: A,C2: A,W2: num] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) % 8.50/4.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2068_le__divide__eq__numeral_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_field @ A ) % 8.50/4.50 => ! [W2: num,B2: A,C2: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) ) % 8.50/4.50 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.50 => ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) ) % 8.50/4.50 & ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.50 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2069_square__le__1,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X3 ) % 8.50/4.50 => ( ( ord_less_eq @ A @ X3 @ ( one_one @ A ) ) % 8.50/4.50 => ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2070_div__pos__geq,axiom, % 8.50/4.50 ! [L: int,K: int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ L ) % 8.50/4.50 => ( ( ord_less_eq @ int @ L @ K ) % 8.50/4.50 => ( ( divide_divide @ int @ K @ L ) % 8.50/4.50 = ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2071_div__pos__neg__trivial,axiom, % 8.50/4.50 ! [K: int,L: int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 8.50/4.50 => ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) ) % 8.50/4.50 => ( ( divide_divide @ int @ K @ L ) % 8.50/4.50 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2072_take__bit__int__greater__eq,axiom, % 8.50/4.50 ! [K: int,N2: nat] : % 8.50/4.50 ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 8.50/4.50 => ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2073_power__minus1__odd,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_1 @ A ) % 8.50/4.50 => ! [N2: nat] : % 8.50/4.50 ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 8.50/4.50 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2074_take__bit__Suc,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [N2: nat,A2: A] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A2 ) % 8.50/4.50 = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2075_signed__take__bit__int__greater__eq,axiom, % 8.50/4.50 ! [K: int,N2: nat] : % 8.50/4.50 ( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 8.50/4.50 => ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2076_stable__imp__take__bit__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [A2: A,N2: nat] : % 8.50/4.50 ( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.50/4.50 = A2 ) % 8.50/4.50 => ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.50/4.50 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) % 8.50/4.50 = ( zero_zero @ A ) ) ) % 8.50/4.50 & ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.50/4.50 => ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) % 8.50/4.50 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2077_signed__take__bit__Suc__minus__bit1,axiom, % 8.50/4.50 ! [N2: nat,K: num] : % 8.50/4.50 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) ) % 8.50/4.50 = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2078_concat__bit__Suc,axiom, % 8.50/4.50 ! [N2: nat,K: int,L: int] : % 8.50/4.50 ( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L ) % 8.50/4.50 = ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2079_Bolzano,axiom, % 8.50/4.50 ! [A2: real,B2: real,P: real > real > $o] : % 8.50/4.50 ( ( ord_less_eq @ real @ A2 @ B2 ) % 8.50/4.50 => ( ! [A5: real,B5: real,C3: real] : % 8.50/4.50 ( ( P @ A5 @ B5 ) % 8.50/4.50 => ( ( P @ B5 @ C3 ) % 8.50/4.50 => ( ( ord_less_eq @ real @ A5 @ B5 ) % 8.50/4.50 => ( ( ord_less_eq @ real @ B5 @ C3 ) % 8.50/4.50 => ( P @ A5 @ C3 ) ) ) ) ) % 8.50/4.50 => ( ! [X5: real] : % 8.50/4.50 ( ( ord_less_eq @ real @ A2 @ X5 ) % 8.50/4.50 => ( ( ord_less_eq @ real @ X5 @ B2 ) % 8.50/4.50 => ? [D4: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 ) % 8.50/4.50 & ! [A5: real,B5: real] : % 8.50/4.50 ( ( ( ord_less_eq @ real @ A5 @ X5 ) % 8.50/4.50 & ( ord_less_eq @ real @ X5 @ B5 ) % 8.50/4.50 & ( ord_less @ real @ ( minus_minus @ real @ B5 @ A5 ) @ D4 ) ) % 8.50/4.50 => ( P @ A5 @ B5 ) ) ) ) ) % 8.50/4.50 => ( P @ A2 @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2080_signed__take__bit__Suc__bit1,axiom, % 8.50/4.50 ! [N2: nat,K: num] : % 8.50/4.50 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N2 ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) % 8.50/4.50 = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2081_and__int_Oelims,axiom, % 8.50/4.50 ! [X3: int,Xa2: int,Y: int] : % 8.50/4.50 ( ( ( bit_se5824344872417868541ns_and @ int @ X3 @ Xa2 ) % 8.50/4.50 = Y ) % 8.50/4.50 => ( ( ( ( member @ int @ X3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 8.50/4.50 & ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 8.50/4.50 => ( Y % 8.50/4.50 = ( uminus_uminus @ int % 8.50/4.50 @ ( zero_neq_one_of_bool @ int % 8.50/4.50 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 ) % 8.50/4.50 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) ) ) ) % 8.50/4.50 & ( ~ ( ( member @ int @ X3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 8.50/4.50 & ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 8.50/4.50 => ( Y % 8.50/4.50 = ( plus_plus @ int % 8.50/4.50 @ ( zero_neq_one_of_bool @ int % 8.50/4.50 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 ) % 8.50/4.50 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) % 8.50/4.50 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2082_and__int_Osimps,axiom, % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ int ) % 8.50/4.50 = ( ^ [K3: int,L2: int] : % 8.50/4.50 ( if @ int % 8.50/4.50 @ ( ( member @ int @ K3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 8.50/4.50 & ( member @ int @ L2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 8.50/4.50 @ ( uminus_uminus @ int % 8.50/4.50 @ ( zero_neq_one_of_bool @ int % 8.50/4.50 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) % 8.50/4.50 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) ) % 8.50/4.50 @ ( plus_plus @ int % 8.50/4.50 @ ( zero_neq_one_of_bool @ int % 8.50/4.50 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) % 8.50/4.50 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) % 8.50/4.50 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2083_Diff__idemp,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.50 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ B3 ) % 8.50/4.50 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2084_Diff__iff,axiom, % 8.50/4.50 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 8.50/4.50 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.50 = ( ( member @ A @ C2 @ A3 ) % 8.50/4.50 & ~ ( member @ A @ C2 @ B3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2085_DiffI,axiom, % 8.50/4.50 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 8.50/4.50 ( ( member @ A @ C2 @ A3 ) % 8.50/4.50 => ( ~ ( member @ A @ C2 @ B3 ) % 8.50/4.50 => ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2086_ComplI,axiom, % 8.50/4.50 ! [A: $tType,C2: A,A3: set @ A] : % 8.50/4.50 ( ~ ( member @ A @ C2 @ A3 ) % 8.50/4.50 => ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2087_Compl__iff,axiom, % 8.50/4.50 ! [A: $tType,C2: A,A3: set @ A] : % 8.50/4.50 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) % 8.50/4.50 = ( ~ ( member @ A @ C2 @ A3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2088_Compl__eq__Compl__iff,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.50 ( ( ( uminus_uminus @ ( set @ A ) @ A3 ) % 8.50/4.50 = ( uminus_uminus @ ( set @ A ) @ B3 ) ) % 8.50/4.50 = ( A3 = B3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2089_and__zero__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( zero_zero @ A ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2090_zero__and__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2091_bit_Oconj__zero__left,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_ri3973907225187159222ations @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ X3 ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2092_bit_Oconj__zero__right,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_ri3973907225187159222ations @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ A @ X3 @ ( zero_zero @ A ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2093_concat__bit__0,axiom, % 8.50/4.50 ! [K: int,L: int] : % 8.50/4.50 ( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L ) % 8.50/4.50 = L ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2094_semiring__norm_I80_J,axiom, % 8.50/4.50 ! [M: num,N2: num] : % 8.50/4.50 ( ( ord_less @ num @ ( bit1 @ M ) @ ( bit1 @ N2 ) ) % 8.50/4.50 = ( ord_less @ num @ M @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2095_and__negative__int__iff,axiom, % 8.50/4.50 ! [K: int,L: int] : % 8.50/4.50 ( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ ( zero_zero @ int ) ) % 8.50/4.50 = ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 8.50/4.50 & ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2096_semiring__norm_I81_J,axiom, % 8.50/4.50 ! [M: num,N2: num] : % 8.50/4.50 ( ( ord_less @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) ) % 8.50/4.50 = ( ord_less @ num @ M @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2097_semiring__norm_I77_J,axiom, % 8.50/4.50 ! [N2: num] : ( ord_less @ num @ one2 @ ( bit1 @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2098_concat__bit__negative__iff,axiom, % 8.50/4.50 ! [N2: nat,K: int,L: int] : % 8.50/4.50 ( ( ord_less @ int @ ( bit_concat_bit @ N2 @ K @ L ) @ ( zero_zero @ int ) ) % 8.50/4.50 = ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2099_semiring__norm_I79_J,axiom, % 8.50/4.50 ! [M: num,N2: num] : % 8.50/4.50 ( ( ord_less @ num @ ( bit0 @ M ) @ ( bit1 @ N2 ) ) % 8.50/4.50 = ( ord_less_eq @ num @ M @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2100_semiring__norm_I74_J,axiom, % 8.50/4.50 ! [M: num,N2: num] : % 8.50/4.50 ( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) ) % 8.50/4.50 = ( ord_less @ num @ M @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2101_and__numerals_I5_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [X3: num] : % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( one_one @ A ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2102_and__numerals_I1_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [Y: num] : % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2103_div__Suc__eq__div__add3,axiom, % 8.50/4.50 ! [M: nat,N2: nat] : % 8.50/4.50 ( ( divide_divide @ nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) ) % 8.50/4.50 = ( divide_divide @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2104_Suc__div__eq__add3__div__numeral,axiom, % 8.50/4.50 ! [M: nat,V2: num] : % 8.50/4.50 ( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V2 ) ) % 8.50/4.50 = ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2105_mod__Suc__eq__mod__add3,axiom, % 8.50/4.50 ! [M: nat,N2: nat] : % 8.50/4.50 ( ( modulo_modulo @ nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) ) % 8.50/4.50 = ( modulo_modulo @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2106_Suc__mod__eq__add3__mod__numeral,axiom, % 8.50/4.50 ! [M: nat,V2: num] : % 8.50/4.50 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V2 ) ) % 8.50/4.50 = ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2107_and__numerals_I7_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [X3: num,Y: num] : % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) % 8.50/4.50 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2108_DiffD2,axiom, % 8.50/4.50 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 8.50/4.50 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.50 => ~ ( member @ A @ C2 @ B3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2109_DiffD1,axiom, % 8.50/4.50 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 8.50/4.50 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.50 => ( member @ A @ C2 @ A3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2110_DiffE,axiom, % 8.50/4.50 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 8.50/4.50 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.50 => ~ ( ( member @ A @ C2 @ A3 ) % 8.50/4.50 => ( member @ A @ C2 @ B3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2111_ComplD,axiom, % 8.50/4.50 ! [A: $tType,C2: A,A3: set @ A] : % 8.50/4.50 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) % 8.50/4.50 => ~ ( member @ A @ C2 @ A3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2112_double__complement,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A] : % 8.50/4.50 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) % 8.50/4.50 = A3 ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2113_concat__bit__assoc,axiom, % 8.50/4.50 ! [N2: nat,K: int,M: nat,L: int,R2: int] : % 8.50/4.50 ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L @ R2 ) ) % 8.50/4.50 = ( bit_concat_bit @ ( plus_plus @ nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2114_xor__num_Ocases,axiom, % 8.50/4.50 ! [X3: product_prod @ num @ num] : % 8.50/4.50 ( ( X3 % 8.50/4.50 != ( product_Pair @ num @ num @ one2 @ one2 ) ) % 8.50/4.50 => ( ! [N3: num] : % 8.50/4.50 ( X3 % 8.50/4.50 != ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) % 8.50/4.50 => ( ! [N3: num] : % 8.50/4.50 ( X3 % 8.50/4.50 != ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) % 8.50/4.50 => ( ! [M3: num] : % 8.50/4.50 ( X3 % 8.50/4.50 != ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ one2 ) ) % 8.50/4.50 => ( ! [M3: num,N3: num] : % 8.50/4.50 ( X3 % 8.50/4.50 != ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) ) % 8.50/4.50 => ( ! [M3: num,N3: num] : % 8.50/4.50 ( X3 % 8.50/4.50 != ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) ) % 8.50/4.50 => ( ! [M3: num] : % 8.50/4.50 ( X3 % 8.50/4.50 != ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ one2 ) ) % 8.50/4.50 => ( ! [M3: num,N3: num] : % 8.50/4.50 ( X3 % 8.50/4.50 != ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) ) % 8.50/4.50 => ~ ! [M3: num,N3: num] : % 8.50/4.50 ( X3 % 8.50/4.50 != ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2115_and__less__eq,axiom, % 8.50/4.50 ! [L: int,K: int] : % 8.50/4.50 ( ( ord_less @ int @ L @ ( zero_zero @ int ) ) % 8.50/4.50 => ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ K ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2116_AND__upper1_H_H,axiom, % 8.50/4.50 ! [Y: int,Z2: int,Ya: int] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y ) % 8.50/4.50 => ( ( ord_less @ int @ Y @ Z2 ) % 8.50/4.50 => ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya ) @ Z2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2117_AND__upper2_H_H,axiom, % 8.50/4.50 ! [Y: int,Z2: int,X3: int] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y ) % 8.50/4.50 => ( ( ord_less @ int @ Y @ Z2 ) % 8.50/4.50 => ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X3 @ Y ) @ Z2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2118_numeral__Bit1,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( numeral @ A ) % 8.50/4.50 => ! [N2: num] : % 8.50/4.50 ( ( numeral_numeral @ A @ ( bit1 @ N2 ) ) % 8.50/4.50 = ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2119_simp__from__to,axiom, % 8.50/4.50 ( ( set_or1337092689740270186AtMost @ int ) % 8.50/4.50 = ( ^ [I3: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I3 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I3 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2120_eval__nat__numeral_I3_J,axiom, % 8.50/4.50 ! [N2: num] : % 8.50/4.50 ( ( numeral_numeral @ nat @ ( bit1 @ N2 ) ) % 8.50/4.50 = ( suc @ ( numeral_numeral @ nat @ ( bit0 @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2121_cong__exp__iff__simps_I3_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( unique1627219031080169319umeral @ A ) % 8.50/4.50 => ! [N2: num,Q4: num] : % 8.50/4.50 ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) % 8.50/4.50 != ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2122_numeral__3__eq__3,axiom, % 8.50/4.50 ( ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) % 8.50/4.50 = ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2123_Suc3__eq__add__3,axiom, % 8.50/4.50 ! [N2: nat] : % 8.50/4.50 ( ( suc @ ( suc @ ( suc @ N2 ) ) ) % 8.50/4.50 = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2124_num_Osize_I6_J,axiom, % 8.50/4.50 ! [X32: num] : % 8.50/4.50 ( ( size_size @ num @ ( bit1 @ X32 ) ) % 8.50/4.50 = ( plus_plus @ nat @ ( size_size @ num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2125_num_Osize__gen_I3_J,axiom, % 8.50/4.50 ! [X32: num] : % 8.50/4.50 ( ( size_num @ ( bit1 @ X32 ) ) % 8.50/4.50 = ( plus_plus @ nat @ ( size_num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2126_cong__exp__iff__simps_I11_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( unique1627219031080169319umeral @ A ) % 8.50/4.50 => ! [M: num,Q4: num] : % 8.50/4.50 ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) % 8.50/4.50 = ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) ) % 8.50/4.50 = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q4 ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2127_cong__exp__iff__simps_I7_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( unique1627219031080169319umeral @ A ) % 8.50/4.50 => ! [Q4: num,N2: num] : % 8.50/4.50 ( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) % 8.50/4.50 = ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) ) % 8.50/4.50 = ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ Q4 ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2128_periodic__finite__ex,axiom, % 8.50/4.50 ! [D2: int,P: int > $o] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 ) % 8.50/4.50 => ( ! [X5: int,K2: int] : % 8.50/4.50 ( ( P @ X5 ) % 8.50/4.50 = ( P @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D2 ) ) ) ) % 8.50/4.50 => ( ( ? [X6: int] : ( P @ X6 ) ) % 8.50/4.50 = ( ? [X: int] : % 8.50/4.50 ( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) ) % 8.50/4.50 & ( P @ X ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2129_bset_I3_J,axiom, % 8.50/4.50 ! [D6: int,T2: int,B3: set @ int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 8.50/4.50 => ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B3 ) % 8.50/4.50 => ! [X2: int] : % 8.50/4.50 ( ! [Xa3: int] : % 8.50/4.50 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 => ! [Xb2: int] : % 8.50/4.50 ( ( member @ int @ Xb2 @ B3 ) % 8.50/4.50 => ( X2 % 8.50/4.50 != ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) ) % 8.50/4.50 => ( ( X2 = T2 ) % 8.50/4.50 => ( ( minus_minus @ int @ X2 @ D6 ) % 8.50/4.50 = T2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2130_bset_I4_J,axiom, % 8.50/4.50 ! [D6: int,T2: int,B3: set @ int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 8.50/4.50 => ( ( member @ int @ T2 @ B3 ) % 8.50/4.50 => ! [X2: int] : % 8.50/4.50 ( ! [Xa3: int] : % 8.50/4.50 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 => ! [Xb2: int] : % 8.50/4.50 ( ( member @ int @ Xb2 @ B3 ) % 8.50/4.50 => ( X2 % 8.50/4.50 != ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) ) % 8.50/4.50 => ( ( X2 != T2 ) % 8.50/4.50 => ( ( minus_minus @ int @ X2 @ D6 ) % 8.50/4.50 != T2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2131_bset_I5_J,axiom, % 8.50/4.50 ! [D6: int,B3: set @ int,T2: int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 8.50/4.50 => ! [X2: int] : % 8.50/4.50 ( ! [Xa3: int] : % 8.50/4.50 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 => ! [Xb2: int] : % 8.50/4.50 ( ( member @ int @ Xb2 @ B3 ) % 8.50/4.50 => ( X2 % 8.50/4.50 != ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) ) % 8.50/4.50 => ( ( ord_less @ int @ X2 @ T2 ) % 8.50/4.50 => ( ord_less @ int @ ( minus_minus @ int @ X2 @ D6 ) @ T2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2132_bset_I7_J,axiom, % 8.50/4.50 ! [D6: int,T2: int,B3: set @ int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 8.50/4.50 => ( ( member @ int @ T2 @ B3 ) % 8.50/4.50 => ! [X2: int] : % 8.50/4.50 ( ! [Xa3: int] : % 8.50/4.50 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 => ! [Xb2: int] : % 8.50/4.50 ( ( member @ int @ Xb2 @ B3 ) % 8.50/4.50 => ( X2 % 8.50/4.50 != ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) ) % 8.50/4.50 => ( ( ord_less @ int @ T2 @ X2 ) % 8.50/4.50 => ( ord_less @ int @ T2 @ ( minus_minus @ int @ X2 @ D6 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2133_aset_I3_J,axiom, % 8.50/4.50 ! [D6: int,T2: int,A3: set @ int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 8.50/4.50 => ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 ) % 8.50/4.50 => ! [X2: int] : % 8.50/4.50 ( ! [Xa3: int] : % 8.50/4.50 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 => ! [Xb2: int] : % 8.50/4.50 ( ( member @ int @ Xb2 @ A3 ) % 8.50/4.50 => ( X2 % 8.50/4.50 != ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) ) % 8.50/4.50 => ( ( X2 = T2 ) % 8.50/4.50 => ( ( plus_plus @ int @ X2 @ D6 ) % 8.50/4.50 = T2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2134_aset_I4_J,axiom, % 8.50/4.50 ! [D6: int,T2: int,A3: set @ int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 8.50/4.50 => ( ( member @ int @ T2 @ A3 ) % 8.50/4.50 => ! [X2: int] : % 8.50/4.50 ( ! [Xa3: int] : % 8.50/4.50 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 => ! [Xb2: int] : % 8.50/4.50 ( ( member @ int @ Xb2 @ A3 ) % 8.50/4.50 => ( X2 % 8.50/4.50 != ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) ) % 8.50/4.50 => ( ( X2 != T2 ) % 8.50/4.50 => ( ( plus_plus @ int @ X2 @ D6 ) % 8.50/4.50 != T2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2135_aset_I5_J,axiom, % 8.50/4.50 ! [D6: int,T2: int,A3: set @ int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 8.50/4.50 => ( ( member @ int @ T2 @ A3 ) % 8.50/4.50 => ! [X2: int] : % 8.50/4.50 ( ! [Xa3: int] : % 8.50/4.50 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 => ! [Xb2: int] : % 8.50/4.50 ( ( member @ int @ Xb2 @ A3 ) % 8.50/4.50 => ( X2 % 8.50/4.50 != ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) ) % 8.50/4.50 => ( ( ord_less @ int @ X2 @ T2 ) % 8.50/4.50 => ( ord_less @ int @ ( plus_plus @ int @ X2 @ D6 ) @ T2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2136_aset_I7_J,axiom, % 8.50/4.50 ! [D6: int,A3: set @ int,T2: int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 8.50/4.50 => ! [X2: int] : % 8.50/4.50 ( ! [Xa3: int] : % 8.50/4.50 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 => ! [Xb2: int] : % 8.50/4.50 ( ( member @ int @ Xb2 @ A3 ) % 8.50/4.50 => ( X2 % 8.50/4.50 != ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) ) % 8.50/4.50 => ( ( ord_less @ int @ T2 @ X2 ) % 8.50/4.50 => ( ord_less @ int @ T2 @ ( plus_plus @ int @ X2 @ D6 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2137_Suc__div__eq__add3__div,axiom, % 8.50/4.50 ! [M: nat,N2: nat] : % 8.50/4.50 ( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 ) % 8.50/4.50 = ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2138_Suc__mod__eq__add3__mod,axiom, % 8.50/4.50 ! [M: nat,N2: nat] : % 8.50/4.50 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 ) % 8.50/4.50 = ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2139_aset_I8_J,axiom, % 8.50/4.50 ! [D6: int,A3: set @ int,T2: int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 8.50/4.50 => ! [X2: int] : % 8.50/4.50 ( ! [Xa3: int] : % 8.50/4.50 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 => ! [Xb2: int] : % 8.50/4.50 ( ( member @ int @ Xb2 @ A3 ) % 8.50/4.50 => ( X2 % 8.50/4.50 != ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) ) % 8.50/4.50 => ( ( ord_less_eq @ int @ T2 @ X2 ) % 8.50/4.50 => ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X2 @ D6 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2140_aset_I6_J,axiom, % 8.50/4.50 ! [D6: int,T2: int,A3: set @ int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 8.50/4.50 => ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 ) % 8.50/4.50 => ! [X2: int] : % 8.50/4.50 ( ! [Xa3: int] : % 8.50/4.50 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 => ! [Xb2: int] : % 8.50/4.50 ( ( member @ int @ Xb2 @ A3 ) % 8.50/4.50 => ( X2 % 8.50/4.50 != ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) ) % 8.50/4.50 => ( ( ord_less_eq @ int @ X2 @ T2 ) % 8.50/4.50 => ( ord_less_eq @ int @ ( plus_plus @ int @ X2 @ D6 ) @ T2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2141_bset_I8_J,axiom, % 8.50/4.50 ! [D6: int,T2: int,B3: set @ int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 8.50/4.50 => ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B3 ) % 8.50/4.50 => ! [X2: int] : % 8.50/4.50 ( ! [Xa3: int] : % 8.50/4.50 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 => ! [Xb2: int] : % 8.50/4.50 ( ( member @ int @ Xb2 @ B3 ) % 8.50/4.50 => ( X2 % 8.50/4.50 != ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) ) % 8.50/4.50 => ( ( ord_less_eq @ int @ T2 @ X2 ) % 8.50/4.50 => ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X2 @ D6 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2142_bset_I6_J,axiom, % 8.50/4.50 ! [D6: int,B3: set @ int,T2: int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 8.50/4.50 => ! [X2: int] : % 8.50/4.50 ( ! [Xa3: int] : % 8.50/4.50 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 => ! [Xb2: int] : % 8.50/4.50 ( ( member @ int @ Xb2 @ B3 ) % 8.50/4.50 => ( X2 % 8.50/4.50 != ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) ) % 8.50/4.50 => ( ( ord_less_eq @ int @ X2 @ T2 ) % 8.50/4.50 => ( ord_less_eq @ int @ ( minus_minus @ int @ X2 @ D6 ) @ T2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2143_cppi,axiom, % 8.50/4.50 ! [D6: int,P: int > $o,P5: int > $o,A3: set @ int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 8.50/4.50 => ( ? [Z4: int] : % 8.50/4.50 ! [X5: int] : % 8.50/4.50 ( ( ord_less @ int @ Z4 @ X5 ) % 8.50/4.50 => ( ( P @ X5 ) % 8.50/4.50 = ( P5 @ X5 ) ) ) % 8.50/4.50 => ( ! [X5: int] : % 8.50/4.50 ( ! [Xa: int] : % 8.50/4.50 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 => ! [Xb3: int] : % 8.50/4.50 ( ( member @ int @ Xb3 @ A3 ) % 8.50/4.50 => ( X5 % 8.50/4.50 != ( minus_minus @ int @ Xb3 @ Xa ) ) ) ) % 8.50/4.50 => ( ( P @ X5 ) % 8.50/4.50 => ( P @ ( plus_plus @ int @ X5 @ D6 ) ) ) ) % 8.50/4.50 => ( ! [X5: int,K2: int] : % 8.50/4.50 ( ( P5 @ X5 ) % 8.50/4.50 = ( P5 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D6 ) ) ) ) % 8.50/4.50 => ( ( ? [X6: int] : ( P @ X6 ) ) % 8.50/4.50 = ( ? [X: int] : % 8.50/4.50 ( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 & ( P5 @ X ) ) % 8.50/4.50 | ? [X: int] : % 8.50/4.50 ( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 & ? [Y2: int] : % 8.50/4.50 ( ( member @ int @ Y2 @ A3 ) % 8.50/4.50 & ( P @ ( minus_minus @ int @ Y2 @ X ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2144_cpmi,axiom, % 8.50/4.50 ! [D6: int,P: int > $o,P5: int > $o,B3: set @ int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ D6 ) % 8.50/4.50 => ( ? [Z4: int] : % 8.50/4.50 ! [X5: int] : % 8.50/4.50 ( ( ord_less @ int @ X5 @ Z4 ) % 8.50/4.50 => ( ( P @ X5 ) % 8.50/4.50 = ( P5 @ X5 ) ) ) % 8.50/4.50 => ( ! [X5: int] : % 8.50/4.50 ( ! [Xa: int] : % 8.50/4.50 ( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 => ! [Xb3: int] : % 8.50/4.50 ( ( member @ int @ Xb3 @ B3 ) % 8.50/4.50 => ( X5 % 8.50/4.50 != ( plus_plus @ int @ Xb3 @ Xa ) ) ) ) % 8.50/4.50 => ( ( P @ X5 ) % 8.50/4.50 => ( P @ ( minus_minus @ int @ X5 @ D6 ) ) ) ) % 8.50/4.50 => ( ! [X5: int,K2: int] : % 8.50/4.50 ( ( P5 @ X5 ) % 8.50/4.50 = ( P5 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K2 @ D6 ) ) ) ) % 8.50/4.50 => ( ( ? [X6: int] : ( P @ X6 ) ) % 8.50/4.50 = ( ? [X: int] : % 8.50/4.50 ( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 & ( P5 @ X ) ) % 8.50/4.50 | ? [X: int] : % 8.50/4.50 ( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) ) % 8.50/4.50 & ? [Y2: int] : % 8.50/4.50 ( ( member @ int @ Y2 @ B3 ) % 8.50/4.50 & ( P @ ( plus_plus @ int @ Y2 @ X ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2145_take__bit__Suc__bit1,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [N2: nat,K: num] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) ) % 8.50/4.50 = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2146_odd__mod__4__div__2,axiom, % 8.50/4.50 ! [N2: nat] : % 8.50/4.50 ( ( ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) % 8.50/4.50 = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) % 8.50/4.50 => ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2147_mod__exhaust__less__4,axiom, % 8.50/4.50 ! [M: nat] : % 8.50/4.50 ( ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 | ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) % 8.50/4.50 = ( one_one @ nat ) ) % 8.50/4.50 | ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) % 8.50/4.50 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.50/4.50 | ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) % 8.50/4.50 = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2148_divmod__algorithm__code_I7_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( unique1627219031080169319umeral @ A ) % 8.50/4.50 => ! [M: num,N2: num] : % 8.50/4.50 ( ( ( ord_less_eq @ num @ M @ N2 ) % 8.50/4.50 => ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N2 ) ) % 8.50/4.50 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M ) ) ) ) ) % 8.50/4.50 & ( ~ ( ord_less_eq @ num @ M @ N2 ) % 8.50/4.50 => ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N2 ) ) % 8.50/4.50 = ( unique1321980374590559556d_step @ A @ ( bit1 @ N2 ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2149_divmod__algorithm__code_I8_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( unique1627219031080169319umeral @ A ) % 8.50/4.50 => ! [M: num,N2: num] : % 8.50/4.50 ( ( ( ord_less @ num @ M @ N2 ) % 8.50/4.50 => ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N2 ) ) % 8.50/4.50 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M ) ) ) ) ) % 8.50/4.50 & ( ~ ( ord_less @ num @ M @ N2 ) % 8.50/4.50 => ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N2 ) ) % 8.50/4.50 = ( unique1321980374590559556d_step @ A @ ( bit1 @ N2 ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2150_take__bit__Suc__minus__bit1,axiom, % 8.50/4.50 ! [N2: nat,K: num] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) ) % 8.50/4.50 = ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2151_and__int_Opsimps,axiom, % 8.50/4.50 ! [K: int,L: int] : % 8.50/4.50 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) ) % 8.50/4.50 => ( ( ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 8.50/4.50 & ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 8.50/4.50 => ( ( bit_se5824344872417868541ns_and @ int @ K @ L ) % 8.50/4.50 = ( uminus_uminus @ int % 8.50/4.50 @ ( zero_neq_one_of_bool @ int % 8.50/4.50 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) % 8.50/4.50 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) ) % 8.50/4.50 & ( ~ ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 8.50/4.50 & ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 8.50/4.50 => ( ( bit_se5824344872417868541ns_and @ int @ K @ L ) % 8.50/4.50 = ( plus_plus @ int % 8.50/4.50 @ ( zero_neq_one_of_bool @ int % 8.50/4.50 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) % 8.50/4.50 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) % 8.50/4.50 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2152_and__int_Opelims,axiom, % 8.50/4.50 ! [X3: int,Xa2: int,Y: int] : % 8.50/4.50 ( ( ( bit_se5824344872417868541ns_and @ int @ X3 @ Xa2 ) % 8.50/4.50 = Y ) % 8.50/4.50 => ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X3 @ Xa2 ) ) % 8.50/4.50 => ~ ( ( ( ( ( member @ int @ X3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 8.50/4.50 & ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 8.50/4.50 => ( Y % 8.50/4.50 = ( uminus_uminus @ int % 8.50/4.50 @ ( zero_neq_one_of_bool @ int % 8.50/4.50 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 ) % 8.50/4.50 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) ) ) ) % 8.50/4.50 & ( ~ ( ( member @ int @ X3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 8.50/4.50 & ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 8.50/4.50 => ( Y % 8.50/4.50 = ( plus_plus @ int % 8.50/4.50 @ ( zero_neq_one_of_bool @ int % 8.50/4.50 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 ) % 8.50/4.50 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) % 8.50/4.50 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) % 8.50/4.50 => ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X3 @ Xa2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2153_finite__atLeastAtMost__int,axiom, % 8.50/4.50 ! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2154_and__nat__numerals_I1_J,axiom, % 8.50/4.50 ! [Y: num] : % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) ) % 8.50/4.50 = ( zero_zero @ nat ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2155_and__nat__numerals_I3_J,axiom, % 8.50/4.50 ! [X3: num] : % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.50 = ( zero_zero @ nat ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2156_and__nat__numerals_I2_J,axiom, % 8.50/4.50 ! [Y: num] : % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) % 8.50/4.50 = ( one_one @ nat ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2157_and__nat__numerals_I4_J,axiom, % 8.50/4.50 ! [X3: num] : % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.50 = ( one_one @ nat ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2158_divmod__algorithm__code_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( unique1627219031080169319umeral @ A ) % 8.50/4.50 => ! [M: num] : % 8.50/4.50 ( ( unique8689654367752047608divmod @ A @ M @ one2 ) % 8.50/4.50 = ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M ) @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2159_add__neg__numeral__special_I6_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( neg_numeral @ A ) % 8.50/4.50 => ! [M: num] : % 8.50/4.50 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 8.50/4.50 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2160_add__neg__numeral__special_I5_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( neg_numeral @ A ) % 8.50/4.50 => ! [N2: num] : % 8.50/4.50 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) % 8.50/4.50 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2161_and__Suc__0__eq,axiom, % 8.50/4.50 ! [N2: nat] : % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.50 = ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2162_Suc__0__and__eq,axiom, % 8.50/4.50 ! [N2: nat] : % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 8.50/4.50 = ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2163_divmod__algorithm__code_I3_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( unique1627219031080169319umeral @ A ) % 8.50/4.50 => ! [N2: num] : % 8.50/4.50 ( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N2 ) ) % 8.50/4.50 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2164_divmod__algorithm__code_I4_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( unique1627219031080169319umeral @ A ) % 8.50/4.50 => ! [N2: num] : % 8.50/4.50 ( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N2 ) ) % 8.50/4.50 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2165_numeral__inc,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( numeral @ A ) % 8.50/4.50 => ! [X3: num] : % 8.50/4.50 ( ( numeral_numeral @ A @ ( inc @ X3 ) ) % 8.50/4.50 = ( plus_plus @ A @ ( numeral_numeral @ A @ X3 ) @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2166_divmod__int__def,axiom, % 8.50/4.50 ( ( unique8689654367752047608divmod @ int ) % 8.50/4.50 = ( ^ [M2: num,N: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2167_divmod__def,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( unique1627219031080169319umeral @ A ) % 8.50/4.50 => ( ( unique8689654367752047608divmod @ A ) % 8.50/4.50 = ( ^ [M2: num,N: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2168_divmod_H__nat__def,axiom, % 8.50/4.50 ( ( unique8689654367752047608divmod @ nat ) % 8.50/4.50 = ( ^ [M2: num,N: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2169_and__nat__unfold,axiom, % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ nat ) % 8.50/4.50 = ( ^ [M2: nat,N: nat] : % 8.50/4.50 ( if @ nat % 8.50/4.50 @ ( ( M2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 | ( N % 8.50/4.50 = ( zero_zero @ nat ) ) ) % 8.50/4.50 @ ( zero_zero @ nat ) % 8.50/4.50 @ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2170_and__nat__rec,axiom, % 8.50/4.50 ( ( bit_se5824344872417868541ns_and @ nat ) % 8.50/4.50 = ( ^ [M2: nat,N: nat] : % 8.50/4.50 ( plus_plus @ nat % 8.50/4.50 @ ( zero_neq_one_of_bool @ nat % 8.50/4.50 @ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) % 8.50/4.50 & ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) % 8.50/4.50 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2171_and__int_Opinduct,axiom, % 8.50/4.50 ! [A0: int,A12: int,P: int > int > $o] : % 8.50/4.50 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) ) % 8.50/4.50 => ( ! [K2: int,L4: int] : % 8.50/4.50 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L4 ) ) % 8.50/4.50 => ( ( ~ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) % 8.50/4.50 & ( member @ int @ L4 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) ) % 8.50/4.50 => ( P @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.50 => ( P @ K2 @ L4 ) ) ) % 8.50/4.50 => ( P @ A0 @ A12 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2172_divmod__divmod__step,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( unique1627219031080169319umeral @ A ) % 8.50/4.50 => ( ( unique8689654367752047608divmod @ A ) % 8.50/4.50 = ( ^ [M2: num,N: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M2 @ N ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M2 ) ) @ ( unique1321980374590559556d_step @ A @ N @ ( unique8689654367752047608divmod @ A @ M2 @ ( bit0 @ N ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2173_upto_Opinduct,axiom, % 8.50/4.50 ! [A0: int,A12: int,P: int > int > $o] : % 8.50/4.50 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) ) % 8.50/4.50 => ( ! [I4: int,J2: int] : % 8.50/4.50 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I4 @ J2 ) ) % 8.50/4.50 => ( ( ( ord_less_eq @ int @ I4 @ J2 ) % 8.50/4.50 => ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J2 ) ) % 8.50/4.50 => ( P @ I4 @ J2 ) ) ) % 8.50/4.50 => ( P @ A0 @ A12 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2174_signed__take__bit__eq__take__bit__minus,axiom, % 8.50/4.50 ( ( bit_ri4674362597316999326ke_bit @ int ) % 8.50/4.50 = ( ^ [N: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K3 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2175_take__bit__numeral__bit1,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [L: num,K: num] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) ) % 8.50/4.50 = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2176_of__int__less__neg__numeral__power__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [A2: int,X3: num,N2: nat] : % 8.50/4.50 ( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) ) % 8.50/4.50 = ( ord_less @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2177_neg__numeral__power__less__of__int__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [X3: num,N2: nat,A2: int] : % 8.50/4.50 ( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) ) % 8.50/4.50 = ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2178_of__int__le__neg__numeral__power__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [A2: int,X3: num,N2: nat] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) ) % 8.50/4.50 = ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2179_neg__numeral__power__le__of__int__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [X3: num,N2: nat,A2: int] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) ) % 8.50/4.50 = ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2180_of__int__eq__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_char_0 @ A ) % 8.50/4.50 => ! [W2: int,Z2: int] : % 8.50/4.50 ( ( ( ring_1_of_int @ A @ W2 ) % 8.50/4.50 = ( ring_1_of_int @ A @ Z2 ) ) % 8.50/4.50 = ( W2 = Z2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2181_bit__0__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_semiring_bits @ A ) % 8.50/4.50 => ( ( bit_se5641148757651400278ts_bit @ A @ ( zero_zero @ A ) ) % 8.50/4.50 = ( bot_bot @ ( nat > $o ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2182_of__int__0,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_1 @ A ) % 8.50/4.50 => ( ( ring_1_of_int @ A @ ( zero_zero @ int ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2183_of__int__0__eq__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_char_0 @ A ) % 8.50/4.50 => ! [Z2: int] : % 8.50/4.50 ( ( ( zero_zero @ A ) % 8.50/4.50 = ( ring_1_of_int @ A @ Z2 ) ) % 8.50/4.50 = ( Z2 % 8.50/4.50 = ( zero_zero @ int ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2184_of__int__eq__0__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_char_0 @ A ) % 8.50/4.50 => ! [Z2: int] : % 8.50/4.50 ( ( ( ring_1_of_int @ A @ Z2 ) % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 = ( Z2 % 8.50/4.50 = ( zero_zero @ int ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2185_of__int__le__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [W2: int,Z2: int] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) % 8.50/4.50 = ( ord_less_eq @ int @ W2 @ Z2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2186_of__int__eq__numeral__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_char_0 @ A ) % 8.50/4.50 => ! [Z2: int,N2: num] : % 8.50/4.50 ( ( ( ring_1_of_int @ A @ Z2 ) % 8.50/4.50 = ( numeral_numeral @ A @ N2 ) ) % 8.50/4.50 = ( Z2 % 8.50/4.50 = ( numeral_numeral @ int @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2187_of__int__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_1 @ A ) % 8.50/4.50 => ! [K: num] : % 8.50/4.50 ( ( ring_1_of_int @ A @ ( numeral_numeral @ int @ K ) ) % 8.50/4.50 = ( numeral_numeral @ A @ K ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2188_of__int__less__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [W2: int,Z2: int] : % 8.50/4.50 ( ( ord_less @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) % 8.50/4.50 = ( ord_less @ int @ W2 @ Z2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2189_of__int__eq__1__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_char_0 @ A ) % 8.50/4.50 => ! [Z2: int] : % 8.50/4.50 ( ( ( ring_1_of_int @ A @ Z2 ) % 8.50/4.50 = ( one_one @ A ) ) % 8.50/4.50 = ( Z2 % 8.50/4.50 = ( one_one @ int ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2190_of__int__1,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_1 @ A ) % 8.50/4.50 => ( ( ring_1_of_int @ A @ ( one_one @ int ) ) % 8.50/4.50 = ( one_one @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2191_of__int__mult,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_1 @ A ) % 8.50/4.50 => ! [W2: int,Z2: int] : % 8.50/4.50 ( ( ring_1_of_int @ A @ ( times_times @ int @ W2 @ Z2 ) ) % 8.50/4.50 = ( times_times @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2192_of__int__add,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_1 @ A ) % 8.50/4.50 => ! [W2: int,Z2: int] : % 8.50/4.50 ( ( ring_1_of_int @ A @ ( plus_plus @ int @ W2 @ Z2 ) ) % 8.50/4.50 = ( plus_plus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2193_of__int__minus,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_1 @ A ) % 8.50/4.50 => ! [Z2: int] : % 8.50/4.50 ( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z2 ) ) % 8.50/4.50 = ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2194_of__int__diff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_1 @ A ) % 8.50/4.50 => ! [W2: int,Z2: int] : % 8.50/4.50 ( ( ring_1_of_int @ A @ ( minus_minus @ int @ W2 @ Z2 ) ) % 8.50/4.50 = ( minus_minus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2195_pred__numeral__simps_I1_J,axiom, % 8.50/4.50 ( ( pred_numeral @ one2 ) % 8.50/4.50 = ( zero_zero @ nat ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2196_eq__numeral__Suc,axiom, % 8.50/4.50 ! [K: num,N2: nat] : % 8.50/4.50 ( ( ( numeral_numeral @ nat @ K ) % 8.50/4.50 = ( suc @ N2 ) ) % 8.50/4.50 = ( ( pred_numeral @ K ) % 8.50/4.50 = N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2197_Suc__eq__numeral,axiom, % 8.50/4.50 ! [N2: nat,K: num] : % 8.50/4.50 ( ( ( suc @ N2 ) % 8.50/4.50 = ( numeral_numeral @ nat @ K ) ) % 8.50/4.50 = ( N2 % 8.50/4.50 = ( pred_numeral @ K ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2198_of__int__power__eq__of__int__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_char_0 @ A ) % 8.50/4.50 => ! [X3: int,B2: int,W2: nat] : % 8.50/4.50 ( ( ( ring_1_of_int @ A @ X3 ) % 8.50/4.50 = ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) ) % 8.50/4.50 = ( X3 % 8.50/4.50 = ( power_power @ int @ B2 @ W2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2199_of__int__eq__of__int__power__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_char_0 @ A ) % 8.50/4.50 => ! [B2: int,W2: nat,X3: int] : % 8.50/4.50 ( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) % 8.50/4.50 = ( ring_1_of_int @ A @ X3 ) ) % 8.50/4.50 = ( ( power_power @ int @ B2 @ W2 ) % 8.50/4.50 = X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2200_of__int__power,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_1 @ A ) % 8.50/4.50 => ! [Z2: int,N2: nat] : % 8.50/4.50 ( ( ring_1_of_int @ A @ ( power_power @ int @ Z2 @ N2 ) ) % 8.50/4.50 = ( power_power @ A @ ( ring_1_of_int @ A @ Z2 ) @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2201_of__int__of__bool,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_1 @ A ) % 8.50/4.50 => ! [P: $o] : % 8.50/4.50 ( ( ring_1_of_int @ A @ ( zero_neq_one_of_bool @ int @ P ) ) % 8.50/4.50 = ( zero_neq_one_of_bool @ A @ P ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2202_bit__numeral__Bit0__Suc__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [M: num,N2: nat] : % 8.50/4.50 ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( suc @ N2 ) ) % 8.50/4.50 = ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2203_bit__numeral__Bit1__Suc__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [M: num,N2: nat] : % 8.50/4.50 ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( suc @ N2 ) ) % 8.50/4.50 = ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2204_less__numeral__Suc,axiom, % 8.50/4.50 ! [K: num,N2: nat] : % 8.50/4.50 ( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) ) % 8.50/4.50 = ( ord_less @ nat @ ( pred_numeral @ K ) @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2205_less__Suc__numeral,axiom, % 8.50/4.50 ! [N2: nat,K: num] : % 8.50/4.50 ( ( ord_less @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) ) % 8.50/4.50 = ( ord_less @ nat @ N2 @ ( pred_numeral @ K ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2206_le__numeral__Suc,axiom, % 8.50/4.50 ! [K: num,N2: nat] : % 8.50/4.50 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) ) % 8.50/4.50 = ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2207_le__Suc__numeral,axiom, % 8.50/4.50 ! [N2: nat,K: num] : % 8.50/4.50 ( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) ) % 8.50/4.50 = ( ord_less_eq @ nat @ N2 @ ( pred_numeral @ K ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2208_diff__Suc__numeral,axiom, % 8.50/4.50 ! [N2: nat,K: num] : % 8.50/4.50 ( ( minus_minus @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) ) % 8.50/4.50 = ( minus_minus @ nat @ N2 @ ( pred_numeral @ K ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2209_diff__numeral__Suc,axiom, % 8.50/4.50 ! [K: num,N2: nat] : % 8.50/4.50 ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) ) % 8.50/4.50 = ( minus_minus @ nat @ ( pred_numeral @ K ) @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2210_signed__take__bit__negative__iff,axiom, % 8.50/4.50 ! [N2: nat,K: int] : % 8.50/4.50 ( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) % 8.50/4.50 = ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2211_of__int__0__le__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [Z2: int] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) % 8.50/4.50 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2212_of__int__le__0__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [Z2: int] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( zero_zero @ A ) ) % 8.50/4.50 = ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2213_of__int__less__0__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [Z2: int] : % 8.50/4.50 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( zero_zero @ A ) ) % 8.50/4.50 = ( ord_less @ int @ Z2 @ ( zero_zero @ int ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2214_of__int__0__less__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [Z2: int] : % 8.50/4.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) % 8.50/4.50 = ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2215_of__int__le__numeral__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [Z2: int,N2: num] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( numeral_numeral @ A @ N2 ) ) % 8.50/4.50 = ( ord_less_eq @ int @ Z2 @ ( numeral_numeral @ int @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2216_of__int__numeral__le__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [N2: num,Z2: int] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_of_int @ A @ Z2 ) ) % 8.50/4.50 = ( ord_less_eq @ int @ ( numeral_numeral @ int @ N2 ) @ Z2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2217_of__int__less__numeral__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [Z2: int,N2: num] : % 8.50/4.50 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( numeral_numeral @ A @ N2 ) ) % 8.50/4.50 = ( ord_less @ int @ Z2 @ ( numeral_numeral @ int @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2218_of__int__numeral__less__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [N2: num,Z2: int] : % 8.50/4.50 ( ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_of_int @ A @ Z2 ) ) % 8.50/4.50 = ( ord_less @ int @ ( numeral_numeral @ int @ N2 ) @ Z2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2219_of__int__le__1__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [Z2: int] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) % 8.50/4.50 = ( ord_less_eq @ int @ Z2 @ ( one_one @ int ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2220_of__int__1__le__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [Z2: int] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) % 8.50/4.50 = ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2221_of__int__less__1__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [Z2: int] : % 8.50/4.50 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) % 8.50/4.50 = ( ord_less @ int @ Z2 @ ( one_one @ int ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2222_of__int__1__less__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [Z2: int] : % 8.50/4.50 ( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) % 8.50/4.50 = ( ord_less @ int @ ( one_one @ int ) @ Z2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2223_of__int__le__of__int__power__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [B2: int,W2: nat,X3: int] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X3 ) ) % 8.50/4.50 = ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W2 ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2224_of__int__power__le__of__int__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [X3: int,B2: int,W2: nat] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) ) % 8.50/4.50 = ( ord_less_eq @ int @ X3 @ ( power_power @ int @ B2 @ W2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2225_of__int__eq__numeral__power__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_char_0 @ A ) % 8.50/4.50 => ! [Y: int,X3: num,N2: nat] : % 8.50/4.50 ( ( ( ring_1_of_int @ A @ Y ) % 8.50/4.50 = ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) ) % 8.50/4.50 = ( Y % 8.50/4.50 = ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2226_numeral__power__eq__of__int__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_char_0 @ A ) % 8.50/4.50 => ! [X3: num,N2: nat,Y: int] : % 8.50/4.50 ( ( ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) % 8.50/4.50 = ( ring_1_of_int @ A @ Y ) ) % 8.50/4.50 = ( ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) % 8.50/4.50 = Y ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2227_of__int__less__of__int__power__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [B2: int,W2: nat,X3: int] : % 8.50/4.50 ( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X3 ) ) % 8.50/4.50 = ( ord_less @ int @ ( power_power @ int @ B2 @ W2 ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2228_of__int__power__less__of__int__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [X3: int,B2: int,W2: nat] : % 8.50/4.50 ( ( ord_less @ A @ ( ring_1_of_int @ A @ X3 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) ) % 8.50/4.50 = ( ord_less @ int @ X3 @ ( power_power @ int @ B2 @ W2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2229_bit__minus__numeral__Bit0__Suc__iff,axiom, % 8.50/4.50 ! [W2: num,N2: nat] : % 8.50/4.50 ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) ) @ ( suc @ N2 ) ) % 8.50/4.50 = ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2230_bit__minus__numeral__Bit1__Suc__iff,axiom, % 8.50/4.50 ! [W2: num,N2: nat] : % 8.50/4.50 ( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) ) @ ( suc @ N2 ) ) % 8.50/4.50 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2231_bit__0,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_semiring_bits @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( zero_zero @ nat ) ) % 8.50/4.50 = ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2232_bit__mod__2__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_semiring_bits @ A ) % 8.50/4.50 => ! [A2: A,N2: nat] : % 8.50/4.50 ( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 ) % 8.50/4.50 = ( ( N2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 & ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2233_numeral__power__le__of__int__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [X3: num,N2: nat,A2: int] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) ) % 8.50/4.50 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2234_of__int__le__numeral__power__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [A2: int,X3: num,N2: nat] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) ) % 8.50/4.50 = ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2235_numeral__power__less__of__int__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [X3: num,N2: nat,A2: int] : % 8.50/4.50 ( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) @ ( ring_1_of_int @ A @ A2 ) ) % 8.50/4.50 = ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2236_of__int__less__numeral__power__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [A2: int,X3: num,N2: nat] : % 8.50/4.50 ( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N2 ) ) % 8.50/4.50 = ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2237_of__int__eq__neg__numeral__power__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_char_0 @ A ) % 8.50/4.50 => ! [Y: int,X3: num,N2: nat] : % 8.50/4.50 ( ( ( ring_1_of_int @ A @ Y ) % 8.50/4.50 = ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) ) % 8.50/4.50 = ( Y % 8.50/4.50 = ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2238_neg__numeral__power__eq__of__int__cancel__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_char_0 @ A ) % 8.50/4.50 => ! [X3: num,N2: nat,Y: int] : % 8.50/4.50 ( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N2 ) % 8.50/4.50 = ( ring_1_of_int @ A @ Y ) ) % 8.50/4.50 = ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N2 ) % 8.50/4.50 = Y ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2239_bit__disjunctive__add__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_semiring_bits @ A ) % 8.50/4.50 => ! [A2: A,B2: A,N2: nat] : % 8.50/4.50 ( ! [N3: nat] : % 8.50/4.50 ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 ) % 8.50/4.50 | ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) ) % 8.50/4.50 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N2 ) % 8.50/4.50 = ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) % 8.50/4.50 | ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2240_ex__le__of__int,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim462609752435547400_field @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ? [Z3: int] : ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2241_ex__less__of__int,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim462609752435547400_field @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ? [Z3: int] : ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ Z3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2242_ex__of__int__less,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim462609752435547400_field @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ? [Z3: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2243_mult__of__int__commute,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_1 @ A ) % 8.50/4.50 => ! [X3: int,Y: A] : % 8.50/4.50 ( ( times_times @ A @ ( ring_1_of_int @ A @ X3 ) @ Y ) % 8.50/4.50 = ( times_times @ A @ Y @ ( ring_1_of_int @ A @ X3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2244_not__bit__1__Suc,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [N2: nat] : % 8.50/4.50 ~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2245_bit__1__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_semiring_bits @ A ) % 8.50/4.50 => ! [N2: nat] : % 8.50/4.50 ( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N2 ) % 8.50/4.50 = ( N2 % 8.50/4.50 = ( zero_zero @ nat ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2246_bit__take__bit__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [M: nat,A2: A,N2: nat] : % 8.50/4.50 ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) @ N2 ) % 8.50/4.50 = ( ( ord_less @ nat @ N2 @ M ) % 8.50/4.50 & ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2247_bit__of__bool__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_semiring_bits @ A ) % 8.50/4.50 => ! [B2: $o,N2: nat] : % 8.50/4.50 ( ( bit_se5641148757651400278ts_bit @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ N2 ) % 8.50/4.50 = ( B2 % 8.50/4.50 & ( N2 % 8.50/4.50 = ( zero_zero @ nat ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2248_numeral__eq__Suc,axiom, % 8.50/4.50 ( ( numeral_numeral @ nat ) % 8.50/4.50 = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2249_of__int__nonneg,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [Z2: int] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 8.50/4.50 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2250_of__int__pos,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( linordered_idom @ A ) % 8.50/4.50 => ! [Z2: int] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) % 8.50/4.50 => ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2251_floor__exists,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim462609752435547400_field @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ? [Z3: int] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X3 ) % 8.50/4.50 & ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2252_floor__exists1,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim462609752435547400_field @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ? [X5: int] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X5 ) @ X3 ) % 8.50/4.50 & ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X5 @ ( one_one @ int ) ) ) ) % 8.50/4.50 & ! [Y5: int] : % 8.50/4.50 ( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y5 ) @ X3 ) % 8.50/4.50 & ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y5 @ ( one_one @ int ) ) ) ) ) % 8.50/4.50 => ( Y5 = X5 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2253_of__int__neg__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_1 @ A ) % 8.50/4.50 => ! [K: num] : % 8.50/4.50 ( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) % 8.50/4.50 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2254_bit__imp__take__bit__positive,axiom, % 8.50/4.50 ! [N2: nat,M: nat,K: int] : % 8.50/4.50 ( ( ord_less @ nat @ N2 @ M ) % 8.50/4.50 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) % 8.50/4.50 => ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2255_int__le__real__less,axiom, % 8.50/4.50 ( ( ord_less_eq @ int ) % 8.50/4.50 = ( ^ [N: int,M2: int] : ( ord_less @ real @ ( ring_1_of_int @ real @ N ) @ ( plus_plus @ real @ ( ring_1_of_int @ real @ M2 ) @ ( one_one @ real ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2256_int__less__real__le,axiom, % 8.50/4.50 ( ( ord_less @ int ) % 8.50/4.50 = ( ^ [N: int,M2: int] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ M2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2257_bit__concat__bit__iff,axiom, % 8.50/4.50 ! [M: nat,K: int,L: int,N2: nat] : % 8.50/4.50 ( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K @ L ) @ N2 ) % 8.50/4.50 = ( ( ( ord_less @ nat @ N2 @ M ) % 8.50/4.50 & ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) % 8.50/4.50 | ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.50 & ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2258_exp__eq__0__imp__not__bit,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_semiring_bits @ A ) % 8.50/4.50 => ! [N2: nat,A2: A] : % 8.50/4.50 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 => ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2259_bit__Suc,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_semiring_bits @ A ) % 8.50/4.50 => ! [A2: A,N2: nat] : % 8.50/4.50 ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ N2 ) ) % 8.50/4.50 = ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2260_int__bit__bound,axiom, % 8.50/4.50 ! [K: int] : % 8.50/4.50 ~ ! [N3: nat] : % 8.50/4.50 ( ! [M6: nat] : % 8.50/4.50 ( ( ord_less_eq @ nat @ N3 @ M6 ) % 8.50/4.50 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ M6 ) % 8.50/4.50 = ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) % 8.50/4.50 => ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 8.50/4.50 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) % 8.50/4.50 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2261_and__exp__eq__0__iff__not__bit,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [A2: A,N2: nat] : % 8.50/4.50 ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 = ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2262_even__bit__succ__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_semiring_bits @ A ) % 8.50/4.50 => ! [A2: A,N2: nat] : % 8.50/4.50 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.50/4.50 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ N2 ) % 8.50/4.50 = ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) % 8.50/4.50 | ( N2 % 8.50/4.50 = ( zero_zero @ nat ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2263_odd__bit__iff__bit__pred,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_semiring_bits @ A ) % 8.50/4.50 => ! [A2: A,N2: nat] : % 8.50/4.50 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) % 8.50/4.50 => ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 ) % 8.50/4.50 = ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ N2 ) % 8.50/4.50 | ( N2 % 8.50/4.50 = ( zero_zero @ nat ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2264_bit__sum__mult__2__cases,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [A2: A,B2: A,N2: nat] : % 8.50/4.50 ( ! [J2: nat] : % 8.50/4.50 ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ J2 ) ) % 8.50/4.50 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N2 ) % 8.50/4.50 = ( ( ( N2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) % 8.50/4.50 & ( ( N2 % 8.50/4.50 != ( zero_zero @ nat ) ) % 8.50/4.50 => ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N2 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2265_bit__rec,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_semiring_bits @ A ) % 8.50/4.50 => ( ( bit_se5641148757651400278ts_bit @ A ) % 8.50/4.50 = ( ^ [A4: A,N: nat] : % 8.50/4.50 ( ( ( N % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) % 8.50/4.50 & ( ( N % 8.50/4.50 != ( zero_zero @ nat ) ) % 8.50/4.50 => ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2266_take__bit__Suc__from__most,axiom, % 8.50/4.50 ! [N2: nat,K: int] : % 8.50/4.50 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ K ) % 8.50/4.50 = ( plus_plus @ int @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2267_round__unique,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A,Y: int] : % 8.50/4.50 ( ( ord_less @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y ) ) % 8.50/4.50 => ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y ) @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.50 => ( ( archimedean_round @ A @ X3 ) % 8.50/4.50 = Y ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2268_of__int__round__gt,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A] : ( ord_less @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2269_of__int__round__ge,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2270_of__int__round__le,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2271_divmod__BitM__2__eq,axiom, % 8.50/4.50 ! [M: num] : % 8.50/4.50 ( ( unique8689654367752047608divmod @ int @ ( bitM @ M ) @ ( bit0 @ one2 ) ) % 8.50/4.50 = ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2272_mask__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [N2: num] : % 8.50/4.50 ( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N2 ) ) % 8.50/4.50 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N2 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2273_neg__numeral__le__ceiling,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [V2: num,X3: A] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X3 ) ) % 8.50/4.50 = ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2274_mask__nat__positive__iff,axiom, % 8.50/4.50 ! [N2: nat] : % 8.50/4.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) % 8.50/4.50 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2275_ceiling__zero,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) ) % 8.50/4.50 = ( zero_zero @ int ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2276_mask__eq__0__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [N2: nat] : % 8.50/4.50 ( ( ( bit_se2239418461657761734s_mask @ A @ N2 ) % 8.50/4.50 = ( zero_zero @ A ) ) % 8.50/4.50 = ( N2 % 8.50/4.50 = ( zero_zero @ nat ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2277_mask__0,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2278_round__0,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ( ( archimedean_round @ A @ ( zero_zero @ A ) ) % 8.50/4.50 = ( zero_zero @ int ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2279_mask__Suc__0,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.50 = ( one_one @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2280_ceiling__add__of__int,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A,Z2: int] : % 8.50/4.50 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) ) % 8.50/4.50 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2281_ceiling__le__zero,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( zero_zero @ int ) ) % 8.50/4.50 = ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2282_zero__less__ceiling,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X3 ) ) % 8.50/4.50 = ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2283_ceiling__le__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A,V2: num] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) % 8.50/4.50 = ( ord_less_eq @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2284_ceiling__less__one,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) ) % 8.50/4.50 = ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2285_one__le__ceiling,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X3 ) ) % 8.50/4.50 = ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2286_numeral__less__ceiling,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [V2: num,X3: A] : % 8.50/4.50 ( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X3 ) ) % 8.50/4.50 = ( ord_less @ A @ ( numeral_numeral @ A @ V2 ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2287_ceiling__le__one,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) ) % 8.50/4.50 = ( ord_less_eq @ A @ X3 @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2288_one__less__ceiling,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X3 ) ) % 8.50/4.50 = ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2289_ceiling__add__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A,V2: num] : % 8.50/4.50 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.50 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2290_ceiling__add__one,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) ) % 8.50/4.50 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2291_ceiling__less__zero,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( zero_zero @ int ) ) % 8.50/4.50 = ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2292_zero__le__ceiling,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X3 ) ) % 8.50/4.50 = ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2293_ceiling__less__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A,V2: num] : % 8.50/4.50 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) % 8.50/4.50 = ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2294_numeral__le__ceiling,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [V2: num,X3: A] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X3 ) ) % 8.50/4.50 = ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2295_ceiling__le__neg__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A,V2: num] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) % 8.50/4.50 = ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2296_neg__numeral__less__ceiling,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [V2: num,X3: A] : % 8.50/4.50 ( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X3 ) ) % 8.50/4.50 = ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2297_ceiling__less__neg__numeral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A,V2: num] : % 8.50/4.50 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) % 8.50/4.50 = ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2298_less__eq__mask,axiom, % 8.50/4.50 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2299_bit__Suc__0__iff,axiom, % 8.50/4.50 ! [N2: nat] : % 8.50/4.50 ( ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 8.50/4.50 = ( N2 % 8.50/4.50 = ( zero_zero @ nat ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2300_not__bit__Suc__0__Suc,axiom, % 8.50/4.50 ! [N2: nat] : % 8.50/4.50 ~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( suc @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2301_ceiling__mono,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [Y: A,X3: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ Y @ X3 ) % 8.50/4.50 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2302_le__of__int__ceiling,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2303_ceiling__less__cancel,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A,Y: A] : % 8.50/4.50 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( archimedean_ceiling @ A @ Y ) ) % 8.50/4.50 => ( ord_less @ A @ X3 @ Y ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2304_not__mask__negative__int,axiom, % 8.50/4.50 ! [N2: nat] : % 8.50/4.50 ~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N2 ) @ ( zero_zero @ int ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2305_not__bit__Suc__0__numeral,axiom, % 8.50/4.50 ! [N2: num] : % 8.50/4.50 ~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2306_ceiling__le__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A,Z2: int] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z2 ) % 8.50/4.50 = ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2307_ceiling__le,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A,A2: int] : % 8.50/4.50 ( ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ A2 ) ) % 8.50/4.50 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X3 ) @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2308_less__ceiling__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [Z2: int,X3: A] : % 8.50/4.50 ( ( ord_less @ int @ Z2 @ ( archimedean_ceiling @ A @ X3 ) ) % 8.50/4.50 = ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2309_ceiling__add__le,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A,Y: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( archimedean_ceiling @ A @ Y ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2310_round__mono,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A,Y: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.50 => ( ord_less_eq @ int @ ( archimedean_round @ A @ X3 ) @ ( archimedean_round @ A @ Y ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2311_less__mask,axiom, % 8.50/4.50 ! [N2: nat] : % 8.50/4.50 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 8.50/4.50 => ( ord_less @ nat @ N2 @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2312_eval__nat__numeral_I2_J,axiom, % 8.50/4.50 ! [N2: num] : % 8.50/4.50 ( ( numeral_numeral @ nat @ ( bit0 @ N2 ) ) % 8.50/4.50 = ( suc @ ( numeral_numeral @ nat @ ( bitM @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2313_of__int__ceiling__le__add__one,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [R2: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R2 ) ) @ ( plus_plus @ A @ R2 @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2314_of__int__ceiling__diff__one__le,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [R2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R2 ) ) @ ( one_one @ A ) ) @ R2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2315_ceiling__split,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [P: int > $o,T2: A] : % 8.50/4.50 ( ( P @ ( archimedean_ceiling @ A @ T2 ) ) % 8.50/4.50 = ( ! [I3: int] : % 8.50/4.50 ( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) @ T2 ) % 8.50/4.50 & ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I3 ) ) ) % 8.50/4.50 => ( P @ I3 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2316_ceiling__eq__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A,A2: int] : % 8.50/4.50 ( ( ( archimedean_ceiling @ A @ X3 ) % 8.50/4.50 = A2 ) % 8.50/4.50 = ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) @ X3 ) % 8.50/4.50 & ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ A2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2317_ceiling__unique,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [Z2: int,X3: A] : % 8.50/4.50 ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X3 ) % 8.50/4.50 => ( ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) % 8.50/4.50 => ( ( archimedean_ceiling @ A @ X3 ) % 8.50/4.50 = Z2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2318_ceiling__correct,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) @ ( one_one @ A ) ) @ X3 ) % 8.50/4.50 & ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2319_mult__ceiling__le,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.50 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.50/4.50 => ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A2 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A2 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2320_ceiling__less__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A,Z2: int] : % 8.50/4.50 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z2 ) % 8.50/4.50 = ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2321_le__ceiling__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [Z2: int,X3: A] : % 8.50/4.50 ( ( ord_less_eq @ int @ Z2 @ ( archimedean_ceiling @ A @ X3 ) ) % 8.50/4.50 = ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2322_ceiling__divide__upper,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [Q4: A,P4: A] : % 8.50/4.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 ) % 8.50/4.50 => ( ord_less_eq @ A @ P4 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2323_Suc__mask__eq__exp,axiom, % 8.50/4.50 ! [N2: nat] : % 8.50/4.50 ( ( suc @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) % 8.50/4.50 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2324_mask__nat__less__exp,axiom, % 8.50/4.50 ! [N2: nat] : ( ord_less @ nat @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2325_semiring__bit__operations__class_Oeven__mask__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [N2: nat] : % 8.50/4.50 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) % 8.50/4.50 = ( N2 % 8.50/4.50 = ( zero_zero @ nat ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2326_ceiling__divide__lower,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [Q4: A,P4: A] : % 8.50/4.50 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 ) % 8.50/4.50 => ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) @ P4 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2327_ceiling__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [N2: int,X3: A] : % 8.50/4.50 ( ( ord_less @ A @ ( ring_1_of_int @ A @ N2 ) @ X3 ) % 8.50/4.50 => ( ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N2 ) @ ( one_one @ A ) ) ) % 8.50/4.50 => ( ( archimedean_ceiling @ A @ X3 ) % 8.50/4.50 = ( plus_plus @ int @ N2 @ ( one_one @ int ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2328_Suc__0__xor__eq,axiom, % 8.50/4.50 ! [N2: nat] : % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 8.50/4.50 = ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 8.50/4.50 @ ( zero_neq_one_of_bool @ nat % 8.50/4.50 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2329_xor__Suc__0__eq,axiom, % 8.50/4.50 ! [N2: nat] : % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.50 = ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 8.50/4.50 @ ( zero_neq_one_of_bool @ nat % 8.50/4.50 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2330_horner__sum__of__bool__2__less,axiom, % 8.50/4.50 ! [Bs: list @ $o] : ( ord_less @ int @ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Bs ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ $o ) @ Bs ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2331_dbl__dec__simps_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( neg_numeral @ A ) % 8.50/4.50 => ( ( neg_numeral_dbl_dec @ A @ ( zero_zero @ A ) ) % 8.50/4.50 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2332_the__elem__eq,axiom, % 8.50/4.50 ! [A: $tType,X3: A] : % 8.50/4.50 ( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.50 = X3 ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2333_vebt__member_Oelims_I1_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 8.50/4.50 ( ( ( vEBT_vebt_member @ X3 @ Xa2 ) % 8.50/4.50 = Y ) % 8.50/4.50 => ( ! [A5: $o,B5: $o] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.50/4.50 => ( Y % 8.50/4.50 = ( ~ ( ( ( Xa2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => A5 ) % 8.50/4.50 & ( ( Xa2 % 8.50/4.50 != ( zero_zero @ nat ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) % 8.50/4.50 => B5 ) % 8.50/4.50 & ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) ) ) ) ) ) ) % 8.50/4.50 => ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 8.50/4.50 => Y ) % 8.50/4.50 => ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) ) % 8.50/4.50 => Y ) % 8.50/4.50 => ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) ) % 8.50/4.50 => Y ) % 8.50/4.50 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT] : % 8.50/4.50 ( ? [Summary2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) % 8.50/4.50 => ( Y % 8.50/4.50 = ( ~ ( ( Xa2 != Mi2 ) % 8.50/4.50 => ( ( Xa2 != Ma2 ) % 8.50/4.50 => ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 8.50/4.50 & ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 8.50/4.50 => ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 8.50/4.50 & ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 8.50/4.50 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2334_set__vebt_H__def,axiom, % 8.50/4.50 ( vEBT_VEBT_set_vebt % 8.50/4.50 = ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2335_finite__Collect__conjI,axiom, % 8.50/4.50 ! [A: $tType,P: A > $o,Q: A > $o] : % 8.50/4.50 ( ( ( finite_finite2 @ A @ ( collect @ A @ P ) ) % 8.50/4.50 | ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) % 8.50/4.50 => ( finite_finite2 @ A % 8.50/4.50 @ ( collect @ A % 8.50/4.50 @ ^ [X: A] : % 8.50/4.50 ( ( P @ X ) % 8.50/4.50 & ( Q @ X ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2336_finite__Collect__disjI,axiom, % 8.50/4.50 ! [A: $tType,P: A > $o,Q: A > $o] : % 8.50/4.50 ( ( finite_finite2 @ A % 8.50/4.50 @ ( collect @ A % 8.50/4.50 @ ^ [X: A] : % 8.50/4.50 ( ( P @ X ) % 8.50/4.50 | ( Q @ X ) ) ) ) % 8.50/4.50 = ( ( finite_finite2 @ A @ ( collect @ A @ P ) ) % 8.50/4.50 & ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2337_bit_Oxor__self,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_ri3973907225187159222ations @ A ) % 8.50/4.50 => ! [X3: A] : % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ A @ X3 @ X3 ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2338_xor__self__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ A @ A2 @ A2 ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2339_xor_Oleft__neutral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.50 = A2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2340_xor_Oright__neutral,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( zero_zero @ A ) ) % 8.50/4.50 = A2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2341_finite__Collect__bounded__ex,axiom, % 8.50/4.50 ! [B: $tType,A: $tType,P: A > $o,Q: B > A > $o] : % 8.50/4.50 ( ( finite_finite2 @ A @ ( collect @ A @ P ) ) % 8.50/4.50 => ( ( finite_finite2 @ B % 8.50/4.50 @ ( collect @ B % 8.50/4.50 @ ^ [X: B] : % 8.50/4.50 ? [Y2: A] : % 8.50/4.50 ( ( P @ Y2 ) % 8.50/4.50 & ( Q @ X @ Y2 ) ) ) ) % 8.50/4.50 = ( ! [Y2: A] : % 8.50/4.50 ( ( P @ Y2 ) % 8.50/4.50 => ( finite_finite2 @ B % 8.50/4.50 @ ( collect @ B % 8.50/4.50 @ ^ [X: B] : ( Q @ X @ Y2 ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2342_finite__Collect__subsets,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A] : % 8.50/4.50 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.50 => ( finite_finite2 @ ( set @ A ) % 8.50/4.50 @ ( collect @ ( set @ A ) % 8.50/4.50 @ ^ [B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2343_singleton__conv2,axiom, % 8.50/4.50 ! [A: $tType,A2: A] : % 8.50/4.50 ( ( collect @ A % 8.50/4.50 @ ( ^ [Y3: A,Z: A] : ( Y3 = Z ) % 8.50/4.50 @ A2 ) ) % 8.50/4.50 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2344_singleton__conv,axiom, % 8.50/4.50 ! [A: $tType,A2: A] : % 8.50/4.50 ( ( collect @ A % 8.50/4.50 @ ^ [X: A] : ( X = A2 ) ) % 8.50/4.50 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2345_finite__Collect__less__nat,axiom, % 8.50/4.50 ! [K: nat] : % 8.50/4.50 ( finite_finite2 @ nat % 8.50/4.50 @ ( collect @ nat % 8.50/4.50 @ ^ [N: nat] : ( ord_less @ nat @ N @ K ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2346_finite__Collect__le__nat,axiom, % 8.50/4.50 ! [K: nat] : % 8.50/4.50 ( finite_finite2 @ nat % 8.50/4.50 @ ( collect @ nat % 8.50/4.50 @ ^ [N: nat] : ( ord_less_eq @ nat @ N @ K ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2347_finite__interval__int1,axiom, % 8.50/4.50 ! [A2: int,B2: int] : % 8.50/4.50 ( finite_finite2 @ int % 8.50/4.50 @ ( collect @ int % 8.50/4.50 @ ^ [I3: int] : % 8.50/4.50 ( ( ord_less_eq @ int @ A2 @ I3 ) % 8.50/4.50 & ( ord_less_eq @ int @ I3 @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2348_finite__interval__int4,axiom, % 8.50/4.50 ! [A2: int,B2: int] : % 8.50/4.50 ( finite_finite2 @ int % 8.50/4.50 @ ( collect @ int % 8.50/4.50 @ ^ [I3: int] : % 8.50/4.50 ( ( ord_less @ int @ A2 @ I3 ) % 8.50/4.50 & ( ord_less @ int @ I3 @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2349_finite__interval__int2,axiom, % 8.50/4.50 ! [A2: int,B2: int] : % 8.50/4.50 ( finite_finite2 @ int % 8.50/4.50 @ ( collect @ int % 8.50/4.50 @ ^ [I3: int] : % 8.50/4.50 ( ( ord_less_eq @ int @ A2 @ I3 ) % 8.50/4.50 & ( ord_less @ int @ I3 @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2350_finite__interval__int3,axiom, % 8.50/4.50 ! [A2: int,B2: int] : % 8.50/4.50 ( finite_finite2 @ int % 8.50/4.50 @ ( collect @ int % 8.50/4.50 @ ^ [I3: int] : % 8.50/4.50 ( ( ord_less @ int @ A2 @ I3 ) % 8.50/4.50 & ( ord_less_eq @ int @ I3 @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2351_xor__nat__numerals_I4_J,axiom, % 8.50/4.50 ! [X3: num] : % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.50 = ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2352_xor__nat__numerals_I3_J,axiom, % 8.50/4.50 ! [X3: num] : % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.50 = ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2353_xor__nat__numerals_I2_J,axiom, % 8.50/4.50 ! [Y: num] : % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) % 8.50/4.50 = ( numeral_numeral @ nat @ ( bit0 @ Y ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2354_xor__nat__numerals_I1_J,axiom, % 8.50/4.50 ! [Y: num] : % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) ) % 8.50/4.50 = ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2355_xor__numerals_I6_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [X3: num,Y: num] : % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) ) % 8.50/4.50 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2356_xor__numerals_I4_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [X3: num,Y: num] : % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) % 8.50/4.50 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2357_empty__def,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bot_bot @ ( set @ A ) ) % 8.50/4.50 = ( collect @ A % 8.50/4.50 @ ^ [X: A] : $false ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2358_Collect__conv__if2,axiom, % 8.50/4.50 ! [A: $tType,P: A > $o,A2: A] : % 8.50/4.50 ( ( ( P @ A2 ) % 8.50/4.50 => ( ( collect @ A % 8.50/4.50 @ ^ [X: A] : % 8.50/4.50 ( ( A2 = X ) % 8.50/4.50 & ( P @ X ) ) ) % 8.50/4.50 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.50 & ( ~ ( P @ A2 ) % 8.50/4.50 => ( ( collect @ A % 8.50/4.50 @ ^ [X: A] : % 8.50/4.50 ( ( A2 = X ) % 8.50/4.50 & ( P @ X ) ) ) % 8.50/4.50 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2359_Collect__conv__if,axiom, % 8.50/4.50 ! [A: $tType,P: A > $o,A2: A] : % 8.50/4.50 ( ( ( P @ A2 ) % 8.50/4.50 => ( ( collect @ A % 8.50/4.50 @ ^ [X: A] : % 8.50/4.50 ( ( X = A2 ) % 8.50/4.50 & ( P @ X ) ) ) % 8.50/4.50 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.50 & ( ~ ( P @ A2 ) % 8.50/4.50 => ( ( collect @ A % 8.50/4.50 @ ^ [X: A] : % 8.50/4.50 ( ( X = A2 ) % 8.50/4.50 & ( P @ X ) ) ) % 8.50/4.50 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2360_pred__subset__eq2,axiom, % 8.50/4.50 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] : % 8.50/4.50 ( ( ord_less_eq @ ( A > B > $o ) % 8.50/4.50 @ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R ) % 8.50/4.50 @ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S3 ) ) % 8.50/4.50 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2361_strict__subset__divisors__dvd,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( comm_monoid_mult @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ord_less @ ( set @ A ) % 8.50/4.50 @ ( collect @ A % 8.50/4.50 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A2 ) ) % 8.50/4.50 @ ( collect @ A % 8.50/4.50 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) ) % 8.50/4.50 = ( ( dvd_dvd @ A @ A2 @ B2 ) % 8.50/4.50 & ~ ( dvd_dvd @ A @ B2 @ A2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2362_pred__equals__eq2,axiom, % 8.50/4.50 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] : % 8.50/4.50 ( ( ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R ) ) % 8.50/4.50 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S3 ) ) ) % 8.50/4.50 = ( R = S3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2363_subset__divisors__dvd,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( comm_monoid_mult @ A ) % 8.50/4.50 => ! [A2: A,B2: A] : % 8.50/4.50 ( ( ord_less_eq @ ( set @ A ) % 8.50/4.50 @ ( collect @ A % 8.50/4.50 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A2 ) ) % 8.50/4.50 @ ( collect @ A % 8.50/4.50 @ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) ) % 8.50/4.50 = ( dvd_dvd @ A @ A2 @ B2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2364_less__set__def,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ord_less @ ( set @ A ) ) % 8.50/4.50 = ( ^ [A6: set @ A,B6: set @ A] : % 8.50/4.50 ( ord_less @ ( A > $o ) % 8.50/4.50 @ ^ [X: A] : ( member @ A @ X @ A6 ) % 8.50/4.50 @ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2365_Collect__subset,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,P: A > $o] : % 8.50/4.50 ( ord_less_eq @ ( set @ A ) % 8.50/4.50 @ ( collect @ A % 8.50/4.50 @ ^ [X: A] : % 8.50/4.50 ( ( member @ A @ X @ A3 ) % 8.50/4.50 & ( P @ X ) ) ) % 8.50/4.50 @ A3 ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2366_less__eq__set__def,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ord_less_eq @ ( set @ A ) ) % 8.50/4.50 = ( ^ [A6: set @ A,B6: set @ A] : % 8.50/4.50 ( ord_less_eq @ ( A > $o ) % 8.50/4.50 @ ^ [X: A] : ( member @ A @ X @ A6 ) % 8.50/4.50 @ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2367_prop__restrict,axiom, % 8.50/4.50 ! [A: $tType,X3: A,Z5: set @ A,X8: set @ A,P: A > $o] : % 8.50/4.50 ( ( member @ A @ X3 @ Z5 ) % 8.50/4.50 => ( ( ord_less_eq @ ( set @ A ) @ Z5 % 8.50/4.50 @ ( collect @ A % 8.50/4.50 @ ^ [X: A] : % 8.50/4.50 ( ( member @ A @ X @ X8 ) % 8.50/4.50 & ( P @ X ) ) ) ) % 8.50/4.50 => ( P @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2368_Collect__restrict,axiom, % 8.50/4.50 ! [A: $tType,X8: set @ A,P: A > $o] : % 8.50/4.50 ( ord_less_eq @ ( set @ A ) % 8.50/4.50 @ ( collect @ A % 8.50/4.50 @ ^ [X: A] : % 8.50/4.50 ( ( member @ A @ X @ X8 ) % 8.50/4.50 & ( P @ X ) ) ) % 8.50/4.50 @ X8 ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2369_pred__subset__eq,axiom, % 8.50/4.50 ! [A: $tType,R: set @ A,S3: set @ A] : % 8.50/4.50 ( ( ord_less_eq @ ( A > $o ) % 8.50/4.50 @ ^ [X: A] : ( member @ A @ X @ R ) % 8.50/4.50 @ ^ [X: A] : ( member @ A @ X @ S3 ) ) % 8.50/4.50 = ( ord_less_eq @ ( set @ A ) @ R @ S3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2370_lambda__one,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( monoid_mult @ A ) % 8.50/4.50 => ( ( ^ [X: A] : X ) % 8.50/4.50 = ( times_times @ A @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2371_lambda__zero,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( mult_zero @ A ) % 8.50/4.50 => ( ( ^ [H: A] : ( zero_zero @ A ) ) % 8.50/4.50 = ( times_times @ A @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2372_uminus__set__def,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( uminus_uminus @ ( set @ A ) ) % 8.50/4.50 = ( ^ [A6: set @ A] : % 8.50/4.50 ( collect @ A % 8.50/4.50 @ ( uminus_uminus @ ( A > $o ) % 8.50/4.50 @ ^ [X: A] : ( member @ A @ X @ A6 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2373_Collect__neg__eq,axiom, % 8.50/4.50 ! [A: $tType,P: A > $o] : % 8.50/4.50 ( ( collect @ A % 8.50/4.50 @ ^ [X: A] : % 8.50/4.50 ~ ( P @ X ) ) % 8.50/4.50 = ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2374_Compl__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( uminus_uminus @ ( set @ A ) ) % 8.50/4.50 = ( ^ [A6: set @ A] : % 8.50/4.50 ( collect @ A % 8.50/4.50 @ ^ [X: A] : % 8.50/4.50 ~ ( member @ A @ X @ A6 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2375_set__diff__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( minus_minus @ ( set @ A ) ) % 8.50/4.50 = ( ^ [A6: set @ A,B6: set @ A] : % 8.50/4.50 ( collect @ A % 8.50/4.50 @ ^ [X: A] : % 8.50/4.50 ( ( member @ A @ X @ A6 ) % 8.50/4.50 & ~ ( member @ A @ X @ B6 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2376_insert__compr,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( insert @ A ) % 8.50/4.50 = ( ^ [A4: A,B6: set @ A] : % 8.50/4.50 ( collect @ A % 8.50/4.50 @ ^ [X: A] : % 8.50/4.50 ( ( X = A4 ) % 8.50/4.50 | ( member @ A @ X @ B6 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2377_minus__set__def,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( minus_minus @ ( set @ A ) ) % 8.50/4.50 = ( ^ [A6: set @ A,B6: set @ A] : % 8.50/4.50 ( collect @ A % 8.50/4.50 @ ( minus_minus @ ( A > $o ) % 8.50/4.50 @ ^ [X: A] : ( member @ A @ X @ A6 ) % 8.50/4.50 @ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2378_insert__Collect,axiom, % 8.50/4.50 ! [A: $tType,A2: A,P: A > $o] : % 8.50/4.50 ( ( insert @ A @ A2 @ ( collect @ A @ P ) ) % 8.50/4.50 = ( collect @ A % 8.50/4.50 @ ^ [U2: A] : % 8.50/4.50 ( ( U2 != A2 ) % 8.50/4.50 => ( P @ U2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2379_not__finite__existsD,axiom, % 8.50/4.50 ! [A: $tType,P: A > $o] : % 8.50/4.50 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) ) % 8.50/4.50 => ? [X_1: A] : ( P @ X_1 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2380_pigeonhole__infinite__rel,axiom, % 8.50/4.50 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,R: A > B > $o] : % 8.50/4.50 ( ~ ( finite_finite2 @ A @ A3 ) % 8.50/4.50 => ( ( finite_finite2 @ B @ B3 ) % 8.50/4.50 => ( ! [X5: A] : % 8.50/4.50 ( ( member @ A @ X5 @ A3 ) % 8.50/4.50 => ? [Xa: B] : % 8.50/4.50 ( ( member @ B @ Xa @ B3 ) % 8.50/4.50 & ( R @ X5 @ Xa ) ) ) % 8.50/4.50 => ? [X5: B] : % 8.50/4.50 ( ( member @ B @ X5 @ B3 ) % 8.50/4.50 & ~ ( finite_finite2 @ A % 8.50/4.50 @ ( collect @ A % 8.50/4.50 @ ^ [A4: A] : % 8.50/4.50 ( ( member @ A @ A4 @ A3 ) % 8.50/4.50 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2381_finite__image__set2,axiom, % 8.50/4.50 ! [A: $tType,B: $tType,C: $tType,P: A > $o,Q: B > $o,F2: A > B > C] : % 8.50/4.50 ( ( finite_finite2 @ A @ ( collect @ A @ P ) ) % 8.50/4.50 => ( ( finite_finite2 @ B @ ( collect @ B @ Q ) ) % 8.50/4.50 => ( finite_finite2 @ C % 8.50/4.50 @ ( collect @ C % 8.50/4.50 @ ^ [Uu3: C] : % 8.50/4.50 ? [X: A,Y2: B] : % 8.50/4.50 ( ( Uu3 % 8.50/4.50 = ( F2 @ X @ Y2 ) ) % 8.50/4.50 & ( P @ X ) % 8.50/4.50 & ( Q @ Y2 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2382_finite__image__set,axiom, % 8.50/4.50 ! [A: $tType,B: $tType,P: A > $o,F2: A > B] : % 8.50/4.50 ( ( finite_finite2 @ A @ ( collect @ A @ P ) ) % 8.50/4.50 => ( finite_finite2 @ B % 8.50/4.50 @ ( collect @ B % 8.50/4.50 @ ^ [Uu3: B] : % 8.50/4.50 ? [X: A] : % 8.50/4.50 ( ( Uu3 % 8.50/4.50 = ( F2 @ X ) ) % 8.50/4.50 & ( P @ X ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2383_bot__empty__eq2,axiom, % 8.50/4.50 ! [B: $tType,A: $tType] : % 8.50/4.50 ( ( bot_bot @ ( A > B > $o ) ) % 8.50/4.50 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2384_finite__M__bounded__by__nat,axiom, % 8.50/4.50 ! [P: nat > $o,I: nat] : % 8.50/4.50 ( finite_finite2 @ nat % 8.50/4.50 @ ( collect @ nat % 8.50/4.50 @ ^ [K3: nat] : % 8.50/4.50 ( ( P @ K3 ) % 8.50/4.50 & ( ord_less @ nat @ K3 @ I ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2385_finite__less__ub,axiom, % 8.50/4.50 ! [F2: nat > nat,U: nat] : % 8.50/4.50 ( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F2 @ N3 ) ) % 8.50/4.50 => ( finite_finite2 @ nat % 8.50/4.50 @ ( collect @ nat % 8.50/4.50 @ ^ [N: nat] : ( ord_less_eq @ nat @ ( F2 @ N ) @ U ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2386_numeral__code_I2_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( numeral @ A ) % 8.50/4.50 => ! [N2: num] : % 8.50/4.50 ( ( numeral_numeral @ A @ ( bit0 @ N2 ) ) % 8.50/4.50 = ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2387_set__vebt__def,axiom, % 8.50/4.50 ( vEBT_set_vebt % 8.50/4.50 = ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2388_finite__divisors__int,axiom, % 8.50/4.50 ! [I: int] : % 8.50/4.50 ( ( I % 8.50/4.50 != ( zero_zero @ int ) ) % 8.50/4.50 => ( finite_finite2 @ int % 8.50/4.50 @ ( collect @ int % 8.50/4.50 @ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ I ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2389_numeral__code_I3_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( numeral @ A ) % 8.50/4.50 => ! [N2: num] : % 8.50/4.50 ( ( numeral_numeral @ A @ ( bit1 @ N2 ) ) % 8.50/4.50 = ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( one_one @ A ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2390_finite__divisors__nat,axiom, % 8.50/4.50 ! [M: nat] : % 8.50/4.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.50/4.50 => ( finite_finite2 @ nat % 8.50/4.50 @ ( collect @ nat % 8.50/4.50 @ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2391_set__conv__nth,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( set2 @ A ) % 8.50/4.50 = ( ^ [Xs2: list @ A] : % 8.50/4.50 ( collect @ A % 8.50/4.50 @ ^ [Uu3: A] : % 8.50/4.50 ? [I3: nat] : % 8.50/4.50 ( ( Uu3 % 8.50/4.50 = ( nth @ A @ Xs2 @ I3 ) ) % 8.50/4.50 & ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2392_finite__lists__length__eq,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,N2: nat] : % 8.50/4.50 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.50 => ( finite_finite2 @ ( list @ A ) % 8.50/4.50 @ ( collect @ ( list @ A ) % 8.50/4.50 @ ^ [Xs2: list @ A] : % 8.50/4.50 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) % 8.50/4.50 & ( ( size_size @ ( list @ A ) @ Xs2 ) % 8.50/4.50 = N2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2393_finite__lists__length__le,axiom, % 8.50/4.50 ! [A: $tType,A3: set @ A,N2: nat] : % 8.50/4.50 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.50 => ( finite_finite2 @ ( list @ A ) % 8.50/4.50 @ ( collect @ ( list @ A ) % 8.50/4.50 @ ^ [Xs2: list @ A] : % 8.50/4.50 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) % 8.50/4.50 & ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2394_vebt__buildup_Osimps_I3_J,axiom, % 8.50/4.50 ! [Va: nat] : % 8.50/4.50 ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) ) % 8.50/4.50 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) ) % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) % 8.50/4.50 & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) ) % 8.50/4.50 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) ) % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2395_vebt__buildup_Oelims,axiom, % 8.50/4.50 ! [X3: nat,Y: vEBT_VEBT] : % 8.50/4.50 ( ( ( vEBT_vebt_buildup @ X3 ) % 8.50/4.50 = Y ) % 8.50/4.50 => ( ( ( X3 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => ( Y % 8.50/4.50 != ( vEBT_Leaf @ $false @ $false ) ) ) % 8.50/4.50 => ( ( ( X3 % 8.50/4.50 = ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.50 => ( Y % 8.50/4.50 != ( vEBT_Leaf @ $false @ $false ) ) ) % 8.50/4.50 => ~ ! [Va2: nat] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( suc @ ( suc @ Va2 ) ) ) % 8.50/4.50 => ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) ) % 8.50/4.50 => ( Y % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) % 8.50/4.50 & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) ) % 8.50/4.50 => ( Y % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2396_VEBT__internal_Onaive__member_Osimps_I3_J,axiom, % 8.50/4.50 ! [Uy: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList2: list @ vEBT_VEBT,S: vEBT_VEBT,X3: nat] : % 8.50/4.50 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList2 @ S ) @ X3 ) % 8.50/4.50 = ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) % 8.50/4.50 => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2397_dbl__dec__def,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( neg_numeral @ A ) % 8.50/4.50 => ( ( neg_numeral_dbl_dec @ A ) % 8.50/4.50 = ( ^ [X: A] : ( minus_minus @ A @ ( plus_plus @ A @ X @ X ) @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2398_VEBT__internal_Omembermima_Osimps_I5_J,axiom, % 8.50/4.50 ! [V2: nat,TreeList2: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] : % 8.50/4.50 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ X3 ) % 8.50/4.50 = ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) % 8.50/4.50 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2399_vebt__member_Osimps_I5_J,axiom, % 8.50/4.50 ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : % 8.50/4.50 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 ) % 8.50/4.50 = ( ( X3 != Mi ) % 8.50/4.50 => ( ( X3 != Ma ) % 8.50/4.50 => ( ~ ( ord_less @ nat @ X3 @ Mi ) % 8.50/4.50 & ( ~ ( ord_less @ nat @ X3 @ Mi ) % 8.50/4.50 => ( ~ ( ord_less @ nat @ Ma @ X3 ) % 8.50/4.50 & ( ~ ( ord_less @ nat @ Ma @ X3 ) % 8.50/4.50 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) % 8.50/4.50 => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2400_VEBT__internal_Omembermima_Osimps_I4_J,axiom, % 8.50/4.50 ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X3: nat] : % 8.50/4.50 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) @ X3 ) % 8.50/4.50 = ( ( X3 = Mi ) % 8.50/4.50 | ( X3 = Ma ) % 8.50/4.50 | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) % 8.50/4.50 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2401_VEBT__internal_Onaive__member_Oelims_I1_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 8.50/4.50 ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 ) % 8.50/4.50 = Y ) % 8.50/4.50 => ( ! [A5: $o,B5: $o] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.50/4.50 => ( Y % 8.50/4.50 = ( ~ ( ( ( Xa2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => A5 ) % 8.50/4.50 & ( ( Xa2 % 8.50/4.50 != ( zero_zero @ nat ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) % 8.50/4.50 => B5 ) % 8.50/4.50 & ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) ) ) ) ) ) ) % 8.50/4.50 => ( ( ? [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) ) % 8.50/4.50 => Y ) % 8.50/4.50 => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT] : % 8.50/4.50 ( ? [S2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) ) % 8.50/4.50 => ( Y % 8.50/4.50 = ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2402_VEBT__internal_Onaive__member_Oelims_I2_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.50 ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 ) % 8.50/4.50 => ( ! [A5: $o,B5: $o] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.50/4.50 => ~ ( ( ( Xa2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => A5 ) % 8.50/4.50 & ( ( Xa2 % 8.50/4.50 != ( zero_zero @ nat ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) % 8.50/4.50 => B5 ) % 8.50/4.50 & ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) ) ) ) ) % 8.50/4.50 => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT] : % 8.50/4.50 ( ? [S2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) ) % 8.50/4.50 => ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2403_VEBT__internal_Onaive__member_Oelims_I3_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.50 ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 ) % 8.50/4.50 => ( ! [A5: $o,B5: $o] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => A5 ) % 8.50/4.50 & ( ( Xa2 % 8.50/4.50 != ( zero_zero @ nat ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) % 8.50/4.50 => B5 ) % 8.50/4.50 & ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) ) ) ) ) % 8.50/4.50 => ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 != ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) ) % 8.50/4.50 => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT] : % 8.50/4.50 ( ? [S2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) ) % 8.50/4.50 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2404_VEBT__internal_Omembermima_Oelims_I2_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.50 ( ( vEBT_VEBT_membermima @ X3 @ Xa2 ) % 8.50/4.50 => ( ! [Mi2: nat,Ma2: nat] : % 8.50/4.50 ( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) ) % 8.50/4.50 => ~ ( ( Xa2 = Mi2 ) % 8.50/4.50 | ( Xa2 = Ma2 ) ) ) % 8.50/4.50 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT] : % 8.50/4.50 ( ? [Vc2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) ) % 8.50/4.50 => ~ ( ( Xa2 = Mi2 ) % 8.50/4.50 | ( Xa2 = Ma2 ) % 8.50/4.50 | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) % 8.50/4.50 => ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT] : % 8.50/4.50 ( ? [Vd: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) ) % 8.50/4.50 => ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2405_xor__nat__unfold,axiom, % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ nat ) % 8.50/4.50 = ( ^ [M2: nat,N: nat] : % 8.50/4.50 ( if @ nat % 8.50/4.50 @ ( M2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 @ N % 8.50/4.50 @ ( if @ nat % 8.50/4.50 @ ( N % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 @ M2 % 8.50/4.50 @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2406_xor__nat__rec,axiom, % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ nat ) % 8.50/4.50 = ( ^ [M2: nat,N: nat] : % 8.50/4.50 ( plus_plus @ nat % 8.50/4.50 @ ( zero_neq_one_of_bool @ nat % 8.50/4.50 @ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) % 8.50/4.50 != ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) % 8.50/4.50 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2407_one__xor__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A2 ) % 8.50/4.50 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) % 8.50/4.50 @ ( zero_neq_one_of_bool @ A % 8.50/4.50 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2408_xor__one__eq,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( one_one @ A ) ) % 8.50/4.50 = ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) % 8.50/4.50 @ ( zero_neq_one_of_bool @ A % 8.50/4.50 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2409_vebt__member_Oelims_I2_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.50 ( ( vEBT_vebt_member @ X3 @ Xa2 ) % 8.50/4.50 => ( ! [A5: $o,B5: $o] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.50/4.50 => ~ ( ( ( Xa2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => A5 ) % 8.50/4.50 & ( ( Xa2 % 8.50/4.50 != ( zero_zero @ nat ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) % 8.50/4.50 => B5 ) % 8.50/4.50 & ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) ) ) ) ) % 8.50/4.50 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT] : % 8.50/4.50 ( ? [Summary2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) % 8.50/4.50 => ~ ( ( Xa2 != Mi2 ) % 8.50/4.50 => ( ( Xa2 != Ma2 ) % 8.50/4.50 => ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 8.50/4.50 & ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 8.50/4.50 => ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 8.50/4.50 & ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 8.50/4.50 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2410_VEBT__internal_Omembermima_Oelims_I3_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.50 ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 ) % 8.50/4.50 => ( ! [Uu2: $o,Uv2: $o] : % 8.50/4.50 ( X3 % 8.50/4.50 != ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 8.50/4.50 => ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) ) % 8.50/4.50 => ( ! [Mi2: nat,Ma2: nat] : % 8.50/4.50 ( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) ) % 8.50/4.50 => ( ( Xa2 = Mi2 ) % 8.50/4.50 | ( Xa2 = Ma2 ) ) ) % 8.50/4.50 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT] : % 8.50/4.50 ( ? [Vc2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) ) % 8.50/4.50 => ( ( Xa2 = Mi2 ) % 8.50/4.50 | ( Xa2 = Ma2 ) % 8.50/4.50 | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) % 8.50/4.50 => ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT] : % 8.50/4.50 ( ? [Vd: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) ) % 8.50/4.50 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2411_VEBT__internal_Omembermima_Oelims_I1_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 8.50/4.50 ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 ) % 8.50/4.50 = Y ) % 8.50/4.50 => ( ( ? [Uu2: $o,Uv2: $o] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 8.50/4.50 => Y ) % 8.50/4.50 => ( ( ? [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) ) % 8.50/4.50 => Y ) % 8.50/4.50 => ( ! [Mi2: nat,Ma2: nat] : % 8.50/4.50 ( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) ) % 8.50/4.50 => ( Y % 8.50/4.50 = ( ~ ( ( Xa2 = Mi2 ) % 8.50/4.50 | ( Xa2 = Ma2 ) ) ) ) ) % 8.50/4.50 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT] : % 8.50/4.50 ( ? [Vc2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) ) % 8.50/4.50 => ( Y % 8.50/4.50 = ( ~ ( ( Xa2 = Mi2 ) % 8.50/4.50 | ( Xa2 = Ma2 ) % 8.50/4.50 | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) % 8.50/4.50 => ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT] : % 8.50/4.50 ( ? [Vd: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) ) % 8.50/4.50 => ( Y % 8.50/4.50 = ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2412_bit__horner__sum__bit__iff,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.50 => ! [Bs: list @ $o,N2: nat] : % 8.50/4.50 ( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) @ N2 ) % 8.50/4.50 = ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ $o ) @ Bs ) ) % 8.50/4.50 & ( nth @ $o @ Bs @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2413_vebt__member_Oelims_I3_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.50 ( ~ ( vEBT_vebt_member @ X3 @ Xa2 ) % 8.50/4.50 => ( ! [A5: $o,B5: $o] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => A5 ) % 8.50/4.50 & ( ( Xa2 % 8.50/4.50 != ( zero_zero @ nat ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) % 8.50/4.50 => B5 ) % 8.50/4.50 & ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) ) ) ) ) % 8.50/4.50 => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 != ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 8.50/4.50 => ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) ) % 8.50/4.50 => ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 != ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) ) % 8.50/4.50 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT] : % 8.50/4.50 ( ? [Summary2: vEBT_VEBT] : % 8.50/4.50 ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) % 8.50/4.50 => ( ( Xa2 != Mi2 ) % 8.50/4.50 => ( ( Xa2 != Ma2 ) % 8.50/4.50 => ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 8.50/4.50 & ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 8.50/4.50 => ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 8.50/4.50 & ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 8.50/4.50 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2414_of__int__code__if,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ring_1 @ A ) % 8.50/4.50 => ( ( ring_1_of_int @ A ) % 8.50/4.50 = ( ^ [K3: int] : % 8.50/4.50 ( if @ A % 8.50/4.50 @ ( K3 % 8.50/4.50 = ( zero_zero @ int ) ) % 8.50/4.50 @ ( zero_zero @ A ) % 8.50/4.50 @ ( if @ A @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ ( uminus_uminus @ int @ K3 ) ) ) % 8.50/4.50 @ ( if @ A % 8.50/4.50 @ ( ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) % 8.50/4.50 = ( zero_zero @ int ) ) % 8.50/4.50 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.50 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ A ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2415_finite__nth__roots,axiom, % 8.50/4.50 ! [N2: nat,C2: complex] : % 8.50/4.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.50 => ( finite_finite2 @ complex % 8.50/4.50 @ ( collect @ complex % 8.50/4.50 @ ^ [Z6: complex] : % 8.50/4.50 ( ( power_power @ complex @ Z6 @ N2 ) % 8.50/4.50 = C2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2416_finite__roots__unity,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.50 & ( idom @ A ) ) % 8.50/4.50 => ! [N2: nat] : % 8.50/4.50 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 ) % 8.50/4.50 => ( finite_finite2 @ A % 8.50/4.50 @ ( collect @ A % 8.50/4.50 @ ^ [Z6: A] : % 8.50/4.50 ( ( power_power @ A @ Z6 @ N2 ) % 8.50/4.50 = ( one_one @ A ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2417_prod_Ofinite__Collect__op,axiom, % 8.50/4.50 ! [A: $tType,B: $tType] : % 8.50/4.50 ( ( comm_monoid_mult @ A ) % 8.50/4.50 => ! [I6: set @ B,X3: B > A,Y: B > A] : % 8.50/4.50 ( ( finite_finite2 @ B % 8.50/4.50 @ ( collect @ B % 8.50/4.50 @ ^ [I3: B] : % 8.50/4.50 ( ( member @ B @ I3 @ I6 ) % 8.50/4.50 & ( ( X3 @ I3 ) % 8.50/4.50 != ( one_one @ A ) ) ) ) ) % 8.50/4.50 => ( ( finite_finite2 @ B % 8.50/4.50 @ ( collect @ B % 8.50/4.50 @ ^ [I3: B] : % 8.50/4.50 ( ( member @ B @ I3 @ I6 ) % 8.50/4.50 & ( ( Y @ I3 ) % 8.50/4.50 != ( one_one @ A ) ) ) ) ) % 8.50/4.50 => ( finite_finite2 @ B % 8.50/4.50 @ ( collect @ B % 8.50/4.50 @ ^ [I3: B] : % 8.50/4.50 ( ( member @ B @ I3 @ I6 ) % 8.50/4.50 & ( ( times_times @ A @ ( X3 @ I3 ) @ ( Y @ I3 ) ) % 8.50/4.50 != ( one_one @ A ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2418_vebt__member_Opelims_I1_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 8.50/4.50 ( ( ( vEBT_vebt_member @ X3 @ Xa2 ) % 8.50/4.50 = Y ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 8.50/4.50 => ( ! [A5: $o,B5: $o] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.50/4.50 => ( ( Y % 8.50/4.50 = ( ( ( Xa2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => A5 ) % 8.50/4.50 & ( ( Xa2 % 8.50/4.50 != ( zero_zero @ nat ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) % 8.50/4.50 => B5 ) % 8.50/4.50 & ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) ) ) ) ) % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) ) % 8.50/4.50 => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 8.50/4.50 => ( ~ Y % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) ) % 8.50/4.50 => ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) ) % 8.50/4.50 => ( ~ Y % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa2 ) ) ) ) % 8.50/4.50 => ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) ) % 8.50/4.50 => ( ~ Y % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) ) % 8.50/4.50 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) % 8.50/4.50 => ( ( Y % 8.50/4.50 = ( ( Xa2 != Mi2 ) % 8.50/4.50 => ( ( Xa2 != Ma2 ) % 8.50/4.50 => ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 8.50/4.50 & ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 8.50/4.50 => ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 8.50/4.50 & ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 8.50/4.50 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2419_vebt__member_Opelims_I3_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.50 ( ~ ( vEBT_vebt_member @ X3 @ Xa2 ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 8.50/4.50 => ( ! [A5: $o,B5: $o] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => A5 ) % 8.50/4.50 & ( ( Xa2 % 8.50/4.50 != ( zero_zero @ nat ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) % 8.50/4.50 => B5 ) % 8.50/4.50 & ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) ) ) ) ) ) % 8.50/4.50 => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) % 8.50/4.50 => ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) ) % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa2 ) ) ) % 8.50/4.50 => ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) ) % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) % 8.50/4.50 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) % 8.50/4.50 => ( ( Xa2 != Mi2 ) % 8.50/4.50 => ( ( Xa2 != Ma2 ) % 8.50/4.50 => ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 8.50/4.50 & ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 8.50/4.50 => ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 8.50/4.50 & ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 8.50/4.50 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2420_xor__negative__int__iff,axiom, % 8.50/4.50 ! [K: int,L: int] : % 8.50/4.50 ( ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) @ ( zero_zero @ int ) ) % 8.50/4.50 = ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 8.50/4.50 != ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2421_bot2E,axiom, % 8.50/4.50 ! [A: $tType,B: $tType,X3: A,Y: B] : % 8.50/4.50 ~ ( bot_bot @ ( A > B > $o ) @ X3 @ Y ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2422_XOR__upper,axiom, % 8.50/4.50 ! [X3: int,N2: nat,Y: int] : % 8.50/4.50 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 ) % 8.50/4.50 => ( ( ord_less @ int @ X3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.50/4.50 => ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.50/4.50 => ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X3 @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2423_vebt__member_Opelims_I2_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.50 ( ( vEBT_vebt_member @ X3 @ Xa2 ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 8.50/4.50 => ( ! [A5: $o,B5: $o] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) % 8.50/4.50 => ~ ( ( ( Xa2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => A5 ) % 8.50/4.50 & ( ( Xa2 % 8.50/4.50 != ( zero_zero @ nat ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) % 8.50/4.50 => B5 ) % 8.50/4.50 & ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) ) ) ) ) ) % 8.50/4.50 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) % 8.50/4.50 => ~ ( ( Xa2 != Mi2 ) % 8.50/4.50 => ( ( Xa2 != Ma2 ) % 8.50/4.50 => ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 8.50/4.50 & ( ~ ( ord_less @ nat @ Xa2 @ Mi2 ) % 8.50/4.50 => ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 8.50/4.50 & ( ~ ( ord_less @ nat @ Ma2 @ Xa2 ) % 8.50/4.50 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2424_sum_Ofinite__Collect__op,axiom, % 8.50/4.50 ! [A: $tType,B: $tType] : % 8.50/4.50 ( ( comm_monoid_add @ A ) % 8.50/4.50 => ! [I6: set @ B,X3: B > A,Y: B > A] : % 8.50/4.50 ( ( finite_finite2 @ B % 8.50/4.50 @ ( collect @ B % 8.50/4.50 @ ^ [I3: B] : % 8.50/4.50 ( ( member @ B @ I3 @ I6 ) % 8.50/4.50 & ( ( X3 @ I3 ) % 8.50/4.50 != ( zero_zero @ A ) ) ) ) ) % 8.50/4.50 => ( ( finite_finite2 @ B % 8.50/4.50 @ ( collect @ B % 8.50/4.50 @ ^ [I3: B] : % 8.50/4.50 ( ( member @ B @ I3 @ I6 ) % 8.50/4.50 & ( ( Y @ I3 ) % 8.50/4.50 != ( zero_zero @ A ) ) ) ) ) % 8.50/4.50 => ( finite_finite2 @ B % 8.50/4.50 @ ( collect @ B % 8.50/4.50 @ ^ [I3: B] : % 8.50/4.50 ( ( member @ B @ I3 @ I6 ) % 8.50/4.50 & ( ( plus_plus @ A @ ( X3 @ I3 ) @ ( Y @ I3 ) ) % 8.50/4.50 != ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2425_VEBT__internal_Onaive__member_Opelims_I1_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 8.50/4.50 ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 ) % 8.50/4.50 = Y ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 8.50/4.50 => ( ! [A5: $o,B5: $o] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.50/4.50 => ( ( Y % 8.50/4.50 = ( ( ( Xa2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => A5 ) % 8.50/4.50 & ( ( Xa2 % 8.50/4.50 != ( zero_zero @ nat ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) % 8.50/4.50 => B5 ) % 8.50/4.50 & ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) ) ) ) ) % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) ) % 8.50/4.50 => ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) ) % 8.50/4.50 => ( ~ Y % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa2 ) ) ) ) % 8.50/4.50 => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT,S2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) ) % 8.50/4.50 => ( ( Y % 8.50/4.50 = ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2426_VEBT__internal_Onaive__member_Opelims_I2_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.50 ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 8.50/4.50 => ( ! [A5: $o,B5: $o] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) % 8.50/4.50 => ~ ( ( ( Xa2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => A5 ) % 8.50/4.50 & ( ( Xa2 % 8.50/4.50 != ( zero_zero @ nat ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) % 8.50/4.50 => B5 ) % 8.50/4.50 & ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) ) ) ) ) ) % 8.50/4.50 => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT,S2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) @ Xa2 ) ) % 8.50/4.50 => ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2427_VEBT__internal_Onaive__member_Opelims_I3_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.50 ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 8.50/4.50 => ( ! [A5: $o,B5: $o] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => A5 ) % 8.50/4.50 & ( ( Xa2 % 8.50/4.50 != ( zero_zero @ nat ) ) % 8.50/4.50 => ( ( ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) % 8.50/4.50 => B5 ) % 8.50/4.50 & ( Xa2 % 8.50/4.50 = ( one_one @ nat ) ) ) ) ) ) ) % 8.50/4.50 => ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) ) % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa2 ) ) ) % 8.50/4.50 => ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList3: list @ vEBT_VEBT,S2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList3 @ S2 ) @ Xa2 ) ) % 8.50/4.50 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2428_VEBT__internal_Omembermima_Opelims_I3_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.50 ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 8.50/4.50 => ( ! [Uu2: $o,Uv2: $o] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) % 8.50/4.50 => ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) ) % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa2 ) ) ) % 8.50/4.50 => ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa2 ) ) % 8.50/4.50 => ( ( Xa2 = Mi2 ) % 8.50/4.50 | ( Xa2 = Ma2 ) ) ) ) % 8.50/4.50 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) % 8.50/4.50 => ( ( Xa2 = Mi2 ) % 8.50/4.50 | ( Xa2 = Ma2 ) % 8.50/4.50 | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) % 8.50/4.50 => ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa2 ) ) % 8.50/4.50 => ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2429_VEBT__internal_Omembermima_Opelims_I1_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 8.50/4.50 ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 ) % 8.50/4.50 = Y ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 8.50/4.50 => ( ! [Uu2: $o,Uv2: $o] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 8.50/4.50 => ( ~ Y % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) ) % 8.50/4.50 => ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) ) % 8.50/4.50 => ( ~ Y % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Xa2 ) ) ) ) % 8.50/4.50 => ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) ) % 8.50/4.50 => ( ( Y % 8.50/4.50 = ( ( Xa2 = Mi2 ) % 8.50/4.50 | ( Xa2 = Ma2 ) ) ) % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa2 ) ) ) ) % 8.50/4.50 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) ) % 8.50/4.50 => ( ( Y % 8.50/4.50 = ( ( Xa2 = Mi2 ) % 8.50/4.50 | ( Xa2 = Ma2 ) % 8.50/4.50 | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) ) % 8.50/4.50 => ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) ) % 8.50/4.50 => ( ( Y % 8.50/4.50 = ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) % 8.50/4.50 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2430_VEBT__internal_Omembermima_Opelims_I2_J,axiom, % 8.50/4.50 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.50 ( ( vEBT_VEBT_membermima @ X3 @ Xa2 ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 8.50/4.50 => ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa2 ) ) % 8.50/4.50 => ~ ( ( Xa2 = Mi2 ) % 8.50/4.50 | ( Xa2 = Ma2 ) ) ) ) % 8.50/4.50 => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList3: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) % 8.50/4.50 => ~ ( ( Xa2 = Mi2 ) % 8.50/4.50 | ( Xa2 = Ma2 ) % 8.50/4.50 | ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) % 8.50/4.50 => ~ ! [V3: nat,TreeList3: list @ vEBT_VEBT,Vd: vEBT_VEBT] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) ) % 8.50/4.50 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList3 @ Vd ) @ Xa2 ) ) % 8.50/4.50 => ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) % 8.50/4.50 => ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.50 & ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2431_arcosh__1,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ln @ A ) % 8.50/4.50 => ( ( arcosh @ A @ ( one_one @ A ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2432_arsinh__0,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ln @ A ) % 8.50/4.50 => ( ( arsinh @ A @ ( zero_zero @ A ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2433_artanh__0,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ( real_V3459762299906320749_field @ A ) % 8.50/4.50 & ( ln @ A ) ) % 8.50/4.50 => ( ( artanh @ A @ ( zero_zero @ A ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2434_vebt__buildup_Opelims,axiom, % 8.50/4.50 ! [X3: nat,Y: vEBT_VEBT] : % 8.50/4.50 ( ( ( vEBT_vebt_buildup @ X3 ) % 8.50/4.50 = Y ) % 8.50/4.50 => ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X3 ) % 8.50/4.50 => ( ( ( X3 % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 => ( ( Y % 8.50/4.50 = ( vEBT_Leaf @ $false @ $false ) ) % 8.50/4.50 => ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) ) % 8.50/4.50 => ( ( ( X3 % 8.50/4.50 = ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.50 => ( ( Y % 8.50/4.50 = ( vEBT_Leaf @ $false @ $false ) ) % 8.50/4.50 => ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) ) % 8.50/4.50 => ~ ! [Va2: nat] : % 8.50/4.50 ( ( X3 % 8.50/4.50 = ( suc @ ( suc @ Va2 ) ) ) % 8.50/4.50 => ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) ) % 8.50/4.50 => ( Y % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) % 8.50/4.50 & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) ) % 8.50/4.50 => ( Y % 8.50/4.50 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) % 8.50/4.50 => ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2435_divmod__algorithm__code_I6_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( unique1627219031080169319umeral @ A ) % 8.50/4.50 => ! [M: num,N2: num] : % 8.50/4.50 ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ N2 ) ) % 8.50/4.50 = ( product_case_prod @ A @ A @ ( product_prod @ A @ A ) % 8.50/4.50 @ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) ) % 8.50/4.50 @ ( unique8689654367752047608divmod @ A @ M @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2436_artanh__def,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ( real_V3459762299906320749_field @ A ) % 8.50/4.50 & ( ln @ A ) ) % 8.50/4.50 => ( ( artanh @ A ) % 8.50/4.50 = ( ^ [X: A] : ( divide_divide @ A @ ( ln_ln @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2437_ln__inj__iff,axiom, % 8.50/4.50 ! [X3: real,Y: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.50 => ( ( ( ln_ln @ real @ X3 ) % 8.50/4.50 = ( ln_ln @ real @ Y ) ) % 8.50/4.50 = ( X3 = Y ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2438_ln__less__cancel__iff,axiom, % 8.50/4.50 ! [X3: real,Y: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.50 => ( ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y ) ) % 8.50/4.50 = ( ord_less @ real @ X3 @ Y ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2439_case__prod__conv,axiom, % 8.50/4.50 ! [B: $tType,A: $tType,C: $tType,F2: B > C > A,A2: B,B2: C] : % 8.50/4.50 ( ( product_case_prod @ B @ C @ A @ F2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) % 8.50/4.50 = ( F2 @ A2 @ B2 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2440_ln__le__cancel__iff,axiom, % 8.50/4.50 ! [X3: real,Y: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.50 => ( ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y ) ) % 8.50/4.50 = ( ord_less_eq @ real @ X3 @ Y ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2441_ln__less__zero__iff,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ ( zero_zero @ real ) ) % 8.50/4.50 = ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2442_ln__gt__zero__iff,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) ) % 8.50/4.50 = ( ord_less @ real @ ( one_one @ real ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2443_ln__eq__zero__iff,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ( ( ln_ln @ real @ X3 ) % 8.50/4.50 = ( zero_zero @ real ) ) % 8.50/4.50 = ( X3 % 8.50/4.50 = ( one_one @ real ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2444_ln__one,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ln @ A ) % 8.50/4.50 => ( ( ln_ln @ A @ ( one_one @ A ) ) % 8.50/4.50 = ( zero_zero @ A ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2445_ln__le__zero__iff,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( zero_zero @ real ) ) % 8.50/4.50 = ( ord_less_eq @ real @ X3 @ ( one_one @ real ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2446_ln__ge__zero__iff,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) ) % 8.50/4.50 = ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2447_divmod__algorithm__code_I5_J,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( unique1627219031080169319umeral @ A ) % 8.50/4.50 => ! [M: num,N2: num] : % 8.50/4.50 ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ N2 ) ) % 8.50/4.50 = ( product_case_prod @ A @ A @ ( product_prod @ A @ A ) % 8.50/4.50 @ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) ) % 8.50/4.50 @ ( unique8689654367752047608divmod @ A @ M @ N2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2448_prod_Ocase__distrib,axiom, % 8.50/4.50 ! [C: $tType,D: $tType,B: $tType,A: $tType,H2: C > D,F2: A > B > C,Prod: product_prod @ A @ B] : % 8.50/4.50 ( ( H2 @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) ) % 8.50/4.50 = ( product_case_prod @ A @ B @ D % 8.50/4.50 @ ^ [X15: A,X24: B] : ( H2 @ ( F2 @ X15 @ X24 ) ) % 8.50/4.50 @ Prod ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2449_old_Oprod_Ocase,axiom, % 8.50/4.50 ! [A: $tType,C: $tType,B: $tType,F2: A > B > C,X1: A,X22: B] : % 8.50/4.50 ( ( product_case_prod @ A @ B @ C @ F2 @ ( product_Pair @ A @ B @ X1 @ X22 ) ) % 8.50/4.50 = ( F2 @ X1 @ X22 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2450_split__cong,axiom, % 8.50/4.50 ! [C: $tType,B: $tType,A: $tType,Q4: product_prod @ A @ B,F2: A > B > C,G: A > B > C,P4: product_prod @ A @ B] : % 8.50/4.50 ( ! [X5: A,Y4: B] : % 8.50/4.50 ( ( ( product_Pair @ A @ B @ X5 @ Y4 ) % 8.50/4.50 = Q4 ) % 8.50/4.50 => ( ( F2 @ X5 @ Y4 ) % 8.50/4.50 = ( G @ X5 @ Y4 ) ) ) % 8.50/4.50 => ( ( P4 = Q4 ) % 8.50/4.50 => ( ( product_case_prod @ A @ B @ C @ F2 @ P4 ) % 8.50/4.50 = ( product_case_prod @ A @ B @ C @ G @ Q4 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2451_cond__case__prod__eta,axiom, % 8.50/4.50 ! [C: $tType,B: $tType,A: $tType,F2: A > B > C,G: ( product_prod @ A @ B ) > C] : % 8.50/4.50 ( ! [X5: A,Y4: B] : % 8.50/4.50 ( ( F2 @ X5 @ Y4 ) % 8.50/4.50 = ( G @ ( product_Pair @ A @ B @ X5 @ Y4 ) ) ) % 8.50/4.50 => ( ( product_case_prod @ A @ B @ C @ F2 ) % 8.50/4.50 = G ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2452_case__prod__eta,axiom, % 8.50/4.50 ! [C: $tType,B: $tType,A: $tType,F2: ( product_prod @ A @ B ) > C] : % 8.50/4.50 ( ( product_case_prod @ A @ B @ C % 8.50/4.50 @ ^ [X: A,Y2: B] : ( F2 @ ( product_Pair @ A @ B @ X @ Y2 ) ) ) % 8.50/4.50 = F2 ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2453_case__prodE2,axiom, % 8.50/4.50 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z2: product_prod @ B @ C] : % 8.50/4.50 ( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) ) % 8.50/4.50 => ~ ! [X5: B,Y4: C] : % 8.50/4.50 ( ( Z2 % 8.50/4.50 = ( product_Pair @ B @ C @ X5 @ Y4 ) ) % 8.50/4.50 => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2454_ln__less__self,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ X3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2455_ln__bound,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ X3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2456_ln__gt__zero__imp__gt__one,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) ) % 8.50/4.50 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ord_less @ real @ ( one_one @ real ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2457_ln__less__zero,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 8.50/4.50 => ( ord_less @ real @ ( ln_ln @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2458_ln__gt__zero,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 8.50/4.50 => ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2459_ln__div,axiom, % 8.50/4.50 ! [X3: real,Y: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.50 => ( ( ln_ln @ real @ ( divide_divide @ real @ X3 @ Y ) ) % 8.50/4.50 = ( minus_minus @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2460_ln__2__less__1,axiom, % 8.50/4.50 ord_less @ real @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2461_ln__ge__zero__imp__ge__one,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X3 ) ) % 8.50/4.50 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2462_ln__diff__le,axiom, % 8.50/4.50 ! [X3: real,Y: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.50 => ( ord_less_eq @ real @ ( minus_minus @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ X3 @ Y ) @ Y ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2463_ln__mult,axiom, % 8.50/4.50 ! [X3: real,Y: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.50 => ( ( ln_ln @ real @ ( times_times @ real @ X3 @ Y ) ) % 8.50/4.50 = ( plus_plus @ real @ ( ln_ln @ real @ X3 ) @ ( ln_ln @ real @ Y ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2464_ln__eq__minus__one,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ( ( ln_ln @ real @ X3 ) % 8.50/4.50 = ( minus_minus @ real @ X3 @ ( one_one @ real ) ) ) % 8.50/4.50 => ( X3 % 8.50/4.50 = ( one_one @ real ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2465_ln__le__minus__one,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( minus_minus @ real @ X3 @ ( one_one @ real ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2466_ln__add__one__self__le__self2,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 8.50/4.50 => ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X3 ) ) @ X3 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2467_ln__one__minus__pos__upper__bound,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 8.50/4.50 => ( ord_less_eq @ real @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X3 ) ) @ ( uminus_uminus @ real @ X3 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2468_divmod__step__nat__def,axiom, % 8.50/4.50 ( ( unique1321980374590559556d_step @ nat ) % 8.50/4.50 = ( ^ [L2: num] : % 8.50/4.50 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 8.50/4.50 @ ^ [Q5: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2469_divmod__step__int__def,axiom, % 8.50/4.50 ( ( unique1321980374590559556d_step @ int ) % 8.50/4.50 = ( ^ [L2: num] : % 8.50/4.50 ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 8.50/4.50 @ ^ [Q5: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L2 ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2470_divmod__step__def,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( unique1627219031080169319umeral @ A ) % 8.50/4.50 => ( ( unique1321980374590559556d_step @ A ) % 8.50/4.50 = ( ^ [L2: num] : % 8.50/4.50 ( product_case_prod @ A @ A @ ( product_prod @ A @ A ) % 8.50/4.50 @ ^ [Q5: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2471_divmod__nat__if,axiom, % 8.50/4.50 ( divmod_nat % 8.50/4.50 = ( ^ [M2: nat,N: nat] : % 8.50/4.50 ( if @ ( product_prod @ nat @ nat ) % 8.50/4.50 @ ( ( N % 8.50/4.50 = ( zero_zero @ nat ) ) % 8.50/4.50 | ( ord_less @ nat @ M2 @ N ) ) % 8.50/4.50 @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M2 ) % 8.50/4.50 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 8.50/4.50 @ ^ [Q5: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q5 ) ) % 8.50/4.50 @ ( divmod_nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2472_tanh__ln__real,axiom, % 8.50/4.50 ! [X3: real] : % 8.50/4.50 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.50 => ( ( tanh @ real @ ( ln_ln @ real @ X3 ) ) % 8.50/4.50 = ( divide_divide @ real @ ( minus_minus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2473_Sum__Icc__nat,axiom, % 8.50/4.50 ! [M: nat,N2: nat] : % 8.50/4.50 ( ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.50 @ ^ [X: nat] : X % 8.50/4.50 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.50 = ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2474_arith__series__nat,axiom, % 8.50/4.50 ! [A2: nat,D2: nat,N2: nat] : % 8.50/4.50 ( ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.50 @ ^ [I3: nat] : ( plus_plus @ nat @ A2 @ ( times_times @ nat @ I3 @ D2 ) ) % 8.50/4.50 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.50/4.50 = ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ nat @ N2 @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2475_case__prod__Pair__iden,axiom, % 8.50/4.50 ! [B: $tType,A: $tType,P4: product_prod @ A @ B] : % 8.50/4.50 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P4 ) % 8.50/4.50 = P4 ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2476_round__unique_H,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.50 => ! [X3: A,N2: int] : % 8.50/4.50 ( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ ( ring_1_of_int @ A @ N2 ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) % 8.50/4.50 => ( ( archimedean_round @ A @ X3 ) % 8.50/4.50 = N2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2477_abs__abs,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( idom_abs_sgn @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) ) % 8.50/4.50 = ( abs_abs @ A @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2478_abs__idempotent,axiom, % 8.50/4.50 ! [A: $tType] : % 8.50/4.50 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.50 => ! [A2: A] : % 8.50/4.50 ( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) ) % 8.50/4.50 = ( abs_abs @ A @ A2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2479_case__prodI,axiom, % 8.50/4.50 ! [A: $tType,B: $tType,F2: A > B > $o,A2: A,B2: B] : % 8.50/4.50 ( ( F2 @ A2 @ B2 ) % 8.50/4.50 => ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2480_case__prodI2,axiom, % 8.50/4.50 ! [B: $tType,A: $tType,P4: product_prod @ A @ B,C2: A > B > $o] : % 8.50/4.50 ( ! [A5: A,B5: B] : % 8.50/4.50 ( ( P4 % 8.50/4.50 = ( product_Pair @ A @ B @ A5 @ B5 ) ) % 8.50/4.50 => ( C2 @ A5 @ B5 ) ) % 8.50/4.50 => ( product_case_prod @ A @ B @ $o @ C2 @ P4 ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2481_mem__case__prodI,axiom, % 8.50/4.50 ! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A2: B,B2: C] : % 8.50/4.50 ( ( member @ A @ Z2 @ ( C2 @ A2 @ B2 ) ) % 8.50/4.50 => ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2482_mem__case__prodI2,axiom, % 8.50/4.50 ! [C: $tType,B: $tType,A: $tType,P4: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] : % 8.50/4.50 ( ! [A5: A,B5: B] : % 8.50/4.50 ( ( P4 % 8.50/4.50 = ( product_Pair @ A @ B @ A5 @ B5 ) ) % 8.50/4.50 => ( member @ C @ Z2 @ ( C2 @ A5 @ B5 ) ) ) % 8.50/4.50 => ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P4 ) ) ) ). % 8.50/4.50 % 8.50/4.50 thf(fact_2483_case__prodI2_H,axiom, % 8.50/4.50 ! [A: $tType,B: $tType,C: $tType,P4: product_prod @ A @ B,C2: A > B > C > $o,X3: C] : % 8.50/4.50 ( ! [A5: A,B5: B] : % 8.50/4.50 ( ( ( product_Pair @ A @ B @ A5 @ B5 ) % 8.50/4.50 = P4 ) % 8.50/4.50 => ( C2 @ A5 @ B5 @ X3 ) ) % 8.50/4.50 => ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ X3 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2484_abs__0__eq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( ( zero_zero @ A ) % 8.50/4.51 = ( abs_abs @ A @ A2 ) ) % 8.50/4.51 = ( A2 % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2485_abs__eq__0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( ( abs_abs @ A @ A2 ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 = ( A2 % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2486_abs__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ( ( abs_abs @ A @ ( zero_zero @ A ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2487_abs__0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( idom_abs_sgn @ A ) % 8.50/4.51 => ( ( abs_abs @ A @ ( zero_zero @ A ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2488_abs__mult__self__eq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ A2 ) ) % 8.50/4.51 = ( times_times @ A @ A2 @ A2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2489_abs__1,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( idom_abs_sgn @ A ) % 8.50/4.51 => ( ( abs_abs @ A @ ( one_one @ A ) ) % 8.50/4.51 = ( one_one @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2490_abs__add__abs,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : % 8.50/4.51 ( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) % 8.50/4.51 = ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2491_abs__divide,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_abs_sgn @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : % 8.50/4.51 ( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 8.50/4.51 = ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2492_abs__minus,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( idom_abs_sgn @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) ) % 8.50/4.51 = ( abs_abs @ A @ A2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2493_abs__minus__cancel,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) ) % 8.50/4.51 = ( abs_abs @ A @ A2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2494_dvd__abs__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [M: A,K: A] : % 8.50/4.51 ( ( dvd_dvd @ A @ M @ ( abs_abs @ A @ K ) ) % 8.50/4.51 = ( dvd_dvd @ A @ M @ K ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2495_abs__dvd__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [M: A,K: A] : % 8.50/4.51 ( ( dvd_dvd @ A @ ( abs_abs @ A @ M ) @ K ) % 8.50/4.51 = ( dvd_dvd @ A @ M @ K ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2496_of__int__abs,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [X3: int] : % 8.50/4.51 ( ( ring_1_of_int @ A @ ( abs_abs @ int @ X3 ) ) % 8.50/4.51 = ( abs_abs @ A @ ( ring_1_of_int @ A @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2497_abs__bool__eq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [P: $o] : % 8.50/4.51 ( ( abs_abs @ A @ ( zero_neq_one_of_bool @ A @ P ) ) % 8.50/4.51 = ( zero_neq_one_of_bool @ A @ P ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2498_zdvd1__eq,axiom, % 8.50/4.51 ! [X3: int] : % 8.50/4.51 ( ( dvd_dvd @ int @ X3 @ ( one_one @ int ) ) % 8.50/4.51 = ( ( abs_abs @ int @ X3 ) % 8.50/4.51 = ( one_one @ int ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2499_tanh__0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ( ( tanh @ A @ ( zero_zero @ A ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2500_tanh__real__less__iff,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( tanh @ real @ X3 ) @ ( tanh @ real @ Y ) ) % 8.50/4.51 = ( ord_less @ real @ X3 @ Y ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2501_sum_Oneutral__const,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [A3: set @ B] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.51 @ ^ [Uu3: B] : ( zero_zero @ A ) % 8.50/4.51 @ A3 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2502_of__int__sum,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( ring_1 @ A ) % 8.50/4.51 => ! [F2: B > int,A3: set @ B] : % 8.50/4.51 ( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F2 @ A3 ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.51 @ ^ [X: B] : ( ring_1_of_int @ A @ ( F2 @ X ) ) % 8.50/4.51 @ A3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2503_abs__of__nonneg,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.51 => ( ( abs_abs @ A @ A2 ) % 8.50/4.51 = A2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2504_abs__le__self__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 ) % 8.50/4.51 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2505_abs__le__zero__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) ) % 8.50/4.51 = ( A2 % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2506_zero__less__abs__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) ) % 8.50/4.51 = ( A2 % 8.50/4.51 != ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2507_sum_Oempty,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [G: B > A] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2508_sum_Oinfinite,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [A3: set @ B,G: B > A] : % 8.50/4.51 ( ~ ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2509_sum__eq__0__iff,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( canoni5634975068530333245id_add @ A ) % 8.50/4.51 => ! [F5: set @ B,F2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ F5 ) % 8.50/4.51 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ F5 ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 = ( ! [X: B] : % 8.50/4.51 ( ( member @ B @ X @ F5 ) % 8.50/4.51 => ( ( F2 @ X ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2510_artanh__minus__real,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 8.50/4.51 => ( ( artanh @ real @ ( uminus_uminus @ real @ X3 ) ) % 8.50/4.51 = ( uminus_uminus @ real @ ( artanh @ real @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2511_zabs__less__one__iff,axiom, % 8.50/4.51 ! [Z2: int] : % 8.50/4.51 ( ( ord_less @ int @ ( abs_abs @ int @ Z2 ) @ ( one_one @ int ) ) % 8.50/4.51 = ( Z2 % 8.50/4.51 = ( zero_zero @ int ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2512_tanh__real__pos__iff,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X3 ) ) % 8.50/4.51 = ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2513_tanh__real__neg__iff,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( tanh @ real @ X3 ) @ ( zero_zero @ real ) ) % 8.50/4.51 = ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2514_sum_Odelta,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [S3: set @ B,A2: B,B2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ S3 ) % 8.50/4.51 => ( ( ( member @ B @ A2 @ S3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.51 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) ) % 8.50/4.51 @ S3 ) % 8.50/4.51 = ( B2 @ A2 ) ) ) % 8.50/4.51 & ( ~ ( member @ B @ A2 @ S3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.51 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) ) % 8.50/4.51 @ S3 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2515_sum_Odelta_H,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [S3: set @ B,A2: B,B2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ S3 ) % 8.50/4.51 => ( ( ( member @ B @ A2 @ S3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.51 @ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) ) % 8.50/4.51 @ S3 ) % 8.50/4.51 = ( B2 @ A2 ) ) ) % 8.50/4.51 & ( ~ ( member @ B @ A2 @ S3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.51 @ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) ) % 8.50/4.51 @ S3 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2516_sum__abs,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ B ) % 8.50/4.51 => ! [F2: A > B,A3: set @ A] : % 8.50/4.51 ( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) ) % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ A @ B % 8.50/4.51 @ ^ [I3: A] : ( abs_abs @ B @ ( F2 @ I3 ) ) % 8.50/4.51 @ A3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2517_divide__le__0__abs__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_field @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) ) % 8.50/4.51 = ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 8.50/4.51 | ( B2 % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2518_zero__le__divide__abs__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_field @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) ) % 8.50/4.51 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.51 | ( B2 % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2519_abs__of__nonpos,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) % 8.50/4.51 => ( ( abs_abs @ A @ A2 ) % 8.50/4.51 = ( uminus_uminus @ A @ A2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2520_sum_Oinsert,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [A3: set @ B,X3: B,G: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ~ ( member @ B @ X3 @ A3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 8.50/4.51 = ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2521_sum__abs__ge__zero,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ B ) % 8.50/4.51 => ! [F2: A > B,A3: set @ A] : % 8.50/4.51 ( ord_less_eq @ B @ ( zero_zero @ B ) % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ A @ B % 8.50/4.51 @ ^ [I3: A] : ( abs_abs @ B @ ( F2 @ I3 ) ) % 8.50/4.51 @ A3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2522_zero__less__power__abs__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [A2: A,N2: nat] : % 8.50/4.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N2 ) ) % 8.50/4.51 = ( ( A2 % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 | ( N2 % 8.50/4.51 = ( zero_zero @ nat ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2523_sum_Ocl__ivl__Suc,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [N2: nat,M: nat,G: nat > A] : % 8.50/4.51 ( ( ( ord_less @ nat @ ( suc @ N2 ) @ M ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 & ( ~ ( ord_less @ nat @ ( suc @ N2 ) @ M ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2524_sum__zero__power,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( division_ring @ A ) % 8.50/4.51 => ! [A3: set @ nat,C2: nat > A] : % 8.50/4.51 ( ( ( ( finite_finite2 @ nat @ A3 ) % 8.50/4.51 & ( member @ nat @ ( zero_zero @ nat ) @ A3 ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) % 8.50/4.51 @ A3 ) % 8.50/4.51 = ( C2 @ ( zero_zero @ nat ) ) ) ) % 8.50/4.51 & ( ~ ( ( finite_finite2 @ nat @ A3 ) % 8.50/4.51 & ( member @ nat @ ( zero_zero @ nat ) @ A3 ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) % 8.50/4.51 @ A3 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2525_sum__zero__power_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field @ A ) % 8.50/4.51 => ! [A3: set @ nat,C2: nat > A,D2: nat > A] : % 8.50/4.51 ( ( ( ( finite_finite2 @ nat @ A3 ) % 8.50/4.51 & ( member @ nat @ ( zero_zero @ nat ) @ A3 ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D2 @ I3 ) ) % 8.50/4.51 @ A3 ) % 8.50/4.51 = ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) ) % 8.50/4.51 & ( ~ ( ( finite_finite2 @ nat @ A3 ) % 8.50/4.51 & ( member @ nat @ ( zero_zero @ nat ) @ A3 ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I3 ) ) @ ( D2 @ I3 ) ) % 8.50/4.51 @ A3 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2526_sum_Onot__neutral__contains__not__neutral,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [G: B > A,A3: set @ B] : % 8.50/4.51 ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ~ ! [A5: B] : % 8.50/4.51 ( ( member @ B @ A5 @ A3 ) % 8.50/4.51 => ( ( G @ A5 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2527_sum_Oneutral,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [A3: set @ B,G: B > A] : % 8.50/4.51 ( ! [X5: B] : % 8.50/4.51 ( ( member @ B @ X5 @ A3 ) % 8.50/4.51 => ( ( G @ X5 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2528_abs__le__D1,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) % 8.50/4.51 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2529_abs__ge__self,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( abs_abs @ A @ A2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2530_abs__eq__0__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( idom_abs_sgn @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( ( abs_abs @ A @ A2 ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 = ( A2 % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2531_abs__mult,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( idom_abs_sgn @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : % 8.50/4.51 ( ( abs_abs @ A @ ( times_times @ A @ A2 @ B2 ) ) % 8.50/4.51 = ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2532_abs__one,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ( ( abs_abs @ A @ ( one_one @ A ) ) % 8.50/4.51 = ( one_one @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2533_abs__minus__commute,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : % 8.50/4.51 ( ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) % 8.50/4.51 = ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2534_abs__eq__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_ring @ A ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( ( abs_abs @ A @ X3 ) % 8.50/4.51 = ( abs_abs @ A @ Y ) ) % 8.50/4.51 = ( ( X3 = Y ) % 8.50/4.51 | ( X3 % 8.50/4.51 = ( uminus_uminus @ A @ Y ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2535_dvd__if__abs__eq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [L: A,K: A] : % 8.50/4.51 ( ( ( abs_abs @ A @ L ) % 8.50/4.51 = ( abs_abs @ A @ K ) ) % 8.50/4.51 => ( dvd_dvd @ A @ L @ K ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2536_mem__case__prodE,axiom, % 8.50/4.51 ! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P4: product_prod @ B @ C] : % 8.50/4.51 ( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P4 ) ) % 8.50/4.51 => ~ ! [X5: B,Y4: C] : % 8.50/4.51 ( ( P4 % 8.50/4.51 = ( product_Pair @ B @ C @ X5 @ Y4 ) ) % 8.50/4.51 => ~ ( member @ A @ Z2 @ ( C2 @ X5 @ Y4 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2537_zdvd__antisym__abs,axiom, % 8.50/4.51 ! [A2: int,B2: int] : % 8.50/4.51 ( ( dvd_dvd @ int @ A2 @ B2 ) % 8.50/4.51 => ( ( dvd_dvd @ int @ B2 @ A2 ) % 8.50/4.51 => ( ( abs_abs @ int @ A2 ) % 8.50/4.51 = ( abs_abs @ int @ B2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2538_case__prodD,axiom, % 8.50/4.51 ! [A: $tType,B: $tType,F2: A > B > $o,A2: A,B2: B] : % 8.50/4.51 ( ( product_case_prod @ A @ B @ $o @ F2 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) % 8.50/4.51 => ( F2 @ A2 @ B2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2539_case__prodE,axiom, % 8.50/4.51 ! [A: $tType,B: $tType,C2: A > B > $o,P4: product_prod @ A @ B] : % 8.50/4.51 ( ( product_case_prod @ A @ B @ $o @ C2 @ P4 ) % 8.50/4.51 => ~ ! [X5: A,Y4: B] : % 8.50/4.51 ( ( P4 % 8.50/4.51 = ( product_Pair @ A @ B @ X5 @ Y4 ) ) % 8.50/4.51 => ~ ( C2 @ X5 @ Y4 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2540_sum__mono,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 => ! [K5: set @ B,F2: B > A,G: B > A] : % 8.50/4.51 ( ! [I4: B] : % 8.50/4.51 ( ( member @ B @ I4 @ K5 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) ) % 8.50/4.51 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ K5 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2541_sum_Odistrib,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [G: B > A,H2: B > A,A3: set @ B] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.51 @ ^ [X: B] : ( plus_plus @ A @ ( G @ X ) @ ( H2 @ X ) ) % 8.50/4.51 @ A3 ) % 8.50/4.51 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ A3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2542_sum_Oswap__restrict,axiom, % 8.50/4.51 ! [A: $tType,B: $tType,C: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [A3: set @ B,B3: set @ C,G: B > C > A,R: B > C > $o] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( finite_finite2 @ C @ B3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.51 @ ^ [X: B] : % 8.50/4.51 ( groups7311177749621191930dd_sum @ C @ A @ ( G @ X ) % 8.50/4.51 @ ( collect @ C % 8.50/4.51 @ ^ [Y2: C] : % 8.50/4.51 ( ( member @ C @ Y2 @ B3 ) % 8.50/4.51 & ( R @ X @ Y2 ) ) ) ) % 8.50/4.51 @ A3 ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ C @ A % 8.50/4.51 @ ^ [Y2: C] : % 8.50/4.51 ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.51 @ ^ [X: B] : ( G @ X @ Y2 ) % 8.50/4.51 @ ( collect @ B % 8.50/4.51 @ ^ [X: B] : % 8.50/4.51 ( ( member @ B @ X @ A3 ) % 8.50/4.51 & ( R @ X @ Y2 ) ) ) ) % 8.50/4.51 @ B3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2543_case__prodD_H,axiom, % 8.50/4.51 ! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A2: A,B2: B,C2: C] : % 8.50/4.51 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 ) % 8.50/4.51 => ( R @ A2 @ B2 @ C2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2544_case__prodE_H,axiom, % 8.50/4.51 ! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P4: product_prod @ A @ B,Z2: C] : % 8.50/4.51 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ Z2 ) % 8.50/4.51 => ~ ! [X5: A,Y4: B] : % 8.50/4.51 ( ( P4 % 8.50/4.51 = ( product_Pair @ A @ B @ X5 @ Y4 ) ) % 8.50/4.51 => ~ ( C2 @ X5 @ Y4 @ Z2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2545_sum__nonneg,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 => ! [A3: set @ B,F2: B > A] : % 8.50/4.51 ( ! [X5: B] : % 8.50/4.51 ( ( member @ B @ X5 @ A3 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) ) % 8.50/4.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2546_sum__nonpos,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 => ! [A3: set @ B,F2: B > A] : % 8.50/4.51 ( ! [X5: B] : % 8.50/4.51 ( ( member @ B @ X5 @ A3 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( zero_zero @ A ) ) ) % 8.50/4.51 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2547_sum__mono__inv,axiom, % 8.50/4.51 ! [A: $tType,I7: $tType] : % 8.50/4.51 ( ( ordere8940638589300402666id_add @ A ) % 8.50/4.51 => ! [F2: I7 > A,I6: set @ I7,G: I7 > A,I: I7] : % 8.50/4.51 ( ( ( groups7311177749621191930dd_sum @ I7 @ A @ F2 @ I6 ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ I7 @ A @ G @ I6 ) ) % 8.50/4.51 => ( ! [I4: I7] : % 8.50/4.51 ( ( member @ I7 @ I4 @ I6 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) ) % 8.50/4.51 => ( ( member @ I7 @ I @ I6 ) % 8.50/4.51 => ( ( finite_finite2 @ I7 @ I6 ) % 8.50/4.51 => ( ( F2 @ I ) % 8.50/4.51 = ( G @ I ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2548_abs__ge__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2549_abs__not__less__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ~ ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2550_abs__of__pos,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.51 => ( ( abs_abs @ A @ A2 ) % 8.50/4.51 = A2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2551_abs__triangle__ineq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2552_abs__mult__less,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [A2: A,C2: A,B2: A,D2: A] : % 8.50/4.51 ( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ C2 ) % 8.50/4.51 => ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D2 ) % 8.50/4.51 => ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2553_abs__triangle__ineq2,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2554_abs__triangle__ineq3,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2555_abs__triangle__ineq2__sym,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2556_nonzero__abs__divide,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_field @ A ) % 8.50/4.51 => ! [B2: A,A2: A] : % 8.50/4.51 ( ( B2 % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 8.50/4.51 = ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2557_abs__ge__minus__self,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( abs_abs @ A @ A2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2558_abs__le__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) % 8.50/4.51 = ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.51 & ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2559_abs__le__D2,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2560_abs__leI,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.51 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2561_abs__less__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : % 8.50/4.51 ( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ B2 ) % 8.50/4.51 = ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.51 & ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2562_sum__cong__Suc,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [A3: set @ nat,F2: nat > A,G: nat > A] : % 8.50/4.51 ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 ) % 8.50/4.51 => ( ! [X5: nat] : % 8.50/4.51 ( ( member @ nat @ ( suc @ X5 ) @ A3 ) % 8.50/4.51 => ( ( F2 @ ( suc @ X5 ) ) % 8.50/4.51 = ( G @ ( suc @ X5 ) ) ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A3 ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ A3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2563_Collect__case__prod__mono,axiom, % 8.50/4.51 ! [B: $tType,A: $tType,A3: A > B > $o,B3: A > B > $o] : % 8.50/4.51 ( ( ord_less_eq @ ( A > B > $o ) @ A3 @ B3 ) % 8.50/4.51 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2564_abs__zmult__eq__1,axiom, % 8.50/4.51 ! [M: int,N2: int] : % 8.50/4.51 ( ( ( abs_abs @ int @ ( times_times @ int @ M @ N2 ) ) % 8.50/4.51 = ( one_one @ int ) ) % 8.50/4.51 => ( ( abs_abs @ int @ M ) % 8.50/4.51 = ( one_one @ int ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2565_sum_Ointer__filter,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [A3: set @ B,G: B > A,P: B > $o] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G % 8.50/4.51 @ ( collect @ B % 8.50/4.51 @ ^ [X: B] : % 8.50/4.51 ( ( member @ B @ X @ A3 ) % 8.50/4.51 & ( P @ X ) ) ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.51 @ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( G @ X ) @ ( zero_zero @ A ) ) % 8.50/4.51 @ A3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2566_sum__subtractf__nat,axiom, % 8.50/4.51 ! [A: $tType,A3: set @ A,G: A > nat,F2: A > nat] : % 8.50/4.51 ( ! [X5: A] : % 8.50/4.51 ( ( member @ A @ X5 @ A3 ) % 8.50/4.51 => ( ord_less_eq @ nat @ ( G @ X5 ) @ ( F2 @ X5 ) ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ A @ nat % 8.50/4.51 @ ^ [X: A] : ( minus_minus @ nat @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.51 @ A3 ) % 8.50/4.51 = ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2567_convex__sum__bound__le,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( linordered_idom @ B ) % 8.50/4.51 => ! [I6: set @ A,X3: A > B,A2: A > B,B2: B,Delta: B] : % 8.50/4.51 ( ! [I4: A] : % 8.50/4.51 ( ( member @ A @ I4 @ I6 ) % 8.50/4.51 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X3 @ I4 ) ) ) % 8.50/4.51 => ( ( ( groups7311177749621191930dd_sum @ A @ B @ X3 @ I6 ) % 8.50/4.51 = ( one_one @ B ) ) % 8.50/4.51 => ( ! [I4: A] : % 8.50/4.51 ( ( member @ A @ I4 @ I6 ) % 8.50/4.51 => ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A2 @ I4 ) @ B2 ) ) @ Delta ) ) % 8.50/4.51 => ( ord_less_eq @ B % 8.50/4.51 @ ( abs_abs @ B % 8.50/4.51 @ ( minus_minus @ B % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ A @ B % 8.50/4.51 @ ^ [I3: A] : ( times_times @ B @ ( A2 @ I3 ) @ ( X3 @ I3 ) ) % 8.50/4.51 @ I6 ) % 8.50/4.51 @ B2 ) ) % 8.50/4.51 @ Delta ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2568_infinite__int__iff__unbounded__le,axiom, % 8.50/4.51 ! [S3: set @ int] : % 8.50/4.51 ( ( ~ ( finite_finite2 @ int @ S3 ) ) % 8.50/4.51 = ( ! [M2: int] : % 8.50/4.51 ? [N: int] : % 8.50/4.51 ( ( ord_less_eq @ int @ M2 @ ( abs_abs @ int @ N ) ) % 8.50/4.51 & ( member @ int @ N @ S3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2569_infinite__int__iff__unbounded,axiom, % 8.50/4.51 ! [S3: set @ int] : % 8.50/4.51 ( ( ~ ( finite_finite2 @ int @ S3 ) ) % 8.50/4.51 = ( ! [M2: int] : % 8.50/4.51 ? [N: int] : % 8.50/4.51 ( ( ord_less @ int @ M2 @ ( abs_abs @ int @ N ) ) % 8.50/4.51 & ( member @ int @ N @ S3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2570_sum_Oshift__bounds__cl__Suc__ivl,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 8.50/4.51 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2571_sum_Oshift__bounds__cl__nat__ivl,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) ) % 8.50/4.51 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2572_tanh__real__lt__1,axiom, % 8.50/4.51 ! [X3: real] : ( ord_less @ real @ ( tanh @ real @ X3 ) @ ( one_one @ real ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2573_sum__le__included,axiom, % 8.50/4.51 ! [B: $tType,A: $tType,C: $tType] : % 8.50/4.51 ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 => ! [S: set @ B,T2: set @ C,G: C > A,I: C > B,F2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ S ) % 8.50/4.51 => ( ( finite_finite2 @ C @ T2 ) % 8.50/4.51 => ( ! [X5: C] : % 8.50/4.51 ( ( member @ C @ X5 @ T2 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X5 ) ) ) % 8.50/4.51 => ( ! [X5: B] : % 8.50/4.51 ( ( member @ B @ X5 @ S ) % 8.50/4.51 => ? [Xa: C] : % 8.50/4.51 ( ( member @ C @ Xa @ T2 ) % 8.50/4.51 & ( ( I @ Xa ) % 8.50/4.51 = X5 ) % 8.50/4.51 & ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ Xa ) ) ) ) % 8.50/4.51 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2574_sum__nonneg__eq__0__iff,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 => ! [A3: set @ B,F2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ! [X5: B] : % 8.50/4.51 ( ( member @ B @ X5 @ A3 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) ) % 8.50/4.51 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 = ( ! [X: B] : % 8.50/4.51 ( ( member @ B @ X @ A3 ) % 8.50/4.51 => ( ( F2 @ X ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2575_sum__strict__mono__ex1,axiom, % 8.50/4.51 ! [A: $tType,I7: $tType] : % 8.50/4.51 ( ( ordere8940638589300402666id_add @ A ) % 8.50/4.51 => ! [A3: set @ I7,F2: I7 > A,G: I7 > A] : % 8.50/4.51 ( ( finite_finite2 @ I7 @ A3 ) % 8.50/4.51 => ( ! [X5: I7] : % 8.50/4.51 ( ( member @ I7 @ X5 @ A3 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 8.50/4.51 => ( ? [X2: I7] : % 8.50/4.51 ( ( member @ I7 @ X2 @ A3 ) % 8.50/4.51 & ( ord_less @ A @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) % 8.50/4.51 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I7 @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ I7 @ A @ G @ A3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2576_sum_Orelated,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [R: A > A > $o,S3: set @ B,H2: B > A,G: B > A] : % 8.50/4.51 ( ( R @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) % 8.50/4.51 => ( ! [X16: A,Y15: A,X23: A,Y23: A] : % 8.50/4.51 ( ( ( R @ X16 @ X23 ) % 8.50/4.51 & ( R @ Y15 @ Y23 ) ) % 8.50/4.51 => ( R @ ( plus_plus @ A @ X16 @ Y15 ) @ ( plus_plus @ A @ X23 @ Y23 ) ) ) % 8.50/4.51 => ( ( finite_finite2 @ B @ S3 ) % 8.50/4.51 => ( ! [X5: B] : % 8.50/4.51 ( ( member @ B @ X5 @ S3 ) % 8.50/4.51 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) ) % 8.50/4.51 => ( R @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2577_dense__eq0__I,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 & ( dense_linorder @ A ) ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ! [E: A] : % 8.50/4.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ E ) % 8.50/4.51 => ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ E ) ) % 8.50/4.51 => ( X3 % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2578_abs__mult__pos,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.50/4.51 => ( ( times_times @ A @ ( abs_abs @ A @ Y ) @ X3 ) % 8.50/4.51 = ( abs_abs @ A @ ( times_times @ A @ Y @ X3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2579_abs__eq__mult,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordered_ring_abs @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : % 8.50/4.51 ( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.51 | ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) % 8.50/4.51 & ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.50/4.51 | ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) % 8.50/4.51 => ( ( abs_abs @ A @ ( times_times @ A @ A2 @ B2 ) ) % 8.50/4.51 = ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2580_sum__strict__mono,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( strict7427464778891057005id_add @ A ) % 8.50/4.51 => ! [A3: set @ B,F2: B > A,G: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( A3 % 8.50/4.51 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.51 => ( ! [X5: B] : % 8.50/4.51 ( ( member @ B @ X5 @ A3 ) % 8.50/4.51 => ( ord_less @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 8.50/4.51 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2581_abs__minus__le__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A2 ) ) @ ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2582_abs__eq__iff_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_ring @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : % 8.50/4.51 ( ( ( abs_abs @ A @ A2 ) % 8.50/4.51 = B2 ) % 8.50/4.51 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.50/4.51 & ( ( A2 = B2 ) % 8.50/4.51 | ( A2 % 8.50/4.51 = ( uminus_uminus @ A @ B2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2583_eq__abs__iff_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_ring @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : % 8.50/4.51 ( ( A2 % 8.50/4.51 = ( abs_abs @ A @ B2 ) ) % 8.50/4.51 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.51 & ( ( B2 = A2 ) % 8.50/4.51 | ( B2 % 8.50/4.51 = ( uminus_uminus @ A @ A2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2584_zero__le__power__abs,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [A2: A,N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2585_abs__div__pos,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_field @ A ) % 8.50/4.51 => ! [Y: A,X3: A] : % 8.50/4.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ Y ) % 8.50/4.51 => ( ( divide_divide @ A @ ( abs_abs @ A @ X3 ) @ Y ) % 8.50/4.51 = ( abs_abs @ A @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2586_abs__if__raw,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( abs_if @ A ) % 8.50/4.51 => ( ( abs_abs @ A ) % 8.50/4.51 = ( ^ [A4: A] : ( if @ A @ ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A4 ) @ A4 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2587_abs__of__neg,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.50/4.51 => ( ( abs_abs @ A @ A2 ) % 8.50/4.51 = ( uminus_uminus @ A @ A2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2588_abs__if,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( abs_if @ A ) % 8.50/4.51 => ( ( abs_abs @ A ) % 8.50/4.51 = ( ^ [A4: A] : ( if @ A @ ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A4 ) @ A4 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2589_abs__diff__triangle__ineq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2590_abs__triangle__ineq4,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ordere166539214618696060dd_abs @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2591_abs__diff__le__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [X3: A,A2: A,R2: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ R2 ) % 8.50/4.51 = ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X3 ) % 8.50/4.51 & ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2592_sum_Oinsert__if,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [A3: set @ B,X3: B,G: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( ( member @ B @ X3 @ A3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) % 8.50/4.51 & ( ~ ( member @ B @ X3 @ A3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 8.50/4.51 = ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2593_abs__diff__less__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [X3: A,A2: A,R2: A] : % 8.50/4.51 ( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ R2 ) % 8.50/4.51 = ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X3 ) % 8.50/4.51 & ( ord_less @ A @ X3 @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2594_sum_Oreindex__bij__witness__not__neutral,axiom, % 8.50/4.51 ! [B: $tType,A: $tType,C: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [S6: set @ B,T6: set @ C,S3: set @ B,I: C > B,J: B > C,T4: set @ C,G: B > A,H2: C > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ S6 ) % 8.50/4.51 => ( ( finite_finite2 @ C @ T6 ) % 8.50/4.51 => ( ! [A5: B] : % 8.50/4.51 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) % 8.50/4.51 => ( ( I @ ( J @ A5 ) ) % 8.50/4.51 = A5 ) ) % 8.50/4.51 => ( ! [A5: B] : % 8.50/4.51 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) % 8.50/4.51 => ( member @ C @ ( J @ A5 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) ) % 8.50/4.51 => ( ! [B5: C] : % 8.50/4.51 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) % 8.50/4.51 => ( ( J @ ( I @ B5 ) ) % 8.50/4.51 = B5 ) ) % 8.50/4.51 => ( ! [B5: C] : % 8.50/4.51 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) % 8.50/4.51 => ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) ) % 8.50/4.51 => ( ! [A5: B] : % 8.50/4.51 ( ( member @ B @ A5 @ S6 ) % 8.50/4.51 => ( ( G @ A5 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 => ( ! [B5: C] : % 8.50/4.51 ( ( member @ C @ B5 @ T6 ) % 8.50/4.51 => ( ( H2 @ B5 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 => ( ! [A5: B] : % 8.50/4.51 ( ( member @ B @ A5 @ S3 ) % 8.50/4.51 => ( ( H2 @ ( J @ A5 ) ) % 8.50/4.51 = ( G @ A5 ) ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2595_sum__eq__Suc0__iff,axiom, % 8.50/4.51 ! [A: $tType,A3: set @ A,F2: A > nat] : % 8.50/4.51 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.51 => ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) % 8.50/4.51 = ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.51 = ( ? [X: A] : % 8.50/4.51 ( ( member @ A @ X @ A3 ) % 8.50/4.51 & ( ( F2 @ X ) % 8.50/4.51 = ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.51 & ! [Y2: A] : % 8.50/4.51 ( ( member @ A @ Y2 @ A3 ) % 8.50/4.51 => ( ( X != Y2 ) % 8.50/4.51 => ( ( F2 @ Y2 ) % 8.50/4.51 = ( zero_zero @ nat ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2596_sum__SucD,axiom, % 8.50/4.51 ! [A: $tType,F2: A > nat,A3: set @ A,N2: nat] : % 8.50/4.51 ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) % 8.50/4.51 = ( suc @ N2 ) ) % 8.50/4.51 => ? [X5: A] : % 8.50/4.51 ( ( member @ A @ X5 @ A3 ) % 8.50/4.51 & ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X5 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2597_sum__eq__1__iff,axiom, % 8.50/4.51 ! [A: $tType,A3: set @ A,F2: A > nat] : % 8.50/4.51 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.51 => ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) % 8.50/4.51 = ( one_one @ nat ) ) % 8.50/4.51 = ( ? [X: A] : % 8.50/4.51 ( ( member @ A @ X @ A3 ) % 8.50/4.51 & ( ( F2 @ X ) % 8.50/4.51 = ( one_one @ nat ) ) % 8.50/4.51 & ! [Y2: A] : % 8.50/4.51 ( ( member @ A @ Y2 @ A3 ) % 8.50/4.51 => ( ( X != Y2 ) % 8.50/4.51 => ( ( F2 @ Y2 ) % 8.50/4.51 = ( zero_zero @ nat ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2598_abs__real__def,axiom, % 8.50/4.51 ( ( abs_abs @ real ) % 8.50/4.51 = ( ^ [A4: real] : ( if @ real @ ( ord_less @ real @ A4 @ ( zero_zero @ real ) ) @ ( uminus_uminus @ real @ A4 ) @ A4 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2599_zabs__def,axiom, % 8.50/4.51 ( ( abs_abs @ int ) % 8.50/4.51 = ( ^ [I3: int] : ( if @ int @ ( ord_less @ int @ I3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I3 ) @ I3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2600_sum__nonneg__leq__bound,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 => ! [S: set @ B,F2: B > A,B3: A,I: B] : % 8.50/4.51 ( ( finite_finite2 @ B @ S ) % 8.50/4.51 => ( ! [I4: B] : % 8.50/4.51 ( ( member @ B @ I4 @ S ) % 8.50/4.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) ) % 8.50/4.51 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S ) % 8.50/4.51 = B3 ) % 8.50/4.51 => ( ( member @ B @ I @ S ) % 8.50/4.51 => ( ord_less_eq @ A @ ( F2 @ I ) @ B3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2601_sum__nonneg__0,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 => ! [S: set @ B,F2: B > A,I: B] : % 8.50/4.51 ( ( finite_finite2 @ B @ S ) % 8.50/4.51 => ( ! [I4: B] : % 8.50/4.51 ( ( member @ B @ I4 @ S ) % 8.50/4.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) ) % 8.50/4.51 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 => ( ( member @ B @ I @ S ) % 8.50/4.51 => ( ( F2 @ I ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2602_dvd__imp__le__int,axiom, % 8.50/4.51 ! [I: int,D2: int] : % 8.50/4.51 ( ( I % 8.50/4.51 != ( zero_zero @ int ) ) % 8.50/4.51 => ( ( dvd_dvd @ int @ D2 @ I ) % 8.50/4.51 => ( ord_less_eq @ int @ ( abs_abs @ int @ D2 ) @ ( abs_abs @ int @ I ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2603_abs__mod__less,axiom, % 8.50/4.51 ! [L: int,K: int] : % 8.50/4.51 ( ( L % 8.50/4.51 != ( zero_zero @ int ) ) % 8.50/4.51 => ( ord_less @ int @ ( abs_abs @ int @ ( modulo_modulo @ int @ K @ L ) ) @ ( abs_abs @ int @ L ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2604_sum_Osetdiff__irrelevant,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [A3: set @ B,G: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G % 8.50/4.51 @ ( minus_minus @ ( set @ B ) @ A3 % 8.50/4.51 @ ( collect @ B % 8.50/4.51 @ ^ [X: B] : % 8.50/4.51 ( ( G @ X ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2605_sum__power__add,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( monoid_mult @ A ) % 8.50/4.51 & ( comm_ring @ A ) ) % 8.50/4.51 => ! [X3: A,M: nat,I6: set @ nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( power_power @ A @ X3 @ ( plus_plus @ nat @ M @ I3 ) ) % 8.50/4.51 @ I6 ) % 8.50/4.51 = ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ I6 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2606_sum_OatLeastAtMost__rev,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [G: nat > A,N2: nat,M: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I3 ) ) % 8.50/4.51 @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2607_tanh__real__gt__neg1,axiom, % 8.50/4.51 ! [X3: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( tanh @ real @ X3 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2608_sum__pos2,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 => ! [I6: set @ B,I: B,F2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ I6 ) % 8.50/4.51 => ( ( member @ B @ I @ I6 ) % 8.50/4.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) ) % 8.50/4.51 => ( ! [I4: B] : % 8.50/4.51 ( ( member @ B @ I4 @ I6 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) ) % 8.50/4.51 => ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2609_sum__pos,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 => ! [I6: set @ B,F2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ I6 ) % 8.50/4.51 => ( ( I6 % 8.50/4.51 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.51 => ( ! [I4: B] : % 8.50/4.51 ( ( member @ B @ I4 @ I6 ) % 8.50/4.51 => ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) ) % 8.50/4.51 => ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I6 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2610_abs__add__one__gt__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [X3: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2611_sum_Omono__neutral__cong__right,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [T4: set @ B,S3: set @ B,G: B > A,H2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ T4 ) % 8.50/4.51 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.51 => ( ! [X5: B] : % 8.50/4.51 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.51 => ( ( G @ X5 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 => ( ! [X5: B] : % 8.50/4.51 ( ( member @ B @ X5 @ S3 ) % 8.50/4.51 => ( ( G @ X5 ) % 8.50/4.51 = ( H2 @ X5 ) ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T4 ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2612_sum_Omono__neutral__cong__left,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [T4: set @ B,S3: set @ B,H2: B > A,G: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ T4 ) % 8.50/4.51 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.51 => ( ! [X5: B] : % 8.50/4.51 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.51 => ( ( H2 @ X5 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 => ( ! [X5: B] : % 8.50/4.51 ( ( member @ B @ X5 @ S3 ) % 8.50/4.51 => ( ( G @ X5 ) % 8.50/4.51 = ( H2 @ X5 ) ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2613_sum_Omono__neutral__right,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [T4: set @ B,S3: set @ B,G: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ T4 ) % 8.50/4.51 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.51 => ( ! [X5: B] : % 8.50/4.51 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.51 => ( ( G @ X5 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T4 ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2614_sum_Omono__neutral__left,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [T4: set @ B,S3: set @ B,G: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ T4 ) % 8.50/4.51 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.51 => ( ! [X5: B] : % 8.50/4.51 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.51 => ( ( G @ X5 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ A @ G @ T4 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2615_sum_Osame__carrierI,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [C5: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ C5 ) % 8.50/4.51 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 ) % 8.50/4.51 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ C5 ) % 8.50/4.51 => ( ! [A5: B] : % 8.50/4.51 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) ) % 8.50/4.51 => ( ( G @ A5 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 => ( ! [B5: B] : % 8.50/4.51 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B3 ) ) % 8.50/4.51 => ( ( H2 @ B5 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B3 ) ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2616_sum_Osame__carrier,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [C5: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ C5 ) % 8.50/4.51 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 ) % 8.50/4.51 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ C5 ) % 8.50/4.51 => ( ! [A5: B] : % 8.50/4.51 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) ) % 8.50/4.51 => ( ( G @ A5 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 => ( ! [B5: B] : % 8.50/4.51 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B3 ) ) % 8.50/4.51 => ( ( H2 @ B5 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 => ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B3 ) ) % 8.50/4.51 = ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2617_sum_Osubset__diff,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [B3: set @ B,A3: set @ B,G: B > A] : % 8.50/4.51 ( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 ) % 8.50/4.51 => ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 8.50/4.51 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2618_sum__diff,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( ab_group_add @ A ) % 8.50/4.51 => ! [A3: set @ B,B3: set @ B,F2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) % 8.50/4.51 = ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2619_of__int__leD,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [N2: int,X3: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N2 ) ) @ X3 ) % 8.50/4.51 => ( ( N2 % 8.50/4.51 = ( zero_zero @ int ) ) % 8.50/4.51 | ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2620_sum__diff__nat,axiom, % 8.50/4.51 ! [A: $tType,B3: set @ A,A3: set @ A,F2: A > nat] : % 8.50/4.51 ( ( finite_finite2 @ A @ B3 ) % 8.50/4.51 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.51 = ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ B3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2621_of__int__lessD,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [N2: int,X3: A] : % 8.50/4.51 ( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N2 ) ) @ X3 ) % 8.50/4.51 => ( ( N2 % 8.50/4.51 = ( zero_zero @ int ) ) % 8.50/4.51 | ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2622_sum__diff1__nat,axiom, % 8.50/4.51 ! [A: $tType,A2: A,A3: set @ A,F2: A > nat] : % 8.50/4.51 ( ( ( member @ A @ A2 @ A3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.51 = ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( F2 @ A2 ) ) ) ) % 8.50/4.51 & ( ~ ( member @ A @ A2 @ A3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2623_sum__shift__lb__Suc0__0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [F2: nat > A,K: nat] : % 8.50/4.51 ( ( ( F2 @ ( zero_zero @ nat ) ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2624_sum_OatLeast0__atMost__Suc,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [G: nat > A,N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2625_sum_OatLeast__Suc__atMost,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [M: nat,N2: nat,G: nat > A] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.51 = ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2626_sum_Onat__ivl__Suc_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [M: nat,N2: nat,G: nat > A] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( plus_plus @ A @ ( G @ ( suc @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2627_round__diff__minimal,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [Z2: A,M: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z2 @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z2 ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z2 @ ( ring_1_of_int @ A @ M ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2628_zdvd__mult__cancel1,axiom, % 8.50/4.51 ! [M: int,N2: int] : % 8.50/4.51 ( ( M % 8.50/4.51 != ( zero_zero @ int ) ) % 8.50/4.51 => ( ( dvd_dvd @ int @ ( times_times @ int @ M @ N2 ) @ M ) % 8.50/4.51 = ( ( abs_abs @ int @ N2 ) % 8.50/4.51 = ( one_one @ int ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2629_sum_OSuc__reindex__ivl,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [M: nat,N2: nat,G: nat > A] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.51 => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( plus_plus @ A @ ( G @ M ) % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 8.50/4.51 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2630_sum__Suc__diff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ab_group_add @ A ) % 8.50/4.51 => ! [M: nat,N2: nat,F2: nat > A] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) % 8.50/4.51 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.51 = ( minus_minus @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ M ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2631_fun__cong__unused__0,axiom, % 8.50/4.51 ! [A: $tType,B: $tType,C: $tType] : % 8.50/4.51 ( ( zero @ B ) % 8.50/4.51 => ! [F2: ( A > B ) > C,G: C] : % 8.50/4.51 ( ( F2 % 8.50/4.51 = ( ^ [X: A > B] : G ) ) % 8.50/4.51 => ( ( F2 % 8.50/4.51 @ ^ [X: A] : ( zero_zero @ B ) ) % 8.50/4.51 = G ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2632_sum__mono2,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 => ! [B3: set @ B,A3: set @ B,F2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ B3 ) % 8.50/4.51 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 ) % 8.50/4.51 => ( ! [B5: B] : % 8.50/4.51 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ B3 @ A3 ) ) % 8.50/4.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ B5 ) ) ) % 8.50/4.51 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2633_abs__le__square__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ ( abs_abs @ A @ Y ) ) % 8.50/4.51 = ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2634_sum_Oremove,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [A3: set @ B,X3: B,G: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( member @ B @ X3 @ A3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 8.50/4.51 = ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2635_sum_Oinsert__remove,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [A3: set @ B,G: B > A,X3: B] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 8.50/4.51 = ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2636_sum__diff1,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( ab_group_add @ A ) % 8.50/4.51 => ! [A3: set @ B,A2: B,F2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( ( member @ B @ A2 @ A3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) % 8.50/4.51 = ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( F2 @ A2 ) ) ) ) % 8.50/4.51 & ( ~ ( member @ B @ A2 @ A3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2637_sum_Oub__add__nat,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [M: nat,N2: nat,G: nat > A,P4: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N2 @ P4 ) ) ) % 8.50/4.51 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ P4 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2638_sum_Odelta__remove,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [S3: set @ B,A2: B,B2: B > A,C2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ S3 ) % 8.50/4.51 => ( ( ( member @ B @ A2 @ S3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.51 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) ) % 8.50/4.51 @ S3 ) % 8.50/4.51 = ( plus_plus @ A @ ( B2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) % 8.50/4.51 & ( ~ ( member @ B @ A2 @ S3 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.51 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) ) % 8.50/4.51 @ S3 ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2639_sum__count__set,axiom, % 8.50/4.51 ! [A: $tType,Xs: list @ A,X8: set @ A] : % 8.50/4.51 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X8 ) % 8.50/4.51 => ( ( finite_finite2 @ A @ X8 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs ) @ X8 ) % 8.50/4.51 = ( size_size @ ( list @ A ) @ Xs ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2640_sum__strict__mono2,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( ordere8940638589300402666id_add @ B ) % 8.50/4.51 => ! [B3: set @ A,A3: set @ A,B2: A,F2: A > B] : % 8.50/4.51 ( ( finite_finite2 @ A @ B3 ) % 8.50/4.51 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.51 => ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) % 8.50/4.51 => ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F2 @ B2 ) ) % 8.50/4.51 => ( ! [X5: A] : % 8.50/4.51 ( ( member @ A @ X5 @ B3 ) % 8.50/4.51 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X5 ) ) ) % 8.50/4.51 => ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ B3 ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2641_member__le__sum,axiom, % 8.50/4.51 ! [B: $tType,C: $tType] : % 8.50/4.51 ( ( ( ordere6911136660526730532id_add @ B ) % 8.50/4.51 & ( semiring_1 @ B ) ) % 8.50/4.51 => ! [I: C,A3: set @ C,F2: C > B] : % 8.50/4.51 ( ( member @ C @ I @ A3 ) % 8.50/4.51 => ( ! [X5: C] : % 8.50/4.51 ( ( member @ C @ X5 @ ( minus_minus @ ( set @ C ) @ A3 @ ( insert @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) ) % 8.50/4.51 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X5 ) ) ) % 8.50/4.51 => ( ( finite_finite2 @ C @ A3 ) % 8.50/4.51 => ( ord_less_eq @ B @ ( F2 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F2 @ A3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2642_power2__le__iff__abs__le,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [Y: A,X3: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y ) % 8.50/4.51 => ( ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 = ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ Y ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2643_abs__sqrt__wlog,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [P: A > A > $o,X3: A] : % 8.50/4.51 ( ! [X5: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 ) % 8.50/4.51 => ( P @ X5 @ ( power_power @ A @ X5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.51 => ( P @ ( abs_abs @ A @ X3 ) @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2644_abs__square__le__1,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) % 8.50/4.51 = ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ ( one_one @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2645_abs__square__less__1,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ord_less @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) % 8.50/4.51 = ( ord_less @ A @ ( abs_abs @ A @ X3 ) @ ( one_one @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2646_power__mono__even,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [N2: nat,A2: A,B2: A] : % 8.50/4.51 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.50/4.51 => ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) % 8.50/4.51 => ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2647_nat__intermed__int__val,axiom, % 8.50/4.51 ! [M: nat,N2: nat,F2: nat > int,K: int] : % 8.50/4.51 ( ! [I4: nat] : % 8.50/4.51 ( ( ( ord_less_eq @ nat @ M @ I4 ) % 8.50/4.51 & ( ord_less @ nat @ I4 @ N2 ) ) % 8.50/4.51 => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I4 ) ) @ ( F2 @ I4 ) ) ) @ ( one_one @ int ) ) ) % 8.50/4.51 => ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.51 => ( ( ord_less_eq @ int @ ( F2 @ M ) @ K ) % 8.50/4.51 => ( ( ord_less_eq @ int @ K @ ( F2 @ N2 ) ) % 8.50/4.51 => ? [I4: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M @ I4 ) % 8.50/4.51 & ( ord_less_eq @ nat @ I4 @ N2 ) % 8.50/4.51 & ( ( F2 @ I4 ) % 8.50/4.51 = K ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2648_incr__lemma,axiom, % 8.50/4.51 ! [D2: int,Z2: int,X3: int] : % 8.50/4.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 ) % 8.50/4.51 => ( ord_less @ int @ Z2 @ ( plus_plus @ int @ X3 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X3 @ Z2 ) ) @ ( one_one @ int ) ) @ D2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2649_decr__lemma,axiom, % 8.50/4.51 ! [D2: int,X3: int,Z2: int] : % 8.50/4.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ D2 ) % 8.50/4.51 => ( ord_less @ int @ ( minus_minus @ int @ X3 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X3 @ Z2 ) ) @ ( one_one @ int ) ) @ D2 ) ) @ Z2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2650_sum__natinterval__diff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ab_group_add @ A ) % 8.50/4.51 => ! [M: nat,N2: nat,F2: nat > A] : % 8.50/4.51 ( ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) ) % 8.50/4.51 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.51 = ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) % 8.50/4.51 & ( ~ ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) ) % 8.50/4.51 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2651_sum__telescope_H_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ab_group_add @ A ) % 8.50/4.51 => ! [M: nat,N2: nat,F2: nat > A] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) ) % 8.50/4.51 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) % 8.50/4.51 = ( minus_minus @ A @ ( F2 @ N2 ) @ ( F2 @ M ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2652_mask__eq__sum__exp,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_parity @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) % 8.50/4.51 @ ( collect @ nat % 8.50/4.51 @ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2653_nat__ivt__aux,axiom, % 8.50/4.51 ! [N2: nat,F2: nat > int,K: int] : % 8.50/4.51 ( ! [I4: nat] : % 8.50/4.51 ( ( ord_less @ nat @ I4 @ N2 ) % 8.50/4.51 => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I4 ) ) @ ( F2 @ I4 ) ) ) @ ( one_one @ int ) ) ) % 8.50/4.51 => ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K ) % 8.50/4.51 => ( ( ord_less_eq @ int @ K @ ( F2 @ N2 ) ) % 8.50/4.51 => ? [I4: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ I4 @ N2 ) % 8.50/4.51 & ( ( F2 @ I4 ) % 8.50/4.51 = K ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2654_divmod__nat__def,axiom, % 8.50/4.51 ( divmod_nat % 8.50/4.51 = ( ^ [M2: nat,N: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M2 @ N ) @ ( modulo_modulo @ nat @ M2 @ N ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2655_sum__gp__multiplied,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( monoid_mult @ A ) % 8.50/4.51 & ( comm_ring @ A ) ) % 8.50/4.51 => ! [M: nat,N2: nat,X3: A] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.51 => ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) % 8.50/4.51 = ( minus_minus @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2656_sum_Oin__pairs,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) ) % 8.50/4.51 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2657_nat0__intermed__int__val,axiom, % 8.50/4.51 ! [N2: nat,F2: nat > int,K: int] : % 8.50/4.51 ( ! [I4: nat] : % 8.50/4.51 ( ( ord_less @ nat @ I4 @ N2 ) % 8.50/4.51 => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) @ ( F2 @ I4 ) ) ) @ ( one_one @ int ) ) ) % 8.50/4.51 => ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K ) % 8.50/4.51 => ( ( ord_less_eq @ int @ K @ ( F2 @ N2 ) ) % 8.50/4.51 => ? [I4: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ I4 @ N2 ) % 8.50/4.51 & ( ( F2 @ I4 ) % 8.50/4.51 = K ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2658_mask__eq__sum__exp__nat,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.50/4.51 @ ( collect @ nat % 8.50/4.51 @ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2659_gauss__sum__nat,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.51 @ ^ [X: nat] : X % 8.50/4.51 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.50/4.51 = ( divide_divide @ nat @ ( times_times @ nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2660_of__int__round__abs__le,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) @ X3 ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2661_listrel1p__def,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( listrel1p @ A ) % 8.50/4.51 = ( ^ [R5: A > A > $o,Xs2: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys3 ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2662_lemma__interval,axiom, % 8.50/4.51 ! [A2: real,X3: real,B2: real] : % 8.50/4.51 ( ( ord_less @ real @ A2 @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ B2 ) % 8.50/4.51 => ? [D3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 8.50/4.51 & ! [Y5: real] : % 8.50/4.51 ( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y5 ) ) @ D3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ A2 @ Y5 ) % 8.50/4.51 & ( ord_less_eq @ real @ Y5 @ B2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2663_sum__gp,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( division_ring @ A ) % 8.50/4.51 & ( comm_ring @ A ) ) % 8.50/4.51 => ! [N2: nat,M: nat,X3: A] : % 8.50/4.51 ( ( ( ord_less @ nat @ N2 @ M ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 & ( ~ ( ord_less @ nat @ N2 @ M ) % 8.50/4.51 => ( ( ( X3 % 8.50/4.51 = ( one_one @ A ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.51 = ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ M ) ) ) ) % 8.50/4.51 & ( ( X3 % 8.50/4.51 != ( one_one @ A ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.51 = ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2664_gauss__sum__from__Suc__0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( euclid5411537665997757685th_nat @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) % 8.50/4.51 = ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2665_lemma__interval__lt,axiom, % 8.50/4.51 ! [A2: real,X3: real,B2: real] : % 8.50/4.51 ( ( ord_less @ real @ A2 @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ B2 ) % 8.50/4.51 => ? [D3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 8.50/4.51 & ! [Y5: real] : % 8.50/4.51 ( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y5 ) ) @ D3 ) % 8.50/4.51 => ( ( ord_less @ real @ A2 @ Y5 ) % 8.50/4.51 & ( ord_less @ real @ Y5 @ B2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2666_arctan__double,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 8.50/4.51 => ( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ X3 ) ) % 8.50/4.51 = ( arctan @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2667_of__nat__eq__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_char_0 @ A ) % 8.50/4.51 => ! [M: nat,N2: nat] : % 8.50/4.51 ( ( ( semiring_1_of_nat @ A @ M ) % 8.50/4.51 = ( semiring_1_of_nat @ A @ N2 ) ) % 8.50/4.51 = ( M = N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2668_split__part,axiom, % 8.50/4.51 ! [B: $tType,A: $tType,P: $o,Q: A > B > $o] : % 8.50/4.51 ( ( product_case_prod @ A @ B @ $o % 8.50/4.51 @ ^ [A4: A,B4: B] : % 8.50/4.51 ( P % 8.50/4.51 & ( Q @ A4 @ B4 ) ) ) % 8.50/4.51 = ( ^ [Ab: product_prod @ A @ B] : % 8.50/4.51 ( P % 8.50/4.51 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2669_int__eq__iff__numeral,axiom, % 8.50/4.51 ! [M: nat,V2: num] : % 8.50/4.51 ( ( ( semiring_1_of_nat @ int @ M ) % 8.50/4.51 = ( numeral_numeral @ int @ V2 ) ) % 8.50/4.51 = ( M % 8.50/4.51 = ( numeral_numeral @ nat @ V2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2670_abs__of__nat,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( abs_abs @ A @ ( semiring_1_of_nat @ A @ N2 ) ) % 8.50/4.51 = ( semiring_1_of_nat @ A @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2671_negative__eq__positive,axiom, % 8.50/4.51 ! [N2: nat,M: nat] : % 8.50/4.51 ( ( ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) % 8.50/4.51 = ( semiring_1_of_nat @ int @ M ) ) % 8.50/4.51 = ( ( N2 % 8.50/4.51 = ( zero_zero @ nat ) ) % 8.50/4.51 & ( M % 8.50/4.51 = ( zero_zero @ nat ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2672_of__int__of__nat__eq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ring_1 @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N2 ) ) % 8.50/4.51 = ( semiring_1_of_nat @ A @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2673_negative__zle,axiom, % 8.50/4.51 ! [N2: nat,M: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( semiring_1_of_nat @ int @ M ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2674_int__dvd__int__iff,axiom, % 8.50/4.51 ! [M: nat,N2: nat] : % 8.50/4.51 ( ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N2 ) ) % 8.50/4.51 = ( dvd_dvd @ nat @ M @ N2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2675_of__nat__eq__0__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_char_0 @ A ) % 8.50/4.51 => ! [M: nat] : % 8.50/4.51 ( ( ( semiring_1_of_nat @ A @ M ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 = ( M % 8.50/4.51 = ( zero_zero @ nat ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2676_of__nat__0__eq__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_char_0 @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( ( zero_zero @ A ) % 8.50/4.51 = ( semiring_1_of_nat @ A @ N2 ) ) % 8.50/4.51 = ( ( zero_zero @ nat ) % 8.50/4.51 = N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2677_of__nat__0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_1 @ A ) % 8.50/4.51 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2678_of__nat__less__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linord181362715937106298miring @ A ) % 8.50/4.51 => ! [M: nat,N2: nat] : % 8.50/4.51 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) % 8.50/4.51 = ( ord_less @ nat @ M @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2679_of__nat__le__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linord181362715937106298miring @ A ) % 8.50/4.51 => ! [M: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) % 8.50/4.51 = ( ord_less_eq @ nat @ M @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2680_of__nat__add,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_1 @ A ) % 8.50/4.51 => ! [M: nat,N2: nat] : % 8.50/4.51 ( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N2 ) ) % 8.50/4.51 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2681_of__nat__mult,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_1 @ A ) % 8.50/4.51 => ! [M: nat,N2: nat] : % 8.50/4.51 ( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N2 ) ) % 8.50/4.51 = ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2682_of__nat__1,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_1 @ A ) % 8.50/4.51 => ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) ) % 8.50/4.51 = ( one_one @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2683_of__nat__1__eq__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_char_0 @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( ( one_one @ A ) % 8.50/4.51 = ( semiring_1_of_nat @ A @ N2 ) ) % 8.50/4.51 = ( N2 % 8.50/4.51 = ( one_one @ nat ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2684_of__nat__eq__1__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_char_0 @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( ( semiring_1_of_nat @ A @ N2 ) % 8.50/4.51 = ( one_one @ A ) ) % 8.50/4.51 = ( N2 % 8.50/4.51 = ( one_one @ nat ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2685_negative__zless,axiom, % 8.50/4.51 ! [N2: nat,M: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) @ ( semiring_1_of_nat @ int @ M ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2686_arctan__less__zero__iff,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( arctan @ X3 ) @ ( zero_zero @ real ) ) % 8.50/4.51 = ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2687_zero__less__arctan__iff,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( arctan @ X3 ) ) % 8.50/4.51 = ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2688_of__nat__of__bool,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_1 @ A ) % 8.50/4.51 => ! [P: $o] : % 8.50/4.51 ( ( semiring_1_of_nat @ A @ ( zero_neq_one_of_bool @ nat @ P ) ) % 8.50/4.51 = ( zero_neq_one_of_bool @ A @ P ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2689_of__nat__sum,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( semiring_1 @ A ) % 8.50/4.51 => ! [F2: B > nat,A3: set @ B] : % 8.50/4.51 ( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A3 ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.51 @ ^ [X: B] : ( semiring_1_of_nat @ A @ ( F2 @ X ) ) % 8.50/4.51 @ A3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2690_of__nat__le__0__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linord181362715937106298miring @ A ) % 8.50/4.51 => ! [M: nat] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) % 8.50/4.51 = ( M % 8.50/4.51 = ( zero_zero @ nat ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2691_of__nat__Suc,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_1 @ A ) % 8.50/4.51 => ! [M: nat] : % 8.50/4.51 ( ( semiring_1_of_nat @ A @ ( suc @ M ) ) % 8.50/4.51 = ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2692_real__of__nat__less__numeral__iff,axiom, % 8.50/4.51 ! [N2: nat,W2: num] : % 8.50/4.51 ( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( numeral_numeral @ real @ W2 ) ) % 8.50/4.51 = ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ W2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2693_numeral__less__real__of__nat__iff,axiom, % 8.50/4.51 ! [W2: num,N2: nat] : % 8.50/4.51 ( ( ord_less @ real @ ( numeral_numeral @ real @ W2 ) @ ( semiring_1_of_nat @ real @ N2 ) ) % 8.50/4.51 = ( ord_less @ nat @ ( numeral_numeral @ nat @ W2 ) @ N2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2694_numeral__le__real__of__nat__iff,axiom, % 8.50/4.51 ! [N2: num,M: nat] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N2 ) @ ( semiring_1_of_nat @ real @ M ) ) % 8.50/4.51 = ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N2 ) @ M ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2695_of__nat__0__less__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linord181362715937106298miring @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) % 8.50/4.51 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2696_of__nat__power__less__of__nat__cancel__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [X3: nat,B2: nat,W2: nat] : % 8.50/4.51 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) ) % 8.50/4.51 = ( ord_less @ nat @ X3 @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2697_of__nat__less__of__nat__power__cancel__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [B2: nat,W2: nat,X3: nat] : % 8.50/4.51 ( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X3 ) ) % 8.50/4.51 = ( ord_less @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2698_of__nat__power__le__of__nat__cancel__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [X3: nat,B2: nat,W2: nat] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) ) % 8.50/4.51 = ( ord_less_eq @ nat @ X3 @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2699_of__nat__le__of__nat__power__cancel__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [B2: nat,W2: nat,X3: nat] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2700_of__nat__zero__less__power__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [X3: nat,N2: nat] : % 8.50/4.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X3 ) @ N2 ) ) % 8.50/4.51 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 ) % 8.50/4.51 | ( N2 % 8.50/4.51 = ( zero_zero @ nat ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2701_numeral__power__less__of__nat__cancel__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [I: num,N2: nat,X3: nat] : % 8.50/4.51 ( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) @ ( semiring_1_of_nat @ A @ X3 ) ) % 8.50/4.51 = ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2702_of__nat__less__numeral__power__cancel__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [X3: nat,I: num,N2: nat] : % 8.50/4.51 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) ) % 8.50/4.51 = ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2703_of__nat__le__numeral__power__cancel__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [X3: nat,I: num,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) ) % 8.50/4.51 = ( ord_less_eq @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2704_numeral__power__le__of__nat__cancel__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [I: num,N2: nat,X3: nat] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N2 ) @ ( semiring_1_of_nat @ A @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N2 ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2705_prod_Odisc__eq__case,axiom, % 8.50/4.51 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] : % 8.50/4.51 ( product_case_prod @ A @ B @ $o % 8.50/4.51 @ ^ [Uu3: A,Uv3: B] : $true % 8.50/4.51 @ Prod ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2706_int__sum,axiom, % 8.50/4.51 ! [B: $tType,F2: B > nat,A3: set @ B] : % 8.50/4.51 ( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A3 ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ B @ int % 8.50/4.51 @ ^ [X: B] : ( semiring_1_of_nat @ int @ ( F2 @ X ) ) % 8.50/4.51 @ A3 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2707_real__arch__simple,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim462609752435547400_field @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ? [N3: nat] : ( ord_less_eq @ A @ X3 @ ( semiring_1_of_nat @ A @ N3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2708_reals__Archimedean2,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim462609752435547400_field @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ? [N3: nat] : ( ord_less @ A @ X3 @ ( semiring_1_of_nat @ A @ N3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2709_mult__of__nat__commute,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_1 @ A ) % 8.50/4.51 => ! [X3: nat,Y: A] : % 8.50/4.51 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ X3 ) @ Y ) % 8.50/4.51 = ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2710_int__cases2,axiom, % 8.50/4.51 ! [Z2: int] : % 8.50/4.51 ( ! [N3: nat] : % 8.50/4.51 ( Z2 % 8.50/4.51 != ( semiring_1_of_nat @ int @ N3 ) ) % 8.50/4.51 => ~ ! [N3: nat] : % 8.50/4.51 ( Z2 % 8.50/4.51 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2711_arctan__less__iff,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( arctan @ X3 ) @ ( arctan @ Y ) ) % 8.50/4.51 = ( ord_less @ real @ X3 @ Y ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2712_arctan__monotone,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ X3 @ Y ) % 8.50/4.51 => ( ord_less @ real @ ( arctan @ X3 ) @ ( arctan @ Y ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2713_int__diff__cases,axiom, % 8.50/4.51 ! [Z2: int] : % 8.50/4.51 ~ ! [M3: nat,N3: nat] : % 8.50/4.51 ( Z2 % 8.50/4.51 != ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M3 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2714_of__nat__less__of__int__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [N2: nat,X3: int] : % 8.50/4.51 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( ring_1_of_int @ A @ X3 ) ) % 8.50/4.51 = ( ord_less @ int @ ( semiring_1_of_nat @ int @ N2 ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2715_of__nat__0__le__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linord181362715937106298miring @ A ) % 8.50/4.51 => ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2716_of__nat__less__0__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linord181362715937106298miring @ A ) % 8.50/4.51 => ! [M: nat] : % 8.50/4.51 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2717_of__nat__neq__0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_char_0 @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) % 8.50/4.51 != ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2718_of__nat__less__imp__less,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linord181362715937106298miring @ A ) % 8.50/4.51 => ! [M: nat,N2: nat] : % 8.50/4.51 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) % 8.50/4.51 => ( ord_less @ nat @ M @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2719_less__imp__of__nat__less,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linord181362715937106298miring @ A ) % 8.50/4.51 => ! [M: nat,N2: nat] : % 8.50/4.51 ( ( ord_less @ nat @ M @ N2 ) % 8.50/4.51 => ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2720_of__nat__mono,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linord181362715937106298miring @ A ) % 8.50/4.51 => ! [I: nat,J: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ I @ J ) % 8.50/4.51 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2721_int__ops_I1_J,axiom, % 8.50/4.51 ( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) ) % 8.50/4.51 = ( zero_zero @ int ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2722_nat__int__comparison_I2_J,axiom, % 8.50/4.51 ( ( ord_less @ nat ) % 8.50/4.51 = ( ^ [A4: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2723_int__of__nat__induct,axiom, % 8.50/4.51 ! [P: int > $o,Z2: int] : % 8.50/4.51 ( ! [N3: nat] : ( P @ ( semiring_1_of_nat @ int @ N3 ) ) % 8.50/4.51 => ( ! [N3: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) % 8.50/4.51 => ( P @ Z2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2724_int__cases,axiom, % 8.50/4.51 ! [Z2: int] : % 8.50/4.51 ( ! [N3: nat] : % 8.50/4.51 ( Z2 % 8.50/4.51 != ( semiring_1_of_nat @ int @ N3 ) ) % 8.50/4.51 => ~ ! [N3: nat] : % 8.50/4.51 ( Z2 % 8.50/4.51 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2725_zle__int,axiom, % 8.50/4.51 ! [M: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N2 ) ) % 8.50/4.51 = ( ord_less_eq @ nat @ M @ N2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2726_nat__int__comparison_I3_J,axiom, % 8.50/4.51 ( ( ord_less_eq @ nat ) % 8.50/4.51 = ( ^ [A4: nat,B4: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2727_zero__le__imp__eq__int,axiom, % 8.50/4.51 ! [K: int] : % 8.50/4.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 8.50/4.51 => ? [N3: nat] : % 8.50/4.51 ( K % 8.50/4.51 = ( semiring_1_of_nat @ int @ N3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2728_nonneg__int__cases,axiom, % 8.50/4.51 ! [K: int] : % 8.50/4.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 8.50/4.51 => ~ ! [N3: nat] : % 8.50/4.51 ( K % 8.50/4.51 != ( semiring_1_of_nat @ int @ N3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2729_zadd__int__left,axiom, % 8.50/4.51 ! [M: nat,N2: nat,Z2: int] : % 8.50/4.51 ( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ Z2 ) ) % 8.50/4.51 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N2 ) ) @ Z2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2730_int__ops_I5_J,axiom, % 8.50/4.51 ! [A2: nat,B2: nat] : % 8.50/4.51 ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A2 @ B2 ) ) % 8.50/4.51 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2731_int__plus,axiom, % 8.50/4.51 ! [N2: nat,M: nat] : % 8.50/4.51 ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N2 @ M ) ) % 8.50/4.51 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( semiring_1_of_nat @ int @ M ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2732_zle__iff__zadd,axiom, % 8.50/4.51 ( ( ord_less_eq @ int ) % 8.50/4.51 = ( ^ [W3: int,Z6: int] : % 8.50/4.51 ? [N: nat] : % 8.50/4.51 ( Z6 % 8.50/4.51 = ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ N ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2733_not__int__zless__negative,axiom, % 8.50/4.51 ! [N2: nat,M: nat] : % 8.50/4.51 ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2734_nat__less__as__int,axiom, % 8.50/4.51 ( ( ord_less @ nat ) % 8.50/4.51 = ( ^ [A4: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2735_nat__leq__as__int,axiom, % 8.50/4.51 ( ( ord_less_eq @ nat ) % 8.50/4.51 = ( ^ [A4: nat,B4: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2736_ex__less__of__nat__mult,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim462609752435547400_field @ A ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 8.50/4.51 => ? [N3: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2737_of__nat__diff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_1_cancel @ A ) % 8.50/4.51 => ! [N2: nat,M: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ N2 @ M ) % 8.50/4.51 => ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N2 ) ) % 8.50/4.51 = ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2738_reals__Archimedean3,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ! [Y5: real] : % 8.50/4.51 ? [N3: nat] : ( ord_less @ real @ Y5 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2739_int__cases4,axiom, % 8.50/4.51 ! [M: int] : % 8.50/4.51 ( ! [N3: nat] : % 8.50/4.51 ( M % 8.50/4.51 != ( semiring_1_of_nat @ int @ N3 ) ) % 8.50/4.51 => ~ ! [N3: nat] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 8.50/4.51 => ( M % 8.50/4.51 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2740_int__zle__neg,axiom, % 8.50/4.51 ! [N2: nat,M: nat] : % 8.50/4.51 ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) ) % 8.50/4.51 = ( ( N2 % 8.50/4.51 = ( zero_zero @ nat ) ) % 8.50/4.51 & ( M % 8.50/4.51 = ( zero_zero @ nat ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2741_int__ops_I4_J,axiom, % 8.50/4.51 ! [A2: nat] : % 8.50/4.51 ( ( semiring_1_of_nat @ int @ ( suc @ A2 ) ) % 8.50/4.51 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( one_one @ int ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2742_int__Suc,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) % 8.50/4.51 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( one_one @ int ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2743_zless__iff__Suc__zadd,axiom, % 8.50/4.51 ( ( ord_less @ int ) % 8.50/4.51 = ( ^ [W3: int,Z6: int] : % 8.50/4.51 ? [N: nat] : % 8.50/4.51 ( Z6 % 8.50/4.51 = ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2744_negative__zle__0,axiom, % 8.50/4.51 ! [N2: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( zero_zero @ int ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2745_nonpos__int__cases,axiom, % 8.50/4.51 ! [K: int] : % 8.50/4.51 ( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) ) % 8.50/4.51 => ~ ! [N3: nat] : % 8.50/4.51 ( K % 8.50/4.51 != ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2746_sum__roots__unity,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( ord_less @ nat @ ( one_one @ nat ) @ N2 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ complex @ complex % 8.50/4.51 @ ^ [X: complex] : X % 8.50/4.51 @ ( collect @ complex % 8.50/4.51 @ ^ [Z6: complex] : % 8.50/4.51 ( ( power_power @ complex @ Z6 @ N2 ) % 8.50/4.51 = ( one_one @ complex ) ) ) ) % 8.50/4.51 = ( zero_zero @ complex ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2747_sum__nth__roots,axiom, % 8.50/4.51 ! [N2: nat,C2: complex] : % 8.50/4.51 ( ( ord_less @ nat @ ( one_one @ nat ) @ N2 ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ complex @ complex % 8.50/4.51 @ ^ [X: complex] : X % 8.50/4.51 @ ( collect @ complex % 8.50/4.51 @ ^ [Z6: complex] : % 8.50/4.51 ( ( power_power @ complex @ Z6 @ N2 ) % 8.50/4.51 = C2 ) ) ) % 8.50/4.51 = ( zero_zero @ complex ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2748_mod__mult2__eq_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( euclid5411537665997757685th_nat @ A ) % 8.50/4.51 => ! [A2: A,M: nat,N2: nat] : % 8.50/4.51 ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) % 8.50/4.51 = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) @ ( modulo_modulo @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2749_pos__int__cases,axiom, % 8.50/4.51 ! [K: int] : % 8.50/4.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 8.50/4.51 => ~ ! [N3: nat] : % 8.50/4.51 ( ( K % 8.50/4.51 = ( semiring_1_of_nat @ int @ N3 ) ) % 8.50/4.51 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2750_zero__less__imp__eq__int,axiom, % 8.50/4.51 ! [K: int] : % 8.50/4.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ K ) % 8.50/4.51 => ? [N3: nat] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 8.50/4.51 & ( K % 8.50/4.51 = ( semiring_1_of_nat @ int @ N3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2751_int__cases3,axiom, % 8.50/4.51 ! [K: int] : % 8.50/4.51 ( ( K % 8.50/4.51 != ( zero_zero @ int ) ) % 8.50/4.51 => ( ! [N3: nat] : % 8.50/4.51 ( ( K % 8.50/4.51 = ( semiring_1_of_nat @ int @ N3 ) ) % 8.50/4.51 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) % 8.50/4.51 => ~ ! [N3: nat] : % 8.50/4.51 ( ( K % 8.50/4.51 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) % 8.50/4.51 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2752_nat__less__real__le,axiom, % 8.50/4.51 ( ( ord_less @ nat ) % 8.50/4.51 = ( ^ [N: nat,M2: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2753_nat__le__real__less,axiom, % 8.50/4.51 ( ( ord_less_eq @ nat ) % 8.50/4.51 = ( ^ [N: nat,M2: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( one_one @ real ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2754_zmult__zless__mono2__lemma,axiom, % 8.50/4.51 ! [I: int,J: int,K: nat] : % 8.50/4.51 ( ( ord_less @ int @ I @ J ) % 8.50/4.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.50/4.51 => ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ I ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ J ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2755_not__zle__0__negative,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2756_negative__zless__0,axiom, % 8.50/4.51 ! [N2: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) @ ( zero_zero @ int ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2757_negD,axiom, % 8.50/4.51 ! [X3: int] : % 8.50/4.51 ( ( ord_less @ int @ X3 @ ( zero_zero @ int ) ) % 8.50/4.51 => ? [N3: nat] : % 8.50/4.51 ( X3 % 8.50/4.51 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2758_int__ops_I6_J,axiom, % 8.50/4.51 ! [A2: nat,B2: nat] : % 8.50/4.51 ( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) % 8.50/4.51 => ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) ) % 8.50/4.51 = ( zero_zero @ int ) ) ) % 8.50/4.51 & ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) % 8.50/4.51 => ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) ) % 8.50/4.51 = ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2759_nat__approx__posE,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [E3: A] : % 8.50/4.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ E3 ) % 8.50/4.51 => ~ ! [N3: nat] : % 8.50/4.51 ~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ E3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2760_of__nat__less__two__power,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_idom @ A ) % 8.50/4.51 => ! [N2: nat] : ( ord_less @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2761_inverse__of__nat__le,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_field @ A ) % 8.50/4.51 => ! [N2: nat,M: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ N2 @ M ) % 8.50/4.51 => ( ( N2 % 8.50/4.51 != ( zero_zero @ nat ) ) % 8.50/4.51 => ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2762_real__archimedian__rdiv__eq__0,axiom, % 8.50/4.51 ! [X3: real,C2: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 ) % 8.50/4.51 => ( ! [M3: nat] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 ) % 8.50/4.51 => ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M3 ) @ X3 ) @ C2 ) ) % 8.50/4.51 => ( X3 % 8.50/4.51 = ( zero_zero @ real ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2763_neg__int__cases,axiom, % 8.50/4.51 ! [K: int] : % 8.50/4.51 ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 8.50/4.51 => ~ ! [N3: nat] : % 8.50/4.51 ( ( K % 8.50/4.51 = ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) % 8.50/4.51 => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2764_zdiff__int__split,axiom, % 8.50/4.51 ! [P: int > $o,X3: nat,Y: nat] : % 8.50/4.51 ( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X3 @ Y ) ) ) % 8.50/4.51 = ( ( ( ord_less_eq @ nat @ Y @ X3 ) % 8.50/4.51 => ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X3 ) @ ( semiring_1_of_nat @ int @ Y ) ) ) ) % 8.50/4.51 & ( ( ord_less @ nat @ X3 @ Y ) % 8.50/4.51 => ( P @ ( zero_zero @ int ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2765_ln__realpow,axiom, % 8.50/4.51 ! [X3: real,N2: nat] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ln_ln @ real @ ( power_power @ real @ X3 @ N2 ) ) % 8.50/4.51 = ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( ln_ln @ real @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2766_arctan__add,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) ) % 8.50/4.51 => ( ( plus_plus @ real @ ( arctan @ X3 ) @ ( arctan @ Y ) ) % 8.50/4.51 = ( arctan @ ( divide_divide @ real @ ( plus_plus @ real @ X3 @ Y ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( times_times @ real @ X3 @ Y ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2767_double__gauss__sum,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_semiring_1 @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) % 8.50/4.51 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2768_double__arith__series,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_semiring_1 @ A ) % 8.50/4.51 => ! [A2: A,D2: A,N2: nat] : % 8.50/4.51 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D2 ) ) % 8.50/4.51 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) % 8.50/4.51 = ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ D2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2769_gauss__sum,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( euclid5411537665997757685th_nat @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.50/4.51 = ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2770_arith__series,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( euclid5411537665997757685th_nat @ A ) % 8.50/4.51 => ! [A2: A,D2: A,N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D2 ) ) % 8.50/4.51 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.50/4.51 = ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2771_double__gauss__sum__from__Suc__0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_semiring_1 @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) ) % 8.50/4.51 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2772_sum__gp__offset,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( division_ring @ A ) % 8.50/4.51 & ( comm_ring @ A ) ) % 8.50/4.51 => ! [X3: A,M: nat,N2: nat] : % 8.50/4.51 ( ( ( X3 % 8.50/4.51 = ( one_one @ A ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N2 ) ) ) % 8.50/4.51 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) % 8.50/4.51 & ( ( X3 % 8.50/4.51 != ( one_one @ A ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N2 ) ) ) % 8.50/4.51 = ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2773_of__nat__code__if,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_1 @ A ) % 8.50/4.51 => ( ( semiring_1_of_nat @ A ) % 8.50/4.51 = ( ^ [N: nat] : % 8.50/4.51 ( if @ A % 8.50/4.51 @ ( N % 8.50/4.51 = ( zero_zero @ nat ) ) % 8.50/4.51 @ ( zero_zero @ A ) % 8.50/4.51 @ ( product_case_prod @ nat @ nat @ A % 8.50/4.51 @ ^ [M2: nat,Q5: nat] : % 8.50/4.51 ( if @ A % 8.50/4.51 @ ( Q5 % 8.50/4.51 = ( zero_zero @ nat ) ) % 8.50/4.51 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) ) % 8.50/4.51 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( one_one @ A ) ) ) % 8.50/4.51 @ ( divmod_nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2774_monoseq__arctan__series,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 8.50/4.51 => ( topological_monoseq @ real % 8.50/4.51 @ ^ [N: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2775_lemma__termdiff3,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.51 => ! [H2: A,Z2: A,K5: real,N2: nat] : % 8.50/4.51 ( ( H2 % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K5 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z2 @ H2 ) ) @ K5 ) % 8.50/4.51 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H2 ) @ N2 ) @ ( power_power @ A @ Z2 @ N2 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) @ ( times_times @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( semiring_1_of_nat @ real @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( power_power @ real @ K5 @ ( minus_minus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2776_ln__series,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) % 8.50/4.51 => ( ( ln_ln @ real @ X3 ) % 8.50/4.51 = ( suminf @ real % 8.50/4.51 @ ^ [N: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X3 @ ( one_one @ real ) ) @ ( suc @ N ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2777_lemma__termdiff2,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field @ A ) % 8.50/4.51 => ! [H2: A,Z2: A,N2: nat] : % 8.50/4.51 ( ( H2 % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H2 ) @ N2 ) @ ( power_power @ A @ Z2 @ N2 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) % 8.50/4.51 = ( times_times @ A @ H2 % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [P6: nat] : % 8.50/4.51 ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [Q5: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H2 ) @ Q5 ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ ( minus_minus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q5 ) ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ P6 ) ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2778_arctan__series,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 8.50/4.51 => ( ( arctan @ X3 ) % 8.50/4.51 = ( suminf @ real % 8.50/4.51 @ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2779_lessThan__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ord @ A ) % 8.50/4.51 => ! [I: A,K: A] : % 8.50/4.51 ( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) ) % 8.50/4.51 = ( ord_less @ A @ I @ K ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2780_finite__lessThan,axiom, % 8.50/4.51 ! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_lessThan @ nat @ K ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2781_lessThan__subset__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linorder @ A ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X3 ) @ ( set_ord_lessThan @ A @ Y ) ) % 8.50/4.51 = ( ord_less_eq @ A @ X3 @ Y ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2782_lessThan__0,axiom, % 8.50/4.51 ( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) ) % 8.50/4.51 = ( bot_bot @ ( set @ nat ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2783_single__Diff__lessThan,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( preorder @ A ) % 8.50/4.51 => ! [K: A] : % 8.50/4.51 ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K ) ) % 8.50/4.51 = ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2784_sum_OlessThan__Suc,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [G: nat > A,N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( G @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2785_powser__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V2822296259951069270ebra_1 @ A ) % 8.50/4.51 => ! [F2: nat > A] : % 8.50/4.51 ( ( suminf @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) ) % 8.50/4.51 = ( F2 @ ( zero_zero @ nat ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2786_int__int__eq,axiom, % 8.50/4.51 ! [M: nat,N2: nat] : % 8.50/4.51 ( ( ( semiring_1_of_nat @ int @ M ) % 8.50/4.51 = ( semiring_1_of_nat @ int @ N2 ) ) % 8.50/4.51 = ( M = N2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2787_lessThan__non__empty,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( no_bot @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( set_ord_lessThan @ A @ X3 ) % 8.50/4.51 != ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2788_infinite__Iio,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( linorder @ A ) % 8.50/4.51 & ( no_bot @ A ) ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ~ ( finite_finite2 @ A @ ( set_ord_lessThan @ A @ A2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2789_lessThan__def,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ord @ A ) % 8.50/4.51 => ( ( set_ord_lessThan @ A ) % 8.50/4.51 = ( ^ [U2: A] : % 8.50/4.51 ( collect @ A % 8.50/4.51 @ ^ [X: A] : ( ord_less @ A @ X @ U2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2790_Iio__eq__empty__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( linorder @ A ) % 8.50/4.51 & ( order_bot @ A ) ) % 8.50/4.51 => ! [N2: A] : % 8.50/4.51 ( ( ( set_ord_lessThan @ A @ N2 ) % 8.50/4.51 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.51 = ( N2 % 8.50/4.51 = ( bot_bot @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2791_lessThan__strict__subset__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linorder @ A ) % 8.50/4.51 => ! [M: A,N2: A] : % 8.50/4.51 ( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N2 ) ) % 8.50/4.51 = ( ord_less @ A @ M @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2792_lessThan__Suc,axiom, % 8.50/4.51 ! [K: nat] : % 8.50/4.51 ( ( set_ord_lessThan @ nat @ ( suc @ K ) ) % 8.50/4.51 = ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2793_lessThan__empty__iff,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( ( set_ord_lessThan @ nat @ N2 ) % 8.50/4.51 = ( bot_bot @ ( set @ nat ) ) ) % 8.50/4.51 = ( N2 % 8.50/4.51 = ( zero_zero @ nat ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2794_finite__nat__iff__bounded,axiom, % 8.50/4.51 ( ( finite_finite2 @ nat ) % 8.50/4.51 = ( ^ [S7: set @ nat] : % 8.50/4.51 ? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S7 @ ( set_ord_lessThan @ nat @ K3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2795_finite__nat__bounded,axiom, % 8.50/4.51 ! [S3: set @ nat] : % 8.50/4.51 ( ( finite_finite2 @ nat @ S3 ) % 8.50/4.51 => ? [K2: nat] : ( ord_less_eq @ ( set @ nat ) @ S3 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2796_sum_Onat__diff__reindex,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [G: nat > A,N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2797_sum__diff__distrib,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ord @ A ) % 8.50/4.51 => ! [Q: A > nat,P: A > nat,N2: A] : % 8.50/4.51 ( ! [X5: A] : ( ord_less_eq @ nat @ ( Q @ X5 ) @ ( P @ X5 ) ) % 8.50/4.51 => ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N2 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N2 ) ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ A @ nat % 8.50/4.51 @ ^ [X: A] : ( minus_minus @ nat @ ( P @ X ) @ ( Q @ X ) ) % 8.50/4.51 @ ( set_ord_lessThan @ A @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2798_lemma__NBseq__def2,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ B ) % 8.50/4.51 => ! [X8: A > B] : % 8.50/4.51 ( ( ? [K6: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ K6 ) % 8.50/4.51 & ! [N: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ K6 ) ) ) % 8.50/4.51 = ( ? [N5: nat] : % 8.50/4.51 ! [N: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2799_lemma__NBseq__def,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ B ) % 8.50/4.51 => ! [X8: A > B] : % 8.50/4.51 ( ( ? [K6: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ K6 ) % 8.50/4.51 & ! [N: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ K6 ) ) ) % 8.50/4.51 = ( ? [N5: nat] : % 8.50/4.51 ! [N: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2800_sum_OlessThan__Suc__shift,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [G: nat > A,N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2801_sum__lessThan__telescope,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ab_group_add @ A ) % 8.50/4.51 => ! [F2: nat > A,M: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ M ) ) % 8.50/4.51 = ( minus_minus @ A @ ( F2 @ M ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2802_sum__lessThan__telescope_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ab_group_add @ A ) % 8.50/4.51 => ! [F2: nat > A,M: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ M ) ) % 8.50/4.51 = ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ M ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2803_sum_OatLeast1__atMost__eq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [G: nat > A,N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2804_lemma__termdiff1,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( monoid_mult @ A ) % 8.50/4.51 & ( comm_ring @ A ) ) % 8.50/4.51 => ! [Z2: A,H2: A,M: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [P6: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H2 ) @ ( minus_minus @ nat @ M @ P6 ) ) @ ( power_power @ A @ Z2 @ P6 ) ) @ ( power_power @ A @ Z2 @ M ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ M ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ Z2 @ P6 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H2 ) @ ( minus_minus @ nat @ M @ P6 ) ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ M @ P6 ) ) ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ M ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2805_diff__power__eq__sum,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( monoid_mult @ A ) % 8.50/4.51 & ( comm_ring @ A ) ) % 8.50/4.51 => ! [X3: A,N2: nat,Y: A] : % 8.50/4.51 ( ( minus_minus @ A @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) @ ( power_power @ A @ Y @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( times_times @ A @ ( minus_minus @ A @ X3 @ Y ) % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X3 @ P6 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N2 @ P6 ) ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2806_power__diff__sumr2,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( monoid_mult @ A ) % 8.50/4.51 & ( comm_ring @ A ) ) % 8.50/4.51 => ! [X3: A,N2: nat,Y: A] : % 8.50/4.51 ( ( minus_minus @ A @ ( power_power @ A @ X3 @ N2 ) @ ( power_power @ A @ Y @ N2 ) ) % 8.50/4.51 = ( times_times @ A @ ( minus_minus @ A @ X3 @ Y ) % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power @ A @ X3 @ I3 ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2807_real__sum__nat__ivl__bounded2,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [N2: nat,F2: nat > A,K5: A,K: nat] : % 8.50/4.51 ( ! [P7: nat] : % 8.50/4.51 ( ( ord_less @ nat @ P7 @ N2 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( F2 @ P7 ) @ K5 ) ) % 8.50/4.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K5 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ K5 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2808_one__diff__power__eq_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( monoid_mult @ A ) % 8.50/4.51 & ( comm_ring @ A ) ) % 8.50/4.51 => ! [X3: A,N2: nat] : % 8.50/4.51 ( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ N2 ) ) % 8.50/4.51 = ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( power_power @ A @ X3 @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2809_sum__split__even__odd,axiom, % 8.50/4.51 ! [F2: nat > real,G: nat > real,N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.51 @ ^ [I3: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( F2 @ I3 ) @ ( G @ I3 ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) % 8.50/4.51 = ( plus_plus @ real % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.51 @ ^ [I3: nat] : ( F2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.51 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( one_one @ nat ) ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2810_norm__le__zero__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( zero_zero @ real ) ) % 8.50/4.51 = ( X3 % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2811_zero__less__norm__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) % 8.50/4.51 = ( X3 % 8.50/4.51 != ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2812_suminf__geometric,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.51 => ! [C2: A] : % 8.50/4.51 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) ) % 8.50/4.51 => ( ( suminf @ A @ ( power_power @ A @ C2 ) ) % 8.50/4.51 = ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2813_norm__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ( ( real_V7770717601297561774m_norm @ A @ ( zero_zero @ A ) ) % 8.50/4.51 = ( zero_zero @ real ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2814_norm__eq__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ( real_V7770717601297561774m_norm @ A @ X3 ) % 8.50/4.51 = ( zero_zero @ real ) ) % 8.50/4.51 = ( X3 % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2815_suminf__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( comm_monoid_add @ A ) % 8.50/4.51 & ( topological_t2_space @ A ) ) % 8.50/4.51 => ( ( suminf @ A % 8.50/4.51 @ ^ [N: nat] : ( zero_zero @ A ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2816_norm__not__less__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ~ ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( zero_zero @ real ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2817_norm__uminus__minus,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) ) % 8.50/4.51 = ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2818_nonzero__norm__divide,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.51 => ! [B2: A,A2: A] : % 8.50/4.51 ( ( B2 % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 8.50/4.51 = ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2819_power__eq__imp__eq__norm,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.51 => ! [W2: A,N2: nat,Z2: A] : % 8.50/4.51 ( ( ( power_power @ A @ W2 @ N2 ) % 8.50/4.51 = ( power_power @ A @ Z2 @ N2 ) ) % 8.50/4.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.51 => ( ( real_V7770717601297561774m_norm @ A @ W2 ) % 8.50/4.51 = ( real_V7770717601297561774m_norm @ A @ Z2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2820_norm__mult__less,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V4412858255891104859lgebra @ A ) % 8.50/4.51 => ! [X3: A,R2: real,Y: A,S: real] : % 8.50/4.51 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ R2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S ) % 8.50/4.51 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X3 @ Y ) ) @ ( times_times @ real @ R2 @ S ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2821_norm__triangle__lt,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [X3: A,Y: A,E3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E3 ) % 8.50/4.51 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ E3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2822_norm__add__less,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [X3: A,R2: real,Y: A,S: real] : % 8.50/4.51 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ R2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S ) % 8.50/4.51 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2823_norm__triangle__mono,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [A2: A,R2: real,B2: A,S: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ R2 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ S ) % 8.50/4.51 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2824_norm__triangle__ineq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [X3: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2825_norm__triangle__le,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [X3: A,Y: A,E3: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E3 ) % 8.50/4.51 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ E3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2826_norm__add__leD,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [A2: A,B2: A,C2: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ C2 ) % 8.50/4.51 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ C2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2827_norm__diff__triangle__less,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [X3: A,Y: A,E1: real,Z2: A,E22: real] : % 8.50/4.51 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ Y ) ) @ E1 ) % 8.50/4.51 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ Z2 ) ) @ E22 ) % 8.50/4.51 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ Z2 ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2828_norm__diff__ineq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2829_suminf__finite,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( comm_monoid_add @ A ) % 8.50/4.51 & ( topological_t2_space @ A ) ) % 8.50/4.51 => ! [N7: set @ nat,F2: nat > A] : % 8.50/4.51 ( ( finite_finite2 @ nat @ N7 ) % 8.50/4.51 => ( ! [N3: nat] : % 8.50/4.51 ( ~ ( member @ nat @ N3 @ N7 ) % 8.50/4.51 => ( ( F2 @ N3 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 => ( ( suminf @ A @ F2 ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N7 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2830_power__eq__1__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.51 => ! [W2: A,N2: nat] : % 8.50/4.51 ( ( ( power_power @ A @ W2 @ N2 ) % 8.50/4.51 = ( one_one @ A ) ) % 8.50/4.51 => ( ( ( real_V7770717601297561774m_norm @ A @ W2 ) % 8.50/4.51 = ( one_one @ real ) ) % 8.50/4.51 | ( N2 % 8.50/4.51 = ( zero_zero @ nat ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2831_norm__diff__triangle__ineq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [A2: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2832_sum__bounds__lt__plus1,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [F2: nat > A,Mm: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [K3: nat] : ( F2 @ ( suc @ K3 ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ Mm ) ) % 8.50/4.51 = ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2833_sumr__cos__zero__one,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.51 @ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ ( zero_zero @ real ) @ M2 ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( one_one @ real ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2834_pi__series,axiom, % 8.50/4.51 ( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 = ( suminf @ real % 8.50/4.51 @ ^ [K3: nat] : ( divide_divide @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2835_summable__arctan__series,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 8.50/4.51 => ( summable @ real % 8.50/4.51 @ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2836_ceiling__log__nat__eq__powr__iff,axiom, % 8.50/4.51 ! [B2: nat,K: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) % 8.50/4.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.50/4.51 => ( ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) ) % 8.50/4.51 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( one_one @ int ) ) ) % 8.50/4.51 = ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K ) % 8.50/4.51 & ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2837_geometric__deriv__sums,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [Z2: A] : % 8.50/4.51 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( one_one @ real ) ) % 8.50/4.51 => ( sums @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( power_power @ A @ Z2 @ N ) ) % 8.50/4.51 @ ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( minus_minus @ A @ ( one_one @ A ) @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2838_summable__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( comm_monoid_add @ A ) % 8.50/4.51 & ( topolo4958980785337419405_space @ A ) ) % 8.50/4.51 => ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2839_summable__single,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( comm_monoid_add @ A ) % 8.50/4.51 & ( topolo4958980785337419405_space @ A ) ) % 8.50/4.51 => ! [I: nat,F2: nat > A] : % 8.50/4.51 ( summable @ A % 8.50/4.51 @ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2840_sums__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( comm_monoid_add @ A ) % 8.50/4.51 & ( topolo4958980785337419405_space @ A ) ) % 8.50/4.51 => ( sums @ A % 8.50/4.51 @ ^ [N: nat] : ( zero_zero @ A ) % 8.50/4.51 @ ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2841_summable__iff__shift,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [F2: nat > A,K: nat] : % 8.50/4.51 ( ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) ) % 8.50/4.51 = ( summable @ A @ F2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2842_cos__coeff__0,axiom, % 8.50/4.51 ( ( cos_coeff @ ( zero_zero @ nat ) ) % 8.50/4.51 = ( one_one @ real ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2843_summable__cmult__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.51 => ! [C2: A,F2: nat > A] : % 8.50/4.51 ( ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) ) ) % 8.50/4.51 = ( ( C2 % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 | ( summable @ A @ F2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2844_summable__divide__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.51 => ! [F2: nat > A,C2: A] : % 8.50/4.51 ( ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( divide_divide @ A @ ( F2 @ N ) @ C2 ) ) % 8.50/4.51 = ( ( C2 % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 | ( summable @ A @ F2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2845_summable__If__finite,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( comm_monoid_add @ A ) % 8.50/4.51 & ( topolo4958980785337419405_space @ A ) ) % 8.50/4.51 => ! [P: nat > $o,F2: nat > A] : % 8.50/4.51 ( ( finite_finite2 @ nat @ ( collect @ nat @ P ) ) % 8.50/4.51 => ( summable @ A % 8.50/4.51 @ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2846_summable__If__finite__set,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( comm_monoid_add @ A ) % 8.50/4.51 & ( topolo4958980785337419405_space @ A ) ) % 8.50/4.51 => ! [A3: set @ nat,F2: nat > A] : % 8.50/4.51 ( ( finite_finite2 @ nat @ A3 ) % 8.50/4.51 => ( summable @ A % 8.50/4.51 @ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A3 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2847_log__eq__one,axiom, % 8.50/4.51 ! [A2: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.51 => ( ( A2 % 8.50/4.51 != ( one_one @ real ) ) % 8.50/4.51 => ( ( log @ A2 @ A2 ) % 8.50/4.51 = ( one_one @ real ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2848_log__less__cancel__iff,axiom, % 8.50/4.51 ! [A2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.51 => ( ( ord_less @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y ) ) % 8.50/4.51 = ( ord_less @ real @ X3 @ Y ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2849_log__less__one__cancel__iff,axiom, % 8.50/4.51 ! [A2: real,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( log @ A2 @ X3 ) @ ( one_one @ real ) ) % 8.50/4.51 = ( ord_less @ real @ X3 @ A2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2850_one__less__log__cancel__iff,axiom, % 8.50/4.51 ! [A2: real,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( one_one @ real ) @ ( log @ A2 @ X3 ) ) % 8.50/4.51 = ( ord_less @ real @ A2 @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2851_log__less__zero__cancel__iff,axiom, % 8.50/4.51 ! [A2: real,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( log @ A2 @ X3 ) @ ( zero_zero @ real ) ) % 8.50/4.51 = ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2852_zero__less__log__cancel__iff,axiom, % 8.50/4.51 ! [A2: real,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X3 ) ) % 8.50/4.51 = ( ord_less @ real @ ( one_one @ real ) @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2853_log__le__cancel__iff,axiom, % 8.50/4.51 ! [A2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y ) ) % 8.50/4.51 = ( ord_less_eq @ real @ X3 @ Y ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2854_log__le__one__cancel__iff,axiom, % 8.50/4.51 ! [A2: real,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( log @ A2 @ X3 ) @ ( one_one @ real ) ) % 8.50/4.51 = ( ord_less_eq @ real @ X3 @ A2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2855_one__le__log__cancel__iff,axiom, % 8.50/4.51 ! [A2: real,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( log @ A2 @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ real @ A2 @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2856_log__le__zero__cancel__iff,axiom, % 8.50/4.51 ! [A2: real,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( log @ A2 @ X3 ) @ ( zero_zero @ real ) ) % 8.50/4.51 = ( ord_less_eq @ real @ X3 @ ( one_one @ real ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2857_zero__le__log__cancel__iff,axiom, % 8.50/4.51 ! [A2: real,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2858_powser__sums__zero__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.51 => ! [A2: nat > A,X3: A] : % 8.50/4.51 ( ( sums @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( A2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) % 8.50/4.51 @ X3 ) % 8.50/4.51 = ( ( A2 @ ( zero_zero @ nat ) ) % 8.50/4.51 = X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2859_summable__geometric__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.51 => ! [C2: A] : % 8.50/4.51 ( ( summable @ A @ ( power_power @ A @ C2 ) ) % 8.50/4.51 = ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2860_log__pow__cancel,axiom, % 8.50/4.51 ! [A2: real,B2: nat] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.51 => ( ( A2 % 8.50/4.51 != ( one_one @ real ) ) % 8.50/4.51 => ( ( log @ A2 @ ( power_power @ real @ A2 @ B2 ) ) % 8.50/4.51 = ( semiring_1_of_nat @ real @ B2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2861_sums__0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( comm_monoid_add @ A ) % 8.50/4.51 & ( topolo4958980785337419405_space @ A ) ) % 8.50/4.51 => ! [F2: nat > A] : % 8.50/4.51 ( ! [N3: nat] : % 8.50/4.51 ( ( F2 @ N3 ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 => ( sums @ A @ F2 @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2862_sums__le,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 & ( topolo1944317154257567458pology @ A ) ) % 8.50/4.51 => ! [F2: nat > A,G: nat > A,S: A,T2: A] : % 8.50/4.51 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ N3 ) ) % 8.50/4.51 => ( ( sums @ A @ F2 @ S ) % 8.50/4.51 => ( ( sums @ A @ G @ T2 ) % 8.50/4.51 => ( ord_less_eq @ A @ S @ T2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2863_sums__single,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( comm_monoid_add @ A ) % 8.50/4.51 & ( topolo4958980785337419405_space @ A ) ) % 8.50/4.51 => ! [I: nat,F2: nat > A] : % 8.50/4.51 ( sums @ A % 8.50/4.51 @ ^ [R5: nat] : ( if @ A @ ( R5 = I ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) % 8.50/4.51 @ ( F2 @ I ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2864_sums__add,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( topolo5987344860129210374id_add @ A ) % 8.50/4.51 & ( topological_t2_space @ A ) ) % 8.50/4.51 => ! [F2: nat > A,A2: A,G: nat > A,B2: A] : % 8.50/4.51 ( ( sums @ A @ F2 @ A2 ) % 8.50/4.51 => ( ( sums @ A @ G @ B2 ) % 8.50/4.51 => ( sums @ A % 8.50/4.51 @ ^ [N: nat] : ( plus_plus @ A @ ( F2 @ N ) @ ( G @ N ) ) % 8.50/4.51 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2865_summable__comparison__test,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_Vector_banach @ A ) % 8.50/4.51 => ! [F2: nat > A,G: nat > real] : % 8.50/4.51 ( ? [N8: nat] : % 8.50/4.51 ! [N3: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ N8 @ N3 ) % 8.50/4.51 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) ) % 8.50/4.51 => ( ( summable @ real @ G ) % 8.50/4.51 => ( summable @ A @ F2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2866_summable__comparison__test_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_Vector_banach @ A ) % 8.50/4.51 => ! [G: nat > real,N7: nat,F2: nat > A] : % 8.50/4.51 ( ( summable @ real @ G ) % 8.50/4.51 => ( ! [N3: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ N7 @ N3 ) % 8.50/4.51 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) ) % 8.50/4.51 => ( summable @ A @ F2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2867_summable__const__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [C2: A] : % 8.50/4.51 ( ( summable @ A % 8.50/4.51 @ ^ [Uu3: nat] : C2 ) % 8.50/4.51 = ( C2 % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2868_summable__add,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( topolo5987344860129210374id_add @ A ) % 8.50/4.51 & ( topological_t2_space @ A ) ) % 8.50/4.51 => ! [F2: nat > A,G: nat > A] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ( summable @ A @ G ) % 8.50/4.51 => ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( plus_plus @ A @ ( F2 @ N ) @ ( G @ N ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2869_summable__Suc__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [F2: nat > A] : % 8.50/4.51 ( ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( F2 @ ( suc @ N ) ) ) % 8.50/4.51 = ( summable @ A @ F2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2870_summable__ignore__initial__segment,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [F2: nat > A,K: nat] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2871_powser__insidea,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.51 => ! [F2: nat > A,X3: A,Z2: A] : % 8.50/4.51 ( ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) % 8.50/4.51 => ( summable @ real % 8.50/4.51 @ ^ [N: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2872_suminf__le,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 & ( topolo1944317154257567458pology @ A ) ) % 8.50/4.51 => ! [F2: nat > A,G: nat > A] : % 8.50/4.51 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ N3 ) ) % 8.50/4.51 => ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ( summable @ A @ G ) % 8.50/4.51 => ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2873_summable__finite,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( comm_monoid_add @ A ) % 8.50/4.51 & ( topolo4958980785337419405_space @ A ) ) % 8.50/4.51 => ! [N7: set @ nat,F2: nat > A] : % 8.50/4.51 ( ( finite_finite2 @ nat @ N7 ) % 8.50/4.51 => ( ! [N3: nat] : % 8.50/4.51 ( ~ ( member @ nat @ N3 @ N7 ) % 8.50/4.51 => ( ( F2 @ N3 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 => ( summable @ A @ F2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2874_sums__mult__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( field @ A ) % 8.50/4.51 & ( real_V4412858255891104859lgebra @ A ) ) % 8.50/4.51 => ! [C2: A,F2: nat > A,D2: A] : % 8.50/4.51 ( ( C2 % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( sums @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) ) % 8.50/4.51 @ ( times_times @ A @ C2 @ D2 ) ) % 8.50/4.51 = ( sums @ A @ F2 @ D2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2875_sums__mult2__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( field @ A ) % 8.50/4.51 & ( real_V4412858255891104859lgebra @ A ) ) % 8.50/4.51 => ! [C2: A,F2: nat > A,D2: A] : % 8.50/4.51 ( ( C2 % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( sums @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ C2 ) % 8.50/4.51 @ ( times_times @ A @ D2 @ C2 ) ) % 8.50/4.51 = ( sums @ A @ F2 @ D2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2876_summable__mult__D,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.51 => ! [C2: A,F2: nat > A] : % 8.50/4.51 ( ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) ) ) % 8.50/4.51 => ( ( C2 % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( summable @ A @ F2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2877_summable__zero__power,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( comm_ring_1 @ A ) % 8.50/4.51 & ( topolo4958980785337419405_space @ A ) ) % 8.50/4.51 => ( summable @ A @ ( power_power @ A @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2878_pi__not__less__zero,axiom, % 8.50/4.51 ~ ( ord_less @ real @ pi @ ( zero_zero @ real ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2879_pi__gt__zero,axiom, % 8.50/4.51 ord_less @ real @ ( zero_zero @ real ) @ pi ). % 8.50/4.51 % 8.50/4.51 thf(fact_2880_suminf__add,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( topolo5987344860129210374id_add @ A ) % 8.50/4.51 & ( topological_t2_space @ A ) ) % 8.50/4.51 => ! [F2: nat > A,G: nat > A] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ( summable @ A @ G ) % 8.50/4.51 => ( ( plus_plus @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) ) % 8.50/4.51 = ( suminf @ A % 8.50/4.51 @ ^ [N: nat] : ( plus_plus @ A @ ( F2 @ N ) @ ( G @ N ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2881_suminf__eq__zero__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 & ( topolo1944317154257567458pology @ A ) ) % 8.50/4.51 => ! [F2: nat > A] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) % 8.50/4.51 => ( ( ( suminf @ A @ F2 ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 = ( ! [N: nat] : % 8.50/4.51 ( ( F2 @ N ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2882_suminf__nonneg,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 & ( topolo1944317154257567458pology @ A ) ) % 8.50/4.51 => ! [F2: nat > A] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) % 8.50/4.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2883_suminf__pos,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 & ( topolo1944317154257567458pology @ A ) ) % 8.50/4.51 => ! [F2: nat > A] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ! [N3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) % 8.50/4.51 => ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2884_sums__mult__D,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.51 => ! [C2: A,F2: nat > A,A2: A] : % 8.50/4.51 ( ( sums @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ C2 @ ( F2 @ N ) ) % 8.50/4.51 @ A2 ) % 8.50/4.51 => ( ( C2 % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( sums @ A @ F2 @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2885_sums__Suc__imp,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [F2: nat > A,S: A] : % 8.50/4.51 ( ( ( F2 @ ( zero_zero @ nat ) ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 => ( ( sums @ A % 8.50/4.51 @ ^ [N: nat] : ( F2 @ ( suc @ N ) ) % 8.50/4.51 @ S ) % 8.50/4.51 => ( sums @ A @ F2 @ S ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2886_sums__Suc,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( topolo5987344860129210374id_add @ A ) % 8.50/4.51 & ( topological_t2_space @ A ) ) % 8.50/4.51 => ! [F2: nat > A,L: A] : % 8.50/4.51 ( ( sums @ A % 8.50/4.51 @ ^ [N: nat] : ( F2 @ ( suc @ N ) ) % 8.50/4.51 @ L ) % 8.50/4.51 => ( sums @ A @ F2 @ ( plus_plus @ A @ L @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2887_sums__Suc__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [F2: nat > A,S: A] : % 8.50/4.51 ( ( sums @ A % 8.50/4.51 @ ^ [N: nat] : ( F2 @ ( suc @ N ) ) % 8.50/4.51 @ S ) % 8.50/4.51 = ( sums @ A @ F2 @ ( plus_plus @ A @ S @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2888_sums__zero__iff__shift,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [N2: nat,F2: nat > A,S: A] : % 8.50/4.51 ( ! [I4: nat] : % 8.50/4.51 ( ( ord_less @ nat @ I4 @ N2 ) % 8.50/4.51 => ( ( F2 @ I4 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 => ( ( sums @ A % 8.50/4.51 @ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) ) % 8.50/4.51 @ S ) % 8.50/4.51 = ( sums @ A @ F2 @ S ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2889_summable__0__powser,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.51 => ! [F2: nat > A] : % 8.50/4.51 ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2890_summable__zero__power_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ring_1 @ A ) % 8.50/4.51 & ( topolo4958980785337419405_space @ A ) ) % 8.50/4.51 => ! [F2: nat > A] : % 8.50/4.51 ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2891_summable__powser__split__head,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.51 => ! [F2: nat > A,Z2: A] : % 8.50/4.51 ( ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z2 @ N ) ) ) % 8.50/4.51 = ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2892_powser__split__head_I3_J,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V8999393235501362500lgebra @ A ) ) % 8.50/4.51 => ! [F2: nat > A,Z2: A] : % 8.50/4.51 ( ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) % 8.50/4.51 => ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z2 @ N ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2893_summable__powser__ignore__initial__segment,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.51 => ! [F2: nat > A,M: nat,Z2: A] : % 8.50/4.51 ( ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( plus_plus @ nat @ N @ M ) ) @ ( power_power @ A @ Z2 @ N ) ) ) % 8.50/4.51 = ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2894_sums__finite,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( comm_monoid_add @ A ) % 8.50/4.51 & ( topolo4958980785337419405_space @ A ) ) % 8.50/4.51 => ! [N7: set @ nat,F2: nat > A] : % 8.50/4.51 ( ( finite_finite2 @ nat @ N7 ) % 8.50/4.51 => ( ! [N3: nat] : % 8.50/4.51 ( ~ ( member @ nat @ N3 @ N7 ) % 8.50/4.51 => ( ( F2 @ N3 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 => ( sums @ A @ F2 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N7 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2895_sums__If__finite,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( comm_monoid_add @ A ) % 8.50/4.51 & ( topolo4958980785337419405_space @ A ) ) % 8.50/4.51 => ! [P: nat > $o,F2: nat > A] : % 8.50/4.51 ( ( finite_finite2 @ nat @ ( collect @ nat @ P ) ) % 8.50/4.51 => ( sums @ A % 8.50/4.51 @ ^ [R5: nat] : ( if @ A @ ( P @ R5 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( collect @ nat @ P ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2896_sums__If__finite__set,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( comm_monoid_add @ A ) % 8.50/4.51 & ( topolo4958980785337419405_space @ A ) ) % 8.50/4.51 => ! [A3: set @ nat,F2: nat > A] : % 8.50/4.51 ( ( finite_finite2 @ nat @ A3 ) % 8.50/4.51 => ( sums @ A % 8.50/4.51 @ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A3 ) @ ( F2 @ R5 ) @ ( zero_zero @ A ) ) % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2897_summable__norm__comparison__test,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [F2: nat > A,G: nat > real] : % 8.50/4.51 ( ? [N8: nat] : % 8.50/4.51 ! [N3: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ N8 @ N3 ) % 8.50/4.51 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) ) % 8.50/4.51 => ( ( summable @ real @ G ) % 8.50/4.51 => ( summable @ real % 8.50/4.51 @ ^ [N: nat] : ( real_V7770717601297561774m_norm @ A @ ( F2 @ N ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2898_summable__rabs__comparison__test,axiom, % 8.50/4.51 ! [F2: nat > real,G: nat > real] : % 8.50/4.51 ( ? [N8: nat] : % 8.50/4.51 ! [N3: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ N8 @ N3 ) % 8.50/4.51 => ( ord_less_eq @ real @ ( abs_abs @ real @ ( F2 @ N3 ) ) @ ( G @ N3 ) ) ) % 8.50/4.51 => ( ( summable @ real @ G ) % 8.50/4.51 => ( summable @ real % 8.50/4.51 @ ^ [N: nat] : ( abs_abs @ real @ ( F2 @ N ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2899_suminf__pos__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 & ( topolo1944317154257567458pology @ A ) ) % 8.50/4.51 => ! [F2: nat > A] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) % 8.50/4.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) % 8.50/4.51 = ( ? [I3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2900_suminf__pos2,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 & ( topolo1944317154257567458pology @ A ) ) % 8.50/4.51 => ! [F2: nat > A,I: nat] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) % 8.50/4.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) ) % 8.50/4.51 => ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2901_suminf__le__const,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 & ( topolo1944317154257567458pology @ A ) ) % 8.50/4.51 => ! [F2: nat > A,X3: A] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X3 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( suminf @ A @ F2 ) @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2902_powser__sums__if,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ring_1 @ A ) % 8.50/4.51 & ( topolo4958980785337419405_space @ A ) ) % 8.50/4.51 => ! [M: nat,Z2: A] : % 8.50/4.51 ( sums @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( if @ A @ ( N = M ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z2 @ N ) ) % 8.50/4.51 @ ( power_power @ A @ Z2 @ M ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2903_powser__sums__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.51 => ! [A2: nat > A] : % 8.50/4.51 ( sums @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( A2 @ N ) @ ( power_power @ A @ ( zero_zero @ A ) @ N ) ) % 8.50/4.51 @ ( A2 @ ( zero_zero @ nat ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2904_log__base__change,axiom, % 8.50/4.51 ! [A2: real,B2: real,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.51 => ( ( A2 % 8.50/4.51 != ( one_one @ real ) ) % 8.50/4.51 => ( ( log @ B2 @ X3 ) % 8.50/4.51 = ( divide_divide @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ B2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2905_log__of__power__eq,axiom, % 8.50/4.51 ! [M: nat,B2: real,N2: nat] : % 8.50/4.51 ( ( ( semiring_1_of_nat @ real @ M ) % 8.50/4.51 = ( power_power @ real @ B2 @ N2 ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 8.50/4.51 => ( ( semiring_1_of_nat @ real @ N2 ) % 8.50/4.51 = ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2906_less__log__of__power,axiom, % 8.50/4.51 ! [B2: real,N2: nat,M: real] : % 8.50/4.51 ( ( ord_less @ real @ ( power_power @ real @ B2 @ N2 ) @ M ) % 8.50/4.51 => ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 8.50/4.51 => ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ M ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2907_powser__inside,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V8999393235501362500lgebra @ A ) ) % 8.50/4.51 => ! [F2: nat > A,X3: A,Z2: A] : % 8.50/4.51 ( ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) % 8.50/4.51 => ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2908_sums__iff__shift,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [F2: nat > A,N2: nat,S: A] : % 8.50/4.51 ( ( sums @ A % 8.50/4.51 @ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) ) % 8.50/4.51 @ S ) % 8.50/4.51 = ( sums @ A @ F2 @ ( plus_plus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2909_summableI__nonneg__bounded,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( condit6923001295902523014norder @ A ) % 8.50/4.51 & ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 & ( topolo1944317154257567458pology @ A ) ) % 8.50/4.51 => ! [F2: nat > A,X3: A] : % 8.50/4.51 ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) % 8.50/4.51 => ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X3 ) % 8.50/4.51 => ( summable @ A @ F2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2910_sums__split__initial__segment,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [F2: nat > A,S: A,N2: nat] : % 8.50/4.51 ( ( sums @ A @ F2 @ S ) % 8.50/4.51 => ( sums @ A % 8.50/4.51 @ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) ) % 8.50/4.51 @ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2911_sums__iff__shift_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [F2: nat > A,N2: nat,S: A] : % 8.50/4.51 ( ( sums @ A % 8.50/4.51 @ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N2 ) ) % 8.50/4.51 @ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) % 8.50/4.51 = ( sums @ A @ F2 @ S ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2912_complete__algebra__summable__geometric,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( one_one @ real ) ) % 8.50/4.51 => ( summable @ A @ ( power_power @ A @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2913_summable__geometric,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.51 => ! [C2: A] : % 8.50/4.51 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) ) % 8.50/4.51 => ( summable @ A @ ( power_power @ A @ C2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2914_sums__If__finite__set_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( topolo1287966508704411220up_add @ A ) % 8.50/4.51 & ( topological_t2_space @ A ) ) % 8.50/4.51 => ! [G: nat > A,S3: A,A3: set @ nat,S6: A,F2: nat > A] : % 8.50/4.51 ( ( sums @ A @ G @ S3 ) % 8.50/4.51 => ( ( finite_finite2 @ nat @ A3 ) % 8.50/4.51 => ( ( S6 % 8.50/4.51 = ( plus_plus @ A @ S3 % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [N: nat] : ( minus_minus @ A @ ( F2 @ N ) @ ( G @ N ) ) % 8.50/4.51 @ A3 ) ) ) % 8.50/4.51 => ( sums @ A % 8.50/4.51 @ ^ [N: nat] : ( if @ A @ ( member @ nat @ N @ A3 ) @ ( F2 @ N ) @ ( G @ N ) ) % 8.50/4.51 @ S6 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2915_suminf__split__head,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [F2: nat > A] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ( suminf @ A % 8.50/4.51 @ ^ [N: nat] : ( F2 @ ( suc @ N ) ) ) % 8.50/4.51 = ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2916_pi__less__4,axiom, % 8.50/4.51 ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2917_sum__le__suminf,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.51 & ( topolo1944317154257567458pology @ A ) ) % 8.50/4.51 => ! [F2: nat > A,I6: set @ nat] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ( finite_finite2 @ nat @ I6 ) % 8.50/4.51 => ( ! [N3: nat] : % 8.50/4.51 ( ( member @ nat @ N3 @ ( uminus_uminus @ ( set @ nat ) @ I6 ) ) % 8.50/4.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N3 ) ) ) % 8.50/4.51 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ I6 ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2918_log__mult,axiom, % 8.50/4.51 ! [A2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.51 => ( ( A2 % 8.50/4.51 != ( one_one @ real ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.51 => ( ( log @ A2 @ ( times_times @ real @ X3 @ Y ) ) % 8.50/4.51 = ( plus_plus @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2919_log__divide,axiom, % 8.50/4.51 ! [A2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.51 => ( ( A2 % 8.50/4.51 != ( one_one @ real ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.51 => ( ( log @ A2 @ ( divide_divide @ real @ X3 @ Y ) ) % 8.50/4.51 = ( minus_minus @ real @ ( log @ A2 @ X3 ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2920_le__log__of__power,axiom, % 8.50/4.51 ! [B2: real,N2: nat,M: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( power_power @ real @ B2 @ N2 ) @ M ) % 8.50/4.51 => ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 8.50/4.51 => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ M ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2921_log__base__pow,axiom, % 8.50/4.51 ! [A2: real,N2: nat,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.51 => ( ( log @ ( power_power @ real @ A2 @ N2 ) @ X3 ) % 8.50/4.51 = ( divide_divide @ real @ ( log @ A2 @ X3 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2922_log__nat__power,axiom, % 8.50/4.51 ! [X3: real,B2: real,N2: nat] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( log @ B2 @ ( power_power @ real @ X3 @ N2 ) ) % 8.50/4.51 = ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2923_suminf__split__initial__segment,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [F2: nat > A,K: nat] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ( suminf @ A @ F2 ) % 8.50/4.51 = ( plus_plus @ A % 8.50/4.51 @ ( suminf @ A % 8.50/4.51 @ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) ) % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2924_suminf__minus__initial__segment,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [F2: nat > A,K: nat] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ( suminf @ A % 8.50/4.51 @ ^ [N: nat] : ( F2 @ ( plus_plus @ nat @ N @ K ) ) ) % 8.50/4.51 = ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2925_pi__half__less__two,axiom, % 8.50/4.51 ord_less @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2926_sum__less__suminf,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ordere8940638589300402666id_add @ A ) % 8.50/4.51 & ( topolo1944317154257567458pology @ A ) ) % 8.50/4.51 => ! [F2: nat > A,N2: nat] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ! [M3: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ N2 @ M3 ) % 8.50/4.51 => ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ M3 ) ) ) % 8.50/4.51 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2927_powser__split__head_I1_J,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V8999393235501362500lgebra @ A ) ) % 8.50/4.51 => ! [F2: nat > A,Z2: A] : % 8.50/4.51 ( ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) % 8.50/4.51 => ( ( suminf @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) % 8.50/4.51 = ( plus_plus @ A @ ( F2 @ ( zero_zero @ nat ) ) % 8.50/4.51 @ ( times_times @ A % 8.50/4.51 @ ( suminf @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z2 @ N ) ) ) % 8.50/4.51 @ Z2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2928_powser__split__head_I2_J,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V8999393235501362500lgebra @ A ) ) % 8.50/4.51 => ! [F2: nat > A,Z2: A] : % 8.50/4.51 ( ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) % 8.50/4.51 => ( ( times_times @ A % 8.50/4.51 @ ( suminf @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ ( suc @ N ) ) @ ( power_power @ A @ Z2 @ N ) ) ) % 8.50/4.51 @ Z2 ) % 8.50/4.51 = ( minus_minus @ A % 8.50/4.51 @ ( suminf @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( F2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) % 8.50/4.51 @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2929_log__of__power__less,axiom, % 8.50/4.51 ! [M: nat,B2: real,N2: nat] : % 8.50/4.51 ( ( ord_less @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N2 ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 8.50/4.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.50/4.51 => ( ord_less @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2930_log__eq__div__ln__mult__log,axiom, % 8.50/4.51 ! [A2: real,B2: real,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.51 => ( ( A2 % 8.50/4.51 != ( one_one @ real ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 8.50/4.51 => ( ( B2 % 8.50/4.51 != ( one_one @ real ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( log @ A2 @ X3 ) % 8.50/4.51 = ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( ln_ln @ real @ A2 ) ) @ ( log @ B2 @ X3 ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2931_summable__partial__sum__bound,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_Vector_banach @ A ) % 8.50/4.51 => ! [F2: nat > A,E3: real] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 ) % 8.50/4.51 => ~ ! [N9: nat] : % 8.50/4.51 ~ ! [M6: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ N9 @ M6 ) % 8.50/4.51 => ! [N6: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ M6 @ N6 ) ) ) @ E3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2932_suminf__exist__split,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [R2: real,F2: nat > A] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 ) % 8.50/4.51 => ( ( summable @ A @ F2 ) % 8.50/4.51 => ? [N9: nat] : % 8.50/4.51 ! [N6: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ N9 @ N6 ) % 8.50/4.51 => ( ord_less @ real % 8.50/4.51 @ ( real_V7770717601297561774m_norm @ A % 8.50/4.51 @ ( suminf @ A % 8.50/4.51 @ ^ [I3: nat] : ( F2 @ ( plus_plus @ nat @ I3 @ N6 ) ) ) ) % 8.50/4.51 @ R2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2933_summable__power__series,axiom, % 8.50/4.51 ! [F2: nat > real,Z2: real] : % 8.50/4.51 ( ! [I4: nat] : ( ord_less_eq @ real @ ( F2 @ I4 ) @ ( one_one @ real ) ) % 8.50/4.51 => ( ! [I4: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ I4 ) ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Z2 ) % 8.50/4.51 => ( ( ord_less @ real @ Z2 @ ( one_one @ real ) ) % 8.50/4.51 => ( summable @ real % 8.50/4.51 @ ^ [I3: nat] : ( times_times @ real @ ( F2 @ I3 ) @ ( power_power @ real @ Z2 @ I3 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2934_Abel__lemma,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 => ! [R2: real,R0: real,A2: nat > A,M7: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 ) % 8.50/4.51 => ( ( ord_less @ real @ R2 @ R0 ) % 8.50/4.51 => ( ! [N3: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N3 ) ) @ ( power_power @ real @ R0 @ N3 ) ) @ M7 ) % 8.50/4.51 => ( summable @ real % 8.50/4.51 @ ^ [N: nat] : ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N ) ) @ ( power_power @ real @ R2 @ N ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2935_pi__half__gt__zero,axiom, % 8.50/4.51 ord_less @ real @ ( zero_zero @ real ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2936_m2pi__less__pi,axiom, % 8.50/4.51 ord_less @ real @ ( uminus_uminus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) @ pi ). % 8.50/4.51 % 8.50/4.51 thf(fact_2937_summable__ratio__test,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_Vector_banach @ A ) % 8.50/4.51 => ! [C2: real,N7: nat,F2: nat > A] : % 8.50/4.51 ( ( ord_less @ real @ C2 @ ( one_one @ real ) ) % 8.50/4.51 => ( ! [N3: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ N7 @ N3 ) % 8.50/4.51 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ ( suc @ N3 ) ) ) @ ( times_times @ real @ C2 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N3 ) ) ) ) ) % 8.50/4.51 => ( summable @ A @ F2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2938_sum__less__suminf2,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ordere8940638589300402666id_add @ A ) % 8.50/4.51 & ( topolo1944317154257567458pology @ A ) ) % 8.50/4.51 => ! [F2: nat > A,N2: nat,I: nat] : % 8.50/4.51 ( ( summable @ A @ F2 ) % 8.50/4.51 => ( ! [M3: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ N2 @ M3 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ M3 ) ) ) % 8.50/4.51 => ( ( ord_less_eq @ nat @ N2 @ I ) % 8.50/4.51 => ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) ) % 8.50/4.51 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2939_arctan__ubound,axiom, % 8.50/4.51 ! [Y: real] : ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2940_geometric__sums,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.51 => ! [C2: A] : % 8.50/4.51 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) ) % 8.50/4.51 => ( sums @ A @ ( power_power @ A @ C2 ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2941_power__half__series,axiom, % 8.50/4.51 ( sums @ real % 8.50/4.51 @ ^ [N: nat] : ( power_power @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( suc @ N ) ) % 8.50/4.51 @ ( one_one @ real ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2942_log__of__power__le,axiom, % 8.50/4.51 ! [M: nat,B2: real,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N2 ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 8.50/4.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.50/4.51 => ( ord_less_eq @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2943_minus__pi__half__less__zero,axiom, % 8.50/4.51 ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ real ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2944_arctan__lbound,axiom, % 8.50/4.51 ! [Y: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2945_arctan__bounded,axiom, % 8.50/4.51 ! [Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) ) % 8.50/4.51 & ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2946_less__log2__of__power,axiom, % 8.50/4.51 ! [N2: nat,M: nat] : % 8.50/4.51 ( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M ) % 8.50/4.51 => ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2947_le__log2__of__power,axiom, % 8.50/4.51 ! [N2: nat,M: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M ) % 8.50/4.51 => ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2948_log2__of__power__less,axiom, % 8.50/4.51 ! [M: nat,N2: nat] : % 8.50/4.51 ( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.50/4.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.50/4.51 => ( ord_less @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2949_log2__of__power__le,axiom, % 8.50/4.51 ! [M: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.50/4.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.50/4.51 => ( ord_less_eq @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2950_sum__pos__lt__pair,axiom, % 8.50/4.51 ! [F2: nat > real,K: nat] : % 8.50/4.51 ( ( summable @ real @ F2 ) % 8.50/4.51 => ( ! [D3: nat] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( F2 @ ( plus_plus @ nat @ K @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D3 ) ) ) @ ( F2 @ ( plus_plus @ nat @ K @ ( plus_plus @ nat @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D3 ) @ ( one_one @ nat ) ) ) ) ) ) % 8.50/4.51 => ( ord_less @ real @ ( groups7311177749621191930dd_sum @ nat @ real @ F2 @ ( set_ord_lessThan @ nat @ K ) ) @ ( suminf @ real @ F2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2951_ceiling__log__nat__eq__if,axiom, % 8.50/4.51 ! [B2: nat,N2: nat,K: nat] : % 8.50/4.51 ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K ) % 8.50/4.51 => ( ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) % 8.50/4.51 => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) % 8.50/4.51 => ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) ) % 8.50/4.51 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( one_one @ int ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2952_ceiling__log2__div2,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.50/4.51 => ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) % 8.50/4.51 = ( plus_plus @ int @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) @ ( one_one @ int ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2953_sin__cos__npi,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( sin @ real @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 = ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2954_diffs__equiv,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_V822414075346904944vector @ A ) % 8.50/4.51 & ( ring_1 @ A ) ) % 8.50/4.51 => ! [C2: nat > A,X3: A] : % 8.50/4.51 ( ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) % 8.50/4.51 => ( sums @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( C2 @ N ) ) @ ( power_power @ A @ X3 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) % 8.50/4.51 @ ( suminf @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2955_cos__pi__eq__zero,axiom, % 8.50/4.51 ! [M: nat] : % 8.50/4.51 ( ( cos @ real @ ( divide_divide @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 = ( zero_zero @ real ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2956_ceiling__log__eq__powr__iff,axiom, % 8.50/4.51 ! [X3: real,B2: real,K: nat] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 8.50/4.51 => ( ( ( archimedean_ceiling @ real @ ( log @ B2 @ X3 ) ) % 8.50/4.51 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) ) ) % 8.50/4.51 = ( ( ord_less @ real @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ K ) ) @ X3 ) % 8.50/4.51 & ( ord_less_eq @ real @ X3 @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2957_floor__log__nat__eq__powr__iff,axiom, % 8.50/4.51 ! [B2: nat,K: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) % 8.50/4.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.50/4.51 => ( ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) ) % 8.50/4.51 = ( semiring_1_of_nat @ int @ N2 ) ) % 8.50/4.51 = ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K ) % 8.50/4.51 & ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2958_sin__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.51 => ( ( sin @ A @ ( zero_zero @ A ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2959_powr__eq__0__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ln @ A ) % 8.50/4.51 => ! [W2: A,Z2: A] : % 8.50/4.51 ( ( ( powr @ A @ W2 @ Z2 ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 = ( W2 % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2960_powr__0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ln @ A ) % 8.50/4.51 => ! [Z2: A] : % 8.50/4.51 ( ( powr @ A @ ( zero_zero @ A ) @ Z2 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2961_cos__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.51 => ( ( cos @ A @ ( zero_zero @ A ) ) % 8.50/4.51 = ( one_one @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2962_powr__zero__eq__one,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ln @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ( X3 % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 => ( ( powr @ A @ X3 @ ( zero_zero @ A ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) % 8.50/4.51 & ( ( X3 % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( powr @ A @ X3 @ ( zero_zero @ A ) ) % 8.50/4.51 = ( one_one @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2963_floor__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) ) % 8.50/4.51 = ( zero_zero @ int ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2964_powr__gt__zero,axiom, % 8.50/4.51 ! [X3: real,A2: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( powr @ real @ X3 @ A2 ) ) % 8.50/4.51 = ( X3 % 8.50/4.51 != ( zero_zero @ real ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2965_powr__less__cancel__iff,axiom, % 8.50/4.51 ! [X3: real,A2: real,B2: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) ) % 8.50/4.51 = ( ord_less @ real @ A2 @ B2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2966_powr__eq__one__iff,axiom, % 8.50/4.51 ! [A2: real,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 8.50/4.51 => ( ( ( powr @ real @ A2 @ X3 ) % 8.50/4.51 = ( one_one @ real ) ) % 8.50/4.51 = ( X3 % 8.50/4.51 = ( zero_zero @ real ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2967_powr__le__cancel__iff,axiom, % 8.50/4.51 ! [X3: real,A2: real,B2: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) ) % 8.50/4.51 = ( ord_less_eq @ real @ A2 @ B2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2968_zero__le__floor,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2969_sin__cos__squared__add3,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ X3 ) ) @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ X3 ) ) ) % 8.50/4.51 = ( one_one @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2970_floor__less__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( zero_zero @ int ) ) % 8.50/4.51 = ( ord_less @ A @ X3 @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2971_numeral__le__floor,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [V2: num,X3: A] : % 8.50/4.51 ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ A @ ( numeral_numeral @ A @ V2 ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2972_zero__less__floor,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2973_floor__le__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( zero_zero @ int ) ) % 8.50/4.51 = ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2974_floor__less__numeral,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A,V2: num] : % 8.50/4.51 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) % 8.50/4.51 = ( ord_less @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2975_one__le__floor,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2976_floor__less__one,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) % 8.50/4.51 = ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2977_powr__log__cancel,axiom, % 8.50/4.51 ! [A2: real,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.51 => ( ( A2 % 8.50/4.51 != ( one_one @ real ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( powr @ real @ A2 @ ( log @ A2 @ X3 ) ) % 8.50/4.51 = X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2978_log__powr__cancel,axiom, % 8.50/4.51 ! [A2: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.51 => ( ( A2 % 8.50/4.51 != ( one_one @ real ) ) % 8.50/4.51 => ( ( log @ A2 @ ( powr @ real @ A2 @ Y ) ) % 8.50/4.51 = Y ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2979_numeral__less__floor,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [V2: num,X3: A] : % 8.50/4.51 ( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2980_floor__le__numeral,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A,V2: num] : % 8.50/4.51 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) % 8.50/4.51 = ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2981_one__less__floor,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2982_floor__le__one,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) % 8.50/4.51 = ( ord_less @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2983_neg__numeral__le__floor,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [V2: num,X3: A] : % 8.50/4.51 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2984_floor__less__neg__numeral,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A,V2: num] : % 8.50/4.51 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) % 8.50/4.51 = ( ord_less @ A @ X3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2985_sin__cos__squared__add2,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( plus_plus @ A @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 = ( one_one @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2986_sin__cos__squared__add,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( plus_plus @ A @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 = ( one_one @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2987_neg__numeral__less__floor,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [V2: num,X3: A] : % 8.50/4.51 ( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2988_floor__le__neg__numeral,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A,V2: num] : % 8.50/4.51 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) % 8.50/4.51 = ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2989_cos__one__sin__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ( cos @ A @ X3 ) % 8.50/4.51 = ( one_one @ A ) ) % 8.50/4.51 => ( ( sin @ A @ X3 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2990_sin__add,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( sin @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 8.50/4.51 = ( plus_plus @ A @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2991_cos__diff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( cos @ A @ ( minus_minus @ A @ X3 @ Y ) ) % 8.50/4.51 = ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2992_cos__add,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 8.50/4.51 = ( minus_minus @ A @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2993_sin__zero__norm__cos__one,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ( sin @ A @ X3 ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 => ( ( real_V7770717601297561774m_norm @ A @ ( cos @ A @ X3 ) ) % 8.50/4.51 = ( one_one @ real ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2994_sincos__principal__value,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ? [Y4: real] : % 8.50/4.51 ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ Y4 ) % 8.50/4.51 & ( ord_less_eq @ real @ Y4 @ pi ) % 8.50/4.51 & ( ( sin @ real @ Y4 ) % 8.50/4.51 = ( sin @ real @ X3 ) ) % 8.50/4.51 & ( ( cos @ real @ Y4 ) % 8.50/4.51 = ( cos @ real @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2995_floor__mono,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.51 => ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2996_of__int__floor__le,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) @ X3 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2997_floor__less__cancel,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) % 8.50/4.51 => ( ord_less @ A @ X3 @ Y ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2998_powr__non__neg,axiom, % 8.50/4.51 ! [A2: real,X3: real] : % 8.50/4.51 ~ ( ord_less @ real @ ( powr @ real @ A2 @ X3 ) @ ( zero_zero @ real ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_2999_powr__less__mono2__neg,axiom, % 8.50/4.51 ! [A2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ Y ) % 8.50/4.51 => ( ord_less @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X3 @ A2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3000_powr__less__cancel,axiom, % 8.50/4.51 ! [X3: real,A2: real,B2: real] : % 8.50/4.51 ( ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 8.50/4.51 => ( ord_less @ real @ A2 @ B2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3001_powr__less__mono,axiom, % 8.50/4.51 ! [A2: real,B2: real,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ A2 @ B2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 8.50/4.51 => ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ X3 @ B2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3002_le__floor__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [Z2: int,X3: A] : % 8.50/4.51 ( ( ord_less_eq @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3003_floor__less__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A,Z2: int] : % 8.50/4.51 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z2 ) % 8.50/4.51 = ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3004_powr__mono2_H,axiom, % 8.50/4.51 ! [A2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ X3 @ Y ) % 8.50/4.51 => ( ord_less_eq @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X3 @ A2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3005_powr__less__mono2,axiom, % 8.50/4.51 ! [A2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ Y ) % 8.50/4.51 => ( ord_less @ real @ ( powr @ real @ X3 @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3006_le__floor__add,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A,Y: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3007_gr__one__powr,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.51 => ( ord_less @ real @ ( one_one @ real ) @ ( powr @ real @ X3 @ Y ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3008_powr__inj,axiom, % 8.50/4.51 ! [A2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.51 => ( ( A2 % 8.50/4.51 != ( one_one @ real ) ) % 8.50/4.51 => ( ( ( powr @ real @ A2 @ X3 ) % 8.50/4.51 = ( powr @ real @ A2 @ Y ) ) % 8.50/4.51 = ( X3 = Y ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3009_int__add__floor,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [Z2: int,X3: A] : % 8.50/4.51 ( ( plus_plus @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 8.50/4.51 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3010_floor__add__int,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A,Z2: int] : % 8.50/4.51 ( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z2 ) % 8.50/4.51 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3011_sin__gt__zero,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ pi ) % 8.50/4.51 => ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3012_sin__times__sin,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [W2: A,Z2: A] : % 8.50/4.51 ( ( times_times @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z2 ) ) % 8.50/4.51 = ( divide_divide @ A @ ( minus_minus @ A @ ( cos @ A @ ( minus_minus @ A @ W2 @ Z2 ) ) @ ( cos @ A @ ( plus_plus @ A @ W2 @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3013_sin__times__cos,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [W2: A,Z2: A] : % 8.50/4.51 ( ( times_times @ A @ ( sin @ A @ W2 ) @ ( cos @ A @ Z2 ) ) % 8.50/4.51 = ( divide_divide @ A @ ( plus_plus @ A @ ( sin @ A @ ( plus_plus @ A @ W2 @ Z2 ) ) @ ( sin @ A @ ( minus_minus @ A @ W2 @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3014_cos__times__sin,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [W2: A,Z2: A] : % 8.50/4.51 ( ( times_times @ A @ ( cos @ A @ W2 ) @ ( sin @ A @ Z2 ) ) % 8.50/4.51 = ( divide_divide @ A @ ( minus_minus @ A @ ( sin @ A @ ( plus_plus @ A @ W2 @ Z2 ) ) @ ( sin @ A @ ( minus_minus @ A @ W2 @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3015_sin__plus__sin,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [W2: A,Z2: A] : % 8.50/4.51 ( ( plus_plus @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z2 ) ) % 8.50/4.51 = ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3016_sin__diff__sin,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [W2: A,Z2: A] : % 8.50/4.51 ( ( minus_minus @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z2 ) ) % 8.50/4.51 = ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3017_cos__diff__cos,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [W2: A,Z2: A] : % 8.50/4.51 ( ( minus_minus @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z2 ) ) % 8.50/4.51 = ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ Z2 @ W2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3018_powr__add,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_V3459762299906320749_field @ A ) % 8.50/4.51 & ( ln @ A ) ) % 8.50/4.51 => ! [X3: A,A2: A,B2: A] : % 8.50/4.51 ( ( powr @ A @ X3 @ ( plus_plus @ A @ A2 @ B2 ) ) % 8.50/4.51 = ( times_times @ A @ ( powr @ A @ X3 @ A2 ) @ ( powr @ A @ X3 @ B2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3019_one__add__floor,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) % 8.50/4.51 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3020_floor__log__eq__powr__iff,axiom, % 8.50/4.51 ! [X3: real,B2: real,K: int] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 8.50/4.51 => ( ( ( archim6421214686448440834_floor @ real @ ( log @ B2 @ X3 ) ) % 8.50/4.51 = K ) % 8.50/4.51 = ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ K ) ) @ X3 ) % 8.50/4.51 & ( ord_less @ real @ X3 @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3021_powr__realpow,axiom, % 8.50/4.51 ! [X3: real,N2: nat] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( powr @ real @ X3 @ ( semiring_1_of_nat @ real @ N2 ) ) % 8.50/4.51 = ( power_power @ real @ X3 @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3022_less__log__iff,axiom, % 8.50/4.51 ! [B2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ Y @ ( log @ B2 @ X3 ) ) % 8.50/4.51 = ( ord_less @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3023_log__less__iff,axiom, % 8.50/4.51 ! [B2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( log @ B2 @ X3 ) @ Y ) % 8.50/4.51 = ( ord_less @ real @ X3 @ ( powr @ real @ B2 @ Y ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3024_less__powr__iff,axiom, % 8.50/4.51 ! [B2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( powr @ real @ B2 @ Y ) ) % 8.50/4.51 = ( ord_less @ real @ ( log @ B2 @ X3 ) @ Y ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3025_powr__less__iff,axiom, % 8.50/4.51 ! [B2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) % 8.50/4.51 = ( ord_less @ real @ Y @ ( log @ B2 @ X3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3026_cos__mono__less__eq,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ X3 @ pi ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.51 => ( ( ord_less_eq @ real @ Y @ pi ) % 8.50/4.51 => ( ( ord_less @ real @ ( cos @ real @ X3 ) @ ( cos @ real @ Y ) ) % 8.50/4.51 = ( ord_less @ real @ Y @ X3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3027_cos__monotone__0__pi,axiom, % 8.50/4.51 ! [Y: real,X3: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.51 => ( ( ord_less @ real @ Y @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ X3 @ pi ) % 8.50/4.51 => ( ord_less @ real @ ( cos @ real @ X3 ) @ ( cos @ real @ Y ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3028_sin__eq__0__pi,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ pi ) % 8.50/4.51 => ( ( ( sin @ real @ X3 ) % 8.50/4.51 = ( zero_zero @ real ) ) % 8.50/4.51 => ( X3 % 8.50/4.51 = ( zero_zero @ real ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3029_real__of__int__floor__add__one__gt,axiom, % 8.50/4.51 ! [R2: real] : ( ord_less @ real @ R2 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R2 ) ) @ ( one_one @ real ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3030_floor__eq,axiom, % 8.50/4.51 ! [N2: int,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( ring_1_of_int @ real @ N2 ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N2 ) @ ( one_one @ real ) ) ) % 8.50/4.51 => ( ( archim6421214686448440834_floor @ real @ X3 ) % 8.50/4.51 = N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3031_sin__zero__pi__iff,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ pi ) % 8.50/4.51 => ( ( ( sin @ real @ X3 ) % 8.50/4.51 = ( zero_zero @ real ) ) % 8.50/4.51 = ( X3 % 8.50/4.51 = ( zero_zero @ real ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3032_real__of__int__floor__gt__diff__one,axiom, % 8.50/4.51 ! [R2: real] : ( ord_less @ real @ ( minus_minus @ real @ R2 @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3033_diffs__def,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ring_1 @ A ) % 8.50/4.51 => ( ( diffs @ A ) % 8.50/4.51 = ( ^ [C4: nat > A,N: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( C4 @ ( suc @ N ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3034_sin__expansion__lemma,axiom, % 8.50/4.51 ! [X3: real,M: nat] : % 8.50/4.51 ( ( sin @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.51 = ( cos @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3035_floor__unique,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [Z2: int,X3: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) % 8.50/4.51 => ( ( archim6421214686448440834_floor @ A @ X3 ) % 8.50/4.51 = Z2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3036_floor__eq__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A,A2: int] : % 8.50/4.51 ( ( ( archim6421214686448440834_floor @ A @ X3 ) % 8.50/4.51 = A2 ) % 8.50/4.51 = ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ X3 ) % 8.50/4.51 & ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3037_floor__split,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [P: int > $o,T2: A] : % 8.50/4.51 ( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) ) % 8.50/4.51 = ( ! [I3: int] : % 8.50/4.51 ( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I3 ) @ T2 ) % 8.50/4.51 & ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) ) ) % 8.50/4.51 => ( P @ I3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3038_le__mult__floor,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [A2: A,B2: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.51 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) % 8.50/4.51 => ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A2 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3039_less__floor__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [Z2: int,X3: A] : % 8.50/4.51 ( ( ord_less @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3040_floor__le__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A,Z2: int] : % 8.50/4.51 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z2 ) % 8.50/4.51 = ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3041_floor__correct,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) @ X3 ) % 8.50/4.51 & ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3042_cos__expansion__lemma,axiom, % 8.50/4.51 ! [X3: real,M: nat] : % 8.50/4.51 ( ( cos @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.51 = ( uminus_uminus @ real @ ( sin @ real @ ( plus_plus @ real @ X3 @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3043_powr__neg__one,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( powr @ real @ X3 @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) % 8.50/4.51 = ( divide_divide @ real @ ( one_one @ real ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3044_sin__gt__zero__02,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) % 8.50/4.51 => ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3045_le__log__iff,axiom, % 8.50/4.51 ! [B2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ Y @ ( log @ B2 @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3046_log__le__iff,axiom, % 8.50/4.51 ! [B2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( log @ B2 @ X3 ) @ Y ) % 8.50/4.51 = ( ord_less_eq @ real @ X3 @ ( powr @ real @ B2 @ Y ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3047_le__powr__iff,axiom, % 8.50/4.51 ! [B2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ X3 @ ( powr @ real @ B2 @ Y ) ) % 8.50/4.51 = ( ord_less_eq @ real @ ( log @ B2 @ X3 ) @ Y ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3048_powr__le__iff,axiom, % 8.50/4.51 ! [B2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ B2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) % 8.50/4.51 = ( ord_less_eq @ real @ Y @ ( log @ B2 @ X3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3049_cos__two__less__zero,axiom, % 8.50/4.51 ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3050_floor__eq2,axiom, % 8.50/4.51 ! [N2: int,X3: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ N2 ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N2 ) @ ( one_one @ real ) ) ) % 8.50/4.51 => ( ( archim6421214686448440834_floor @ real @ X3 ) % 8.50/4.51 = N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3051_cos__monotone__minus__pi__0,axiom, % 8.50/4.51 ! [Y: real,X3: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y ) % 8.50/4.51 => ( ( ord_less @ real @ Y @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) % 8.50/4.51 => ( ord_less @ real @ ( cos @ real @ Y ) @ ( cos @ real @ X3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3052_floor__divide__lower,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [Q4: A,P4: A] : % 8.50/4.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3053_sincos__total__2pi,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 = ( one_one @ real ) ) % 8.50/4.51 => ~ ! [T7: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 ) % 8.50/4.51 => ( ( ord_less @ real @ T7 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) % 8.50/4.51 => ( ( X3 % 8.50/4.51 = ( cos @ real @ T7 ) ) % 8.50/4.51 => ( Y % 8.50/4.51 != ( sin @ real @ T7 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3054_ln__powr__bound,axiom, % 8.50/4.51 ! [X3: real,A2: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.51 => ( ord_less_eq @ real @ ( ln_ln @ real @ X3 ) @ ( divide_divide @ real @ ( powr @ real @ X3 @ A2 ) @ A2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3055_ln__powr__bound2,axiom, % 8.50/4.51 ! [X3: real,A2: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.51 => ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X3 ) @ A2 ) @ ( times_times @ real @ ( powr @ real @ A2 @ A2 ) @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3056_log__add__eq__powr,axiom, % 8.50/4.51 ! [B2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 8.50/4.51 => ( ( B2 % 8.50/4.51 != ( one_one @ real ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( plus_plus @ real @ ( log @ B2 @ X3 ) @ Y ) % 8.50/4.51 = ( log @ B2 @ ( times_times @ real @ X3 @ ( powr @ real @ B2 @ Y ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3057_add__log__eq__powr,axiom, % 8.50/4.51 ! [B2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 8.50/4.51 => ( ( B2 % 8.50/4.51 != ( one_one @ real ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( plus_plus @ real @ Y @ ( log @ B2 @ X3 ) ) % 8.50/4.51 = ( log @ B2 @ ( times_times @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3058_minus__log__eq__powr,axiom, % 8.50/4.51 ! [B2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 8.50/4.51 => ( ( B2 % 8.50/4.51 != ( one_one @ real ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( minus_minus @ real @ Y @ ( log @ B2 @ X3 ) ) % 8.50/4.51 = ( log @ B2 @ ( divide_divide @ real @ ( powr @ real @ B2 @ Y ) @ X3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3059_sin__pi__divide__n__ge__0,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( N2 % 8.50/4.51 != ( zero_zero @ nat ) ) % 8.50/4.51 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3060_cos__times__cos,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [W2: A,Z2: A] : % 8.50/4.51 ( ( times_times @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z2 ) ) % 8.50/4.51 = ( divide_divide @ A @ ( plus_plus @ A @ ( cos @ A @ ( minus_minus @ A @ W2 @ Z2 ) ) @ ( cos @ A @ ( plus_plus @ A @ W2 @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3061_cos__plus__cos,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [W2: A,Z2: A] : % 8.50/4.51 ( ( plus_plus @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z2 ) ) % 8.50/4.51 = ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3062_floor__divide__upper,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [Q4: A,P4: A] : % 8.50/4.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 ) % 8.50/4.51 => ( ord_less @ A @ P4 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3063_round__def,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ( ( archimedean_round @ A ) % 8.50/4.51 = ( ^ [X: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3064_sin__gt__zero2,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3065_sin__lt__zero,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ pi @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) % 8.50/4.51 => ( ord_less @ real @ ( sin @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3066_cos__double__less__one,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) % 8.50/4.51 => ( ord_less @ real @ ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) ) @ ( one_one @ real ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3067_cos__gt__zero,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3068_log__minus__eq__powr,axiom, % 8.50/4.51 ! [B2: real,X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 8.50/4.51 => ( ( B2 % 8.50/4.51 != ( one_one @ real ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( minus_minus @ real @ ( log @ B2 @ X3 ) @ Y ) % 8.50/4.51 = ( log @ B2 @ ( times_times @ real @ X3 @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ Y ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3069_termdiff__converges,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A,K5: real,C2: nat > A] : % 8.50/4.51 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K5 ) % 8.50/4.51 => ( ! [X5: A] : % 8.50/4.51 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K5 ) % 8.50/4.51 => ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ X5 @ N ) ) ) ) % 8.50/4.51 => ( summable @ A % 8.50/4.51 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3070_powr__neg__numeral,axiom, % 8.50/4.51 ! [X3: real,N2: num] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( powr @ real @ X3 @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ N2 ) ) ) % 8.50/4.51 = ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3071_sin__le__zero,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ pi @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) % 8.50/4.51 => ( ord_less_eq @ real @ ( sin @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3072_sin__less__zero,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) % 8.50/4.51 => ( ord_less @ real @ ( sin @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3073_sin__mono__less__eq,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y ) % 8.50/4.51 => ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( sin @ real @ X3 ) @ ( sin @ real @ Y ) ) % 8.50/4.51 = ( ord_less @ real @ X3 @ Y ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3074_sin__monotone__2pi,axiom, % 8.50/4.51 ! [Y: real,X3: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y ) % 8.50/4.51 => ( ( ord_less @ real @ Y @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ord_less @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3075_cos__gt__zero__pi,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3076_sin__pi__divide__n__gt__0,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.50/4.51 => ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3077_floor__log2__div2,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.50/4.51 => ( ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) % 8.50/4.51 = ( plus_plus @ int @ ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ int ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3078_floor__log__nat__eq__if,axiom, % 8.50/4.51 ! [B2: nat,N2: nat,K: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K ) % 8.50/4.51 => ( ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) % 8.50/4.51 => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) % 8.50/4.51 => ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) ) % 8.50/4.51 = ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3079_Maclaurin__minus__cos__expansion,axiom, % 8.50/4.51 ! [N2: nat,X3: real] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) % 8.50/4.51 => ? [T7: real] : % 8.50/4.51 ( ( ord_less @ real @ X3 @ T7 ) % 8.50/4.51 & ( ord_less @ real @ T7 @ ( zero_zero @ real ) ) % 8.50/4.51 & ( ( cos @ real @ X3 ) % 8.50/4.51 = ( plus_plus @ real % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.51 @ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ X3 @ M2 ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.51 @ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3080_Maclaurin__cos__expansion2,axiom, % 8.50/4.51 ! [X3: real,N2: nat] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.51 => ? [T7: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ T7 ) % 8.50/4.51 & ( ord_less @ real @ T7 @ X3 ) % 8.50/4.51 & ( ( cos @ real @ X3 ) % 8.50/4.51 = ( plus_plus @ real % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.51 @ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ X3 @ M2 ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.51 @ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3081_sin__paired,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( sums @ real % 8.50/4.51 @ ^ [N: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X3 @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) % 8.50/4.51 @ ( sin @ real @ X3 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3082_tan__double,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ( cos @ A @ X3 ) % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) ) % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( tan @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) ) % 8.50/4.51 = ( divide_divide @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( tan @ A @ X3 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3083_arcosh__def,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ln @ A ) % 8.50/4.51 => ( ( arcosh @ A ) % 8.50/4.51 = ( ^ [X: A] : ( ln_ln @ A @ ( plus_plus @ A @ X @ ( powr @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3084_tan__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ( ( tan @ A @ ( zero_zero @ A ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3085_of__real__eq__0__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V2191834092415804123ebra_1 @ A ) % 8.50/4.51 => ! [X3: real] : % 8.50/4.51 ( ( ( real_Vector_of_real @ A @ X3 ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 = ( X3 % 8.50/4.51 = ( zero_zero @ real ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3086_of__real__0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V2191834092415804123ebra_1 @ A ) % 8.50/4.51 => ( ( real_Vector_of_real @ A @ ( zero_zero @ real ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3087_fact__0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_char_0 @ A ) % 8.50/4.51 => ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) ) % 8.50/4.51 = ( one_one @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3088_of__real__add,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V2191834092415804123ebra_1 @ A ) % 8.50/4.51 => ! [X3: real,Y: real] : % 8.50/4.51 ( ( real_Vector_of_real @ A @ ( plus_plus @ real @ X3 @ Y ) ) % 8.50/4.51 = ( plus_plus @ A @ ( real_Vector_of_real @ A @ X3 ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3089_fact__Suc__0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_char_0 @ A ) % 8.50/4.51 => ( ( semiring_char_0_fact @ A @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.51 = ( one_one @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3090_fact__Suc,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_char_0 @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( semiring_char_0_fact @ A @ ( suc @ N2 ) ) % 8.50/4.51 = ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3091_sin__of__real__pi,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.51 => ( ( sin @ A @ ( real_Vector_of_real @ A @ pi ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3092_norm__of__real__add1,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.51 => ! [X3: real] : % 8.50/4.51 ( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X3 ) @ ( one_one @ A ) ) ) % 8.50/4.51 = ( abs_abs @ real @ ( plus_plus @ real @ X3 @ ( one_one @ real ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3093_norm__of__real__addn,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.51 => ! [X3: real,B2: num] : % 8.50/4.51 ( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X3 ) @ ( numeral_numeral @ A @ B2 ) ) ) % 8.50/4.51 = ( abs_abs @ real @ ( plus_plus @ real @ X3 @ ( numeral_numeral @ real @ B2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3094_cos__of__real__pi__half,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V7773925162809079976_field @ A ) % 8.50/4.51 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.51 => ( ( cos @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3095_fact__nonzero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( semiring_char_0 @ A ) % 8.50/4.51 & ( semiri3467727345109120633visors @ A ) ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( semiring_char_0_fact @ A @ N2 ) % 8.50/4.51 != ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3096_fact__ge__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3097_fact__not__neg,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ~ ( ord_less @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3098_fact__gt__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [N2: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3099_fact__ge__1,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [N2: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3100_fact__mono,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [M: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3101_fact__dvd,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [N2: nat,M: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ N2 @ M ) % 8.50/4.51 => ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ M ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3102_fact__less__mono,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [M: nat,N2: nat] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.50/4.51 => ( ( ord_less @ nat @ M @ N2 ) % 8.50/4.51 => ( ord_less @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3103_fact__fact__dvd__fact,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [K: nat,N2: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N2 ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3104_fact__mod,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( linordered_semidom @ A ) % 8.50/4.51 & ( semidom_modulo @ A ) ) % 8.50/4.51 => ! [M: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.51 => ( ( modulo_modulo @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ M ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3105_fact__le__power,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [N2: nat] : ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ N2 @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3106_norm__less__p1,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( real_V2822296259951069270ebra_1 @ A ) % 8.50/4.51 => ! [X3: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3107_choose__dvd,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [K: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.51 => ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3108_fact__num__eq__if,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_char_0 @ A ) % 8.50/4.51 => ( ( semiring_char_0_fact @ A ) % 8.50/4.51 = ( ^ [M2: nat] : % 8.50/4.51 ( if @ A % 8.50/4.51 @ ( M2 % 8.50/4.51 = ( zero_zero @ nat ) ) % 8.50/4.51 @ ( one_one @ A ) % 8.50/4.51 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3109_fact__reduce,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_char_0 @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.51 => ( ( semiring_char_0_fact @ A @ N2 ) % 8.50/4.51 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3110_tan__gt__zero,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ord_less @ real @ ( zero_zero @ real ) @ ( tan @ real @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3111_lemma__tan__total,axiom, % 8.50/4.51 ! [Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.51 => ? [X5: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X5 ) % 8.50/4.51 & ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 & ( ord_less @ real @ Y @ ( tan @ real @ X5 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3112_lemma__tan__total1,axiom, % 8.50/4.51 ! [Y: real] : % 8.50/4.51 ? [X5: real] : % 8.50/4.51 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X5 ) % 8.50/4.51 & ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 & ( ( tan @ real @ X5 ) % 8.50/4.51 = Y ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3113_tan__mono__lt__eq,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y ) % 8.50/4.51 => ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( tan @ real @ X3 ) @ ( tan @ real @ Y ) ) % 8.50/4.51 = ( ord_less @ real @ X3 @ Y ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3114_tan__monotone_H,axiom, % 8.50/4.51 ! [Y: real,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y ) % 8.50/4.51 => ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ( ord_less @ real @ Y @ X3 ) % 8.50/4.51 = ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X3 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3115_tan__monotone,axiom, % 8.50/4.51 ! [Y: real,X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y ) % 8.50/4.51 => ( ( ord_less @ real @ Y @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3116_tan__total,axiom, % 8.50/4.51 ! [Y: real] : % 8.50/4.51 ? [X5: real] : % 8.50/4.51 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X5 ) % 8.50/4.51 & ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 & ( ( tan @ real @ X5 ) % 8.50/4.51 = Y ) % 8.50/4.51 & ! [Y5: real] : % 8.50/4.51 ( ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y5 ) % 8.50/4.51 & ( ord_less @ real @ Y5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 & ( ( tan @ real @ Y5 ) % 8.50/4.51 = Y ) ) % 8.50/4.51 => ( Y5 = X5 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3117_add__tan__eq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( ( cos @ A @ X3 ) % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( ( cos @ A @ Y ) % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( plus_plus @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) % 8.50/4.51 = ( divide_divide @ A @ ( sin @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3118_tan__total__pos,axiom, % 8.50/4.51 ! [Y: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.51 => ? [X5: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X5 ) % 8.50/4.51 & ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 & ( ( tan @ real @ X5 ) % 8.50/4.51 = Y ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3119_tan__pos__pi2__le,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tan @ real @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3120_tan__less__zero,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) % 8.50/4.51 => ( ord_less @ real @ ( tan @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3121_tan__mono__le__eq,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y ) % 8.50/4.51 => ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( tan @ real @ X3 ) @ ( tan @ real @ Y ) ) % 8.50/4.51 = ( ord_less_eq @ real @ X3 @ Y ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3122_tan__mono__le,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ X3 @ Y ) % 8.50/4.51 => ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ord_less_eq @ real @ ( tan @ real @ X3 ) @ ( tan @ real @ Y ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3123_tan__bound__pi2,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.51 => ( ord_less @ real @ ( abs_abs @ real @ ( tan @ real @ X3 ) ) @ ( one_one @ real ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3124_arctan,axiom, % 8.50/4.51 ! [Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) ) % 8.50/4.51 & ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 & ( ( tan @ real @ ( arctan @ Y ) ) % 8.50/4.51 = Y ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3125_arctan__tan,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ( arctan @ ( tan @ real @ X3 ) ) % 8.50/4.51 = X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3126_arctan__unique,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ( ( tan @ real @ X3 ) % 8.50/4.51 = Y ) % 8.50/4.51 => ( ( arctan @ Y ) % 8.50/4.51 = X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3127_Maclaurin__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( zero @ A ) % 8.50/4.51 => ! [X3: real,N2: nat,Diff: nat > A > real] : % 8.50/4.51 ( ( X3 % 8.50/4.51 = ( zero_zero @ real ) ) % 8.50/4.51 => ( ( N2 % 8.50/4.51 != ( zero_zero @ nat ) ) % 8.50/4.51 => ( ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.51 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X3 @ M2 ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.51 = ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3128_Maclaurin__lemma,axiom, % 8.50/4.51 ! [H2: real,F2: real > real,J: nat > real,N2: nat] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ H2 ) % 8.50/4.51 => ? [B10: real] : % 8.50/4.51 ( ( F2 @ H2 ) % 8.50/4.51 = ( plus_plus @ real % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.51 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J @ M2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.51 @ ( times_times @ real @ B10 @ ( divide_divide @ real @ ( power_power @ real @ H2 @ N2 ) @ ( semiring_char_0_fact @ real @ N2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3129_tan__add,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( ( cos @ A @ X3 ) % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( ( cos @ A @ Y ) % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( tan @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 8.50/4.51 = ( divide_divide @ A @ ( plus_plus @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3130_tan__diff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( ( cos @ A @ X3 ) % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( ( cos @ A @ Y ) % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( ( cos @ A @ ( minus_minus @ A @ X3 @ Y ) ) % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( tan @ A @ ( minus_minus @ A @ X3 @ Y ) ) % 8.50/4.51 = ( divide_divide @ A @ ( minus_minus @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3131_lemma__tan__add1,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( ( cos @ A @ X3 ) % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( ( cos @ A @ Y ) % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X3 ) @ ( tan @ A @ Y ) ) ) % 8.50/4.51 = ( divide_divide @ A @ ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3132_minus__sin__cos__eq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( uminus_uminus @ A @ ( sin @ A @ X3 ) ) % 8.50/4.51 = ( cos @ A @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3133_tan__total__pi4,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 8.50/4.51 => ? [Z3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) @ Z3 ) % 8.50/4.51 & ( ord_less @ real @ Z3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.51 & ( ( tan @ real @ Z3 ) % 8.50/4.51 = X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3134_tan__half,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ( ( tan @ A ) % 8.50/4.51 = ( ^ [X: A] : ( divide_divide @ A @ ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) @ ( plus_plus @ A @ ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) @ ( one_one @ A ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3135_arsinh__def,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ln @ A ) % 8.50/4.51 => ( ( arsinh @ A ) % 8.50/4.51 = ( ^ [X: A] : ( ln_ln @ A @ ( plus_plus @ A @ X @ ( powr @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3136_Maclaurin__sin__expansion3,axiom, % 8.50/4.51 ! [N2: nat,X3: real] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.51 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ? [T7: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ T7 ) % 8.50/4.51 & ( ord_less @ real @ T7 @ X3 ) % 8.50/4.51 & ( ( sin @ real @ X3 ) % 8.50/4.51 = ( plus_plus @ real % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.51 @ ^ [M2: nat] : ( times_times @ real @ ( sin_coeff @ M2 ) @ ( power_power @ real @ X3 @ M2 ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.51 @ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3137_Maclaurin__sin__expansion4,axiom, % 8.50/4.51 ! [X3: real,N2: nat] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ? [T7: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ T7 ) % 8.50/4.51 & ( ord_less_eq @ real @ T7 @ X3 ) % 8.50/4.51 & ( ( sin @ real @ X3 ) % 8.50/4.51 = ( plus_plus @ real % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.51 @ ^ [M2: nat] : ( times_times @ real @ ( sin_coeff @ M2 ) @ ( power_power @ real @ X3 @ M2 ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.51 @ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3138_sin__coeff__def,axiom, % 8.50/4.51 ( sin_coeff % 8.50/4.51 = ( ^ [N: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3139_Maclaurin__exp__lt,axiom, % 8.50/4.51 ! [X3: real,N2: nat] : % 8.50/4.51 ( ( X3 % 8.50/4.51 != ( zero_zero @ real ) ) % 8.50/4.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.51 => ? [T7: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T7 ) ) % 8.50/4.51 & ( ord_less @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) ) % 8.50/4.51 & ( ( exp @ real @ X3 ) % 8.50/4.51 = ( plus_plus @ real % 8.50/4.51 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.51 @ ^ [M2: nat] : ( divide_divide @ real @ ( power_power @ real @ X3 @ M2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) % 8.50/4.51 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.51 @ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3140_round__altdef,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ( ( archimedean_round @ A ) % 8.50/4.51 = ( ^ [X: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X ) ) @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3141_exp__less__mono,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ X3 @ Y ) % 8.50/4.51 => ( ord_less @ real @ ( exp @ real @ X3 ) @ ( exp @ real @ Y ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3142_exp__less__cancel__iff,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( exp @ real @ X3 ) @ ( exp @ real @ Y ) ) % 8.50/4.51 = ( ord_less @ real @ X3 @ Y ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3143_exp__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.51 => ( ( exp @ A @ ( zero_zero @ A ) ) % 8.50/4.51 = ( one_one @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3144_frac__of__int,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [Z2: int] : % 8.50/4.51 ( ( archimedean_frac @ A @ ( ring_1_of_int @ A @ Z2 ) ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3145_sin__coeff__0,axiom, % 8.50/4.51 ( ( sin_coeff @ ( zero_zero @ nat ) ) % 8.50/4.51 = ( zero_zero @ real ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3146_one__less__exp__iff,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X3 ) ) % 8.50/4.51 = ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3147_exp__less__one__iff,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( exp @ real @ X3 ) @ ( one_one @ real ) ) % 8.50/4.51 = ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3148_exp__ln__iff,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ( exp @ real @ ( ln_ln @ real @ X3 ) ) % 8.50/4.51 = X3 ) % 8.50/4.51 = ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3149_exp__ln,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( exp @ real @ ( ln_ln @ real @ X3 ) ) % 8.50/4.51 = X3 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3150_fact__mono__nat,axiom, % 8.50/4.51 ! [M: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.51 => ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3151_fact__ge__self,axiom, % 8.50/4.51 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( semiring_char_0_fact @ nat @ N2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3152_exp__less__cancel,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( exp @ real @ X3 ) @ ( exp @ real @ Y ) ) % 8.50/4.51 => ( ord_less @ real @ X3 @ Y ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3153_exp__not__eq__zero,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( exp @ A @ X3 ) % 8.50/4.51 != ( zero_zero @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3154_fact__less__mono__nat,axiom, % 8.50/4.51 ! [M: nat,N2: nat] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.50/4.51 => ( ( ord_less @ nat @ M @ N2 ) % 8.50/4.51 => ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3155_exp__total,axiom, % 8.50/4.51 ! [Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.51 => ? [X5: real] : % 8.50/4.51 ( ( exp @ real @ X5 ) % 8.50/4.51 = Y ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3156_exp__gt__zero,axiom, % 8.50/4.51 ! [X3: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( exp @ real @ X3 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3157_not__exp__less__zero,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ~ ( ord_less @ real @ ( exp @ real @ X3 ) @ ( zero_zero @ real ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3158_mult__exp__exp,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( times_times @ A @ ( exp @ A @ X3 ) @ ( exp @ A @ Y ) ) % 8.50/4.51 = ( exp @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3159_exp__add__commuting,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( ( times_times @ A @ X3 @ Y ) % 8.50/4.51 = ( times_times @ A @ Y @ X3 ) ) % 8.50/4.51 => ( ( exp @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 8.50/4.51 = ( times_times @ A @ ( exp @ A @ X3 ) @ ( exp @ A @ Y ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3160_frac__ge__0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3161_frac__lt__1,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X3 ) @ ( one_one @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3162_frac__1__eq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) ) % 8.50/4.51 = ( archimedean_frac @ A @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3163_fact__ge__Suc__0__nat,axiom, % 8.50/4.51 ! [N2: nat] : ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3164_exp__gt__one,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3165_dvd__fact,axiom, % 8.50/4.51 ! [M: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M ) % 8.50/4.51 => ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.51 => ( dvd_dvd @ nat @ M @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3166_fact__diff__Suc,axiom, % 8.50/4.51 ! [N2: nat,M: nat] : % 8.50/4.51 ( ( ord_less @ nat @ N2 @ ( suc @ M ) ) % 8.50/4.51 => ( ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N2 ) ) % 8.50/4.51 = ( times_times @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3167_ln__ge__iff,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ Y @ ( ln_ln @ real @ X3 ) ) % 8.50/4.51 = ( ord_less_eq @ real @ ( exp @ real @ Y ) @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3168_fact__div__fact__le__pow,axiom, % 8.50/4.51 ! [R2: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ R2 @ N2 ) % 8.50/4.51 => ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ R2 ) ) ) @ ( power_power @ nat @ N2 @ R2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3169_powr__def,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ln @ A ) % 8.50/4.51 => ( ( powr @ A ) % 8.50/4.51 = ( ^ [X: A,A4: A] : % 8.50/4.51 ( if @ A % 8.50/4.51 @ ( X % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 @ ( zero_zero @ A ) % 8.50/4.51 @ ( exp @ A @ ( times_times @ A @ A4 @ ( ln_ln @ A @ X ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3170_exp__divide__power__eq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ! [N2: nat,X3: A] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.51 => ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X3 @ ( semiring_1_of_nat @ A @ N2 ) ) ) @ N2 ) % 8.50/4.51 = ( exp @ A @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3171_frac__eq,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A] : % 8.50/4.51 ( ( ( archimedean_frac @ A @ X3 ) % 8.50/4.51 = X3 ) % 8.50/4.51 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.50/4.51 & ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3172_frac__add,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) % 8.50/4.51 => ( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 8.50/4.51 = ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) ) ) % 8.50/4.51 & ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) % 8.50/4.51 => ( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 8.50/4.51 = ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3173_tanh__altdef,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( real_Vector_banach @ A ) % 8.50/4.51 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.51 => ( ( tanh @ A ) % 8.50/4.51 = ( ^ [X: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3174_sin__coeff__Suc,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( sin_coeff @ ( suc @ N2 ) ) % 8.50/4.51 = ( divide_divide @ real @ ( cos_coeff @ N2 ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3175_floor__add,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) % 8.50/4.51 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 8.50/4.51 = ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) % 8.50/4.51 & ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) % 8.50/4.51 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 8.50/4.51 = ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3176_cos__coeff__Suc,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( cos_coeff @ ( suc @ N2 ) ) % 8.50/4.51 = ( divide_divide @ real @ ( uminus_uminus @ real @ ( sin_coeff @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3177_exp__ge__one__plus__x__over__n__power__n,axiom, % 8.50/4.51 ! [N2: nat,X3: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N2 ) ) @ X3 ) % 8.50/4.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.51 => ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X3 @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ N2 ) @ ( exp @ real @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3178_exp__ge__one__minus__x__over__n__power__n,axiom, % 8.50/4.51 ! [X3: real,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ real @ X3 @ ( semiring_1_of_nat @ real @ N2 ) ) % 8.50/4.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.51 => ( ord_less_eq @ real @ ( power_power @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X3 @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ N2 ) @ ( exp @ real @ ( uminus_uminus @ real @ X3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3179_log__base__10__eq2,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 8.50/4.51 = ( times_times @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3180_log__base__10__eq1,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X3 ) % 8.50/4.51 = ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( ln_ln @ real @ X3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3181_sin__tan,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ( sin @ real @ X3 ) % 8.50/4.51 = ( divide_divide @ real @ ( tan @ real @ X3 ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3182_cos__tan,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ( cos @ real @ X3 ) % 8.50/4.51 = ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3183_monoseq__def,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( order @ A ) % 8.50/4.51 => ( ( topological_monoseq @ A ) % 8.50/4.51 = ( ^ [X6: nat > A] : % 8.50/4.51 ( ! [M2: nat,N: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M2 @ N ) % 8.50/4.51 => ( ord_less_eq @ A @ ( X6 @ M2 ) @ ( X6 @ N ) ) ) % 8.50/4.51 | ! [M2: nat,N: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M2 @ N ) % 8.50/4.51 => ( ord_less_eq @ A @ ( X6 @ N ) @ ( X6 @ M2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3184_monoI2,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( order @ A ) % 8.50/4.51 => ! [X8: nat > A] : % 8.50/4.51 ( ! [M3: nat,N3: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M3 @ N3 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ M3 ) ) ) % 8.50/4.51 => ( topological_monoseq @ A @ X8 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3185_monoI1,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( order @ A ) % 8.50/4.51 => ! [X8: nat > A] : % 8.50/4.51 ( ! [M3: nat,N3: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M3 @ N3 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) % 8.50/4.51 => ( topological_monoseq @ A @ X8 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3186_real__sqrt__less__iff,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( sqrt @ X3 ) @ ( sqrt @ Y ) ) % 8.50/4.51 = ( ord_less @ real @ X3 @ Y ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3187_real__sqrt__lt__0__iff,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( sqrt @ X3 ) @ ( zero_zero @ real ) ) % 8.50/4.51 = ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3188_real__sqrt__gt__0__iff,axiom, % 8.50/4.51 ! [Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ Y ) ) % 8.50/4.51 = ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3189_real__sqrt__lt__1__iff,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( sqrt @ X3 ) @ ( one_one @ real ) ) % 8.50/4.51 = ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3190_real__sqrt__gt__1__iff,axiom, % 8.50/4.51 ! [Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( one_one @ real ) @ ( sqrt @ Y ) ) % 8.50/4.51 = ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3191_real__sqrt__less__mono,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ X3 @ Y ) % 8.50/4.51 => ( ord_less @ real @ ( sqrt @ X3 ) @ ( sqrt @ Y ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3192_real__sqrt__gt__zero,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ X3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3193_sqrt2__less__2,axiom, % 8.50/4.51 ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3194_real__less__rsqrt,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y ) % 8.50/4.51 => ( ord_less @ real @ X3 @ ( sqrt @ Y ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3195_lemma__real__divide__sqrt__less,axiom, % 8.50/4.51 ! [U: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ U ) % 8.50/4.51 => ( ord_less @ real @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ U ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3196_real__less__lsqrt,axiom, % 8.50/4.51 ! [X3: real,Y: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.51 => ( ( ord_less @ real @ X3 @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ord_less @ real @ ( sqrt @ X3 ) @ Y ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3197_ln__sqrt,axiom, % 8.50/4.51 ! [X3: real] : % 8.50/4.51 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ln_ln @ real @ ( sqrt @ X3 ) ) % 8.50/4.51 = ( divide_divide @ real @ ( ln_ln @ real @ X3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3198_arsinh__real__aux,axiom, % 8.50/4.51 ! [X3: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X3 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3199_real__sqrt__sum__squares__less,axiom, % 8.50/4.51 ! [X3: real,U: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.51 => ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.51 => ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3200_sqrt__sum__squares__half__less,axiom, % 8.50/4.51 ! [X3: real,U: real,Y: real] : % 8.50/4.51 ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ( ord_less @ real @ Y @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.51 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.51 => ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3201_monoseq__Suc,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( order @ A ) % 8.50/4.51 => ( ( topological_monoseq @ A ) % 8.50/4.51 = ( ^ [X6: nat > A] : % 8.50/4.51 ( ! [N: nat] : ( ord_less_eq @ A @ ( X6 @ N ) @ ( X6 @ ( suc @ N ) ) ) % 8.50/4.51 | ! [N: nat] : ( ord_less_eq @ A @ ( X6 @ ( suc @ N ) ) @ ( X6 @ N ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3202_mono__SucI2,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( order @ A ) % 8.50/4.51 => ! [X8: nat > A] : % 8.50/4.51 ( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) ) % 8.50/4.51 => ( topological_monoseq @ A @ X8 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3203_mono__SucI1,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( order @ A ) % 8.50/4.51 => ! [X8: nat > A] : % 8.50/4.51 ( ! [N3: nat] : ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) ) % 8.50/4.51 => ( topological_monoseq @ A @ X8 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3204_pochhammer__double,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [Z2: A,N2: nat] : % 8.50/4.51 ( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.50/4.51 = ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3205_of__nat__code,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_1 @ A ) % 8.50/4.51 => ( ( semiring_1_of_nat @ A ) % 8.50/4.51 = ( ^ [N: nat] : % 8.50/4.51 ( semiri8178284476397505188at_aux @ A % 8.50/4.51 @ ^ [I3: A] : ( plus_plus @ A @ I3 @ ( one_one @ A ) ) % 8.50/4.51 @ N % 8.50/4.51 @ ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3206_gchoose__row__sum__weighted,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [R2: A,M: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ R2 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) ) % 8.50/4.51 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M ) ) % 8.50/4.51 = ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R2 @ ( suc @ M ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3207_central__binomial__lower__bound,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.51 => ( ord_less_eq @ real @ ( divide_divide @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ N2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ ( semiring_1_of_nat @ real @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3208_complex__unimodular__polar,axiom, % 8.50/4.51 ! [Z2: complex] : % 8.50/4.51 ( ( ( real_V7770717601297561774m_norm @ complex @ Z2 ) % 8.50/4.51 = ( one_one @ real ) ) % 8.50/4.51 => ~ ! [T7: real] : % 8.50/4.51 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 ) % 8.50/4.51 => ( ( ord_less @ real @ T7 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) % 8.50/4.51 => ( Z2 % 8.50/4.51 != ( complex2 @ ( cos @ real @ T7 ) @ ( sin @ real @ T7 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3209_binomial__Suc__n,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( binomial @ ( suc @ N2 ) @ N2 ) % 8.50/4.51 = ( suc @ N2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3210_gbinomial__0_I2_J,axiom, % 8.50/4.51 ! [B: $tType] : % 8.50/4.51 ( ( ( semiring_char_0 @ B ) % 8.50/4.51 & ( semidom_divide @ B ) ) % 8.50/4.51 => ! [K: nat] : % 8.50/4.51 ( ( gbinomial @ B @ ( zero_zero @ B ) @ ( suc @ K ) ) % 8.50/4.51 = ( zero_zero @ B ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3211_binomial__1,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( binomial @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.51 = N2 ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3212_binomial__0__Suc,axiom, % 8.50/4.51 ! [K: nat] : % 8.50/4.51 ( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) ) % 8.50/4.51 = ( zero_zero @ nat ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3213_binomial__eq__0__iff,axiom, % 8.50/4.51 ! [N2: nat,K: nat] : % 8.50/4.51 ( ( ( binomial @ N2 @ K ) % 8.50/4.51 = ( zero_zero @ nat ) ) % 8.50/4.51 = ( ord_less @ nat @ N2 @ K ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3214_gbinomial__0_I1_J,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( semiring_char_0 @ A ) % 8.50/4.51 & ( semidom_divide @ A ) ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( gbinomial @ A @ A2 @ ( zero_zero @ nat ) ) % 8.50/4.51 = ( one_one @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3215_binomial__Suc__Suc,axiom, % 8.50/4.51 ! [N2: nat,K: nat] : % 8.50/4.51 ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) % 8.50/4.51 = ( plus_plus @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3216_pochhammer__0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_semiring_1 @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( zero_zero @ nat ) ) % 8.50/4.51 = ( one_one @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3217_gbinomial__Suc0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( semiring_char_0 @ A ) % 8.50/4.51 & ( semidom_divide @ A ) ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( gbinomial @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.51 = A2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3218_pochhammer__Suc0,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_semiring_1 @ A ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.51 = A2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3219_binomial__n__0,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ( binomial @ N2 @ ( zero_zero @ nat ) ) % 8.50/4.51 = ( one_one @ nat ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3220_zero__less__binomial__iff,axiom, % 8.50/4.51 ! [N2: nat,K: nat] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N2 @ K ) ) % 8.50/4.51 = ( ord_less_eq @ nat @ K @ N2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3221_binomial__eq__0,axiom, % 8.50/4.51 ! [N2: nat,K: nat] : % 8.50/4.51 ( ( ord_less @ nat @ N2 @ K ) % 8.50/4.51 => ( ( binomial @ N2 @ K ) % 8.50/4.51 = ( zero_zero @ nat ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3222_Suc__times__binomial,axiom, % 8.50/4.51 ! [K: nat,N2: nat] : % 8.50/4.51 ( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) ) % 8.50/4.51 = ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3223_Suc__times__binomial__eq,axiom, % 8.50/4.51 ! [N2: nat,K: nat] : % 8.50/4.51 ( ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) % 8.50/4.51 = ( times_times @ nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3224_binomial__symmetric,axiom, % 8.50/4.51 ! [K: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.51 => ( ( binomial @ N2 @ K ) % 8.50/4.51 = ( binomial @ N2 @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3225_choose__mult__lemma,axiom, % 8.50/4.51 ! [M: nat,R2: nat,K: nat] : % 8.50/4.51 ( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R2 ) @ K ) @ ( plus_plus @ nat @ M @ K ) ) @ ( binomial @ ( plus_plus @ nat @ M @ K ) @ K ) ) % 8.50/4.51 = ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus @ nat @ M @ R2 ) @ M ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3226_binomial__le__pow,axiom, % 8.50/4.51 ! [R2: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ R2 @ N2 ) % 8.50/4.51 => ( ord_less_eq @ nat @ ( binomial @ N2 @ R2 ) @ ( power_power @ nat @ N2 @ R2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3227_pochhammer__pos,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [X3: A,N2: nat] : % 8.50/4.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 8.50/4.51 => ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X3 @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3228_pochhammer__neq__0__mono,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [A2: A,M: nat,N2: nat] : % 8.50/4.51 ( ( ( comm_s3205402744901411588hammer @ A @ A2 @ M ) % 8.50/4.51 != ( zero_zero @ A ) ) % 8.50/4.51 => ( ( ord_less_eq @ nat @ N2 @ M ) % 8.50/4.51 => ( ( comm_s3205402744901411588hammer @ A @ A2 @ N2 ) % 8.50/4.51 != ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3229_pochhammer__eq__0__mono,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [A2: A,N2: nat,M: nat] : % 8.50/4.51 ( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N2 ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 => ( ( ord_less_eq @ nat @ N2 @ M ) % 8.50/4.51 => ( ( comm_s3205402744901411588hammer @ A @ A2 @ M ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3230_zero__less__binomial,axiom, % 8.50/4.51 ! [K: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.51 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N2 @ K ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3231_Suc__times__binomial__add,axiom, % 8.50/4.51 ! [A2: nat,B2: nat] : % 8.50/4.51 ( ( times_times @ nat @ ( suc @ A2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( suc @ A2 ) ) ) % 8.50/4.51 = ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ A2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3232_choose__mult,axiom, % 8.50/4.51 ! [K: nat,M: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ K @ M ) % 8.50/4.51 => ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.51 => ( ( times_times @ nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) ) % 8.50/4.51 = ( times_times @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus @ nat @ N2 @ K ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3233_binomial__Suc__Suc__eq__times,axiom, % 8.50/4.51 ! [N2: nat,K: nat] : % 8.50/4.51 ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) % 8.50/4.51 = ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3234_gbinomial__pochhammer_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ( ( gbinomial @ A ) % 8.50/4.51 = ( ^ [A4: A,K3: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3235_gbinomial__Suc__Suc,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [A2: A,K: nat] : % 8.50/4.51 ( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) % 8.50/4.51 = ( plus_plus @ A @ ( gbinomial @ A @ A2 @ K ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3236_gbinomial__of__nat__symmetric,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [K: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.51 => ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N2 ) @ K ) % 8.50/4.51 = ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3237_pochhammer__nonneg,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_semidom @ A ) % 8.50/4.51 => ! [X3: A,N2: nat] : % 8.50/4.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X3 @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3238_of__nat__aux_Osimps_I2_J,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_1 @ A ) % 8.50/4.51 => ! [Inc: A > A,N2: nat,I: A] : % 8.50/4.51 ( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N2 ) @ I ) % 8.50/4.51 = ( semiri8178284476397505188at_aux @ A @ Inc @ N2 @ ( Inc @ I ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3239_of__nat__aux_Osimps_I1_J,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( semiring_1 @ A ) % 8.50/4.51 => ! [Inc: A > A,I: A] : % 8.50/4.51 ( ( semiri8178284476397505188at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I ) % 8.50/4.51 = I ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3240_pochhammer__0__left,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_semiring_1 @ A ) % 8.50/4.51 => ! [N2: nat] : % 8.50/4.51 ( ( ( N2 % 8.50/4.51 = ( zero_zero @ nat ) ) % 8.50/4.51 => ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N2 ) % 8.50/4.51 = ( one_one @ A ) ) ) % 8.50/4.51 & ( ( N2 % 8.50/4.51 != ( zero_zero @ nat ) ) % 8.50/4.51 => ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N2 ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3241_binomial__absorption,axiom, % 8.50/4.51 ! [K: nat,N2: nat] : % 8.50/4.51 ( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) % 8.50/4.51 = ( times_times @ nat @ N2 @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3242_gbinomial__addition__formula,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [A2: A,K: nat] : % 8.50/4.51 ( ( gbinomial @ A @ A2 @ ( suc @ K ) ) % 8.50/4.51 = ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3243_gbinomial__ge__n__over__k__pow__k,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_field @ A ) % 8.50/4.51 => ! [K: nat,A2: A] : % 8.50/4.51 ( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K ) @ A2 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3244_gbinomial__mult__1_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [A2: A,K: nat] : % 8.50/4.51 ( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ A2 ) % 8.50/4.51 = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3245_gbinomial__mult__1,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [A2: A,K: nat] : % 8.50/4.51 ( ( times_times @ A @ A2 @ ( gbinomial @ A @ A2 @ K ) ) % 8.50/4.51 = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3246_binomial__fact__lemma,axiom, % 8.50/4.51 ! [K: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.51 => ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) ) % 8.50/4.51 = ( semiring_char_0_fact @ nat @ N2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3247_pochhammer__rec,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_semiring_1 @ A ) % 8.50/4.51 => ! [A2: A,N2: nat] : % 8.50/4.51 ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) ) % 8.50/4.51 = ( times_times @ A @ A2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3248_pochhammer__rec_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_semiring_1 @ A ) % 8.50/4.51 => ! [Z2: A,N2: nat] : % 8.50/4.51 ( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N2 ) ) % 8.50/4.51 = ( times_times @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3249_pochhammer__Suc,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_semiring_1 @ A ) % 8.50/4.51 => ! [A2: A,N2: nat] : % 8.50/4.51 ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) ) % 8.50/4.51 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A2 @ N2 ) @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3250_pochhammer__eq__0__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [A2: A,N2: nat] : % 8.50/4.51 ( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N2 ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 = ( ? [K3: nat] : % 8.50/4.51 ( ( ord_less @ nat @ K3 @ N2 ) % 8.50/4.51 & ( A2 % 8.50/4.51 = ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3251_pochhammer__of__nat__eq__0__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ring_char_0 @ A ) % 8.50/4.51 & ( idom @ A ) ) % 8.50/4.51 => ! [N2: nat,K: nat] : % 8.50/4.51 ( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 = ( ord_less @ nat @ N2 @ K ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3252_pochhammer__of__nat__eq__0__lemma,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( idom @ A ) % 8.50/4.51 => ! [N2: nat,K: nat] : % 8.50/4.51 ( ( ord_less @ nat @ N2 @ K ) % 8.50/4.51 => ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3253_pochhammer__of__nat__eq__0__lemma_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( ring_char_0 @ A ) % 8.50/4.51 & ( idom @ A ) ) % 8.50/4.51 => ! [K: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.51 => ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K ) % 8.50/4.51 != ( zero_zero @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3254_pochhammer__product_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_semiring_1 @ A ) % 8.50/4.51 => ! [Z2: A,N2: nat,M: nat] : % 8.50/4.51 ( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( plus_plus @ nat @ N2 @ M ) ) % 8.50/4.51 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N2 ) ) @ M ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3255_binomial__ge__n__over__k__pow__k,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( linordered_field @ A ) % 8.50/4.51 => ! [K: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.51 => ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3256_binomial__maximum_H,axiom, % 8.50/4.51 ! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ N2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3257_binomial__mono,axiom, % 8.50/4.51 ! [K: nat,K7: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ K @ K7 ) % 8.50/4.51 => ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N2 ) % 8.50/4.51 => ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K7 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3258_binomial__antimono,axiom, % 8.50/4.51 ! [K: nat,K7: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ K @ K7 ) % 8.50/4.51 => ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K ) % 8.50/4.51 => ( ( ord_less_eq @ nat @ K7 @ N2 ) % 8.50/4.51 => ( ord_less_eq @ nat @ ( binomial @ N2 @ K7 ) @ ( binomial @ N2 @ K ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3259_binomial__maximum,axiom, % 8.50/4.51 ! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3260_binomial__le__pow2,axiom, % 8.50/4.51 ! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3261_choose__reduce__nat,axiom, % 8.50/4.51 ! [N2: nat,K: nat] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.51 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.50/4.51 => ( ( binomial @ N2 @ K ) % 8.50/4.51 = ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3262_times__binomial__minus1__eq,axiom, % 8.50/4.51 ! [K: nat,N2: nat] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.50/4.51 => ( ( times_times @ nat @ K @ ( binomial @ N2 @ K ) ) % 8.50/4.51 = ( times_times @ nat @ N2 @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3263_Suc__times__gbinomial,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [K: nat,A2: A] : % 8.50/4.51 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) ) % 8.50/4.51 = ( times_times @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3264_gbinomial__absorption,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [K: nat,A2: A] : % 8.50/4.51 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) % 8.50/4.51 = ( times_times @ A @ A2 @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3265_binomial__altdef__nat,axiom, % 8.50/4.51 ! [K: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.51 => ( ( binomial @ N2 @ K ) % 8.50/4.51 = ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N2 ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3266_gbinomial__trinomial__revision,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [K: nat,M: nat,A2: A] : % 8.50/4.51 ( ( ord_less_eq @ nat @ K @ M ) % 8.50/4.51 => ( ( times_times @ A @ ( gbinomial @ A @ A2 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) ) % 8.50/4.51 = ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3267_pochhammer__product,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_semiring_1 @ A ) % 8.50/4.51 => ! [M: nat,N2: nat,Z2: A] : % 8.50/4.51 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.51 => ( ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) % 8.50/4.51 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3268_binomial__strict__antimono,axiom, % 8.50/4.51 ! [K: nat,K7: nat,N2: nat] : % 8.50/4.51 ( ( ord_less @ nat @ K @ K7 ) % 8.50/4.51 => ( ( ord_less_eq @ nat @ N2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) ) % 8.50/4.51 => ( ( ord_less_eq @ nat @ K7 @ N2 ) % 8.50/4.51 => ( ord_less @ nat @ ( binomial @ N2 @ K7 ) @ ( binomial @ N2 @ K ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3269_binomial__strict__mono,axiom, % 8.50/4.51 ! [K: nat,K7: nat,N2: nat] : % 8.50/4.51 ( ( ord_less @ nat @ K @ K7 ) % 8.50/4.51 => ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N2 ) % 8.50/4.51 => ( ord_less @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K7 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3270_binomial__less__binomial__Suc,axiom, % 8.50/4.51 ! [K: nat,N2: nat] : % 8.50/4.51 ( ( ord_less @ nat @ K @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 => ( ord_less @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3271_central__binomial__odd,axiom, % 8.50/4.51 ! [N2: nat] : % 8.50/4.51 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.50/4.51 => ( ( binomial @ N2 @ ( suc @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.51 = ( binomial @ N2 @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3272_binomial__addition__formula,axiom, % 8.50/4.51 ! [N2: nat,K: nat] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.51 => ( ( binomial @ N2 @ ( suc @ K ) ) % 8.50/4.51 = ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3273_binomial__fact,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [K: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.51 => ( ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) % 8.50/4.51 = ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3274_fact__binomial,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [K: nat,N2: nat] : % 8.50/4.51 ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.51 => ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) ) % 8.50/4.51 = ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3275_gbinomial__rec,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [A2: A,K: nat] : % 8.50/4.51 ( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) % 8.50/4.51 = ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3276_gbinomial__factors,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [A2: A,K: nat] : % 8.50/4.51 ( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) % 8.50/4.51 = ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3277_pochhammer__absorb__comp,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_ring_1 @ A ) % 8.50/4.51 => ! [R2: A,K: nat] : % 8.50/4.51 ( ( times_times @ A @ ( minus_minus @ A @ R2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R2 ) @ K ) ) % 8.50/4.51 = ( times_times @ A @ R2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R2 ) @ ( one_one @ A ) ) @ K ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3278_gbinomial__minus,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [A2: A,K: nat] : % 8.50/4.51 ( ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K ) % 8.50/4.51 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3279_gbinomial__reduce__nat,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [K: nat,A2: A] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.50/4.51 => ( ( gbinomial @ A @ A2 @ K ) % 8.50/4.51 = ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3280_pochhammer__minus,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_ring_1 @ A ) % 8.50/4.51 => ! [B2: A,K: nat] : % 8.50/4.51 ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) % 8.50/4.51 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3281_pochhammer__minus_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_ring_1 @ A ) % 8.50/4.51 => ! [B2: A,K: nat] : % 8.50/4.51 ( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) % 8.50/4.51 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3282_gbinomial__sum__up__index,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [K: nat,N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K ) % 8.50/4.51 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.50/4.51 = ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3283_gbinomial__absorption_H,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [K: nat,A2: A] : % 8.50/4.51 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.50/4.51 => ( ( gbinomial @ A @ A2 @ K ) % 8.50/4.51 = ( times_times @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3284_binomial__code,axiom, % 8.50/4.51 ( binomial % 8.50/4.51 = ( ^ [N: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N @ ( minus_minus @ nat @ N @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N @ K3 ) @ ( one_one @ nat ) ) @ N @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3285_gbinomial__code,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ( ( gbinomial @ A ) % 8.50/4.51 = ( ^ [A4: A,K3: nat] : % 8.50/4.51 ( if @ A % 8.50/4.51 @ ( K3 % 8.50/4.51 = ( zero_zero @ nat ) ) % 8.50/4.51 @ ( one_one @ A ) % 8.50/4.51 @ ( divide_divide @ A % 8.50/4.51 @ ( set_fo6178422350223883121st_nat @ A % 8.50/4.51 @ ^ [L2: nat] : ( times_times @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ L2 ) ) ) % 8.50/4.51 @ ( zero_zero @ nat ) % 8.50/4.51 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) % 8.50/4.51 @ ( one_one @ A ) ) % 8.50/4.51 @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3286_pochhammer__times__pochhammer__half,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [Z2: A,N2: nat] : % 8.50/4.51 ( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.51 @ ^ [K3: nat] : ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) % 8.50/4.51 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3287_pochhammer__code,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_semiring_1 @ A ) % 8.50/4.51 => ( ( comm_s3205402744901411588hammer @ A ) % 8.50/4.51 = ( ^ [A4: A,N: nat] : % 8.50/4.51 ( if @ A % 8.50/4.51 @ ( N % 8.50/4.51 = ( zero_zero @ nat ) ) % 8.50/4.51 @ ( one_one @ A ) % 8.50/4.51 @ ( set_fo6178422350223883121st_nat @ A % 8.50/4.51 @ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ O ) ) ) % 8.50/4.51 @ ( zero_zero @ nat ) % 8.50/4.51 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) % 8.50/4.51 @ ( one_one @ A ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3288_gbinomial__partial__row__sum,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( field_char_0 @ A ) % 8.50/4.51 => ! [A2: A,M: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.51 @ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) ) % 8.50/4.51 @ ( set_ord_atMost @ nat @ M ) ) % 8.50/4.51 = ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A2 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3289_atMost__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ord @ A ) % 8.50/4.51 => ! [I: A,K: A] : % 8.50/4.51 ( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) ) % 8.50/4.51 = ( ord_less_eq @ A @ I @ K ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3290_finite__atMost,axiom, % 8.50/4.51 ! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_atMost @ nat @ K ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3291_of__nat__prod,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_semiring_1 @ A ) % 8.50/4.51 => ! [F2: B > nat,A3: set @ B] : % 8.50/4.51 ( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A3 ) ) % 8.50/4.51 = ( groups7121269368397514597t_prod @ B @ A % 8.50/4.51 @ ^ [X: B] : ( semiring_1_of_nat @ A @ ( F2 @ X ) ) % 8.50/4.51 @ A3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3292_of__int__prod,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_ring_1 @ A ) % 8.50/4.51 => ! [F2: B > int,A3: set @ B] : % 8.50/4.51 ( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F2 @ A3 ) ) % 8.50/4.51 = ( groups7121269368397514597t_prod @ B @ A % 8.50/4.51 @ ^ [X: B] : ( ring_1_of_int @ A @ ( F2 @ X ) ) % 8.50/4.51 @ A3 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3293_prod__zero__iff,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( semidom @ A ) % 8.50/4.51 => ! [A3: set @ B,F2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) % 8.50/4.51 = ( zero_zero @ A ) ) % 8.50/4.51 = ( ? [X: B] : % 8.50/4.51 ( ( member @ B @ X @ A3 ) % 8.50/4.51 & ( ( F2 @ X ) % 8.50/4.51 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3294_prod_Oempty,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( comm_monoid_mult @ A ) % 8.50/4.51 => ! [G: B > A] : % 8.50/4.51 ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) ) % 8.50/4.51 = ( one_one @ A ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3295_prod_Oinfinite,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( comm_monoid_mult @ A ) % 8.50/4.51 => ! [A3: set @ B,G: B > A] : % 8.50/4.51 ( ~ ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) % 8.50/4.51 = ( one_one @ A ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3296_atMost__subset__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( preorder @ A ) % 8.50/4.51 => ! [X3: A,Y: A] : % 8.50/4.51 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X3 ) @ ( set_ord_atMost @ A @ Y ) ) % 8.50/4.51 = ( ord_less_eq @ A @ X3 @ Y ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3297_dvd__prod__eqI,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_semiring_1 @ A ) % 8.50/4.51 => ! [A3: set @ B,A2: B,B2: A,F2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( member @ B @ A2 @ A3 ) % 8.50/4.51 => ( ( B2 % 8.50/4.51 = ( F2 @ A2 ) ) % 8.50/4.51 => ( dvd_dvd @ A @ B2 @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3298_dvd__prodI,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_semiring_1 @ A ) % 8.50/4.51 => ! [A3: set @ B,A2: B,F2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( member @ B @ A2 @ A3 ) % 8.50/4.51 => ( dvd_dvd @ A @ ( F2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3299_prod_Odelta,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( comm_monoid_mult @ A ) % 8.50/4.51 => ! [S3: set @ B,A2: B,B2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ S3 ) % 8.50/4.51 => ( ( ( member @ B @ A2 @ S3 ) % 8.50/4.51 => ( ( groups7121269368397514597t_prod @ B @ A % 8.50/4.51 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) ) % 8.50/4.51 @ S3 ) % 8.50/4.51 = ( B2 @ A2 ) ) ) % 8.50/4.51 & ( ~ ( member @ B @ A2 @ S3 ) % 8.50/4.51 => ( ( groups7121269368397514597t_prod @ B @ A % 8.50/4.51 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) ) % 8.50/4.51 @ S3 ) % 8.50/4.51 = ( one_one @ A ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3300_prod_Odelta_H,axiom, % 8.50/4.51 ! [B: $tType,A: $tType] : % 8.50/4.51 ( ( comm_monoid_mult @ A ) % 8.50/4.51 => ! [S3: set @ B,A2: B,B2: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ S3 ) % 8.50/4.51 => ( ( ( member @ B @ A2 @ S3 ) % 8.50/4.51 => ( ( groups7121269368397514597t_prod @ B @ A % 8.50/4.51 @ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) ) % 8.50/4.51 @ S3 ) % 8.50/4.51 = ( B2 @ A2 ) ) ) % 8.50/4.51 & ( ~ ( member @ B @ A2 @ S3 ) % 8.50/4.51 => ( ( groups7121269368397514597t_prod @ B @ A % 8.50/4.51 @ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) ) % 8.50/4.51 @ S3 ) % 8.50/4.51 = ( one_one @ A ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3301_prod_Oinsert,axiom, % 8.50/4.51 ! [A: $tType,B: $tType] : % 8.50/4.51 ( ( comm_monoid_mult @ A ) % 8.50/4.51 => ! [A3: set @ B,X3: B,G: B > A] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ~ ( member @ B @ X3 @ A3 ) % 8.50/4.51 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 8.50/4.51 = ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3302_Icc__subset__Iic__iff,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( preorder @ A ) % 8.50/4.51 => ! [L: A,H2: A,H3: A] : % 8.50/4.51 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atMost @ A @ H3 ) ) % 8.50/4.51 = ( ~ ( ord_less_eq @ A @ L @ H2 ) % 8.50/4.51 | ( ord_less_eq @ A @ H2 @ H3 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3303_sum_OatMost__Suc,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_add @ A ) % 8.50/4.51 => ! [G: nat > A,N2: nat] : % 8.50/4.51 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3304_prod_OlessThan__Suc,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_mult @ A ) % 8.50/4.51 => ! [G: nat > A,N2: nat] : % 8.50/4.51 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( G @ N2 ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3305_prod_OatMost__Suc,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_mult @ A ) % 8.50/4.51 => ! [G: nat > A,N2: nat] : % 8.50/4.51 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3306_atMost__0,axiom, % 8.50/4.51 ( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) ) % 8.50/4.51 = ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3307_prod_Ocl__ivl__Suc,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_mult @ A ) % 8.50/4.51 => ! [N2: nat,M: nat,G: nat > A] : % 8.50/4.51 ( ( ( ord_less @ nat @ ( suc @ N2 ) @ M ) % 8.50/4.51 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( one_one @ A ) ) ) % 8.50/4.51 & ( ~ ( ord_less @ nat @ ( suc @ N2 ) @ M ) % 8.50/4.51 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3308_not__empty__eq__Iic__eq__empty,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( preorder @ A ) % 8.50/4.51 => ! [H2: A] : % 8.50/4.51 ( ( bot_bot @ ( set @ A ) ) % 8.50/4.51 != ( set_ord_atMost @ A @ H2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3309_infinite__Iic,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( ( linorder @ A ) % 8.50/4.51 & ( no_bot @ A ) ) % 8.50/4.51 => ! [A2: A] : % 8.50/4.51 ~ ( finite_finite2 @ A @ ( set_ord_atMost @ A @ A2 ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3310_prod_Oswap__restrict,axiom, % 8.50/4.51 ! [A: $tType,B: $tType,C: $tType] : % 8.50/4.51 ( ( comm_monoid_mult @ A ) % 8.50/4.51 => ! [A3: set @ B,B3: set @ C,G: B > C > A,R: B > C > $o] : % 8.50/4.51 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.51 => ( ( finite_finite2 @ C @ B3 ) % 8.50/4.51 => ( ( groups7121269368397514597t_prod @ B @ A % 8.50/4.51 @ ^ [X: B] : % 8.50/4.51 ( groups7121269368397514597t_prod @ C @ A @ ( G @ X ) % 8.50/4.51 @ ( collect @ C % 8.50/4.51 @ ^ [Y2: C] : % 8.50/4.51 ( ( member @ C @ Y2 @ B3 ) % 8.50/4.51 & ( R @ X @ Y2 ) ) ) ) % 8.50/4.51 @ A3 ) % 8.50/4.51 = ( groups7121269368397514597t_prod @ C @ A % 8.50/4.51 @ ^ [Y2: C] : % 8.50/4.51 ( groups7121269368397514597t_prod @ B @ A % 8.50/4.51 @ ^ [X: B] : ( G @ X @ Y2 ) % 8.50/4.51 @ ( collect @ B % 8.50/4.51 @ ^ [X: B] : % 8.50/4.51 ( ( member @ B @ X @ A3 ) % 8.50/4.51 & ( R @ X @ Y2 ) ) ) ) % 8.50/4.51 @ B3 ) ) ) ) ) ). % 8.50/4.51 % 8.50/4.51 thf(fact_3311_prod_OatMost__Suc__shift,axiom, % 8.50/4.51 ! [A: $tType] : % 8.50/4.51 ( ( comm_monoid_mult @ A ) % 8.50/4.51 => ! [G: nat > A,N2: nat] : % 8.50/4.51 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) ) % 8.50/4.51 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) % 8.50/4.51 @ ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.51 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 8.50/4.51 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3312_atMost__def,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ord @ A ) % 8.50/4.52 => ( ( set_ord_atMost @ A ) % 8.50/4.52 = ( ^ [U2: A] : % 8.50/4.52 ( collect @ A % 8.50/4.52 @ ^ [X: A] : ( ord_less_eq @ A @ X @ U2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3313_prod_Onested__swap_H,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [A2: nat > nat > A,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [J3: nat] : % 8.50/4.52 ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( A2 @ I3 @ J3 ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3314_prod__mono,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( linordered_semidom @ A ) % 8.50/4.52 => ! [A3: set @ B,F2: B > A,G: B > A] : % 8.50/4.52 ( ! [I4: B] : % 8.50/4.52 ( ( member @ B @ I4 @ A3 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) % 8.50/4.52 & ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3315_prod__nonneg,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( linordered_semidom @ A ) % 8.50/4.52 => ! [A3: set @ B,F2: B > A] : % 8.50/4.52 ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ A3 ) % 8.50/4.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3316_prod__pos,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( linordered_semidom @ A ) % 8.50/4.52 => ! [A3: set @ B,F2: B > A] : % 8.50/4.52 ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ A3 ) % 8.50/4.52 => ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) ) % 8.50/4.52 => ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3317_prod__ge__1,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( linord181362715937106298miring @ A ) % 8.50/4.52 => ! [A3: set @ B,F2: B > A] : % 8.50/4.52 ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ A3 ) % 8.50/4.52 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( F2 @ X5 ) ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3318_prod__zero,axiom, % 8.50/4.52 ! [B: $tType,A: $tType] : % 8.50/4.52 ( ( comm_semiring_1 @ A ) % 8.50/4.52 => ! [A3: set @ B,F2: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.52 => ( ? [X2: B] : % 8.50/4.52 ( ( member @ B @ X2 @ A3 ) % 8.50/4.52 & ( ( F2 @ X2 ) % 8.50/4.52 = ( zero_zero @ A ) ) ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3319_prod_OatMost__shift,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) % 8.50/4.52 @ ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3320_atMost__atLeast0,axiom, % 8.50/4.52 ( ( set_ord_atMost @ nat ) % 8.50/4.52 = ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3321_lessThan__Suc__atMost,axiom, % 8.50/4.52 ! [K: nat] : % 8.50/4.52 ( ( set_ord_lessThan @ nat @ ( suc @ K ) ) % 8.50/4.52 = ( set_ord_atMost @ nat @ K ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3322_atMost__Suc,axiom, % 8.50/4.52 ! [K: nat] : % 8.50/4.52 ( ( set_ord_atMost @ nat @ ( suc @ K ) ) % 8.50/4.52 = ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3323_prod_Ointer__filter,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [A3: set @ B,G: B > A,P: B > $o] : % 8.50/4.52 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ G % 8.50/4.52 @ ( collect @ B % 8.50/4.52 @ ^ [X: B] : % 8.50/4.52 ( ( member @ B @ X @ A3 ) % 8.50/4.52 & ( P @ X ) ) ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ B @ A % 8.50/4.52 @ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( G @ X ) @ ( one_one @ A ) ) % 8.50/4.52 @ A3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3324_not__Iic__le__Icc,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( no_bot @ A ) % 8.50/4.52 => ! [H2: A,L3: A,H3: A] : % 8.50/4.52 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3325_prod_Oshift__bounds__cl__Suc__ivl,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3326_prod_Oshift__bounds__cl__nat__ivl,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3327_finite__nat__iff__bounded__le,axiom, % 8.50/4.52 ( ( finite_finite2 @ nat ) % 8.50/4.52 = ( ^ [S7: set @ nat] : % 8.50/4.52 ? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S7 @ ( set_ord_atMost @ nat @ K3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3328_prod__le__1,axiom, % 8.50/4.52 ! [B: $tType,A: $tType] : % 8.50/4.52 ( ( linord181362715937106298miring @ A ) % 8.50/4.52 => ! [A3: set @ B,F2: B > A] : % 8.50/4.52 ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ A3 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X5 ) ) % 8.50/4.52 & ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( one_one @ A ) ) ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( one_one @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3329_prod_Orelated,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [R: A > A > $o,S3: set @ B,H2: B > A,G: B > A] : % 8.50/4.52 ( ( R @ ( one_one @ A ) @ ( one_one @ A ) ) % 8.50/4.52 => ( ! [X16: A,Y15: A,X23: A,Y23: A] : % 8.50/4.52 ( ( ( R @ X16 @ X23 ) % 8.50/4.52 & ( R @ Y15 @ Y23 ) ) % 8.50/4.52 => ( R @ ( times_times @ A @ X16 @ Y15 ) @ ( times_times @ A @ X23 @ Y23 ) ) ) % 8.50/4.52 => ( ( finite_finite2 @ B @ S3 ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ S3 ) % 8.50/4.52 => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) ) % 8.50/4.52 => ( R @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3330_prod_Oinsert__if,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [A3: set @ B,X3: B,G: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.52 => ( ( ( member @ B @ X3 @ A3 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) % 8.50/4.52 & ( ~ ( member @ B @ X3 @ A3 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 8.50/4.52 = ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3331_prod__dvd__prod__subset,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [B3: set @ B,A3: set @ B,F2: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ B3 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 ) % 8.50/4.52 => ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B3 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3332_prod__dvd__prod__subset2,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_semiring_1 @ A ) % 8.50/4.52 => ! [B3: set @ B,A3: set @ B,F2: B > A,G: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ B3 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 ) % 8.50/4.52 => ( ! [A5: B] : % 8.50/4.52 ( ( member @ B @ A5 @ A3 ) % 8.50/4.52 => ( dvd_dvd @ A @ ( F2 @ A5 ) @ ( G @ A5 ) ) ) % 8.50/4.52 => ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3333_prod_Oreindex__bij__witness__not__neutral,axiom, % 8.50/4.52 ! [B: $tType,A: $tType,C: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [S6: set @ B,T6: set @ C,S3: set @ B,I: C > B,J: B > C,T4: set @ C,G: B > A,H2: C > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ S6 ) % 8.50/4.52 => ( ( finite_finite2 @ C @ T6 ) % 8.50/4.52 => ( ! [A5: B] : % 8.50/4.52 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) % 8.50/4.52 => ( ( I @ ( J @ A5 ) ) % 8.50/4.52 = A5 ) ) % 8.50/4.52 => ( ! [A5: B] : % 8.50/4.52 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) % 8.50/4.52 => ( member @ C @ ( J @ A5 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) ) % 8.50/4.52 => ( ! [B5: C] : % 8.50/4.52 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) % 8.50/4.52 => ( ( J @ ( I @ B5 ) ) % 8.50/4.52 = B5 ) ) % 8.50/4.52 => ( ! [B5: C] : % 8.50/4.52 ( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) % 8.50/4.52 => ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) ) % 8.50/4.52 => ( ! [A5: B] : % 8.50/4.52 ( ( member @ B @ A5 @ S6 ) % 8.50/4.52 => ( ( G @ A5 ) % 8.50/4.52 = ( one_one @ A ) ) ) % 8.50/4.52 => ( ! [B5: C] : % 8.50/4.52 ( ( member @ C @ B5 @ T6 ) % 8.50/4.52 => ( ( H2 @ B5 ) % 8.50/4.52 = ( one_one @ A ) ) ) % 8.50/4.52 => ( ! [A5: B] : % 8.50/4.52 ( ( member @ B @ A5 @ S3 ) % 8.50/4.52 => ( ( H2 @ ( J @ A5 ) ) % 8.50/4.52 = ( G @ A5 ) ) ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3334_prod_Oin__pairs__0,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3335_prod_Osetdiff__irrelevant,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [A3: set @ B,G: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ G % 8.50/4.52 @ ( minus_minus @ ( set @ B ) @ A3 % 8.50/4.52 @ ( collect @ B % 8.50/4.52 @ ^ [X: B] : % 8.50/4.52 ( ( G @ X ) % 8.50/4.52 = ( one_one @ A ) ) ) ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3336_exp__sum,axiom, % 8.50/4.52 ! [B: $tType,A: $tType] : % 8.50/4.52 ( ( ( comm_monoid_mult @ B ) % 8.50/4.52 & ( real_Vector_banach @ B ) % 8.50/4.52 & ( real_V2822296259951069270ebra_1 @ B ) ) % 8.50/4.52 => ! [I6: set @ A,F2: A > B] : % 8.50/4.52 ( ( finite_finite2 @ A @ I6 ) % 8.50/4.52 => ( ( exp @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ I6 ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ A @ B % 8.50/4.52 @ ^ [X: A] : ( exp @ B @ ( F2 @ X ) ) % 8.50/4.52 @ I6 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3337_prod_Onat__diff__reindex,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N2 @ ( suc @ I3 ) ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3338_Iic__subset__Iio__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linorder @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A2 ) @ ( set_ord_lessThan @ A @ B2 ) ) % 8.50/4.52 = ( ord_less @ A @ A2 @ B2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3339_prod_OatLeastAtMost__rev,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat,M: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I3 ) ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3340_prod_Ozero__middle,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [P4: nat,K: nat,G: nat > A,H2: nat > A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ K @ P4 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ P4 ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3341_less__1__prod2,axiom, % 8.50/4.52 ! [B: $tType,A: $tType] : % 8.50/4.52 ( ( linordered_idom @ B ) % 8.50/4.52 => ! [I6: set @ A,I: A,F2: A > B] : % 8.50/4.52 ( ( finite_finite2 @ A @ I6 ) % 8.50/4.52 => ( ( member @ A @ I @ I6 ) % 8.50/4.52 => ( ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I ) ) % 8.50/4.52 => ( ! [I4: A] : % 8.50/4.52 ( ( member @ A @ I4 @ I6 ) % 8.50/4.52 => ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ I4 ) ) ) % 8.50/4.52 => ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3342_sum__choose__upper,axiom, % 8.50/4.52 ! [M: nat,N2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.52 @ ^ [K3: nat] : ( binomial @ K3 @ M ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3343_less__1__prod,axiom, % 8.50/4.52 ! [B: $tType,A: $tType] : % 8.50/4.52 ( ( linordered_idom @ B ) % 8.50/4.52 => ! [I6: set @ A,F2: A > B] : % 8.50/4.52 ( ( finite_finite2 @ A @ I6 ) % 8.50/4.52 => ( ( I6 % 8.50/4.52 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.52 => ( ! [I4: A] : % 8.50/4.52 ( ( member @ A @ I4 @ I6 ) % 8.50/4.52 => ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I4 ) ) ) % 8.50/4.52 => ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I6 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3344_prod_Osubset__diff,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [B3: set @ B,A3: set @ B,G: B > A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 ) % 8.50/4.52 => ( ( finite_finite2 @ B @ A3 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) % 8.50/4.52 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3345_prod_Omono__neutral__cong__right,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [T4: set @ B,S3: set @ B,G: B > A,H2: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ T4 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.52 => ( ( G @ X5 ) % 8.50/4.52 = ( one_one @ A ) ) ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ S3 ) % 8.50/4.52 => ( ( G @ X5 ) % 8.50/4.52 = ( H2 @ X5 ) ) ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ T4 ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3346_prod_Omono__neutral__cong__left,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [T4: set @ B,S3: set @ B,H2: B > A,G: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ T4 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.52 => ( ( H2 @ X5 ) % 8.50/4.52 = ( one_one @ A ) ) ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ S3 ) % 8.50/4.52 => ( ( G @ X5 ) % 8.50/4.52 = ( H2 @ X5 ) ) ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3347_prod_Omono__neutral__right,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [T4: set @ B,S3: set @ B,G: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ T4 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.52 => ( ( G @ X5 ) % 8.50/4.52 = ( one_one @ A ) ) ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ T4 ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3348_prod_Omono__neutral__left,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [T4: set @ B,S3: set @ B,G: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ T4 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.52 => ( ( G @ X5 ) % 8.50/4.52 = ( one_one @ A ) ) ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ B @ A @ G @ T4 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3349_prod_Osame__carrierI,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [C5: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ C5 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ C5 ) % 8.50/4.52 => ( ! [A5: B] : % 8.50/4.52 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) ) % 8.50/4.52 => ( ( G @ A5 ) % 8.50/4.52 = ( one_one @ A ) ) ) % 8.50/4.52 => ( ! [B5: B] : % 8.50/4.52 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B3 ) ) % 8.50/4.52 => ( ( H2 @ B5 ) % 8.50/4.52 = ( one_one @ A ) ) ) % 8.50/4.52 => ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ B3 ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3350_prod_Osame__carrier,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [C5: set @ B,A3: set @ B,B3: set @ B,G: B > A,H2: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ C5 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ C5 ) % 8.50/4.52 => ( ! [A5: B] : % 8.50/4.52 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) ) % 8.50/4.52 => ( ( G @ A5 ) % 8.50/4.52 = ( one_one @ A ) ) ) % 8.50/4.52 => ( ! [B5: B] : % 8.50/4.52 ( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C5 @ B3 ) ) % 8.50/4.52 => ( ( H2 @ B5 ) % 8.50/4.52 = ( one_one @ A ) ) ) % 8.50/4.52 => ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ B3 ) ) % 8.50/4.52 = ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3351_prod_OatLeast0__atMost__Suc,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 8.50/4.52 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3352_prod_OatLeast__Suc__atMost,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [M: nat,N2: nat,G: nat > A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.52 = ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3353_prod_Onat__ivl__Suc_H,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [M: nat,N2: nat,G: nat > A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) ) % 8.50/4.52 = ( times_times @ A @ ( G @ ( suc @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3354_prod_OlessThan__Suc__shift,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) % 8.50/4.52 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) % 8.50/4.52 @ ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3355_prod_OSuc__reindex__ivl,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [M: nat,N2: nat,G: nat > A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.52 => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) % 8.50/4.52 = ( times_times @ A @ ( G @ M ) % 8.50/4.52 @ ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3356_sum_OatMost__Suc__shift,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) ) % 8.50/4.52 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3357_sum__telescope,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ab_group_add @ A ) % 8.50/4.52 => ! [F2: nat > A,I: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ I ) ) % 8.50/4.52 = ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ ( suc @ I ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3358_polyfun__eq__coeffs,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.52 & ( idom @ A ) ) % 8.50/4.52 => ! [C2: nat > A,N2: nat,D2: nat > A] : % 8.50/4.52 ( ( ! [X: A] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( D2 @ I3 ) @ ( power_power @ A @ X @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) % 8.50/4.52 = ( ! [I3: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ I3 @ N2 ) % 8.50/4.52 => ( ( C2 @ I3 ) % 8.50/4.52 = ( D2 @ I3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3359_prod_OatLeast1__atMost__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3360_bounded__imp__summable,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( condit6923001295902523014norder @ A ) % 8.50/4.52 & ( linord2810124833399127020strict @ A ) % 8.50/4.52 & ( topolo1944317154257567458pology @ A ) ) % 8.50/4.52 => ! [A2: nat > A,B3: A] : % 8.50/4.52 ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A2 @ N3 ) ) % 8.50/4.52 => ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_ord_atMost @ nat @ N3 ) ) @ B3 ) % 8.50/4.52 => ( summable @ A @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3361_sum_Onested__swap_H,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [A2: nat > nat > A,N2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [J3: nat] : % 8.50/4.52 ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( A2 @ I3 @ J3 ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3362_prod__mono__strict,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( linordered_semidom @ A ) % 8.50/4.52 => ! [A3: set @ B,F2: B > A,G: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.52 => ( ! [I4: B] : % 8.50/4.52 ( ( member @ B @ I4 @ A3 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) % 8.50/4.52 & ( ord_less @ A @ ( F2 @ I4 ) @ ( G @ I4 ) ) ) ) % 8.50/4.52 => ( ( A3 % 8.50/4.52 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.52 => ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3363_even__prod__iff,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( semiring_parity @ A ) % 8.50/4.52 => ! [A3: set @ B,F2: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.52 => ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) % 8.50/4.52 = ( ? [X: B] : % 8.50/4.52 ( ( member @ B @ X @ A3 ) % 8.50/4.52 & ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3364_sum__choose__lower,axiom, % 8.50/4.52 ! [R2: nat,N2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.52 @ ^ [K3: nat] : ( binomial @ ( plus_plus @ nat @ R2 @ K3 ) @ K3 ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( binomial @ ( suc @ ( plus_plus @ nat @ R2 @ N2 ) ) @ N2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3365_prod_Oremove,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [A3: set @ B,X3: B,G: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.52 => ( ( member @ B @ X3 @ A3 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) % 8.50/4.52 = ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3366_prod_Oinsert__remove,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [A3: set @ B,G: B > A,X3: B] : % 8.50/4.52 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X3 @ A3 ) ) % 8.50/4.52 = ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3367_choose__rising__sum_I1_J,axiom, % 8.50/4.52 ! [N2: nat,M: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.52 @ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N2 @ J3 ) @ N2 ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ M ) ) % 8.50/4.52 = ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3368_choose__rising__sum_I2_J,axiom, % 8.50/4.52 ! [N2: nat,M: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.52 @ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N2 @ J3 ) @ N2 ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ M ) ) % 8.50/4.52 = ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ ( one_one @ nat ) ) @ M ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3369_prod_Oub__add__nat,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [M: nat,N2: nat,G: nat > A,P4: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N2 @ P4 ) ) ) % 8.50/4.52 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ P4 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3370_prod_Odelta__remove,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [S3: set @ B,A2: B,B2: B > A,C2: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ S3 ) % 8.50/4.52 => ( ( ( member @ B @ A2 @ S3 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A % 8.50/4.52 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) ) % 8.50/4.52 @ S3 ) % 8.50/4.52 = ( times_times @ A @ ( B2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) % 8.50/4.52 & ( ~ ( member @ B @ A2 @ S3 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A % 8.50/4.52 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) ) % 8.50/4.52 @ S3 ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3371_fold__atLeastAtMost__nat_Osimps,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( set_fo6178422350223883121st_nat @ A ) % 8.50/4.52 = ( ^ [F3: nat > A > A,A4: nat,B4: nat,Acc2: A] : ( if @ A @ ( ord_less @ nat @ B4 @ A4 ) @ Acc2 @ ( set_fo6178422350223883121st_nat @ A @ F3 @ ( plus_plus @ nat @ A4 @ ( one_one @ nat ) ) @ B4 @ ( F3 @ A4 @ Acc2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3372_fold__atLeastAtMost__nat_Oelims,axiom, % 8.50/4.52 ! [A: $tType,X3: nat > A > A,Xa2: nat,Xb: nat,Xc: A,Y: A] : % 8.50/4.52 ( ( ( set_fo6178422350223883121st_nat @ A @ X3 @ Xa2 @ Xb @ Xc ) % 8.50/4.52 = Y ) % 8.50/4.52 => ( ( ( ord_less @ nat @ Xb @ Xa2 ) % 8.50/4.52 => ( Y = Xc ) ) % 8.50/4.52 & ( ~ ( ord_less @ nat @ Xb @ Xa2 ) % 8.50/4.52 => ( Y % 8.50/4.52 = ( set_fo6178422350223883121st_nat @ A @ X3 @ ( plus_plus @ nat @ Xa2 @ ( one_one @ nat ) ) @ Xb @ ( X3 @ Xa2 @ Xc ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3373_polyfun__eq__0,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.52 & ( idom @ A ) ) % 8.50/4.52 => ! [C2: nat > A,N2: nat] : % 8.50/4.52 ( ( ! [X: A] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) % 8.50/4.52 = ( ! [I3: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ I3 @ N2 ) % 8.50/4.52 => ( ( C2 @ I3 ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3374_zero__polynom__imp__zero__coeffs,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( ab_semigroup_mult @ A ) % 8.50/4.52 & ( real_V8999393235501362500lgebra @ A ) ) % 8.50/4.52 => ! [C2: nat > A,N2: nat,K: nat] : % 8.50/4.52 ( ! [W: A] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ W @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( zero_zero @ A ) ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.52 => ( ( C2 @ K ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3375_sum_OatMost__shift,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3376_sum__up__index__split,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [F2: nat > A,M: nat,N2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N2 ) ) ) % 8.50/4.52 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ M ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3377_atLeast1__atMost__eq__remove0,axiom, % 8.50/4.52 ! [N2: nat] : % 8.50/4.52 ( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 8.50/4.52 = ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N2 ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3378_gbinomial__parallel__sum,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field_char_0 @ A ) % 8.50/4.52 => ! [A2: A,N2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K3 ) ) @ K3 ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( one_one @ A ) ) @ N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3379_sum_Otriangle__reindex__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [G: nat > nat > A,N2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G ) % 8.50/4.52 @ ( collect @ ( product_prod @ nat @ nat ) % 8.50/4.52 @ ( product_case_prod @ nat @ nat @ $o % 8.50/4.52 @ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N2 ) ) ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : % 8.50/4.52 ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ K3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3380_fact__eq__fact__times,axiom, % 8.50/4.52 ! [N2: nat,M: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ N2 @ M ) % 8.50/4.52 => ( ( semiring_char_0_fact @ nat @ M ) % 8.50/4.52 = ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N2 ) % 8.50/4.52 @ ( groups7121269368397514597t_prod @ nat @ nat % 8.50/4.52 @ ^ [X: nat] : X % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3381_prod__mono2,axiom, % 8.50/4.52 ! [B: $tType,A: $tType] : % 8.50/4.52 ( ( linordered_idom @ B ) % 8.50/4.52 => ! [B3: set @ A,A3: set @ A,F2: A > B] : % 8.50/4.52 ( ( finite_finite2 @ A @ B3 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.52 => ( ! [B5: A] : % 8.50/4.52 ( ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) % 8.50/4.52 => ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ B5 ) ) ) % 8.50/4.52 => ( ! [A5: A] : % 8.50/4.52 ( ( member @ A @ A5 @ A3 ) % 8.50/4.52 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ A5 ) ) ) % 8.50/4.52 => ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A3 ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ B3 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3382_prod__diff1,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( semidom_divide @ A ) % 8.50/4.52 => ! [A3: set @ B,F2: B > A,A2: B] : % 8.50/4.52 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.52 => ( ( ( F2 @ A2 ) % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( ( member @ B @ A2 @ A3 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) % 8.50/4.52 = ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( F2 @ A2 ) ) ) ) % 8.50/4.52 & ( ~ ( member @ B @ A2 @ A3 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3383_sum__choose__diagonal,axiom, % 8.50/4.52 ! [M: nat,N2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.52 @ ^ [K3: nat] : ( binomial @ ( minus_minus @ nat @ N2 @ K3 ) @ ( minus_minus @ nat @ M @ K3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ M ) ) % 8.50/4.52 = ( binomial @ ( suc @ N2 ) @ M ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3384_vandermonde,axiom, % 8.50/4.52 ! [M: nat,N2: nat,R2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.52 @ ^ [K3: nat] : ( times_times @ nat @ ( binomial @ M @ K3 ) @ ( binomial @ N2 @ ( minus_minus @ nat @ R2 @ K3 ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ R2 ) ) % 8.50/4.52 = ( binomial @ ( plus_plus @ nat @ M @ N2 ) @ R2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3385_sum__gp__basic,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( monoid_mult @ A ) % 8.50/4.52 & ( comm_ring @ A ) ) % 8.50/4.52 => ! [X3: A,N2: nat] : % 8.50/4.52 ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N2 ) ) ) % 8.50/4.52 = ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3386_polyfun__finite__roots,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.52 & ( idom @ A ) ) % 8.50/4.52 => ! [C2: nat > A,N2: nat] : % 8.50/4.52 ( ( finite_finite2 @ A % 8.50/4.52 @ ( collect @ A % 8.50/4.52 @ ^ [X: A] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ) % 8.50/4.52 = ( ? [I3: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ I3 @ N2 ) % 8.50/4.52 & ( ( C2 @ I3 ) % 8.50/4.52 != ( zero_zero @ A ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3387_polyfun__roots__finite,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.52 & ( idom @ A ) ) % 8.50/4.52 => ! [C2: nat > A,K: nat,N2: nat] : % 8.50/4.52 ( ( ( C2 @ K ) % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.52 => ( finite_finite2 @ A % 8.50/4.52 @ ( collect @ A % 8.50/4.52 @ ^ [Z6: A] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3388_pochhammer__Suc__prod,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_semiring_1 @ A ) % 8.50/4.52 => ! [A2: A,N2: nat] : % 8.50/4.52 ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3389_polyfun__linear__factor__root,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom @ A ) % 8.50/4.52 => ! [C2: nat > A,A2: A,N2: nat] : % 8.50/4.52 ( ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ A2 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( zero_zero @ A ) ) % 8.50/4.52 => ~ ! [B5: nat > A] : % 8.50/4.52 ~ ! [Z4: A] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( B5 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3390_polyfun__linear__factor,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom @ A ) % 8.50/4.52 => ! [C2: nat > A,N2: nat,A2: A] : % 8.50/4.52 ? [B5: nat > A] : % 8.50/4.52 ! [Z4: A] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( plus_plus @ A % 8.50/4.52 @ ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( B5 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ N2 ) ) ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ A2 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3391_pochhammer__prod__rev,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_semiring_1 @ A ) % 8.50/4.52 => ( ( comm_s3205402744901411588hammer @ A ) % 8.50/4.52 = ( ^ [A4: A,N: nat] : % 8.50/4.52 ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I3 ) ) ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3392_sum__power__shift,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( monoid_mult @ A ) % 8.50/4.52 & ( comm_ring @ A ) ) % 8.50/4.52 => ! [M: nat,N2: nat,X3: A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.52 = ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3393_sum_Otriangle__reindex,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [G: nat > nat > A,N2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G ) % 8.50/4.52 @ ( collect @ ( product_prod @ nat @ nat ) % 8.50/4.52 @ ( product_case_prod @ nat @ nat @ $o % 8.50/4.52 @ ^ [I3: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N2 ) ) ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : % 8.50/4.52 ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ K3 ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3394_fact__div__fact,axiom, % 8.50/4.52 ! [N2: nat,M: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ N2 @ M ) % 8.50/4.52 => ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N2 ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ nat % 8.50/4.52 @ ^ [X: nat] : X % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ M ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3395_binomial,axiom, % 8.50/4.52 ! [A2: nat,B2: nat,N2: nat] : % 8.50/4.52 ( ( power_power @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ N2 ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.52 @ ^ [K3: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power @ nat @ A2 @ K3 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N2 @ K3 ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3396_prod_Oin__pairs,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3397_sum__atLeastAtMost__code,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [F2: nat > A,A2: nat,B2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) ) % 8.50/4.52 = ( set_fo6178422350223883121st_nat @ A % 8.50/4.52 @ ^ [A4: nat] : ( plus_plus @ A @ ( F2 @ A4 ) ) % 8.50/4.52 @ A2 % 8.50/4.52 @ B2 % 8.50/4.52 @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3398_sum_Oin__pairs__0,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3399_polynomial__product,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom @ A ) % 8.50/4.52 => ! [M: nat,A2: nat > A,N2: nat,B2: nat > A,X3: A] : % 8.50/4.52 ( ! [I4: nat] : % 8.50/4.52 ( ( ord_less @ nat @ M @ I4 ) % 8.50/4.52 => ( ( A2 @ I4 ) % 8.50/4.52 = ( zero_zero @ A ) ) ) % 8.50/4.52 => ( ! [J2: nat] : % 8.50/4.52 ( ( ord_less @ nat @ N2 @ J2 ) % 8.50/4.52 => ( ( B2 @ J2 ) % 8.50/4.52 = ( zero_zero @ A ) ) ) % 8.50/4.52 => ( ( times_times @ A % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X3 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ M ) ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [J3: nat] : ( times_times @ A @ ( B2 @ J3 ) @ ( power_power @ A @ X3 @ J3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [R5: nat] : % 8.50/4.52 ( times_times @ A % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : ( times_times @ A @ ( A2 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ R5 ) ) % 8.50/4.52 @ ( power_power @ A @ X3 @ R5 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3400_pochhammer__Suc__prod__rev,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_semiring_1 @ A ) % 8.50/4.52 => ! [A2: A,N2: nat] : % 8.50/4.52 ( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N2 ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I3 ) ) ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3401_polyfun__eq__const,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.52 & ( idom @ A ) ) % 8.50/4.52 => ! [C2: nat > A,N2: nat,K: A] : % 8.50/4.52 ( ( ! [X: A] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ X @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = K ) ) % 8.50/4.52 = ( ( ( C2 @ ( zero_zero @ nat ) ) % 8.50/4.52 = K ) % 8.50/4.52 & ! [X: nat] : % 8.50/4.52 ( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) % 8.50/4.52 => ( ( C2 @ X ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3402_binomial__ring,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_semiring_1 @ A ) % 8.50/4.52 => ! [A2: A,B2: A,N2: nat] : % 8.50/4.52 ( ( power_power @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N2 ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K3 ) ) @ ( power_power @ A @ A2 @ K3 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N2 @ K3 ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3403_pochhammer__binomial__sum,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_ring_1 @ A ) % 8.50/4.52 => ! [A2: A,B2: A,N2: nat] : % 8.50/4.52 ( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N2 ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ A2 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N2 @ K3 ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3404_polynomial__product__nat,axiom, % 8.50/4.52 ! [M: nat,A2: nat > nat,N2: nat,B2: nat > nat,X3: nat] : % 8.50/4.52 ( ! [I4: nat] : % 8.50/4.52 ( ( ord_less @ nat @ M @ I4 ) % 8.50/4.52 => ( ( A2 @ I4 ) % 8.50/4.52 = ( zero_zero @ nat ) ) ) % 8.50/4.52 => ( ! [J2: nat] : % 8.50/4.52 ( ( ord_less @ nat @ N2 @ J2 ) % 8.50/4.52 => ( ( B2 @ J2 ) % 8.50/4.52 = ( zero_zero @ nat ) ) ) % 8.50/4.52 => ( ( times_times @ nat % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ nat @ ( A2 @ I3 ) @ ( power_power @ nat @ X3 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ M ) ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.52 @ ^ [J3: nat] : ( times_times @ nat @ ( B2 @ J3 ) @ ( power_power @ nat @ X3 @ J3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.52 @ ^ [R5: nat] : % 8.50/4.52 ( times_times @ nat % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.52 @ ^ [K3: nat] : ( times_times @ nat @ ( A2 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ R5 ) ) % 8.50/4.52 @ ( power_power @ nat @ X3 @ R5 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3405_gbinomial__Suc,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( semiring_char_0 @ A ) % 8.50/4.52 & ( semidom_divide @ A ) ) % 8.50/4.52 => ! [A2: A,K: nat] : % 8.50/4.52 ( ( gbinomial @ A @ A2 @ ( suc @ K ) ) % 8.50/4.52 = ( divide_divide @ A % 8.50/4.52 @ ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) % 8.50/4.52 @ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3406_sum_Ozero__middle,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [P4: nat,K: nat,G: nat > A,H2: nat > A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ K @ P4 ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ P4 ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3407_gbinomial__partial__sum__poly,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field_char_0 @ A ) % 8.50/4.52 => ! [M: nat,A2: A,X3: A,Y: A] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K3 ) @ ( power_power @ A @ X3 @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K3 ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ M ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( minus_minus @ nat @ M @ K3 ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ M ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3408_root__polyfun,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom @ A ) % 8.50/4.52 => ! [N2: nat,Z2: A,A2: A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 ) % 8.50/4.52 => ( ( ( power_power @ A @ Z2 @ N2 ) % 8.50/4.52 = A2 ) % 8.50/4.52 = ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : % 8.50/4.52 ( times_times @ A % 8.50/4.52 @ ( if @ A % 8.50/4.52 @ ( I3 % 8.50/4.52 = ( zero_zero @ nat ) ) % 8.50/4.52 @ ( uminus_uminus @ A @ A2 ) % 8.50/4.52 @ ( if @ A @ ( I3 = N2 ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) % 8.50/4.52 @ ( power_power @ A @ Z2 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3409_sum__gp0,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( division_ring @ A ) % 8.50/4.52 & ( comm_ring @ A ) ) % 8.50/4.52 => ! [X3: A,N2: nat] : % 8.50/4.52 ( ( ( X3 % 8.50/4.52 = ( one_one @ A ) ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) % 8.50/4.52 & ( ( X3 % 8.50/4.52 != ( one_one @ A ) ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N2 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3410_choose__alternating__linear__sum,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_ring_1 @ A ) % 8.50/4.52 => ! [N2: nat] : % 8.50/4.52 ( ( N2 % 8.50/4.52 != ( one_one @ nat ) ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3411_gbinomial__sum__nat__pow2,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field_char_0 @ A ) % 8.50/4.52 => ! [M: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K3 ) ) @ K3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ M ) ) % 8.50/4.52 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3412_gbinomial__partial__sum__poly__xpos,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field_char_0 @ A ) % 8.50/4.52 => ! [M: nat,A2: A,X3: A,Y: A] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K3 ) @ ( power_power @ A @ X3 @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K3 ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ M ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A2 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X3 @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( minus_minus @ nat @ M @ K3 ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ M ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3413_polyfun__diff__alt,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom @ A ) % 8.50/4.52 => ! [N2: nat,A2: nat > A,X3: A,Y: A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 ) % 8.50/4.52 => ( ( minus_minus @ A % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X3 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 8.50/4.52 = ( times_times @ A @ ( minus_minus @ A @ X3 @ Y ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [J3: nat] : % 8.50/4.52 ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( A2 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J3 @ K3 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y @ K3 ) ) @ ( power_power @ A @ X3 @ J3 ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ J3 ) ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3414_binomial__r__part__sum,axiom, % 8.50/4.52 ! [M: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M ) ) % 8.50/4.52 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3415_choose__alternating__sum,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_ring_1 @ A ) % 8.50/4.52 => ! [N2: nat] : % 8.50/4.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3416_polyfun__extremal__lemma,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.52 => ! [E3: real,C2: nat > A,N2: nat] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 ) % 8.50/4.52 => ? [M8: real] : % 8.50/4.52 ! [Z4: A] : % 8.50/4.52 ( ( ord_less_eq @ real @ M8 @ ( real_V7770717601297561774m_norm @ A @ Z4 ) ) % 8.50/4.52 => ( ord_less_eq @ real % 8.50/4.52 @ ( real_V7770717601297561774m_norm @ A % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z4 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 8.50/4.52 @ ( times_times @ real @ E3 @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z4 ) @ ( suc @ N2 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3417_polyfun__diff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom @ A ) % 8.50/4.52 => ! [N2: nat,A2: nat > A,X3: A,Y: A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 ) % 8.50/4.52 => ( ( minus_minus @ A % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ X3 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 8.50/4.52 = ( times_times @ A @ ( minus_minus @ A @ X3 @ Y ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [J3: nat] : % 8.50/4.52 ( times_times @ A % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( A2 @ I3 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ J3 ) @ ( one_one @ nat ) ) ) ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) ) % 8.50/4.52 @ ( power_power @ A @ X3 @ J3 ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3418_gbinomial__r__part__sum,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field_char_0 @ A ) % 8.50/4.52 => ! [M: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M ) ) % 8.50/4.52 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3419_choose__odd__sum,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_ring_1 @ A ) % 8.50/4.52 => ! [N2: nat] : % 8.50/4.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.52 => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : % 8.50/4.52 ( if @ A % 8.50/4.52 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) % 8.50/4.52 @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) ) % 8.50/4.52 @ ( zero_zero @ A ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 8.50/4.52 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3420_choose__even__sum,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_ring_1 @ A ) % 8.50/4.52 => ! [N2: nat] : % 8.50/4.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.52 => ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I3 ) ) @ ( zero_zero @ A ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 8.50/4.52 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3421_sin__x__sin__y,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.52 => ! [X3: A,Y: A] : % 8.50/4.52 ( sums @ A % 8.50/4.52 @ ^ [P6: nat] : % 8.50/4.52 ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [N: nat] : % 8.50/4.52 ( if @ A % 8.50/4.52 @ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 ) % 8.50/4.52 & ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) % 8.50/4.52 @ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) ) @ ( power_power @ A @ X3 @ N ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N ) ) ) % 8.50/4.52 @ ( zero_zero @ A ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ P6 ) ) % 8.50/4.52 @ ( times_times @ A @ ( sin @ A @ X3 ) @ ( sin @ A @ Y ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3422_sums__cos__x__plus__y,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.52 => ! [X3: A,Y: A] : % 8.50/4.52 ( sums @ A % 8.50/4.52 @ ^ [P6: nat] : % 8.50/4.52 ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [N: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 ) @ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X3 @ N ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N ) ) ) @ ( zero_zero @ A ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ P6 ) ) % 8.50/4.52 @ ( cos @ A @ ( plus_plus @ A @ X3 @ Y ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3423_cos__x__cos__y,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.52 => ! [X3: A,Y: A] : % 8.50/4.52 ( sums @ A % 8.50/4.52 @ ^ [P6: nat] : % 8.50/4.52 ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [N: nat] : % 8.50/4.52 ( if @ A % 8.50/4.52 @ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 ) % 8.50/4.52 & ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) % 8.50/4.52 @ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X3 @ N ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N ) ) ) % 8.50/4.52 @ ( zero_zero @ A ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ P6 ) ) % 8.50/4.52 @ ( times_times @ A @ ( cos @ A @ X3 ) @ ( cos @ A @ Y ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3424_cot__less__zero,axiom, % 8.50/4.52 ! [X3: real] : % 8.50/4.52 ( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X3 ) % 8.50/4.52 => ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ord_less @ real @ ( cot @ real @ X3 ) @ ( zero_zero @ real ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3425_int__ge__less__than2__def,axiom, % 8.50/4.52 ( int_ge_less_than2 % 8.50/4.52 = ( ^ [D5: int] : % 8.50/4.52 ( collect @ ( product_prod @ int @ int ) % 8.50/4.52 @ ( product_case_prod @ int @ int @ $o % 8.50/4.52 @ ^ [Z7: int,Z6: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ D5 @ Z6 ) % 8.50/4.52 & ( ord_less @ int @ Z7 @ Z6 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3426_of__nat__id,axiom, % 8.50/4.52 ( ( semiring_1_of_nat @ nat ) % 8.50/4.52 = ( ^ [N: nat] : N ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3427_scaleR__cancel__right,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [A2: real,X3: A,B2: real] : % 8.50/4.52 ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) % 8.50/4.52 = ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) % 8.50/4.52 = ( ( A2 = B2 ) % 8.50/4.52 | ( X3 % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3428_scaleR__zero__right,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [A2: real] : % 8.50/4.52 ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( zero_zero @ A ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3429_cot__zero,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.52 => ( ( cot @ A @ ( zero_zero @ A ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3430_scaleR__zero__left,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( real_V8093663219630862766scaleR @ A @ ( zero_zero @ real ) @ X3 ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3431_scaleR__eq__0__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [A2: real,X3: A] : % 8.50/4.52 ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) % 8.50/4.52 = ( zero_zero @ A ) ) % 8.50/4.52 = ( ( A2 % 8.50/4.52 = ( zero_zero @ real ) ) % 8.50/4.52 | ( X3 % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3432_scaleR__eq__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [B2: A,U: real,A2: A] : % 8.50/4.52 ( ( ( plus_plus @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ U @ A2 ) ) % 8.50/4.52 = ( plus_plus @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ U @ B2 ) ) ) % 8.50/4.52 = ( ( A2 = B2 ) % 8.50/4.52 | ( U % 8.50/4.52 = ( one_one @ real ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3433_prod__eq__1__iff,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,F2: A > nat] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A3 ) % 8.50/4.52 = ( one_one @ nat ) ) % 8.50/4.52 = ( ! [X: A] : % 8.50/4.52 ( ( member @ A @ X @ A3 ) % 8.50/4.52 => ( ( F2 @ X ) % 8.50/4.52 = ( one_one @ nat ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3434_scaleR__collapse,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [U: real,A2: A] : % 8.50/4.52 ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ U @ A2 ) ) % 8.50/4.52 = A2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3435_prod__pos__nat__iff,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,F2: A > nat] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A3 ) ) % 8.50/4.52 = ( ! [X: A] : % 8.50/4.52 ( ( member @ A @ X @ A3 ) % 8.50/4.52 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3436_scaleR__half__double,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ A2 @ A2 ) ) % 8.50/4.52 = A2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3437_int__prod,axiom, % 8.50/4.52 ! [B: $tType,F2: B > nat,A3: set @ B] : % 8.50/4.52 ( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A3 ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ B @ int % 8.50/4.52 @ ^ [X: B] : ( semiring_1_of_nat @ int @ ( F2 @ X ) ) % 8.50/4.52 @ A3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3438_scaleR__right__distrib,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [A2: real,X3: A,Y: A] : % 8.50/4.52 ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( plus_plus @ A @ X3 @ Y ) ) % 8.50/4.52 = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3439_scaleR__right__imp__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [X3: A,A2: real,B2: real] : % 8.50/4.52 ( ( X3 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) % 8.50/4.52 = ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) % 8.50/4.52 => ( A2 = B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3440_scaleR__left__distrib,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [A2: real,B2: real,X3: A] : % 8.50/4.52 ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X3 ) % 8.50/4.52 = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3441_scaleR__left_Oadd,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [X3: real,Y: real,Xa2: A] : % 8.50/4.52 ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ X3 @ Y ) @ Xa2 ) % 8.50/4.52 = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ X3 @ Xa2 ) @ ( real_V8093663219630862766scaleR @ A @ Y @ Xa2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3442_scaleR__right__mono__neg,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [B2: real,A2: real,C2: A] : % 8.50/4.52 ( ( ord_less_eq @ real @ B2 @ A2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ C2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3443_scaleR__right__mono,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [A2: real,B2: real,X3: A] : % 8.50/4.52 ( ( ord_less_eq @ real @ A2 @ B2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.50/4.52 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3444_scaleR__le__cancel__left,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,A2: A,B2: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) ) % 8.50/4.52 = ( ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 8.50/4.52 => ( ord_less_eq @ A @ A2 @ B2 ) ) % 8.50/4.52 & ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3445_scaleR__le__cancel__left__neg,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) ) % 8.50/4.52 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3446_scaleR__le__cancel__left__pos,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) ) % 8.50/4.52 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3447_scaleR__left__mono__neg,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [B2: A,A2: A,C2: real] : % 8.50/4.52 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.50/4.52 => ( ( ord_less_eq @ real @ C2 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3448_scaleR__left__mono,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [X3: A,Y: A,A2: real] : % 8.50/4.52 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.52 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.52 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3449_eq__vector__fraction__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [X3: A,U: real,V2: real,A2: A] : % 8.50/4.52 ( ( X3 % 8.50/4.52 = ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V2 ) @ A2 ) ) % 8.50/4.52 = ( ( ( V2 % 8.50/4.52 = ( zero_zero @ real ) ) % 8.50/4.52 => ( X3 % 8.50/4.52 = ( zero_zero @ A ) ) ) % 8.50/4.52 & ( ( V2 % 8.50/4.52 != ( zero_zero @ real ) ) % 8.50/4.52 => ( ( real_V8093663219630862766scaleR @ A @ V2 @ X3 ) % 8.50/4.52 = ( real_V8093663219630862766scaleR @ A @ U @ A2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3450_vector__fraction__eq__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [U: real,V2: real,A2: A,X3: A] : % 8.50/4.52 ( ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V2 ) @ A2 ) % 8.50/4.52 = X3 ) % 8.50/4.52 = ( ( ( V2 % 8.50/4.52 = ( zero_zero @ real ) ) % 8.50/4.52 => ( X3 % 8.50/4.52 = ( zero_zero @ A ) ) ) % 8.50/4.52 & ( ( V2 % 8.50/4.52 != ( zero_zero @ real ) ) % 8.50/4.52 => ( ( real_V8093663219630862766scaleR @ A @ U @ A2 ) % 8.50/4.52 = ( real_V8093663219630862766scaleR @ A @ V2 @ X3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3451_Real__Vector__Spaces_Ole__add__iff1,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [A2: real,E3: A,C2: A,B2: real,D2: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E3 ) @ D2 ) ) % 8.50/4.52 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ E3 ) @ C2 ) @ D2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3452_Real__Vector__Spaces_Ole__add__iff2,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [A2: real,E3: A,C2: A,B2: real,D2: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E3 ) @ D2 ) ) % 8.50/4.52 = ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B2 @ A2 ) @ E3 ) @ D2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3453_scaleR__le__0__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [A2: real,B2: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) % 8.50/4.52 = ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.52 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) % 8.50/4.52 | ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) ) % 8.50/4.52 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) % 8.50/4.52 | ( A2 % 8.50/4.52 = ( zero_zero @ real ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3454_zero__le__scaleR__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [A2: real,B2: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) % 8.50/4.52 = ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.52 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) % 8.50/4.52 | ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) ) % 8.50/4.52 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) % 8.50/4.52 | ( A2 % 8.50/4.52 = ( zero_zero @ real ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3455_scaleR__nonpos__nonpos,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [A2: real,B2: A] : % 8.50/4.52 ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3456_scaleR__nonpos__nonneg,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [A2: real,X3: A] : % 8.50/4.52 ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.50/4.52 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3457_scaleR__nonneg__nonpos,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [A2: real,X3: A] : % 8.50/4.52 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3458_scaleR__nonneg__nonneg,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [A2: real,X3: A] : % 8.50/4.52 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.50/4.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3459_split__scaleR__pos__le,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [A2: real,B2: A] : % 8.50/4.52 ( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.52 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) % 8.50/4.52 | ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) ) % 8.50/4.52 & ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3460_split__scaleR__neg__le,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [A2: real,X3: A] : % 8.50/4.52 ( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.52 & ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) ) % 8.50/4.52 | ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) ) % 8.50/4.52 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3461_scaleR__mono_H,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [A2: real,B2: real,C2: A,D2: A] : % 8.50/4.52 ( ( ord_less_eq @ real @ A2 @ B2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ C2 @ D2 ) % 8.50/4.52 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.52 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ D2 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3462_scaleR__mono,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [A2: real,B2: real,X3: A,Y: A] : % 8.50/4.52 ( ( ord_less_eq @ real @ A2 @ B2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.52 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.50/4.52 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ Y ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3463_scaleR__left__le__one__le,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [X3: A,A2: real] : % 8.50/4.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) % 8.50/4.52 => ( ( ord_less_eq @ real @ A2 @ ( one_one @ real ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ X3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3464_scaleR__2,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X3 ) % 8.50/4.52 = ( plus_plus @ A @ X3 @ X3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3465_prod__int__plus__eq,axiom, % 8.50/4.52 ! [I: nat,J: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ int @ int % 8.50/4.52 @ ^ [X: int] : X % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3466_ln__prod,axiom, % 8.50/4.52 ! [A: $tType,I6: set @ A,F2: A > real] : % 8.50/4.52 ( ( finite_finite2 @ A @ I6 ) % 8.50/4.52 => ( ! [I4: A] : % 8.50/4.52 ( ( member @ A @ I4 @ I6 ) % 8.50/4.52 => ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ I4 ) ) ) % 8.50/4.52 => ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F2 @ I6 ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ A @ real % 8.50/4.52 @ ^ [X: A] : ( ln_ln @ real @ ( F2 @ X ) ) % 8.50/4.52 @ I6 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3467_prod_Otriangle__reindex__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > nat > A,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G ) % 8.50/4.52 @ ( collect @ ( product_prod @ nat @ nat ) % 8.50/4.52 @ ( product_case_prod @ nat @ nat @ $o % 8.50/4.52 @ ^ [I3: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N2 ) ) ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : % 8.50/4.52 ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ K3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3468_prod_Otriangle__reindex,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > nat > A,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G ) % 8.50/4.52 @ ( collect @ ( product_prod @ nat @ nat ) % 8.50/4.52 @ ( product_case_prod @ nat @ nat @ $o % 8.50/4.52 @ ^ [I3: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J3 ) @ N2 ) ) ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : % 8.50/4.52 ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ K3 ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3469_cot__gt__zero,axiom, % 8.50/4.52 ! [X3: real] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.52 => ( ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.52 => ( ord_less @ real @ ( zero_zero @ real ) @ ( cot @ real @ X3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3470_int__ge__less__than__def,axiom, % 8.50/4.52 ( int_ge_less_than % 8.50/4.52 = ( ^ [D5: int] : % 8.50/4.52 ( collect @ ( product_prod @ int @ int ) % 8.50/4.52 @ ( product_case_prod @ int @ int @ $o % 8.50/4.52 @ ^ [Z7: int,Z6: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ D5 @ Z7 ) % 8.50/4.52 & ( ord_less @ int @ Z7 @ Z6 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3471_exp__first__two__terms,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.52 => ( ( exp @ A ) % 8.50/4.52 = ( ^ [X: A] : % 8.50/4.52 ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X ) % 8.50/4.52 @ ( suminf @ A % 8.50/4.52 @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ A @ X @ ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3472_arctan__def,axiom, % 8.50/4.52 ( arctan % 8.50/4.52 = ( ^ [Y2: real] : % 8.50/4.52 ( the @ real % 8.50/4.52 @ ^ [X: real] : % 8.50/4.52 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X ) % 8.50/4.52 & ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.52 & ( ( tan @ real @ X ) % 8.50/4.52 = Y2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3473_modulo__int__unfold,axiom, % 8.50/4.52 ! [L: int,K: int,N2: nat,M: nat] : % 8.50/4.52 ( ( ( ( ( sgn_sgn @ int @ L ) % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 | ( ( sgn_sgn @ int @ K ) % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 | ( N2 % 8.50/4.52 = ( zero_zero @ nat ) ) ) % 8.50/4.52 => ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) % 8.50/4.52 = ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) ) ) % 8.50/4.52 & ( ~ ( ( ( sgn_sgn @ int @ L ) % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 | ( ( sgn_sgn @ int @ K ) % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 | ( N2 % 8.50/4.52 = ( zero_zero @ nat ) ) ) % 8.50/4.52 => ( ( ( ( sgn_sgn @ int @ K ) % 8.50/4.52 = ( sgn_sgn @ int @ L ) ) % 8.50/4.52 => ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) % 8.50/4.52 = ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) ) % 8.50/4.52 & ( ( ( sgn_sgn @ int @ K ) % 8.50/4.52 != ( sgn_sgn @ int @ L ) ) % 8.50/4.52 => ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) % 8.50/4.52 = ( times_times @ int @ ( sgn_sgn @ int @ L ) % 8.50/4.52 @ ( minus_minus @ int % 8.50/4.52 @ ( semiring_1_of_nat @ int % 8.50/4.52 @ ( times_times @ nat @ N2 % 8.50/4.52 @ ( zero_neq_one_of_bool @ nat % 8.50/4.52 @ ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ) % 8.50/4.52 @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3474_powr__int,axiom, % 8.50/4.52 ! [X3: real,I: int] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.52 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I ) % 8.50/4.52 => ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ I ) ) % 8.50/4.52 = ( power_power @ real @ X3 @ ( nat2 @ I ) ) ) ) % 8.50/4.52 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I ) % 8.50/4.52 => ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ I ) ) % 8.50/4.52 = ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( nat2 @ ( uminus_uminus @ int @ I ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3475_inverse__inverse__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) ) % 8.50/4.52 = A2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3476_inverse__eq__iff__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ( inverse_inverse @ A @ A2 ) % 8.50/4.52 = ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 = ( A2 = B2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3477_sgn__sgn,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom_abs_sgn @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) ) % 8.50/4.52 = ( sgn_sgn @ A @ A2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3478_inverse__zero,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3479_inverse__nonzero__iff__nonzero,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ( inverse_inverse @ A @ A2 ) % 8.50/4.52 = ( zero_zero @ A ) ) % 8.50/4.52 = ( A2 % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3480_inverse__mult__distrib,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) ) % 8.50/4.52 = ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3481_inverse__1,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ( ( inverse_inverse @ A @ ( one_one @ A ) ) % 8.50/4.52 = ( one_one @ A ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3482_inverse__eq__1__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field @ A ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( ( inverse_inverse @ A @ X3 ) % 8.50/4.52 = ( one_one @ A ) ) % 8.50/4.52 = ( X3 % 8.50/4.52 = ( one_one @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3483_inverse__divide,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( inverse_inverse @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 8.50/4.52 = ( divide_divide @ A @ B2 @ A2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3484_inverse__minus__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A2 ) ) % 8.50/4.52 = ( uminus_uminus @ A @ ( inverse_inverse @ A @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3485_sgn__0,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom_abs_sgn @ A ) % 8.50/4.52 => ( ( sgn_sgn @ A @ ( zero_zero @ A ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3486_sgn__zero,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V822414075346904944vector @ A ) % 8.50/4.52 => ( ( sgn_sgn @ A @ ( zero_zero @ A ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3487_sgn__1,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom_abs_sgn @ A ) % 8.50/4.52 => ( ( sgn_sgn @ A @ ( one_one @ A ) ) % 8.50/4.52 = ( one_one @ A ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3488_sgn__divide,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field_abs_sgn @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( sgn_sgn @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 8.50/4.52 = ( divide_divide @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3489_idom__abs__sgn__class_Osgn__minus,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom_abs_sgn @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ A2 ) ) % 8.50/4.52 = ( uminus_uminus @ A @ ( sgn_sgn @ A @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3490_abs__inverse,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field_abs_sgn @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( abs_abs @ A @ ( inverse_inverse @ A @ A2 ) ) % 8.50/4.52 = ( inverse_inverse @ A @ ( abs_abs @ A @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3491_inverse__sgn,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( inverse_inverse @ A @ ( sgn_sgn @ A @ A2 ) ) % 8.50/4.52 = ( sgn_sgn @ A @ A2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3492_sgn__inverse,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field_abs_sgn @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( sgn_sgn @ A @ ( inverse_inverse @ A @ A2 ) ) % 8.50/4.52 = ( inverse_inverse @ A @ ( sgn_sgn @ A @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3493_nat__int,axiom, % 8.50/4.52 ! [N2: nat] : % 8.50/4.52 ( ( nat2 @ ( semiring_1_of_nat @ int @ N2 ) ) % 8.50/4.52 = N2 ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3494_inverse__nonnegative__iff__nonnegative,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) % 8.50/4.52 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3495_inverse__nonpositive__iff__nonpositive,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) % 8.50/4.52 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3496_inverse__positive__iff__positive,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) % 8.50/4.52 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3497_inverse__negative__iff__negative,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) % 8.50/4.52 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3498_inverse__less__iff__less__neg,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3499_inverse__less__iff__less,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.52 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.50/4.52 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3500_sgn__greater,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) ) % 8.50/4.52 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3501_sgn__less,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) ) % 8.50/4.52 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3502_divide__sgn,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( divide_divide @ A @ A2 @ ( sgn_sgn @ A @ B2 ) ) % 8.50/4.52 = ( times_times @ A @ A2 @ ( sgn_sgn @ A @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3503_nat__numeral,axiom, % 8.50/4.52 ! [K: num] : % 8.50/4.52 ( ( nat2 @ ( numeral_numeral @ int @ K ) ) % 8.50/4.52 = ( numeral_numeral @ nat @ K ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3504_nat__of__bool,axiom, % 8.50/4.52 ! [P: $o] : % 8.50/4.52 ( ( nat2 @ ( zero_neq_one_of_bool @ int @ P ) ) % 8.50/4.52 = ( zero_neq_one_of_bool @ nat @ P ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3505_inverse__le__iff__le__neg,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3506_inverse__le__iff__le,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.52 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3507_right__inverse,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( times_times @ A @ A2 @ ( inverse_inverse @ A @ A2 ) ) % 8.50/4.52 = ( one_one @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3508_left__inverse,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 ) % 8.50/4.52 = ( one_one @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3509_sgn__pos,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.52 => ( ( sgn_sgn @ A @ A2 ) % 8.50/4.52 = ( one_one @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3510_abs__sgn__eq__1,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) ) % 8.50/4.52 = ( one_one @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3511_nat__1,axiom, % 8.50/4.52 ( ( nat2 @ ( one_one @ int ) ) % 8.50/4.52 = ( suc @ ( zero_zero @ nat ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3512_sgn__mult__self__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ A2 ) ) % 8.50/4.52 = ( zero_neq_one_of_bool @ A % 8.50/4.52 @ ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3513_nat__0__iff,axiom, % 8.50/4.52 ! [I: int] : % 8.50/4.52 ( ( ( nat2 @ I ) % 8.50/4.52 = ( zero_zero @ nat ) ) % 8.50/4.52 = ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3514_nat__le__0,axiom, % 8.50/4.52 ! [Z2: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) ) % 8.50/4.52 => ( ( nat2 @ Z2 ) % 8.50/4.52 = ( zero_zero @ nat ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3515_zless__nat__conj,axiom, % 8.50/4.52 ! [W2: int,Z2: int] : % 8.50/4.52 ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) ) % 8.50/4.52 = ( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) % 8.50/4.52 & ( ord_less @ int @ W2 @ Z2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3516_nat__neg__numeral,axiom, % 8.50/4.52 ! [K: num] : % 8.50/4.52 ( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) % 8.50/4.52 = ( zero_zero @ nat ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3517_sgn__abs,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom_abs_sgn @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) ) % 8.50/4.52 = ( zero_neq_one_of_bool @ A % 8.50/4.52 @ ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3518_idom__abs__sgn__class_Oabs__sgn,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom_abs_sgn @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( sgn_sgn @ A @ ( abs_abs @ A @ A2 ) ) % 8.50/4.52 = ( zero_neq_one_of_bool @ A % 8.50/4.52 @ ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3519_nat__zminus__int,axiom, % 8.50/4.52 ! [N2: nat] : % 8.50/4.52 ( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) % 8.50/4.52 = ( zero_zero @ nat ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3520_int__nat__eq,axiom, % 8.50/4.52 ! [Z2: int] : % 8.50/4.52 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 8.50/4.52 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) ) % 8.50/4.52 = Z2 ) ) % 8.50/4.52 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 8.50/4.52 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) ) % 8.50/4.52 = ( zero_zero @ int ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3521_sgn__mult__dvd__iff,axiom, % 8.50/4.52 ! [R2: int,L: int,K: int] : % 8.50/4.52 ( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ L ) @ K ) % 8.50/4.52 = ( ( dvd_dvd @ int @ L @ K ) % 8.50/4.52 & ( ( R2 % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 => ( K % 8.50/4.52 = ( zero_zero @ int ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3522_mult__sgn__dvd__iff,axiom, % 8.50/4.52 ! [L: int,R2: int,K: int] : % 8.50/4.52 ( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R2 ) ) @ K ) % 8.50/4.52 = ( ( dvd_dvd @ int @ L @ K ) % 8.50/4.52 & ( ( R2 % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 => ( K % 8.50/4.52 = ( zero_zero @ int ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3523_dvd__sgn__mult__iff,axiom, % 8.50/4.52 ! [L: int,R2: int,K: int] : % 8.50/4.52 ( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ K ) ) % 8.50/4.52 = ( ( dvd_dvd @ int @ L @ K ) % 8.50/4.52 | ( R2 % 8.50/4.52 = ( zero_zero @ int ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3524_dvd__mult__sgn__iff,axiom, % 8.50/4.52 ! [L: int,K: int,R2: int] : % 8.50/4.52 ( ( dvd_dvd @ int @ L @ ( times_times @ int @ K @ ( sgn_sgn @ int @ R2 ) ) ) % 8.50/4.52 = ( ( dvd_dvd @ int @ L @ K ) % 8.50/4.52 | ( R2 % 8.50/4.52 = ( zero_zero @ int ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3525_sgn__neg,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ( sgn_sgn @ A @ A2 ) % 8.50/4.52 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3526_zero__less__nat__eq,axiom, % 8.50/4.52 ! [Z2: int] : % 8.50/4.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z2 ) ) % 8.50/4.52 = ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3527_of__nat__nat,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ring_1 @ A ) % 8.50/4.52 => ! [Z2: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 8.50/4.52 => ( ( semiring_1_of_nat @ A @ ( nat2 @ Z2 ) ) % 8.50/4.52 = ( ring_1_of_int @ A @ Z2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3528_sgn__of__nat,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [N2: nat] : % 8.50/4.52 ( ( sgn_sgn @ A @ ( semiring_1_of_nat @ A @ N2 ) ) % 8.50/4.52 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3529_diff__nat__numeral,axiom, % 8.50/4.52 ! [V2: num,V4: num] : % 8.50/4.52 ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( numeral_numeral @ nat @ V4 ) ) % 8.50/4.52 = ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ V4 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3530_numeral__power__eq__nat__cancel__iff,axiom, % 8.50/4.52 ! [X3: num,N2: nat,Y: int] : % 8.50/4.52 ( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) % 8.50/4.52 = ( nat2 @ Y ) ) % 8.50/4.52 = ( ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) % 8.50/4.52 = Y ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3531_nat__eq__numeral__power__cancel__iff,axiom, % 8.50/4.52 ! [Y: int,X3: num,N2: nat] : % 8.50/4.52 ( ( ( nat2 @ Y ) % 8.50/4.52 = ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) ) % 8.50/4.52 = ( Y % 8.50/4.52 = ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3532_nat__abs__dvd__iff,axiom, % 8.50/4.52 ! [K: int,N2: nat] : % 8.50/4.52 ( ( dvd_dvd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N2 ) % 8.50/4.52 = ( dvd_dvd @ int @ K @ ( semiring_1_of_nat @ int @ N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3533_dvd__nat__abs__iff,axiom, % 8.50/4.52 ! [N2: nat,K: int] : % 8.50/4.52 ( ( dvd_dvd @ nat @ N2 @ ( nat2 @ ( abs_abs @ int @ K ) ) ) % 8.50/4.52 = ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3534_nat__ceiling__le__eq,axiom, % 8.50/4.52 ! [X3: real,A2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X3 ) ) @ A2 ) % 8.50/4.52 = ( ord_less_eq @ real @ X3 @ ( semiring_1_of_nat @ real @ A2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3535_one__less__nat__eq,axiom, % 8.50/4.52 ! [Z2: int] : % 8.50/4.52 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z2 ) ) % 8.50/4.52 = ( ord_less @ int @ ( one_one @ int ) @ Z2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3536_nat__numeral__diff__1,axiom, % 8.50/4.52 ! [V2: num] : % 8.50/4.52 ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) % 8.50/4.52 = ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( one_one @ int ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3537_nat__less__numeral__power__cancel__iff,axiom, % 8.50/4.52 ! [A2: int,X3: num,N2: nat] : % 8.50/4.52 ( ( ord_less @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) ) % 8.50/4.52 = ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3538_numeral__power__less__nat__cancel__iff,axiom, % 8.50/4.52 ! [X3: num,N2: nat,A2: int] : % 8.50/4.52 ( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) @ ( nat2 @ A2 ) ) % 8.50/4.52 = ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) @ A2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3539_nat__le__numeral__power__cancel__iff,axiom, % 8.50/4.52 ! [A2: int,X3: num,N2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) ) % 8.50/4.52 = ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3540_numeral__power__le__nat__cancel__iff,axiom, % 8.50/4.52 ! [X3: num,N2: nat,A2: int] : % 8.50/4.52 ( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N2 ) @ ( nat2 @ A2 ) ) % 8.50/4.52 = ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N2 ) @ A2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3541_nonzero__norm__inverse,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ A2 ) ) % 8.50/4.52 = ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3542_sgn__zero__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V822414075346904944vector @ A ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( ( sgn_sgn @ A @ X3 ) % 8.50/4.52 = ( zero_zero @ A ) ) % 8.50/4.52 = ( X3 % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3543_sgn__eq__0__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom_abs_sgn @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ( sgn_sgn @ A @ A2 ) % 8.50/4.52 = ( zero_zero @ A ) ) % 8.50/4.52 = ( A2 % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3544_sgn__0__0,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ( sgn_sgn @ A @ A2 ) % 8.50/4.52 = ( zero_zero @ A ) ) % 8.50/4.52 = ( A2 % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3545_sgn__mult,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom_abs_sgn @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( sgn_sgn @ A @ ( times_times @ A @ A2 @ B2 ) ) % 8.50/4.52 = ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3546_same__sgn__sgn__add,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [B2: A,A2: A] : % 8.50/4.52 ( ( ( sgn_sgn @ A @ B2 ) % 8.50/4.52 = ( sgn_sgn @ A @ A2 ) ) % 8.50/4.52 => ( ( sgn_sgn @ A @ ( plus_plus @ A @ A2 @ B2 ) ) % 8.50/4.52 = ( sgn_sgn @ A @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3547_mult__commute__imp__mult__inverse__commute,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [Y: A,X3: A] : % 8.50/4.52 ( ( ( times_times @ A @ Y @ X3 ) % 8.50/4.52 = ( times_times @ A @ X3 @ Y ) ) % 8.50/4.52 => ( ( times_times @ A @ ( inverse_inverse @ A @ Y ) @ X3 ) % 8.50/4.52 = ( times_times @ A @ X3 @ ( inverse_inverse @ A @ Y ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3548_inverse__eq__imp__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ( inverse_inverse @ A @ A2 ) % 8.50/4.52 = ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 => ( A2 = B2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3549_nonzero__imp__inverse__nonzero,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( inverse_inverse @ A @ A2 ) % 8.50/4.52 != ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3550_nonzero__inverse__inverse__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) ) % 8.50/4.52 = A2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3551_nonzero__inverse__eq__imp__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ( inverse_inverse @ A @ A2 ) % 8.50/4.52 = ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 => ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( B2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( A2 = B2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3552_inverse__zero__imp__zero,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ( inverse_inverse @ A @ A2 ) % 8.50/4.52 = ( zero_zero @ A ) ) % 8.50/4.52 => ( A2 % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3553_field__class_Ofield__inverse__zero,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field @ A ) % 8.50/4.52 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3554_nonzero__inverse__scaleR__distrib,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5047593784448816457lgebra @ A ) % 8.50/4.52 => ! [A2: real,X3: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ real ) ) % 8.50/4.52 => ( ( X3 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( inverse_inverse @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) ) % 8.50/4.52 = ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ A2 ) @ ( inverse_inverse @ A @ X3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3555_norm__inverse__le__norm,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.52 => ! [R2: real,X3: A] : % 8.50/4.52 ( ( ord_less_eq @ real @ R2 @ ( real_V7770717601297561774m_norm @ A @ X3 ) ) % 8.50/4.52 => ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 ) % 8.50/4.52 => ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ X3 ) ) @ ( inverse_inverse @ real @ R2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3556_positive__imp__inverse__positive,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.52 => ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3557_negative__imp__inverse__negative,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3558_inverse__positive__imp__positive,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) % 8.50/4.52 => ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3559_inverse__negative__imp__negative,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3560_less__imp__inverse__less__neg,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.52 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3561_inverse__less__imp__less__neg,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3562_less__imp__inverse__less,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.52 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.52 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3563_inverse__less__imp__less,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.52 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3564_nonzero__inverse__mult__distrib,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( B2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) ) % 8.50/4.52 = ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3565_nonzero__inverse__minus__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A2 ) ) % 8.50/4.52 = ( uminus_uminus @ A @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3566_inverse__unique,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ( times_times @ A @ A2 @ B2 ) % 8.50/4.52 = ( one_one @ A ) ) % 8.50/4.52 => ( ( inverse_inverse @ A @ A2 ) % 8.50/4.52 = B2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3567_divide__inverse__commute,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field @ A ) % 8.50/4.52 => ( ( divide_divide @ A ) % 8.50/4.52 = ( ^ [A4: A,B4: A] : ( times_times @ A @ ( inverse_inverse @ A @ B4 ) @ A4 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3568_divide__inverse,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ( ( divide_divide @ A ) % 8.50/4.52 = ( ^ [A4: A,B4: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B4 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3569_field__class_Ofield__divide__inverse,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field @ A ) % 8.50/4.52 => ( ( divide_divide @ A ) % 8.50/4.52 = ( ^ [A4: A,B4: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B4 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3570_inverse__eq__divide,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ( ( inverse_inverse @ A ) % 8.50/4.52 = ( divide_divide @ A @ ( one_one @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3571_power__mult__power__inverse__commute,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [X3: A,M: nat,N2: nat] : % 8.50/4.52 ( ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ N2 ) ) % 8.50/4.52 = ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ N2 ) @ ( power_power @ A @ X3 @ M ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3572_power__mult__inverse__distrib,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [X3: A,M: nat] : % 8.50/4.52 ( ( times_times @ A @ ( power_power @ A @ X3 @ M ) @ ( inverse_inverse @ A @ X3 ) ) % 8.50/4.52 = ( times_times @ A @ ( inverse_inverse @ A @ X3 ) @ ( power_power @ A @ X3 @ M ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3573_mult__inverse__of__nat__commute,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [Xa2: nat,X3: A] : % 8.50/4.52 ( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa2 ) ) @ X3 ) % 8.50/4.52 = ( times_times @ A @ X3 @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3574_nonzero__abs__inverse,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( abs_abs @ A @ ( inverse_inverse @ A @ A2 ) ) % 8.50/4.52 = ( inverse_inverse @ A @ ( abs_abs @ A @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3575_sgn__not__eq__imp,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [B2: A,A2: A] : % 8.50/4.52 ( ( ( sgn_sgn @ A @ B2 ) % 8.50/4.52 != ( sgn_sgn @ A @ A2 ) ) % 8.50/4.52 => ( ( ( sgn_sgn @ A @ A2 ) % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( ( sgn_sgn @ A @ B2 ) % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( sgn_sgn @ A @ A2 ) % 8.50/4.52 = ( uminus_uminus @ A @ ( sgn_sgn @ A @ B2 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3576_mult__inverse__of__int__commute,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [Xa2: int,X3: A] : % 8.50/4.52 ( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa2 ) ) @ X3 ) % 8.50/4.52 = ( times_times @ A @ X3 @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3577_sgn__minus__1,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom_abs_sgn @ A ) % 8.50/4.52 => ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 8.50/4.52 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3578_nat__zero__as__int,axiom, % 8.50/4.52 ( ( zero_zero @ nat ) % 8.50/4.52 = ( nat2 @ ( zero_zero @ int ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3579_mult__sgn__abs,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( times_times @ A @ ( sgn_sgn @ A @ X3 ) @ ( abs_abs @ A @ X3 ) ) % 8.50/4.52 = X3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3580_sgn__mult__abs,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom_abs_sgn @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( abs_abs @ A @ A2 ) ) % 8.50/4.52 = A2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3581_abs__mult__sgn,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( idom_abs_sgn @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( sgn_sgn @ A @ A2 ) ) % 8.50/4.52 = A2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3582_linordered__idom__class_Oabs__sgn,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ( ( abs_abs @ A ) % 8.50/4.52 = ( ^ [K3: A] : ( times_times @ A @ K3 @ ( sgn_sgn @ A @ K3 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3583_nat__mono,axiom, % 8.50/4.52 ! [X3: int,Y: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ X3 @ Y ) % 8.50/4.52 => ( ord_less_eq @ nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3584_int__sgnE,axiom, % 8.50/4.52 ! [K: int] : % 8.50/4.52 ~ ! [N3: nat,L4: int] : % 8.50/4.52 ( K % 8.50/4.52 != ( times_times @ int @ ( sgn_sgn @ int @ L4 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3585_same__sgn__abs__add,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [B2: A,A2: A] : % 8.50/4.52 ( ( ( sgn_sgn @ A @ B2 ) % 8.50/4.52 = ( sgn_sgn @ A @ A2 ) ) % 8.50/4.52 => ( ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) ) % 8.50/4.52 = ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3586_ex__nat,axiom, % 8.50/4.52 ( ( ^ [P2: nat > $o] : % 8.50/4.52 ? [X7: nat] : ( P2 @ X7 ) ) % 8.50/4.52 = ( ^ [P3: nat > $o] : % 8.50/4.52 ? [X: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X ) % 8.50/4.52 & ( P3 @ ( nat2 @ X ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3587_all__nat,axiom, % 8.50/4.52 ( ( ^ [P2: nat > $o] : % 8.50/4.52 ! [X7: nat] : ( P2 @ X7 ) ) % 8.50/4.52 = ( ^ [P3: nat > $o] : % 8.50/4.52 ! [X: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X ) % 8.50/4.52 => ( P3 @ ( nat2 @ X ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3588_eq__nat__nat__iff,axiom, % 8.50/4.52 ! [Z2: int,Z8: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 8.50/4.52 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 ) % 8.50/4.52 => ( ( ( nat2 @ Z2 ) % 8.50/4.52 = ( nat2 @ Z8 ) ) % 8.50/4.52 = ( Z2 = Z8 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3589_le__imp__inverse__le__neg,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.52 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3590_inverse__le__imp__le__neg,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3591_le__imp__inverse__le,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.52 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.52 => ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3592_inverse__le__imp__le,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.52 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3593_inverse__le__1__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X3 ) @ ( one_one @ A ) ) % 8.50/4.52 = ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 8.50/4.52 | ( ord_less_eq @ A @ ( one_one @ A ) @ X3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3594_one__less__inverse__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X3 ) ) % 8.50/4.52 = ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 8.50/4.52 & ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3595_one__less__inverse,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.52 => ( ( ord_less @ A @ A2 @ ( one_one @ A ) ) % 8.50/4.52 => ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3596_field__class_Ofield__inverse,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 ) % 8.50/4.52 = ( one_one @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3597_division__ring__inverse__add,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( B2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 = ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3598_inverse__add,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( B2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 = ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3599_division__ring__inverse__diff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( B2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( minus_minus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 = ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( minus_minus @ A @ B2 @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3600_nonzero__inverse__eq__divide,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( inverse_inverse @ A @ A2 ) % 8.50/4.52 = ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3601_sgn__1__pos,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ( sgn_sgn @ A @ A2 ) % 8.50/4.52 = ( one_one @ A ) ) % 8.50/4.52 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3602_abs__sgn__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ( A2 % 8.50/4.52 = ( zero_zero @ A ) ) % 8.50/4.52 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) % 8.50/4.52 & ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) ) % 8.50/4.52 = ( one_one @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3603_nat__mono__iff,axiom, % 8.50/4.52 ! [Z2: int,W2: int] : % 8.50/4.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) % 8.50/4.52 => ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) ) % 8.50/4.52 = ( ord_less @ int @ W2 @ Z2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3604_of__nat__ceiling,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.52 => ! [R2: A] : ( ord_less_eq @ A @ R2 @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3605_zless__nat__eq__int__zless,axiom, % 8.50/4.52 ! [M: nat,Z2: int] : % 8.50/4.52 ( ( ord_less @ nat @ M @ ( nat2 @ Z2 ) ) % 8.50/4.52 = ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3606_nat__le__iff,axiom, % 8.50/4.52 ! [X3: int,N2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ ( nat2 @ X3 ) @ N2 ) % 8.50/4.52 = ( ord_less_eq @ int @ X3 @ ( semiring_1_of_nat @ int @ N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3607_nat__0__le,axiom, % 8.50/4.52 ! [Z2: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 8.50/4.52 => ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) ) % 8.50/4.52 = Z2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3608_int__eq__iff,axiom, % 8.50/4.52 ! [M: nat,Z2: int] : % 8.50/4.52 ( ( ( semiring_1_of_nat @ int @ M ) % 8.50/4.52 = Z2 ) % 8.50/4.52 = ( ( M % 8.50/4.52 = ( nat2 @ Z2 ) ) % 8.50/4.52 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3609_nat__int__add,axiom, % 8.50/4.52 ! [A2: nat,B2: nat] : % 8.50/4.52 ( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) % 8.50/4.52 = ( plus_plus @ nat @ A2 @ B2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3610_nat__abs__mult__distrib,axiom, % 8.50/4.52 ! [W2: int,Z2: int] : % 8.50/4.52 ( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W2 @ Z2 ) ) ) % 8.50/4.52 = ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W2 ) ) @ ( nat2 @ ( abs_abs @ int @ Z2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3611_nat__plus__as__int,axiom, % 8.50/4.52 ( ( plus_plus @ nat ) % 8.50/4.52 = ( ^ [A4: nat,B4: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3612_inverse__less__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 8.50/4.52 => ( ord_less @ A @ B2 @ A2 ) ) % 8.50/4.52 & ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3613_inverse__le__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) % 8.50/4.52 => ( ord_less_eq @ A @ B2 @ A2 ) ) % 8.50/4.52 & ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) % 8.50/4.52 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3614_one__le__inverse__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X3 ) ) % 8.50/4.52 = ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 8.50/4.52 & ( ord_less_eq @ A @ X3 @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3615_inverse__less__1__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( ord_less @ A @ ( inverse_inverse @ A @ X3 ) @ ( one_one @ A ) ) % 8.50/4.52 = ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) % 8.50/4.52 | ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3616_one__le__inverse,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3617_inverse__diff__inverse,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( A2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( B2 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( minus_minus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) % 8.50/4.52 = ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3618_reals__Archimedean,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( archim462609752435547400_field @ A ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 8.50/4.52 => ? [N3: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ X3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3619_sgn__if,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ( ( sgn_sgn @ A ) % 8.50/4.52 = ( ^ [X: A] : % 8.50/4.52 ( if @ A % 8.50/4.52 @ ( X % 8.50/4.52 = ( zero_zero @ A ) ) % 8.50/4.52 @ ( zero_zero @ A ) % 8.50/4.52 @ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3620_sgn__1__neg,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [A2: A] : % 8.50/4.52 ( ( ( sgn_sgn @ A @ A2 ) % 8.50/4.52 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 8.50/4.52 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3621_of__nat__floor,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.52 => ! [R2: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 ) % 8.50/4.52 => ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archim6421214686448440834_floor @ A @ R2 ) ) ) @ R2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3622_zsgn__def,axiom, % 8.50/4.52 ( ( sgn_sgn @ int ) % 8.50/4.52 = ( ^ [I3: int] : % 8.50/4.52 ( if @ int % 8.50/4.52 @ ( I3 % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 @ ( zero_zero @ int ) % 8.50/4.52 @ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I3 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3623_powr__real__of__int,axiom, % 8.50/4.52 ! [X3: real,N2: int] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.52 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) % 8.50/4.52 => ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ N2 ) ) % 8.50/4.52 = ( power_power @ real @ X3 @ ( nat2 @ N2 ) ) ) ) % 8.50/4.52 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) % 8.50/4.52 => ( ( powr @ real @ X3 @ ( ring_1_of_int @ real @ N2 ) ) % 8.50/4.52 = ( inverse_inverse @ real @ ( power_power @ real @ X3 @ ( nat2 @ ( uminus_uminus @ int @ N2 ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3624_real__vector__eq__affinity,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [M: real,Y: A,X3: A,C2: A] : % 8.50/4.52 ( ( M % 8.50/4.52 != ( zero_zero @ real ) ) % 8.50/4.52 => ( ( Y % 8.50/4.52 = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X3 ) @ C2 ) ) % 8.50/4.52 = ( ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) ) % 8.50/4.52 = X3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3625_real__vector__affinity__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.52 => ! [M: real,X3: A,C2: A,Y: A] : % 8.50/4.52 ( ( M % 8.50/4.52 != ( zero_zero @ real ) ) % 8.50/4.52 => ( ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X3 ) @ C2 ) % 8.50/4.52 = Y ) % 8.50/4.52 = ( X3 % 8.50/4.52 = ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3626_pos__divideR__le__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,B2: A,A2: A] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 ) % 8.50/4.52 = ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3627_pos__le__divideR__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) % 8.50/4.52 = ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3628_neg__divideR__le__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,B2: A,A2: A] : % 8.50/4.52 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 ) % 8.50/4.52 = ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3629_neg__le__divideR__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) % 8.50/4.52 = ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3630_pos__divideR__less__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,B2: A,A2: A] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 8.50/4.52 => ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 ) % 8.50/4.52 = ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3631_pos__less__divideR__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 8.50/4.52 => ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) % 8.50/4.52 = ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3632_neg__divideR__less__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,B2: A,A2: A] : % 8.50/4.52 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 ) % 8.50/4.52 = ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3633_neg__less__divideR__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) % 8.50/4.52 = ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3634_nat__less__eq__zless,axiom, % 8.50/4.52 ! [W2: int,Z2: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 ) % 8.50/4.52 => ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) ) % 8.50/4.52 = ( ord_less @ int @ W2 @ Z2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3635_nat__le__eq__zle,axiom, % 8.50/4.52 ! [W2: int,Z2: int] : % 8.50/4.52 ( ( ( ord_less @ int @ ( zero_zero @ int ) @ W2 ) % 8.50/4.52 | ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) ) % 8.50/4.52 = ( ord_less_eq @ int @ W2 @ Z2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3636_norm__sgn,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V822414075346904944vector @ A ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( ( X3 % 8.50/4.52 = ( zero_zero @ A ) ) % 8.50/4.52 => ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X3 ) ) % 8.50/4.52 = ( zero_zero @ real ) ) ) % 8.50/4.52 & ( ( X3 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X3 ) ) % 8.50/4.52 = ( one_one @ real ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3637_nat__eq__iff,axiom, % 8.50/4.52 ! [W2: int,M: nat] : % 8.50/4.52 ( ( ( nat2 @ W2 ) % 8.50/4.52 = M ) % 8.50/4.52 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 ) % 8.50/4.52 => ( W2 % 8.50/4.52 = ( semiring_1_of_nat @ int @ M ) ) ) % 8.50/4.52 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 ) % 8.50/4.52 => ( M % 8.50/4.52 = ( zero_zero @ nat ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3638_nat__eq__iff2,axiom, % 8.50/4.52 ! [M: nat,W2: int] : % 8.50/4.52 ( ( M % 8.50/4.52 = ( nat2 @ W2 ) ) % 8.50/4.52 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 ) % 8.50/4.52 => ( W2 % 8.50/4.52 = ( semiring_1_of_nat @ int @ M ) ) ) % 8.50/4.52 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 ) % 8.50/4.52 => ( M % 8.50/4.52 = ( zero_zero @ nat ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3639_split__nat,axiom, % 8.50/4.52 ! [P: nat > $o,I: int] : % 8.50/4.52 ( ( P @ ( nat2 @ I ) ) % 8.50/4.52 = ( ! [N: nat] : % 8.50/4.52 ( ( I % 8.50/4.52 = ( semiring_1_of_nat @ int @ N ) ) % 8.50/4.52 => ( P @ N ) ) % 8.50/4.52 & ( ( ord_less @ int @ I @ ( zero_zero @ int ) ) % 8.50/4.52 => ( P @ ( zero_zero @ nat ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3640_le__mult__nat__floor,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( archim2362893244070406136eiling @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( nat2 @ ( archim6421214686448440834_floor @ A @ A2 ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ B2 ) ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3641_le__nat__iff,axiom, % 8.50/4.52 ! [K: int,N2: nat] : % 8.50/4.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ N2 @ ( nat2 @ K ) ) % 8.50/4.52 = ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3642_nat__add__distrib,axiom, % 8.50/4.52 ! [Z2: int,Z8: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 8.50/4.52 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 ) % 8.50/4.52 => ( ( nat2 @ ( plus_plus @ int @ Z2 @ Z8 ) ) % 8.50/4.52 = ( plus_plus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3643_nat__mult__distrib,axiom, % 8.50/4.52 ! [Z2: int,Z8: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 8.50/4.52 => ( ( nat2 @ ( times_times @ int @ Z2 @ Z8 ) ) % 8.50/4.52 = ( times_times @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3644_Suc__as__int,axiom, % 8.50/4.52 ( suc % 8.50/4.52 = ( ^ [A4: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A4 ) @ ( one_one @ int ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3645_forall__pos__mono__1,axiom, % 8.50/4.52 ! [P: real > $o,E3: real] : % 8.50/4.52 ( ! [D3: real,E: real] : % 8.50/4.52 ( ( ord_less @ real @ D3 @ E ) % 8.50/4.52 => ( ( P @ D3 ) % 8.50/4.52 => ( P @ E ) ) ) % 8.50/4.52 => ( ! [N3: nat] : ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) ) % 8.50/4.52 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 ) % 8.50/4.52 => ( P @ E3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3646_forall__pos__mono,axiom, % 8.50/4.52 ! [P: real > $o,E3: real] : % 8.50/4.52 ( ! [D3: real,E: real] : % 8.50/4.52 ( ( ord_less @ real @ D3 @ E ) % 8.50/4.52 => ( ( P @ D3 ) % 8.50/4.52 => ( P @ E ) ) ) % 8.50/4.52 => ( ! [N3: nat] : % 8.50/4.52 ( ( N3 % 8.50/4.52 != ( zero_zero @ nat ) ) % 8.50/4.52 => ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) ) ) % 8.50/4.52 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 ) % 8.50/4.52 => ( P @ E3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3647_real__arch__inverse,axiom, % 8.50/4.52 ! [E3: real] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 ) % 8.50/4.52 = ( ? [N: nat] : % 8.50/4.52 ( ( N % 8.50/4.52 != ( zero_zero @ nat ) ) % 8.50/4.52 & ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N ) ) ) % 8.50/4.52 & ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N ) ) @ E3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3648_nat__diff__distrib,axiom, % 8.50/4.52 ! [Z8: int,Z2: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 ) % 8.50/4.52 => ( ( ord_less_eq @ int @ Z8 @ Z2 ) % 8.50/4.52 => ( ( nat2 @ ( minus_minus @ int @ Z2 @ Z8 ) ) % 8.50/4.52 = ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3649_nat__diff__distrib_H,axiom, % 8.50/4.52 ! [X3: int,Y: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 ) % 8.50/4.52 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y ) % 8.50/4.52 => ( ( nat2 @ ( minus_minus @ int @ X3 @ Y ) ) % 8.50/4.52 = ( minus_minus @ nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3650_nat__abs__triangle__ineq,axiom, % 8.50/4.52 ! [K: int,L: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K @ L ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3651_nat__floor__neg,axiom, % 8.50/4.52 ! [X3: real] : % 8.50/4.52 ( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X3 ) ) % 8.50/4.52 = ( zero_zero @ nat ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3652_nat__power__eq,axiom, % 8.50/4.52 ! [Z2: int,N2: nat] : % 8.50/4.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 8.50/4.52 => ( ( nat2 @ ( power_power @ int @ Z2 @ N2 ) ) % 8.50/4.52 = ( power_power @ nat @ ( nat2 @ Z2 ) @ N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3653_ln__inverse,axiom, % 8.50/4.52 ! [X3: real] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.52 => ( ( ln_ln @ real @ ( inverse_inverse @ real @ X3 ) ) % 8.50/4.52 = ( uminus_uminus @ real @ ( ln_ln @ real @ X3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3654_floor__eq3,axiom, % 8.50/4.52 ! [N2: nat,X3: real] : % 8.50/4.52 ( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X3 ) % 8.50/4.52 => ( ( ord_less @ real @ X3 @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) % 8.50/4.52 => ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X3 ) ) % 8.50/4.52 = N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3655_le__nat__floor,axiom, % 8.50/4.52 ! [X3: nat,A2: real] : % 8.50/4.52 ( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X3 ) @ A2 ) % 8.50/4.52 => ( ord_less_eq @ nat @ X3 @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3656_the__elem__def,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( the_elem @ A ) % 8.50/4.52 = ( ^ [X6: set @ A] : % 8.50/4.52 ( the @ A % 8.50/4.52 @ ^ [X: A] : % 8.50/4.52 ( X6 % 8.50/4.52 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3657_divide__int__def,axiom, % 8.50/4.52 ( ( divide_divide @ int ) % 8.50/4.52 = ( ^ [K3: int,L2: int] : % 8.50/4.52 ( if @ int % 8.50/4.52 @ ( L2 % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 @ ( zero_zero @ int ) % 8.50/4.52 @ ( if @ int % 8.50/4.52 @ ( ( sgn_sgn @ int @ K3 ) % 8.50/4.52 = ( sgn_sgn @ int @ L2 ) ) % 8.50/4.52 @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) % 8.50/4.52 @ ( uminus_uminus @ int % 8.50/4.52 @ ( semiring_1_of_nat @ int % 8.50/4.52 @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) % 8.50/4.52 @ ( zero_neq_one_of_bool @ nat % 8.50/4.52 @ ~ ( dvd_dvd @ int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3658_ex__inverse__of__nat__less,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( archim462609752435547400_field @ A ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) % 8.50/4.52 => ? [N3: nat] : % 8.50/4.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) % 8.50/4.52 & ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N3 ) ) @ X3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3659_power__diff__conv__inverse,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( division_ring @ A ) % 8.50/4.52 => ! [X3: A,M: nat,N2: nat] : % 8.50/4.52 ( ( X3 % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.52 => ( ( power_power @ A @ X3 @ ( minus_minus @ nat @ N2 @ M ) ) % 8.50/4.52 = ( times_times @ A @ ( power_power @ A @ X3 @ N2 ) @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ M ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3660_nat__2,axiom, % 8.50/4.52 ( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) % 8.50/4.52 = ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3661_pos__le__minus__divideR__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) ) % 8.50/4.52 = ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3662_pos__minus__divideR__le__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,B2: A,A2: A] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 ) % 8.50/4.52 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3663_neg__le__minus__divideR__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) ) % 8.50/4.52 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3664_neg__minus__divideR__le__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,B2: A,A2: A] : % 8.50/4.52 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 ) % 8.50/4.52 = ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3665_pos__less__minus__divideR__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 8.50/4.52 => ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) ) % 8.50/4.52 = ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3666_pos__minus__divideR__less__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,B2: A,A2: A] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 8.50/4.52 => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 ) % 8.50/4.52 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3667_neg__less__minus__divideR__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) ) % 8.50/4.52 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3668_neg__minus__divideR__less__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.52 => ! [C2: real,B2: A,A2: A] : % 8.50/4.52 ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 ) % 8.50/4.52 = ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3669_Suc__nat__eq__nat__zadd1,axiom, % 8.50/4.52 ! [Z2: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 8.50/4.52 => ( ( suc @ ( nat2 @ Z2 ) ) % 8.50/4.52 = ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3670_nat__less__iff,axiom, % 8.50/4.52 ! [W2: int,M: nat] : % 8.50/4.52 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 ) % 8.50/4.52 => ( ( ord_less @ nat @ ( nat2 @ W2 ) @ M ) % 8.50/4.52 = ( ord_less @ int @ W2 @ ( semiring_1_of_nat @ int @ M ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3671_nat__mult__distrib__neg,axiom, % 8.50/4.52 ! [Z2: int,Z8: int] : % 8.50/4.52 ( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) ) % 8.50/4.52 => ( ( nat2 @ ( times_times @ int @ Z2 @ Z8 ) ) % 8.50/4.52 = ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z8 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3672_nat__abs__int__diff,axiom, % 8.50/4.52 ! [A2: nat,B2: nat] : % 8.50/4.52 ( ( ( ord_less_eq @ nat @ A2 @ B2 ) % 8.50/4.52 => ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) % 8.50/4.52 = ( minus_minus @ nat @ B2 @ A2 ) ) ) % 8.50/4.52 & ( ~ ( ord_less_eq @ nat @ A2 @ B2 ) % 8.50/4.52 => ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) % 8.50/4.52 = ( minus_minus @ nat @ A2 @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3673_log__inverse,axiom, % 8.50/4.52 ! [A2: real,X3: real] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.52 => ( ( A2 % 8.50/4.52 != ( one_one @ real ) ) % 8.50/4.52 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.52 => ( ( log @ A2 @ ( inverse_inverse @ real @ X3 ) ) % 8.50/4.52 = ( uminus_uminus @ real @ ( log @ A2 @ X3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3674_floor__eq4,axiom, % 8.50/4.52 ! [N2: nat,X3: real] : % 8.50/4.52 ( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X3 ) % 8.50/4.52 => ( ( ord_less @ real @ X3 @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) % 8.50/4.52 => ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X3 ) ) % 8.50/4.52 = N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3675_diff__nat__eq__if,axiom, % 8.50/4.52 ! [Z8: int,Z2: int] : % 8.50/4.52 ( ( ( ord_less @ int @ Z8 @ ( zero_zero @ int ) ) % 8.50/4.52 => ( ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) % 8.50/4.52 = ( nat2 @ Z2 ) ) ) % 8.50/4.52 & ( ~ ( ord_less @ int @ Z8 @ ( zero_zero @ int ) ) % 8.50/4.52 => ( ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) % 8.50/4.52 = ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z2 @ Z8 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z2 @ Z8 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3676_of__int__of__nat,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ring_1 @ A ) % 8.50/4.52 => ( ( ring_1_of_int @ A ) % 8.50/4.52 = ( ^ [K3: int] : ( if @ A @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K3 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3677_nat__dvd__iff,axiom, % 8.50/4.52 ! [Z2: int,M: nat] : % 8.50/4.52 ( ( dvd_dvd @ nat @ ( nat2 @ Z2 ) @ M ) % 8.50/4.52 = ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 8.50/4.52 => ( dvd_dvd @ int @ Z2 @ ( semiring_1_of_nat @ int @ M ) ) ) % 8.50/4.52 & ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) % 8.50/4.52 => ( M % 8.50/4.52 = ( zero_zero @ nat ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3678_eucl__rel__int__remainderI,axiom, % 8.50/4.52 ! [R2: int,L: int,K: int,Q4: int] : % 8.50/4.52 ( ( ( sgn_sgn @ int @ R2 ) % 8.50/4.52 = ( sgn_sgn @ int @ L ) ) % 8.50/4.52 => ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ L ) ) % 8.50/4.52 => ( ( K % 8.50/4.52 = ( plus_plus @ int @ ( times_times @ int @ Q4 @ L ) @ R2 ) ) % 8.50/4.52 => ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3679_plus__inverse__ge__2,axiom, % 8.50/4.52 ! [X3: real] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.52 => ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ X3 @ ( inverse_inverse @ real @ X3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3680_real__inv__sqrt__pow2,axiom, % 8.50/4.52 ! [X3: real] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.52 => ( ( power_power @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.50/4.52 = ( inverse_inverse @ real @ X3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3681_eucl__rel__int_Ocases,axiom, % 8.50/4.52 ! [A12: int,A23: int,A32: product_prod @ int @ int] : % 8.50/4.52 ( ( eucl_rel_int @ A12 @ A23 @ A32 ) % 8.50/4.52 => ( ( ( A23 % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 => ( A32 % 8.50/4.52 != ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A12 ) ) ) % 8.50/4.52 => ( ! [Q2: int] : % 8.50/4.52 ( ( A32 % 8.50/4.52 = ( product_Pair @ int @ int @ Q2 @ ( zero_zero @ int ) ) ) % 8.50/4.52 => ( ( A23 % 8.50/4.52 != ( zero_zero @ int ) ) % 8.50/4.52 => ( A12 % 8.50/4.52 != ( times_times @ int @ Q2 @ A23 ) ) ) ) % 8.50/4.52 => ~ ! [R4: int,Q2: int] : % 8.50/4.52 ( ( A32 % 8.50/4.52 = ( product_Pair @ int @ int @ Q2 @ R4 ) ) % 8.50/4.52 => ( ( ( sgn_sgn @ int @ R4 ) % 8.50/4.52 = ( sgn_sgn @ int @ A23 ) ) % 8.50/4.52 => ( ( ord_less @ int @ ( abs_abs @ int @ R4 ) @ ( abs_abs @ int @ A23 ) ) % 8.50/4.52 => ( A12 % 8.50/4.52 != ( plus_plus @ int @ ( times_times @ int @ Q2 @ A23 ) @ R4 ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3682_eucl__rel__int_Osimps,axiom, % 8.50/4.52 ( eucl_rel_int % 8.50/4.52 = ( ^ [A1: int,A22: int,A33: product_prod @ int @ int] : % 8.50/4.52 ( ? [K3: int] : % 8.50/4.52 ( ( A1 = K3 ) % 8.50/4.52 & ( A22 % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 & ( A33 % 8.50/4.52 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) ) % 8.50/4.52 | ? [L2: int,K3: int,Q5: int] : % 8.50/4.52 ( ( A1 = K3 ) % 8.50/4.52 & ( A22 = L2 ) % 8.50/4.52 & ( A33 % 8.50/4.52 = ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) ) % 8.50/4.52 & ( L2 % 8.50/4.52 != ( zero_zero @ int ) ) % 8.50/4.52 & ( K3 % 8.50/4.52 = ( times_times @ int @ Q5 @ L2 ) ) ) % 8.50/4.52 | ? [R5: int,L2: int,K3: int,Q5: int] : % 8.50/4.52 ( ( A1 = K3 ) % 8.50/4.52 & ( A22 = L2 ) % 8.50/4.52 & ( A33 % 8.50/4.52 = ( product_Pair @ int @ int @ Q5 @ R5 ) ) % 8.50/4.52 & ( ( sgn_sgn @ int @ R5 ) % 8.50/4.52 = ( sgn_sgn @ int @ L2 ) ) % 8.50/4.52 & ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L2 ) ) % 8.50/4.52 & ( K3 % 8.50/4.52 = ( plus_plus @ int @ ( times_times @ int @ Q5 @ L2 ) @ R5 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3683_exp__series__add__commuting,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.52 => ! [X3: A,Y: A,N2: nat] : % 8.50/4.52 ( ( ( times_times @ A @ X3 @ Y ) % 8.50/4.52 = ( times_times @ A @ Y @ X3 ) ) % 8.50/4.52 => ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ N2 ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I3 ) ) @ ( power_power @ A @ X3 @ I3 ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N2 @ I3 ) ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N2 @ I3 ) ) ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3684_exp__first__term,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.52 => ( ( exp @ A ) % 8.50/4.52 = ( ^ [X: A] : % 8.50/4.52 ( plus_plus @ A @ ( one_one @ A ) % 8.50/4.52 @ ( suminf @ A % 8.50/4.52 @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N ) ) ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3685_tan__sec,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( ( cos @ A @ X3 ) % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) % 8.50/4.52 = ( power_power @ A @ ( inverse_inverse @ A @ ( cos @ A @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3686_exp__first__terms,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.52 => ! [K: nat] : % 8.50/4.52 ( ( exp @ A ) % 8.50/4.52 = ( ^ [X: A] : % 8.50/4.52 ( plus_plus @ A % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X @ N ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ K ) ) % 8.50/4.52 @ ( suminf @ A % 8.50/4.52 @ ^ [N: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N @ K ) ) ) @ ( power_power @ A @ X @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3687_divide__int__unfold,axiom, % 8.50/4.52 ! [L: int,K: int,N2: nat,M: nat] : % 8.50/4.52 ( ( ( ( ( sgn_sgn @ int @ L ) % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 | ( ( sgn_sgn @ int @ K ) % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 | ( N2 % 8.50/4.52 = ( zero_zero @ nat ) ) ) % 8.50/4.52 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) % 8.50/4.52 = ( zero_zero @ int ) ) ) % 8.50/4.52 & ( ~ ( ( ( sgn_sgn @ int @ L ) % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 | ( ( sgn_sgn @ int @ K ) % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 | ( N2 % 8.50/4.52 = ( zero_zero @ nat ) ) ) % 8.50/4.52 => ( ( ( ( sgn_sgn @ int @ K ) % 8.50/4.52 = ( sgn_sgn @ int @ L ) ) % 8.50/4.52 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) % 8.50/4.52 = ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N2 ) ) ) ) % 8.50/4.52 & ( ( ( sgn_sgn @ int @ K ) % 8.50/4.52 != ( sgn_sgn @ int @ L ) ) % 8.50/4.52 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) % 8.50/4.52 = ( uminus_uminus @ int % 8.50/4.52 @ ( semiring_1_of_nat @ int % 8.50/4.52 @ ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N2 ) % 8.50/4.52 @ ( zero_neq_one_of_bool @ nat % 8.50/4.52 @ ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3688_old_Orec__prod__def,axiom, % 8.50/4.52 ! [T: $tType,B: $tType,A: $tType] : % 8.50/4.52 ( ( product_rec_prod @ A @ B @ T ) % 8.50/4.52 = ( ^ [F12: A > B > T,X: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3689_sinh__converges,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( sums @ A % 8.50/4.52 @ ^ [N: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X3 @ N ) ) ) % 8.50/4.52 @ ( sinh @ A @ X3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3690_cosh__converges,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( sums @ A % 8.50/4.52 @ ^ [N: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ X3 @ N ) ) @ ( zero_zero @ A ) ) % 8.50/4.52 @ ( cosh @ A @ X3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3691_sinh__ln__real,axiom, % 8.50/4.52 ! [X3: real] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.52 => ( ( sinh @ real @ ( ln_ln @ real @ X3 ) ) % 8.50/4.52 = ( divide_divide @ real @ ( minus_minus @ real @ X3 @ ( inverse_inverse @ real @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3692_arcsin__lt__bounded,axiom, % 8.50/4.52 ! [Y: real] : % 8.50/4.52 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y ) % 8.50/4.52 => ( ( ord_less @ real @ Y @ ( one_one @ real ) ) % 8.50/4.52 => ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) ) % 8.50/4.52 & ( ord_less @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3693_sinh__0,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.52 => ( ( sinh @ A @ ( zero_zero @ A ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3694_sinh__real__less__iff,axiom, % 8.50/4.52 ! [X3: real,Y: real] : % 8.50/4.52 ( ( ord_less @ real @ ( sinh @ real @ X3 ) @ ( sinh @ real @ Y ) ) % 8.50/4.52 = ( ord_less @ real @ X3 @ Y ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3695_The__split__eq,axiom, % 8.50/4.52 ! [A: $tType,B: $tType,X3: A,Y: B] : % 8.50/4.52 ( ( the @ ( product_prod @ A @ B ) % 8.50/4.52 @ ( product_case_prod @ A @ B @ $o % 8.50/4.52 @ ^ [X9: A,Y6: B] : % 8.50/4.52 ( ( X3 = X9 ) % 8.50/4.52 & ( Y = Y6 ) ) ) ) % 8.50/4.52 = ( product_Pair @ A @ B @ X3 @ Y ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3696_cosh__0,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.52 => ( ( cosh @ A @ ( zero_zero @ A ) ) % 8.50/4.52 = ( one_one @ A ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3697_sinh__real__neg__iff,axiom, % 8.50/4.52 ! [X3: real] : % 8.50/4.52 ( ( ord_less @ real @ ( sinh @ real @ X3 ) @ ( zero_zero @ real ) ) % 8.50/4.52 = ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3698_sinh__real__pos__iff,axiom, % 8.50/4.52 ! [X3: real] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X3 ) ) % 8.50/4.52 = ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3699_sinh__plus__cosh,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( plus_plus @ A @ ( sinh @ A @ X3 ) @ ( cosh @ A @ X3 ) ) % 8.50/4.52 = ( exp @ A @ X3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3700_cosh__plus__sinh,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( plus_plus @ A @ ( cosh @ A @ X3 ) @ ( sinh @ A @ X3 ) ) % 8.50/4.52 = ( exp @ A @ X3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3701_sinh__less__cosh__real,axiom, % 8.50/4.52 ! [X3: real] : ( ord_less @ real @ ( sinh @ real @ X3 ) @ ( cosh @ real @ X3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3702_sinh__add,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.52 => ! [X3: A,Y: A] : % 8.50/4.52 ( ( sinh @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 8.50/4.52 = ( plus_plus @ A @ ( times_times @ A @ ( sinh @ A @ X3 ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( cosh @ A @ X3 ) @ ( sinh @ A @ Y ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3703_cosh__add,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.52 => ! [X3: A,Y: A] : % 8.50/4.52 ( ( cosh @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 8.50/4.52 = ( plus_plus @ A @ ( times_times @ A @ ( cosh @ A @ X3 ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( sinh @ A @ X3 ) @ ( sinh @ A @ Y ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3704_cosh__real__pos,axiom, % 8.50/4.52 ! [X3: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3705_cosh__real__strict__mono,axiom, % 8.50/4.52 ! [X3: real,Y: real] : % 8.50/4.52 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.52 => ( ( ord_less @ real @ X3 @ Y ) % 8.50/4.52 => ( ord_less @ real @ ( cosh @ real @ X3 ) @ ( cosh @ real @ Y ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3706_cosh__real__nonneg__less__iff,axiom, % 8.50/4.52 ! [X3: real,Y: real] : % 8.50/4.52 ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.52 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.52 => ( ( ord_less @ real @ ( cosh @ real @ X3 ) @ ( cosh @ real @ Y ) ) % 8.50/4.52 = ( ord_less @ real @ X3 @ Y ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3707_cosh__real__nonpos__less__iff,axiom, % 8.50/4.52 ! [X3: real,Y: real] : % 8.50/4.52 ( ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ( ord_less_eq @ real @ Y @ ( zero_zero @ real ) ) % 8.50/4.52 => ( ( ord_less @ real @ ( cosh @ real @ X3 ) @ ( cosh @ real @ Y ) ) % 8.50/4.52 = ( ord_less @ real @ Y @ X3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3708_cosh__square__eq,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( power_power @ A @ ( cosh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.50/4.52 = ( plus_plus @ A @ ( power_power @ A @ ( sinh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3709_cosh__double,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( cosh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X3 ) ) % 8.50/4.52 = ( plus_plus @ A @ ( power_power @ A @ ( cosh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3710_sgn__real__def,axiom, % 8.50/4.52 ( ( sgn_sgn @ real ) % 8.50/4.52 = ( ^ [A4: real] : % 8.50/4.52 ( if @ real % 8.50/4.52 @ ( A4 % 8.50/4.52 = ( zero_zero @ real ) ) % 8.50/4.52 @ ( zero_zero @ real ) % 8.50/4.52 @ ( if @ real @ ( ord_less @ real @ ( zero_zero @ real ) @ A4 ) @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3711_arcsin__less__arcsin,axiom, % 8.50/4.52 ! [X3: real,Y: real] : % 8.50/4.52 ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 8.50/4.52 => ( ( ord_less @ real @ X3 @ Y ) % 8.50/4.52 => ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) ) % 8.50/4.52 => ( ord_less @ real @ ( arcsin @ X3 ) @ ( arcsin @ Y ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3712_arcsin__less__mono,axiom, % 8.50/4.52 ! [X3: real,Y: real] : % 8.50/4.52 ( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 8.50/4.52 => ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) ) % 8.50/4.52 => ( ( ord_less @ real @ ( arcsin @ X3 ) @ ( arcsin @ Y ) ) % 8.50/4.52 = ( ord_less @ real @ X3 @ Y ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3713_sgn__power__injE,axiom, % 8.50/4.52 ! [A2: real,N2: nat,X3: real,B2: real] : % 8.50/4.52 ( ( ( times_times @ real @ ( sgn_sgn @ real @ A2 ) @ ( power_power @ real @ ( abs_abs @ real @ A2 ) @ N2 ) ) % 8.50/4.52 = X3 ) % 8.50/4.52 => ( ( X3 % 8.50/4.52 = ( times_times @ real @ ( sgn_sgn @ real @ B2 ) @ ( power_power @ real @ ( abs_abs @ real @ B2 ) @ N2 ) ) ) % 8.50/4.52 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.52 => ( A2 = B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3714_cos__arcsin__nonzero,axiom, % 8.50/4.52 ! [X3: real] : % 8.50/4.52 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 8.50/4.52 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 8.50/4.52 => ( ( cos @ real @ ( arcsin @ X3 ) ) % 8.50/4.52 != ( zero_zero @ real ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3715_tanh__add,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.52 => ! [X3: A,Y: A] : % 8.50/4.52 ( ( ( cosh @ A @ X3 ) % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( ( cosh @ A @ Y ) % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( tanh @ A @ ( plus_plus @ A @ X3 @ Y ) ) % 8.50/4.52 = ( divide_divide @ A @ ( plus_plus @ A @ ( tanh @ A @ X3 ) @ ( tanh @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tanh @ A @ X3 ) @ ( tanh @ A @ Y ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3716_sinh__zero__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( ( sinh @ A @ X3 ) % 8.50/4.52 = ( zero_zero @ A ) ) % 8.50/4.52 = ( member @ A @ ( exp @ A @ X3 ) @ ( insert @ A @ ( one_one @ A ) @ ( insert @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3717_floor__real__def,axiom, % 8.50/4.52 ( ( archim6421214686448440834_floor @ real ) % 8.50/4.52 = ( ^ [X: real] : % 8.50/4.52 ( the @ int % 8.50/4.52 @ ^ [Z6: int] : % 8.50/4.52 ( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ Z6 ) @ X ) % 8.50/4.52 & ( ord_less @ real @ X @ ( ring_1_of_int @ real @ ( plus_plus @ int @ Z6 @ ( one_one @ int ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3718_cosh__field__def,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.52 => ( ( cosh @ A ) % 8.50/4.52 = ( ^ [Z6: A] : ( divide_divide @ A @ ( plus_plus @ A @ ( exp @ A @ Z6 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z6 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3719_cosh__zero__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( ( cosh @ A @ X3 ) % 8.50/4.52 = ( zero_zero @ A ) ) % 8.50/4.52 = ( ( power_power @ A @ ( exp @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.50/4.52 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3720_cosh__def,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_Vector_banach @ A ) % 8.50/4.52 & ( real_V2822296259951069270ebra_1 @ A ) ) % 8.50/4.52 => ( ( cosh @ A ) % 8.50/4.52 = ( ^ [X: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3721_cosh__ln__real,axiom, % 8.50/4.52 ! [X3: real] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.52 => ( ( cosh @ real @ ( ln_ln @ real @ X3 ) ) % 8.50/4.52 = ( divide_divide @ real @ ( plus_plus @ real @ X3 @ ( inverse_inverse @ real @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3722_floor__rat__def,axiom, % 8.50/4.52 ( ( archim6421214686448440834_floor @ rat ) % 8.50/4.52 = ( ^ [X: rat] : % 8.50/4.52 ( the @ int % 8.50/4.52 @ ^ [Z6: int] : % 8.50/4.52 ( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z6 ) @ X ) % 8.50/4.52 & ( ord_less @ rat @ X @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z6 @ ( one_one @ int ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3723_sum__diff1_H__aux,axiom, % 8.50/4.52 ! [B: $tType,A: $tType] : % 8.50/4.52 ( ( ab_group_add @ B ) % 8.50/4.52 => ! [F5: set @ A,I6: set @ A,F2: A > B,I: A] : % 8.50/4.52 ( ( finite_finite2 @ A @ F5 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ A ) % 8.50/4.52 @ ( collect @ A % 8.50/4.52 @ ^ [I3: A] : % 8.50/4.52 ( ( member @ A @ I3 @ I6 ) % 8.50/4.52 & ( ( F2 @ I3 ) % 8.50/4.52 != ( zero_zero @ B ) ) ) ) % 8.50/4.52 @ F5 ) % 8.50/4.52 => ( ( ( member @ A @ I @ I6 ) % 8.50/4.52 => ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.52 = ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) @ ( F2 @ I ) ) ) ) % 8.50/4.52 & ( ~ ( member @ A @ I @ I6 ) % 8.50/4.52 => ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.52 = ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3724_VEBT_Osize_I3_J,axiom, % 8.50/4.52 ! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] : % 8.50/4.52 ( ( size_size @ vEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) ) % 8.50/4.52 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_list @ vEBT_VEBT @ ( size_size @ vEBT_VEBT ) @ X13 ) @ ( size_size @ vEBT_VEBT @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3725_sum__power2,axiom, % 8.50/4.52 ! [K: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) % 8.50/4.52 = ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ ( one_one @ nat ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3726_finite__atLeastLessThan,axiom, % 8.50/4.52 ! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3727_atLeastLessThan__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ord @ A ) % 8.50/4.52 => ! [I: A,L: A,U: A] : % 8.50/4.52 ( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) ) % 8.50/4.52 = ( ( ord_less_eq @ A @ L @ I ) % 8.50/4.52 & ( ord_less @ A @ I @ U ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3728_atLeastLessThan__empty,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( order @ A ) % 8.50/4.52 => ! [B2: A,A2: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.50/4.52 => ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) % 8.50/4.52 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3729_ivl__subset,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linorder @ A ) % 8.50/4.52 => ! [I: A,J: A,M: A,N2: A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M @ N2 ) ) % 8.50/4.52 = ( ( ord_less_eq @ A @ J @ I ) % 8.50/4.52 | ( ( ord_less_eq @ A @ M @ I ) % 8.50/4.52 & ( ord_less_eq @ A @ J @ N2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3730_atLeastLessThan__empty__iff2,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( preorder @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ( bot_bot @ ( set @ A ) ) % 8.50/4.52 = ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) % 8.50/4.52 = ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3731_atLeastLessThan__empty__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( preorder @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) % 8.50/4.52 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.52 = ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3732_infinite__Ico__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( dense_linorder @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) ) % 8.50/4.52 = ( ord_less @ A @ A2 @ B2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3733_ivl__diff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linorder @ A ) % 8.50/4.52 => ! [I: A,N2: A,M: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ I @ N2 ) % 8.50/4.52 => ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M ) @ ( set_or7035219750837199246ssThan @ A @ I @ N2 ) ) % 8.50/4.52 = ( set_or7035219750837199246ssThan @ A @ N2 @ M ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3734_sum_Oempty_H,axiom, % 8.50/4.52 ! [B: $tType,A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [P4: B > A] : % 8.50/4.52 ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3735_sum_Oeq__sum,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [I6: set @ B,P4: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ I6 ) % 8.50/4.52 => ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I6 ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ B @ A @ P4 @ I6 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3736_atLeastLessThan__singleton,axiom, % 8.50/4.52 ! [M: nat] : % 8.50/4.52 ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) ) % 8.50/4.52 = ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3737_sum_Oinsert_H,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [I6: set @ B,P4: B > A,I: B] : % 8.50/4.52 ( ( finite_finite2 @ B % 8.50/4.52 @ ( collect @ B % 8.50/4.52 @ ^ [X: B] : % 8.50/4.52 ( ( member @ B @ X @ I6 ) % 8.50/4.52 & ( ( P4 @ X ) % 8.50/4.52 != ( zero_zero @ A ) ) ) ) ) % 8.50/4.52 => ( ( ( member @ B @ I @ I6 ) % 8.50/4.52 => ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) ) % 8.50/4.52 = ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I6 ) ) ) % 8.50/4.52 & ( ~ ( member @ B @ I @ I6 ) % 8.50/4.52 => ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) ) % 8.50/4.52 = ( plus_plus @ A @ ( P4 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I6 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3738_sum_Oop__ivl__Suc,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [N2: nat,M: nat,G: nat > A] : % 8.50/4.52 ( ( ( ord_less @ nat @ N2 @ M ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) % 8.50/4.52 & ( ~ ( ord_less @ nat @ N2 @ M ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) ) % 8.50/4.52 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3739_prod_Oop__ivl__Suc,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [N2: nat,M: nat,G: nat > A] : % 8.50/4.52 ( ( ( ord_less @ nat @ N2 @ M ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) ) % 8.50/4.52 = ( one_one @ A ) ) ) % 8.50/4.52 & ( ~ ( ord_less @ nat @ N2 @ M ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) ) % 8.50/4.52 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3740_sgn__rat__def,axiom, % 8.50/4.52 ( ( sgn_sgn @ rat ) % 8.50/4.52 = ( ^ [A4: rat] : % 8.50/4.52 ( if @ rat % 8.50/4.52 @ ( A4 % 8.50/4.52 = ( zero_zero @ rat ) ) % 8.50/4.52 @ ( zero_zero @ rat ) % 8.50/4.52 @ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A4 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3741_abs__rat__def,axiom, % 8.50/4.52 ( ( abs_abs @ rat ) % 8.50/4.52 = ( ^ [A4: rat] : ( if @ rat @ ( ord_less @ rat @ A4 @ ( zero_zero @ rat ) ) @ ( uminus_uminus @ rat @ A4 ) @ A4 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3742_atLeastLessThan__eq__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linorder @ A ) % 8.50/4.52 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.50/4.52 ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.52 => ( ( ord_less @ A @ C2 @ D2 ) % 8.50/4.52 => ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) % 8.50/4.52 = ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) ) % 8.50/4.52 = ( ( A2 = C2 ) % 8.50/4.52 & ( B2 = D2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3743_atLeastLessThan__inj_I1_J,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linorder @ A ) % 8.50/4.52 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.50/4.52 ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) % 8.50/4.52 = ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) ) % 8.50/4.52 => ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.52 => ( ( ord_less @ A @ C2 @ D2 ) % 8.50/4.52 => ( A2 = C2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3744_atLeastLessThan__inj_I2_J,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linorder @ A ) % 8.50/4.52 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.50/4.52 ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) % 8.50/4.52 = ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) ) % 8.50/4.52 => ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.52 => ( ( ord_less @ A @ C2 @ D2 ) % 8.50/4.52 => ( B2 = D2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3745_less__eq__rat__def,axiom, % 8.50/4.52 ( ( ord_less_eq @ rat ) % 8.50/4.52 = ( ^ [X: rat,Y2: rat] : % 8.50/4.52 ( ( ord_less @ rat @ X @ Y2 ) % 8.50/4.52 | ( X = Y2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3746_obtain__pos__sum,axiom, % 8.50/4.52 ! [R2: rat] : % 8.50/4.52 ( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 ) % 8.50/4.52 => ~ ! [S2: rat] : % 8.50/4.52 ( ( ord_less @ rat @ ( zero_zero @ rat ) @ S2 ) % 8.50/4.52 => ! [T7: rat] : % 8.50/4.52 ( ( ord_less @ rat @ ( zero_zero @ rat ) @ T7 ) % 8.50/4.52 => ( R2 % 8.50/4.52 != ( plus_plus @ rat @ S2 @ T7 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3747_sum_Onon__neutral_H,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [G: B > A,I6: set @ B] : % 8.50/4.52 ( ( groups1027152243600224163dd_sum @ B @ A @ G % 8.50/4.52 @ ( collect @ B % 8.50/4.52 @ ^ [X: B] : % 8.50/4.52 ( ( member @ B @ X @ I6 ) % 8.50/4.52 & ( ( G @ X ) % 8.50/4.52 != ( zero_zero @ A ) ) ) ) ) % 8.50/4.52 = ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3748_atLeastLessThan__subset__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linorder @ A ) % 8.50/4.52 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) ) % 8.50/4.52 => ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.50/4.52 | ( ( ord_less_eq @ A @ C2 @ A2 ) % 8.50/4.52 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3749_infinite__Ico,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( dense_linorder @ A ) % 8.50/4.52 => ! [A2: A,B2: A] : % 8.50/4.52 ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.52 => ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3750_ex__nat__less__eq,axiom, % 8.50/4.52 ! [N2: nat,P: nat > $o] : % 8.50/4.52 ( ( ? [M2: nat] : % 8.50/4.52 ( ( ord_less @ nat @ M2 @ N2 ) % 8.50/4.52 & ( P @ M2 ) ) ) % 8.50/4.52 = ( ? [X: nat] : % 8.50/4.52 ( ( member @ nat @ X @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.50/4.52 & ( P @ X ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3751_all__nat__less__eq,axiom, % 8.50/4.52 ! [N2: nat,P: nat > $o] : % 8.50/4.52 ( ( ! [M2: nat] : % 8.50/4.52 ( ( ord_less @ nat @ M2 @ N2 ) % 8.50/4.52 => ( P @ M2 ) ) ) % 8.50/4.52 = ( ! [X: nat] : % 8.50/4.52 ( ( member @ nat @ X @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.50/4.52 => ( P @ X ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3752_atLeastLessThanSuc__atLeastAtMost,axiom, % 8.50/4.52 ! [L: nat,U: nat] : % 8.50/4.52 ( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) ) % 8.50/4.52 = ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3753_lessThan__atLeast0,axiom, % 8.50/4.52 ( ( set_ord_lessThan @ nat ) % 8.50/4.52 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3754_atLeastLessThan0,axiom, % 8.50/4.52 ! [M: nat] : % 8.50/4.52 ( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) ) % 8.50/4.52 = ( bot_bot @ ( set @ nat ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3755_sum_Oshift__bounds__Suc__ivl,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3756_sum_Oshift__bounds__nat__ivl,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3757_prod_Oshift__bounds__Suc__ivl,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3758_prod_Oshift__bounds__nat__ivl,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3759_sum_Odistrib__triv_H,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [I6: set @ B,G: B > A,H2: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ I6 ) % 8.50/4.52 => ( ( groups1027152243600224163dd_sum @ B @ A % 8.50/4.52 @ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H2 @ I3 ) ) % 8.50/4.52 @ I6 ) % 8.50/4.52 = ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3760_sum_Oivl__cong,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( ( ord @ B ) % 8.50/4.52 & ( comm_monoid_add @ A ) ) % 8.50/4.52 => ! [A2: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] : % 8.50/4.52 ( ( A2 = C2 ) % 8.50/4.52 => ( ( B2 = D2 ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( ord_less_eq @ B @ C2 @ X5 ) % 8.50/4.52 => ( ( ord_less @ B @ X5 @ D2 ) % 8.50/4.52 => ( ( G @ X5 ) % 8.50/4.52 = ( H2 @ X5 ) ) ) ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3761_prod_Oivl__cong,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( ( ord @ B ) % 8.50/4.52 & ( comm_monoid_mult @ A ) ) % 8.50/4.52 => ! [A2: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] : % 8.50/4.52 ( ( A2 = C2 ) % 8.50/4.52 => ( ( B2 = D2 ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( ord_less_eq @ B @ C2 @ X5 ) % 8.50/4.52 => ( ( ord_less @ B @ X5 @ D2 ) % 8.50/4.52 => ( ( G @ X5 ) % 8.50/4.52 = ( H2 @ X5 ) ) ) ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3762_sum_OatLeastLessThan__concat,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [M: nat,N2: nat,P4: nat,G: nat > A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ N2 @ P4 ) % 8.50/4.52 => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P4 ) ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3763_sum__diff__nat__ivl,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ab_group_add @ A ) % 8.50/4.52 => ! [M: nat,N2: nat,P4: nat,F2: nat > A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ N2 @ P4 ) % 8.50/4.52 => ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P4 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3764_size__list__estimation,axiom, % 8.50/4.52 ! [A: $tType,X3: A,Xs: list @ A,Y: nat,F2: A > nat] : % 8.50/4.52 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.50/4.52 => ( ( ord_less @ nat @ Y @ ( F2 @ X3 ) ) % 8.50/4.52 => ( ord_less @ nat @ Y @ ( size_list @ A @ F2 @ Xs ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3765_size__list__pointwise,axiom, % 8.50/4.52 ! [A: $tType,Xs: list @ A,F2: A > nat,G: A > nat] : % 8.50/4.52 ( ! [X5: A] : % 8.50/4.52 ( ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) % 8.50/4.52 => ( ord_less_eq @ nat @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 8.50/4.52 => ( ord_less_eq @ nat @ ( size_list @ A @ F2 @ Xs ) @ ( size_list @ A @ G @ Xs ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3766_size__list__estimation_H,axiom, % 8.50/4.52 ! [A: $tType,X3: A,Xs: list @ A,Y: nat,F2: A > nat] : % 8.50/4.52 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ Y @ ( F2 @ X3 ) ) % 8.50/4.52 => ( ord_less_eq @ nat @ Y @ ( size_list @ A @ F2 @ Xs ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3767_prod_OatLeastLessThan__concat,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [M: nat,N2: nat,P4: nat,G: nat > A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ N2 @ P4 ) % 8.50/4.52 => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P4 ) ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3768_atLeast0__lessThan__Suc,axiom, % 8.50/4.52 ! [N2: nat] : % 8.50/4.52 ( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) % 8.50/4.52 = ( insert @ nat @ N2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3769_subset__eq__atLeast0__lessThan__finite,axiom, % 8.50/4.52 ! [N7: set @ nat,N2: nat] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.50/4.52 => ( finite_finite2 @ nat @ N7 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3770_sum_Omono__neutral__left_H,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [S3: set @ B,T4: set @ B,G: B > A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.52 => ( ( G @ X5 ) % 8.50/4.52 = ( zero_zero @ A ) ) ) % 8.50/4.52 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 ) % 8.50/4.52 = ( groups1027152243600224163dd_sum @ B @ A @ G @ T4 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3771_sum_Omono__neutral__right_H,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [S3: set @ B,T4: set @ B,G: B > A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.52 => ( ( G @ X5 ) % 8.50/4.52 = ( zero_zero @ A ) ) ) % 8.50/4.52 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T4 ) % 8.50/4.52 = ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3772_sum_Omono__neutral__cong__left_H,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [S3: set @ B,T4: set @ B,H2: B > A,G: B > A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.52 => ( ! [I4: B] : % 8.50/4.52 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.52 => ( ( H2 @ I4 ) % 8.50/4.52 = ( zero_zero @ A ) ) ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ S3 ) % 8.50/4.52 => ( ( G @ X5 ) % 8.50/4.52 = ( H2 @ X5 ) ) ) % 8.50/4.52 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 ) % 8.50/4.52 = ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3773_sum_Omono__neutral__cong__right_H,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [S3: set @ B,T4: set @ B,G: B > A,H2: B > A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.52 => ( ( G @ X5 ) % 8.50/4.52 = ( zero_zero @ A ) ) ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ S3 ) % 8.50/4.52 => ( ( G @ X5 ) % 8.50/4.52 = ( H2 @ X5 ) ) ) % 8.50/4.52 => ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T4 ) % 8.50/4.52 = ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3774_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( dense_linorder @ A ) % 8.50/4.52 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) ) % 8.50/4.52 = ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 8.50/4.52 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3775_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( preorder @ A ) % 8.50/4.52 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) ) % 8.50/4.52 = ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 8.50/4.52 & ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3776_sum__shift__lb__Suc0__0__upt,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [F2: nat > A,K: nat] : % 8.50/4.52 ( ( ( F2 @ ( zero_zero @ nat ) ) % 8.50/4.52 = ( zero_zero @ A ) ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3777_sum_OatLeast0__lessThan__Suc,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 8.50/4.52 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3778_sum_OatLeast__Suc__lessThan,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [M: nat,N2: nat,G: nat > A] : % 8.50/4.52 ( ( ord_less @ nat @ M @ N2 ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) % 8.50/4.52 = ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3779_sum_OatLeastLessThan__Suc,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [A2: nat,B2: nat,G: nat > A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ A2 @ B2 ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) ) % 8.50/4.52 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3780_sum_Odistrib_H,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [I6: set @ B,G: B > A,H2: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B % 8.50/4.52 @ ( collect @ B % 8.50/4.52 @ ^ [X: B] : % 8.50/4.52 ( ( member @ B @ X @ I6 ) % 8.50/4.52 & ( ( G @ X ) % 8.50/4.52 != ( zero_zero @ A ) ) ) ) ) % 8.50/4.52 => ( ( finite_finite2 @ B % 8.50/4.52 @ ( collect @ B % 8.50/4.52 @ ^ [X: B] : % 8.50/4.52 ( ( member @ B @ X @ I6 ) % 8.50/4.52 & ( ( H2 @ X ) % 8.50/4.52 != ( zero_zero @ A ) ) ) ) ) % 8.50/4.52 => ( ( groups1027152243600224163dd_sum @ B @ A % 8.50/4.52 @ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H2 @ I3 ) ) % 8.50/4.52 @ I6 ) % 8.50/4.52 = ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3781_atLeastLessThan__eq__atLeastAtMost__diff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( order @ A ) % 8.50/4.52 => ( ( set_or7035219750837199246ssThan @ A ) % 8.50/4.52 = ( ^ [A4: A,B4: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A4 @ B4 ) @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3782_prod_OatLeast0__lessThan__Suc,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 8.50/4.52 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3783_prod_OatLeast__Suc__lessThan,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [M: nat,N2: nat,G: nat > A] : % 8.50/4.52 ( ( ord_less @ nat @ M @ N2 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) % 8.50/4.52 = ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3784_prod_OatLeastLessThan__Suc,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [A2: nat,B2: nat,G: nat > A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ A2 @ B2 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) ) % 8.50/4.52 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3785_sum_OG__def,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ( ( groups1027152243600224163dd_sum @ B @ A ) % 8.50/4.52 = ( ^ [P6: B > A,I8: set @ B] : % 8.50/4.52 ( if @ A % 8.50/4.52 @ ( finite_finite2 @ B % 8.50/4.52 @ ( collect @ B % 8.50/4.52 @ ^ [X: B] : % 8.50/4.52 ( ( member @ B @ X @ I8 ) % 8.50/4.52 & ( ( P6 @ X ) % 8.50/4.52 != ( zero_zero @ A ) ) ) ) ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ B @ A @ P6 % 8.50/4.52 @ ( collect @ B % 8.50/4.52 @ ^ [X: B] : % 8.50/4.52 ( ( member @ B @ X @ I8 ) % 8.50/4.52 & ( ( P6 @ X ) % 8.50/4.52 != ( zero_zero @ A ) ) ) ) ) % 8.50/4.52 @ ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3786_sum_Olast__plus,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [M: nat,N2: nat,G: nat > A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.52 = ( plus_plus @ A @ ( G @ N2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3787_prod_Olast__plus,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [M: nat,N2: nat,G: nat > A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.52 = ( times_times @ A @ ( G @ N2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3788_sum__Suc__diff_H,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ab_group_add @ A ) % 8.50/4.52 => ! [M: nat,N2: nat,F2: nat > A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) % 8.50/4.52 = ( minus_minus @ A @ ( F2 @ N2 ) @ ( F2 @ M ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3789_atLeastLessThanSuc,axiom, % 8.50/4.52 ! [M: nat,N2: nat] : % 8.50/4.52 ( ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.52 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) % 8.50/4.52 = ( insert @ nat @ N2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) % 8.50/4.52 & ( ~ ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.52 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) % 8.50/4.52 = ( bot_bot @ ( set @ nat ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3790_sum_OatLeastLessThan__rev,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat,M: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( suc @ I3 ) ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3791_sum_Onested__swap,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [A2: nat > nat > A,N2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [J3: nat] : % 8.50/4.52 ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( A2 @ I3 @ J3 ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3792_prod_OatLeastLessThan__rev,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat,M: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( suc @ I3 ) ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3793_prod_Onested__swap,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [A2: nat > nat > A,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [J3: nat] : % 8.50/4.52 ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( A2 @ I3 @ J3 ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3794_sum_Onat__group,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [G: nat > A,K: nat,N2: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [M2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N2 @ K ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3795_prod_Onat__group,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,K: nat,N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [M2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) ) % 8.50/4.52 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N2 @ K ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3796_prod__Suc__Suc__fact,axiom, % 8.50/4.52 ! [N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) % 8.50/4.52 = ( semiring_char_0_fact @ nat @ N2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3797_prod__Suc__fact,axiom, % 8.50/4.52 ! [N2: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.50/4.52 = ( semiring_char_0_fact @ nat @ N2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3798_sum_Ohead__if,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [N2: nat,M: nat,G: nat > A] : % 8.50/4.52 ( ( ( ord_less @ nat @ N2 @ M ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) % 8.50/4.52 & ( ~ ( ord_less @ nat @ N2 @ M ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.52 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3799_prod_Ohead__if,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [N2: nat,M: nat,G: nat > A] : % 8.50/4.52 ( ( ( ord_less @ nat @ N2 @ M ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.52 = ( one_one @ A ) ) ) % 8.50/4.52 & ( ~ ( ord_less @ nat @ N2 @ M ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.52 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3800_fact__prod__Suc,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( semiring_char_0 @ A ) % 8.50/4.52 => ( ( semiring_char_0_fact @ A ) % 8.50/4.52 = ( ^ [N: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3801_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_add @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat,M: nat] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I3 ) ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3802_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [G: nat > A,N2: nat,M: nat] : % 8.50/4.52 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I3 ) ) % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3803_pochhammer__prod,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comm_semiring_1 @ A ) % 8.50/4.52 => ( ( comm_s3205402744901411588hammer @ A ) % 8.50/4.52 = ( ^ [A4: A,N: nat] : % 8.50/4.52 ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ I3 ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3804_atLeastLessThan__nat__numeral,axiom, % 8.50/4.52 ! [M: nat,K: num] : % 8.50/4.52 ( ( ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) ) % 8.50/4.52 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) ) % 8.50/4.52 = ( insert @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M @ ( pred_numeral @ K ) ) ) ) ) % 8.50/4.52 & ( ~ ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) ) % 8.50/4.52 => ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) ) % 8.50/4.52 = ( bot_bot @ ( set @ nat ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3805_fact__prod__rev,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( semiring_char_0 @ A ) % 8.50/4.52 => ( ( semiring_char_0_fact @ A ) % 8.50/4.52 = ( ^ [N: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3806_summable__Cauchy,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_Vector_banach @ A ) % 8.50/4.52 => ( ( summable @ A ) % 8.50/4.52 = ( ^ [F3: nat > A] : % 8.50/4.52 ! [E4: real] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 ) % 8.50/4.52 => ? [N5: nat] : % 8.50/4.52 ! [M2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ N5 @ M2 ) % 8.50/4.52 => ! [N: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) @ E4 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3807_sums__group,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( comm_monoid_add @ A ) % 8.50/4.52 & ( topolo4958980785337419405_space @ A ) ) % 8.50/4.52 => ! [F2: nat > A,S: A,K: nat] : % 8.50/4.52 ( ( sums @ A @ F2 @ S ) % 8.50/4.52 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K ) % 8.50/4.52 => ( sums @ A % 8.50/4.52 @ ^ [N: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K ) @ K ) ) ) % 8.50/4.52 @ S ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3808_atLeast1__lessThan__eq__remove0,axiom, % 8.50/4.52 ! [N2: nat] : % 8.50/4.52 ( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 8.50/4.52 = ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N2 ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3809_fact__split,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( semiring_char_0 @ A ) % 8.50/4.52 => ! [K: nat,N2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.52 => ( ( semiring_char_0_fact @ A @ N2 ) % 8.50/4.52 = ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N2 @ K ) @ N2 ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3810_binomial__altdef__of__nat,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field_char_0 @ A ) % 8.50/4.52 => ! [K: nat,N2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.52 => ( ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I3 ) ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3811_gbinomial__altdef__of__nat,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field_char_0 @ A ) % 8.50/4.52 => ( ( gbinomial @ A ) % 8.50/4.52 = ( ^ [A4: A,K3: nat] : % 8.50/4.52 ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I3 ) ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3812_gbinomial__mult__fact_H,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field_char_0 @ A ) % 8.50/4.52 => ! [A2: A,K: nat] : % 8.50/4.52 ( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( semiring_char_0_fact @ A @ K ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3813_gbinomial__mult__fact,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( field_char_0 @ A ) % 8.50/4.52 => ! [K: nat,A2: A] : % 8.50/4.52 ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I3 ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3814_gbinomial__prod__rev,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( semiring_char_0 @ A ) % 8.50/4.52 & ( semidom_divide @ A ) ) % 8.50/4.52 => ( ( gbinomial @ A ) % 8.50/4.52 = ( ^ [A4: A,K3: nat] : % 8.50/4.52 ( divide_divide @ A % 8.50/4.52 @ ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I3 ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) % 8.50/4.52 @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3815_sum__diff1_H,axiom, % 8.50/4.52 ! [B: $tType,A: $tType] : % 8.50/4.52 ( ( ab_group_add @ B ) % 8.50/4.52 => ! [I6: set @ A,F2: A > B,I: A] : % 8.50/4.52 ( ( finite_finite2 @ A % 8.50/4.52 @ ( collect @ A % 8.50/4.52 @ ^ [I3: A] : % 8.50/4.52 ( ( member @ A @ I3 @ I6 ) % 8.50/4.52 & ( ( F2 @ I3 ) % 8.50/4.52 != ( zero_zero @ B ) ) ) ) ) % 8.50/4.52 => ( ( ( member @ A @ I @ I6 ) % 8.50/4.52 => ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.52 = ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) @ ( F2 @ I ) ) ) ) % 8.50/4.52 & ( ~ ( member @ A @ I @ I6 ) % 8.50/4.52 => ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.52 = ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I6 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3816_horner__sum__eq__sum,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_semiring_1 @ A ) % 8.50/4.52 => ( ( groups4207007520872428315er_sum @ B @ A ) % 8.50/4.52 = ( ^ [F3: B > A,A4: A,Xs2: list @ B] : % 8.50/4.52 ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [N: nat] : ( times_times @ A @ ( F3 @ ( nth @ B @ Xs2 @ N ) ) @ ( power_power @ A @ A4 @ N ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3817_Chebyshev__sum__upper,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linordered_idom @ A ) % 8.50/4.52 => ! [N2: nat,A2: nat > A,B2: nat > A] : % 8.50/4.52 ( ! [I4: nat,J2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ I4 @ J2 ) % 8.50/4.52 => ( ( ord_less @ nat @ J2 @ N2 ) % 8.50/4.52 => ( ord_less_eq @ A @ ( A2 @ I4 ) @ ( A2 @ J2 ) ) ) ) % 8.50/4.52 => ( ! [I4: nat,J2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ I4 @ J2 ) % 8.50/4.52 => ( ( ord_less @ nat @ J2 @ N2 ) % 8.50/4.52 => ( ord_less_eq @ A @ ( B2 @ J2 ) @ ( B2 @ I4 ) ) ) ) % 8.50/4.52 => ( ord_less_eq @ A % 8.50/4.52 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : ( times_times @ A @ ( A2 @ K3 ) @ ( B2 @ K3 ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) % 8.50/4.52 @ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3818_Chebyshev__sum__upper__nat,axiom, % 8.50/4.52 ! [N2: nat,A2: nat > nat,B2: nat > nat] : % 8.50/4.52 ( ! [I4: nat,J2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ I4 @ J2 ) % 8.50/4.52 => ( ( ord_less @ nat @ J2 @ N2 ) % 8.50/4.52 => ( ord_less_eq @ nat @ ( A2 @ I4 ) @ ( A2 @ J2 ) ) ) ) % 8.50/4.52 => ( ! [I4: nat,J2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ I4 @ J2 ) % 8.50/4.52 => ( ( ord_less @ nat @ J2 @ N2 ) % 8.50/4.52 => ( ord_less_eq @ nat @ ( B2 @ J2 ) @ ( B2 @ I4 ) ) ) ) % 8.50/4.52 => ( ord_less_eq @ nat % 8.50/4.52 @ ( times_times @ nat @ N2 % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ nat @ ( A2 @ I3 ) @ ( B2 @ I3 ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) % 8.50/4.52 @ ( times_times @ nat @ ( groups7311177749621191930dd_sum @ nat @ nat @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ nat @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3819_VEBT_Osize__gen_I1_J,axiom, % 8.50/4.52 ! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] : % 8.50/4.52 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) ) % 8.50/4.52 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_list @ vEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3820_finite__atLeastLessThan__int,axiom, % 8.50/4.52 ! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3821_finite__atLeastZeroLessThan__int,axiom, % 8.50/4.52 ! [U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3822_VEBT_Osize__gen_I2_J,axiom, % 8.50/4.52 ! [X21: $o,X222: $o] : % 8.50/4.52 ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) ) % 8.50/4.52 = ( zero_zero @ nat ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3823_rat__inverse__code,axiom, % 8.50/4.52 ! [P4: rat] : % 8.50/4.52 ( ( quotient_of @ ( inverse_inverse @ rat @ P4 ) ) % 8.50/4.52 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 8.50/4.52 @ ^ [A4: int,B4: int] : % 8.50/4.52 ( if @ ( product_prod @ int @ int ) % 8.50/4.52 @ ( A4 % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) % 8.50/4.52 @ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A4 ) @ B4 ) @ ( abs_abs @ int @ A4 ) ) ) % 8.50/4.52 @ ( quotient_of @ P4 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3824_normalize__negative,axiom, % 8.50/4.52 ! [Q4: int,P4: int] : % 8.50/4.52 ( ( ord_less @ int @ Q4 @ ( zero_zero @ int ) ) % 8.50/4.52 => ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q4 ) ) % 8.50/4.52 = ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P4 ) @ ( uminus_uminus @ int @ Q4 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3825_Cauchy__iff2,axiom, % 8.50/4.52 ( ( topolo3814608138187158403Cauchy @ real ) % 8.50/4.52 = ( ^ [X6: nat > real] : % 8.50/4.52 ! [J3: nat] : % 8.50/4.52 ? [M9: nat] : % 8.50/4.52 ! [M2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M9 @ M2 ) % 8.50/4.52 => ! [N: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M9 @ N ) % 8.50/4.52 => ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X6 @ M2 ) @ ( X6 @ N ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3826_vebt__mint_Opelims,axiom, % 8.50/4.52 ! [X3: vEBT_VEBT,Y: option @ nat] : % 8.50/4.52 ( ( ( vEBT_vebt_mint @ X3 ) % 8.50/4.52 = Y ) % 8.50/4.52 => ( ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ X3 ) % 8.50/4.52 => ( ! [A5: $o,B5: $o] : % 8.50/4.52 ( ( X3 % 8.50/4.52 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.50/4.52 => ( ( ( A5 % 8.50/4.52 => ( Y % 8.50/4.52 = ( some @ nat @ ( zero_zero @ nat ) ) ) ) % 8.50/4.52 & ( ~ A5 % 8.50/4.52 => ( ( B5 % 8.50/4.52 => ( Y % 8.50/4.52 = ( some @ nat @ ( one_one @ nat ) ) ) ) % 8.50/4.52 & ( ~ B5 % 8.50/4.52 => ( Y % 8.50/4.52 = ( none @ nat ) ) ) ) ) ) % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) ) % 8.50/4.52 => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 8.50/4.52 ( ( X3 % 8.50/4.52 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 8.50/4.52 => ( ( Y % 8.50/4.52 = ( none @ nat ) ) % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) ) % 8.50/4.52 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 8.50/4.52 ( ( X3 % 8.50/4.52 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) % 8.50/4.52 => ( ( Y % 8.50/4.52 = ( some @ nat @ Mi2 ) ) % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3827_quotient__of__number_I3_J,axiom, % 8.50/4.52 ! [K: num] : % 8.50/4.52 ( ( quotient_of @ ( numeral_numeral @ rat @ K ) ) % 8.50/4.52 = ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3828_normalize__denom__zero,axiom, % 8.50/4.52 ! [P4: int] : % 8.50/4.52 ( ( normalize @ ( product_Pair @ int @ int @ P4 @ ( zero_zero @ int ) ) ) % 8.50/4.52 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3829_rat__one__code,axiom, % 8.50/4.52 ( ( quotient_of @ ( one_one @ rat ) ) % 8.50/4.52 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3830_rat__zero__code,axiom, % 8.50/4.52 ( ( quotient_of @ ( zero_zero @ rat ) ) % 8.50/4.52 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3831_quotient__of__number_I5_J,axiom, % 8.50/4.52 ! [K: num] : % 8.50/4.52 ( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) ) ) % 8.50/4.52 = ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3832_quotient__of__number_I4_J,axiom, % 8.50/4.52 ( ( quotient_of @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) % 8.50/4.52 = ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3833_rat__divide__code,axiom, % 8.50/4.52 ! [P4: rat,Q4: rat] : % 8.50/4.52 ( ( quotient_of @ ( divide_divide @ rat @ P4 @ Q4 ) ) % 8.50/4.52 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 8.50/4.52 @ ^ [A4: int,C4: int] : % 8.50/4.52 ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 8.50/4.52 @ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A4 @ D5 ) @ ( times_times @ int @ C4 @ B4 ) ) ) % 8.50/4.52 @ ( quotient_of @ Q4 ) ) % 8.50/4.52 @ ( quotient_of @ P4 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3834_rat__times__code,axiom, % 8.50/4.52 ! [P4: rat,Q4: rat] : % 8.50/4.52 ( ( quotient_of @ ( times_times @ rat @ P4 @ Q4 ) ) % 8.50/4.52 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 8.50/4.52 @ ^ [A4: int,C4: int] : % 8.50/4.52 ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 8.50/4.52 @ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A4 @ B4 ) @ ( times_times @ int @ C4 @ D5 ) ) ) % 8.50/4.52 @ ( quotient_of @ Q4 ) ) % 8.50/4.52 @ ( quotient_of @ P4 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3835_quotient__of__div,axiom, % 8.50/4.52 ! [R2: rat,N2: int,D2: int] : % 8.50/4.52 ( ( ( quotient_of @ R2 ) % 8.50/4.52 = ( product_Pair @ int @ int @ N2 @ D2 ) ) % 8.50/4.52 => ( R2 % 8.50/4.52 = ( divide_divide @ rat @ ( ring_1_of_int @ rat @ N2 ) @ ( ring_1_of_int @ rat @ D2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3836_rat__minus__code,axiom, % 8.50/4.52 ! [P4: rat,Q4: rat] : % 8.50/4.52 ( ( quotient_of @ ( minus_minus @ rat @ P4 @ Q4 ) ) % 8.50/4.52 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 8.50/4.52 @ ^ [A4: int,C4: int] : % 8.50/4.52 ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 8.50/4.52 @ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A4 @ D5 ) @ ( times_times @ int @ B4 @ C4 ) ) @ ( times_times @ int @ C4 @ D5 ) ) ) % 8.50/4.52 @ ( quotient_of @ Q4 ) ) % 8.50/4.52 @ ( quotient_of @ P4 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3837_quotient__of__denom__pos,axiom, % 8.50/4.52 ! [R2: rat,P4: int,Q4: int] : % 8.50/4.52 ( ( ( quotient_of @ R2 ) % 8.50/4.52 = ( product_Pair @ int @ int @ P4 @ Q4 ) ) % 8.50/4.52 => ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3838_rat__plus__code,axiom, % 8.50/4.52 ! [P4: rat,Q4: rat] : % 8.50/4.52 ( ( quotient_of @ ( plus_plus @ rat @ P4 @ Q4 ) ) % 8.50/4.52 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 8.50/4.52 @ ^ [A4: int,C4: int] : % 8.50/4.52 ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 8.50/4.52 @ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A4 @ D5 ) @ ( times_times @ int @ B4 @ C4 ) ) @ ( times_times @ int @ C4 @ D5 ) ) ) % 8.50/4.52 @ ( quotient_of @ Q4 ) ) % 8.50/4.52 @ ( quotient_of @ P4 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3839_normalize__denom__pos,axiom, % 8.50/4.52 ! [R2: product_prod @ int @ int,P4: int,Q4: int] : % 8.50/4.52 ( ( ( normalize @ R2 ) % 8.50/4.52 = ( product_Pair @ int @ int @ P4 @ Q4 ) ) % 8.50/4.52 => ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3840_normalize__crossproduct,axiom, % 8.50/4.52 ! [Q4: int,S: int,P4: int,R2: int] : % 8.50/4.52 ( ( Q4 % 8.50/4.52 != ( zero_zero @ int ) ) % 8.50/4.52 => ( ( S % 8.50/4.52 != ( zero_zero @ int ) ) % 8.50/4.52 => ( ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q4 ) ) % 8.50/4.52 = ( normalize @ ( product_Pair @ int @ int @ R2 @ S ) ) ) % 8.50/4.52 => ( ( times_times @ int @ P4 @ S ) % 8.50/4.52 = ( times_times @ int @ R2 @ Q4 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3841_rat__uminus__code,axiom, % 8.50/4.52 ! [P4: rat] : % 8.50/4.52 ( ( quotient_of @ ( uminus_uminus @ rat @ P4 ) ) % 8.50/4.52 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 8.50/4.52 @ ^ [A4: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A4 ) ) % 8.50/4.52 @ ( quotient_of @ P4 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3842_rat__less__code,axiom, % 8.50/4.52 ( ( ord_less @ rat ) % 8.50/4.52 = ( ^ [P6: rat,Q5: rat] : % 8.50/4.52 ( product_case_prod @ int @ int @ $o % 8.50/4.52 @ ^ [A4: int,C4: int] : % 8.50/4.52 ( product_case_prod @ int @ int @ $o % 8.50/4.52 @ ^ [B4: int,D5: int] : ( ord_less @ int @ ( times_times @ int @ A4 @ D5 ) @ ( times_times @ int @ C4 @ B4 ) ) % 8.50/4.52 @ ( quotient_of @ Q5 ) ) % 8.50/4.52 @ ( quotient_of @ P6 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3843_rat__abs__code,axiom, % 8.50/4.52 ! [P4: rat] : % 8.50/4.52 ( ( quotient_of @ ( abs_abs @ rat @ P4 ) ) % 8.50/4.52 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int ) % 8.50/4.52 @ ^ [A4: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A4 ) ) % 8.50/4.52 @ ( quotient_of @ P4 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3844_Cauchy__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V822414075346904944vector @ A ) % 8.50/4.52 => ( ( topolo3814608138187158403Cauchy @ A ) % 8.50/4.52 = ( ^ [X6: nat > A] : % 8.50/4.52 ! [E4: real] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ E4 ) % 8.50/4.52 => ? [M9: nat] : % 8.50/4.52 ! [M2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M9 @ M2 ) % 8.50/4.52 => ! [N: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M9 @ N ) % 8.50/4.52 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ M2 ) @ ( X6 @ N ) ) ) @ E4 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3845_CauchyI,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V822414075346904944vector @ A ) % 8.50/4.52 => ! [X8: nat > A] : % 8.50/4.52 ( ! [E: real] : % 8.50/4.52 ( ( ord_less @ real @ ( zero_zero @ real ) @ E ) % 8.50/4.52 => ? [M10: nat] : % 8.50/4.52 ! [M3: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M10 @ M3 ) % 8.50/4.52 => ! [N3: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M10 @ N3 ) % 8.50/4.52 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) ) @ E ) ) ) ) % 8.50/4.52 => ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3846_CauchyD,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( real_V822414075346904944vector @ A ) % 8.50/4.52 => ! [X8: nat > A,E3: real] : % 8.50/4.52 ( ( topolo3814608138187158403Cauchy @ A @ X8 ) % 8.50/4.52 => ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 ) % 8.50/4.52 => ? [M8: nat] : % 8.50/4.52 ! [M6: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M8 @ M6 ) % 8.50/4.52 => ! [N6: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ M8 @ N6 ) % 8.50/4.52 => ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M6 ) @ ( X8 @ N6 ) ) ) @ E3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3847_vebt__maxt_Opelims,axiom, % 8.50/4.52 ! [X3: vEBT_VEBT,Y: option @ nat] : % 8.50/4.52 ( ( ( vEBT_vebt_maxt @ X3 ) % 8.50/4.52 = Y ) % 8.50/4.52 => ( ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ X3 ) % 8.50/4.52 => ( ! [A5: $o,B5: $o] : % 8.50/4.52 ( ( X3 % 8.50/4.52 = ( vEBT_Leaf @ A5 @ B5 ) ) % 8.50/4.52 => ( ( ( B5 % 8.50/4.52 => ( Y % 8.50/4.52 = ( some @ nat @ ( one_one @ nat ) ) ) ) % 8.50/4.52 & ( ~ B5 % 8.50/4.52 => ( ( A5 % 8.50/4.52 => ( Y % 8.50/4.52 = ( some @ nat @ ( zero_zero @ nat ) ) ) ) % 8.50/4.52 & ( ~ A5 % 8.50/4.52 => ( Y % 8.50/4.52 = ( none @ nat ) ) ) ) ) ) % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) ) % 8.50/4.52 => ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] : % 8.50/4.52 ( ( X3 % 8.50/4.52 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) % 8.50/4.52 => ( ( Y % 8.50/4.52 = ( none @ nat ) ) % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) ) ) ) % 8.50/4.52 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] : % 8.50/4.52 ( ( X3 % 8.50/4.52 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) % 8.50/4.52 => ( ( Y % 8.50/4.52 = ( some @ nat @ Ma2 ) ) % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3848_quotient__of__int,axiom, % 8.50/4.52 ! [A2: int] : % 8.50/4.52 ( ( quotient_of @ ( of_int @ A2 ) ) % 8.50/4.52 = ( product_Pair @ int @ int @ A2 @ ( one_one @ int ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3849_VEBT__internal_OminNull_Opelims_I1_J,axiom, % 8.50/4.52 ! [X3: vEBT_VEBT,Y: $o] : % 8.50/4.52 ( ( ( vEBT_VEBT_minNull @ X3 ) % 8.50/4.52 = Y ) % 8.50/4.52 => ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 ) % 8.50/4.52 => ( ( ( X3 % 8.50/4.52 = ( vEBT_Leaf @ $false @ $false ) ) % 8.50/4.52 => ( Y % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) ) % 8.50/4.52 => ( ! [Uv2: $o] : % 8.50/4.52 ( ( X3 % 8.50/4.52 = ( vEBT_Leaf @ $true @ Uv2 ) ) % 8.50/4.52 => ( ~ Y % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) ) % 8.50/4.52 => ( ! [Uu2: $o] : % 8.50/4.52 ( ( X3 % 8.50/4.52 = ( vEBT_Leaf @ Uu2 @ $true ) ) % 8.50/4.52 => ( ~ Y % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) ) % 8.50/4.52 => ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 8.50/4.52 ( ( X3 % 8.50/4.52 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) % 8.50/4.52 => ( Y % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) % 8.50/4.52 => ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 8.50/4.52 ( ( X3 % 8.50/4.52 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) % 8.50/4.52 => ( ~ Y % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3850_VEBT__internal_OminNull_Opelims_I2_J,axiom, % 8.50/4.52 ! [X3: vEBT_VEBT] : % 8.50/4.52 ( ( vEBT_VEBT_minNull @ X3 ) % 8.50/4.52 => ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 ) % 8.50/4.52 => ( ( ( X3 % 8.50/4.52 = ( vEBT_Leaf @ $false @ $false ) ) % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) % 8.50/4.52 => ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : % 8.50/4.52 ( ( X3 % 8.50/4.52 = ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3851_VEBT__internal_OminNull_Opelims_I3_J,axiom, % 8.50/4.52 ! [X3: vEBT_VEBT] : % 8.50/4.52 ( ~ ( vEBT_VEBT_minNull @ X3 ) % 8.50/4.52 => ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 ) % 8.50/4.52 => ( ! [Uv2: $o] : % 8.50/4.52 ( ( X3 % 8.50/4.52 = ( vEBT_Leaf @ $true @ Uv2 ) ) % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) % 8.50/4.52 => ( ! [Uu2: $o] : % 8.50/4.52 ( ( X3 % 8.50/4.52 = ( vEBT_Leaf @ Uu2 @ $true ) ) % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) % 8.50/4.52 => ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] : % 8.50/4.52 ( ( X3 % 8.50/4.52 = ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) % 8.50/4.52 => ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3852_Frct__code__post_I5_J,axiom, % 8.50/4.52 ! [K: num] : % 8.50/4.52 ( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K ) ) ) % 8.50/4.52 = ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3853_is__singleton__the__elem,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( is_singleton @ A ) % 8.50/4.52 = ( ^ [A6: set @ A] : % 8.50/4.52 ( A6 % 8.50/4.52 = ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3854_length__subseqs,axiom, % 8.50/4.52 ! [A: $tType,Xs: list @ A] : % 8.50/4.52 ( ( size_size @ ( list @ ( list @ A ) ) @ ( subseqs @ A @ Xs ) ) % 8.50/4.52 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3855_divmod__step__integer__def,axiom, % 8.50/4.52 ( ( unique1321980374590559556d_step @ code_integer ) % 8.50/4.52 = ( ^ [L2: num] : % 8.50/4.52 ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer ) % 8.50/4.52 @ ^ [Q5: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L2 ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L2 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3856_is__singletonI,axiom, % 8.50/4.52 ! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3857_divmod__integer_H__def,axiom, % 8.50/4.52 ( ( unique8689654367752047608divmod @ code_integer ) % 8.50/4.52 = ( ^ [M2: num,N: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3858_sgn__integer__code,axiom, % 8.50/4.52 ( ( sgn_sgn @ code_integer ) % 8.50/4.52 = ( ^ [K3: code_integer] : % 8.50/4.52 ( if @ code_integer % 8.50/4.52 @ ( K3 % 8.50/4.52 = ( zero_zero @ code_integer ) ) % 8.50/4.52 @ ( zero_zero @ code_integer ) % 8.50/4.52 @ ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3859_exhaustive__integer_H_Ocases,axiom, % 8.50/4.52 ! [X3: product_prod @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] : % 8.50/4.52 ~ ! [F4: code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: code_integer,I4: code_integer] : % 8.50/4.52 ( X3 % 8.50/4.52 != ( product_Pair @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D3 @ I4 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3860_full__exhaustive__integer_H_Ocases,axiom, % 8.50/4.52 ! [X3: product_prod @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] : % 8.50/4.52 ~ ! [F4: ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D3: code_integer,I4: code_integer] : % 8.50/4.52 ( X3 % 8.50/4.52 != ( product_Pair @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D3 @ I4 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3861_subseqs__refl,axiom, % 8.50/4.52 ! [A: $tType,Xs: list @ A] : ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3862_is__singletonI_H,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ( A3 % 8.50/4.52 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.52 => ( ! [X5: A,Y4: A] : % 8.50/4.52 ( ( member @ A @ X5 @ A3 ) % 8.50/4.52 => ( ( member @ A @ Y4 @ A3 ) % 8.50/4.52 => ( X5 = Y4 ) ) ) % 8.50/4.52 => ( is_singleton @ A @ A3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3863_zero__natural_Orsp,axiom, % 8.50/4.52 ( ( zero_zero @ nat ) % 8.50/4.52 = ( zero_zero @ nat ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3864_is__singleton__def,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( is_singleton @ A ) % 8.50/4.52 = ( ^ [A6: set @ A] : % 8.50/4.52 ? [X: A] : % 8.50/4.52 ( A6 % 8.50/4.52 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3865_is__singletonE,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ( is_singleton @ A @ A3 ) % 8.50/4.52 => ~ ! [X5: A] : % 8.50/4.52 ( A3 % 8.50/4.52 != ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3866_Frct__code__post_I1_J,axiom, % 8.50/4.52 ! [A2: int] : % 8.50/4.52 ( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A2 ) ) % 8.50/4.52 = ( zero_zero @ rat ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3867_Frct__code__post_I2_J,axiom, % 8.50/4.52 ! [A2: int] : % 8.50/4.52 ( ( frct @ ( product_Pair @ int @ int @ A2 @ ( zero_zero @ int ) ) ) % 8.50/4.52 = ( zero_zero @ rat ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3868_Frct__code__post_I8_J,axiom, % 8.50/4.52 ! [A2: int,B2: int] : % 8.50/4.52 ( ( frct @ ( product_Pair @ int @ int @ A2 @ ( uminus_uminus @ int @ B2 ) ) ) % 8.50/4.52 = ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A2 @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3869_Frct__code__post_I7_J,axiom, % 8.50/4.52 ! [A2: int,B2: int] : % 8.50/4.52 ( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A2 ) @ B2 ) ) % 8.50/4.52 = ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A2 @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3870_Frct__code__post_I3_J,axiom, % 8.50/4.52 ( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) % 8.50/4.52 = ( one_one @ rat ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3871_Frct__code__post_I4_J,axiom, % 8.50/4.52 ! [K: num] : % 8.50/4.52 ( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) ) % 8.50/4.52 = ( numeral_numeral @ rat @ K ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3872_Frct__code__post_I6_J,axiom, % 8.50/4.52 ! [K: num,L: num] : % 8.50/4.52 ( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( numeral_numeral @ int @ L ) ) ) % 8.50/4.52 = ( divide_divide @ rat @ ( numeral_numeral @ rat @ K ) @ ( numeral_numeral @ rat @ L ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3873_integer__of__int__code,axiom, % 8.50/4.52 ( code_integer_of_int % 8.50/4.52 = ( ^ [K3: int] : % 8.50/4.52 ( if @ code_integer @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K3 ) ) ) % 8.50/4.52 @ ( if @ code_integer % 8.50/4.52 @ ( K3 % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 @ ( zero_zero @ code_integer ) % 8.50/4.52 @ ( if @ code_integer % 8.50/4.52 @ ( ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) % 8.50/4.52 = ( zero_zero @ int ) ) % 8.50/4.52 @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.52 @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3874_length__mul__elem,axiom, % 8.50/4.52 ! [A: $tType,Xs: list @ ( list @ A ),N2: nat] : % 8.50/4.52 ( ! [X5: list @ A] : % 8.50/4.52 ( ( member @ ( list @ A ) @ X5 @ ( set2 @ ( list @ A ) @ Xs ) ) % 8.50/4.52 => ( ( size_size @ ( list @ A ) @ X5 ) % 8.50/4.52 = N2 ) ) % 8.50/4.52 => ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs ) ) % 8.50/4.52 = ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) @ N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3875_even__sum__iff,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( semiring_parity @ A ) % 8.50/4.52 => ! [A3: set @ B,F2: B > A] : % 8.50/4.52 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.52 => ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) % 8.50/4.52 = ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) % 8.50/4.52 @ ( finite_card @ B % 8.50/4.52 @ ( collect @ B % 8.50/4.52 @ ^ [A4: B] : % 8.50/4.52 ( ( member @ B @ A4 @ A3 ) % 8.50/4.52 & ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F2 @ A4 ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3876_pair__leqI2,axiom, % 8.50/4.52 ! [A2: nat,B2: nat,S: nat,T2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ A2 @ B2 ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ S @ T2 ) % 8.50/4.52 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3877_card__Collect__less__nat,axiom, % 8.50/4.52 ! [N2: nat] : % 8.50/4.52 ( ( finite_card @ nat % 8.50/4.52 @ ( collect @ nat % 8.50/4.52 @ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) ) % 8.50/4.52 = N2 ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3878_card__atMost,axiom, % 8.50/4.52 ! [U: nat] : % 8.50/4.52 ( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) ) % 8.50/4.52 = ( suc @ U ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3879_card__Collect__le__nat,axiom, % 8.50/4.52 ! [N2: nat] : % 8.50/4.52 ( ( finite_card @ nat % 8.50/4.52 @ ( collect @ nat % 8.50/4.52 @ ^ [I3: nat] : ( ord_less_eq @ nat @ I3 @ N2 ) ) ) % 8.50/4.52 = ( suc @ N2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3880_card_Oempty,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.52 = ( zero_zero @ nat ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3881_card_Oinfinite,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ~ ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( finite_card @ A @ A3 ) % 8.50/4.52 = ( zero_zero @ nat ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3882_card__atLeastAtMost,axiom, % 8.50/4.52 ! [L: nat,U: nat] : % 8.50/4.52 ( ( finite_card @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) % 8.50/4.52 = ( minus_minus @ nat @ ( suc @ U ) @ L ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3883_card__0__eq,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( ( finite_card @ A @ A3 ) % 8.50/4.52 = ( zero_zero @ nat ) ) % 8.50/4.52 = ( A3 % 8.50/4.52 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3884_card__insert__disjoint,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,X3: A] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ~ ( member @ A @ X3 @ A3 ) % 8.50/4.52 => ( ( finite_card @ A @ ( insert @ A @ X3 @ A3 ) ) % 8.50/4.52 = ( suc @ ( finite_card @ A @ A3 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3885_abs__integer__code,axiom, % 8.50/4.52 ( ( abs_abs @ code_integer ) % 8.50/4.52 = ( ^ [K3: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K3 ) @ K3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3886_less__integer__code_I1_J,axiom, % 8.50/4.52 ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3887_less__integer_Oabs__eq,axiom, % 8.50/4.52 ! [Xa2: int,X3: int] : % 8.50/4.52 ( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) ) % 8.50/4.52 = ( ord_less @ int @ Xa2 @ X3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3888_n__subsets,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,K: nat] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( finite_card @ ( set @ A ) % 8.50/4.52 @ ( collect @ ( set @ A ) % 8.50/4.52 @ ^ [B6: set @ A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ A ) @ B6 @ A3 ) % 8.50/4.52 & ( ( finite_card @ A @ B6 ) % 8.50/4.52 = K ) ) ) ) % 8.50/4.52 = ( binomial @ ( finite_card @ A @ A3 ) @ K ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3889_card__subset__eq,axiom, % 8.50/4.52 ! [A: $tType,B3: set @ A,A3: set @ A] : % 8.50/4.52 ( ( finite_finite2 @ A @ B3 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.52 => ( ( ( finite_card @ A @ A3 ) % 8.50/4.52 = ( finite_card @ A @ B3 ) ) % 8.50/4.52 => ( A3 = B3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3890_infinite__arbitrarily__large,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,N2: nat] : % 8.50/4.52 ( ~ ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ? [B10: set @ A] : % 8.50/4.52 ( ( finite_finite2 @ A @ B10 ) % 8.50/4.52 & ( ( finite_card @ A @ B10 ) % 8.50/4.52 = N2 ) % 8.50/4.52 & ( ord_less_eq @ ( set @ A ) @ B10 @ A3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3891_card__le__if__inj__on__rel,axiom, % 8.50/4.52 ! [B: $tType,A: $tType,B3: set @ A,A3: set @ B,R2: B > A > $o] : % 8.50/4.52 ( ( finite_finite2 @ A @ B3 ) % 8.50/4.52 => ( ! [A5: B] : % 8.50/4.52 ( ( member @ B @ A5 @ A3 ) % 8.50/4.52 => ? [B11: A] : % 8.50/4.52 ( ( member @ A @ B11 @ B3 ) % 8.50/4.52 & ( R2 @ A5 @ B11 ) ) ) % 8.50/4.52 => ( ! [A13: B,A24: B,B5: A] : % 8.50/4.52 ( ( member @ B @ A13 @ A3 ) % 8.50/4.52 => ( ( member @ B @ A24 @ A3 ) % 8.50/4.52 => ( ( member @ A @ B5 @ B3 ) % 8.50/4.52 => ( ( R2 @ A13 @ B5 ) % 8.50/4.52 => ( ( R2 @ A24 @ B5 ) % 8.50/4.52 => ( A13 = A24 ) ) ) ) ) ) % 8.50/4.52 => ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3892_card__insert__le,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,X3: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ ( insert @ A @ X3 @ A3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3893_sum__multicount__gen,axiom, % 8.50/4.52 ! [A: $tType,B: $tType,S: set @ A,T2: set @ B,R: A > B > $o,K: B > nat] : % 8.50/4.52 ( ( finite_finite2 @ A @ S ) % 8.50/4.52 => ( ( finite_finite2 @ B @ T2 ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ T2 ) % 8.50/4.52 => ( ( finite_card @ A % 8.50/4.52 @ ( collect @ A % 8.50/4.52 @ ^ [I3: A] : % 8.50/4.52 ( ( member @ A @ I3 @ S ) % 8.50/4.52 & ( R @ I3 @ X5 ) ) ) ) % 8.50/4.52 = ( K @ X5 ) ) ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ A @ nat % 8.50/4.52 @ ^ [I3: A] : % 8.50/4.52 ( finite_card @ B % 8.50/4.52 @ ( collect @ B % 8.50/4.52 @ ^ [J3: B] : % 8.50/4.52 ( ( member @ B @ J3 @ T2 ) % 8.50/4.52 & ( R @ I3 @ J3 ) ) ) ) % 8.50/4.52 @ S ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ B @ nat @ K @ T2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3894_card__lists__length__eq,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,N2: nat] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( finite_card @ ( list @ A ) % 8.50/4.52 @ ( collect @ ( list @ A ) % 8.50/4.52 @ ^ [Xs2: list @ A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) % 8.50/4.52 & ( ( size_size @ ( list @ A ) @ Xs2 ) % 8.50/4.52 = N2 ) ) ) ) % 8.50/4.52 = ( power_power @ nat @ ( finite_card @ A @ A3 ) @ N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3895_card__eq__0__iff,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ( ( finite_card @ A @ A3 ) % 8.50/4.52 = ( zero_zero @ nat ) ) % 8.50/4.52 = ( ( A3 % 8.50/4.52 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.52 | ~ ( finite_finite2 @ A @ A3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3896_card__ge__0__finite,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) ) % 8.50/4.52 => ( finite_finite2 @ A @ A3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3897_card__insert__if,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,X3: A] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( ( member @ A @ X3 @ A3 ) % 8.50/4.52 => ( ( finite_card @ A @ ( insert @ A @ X3 @ A3 ) ) % 8.50/4.52 = ( finite_card @ A @ A3 ) ) ) % 8.50/4.52 & ( ~ ( member @ A @ X3 @ A3 ) % 8.50/4.52 => ( ( finite_card @ A @ ( insert @ A @ X3 @ A3 ) ) % 8.50/4.52 = ( suc @ ( finite_card @ A @ A3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3898_card__Suc__eq__finite,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,K: nat] : % 8.50/4.52 ( ( ( finite_card @ A @ A3 ) % 8.50/4.52 = ( suc @ K ) ) % 8.50/4.52 = ( ? [B4: A,B6: set @ A] : % 8.50/4.52 ( ( A3 % 8.50/4.52 = ( insert @ A @ B4 @ B6 ) ) % 8.50/4.52 & ~ ( member @ A @ B4 @ B6 ) % 8.50/4.52 & ( ( finite_card @ A @ B6 ) % 8.50/4.52 = K ) % 8.50/4.52 & ( finite_finite2 @ A @ B6 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3899_finite__if__finite__subsets__card__bdd,axiom, % 8.50/4.52 ! [A: $tType,F5: set @ A,C5: nat] : % 8.50/4.52 ( ! [G5: set @ A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ A ) @ G5 @ F5 ) % 8.50/4.52 => ( ( finite_finite2 @ A @ G5 ) % 8.50/4.52 => ( ord_less_eq @ nat @ ( finite_card @ A @ G5 ) @ C5 ) ) ) % 8.50/4.52 => ( ( finite_finite2 @ A @ F5 ) % 8.50/4.52 & ( ord_less_eq @ nat @ ( finite_card @ A @ F5 ) @ C5 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3900_obtain__subset__with__card__n,axiom, % 8.50/4.52 ! [A: $tType,N2: nat,S3: set @ A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ N2 @ ( finite_card @ A @ S3 ) ) % 8.50/4.52 => ~ ! [T5: set @ A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ A ) @ T5 @ S3 ) % 8.50/4.52 => ( ( ( finite_card @ A @ T5 ) % 8.50/4.52 = N2 ) % 8.50/4.52 => ~ ( finite_finite2 @ A @ T5 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3901_card__seteq,axiom, % 8.50/4.52 ! [A: $tType,B3: set @ A,A3: set @ A] : % 8.50/4.52 ( ( finite_finite2 @ A @ B3 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ B3 ) @ ( finite_card @ A @ A3 ) ) % 8.50/4.52 => ( A3 = B3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3902_card__mono,axiom, % 8.50/4.52 ! [A: $tType,B3: set @ A,A3: set @ A] : % 8.50/4.52 ( ( finite_finite2 @ A @ B3 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.52 => ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3903_card__less__sym__Diff,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( finite_finite2 @ A @ B3 ) % 8.50/4.52 => ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) % 8.50/4.52 => ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3904_card__le__sym__Diff,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( finite_finite2 @ A @ B3 ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) % 8.50/4.52 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3905_card__length,axiom, % 8.50/4.52 ! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( set2 @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3906_card__1__singletonE,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ( ( finite_card @ A @ A3 ) % 8.50/4.52 = ( one_one @ nat ) ) % 8.50/4.52 => ~ ! [X5: A] : % 8.50/4.52 ( A3 % 8.50/4.52 != ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3907_psubset__card__mono,axiom, % 8.50/4.52 ! [A: $tType,B3: set @ A,A3: set @ A] : % 8.50/4.52 ( ( finite_finite2 @ A @ B3 ) % 8.50/4.52 => ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.52 => ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3908_card__less,axiom, % 8.50/4.52 ! [M7: set @ nat,I: nat] : % 8.50/4.52 ( ( member @ nat @ ( zero_zero @ nat ) @ M7 ) % 8.50/4.52 => ( ( finite_card @ nat % 8.50/4.52 @ ( collect @ nat % 8.50/4.52 @ ^ [K3: nat] : % 8.50/4.52 ( ( member @ nat @ K3 @ M7 ) % 8.50/4.52 & ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) % 8.50/4.52 != ( zero_zero @ nat ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3909_card__less__Suc,axiom, % 8.50/4.52 ! [M7: set @ nat,I: nat] : % 8.50/4.52 ( ( member @ nat @ ( zero_zero @ nat ) @ M7 ) % 8.50/4.52 => ( ( suc % 8.50/4.52 @ ( finite_card @ nat % 8.50/4.52 @ ( collect @ nat % 8.50/4.52 @ ^ [K3: nat] : % 8.50/4.52 ( ( member @ nat @ ( suc @ K3 ) @ M7 ) % 8.50/4.52 & ( ord_less @ nat @ K3 @ I ) ) ) ) ) % 8.50/4.52 = ( finite_card @ nat % 8.50/4.52 @ ( collect @ nat % 8.50/4.52 @ ^ [K3: nat] : % 8.50/4.52 ( ( member @ nat @ K3 @ M7 ) % 8.50/4.52 & ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3910_card__less__Suc2,axiom, % 8.50/4.52 ! [M7: set @ nat,I: nat] : % 8.50/4.52 ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 ) % 8.50/4.52 => ( ( finite_card @ nat % 8.50/4.52 @ ( collect @ nat % 8.50/4.52 @ ^ [K3: nat] : % 8.50/4.52 ( ( member @ nat @ ( suc @ K3 ) @ M7 ) % 8.50/4.52 & ( ord_less @ nat @ K3 @ I ) ) ) ) % 8.50/4.52 = ( finite_card @ nat % 8.50/4.52 @ ( collect @ nat % 8.50/4.52 @ ^ [K3: nat] : % 8.50/4.52 ( ( member @ nat @ K3 @ M7 ) % 8.50/4.52 & ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3911_sum__Suc,axiom, % 8.50/4.52 ! [A: $tType,F2: A > nat,A3: set @ A] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ A @ nat % 8.50/4.52 @ ^ [X: A] : ( suc @ ( F2 @ X ) ) % 8.50/4.52 @ A3 ) % 8.50/4.52 = ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A3 ) @ ( finite_card @ A @ A3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3912_subset__card__intvl__is__intvl,axiom, % 8.50/4.52 ! [A3: set @ nat,K: nat] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ nat ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A3 ) ) ) ) % 8.50/4.52 => ( A3 % 8.50/4.52 = ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A3 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3913_sum__multicount,axiom, % 8.50/4.52 ! [A: $tType,B: $tType,S3: set @ A,T4: set @ B,R: A > B > $o,K: nat] : % 8.50/4.52 ( ( finite_finite2 @ A @ S3 ) % 8.50/4.52 => ( ( finite_finite2 @ B @ T4 ) % 8.50/4.52 => ( ! [X5: B] : % 8.50/4.52 ( ( member @ B @ X5 @ T4 ) % 8.50/4.52 => ( ( finite_card @ A % 8.50/4.52 @ ( collect @ A % 8.50/4.52 @ ^ [I3: A] : % 8.50/4.52 ( ( member @ A @ I3 @ S3 ) % 8.50/4.52 & ( R @ I3 @ X5 ) ) ) ) % 8.50/4.52 = K ) ) % 8.50/4.52 => ( ( groups7311177749621191930dd_sum @ A @ nat % 8.50/4.52 @ ^ [I3: A] : % 8.50/4.52 ( finite_card @ B % 8.50/4.52 @ ( collect @ B % 8.50/4.52 @ ^ [J3: B] : % 8.50/4.52 ( ( member @ B @ J3 @ T4 ) % 8.50/4.52 & ( R @ I3 @ J3 ) ) ) ) % 8.50/4.52 @ S3 ) % 8.50/4.52 = ( times_times @ nat @ K @ ( finite_card @ B @ T4 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3914_sum__bounded__below,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.52 & ( semiring_1 @ A ) ) % 8.50/4.52 => ! [A3: set @ B,K5: A,F2: B > A] : % 8.50/4.52 ( ! [I4: B] : % 8.50/4.52 ( ( member @ B @ I4 @ A3 ) % 8.50/4.52 => ( ord_less_eq @ A @ K5 @ ( F2 @ I4 ) ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3915_sum__bounded__above,axiom, % 8.50/4.52 ! [B: $tType,A: $tType] : % 8.50/4.52 ( ( ( ordere6911136660526730532id_add @ A ) % 8.50/4.52 & ( semiring_1 @ A ) ) % 8.50/4.52 => ! [A3: set @ B,F2: B > A,K5: A] : % 8.50/4.52 ( ! [I4: B] : % 8.50/4.52 ( ( member @ B @ I4 @ A3 ) % 8.50/4.52 => ( ord_less_eq @ A @ ( F2 @ I4 ) @ K5 ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K5 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3916_card__gt__0__iff,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) ) % 8.50/4.52 = ( ( A3 % 8.50/4.52 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.52 & ( finite_finite2 @ A @ A3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3917_card__Suc__eq,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,K: nat] : % 8.50/4.52 ( ( ( finite_card @ A @ A3 ) % 8.50/4.52 = ( suc @ K ) ) % 8.50/4.52 = ( ? [B4: A,B6: set @ A] : % 8.50/4.52 ( ( A3 % 8.50/4.52 = ( insert @ A @ B4 @ B6 ) ) % 8.50/4.52 & ~ ( member @ A @ B4 @ B6 ) % 8.50/4.52 & ( ( finite_card @ A @ B6 ) % 8.50/4.52 = K ) % 8.50/4.52 & ( ( K % 8.50/4.52 = ( zero_zero @ nat ) ) % 8.50/4.52 => ( B6 % 8.50/4.52 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3918_card__eq__SucD,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,K: nat] : % 8.50/4.52 ( ( ( finite_card @ A @ A3 ) % 8.50/4.52 = ( suc @ K ) ) % 8.50/4.52 => ? [B5: A,B10: set @ A] : % 8.50/4.52 ( ( A3 % 8.50/4.52 = ( insert @ A @ B5 @ B10 ) ) % 8.50/4.52 & ~ ( member @ A @ B5 @ B10 ) % 8.50/4.52 & ( ( finite_card @ A @ B10 ) % 8.50/4.52 = K ) % 8.50/4.52 & ( ( K % 8.50/4.52 = ( zero_zero @ nat ) ) % 8.50/4.52 => ( B10 % 8.50/4.52 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3919_card__1__singleton__iff,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ( ( finite_card @ A @ A3 ) % 8.50/4.52 = ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.52 = ( ? [X: A] : % 8.50/4.52 ( A3 % 8.50/4.52 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3920_card__le__Suc0__iff__eq,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.52 = ( ! [X: A] : % 8.50/4.52 ( ( member @ A @ X @ A3 ) % 8.50/4.52 => ! [Y2: A] : % 8.50/4.52 ( ( member @ A @ Y2 @ A3 ) % 8.50/4.52 => ( X = Y2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3921_card__le__Suc__iff,axiom, % 8.50/4.52 ! [A: $tType,N2: nat,A3: set @ A] : % 8.50/4.52 ( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( finite_card @ A @ A3 ) ) % 8.50/4.52 = ( ? [A4: A,B6: set @ A] : % 8.50/4.52 ( ( A3 % 8.50/4.52 = ( insert @ A @ A4 @ B6 ) ) % 8.50/4.52 & ~ ( member @ A @ A4 @ B6 ) % 8.50/4.52 & ( ord_less_eq @ nat @ N2 @ ( finite_card @ A @ B6 ) ) % 8.50/4.52 & ( finite_finite2 @ A @ B6 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3922_card__Diff1__le,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,X3: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3923_card__Diff__subset,axiom, % 8.50/4.52 ! [A: $tType,B3: set @ A,A3: set @ A] : % 8.50/4.52 ( ( finite_finite2 @ A @ B3 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 8.50/4.52 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.52 = ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3924_card__psubset,axiom, % 8.50/4.52 ! [A: $tType,B3: set @ A,A3: set @ A] : % 8.50/4.52 ( ( finite_finite2 @ A @ B3 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.52 => ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) % 8.50/4.52 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3925_diff__card__le__card__Diff,axiom, % 8.50/4.52 ! [A: $tType,B3: set @ A,A3: set @ A] : % 8.50/4.52 ( ( finite_finite2 @ A @ B3 ) % 8.50/4.52 => ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3926_card__lists__length__le,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,N2: nat] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( finite_card @ ( list @ A ) % 8.50/4.52 @ ( collect @ ( list @ A ) % 8.50/4.52 @ ^ [Xs2: list @ A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) % 8.50/4.52 & ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N2 ) ) ) ) % 8.50/4.52 = ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A3 ) ) @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3927_card__roots__unity,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.52 & ( idom @ A ) ) % 8.50/4.52 => ! [N2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N2 ) % 8.50/4.52 => ( ord_less_eq @ nat % 8.50/4.52 @ ( finite_card @ A % 8.50/4.52 @ ( collect @ A % 8.50/4.52 @ ^ [Z6: A] : % 8.50/4.52 ( ( power_power @ A @ Z6 @ N2 ) % 8.50/4.52 = ( one_one @ A ) ) ) ) % 8.50/4.52 @ N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3928_subset__eq__atLeast0__lessThan__card,axiom, % 8.50/4.52 ! [N7: set @ nat,N2: nat] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) % 8.50/4.52 => ( ord_less_eq @ nat @ ( finite_card @ nat @ N7 ) @ N2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3929_card__sum__le__nat__sum,axiom, % 8.50/4.52 ! [S3: set @ nat] : % 8.50/4.52 ( ord_less_eq @ nat % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.52 @ ^ [X: nat] : X % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S3 ) ) ) % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ nat @ nat % 8.50/4.52 @ ^ [X: nat] : X % 8.50/4.52 @ S3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3930_card__nth__roots,axiom, % 8.50/4.52 ! [C2: complex,N2: nat] : % 8.50/4.52 ( ( C2 % 8.50/4.52 != ( zero_zero @ complex ) ) % 8.50/4.52 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.52 => ( ( finite_card @ complex % 8.50/4.52 @ ( collect @ complex % 8.50/4.52 @ ^ [Z6: complex] : % 8.50/4.52 ( ( power_power @ complex @ Z6 @ N2 ) % 8.50/4.52 = C2 ) ) ) % 8.50/4.52 = N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3931_card__roots__unity__eq,axiom, % 8.50/4.52 ! [N2: nat] : % 8.50/4.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.52 => ( ( finite_card @ complex % 8.50/4.52 @ ( collect @ complex % 8.50/4.52 @ ^ [Z6: complex] : % 8.50/4.52 ( ( power_power @ complex @ Z6 @ N2 ) % 8.50/4.52 = ( one_one @ complex ) ) ) ) % 8.50/4.52 = N2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3932_card__2__iff,axiom, % 8.50/4.52 ! [A: $tType,S3: set @ A] : % 8.50/4.52 ( ( ( finite_card @ A @ S3 ) % 8.50/4.52 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) % 8.50/4.52 = ( ? [X: A,Y2: A] : % 8.50/4.52 ( ( S3 % 8.50/4.52 = ( insert @ A @ X @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.52 & ( X != Y2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3933_card__3__iff,axiom, % 8.50/4.52 ! [A: $tType,S3: set @ A] : % 8.50/4.52 ( ( ( finite_card @ A @ S3 ) % 8.50/4.52 = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) % 8.50/4.52 = ( ? [X: A,Y2: A,Z6: A] : % 8.50/4.52 ( ( S3 % 8.50/4.52 = ( insert @ A @ X @ ( insert @ A @ Y2 @ ( insert @ A @ Z6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) % 8.50/4.52 & ( X != Y2 ) % 8.50/4.52 & ( Y2 != Z6 ) % 8.50/4.52 & ( X != Z6 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3934_odd__card__imp__not__empty,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A3 ) ) % 8.50/4.52 => ( A3 % 8.50/4.52 != ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3935_card__Suc__Diff1,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,X3: A] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( member @ A @ X3 @ A3 ) % 8.50/4.52 => ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) % 8.50/4.52 = ( finite_card @ A @ A3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3936_card_Oinsert__remove,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,X3: A] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( finite_card @ A @ ( insert @ A @ X3 @ A3 ) ) % 8.50/4.52 = ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3937_card_Oremove,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,X3: A] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( member @ A @ X3 @ A3 ) % 8.50/4.52 => ( ( finite_card @ A @ A3 ) % 8.50/4.52 = ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3938_card__Diff1__less,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,X3: A] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( member @ A @ X3 @ A3 ) % 8.50/4.52 => ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3939_card__Diff2__less,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,X3: A,Y: A] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( member @ A @ X3 @ A3 ) % 8.50/4.52 => ( ( member @ A @ Y @ A3 ) % 8.50/4.52 => ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3940_card__Diff1__less__iff,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,X3: A] : % 8.50/4.52 ( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) % 8.50/4.52 = ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 & ( member @ A @ X3 @ A3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3941_card__Diff__singleton__if,axiom, % 8.50/4.52 ! [A: $tType,X3: A,A3: set @ A] : % 8.50/4.52 ( ( ( member @ A @ X3 @ A3 ) % 8.50/4.52 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.52 = ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) ) % 8.50/4.52 & ( ~ ( member @ A @ X3 @ A3 ) % 8.50/4.52 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.52 = ( finite_card @ A @ A3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3942_card__Diff__singleton,axiom, % 8.50/4.52 ! [A: $tType,X3: A,A3: set @ A] : % 8.50/4.52 ( ( member @ A @ X3 @ A3 ) % 8.50/4.52 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.52 = ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3943_prod__le__power,axiom, % 8.50/4.52 ! [B: $tType,A: $tType] : % 8.50/4.52 ( ( linordered_semidom @ A ) % 8.50/4.52 => ! [A3: set @ B,F2: B > A,N2: A,K: nat] : % 8.50/4.52 ( ! [I4: B] : % 8.50/4.52 ( ( member @ B @ I4 @ A3 ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I4 ) ) % 8.50/4.52 & ( ord_less_eq @ A @ ( F2 @ I4 ) @ N2 ) ) ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ K ) % 8.50/4.52 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ N2 ) % 8.50/4.52 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A3 ) @ ( power_power @ A @ N2 @ K ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3944_sum__bounded__above__strict,axiom, % 8.50/4.52 ! [B: $tType,A: $tType] : % 8.50/4.52 ( ( ( ordere8940638589300402666id_add @ A ) % 8.50/4.52 & ( semiring_1 @ A ) ) % 8.50/4.52 => ! [A3: set @ B,F2: B > A,K5: A] : % 8.50/4.52 ( ! [I4: B] : % 8.50/4.52 ( ( member @ B @ I4 @ A3 ) % 8.50/4.52 => ( ord_less @ A @ ( F2 @ I4 ) @ K5 ) ) % 8.50/4.52 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A3 ) ) % 8.50/4.52 => ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K5 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3945_sum__bounded__above__divide,axiom, % 8.50/4.52 ! [B: $tType,A: $tType] : % 8.50/4.52 ( ( linordered_field @ A ) % 8.50/4.52 => ! [A3: set @ B,F2: B > A,K5: A] : % 8.50/4.52 ( ! [I4: B] : % 8.50/4.52 ( ( member @ B @ I4 @ A3 ) % 8.50/4.52 => ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( divide_divide @ A @ K5 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) ) ) ) % 8.50/4.52 => ( ( finite_finite2 @ B @ A3 ) % 8.50/4.52 => ( ( A3 % 8.50/4.52 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.52 => ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ K5 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3946_card__insert__le__m1,axiom, % 8.50/4.52 ! [A: $tType,N2: nat,Y: set @ A,X3: A] : % 8.50/4.52 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) % 8.50/4.52 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X3 @ Y ) ) @ N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3947_polyfun__roots__card,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.52 & ( idom @ A ) ) % 8.50/4.52 => ! [C2: nat > A,K: nat,N2: nat] : % 8.50/4.52 ( ( ( C2 @ K ) % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.52 => ( ord_less_eq @ nat % 8.50/4.52 @ ( finite_card @ A % 8.50/4.52 @ ( collect @ A % 8.50/4.52 @ ^ [Z6: A] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ) % 8.50/4.52 @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3948_prod__gen__delta,axiom, % 8.50/4.52 ! [A: $tType,B: $tType] : % 8.50/4.52 ( ( comm_monoid_mult @ A ) % 8.50/4.52 => ! [S3: set @ B,A2: B,B2: B > A,C2: A] : % 8.50/4.52 ( ( finite_finite2 @ B @ S3 ) % 8.50/4.52 => ( ( ( member @ B @ A2 @ S3 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A % 8.50/4.52 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ C2 ) % 8.50/4.52 @ S3 ) % 8.50/4.52 = ( times_times @ A @ ( B2 @ A2 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S3 ) @ ( one_one @ nat ) ) ) ) ) ) % 8.50/4.52 & ( ~ ( member @ B @ A2 @ S3 ) % 8.50/4.52 => ( ( groups7121269368397514597t_prod @ B @ A % 8.50/4.52 @ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ C2 ) % 8.50/4.52 @ S3 ) % 8.50/4.52 = ( power_power @ A @ C2 @ ( finite_card @ B @ S3 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3949_polyfun__rootbound,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.52 & ( idom @ A ) ) % 8.50/4.52 => ! [C2: nat > A,K: nat,N2: nat] : % 8.50/4.52 ( ( ( C2 @ K ) % 8.50/4.52 != ( zero_zero @ A ) ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ K @ N2 ) % 8.50/4.52 => ( ( finite_finite2 @ A % 8.50/4.52 @ ( collect @ A % 8.50/4.52 @ ^ [Z6: A] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ) % 8.50/4.52 & ( ord_less_eq @ nat % 8.50/4.52 @ ( finite_card @ A % 8.50/4.52 @ ( collect @ A % 8.50/4.52 @ ^ [Z6: A] : % 8.50/4.52 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [I3: nat] : ( times_times @ A @ ( C2 @ I3 ) @ ( power_power @ A @ Z6 @ I3 ) ) % 8.50/4.52 @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ) % 8.50/4.52 @ N2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3950_pair__leqI1,axiom, % 8.50/4.52 ! [A2: nat,B2: nat,S: nat,T2: nat] : % 8.50/4.52 ( ( ord_less @ nat @ A2 @ B2 ) % 8.50/4.52 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3951_card__lists__distinct__length__eq,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,K: nat] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A3 ) ) % 8.50/4.52 => ( ( finite_card @ ( list @ A ) % 8.50/4.52 @ ( collect @ ( list @ A ) % 8.50/4.52 @ ^ [Xs2: list @ A] : % 8.50/4.52 ( ( ( size_size @ ( list @ A ) @ Xs2 ) % 8.50/4.52 = K ) % 8.50/4.52 & ( distinct @ A @ Xs2 ) % 8.50/4.52 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) ) ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ nat % 8.50/4.52 @ ^ [X: nat] : X % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3952_card__lists__distinct__length__eq_H,axiom, % 8.50/4.52 ! [A: $tType,K: nat,A3: set @ A] : % 8.50/4.52 ( ( ord_less @ nat @ K @ ( finite_card @ A @ A3 ) ) % 8.50/4.52 => ( ( finite_card @ ( list @ A ) % 8.50/4.52 @ ( collect @ ( list @ A ) % 8.50/4.52 @ ^ [Xs2: list @ A] : % 8.50/4.52 ( ( ( size_size @ ( list @ A ) @ Xs2 ) % 8.50/4.52 = K ) % 8.50/4.52 & ( distinct @ A @ Xs2 ) % 8.50/4.52 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) ) ) ) % 8.50/4.52 = ( groups7121269368397514597t_prod @ nat @ nat % 8.50/4.52 @ ^ [X: nat] : X % 8.50/4.52 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3953_set__n__lists,axiom, % 8.50/4.52 ! [A: $tType,N2: nat,Xs: list @ A] : % 8.50/4.52 ( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N2 @ Xs ) ) % 8.50/4.52 = ( collect @ ( list @ A ) % 8.50/4.52 @ ^ [Ys3: list @ A] : % 8.50/4.52 ( ( ( size_size @ ( list @ A ) @ Ys3 ) % 8.50/4.52 = N2 ) % 8.50/4.52 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Xs ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3954_int__of__integer__code,axiom, % 8.50/4.52 ( code_int_of_integer % 8.50/4.52 = ( ^ [K3: code_integer] : % 8.50/4.52 ( if @ int @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K3 ) ) ) % 8.50/4.52 @ ( if @ int % 8.50/4.52 @ ( K3 % 8.50/4.52 = ( zero_zero @ code_integer ) ) % 8.50/4.52 @ ( zero_zero @ int ) % 8.50/4.52 @ ( product_case_prod @ code_integer @ code_integer @ int % 8.50/4.52 @ ^ [L2: code_integer,J3: code_integer] : % 8.50/4.52 ( if @ int % 8.50/4.52 @ ( J3 % 8.50/4.52 = ( zero_zero @ code_integer ) ) % 8.50/4.52 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) ) % 8.50/4.52 @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) ) @ ( one_one @ int ) ) ) % 8.50/4.52 @ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3955_distinct__swap,axiom, % 8.50/4.52 ! [A: $tType,I: nat,Xs: list @ A,J: nat] : % 8.50/4.52 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.52 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.52 => ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) ) % 8.50/4.52 = ( distinct @ A @ Xs ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3956_finite__lists__distinct__length__eq,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,N2: nat] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( finite_finite2 @ ( list @ A ) % 8.50/4.52 @ ( collect @ ( list @ A ) % 8.50/4.52 @ ^ [Xs2: list @ A] : % 8.50/4.52 ( ( ( size_size @ ( list @ A ) @ Xs2 ) % 8.50/4.52 = N2 ) % 8.50/4.52 & ( distinct @ A @ Xs2 ) % 8.50/4.52 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3957_distinct__enumerate,axiom, % 8.50/4.52 ! [A: $tType,N2: nat,Xs: list @ A] : ( distinct @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3958_distinct__product,axiom, % 8.50/4.52 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 8.50/4.52 ( ( distinct @ A @ Xs ) % 8.50/4.52 => ( ( distinct @ B @ Ys ) % 8.50/4.52 => ( distinct @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3959_sorted__list__of__set_Odistinct__if__distinct__map,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( linorder @ A ) % 8.50/4.52 => ! [Xs: list @ A] : % 8.50/4.52 ( ( distinct @ A @ Xs ) % 8.50/4.52 => ( distinct @ A @ Xs ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3960_distinct__zipI1,axiom, % 8.50/4.52 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 8.50/4.52 ( ( distinct @ A @ Xs ) % 8.50/4.52 => ( distinct @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3961_distinct__zipI2,axiom, % 8.50/4.52 ! [B: $tType,A: $tType,Ys: list @ A,Xs: list @ B] : % 8.50/4.52 ( ( distinct @ A @ Ys ) % 8.50/4.52 => ( distinct @ ( product_prod @ B @ A ) @ ( zip @ B @ A @ Xs @ Ys ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3962_finite__distinct__list,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ? [Xs3: list @ A] : % 8.50/4.52 ( ( ( set2 @ A @ Xs3 ) % 8.50/4.52 = A3 ) % 8.50/4.52 & ( distinct @ A @ Xs3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3963_less__integer_Orep__eq,axiom, % 8.50/4.52 ( ( ord_less @ code_integer ) % 8.50/4.52 = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3964_integer__less__iff,axiom, % 8.50/4.52 ( ( ord_less @ code_integer ) % 8.50/4.52 = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3965_subseqs__distinctD,axiom, % 8.50/4.52 ! [A: $tType,Ys: list @ A,Xs: list @ A] : % 8.50/4.52 ( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) % 8.50/4.52 => ( ( distinct @ A @ Xs ) % 8.50/4.52 => ( distinct @ A @ Ys ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3966_distinct__card,axiom, % 8.50/4.52 ! [A: $tType,Xs: list @ A] : % 8.50/4.52 ( ( distinct @ A @ Xs ) % 8.50/4.52 => ( ( finite_card @ A @ ( set2 @ A @ Xs ) ) % 8.50/4.52 = ( size_size @ ( list @ A ) @ Xs ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3967_card__distinct,axiom, % 8.50/4.52 ! [A: $tType,Xs: list @ A] : % 8.50/4.52 ( ( ( finite_card @ A @ ( set2 @ A @ Xs ) ) % 8.50/4.52 = ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.52 => ( distinct @ A @ Xs ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3968_distinct__conv__nth,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( distinct @ A ) % 8.50/4.52 = ( ^ [Xs2: list @ A] : % 8.50/4.52 ! [I3: nat] : % 8.50/4.52 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) % 8.50/4.52 => ! [J3: nat] : % 8.50/4.52 ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) % 8.50/4.52 => ( ( I3 != J3 ) % 8.50/4.52 => ( ( nth @ A @ Xs2 @ I3 ) % 8.50/4.52 != ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3969_nth__eq__iff__index__eq,axiom, % 8.50/4.52 ! [A: $tType,Xs: list @ A,I: nat,J: nat] : % 8.50/4.52 ( ( distinct @ A @ Xs ) % 8.50/4.52 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.52 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.52 => ( ( ( nth @ A @ Xs @ I ) % 8.50/4.52 = ( nth @ A @ Xs @ J ) ) % 8.50/4.52 = ( I = J ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3970_distinct__Ex1,axiom, % 8.50/4.52 ! [A: $tType,Xs: list @ A,X3: A] : % 8.50/4.52 ( ( distinct @ A @ Xs ) % 8.50/4.52 => ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.50/4.52 => ? [X5: nat] : % 8.50/4.52 ( ( ord_less @ nat @ X5 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.52 & ( ( nth @ A @ Xs @ X5 ) % 8.50/4.52 = X3 ) % 8.50/4.52 & ! [Y5: nat] : % 8.50/4.52 ( ( ( ord_less @ nat @ Y5 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.52 & ( ( nth @ A @ Xs @ Y5 ) % 8.50/4.52 = X3 ) ) % 8.50/4.52 => ( Y5 = X5 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3971_length__n__lists__elem,axiom, % 8.50/4.52 ! [A: $tType,Ys: list @ A,N2: nat,Xs: list @ A] : % 8.50/4.52 ( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N2 @ Xs ) ) ) % 8.50/4.52 => ( ( size_size @ ( list @ A ) @ Ys ) % 8.50/4.52 = N2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3972_distinct__list__update,axiom, % 8.50/4.52 ! [A: $tType,Xs: list @ A,A2: A,I: nat] : % 8.50/4.52 ( ( distinct @ A @ Xs ) % 8.50/4.52 => ( ~ ( member @ A @ A2 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ I ) @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.52 => ( distinct @ A @ ( list_update @ A @ Xs @ I @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3973_length__n__lists,axiom, % 8.50/4.52 ! [A: $tType,N2: nat,Xs: list @ A] : % 8.50/4.52 ( ( size_size @ ( list @ ( list @ A ) ) @ ( n_lists @ A @ N2 @ Xs ) ) % 8.50/4.52 = ( power_power @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3974_set__update__distinct,axiom, % 8.50/4.52 ! [A: $tType,Xs: list @ A,N2: nat,X3: A] : % 8.50/4.52 ( ( distinct @ A @ Xs ) % 8.50/4.52 => ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.52 => ( ( set2 @ A @ ( list_update @ A @ Xs @ N2 @ X3 ) ) % 8.50/4.52 = ( insert @ A @ X3 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ N2 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3975_divmod__integer__def,axiom, % 8.50/4.52 ( code_divmod_integer % 8.50/4.52 = ( ^ [K3: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K3 @ L2 ) @ ( modulo_modulo @ code_integer @ K3 @ L2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3976_nat__of__integer__code,axiom, % 8.50/4.52 ( code_nat_of_integer % 8.50/4.52 = ( ^ [K3: code_integer] : % 8.50/4.52 ( if @ nat @ ( ord_less_eq @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat ) % 8.50/4.52 @ ( product_case_prod @ code_integer @ code_integer @ nat % 8.50/4.52 @ ^ [L2: code_integer,J3: code_integer] : % 8.50/4.52 ( if @ nat % 8.50/4.52 @ ( J3 % 8.50/4.52 = ( zero_zero @ code_integer ) ) % 8.50/4.52 @ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) % 8.50/4.52 @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( one_one @ nat ) ) ) % 8.50/4.52 @ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3977_distinct__union,axiom, % 8.50/4.52 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 8.50/4.52 ( ( distinct @ A @ ( union @ A @ Xs @ Ys ) ) % 8.50/4.52 = ( distinct @ A @ Ys ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3978_bit__cut__integer__def,axiom, % 8.50/4.52 ( code_bit_cut_integer % 8.50/4.52 = ( ^ [K3: code_integer] : % 8.50/4.52 ( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) % 8.50/4.52 @ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3979_divmod__abs__def,axiom, % 8.50/4.52 ( code_divmod_abs % 8.50/4.52 = ( ^ [K3: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3980_nat__of__integer__non__positive,axiom, % 8.50/4.52 ! [K: code_integer] : % 8.50/4.52 ( ( ord_less_eq @ code_integer @ K @ ( zero_zero @ code_integer ) ) % 8.50/4.52 => ( ( code_nat_of_integer @ K ) % 8.50/4.52 = ( zero_zero @ nat ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3981_nat__of__integer__code__post_I1_J,axiom, % 8.50/4.52 ( ( code_nat_of_integer @ ( zero_zero @ code_integer ) ) % 8.50/4.52 = ( zero_zero @ nat ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3982_divmod__abs__code_I6_J,axiom, % 8.50/4.52 ! [J: code_integer] : % 8.50/4.52 ( ( code_divmod_abs @ ( zero_zero @ code_integer ) @ J ) % 8.50/4.52 = ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3983_divmod__abs__code_I5_J,axiom, % 8.50/4.52 ! [J: code_integer] : % 8.50/4.52 ( ( code_divmod_abs @ J @ ( zero_zero @ code_integer ) ) % 8.50/4.52 = ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( abs_abs @ code_integer @ J ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3984_bit__cut__integer__code,axiom, % 8.50/4.52 ( code_bit_cut_integer % 8.50/4.52 = ( ^ [K3: code_integer] : % 8.50/4.52 ( if @ ( product_prod @ code_integer @ $o ) % 8.50/4.52 @ ( K3 % 8.50/4.52 = ( zero_zero @ code_integer ) ) % 8.50/4.52 @ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false ) % 8.50/4.52 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o ) % 8.50/4.52 @ ^ [R5: code_integer,S8: code_integer] : % 8.50/4.52 ( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ R5 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ S8 ) ) % 8.50/4.52 @ ( S8 % 8.50/4.52 = ( one_one @ code_integer ) ) ) % 8.50/4.52 @ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3985_divmod__integer__code,axiom, % 8.50/4.52 ( code_divmod_integer % 8.50/4.52 = ( ^ [K3: code_integer,L2: code_integer] : % 8.50/4.52 ( if @ ( product_prod @ code_integer @ code_integer ) % 8.50/4.52 @ ( K3 % 8.50/4.52 = ( zero_zero @ code_integer ) ) % 8.50/4.52 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) % 8.50/4.52 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L2 ) % 8.50/4.52 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ ( code_divmod_abs @ K3 @ L2 ) % 8.50/4.52 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer ) % 8.50/4.52 @ ^ [R5: code_integer,S8: code_integer] : % 8.50/4.52 ( if @ ( product_prod @ code_integer @ code_integer ) % 8.50/4.52 @ ( S8 % 8.50/4.52 = ( zero_zero @ code_integer ) ) % 8.50/4.52 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) ) % 8.50/4.52 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L2 @ S8 ) ) ) % 8.50/4.52 @ ( code_divmod_abs @ K3 @ L2 ) ) ) % 8.50/4.52 @ ( if @ ( product_prod @ code_integer @ code_integer ) % 8.50/4.52 @ ( L2 % 8.50/4.52 = ( zero_zero @ code_integer ) ) % 8.50/4.52 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) % 8.50/4.52 @ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer ) % 8.50/4.52 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K3 @ L2 ) % 8.50/4.52 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer ) % 8.50/4.52 @ ^ [R5: code_integer,S8: code_integer] : % 8.50/4.52 ( if @ ( product_prod @ code_integer @ code_integer ) % 8.50/4.52 @ ( S8 % 8.50/4.52 = ( zero_zero @ code_integer ) ) % 8.50/4.52 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) ) % 8.50/4.52 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L2 ) @ S8 ) ) ) % 8.50/4.52 @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3986_card__Pow,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.52 => ( ( finite_card @ ( set @ A ) @ ( pow2 @ A @ A3 ) ) % 8.50/4.52 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3987_rec__nat__add__eq__if,axiom, % 8.50/4.52 ! [A: $tType,A2: A,F2: nat > A > A,V2: num,N2: nat] : % 8.50/4.52 ( ( rec_nat @ A @ A2 @ F2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N2 ) ) % 8.50/4.52 = ( F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N2 ) @ ( rec_nat @ A @ A2 @ F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3988_case__nat__add__eq__if,axiom, % 8.50/4.52 ! [A: $tType,A2: A,F2: nat > A,V2: num,N2: nat] : % 8.50/4.52 ( ( case_nat @ A @ A2 @ F2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N2 ) ) % 8.50/4.52 = ( F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3989_Pow__empty,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( pow2 @ A @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.52 = ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3990_Pow__singleton__iff,axiom, % 8.50/4.52 ! [A: $tType,X8: set @ A,Y7: set @ A] : % 8.50/4.52 ( ( ( pow2 @ A @ X8 ) % 8.50/4.52 = ( insert @ ( set @ A ) @ Y7 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) % 8.50/4.52 = ( ( X8 % 8.50/4.52 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.52 & ( Y7 % 8.50/4.52 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3991_PowI,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.52 => ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3992_Pow__iff,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.52 ( ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B3 ) ) % 8.50/4.52 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3993_apsnd__conv,axiom, % 8.50/4.52 ! [A: $tType,B: $tType,C: $tType,F2: C > B,X3: A,Y: C] : % 8.50/4.52 ( ( product_apsnd @ C @ B @ A @ F2 @ ( product_Pair @ A @ C @ X3 @ Y ) ) % 8.50/4.52 = ( product_Pair @ A @ B @ X3 @ ( F2 @ Y ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3994_old_Onat_Osimps_I7_J,axiom, % 8.50/4.52 ! [T: $tType,F1: T,F22: nat > T > T,Nat: nat] : % 8.50/4.52 ( ( rec_nat @ T @ F1 @ F22 @ ( suc @ Nat ) ) % 8.50/4.52 = ( F22 @ Nat @ ( rec_nat @ T @ F1 @ F22 @ Nat ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3995_old_Onat_Osimps_I6_J,axiom, % 8.50/4.52 ! [T: $tType,F1: T,F22: nat > T > T] : % 8.50/4.52 ( ( rec_nat @ T @ F1 @ F22 @ ( zero_zero @ nat ) ) % 8.50/4.52 = F1 ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3996_finite__Pow__iff,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ( finite_finite2 @ ( set @ A ) @ ( pow2 @ A @ A3 ) ) % 8.50/4.52 = ( finite_finite2 @ A @ A3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3997_Pow__bottom,axiom, % 8.50/4.52 ! [A: $tType,B3: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow2 @ A @ B3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3998_nat_Ocase__distrib,axiom, % 8.50/4.52 ! [B: $tType,A: $tType,H2: A > B,F1: A,F22: nat > A,Nat: nat] : % 8.50/4.52 ( ( H2 @ ( case_nat @ A @ F1 @ F22 @ Nat ) ) % 8.50/4.52 = ( case_nat @ B @ ( H2 @ F1 ) % 8.50/4.52 @ ^ [X: nat] : ( H2 @ ( F22 @ X ) ) % 8.50/4.52 @ Nat ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_3999_Pow__not__empty,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ( pow2 @ A @ A3 ) % 8.50/4.52 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4000_Pow__top,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ A3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4001_Pow__mono,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.52 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4002_PowD,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.52 ( ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B3 ) ) % 8.50/4.52 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4003_Pow__def,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( pow2 @ A ) % 8.50/4.52 = ( ^ [A6: set @ A] : % 8.50/4.52 ( collect @ ( set @ A ) % 8.50/4.52 @ ^ [B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4004_Pow__Compl,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ A] : % 8.50/4.52 ( ( pow2 @ A @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) % 8.50/4.52 = ( collect @ ( set @ A ) % 8.50/4.52 @ ^ [Uu3: set @ A] : % 8.50/4.52 ? [B6: set @ A] : % 8.50/4.52 ( ( Uu3 % 8.50/4.52 = ( uminus_uminus @ ( set @ A ) @ B6 ) ) % 8.50/4.52 & ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B6 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4005_old_Onat_Osimps_I5_J,axiom, % 8.50/4.52 ! [A: $tType,F1: A,F22: nat > A,X22: nat] : % 8.50/4.52 ( ( case_nat @ A @ F1 @ F22 @ ( suc @ X22 ) ) % 8.50/4.52 = ( F22 @ X22 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4006_old_Onat_Osimps_I4_J,axiom, % 8.50/4.52 ! [A: $tType,F1: A,F22: nat > A] : % 8.50/4.52 ( ( case_nat @ A @ F1 @ F22 @ ( zero_zero @ nat ) ) % 8.50/4.52 = F1 ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4007_nat_Odisc__eq__case_I2_J,axiom, % 8.50/4.52 ! [Nat: nat] : % 8.50/4.52 ( ( Nat % 8.50/4.52 != ( zero_zero @ nat ) ) % 8.50/4.52 = ( case_nat @ $o @ $false % 8.50/4.52 @ ^ [Uu3: nat] : $true % 8.50/4.52 @ Nat ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4008_nat_Odisc__eq__case_I1_J,axiom, % 8.50/4.52 ! [Nat: nat] : % 8.50/4.52 ( ( Nat % 8.50/4.52 = ( zero_zero @ nat ) ) % 8.50/4.52 = ( case_nat @ $o @ $true % 8.50/4.52 @ ^ [Uu3: nat] : $false % 8.50/4.52 @ Nat ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4009_less__eq__nat_Osimps_I2_J,axiom, % 8.50/4.52 ! [M: nat,N2: nat] : % 8.50/4.52 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 ) % 8.50/4.52 = ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M ) @ N2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4010_diff__Suc,axiom, % 8.50/4.52 ! [M: nat,N2: nat] : % 8.50/4.52 ( ( minus_minus @ nat @ M @ ( suc @ N2 ) ) % 8.50/4.52 = ( case_nat @ nat @ ( zero_zero @ nat ) % 8.50/4.52 @ ^ [K3: nat] : K3 % 8.50/4.52 @ ( minus_minus @ nat @ M @ N2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4011_Nitpick_Ocase__nat__unfold,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( case_nat @ A ) % 8.50/4.52 = ( ^ [X: A,F3: nat > A,N: nat] : % 8.50/4.52 ( if @ A % 8.50/4.52 @ ( N % 8.50/4.52 = ( zero_zero @ nat ) ) % 8.50/4.52 @ X % 8.50/4.52 @ ( F3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4012_binomial__def,axiom, % 8.50/4.52 ( binomial % 8.50/4.52 = ( ^ [N: nat,K3: nat] : % 8.50/4.52 ( finite_card @ ( set @ nat ) % 8.50/4.52 @ ( collect @ ( set @ nat ) % 8.50/4.52 @ ^ [K6: set @ nat] : % 8.50/4.52 ( ( member @ ( set @ nat ) @ K6 @ ( pow2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) % 8.50/4.52 & ( ( finite_card @ nat @ K6 ) % 8.50/4.52 = K3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4013_old_Orec__nat__def,axiom, % 8.50/4.52 ! [T: $tType] : % 8.50/4.52 ( ( rec_nat @ T ) % 8.50/4.52 = ( ^ [F12: T,F23: nat > T > T,X: nat] : ( the @ T @ ( rec_set_nat @ T @ F12 @ F23 @ X ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4014_rec__nat__0__imp,axiom, % 8.50/4.52 ! [A: $tType,F2: nat > A,F1: A,F22: nat > A > A] : % 8.50/4.52 ( ( F2 % 8.50/4.52 = ( rec_nat @ A @ F1 @ F22 ) ) % 8.50/4.52 => ( ( F2 @ ( zero_zero @ nat ) ) % 8.50/4.52 = F1 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4015_rec__nat__Suc__imp,axiom, % 8.50/4.52 ! [A: $tType,F2: nat > A,F1: A,F22: nat > A > A,N2: nat] : % 8.50/4.52 ( ( F2 % 8.50/4.52 = ( rec_nat @ A @ F1 @ F22 ) ) % 8.50/4.52 => ( ( F2 @ ( suc @ N2 ) ) % 8.50/4.52 = ( F22 @ N2 @ ( F2 @ N2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4016_nat_Osplit__sels_I1_J,axiom, % 8.50/4.52 ! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] : % 8.50/4.52 ( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) ) % 8.50/4.52 = ( ( ( Nat % 8.50/4.52 = ( zero_zero @ nat ) ) % 8.50/4.52 => ( P @ F1 ) ) % 8.50/4.52 & ( ( Nat % 8.50/4.52 = ( suc @ ( pred @ Nat ) ) ) % 8.50/4.52 => ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4017_pred__def,axiom, % 8.50/4.52 ( pred % 8.50/4.52 = ( case_nat @ nat @ ( zero_zero @ nat ) % 8.50/4.52 @ ^ [X24: nat] : X24 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4018_subset__Collect__iff,axiom, % 8.50/4.52 ! [A: $tType,B3: set @ A,A3: set @ A,P: A > $o] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 8.50/4.52 => ( ( ord_less_eq @ ( set @ A ) @ B3 % 8.50/4.52 @ ( collect @ A % 8.50/4.52 @ ^ [X: A] : % 8.50/4.52 ( ( member @ A @ X @ A3 ) % 8.50/4.52 & ( P @ X ) ) ) ) % 8.50/4.52 = ( ! [X: A] : % 8.50/4.52 ( ( member @ A @ X @ B3 ) % 8.50/4.52 => ( P @ X ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4019_subset__CollectI,axiom, % 8.50/4.52 ! [A: $tType,B3: set @ A,A3: set @ A,Q: A > $o,P: A > $o] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 8.50/4.52 => ( ! [X5: A] : % 8.50/4.52 ( ( member @ A @ X5 @ B3 ) % 8.50/4.52 => ( ( Q @ X5 ) % 8.50/4.52 => ( P @ X5 ) ) ) % 8.50/4.52 => ( ord_less_eq @ ( set @ A ) % 8.50/4.52 @ ( collect @ A % 8.50/4.52 @ ^ [X: A] : % 8.50/4.52 ( ( member @ A @ X @ B3 ) % 8.50/4.52 & ( Q @ X ) ) ) % 8.50/4.52 @ ( collect @ A % 8.50/4.52 @ ^ [X: A] : % 8.50/4.52 ( ( member @ A @ X @ A3 ) % 8.50/4.52 & ( P @ X ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4020_nat_Osplit__sels_I2_J,axiom, % 8.50/4.52 ! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] : % 8.50/4.52 ( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) ) % 8.50/4.52 = ( ~ ( ( ( Nat % 8.50/4.52 = ( zero_zero @ nat ) ) % 8.50/4.52 & ~ ( P @ F1 ) ) % 8.50/4.52 | ( ( Nat % 8.50/4.52 = ( suc @ ( pred @ Nat ) ) ) % 8.50/4.52 & ~ ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4021_signed__take__bit__code,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( bit_ri3973907225187159222ations @ A ) % 8.50/4.52 => ( ( bit_ri4674362597316999326ke_bit @ A ) % 8.50/4.52 = ( ^ [N: nat,A4: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A4 ) @ N ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A4 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A4 ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4022_wmin__insertI,axiom, % 8.50/4.52 ! [X3: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] : % 8.50/4.52 ( ( member @ ( product_prod @ nat @ nat ) @ X3 @ XS ) % 8.50/4.52 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_leq ) % 8.50/4.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_min_weak ) % 8.50/4.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ ( insert @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4023_wmax__insertI,axiom, % 8.50/4.52 ! [Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X3: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat )] : % 8.50/4.52 ( ( member @ ( product_prod @ nat @ nat ) @ Y @ YS ) % 8.50/4.52 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_leq ) % 8.50/4.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_max_weak ) % 8.50/4.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X3 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4024_bezw__0,axiom, % 8.50/4.52 ! [X3: nat] : % 8.50/4.52 ( ( bezw @ X3 @ ( zero_zero @ nat ) ) % 8.50/4.52 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4025_push__bit__negative__int__iff,axiom, % 8.50/4.52 ! [N2: nat,K: int] : % 8.50/4.52 ( ( ord_less @ int @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) % 8.50/4.52 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4026_push__bit__eq__0__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.52 => ! [N2: nat,A2: A] : % 8.50/4.52 ( ( ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) % 8.50/4.52 = ( zero_zero @ A ) ) % 8.50/4.52 = ( A2 % 8.50/4.52 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4027_push__bit__of__0,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.52 => ! [N2: nat] : % 8.50/4.52 ( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( zero_zero @ A ) ) % 8.50/4.52 = ( zero_zero @ A ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4028_push__bit__push__bit,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.52 => ! [M: nat,N2: nat,A2: A] : % 8.50/4.52 ( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) ) % 8.50/4.52 = ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ A2 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4029_push__bit__Suc__numeral,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.52 => ! [N2: nat,K: num] : % 8.50/4.52 ( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ K ) ) % 8.50/4.52 = ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4030_push__bit__Suc__minus__numeral,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( bit_ri3973907225187159222ations @ A ) % 8.50/4.52 => ! [N2: nat,K: num] : % 8.50/4.52 ( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) % 8.50/4.52 = ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4031_push__bit__Suc,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.52 => ! [N2: nat,A2: A] : % 8.50/4.52 ( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ A2 ) % 8.50/4.52 = ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4032_push__bit__of__Suc__0,axiom, % 8.50/4.52 ! [N2: nat] : % 8.50/4.52 ( ( bit_se4730199178511100633sh_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.52 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4033_even__push__bit__iff,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.52 => ! [N2: nat,A2: A] : % 8.50/4.52 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) ) % 8.50/4.52 = ( ( N2 % 8.50/4.52 != ( zero_zero @ nat ) ) % 8.50/4.52 | ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4034_wmin__emptyI,axiom, % 8.50/4.52 ! [X8: set @ ( product_prod @ nat @ nat )] : ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X8 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4035_wmax__emptyI,axiom, % 8.50/4.52 ! [X8: set @ ( product_prod @ nat @ nat )] : % 8.50/4.52 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X8 ) % 8.50/4.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ X8 ) @ fun_max_weak ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4036_push__bit__add,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.52 => ! [N2: nat,A2: A,B2: A] : % 8.50/4.52 ( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( plus_plus @ A @ A2 @ B2 ) ) % 8.50/4.52 = ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ B2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4037_push__bit__take__bit,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.52 => ! [M: nat,N2: nat,A2: A] : % 8.50/4.52 ( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) % 8.50/4.52 = ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ ( bit_se4730199178511100633sh_bit @ A @ M @ A2 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4038_bit__push__bit__iff__int,axiom, % 8.50/4.52 ! [M: nat,K: int,N2: nat] : % 8.50/4.52 ( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M @ K ) @ N2 ) % 8.50/4.52 = ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.52 & ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4039_bit__push__bit__iff__nat,axiom, % 8.50/4.52 ! [M: nat,Q4: nat,N2: nat] : % 8.50/4.52 ( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M @ Q4 ) @ N2 ) % 8.50/4.52 = ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.52 & ( bit_se5641148757651400278ts_bit @ nat @ Q4 @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4040_bit__iff__and__push__bit__not__eq__0,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.52 => ( ( bit_se5641148757651400278ts_bit @ A ) % 8.50/4.52 = ( ^ [A4: A,N: nat] : % 8.50/4.52 ( ( bit_se5824344872417868541ns_and @ A @ A4 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) ) % 8.50/4.52 != ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4041_take__bit__sum,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.52 => ( ( bit_se2584673776208193580ke_bit @ A ) % 8.50/4.52 = ( ^ [N: nat,A4: A] : % 8.50/4.52 ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.52 @ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A4 @ K3 ) ) ) % 8.50/4.52 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4042_smax__insertI,axiom, % 8.50/4.52 ! [Y: product_prod @ nat @ nat,Y7: set @ ( product_prod @ nat @ nat ),X3: product_prod @ nat @ nat,X8: set @ ( product_prod @ nat @ nat )] : % 8.50/4.52 ( ( member @ ( product_prod @ nat @ nat ) @ Y @ Y7 ) % 8.50/4.52 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_less ) % 8.50/4.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X8 @ Y7 ) @ fun_max_strict ) % 8.50/4.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X3 @ X8 ) @ Y7 ) @ fun_max_strict ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4043_smin__insertI,axiom, % 8.50/4.52 ! [X3: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] : % 8.50/4.52 ( ( member @ ( product_prod @ nat @ nat ) @ X3 @ XS ) % 8.50/4.52 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_less ) % 8.50/4.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_min_strict ) % 8.50/4.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ ( insert @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4044_prod__decode__aux_Osimps,axiom, % 8.50/4.52 ( nat_prod_decode_aux % 8.50/4.52 = ( ^ [K3: nat,M2: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M2 @ K3 ) @ ( product_Pair @ nat @ nat @ M2 @ ( minus_minus @ nat @ K3 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus @ nat @ M2 @ ( suc @ K3 ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4045_prod__decode__aux_Oelims,axiom, % 8.50/4.52 ! [X3: nat,Xa2: nat,Y: product_prod @ nat @ nat] : % 8.50/4.52 ( ( ( nat_prod_decode_aux @ X3 @ Xa2 ) % 8.50/4.52 = Y ) % 8.50/4.52 => ( ( ( ord_less_eq @ nat @ Xa2 @ X3 ) % 8.50/4.52 => ( Y % 8.50/4.52 = ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X3 @ Xa2 ) ) ) ) % 8.50/4.52 & ( ~ ( ord_less_eq @ nat @ Xa2 @ X3 ) % 8.50/4.52 => ( Y % 8.50/4.52 = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4046_smax__emptyI,axiom, % 8.50/4.52 ! [Y7: set @ ( product_prod @ nat @ nat )] : % 8.50/4.52 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y7 ) % 8.50/4.52 => ( ( Y7 % 8.50/4.52 != ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) % 8.50/4.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ Y7 ) @ fun_max_strict ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4047_smin__emptyI,axiom, % 8.50/4.52 ! [X8: set @ ( product_prod @ nat @ nat )] : % 8.50/4.52 ( ( X8 % 8.50/4.52 != ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) % 8.50/4.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X8 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4048_Suc__0__div__numeral,axiom, % 8.50/4.52 ! [K: num] : % 8.50/4.52 ( ( divide_divide @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) ) % 8.50/4.52 = ( product_fst @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4049_Suc__0__mod__numeral,axiom, % 8.50/4.52 ! [K: num] : % 8.50/4.52 ( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) ) % 8.50/4.52 = ( product_snd @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4050_finite__enumerate,axiom, % 8.50/4.52 ! [S3: set @ nat] : % 8.50/4.52 ( ( finite_finite2 @ nat @ S3 ) % 8.50/4.52 => ? [R4: nat > nat] : % 8.50/4.52 ( ( strict_mono_on @ nat @ nat @ R4 @ ( set_ord_lessThan @ nat @ ( finite_card @ nat @ S3 ) ) ) % 8.50/4.52 & ! [N6: nat] : % 8.50/4.52 ( ( ord_less @ nat @ N6 @ ( finite_card @ nat @ S3 ) ) % 8.50/4.52 => ( member @ nat @ ( R4 @ N6 ) @ S3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4051_card__UNION,axiom, % 8.50/4.52 ! [A: $tType,A3: set @ ( set @ A )] : % 8.50/4.52 ( ( finite_finite2 @ ( set @ A ) @ A3 ) % 8.50/4.52 => ( ! [X5: set @ A] : % 8.50/4.52 ( ( member @ ( set @ A ) @ X5 @ A3 ) % 8.50/4.52 => ( finite_finite2 @ A @ X5 ) ) % 8.50/4.52 => ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) % 8.50/4.52 = ( nat2 % 8.50/4.52 @ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int % 8.50/4.52 @ ^ [I8: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I8 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I8 ) ) ) ) % 8.50/4.52 @ ( collect @ ( set @ ( set @ A ) ) % 8.50/4.52 @ ^ [I8: set @ ( set @ A )] : % 8.50/4.52 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I8 @ A3 ) % 8.50/4.52 & ( I8 % 8.50/4.52 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4052_finite__Inter,axiom, % 8.50/4.52 ! [A: $tType,M7: set @ ( set @ A )] : % 8.50/4.52 ( ? [X2: set @ A] : % 8.50/4.52 ( ( member @ ( set @ A ) @ X2 @ M7 ) % 8.50/4.52 & ( finite_finite2 @ A @ X2 ) ) % 8.50/4.52 => ( finite_finite2 @ A @ ( complete_Inf_Inf @ ( set @ A ) @ M7 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4053_fst__apsnd,axiom, % 8.50/4.52 ! [B: $tType,C: $tType,A: $tType,F2: C > B,X3: product_prod @ A @ C] : % 8.50/4.52 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F2 @ X3 ) ) % 8.50/4.52 = ( product_fst @ A @ C @ X3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4054_apsnd__eq__conv,axiom, % 8.50/4.52 ! [B: $tType,C: $tType,A: $tType,F2: C > B,X3: product_prod @ A @ C,G: C > B] : % 8.50/4.52 ( ( ( product_apsnd @ C @ B @ A @ F2 @ X3 ) % 8.50/4.52 = ( product_apsnd @ C @ B @ A @ G @ X3 ) ) % 8.50/4.52 = ( ( F2 @ ( product_snd @ A @ C @ X3 ) ) % 8.50/4.52 = ( G @ ( product_snd @ A @ C @ X3 ) ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4055_snd__apsnd,axiom, % 8.50/4.52 ! [A: $tType,C: $tType,B: $tType,F2: C > A,X3: product_prod @ B @ C] : % 8.50/4.52 ( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F2 @ X3 ) ) % 8.50/4.52 = ( F2 @ ( product_snd @ B @ C @ X3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4056_cSup__atLeastAtMost,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( condit1219197933456340205attice @ A ) % 8.50/4.52 => ! [Y: A,X3: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ Y @ X3 ) % 8.50/4.52 => ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X3 ) ) % 8.50/4.52 = X3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4057_Sup__atLeastAtMost,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comple6319245703460814977attice @ A ) % 8.50/4.52 => ! [X3: A,Y: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.52 => ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 8.50/4.52 = Y ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4058_cSup__singleton,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( condit1219197933456340205attice @ A ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( complete_Sup_Sup @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.52 = X3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4059_cInf__atLeastAtMost,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( condit1219197933456340205attice @ A ) % 8.50/4.52 => ! [Y: A,X3: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ Y @ X3 ) % 8.50/4.52 => ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X3 ) ) % 8.50/4.52 = Y ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4060_Inf__atLeastAtMost,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( comple6319245703460814977attice @ A ) % 8.50/4.52 => ! [X3: A,Y: A] : % 8.50/4.52 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.52 => ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 8.50/4.52 = X3 ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4061_cInf__singleton,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( condit1219197933456340205attice @ A ) % 8.50/4.52 => ! [X3: A] : % 8.50/4.52 ( ( complete_Inf_Inf @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.52 = X3 ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4062_Sup__atLeastLessThan,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( comple6319245703460814977attice @ A ) % 8.50/4.52 & ( dense_linorder @ A ) ) % 8.50/4.52 => ! [X3: A,Y: A] : % 8.50/4.52 ( ( ord_less @ A @ X3 @ Y ) % 8.50/4.52 => ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X3 @ Y ) ) % 8.50/4.52 = Y ) ) ) ). % 8.50/4.52 % 8.50/4.52 thf(fact_4063_cSup__atLeastLessThan,axiom, % 8.50/4.52 ! [A: $tType] : % 8.50/4.52 ( ( ( condit6923001295902523014norder @ A ) % 8.50/4.52 & ( dense_linorder @ A ) ) % 8.50/4.52 => ! [Y: A,X3: A] : % 8.50/4.53 ( ( ord_less @ A @ Y @ X3 ) % 8.50/4.53 => ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X3 ) ) % 8.50/4.53 = X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4064_Inf__atLeastLessThan,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : % 8.50/4.53 ( ( ord_less @ A @ X3 @ Y ) % 8.50/4.53 => ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X3 @ Y ) ) % 8.50/4.53 = X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4065_cInf__atLeastLessThan,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit1219197933456340205attice @ A ) % 8.50/4.53 => ! [Y: A,X3: A] : % 8.50/4.53 ( ( ord_less @ A @ Y @ X3 ) % 8.50/4.53 => ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X3 ) ) % 8.50/4.53 = Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4066_Inf__atMost,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [X3: A] : % 8.50/4.53 ( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X3 ) ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4067_prod_Ocollapse,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] : % 8.50/4.53 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) % 8.50/4.53 = Prod ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4068_finite__Union,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ ( set @ A )] : % 8.50/4.53 ( ( finite_finite2 @ ( set @ A ) @ A3 ) % 8.50/4.53 => ( ! [M8: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ M8 @ A3 ) % 8.50/4.53 => ( finite_finite2 @ A @ M8 ) ) % 8.50/4.53 => ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4069_split__comp__eq,axiom, % 8.50/4.53 ! [A: $tType,C: $tType,B: $tType,D: $tType,F2: A > B > C,G: D > A] : % 8.50/4.53 ( ( ^ [U2: product_prod @ D @ B] : ( F2 @ ( G @ ( product_fst @ D @ B @ U2 ) ) @ ( product_snd @ D @ B @ U2 ) ) ) % 8.50/4.53 = ( product_case_prod @ D @ B @ C % 8.50/4.53 @ ^ [X: D] : ( F2 @ ( G @ X ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4070_case__prod__beta_H,axiom, % 8.50/4.53 ! [C: $tType,B: $tType,A: $tType] : % 8.50/4.53 ( ( product_case_prod @ A @ B @ C ) % 8.50/4.53 = ( ^ [F3: A > B > C,X: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4071_case__prod__unfold,axiom, % 8.50/4.53 ! [C: $tType,B: $tType,A: $tType] : % 8.50/4.53 ( ( product_case_prod @ A @ B @ C ) % 8.50/4.53 = ( ^ [C4: A > B > C,P6: product_prod @ A @ B] : ( C4 @ ( product_fst @ A @ B @ P6 ) @ ( product_snd @ A @ B @ P6 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4072_The__case__prod,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,P: A > B > $o] : % 8.50/4.53 ( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) % 8.50/4.53 = ( the @ ( product_prod @ A @ B ) % 8.50/4.53 @ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4073_cSup__eq__maximum,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit1219197933456340205attice @ A ) % 8.50/4.53 => ! [Z2: A,X8: set @ A] : % 8.50/4.53 ( ( member @ A @ Z2 @ X8 ) % 8.50/4.53 => ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ X8 ) % 8.50/4.53 => ( ord_less_eq @ A @ X5 @ Z2 ) ) % 8.50/4.53 => ( ( complete_Sup_Sup @ A @ X8 ) % 8.50/4.53 = Z2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4074_cSup__eq,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ( condit1219197933456340205attice @ A ) % 8.50/4.53 & ( no_bot @ A ) ) % 8.50/4.53 => ! [X8: set @ A,A2: A] : % 8.50/4.53 ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ X8 ) % 8.50/4.53 => ( ord_less_eq @ A @ X5 @ A2 ) ) % 8.50/4.53 => ( ! [Y4: A] : % 8.50/4.53 ( ! [X2: A] : % 8.50/4.53 ( ( member @ A @ X2 @ X8 ) % 8.50/4.53 => ( ord_less_eq @ A @ X2 @ Y4 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ A2 @ Y4 ) ) % 8.50/4.53 => ( ( complete_Sup_Sup @ A @ X8 ) % 8.50/4.53 = A2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4075_cInf__eq__minimum,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit1219197933456340205attice @ A ) % 8.50/4.53 => ! [Z2: A,X8: set @ A] : % 8.50/4.53 ( ( member @ A @ Z2 @ X8 ) % 8.50/4.53 => ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ X8 ) % 8.50/4.53 => ( ord_less_eq @ A @ Z2 @ X5 ) ) % 8.50/4.53 => ( ( complete_Inf_Inf @ A @ X8 ) % 8.50/4.53 = Z2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4076_cInf__eq,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ( condit1219197933456340205attice @ A ) % 8.50/4.53 & ( no_top @ A ) ) % 8.50/4.53 => ! [X8: set @ A,A2: A] : % 8.50/4.53 ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ X8 ) % 8.50/4.53 => ( ord_less_eq @ A @ A2 @ X5 ) ) % 8.50/4.53 => ( ! [Y4: A] : % 8.50/4.53 ( ! [X2: A] : % 8.50/4.53 ( ( member @ A @ X2 @ X8 ) % 8.50/4.53 => ( ord_less_eq @ A @ Y4 @ X2 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ Y4 @ A2 ) ) % 8.50/4.53 => ( ( complete_Inf_Inf @ A @ X8 ) % 8.50/4.53 = A2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4077_snd__def,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( product_snd @ A @ B ) % 8.50/4.53 = ( product_case_prod @ A @ B @ B % 8.50/4.53 @ ^ [X15: A,X24: B] : X24 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4078_fst__def,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( product_fst @ A @ B ) % 8.50/4.53 = ( product_case_prod @ A @ B @ A % 8.50/4.53 @ ^ [X15: A,X24: B] : X15 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4079_prod__eq__iff,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( ^ [Y3: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y3 = Z ) ) % 8.50/4.53 = ( ^ [S8: product_prod @ A @ B,T3: product_prod @ A @ B] : % 8.50/4.53 ( ( ( product_fst @ A @ B @ S8 ) % 8.50/4.53 = ( product_fst @ A @ B @ T3 ) ) % 8.50/4.53 & ( ( product_snd @ A @ B @ S8 ) % 8.50/4.53 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4080_prod__eqI,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,P4: product_prod @ A @ B,Q4: product_prod @ A @ B] : % 8.50/4.53 ( ( ( product_fst @ A @ B @ P4 ) % 8.50/4.53 = ( product_fst @ A @ B @ Q4 ) ) % 8.50/4.53 => ( ( ( product_snd @ A @ B @ P4 ) % 8.50/4.53 = ( product_snd @ A @ B @ Q4 ) ) % 8.50/4.53 => ( P4 = Q4 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4081_prod_Oexpand,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] : % 8.50/4.53 ( ( ( ( product_fst @ A @ B @ Prod ) % 8.50/4.53 = ( product_fst @ A @ B @ Prod2 ) ) % 8.50/4.53 & ( ( product_snd @ A @ B @ Prod ) % 8.50/4.53 = ( product_snd @ A @ B @ Prod2 ) ) ) % 8.50/4.53 => ( Prod = Prod2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4082_split__beta,axiom, % 8.50/4.53 ! [C: $tType,B: $tType,A: $tType] : % 8.50/4.53 ( ( product_case_prod @ A @ B @ C ) % 8.50/4.53 = ( ^ [F3: A > B > C,Prod3: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4083_case__prod__beta,axiom, % 8.50/4.53 ! [A: $tType,C: $tType,B: $tType] : % 8.50/4.53 ( ( product_case_prod @ B @ C @ A ) % 8.50/4.53 = ( ^ [F3: B > C > A,P6: product_prod @ B @ C] : ( F3 @ ( product_fst @ B @ C @ P6 ) @ ( product_snd @ B @ C @ P6 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4084_Product__Type_OCollect__case__prodD,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,X3: product_prod @ A @ B,A3: A > B > $o] : % 8.50/4.53 ( ( member @ ( product_prod @ A @ B ) @ X3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) ) % 8.50/4.53 => ( A3 @ ( product_fst @ A @ B @ X3 ) @ ( product_snd @ A @ B @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4085_prod_Osplit__sel__asm,axiom, % 8.50/4.53 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F2: A > B > C,Prod: product_prod @ A @ B] : % 8.50/4.53 ( ( P @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) ) % 8.50/4.53 = ( ~ ( ( Prod % 8.50/4.53 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) % 8.50/4.53 & ~ ( P @ ( F2 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4086_prod_Osplit__sel,axiom, % 8.50/4.53 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F2: A > B > C,Prod: product_prod @ A @ B] : % 8.50/4.53 ( ( P @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) ) % 8.50/4.53 = ( ( Prod % 8.50/4.53 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) % 8.50/4.53 => ( P @ ( F2 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4087_fst__conv,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,X1: A,X22: B] : % 8.50/4.53 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) ) % 8.50/4.53 = X1 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4088_fst__eqD,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,X3: A,Y: B,A2: A] : % 8.50/4.53 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y ) ) % 8.50/4.53 = A2 ) % 8.50/4.53 => ( X3 = A2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4089_prod_Oexhaust__sel,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] : % 8.50/4.53 ( Prod % 8.50/4.53 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4090_surjective__pairing,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] : % 8.50/4.53 ( T2 % 8.50/4.53 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4091_snd__conv,axiom, % 8.50/4.53 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] : % 8.50/4.53 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) ) % 8.50/4.53 = X22 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4092_snd__eqD,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,X3: B,Y: A,A2: A] : % 8.50/4.53 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) ) % 8.50/4.53 = A2 ) % 8.50/4.53 => ( Y = A2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4093_BNF__Greatest__Fixpoint_Osubst__Pair,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,P: A > B > $o,X3: A,Y: B,A2: product_prod @ A @ B] : % 8.50/4.53 ( ( P @ X3 @ Y ) % 8.50/4.53 => ( ( A2 % 8.50/4.53 = ( product_Pair @ A @ B @ X3 @ Y ) ) % 8.50/4.53 => ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4094_cInf__eq__non__empty,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit1219197933456340205attice @ A ) % 8.50/4.53 => ! [X8: set @ A,A2: A] : % 8.50/4.53 ( ( X8 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ X8 ) % 8.50/4.53 => ( ord_less_eq @ A @ A2 @ X5 ) ) % 8.50/4.53 => ( ! [Y4: A] : % 8.50/4.53 ( ! [X2: A] : % 8.50/4.53 ( ( member @ A @ X2 @ X8 ) % 8.50/4.53 => ( ord_less_eq @ A @ Y4 @ X2 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ Y4 @ A2 ) ) % 8.50/4.53 => ( ( complete_Inf_Inf @ A @ X8 ) % 8.50/4.53 = A2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4095_cInf__greatest,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit1219197933456340205attice @ A ) % 8.50/4.53 => ! [X8: set @ A,Z2: A] : % 8.50/4.53 ( ( X8 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ X8 ) % 8.50/4.53 => ( ord_less_eq @ A @ Z2 @ X5 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4096_cInf__le__finite,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit1219197933456340205attice @ A ) % 8.50/4.53 => ! [X8: set @ A,X3: A] : % 8.50/4.53 ( ( finite_finite2 @ A @ X8 ) % 8.50/4.53 => ( ( member @ A @ X3 @ X8 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4097_cInf__lessD,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit6923001295902523014norder @ A ) % 8.50/4.53 => ! [X8: set @ A,Z2: A] : % 8.50/4.53 ( ( X8 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Z2 ) % 8.50/4.53 => ? [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ X8 ) % 8.50/4.53 & ( ord_less @ A @ X5 @ Z2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4098_finite__imp__less__Inf,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit6923001295902523014norder @ A ) % 8.50/4.53 => ! [X8: set @ A,X3: A,A2: A] : % 8.50/4.53 ( ( finite_finite2 @ A @ X8 ) % 8.50/4.53 => ( ( member @ A @ X3 @ X8 ) % 8.50/4.53 => ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ X8 ) % 8.50/4.53 => ( ord_less @ A @ A2 @ X5 ) ) % 8.50/4.53 => ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4099_cSup__eq__non__empty,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit1219197933456340205attice @ A ) % 8.50/4.53 => ! [X8: set @ A,A2: A] : % 8.50/4.53 ( ( X8 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ X8 ) % 8.50/4.53 => ( ord_less_eq @ A @ X5 @ A2 ) ) % 8.50/4.53 => ( ! [Y4: A] : % 8.50/4.53 ( ! [X2: A] : % 8.50/4.53 ( ( member @ A @ X2 @ X8 ) % 8.50/4.53 => ( ord_less_eq @ A @ X2 @ Y4 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ A2 @ Y4 ) ) % 8.50/4.53 => ( ( complete_Sup_Sup @ A @ X8 ) % 8.50/4.53 = A2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4100_cSup__least,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit1219197933456340205attice @ A ) % 8.50/4.53 => ! [X8: set @ A,Z2: A] : % 8.50/4.53 ( ( X8 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ X8 ) % 8.50/4.53 => ( ord_less_eq @ A @ X5 @ Z2 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X8 ) @ Z2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4101_le__cSup__finite,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit1219197933456340205attice @ A ) % 8.50/4.53 => ! [X8: set @ A,X3: A] : % 8.50/4.53 ( ( finite_finite2 @ A @ X8 ) % 8.50/4.53 => ( ( member @ A @ X3 @ X8 ) % 8.50/4.53 => ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4102_update__zip,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,I: nat,Xy2: product_prod @ A @ B] : % 8.50/4.53 ( ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I @ Xy2 ) % 8.50/4.53 = ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ ( product_fst @ A @ B @ Xy2 ) ) @ ( list_update @ B @ Ys @ I @ ( product_snd @ A @ B @ Xy2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4103_less__cSupE,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit6923001295902523014norder @ A ) % 8.50/4.53 => ! [Y: A,X8: set @ A] : % 8.50/4.53 ( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X8 ) ) % 8.50/4.53 => ( ( X8 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ~ ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ X8 ) % 8.50/4.53 => ~ ( ord_less @ A @ Y @ X5 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4104_less__cSupD,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit6923001295902523014norder @ A ) % 8.50/4.53 => ! [X8: set @ A,Z2: A] : % 8.50/4.53 ( ( X8 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ( ord_less @ A @ Z2 @ ( complete_Sup_Sup @ A @ X8 ) ) % 8.50/4.53 => ? [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ X8 ) % 8.50/4.53 & ( ord_less @ A @ Z2 @ X5 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4105_finite__imp__Sup__less,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit6923001295902523014norder @ A ) % 8.50/4.53 => ! [X8: set @ A,X3: A,A2: A] : % 8.50/4.53 ( ( finite_finite2 @ A @ X8 ) % 8.50/4.53 => ( ( member @ A @ X3 @ X8 ) % 8.50/4.53 => ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ X8 ) % 8.50/4.53 => ( ord_less @ A @ X5 @ A2 ) ) % 8.50/4.53 => ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4106_card__Union__le__sum__card,axiom, % 8.50/4.53 ! [A: $tType,U3: set @ ( set @ A )] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U3 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4107_finite__UnionD,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ ( set @ A )] : % 8.50/4.53 ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) % 8.50/4.53 => ( finite_finite2 @ ( set @ A ) @ A3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4108_in__set__zip,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,P4: product_prod @ A @ B,Xs: list @ A,Ys: list @ B] : % 8.50/4.53 ( ( member @ ( product_prod @ A @ B ) @ P4 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) % 8.50/4.53 = ( ? [N: nat] : % 8.50/4.53 ( ( ( nth @ A @ Xs @ N ) % 8.50/4.53 = ( product_fst @ A @ B @ P4 ) ) % 8.50/4.53 & ( ( nth @ B @ Ys @ N ) % 8.50/4.53 = ( product_snd @ A @ B @ P4 ) ) % 8.50/4.53 & ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.53 & ( ord_less @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4109_divides__aux__def,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( unique1627219031080169319umeral @ A ) % 8.50/4.53 => ( ( unique5940410009612947441es_aux @ A ) % 8.50/4.53 = ( ^ [Qr: product_prod @ A @ A] : % 8.50/4.53 ( ( product_snd @ A @ A @ Qr ) % 8.50/4.53 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4110_finite__less__Inf__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit6923001295902523014norder @ A ) % 8.50/4.53 => ! [X8: set @ A,A2: A] : % 8.50/4.53 ( ( finite_finite2 @ A @ X8 ) % 8.50/4.53 => ( ( X8 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X8 ) ) % 8.50/4.53 = ( ! [X: A] : % 8.50/4.53 ( ( member @ A @ X @ X8 ) % 8.50/4.53 => ( ord_less @ A @ A2 @ X ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4111_finite__Sup__less__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit6923001295902523014norder @ A ) % 8.50/4.53 => ! [X8: set @ A,A2: A] : % 8.50/4.53 ( ( finite_finite2 @ A @ X8 ) % 8.50/4.53 => ( ( X8 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A2 ) % 8.50/4.53 = ( ! [X: A] : % 8.50/4.53 ( ( member @ A @ X @ X8 ) % 8.50/4.53 => ( ord_less @ A @ X @ A2 ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4112_cInf__abs__ge,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ( condit6923001295902523014norder @ A ) % 8.50/4.53 & ( linordered_idom @ A ) ) % 8.50/4.53 => ! [S3: set @ A,A2: A] : % 8.50/4.53 ( ( S3 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ S3 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( abs_abs @ A @ X5 ) @ A2 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S3 ) ) @ A2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4113_cSup__abs__le,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ( condit6923001295902523014norder @ A ) % 8.50/4.53 & ( linordered_idom @ A ) ) % 8.50/4.53 => ! [S3: set @ A,A2: A] : % 8.50/4.53 ( ( S3 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ S3 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( abs_abs @ A @ X5 ) @ A2 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S3 ) ) @ A2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4114_card__Union__le__sum__card__weak,axiom, % 8.50/4.53 ! [A: $tType,U3: set @ ( set @ A )] : % 8.50/4.53 ( ! [X5: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ X5 @ U3 ) % 8.50/4.53 => ( finite_finite2 @ A @ X5 ) ) % 8.50/4.53 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U3 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4115_in__set__enumerate__eq,axiom, % 8.50/4.53 ! [A: $tType,P4: product_prod @ nat @ A,N2: nat,Xs: list @ A] : % 8.50/4.53 ( ( member @ ( product_prod @ nat @ A ) @ P4 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) ) ) % 8.50/4.53 = ( ( ord_less_eq @ nat @ N2 @ ( product_fst @ nat @ A @ P4 ) ) % 8.50/4.53 & ( ord_less @ nat @ ( product_fst @ nat @ A @ P4 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) % 8.50/4.53 & ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P4 ) @ N2 ) ) % 8.50/4.53 = ( product_snd @ nat @ A @ P4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4116_cInf__asclose,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ( condit6923001295902523014norder @ A ) % 8.50/4.53 & ( linordered_idom @ A ) ) % 8.50/4.53 => ! [S3: set @ A,L: A,E3: A] : % 8.50/4.53 ( ( S3 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ S3 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X5 @ L ) ) @ E3 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S3 ) @ L ) ) @ E3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4117_cSup__asclose,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ( condit6923001295902523014norder @ A ) % 8.50/4.53 & ( linordered_idom @ A ) ) % 8.50/4.53 => ! [S3: set @ A,L: A,E3: A] : % 8.50/4.53 ( ( S3 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ S3 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X5 @ L ) ) @ E3 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S3 ) @ L ) ) @ E3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4118_finite__subset__Union,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B12: set @ ( set @ A )] : % 8.50/4.53 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.53 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ B12 ) ) % 8.50/4.53 => ~ ! [F7: set @ ( set @ A )] : % 8.50/4.53 ( ( finite_finite2 @ ( set @ A ) @ F7 ) % 8.50/4.53 => ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F7 @ B12 ) % 8.50/4.53 => ~ ( ord_less_eq @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ F7 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4119_ccpo__Sup__singleton,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple9053668089753744459l_ccpo @ A ) % 8.50/4.53 => ! [X3: A] : % 8.50/4.53 ( ( complete_Sup_Sup @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.53 = X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4120_Sup__empty,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4121_ccSup__empty,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( counta3822494911875563373attice @ A ) % 8.50/4.53 => ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4122_Inf__eq__bot__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple5582772986160207858norder @ A ) % 8.50/4.53 => ! [A3: set @ A] : % 8.50/4.53 ( ( ( complete_Inf_Inf @ A @ A3 ) % 8.50/4.53 = ( bot_bot @ A ) ) % 8.50/4.53 = ( ! [X: A] : % 8.50/4.53 ( ( ord_less @ A @ ( bot_bot @ A ) @ X ) % 8.50/4.53 => ? [Y2: A] : % 8.50/4.53 ( ( member @ A @ Y2 @ A3 ) % 8.50/4.53 & ( ord_less @ A @ Y2 @ X ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4123_Sup__bot__conv_I1_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [A3: set @ A] : % 8.50/4.53 ( ( ( complete_Sup_Sup @ A @ A3 ) % 8.50/4.53 = ( bot_bot @ A ) ) % 8.50/4.53 = ( ! [X: A] : % 8.50/4.53 ( ( member @ A @ X @ A3 ) % 8.50/4.53 => ( X % 8.50/4.53 = ( bot_bot @ A ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4124_Sup__bot__conv_I2_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [A3: set @ A] : % 8.50/4.53 ( ( ( bot_bot @ A ) % 8.50/4.53 = ( complete_Sup_Sup @ A @ A3 ) ) % 8.50/4.53 = ( ! [X: A] : % 8.50/4.53 ( ( member @ A @ X @ A3 ) % 8.50/4.53 => ( X % 8.50/4.53 = ( bot_bot @ A ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4125_Sup__nat__empty,axiom, % 8.50/4.53 ( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) ) % 8.50/4.53 = ( zero_zero @ nat ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4126_Inf__nat__def1,axiom, % 8.50/4.53 ! [K5: set @ nat] : % 8.50/4.53 ( ( K5 % 8.50/4.53 != ( bot_bot @ ( set @ nat ) ) ) % 8.50/4.53 => ( member @ nat @ ( complete_Inf_Inf @ nat @ K5 ) @ K5 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4127_quotient__of__denom__pos_H,axiom, % 8.50/4.53 ! [R2: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ ( quotient_of @ R2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4128_bezw__non__0,axiom, % 8.50/4.53 ! [Y: nat,X3: nat] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y ) % 8.50/4.53 => ( ( bezw @ X3 @ Y ) % 8.50/4.53 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Y ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4129_bezw_Oelims,axiom, % 8.50/4.53 ! [X3: nat,Xa2: nat,Y: product_prod @ int @ int] : % 8.50/4.53 ( ( ( bezw @ X3 @ Xa2 ) % 8.50/4.53 = Y ) % 8.50/4.53 => ( ( ( Xa2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 => ( Y % 8.50/4.53 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ) % 8.50/4.53 & ( ( Xa2 % 8.50/4.53 != ( zero_zero @ nat ) ) % 8.50/4.53 => ( Y % 8.50/4.53 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Xa2 ) ) ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4130_bezw_Osimps,axiom, % 8.50/4.53 ( bezw % 8.50/4.53 = ( ^ [X: nat,Y2: nat] : % 8.50/4.53 ( if @ ( product_prod @ int @ int ) % 8.50/4.53 @ ( Y2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) % 8.50/4.53 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y2 ) ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4131_rat__sgn__code,axiom, % 8.50/4.53 ! [P4: rat] : % 8.50/4.53 ( ( quotient_of @ ( sgn_sgn @ rat @ P4 ) ) % 8.50/4.53 = ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P4 ) ) ) @ ( one_one @ int ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4132_Sup__upper2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [U: A,A3: set @ A,V2: A] : % 8.50/4.53 ( ( member @ A @ U @ A3 ) % 8.50/4.53 => ( ( ord_less_eq @ A @ V2 @ U ) % 8.50/4.53 => ( ord_less_eq @ A @ V2 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4133_Sup__le__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [A3: set @ A,B2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ B2 ) % 8.50/4.53 = ( ! [X: A] : % 8.50/4.53 ( ( member @ A @ X @ A3 ) % 8.50/4.53 => ( ord_less_eq @ A @ X @ B2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4134_Sup__upper,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [X3: A,A3: set @ A] : % 8.50/4.53 ( ( member @ A @ X3 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4135_Sup__least,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [A3: set @ A,Z2: A] : % 8.50/4.53 ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ A @ X5 @ Z2 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ Z2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4136_Sup__mono,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ! [A5: A] : % 8.50/4.53 ( ( member @ A @ A5 @ A3 ) % 8.50/4.53 => ? [X2: A] : % 8.50/4.53 ( ( member @ A @ X2 @ B3 ) % 8.50/4.53 & ( ord_less_eq @ A @ A5 @ X2 ) ) ) % 8.50/4.53 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4137_Sup__eqI,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [A3: set @ A,X3: A] : % 8.50/4.53 ( ! [Y4: A] : % 8.50/4.53 ( ( member @ A @ Y4 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ A @ Y4 @ X3 ) ) % 8.50/4.53 => ( ! [Y4: A] : % 8.50/4.53 ( ! [Z4: A] : % 8.50/4.53 ( ( member @ A @ Z4 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ A @ Z4 @ Y4 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ X3 @ Y4 ) ) % 8.50/4.53 => ( ( complete_Sup_Sup @ A @ A3 ) % 8.50/4.53 = X3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4138_less__Sup__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple5582772986160207858norder @ A ) % 8.50/4.53 => ! [A2: A,S3: set @ A] : % 8.50/4.53 ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ S3 ) ) % 8.50/4.53 = ( ? [X: A] : % 8.50/4.53 ( ( member @ A @ X @ S3 ) % 8.50/4.53 & ( ord_less @ A @ A2 @ X ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4139_Inf__greatest,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [A3: set @ A,Z2: A] : % 8.50/4.53 ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ A @ Z2 @ X5 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4140_le__Inf__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [B2: A,A3: set @ A] : % 8.50/4.53 ( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A3 ) ) % 8.50/4.53 = ( ! [X: A] : % 8.50/4.53 ( ( member @ A @ X @ A3 ) % 8.50/4.53 => ( ord_less_eq @ A @ B2 @ X ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4141_Inf__lower2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [U: A,A3: set @ A,V2: A] : % 8.50/4.53 ( ( member @ A @ U @ A3 ) % 8.50/4.53 => ( ( ord_less_eq @ A @ U @ V2 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ V2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4142_Inf__lower,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [X3: A,A3: set @ A] : % 8.50/4.53 ( ( member @ A @ X3 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4143_Inf__mono,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [B3: set @ A,A3: set @ A] : % 8.50/4.53 ( ! [B5: A] : % 8.50/4.53 ( ( member @ A @ B5 @ B3 ) % 8.50/4.53 => ? [X2: A] : % 8.50/4.53 ( ( member @ A @ X2 @ A3 ) % 8.50/4.53 & ( ord_less_eq @ A @ X2 @ B5 ) ) ) % 8.50/4.53 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4144_Inf__eqI,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [A3: set @ A,X3: A] : % 8.50/4.53 ( ! [I4: A] : % 8.50/4.53 ( ( member @ A @ I4 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ A @ X3 @ I4 ) ) % 8.50/4.53 => ( ! [Y4: A] : % 8.50/4.53 ( ! [I5: A] : % 8.50/4.53 ( ( member @ A @ I5 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ A @ Y4 @ I5 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ Y4 @ X3 ) ) % 8.50/4.53 => ( ( complete_Inf_Inf @ A @ A3 ) % 8.50/4.53 = X3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4145_Inf__less__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple5582772986160207858norder @ A ) % 8.50/4.53 => ! [S3: set @ A,A2: A] : % 8.50/4.53 ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S3 ) @ A2 ) % 8.50/4.53 = ( ? [X: A] : % 8.50/4.53 ( ( member @ A @ X @ S3 ) % 8.50/4.53 & ( ord_less @ A @ X @ A2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4146_Union__empty__conv,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ ( set @ A )] : % 8.50/4.53 ( ( ( complete_Sup_Sup @ ( set @ A ) @ A3 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 = ( ! [X: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ X @ A3 ) % 8.50/4.53 => ( X % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4147_empty__Union__conv,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ ( set @ A )] : % 8.50/4.53 ( ( ( bot_bot @ ( set @ A ) ) % 8.50/4.53 = ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) % 8.50/4.53 = ( ! [X: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ X @ A3 ) % 8.50/4.53 => ( X % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4148_Union__least,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ ( set @ A ),C5: set @ A] : % 8.50/4.53 ( ! [X10: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ X10 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ ( set @ A ) @ X10 @ C5 ) ) % 8.50/4.53 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ C5 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4149_Union__upper,axiom, % 8.50/4.53 ! [A: $tType,B3: set @ A,A3: set @ ( set @ A )] : % 8.50/4.53 ( ( member @ ( set @ A ) @ B3 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ ( set @ A ) @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4150_Union__subsetI,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] : % 8.50/4.53 ( ! [X5: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ X5 @ A3 ) % 8.50/4.53 => ? [Y5: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ Y5 @ B3 ) % 8.50/4.53 & ( ord_less_eq @ ( set @ A ) @ X5 @ Y5 ) ) ) % 8.50/4.53 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4151_Inter__lower,axiom, % 8.50/4.53 ! [A: $tType,B3: set @ A,A3: set @ ( set @ A )] : % 8.50/4.53 ( ( member @ ( set @ A ) @ B3 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ B3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4152_Inter__greatest,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ ( set @ A ),C5: set @ A] : % 8.50/4.53 ( ! [X10: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ X10 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ ( set @ A ) @ C5 @ X10 ) ) % 8.50/4.53 => ( ord_less_eq @ ( set @ A ) @ C5 @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4153_Union__SetCompr__eq,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,F2: B > ( set @ A ),P: B > $o] : % 8.50/4.53 ( ( complete_Sup_Sup @ ( set @ A ) % 8.50/4.53 @ ( collect @ ( set @ A ) % 8.50/4.53 @ ^ [Uu3: set @ A] : % 8.50/4.53 ? [X: B] : % 8.50/4.53 ( ( Uu3 % 8.50/4.53 = ( F2 @ X ) ) % 8.50/4.53 & ( P @ X ) ) ) ) % 8.50/4.53 = ( collect @ A % 8.50/4.53 @ ^ [A4: A] : % 8.50/4.53 ? [X: B] : % 8.50/4.53 ( ( P @ X ) % 8.50/4.53 & ( member @ A @ A4 @ ( F2 @ X ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4154_le__Sup__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple5582772986160207858norder @ A ) % 8.50/4.53 => ! [X3: A,A3: set @ A] : % 8.50/4.53 ( ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A3 ) ) % 8.50/4.53 = ( ! [Y2: A] : % 8.50/4.53 ( ( ord_less @ A @ Y2 @ X3 ) % 8.50/4.53 => ? [X: A] : % 8.50/4.53 ( ( member @ A @ X @ A3 ) % 8.50/4.53 & ( ord_less @ A @ Y2 @ X ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4155_Inf__le__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple5582772986160207858norder @ A ) % 8.50/4.53 => ! [A3: set @ A,X3: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ X3 ) % 8.50/4.53 = ( ! [Y2: A] : % 8.50/4.53 ( ( ord_less @ A @ X3 @ Y2 ) % 8.50/4.53 => ? [X: A] : % 8.50/4.53 ( ( member @ A @ X @ A3 ) % 8.50/4.53 & ( ord_less @ A @ X @ Y2 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4156_less__eq__Sup,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [A3: set @ A,U: A] : % 8.50/4.53 ( ! [V3: A] : % 8.50/4.53 ( ( member @ A @ V3 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ A @ U @ V3 ) ) % 8.50/4.53 => ( ( A3 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4157_Sup__subset__mono,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4158_Inf__less__eq,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [A3: set @ A,U: A] : % 8.50/4.53 ( ! [V3: A] : % 8.50/4.53 ( ( member @ A @ V3 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ A @ V3 @ U ) ) % 8.50/4.53 => ( ( A3 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ U ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4159_Inf__superset__mono,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [B3: set @ A,A3: set @ A] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4160_Union__mono,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A3 @ B3 ) % 8.50/4.53 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4161_Inter__anti__mono,axiom, % 8.50/4.53 ! [A: $tType,B3: set @ ( set @ A ),A3: set @ ( set @ A )] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B3 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4162_Union__empty,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4163_Inter__subset,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ ( set @ A ),B3: set @ A] : % 8.50/4.53 ( ! [X10: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ X10 @ A3 ) % 8.50/4.53 => ( ord_less_eq @ ( set @ A ) @ X10 @ B3 ) ) % 8.50/4.53 => ( ( A3 % 8.50/4.53 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 8.50/4.53 => ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ B3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4164_Inf__le__Sup,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [A3: set @ A] : % 8.50/4.53 ( ( A3 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4165_size__prod__simp,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( basic_BNF_size_prod @ A @ B ) % 8.50/4.53 = ( ^ [F3: A > nat,G2: B > nat,P6: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F3 @ ( product_fst @ A @ B @ P6 ) ) @ ( G2 @ ( product_snd @ A @ B @ P6 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4166_conjI__realizer,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,P: A > $o,P4: A,Q: B > $o,Q4: B] : % 8.50/4.53 ( ( P @ P4 ) % 8.50/4.53 => ( ( Q @ Q4 ) % 8.50/4.53 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P4 @ Q4 ) ) ) % 8.50/4.53 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P4 @ Q4 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4167_exI__realizer,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X3: B] : % 8.50/4.53 ( ( P @ Y @ X3 ) % 8.50/4.53 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4168_bezw_Opelims,axiom, % 8.50/4.53 ! [X3: nat,Xa2: nat,Y: product_prod @ int @ int] : % 8.50/4.53 ( ( ( bezw @ X3 @ Xa2 ) % 8.50/4.53 = Y ) % 8.50/4.53 => ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) % 8.50/4.53 => ~ ( ( ( ( Xa2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 => ( Y % 8.50/4.53 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ) % 8.50/4.53 & ( ( Xa2 % 8.50/4.53 != ( zero_zero @ nat ) ) % 8.50/4.53 => ( Y % 8.50/4.53 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Xa2 ) ) ) ) ) ) ) ) % 8.50/4.53 => ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4169_prod__decode__aux_Opelims,axiom, % 8.50/4.53 ! [X3: nat,Xa2: nat,Y: product_prod @ nat @ nat] : % 8.50/4.53 ( ( ( nat_prod_decode_aux @ X3 @ Xa2 ) % 8.50/4.53 = Y ) % 8.50/4.53 => ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) % 8.50/4.53 => ~ ( ( ( ( ord_less_eq @ nat @ Xa2 @ X3 ) % 8.50/4.53 => ( Y % 8.50/4.53 = ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X3 @ Xa2 ) ) ) ) % 8.50/4.53 & ( ~ ( ord_less_eq @ nat @ Xa2 @ X3 ) % 8.50/4.53 => ( Y % 8.50/4.53 = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) % 8.50/4.53 => ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4170_normalize__def,axiom, % 8.50/4.53 ( normalize % 8.50/4.53 = ( ^ [P6: product_prod @ int @ int] : % 8.50/4.53 ( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P6 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) % 8.50/4.53 @ ( if @ ( product_prod @ int @ int ) % 8.50/4.53 @ ( ( product_snd @ int @ int @ P6 ) % 8.50/4.53 = ( zero_zero @ int ) ) % 8.50/4.53 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) % 8.50/4.53 @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4171_strict__mono__on__def,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( ( ord @ A ) % 8.50/4.53 & ( ord @ B ) ) % 8.50/4.53 => ( ( strict_mono_on @ A @ B ) % 8.50/4.53 = ( ^ [F3: A > B,A6: set @ A] : % 8.50/4.53 ! [R5: A,S8: A] : % 8.50/4.53 ( ( ( member @ A @ R5 @ A6 ) % 8.50/4.53 & ( member @ A @ S8 @ A6 ) % 8.50/4.53 & ( ord_less @ A @ R5 @ S8 ) ) % 8.50/4.53 => ( ord_less @ B @ ( F3 @ R5 ) @ ( F3 @ S8 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4172_strict__mono__onI,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( ( ord @ A ) % 8.50/4.53 & ( ord @ B ) ) % 8.50/4.53 => ! [A3: set @ A,F2: A > B] : % 8.50/4.53 ( ! [R4: A,S2: A] : % 8.50/4.53 ( ( member @ A @ R4 @ A3 ) % 8.50/4.53 => ( ( member @ A @ S2 @ A3 ) % 8.50/4.53 => ( ( ord_less @ A @ R4 @ S2 ) % 8.50/4.53 => ( ord_less @ B @ ( F2 @ R4 ) @ ( F2 @ S2 ) ) ) ) ) % 8.50/4.53 => ( strict_mono_on @ A @ B @ F2 @ A3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4173_gcd__left__idem,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( gcd_gcd @ A @ A2 @ ( gcd_gcd @ A @ A2 @ B2 ) ) % 8.50/4.53 = ( gcd_gcd @ A @ A2 @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4174_gcd__right__idem,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( gcd_gcd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ B2 ) % 8.50/4.53 = ( gcd_gcd @ A @ A2 @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4175_gcd__eq__0__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( ( gcd_gcd @ A @ A2 @ B2 ) % 8.50/4.53 = ( zero_zero @ A ) ) % 8.50/4.53 = ( ( A2 % 8.50/4.53 = ( zero_zero @ A ) ) % 8.50/4.53 & ( B2 % 8.50/4.53 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4176_gcd_Obottom__right__bottom,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A] : % 8.50/4.53 ( ( gcd_gcd @ A @ A2 @ ( one_one @ A ) ) % 8.50/4.53 = ( one_one @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4177_gcd_Obottom__left__bottom,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A] : % 8.50/4.53 ( ( gcd_gcd @ A @ ( one_one @ A ) @ A2 ) % 8.50/4.53 = ( one_one @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4178_gcd__add1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [M: A,N2: A] : % 8.50/4.53 ( ( gcd_gcd @ A @ ( plus_plus @ A @ M @ N2 ) @ N2 ) % 8.50/4.53 = ( gcd_gcd @ A @ M @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4179_gcd__add2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [M: A,N2: A] : % 8.50/4.53 ( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ M @ N2 ) ) % 8.50/4.53 = ( gcd_gcd @ A @ M @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4180_gcd__neg2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( gcd_gcd @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) % 8.50/4.53 = ( gcd_gcd @ A @ A2 @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4181_gcd__neg1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( gcd_gcd @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) % 8.50/4.53 = ( gcd_gcd @ A @ A2 @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4182_gcd__exp,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiri6843258321239162965malize @ A ) % 8.50/4.53 => ! [A2: A,N2: nat,B2: A] : % 8.50/4.53 ( ( gcd_gcd @ A @ ( power_power @ A @ A2 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) % 8.50/4.53 = ( power_power @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4183_gcd__dvd1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ A2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4184_gcd__dvd2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ B2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4185_gcd__greatest__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( dvd_dvd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) % 8.50/4.53 = ( ( dvd_dvd @ A @ A2 @ B2 ) % 8.50/4.53 & ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4186_gcd__1__int,axiom, % 8.50/4.53 ! [M: int] : % 8.50/4.53 ( ( gcd_gcd @ int @ M @ ( one_one @ int ) ) % 8.50/4.53 = ( one_one @ int ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4187_gcd__neg2__int,axiom, % 8.50/4.53 ! [X3: int,Y: int] : % 8.50/4.53 ( ( gcd_gcd @ int @ X3 @ ( uminus_uminus @ int @ Y ) ) % 8.50/4.53 = ( gcd_gcd @ int @ X3 @ Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4188_gcd__neg1__int,axiom, % 8.50/4.53 ! [X3: int,Y: int] : % 8.50/4.53 ( ( gcd_gcd @ int @ ( uminus_uminus @ int @ X3 ) @ Y ) % 8.50/4.53 = ( gcd_gcd @ int @ X3 @ Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4189_abs__gcd__int,axiom, % 8.50/4.53 ! [X3: int,Y: int] : % 8.50/4.53 ( ( abs_abs @ int @ ( gcd_gcd @ int @ X3 @ Y ) ) % 8.50/4.53 = ( gcd_gcd @ int @ X3 @ Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4190_gcd__abs1__int,axiom, % 8.50/4.53 ! [X3: int,Y: int] : % 8.50/4.53 ( ( gcd_gcd @ int @ ( abs_abs @ int @ X3 ) @ Y ) % 8.50/4.53 = ( gcd_gcd @ int @ X3 @ Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4191_gcd__abs2__int,axiom, % 8.50/4.53 ! [X3: int,Y: int] : % 8.50/4.53 ( ( gcd_gcd @ int @ X3 @ ( abs_abs @ int @ Y ) ) % 8.50/4.53 = ( gcd_gcd @ int @ X3 @ Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4192_gcd__neg__numeral__1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ring_gcd @ A ) % 8.50/4.53 => ! [N2: num,A2: A] : % 8.50/4.53 ( ( gcd_gcd @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ A2 ) % 8.50/4.53 = ( gcd_gcd @ A @ ( numeral_numeral @ A @ N2 ) @ A2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4193_gcd__neg__numeral__2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ring_gcd @ A ) % 8.50/4.53 => ! [A2: A,N2: num] : % 8.50/4.53 ( ( gcd_gcd @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) % 8.50/4.53 = ( gcd_gcd @ A @ A2 @ ( numeral_numeral @ A @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4194_is__unit__gcd__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( one_one @ A ) ) % 8.50/4.53 = ( ( gcd_gcd @ A @ A2 @ B2 ) % 8.50/4.53 = ( one_one @ A ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4195_gcd__pos__int,axiom, % 8.50/4.53 ! [M: int,N2: int] : % 8.50/4.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ M @ N2 ) ) % 8.50/4.53 = ( ( M % 8.50/4.53 != ( zero_zero @ int ) ) % 8.50/4.53 | ( N2 % 8.50/4.53 != ( zero_zero @ int ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4196_gcd__neg__numeral__1__int,axiom, % 8.50/4.53 ! [N2: num,X3: int] : % 8.50/4.53 ( ( gcd_gcd @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) @ X3 ) % 8.50/4.53 = ( gcd_gcd @ int @ ( numeral_numeral @ int @ N2 ) @ X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4197_gcd__neg__numeral__2__int,axiom, % 8.50/4.53 ! [X3: int,N2: num] : % 8.50/4.53 ( ( gcd_gcd @ int @ X3 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) % 8.50/4.53 = ( gcd_gcd @ int @ X3 @ ( numeral_numeral @ int @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4198_gcd__0__int,axiom, % 8.50/4.53 ! [X3: int] : % 8.50/4.53 ( ( gcd_gcd @ int @ X3 @ ( zero_zero @ int ) ) % 8.50/4.53 = ( abs_abs @ int @ X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4199_gcd__0__left__int,axiom, % 8.50/4.53 ! [X3: int] : % 8.50/4.53 ( ( gcd_gcd @ int @ ( zero_zero @ int ) @ X3 ) % 8.50/4.53 = ( abs_abs @ int @ X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4200_gcd__proj1__if__dvd__int,axiom, % 8.50/4.53 ! [X3: int,Y: int] : % 8.50/4.53 ( ( dvd_dvd @ int @ X3 @ Y ) % 8.50/4.53 => ( ( gcd_gcd @ int @ X3 @ Y ) % 8.50/4.53 = ( abs_abs @ int @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4201_gcd__proj2__if__dvd__int,axiom, % 8.50/4.53 ! [Y: int,X3: int] : % 8.50/4.53 ( ( dvd_dvd @ int @ Y @ X3 ) % 8.50/4.53 => ( ( gcd_gcd @ int @ X3 @ Y ) % 8.50/4.53 = ( abs_abs @ int @ Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4202_gcd__idem__int,axiom, % 8.50/4.53 ! [X3: int] : % 8.50/4.53 ( ( gcd_gcd @ int @ X3 @ X3 ) % 8.50/4.53 = ( abs_abs @ int @ X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4203_gcd__diff1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ring_gcd @ A ) % 8.50/4.53 => ! [M: A,N2: A] : % 8.50/4.53 ( ( gcd_gcd @ A @ ( minus_minus @ A @ M @ N2 ) @ N2 ) % 8.50/4.53 = ( gcd_gcd @ A @ M @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4204_gcd__diff2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ring_gcd @ A ) % 8.50/4.53 => ! [N2: A,M: A] : % 8.50/4.53 ( ( gcd_gcd @ A @ ( minus_minus @ A @ N2 @ M ) @ N2 ) % 8.50/4.53 = ( gcd_gcd @ A @ M @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4205_gcd_Oassoc,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( gcd_gcd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ C2 ) % 8.50/4.53 = ( gcd_gcd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4206_gcd_Ocommute,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ( ( gcd_gcd @ A ) % 8.50/4.53 = ( ^ [A4: A,B4: A] : ( gcd_gcd @ A @ B4 @ A4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4207_gcd_Oleft__commute,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [B2: A,A2: A,C2: A] : % 8.50/4.53 ( ( gcd_gcd @ A @ B2 @ ( gcd_gcd @ A @ A2 @ C2 ) ) % 8.50/4.53 = ( gcd_gcd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4208_gcd__red__int,axiom, % 8.50/4.53 ( ( gcd_gcd @ int ) % 8.50/4.53 = ( ^ [X: int,Y2: int] : ( gcd_gcd @ int @ Y2 @ ( modulo_modulo @ int @ X @ Y2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4209_gcd__mono,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,C2: A,B2: A,D2: A] : % 8.50/4.53 ( ( dvd_dvd @ A @ A2 @ C2 ) % 8.50/4.53 => ( ( dvd_dvd @ A @ B2 @ D2 ) % 8.50/4.53 => ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( gcd_gcd @ A @ C2 @ D2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4210_dvd__gcdD1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( dvd_dvd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) % 8.50/4.53 => ( dvd_dvd @ A @ A2 @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4211_dvd__gcdD2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( dvd_dvd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) % 8.50/4.53 => ( dvd_dvd @ A @ A2 @ C2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4212_gcd__dvdI1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,C2: A,B2: A] : % 8.50/4.53 ( ( dvd_dvd @ A @ A2 @ C2 ) % 8.50/4.53 => ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4213_gcd__dvdI2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [B2: A,C2: A,A2: A] : % 8.50/4.53 ( ( dvd_dvd @ A @ B2 @ C2 ) % 8.50/4.53 => ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4214_gcd__greatest,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [C2: A,A2: A,B2: A] : % 8.50/4.53 ( ( dvd_dvd @ A @ C2 @ A2 ) % 8.50/4.53 => ( ( dvd_dvd @ A @ C2 @ B2 ) % 8.50/4.53 => ( dvd_dvd @ A @ C2 @ ( gcd_gcd @ A @ A2 @ B2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4215_gcd__dvd__antisym,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.50/4.53 ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( gcd_gcd @ A @ C2 @ D2 ) ) % 8.50/4.53 => ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ C2 @ D2 ) @ ( gcd_gcd @ A @ A2 @ B2 ) ) % 8.50/4.53 => ( ( gcd_gcd @ A @ A2 @ B2 ) % 8.50/4.53 = ( gcd_gcd @ A @ C2 @ D2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4216_gcd__add__mult,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [M: A,K: A,N2: A] : % 8.50/4.53 ( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ ( times_times @ A @ K @ M ) @ N2 ) ) % 8.50/4.53 = ( gcd_gcd @ A @ M @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4217_gcd__dvd__prod,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A,K: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( times_times @ A @ K @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4218_gcd__ge__0__int,axiom, % 8.50/4.53 ! [X3: int,Y: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ X3 @ Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4219_bezout__int,axiom, % 8.50/4.53 ! [X3: int,Y: int] : % 8.50/4.53 ? [U4: int,V3: int] : % 8.50/4.53 ( ( plus_plus @ int @ ( times_times @ int @ U4 @ X3 ) @ ( times_times @ int @ V3 @ Y ) ) % 8.50/4.53 = ( gcd_gcd @ int @ X3 @ Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4220_gcd__mult__distrib__int,axiom, % 8.50/4.53 ! [K: int,M: int,N2: int] : % 8.50/4.53 ( ( times_times @ int @ ( abs_abs @ int @ K ) @ ( gcd_gcd @ int @ M @ N2 ) ) % 8.50/4.53 = ( gcd_gcd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4221_gcd__mult__unit2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.50/4.53 => ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) ) % 8.50/4.53 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4222_gcd__mult__unit1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.50/4.53 => ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) % 8.50/4.53 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4223_gcd__div__unit2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.50/4.53 => ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C2 @ A2 ) ) % 8.50/4.53 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4224_gcd__div__unit1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.50/4.53 => ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A2 ) @ C2 ) % 8.50/4.53 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4225_gcd__le2__int,axiom, % 8.50/4.53 ! [B2: int,A2: int] : % 8.50/4.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.50/4.53 => ( ord_less_eq @ int @ ( gcd_gcd @ int @ A2 @ B2 ) @ B2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4226_gcd__le1__int,axiom, % 8.50/4.53 ! [A2: int,B2: int] : % 8.50/4.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) % 8.50/4.53 => ( ord_less_eq @ int @ ( gcd_gcd @ int @ A2 @ B2 ) @ A2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4227_gcd__cases__int,axiom, % 8.50/4.53 ! [X3: int,Y: int,P: int > $o] : % 8.50/4.53 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 ) % 8.50/4.53 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y ) % 8.50/4.53 => ( P @ ( gcd_gcd @ int @ X3 @ Y ) ) ) ) % 8.50/4.53 => ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 ) % 8.50/4.53 => ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) ) % 8.50/4.53 => ( P @ ( gcd_gcd @ int @ X3 @ ( uminus_uminus @ int @ Y ) ) ) ) ) % 8.50/4.53 => ( ( ( ord_less_eq @ int @ X3 @ ( zero_zero @ int ) ) % 8.50/4.53 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y ) % 8.50/4.53 => ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X3 ) @ Y ) ) ) ) % 8.50/4.53 => ( ( ( ord_less_eq @ int @ X3 @ ( zero_zero @ int ) ) % 8.50/4.53 => ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) ) % 8.50/4.53 => ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X3 ) @ ( uminus_uminus @ int @ Y ) ) ) ) ) % 8.50/4.53 => ( P @ ( gcd_gcd @ int @ X3 @ Y ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4228_gcd__unique__int,axiom, % 8.50/4.53 ! [D2: int,A2: int,B2: int] : % 8.50/4.53 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 ) % 8.50/4.53 & ( dvd_dvd @ int @ D2 @ A2 ) % 8.50/4.53 & ( dvd_dvd @ int @ D2 @ B2 ) % 8.50/4.53 & ! [E4: int] : % 8.50/4.53 ( ( ( dvd_dvd @ int @ E4 @ A2 ) % 8.50/4.53 & ( dvd_dvd @ int @ E4 @ B2 ) ) % 8.50/4.53 => ( dvd_dvd @ int @ E4 @ D2 ) ) ) % 8.50/4.53 = ( D2 % 8.50/4.53 = ( gcd_gcd @ int @ A2 @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4229_gcd__non__0__int,axiom, % 8.50/4.53 ! [Y: int,X3: int] : % 8.50/4.53 ( ( ord_less @ int @ ( zero_zero @ int ) @ Y ) % 8.50/4.53 => ( ( gcd_gcd @ int @ X3 @ Y ) % 8.50/4.53 = ( gcd_gcd @ int @ Y @ ( modulo_modulo @ int @ X3 @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4230_gcd__code__int,axiom, % 8.50/4.53 ( ( gcd_gcd @ int ) % 8.50/4.53 = ( ^ [K3: int,L2: int] : % 8.50/4.53 ( abs_abs @ int % 8.50/4.53 @ ( if @ int % 8.50/4.53 @ ( L2 % 8.50/4.53 = ( zero_zero @ int ) ) % 8.50/4.53 @ K3 % 8.50/4.53 @ ( gcd_gcd @ int @ L2 @ ( modulo_modulo @ int @ ( abs_abs @ int @ K3 ) @ ( abs_abs @ int @ L2 ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4231_strict__mono__on__leD,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( ( linorder @ A ) % 8.50/4.53 & ( preorder @ B ) ) % 8.50/4.53 => ! [F2: A > B,A3: set @ A,X3: A,Y: A] : % 8.50/4.53 ( ( strict_mono_on @ A @ B @ F2 @ A3 ) % 8.50/4.53 => ( ( member @ A @ X3 @ A3 ) % 8.50/4.53 => ( ( member @ A @ Y @ A3 ) % 8.50/4.53 => ( ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.53 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4232_strict__mono__onD,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( ( ord @ A ) % 8.50/4.53 & ( ord @ B ) ) % 8.50/4.53 => ! [F2: A > B,A3: set @ A,R2: A,S: A] : % 8.50/4.53 ( ( strict_mono_on @ A @ B @ F2 @ A3 ) % 8.50/4.53 => ( ( member @ A @ R2 @ A3 ) % 8.50/4.53 => ( ( member @ A @ S @ A3 ) % 8.50/4.53 => ( ( ord_less @ A @ R2 @ S ) % 8.50/4.53 => ( ord_less @ B @ ( F2 @ R2 ) @ ( F2 @ S ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4233_card__partition,axiom, % 8.50/4.53 ! [A: $tType,C5: set @ ( set @ A ),K: nat] : % 8.50/4.53 ( ( finite_finite2 @ ( set @ A ) @ C5 ) % 8.50/4.53 => ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) % 8.50/4.53 => ( ! [C3: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ C3 @ C5 ) % 8.50/4.53 => ( ( finite_card @ A @ C3 ) % 8.50/4.53 = K ) ) % 8.50/4.53 => ( ! [C1: set @ A,C22: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ C1 @ C5 ) % 8.50/4.53 => ( ( member @ ( set @ A ) @ C22 @ C5 ) % 8.50/4.53 => ( ( C1 != C22 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ C1 @ C22 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ) ) % 8.50/4.53 => ( ( times_times @ nat @ K @ ( finite_card @ ( set @ A ) @ C5 ) ) % 8.50/4.53 = ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4234_or__numerals_I4_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.53 => ! [X3: num,Y: num] : % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) % 8.50/4.53 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4235_or__numerals_I6_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.53 => ! [X3: num,Y: num] : % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) ) % 8.50/4.53 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4236_or__numerals_I7_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.53 => ! [X3: num,Y: num] : % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) % 8.50/4.53 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4237_gcd__nat_Oeq__neutr__iff,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : % 8.50/4.53 ( ( ( gcd_gcd @ nat @ A2 @ B2 ) % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 = ( ( A2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 & ( B2 % 8.50/4.53 = ( zero_zero @ nat ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4238_gcd__nat_Oleft__neutral,axiom, % 8.50/4.53 ! [A2: nat] : % 8.50/4.53 ( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ A2 ) % 8.50/4.53 = A2 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4239_gcd__nat_Oneutr__eq__iff,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : % 8.50/4.53 ( ( ( zero_zero @ nat ) % 8.50/4.53 = ( gcd_gcd @ nat @ A2 @ B2 ) ) % 8.50/4.53 = ( ( A2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 & ( B2 % 8.50/4.53 = ( zero_zero @ nat ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4240_gcd__nat_Oright__neutral,axiom, % 8.50/4.53 ! [A2: nat] : % 8.50/4.53 ( ( gcd_gcd @ nat @ A2 @ ( zero_zero @ nat ) ) % 8.50/4.53 = A2 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4241_gcd__0__nat,axiom, % 8.50/4.53 ! [X3: nat] : % 8.50/4.53 ( ( gcd_gcd @ nat @ X3 @ ( zero_zero @ nat ) ) % 8.50/4.53 = X3 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4242_gcd__0__left__nat,axiom, % 8.50/4.53 ! [X3: nat] : % 8.50/4.53 ( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ X3 ) % 8.50/4.53 = X3 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4243_Int__iff,axiom, % 8.50/4.53 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.53 = ( ( member @ A @ C2 @ A3 ) % 8.50/4.53 & ( member @ A @ C2 @ B3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4244_IntI,axiom, % 8.50/4.53 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( member @ A @ C2 @ A3 ) % 8.50/4.53 => ( ( member @ A @ C2 @ B3 ) % 8.50/4.53 => ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4245_gcd__1__nat,axiom, % 8.50/4.53 ! [M: nat] : % 8.50/4.53 ( ( gcd_gcd @ nat @ M @ ( one_one @ nat ) ) % 8.50/4.53 = ( one_one @ nat ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4246_gcd__proj2__if__dvd__nat,axiom, % 8.50/4.53 ! [Y: nat,X3: nat] : % 8.50/4.53 ( ( dvd_dvd @ nat @ Y @ X3 ) % 8.50/4.53 => ( ( gcd_gcd @ nat @ X3 @ Y ) % 8.50/4.53 = Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4247_gcd__proj1__if__dvd__nat,axiom, % 8.50/4.53 ! [X3: nat,Y: nat] : % 8.50/4.53 ( ( dvd_dvd @ nat @ X3 @ Y ) % 8.50/4.53 => ( ( gcd_gcd @ nat @ X3 @ Y ) % 8.50/4.53 = X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4248_gcd__nat_Obounded__iff,axiom, % 8.50/4.53 ! [A2: nat,B2: nat,C2: nat] : % 8.50/4.53 ( ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) ) % 8.50/4.53 = ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.50/4.53 & ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4249_gcd__nat_Oabsorb2,axiom, % 8.50/4.53 ! [B2: nat,A2: nat] : % 8.50/4.53 ( ( dvd_dvd @ nat @ B2 @ A2 ) % 8.50/4.53 => ( ( gcd_gcd @ nat @ A2 @ B2 ) % 8.50/4.53 = B2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4250_gcd__nat_Oabsorb1,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : % 8.50/4.53 ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.50/4.53 => ( ( gcd_gcd @ nat @ A2 @ B2 ) % 8.50/4.53 = A2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4251_gcd__Suc__0,axiom, % 8.50/4.53 ! [M: nat] : % 8.50/4.53 ( ( gcd_gcd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.53 = ( suc @ ( zero_zero @ nat ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4252_gcd__pos__nat,axiom, % 8.50/4.53 ! [M: nat,N2: nat] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_gcd @ nat @ M @ N2 ) ) % 8.50/4.53 = ( ( M % 8.50/4.53 != ( zero_zero @ nat ) ) % 8.50/4.53 | ( N2 % 8.50/4.53 != ( zero_zero @ nat ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4253_boolean__algebra_Oconj__zero__left,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.53 => ! [X3: A] : % 8.50/4.53 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X3 ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4254_boolean__algebra_Oconj__zero__right,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.53 => ! [X3: A] : % 8.50/4.53 ( ( inf_inf @ A @ X3 @ ( bot_bot @ A ) ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4255_finite__Int,axiom, % 8.50/4.53 ! [A: $tType,F5: set @ A,G6: set @ A] : % 8.50/4.53 ( ( ( finite_finite2 @ A @ F5 ) % 8.50/4.53 | ( finite_finite2 @ A @ G6 ) ) % 8.50/4.53 => ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F5 @ G6 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4256_Int__subset__iff,axiom, % 8.50/4.53 ! [A: $tType,C5: set @ A,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ A ) @ C5 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.53 = ( ( ord_less_eq @ ( set @ A ) @ C5 @ A3 ) % 8.50/4.53 & ( ord_less_eq @ ( set @ A ) @ C5 @ B3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4257_Int__insert__left__if0,axiom, % 8.50/4.53 ! [A: $tType,A2: A,C5: set @ A,B3: set @ A] : % 8.50/4.53 ( ~ ( member @ A @ A2 @ C5 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C5 ) % 8.50/4.53 = ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4258_Int__insert__left__if1,axiom, % 8.50/4.53 ! [A: $tType,A2: A,C5: set @ A,B3: set @ A] : % 8.50/4.53 ( ( member @ A @ A2 @ C5 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C5 ) % 8.50/4.53 = ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4259_insert__inter__insert,axiom, % 8.50/4.53 ! [A: $tType,A2: A,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ ( insert @ A @ A2 @ B3 ) ) % 8.50/4.53 = ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4260_Int__insert__right__if0,axiom, % 8.50/4.53 ! [A: $tType,A2: A,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ~ ( member @ A @ A2 @ A3 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) % 8.50/4.53 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4261_Int__insert__right__if1,axiom, % 8.50/4.53 ! [A: $tType,A2: A,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( member @ A @ A2 @ A3 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) % 8.50/4.53 = ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4262_or_Oright__neutral,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [A2: A] : % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( zero_zero @ A ) ) % 8.50/4.53 = A2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4263_or_Oleft__neutral,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [A2: A] : % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ A @ ( zero_zero @ A ) @ A2 ) % 8.50/4.53 = A2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4264_or__negative__int__iff,axiom, % 8.50/4.53 ! [K: int,L: int] : % 8.50/4.53 ( ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) @ ( zero_zero @ int ) ) % 8.50/4.53 = ( ( ord_less @ int @ K @ ( zero_zero @ int ) ) % 8.50/4.53 | ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4265_Pow__Int__eq,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( pow2 @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.53 = ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4266_gcd__int__int__eq,axiom, % 8.50/4.53 ! [M: nat,N2: nat] : % 8.50/4.53 ( ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N2 ) ) % 8.50/4.53 = ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ M @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4267_boolean__algebra_Oconj__cancel__right,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.53 => ! [X3: A] : % 8.50/4.53 ( ( inf_inf @ A @ X3 @ ( uminus_uminus @ A @ X3 ) ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4268_boolean__algebra_Oconj__cancel__left,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.53 => ! [X3: A] : % 8.50/4.53 ( ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ X3 ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4269_inf__compl__bot__right,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : % 8.50/4.53 ( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y @ ( uminus_uminus @ A @ X3 ) ) ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4270_inf__compl__bot__left2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : % 8.50/4.53 ( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4271_inf__compl__bot__left1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : % 8.50/4.53 ( ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ ( inf_inf @ A @ X3 @ Y ) ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4272_insert__disjoint_I1_J,axiom, % 8.50/4.53 ! [A: $tType,A2: A,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ B3 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 = ( ~ ( member @ A @ A2 @ B3 ) % 8.50/4.53 & ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4273_insert__disjoint_I2_J,axiom, % 8.50/4.53 ! [A: $tType,A2: A,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( ( bot_bot @ ( set @ A ) ) % 8.50/4.53 = ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ B3 ) ) % 8.50/4.53 = ( ~ ( member @ A @ A2 @ B3 ) % 8.50/4.53 & ( ( bot_bot @ ( set @ A ) ) % 8.50/4.53 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4274_disjoint__insert_I1_J,axiom, % 8.50/4.53 ! [A: $tType,B3: set @ A,A2: A,A3: set @ A] : % 8.50/4.53 ( ( ( inf_inf @ ( set @ A ) @ B3 @ ( insert @ A @ A2 @ A3 ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 = ( ~ ( member @ A @ A2 @ B3 ) % 8.50/4.53 & ( ( inf_inf @ ( set @ A ) @ B3 @ A3 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4275_disjoint__insert_I2_J,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B2: A,B3: set @ A] : % 8.50/4.53 ( ( ( bot_bot @ ( set @ A ) ) % 8.50/4.53 = ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B3 ) ) ) % 8.50/4.53 = ( ~ ( member @ A @ B2 @ A3 ) % 8.50/4.53 & ( ( bot_bot @ ( set @ A ) ) % 8.50/4.53 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4276_Diff__disjoint,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4277_Compl__disjoint2,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ A3 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4278_Compl__disjoint,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4279_Diff__Compl,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( minus_minus @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) % 8.50/4.53 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4280_or__nat__numerals_I4_J,axiom, % 8.50/4.53 ! [X3: num] : % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.53 = ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4281_or__nat__numerals_I2_J,axiom, % 8.50/4.53 ! [Y: num] : % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) % 8.50/4.53 = ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4282_gcd__nat__abs__right__eq,axiom, % 8.50/4.53 ! [N2: nat,K: int] : % 8.50/4.53 ( ( gcd_gcd @ nat @ N2 @ ( nat2 @ ( abs_abs @ int @ K ) ) ) % 8.50/4.53 = ( nat2 @ ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4283_gcd__nat__abs__left__eq,axiom, % 8.50/4.53 ! [K: int,N2: nat] : % 8.50/4.53 ( ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N2 ) % 8.50/4.53 = ( nat2 @ ( gcd_gcd @ int @ K @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4284_or__nat__numerals_I3_J,axiom, % 8.50/4.53 ! [X3: num] : % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X3 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.53 = ( numeral_numeral @ nat @ ( bit1 @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4285_or__nat__numerals_I1_J,axiom, % 8.50/4.53 ! [Y: num] : % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) ) % 8.50/4.53 = ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4286_sum__of__bool__mult__eq,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( semiring_1 @ A ) % 8.50/4.53 => ! [A3: set @ B,P: B > $o,F2: B > A] : % 8.50/4.53 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.53 => ( ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.53 @ ^ [X: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X ) ) @ ( F2 @ X ) ) % 8.50/4.53 @ A3 ) % 8.50/4.53 = ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4287_sum__mult__of__bool__eq,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( semiring_1 @ A ) % 8.50/4.53 => ! [A3: set @ B,F2: B > A,P: B > $o] : % 8.50/4.53 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.53 => ( ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.53 @ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ ( zero_neq_one_of_bool @ A @ ( P @ X ) ) ) % 8.50/4.53 @ A3 ) % 8.50/4.53 = ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4288_sum__of__bool__eq,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( semiring_1 @ A ) % 8.50/4.53 => ! [A3: set @ B,P: B > $o] : % 8.50/4.53 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.53 => ( ( finite_finite2 @ B @ A3 ) % 8.50/4.53 => ( ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.53 @ ^ [X: B] : ( zero_neq_one_of_bool @ A @ ( P @ X ) ) % 8.50/4.53 @ A3 ) % 8.50/4.53 = ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4289_gcd__mult__distrib__nat,axiom, % 8.50/4.53 ! [K: nat,M: nat,N2: nat] : % 8.50/4.53 ( ( times_times @ nat @ K @ ( gcd_gcd @ nat @ M @ N2 ) ) % 8.50/4.53 = ( gcd_gcd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4290_gcd__red__nat,axiom, % 8.50/4.53 ( ( gcd_gcd @ nat ) % 8.50/4.53 = ( ^ [X: nat,Y2: nat] : ( gcd_gcd @ nat @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4291_gcd__unique__nat,axiom, % 8.50/4.53 ! [D2: nat,A2: nat,B2: nat] : % 8.50/4.53 ( ( ( dvd_dvd @ nat @ D2 @ A2 ) % 8.50/4.53 & ( dvd_dvd @ nat @ D2 @ B2 ) % 8.50/4.53 & ! [E4: nat] : % 8.50/4.53 ( ( ( dvd_dvd @ nat @ E4 @ A2 ) % 8.50/4.53 & ( dvd_dvd @ nat @ E4 @ B2 ) ) % 8.50/4.53 => ( dvd_dvd @ nat @ E4 @ D2 ) ) ) % 8.50/4.53 = ( D2 % 8.50/4.53 = ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4292_gcd__nat_Ostrict__coboundedI2,axiom, % 8.50/4.53 ! [B2: nat,C2: nat,A2: nat] : % 8.50/4.53 ( ( ( dvd_dvd @ nat @ B2 @ C2 ) % 8.50/4.53 & ( B2 != C2 ) ) % 8.50/4.53 => ( ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 ) % 8.50/4.53 & ( ( gcd_gcd @ nat @ A2 @ B2 ) % 8.50/4.53 != C2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4293_gcd__nat_Ostrict__coboundedI1,axiom, % 8.50/4.53 ! [A2: nat,C2: nat,B2: nat] : % 8.50/4.53 ( ( ( dvd_dvd @ nat @ A2 @ C2 ) % 8.50/4.53 & ( A2 != C2 ) ) % 8.50/4.53 => ( ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 ) % 8.50/4.53 & ( ( gcd_gcd @ nat @ A2 @ B2 ) % 8.50/4.53 != C2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4294_gcd__nat_Ostrict__order__iff,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : % 8.50/4.53 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.50/4.53 & ( A2 != B2 ) ) % 8.50/4.53 = ( ( A2 % 8.50/4.53 = ( gcd_gcd @ nat @ A2 @ B2 ) ) % 8.50/4.53 & ( A2 != B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4295_gcd__nat_Ostrict__boundedE,axiom, % 8.50/4.53 ! [A2: nat,B2: nat,C2: nat] : % 8.50/4.53 ( ( ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) ) % 8.50/4.53 & ( A2 % 8.50/4.53 != ( gcd_gcd @ nat @ B2 @ C2 ) ) ) % 8.50/4.53 => ~ ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.50/4.53 & ( A2 != B2 ) ) % 8.50/4.53 => ~ ( ( dvd_dvd @ nat @ A2 @ C2 ) % 8.50/4.53 & ( A2 != C2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4296_gcd__nat_OcoboundedI2,axiom, % 8.50/4.53 ! [B2: nat,C2: nat,A2: nat] : % 8.50/4.53 ( ( dvd_dvd @ nat @ B2 @ C2 ) % 8.50/4.53 => ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4297_gcd__nat_OcoboundedI1,axiom, % 8.50/4.53 ! [A2: nat,C2: nat,B2: nat] : % 8.50/4.53 ( ( dvd_dvd @ nat @ A2 @ C2 ) % 8.50/4.53 => ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4298_gcd__nat_Oabsorb__iff2,axiom, % 8.50/4.53 ( ( dvd_dvd @ nat ) % 8.50/4.53 = ( ^ [B4: nat,A4: nat] : % 8.50/4.53 ( ( gcd_gcd @ nat @ A4 @ B4 ) % 8.50/4.53 = B4 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4299_gcd__nat_Oabsorb__iff1,axiom, % 8.50/4.53 ( ( dvd_dvd @ nat ) % 8.50/4.53 = ( ^ [A4: nat,B4: nat] : % 8.50/4.53 ( ( gcd_gcd @ nat @ A4 @ B4 ) % 8.50/4.53 = A4 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4300_gcd__nat_Ocobounded2,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4301_gcd__nat_Ocobounded1,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ A2 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4302_gcd__nat_Oorder__iff,axiom, % 8.50/4.53 ( ( dvd_dvd @ nat ) % 8.50/4.53 = ( ^ [A4: nat,B4: nat] : % 8.50/4.53 ( A4 % 8.50/4.53 = ( gcd_gcd @ nat @ A4 @ B4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4303_gcd__nat_OboundedI,axiom, % 8.50/4.53 ! [A2: nat,B2: nat,C2: nat] : % 8.50/4.53 ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.50/4.53 => ( ( dvd_dvd @ nat @ A2 @ C2 ) % 8.50/4.53 => ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4304_gcd__nat_OboundedE,axiom, % 8.50/4.53 ! [A2: nat,B2: nat,C2: nat] : % 8.50/4.53 ( ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) ) % 8.50/4.53 => ~ ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.50/4.53 => ~ ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4305_gcd__nat_Oabsorb4,axiom, % 8.50/4.53 ! [B2: nat,A2: nat] : % 8.50/4.53 ( ( ( dvd_dvd @ nat @ B2 @ A2 ) % 8.50/4.53 & ( B2 != A2 ) ) % 8.50/4.53 => ( ( gcd_gcd @ nat @ A2 @ B2 ) % 8.50/4.53 = B2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4306_gcd__nat_Oabsorb3,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : % 8.50/4.53 ( ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.50/4.53 & ( A2 != B2 ) ) % 8.50/4.53 => ( ( gcd_gcd @ nat @ A2 @ B2 ) % 8.50/4.53 = A2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4307_gcd__nat_OorderI,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : % 8.50/4.53 ( ( A2 % 8.50/4.53 = ( gcd_gcd @ nat @ A2 @ B2 ) ) % 8.50/4.53 => ( dvd_dvd @ nat @ A2 @ B2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4308_gcd__nat_OorderE,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : % 8.50/4.53 ( ( dvd_dvd @ nat @ A2 @ B2 ) % 8.50/4.53 => ( A2 % 8.50/4.53 = ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4309_gcd__nat_Omono,axiom, % 8.50/4.53 ! [A2: nat,C2: nat,B2: nat,D2: nat] : % 8.50/4.53 ( ( dvd_dvd @ nat @ A2 @ C2 ) % 8.50/4.53 => ( ( dvd_dvd @ nat @ B2 @ D2 ) % 8.50/4.53 => ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ ( gcd_gcd @ nat @ C2 @ D2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4310_disjoint__iff__not__equal,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 = ( ! [X: A] : % 8.50/4.53 ( ( member @ A @ X @ A3 ) % 8.50/4.53 => ! [Y2: A] : % 8.50/4.53 ( ( member @ A @ Y2 @ B3 ) % 8.50/4.53 => ( X != Y2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4311_Int__empty__right,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4312_Int__empty__left,axiom, % 8.50/4.53 ! [A: $tType,B3: set @ A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4313_disjoint__iff,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 = ( ! [X: A] : % 8.50/4.53 ( ( member @ A @ X @ A3 ) % 8.50/4.53 => ~ ( member @ A @ X @ B3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4314_Int__emptyI,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ A3 ) % 8.50/4.53 => ~ ( member @ A @ X5 @ B3 ) ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4315_Int__mono,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,C5: set @ A,B3: set @ A,D6: set @ A] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) % 8.50/4.53 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ D6 ) % 8.50/4.53 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( inf_inf @ ( set @ A ) @ C5 @ D6 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4316_Int__lower1,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ A3 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4317_Int__lower2,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ B3 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4318_Int__absorb1,axiom, % 8.50/4.53 ! [A: $tType,B3: set @ A,A3: set @ A] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.53 = B3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4319_Int__absorb2,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.53 = A3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4320_Int__greatest,axiom, % 8.50/4.53 ! [A: $tType,C5: set @ A,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ A ) @ C5 @ A3 ) % 8.50/4.53 => ( ( ord_less_eq @ ( set @ A ) @ C5 @ B3 ) % 8.50/4.53 => ( ord_less_eq @ ( set @ A ) @ C5 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4321_Int__Collect__mono,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o,Q: A > $o] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.53 => ( ! [X5: A] : % 8.50/4.53 ( ( member @ A @ X5 @ A3 ) % 8.50/4.53 => ( ( P @ X5 ) % 8.50/4.53 => ( Q @ X5 ) ) ) % 8.50/4.53 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B3 @ ( collect @ A @ Q ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4322_Collect__conj__eq,axiom, % 8.50/4.53 ! [A: $tType,P: A > $o,Q: A > $o] : % 8.50/4.53 ( ( collect @ A % 8.50/4.53 @ ^ [X: A] : % 8.50/4.53 ( ( P @ X ) % 8.50/4.53 & ( Q @ X ) ) ) % 8.50/4.53 = ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4323_Int__Collect,axiom, % 8.50/4.53 ! [A: $tType,X3: A,A3: set @ A,P: A > $o] : % 8.50/4.53 ( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) ) % 8.50/4.53 = ( ( member @ A @ X3 @ A3 ) % 8.50/4.53 & ( P @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4324_Int__def,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) ) % 8.50/4.53 = ( ^ [A6: set @ A,B6: set @ A] : % 8.50/4.53 ( collect @ A % 8.50/4.53 @ ^ [X: A] : % 8.50/4.53 ( ( member @ A @ X @ A6 ) % 8.50/4.53 & ( member @ A @ X @ B6 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4325_or__eq__0__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) % 8.50/4.53 = ( zero_zero @ A ) ) % 8.50/4.53 = ( ( A2 % 8.50/4.53 = ( zero_zero @ A ) ) % 8.50/4.53 & ( B2 % 8.50/4.53 = ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4326_bit_Odisj__zero__right,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_ri3973907225187159222ations @ A ) % 8.50/4.53 => ! [X3: A] : % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ A @ X3 @ ( zero_zero @ A ) ) % 8.50/4.53 = X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4327_Int__left__commute,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) % 8.50/4.53 = ( inf_inf @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ A ) @ A3 @ C5 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4328_Int__left__absorb,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.53 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4329_Int__commute,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) ) % 8.50/4.53 = ( ^ [A6: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ B6 @ A6 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4330_Int__absorb,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ A3 @ A3 ) % 8.50/4.53 = A3 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4331_Int__assoc,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C5 ) % 8.50/4.53 = ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4332_IntD2,axiom, % 8.50/4.53 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.53 => ( member @ A @ C2 @ B3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4333_IntD1,axiom, % 8.50/4.53 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.53 => ( member @ A @ C2 @ A3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4334_IntE,axiom, % 8.50/4.53 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.53 => ~ ( ( member @ A @ C2 @ A3 ) % 8.50/4.53 => ~ ( member @ A @ C2 @ B3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4335_Diff__Int__distrib2,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ C5 ) % 8.50/4.53 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C5 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4336_Diff__Int__distrib,axiom, % 8.50/4.53 ! [A: $tType,C5: set @ A,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ C5 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.53 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C5 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C5 @ B3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4337_Diff__Diff__Int,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( minus_minus @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.53 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4338_Diff__Int2,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,C5: set @ A,B3: set @ A] : % 8.50/4.53 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C5 ) @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) % 8.50/4.53 = ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C5 ) @ B3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4339_Int__Diff,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A,C5: set @ A] : % 8.50/4.53 ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C5 ) % 8.50/4.53 = ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ C5 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4340_Int__insert__left,axiom, % 8.50/4.53 ! [A: $tType,A2: A,C5: set @ A,B3: set @ A] : % 8.50/4.53 ( ( ( member @ A @ A2 @ C5 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C5 ) % 8.50/4.53 = ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ) % 8.50/4.53 & ( ~ ( member @ A @ A2 @ C5 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C5 ) % 8.50/4.53 = ( inf_inf @ ( set @ A ) @ B3 @ C5 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4341_Int__insert__right,axiom, % 8.50/4.53 ! [A: $tType,A2: A,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( ( member @ A @ A2 @ A3 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) % 8.50/4.53 = ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) % 8.50/4.53 & ( ~ ( member @ A @ A2 @ A3 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) % 8.50/4.53 = ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4342_Union__Int__subset,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ ( set @ A ),B3: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A3 @ B3 ) ) @ ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4343_Sup__inter__less__eq,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [A3: set @ A,B3: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4344_inf__cancel__left2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.53 => ! [X3: A,A2: A,B2: A] : % 8.50/4.53 ( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ A2 ) @ ( inf_inf @ A @ X3 @ B2 ) ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4345_inf__cancel__left1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.53 => ! [X3: A,A2: A,B2: A] : % 8.50/4.53 ( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ A2 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ B2 ) ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4346_gcd__le1__nat,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : % 8.50/4.53 ( ( A2 % 8.50/4.53 != ( zero_zero @ nat ) ) % 8.50/4.53 => ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ A2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4347_gcd__le2__nat,axiom, % 8.50/4.53 ! [B2: nat,A2: nat] : % 8.50/4.53 ( ( B2 % 8.50/4.53 != ( zero_zero @ nat ) ) % 8.50/4.53 => ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4348_Sup__inf__eq__bot__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple592849572758109894attice @ A ) % 8.50/4.53 => ! [B3: set @ A,A2: A] : % 8.50/4.53 ( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B3 ) @ A2 ) % 8.50/4.53 = ( bot_bot @ A ) ) % 8.50/4.53 = ( ! [X: A] : % 8.50/4.53 ( ( member @ A @ X @ B3 ) % 8.50/4.53 => ( ( inf_inf @ A @ X @ A2 ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4349_disjunctive__add,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ! [N3: nat] : % 8.50/4.53 ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 ) % 8.50/4.53 | ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) ) % 8.50/4.53 => ( ( plus_plus @ A @ A2 @ B2 ) % 8.50/4.53 = ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4350_gcd__diff1__nat,axiom, % 8.50/4.53 ! [N2: nat,M: nat] : % 8.50/4.53 ( ( ord_less_eq @ nat @ N2 @ M ) % 8.50/4.53 => ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) % 8.50/4.53 = ( gcd_gcd @ nat @ M @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4351_gcd__diff2__nat,axiom, % 8.50/4.53 ! [M: nat,N2: nat] : % 8.50/4.53 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.53 => ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ N2 @ M ) @ N2 ) % 8.50/4.53 = ( gcd_gcd @ nat @ M @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4352_gcd__nat_Oelims,axiom, % 8.50/4.53 ! [X3: nat,Xa2: nat,Y: nat] : % 8.50/4.53 ( ( ( gcd_gcd @ nat @ X3 @ Xa2 ) % 8.50/4.53 = Y ) % 8.50/4.53 => ( ( ( Xa2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 => ( Y = X3 ) ) % 8.50/4.53 & ( ( Xa2 % 8.50/4.53 != ( zero_zero @ nat ) ) % 8.50/4.53 => ( Y % 8.50/4.53 = ( gcd_gcd @ nat @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4353_gcd__nat_Osimps,axiom, % 8.50/4.53 ( ( gcd_gcd @ nat ) % 8.50/4.53 = ( ^ [X: nat,Y2: nat] : % 8.50/4.53 ( if @ nat % 8.50/4.53 @ ( Y2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 @ X % 8.50/4.53 @ ( gcd_gcd @ nat @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4354_gcd__non__0__nat,axiom, % 8.50/4.53 ! [Y: nat,X3: nat] : % 8.50/4.53 ( ( Y % 8.50/4.53 != ( zero_zero @ nat ) ) % 8.50/4.53 => ( ( gcd_gcd @ nat @ X3 @ Y ) % 8.50/4.53 = ( gcd_gcd @ nat @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4355_Int__Diff__disjoint,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4356_Diff__triv,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ( minus_minus @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.53 = A3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4357_Union__disjoint,axiom, % 8.50/4.53 ! [A: $tType,C5: set @ ( set @ A ),A3: set @ A] : % 8.50/4.53 ( ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) @ A3 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 = ( ! [X: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ X @ C5 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ X @ A3 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4358_ivl__disj__int__two_I3_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( order @ A ) % 8.50/4.53 => ! [L: A,M: A,U: A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4359_gcd__integer_Orep__eq,axiom, % 8.50/4.53 ! [X3: code_integer,Xa2: code_integer] : % 8.50/4.53 ( ( code_int_of_integer @ ( gcd_gcd @ code_integer @ X3 @ Xa2 ) ) % 8.50/4.53 = ( gcd_gcd @ int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4360_gcd__integer_Oabs__eq,axiom, % 8.50/4.53 ! [Xa2: int,X3: int] : % 8.50/4.53 ( ( gcd_gcd @ code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) ) % 8.50/4.53 = ( code_integer_of_int @ ( gcd_gcd @ int @ Xa2 @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4361_Diff__eq,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( minus_minus @ ( set @ A ) ) % 8.50/4.53 = ( ^ [A6: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ B6 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4362_inf__shunt,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : % 8.50/4.53 ( ( ( inf_inf @ A @ X3 @ Y ) % 8.50/4.53 = ( bot_bot @ A ) ) % 8.50/4.53 = ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4363_finite__Inf__in,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ! [A3: set @ A] : % 8.50/4.53 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.53 => ( ( A3 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ! [X5: A,Y4: A] : % 8.50/4.53 ( ( member @ A @ X5 @ A3 ) % 8.50/4.53 => ( ( member @ A @ Y4 @ A3 ) % 8.50/4.53 => ( member @ A @ ( inf_inf @ A @ X5 @ Y4 ) @ A3 ) ) ) % 8.50/4.53 => ( member @ A @ ( complete_Inf_Inf @ A @ A3 ) @ A3 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4364_bezout__nat,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : % 8.50/4.53 ( ( A2 % 8.50/4.53 != ( zero_zero @ nat ) ) % 8.50/4.53 => ? [X5: nat,Y4: nat] : % 8.50/4.53 ( ( times_times @ nat @ A2 @ X5 ) % 8.50/4.53 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4365_bezout__gcd__nat_H,axiom, % 8.50/4.53 ! [B2: nat,A2: nat] : % 8.50/4.53 ? [X5: nat,Y4: nat] : % 8.50/4.53 ( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( times_times @ nat @ A2 @ X5 ) ) % 8.50/4.53 & ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X5 ) @ ( times_times @ nat @ B2 @ Y4 ) ) % 8.50/4.53 = ( gcd_gcd @ nat @ A2 @ B2 ) ) ) % 8.50/4.53 | ( ( ord_less_eq @ nat @ ( times_times @ nat @ A2 @ Y4 ) @ ( times_times @ nat @ B2 @ X5 ) ) % 8.50/4.53 & ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X5 ) @ ( times_times @ nat @ A2 @ Y4 ) ) % 8.50/4.53 = ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4366_ivl__disj__int__two_I7_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( order @ A ) % 8.50/4.53 => ! [L: A,M: A,U: A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4367_ivl__disj__int__one_I4_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( order @ A ) % 8.50/4.53 => ! [L: A,U: A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4368_ivl__disj__int__one_I2_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( order @ A ) % 8.50/4.53 => ! [L: A,U: A] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4369_disjoint__eq__subset__Compl,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 = ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4370_insert__partition,axiom, % 8.50/4.53 ! [A: $tType,X3: set @ A,F5: set @ ( set @ A )] : % 8.50/4.53 ( ~ ( member @ ( set @ A ) @ X3 @ F5 ) % 8.50/4.53 => ( ! [X5: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ X5 @ ( insert @ ( set @ A ) @ X3 @ F5 ) ) % 8.50/4.53 => ! [Xa3: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ Xa3 @ ( insert @ ( set @ A ) @ X3 @ F5 ) ) % 8.50/4.53 => ( ( X5 != Xa3 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ X5 @ Xa3 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ) ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ X3 @ ( complete_Sup_Sup @ ( set @ A ) @ F5 ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4371_sum_Ointer__restrict,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [A3: set @ B,G: B > A,B3: set @ B] : % 8.50/4.53 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.53 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) % 8.50/4.53 = ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.53 @ ^ [X: B] : ( if @ A @ ( member @ B @ X @ B3 ) @ ( G @ X ) @ ( zero_zero @ A ) ) % 8.50/4.53 @ A3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4372_prod_Ointer__restrict,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [A3: set @ B,G: B > A,B3: set @ B] : % 8.50/4.53 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.53 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) % 8.50/4.53 = ( groups7121269368397514597t_prod @ B @ A % 8.50/4.53 @ ^ [X: B] : ( if @ A @ ( member @ B @ X @ B3 ) @ ( G @ X ) @ ( one_one @ A ) ) % 8.50/4.53 @ A3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4373_gcd__code__integer,axiom, % 8.50/4.53 ( ( gcd_gcd @ code_integer ) % 8.50/4.53 = ( ^ [K3: code_integer,L2: code_integer] : % 8.50/4.53 ( abs_abs @ code_integer % 8.50/4.53 @ ( if @ code_integer % 8.50/4.53 @ ( L2 % 8.50/4.53 = ( zero_zero @ code_integer ) ) % 8.50/4.53 @ K3 % 8.50/4.53 @ ( gcd_gcd @ code_integer @ L2 @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4374_bit_Ocomplement__unique,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_ri3973907225187159222ations @ A ) % 8.50/4.53 => ! [A2: A,X3: A,Y: A] : % 8.50/4.53 ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ X3 ) % 8.50/4.53 = ( zero_zero @ A ) ) % 8.50/4.53 => ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ X3 ) % 8.50/4.53 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 8.50/4.53 => ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ Y ) % 8.50/4.53 = ( zero_zero @ A ) ) % 8.50/4.53 => ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ Y ) % 8.50/4.53 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 8.50/4.53 => ( X3 = Y ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4375_sum_Omono__neutral__cong,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [T4: set @ B,S3: set @ B,H2: B > A,G: B > A] : % 8.50/4.53 ( ( finite_finite2 @ B @ T4 ) % 8.50/4.53 => ( ( finite_finite2 @ B @ S3 ) % 8.50/4.53 => ( ! [I4: B] : % 8.50/4.53 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.53 => ( ( H2 @ I4 ) % 8.50/4.53 = ( zero_zero @ A ) ) ) % 8.50/4.53 => ( ! [I4: B] : % 8.50/4.53 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ S3 @ T4 ) ) % 8.50/4.53 => ( ( G @ I4 ) % 8.50/4.53 = ( zero_zero @ A ) ) ) % 8.50/4.53 => ( ! [X5: B] : % 8.50/4.53 ( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ S3 @ T4 ) ) % 8.50/4.53 => ( ( G @ X5 ) % 8.50/4.53 = ( H2 @ X5 ) ) ) % 8.50/4.53 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) % 8.50/4.53 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4376_Iio__Int__singleton,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( order @ A ) % 8.50/4.53 => ! [X3: A,K: A] : % 8.50/4.53 ( ( ( ord_less @ A @ X3 @ K ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.53 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.53 & ( ~ ( ord_less @ A @ X3 @ K ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4377_sum_OInt__Diff,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [A3: set @ B,G: B > A,B3: set @ B] : % 8.50/4.53 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.53 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) % 8.50/4.53 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4378_prod_OInt__Diff,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [A3: set @ B,G: B > A,B3: set @ B] : % 8.50/4.53 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.53 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) % 8.50/4.53 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4379_prod_Omono__neutral__cong,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [T4: set @ B,S3: set @ B,H2: B > A,G: B > A] : % 8.50/4.53 ( ( finite_finite2 @ B @ T4 ) % 8.50/4.53 => ( ( finite_finite2 @ B @ S3 ) % 8.50/4.53 => ( ! [I4: B] : % 8.50/4.53 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.53 => ( ( H2 @ I4 ) % 8.50/4.53 = ( one_one @ A ) ) ) % 8.50/4.53 => ( ! [I4: B] : % 8.50/4.53 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ S3 @ T4 ) ) % 8.50/4.53 => ( ( G @ I4 ) % 8.50/4.53 = ( one_one @ A ) ) ) % 8.50/4.53 => ( ! [X5: B] : % 8.50/4.53 ( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ S3 @ T4 ) ) % 8.50/4.53 => ( ( G @ X5 ) % 8.50/4.53 = ( H2 @ X5 ) ) ) % 8.50/4.53 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) % 8.50/4.53 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4380_card__Diff__subset__Int,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.53 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) % 8.50/4.53 = ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4381_gcd__int__def,axiom, % 8.50/4.53 ( ( gcd_gcd @ int ) % 8.50/4.53 = ( ^ [X: int,Y2: int] : ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ X ) ) @ ( nat2 @ ( abs_abs @ int @ Y2 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4382_sum_OIf__cases,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [A3: set @ B,P: B > $o,H2: B > A,G: B > A] : % 8.50/4.53 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.53 => ( ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.53 @ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( H2 @ X ) @ ( G @ X ) ) % 8.50/4.53 @ A3 ) % 8.50/4.53 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4383_prod_OIf__cases,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [A3: set @ B,P: B > $o,H2: B > A,G: B > A] : % 8.50/4.53 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.53 => ( ( groups7121269368397514597t_prod @ B @ A % 8.50/4.53 @ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( H2 @ X ) @ ( G @ X ) ) % 8.50/4.53 @ A3 ) % 8.50/4.53 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4384_dvd__partition,axiom, % 8.50/4.53 ! [A: $tType,C5: set @ ( set @ A ),K: nat] : % 8.50/4.53 ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) % 8.50/4.53 => ( ! [X5: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ X5 @ C5 ) % 8.50/4.53 => ( dvd_dvd @ nat @ K @ ( finite_card @ A @ X5 ) ) ) % 8.50/4.53 => ( ! [X5: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ X5 @ C5 ) % 8.50/4.53 => ! [Xa3: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ Xa3 @ C5 ) % 8.50/4.53 => ( ( X5 != Xa3 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ X5 @ Xa3 ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ) ) % 8.50/4.53 => ( dvd_dvd @ nat @ K @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4385_sum__div__partition,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( euclid4440199948858584721cancel @ A ) % 8.50/4.53 => ! [A3: set @ B,F2: B > A,B2: A] : % 8.50/4.53 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.53 => ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A3 ) @ B2 ) % 8.50/4.53 = ( plus_plus @ A % 8.50/4.53 @ ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.53 @ ^ [A4: B] : ( divide_divide @ A @ ( F2 @ A4 ) @ B2 ) % 8.50/4.53 @ ( inf_inf @ ( set @ B ) @ A3 % 8.50/4.53 @ ( collect @ B % 8.50/4.53 @ ^ [A4: B] : ( dvd_dvd @ A @ B2 @ ( F2 @ A4 ) ) ) ) ) % 8.50/4.53 @ ( divide_divide @ A % 8.50/4.53 @ ( groups7311177749621191930dd_sum @ B @ A @ F2 % 8.50/4.53 @ ( inf_inf @ ( set @ B ) @ A3 % 8.50/4.53 @ ( collect @ B % 8.50/4.53 @ ^ [A4: B] : % 8.50/4.53 ~ ( dvd_dvd @ A @ B2 @ ( F2 @ A4 ) ) ) ) ) % 8.50/4.53 @ B2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4386_distinct__concat,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ ( list @ A )] : % 8.50/4.53 ( ( distinct @ ( list @ A ) @ Xs ) % 8.50/4.53 => ( ! [Ys4: list @ A] : % 8.50/4.53 ( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) ) % 8.50/4.53 => ( distinct @ A @ Ys4 ) ) % 8.50/4.53 => ( ! [Ys4: list @ A,Zs: list @ A] : % 8.50/4.53 ( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) ) % 8.50/4.53 => ( ( member @ ( list @ A ) @ Zs @ ( set2 @ ( list @ A ) @ Xs ) ) % 8.50/4.53 => ( ( Ys4 != Zs ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys4 ) @ ( set2 @ A @ Zs ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ) ) % 8.50/4.53 => ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4387_mask__Suc__exp,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [N2: nat] : % 8.50/4.53 ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N2 ) ) % 8.50/4.53 = ( bit_se1065995026697491101ons_or @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4388_bezw__aux,axiom, % 8.50/4.53 ! [X3: nat,Y: nat] : % 8.50/4.53 ( ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ X3 @ Y ) ) % 8.50/4.53 = ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ ( bezw @ X3 @ Y ) ) @ ( semiring_1_of_nat @ int @ X3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ X3 @ Y ) ) @ ( semiring_1_of_nat @ int @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4389_one__or__eq,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [A2: A] : % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A2 ) % 8.50/4.53 = ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4390_or__one__eq,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [A2: A] : % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( one_one @ A ) ) % 8.50/4.53 = ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4391_mask__Suc__double,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [N2: nat] : % 8.50/4.53 ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N2 ) ) % 8.50/4.53 = ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4392_OR__upper,axiom, % 8.50/4.53 ! [X3: int,N2: nat,Y: int] : % 8.50/4.53 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 ) % 8.50/4.53 => ( ( ord_less @ int @ X3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.50/4.53 => ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) % 8.50/4.53 => ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X3 @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4393_or__Suc__0__eq,axiom, % 8.50/4.53 ! [N2: nat] : % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.53 = ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4394_Suc__0__or__eq,axiom, % 8.50/4.53 ! [N2: nat] : % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) % 8.50/4.53 = ( plus_plus @ nat @ N2 @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4395_or__nat__rec,axiom, % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ nat ) % 8.50/4.53 = ( ^ [M2: nat,N: nat] : % 8.50/4.53 ( plus_plus @ nat % 8.50/4.53 @ ( zero_neq_one_of_bool @ nat % 8.50/4.53 @ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) % 8.50/4.53 | ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) % 8.50/4.53 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4396_gcd__nat_Opelims,axiom, % 8.50/4.53 ! [X3: nat,Xa2: nat,Y: nat] : % 8.50/4.53 ( ( ( gcd_gcd @ nat @ X3 @ Xa2 ) % 8.50/4.53 = Y ) % 8.50/4.53 => ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) % 8.50/4.53 => ~ ( ( ( ( Xa2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 => ( Y = X3 ) ) % 8.50/4.53 & ( ( Xa2 % 8.50/4.53 != ( zero_zero @ nat ) ) % 8.50/4.53 => ( Y % 8.50/4.53 = ( gcd_gcd @ nat @ Xa2 @ ( modulo_modulo @ nat @ X3 @ Xa2 ) ) ) ) ) % 8.50/4.53 => ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4397_inf__bot__left,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bounded_lattice_bot @ A ) % 8.50/4.53 => ! [X3: A] : % 8.50/4.53 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X3 ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4398_inf__bot__right,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bounded_lattice_bot @ A ) % 8.50/4.53 => ! [X3: A] : % 8.50/4.53 ( ( inf_inf @ A @ X3 @ ( bot_bot @ A ) ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4399_inf_Obounded__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) % 8.50/4.53 = ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.53 & ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4400_gcd__nat_Oidem,axiom, % 8.50/4.53 ! [A2: nat] : % 8.50/4.53 ( ( gcd_gcd @ nat @ A2 @ A2 ) % 8.50/4.53 = A2 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4401_gcd__nat_Oleft__idem,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : % 8.50/4.53 ( ( gcd_gcd @ nat @ A2 @ ( gcd_gcd @ nat @ A2 @ B2 ) ) % 8.50/4.53 = ( gcd_gcd @ nat @ A2 @ B2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4402_gcd__nat_Oright__idem,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : % 8.50/4.53 ( ( gcd_gcd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 ) % 8.50/4.53 = ( gcd_gcd @ nat @ A2 @ B2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4403_gcd__idem__nat,axiom, % 8.50/4.53 ! [X3: nat] : % 8.50/4.53 ( ( gcd_gcd @ nat @ X3 @ X3 ) % 8.50/4.53 = X3 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4404_le__inf__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [X3: A,Y: A,Z2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) ) % 8.50/4.53 = ( ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.53 & ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4405_inf__set__def,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( inf_inf @ ( set @ A ) ) % 8.50/4.53 = ( ^ [A6: set @ A,B6: set @ A] : % 8.50/4.53 ( collect @ A % 8.50/4.53 @ ( inf_inf @ ( A > $o ) % 8.50/4.53 @ ^ [X: A] : ( member @ A @ X @ A6 ) % 8.50/4.53 @ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4406_inf__Int__eq2,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] : % 8.50/4.53 ( ( inf_inf @ ( A > B > $o ) % 8.50/4.53 @ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R ) % 8.50/4.53 @ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S3 ) ) % 8.50/4.53 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4407_inf_OcoboundedI2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [B2: A,C2: A,A2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ B2 @ C2 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4408_inf_OcoboundedI1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [A2: A,C2: A,B2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ C2 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4409_inf_Oabsorb__iff2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ( ( ord_less_eq @ A ) % 8.50/4.53 = ( ^ [B4: A,A4: A] : % 8.50/4.53 ( ( inf_inf @ A @ A4 @ B4 ) % 8.50/4.53 = B4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4410_inf_Oabsorb__iff1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ( ( ord_less_eq @ A ) % 8.50/4.53 = ( ^ [A4: A,B4: A] : % 8.50/4.53 ( ( inf_inf @ A @ A4 @ B4 ) % 8.50/4.53 = A4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4411_inf_Ocobounded2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4412_inf_Ocobounded1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ A2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4413_inf_Oorder__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ( ( ord_less_eq @ A ) % 8.50/4.53 = ( ^ [A4: A,B4: A] : % 8.50/4.53 ( A4 % 8.50/4.53 = ( inf_inf @ A @ A4 @ B4 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4414_inf__greatest,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [X3: A,Y: A,Z2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.53 => ( ( ord_less_eq @ A @ X3 @ Z2 ) % 8.50/4.53 => ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4415_inf_OboundedI,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.53 => ( ( ord_less_eq @ A @ A2 @ C2 ) % 8.50/4.53 => ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4416_inf_OboundedE,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) % 8.50/4.53 => ~ ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.53 => ~ ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4417_inf__absorb2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [Y: A,X3: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ Y @ X3 ) % 8.50/4.53 => ( ( inf_inf @ A @ X3 @ Y ) % 8.50/4.53 = Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4418_inf__absorb1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.53 => ( ( inf_inf @ A @ X3 @ Y ) % 8.50/4.53 = X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4419_inf_Oabsorb2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [B2: A,A2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.50/4.53 => ( ( inf_inf @ A @ A2 @ B2 ) % 8.50/4.53 = B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4420_inf_Oabsorb1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.53 => ( ( inf_inf @ A @ A2 @ B2 ) % 8.50/4.53 = A2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4421_le__iff__inf,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ( ( ord_less_eq @ A ) % 8.50/4.53 = ( ^ [X: A,Y2: A] : % 8.50/4.53 ( ( inf_inf @ A @ X @ Y2 ) % 8.50/4.53 = X ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4422_inf__unique,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [F2: A > A > A,X3: A,Y: A] : % 8.50/4.53 ( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ ( F2 @ X5 @ Y4 ) @ X5 ) % 8.50/4.53 => ( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ ( F2 @ X5 @ Y4 ) @ Y4 ) % 8.50/4.53 => ( ! [X5: A,Y4: A,Z3: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ X5 @ Y4 ) % 8.50/4.53 => ( ( ord_less_eq @ A @ X5 @ Z3 ) % 8.50/4.53 => ( ord_less_eq @ A @ X5 @ ( F2 @ Y4 @ Z3 ) ) ) ) % 8.50/4.53 => ( ( inf_inf @ A @ X3 @ Y ) % 8.50/4.53 = ( F2 @ X3 @ Y ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4423_inf_OorderI,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( A2 % 8.50/4.53 = ( inf_inf @ A @ A2 @ B2 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4424_inf_OorderE,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.53 => ( A2 % 8.50/4.53 = ( inf_inf @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4425_le__infI2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [B2: A,X3: A,A2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ B2 @ X3 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4426_le__infI1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [A2: A,X3: A,B2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ X3 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4427_inf__mono,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [A2: A,C2: A,B2: A,D2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ C2 ) % 8.50/4.53 => ( ( ord_less_eq @ A @ B2 @ D2 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ ( inf_inf @ A @ C2 @ D2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4428_le__infI,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [X3: A,A2: A,B2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ X3 @ A2 ) % 8.50/4.53 => ( ( ord_less_eq @ A @ X3 @ B2 ) % 8.50/4.53 => ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4429_le__infE,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [X3: A,A2: A,B2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A2 @ B2 ) ) % 8.50/4.53 => ~ ( ( ord_less_eq @ A @ X3 @ A2 ) % 8.50/4.53 => ~ ( ord_less_eq @ A @ X3 @ B2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4430_inf__le2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4431_inf__le1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4432_inf__sup__ord_I1_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( lattice @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4433_inf__sup__ord_I2_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( lattice @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4434_less__infI1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [A2: A,X3: A,B2: A] : % 8.50/4.53 ( ( ord_less @ A @ A2 @ X3 ) % 8.50/4.53 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4435_less__infI2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [B2: A,X3: A,A2: A] : % 8.50/4.53 ( ( ord_less @ A @ B2 @ X3 ) % 8.50/4.53 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4436_inf_Oabsorb3,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.53 => ( ( inf_inf @ A @ A2 @ B2 ) % 8.50/4.53 = A2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4437_inf_Oabsorb4,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [B2: A,A2: A] : % 8.50/4.53 ( ( ord_less @ A @ B2 @ A2 ) % 8.50/4.53 => ( ( inf_inf @ A @ A2 @ B2 ) % 8.50/4.53 = B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4438_inf_Ostrict__boundedE,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( ord_less @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) % 8.50/4.53 => ~ ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.53 => ~ ( ord_less @ A @ A2 @ C2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4439_inf_Ostrict__order__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ( ( ord_less @ A ) % 8.50/4.53 = ( ^ [A4: A,B4: A] : % 8.50/4.53 ( ( A4 % 8.50/4.53 = ( inf_inf @ A @ A4 @ B4 ) ) % 8.50/4.53 & ( A4 != B4 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4440_inf_Ostrict__coboundedI1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [A2: A,C2: A,B2: A] : % 8.50/4.53 ( ( ord_less @ A @ A2 @ C2 ) % 8.50/4.53 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4441_inf_Ostrict__coboundedI2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semilattice_inf @ A ) % 8.50/4.53 => ! [B2: A,C2: A,A2: A] : % 8.50/4.53 ( ( ord_less @ A @ B2 @ C2 ) % 8.50/4.53 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4442_finite__inf__Sup,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( finite8700451911770168679attice @ A ) % 8.50/4.53 => ! [A2: A,A3: set @ A] : % 8.50/4.53 ( ( inf_inf @ A @ A2 @ ( complete_Sup_Sup @ A @ A3 ) ) % 8.50/4.53 = ( complete_Sup_Sup @ A % 8.50/4.53 @ ( collect @ A % 8.50/4.53 @ ^ [Uu3: A] : % 8.50/4.53 ? [B4: A] : % 8.50/4.53 ( ( Uu3 % 8.50/4.53 = ( inf_inf @ A @ A2 @ B4 ) ) % 8.50/4.53 & ( member @ A @ B4 @ A3 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4443_or__nat__unfold,axiom, % 8.50/4.53 ( ( bit_se1065995026697491101ons_or @ nat ) % 8.50/4.53 = ( ^ [M2: nat,N: nat] : % 8.50/4.53 ( if @ nat % 8.50/4.53 @ ( M2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 @ N % 8.50/4.53 @ ( if @ nat % 8.50/4.53 @ ( N % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 @ M2 % 8.50/4.53 @ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4444_card__disjoint__shuffles,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 8.50/4.53 ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) % 8.50/4.53 = ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4445_distinct__product__lists,axiom, % 8.50/4.53 ! [A: $tType,Xss: list @ ( list @ A )] : % 8.50/4.53 ( ! [X5: list @ A] : % 8.50/4.53 ( ( member @ ( list @ A ) @ X5 @ ( set2 @ ( list @ A ) @ Xss ) ) % 8.50/4.53 => ( distinct @ A @ X5 ) ) % 8.50/4.53 => ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4446_max_Oabsorb1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [B2: A,A2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.50/4.53 => ( ( ord_max @ A @ A2 @ B2 ) % 8.50/4.53 = A2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4447_max_Oabsorb2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.53 => ( ( ord_max @ A @ A2 @ B2 ) % 8.50/4.53 = B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4448_max_Obounded__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [B2: A,C2: A,A2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 ) % 8.50/4.53 = ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.50/4.53 & ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4449_max_Oabsorb3,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [B2: A,A2: A] : % 8.50/4.53 ( ( ord_less @ A @ B2 @ A2 ) % 8.50/4.53 => ( ( ord_max @ A @ A2 @ B2 ) % 8.50/4.53 = A2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4450_max_Oabsorb4,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.53 => ( ( ord_max @ A @ A2 @ B2 ) % 8.50/4.53 = B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4451_max__less__iff__conj,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [X3: A,Y: A,Z2: A] : % 8.50/4.53 ( ( ord_less @ A @ ( ord_max @ A @ X3 @ Y ) @ Z2 ) % 8.50/4.53 = ( ( ord_less @ A @ X3 @ Z2 ) % 8.50/4.53 & ( ord_less @ A @ Y @ Z2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4452_max__bot2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( order_bot @ A ) % 8.50/4.53 => ! [X3: A] : % 8.50/4.53 ( ( ord_max @ A @ X3 @ ( bot_bot @ A ) ) % 8.50/4.53 = X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4453_max__bot,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( order_bot @ A ) % 8.50/4.53 => ! [X3: A] : % 8.50/4.53 ( ( ord_max @ A @ ( bot_bot @ A ) @ X3 ) % 8.50/4.53 = X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4454_of__bool__or__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linordered_semidom @ A ) % 8.50/4.53 => ! [P: $o,Q: $o] : % 8.50/4.53 ( ( zero_neq_one_of_bool @ A % 8.50/4.53 @ ( P % 8.50/4.53 | Q ) ) % 8.50/4.53 = ( ord_max @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4455_max__Suc__Suc,axiom, % 8.50/4.53 ! [M: nat,N2: nat] : % 8.50/4.53 ( ( ord_max @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) % 8.50/4.53 = ( suc @ ( ord_max @ nat @ M @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4456_max__0R,axiom, % 8.50/4.53 ! [N2: nat] : % 8.50/4.53 ( ( ord_max @ nat @ N2 @ ( zero_zero @ nat ) ) % 8.50/4.53 = N2 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4457_max__0L,axiom, % 8.50/4.53 ! [N2: nat] : % 8.50/4.53 ( ( ord_max @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 = N2 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4458_max__nat_Oright__neutral,axiom, % 8.50/4.53 ! [A2: nat] : % 8.50/4.53 ( ( ord_max @ nat @ A2 @ ( zero_zero @ nat ) ) % 8.50/4.53 = A2 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4459_max__nat_Oneutr__eq__iff,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : % 8.50/4.53 ( ( ( zero_zero @ nat ) % 8.50/4.53 = ( ord_max @ nat @ A2 @ B2 ) ) % 8.50/4.53 = ( ( A2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 & ( B2 % 8.50/4.53 = ( zero_zero @ nat ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4460_max__nat_Oleft__neutral,axiom, % 8.50/4.53 ! [A2: nat] : % 8.50/4.53 ( ( ord_max @ nat @ ( zero_zero @ nat ) @ A2 ) % 8.50/4.53 = A2 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4461_max__nat_Oeq__neutr__iff,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : % 8.50/4.53 ( ( ( ord_max @ nat @ A2 @ B2 ) % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 = ( ( A2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 & ( B2 % 8.50/4.53 = ( zero_zero @ nat ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4462_finite__shuffles,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A,Ys: list @ A] : ( finite_finite2 @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4463_max__number__of_I1_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ( numeral @ A ) % 8.50/4.53 & ( ord @ A ) ) % 8.50/4.53 => ! [U: num,V2: num] : % 8.50/4.53 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 = ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 = ( numeral_numeral @ A @ U ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4464_max__0__1_I4_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linord181362715937106298miring @ A ) % 8.50/4.53 => ! [X3: num] : % 8.50/4.53 ( ( ord_max @ A @ ( numeral_numeral @ A @ X3 ) @ ( zero_zero @ A ) ) % 8.50/4.53 = ( numeral_numeral @ A @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4465_max__0__1_I3_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linord181362715937106298miring @ A ) % 8.50/4.53 => ! [X3: num] : % 8.50/4.53 ( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X3 ) ) % 8.50/4.53 = ( numeral_numeral @ A @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4466_max__0__1_I1_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linord181362715937106298miring @ A ) % 8.50/4.53 => ( ( ord_max @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) % 8.50/4.53 = ( one_one @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4467_max__0__1_I2_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linord181362715937106298miring @ A ) % 8.50/4.53 => ( ( ord_max @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) % 8.50/4.53 = ( one_one @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4468_max__number__of_I2_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ( uminus @ A ) % 8.50/4.53 & ( numeral @ A ) % 8.50/4.53 & ( ord @ A ) ) % 8.50/4.53 => ! [U: num,V2: num] : % 8.50/4.53 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) % 8.50/4.53 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 => ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 = ( numeral_numeral @ A @ U ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4469_max__number__of_I3_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ( uminus @ A ) % 8.50/4.53 & ( numeral @ A ) % 8.50/4.53 & ( ord @ A ) ) % 8.50/4.53 => ! [U: num,V2: num] : % 8.50/4.53 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 = ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4470_max__number__of_I4_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ( uminus @ A ) % 8.50/4.53 & ( numeral @ A ) % 8.50/4.53 & ( ord @ A ) ) % 8.50/4.53 => ! [U: num,V2: num] : % 8.50/4.53 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) % 8.50/4.53 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 => ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4471_max__Suc__numeral,axiom, % 8.50/4.53 ! [N2: nat,K: num] : % 8.50/4.53 ( ( ord_max @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) ) % 8.50/4.53 = ( suc @ ( ord_max @ nat @ N2 @ ( pred_numeral @ K ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4472_max__numeral__Suc,axiom, % 8.50/4.53 ! [K: num,N2: nat] : % 8.50/4.53 ( ( ord_max @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) ) % 8.50/4.53 = ( suc @ ( ord_max @ nat @ ( pred_numeral @ K ) @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4473_max_Ostrict__coboundedI2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [C2: A,B2: A,A2: A] : % 8.50/4.53 ( ( ord_less @ A @ C2 @ B2 ) % 8.50/4.53 => ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4474_max_Ostrict__coboundedI1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [C2: A,A2: A,B2: A] : % 8.50/4.53 ( ( ord_less @ A @ C2 @ A2 ) % 8.50/4.53 => ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4475_max_Ostrict__order__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ( ( ord_less @ A ) % 8.50/4.53 = ( ^ [B4: A,A4: A] : % 8.50/4.53 ( ( A4 % 8.50/4.53 = ( ord_max @ A @ A4 @ B4 ) ) % 8.50/4.53 & ( A4 != B4 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4476_max_Ostrict__boundedE,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [B2: A,C2: A,A2: A] : % 8.50/4.53 ( ( ord_less @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 ) % 8.50/4.53 => ~ ( ( ord_less @ A @ B2 @ A2 ) % 8.50/4.53 => ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4477_less__max__iff__disj,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [Z2: A,X3: A,Y: A] : % 8.50/4.53 ( ( ord_less @ A @ Z2 @ ( ord_max @ A @ X3 @ Y ) ) % 8.50/4.53 = ( ( ord_less @ A @ Z2 @ X3 ) % 8.50/4.53 | ( ord_less @ A @ Z2 @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4478_max_Omono,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [C2: A,A2: A,D2: A,B2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ C2 @ A2 ) % 8.50/4.53 => ( ( ord_less_eq @ A @ D2 @ B2 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D2 ) @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4479_max_OorderE,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [B2: A,A2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.50/4.53 => ( A2 % 8.50/4.53 = ( ord_max @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4480_max_OorderI,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( A2 % 8.50/4.53 = ( ord_max @ A @ A2 @ B2 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4481_max_OboundedE,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [B2: A,C2: A,A2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 ) % 8.50/4.53 => ~ ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.50/4.53 => ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4482_max_OboundedI,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [B2: A,A2: A,C2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.50/4.53 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4483_max_Oorder__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ( ( ord_less_eq @ A ) % 8.50/4.53 = ( ^ [B4: A,A4: A] : % 8.50/4.53 ( A4 % 8.50/4.53 = ( ord_max @ A @ A4 @ B4 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4484_max_Ocobounded1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4485_max_Ocobounded2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4486_le__max__iff__disj,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [Z2: A,X3: A,Y: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ Z2 @ ( ord_max @ A @ X3 @ Y ) ) % 8.50/4.53 = ( ( ord_less_eq @ A @ Z2 @ X3 ) % 8.50/4.53 | ( ord_less_eq @ A @ Z2 @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4487_max_Oabsorb__iff1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ( ( ord_less_eq @ A ) % 8.50/4.53 = ( ^ [B4: A,A4: A] : % 8.50/4.53 ( ( ord_max @ A @ A4 @ B4 ) % 8.50/4.53 = A4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4488_max_Oabsorb__iff2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ( ( ord_less_eq @ A ) % 8.50/4.53 = ( ^ [A4: A,B4: A] : % 8.50/4.53 ( ( ord_max @ A @ A4 @ B4 ) % 8.50/4.53 = B4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4489_max_OcoboundedI1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [C2: A,A2: A,B2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ C2 @ A2 ) % 8.50/4.53 => ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4490_max_OcoboundedI2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [C2: A,B2: A,A2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ C2 @ B2 ) % 8.50/4.53 => ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4491_max__def__raw,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ord @ A ) % 8.50/4.53 => ( ( ord_max @ A ) % 8.50/4.53 = ( ^ [A4: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B4 ) @ B4 @ A4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4492_max__def,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ord @ A ) % 8.50/4.53 => ( ( ord_max @ A ) % 8.50/4.53 = ( ^ [A4: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B4 ) @ B4 @ A4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4493_max__absorb1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( order @ A ) % 8.50/4.53 => ! [Y: A,X3: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ Y @ X3 ) % 8.50/4.53 => ( ( ord_max @ A @ X3 @ Y ) % 8.50/4.53 = X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4494_max__absorb2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ord @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.53 => ( ( ord_max @ A @ X3 @ Y ) % 8.50/4.53 = Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4495_of__nat__max,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linord181362715937106298miring @ A ) % 8.50/4.53 => ! [X3: nat,Y: nat] : % 8.50/4.53 ( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X3 @ Y ) ) % 8.50/4.53 = ( ord_max @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4496_shuffles__commutes,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( shuffles @ A ) % 8.50/4.53 = ( ^ [Xs2: list @ A,Ys3: list @ A] : ( shuffles @ A @ Ys3 @ Xs2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4497_of__int__max,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linordered_idom @ A ) % 8.50/4.53 => ! [X3: int,Y: int] : % 8.50/4.53 ( ( ring_1_of_int @ A @ ( ord_max @ int @ X3 @ Y ) ) % 8.50/4.53 = ( ord_max @ A @ ( ring_1_of_int @ A @ X3 ) @ ( ring_1_of_int @ A @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4498_nat__add__max__right,axiom, % 8.50/4.53 ! [M: nat,N2: nat,Q4: nat] : % 8.50/4.53 ( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N2 @ Q4 ) ) % 8.50/4.53 = ( ord_max @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( plus_plus @ nat @ M @ Q4 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4499_nat__add__max__left,axiom, % 8.50/4.53 ! [M: nat,N2: nat,Q4: nat] : % 8.50/4.53 ( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N2 ) @ Q4 ) % 8.50/4.53 = ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q4 ) @ ( plus_plus @ nat @ N2 @ Q4 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4500_max__add__distrib__right,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ordere2412721322843649153imp_le @ A ) % 8.50/4.53 => ! [X3: A,Y: A,Z2: A] : % 8.50/4.53 ( ( plus_plus @ A @ X3 @ ( ord_max @ A @ Y @ Z2 ) ) % 8.50/4.53 = ( ord_max @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( plus_plus @ A @ X3 @ Z2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4501_max__add__distrib__left,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ordere2412721322843649153imp_le @ A ) % 8.50/4.53 => ! [X3: A,Y: A,Z2: A] : % 8.50/4.53 ( ( plus_plus @ A @ ( ord_max @ A @ X3 @ Y ) @ Z2 ) % 8.50/4.53 = ( ord_max @ A @ ( plus_plus @ A @ X3 @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4502_max__diff__distrib__left,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ordered_ab_group_add @ A ) % 8.50/4.53 => ! [X3: A,Y: A,Z2: A] : % 8.50/4.53 ( ( minus_minus @ A @ ( ord_max @ A @ X3 @ Y ) @ Z2 ) % 8.50/4.53 = ( ord_max @ A @ ( minus_minus @ A @ X3 @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4503_nat__mult__max__left,axiom, % 8.50/4.53 ! [M: nat,N2: nat,Q4: nat] : % 8.50/4.53 ( ( times_times @ nat @ ( ord_max @ nat @ M @ N2 ) @ Q4 ) % 8.50/4.53 = ( ord_max @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N2 @ Q4 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4504_nat__mult__max__right,axiom, % 8.50/4.53 ! [M: nat,N2: nat,Q4: nat] : % 8.50/4.53 ( ( times_times @ nat @ M @ ( ord_max @ nat @ N2 @ Q4 ) ) % 8.50/4.53 = ( ord_max @ nat @ ( times_times @ nat @ M @ N2 ) @ ( times_times @ nat @ M @ Q4 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4505_nat__minus__add__max,axiom, % 8.50/4.53 ! [N2: nat,M: nat] : % 8.50/4.53 ( ( plus_plus @ nat @ ( minus_minus @ nat @ N2 @ M ) @ M ) % 8.50/4.53 = ( ord_max @ nat @ N2 @ M ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4506_length__shuffles,axiom, % 8.50/4.53 ! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys: list @ A] : % 8.50/4.53 ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) ) % 8.50/4.53 => ( ( size_size @ ( list @ A ) @ Zs2 ) % 8.50/4.53 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4507_max__Suc2,axiom, % 8.50/4.53 ! [M: nat,N2: nat] : % 8.50/4.53 ( ( ord_max @ nat @ M @ ( suc @ N2 ) ) % 8.50/4.53 = ( case_nat @ nat @ ( suc @ N2 ) % 8.50/4.53 @ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ M4 @ N2 ) ) % 8.50/4.53 @ M ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4508_max__Suc1,axiom, % 8.50/4.53 ! [N2: nat,M: nat] : % 8.50/4.53 ( ( ord_max @ nat @ ( suc @ N2 ) @ M ) % 8.50/4.53 = ( case_nat @ nat @ ( suc @ N2 ) % 8.50/4.53 @ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ N2 @ M4 ) ) % 8.50/4.53 @ M ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4509_Sup__insert__finite,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit6923001295902523014norder @ A ) % 8.50/4.53 => ! [S3: set @ A,X3: A] : % 8.50/4.53 ( ( finite_finite2 @ A @ S3 ) % 8.50/4.53 => ( ( ( S3 % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ( complete_Sup_Sup @ A @ ( insert @ A @ X3 @ S3 ) ) % 8.50/4.53 = X3 ) ) % 8.50/4.53 & ( ( S3 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ( complete_Sup_Sup @ A @ ( insert @ A @ X3 @ S3 ) ) % 8.50/4.53 = ( ord_max @ A @ X3 @ ( complete_Sup_Sup @ A @ S3 ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4510_distinct__disjoint__shuffles,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] : % 8.50/4.53 ( ( distinct @ A @ Xs ) % 8.50/4.53 => ( ( distinct @ A @ Ys ) % 8.50/4.53 => ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ( member @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys ) ) % 8.50/4.53 => ( distinct @ A @ Zs2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4511_in__set__product__lists__length,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A,Xss: list @ ( list @ A )] : % 8.50/4.53 ( ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) ) % 8.50/4.53 => ( ( size_size @ ( list @ A ) @ Xs ) % 8.50/4.53 = ( size_size @ ( list @ ( list @ A ) ) @ Xss ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4512_mlex__eq,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( mlex_prod @ A ) % 8.50/4.53 = ( ^ [F3: A > nat,R6: set @ ( product_prod @ A @ A )] : % 8.50/4.53 ( collect @ ( product_prod @ A @ A ) % 8.50/4.53 @ ( product_case_prod @ A @ A @ $o % 8.50/4.53 @ ^ [X: A,Y2: A] : % 8.50/4.53 ( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y2 ) ) % 8.50/4.53 | ( ( ord_less_eq @ nat @ ( F3 @ X ) @ ( F3 @ Y2 ) ) % 8.50/4.53 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R6 ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4513_divmod__integer__eq__cases,axiom, % 8.50/4.53 ( code_divmod_integer % 8.50/4.53 = ( ^ [K3: code_integer,L2: code_integer] : % 8.50/4.53 ( if @ ( product_prod @ code_integer @ code_integer ) % 8.50/4.53 @ ( K3 % 8.50/4.53 = ( zero_zero @ code_integer ) ) % 8.50/4.53 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) % 8.50/4.53 @ ( if @ ( product_prod @ code_integer @ code_integer ) % 8.50/4.53 @ ( L2 % 8.50/4.53 = ( zero_zero @ code_integer ) ) % 8.50/4.53 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) % 8.50/4.53 @ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( product_apsnd @ code_integer @ code_integer @ code_integer ) @ ( times_times @ code_integer ) ) @ ( sgn_sgn @ code_integer ) @ L2 % 8.50/4.53 @ ( if @ ( product_prod @ code_integer @ code_integer ) % 8.50/4.53 @ ( ( sgn_sgn @ code_integer @ K3 ) % 8.50/4.53 = ( sgn_sgn @ code_integer @ L2 ) ) % 8.50/4.53 @ ( code_divmod_abs @ K3 @ L2 ) % 8.50/4.53 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer ) % 8.50/4.53 @ ^ [R5: code_integer,S8: code_integer] : % 8.50/4.53 ( if @ ( product_prod @ code_integer @ code_integer ) % 8.50/4.53 @ ( S8 % 8.50/4.53 = ( zero_zero @ code_integer ) ) % 8.50/4.53 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) ) % 8.50/4.53 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L2 ) @ S8 ) ) ) % 8.50/4.53 @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4514_totally__bounded__Union,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( topolo7287701948861334536_space @ A ) % 8.50/4.53 => ! [M7: set @ ( set @ A )] : % 8.50/4.53 ( ( finite_finite2 @ ( set @ A ) @ M7 ) % 8.50/4.53 => ( ! [S5: set @ A] : % 8.50/4.53 ( ( member @ ( set @ A ) @ S5 @ M7 ) % 8.50/4.53 => ( topolo6688025880775521714ounded @ A @ S5 ) ) % 8.50/4.53 => ( topolo6688025880775521714ounded @ A @ ( complete_Sup_Sup @ ( set @ A ) @ M7 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4515_nth__rotate1,axiom, % 8.50/4.53 ! [A: $tType,N2: nat,Xs: list @ A] : % 8.50/4.53 ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.53 => ( ( nth @ A @ ( rotate1 @ A @ Xs ) @ N2 ) % 8.50/4.53 = ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( suc @ N2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4516_set__rotate1,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A] : % 8.50/4.53 ( ( set2 @ A @ ( rotate1 @ A @ Xs ) ) % 8.50/4.53 = ( set2 @ A @ Xs ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4517_length__rotate1,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A] : % 8.50/4.53 ( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) ) % 8.50/4.53 = ( size_size @ ( list @ A ) @ Xs ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4518_distinct1__rotate,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A] : % 8.50/4.53 ( ( distinct @ A @ ( rotate1 @ A @ Xs ) ) % 8.50/4.53 = ( distinct @ A @ Xs ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4519_totally__bounded__empty,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( topolo7287701948861334536_space @ A ) % 8.50/4.53 => ( topolo6688025880775521714ounded @ A @ ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4520_rotate1__length01,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A] : % 8.50/4.53 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) % 8.50/4.53 => ( ( rotate1 @ A @ Xs ) % 8.50/4.53 = Xs ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4521_apsnd__compose,axiom, % 8.50/4.53 ! [C: $tType,B: $tType,D: $tType,A: $tType,F2: C > B,G: D > C,X3: product_prod @ A @ D] : % 8.50/4.53 ( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apsnd @ D @ C @ A @ G @ X3 ) ) % 8.50/4.53 = ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F2 @ G ) @ X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4522_totally__bounded__subset,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( topolo7287701948861334536_space @ A ) % 8.50/4.53 => ! [S3: set @ A,T4: set @ A] : % 8.50/4.53 ( ( topolo6688025880775521714ounded @ A @ S3 ) % 8.50/4.53 => ( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 ) % 8.50/4.53 => ( topolo6688025880775521714ounded @ A @ T4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4523_sum__comp__morphism,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,C: $tType] : % 8.50/4.53 ( ( ( comm_monoid_add @ B ) % 8.50/4.53 & ( comm_monoid_add @ A ) ) % 8.50/4.53 => ! [H2: B > A,G: C > B,A3: set @ C] : % 8.50/4.53 ( ( ( H2 @ ( zero_zero @ B ) ) % 8.50/4.53 = ( zero_zero @ A ) ) % 8.50/4.53 => ( ! [X5: B,Y4: B] : % 8.50/4.53 ( ( H2 @ ( plus_plus @ B @ X5 @ Y4 ) ) % 8.50/4.53 = ( plus_plus @ A @ ( H2 @ X5 ) @ ( H2 @ Y4 ) ) ) % 8.50/4.53 => ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H2 @ G ) @ A3 ) % 8.50/4.53 = ( H2 @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A3 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4524_case__prod__comp,axiom, % 8.50/4.53 ! [D: $tType,A: $tType,C: $tType,B: $tType,F2: D > C > A,G: B > D,X3: product_prod @ B @ C] : % 8.50/4.53 ( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F2 @ G ) @ X3 ) % 8.50/4.53 = ( F2 @ ( G @ ( product_fst @ B @ C @ X3 ) ) @ ( product_snd @ B @ C @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4525_mlex__leq,axiom, % 8.50/4.53 ! [A: $tType,F2: A > nat,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] : % 8.50/4.53 ( ( ord_less_eq @ nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) % 8.50/4.53 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R ) % 8.50/4.53 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( mlex_prod @ A @ F2 @ R ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4526_mlex__less,axiom, % 8.50/4.53 ! [A: $tType,F2: A > nat,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] : % 8.50/4.53 ( ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) % 8.50/4.53 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( mlex_prod @ A @ F2 @ R ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4527_mlex__iff,axiom, % 8.50/4.53 ! [A: $tType,X3: A,Y: A,F2: A > nat,R: set @ ( product_prod @ A @ A )] : % 8.50/4.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( mlex_prod @ A @ F2 @ R ) ) % 8.50/4.53 = ( ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) % 8.50/4.53 | ( ( ( F2 @ X3 ) % 8.50/4.53 = ( F2 @ Y ) ) % 8.50/4.53 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4528_set__remove1__eq,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A,X3: A] : % 8.50/4.53 ( ( distinct @ A @ Xs ) % 8.50/4.53 => ( ( set2 @ A @ ( remove1 @ A @ X3 @ Xs ) ) % 8.50/4.53 = ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4529_in__set__remove1,axiom, % 8.50/4.53 ! [A: $tType,A2: A,B2: A,Xs: list @ A] : % 8.50/4.53 ( ( A2 != B2 ) % 8.50/4.53 => ( ( member @ A @ A2 @ ( set2 @ A @ ( remove1 @ A @ B2 @ Xs ) ) ) % 8.50/4.53 = ( member @ A @ A2 @ ( set2 @ A @ Xs ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4530_fst__comp__apsnd,axiom, % 8.50/4.53 ! [C: $tType,B: $tType,A: $tType,F2: B > C] : % 8.50/4.53 ( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 ) ) % 8.50/4.53 = ( product_fst @ A @ B ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4531_snd__comp__apsnd,axiom, % 8.50/4.53 ! [C: $tType,B: $tType,A: $tType,F2: B > C] : % 8.50/4.53 ( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F2 ) ) % 8.50/4.53 = ( comp @ B @ C @ ( product_prod @ A @ B ) @ F2 @ ( product_snd @ A @ B ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4532_fst__diag__fst,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A ) % 8.50/4.53 @ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B ) % 8.50/4.53 @ ^ [X: A] : ( product_Pair @ A @ A @ X @ X ) % 8.50/4.53 @ ( product_fst @ A @ B ) ) ) % 8.50/4.53 = ( product_fst @ A @ B ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4533_snd__diag__snd,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B ) % 8.50/4.53 @ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B ) % 8.50/4.53 @ ^ [X: B] : ( product_Pair @ B @ B @ X @ X ) % 8.50/4.53 @ ( product_snd @ A @ B ) ) ) % 8.50/4.53 = ( product_snd @ A @ B ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4534_card_Ocomp__fun__commute__on,axiom, % 8.50/4.53 ( ( comp @ nat @ nat @ nat @ suc @ suc ) % 8.50/4.53 = ( comp @ nat @ nat @ nat @ suc @ suc ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4535_fst__diag__snd,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B ) % 8.50/4.53 @ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B ) % 8.50/4.53 @ ^ [X: B] : ( product_Pair @ B @ B @ X @ X ) % 8.50/4.53 @ ( product_snd @ A @ B ) ) ) % 8.50/4.53 = ( product_snd @ A @ B ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4536_snd__diag__fst,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A ) % 8.50/4.53 @ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B ) % 8.50/4.53 @ ^ [X: A] : ( product_Pair @ A @ A @ X @ X ) % 8.50/4.53 @ ( product_fst @ A @ B ) ) ) % 8.50/4.53 = ( product_fst @ A @ B ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4537_distinct__remove1,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A,X3: A] : % 8.50/4.53 ( ( distinct @ A @ Xs ) % 8.50/4.53 => ( distinct @ A @ ( remove1 @ A @ X3 @ Xs ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4538_notin__set__remove1,axiom, % 8.50/4.53 ! [A: $tType,X3: A,Xs: list @ A,Y: A] : % 8.50/4.53 ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.50/4.53 => ~ ( member @ A @ X3 @ ( set2 @ A @ ( remove1 @ A @ Y @ Xs ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4539_remove1__idem,axiom, % 8.50/4.53 ! [A: $tType,X3: A,Xs: list @ A] : % 8.50/4.53 ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.50/4.53 => ( ( remove1 @ A @ X3 @ Xs ) % 8.50/4.53 = Xs ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4540_remove1__commute,axiom, % 8.50/4.53 ! [A: $tType,X3: A,Y: A,Zs2: list @ A] : % 8.50/4.53 ( ( remove1 @ A @ X3 @ ( remove1 @ A @ Y @ Zs2 ) ) % 8.50/4.53 = ( remove1 @ A @ Y @ ( remove1 @ A @ X3 @ Zs2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4541_set__remove1__subset,axiom, % 8.50/4.53 ! [A: $tType,X3: A,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( remove1 @ A @ X3 @ Xs ) ) @ ( set2 @ A @ Xs ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4542_sum_OatLeast__Suc__atMost__Suc__shift,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.53 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 8.50/4.53 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4543_sum_OatLeast__Suc__lessThan__Suc__shift,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.53 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 8.50/4.53 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4544_sum_OatLeastAtMost__shift__bounds,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 8.50/4.53 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 8.50/4.53 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4545_sum_OatLeastLessThan__shift__bounds,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 8.50/4.53 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 8.50/4.53 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4546_prod_OatLeast__Suc__atMost__Suc__shift,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.53 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 8.50/4.53 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4547_prod_OatLeast__Suc__lessThan__Suc__shift,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.53 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 8.50/4.53 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4548_prod_OatLeastAtMost__shift__bounds,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 8.50/4.53 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 8.50/4.53 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4549_prod_OatLeastLessThan__shift__bounds,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [G: nat > A,M: nat,K: nat,N2: nat] : % 8.50/4.53 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) ) % 8.50/4.53 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4550_prod_OUnion__comp,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [B3: set @ ( set @ B ),G: B > A] : % 8.50/4.53 ( ! [X5: set @ B] : % 8.50/4.53 ( ( member @ ( set @ B ) @ X5 @ B3 ) % 8.50/4.53 => ( finite_finite2 @ B @ X5 ) ) % 8.50/4.53 => ( ! [A14: set @ B] : % 8.50/4.53 ( ( member @ ( set @ B ) @ A14 @ B3 ) % 8.50/4.53 => ! [A25: set @ B] : % 8.50/4.53 ( ( member @ ( set @ B ) @ A25 @ B3 ) % 8.50/4.53 => ( ( A14 != A25 ) % 8.50/4.53 => ! [X5: B] : % 8.50/4.53 ( ( member @ B @ X5 @ A14 ) % 8.50/4.53 => ( ( member @ B @ X5 @ A25 ) % 8.50/4.53 => ( ( G @ X5 ) % 8.50/4.53 = ( one_one @ A ) ) ) ) ) ) ) % 8.50/4.53 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B3 ) ) % 8.50/4.53 = ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B3 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4551_prod_OUnion__disjoint,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [C5: set @ ( set @ B ),G: B > A] : % 8.50/4.53 ( ! [X5: set @ B] : % 8.50/4.53 ( ( member @ ( set @ B ) @ X5 @ C5 ) % 8.50/4.53 => ( finite_finite2 @ B @ X5 ) ) % 8.50/4.53 => ( ! [X5: set @ B] : % 8.50/4.53 ( ( member @ ( set @ B ) @ X5 @ C5 ) % 8.50/4.53 => ! [Xa3: set @ B] : % 8.50/4.53 ( ( member @ ( set @ B ) @ Xa3 @ C5 ) % 8.50/4.53 => ( ( X5 != Xa3 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ B ) @ X5 @ Xa3 ) % 8.50/4.53 = ( bot_bot @ ( set @ B ) ) ) ) ) ) % 8.50/4.53 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C5 ) ) % 8.50/4.53 = ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ C5 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4552_sum_OatLeast0__atMost__Suc__shift,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [G: nat > A,N2: nat] : % 8.50/4.53 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 8.50/4.53 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4553_sum_OatLeast0__lessThan__Suc__shift,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [G: nat > A,N2: nat] : % 8.50/4.53 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 8.50/4.53 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4554_prod_OatLeast0__atMost__Suc__shift,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [G: nat > A,N2: nat] : % 8.50/4.53 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 8.50/4.53 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4555_prod_OatLeast0__lessThan__Suc__shift,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [G: nat > A,N2: nat] : % 8.50/4.53 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) % 8.50/4.53 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4556_sum_OatLeastLessThan__shift__0,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.53 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) % 8.50/4.53 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4557_prod_OatLeastLessThan__shift__0,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.53 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) % 8.50/4.53 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4558_length__remove1,axiom, % 8.50/4.53 ! [A: $tType,X3: A,Xs: list @ A] : % 8.50/4.53 ( ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.50/4.53 => ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X3 @ Xs ) ) % 8.50/4.53 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) % 8.50/4.53 & ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.50/4.53 => ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X3 @ Xs ) ) % 8.50/4.53 = ( size_size @ ( list @ A ) @ Xs ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4559_sum_OUnion__comp,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [B3: set @ ( set @ B ),G: B > A] : % 8.50/4.53 ( ! [X5: set @ B] : % 8.50/4.53 ( ( member @ ( set @ B ) @ X5 @ B3 ) % 8.50/4.53 => ( finite_finite2 @ B @ X5 ) ) % 8.50/4.53 => ( ! [A14: set @ B] : % 8.50/4.53 ( ( member @ ( set @ B ) @ A14 @ B3 ) % 8.50/4.53 => ! [A25: set @ B] : % 8.50/4.53 ( ( member @ ( set @ B ) @ A25 @ B3 ) % 8.50/4.53 => ( ( A14 != A25 ) % 8.50/4.53 => ! [X5: B] : % 8.50/4.53 ( ( member @ B @ X5 @ A14 ) % 8.50/4.53 => ( ( member @ B @ X5 @ A25 ) % 8.50/4.53 => ( ( G @ X5 ) % 8.50/4.53 = ( zero_zero @ A ) ) ) ) ) ) ) % 8.50/4.53 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B3 ) ) % 8.50/4.53 = ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ B3 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4560_sum_OatLeast__atMost__pred__shift,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.53 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.53 @ ( comp @ nat @ A @ nat @ G % 8.50/4.53 @ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) % 8.50/4.53 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 8.50/4.53 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4561_sum_OatLeast__lessThan__pred__shift,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.53 ( ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.53 @ ( comp @ nat @ A @ nat @ G % 8.50/4.53 @ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) % 8.50/4.53 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 8.50/4.53 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4562_prod_OatLeast__atMost__pred__shift,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.53 ( ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.53 @ ( comp @ nat @ A @ nat @ G % 8.50/4.53 @ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) % 8.50/4.53 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 8.50/4.53 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4563_prod_OatLeast__lessThan__pred__shift,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [G: nat > A,M: nat,N2: nat] : % 8.50/4.53 ( ( groups7121269368397514597t_prod @ nat @ A % 8.50/4.53 @ ( comp @ nat @ A @ nat @ G % 8.50/4.53 @ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) % 8.50/4.53 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) % 8.50/4.53 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4564_sum_OatLeastAtMost__shift__0,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [M: nat,N2: nat,G: nat > A] : % 8.50/4.53 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.53 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.53 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4565_prod_OatLeastAtMost__shift__0,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [M: nat,N2: nat,G: nat > A] : % 8.50/4.53 ( ( ord_less_eq @ nat @ M @ N2 ) % 8.50/4.53 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) % 8.50/4.53 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4566_sum_OUnion__disjoint,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_add @ A ) % 8.50/4.53 => ! [C5: set @ ( set @ B ),G: B > A] : % 8.50/4.53 ( ! [X5: set @ B] : % 8.50/4.53 ( ( member @ ( set @ B ) @ X5 @ C5 ) % 8.50/4.53 => ( finite_finite2 @ B @ X5 ) ) % 8.50/4.53 => ( ! [X5: set @ B] : % 8.50/4.53 ( ( member @ ( set @ B ) @ X5 @ C5 ) % 8.50/4.53 => ! [Xa3: set @ B] : % 8.50/4.53 ( ( member @ ( set @ B ) @ Xa3 @ C5 ) % 8.50/4.53 => ( ( X5 != Xa3 ) % 8.50/4.53 => ( ( inf_inf @ ( set @ B ) @ X5 @ Xa3 ) % 8.50/4.53 = ( bot_bot @ ( set @ B ) ) ) ) ) ) % 8.50/4.53 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C5 ) ) % 8.50/4.53 = ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ C5 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4567_in__finite__psubset,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A3 @ B3 ) @ ( finite_psubset @ A ) ) % 8.50/4.53 = ( ( ord_less @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.53 & ( finite_finite2 @ A @ B3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4568_snd__fst__flip,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( product_snd @ B @ A ) % 8.50/4.53 = ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B ) % 8.50/4.53 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B ) % 8.50/4.53 @ ^ [X: B,Y2: A] : ( product_Pair @ A @ B @ Y2 @ X ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4569_fst__snd__flip,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( product_fst @ A @ B ) % 8.50/4.53 = ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A ) % 8.50/4.53 @ ( product_case_prod @ A @ B @ ( product_prod @ B @ A ) % 8.50/4.53 @ ^ [X: A,Y2: B] : ( product_Pair @ B @ A @ Y2 @ X ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4570_in__measure,axiom, % 8.50/4.53 ! [A: $tType,X3: A,Y: A,F2: A > nat] : % 8.50/4.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measure @ A @ F2 ) ) % 8.50/4.53 = ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4571_fstI,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,X3: product_prod @ A @ B,Y: A,Z2: B] : % 8.50/4.53 ( ( X3 % 8.50/4.53 = ( product_Pair @ A @ B @ Y @ Z2 ) ) % 8.50/4.53 => ( ( product_fst @ A @ B @ X3 ) % 8.50/4.53 = Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4572_sndI,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,X3: product_prod @ A @ B,Y: A,Z2: B] : % 8.50/4.53 ( ( X3 % 8.50/4.53 = ( product_Pair @ A @ B @ Y @ Z2 ) ) % 8.50/4.53 => ( ( product_snd @ A @ B @ X3 ) % 8.50/4.53 = Z2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4573_finite__psubset__def,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( finite_psubset @ A ) % 8.50/4.53 = ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) % 8.50/4.53 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o % 8.50/4.53 @ ^ [A6: set @ A,B6: set @ A] : % 8.50/4.53 ( ( ord_less @ ( set @ A ) @ A6 @ B6 ) % 8.50/4.53 & ( finite_finite2 @ A @ B6 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4574_drop__bit__rec,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ( ( bit_se4197421643247451524op_bit @ A ) % 8.50/4.53 = ( ^ [N: nat,A4: A] : % 8.50/4.53 ( if @ A % 8.50/4.53 @ ( N % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 @ A4 % 8.50/4.53 @ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4575_root__powr__inverse,axiom, % 8.50/4.53 ! [N2: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.53 => ( ( root @ N2 @ X3 ) % 8.50/4.53 = ( powr @ real @ X3 @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4576_image2__def,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,C: $tType] : % 8.50/4.53 ( ( bNF_Greatest_image2 @ C @ A @ B ) % 8.50/4.53 = ( ^ [A6: set @ C,F3: C > A,G2: C > B] : % 8.50/4.53 ( collect @ ( product_prod @ A @ B ) % 8.50/4.53 @ ^ [Uu3: product_prod @ A @ B] : % 8.50/4.53 ? [A4: C] : % 8.50/4.53 ( ( Uu3 % 8.50/4.53 = ( product_Pair @ A @ B @ ( F3 @ A4 ) @ ( G2 @ A4 ) ) ) % 8.50/4.53 & ( member @ C @ A4 @ A6 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4577_drop__bit__negative__int__iff,axiom, % 8.50/4.53 ! [N2: nat,K: int] : % 8.50/4.53 ( ( ord_less @ int @ ( bit_se4197421643247451524op_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) % 8.50/4.53 = ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4578_drop__bit__of__0,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [N2: nat] : % 8.50/4.53 ( ( bit_se4197421643247451524op_bit @ A @ N2 @ ( zero_zero @ A ) ) % 8.50/4.53 = ( zero_zero @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4579_drop__bit__drop__bit,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [M: nat,N2: nat,A2: A] : % 8.50/4.53 ( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) ) % 8.50/4.53 = ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ A2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4580_real__root__Suc__0,axiom, % 8.50/4.53 ! [X3: real] : % 8.50/4.53 ( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X3 ) % 8.50/4.53 = X3 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4581_root__0,axiom, % 8.50/4.53 ! [X3: real] : % 8.50/4.53 ( ( root @ ( zero_zero @ nat ) @ X3 ) % 8.50/4.53 = ( zero_zero @ real ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4582_real__root__eq__iff,axiom, % 8.50/4.53 ! [N2: nat,X3: real,Y: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ( root @ N2 @ X3 ) % 8.50/4.53 = ( root @ N2 @ Y ) ) % 8.50/4.53 = ( X3 = Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4583_drop__bit__of__bool,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [N2: nat,B2: $o] : % 8.50/4.53 ( ( bit_se4197421643247451524op_bit @ A @ N2 @ ( zero_neq_one_of_bool @ A @ B2 ) ) % 8.50/4.53 = ( zero_neq_one_of_bool @ A % 8.50/4.53 @ ( ( N2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 & B2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4584_drop__bit__of__Suc__0,axiom, % 8.50/4.53 ! [N2: nat] : % 8.50/4.53 ( ( bit_se4197421643247451524op_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.53 = ( zero_neq_one_of_bool @ nat % 8.50/4.53 @ ( N2 % 8.50/4.53 = ( zero_zero @ nat ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4585_drop__bit__Suc__minus__bit0,axiom, % 8.50/4.53 ! [N2: nat,K: num] : % 8.50/4.53 ( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) ) % 8.50/4.53 = ( bit_se4197421643247451524op_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4586_drop__bit__Suc__bit0,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.53 => ! [N2: nat,K: num] : % 8.50/4.53 ( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) % 8.50/4.53 = ( bit_se4197421643247451524op_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4587_drop__bit__Suc__bit1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_un5681908812861735899ations @ A ) % 8.50/4.53 => ! [N2: nat,K: num] : % 8.50/4.53 ( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) ) % 8.50/4.53 = ( bit_se4197421643247451524op_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4588_real__root__eq__0__iff,axiom, % 8.50/4.53 ! [N2: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ( root @ N2 @ X3 ) % 8.50/4.53 = ( zero_zero @ real ) ) % 8.50/4.53 = ( X3 % 8.50/4.53 = ( zero_zero @ real ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4589_real__root__less__iff,axiom, % 8.50/4.53 ! [N2: nat,X3: real,Y: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less @ real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y ) ) % 8.50/4.53 = ( ord_less @ real @ X3 @ Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4590_real__root__le__iff,axiom, % 8.50/4.53 ! [N2: nat,X3: real,Y: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less_eq @ real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y ) ) % 8.50/4.53 = ( ord_less_eq @ real @ X3 @ Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4591_real__root__eq__1__iff,axiom, % 8.50/4.53 ! [N2: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ( root @ N2 @ X3 ) % 8.50/4.53 = ( one_one @ real ) ) % 8.50/4.53 = ( X3 % 8.50/4.53 = ( one_one @ real ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4592_real__root__one,axiom, % 8.50/4.53 ! [N2: nat] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( root @ N2 @ ( one_one @ real ) ) % 8.50/4.53 = ( one_one @ real ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4593_drop__bit__of__1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [N2: nat] : % 8.50/4.53 ( ( bit_se4197421643247451524op_bit @ A @ N2 @ ( one_one @ A ) ) % 8.50/4.53 = ( zero_neq_one_of_bool @ A % 8.50/4.53 @ ( N2 % 8.50/4.53 = ( zero_zero @ nat ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4594_real__root__lt__0__iff,axiom, % 8.50/4.53 ! [N2: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less @ real @ ( root @ N2 @ X3 ) @ ( zero_zero @ real ) ) % 8.50/4.53 = ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4595_real__root__gt__0__iff,axiom, % 8.50/4.53 ! [N2: nat,Y: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N2 @ Y ) ) % 8.50/4.53 = ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4596_real__root__le__0__iff,axiom, % 8.50/4.53 ! [N2: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less_eq @ real @ ( root @ N2 @ X3 ) @ ( zero_zero @ real ) ) % 8.50/4.53 = ( ord_less_eq @ real @ X3 @ ( zero_zero @ real ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4597_real__root__ge__0__iff,axiom, % 8.50/4.53 ! [N2: nat,Y: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N2 @ Y ) ) % 8.50/4.53 = ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4598_real__root__lt__1__iff,axiom, % 8.50/4.53 ! [N2: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less @ real @ ( root @ N2 @ X3 ) @ ( one_one @ real ) ) % 8.50/4.53 = ( ord_less @ real @ X3 @ ( one_one @ real ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4599_real__root__gt__1__iff,axiom, % 8.50/4.53 ! [N2: nat,Y: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N2 @ Y ) ) % 8.50/4.53 = ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4600_real__root__le__1__iff,axiom, % 8.50/4.53 ! [N2: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less_eq @ real @ ( root @ N2 @ X3 ) @ ( one_one @ real ) ) % 8.50/4.53 = ( ord_less_eq @ real @ X3 @ ( one_one @ real ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4601_real__root__ge__1__iff,axiom, % 8.50/4.53 ! [N2: nat,Y: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N2 @ Y ) ) % 8.50/4.53 = ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4602_drop__bit__Suc__minus__bit1,axiom, % 8.50/4.53 ! [N2: nat,K: num] : % 8.50/4.53 ( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) ) % 8.50/4.53 = ( bit_se4197421643247451524op_bit @ int @ N2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4603_real__root__pow__pos2,axiom, % 8.50/4.53 ! [N2: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.53 => ( ( power_power @ real @ ( root @ N2 @ X3 ) @ N2 ) % 8.50/4.53 = X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4604_take__bit__eq__self__iff__drop__bit__eq__0,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [N2: nat,A2: A] : % 8.50/4.53 ( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) % 8.50/4.53 = A2 ) % 8.50/4.53 = ( ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) % 8.50/4.53 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4605_take__bit__drop__bit,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [M: nat,N2: nat,A2: A] : % 8.50/4.53 ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) ) % 8.50/4.53 = ( bit_se4197421643247451524op_bit @ A @ N2 @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N2 ) @ A2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4606_bit__drop__bit__eq,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [N2: nat,A2: A] : % 8.50/4.53 ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) ) % 8.50/4.53 = ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( plus_plus @ nat @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4607_real__root__less__mono,axiom, % 8.50/4.53 ! [N2: nat,X3: real,Y: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less @ real @ X3 @ Y ) % 8.50/4.53 => ( ord_less @ real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4608_real__root__le__mono,axiom, % 8.50/4.53 ! [N2: nat,X3: real,Y: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less_eq @ real @ X3 @ Y ) % 8.50/4.53 => ( ord_less_eq @ real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4609_real__root__power,axiom, % 8.50/4.53 ! [N2: nat,X3: real,K: nat] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( root @ N2 @ ( power_power @ real @ X3 @ K ) ) % 8.50/4.53 = ( power_power @ real @ ( root @ N2 @ X3 ) @ K ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4610_real__root__abs,axiom, % 8.50/4.53 ! [N2: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( root @ N2 @ ( abs_abs @ real @ X3 ) ) % 8.50/4.53 = ( abs_abs @ real @ ( root @ N2 @ X3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4611_sgn__root,axiom, % 8.50/4.53 ! [N2: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( sgn_sgn @ real @ ( root @ N2 @ X3 ) ) % 8.50/4.53 = ( sgn_sgn @ real @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4612_bits__ident,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [N2: nat,A2: A] : % 8.50/4.53 ( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A2 ) ) % 8.50/4.53 = A2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4613_image2__eqI,axiom, % 8.50/4.53 ! [A: $tType,C: $tType,B: $tType,B2: A,F2: B > A,X3: B,C2: C,G: B > C,A3: set @ B] : % 8.50/4.53 ( ( B2 % 8.50/4.53 = ( F2 @ X3 ) ) % 8.50/4.53 => ( ( C2 % 8.50/4.53 = ( G @ X3 ) ) % 8.50/4.53 => ( ( member @ B @ X3 @ A3 ) % 8.50/4.53 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A3 @ F2 @ G ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4614_real__root__gt__zero,axiom, % 8.50/4.53 ! [N2: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.53 => ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N2 @ X3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4615_real__root__strict__decreasing,axiom, % 8.50/4.53 ! [N2: nat,N7: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less @ nat @ N2 @ N7 ) % 8.50/4.53 => ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 8.50/4.53 => ( ord_less @ real @ ( root @ N7 @ X3 ) @ ( root @ N2 @ X3 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4616_root__abs__power,axiom, % 8.50/4.53 ! [N2: nat,Y: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( abs_abs @ real @ ( root @ N2 @ ( power_power @ real @ Y @ N2 ) ) ) % 8.50/4.53 = ( abs_abs @ real @ Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4617_real__root__pos__pos,axiom, % 8.50/4.53 ! [N2: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.53 => ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N2 @ X3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4618_real__root__strict__increasing,axiom, % 8.50/4.53 ! [N2: nat,N7: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less @ nat @ N2 @ N7 ) % 8.50/4.53 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.53 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 8.50/4.53 => ( ord_less @ real @ ( root @ N2 @ X3 ) @ ( root @ N7 @ X3 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4619_real__root__decreasing,axiom, % 8.50/4.53 ! [N2: nat,N7: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less_eq @ nat @ N2 @ N7 ) % 8.50/4.53 => ( ( ord_less_eq @ real @ ( one_one @ real ) @ X3 ) % 8.50/4.53 => ( ord_less_eq @ real @ ( root @ N7 @ X3 ) @ ( root @ N2 @ X3 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4620_real__root__pow__pos,axiom, % 8.50/4.53 ! [N2: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.53 => ( ( power_power @ real @ ( root @ N2 @ X3 ) @ N2 ) % 8.50/4.53 = X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4621_real__root__pos__unique,axiom, % 8.50/4.53 ! [N2: nat,Y: real,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) % 8.50/4.53 => ( ( ( power_power @ real @ Y @ N2 ) % 8.50/4.53 = X3 ) % 8.50/4.53 => ( ( root @ N2 @ X3 ) % 8.50/4.53 = Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4622_real__root__power__cancel,axiom, % 8.50/4.53 ! [N2: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.53 => ( ( root @ N2 @ ( power_power @ real @ X3 @ N2 ) ) % 8.50/4.53 = X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4623_drop__bit__Suc,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.53 => ! [N2: nat,A2: A] : % 8.50/4.53 ( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N2 ) @ A2 ) % 8.50/4.53 = ( bit_se4197421643247451524op_bit @ A @ N2 @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4624_real__root__increasing,axiom, % 8.50/4.53 ! [N2: nat,N7: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less_eq @ nat @ N2 @ N7 ) % 8.50/4.53 => ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.53 => ( ( ord_less_eq @ real @ X3 @ ( one_one @ real ) ) % 8.50/4.53 => ( ord_less_eq @ real @ ( root @ N2 @ X3 ) @ ( root @ N7 @ X3 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4625_root__sgn__power,axiom, % 8.50/4.53 ! [N2: nat,Y: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( root @ N2 @ ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N2 ) ) ) % 8.50/4.53 = Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4626_sgn__power__root,axiom, % 8.50/4.53 ! [N2: nat,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N2 @ X3 ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N2 @ X3 ) ) @ N2 ) ) % 8.50/4.53 = X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4627_ln__root,axiom, % 8.50/4.53 ! [N2: nat,B2: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 8.50/4.53 => ( ( ln_ln @ real @ ( root @ N2 @ B2 ) ) % 8.50/4.53 = ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4628_log__root,axiom, % 8.50/4.53 ! [N2: nat,A2: real,B2: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 ) % 8.50/4.53 => ( ( log @ B2 @ ( root @ N2 @ A2 ) ) % 8.50/4.53 = ( divide_divide @ real @ ( log @ B2 @ A2 ) @ ( semiring_1_of_nat @ real @ N2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4629_log__base__root,axiom, % 8.50/4.53 ! [N2: nat,B2: real,X3: real] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 ) % 8.50/4.53 => ( ( log @ ( root @ N2 @ B2 ) @ X3 ) % 8.50/4.53 = ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( log @ B2 @ X3 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4630_split__root,axiom, % 8.50/4.53 ! [P: real > $o,N2: nat,X3: real] : % 8.50/4.53 ( ( P @ ( root @ N2 @ X3 ) ) % 8.50/4.53 = ( ( ( N2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 => ( P @ ( zero_zero @ real ) ) ) % 8.50/4.53 & ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ! [Y2: real] : % 8.50/4.53 ( ( ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N2 ) ) % 8.50/4.53 = X3 ) % 8.50/4.53 => ( P @ Y2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4631_in__lex__prod,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] : % 8.50/4.53 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ A7 @ B7 ) ) @ ( lex_prod @ A @ B @ R2 @ S ) ) % 8.50/4.53 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A7 ) @ R2 ) % 8.50/4.53 | ( ( A2 = A7 ) % 8.50/4.53 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B7 ) @ S ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4632_div__add__self1__no__field,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( ( euclid4440199948858584721cancel @ A ) % 8.50/4.53 & ( field @ B ) ) % 8.50/4.53 => ! [X3: B,B2: A,A2: A] : % 8.50/4.53 ( ( nO_MATCH @ B @ A @ X3 @ B2 ) % 8.50/4.53 => ( ( B2 % 8.50/4.53 != ( zero_zero @ A ) ) % 8.50/4.53 => ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 ) % 8.50/4.53 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4633_div__add__self2__no__field,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( ( euclid4440199948858584721cancel @ A ) % 8.50/4.53 & ( field @ B ) ) % 8.50/4.53 => ! [X3: B,B2: A,A2: A] : % 8.50/4.53 ( ( nO_MATCH @ B @ A @ X3 @ B2 ) % 8.50/4.53 => ( ( B2 % 8.50/4.53 != ( zero_zero @ A ) ) % 8.50/4.53 => ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 ) % 8.50/4.53 = ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4634_scale__right__distrib__NO__MATCH,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.53 => ! [X3: A,Y: A,A2: real] : % 8.50/4.53 ( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X3 @ Y ) @ A2 ) % 8.50/4.53 => ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( plus_plus @ A @ X3 @ Y ) ) % 8.50/4.53 = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4635_distrib__right__NO__MATCH,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( semiring @ A ) % 8.50/4.53 => ! [X3: B,Y: B,C2: A,A2: A,B2: A] : % 8.50/4.53 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ C2 ) % 8.50/4.53 => ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) % 8.50/4.53 = ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4636_distrib__left__NO__MATCH,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( semiring @ A ) % 8.50/4.53 => ! [X3: B,Y: B,A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ A2 ) % 8.50/4.53 => ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) % 8.50/4.53 = ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4637_left__diff__distrib__NO__MATCH,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( ring @ A ) % 8.50/4.53 => ! [X3: B,Y: B,C2: A,A2: A,B2: A] : % 8.50/4.53 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ C2 ) % 8.50/4.53 => ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) % 8.50/4.53 = ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4638_right__diff__distrib__NO__MATCH,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( ring @ A ) % 8.50/4.53 => ! [X3: B,Y: B,A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ A2 ) % 8.50/4.53 => ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) ) % 8.50/4.53 = ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4639_lex__prod__def,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( lex_prod @ A @ B ) % 8.50/4.53 = ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] : % 8.50/4.53 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) % 8.50/4.53 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o % 8.50/4.53 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o ) % 8.50/4.53 @ ^ [A4: A,B4: B] : % 8.50/4.53 ( product_case_prod @ A @ B @ $o % 8.50/4.53 @ ^ [A9: A,B13: B] : % 8.50/4.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A9 ) @ Ra ) % 8.50/4.53 | ( ( A4 = A9 ) % 8.50/4.53 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B4 @ B13 ) @ Rb ) ) ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4640_scale__left__distrib__NO__MATCH,axiom, % 8.50/4.53 ! [C: $tType,A: $tType] : % 8.50/4.53 ( ( real_V4867850818363320053vector @ A ) % 8.50/4.53 => ! [X3: A,Y: A,C2: C,A2: real,B2: real] : % 8.50/4.53 ( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X3 @ Y ) @ C2 ) % 8.50/4.53 => ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X3 ) % 8.50/4.53 = ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X3 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4641_horner__sum__eq__sum__funpow,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_semiring_0 @ A ) % 8.50/4.53 => ( ( groups4207007520872428315er_sum @ B @ A ) % 8.50/4.53 = ( ^ [F3: B > A,A4: A,Xs2: list @ B] : % 8.50/4.53 ( groups7311177749621191930dd_sum @ nat @ A % 8.50/4.53 @ ^ [N: nat] : ( compow @ ( A > A ) @ N @ ( times_times @ A @ A4 ) @ ( F3 @ ( nth @ B @ Xs2 @ N ) ) ) % 8.50/4.53 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4642_set__zip,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] : % 8.50/4.53 ( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) % 8.50/4.53 = ( collect @ ( product_prod @ A @ B ) % 8.50/4.53 @ ^ [Uu3: product_prod @ A @ B] : % 8.50/4.53 ? [I3: nat] : % 8.50/4.53 ( ( Uu3 % 8.50/4.53 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I3 ) @ ( nth @ B @ Ys @ I3 ) ) ) % 8.50/4.53 & ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4643_Fpow__Pow__finite,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( finite_Fpow @ A ) % 8.50/4.53 = ( ^ [A6: set @ A] : ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A6 ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4644_Suc__funpow,axiom, % 8.50/4.53 ! [N2: nat] : % 8.50/4.53 ( ( compow @ ( nat > nat ) @ N2 @ suc ) % 8.50/4.53 = ( plus_plus @ nat @ N2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4645_min_Oabsorb1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.53 => ( ( ord_min @ A @ A2 @ B2 ) % 8.50/4.53 = A2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4646_min_Oabsorb2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [B2: A,A2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.50/4.53 => ( ( ord_min @ A @ A2 @ B2 ) % 8.50/4.53 = B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4647_min_Obounded__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) ) % 8.50/4.53 = ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.53 & ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4648_min_Oabsorb3,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.53 => ( ( ord_min @ A @ A2 @ B2 ) % 8.50/4.53 = A2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4649_min_Oabsorb4,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [B2: A,A2: A] : % 8.50/4.53 ( ( ord_less @ A @ B2 @ A2 ) % 8.50/4.53 => ( ( ord_min @ A @ A2 @ B2 ) % 8.50/4.53 = B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4650_min__less__iff__conj,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [Z2: A,X3: A,Y: A] : % 8.50/4.53 ( ( ord_less @ A @ Z2 @ ( ord_min @ A @ X3 @ Y ) ) % 8.50/4.53 = ( ( ord_less @ A @ Z2 @ X3 ) % 8.50/4.53 & ( ord_less @ A @ Z2 @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4651_min__bot2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( order_bot @ A ) % 8.50/4.53 => ! [X3: A] : % 8.50/4.53 ( ( ord_min @ A @ X3 @ ( bot_bot @ A ) ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4652_min__bot,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( order_bot @ A ) % 8.50/4.53 => ! [X3: A] : % 8.50/4.53 ( ( ord_min @ A @ ( bot_bot @ A ) @ X3 ) % 8.50/4.53 = ( bot_bot @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4653_funpow__0,axiom, % 8.50/4.53 ! [A: $tType,F2: A > A,X3: A] : % 8.50/4.53 ( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F2 @ X3 ) % 8.50/4.53 = X3 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4654_max__min__same_I1_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : % 8.50/4.53 ( ( ord_max @ A @ X3 @ ( ord_min @ A @ X3 @ Y ) ) % 8.50/4.53 = X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4655_max__min__same_I2_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : % 8.50/4.53 ( ( ord_max @ A @ ( ord_min @ A @ X3 @ Y ) @ X3 ) % 8.50/4.53 = X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4656_max__min__same_I3_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : % 8.50/4.53 ( ( ord_max @ A @ ( ord_min @ A @ X3 @ Y ) @ Y ) % 8.50/4.53 = Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4657_max__min__same_I4_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [Y: A,X3: A] : % 8.50/4.53 ( ( ord_max @ A @ Y @ ( ord_min @ A @ X3 @ Y ) ) % 8.50/4.53 = Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4658_min__Suc__Suc,axiom, % 8.50/4.53 ! [M: nat,N2: nat] : % 8.50/4.53 ( ( ord_min @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) % 8.50/4.53 = ( suc @ ( ord_min @ nat @ M @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4659_min__0R,axiom, % 8.50/4.53 ! [N2: nat] : % 8.50/4.53 ( ( ord_min @ nat @ N2 @ ( zero_zero @ nat ) ) % 8.50/4.53 = ( zero_zero @ nat ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4660_min__0L,axiom, % 8.50/4.53 ! [N2: nat] : % 8.50/4.53 ( ( ord_min @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 = ( zero_zero @ nat ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4661_min__number__of_I1_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ( numeral @ A ) % 8.50/4.53 & ( ord @ A ) ) % 8.50/4.53 => ! [U: num,V2: num] : % 8.50/4.53 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 = ( numeral_numeral @ A @ U ) ) ) % 8.50/4.53 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 = ( numeral_numeral @ A @ V2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4662_min__0__1_I4_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linord181362715937106298miring @ A ) % 8.50/4.53 => ! [X3: num] : % 8.50/4.53 ( ( ord_min @ A @ ( numeral_numeral @ A @ X3 ) @ ( zero_zero @ A ) ) % 8.50/4.53 = ( zero_zero @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4663_min__0__1_I3_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linord181362715937106298miring @ A ) % 8.50/4.53 => ! [X3: num] : % 8.50/4.53 ( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X3 ) ) % 8.50/4.53 = ( zero_zero @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4664_min__0__1_I1_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linord181362715937106298miring @ A ) % 8.50/4.53 => ( ( ord_min @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) % 8.50/4.53 = ( zero_zero @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4665_min__0__1_I2_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linord181362715937106298miring @ A ) % 8.50/4.53 => ( ( ord_min @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) % 8.50/4.53 = ( zero_zero @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4666_min__number__of_I2_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ( uminus @ A ) % 8.50/4.53 & ( numeral @ A ) % 8.50/4.53 & ( ord @ A ) ) % 8.50/4.53 => ! [U: num,V2: num] : % 8.50/4.53 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 = ( numeral_numeral @ A @ U ) ) ) % 8.50/4.53 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 => ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4667_min__number__of_I3_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ( uminus @ A ) % 8.50/4.53 & ( numeral @ A ) % 8.50/4.53 & ( ord @ A ) ) % 8.50/4.53 => ! [U: num,V2: num] : % 8.50/4.53 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) % 8.50/4.53 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) ) % 8.50/4.53 = ( numeral_numeral @ A @ V2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4668_min__number__of_I4_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ( uminus @ A ) % 8.50/4.53 & ( numeral @ A ) % 8.50/4.53 & ( ord @ A ) ) % 8.50/4.53 => ! [U: num,V2: num] : % 8.50/4.53 ( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) % 8.50/4.53 & ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 => ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) % 8.50/4.53 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4669_min__Suc__numeral,axiom, % 8.50/4.53 ! [N2: nat,K: num] : % 8.50/4.53 ( ( ord_min @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) ) % 8.50/4.53 = ( suc @ ( ord_min @ nat @ N2 @ ( pred_numeral @ K ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4670_min__numeral__Suc,axiom, % 8.50/4.53 ! [K: num,N2: nat] : % 8.50/4.53 ( ( ord_min @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) ) % 8.50/4.53 = ( suc @ ( ord_min @ nat @ ( pred_numeral @ K ) @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4671_zip__replicate,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,I: nat,X3: A,J: nat,Y: B] : % 8.50/4.53 ( ( zip @ A @ B @ ( replicate @ A @ I @ X3 ) @ ( replicate @ B @ J @ Y ) ) % 8.50/4.53 = ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I @ J ) @ ( product_Pair @ A @ B @ X3 @ Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4672_length__zip,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 8.50/4.53 ( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) ) % 8.50/4.53 = ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4673_comp__funpow,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,N2: nat,F2: A > A] : % 8.50/4.53 ( ( compow @ ( ( B > A ) > B > A ) @ N2 @ ( comp @ A @ A @ B @ F2 ) ) % 8.50/4.53 = ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N2 @ F2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4674_min_Ostrict__coboundedI2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [B2: A,C2: A,A2: A] : % 8.50/4.53 ( ( ord_less @ A @ B2 @ C2 ) % 8.50/4.53 => ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4675_min_Ostrict__coboundedI1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,C2: A,B2: A] : % 8.50/4.53 ( ( ord_less @ A @ A2 @ C2 ) % 8.50/4.53 => ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4676_min_Ostrict__order__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ( ( ord_less @ A ) % 8.50/4.53 = ( ^ [A4: A,B4: A] : % 8.50/4.53 ( ( A4 % 8.50/4.53 = ( ord_min @ A @ A4 @ B4 ) ) % 8.50/4.53 & ( A4 != B4 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4677_min_Ostrict__boundedE,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( ord_less @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) ) % 8.50/4.53 => ~ ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.53 => ~ ( ord_less @ A @ A2 @ C2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4678_min__less__iff__disj,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [X3: A,Y: A,Z2: A] : % 8.50/4.53 ( ( ord_less @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 ) % 8.50/4.53 = ( ( ord_less @ A @ X3 @ Z2 ) % 8.50/4.53 | ( ord_less @ A @ Y @ Z2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4679_min_Omono,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,C2: A,B2: A,D2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ C2 ) % 8.50/4.53 => ( ( ord_less_eq @ A @ B2 @ D2 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ ( ord_min @ A @ C2 @ D2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4680_min_OorderE,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.53 => ( A2 % 8.50/4.53 = ( ord_min @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4681_min_OorderI,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( A2 % 8.50/4.53 = ( ord_min @ A @ A2 @ B2 ) ) % 8.50/4.53 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4682_min_OboundedE,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) ) % 8.50/4.53 => ~ ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.53 => ~ ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4683_min_OboundedI,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,B2: A,C2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ B2 ) % 8.50/4.53 => ( ( ord_less_eq @ A @ A2 @ C2 ) % 8.50/4.53 => ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4684_min_Oorder__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ( ( ord_less_eq @ A ) % 8.50/4.53 = ( ^ [A4: A,B4: A] : % 8.50/4.53 ( A4 % 8.50/4.53 = ( ord_min @ A @ A4 @ B4 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4685_min_Ocobounded1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ A2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4686_min_Ocobounded2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ B2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4687_min_Oabsorb__iff1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ( ( ord_less_eq @ A ) % 8.50/4.53 = ( ^ [A4: A,B4: A] : % 8.50/4.53 ( ( ord_min @ A @ A4 @ B4 ) % 8.50/4.53 = A4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4688_min_Oabsorb__iff2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ( ( ord_less_eq @ A ) % 8.50/4.53 = ( ^ [B4: A,A4: A] : % 8.50/4.53 ( ( ord_min @ A @ A4 @ B4 ) % 8.50/4.53 = B4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4689_min_OcoboundedI1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A2: A,C2: A,B2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ A2 @ C2 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4690_min_OcoboundedI2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [B2: A,C2: A,A2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ B2 @ C2 ) % 8.50/4.53 => ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4691_min__le__iff__disj,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [X3: A,Y: A,Z2: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 ) % 8.50/4.53 = ( ( ord_less_eq @ A @ X3 @ Z2 ) % 8.50/4.53 | ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4692_min__def__raw,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ord @ A ) % 8.50/4.53 => ( ( ord_min @ A ) % 8.50/4.53 = ( ^ [A4: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B4 ) @ A4 @ B4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4693_min__def,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ord @ A ) % 8.50/4.53 => ( ( ord_min @ A ) % 8.50/4.53 = ( ^ [A4: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B4 ) @ A4 @ B4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4694_min__absorb1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ord @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.53 => ( ( ord_min @ A @ X3 @ Y ) % 8.50/4.53 = X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4695_min__absorb2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( order @ A ) % 8.50/4.53 => ! [Y: A,X3: A] : % 8.50/4.53 ( ( ord_less_eq @ A @ Y @ X3 ) % 8.50/4.53 => ( ( ord_min @ A @ X3 @ Y ) % 8.50/4.53 = Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4696_of__nat__min,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linord181362715937106298miring @ A ) % 8.50/4.53 => ! [X3: nat,Y: nat] : % 8.50/4.53 ( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X3 @ Y ) ) % 8.50/4.53 = ( ord_min @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4697_funpow__swap1,axiom, % 8.50/4.53 ! [A: $tType,F2: A > A,N2: nat,X3: A] : % 8.50/4.53 ( ( F2 @ ( compow @ ( A > A ) @ N2 @ F2 @ X3 ) ) % 8.50/4.53 = ( compow @ ( A > A ) @ N2 @ F2 @ ( F2 @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4698_of__int__min,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linordered_idom @ A ) % 8.50/4.53 => ! [X3: int,Y: int] : % 8.50/4.53 ( ( ring_1_of_int @ A @ ( ord_min @ int @ X3 @ Y ) ) % 8.50/4.53 = ( ord_min @ A @ ( ring_1_of_int @ A @ X3 ) @ ( ring_1_of_int @ A @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4699_min__add__distrib__right,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ordere2412721322843649153imp_le @ A ) % 8.50/4.53 => ! [X3: A,Y: A,Z2: A] : % 8.50/4.53 ( ( plus_plus @ A @ X3 @ ( ord_min @ A @ Y @ Z2 ) ) % 8.50/4.53 = ( ord_min @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( plus_plus @ A @ X3 @ Z2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4700_min__add__distrib__left,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ordere2412721322843649153imp_le @ A ) % 8.50/4.53 => ! [X3: A,Y: A,Z2: A] : % 8.50/4.53 ( ( plus_plus @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 ) % 8.50/4.53 = ( ord_min @ A @ ( plus_plus @ A @ X3 @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4701_min__diff__distrib__left,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ordered_ab_group_add @ A ) % 8.50/4.53 => ! [X3: A,Y: A,Z2: A] : % 8.50/4.53 ( ( minus_minus @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 ) % 8.50/4.53 = ( ord_min @ A @ ( minus_minus @ A @ X3 @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4702_min__diff,axiom, % 8.50/4.53 ! [M: nat,I: nat,N2: nat] : % 8.50/4.53 ( ( ord_min @ nat @ ( minus_minus @ nat @ M @ I ) @ ( minus_minus @ nat @ N2 @ I ) ) % 8.50/4.53 = ( minus_minus @ nat @ ( ord_min @ nat @ M @ N2 ) @ I ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4703_funpow__mult,axiom, % 8.50/4.53 ! [A: $tType,N2: nat,M: nat,F2: A > A] : % 8.50/4.53 ( ( compow @ ( A > A ) @ N2 @ ( compow @ ( A > A ) @ M @ F2 ) ) % 8.50/4.53 = ( compow @ ( A > A ) @ ( times_times @ nat @ M @ N2 ) @ F2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4704_nat__mult__min__left,axiom, % 8.50/4.53 ! [M: nat,N2: nat,Q4: nat] : % 8.50/4.53 ( ( times_times @ nat @ ( ord_min @ nat @ M @ N2 ) @ Q4 ) % 8.50/4.53 = ( ord_min @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N2 @ Q4 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4705_nat__mult__min__right,axiom, % 8.50/4.53 ! [M: nat,N2: nat,Q4: nat] : % 8.50/4.53 ( ( times_times @ nat @ M @ ( ord_min @ nat @ N2 @ Q4 ) ) % 8.50/4.53 = ( ord_min @ nat @ ( times_times @ nat @ M @ N2 ) @ ( times_times @ nat @ M @ Q4 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4706_inf__nat__def,axiom, % 8.50/4.53 ( ( inf_inf @ nat ) % 8.50/4.53 = ( ord_min @ nat ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4707_funpow__Suc__right,axiom, % 8.50/4.53 ! [A: $tType,N2: nat,F2: A > A] : % 8.50/4.53 ( ( compow @ ( A > A ) @ ( suc @ N2 ) @ F2 ) % 8.50/4.53 = ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ F2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4708_funpow_Osimps_I2_J,axiom, % 8.50/4.53 ! [A: $tType,N2: nat,F2: A > A] : % 8.50/4.53 ( ( compow @ ( A > A ) @ ( suc @ N2 ) @ F2 ) % 8.50/4.53 = ( comp @ A @ A @ A @ F2 @ ( compow @ ( A > A ) @ N2 @ F2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4709_funpow__add,axiom, % 8.50/4.53 ! [A: $tType,M: nat,N2: nat,F2: A > A] : % 8.50/4.53 ( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M @ N2 ) @ F2 ) % 8.50/4.53 = ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M @ F2 ) @ ( compow @ ( A > A ) @ N2 @ F2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4710_minus__min__eq__max,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linord5086331880401160121up_add @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : % 8.50/4.53 ( ( uminus_uminus @ A @ ( ord_min @ A @ X3 @ Y ) ) % 8.50/4.53 = ( ord_max @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4711_minus__max__eq__min,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linord5086331880401160121up_add @ A ) % 8.50/4.53 => ! [X3: A,Y: A] : % 8.50/4.53 ( ( uminus_uminus @ A @ ( ord_max @ A @ X3 @ Y ) ) % 8.50/4.53 = ( ord_min @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4712_empty__in__Fpow,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4713_Fpow__not__empty,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A] : % 8.50/4.53 ( ( finite_Fpow @ A @ A3 ) % 8.50/4.53 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4714_max__mult__distrib__left,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linordered_idom @ A ) % 8.50/4.53 => ! [P4: A,X3: A,Y: A] : % 8.50/4.53 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 8.50/4.53 => ( ( times_times @ A @ P4 @ ( ord_max @ A @ X3 @ Y ) ) % 8.50/4.53 = ( ord_max @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) ) % 8.50/4.53 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 8.50/4.53 => ( ( times_times @ A @ P4 @ ( ord_max @ A @ X3 @ Y ) ) % 8.50/4.53 = ( ord_min @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4715_min__mult__distrib__left,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linordered_idom @ A ) % 8.50/4.53 => ! [P4: A,X3: A,Y: A] : % 8.50/4.53 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 8.50/4.53 => ( ( times_times @ A @ P4 @ ( ord_min @ A @ X3 @ Y ) ) % 8.50/4.53 = ( ord_min @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) ) % 8.50/4.53 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 8.50/4.53 => ( ( times_times @ A @ P4 @ ( ord_min @ A @ X3 @ Y ) ) % 8.50/4.53 = ( ord_max @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4716_max__mult__distrib__right,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linordered_idom @ A ) % 8.50/4.53 => ! [P4: A,X3: A,Y: A] : % 8.50/4.53 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 8.50/4.53 => ( ( times_times @ A @ ( ord_max @ A @ X3 @ Y ) @ P4 ) % 8.50/4.53 = ( ord_max @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) % 8.50/4.53 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 8.50/4.53 => ( ( times_times @ A @ ( ord_max @ A @ X3 @ Y ) @ P4 ) % 8.50/4.53 = ( ord_min @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4717_min__mult__distrib__right,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linordered_idom @ A ) % 8.50/4.53 => ! [P4: A,X3: A,Y: A] : % 8.50/4.53 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 8.50/4.53 => ( ( times_times @ A @ ( ord_min @ A @ X3 @ Y ) @ P4 ) % 8.50/4.53 = ( ord_min @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) % 8.50/4.53 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 8.50/4.53 => ( ( times_times @ A @ ( ord_min @ A @ X3 @ Y ) @ P4 ) % 8.50/4.53 = ( ord_max @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4718_of__nat__def,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_1 @ A ) % 8.50/4.53 => ( ( semiring_1_of_nat @ A ) % 8.50/4.53 = ( ^ [N: nat] : ( compow @ ( A > A ) @ N @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4719_numeral__add__unfold__funpow,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_numeral @ A ) % 8.50/4.53 => ! [K: num,A2: A] : % 8.50/4.53 ( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A2 ) % 8.50/4.53 = ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4720_min__divide__distrib__right,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linordered_field @ A ) % 8.50/4.53 => ! [P4: A,X3: A,Y: A] : % 8.50/4.53 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 8.50/4.53 => ( ( divide_divide @ A @ ( ord_min @ A @ X3 @ Y ) @ P4 ) % 8.50/4.53 = ( ord_min @ A @ ( divide_divide @ A @ X3 @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) ) % 8.50/4.53 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 8.50/4.53 => ( ( divide_divide @ A @ ( ord_min @ A @ X3 @ Y ) @ P4 ) % 8.50/4.53 = ( ord_max @ A @ ( divide_divide @ A @ X3 @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4721_max__divide__distrib__right,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linordered_field @ A ) % 8.50/4.53 => ! [P4: A,X3: A,Y: A] : % 8.50/4.53 ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 8.50/4.53 => ( ( divide_divide @ A @ ( ord_max @ A @ X3 @ Y ) @ P4 ) % 8.50/4.53 = ( ord_max @ A @ ( divide_divide @ A @ X3 @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) ) % 8.50/4.53 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 ) % 8.50/4.53 => ( ( divide_divide @ A @ ( ord_max @ A @ X3 @ Y ) @ P4 ) % 8.50/4.53 = ( ord_min @ A @ ( divide_divide @ A @ X3 @ P4 ) @ ( divide_divide @ A @ Y @ P4 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4722_min__Suc2,axiom, % 8.50/4.53 ! [M: nat,N2: nat] : % 8.50/4.53 ( ( ord_min @ nat @ M @ ( suc @ N2 ) ) % 8.50/4.53 = ( case_nat @ nat @ ( zero_zero @ nat ) % 8.50/4.53 @ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ M4 @ N2 ) ) % 8.50/4.53 @ M ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4723_min__Suc1,axiom, % 8.50/4.53 ! [N2: nat,M: nat] : % 8.50/4.53 ( ( ord_min @ nat @ ( suc @ N2 ) @ M ) % 8.50/4.53 = ( case_nat @ nat @ ( zero_zero @ nat ) % 8.50/4.53 @ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ N2 @ M4 ) ) % 8.50/4.53 @ M ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4724_Inf__insert__finite,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( condit6923001295902523014norder @ A ) % 8.50/4.53 => ! [S3: set @ A,X3: A] : % 8.50/4.53 ( ( finite_finite2 @ A @ S3 ) % 8.50/4.53 => ( ( ( S3 % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ( complete_Inf_Inf @ A @ ( insert @ A @ X3 @ S3 ) ) % 8.50/4.53 = X3 ) ) % 8.50/4.53 & ( ( S3 % 8.50/4.53 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( ( complete_Inf_Inf @ A @ ( insert @ A @ X3 @ S3 ) ) % 8.50/4.53 = ( ord_min @ A @ X3 @ ( complete_Inf_Inf @ A @ S3 ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4725_Fpow__mono,axiom, % 8.50/4.53 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.53 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) @ ( finite_Fpow @ A @ B3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4726_Fpow__def,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( finite_Fpow @ A ) % 8.50/4.53 = ( ^ [A6: set @ A] : % 8.50/4.53 ( collect @ ( set @ A ) % 8.50/4.53 @ ^ [X6: set @ A] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ A ) @ X6 @ A6 ) % 8.50/4.53 & ( finite_finite2 @ A @ X6 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4727_numeral__unfold__funpow,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_1 @ A ) % 8.50/4.53 => ( ( numeral_numeral @ A ) % 8.50/4.53 = ( ^ [K3: num] : ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K3 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4728_relpowp__bot,axiom, % 8.50/4.53 ! [A: $tType,N2: nat] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( compow @ ( A > A > $o ) @ N2 @ ( bot_bot @ ( A > A > $o ) ) ) % 8.50/4.53 = ( bot_bot @ ( A > A > $o ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4729_relpowp__fun__conv,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( compow @ ( A > A > $o ) ) % 8.50/4.53 = ( ^ [N: nat,P3: A > A > $o,X: A,Y2: A] : % 8.50/4.53 ? [F3: nat > A] : % 8.50/4.53 ( ( ( F3 @ ( zero_zero @ nat ) ) % 8.50/4.53 = X ) % 8.50/4.53 & ( ( F3 @ N ) % 8.50/4.53 = Y2 ) % 8.50/4.53 & ! [I3: nat] : % 8.50/4.53 ( ( ord_less @ nat @ I3 @ N ) % 8.50/4.53 => ( P3 @ ( F3 @ I3 ) @ ( F3 @ ( suc @ I3 ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4730_same__fst__def,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( same_fst @ A @ B ) % 8.50/4.53 = ( ^ [P3: A > $o,R6: A > ( set @ ( product_prod @ B @ B ) )] : % 8.50/4.53 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) % 8.50/4.53 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o % 8.50/4.53 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o ) % 8.50/4.53 @ ^ [X9: A,Y6: B] : % 8.50/4.53 ( product_case_prod @ A @ B @ $o % 8.50/4.53 @ ^ [X: A,Y2: B] : % 8.50/4.53 ( ( X9 = X ) % 8.50/4.53 & ( P3 @ X ) % 8.50/4.53 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y2 ) @ ( R6 @ X ) ) ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4731_same__fstI,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,P: A > $o,X3: A,Y8: B,Y: B,R: A > ( set @ ( product_prod @ B @ B ) )] : % 8.50/4.53 ( ( P @ X3 ) % 8.50/4.53 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y ) @ ( R @ X3 ) ) % 8.50/4.53 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y8 ) @ ( product_Pair @ A @ B @ X3 @ Y ) ) @ ( same_fst @ A @ B @ P @ R ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4732_inf__int__def,axiom, % 8.50/4.53 ( ( inf_inf @ int ) % 8.50/4.53 = ( ord_min @ int ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4733_relpowp__Suc__E,axiom, % 8.50/4.53 ! [A: $tType,N2: nat,P: A > A > $o,X3: A,Z2: A] : % 8.50/4.53 ( ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X3 @ Z2 ) % 8.50/4.53 => ~ ! [Y4: A] : % 8.50/4.53 ( ( compow @ ( A > A > $o ) @ N2 @ P @ X3 @ Y4 ) % 8.50/4.53 => ~ ( P @ Y4 @ Z2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4734_relpowp__Suc__I,axiom, % 8.50/4.53 ! [A: $tType,N2: nat,P: A > A > $o,X3: A,Y: A,Z2: A] : % 8.50/4.53 ( ( compow @ ( A > A > $o ) @ N2 @ P @ X3 @ Y ) % 8.50/4.53 => ( ( P @ Y @ Z2 ) % 8.50/4.53 => ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X3 @ Z2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4735_relpowp__Suc__D2,axiom, % 8.50/4.53 ! [A: $tType,N2: nat,P: A > A > $o,X3: A,Z2: A] : % 8.50/4.53 ( ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X3 @ Z2 ) % 8.50/4.53 => ? [Y4: A] : % 8.50/4.53 ( ( P @ X3 @ Y4 ) % 8.50/4.53 & ( compow @ ( A > A > $o ) @ N2 @ P @ Y4 @ Z2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4736_relpowp__Suc__E2,axiom, % 8.50/4.53 ! [A: $tType,N2: nat,P: A > A > $o,X3: A,Z2: A] : % 8.50/4.53 ( ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X3 @ Z2 ) % 8.50/4.53 => ~ ! [Y4: A] : % 8.50/4.53 ( ( P @ X3 @ Y4 ) % 8.50/4.53 => ~ ( compow @ ( A > A > $o ) @ N2 @ P @ Y4 @ Z2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4737_relpowp__Suc__I2,axiom, % 8.50/4.53 ! [A: $tType,P: A > A > $o,X3: A,Y: A,N2: nat,Z2: A] : % 8.50/4.53 ( ( P @ X3 @ Y ) % 8.50/4.53 => ( ( compow @ ( A > A > $o ) @ N2 @ P @ Y @ Z2 ) % 8.50/4.53 => ( compow @ ( A > A > $o ) @ ( suc @ N2 ) @ P @ X3 @ Z2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4738_relpowp__0__I,axiom, % 8.50/4.53 ! [A: $tType,P: A > A > $o,X3: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X3 @ X3 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4739_relpowp__0__E,axiom, % 8.50/4.53 ! [A: $tType,P: A > A > $o,X3: A,Y: A] : % 8.50/4.53 ( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X3 @ Y ) % 8.50/4.53 => ( X3 = Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4740_relpowp_Osimps_I1_J,axiom, % 8.50/4.53 ! [A: $tType,R: A > A > $o] : % 8.50/4.53 ( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R ) % 8.50/4.53 = ( ^ [Y3: A,Z: A] : ( Y3 = Z ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4741_relpowp__E2,axiom, % 8.50/4.53 ! [A: $tType,N2: nat,P: A > A > $o,X3: A,Z2: A] : % 8.50/4.53 ( ( compow @ ( A > A > $o ) @ N2 @ P @ X3 @ Z2 ) % 8.50/4.53 => ( ( ( N2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 => ( X3 != Z2 ) ) % 8.50/4.53 => ~ ! [Y4: A,M3: nat] : % 8.50/4.53 ( ( N2 % 8.50/4.53 = ( suc @ M3 ) ) % 8.50/4.53 => ( ( P @ X3 @ Y4 ) % 8.50/4.53 => ~ ( compow @ ( A > A > $o ) @ M3 @ P @ Y4 @ Z2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4742_relpowp__E,axiom, % 8.50/4.53 ! [A: $tType,N2: nat,P: A > A > $o,X3: A,Z2: A] : % 8.50/4.53 ( ( compow @ ( A > A > $o ) @ N2 @ P @ X3 @ Z2 ) % 8.50/4.53 => ( ( ( N2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 => ( X3 != Z2 ) ) % 8.50/4.53 => ~ ! [Y4: A,M3: nat] : % 8.50/4.53 ( ( N2 % 8.50/4.53 = ( suc @ M3 ) ) % 8.50/4.53 => ( ( compow @ ( A > A > $o ) @ M3 @ P @ X3 @ Y4 ) % 8.50/4.53 => ~ ( P @ Y4 @ Z2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4743_Nat_Ofunpow__code__def,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( funpow @ A ) % 8.50/4.53 = ( compow @ ( A > A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4744_max__nat_Osemilattice__neutr__order__axioms,axiom, % 8.50/4.53 ( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) % 8.50/4.53 @ ^ [X: nat,Y2: nat] : ( ord_less_eq @ nat @ Y2 @ X ) % 8.50/4.53 @ ^ [X: nat,Y2: nat] : ( ord_less @ nat @ Y2 @ X ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4745_apsnd__apfst,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,C: $tType,D: $tType,F2: C > B,G: D > A,X3: product_prod @ D @ C] : % 8.50/4.53 ( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apfst @ D @ A @ C @ G @ X3 ) ) % 8.50/4.53 = ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X3 ) ) @ ( F2 @ ( product_snd @ D @ C @ X3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4746_apfst__conv,axiom, % 8.50/4.53 ! [C: $tType,A: $tType,B: $tType,F2: C > A,X3: C,Y: B] : % 8.50/4.53 ( ( product_apfst @ C @ A @ B @ F2 @ ( product_Pair @ C @ B @ X3 @ Y ) ) % 8.50/4.53 = ( product_Pair @ A @ B @ ( F2 @ X3 ) @ Y ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4747_apfst__eq__conv,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,C: $tType,F2: C > A,X3: product_prod @ C @ B,G: C > A] : % 8.50/4.53 ( ( ( product_apfst @ C @ A @ B @ F2 @ X3 ) % 8.50/4.53 = ( product_apfst @ C @ A @ B @ G @ X3 ) ) % 8.50/4.53 = ( ( F2 @ ( product_fst @ C @ B @ X3 ) ) % 8.50/4.53 = ( G @ ( product_fst @ C @ B @ X3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4748_fst__apfst,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,C: $tType,F2: C > A,X3: product_prod @ C @ B] : % 8.50/4.53 ( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F2 @ X3 ) ) % 8.50/4.53 = ( F2 @ ( product_fst @ C @ B @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4749_snd__apfst,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,C: $tType,F2: C > B,X3: product_prod @ C @ A] : % 8.50/4.53 ( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F2 @ X3 ) ) % 8.50/4.53 = ( product_snd @ C @ A @ X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4750_snd__comp__apfst,axiom, % 8.50/4.53 ! [C: $tType,B: $tType,A: $tType,F2: A > C] : % 8.50/4.53 ( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 ) ) % 8.50/4.53 = ( product_snd @ A @ B ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4751_fst__comp__apfst,axiom, % 8.50/4.53 ! [C: $tType,B: $tType,A: $tType,F2: A > C] : % 8.50/4.53 ( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F2 ) ) % 8.50/4.53 = ( comp @ A @ C @ ( product_prod @ A @ B ) @ F2 @ ( product_fst @ A @ B ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4752_apfst__apsnd,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,D: $tType,C: $tType,F2: C > A,G: D > B,X3: product_prod @ C @ D] : % 8.50/4.53 ( ( product_apfst @ C @ A @ B @ F2 @ ( product_apsnd @ D @ B @ C @ G @ X3 ) ) % 8.50/4.53 = ( product_Pair @ A @ B @ ( F2 @ ( product_fst @ C @ D @ X3 ) ) @ ( G @ ( product_snd @ C @ D @ X3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4753_apfst__compose,axiom, % 8.50/4.53 ! [C: $tType,A: $tType,B: $tType,D: $tType,F2: C > A,G: D > C,X3: product_prod @ D @ B] : % 8.50/4.53 ( ( product_apfst @ C @ A @ B @ F2 @ ( product_apfst @ D @ C @ B @ G @ X3 ) ) % 8.50/4.53 = ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F2 @ G ) @ X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4754_apsnd__apfst__commute,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,C: $tType,D: $tType,F2: C > B,G: D > A,P4: product_prod @ D @ C] : % 8.50/4.53 ( ( product_apsnd @ C @ B @ A @ F2 @ ( product_apfst @ D @ A @ C @ G @ P4 ) ) % 8.50/4.53 = ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F2 @ P4 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4755_gcd__nat_Osemilattice__neutr__order__axioms,axiom, % 8.50/4.53 ( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat ) % 8.50/4.53 @ ^ [M2: nat,N: nat] : % 8.50/4.53 ( ( dvd_dvd @ nat @ M2 @ N ) % 8.50/4.53 & ( M2 != N ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4756_apfst__convE,axiom, % 8.50/4.53 ! [C: $tType,A: $tType,B: $tType,Q4: product_prod @ A @ B,F2: C > A,P4: product_prod @ C @ B] : % 8.50/4.53 ( ( Q4 % 8.50/4.53 = ( product_apfst @ C @ A @ B @ F2 @ P4 ) ) % 8.50/4.53 => ~ ! [X5: C,Y4: B] : % 8.50/4.53 ( ( P4 % 8.50/4.53 = ( product_Pair @ C @ B @ X5 @ Y4 ) ) % 8.50/4.53 => ( Q4 % 8.50/4.53 != ( product_Pair @ A @ B @ ( F2 @ X5 ) @ Y4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4757_set__removeAll,axiom, % 8.50/4.53 ! [A: $tType,X3: A,Xs: list @ A] : % 8.50/4.53 ( ( set2 @ A @ ( removeAll @ A @ X3 @ Xs ) ) % 8.50/4.53 = ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4758_times__int_Oabs__eq,axiom, % 8.50/4.53 ! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] : % 8.50/4.53 ( ( times_times @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) ) % 8.50/4.53 = ( abs_Integ % 8.50/4.53 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) % 8.50/4.53 @ ^ [X: nat,Y2: nat] : % 8.50/4.53 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 8.50/4.53 @ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y2 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V5 ) @ ( times_times @ nat @ Y2 @ U2 ) ) ) ) % 8.50/4.53 @ Xa2 % 8.50/4.53 @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4759_removeAll__id,axiom, % 8.50/4.53 ! [A: $tType,X3: A,Xs: list @ A] : % 8.50/4.53 ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.50/4.53 => ( ( removeAll @ A @ X3 @ Xs ) % 8.50/4.53 = Xs ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4760_distinct__removeAll,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A,X3: A] : % 8.50/4.53 ( ( distinct @ A @ Xs ) % 8.50/4.53 => ( distinct @ A @ ( removeAll @ A @ X3 @ Xs ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4761_int_Oabs__induct,axiom, % 8.50/4.53 ! [P: int > $o,X3: int] : % 8.50/4.53 ( ! [Y4: product_prod @ nat @ nat] : ( P @ ( abs_Integ @ Y4 ) ) % 8.50/4.53 => ( P @ X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4762_eq__Abs__Integ,axiom, % 8.50/4.53 ! [Z2: int] : % 8.50/4.53 ~ ! [X5: nat,Y4: nat] : % 8.50/4.53 ( Z2 % 8.50/4.53 != ( abs_Integ @ ( product_Pair @ nat @ nat @ X5 @ Y4 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4763_nat_Oabs__eq,axiom, % 8.50/4.53 ! [X3: product_prod @ nat @ nat] : % 8.50/4.53 ( ( nat2 @ ( abs_Integ @ X3 ) ) % 8.50/4.53 = ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4764_length__removeAll__less__eq,axiom, % 8.50/4.53 ! [A: $tType,X3: A,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4765_distinct__remove1__removeAll,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A,X3: A] : % 8.50/4.53 ( ( distinct @ A @ Xs ) % 8.50/4.53 => ( ( remove1 @ A @ X3 @ Xs ) % 8.50/4.53 = ( removeAll @ A @ X3 @ Xs ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4766_zero__int__def,axiom, % 8.50/4.53 ( ( zero_zero @ int ) % 8.50/4.53 = ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4767_int__def,axiom, % 8.50/4.53 ( ( semiring_1_of_nat @ int ) % 8.50/4.53 = ( ^ [N: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N @ ( zero_zero @ nat ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4768_length__removeAll__less,axiom, % 8.50/4.53 ! [A: $tType,X3: A,Xs: list @ A] : % 8.50/4.53 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.50/4.53 => ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4769_uminus__int_Oabs__eq,axiom, % 8.50/4.53 ! [X3: product_prod @ nat @ nat] : % 8.50/4.53 ( ( uminus_uminus @ int @ ( abs_Integ @ X3 ) ) % 8.50/4.53 = ( abs_Integ % 8.50/4.53 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 8.50/4.53 @ ^ [X: nat,Y2: nat] : ( product_Pair @ nat @ nat @ Y2 @ X ) % 8.50/4.53 @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4770_one__int__def,axiom, % 8.50/4.53 ( ( one_one @ int ) % 8.50/4.53 = ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4771_of__int_Oabs__eq,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ring_1 @ A ) % 8.50/4.53 => ! [X3: product_prod @ nat @ nat] : % 8.50/4.53 ( ( ring_1_of_int @ A @ ( abs_Integ @ X3 ) ) % 8.50/4.53 = ( product_case_prod @ nat @ nat @ A % 8.50/4.53 @ ^ [I3: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J3 ) ) % 8.50/4.53 @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4772_less__int_Oabs__eq,axiom, % 8.50/4.53 ! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] : % 8.50/4.53 ( ( ord_less @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) ) % 8.50/4.53 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o ) % 8.50/4.53 @ ^ [X: nat,Y2: nat] : % 8.50/4.53 ( product_case_prod @ nat @ nat @ $o % 8.50/4.53 @ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y2 ) ) ) % 8.50/4.53 @ Xa2 % 8.50/4.53 @ X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4773_less__eq__int_Oabs__eq,axiom, % 8.50/4.53 ! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] : % 8.50/4.53 ( ( ord_less_eq @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) ) % 8.50/4.53 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o ) % 8.50/4.53 @ ^ [X: nat,Y2: nat] : % 8.50/4.53 ( product_case_prod @ nat @ nat @ $o % 8.50/4.53 @ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ U2 @ Y2 ) ) ) % 8.50/4.53 @ Xa2 % 8.50/4.53 @ X3 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4774_plus__int_Oabs__eq,axiom, % 8.50/4.53 ! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] : % 8.50/4.53 ( ( plus_plus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) ) % 8.50/4.53 = ( abs_Integ % 8.50/4.53 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) % 8.50/4.53 @ ^ [X: nat,Y2: nat] : % 8.50/4.53 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 8.50/4.53 @ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y2 @ V5 ) ) ) % 8.50/4.53 @ Xa2 % 8.50/4.53 @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4775_minus__int_Oabs__eq,axiom, % 8.50/4.53 ! [Xa2: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] : % 8.50/4.53 ( ( minus_minus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) ) % 8.50/4.53 = ( abs_Integ % 8.50/4.53 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) % 8.50/4.53 @ ^ [X: nat,Y2: nat] : % 8.50/4.53 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 8.50/4.53 @ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ Y2 @ U2 ) ) ) % 8.50/4.53 @ Xa2 % 8.50/4.53 @ X3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4776_distinct__concat__iff,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ ( list @ A )] : % 8.50/4.53 ( ( distinct @ A @ ( concat @ A @ Xs ) ) % 8.50/4.53 = ( ( distinct @ ( list @ A ) @ ( removeAll @ ( list @ A ) @ ( nil @ A ) @ Xs ) ) % 8.50/4.53 & ! [Ys3: list @ A] : % 8.50/4.53 ( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) ) % 8.50/4.53 => ( distinct @ A @ Ys3 ) ) % 8.50/4.53 & ! [Ys3: list @ A,Zs3: list @ A] : % 8.50/4.53 ( ( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) ) % 8.50/4.53 & ( member @ ( list @ A ) @ Zs3 @ ( set2 @ ( list @ A ) @ Xs ) ) % 8.50/4.53 & ( Ys3 != Zs3 ) ) % 8.50/4.53 => ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Zs3 ) ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4777_max__rpair__set,axiom, % 8.50/4.53 fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_max_strict @ fun_max_weak ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4778_min__rpair__set,axiom, % 8.50/4.53 fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict @ fun_min_weak ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4779_zip__Nil,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,Ys: list @ B] : % 8.50/4.53 ( ( zip @ A @ B @ ( nil @ A ) @ Ys ) % 8.50/4.53 = ( nil @ ( product_prod @ A @ B ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4780_Nil__eq__zip__iff,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 8.50/4.53 ( ( ( nil @ ( product_prod @ A @ B ) ) % 8.50/4.53 = ( zip @ A @ B @ Xs @ Ys ) ) % 8.50/4.53 = ( ( Xs % 8.50/4.53 = ( nil @ A ) ) % 8.50/4.53 | ( Ys % 8.50/4.53 = ( nil @ B ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4781_zip__eq__Nil__iff,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] : % 8.50/4.53 ( ( ( zip @ A @ B @ Xs @ Ys ) % 8.50/4.53 = ( nil @ ( product_prod @ A @ B ) ) ) % 8.50/4.53 = ( ( Xs % 8.50/4.53 = ( nil @ A ) ) % 8.50/4.53 | ( Ys % 8.50/4.53 = ( nil @ B ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4782_list__update__nonempty,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A,K: nat,X3: A] : % 8.50/4.53 ( ( ( list_update @ A @ Xs @ K @ X3 ) % 8.50/4.53 = ( nil @ A ) ) % 8.50/4.53 = ( Xs % 8.50/4.53 = ( nil @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4783_concat__replicate__trivial,axiom, % 8.50/4.53 ! [A: $tType,I: nat] : % 8.50/4.53 ( ( concat @ A @ ( replicate @ ( list @ A ) @ I @ ( nil @ A ) ) ) % 8.50/4.53 = ( nil @ A ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4784_Nil__in__shuffles,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 8.50/4.53 ( ( member @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) % 8.50/4.53 = ( ( Xs % 8.50/4.53 = ( nil @ A ) ) % 8.50/4.53 & ( Ys % 8.50/4.53 = ( nil @ A ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4785_enumerate__simps_I1_J,axiom, % 8.50/4.53 ! [A: $tType,N2: nat] : % 8.50/4.53 ( ( enumerate @ A @ N2 @ ( nil @ A ) ) % 8.50/4.53 = ( nil @ ( product_prod @ nat @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4786_rotate1__is__Nil__conv,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A] : % 8.50/4.53 ( ( ( rotate1 @ A @ Xs ) % 8.50/4.53 = ( nil @ A ) ) % 8.50/4.53 = ( Xs % 8.50/4.53 = ( nil @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4787_set__empty,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A] : % 8.50/4.53 ( ( ( set2 @ A @ Xs ) % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 = ( Xs % 8.50/4.53 = ( nil @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4788_set__empty2,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A] : % 8.50/4.53 ( ( ( bot_bot @ ( set @ A ) ) % 8.50/4.53 = ( set2 @ A @ Xs ) ) % 8.50/4.53 = ( Xs % 8.50/4.53 = ( nil @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4789_length__0__conv,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A] : % 8.50/4.53 ( ( ( size_size @ ( list @ A ) @ Xs ) % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 = ( Xs % 8.50/4.53 = ( nil @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4790_empty__replicate,axiom, % 8.50/4.53 ! [A: $tType,N2: nat,X3: A] : % 8.50/4.53 ( ( ( nil @ A ) % 8.50/4.53 = ( replicate @ A @ N2 @ X3 ) ) % 8.50/4.53 = ( N2 % 8.50/4.53 = ( zero_zero @ nat ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4791_replicate__empty,axiom, % 8.50/4.53 ! [A: $tType,N2: nat,X3: A] : % 8.50/4.53 ( ( ( replicate @ A @ N2 @ X3 ) % 8.50/4.53 = ( nil @ A ) ) % 8.50/4.53 = ( N2 % 8.50/4.53 = ( zero_zero @ nat ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4792_horner__sum__simps_I1_J,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( comm_semiring_0 @ A ) % 8.50/4.53 => ! [F2: B > A,A2: A] : % 8.50/4.53 ( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A2 @ ( nil @ B ) ) % 8.50/4.53 = ( zero_zero @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4793_concat__eq__Nil__conv,axiom, % 8.50/4.53 ! [A: $tType,Xss: list @ ( list @ A )] : % 8.50/4.53 ( ( ( concat @ A @ Xss ) % 8.50/4.53 = ( nil @ A ) ) % 8.50/4.53 = ( ! [X: list @ A] : % 8.50/4.53 ( ( member @ ( list @ A ) @ X @ ( set2 @ ( list @ A ) @ Xss ) ) % 8.50/4.53 => ( X % 8.50/4.53 = ( nil @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4794_Nil__eq__concat__conv,axiom, % 8.50/4.53 ! [A: $tType,Xss: list @ ( list @ A )] : % 8.50/4.53 ( ( ( nil @ A ) % 8.50/4.53 = ( concat @ A @ Xss ) ) % 8.50/4.53 = ( ! [X: list @ A] : % 8.50/4.53 ( ( member @ ( list @ A ) @ X @ ( set2 @ ( list @ A ) @ Xss ) ) % 8.50/4.53 => ( X % 8.50/4.53 = ( nil @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4795_length__greater__0__conv,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.53 = ( Xs % 8.50/4.53 != ( nil @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4796_removeAll_Osimps_I1_J,axiom, % 8.50/4.53 ! [A: $tType,X3: A] : % 8.50/4.53 ( ( removeAll @ A @ X3 @ ( nil @ A ) ) % 8.50/4.53 = ( nil @ A ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4797_rotate1_Osimps_I1_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( rotate1 @ A @ ( nil @ A ) ) % 8.50/4.53 = ( nil @ A ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4798_distinct_Osimps_I1_J,axiom, % 8.50/4.53 ! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4799_concat_Osimps_I1_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( concat @ A @ ( nil @ ( list @ A ) ) ) % 8.50/4.53 = ( nil @ A ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4800_list__update_Osimps_I1_J,axiom, % 8.50/4.53 ! [A: $tType,I: nat,V2: A] : % 8.50/4.53 ( ( list_update @ A @ ( nil @ A ) @ I @ V2 ) % 8.50/4.53 = ( nil @ A ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4801_list__update__code_I1_J,axiom, % 8.50/4.53 ! [A: $tType,I: nat,Y: A] : % 8.50/4.53 ( ( list_update @ A @ ( nil @ A ) @ I @ Y ) % 8.50/4.53 = ( nil @ A ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4802_product_Osimps_I1_J,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,Uu: list @ B] : % 8.50/4.53 ( ( product @ A @ B @ ( nil @ A ) @ Uu ) % 8.50/4.53 = ( nil @ ( product_prod @ A @ B ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4803_zip_Osimps_I1_J,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,Xs: list @ A] : % 8.50/4.53 ( ( zip @ A @ B @ Xs @ ( nil @ B ) ) % 8.50/4.53 = ( nil @ ( product_prod @ A @ B ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4804_Nil__in__shufflesI,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A,Ys: list @ A] : % 8.50/4.53 ( ( Xs % 8.50/4.53 = ( nil @ A ) ) % 8.50/4.53 => ( ( Ys % 8.50/4.53 = ( nil @ A ) ) % 8.50/4.53 => ( member @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4805_shuffles_Osimps_I1_J,axiom, % 8.50/4.53 ! [A: $tType,Ys: list @ A] : % 8.50/4.53 ( ( shuffles @ A @ ( nil @ A ) @ Ys ) % 8.50/4.53 = ( insert @ ( list @ A ) @ Ys @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4806_shuffles_Osimps_I2_J,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A] : % 8.50/4.53 ( ( shuffles @ A @ Xs @ ( nil @ A ) ) % 8.50/4.53 = ( insert @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4807_remove1_Osimps_I1_J,axiom, % 8.50/4.53 ! [A: $tType,X3: A] : % 8.50/4.53 ( ( remove1 @ A @ X3 @ ( nil @ A ) ) % 8.50/4.53 = ( nil @ A ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4808_gen__length__code_I1_J,axiom, % 8.50/4.53 ! [A: $tType,N2: nat] : % 8.50/4.53 ( ( gen_length @ A @ N2 @ ( nil @ A ) ) % 8.50/4.53 = N2 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4809_empty__set,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( bot_bot @ ( set @ A ) ) % 8.50/4.53 = ( set2 @ A @ ( nil @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4810_list_Osize_I3_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( size_size @ ( list @ A ) @ ( nil @ A ) ) % 8.50/4.53 = ( zero_zero @ nat ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4811_replicate__0,axiom, % 8.50/4.53 ! [A: $tType,X3: A] : % 8.50/4.53 ( ( replicate @ A @ ( zero_zero @ nat ) @ X3 ) % 8.50/4.53 = ( nil @ A ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4812_not__listrel1__Nil,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] : % 8.50/4.53 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( listrel1 @ A @ R2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4813_not__Nil__listrel1,axiom, % 8.50/4.53 ! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] : % 8.50/4.53 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs ) @ ( listrel1 @ A @ R2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4814_list_Osize__gen_I1_J,axiom, % 8.50/4.53 ! [A: $tType,X3: A > nat] : % 8.50/4.53 ( ( size_list @ A @ X3 @ ( nil @ A ) ) % 8.50/4.53 = ( zero_zero @ nat ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4815_find_Osimps_I1_J,axiom, % 8.50/4.53 ! [A: $tType,Uu: A > $o] : % 8.50/4.53 ( ( find @ A @ Uu @ ( nil @ A ) ) % 8.50/4.53 = ( none @ A ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4816_count__list_Osimps_I1_J,axiom, % 8.50/4.53 ! [A: $tType,Y: A] : % 8.50/4.53 ( ( count_list @ A @ ( nil @ A ) @ Y ) % 8.50/4.53 = ( zero_zero @ nat ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4817_Pow__set_I1_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( pow2 @ A @ ( set2 @ A @ ( nil @ A ) ) ) % 8.50/4.53 = ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4818_less__eq__int_Orep__eq,axiom, % 8.50/4.53 ( ( ord_less_eq @ int ) % 8.50/4.53 = ( ^ [X: int,Xa4: int] : % 8.50/4.53 ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o ) % 8.50/4.53 @ ^ [Y2: nat,Z6: nat] : % 8.50/4.53 ( product_case_prod @ nat @ nat @ $o % 8.50/4.53 @ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) ) % 8.50/4.53 @ ( rep_Integ @ X ) % 8.50/4.53 @ ( rep_Integ @ Xa4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4819_less__int_Orep__eq,axiom, % 8.50/4.53 ( ( ord_less @ int ) % 8.50/4.53 = ( ^ [X: int,Xa4: int] : % 8.50/4.53 ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o ) % 8.50/4.53 @ ^ [Y2: nat,Z6: nat] : % 8.50/4.53 ( product_case_prod @ nat @ nat @ $o % 8.50/4.53 @ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) ) % 8.50/4.53 @ ( rep_Integ @ X ) % 8.50/4.53 @ ( rep_Integ @ Xa4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4820_prod__encode__def,axiom, % 8.50/4.53 ( nat_prod_encode % 8.50/4.53 = ( product_case_prod @ nat @ nat @ nat % 8.50/4.53 @ ^ [M2: nat,N: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M2 @ N ) ) @ M2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4821_le__prod__encode__2,axiom, % 8.50/4.53 ! [B2: nat,A2: nat] : ( ord_less_eq @ nat @ B2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A2 @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4822_le__prod__encode__1,axiom, % 8.50/4.53 ! [A2: nat,B2: nat] : ( ord_less_eq @ nat @ A2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A2 @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4823_nat_Orep__eq,axiom, % 8.50/4.53 ( nat2 % 8.50/4.53 = ( ^ [X: int] : ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ ( rep_Integ @ X ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4824_of__int_Orep__eq,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ring_1 @ A ) % 8.50/4.53 => ( ( ring_1_of_int @ A ) % 8.50/4.53 = ( ^ [X: int] : % 8.50/4.53 ( product_case_prod @ nat @ nat @ A % 8.50/4.53 @ ^ [I3: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J3 ) ) % 8.50/4.53 @ ( rep_Integ @ X ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4825_prod__encode__prod__decode__aux,axiom, % 8.50/4.53 ! [K: nat,M: nat] : % 8.50/4.53 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) ) % 8.50/4.53 = ( plus_plus @ nat @ ( nat_triangle @ K ) @ M ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4826_listset_Osimps_I1_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( listset @ A @ ( nil @ ( set @ A ) ) ) % 8.50/4.53 = ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4827_uminus__int__def,axiom, % 8.50/4.53 ( ( uminus_uminus @ int ) % 8.50/4.53 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ % 8.50/4.53 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 8.50/4.53 @ ^ [X: nat,Y2: nat] : ( product_Pair @ nat @ nat @ Y2 @ X ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4828_eq__snd__iff,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,B2: A,P4: product_prod @ B @ A] : % 8.50/4.53 ( ( B2 % 8.50/4.53 = ( product_snd @ B @ A @ P4 ) ) % 8.50/4.53 = ( ? [A4: B] : % 8.50/4.53 ( P4 % 8.50/4.53 = ( product_Pair @ B @ A @ A4 @ B2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4829_prod_Osize__neq,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,X3: product_prod @ A @ B] : % 8.50/4.53 ( ( size_size @ ( product_prod @ A @ B ) @ X3 ) % 8.50/4.53 != ( zero_zero @ nat ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4830_sum_Osize__neq,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,X3: sum_sum @ A @ B] : % 8.50/4.53 ( ( size_size @ ( sum_sum @ A @ B ) @ X3 ) % 8.50/4.53 != ( zero_zero @ nat ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4831_eq__fst__iff,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,A2: A,P4: product_prod @ A @ B] : % 8.50/4.53 ( ( A2 % 8.50/4.53 = ( product_fst @ A @ B @ P4 ) ) % 8.50/4.53 = ( ? [B4: B] : % 8.50/4.53 ( P4 % 8.50/4.53 = ( product_Pair @ A @ B @ A2 @ B4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4832_times__int__def,axiom, % 8.50/4.53 ( ( times_times @ int ) % 8.50/4.53 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ ) % 8.50/4.53 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) % 8.50/4.53 @ ^ [X: nat,Y2: nat] : % 8.50/4.53 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 8.50/4.53 @ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y2 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V5 ) @ ( times_times @ nat @ Y2 @ U2 ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4833_minus__int__def,axiom, % 8.50/4.53 ( ( minus_minus @ int ) % 8.50/4.53 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ ) % 8.50/4.53 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) % 8.50/4.53 @ ^ [X: nat,Y2: nat] : % 8.50/4.53 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 8.50/4.53 @ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V5 ) @ ( plus_plus @ nat @ Y2 @ U2 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4834_plus__int__def,axiom, % 8.50/4.53 ( ( plus_plus @ int ) % 8.50/4.53 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ ) % 8.50/4.53 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) % 8.50/4.53 @ ^ [X: nat,Y2: nat] : % 8.50/4.53 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat ) % 8.50/4.53 @ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y2 @ V5 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4835_Gcd__remove0__nat,axiom, % 8.50/4.53 ! [M7: set @ nat] : % 8.50/4.53 ( ( finite_finite2 @ nat @ M7 ) % 8.50/4.53 => ( ( gcd_Gcd @ nat @ M7 ) % 8.50/4.53 = ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M7 @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4836_add__neg__numeral__special_I4_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( neg_numeral @ A ) % 8.50/4.53 => ! [N2: num] : % 8.50/4.53 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N2 ) ) % 8.50/4.53 = ( neg_numeral_sub @ A @ N2 @ one2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4837_add__neg__numeral__special_I3_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( neg_numeral @ A ) % 8.50/4.53 => ! [M: num] : % 8.50/4.53 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) % 8.50/4.53 = ( neg_numeral_sub @ A @ M @ one2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4838_sub__num__simps_I1_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( neg_numeral @ A ) % 8.50/4.53 => ( ( neg_numeral_sub @ A @ one2 @ one2 ) % 8.50/4.53 = ( zero_zero @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4839_Gcd__empty,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_Gcd @ A ) % 8.50/4.53 => ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 = ( zero_zero @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4840_Gcd__insert,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_Gcd @ A ) % 8.50/4.53 => ! [A2: A,A3: set @ A] : % 8.50/4.53 ( ( gcd_Gcd @ A @ ( insert @ A @ A2 @ A3 ) ) % 8.50/4.53 = ( gcd_gcd @ A @ A2 @ ( gcd_Gcd @ A @ A3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4841_semiring__norm_I167_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( neg_numeral @ A ) % 8.50/4.53 => ! [V2: num,W2: num,Y: A] : % 8.50/4.53 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) ) % 8.50/4.53 = ( plus_plus @ A @ ( neg_numeral_sub @ A @ W2 @ V2 ) @ Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4842_semiring__norm_I166_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( neg_numeral @ A ) % 8.50/4.53 => ! [V2: num,W2: num,Y: A] : % 8.50/4.53 ( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) ) % 8.50/4.53 = ( plus_plus @ A @ ( neg_numeral_sub @ A @ V2 @ W2 ) @ Y ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4843_add__neg__numeral__simps_I1_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( neg_numeral @ A ) % 8.50/4.53 => ! [M: num,N2: num] : % 8.50/4.53 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) % 8.50/4.53 = ( neg_numeral_sub @ A @ M @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4844_add__neg__numeral__simps_I2_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( neg_numeral @ A ) % 8.50/4.53 => ! [M: num,N2: num] : % 8.50/4.53 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) ) % 8.50/4.53 = ( neg_numeral_sub @ A @ N2 @ M ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4845_Gcd__2,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_Gcd @ A ) % 8.50/4.53 => ! [A2: A,B2: A] : % 8.50/4.53 ( ( gcd_Gcd @ A @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.53 = ( gcd_gcd @ A @ A2 @ B2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4846_Gcd__0__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_Gcd @ A ) % 8.50/4.53 => ! [A3: set @ A] : % 8.50/4.53 ( ( ( gcd_Gcd @ A @ A3 ) % 8.50/4.53 = ( zero_zero @ A ) ) % 8.50/4.53 = ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4847_add__neg__numeral__special_I1_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( neg_numeral @ A ) % 8.50/4.53 => ! [M: num] : % 8.50/4.53 ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) % 8.50/4.53 = ( neg_numeral_sub @ A @ one2 @ M ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4848_add__neg__numeral__special_I2_J,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( neg_numeral @ A ) % 8.50/4.53 => ! [M: num] : % 8.50/4.53 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) % 8.50/4.53 = ( neg_numeral_sub @ A @ one2 @ M ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4849_Gcd__dvd,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_Gcd @ A ) % 8.50/4.53 => ! [A2: A,A3: set @ A] : % 8.50/4.53 ( ( member @ A @ A2 @ A3 ) % 8.50/4.53 => ( dvd_dvd @ A @ ( gcd_Gcd @ A @ A3 ) @ A2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4850_dvd__GcdD,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_Gcd @ A ) % 8.50/4.53 => ! [X3: A,A3: set @ A,Y: A] : % 8.50/4.53 ( ( dvd_dvd @ A @ X3 @ ( gcd_Gcd @ A @ A3 ) ) % 8.50/4.53 => ( ( member @ A @ Y @ A3 ) % 8.50/4.53 => ( dvd_dvd @ A @ X3 @ Y ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4851_dvd__Gcd__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_Gcd @ A ) % 8.50/4.53 => ! [X3: A,A3: set @ A] : % 8.50/4.53 ( ( dvd_dvd @ A @ X3 @ ( gcd_Gcd @ A @ A3 ) ) % 8.50/4.53 = ( ! [X: A] : % 8.50/4.53 ( ( member @ A @ X @ A3 ) % 8.50/4.53 => ( dvd_dvd @ A @ X3 @ X ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4852_Gcd__greatest,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_Gcd @ A ) % 8.50/4.53 => ! [A3: set @ A,A2: A] : % 8.50/4.53 ( ! [B5: A] : % 8.50/4.53 ( ( member @ A @ B5 @ A3 ) % 8.50/4.53 => ( dvd_dvd @ A @ A2 @ B5 ) ) % 8.50/4.53 => ( dvd_dvd @ A @ A2 @ ( gcd_Gcd @ A @ A3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4853_Gcd__dvd__nat,axiom, % 8.50/4.53 ! [A2: nat,A3: set @ nat] : % 8.50/4.53 ( ( member @ nat @ A2 @ A3 ) % 8.50/4.53 => ( dvd_dvd @ nat @ ( gcd_Gcd @ nat @ A3 ) @ A2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4854_Gcd__greatest__nat,axiom, % 8.50/4.53 ! [A3: set @ nat,A2: nat] : % 8.50/4.53 ( ! [B5: nat] : % 8.50/4.53 ( ( member @ nat @ B5 @ A3 ) % 8.50/4.53 => ( dvd_dvd @ nat @ A2 @ B5 ) ) % 8.50/4.53 => ( dvd_dvd @ nat @ A2 @ ( gcd_Gcd @ nat @ A3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4855_Gcd__1,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_Gcd @ A ) % 8.50/4.53 => ! [A3: set @ A] : % 8.50/4.53 ( ( member @ A @ ( one_one @ A ) @ A3 ) % 8.50/4.53 => ( ( gcd_Gcd @ A @ A3 ) % 8.50/4.53 = ( one_one @ A ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4856_Gcd__nat__eq__one,axiom, % 8.50/4.53 ! [N7: set @ nat] : % 8.50/4.53 ( ( member @ nat @ ( one_one @ nat ) @ N7 ) % 8.50/4.53 => ( ( gcd_Gcd @ nat @ N7 ) % 8.50/4.53 = ( one_one @ nat ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4857_Gcd__eq__1__I,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_Gcd @ A ) % 8.50/4.53 => ! [A2: A,A3: set @ A] : % 8.50/4.53 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) % 8.50/4.53 => ( ( member @ A @ A2 @ A3 ) % 8.50/4.53 => ( ( gcd_Gcd @ A @ A3 ) % 8.50/4.53 = ( one_one @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4858_Gcd__in,axiom, % 8.50/4.53 ! [A3: set @ nat] : % 8.50/4.53 ( ! [A5: nat,B5: nat] : % 8.50/4.53 ( ( member @ nat @ A5 @ A3 ) % 8.50/4.53 => ( ( member @ nat @ B5 @ A3 ) % 8.50/4.53 => ( member @ nat @ ( gcd_gcd @ nat @ A5 @ B5 ) @ A3 ) ) ) % 8.50/4.53 => ( ( A3 % 8.50/4.53 != ( bot_bot @ ( set @ nat ) ) ) % 8.50/4.53 => ( member @ nat @ ( gcd_Gcd @ nat @ A3 ) @ A3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4859_sub__non__negative,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linordered_idom @ A ) % 8.50/4.53 => ! [N2: num,M: num] : % 8.50/4.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N2 @ M ) ) % 8.50/4.53 = ( ord_less_eq @ num @ M @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4860_sub__non__positive,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linordered_idom @ A ) % 8.50/4.53 => ! [N2: num,M: num] : % 8.50/4.53 ( ( ord_less_eq @ A @ ( neg_numeral_sub @ A @ N2 @ M ) @ ( zero_zero @ A ) ) % 8.50/4.53 = ( ord_less_eq @ num @ N2 @ M ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4861_sub__positive,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linordered_idom @ A ) % 8.50/4.53 => ! [N2: num,M: num] : % 8.50/4.53 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N2 @ M ) ) % 8.50/4.53 = ( ord_less @ num @ M @ N2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4862_sub__negative,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linordered_idom @ A ) % 8.50/4.53 => ! [N2: num,M: num] : % 8.50/4.53 ( ( ord_less @ A @ ( neg_numeral_sub @ A @ N2 @ M ) @ ( zero_zero @ A ) ) % 8.50/4.53 = ( ord_less @ num @ N2 @ M ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4863_sub__BitM__One__eq,axiom, % 8.50/4.53 ! [N2: num] : % 8.50/4.53 ( ( neg_numeral_sub @ int @ ( bitM @ N2 ) @ one2 ) % 8.50/4.53 = ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( neg_numeral_sub @ int @ N2 @ one2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4864_semiring__char__def,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_1 @ A ) % 8.50/4.53 => ( ( semiri4206861660011772517g_char @ A ) % 8.50/4.53 = ( ^ [Uu4: itself @ A] : % 8.50/4.53 ( gcd_Gcd @ nat % 8.50/4.53 @ ( collect @ nat % 8.50/4.53 @ ^ [N: nat] : % 8.50/4.53 ( ( semiring_1_of_nat @ A @ N ) % 8.50/4.53 = ( zero_zero @ A ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4865_num__of__nat_Osimps_I2_J,axiom, % 8.50/4.53 ! [N2: nat] : % 8.50/4.53 ( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( num_of_nat @ ( suc @ N2 ) ) % 8.50/4.53 = ( inc @ ( num_of_nat @ N2 ) ) ) ) % 8.50/4.53 & ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( num_of_nat @ ( suc @ N2 ) ) % 8.50/4.53 = one2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4866_pred__nat__def,axiom, % 8.50/4.53 ( pred_nat % 8.50/4.53 = ( collect @ ( product_prod @ nat @ nat ) % 8.50/4.53 @ ( product_case_prod @ nat @ nat @ $o % 8.50/4.53 @ ^ [M2: nat,N: nat] : % 8.50/4.53 ( N % 8.50/4.53 = ( suc @ M2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4867_abs__Gcd__eq,axiom, % 8.50/4.53 ! [K5: set @ int] : % 8.50/4.53 ( ( abs_abs @ int @ ( gcd_Gcd @ int @ K5 ) ) % 8.50/4.53 = ( gcd_Gcd @ int @ K5 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4868_Gcd__dvd__int,axiom, % 8.50/4.53 ! [A2: int,A3: set @ int] : % 8.50/4.53 ( ( member @ int @ A2 @ A3 ) % 8.50/4.53 => ( dvd_dvd @ int @ ( gcd_Gcd @ int @ A3 ) @ A2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4869_Gcd__greatest__int,axiom, % 8.50/4.53 ! [A3: set @ int,A2: int] : % 8.50/4.53 ( ! [B5: int] : % 8.50/4.53 ( ( member @ int @ B5 @ A3 ) % 8.50/4.53 => ( dvd_dvd @ int @ A2 @ B5 ) ) % 8.50/4.53 => ( dvd_dvd @ int @ A2 @ ( gcd_Gcd @ int @ A3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4870_Gcd__int__greater__eq__0,axiom, % 8.50/4.53 ! [K5: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Gcd @ int @ K5 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4871_num__of__nat_Osimps_I1_J,axiom, % 8.50/4.53 ( ( num_of_nat @ ( zero_zero @ nat ) ) % 8.50/4.53 = one2 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4872_numeral__num__of__nat,axiom, % 8.50/4.53 ! [N2: nat] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( numeral_numeral @ nat @ ( num_of_nat @ N2 ) ) % 8.50/4.53 = N2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4873_num__of__nat__One,axiom, % 8.50/4.53 ! [N2: nat] : % 8.50/4.53 ( ( ord_less_eq @ nat @ N2 @ ( one_one @ nat ) ) % 8.50/4.53 => ( ( num_of_nat @ N2 ) % 8.50/4.53 = one2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4874_numeral__num__of__nat__unfold,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( semiring_1 @ A ) % 8.50/4.53 => ! [N2: nat] : % 8.50/4.53 ( ( ( N2 % 8.50/4.53 = ( zero_zero @ nat ) ) % 8.50/4.53 => ( ( numeral_numeral @ A @ ( num_of_nat @ N2 ) ) % 8.50/4.53 = ( one_one @ A ) ) ) % 8.50/4.53 & ( ( N2 % 8.50/4.53 != ( zero_zero @ nat ) ) % 8.50/4.53 => ( ( numeral_numeral @ A @ ( num_of_nat @ N2 ) ) % 8.50/4.53 = ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4875_num__of__nat__double,axiom, % 8.50/4.53 ! [N2: nat] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( num_of_nat @ ( plus_plus @ nat @ N2 @ N2 ) ) % 8.50/4.53 = ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4876_num__of__nat__plus__distrib,axiom, % 8.50/4.53 ! [M: nat,N2: nat] : % 8.50/4.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) % 8.50/4.53 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.53 => ( ( num_of_nat @ ( plus_plus @ nat @ M @ N2 ) ) % 8.50/4.53 = ( plus_plus @ num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4877_prod_Oinsert_H,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [I6: set @ B,P4: B > A,I: B] : % 8.50/4.53 ( ( finite_finite2 @ B % 8.50/4.53 @ ( collect @ B % 8.50/4.53 @ ^ [X: B] : % 8.50/4.53 ( ( member @ B @ X @ I6 ) % 8.50/4.53 & ( ( P4 @ X ) % 8.50/4.53 != ( one_one @ A ) ) ) ) ) % 8.50/4.53 => ( ( ( member @ B @ I @ I6 ) % 8.50/4.53 => ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) ) % 8.50/4.53 = ( groups1962203154675924110t_prod @ B @ A @ P4 @ I6 ) ) ) % 8.50/4.53 & ( ~ ( member @ B @ I @ I6 ) % 8.50/4.53 => ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert @ B @ I @ I6 ) ) % 8.50/4.53 = ( times_times @ A @ ( P4 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P4 @ I6 ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4878_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A3: set @ A,X3: A] : % 8.50/4.53 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.53 => ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.53 = ( remove1 @ A @ X3 @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4879_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom, % 8.50/4.53 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.53 ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 ) % 8.50/4.53 => ( ( ? [Uu2: $o,Uv2: $o] : % 8.50/4.53 ( X3 % 8.50/4.53 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 8.50/4.53 => ( Xa2 % 8.50/4.53 = ( one_one @ nat ) ) ) % 8.50/4.53 => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 8.50/4.53 ( ( X3 % 8.50/4.53 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) ) % 8.50/4.53 => ( ( Deg2 = Xa2 ) % 8.50/4.53 & ! [X5: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.53 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.53 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 8.50/4.53 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.53 & ( case_option @ $o @ ( product_prod @ nat @ nat ) % 8.50/4.53 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 ) % 8.50/4.53 & ! [X: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.50/4.53 @ ( product_case_prod @ nat @ nat @ $o % 8.50/4.53 @ ^ [Mi3: nat,Ma3: nat] : % 8.50/4.53 ( ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 8.50/4.53 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 8.50/4.53 & ! [I3: nat] : % 8.50/4.53 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.53 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) ) % 8.50/4.53 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) ) % 8.50/4.53 & ( ( Mi3 = Ma3 ) % 8.50/4.53 => ! [X: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.50/4.53 & ( ( Mi3 != Ma3 ) % 8.50/4.53 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 ) % 8.50/4.53 & ! [X: nat] : % 8.50/4.53 ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 8.50/4.53 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X ) % 8.50/4.53 => ( ( ord_less @ nat @ Mi3 @ X ) % 8.50/4.53 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) % 8.50/4.53 @ Mima ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4880_ball__empty,axiom, % 8.50/4.53 ! [A: $tType,P: A > $o,X2: A] : % 8.50/4.53 ( ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 => ( P @ X2 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4881_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ( ( linord4507533701916653071of_set @ A @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.53 = ( nil @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4882_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A3: set @ A] : % 8.50/4.53 ( ~ ( finite_finite2 @ A @ A3 ) % 8.50/4.53 => ( ( linord4507533701916653071of_set @ A @ A3 ) % 8.50/4.53 = ( nil @ A ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4883_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A3: set @ A] : % 8.50/4.53 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.53 => ( ( set2 @ A @ ( linord4507533701916653071of_set @ A @ A3 ) ) % 8.50/4.53 = A3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4884_prod_Oempty_H,axiom, % 8.50/4.53 ! [B: $tType,A: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [P4: B > A] : % 8.50/4.53 ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) ) % 8.50/4.53 = ( one_one @ A ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4885_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A3: set @ A] : % 8.50/4.53 ( ( size_size @ ( list @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) ) % 8.50/4.53 = ( finite_card @ A @ A3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4886_prod_Oeq__sum,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [I6: set @ B,P4: B > A] : % 8.50/4.53 ( ( finite_finite2 @ B @ I6 ) % 8.50/4.53 => ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ I6 ) % 8.50/4.53 = ( groups7121269368397514597t_prod @ B @ A @ P4 @ I6 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4887_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A3: set @ A] : % 8.50/4.53 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.53 => ( ( ( linord4507533701916653071of_set @ A @ A3 ) % 8.50/4.53 = ( nil @ A ) ) % 8.50/4.53 = ( A3 % 8.50/4.53 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4888_finite__set__of__finite__funs,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,D2: B] : % 8.50/4.53 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.53 => ( ( finite_finite2 @ B @ B3 ) % 8.50/4.53 => ( finite_finite2 @ ( A > B ) % 8.50/4.53 @ ( collect @ ( A > B ) % 8.50/4.53 @ ^ [F3: A > B] : % 8.50/4.53 ! [X: A] : % 8.50/4.53 ( ( ( member @ A @ X @ A3 ) % 8.50/4.53 => ( member @ B @ ( F3 @ X ) @ B3 ) ) % 8.50/4.53 & ( ~ ( member @ A @ X @ A3 ) % 8.50/4.53 => ( ( F3 @ X ) % 8.50/4.53 = D2 ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4889_option_Osimps_I5_J,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,F1: B,F22: A > B,X22: A] : % 8.50/4.53 ( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X22 ) ) % 8.50/4.53 = ( F22 @ X22 ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4890_option_Osimps_I4_J,axiom, % 8.50/4.53 ! [A: $tType,B: $tType,F1: B,F22: A > B] : % 8.50/4.53 ( ( case_option @ B @ A @ F1 @ F22 @ ( none @ A ) ) % 8.50/4.53 = F1 ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4891_Ball__Collect,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ball @ A ) % 8.50/4.53 = ( ^ [A6: set @ A,P3: A > $o] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( collect @ A @ P3 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4892_option_Ocase__distrib,axiom, % 8.50/4.53 ! [C: $tType,B: $tType,A: $tType,H2: B > C,F1: B,F22: A > B,Option: option @ A] : % 8.50/4.53 ( ( H2 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) ) % 8.50/4.53 = ( case_option @ C @ A @ ( H2 @ F1 ) % 8.50/4.53 @ ^ [X: A] : ( H2 @ ( F22 @ X ) ) % 8.50/4.53 @ Option ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4893_Ball__def,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( ball @ A ) % 8.50/4.53 = ( ^ [A6: set @ A,P3: A > $o] : % 8.50/4.53 ! [X: A] : % 8.50/4.53 ( ( member @ A @ X @ A6 ) % 8.50/4.53 => ( P3 @ X ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4894_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A3: set @ A,B3: set @ A] : % 8.50/4.53 ( ( ( linord4507533701916653071of_set @ A @ A3 ) % 8.50/4.53 = ( linord4507533701916653071of_set @ A @ B3 ) ) % 8.50/4.53 => ( ( finite_finite2 @ A @ A3 ) % 8.50/4.53 => ( ( finite_finite2 @ A @ B3 ) % 8.50/4.53 => ( A3 = B3 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4895_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( linorder @ A ) % 8.50/4.53 => ! [A3: set @ A] : ( distinct @ A @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4896_option_Odisc__eq__case_I2_J,axiom, % 8.50/4.53 ! [A: $tType,Option: option @ A] : % 8.50/4.53 ( ( Option % 8.50/4.53 != ( none @ A ) ) % 8.50/4.53 = ( case_option @ $o @ A @ $false % 8.50/4.53 @ ^ [Uu3: A] : $true % 8.50/4.53 @ Option ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4897_option_Odisc__eq__case_I1_J,axiom, % 8.50/4.53 ! [A: $tType,Option: option @ A] : % 8.50/4.53 ( ( Option % 8.50/4.53 = ( none @ A ) ) % 8.50/4.53 = ( case_option @ $o @ A @ $true % 8.50/4.53 @ ^ [Uu3: A] : $false % 8.50/4.53 @ Option ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4898_prod_Odistrib__triv_H,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [I6: set @ B,G: B > A,H2: B > A] : % 8.50/4.53 ( ( finite_finite2 @ B @ I6 ) % 8.50/4.53 => ( ( groups1962203154675924110t_prod @ B @ A % 8.50/4.53 @ ^ [I3: B] : ( times_times @ A @ ( G @ I3 ) @ ( H2 @ I3 ) ) % 8.50/4.53 @ I6 ) % 8.50/4.53 = ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4899_case__optionE,axiom, % 8.50/4.53 ! [A: $tType,P: $o,Q: A > $o,X3: option @ A] : % 8.50/4.53 ( ( case_option @ $o @ A @ P @ Q @ X3 ) % 8.50/4.53 => ( ( ( X3 % 8.50/4.53 = ( none @ A ) ) % 8.50/4.53 => ~ P ) % 8.50/4.53 => ~ ! [Y4: A] : % 8.50/4.53 ( ( X3 % 8.50/4.53 = ( some @ A @ Y4 ) ) % 8.50/4.53 => ~ ( Q @ Y4 ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4900_Sup__eq__Inf,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ( ( complete_Sup_Sup @ A ) % 8.50/4.53 = ( ^ [A6: set @ A] : % 8.50/4.53 ( complete_Inf_Inf @ A % 8.50/4.53 @ ( collect @ A % 8.50/4.53 @ ^ [B4: A] : % 8.50/4.53 ! [X: A] : % 8.50/4.53 ( ( member @ A @ X @ A6 ) % 8.50/4.53 => ( ord_less_eq @ A @ X @ B4 ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4901_Inf__eq__Sup,axiom, % 8.50/4.53 ! [A: $tType] : % 8.50/4.53 ( ( comple6319245703460814977attice @ A ) % 8.50/4.53 => ( ( complete_Inf_Inf @ A ) % 8.50/4.53 = ( ^ [A6: set @ A] : % 8.50/4.53 ( complete_Sup_Sup @ A % 8.50/4.53 @ ( collect @ A % 8.50/4.53 @ ^ [B4: A] : % 8.50/4.53 ! [X: A] : % 8.50/4.53 ( ( member @ A @ X @ A6 ) % 8.50/4.53 => ( ord_less_eq @ A @ B4 @ X ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4902_option_Ocase__eq__if,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( case_option @ B @ A ) % 8.50/4.53 = ( ^ [F12: B,F23: A > B,Option3: option @ A] : % 8.50/4.53 ( if @ B % 8.50/4.53 @ ( Option3 % 8.50/4.53 = ( none @ A ) ) % 8.50/4.53 @ F12 % 8.50/4.53 @ ( F23 @ ( the2 @ A @ Option3 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4903_prod_Omono__neutral__cong__right_H,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [S3: set @ B,T4: set @ B,G: B > A,H2: B > A] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.53 => ( ! [X5: B] : % 8.50/4.53 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.53 => ( ( G @ X5 ) % 8.50/4.53 = ( one_one @ A ) ) ) % 8.50/4.53 => ( ! [X5: B] : % 8.50/4.53 ( ( member @ B @ X5 @ S3 ) % 8.50/4.53 => ( ( G @ X5 ) % 8.50/4.53 = ( H2 @ X5 ) ) ) % 8.50/4.53 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ T4 ) % 8.50/4.53 = ( groups1962203154675924110t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4904_prod_Omono__neutral__cong__left_H,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [S3: set @ B,T4: set @ B,H2: B > A,G: B > A] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.53 => ( ! [I4: B] : % 8.50/4.53 ( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.53 => ( ( H2 @ I4 ) % 8.50/4.53 = ( one_one @ A ) ) ) % 8.50/4.53 => ( ! [X5: B] : % 8.50/4.53 ( ( member @ B @ X5 @ S3 ) % 8.50/4.53 => ( ( G @ X5 ) % 8.50/4.53 = ( H2 @ X5 ) ) ) % 8.50/4.53 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 ) % 8.50/4.53 = ( groups1962203154675924110t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4905_prod_Omono__neutral__right_H,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [S3: set @ B,T4: set @ B,G: B > A] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.53 => ( ! [X5: B] : % 8.50/4.53 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.53 => ( ( G @ X5 ) % 8.50/4.53 = ( one_one @ A ) ) ) % 8.50/4.53 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ T4 ) % 8.50/4.53 = ( groups1962203154675924110t_prod @ B @ A @ G @ S3 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4906_prod_Omono__neutral__left_H,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [S3: set @ B,T4: set @ B,G: B > A] : % 8.50/4.53 ( ( ord_less_eq @ ( set @ B ) @ S3 @ T4 ) % 8.50/4.53 => ( ! [X5: B] : % 8.50/4.53 ( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T4 @ S3 ) ) % 8.50/4.53 => ( ( G @ X5 ) % 8.50/4.53 = ( one_one @ A ) ) ) % 8.50/4.53 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 ) % 8.50/4.53 = ( groups1962203154675924110t_prod @ B @ A @ G @ T4 ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4907_prod_Odistrib_H,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ! [I6: set @ B,G: B > A,H2: B > A] : % 8.50/4.53 ( ( finite_finite2 @ B % 8.50/4.53 @ ( collect @ B % 8.50/4.53 @ ^ [X: B] : % 8.50/4.53 ( ( member @ B @ X @ I6 ) % 8.50/4.53 & ( ( G @ X ) % 8.50/4.53 != ( one_one @ A ) ) ) ) ) % 8.50/4.53 => ( ( finite_finite2 @ B % 8.50/4.53 @ ( collect @ B % 8.50/4.53 @ ^ [X: B] : % 8.50/4.53 ( ( member @ B @ X @ I6 ) % 8.50/4.53 & ( ( H2 @ X ) % 8.50/4.53 != ( one_one @ A ) ) ) ) ) % 8.50/4.53 => ( ( groups1962203154675924110t_prod @ B @ A % 8.50/4.53 @ ^ [I3: B] : ( times_times @ A @ ( G @ I3 ) @ ( H2 @ I3 ) ) % 8.50/4.53 @ I6 ) % 8.50/4.53 = ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4908_prod_OG__def,axiom, % 8.50/4.53 ! [A: $tType,B: $tType] : % 8.50/4.53 ( ( comm_monoid_mult @ A ) % 8.50/4.53 => ( ( groups1962203154675924110t_prod @ B @ A ) % 8.50/4.53 = ( ^ [P6: B > A,I8: set @ B] : % 8.50/4.53 ( if @ A % 8.50/4.53 @ ( finite_finite2 @ B % 8.50/4.53 @ ( collect @ B % 8.50/4.53 @ ^ [X: B] : % 8.50/4.53 ( ( member @ B @ X @ I8 ) % 8.50/4.53 & ( ( P6 @ X ) % 8.50/4.53 != ( one_one @ A ) ) ) ) ) % 8.50/4.53 @ ( groups7121269368397514597t_prod @ B @ A @ P6 % 8.50/4.53 @ ( collect @ B % 8.50/4.53 @ ^ [X: B] : % 8.50/4.53 ( ( member @ B @ X @ I8 ) % 8.50/4.53 & ( ( P6 @ X ) % 8.50/4.53 != ( one_one @ A ) ) ) ) ) % 8.50/4.53 @ ( one_one @ A ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4909_option_Osplit__sel__asm,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] : % 8.50/4.53 ( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) ) % 8.50/4.53 = ( ~ ( ( ( Option % 8.50/4.53 = ( none @ A ) ) % 8.50/4.53 & ~ ( P @ F1 ) ) % 8.50/4.53 | ( ( Option % 8.50/4.53 = ( some @ A @ ( the2 @ A @ Option ) ) ) % 8.50/4.53 & ~ ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4910_option_Osplit__sel,axiom, % 8.50/4.53 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] : % 8.50/4.53 ( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) ) % 8.50/4.53 = ( ( ( Option % 8.50/4.53 = ( none @ A ) ) % 8.50/4.53 => ( P @ F1 ) ) % 8.50/4.53 & ( ( Option % 8.50/4.53 = ( some @ A @ ( the2 @ A @ Option ) ) ) % 8.50/4.53 => ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4911_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom, % 8.50/4.53 ! [Mima2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList2: list @ vEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] : % 8.50/4.53 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 ) % 8.50/4.53 = ( ( Deg = Deg4 ) % 8.50/4.53 & ! [X: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 8.50/4.53 => ( vEBT_VEBT_valid @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.53 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.53 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) % 8.50/4.53 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.53 & ( case_option @ $o @ ( product_prod @ nat @ nat ) % 8.50/4.53 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 ) % 8.50/4.53 & ! [X: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 8.50/4.53 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.50/4.53 @ ( product_case_prod @ nat @ nat @ $o % 8.50/4.53 @ ^ [Mi3: nat,Ma3: nat] : % 8.50/4.53 ( ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 8.50/4.53 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) % 8.50/4.53 & ! [I3: nat] : % 8.50/4.53 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.53 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) ) % 8.50/4.53 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) ) % 8.50/4.53 & ( ( Mi3 = Ma3 ) % 8.50/4.53 => ! [X: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList2 ) ) % 8.50/4.53 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.50/4.53 & ( ( Mi3 != Ma3 ) % 8.50/4.53 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 ) % 8.50/4.53 & ! [X: nat] : % 8.50/4.53 ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) % 8.50/4.53 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X ) % 8.50/4.53 => ( ( ord_less @ nat @ Mi3 @ X ) % 8.50/4.53 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) % 8.50/4.53 @ Mima2 ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4912_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom, % 8.50/4.53 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 8.50/4.53 ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 ) % 8.50/4.53 = Y ) % 8.50/4.53 => ( ( ? [Uu2: $o,Uv2: $o] : % 8.50/4.53 ( X3 % 8.50/4.53 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 8.50/4.53 => ( Y % 8.50/4.53 = ( Xa2 % 8.50/4.53 != ( one_one @ nat ) ) ) ) % 8.50/4.53 => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 8.50/4.53 ( ( X3 % 8.50/4.53 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) ) % 8.50/4.53 => ( Y % 8.50/4.53 = ( ~ ( ( Deg2 = Xa2 ) % 8.50/4.53 & ! [X: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ( vEBT_VEBT_valid @ X @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.53 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.53 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 8.50/4.53 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.53 & ( case_option @ $o @ ( product_prod @ nat @ nat ) % 8.50/4.53 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 ) % 8.50/4.53 & ! [X: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.50/4.53 @ ( product_case_prod @ nat @ nat @ $o % 8.50/4.53 @ ^ [Mi3: nat,Ma3: nat] : % 8.50/4.53 ( ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 8.50/4.53 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 8.50/4.53 & ! [I3: nat] : % 8.50/4.53 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.53 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) ) % 8.50/4.53 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) ) % 8.50/4.53 & ( ( Mi3 = Ma3 ) % 8.50/4.53 => ! [X: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.50/4.53 & ( ( Mi3 != Ma3 ) % 8.50/4.53 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 ) % 8.50/4.53 & ! [X: nat] : % 8.50/4.53 ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 8.50/4.53 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X ) % 8.50/4.53 => ( ( ord_less @ nat @ Mi3 @ X ) % 8.50/4.53 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) % 8.50/4.53 @ Mima ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4913_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom, % 8.50/4.53 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.53 ( ( vEBT_VEBT_valid @ X3 @ Xa2 ) % 8.50/4.53 => ( ( ? [Uu2: $o,Uv2: $o] : % 8.50/4.53 ( X3 % 8.50/4.53 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 8.50/4.53 => ( Xa2 % 8.50/4.53 != ( one_one @ nat ) ) ) % 8.50/4.53 => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 8.50/4.53 ( ( X3 % 8.50/4.53 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) ) % 8.50/4.53 => ~ ( ( Deg2 = Xa2 ) % 8.50/4.53 & ! [X2: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.53 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.53 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 8.50/4.53 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.53 & ( case_option @ $o @ ( product_prod @ nat @ nat ) % 8.50/4.53 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 ) % 8.50/4.53 & ! [X: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.50/4.53 @ ( product_case_prod @ nat @ nat @ $o % 8.50/4.53 @ ^ [Mi3: nat,Ma3: nat] : % 8.50/4.53 ( ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 8.50/4.53 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 8.50/4.53 & ! [I3: nat] : % 8.50/4.53 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.53 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) ) % 8.50/4.53 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) ) % 8.50/4.53 & ( ( Mi3 = Ma3 ) % 8.50/4.53 => ! [X: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.50/4.53 & ( ( Mi3 != Ma3 ) % 8.50/4.53 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 ) % 8.50/4.53 & ! [X: nat] : % 8.50/4.53 ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 8.50/4.53 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X ) % 8.50/4.53 => ( ( ord_less @ nat @ Mi3 @ X ) % 8.50/4.53 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) % 8.50/4.53 @ Mima ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4914_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom, % 8.50/4.53 ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] : % 8.50/4.53 ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 ) % 8.50/4.53 = Y ) % 8.50/4.53 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 8.50/4.53 => ( ! [Uu2: $o,Uv2: $o] : % 8.50/4.53 ( ( X3 % 8.50/4.53 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 8.50/4.53 => ( ( Y % 8.50/4.53 = ( Xa2 % 8.50/4.53 = ( one_one @ nat ) ) ) % 8.50/4.53 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) ) % 8.50/4.53 => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 8.50/4.53 ( ( X3 % 8.50/4.53 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) ) % 8.50/4.53 => ( ( Y % 8.50/4.53 = ( ( Deg2 = Xa2 ) % 8.50/4.53 & ! [X: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ( vEBT_VEBT_valid @ X @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.53 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.53 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 8.50/4.53 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.53 & ( case_option @ $o @ ( product_prod @ nat @ nat ) % 8.50/4.53 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 ) % 8.50/4.53 & ! [X: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.50/4.53 @ ( product_case_prod @ nat @ nat @ $o % 8.50/4.53 @ ^ [Mi3: nat,Ma3: nat] : % 8.50/4.53 ( ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 8.50/4.53 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 8.50/4.53 & ! [I3: nat] : % 8.50/4.53 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.53 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) ) % 8.50/4.53 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) ) % 8.50/4.53 & ( ( Mi3 = Ma3 ) % 8.50/4.53 => ! [X: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.50/4.53 & ( ( Mi3 != Ma3 ) % 8.50/4.53 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 ) % 8.50/4.53 & ! [X: nat] : % 8.50/4.53 ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 8.50/4.53 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X ) % 8.50/4.53 => ( ( ord_less @ nat @ Mi3 @ X ) % 8.50/4.53 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) % 8.50/4.53 @ Mima ) ) ) % 8.50/4.53 => ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4915_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom, % 8.50/4.53 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.53 ( ( vEBT_VEBT_valid @ X3 @ Xa2 ) % 8.50/4.53 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 8.50/4.53 => ( ! [Uu2: $o,Uv2: $o] : % 8.50/4.53 ( ( X3 % 8.50/4.53 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 8.50/4.53 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) % 8.50/4.53 => ( Xa2 % 8.50/4.53 != ( one_one @ nat ) ) ) ) % 8.50/4.53 => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 8.50/4.53 ( ( X3 % 8.50/4.53 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) ) % 8.50/4.53 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) % 8.50/4.53 => ~ ( ( Deg2 = Xa2 ) % 8.50/4.53 & ! [X2: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.53 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.53 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 8.50/4.53 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.53 & ( case_option @ $o @ ( product_prod @ nat @ nat ) % 8.50/4.53 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 ) % 8.50/4.53 & ! [X: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.50/4.53 @ ( product_case_prod @ nat @ nat @ $o % 8.50/4.53 @ ^ [Mi3: nat,Ma3: nat] : % 8.50/4.53 ( ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 8.50/4.53 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 8.50/4.53 & ! [I3: nat] : % 8.50/4.53 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.53 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) ) % 8.50/4.53 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) ) % 8.50/4.53 & ( ( Mi3 = Ma3 ) % 8.50/4.53 => ! [X: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.50/4.53 & ( ( Mi3 != Ma3 ) % 8.50/4.53 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 ) % 8.50/4.53 & ! [X: nat] : % 8.50/4.53 ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 8.50/4.53 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X ) % 8.50/4.53 => ( ( ord_less @ nat @ Mi3 @ X ) % 8.50/4.53 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) % 8.50/4.53 @ Mima ) ) ) ) ) ) ) ). % 8.50/4.53 % 8.50/4.53 thf(fact_4916_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom, % 8.50/4.53 ! [X3: vEBT_VEBT,Xa2: nat] : % 8.50/4.53 ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 ) % 8.50/4.53 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X3 @ Xa2 ) ) % 8.50/4.53 => ( ! [Uu2: $o,Uv2: $o] : % 8.50/4.53 ( ( X3 % 8.50/4.53 = ( vEBT_Leaf @ Uu2 @ Uv2 ) ) % 8.50/4.53 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) % 8.50/4.53 => ( Xa2 % 8.50/4.53 = ( one_one @ nat ) ) ) ) % 8.50/4.53 => ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList3: list @ vEBT_VEBT,Summary2: vEBT_VEBT] : % 8.50/4.53 ( ( X3 % 8.50/4.53 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) ) % 8.50/4.53 => ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) % 8.50/4.53 => ( ( Deg2 = Xa2 ) % 8.50/4.53 & ! [X5: vEBT_VEBT] : % 8.50/4.53 ( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.53 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.53 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.54 & ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 ) % 8.50/4.54 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.54 & ( case_option @ $o @ ( product_prod @ nat @ nat ) % 8.50/4.54 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 ) % 8.50/4.54 & ! [X: vEBT_VEBT] : % 8.50/4.54 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.54 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.50/4.54 @ ( product_case_prod @ nat @ nat @ $o % 8.50/4.54 @ ^ [Mi3: nat,Ma3: nat] : % 8.50/4.54 ( ( ord_less_eq @ nat @ Mi3 @ Ma3 ) % 8.50/4.54 & ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 8.50/4.54 & ! [I3: nat] : % 8.50/4.54 ( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.54 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) ) % 8.50/4.54 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) ) % 8.50/4.54 & ( ( Mi3 = Ma3 ) % 8.50/4.54 => ! [X: vEBT_VEBT] : % 8.50/4.54 ( ( member @ vEBT_VEBT @ X @ ( set2 @ vEBT_VEBT @ TreeList3 ) ) % 8.50/4.54 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) ) % 8.50/4.54 & ( ( Mi3 != Ma3 ) % 8.50/4.54 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ Ma3 ) % 8.50/4.54 & ! [X: nat] : % 8.50/4.54 ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) ) % 8.50/4.54 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList3 @ X ) % 8.50/4.54 => ( ( ord_less @ nat @ Mi3 @ X ) % 8.50/4.54 & ( ord_less_eq @ nat @ X @ Ma3 ) ) ) ) ) ) ) ) % 8.50/4.54 @ Mima ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4917_Union__maximal__sets,axiom, % 8.50/4.54 ! [A: $tType,F8: set @ ( set @ A )] : % 8.50/4.54 ( ( finite_finite2 @ ( set @ A ) @ F8 ) % 8.50/4.54 => ( ( complete_Sup_Sup @ ( set @ A ) % 8.50/4.54 @ ( collect @ ( set @ A ) % 8.50/4.54 @ ^ [T8: set @ A] : % 8.50/4.54 ( ( member @ ( set @ A ) @ T8 @ F8 ) % 8.50/4.54 & ! [X: set @ A] : % 8.50/4.54 ( ( member @ ( set @ A ) @ X @ F8 ) % 8.50/4.54 => ~ ( ord_less @ ( set @ A ) @ T8 @ X ) ) ) ) ) % 8.50/4.54 = ( complete_Sup_Sup @ ( set @ A ) @ F8 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4918_concat__injective,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] : % 8.50/4.54 ( ( ( concat @ A @ Xs ) % 8.50/4.54 = ( concat @ A @ Ys ) ) % 8.50/4.54 => ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs ) % 8.50/4.54 = ( size_size @ ( list @ ( list @ A ) ) @ Ys ) ) % 8.50/4.54 => ( ! [X5: product_prod @ ( list @ A ) @ ( list @ A )] : % 8.50/4.54 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X5 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) ) % 8.50/4.54 => ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o % 8.50/4.54 @ ^ [Y2: list @ A,Z6: list @ A] : % 8.50/4.54 ( ( size_size @ ( list @ A ) @ Y2 ) % 8.50/4.54 = ( size_size @ ( list @ A ) @ Z6 ) ) % 8.50/4.54 @ X5 ) ) % 8.50/4.54 => ( Xs = Ys ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4919_concat__eq__concat__iff,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] : % 8.50/4.54 ( ! [X5: product_prod @ ( list @ A ) @ ( list @ A )] : % 8.50/4.54 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X5 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) ) % 8.50/4.54 => ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o % 8.50/4.54 @ ^ [Y2: list @ A,Z6: list @ A] : % 8.50/4.54 ( ( size_size @ ( list @ A ) @ Y2 ) % 8.50/4.54 = ( size_size @ ( list @ A ) @ Z6 ) ) % 8.50/4.54 @ X5 ) ) % 8.50/4.54 => ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs ) % 8.50/4.54 = ( size_size @ ( list @ ( list @ A ) ) @ Ys ) ) % 8.50/4.54 => ( ( ( concat @ A @ Xs ) % 8.50/4.54 = ( concat @ A @ Ys ) ) % 8.50/4.54 = ( Xs = Ys ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4920_list__eq__iff__zip__eq,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ^ [Y3: list @ A,Z: list @ A] : ( Y3 = Z ) ) % 8.50/4.54 = ( ^ [Xs2: list @ A,Ys3: list @ A] : % 8.50/4.54 ( ( ( size_size @ ( list @ A ) @ Xs2 ) % 8.50/4.54 = ( size_size @ ( list @ A ) @ Ys3 ) ) % 8.50/4.54 & ! [X: product_prod @ A @ A] : % 8.50/4.54 ( ( member @ ( product_prod @ A @ A ) @ X @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs2 @ Ys3 ) ) ) % 8.50/4.54 => ( product_case_prod @ A @ A @ $o % 8.50/4.54 @ ^ [Y3: A,Z: A] : ( Y3 = Z ) % 8.50/4.54 @ X ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4921_Greatest__def,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order @ A ) % 8.50/4.54 => ( ( order_Greatest @ A ) % 8.50/4.54 = ( ^ [P3: A > $o] : % 8.50/4.54 ( the @ A % 8.50/4.54 @ ^ [X: A] : % 8.50/4.54 ( ( P3 @ X ) % 8.50/4.54 & ! [Y2: A] : % 8.50/4.54 ( ( P3 @ Y2 ) % 8.50/4.54 => ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4922_ord_OLeast__def,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( least @ A ) % 8.50/4.54 = ( ^ [Less_eq: A > A > $o,P3: A > $o] : % 8.50/4.54 ( the @ A % 8.50/4.54 @ ^ [X: A] : % 8.50/4.54 ( ( P3 @ X ) % 8.50/4.54 & ! [Y2: A] : % 8.50/4.54 ( ( P3 @ Y2 ) % 8.50/4.54 => ( Less_eq @ X @ Y2 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4923_listrel__iff__zip,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] : % 8.50/4.54 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) ) % 8.50/4.54 = ( ( ( size_size @ ( list @ A ) @ Xs ) % 8.50/4.54 = ( size_size @ ( list @ B ) @ Ys ) ) % 8.50/4.54 & ! [X: product_prod @ A @ B] : % 8.50/4.54 ( ( member @ ( product_prod @ A @ B ) @ X @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) % 8.50/4.54 => ( product_case_prod @ A @ B @ $o % 8.50/4.54 @ ^ [Y2: A,Z6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y2 @ Z6 ) @ R2 ) % 8.50/4.54 @ X ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4924_ord_OLeast_Ocong,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( least @ A ) % 8.50/4.54 = ( least @ A ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4925_listrel__Nil2,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ B )] : % 8.50/4.54 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( nil @ B ) ) @ ( listrel @ A @ B @ R2 ) ) % 8.50/4.54 => ( Xs % 8.50/4.54 = ( nil @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4926_listrel__Nil1,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,Xs: list @ B,R2: set @ ( product_prod @ A @ B )] : % 8.50/4.54 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs ) @ ( listrel @ A @ B @ R2 ) ) % 8.50/4.54 => ( Xs % 8.50/4.54 = ( nil @ B ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4927_listrel_ONil,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R2 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4928_listrel__eq__len,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] : % 8.50/4.54 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) ) % 8.50/4.54 => ( ( size_size @ ( list @ A ) @ Xs ) % 8.50/4.54 = ( size_size @ ( list @ B ) @ Ys ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4929_GreatestI__nat,axiom, % 8.50/4.54 ! [P: nat > $o,K: nat,B2: nat] : % 8.50/4.54 ( ( P @ K ) % 8.50/4.54 => ( ! [Y4: nat] : % 8.50/4.54 ( ( P @ Y4 ) % 8.50/4.54 => ( ord_less_eq @ nat @ Y4 @ B2 ) ) % 8.50/4.54 => ( P @ ( order_Greatest @ nat @ P ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4930_Greatest__le__nat,axiom, % 8.50/4.54 ! [P: nat > $o,K: nat,B2: nat] : % 8.50/4.54 ( ( P @ K ) % 8.50/4.54 => ( ! [Y4: nat] : % 8.50/4.54 ( ( P @ Y4 ) % 8.50/4.54 => ( ord_less_eq @ nat @ Y4 @ B2 ) ) % 8.50/4.54 => ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4931_GreatestI__ex__nat,axiom, % 8.50/4.54 ! [P: nat > $o,B2: nat] : % 8.50/4.54 ( ? [X_12: nat] : ( P @ X_12 ) % 8.50/4.54 => ( ! [Y4: nat] : % 8.50/4.54 ( ( P @ Y4 ) % 8.50/4.54 => ( ord_less_eq @ nat @ Y4 @ B2 ) ) % 8.50/4.54 => ( P @ ( order_Greatest @ nat @ P ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4932_listrel__mono,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) % 8.50/4.54 => ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( listrel @ A @ B @ R2 ) @ ( listrel @ A @ B @ S ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4933_GreatestI2__order,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order @ A ) % 8.50/4.54 => ! [P: A > $o,X3: A,Q: A > $o] : % 8.50/4.54 ( ( P @ X3 ) % 8.50/4.54 => ( ! [Y4: A] : % 8.50/4.54 ( ( P @ Y4 ) % 8.50/4.54 => ( ord_less_eq @ A @ Y4 @ X3 ) ) % 8.50/4.54 => ( ! [X5: A] : % 8.50/4.54 ( ( P @ X5 ) % 8.50/4.54 => ( ! [Y5: A] : % 8.50/4.54 ( ( P @ Y5 ) % 8.50/4.54 => ( ord_less_eq @ A @ Y5 @ X5 ) ) % 8.50/4.54 => ( Q @ X5 ) ) ) % 8.50/4.54 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4934_Greatest__equality,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order @ A ) % 8.50/4.54 => ! [P: A > $o,X3: A] : % 8.50/4.54 ( ( P @ X3 ) % 8.50/4.54 => ( ! [Y4: A] : % 8.50/4.54 ( ( P @ Y4 ) % 8.50/4.54 => ( ord_less_eq @ A @ Y4 @ X3 ) ) % 8.50/4.54 => ( ( order_Greatest @ A @ P ) % 8.50/4.54 = X3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4935_listrel__iff__nth,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] : % 8.50/4.54 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) ) % 8.50/4.54 = ( ( ( size_size @ ( list @ A ) @ Xs ) % 8.50/4.54 = ( size_size @ ( list @ B ) @ Ys ) ) % 8.50/4.54 & ! [N: nat] : % 8.50/4.54 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.54 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N ) @ ( nth @ B @ Ys @ N ) ) @ R2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4936_take__bit__numeral__numeral,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.54 => ! [M: num,N2: num] : % 8.50/4.54 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ A @ N2 ) ) % 8.50/4.54 = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4937_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linorder @ A ) % 8.50/4.54 => ! [A3: set @ A,X3: A] : % 8.50/4.54 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.54 => ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X3 @ A3 ) ) % 8.50/4.54 = ( linorder_insort_key @ A @ A % 8.50/4.54 @ ^ [X: A] : X % 8.50/4.54 @ X3 % 8.50/4.54 @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4938_sorted__list__of__set__def,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linorder @ A ) % 8.50/4.54 => ( ( linord4507533701916653071of_set @ A ) % 8.50/4.54 = ( linord144544945434240204of_set @ A @ A % 8.50/4.54 @ ^ [X: A] : X ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4939_take__bit__num__simps_I1_J,axiom, % 8.50/4.54 ! [M: num] : % 8.50/4.54 ( ( bit_take_bit_num @ ( zero_zero @ nat ) @ M ) % 8.50/4.54 = ( none @ num ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4940_remove1__insort__key,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( linorder @ A ) % 8.50/4.54 => ! [X3: B,F2: B > A,Xs: list @ B] : % 8.50/4.54 ( ( remove1 @ B @ X3 @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) % 8.50/4.54 = Xs ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4941_length__insort,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( linorder @ A ) % 8.50/4.54 => ! [F2: B > A,X3: B,Xs: list @ B] : % 8.50/4.54 ( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) % 8.50/4.54 = ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4942_take__bit__num__simps_I2_J,axiom, % 8.50/4.54 ! [N2: nat] : % 8.50/4.54 ( ( bit_take_bit_num @ ( suc @ N2 ) @ one2 ) % 8.50/4.54 = ( some @ num @ one2 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4943_take__bit__num__simps_I3_J,axiom, % 8.50/4.54 ! [N2: nat,M: num] : % 8.50/4.54 ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) ) % 8.50/4.54 = ( case_option @ ( option @ num ) @ num @ ( none @ num ) % 8.50/4.54 @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) ) % 8.50/4.54 @ ( bit_take_bit_num @ N2 @ M ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4944_take__bit__num__simps_I4_J,axiom, % 8.50/4.54 ! [N2: nat,M: num] : % 8.50/4.54 ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) ) % 8.50/4.54 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4945_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linorder @ A ) % 8.50/4.54 => ! [A3: set @ A,X3: A] : % 8.50/4.54 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.54 => ( ~ ( member @ A @ X3 @ A3 ) % 8.50/4.54 => ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X3 @ A3 ) ) % 8.50/4.54 = ( linorder_insort_key @ A @ A % 8.50/4.54 @ ^ [X: A] : X % 8.50/4.54 @ X3 % 8.50/4.54 @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4946_take__bit__num__simps_I6_J,axiom, % 8.50/4.54 ! [R2: num,M: num] : % 8.50/4.54 ( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R2 ) @ ( bit0 @ M ) ) % 8.50/4.54 = ( case_option @ ( option @ num ) @ num @ ( none @ num ) % 8.50/4.54 @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) ) % 8.50/4.54 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4947_insort__not__Nil,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( linorder @ A ) % 8.50/4.54 => ! [F2: B > A,A2: B,Xs: list @ B] : % 8.50/4.54 ( ( linorder_insort_key @ B @ A @ F2 @ A2 @ Xs ) % 8.50/4.54 != ( nil @ B ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4948_insort__key__left__comm,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( linorder @ A ) % 8.50/4.54 => ! [F2: B > A,X3: B,Y: B,Xs: list @ B] : % 8.50/4.54 ( ( ( F2 @ X3 ) % 8.50/4.54 != ( F2 @ Y ) ) % 8.50/4.54 => ( ( linorder_insort_key @ B @ A @ F2 @ Y @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) % 8.50/4.54 = ( linorder_insort_key @ B @ A @ F2 @ X3 @ ( linorder_insort_key @ B @ A @ F2 @ Y @ Xs ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4949_insort__left__comm,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linorder @ A ) % 8.50/4.54 => ! [X3: A,Y: A,Xs: list @ A] : % 8.50/4.54 ( ( linorder_insort_key @ A @ A % 8.50/4.54 @ ^ [X: A] : X % 8.50/4.54 @ X3 % 8.50/4.54 @ ( linorder_insort_key @ A @ A % 8.50/4.54 @ ^ [X: A] : X % 8.50/4.54 @ Y % 8.50/4.54 @ Xs ) ) % 8.50/4.54 = ( linorder_insort_key @ A @ A % 8.50/4.54 @ ^ [X: A] : X % 8.50/4.54 @ Y % 8.50/4.54 @ ( linorder_insort_key @ A @ A % 8.50/4.54 @ ^ [X: A] : X % 8.50/4.54 @ X3 % 8.50/4.54 @ Xs ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4950_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linorder @ A ) % 8.50/4.54 => ! [Y: A,X3: A] : % 8.50/4.54 ( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) % 8.50/4.54 @ ( linorder_insort_key @ A @ A % 8.50/4.54 @ ^ [X: A] : X % 8.50/4.54 @ Y ) % 8.50/4.54 @ ( linorder_insort_key @ A @ A % 8.50/4.54 @ ^ [X: A] : X % 8.50/4.54 @ X3 ) ) % 8.50/4.54 = ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) % 8.50/4.54 @ ( linorder_insort_key @ A @ A % 8.50/4.54 @ ^ [X: A] : X % 8.50/4.54 @ X3 ) % 8.50/4.54 @ ( linorder_insort_key @ A @ A % 8.50/4.54 @ ^ [X: A] : X % 8.50/4.54 @ Y ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4951_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom, % 8.50/4.54 ! [N2: nat,M: num] : % 8.50/4.54 ( ( bit_take_bit_num @ N2 @ ( bit0 @ M ) ) % 8.50/4.54 = ( case_nat @ ( option @ num ) @ ( none @ num ) % 8.50/4.54 @ ^ [N: nat] : % 8.50/4.54 ( case_option @ ( option @ num ) @ num @ ( none @ num ) % 8.50/4.54 @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) ) % 8.50/4.54 @ ( bit_take_bit_num @ N @ M ) ) % 8.50/4.54 @ N2 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4952_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom, % 8.50/4.54 ! [N2: nat] : % 8.50/4.54 ( ( bit_take_bit_num @ N2 @ one2 ) % 8.50/4.54 = ( case_nat @ ( option @ num ) @ ( none @ num ) % 8.50/4.54 @ ^ [N: nat] : ( some @ num @ one2 ) % 8.50/4.54 @ N2 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4953_set__insort__key,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( linorder @ A ) % 8.50/4.54 => ! [F2: B > A,X3: B,Xs: list @ B] : % 8.50/4.54 ( ( set2 @ B @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) % 8.50/4.54 = ( insert @ B @ X3 @ ( set2 @ B @ Xs ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4954_distinct__insort,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( linorder @ A ) % 8.50/4.54 => ! [F2: B > A,X3: B,Xs: list @ B] : % 8.50/4.54 ( ( distinct @ B @ ( linorder_insort_key @ B @ A @ F2 @ X3 @ Xs ) ) % 8.50/4.54 = ( ~ ( member @ B @ X3 @ ( set2 @ B @ Xs ) ) % 8.50/4.54 & ( distinct @ B @ Xs ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4955_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom, % 8.50/4.54 ! [N2: nat,M: num] : % 8.50/4.54 ( ( bit_take_bit_num @ N2 @ ( bit1 @ M ) ) % 8.50/4.54 = ( case_nat @ ( option @ num ) @ ( none @ num ) % 8.50/4.54 @ ^ [N: nat] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) % 8.50/4.54 @ N2 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4956_take__bit__num__eq__None__imp,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( bit_se359711467146920520ations @ A ) % 8.50/4.54 => ! [M: nat,N2: num] : % 8.50/4.54 ( ( ( bit_take_bit_num @ M @ N2 ) % 8.50/4.54 = ( none @ num ) ) % 8.50/4.54 => ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N2 ) ) % 8.50/4.54 = ( zero_zero @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4957_sorted__list__of__set_Ofold__insort__key_Oremove,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linorder @ A ) % 8.50/4.54 => ! [A3: set @ A,X3: A] : % 8.50/4.54 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.54 => ( ( member @ A @ X3 @ A3 ) % 8.50/4.54 => ( ( linord4507533701916653071of_set @ A @ A3 ) % 8.50/4.54 = ( linorder_insort_key @ A @ A % 8.50/4.54 @ ^ [X: A] : X % 8.50/4.54 @ X3 % 8.50/4.54 @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4958_take__bit__num__def,axiom, % 8.50/4.54 ( bit_take_bit_num % 8.50/4.54 = ( ^ [N: nat,M2: num] : % 8.50/4.54 ( if @ ( option @ num ) % 8.50/4.54 @ ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( numeral_numeral @ nat @ M2 ) ) % 8.50/4.54 = ( zero_zero @ nat ) ) % 8.50/4.54 @ ( none @ num ) % 8.50/4.54 @ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ ( numeral_numeral @ nat @ M2 ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4959_Bit__Operations_Otake__bit__num__code,axiom, % 8.50/4.54 ( bit_take_bit_num % 8.50/4.54 = ( ^ [N: nat,M2: num] : % 8.50/4.54 ( product_case_prod @ nat @ num @ ( option @ num ) % 8.50/4.54 @ ^ [A4: nat,X: num] : % 8.50/4.54 ( case_nat @ ( option @ num ) @ ( none @ num ) % 8.50/4.54 @ ^ [O: nat] : % 8.50/4.54 ( case_num @ ( option @ num ) @ ( some @ num @ one2 ) % 8.50/4.54 @ ^ [P6: num] : % 8.50/4.54 ( case_option @ ( option @ num ) @ num @ ( none @ num ) % 8.50/4.54 @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) ) % 8.50/4.54 @ ( bit_take_bit_num @ O @ P6 ) ) % 8.50/4.54 @ ^ [P6: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) ) % 8.50/4.54 @ X ) % 8.50/4.54 @ A4 ) % 8.50/4.54 @ ( product_Pair @ nat @ num @ N @ M2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4960_set__nths,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A,I6: set @ nat] : % 8.50/4.54 ( ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) % 8.50/4.54 = ( collect @ A % 8.50/4.54 @ ^ [Uu3: A] : % 8.50/4.54 ? [I3: nat] : % 8.50/4.54 ( ( Uu3 % 8.50/4.54 = ( nth @ A @ Xs @ I3 ) ) % 8.50/4.54 & ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.54 & ( member @ nat @ I3 @ I6 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4961_Rats__eq__int__div__nat,axiom, % 8.50/4.54 ( ( field_char_0_Rats @ real ) % 8.50/4.54 = ( collect @ real % 8.50/4.54 @ ^ [Uu3: real] : % 8.50/4.54 ? [I3: int,N: nat] : % 8.50/4.54 ( ( Uu3 % 8.50/4.54 = ( divide_divide @ real @ ( ring_1_of_int @ real @ I3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) % 8.50/4.54 & ( N % 8.50/4.54 != ( zero_zero @ nat ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4962_nths__nil,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ nat] : % 8.50/4.54 ( ( nths @ A @ ( nil @ A ) @ A3 ) % 8.50/4.54 = ( nil @ A ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4963_nths__empty,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A] : % 8.50/4.54 ( ( nths @ A @ Xs @ ( bot_bot @ ( set @ nat ) ) ) % 8.50/4.54 = ( nil @ A ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4964_Rats__infinite,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( field_char_0 @ A ) % 8.50/4.54 => ~ ( finite_finite2 @ A @ ( field_char_0_Rats @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4965_distinct__nthsI,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A,I6: set @ nat] : % 8.50/4.54 ( ( distinct @ A @ Xs ) % 8.50/4.54 => ( distinct @ A @ ( nths @ A @ Xs @ I6 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4966_Rats__add,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( field_char_0 @ A ) % 8.50/4.54 => ! [A2: A,B2: A] : % 8.50/4.54 ( ( member @ A @ A2 @ ( field_char_0_Rats @ A ) ) % 8.50/4.54 => ( ( member @ A @ B2 @ ( field_char_0_Rats @ A ) ) % 8.50/4.54 => ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( field_char_0_Rats @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4967_Rats__dense__in__real,axiom, % 8.50/4.54 ! [X3: real,Y: real] : % 8.50/4.54 ( ( ord_less @ real @ X3 @ Y ) % 8.50/4.54 => ? [X5: real] : % 8.50/4.54 ( ( member @ real @ X5 @ ( field_char_0_Rats @ real ) ) % 8.50/4.54 & ( ord_less @ real @ X3 @ X5 ) % 8.50/4.54 & ( ord_less @ real @ X5 @ Y ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4968_Rats__no__bot__less,axiom, % 8.50/4.54 ! [X3: real] : % 8.50/4.54 ? [X5: real] : % 8.50/4.54 ( ( member @ real @ X5 @ ( field_char_0_Rats @ real ) ) % 8.50/4.54 & ( ord_less @ real @ X5 @ X3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4969_Rats__0,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( field_char_0 @ A ) % 8.50/4.54 => ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4970_notin__set__nthsI,axiom, % 8.50/4.54 ! [A: $tType,X3: A,Xs: list @ A,I6: set @ nat] : % 8.50/4.54 ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) % 8.50/4.54 => ~ ( member @ A @ X3 @ ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4971_in__set__nthsD,axiom, % 8.50/4.54 ! [A: $tType,X3: A,Xs: list @ A,I6: set @ nat] : % 8.50/4.54 ( ( member @ A @ X3 @ ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) ) % 8.50/4.54 => ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4972_set__nths__subset,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A,I6: set @ nat] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) @ ( set2 @ A @ Xs ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4973_nths__all,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A,I6: set @ nat] : % 8.50/4.54 ( ! [I4: nat] : % 8.50/4.54 ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.54 => ( member @ nat @ I4 @ I6 ) ) % 8.50/4.54 => ( ( nths @ A @ Xs @ I6 ) % 8.50/4.54 = Xs ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4974_length__nths,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A,I6: set @ nat] : % 8.50/4.54 ( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I6 ) ) % 8.50/4.54 = ( finite_card @ nat % 8.50/4.54 @ ( collect @ nat % 8.50/4.54 @ ^ [I3: nat] : % 8.50/4.54 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.54 & ( member @ nat @ I3 @ I6 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4975_Nats__altdef1,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ring_1 @ A ) % 8.50/4.54 => ( ( semiring_1_Nats @ A ) % 8.50/4.54 = ( collect @ A % 8.50/4.54 @ ^ [Uu3: A] : % 8.50/4.54 ? [N: int] : % 8.50/4.54 ( ( Uu3 % 8.50/4.54 = ( ring_1_of_int @ A @ N ) ) % 8.50/4.54 & ( ord_less_eq @ int @ ( zero_zero @ int ) @ N ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4976_rat__floor__lemma,axiom, % 8.50/4.54 ! [A2: int,B2: int] : % 8.50/4.54 ( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ ( divide_divide @ int @ A2 @ B2 ) ) @ ( fract @ A2 @ B2 ) ) % 8.50/4.54 & ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( one_one @ int ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4977_image__minus__const__atLeastLessThan__nat,axiom, % 8.50/4.54 ! [C2: nat,Y: nat,X3: nat] : % 8.50/4.54 ( ( ( ord_less @ nat @ C2 @ Y ) % 8.50/4.54 => ( ( image @ nat @ nat % 8.50/4.54 @ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 ) % 8.50/4.54 @ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y ) ) % 8.50/4.54 = ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X3 @ C2 ) @ ( minus_minus @ nat @ Y @ C2 ) ) ) ) % 8.50/4.54 & ( ~ ( ord_less @ nat @ C2 @ Y ) % 8.50/4.54 => ( ( ( ord_less @ nat @ X3 @ Y ) % 8.50/4.54 => ( ( image @ nat @ nat % 8.50/4.54 @ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 ) % 8.50/4.54 @ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y ) ) % 8.50/4.54 = ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) % 8.50/4.54 & ( ~ ( ord_less @ nat @ X3 @ Y ) % 8.50/4.54 => ( ( image @ nat @ nat % 8.50/4.54 @ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 ) % 8.50/4.54 @ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y ) ) % 8.50/4.54 = ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4978_image__eqI,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,B2: A,F2: B > A,X3: B,A3: set @ B] : % 8.50/4.54 ( ( B2 % 8.50/4.54 = ( F2 @ X3 ) ) % 8.50/4.54 => ( ( member @ B @ X3 @ A3 ) % 8.50/4.54 => ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4979_image__ident,axiom, % 8.50/4.54 ! [A: $tType,Y7: set @ A] : % 8.50/4.54 ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : X % 8.50/4.54 @ Y7 ) % 8.50/4.54 = Y7 ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4980_image__is__empty,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] : % 8.50/4.54 ( ( ( image @ B @ A @ F2 @ A3 ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 = ( A3 % 8.50/4.54 = ( bot_bot @ ( set @ B ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4981_empty__is__image,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] : % 8.50/4.54 ( ( ( bot_bot @ ( set @ A ) ) % 8.50/4.54 = ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 = ( A3 % 8.50/4.54 = ( bot_bot @ ( set @ B ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4982_image__empty,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,F2: B > A] : % 8.50/4.54 ( ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4983_finite__imageI,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,F5: set @ A,H2: A > B] : % 8.50/4.54 ( ( finite_finite2 @ A @ F5 ) % 8.50/4.54 => ( finite_finite2 @ B @ ( image @ A @ B @ H2 @ F5 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4984_image__insert,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,A2: B,B3: set @ B] : % 8.50/4.54 ( ( image @ B @ A @ F2 @ ( insert @ B @ A2 @ B3 ) ) % 8.50/4.54 = ( insert @ A @ ( F2 @ A2 ) @ ( image @ B @ A @ F2 @ B3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4985_insert__image,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,X3: A,A3: set @ A,F2: A > B] : % 8.50/4.54 ( ( member @ A @ X3 @ A3 ) % 8.50/4.54 => ( ( insert @ B @ ( F2 @ X3 ) @ ( image @ A @ B @ F2 @ A3 ) ) % 8.50/4.54 = ( image @ A @ B @ F2 @ A3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4986_image__add__0,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( comm_monoid_add @ A ) % 8.50/4.54 => ! [S3: set @ A] : % 8.50/4.54 ( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S3 ) % 8.50/4.54 = S3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4987_image__add__atLeastAtMost,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linordered_semidom @ A ) % 8.50/4.54 => ! [K: A,I: A,J: A] : % 8.50/4.54 ( ( image @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or1337092689740270186AtMost @ A @ I @ J ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4988_image__add__atLeastLessThan,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linordered_semidom @ A ) % 8.50/4.54 => ! [K: A,I: A,J: A] : % 8.50/4.54 ( ( image @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) ) % 8.50/4.54 = ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4989_image__add__atMost,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ordered_ab_group_add @ A ) % 8.50/4.54 => ! [C2: A,A2: A] : % 8.50/4.54 ( ( image @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_ord_atMost @ A @ A2 ) ) % 8.50/4.54 = ( set_ord_atMost @ A @ ( plus_plus @ A @ C2 @ A2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4990_image__Suc__atLeastAtMost,axiom, % 8.50/4.54 ! [I: nat,J: nat] : % 8.50/4.54 ( ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4991_image__Suc__atLeastLessThan,axiom, % 8.50/4.54 ! [I: nat,J: nat] : % 8.50/4.54 ( ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) ) % 8.50/4.54 = ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4992_SUP__bot__conv_I2_J,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [B3: B > A,A3: set @ B] : % 8.50/4.54 ( ( ( bot_bot @ A ) % 8.50/4.54 = ( complete_Sup_Sup @ A @ ( image @ B @ A @ B3 @ A3 ) ) ) % 8.50/4.54 = ( ! [X: B] : % 8.50/4.54 ( ( member @ B @ X @ A3 ) % 8.50/4.54 => ( ( B3 @ X ) % 8.50/4.54 = ( bot_bot @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4993_SUP__bot__conv_I1_J,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [B3: B > A,A3: set @ B] : % 8.50/4.54 ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ B3 @ A3 ) ) % 8.50/4.54 = ( bot_bot @ A ) ) % 8.50/4.54 = ( ! [X: B] : % 8.50/4.54 ( ( member @ B @ X @ A3 ) % 8.50/4.54 => ( ( B3 @ X ) % 8.50/4.54 = ( bot_bot @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4994_SUP__bot,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [A3: set @ B] : % 8.50/4.54 ( ( complete_Sup_Sup @ A % 8.50/4.54 @ ( image @ B @ A % 8.50/4.54 @ ^ [X: B] : ( bot_bot @ A ) % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = ( bot_bot @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4995_ccSUP__bot,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( counta3822494911875563373attice @ A ) % 8.50/4.54 => ! [A3: set @ B] : % 8.50/4.54 ( ( complete_Sup_Sup @ A % 8.50/4.54 @ ( image @ B @ A % 8.50/4.54 @ ^ [X: B] : ( bot_bot @ A ) % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = ( bot_bot @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4996_SUP__const,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [A3: set @ B,F2: A] : % 8.50/4.54 ( ( A3 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Sup_Sup @ A % 8.50/4.54 @ ( image @ B @ A % 8.50/4.54 @ ^ [I3: B] : F2 % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = F2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4997_ccSUP__const,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( counta3822494911875563373attice @ A ) % 8.50/4.54 => ! [A3: set @ B,F2: A] : % 8.50/4.54 ( ( A3 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Sup_Sup @ A % 8.50/4.54 @ ( image @ B @ A % 8.50/4.54 @ ^ [I3: B] : F2 % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = F2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4998_cSUP__const,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( condit1219197933456340205attice @ A ) % 8.50/4.54 => ! [A3: set @ B,C2: A] : % 8.50/4.54 ( ( A3 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Sup_Sup @ A % 8.50/4.54 @ ( image @ B @ A % 8.50/4.54 @ ^ [X: B] : C2 % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = C2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_4999_image__add__atLeastAtMost_H,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linordered_semidom @ A ) % 8.50/4.54 => ! [K: A,I: A,J: A] : % 8.50/4.54 ( ( image @ A @ A % 8.50/4.54 @ ^ [N: A] : ( plus_plus @ A @ N @ K ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ I @ J ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5000_INF__const,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [A3: set @ B,F2: A] : % 8.50/4.54 ( ( A3 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A % 8.50/4.54 @ ( image @ B @ A % 8.50/4.54 @ ^ [I3: B] : F2 % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = F2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5001_ccINF__const,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( counta3822494911875563373attice @ A ) % 8.50/4.54 => ! [A3: set @ B,F2: A] : % 8.50/4.54 ( ( A3 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A % 8.50/4.54 @ ( image @ B @ A % 8.50/4.54 @ ^ [I3: B] : F2 % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = F2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5002_cINF__const,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( condit1219197933456340205attice @ A ) % 8.50/4.54 => ! [A3: set @ B,C2: A] : % 8.50/4.54 ( ( A3 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A % 8.50/4.54 @ ( image @ B @ A % 8.50/4.54 @ ^ [X: B] : C2 % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = C2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5003_image__add__atLeastLessThan_H,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linordered_semidom @ A ) % 8.50/4.54 => ! [K: A,I: A,J: A] : % 8.50/4.54 ( ( image @ A @ A % 8.50/4.54 @ ^ [N: A] : ( plus_plus @ A @ N @ K ) % 8.50/4.54 @ ( set_or7035219750837199246ssThan @ A @ I @ J ) ) % 8.50/4.54 = ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5004_INF__eq__bot__iff,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple5582772986160207858norder @ A ) % 8.50/4.54 => ! [F2: B > A,A3: set @ B] : % 8.50/4.54 ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 = ( bot_bot @ A ) ) % 8.50/4.54 = ( ! [X: A] : % 8.50/4.54 ( ( ord_less @ A @ ( bot_bot @ A ) @ X ) % 8.50/4.54 => ? [Y2: B] : % 8.50/4.54 ( ( member @ B @ Y2 @ A3 ) % 8.50/4.54 & ( ord_less @ A @ ( F2 @ Y2 ) @ X ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5005_ccSUP__empty,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( counta3822494911875563373attice @ A ) % 8.50/4.54 => ! [F2: B > A] : % 8.50/4.54 ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) ) % 8.50/4.54 = ( bot_bot @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5006_image__mult__atLeastAtMost,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linordered_field @ A ) % 8.50/4.54 => ! [D2: A,A2: A,B2: A] : % 8.50/4.54 ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 ) % 8.50/4.54 => ( ( image @ A @ A @ ( times_times @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D2 @ A2 ) @ ( times_times @ A @ D2 @ B2 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5007_image__divide__atLeastAtMost,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linordered_field @ A ) % 8.50/4.54 => ! [D2: A,A2: A,B2: A] : % 8.50/4.54 ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [C4: A] : ( divide_divide @ A @ C4 @ D2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A2 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5008_less__rat,axiom, % 8.50/4.54 ! [B2: int,D2: int,A2: int,C2: int] : % 8.50/4.54 ( ( B2 % 8.50/4.54 != ( zero_zero @ int ) ) % 8.50/4.54 => ( ( D2 % 8.50/4.54 != ( zero_zero @ int ) ) % 8.50/4.54 => ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( fract @ C2 @ D2 ) ) % 8.50/4.54 = ( ord_less @ int @ ( times_times @ int @ ( times_times @ int @ A2 @ D2 ) @ ( times_times @ int @ B2 @ D2 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5009_setcompr__eq__image,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,P: B > $o] : % 8.50/4.54 ( ( collect @ A % 8.50/4.54 @ ^ [Uu3: A] : % 8.50/4.54 ? [X: B] : % 8.50/4.54 ( ( Uu3 % 8.50/4.54 = ( F2 @ X ) ) % 8.50/4.54 & ( P @ X ) ) ) % 8.50/4.54 = ( image @ B @ A @ F2 @ ( collect @ B @ P ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5010_Setcompr__eq__image,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] : % 8.50/4.54 ( ( collect @ A % 8.50/4.54 @ ^ [Uu3: A] : % 8.50/4.54 ? [X: B] : % 8.50/4.54 ( ( Uu3 % 8.50/4.54 = ( F2 @ X ) ) % 8.50/4.54 & ( member @ B @ X @ A3 ) ) ) % 8.50/4.54 = ( image @ B @ A @ F2 @ A3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5011_of__nat__in__Nats,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( semiring_1 @ A ) % 8.50/4.54 => ! [N2: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_1_Nats @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5012_Nats__induct,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( semiring_1 @ A ) % 8.50/4.54 => ! [X3: A,P: A > $o] : % 8.50/4.54 ( ( member @ A @ X3 @ ( semiring_1_Nats @ A ) ) % 8.50/4.54 => ( ! [N3: nat] : ( P @ ( semiring_1_of_nat @ A @ N3 ) ) % 8.50/4.54 => ( P @ X3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5013_Nats__cases,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( semiring_1 @ A ) % 8.50/4.54 => ! [X3: A] : % 8.50/4.54 ( ( member @ A @ X3 @ ( semiring_1_Nats @ A ) ) % 8.50/4.54 => ~ ! [N3: nat] : % 8.50/4.54 ( X3 % 8.50/4.54 != ( semiring_1_of_nat @ A @ N3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5014_image__Fpow__mono,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B3: set @ A] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B3 ) % 8.50/4.54 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F2 ) @ ( finite_Fpow @ B @ A3 ) ) @ ( finite_Fpow @ A @ B3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5015_imageI,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,X3: A,A3: set @ A,F2: A > B] : % 8.50/4.54 ( ( member @ A @ X3 @ A3 ) % 8.50/4.54 => ( member @ B @ ( F2 @ X3 ) @ ( image @ A @ B @ F2 @ A3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5016_image__iff,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,Z2: A,F2: B > A,A3: set @ B] : % 8.50/4.54 ( ( member @ A @ Z2 @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 = ( ? [X: B] : % 8.50/4.54 ( ( member @ B @ X @ A3 ) % 8.50/4.54 & ( Z2 % 8.50/4.54 = ( F2 @ X ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5017_bex__imageD,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] : % 8.50/4.54 ( ? [X2: A] : % 8.50/4.54 ( ( member @ A @ X2 @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 & ( P @ X2 ) ) % 8.50/4.54 => ? [X5: B] : % 8.50/4.54 ( ( member @ B @ X5 @ A3 ) % 8.50/4.54 & ( P @ ( F2 @ X5 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5018_image__cong,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,M7: set @ A,N7: set @ A,F2: A > B,G: A > B] : % 8.50/4.54 ( ( M7 = N7 ) % 8.50/4.54 => ( ! [X5: A] : % 8.50/4.54 ( ( member @ A @ X5 @ N7 ) % 8.50/4.54 => ( ( F2 @ X5 ) % 8.50/4.54 = ( G @ X5 ) ) ) % 8.50/4.54 => ( ( image @ A @ B @ F2 @ M7 ) % 8.50/4.54 = ( image @ A @ B @ G @ N7 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5019_ball__imageD,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] : % 8.50/4.54 ( ! [X5: A] : % 8.50/4.54 ( ( member @ A @ X5 @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 => ( P @ X5 ) ) % 8.50/4.54 => ! [X2: B] : % 8.50/4.54 ( ( member @ B @ X2 @ A3 ) % 8.50/4.54 => ( P @ ( F2 @ X2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5020_rev__image__eqI,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,X3: A,A3: set @ A,B2: B,F2: A > B] : % 8.50/4.54 ( ( member @ A @ X3 @ A3 ) % 8.50/4.54 => ( ( B2 % 8.50/4.54 = ( F2 @ X3 ) ) % 8.50/4.54 => ( member @ B @ B2 @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5021_imageE,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,B2: A,F2: B > A,A3: set @ B] : % 8.50/4.54 ( ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 => ~ ! [X5: B] : % 8.50/4.54 ( ( B2 % 8.50/4.54 = ( F2 @ X5 ) ) % 8.50/4.54 => ~ ( member @ B @ X5 @ A3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5022_image__image,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,C: $tType,F2: B > A,G: C > B,A3: set @ C] : % 8.50/4.54 ( ( image @ B @ A @ F2 @ ( image @ C @ B @ G @ A3 ) ) % 8.50/4.54 = ( image @ C @ A % 8.50/4.54 @ ^ [X: C] : ( F2 @ ( G @ X ) ) % 8.50/4.54 @ A3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5023_Compr__image__eq,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] : % 8.50/4.54 ( ( collect @ A % 8.50/4.54 @ ^ [X: A] : % 8.50/4.54 ( ( member @ A @ X @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 & ( P @ X ) ) ) % 8.50/4.54 = ( image @ B @ A @ F2 % 8.50/4.54 @ ( collect @ B % 8.50/4.54 @ ^ [X: B] : % 8.50/4.54 ( ( member @ B @ X @ A3 ) % 8.50/4.54 & ( P @ ( F2 @ X ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5024_all__subset__image,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] : % 8.50/4.54 ( ( ! [B6: set @ A] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 => ( P @ B6 ) ) ) % 8.50/4.54 = ( ! [B6: set @ B] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ B ) @ B6 @ A3 ) % 8.50/4.54 => ( P @ ( image @ B @ A @ F2 @ B6 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5025_subset__image__iff,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,B3: set @ A,F2: B > A,A3: set @ B] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 = ( ? [AA: set @ B] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ B ) @ AA @ A3 ) % 8.50/4.54 & ( B3 % 8.50/4.54 = ( image @ B @ A @ F2 @ AA ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5026_image__subset__iff,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B3: set @ A] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B3 ) % 8.50/4.54 = ( ! [X: B] : % 8.50/4.54 ( ( member @ B @ X @ A3 ) % 8.50/4.54 => ( member @ A @ ( F2 @ X ) @ B3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5027_subset__imageE,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,B3: set @ A,F2: B > A,A3: set @ B] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 => ~ ! [C8: set @ B] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ B ) @ C8 @ A3 ) % 8.50/4.54 => ( B3 % 8.50/4.54 != ( image @ B @ A @ F2 @ C8 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5028_image__subsetI,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,A3: set @ A,F2: A > B,B3: set @ B] : % 8.50/4.54 ( ! [X5: A] : % 8.50/4.54 ( ( member @ A @ X5 @ A3 ) % 8.50/4.54 => ( member @ B @ ( F2 @ X5 ) @ B3 ) ) % 8.50/4.54 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ B3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5029_image__mono,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,F2: A > B] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.54 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A3 ) @ ( image @ A @ B @ F2 @ B3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5030_image__Pow__surj,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B3: set @ A] : % 8.50/4.54 ( ( ( image @ B @ A @ F2 @ A3 ) % 8.50/4.54 = B3 ) % 8.50/4.54 => ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F2 ) @ ( pow2 @ B @ A3 ) ) % 8.50/4.54 = ( pow2 @ A @ B3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5031_image__Collect__subsetI,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,P: A > $o,F2: A > B,B3: set @ B] : % 8.50/4.54 ( ! [X5: A] : % 8.50/4.54 ( ( P @ X5 ) % 8.50/4.54 => ( member @ B @ ( F2 @ X5 ) @ B3 ) ) % 8.50/4.54 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( collect @ A @ P ) ) @ B3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5032_image__Pow__mono,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,F2: B > A,A3: set @ B,B3: set @ A] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ B3 ) % 8.50/4.54 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F2 ) @ ( pow2 @ B @ A3 ) ) @ ( pow2 @ A @ B3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5033_Nats__0,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( semiring_1 @ A ) % 8.50/4.54 => ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5034_Nats__add,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( semiring_1 @ A ) % 8.50/4.54 => ! [A2: A,B2: A] : % 8.50/4.54 ( ( member @ A @ A2 @ ( semiring_1_Nats @ A ) ) % 8.50/4.54 => ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) ) % 8.50/4.54 => ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5035_Nats__1,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( semiring_1 @ A ) % 8.50/4.54 => ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5036_Nats__mult,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( semiring_1 @ A ) % 8.50/4.54 => ! [A2: A,B2: A] : % 8.50/4.54 ( ( member @ A @ A2 @ ( semiring_1_Nats @ A ) ) % 8.50/4.54 => ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) ) % 8.50/4.54 => ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5037_Nats__numeral,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( semiring_1 @ A ) % 8.50/4.54 => ! [W2: num] : ( member @ A @ ( numeral_numeral @ A @ W2 ) @ ( semiring_1_Nats @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5038_UNION__singleton__eq__range,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] : % 8.50/4.54 ( ( complete_Sup_Sup @ ( set @ A ) % 8.50/4.54 @ ( image @ B @ ( set @ A ) % 8.50/4.54 @ ^ [X: B] : ( insert @ A @ ( F2 @ X ) @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = ( image @ B @ A @ F2 @ A3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5039_pigeonhole__infinite,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A3: set @ A,F2: A > B] : % 8.50/4.54 ( ~ ( finite_finite2 @ A @ A3 ) % 8.50/4.54 => ( ( finite_finite2 @ B @ ( image @ A @ B @ F2 @ A3 ) ) % 8.50/4.54 => ? [X5: A] : % 8.50/4.54 ( ( member @ A @ X5 @ A3 ) % 8.50/4.54 & ~ ( finite_finite2 @ A % 8.50/4.54 @ ( collect @ A % 8.50/4.54 @ ^ [A4: A] : % 8.50/4.54 ( ( member @ A @ A4 @ A3 ) % 8.50/4.54 & ( ( F2 @ A4 ) % 8.50/4.54 = ( F2 @ X5 ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5040_SUP__eq,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,C: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [A3: set @ B,B3: set @ C,F2: B > A,G: C > A] : % 8.50/4.54 ( ! [I4: B] : % 8.50/4.54 ( ( member @ B @ I4 @ A3 ) % 8.50/4.54 => ? [X2: C] : % 8.50/4.54 ( ( member @ C @ X2 @ B3 ) % 8.50/4.54 & ( ord_less_eq @ A @ ( F2 @ I4 ) @ ( G @ X2 ) ) ) ) % 8.50/4.54 => ( ! [J2: C] : % 8.50/4.54 ( ( member @ C @ J2 @ B3 ) % 8.50/4.54 => ? [X2: B] : % 8.50/4.54 ( ( member @ B @ X2 @ A3 ) % 8.50/4.54 & ( ord_less_eq @ A @ ( G @ J2 ) @ ( F2 @ X2 ) ) ) ) % 8.50/4.54 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 = ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B3 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5041_INF__eq,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,C: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [A3: set @ B,B3: set @ C,G: C > A,F2: B > A] : % 8.50/4.54 ( ! [I4: B] : % 8.50/4.54 ( ( member @ B @ I4 @ A3 ) % 8.50/4.54 => ? [X2: C] : % 8.50/4.54 ( ( member @ C @ X2 @ B3 ) % 8.50/4.54 & ( ord_less_eq @ A @ ( G @ X2 ) @ ( F2 @ I4 ) ) ) ) % 8.50/4.54 => ( ! [J2: C] : % 8.50/4.54 ( ( member @ C @ J2 @ B3 ) % 8.50/4.54 => ? [X2: B] : % 8.50/4.54 ( ( member @ B @ X2 @ A3 ) % 8.50/4.54 & ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G @ J2 ) ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 = ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B3 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5042_zero__notin__Suc__image,axiom, % 8.50/4.54 ! [A3: set @ nat] : % 8.50/4.54 ~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5043_finite__surj,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,A3: set @ A,B3: set @ B,F2: A > B] : % 8.50/4.54 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.54 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ ( image @ A @ B @ F2 @ A3 ) ) % 8.50/4.54 => ( finite_finite2 @ B @ B3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5044_finite__subset__image,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,B3: set @ A,F2: B > A,A3: set @ B] : % 8.50/4.54 ( ( finite_finite2 @ A @ B3 ) % 8.50/4.54 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 => ? [C8: set @ B] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ B ) @ C8 @ A3 ) % 8.50/4.54 & ( finite_finite2 @ B @ C8 ) % 8.50/4.54 & ( B3 % 8.50/4.54 = ( image @ B @ A @ F2 @ C8 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5045_ex__finite__subset__image,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] : % 8.50/4.54 ( ( ? [B6: set @ A] : % 8.50/4.54 ( ( finite_finite2 @ A @ B6 ) % 8.50/4.54 & ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 & ( P @ B6 ) ) ) % 8.50/4.54 = ( ? [B6: set @ B] : % 8.50/4.54 ( ( finite_finite2 @ B @ B6 ) % 8.50/4.54 & ( ord_less_eq @ ( set @ B ) @ B6 @ A3 ) % 8.50/4.54 & ( P @ ( image @ B @ A @ F2 @ B6 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5046_all__finite__subset__image,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: ( set @ A ) > $o] : % 8.50/4.54 ( ( ! [B6: set @ A] : % 8.50/4.54 ( ( ( finite_finite2 @ A @ B6 ) % 8.50/4.54 & ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F2 @ A3 ) ) ) % 8.50/4.54 => ( P @ B6 ) ) ) % 8.50/4.54 = ( ! [B6: set @ B] : % 8.50/4.54 ( ( ( finite_finite2 @ B @ B6 ) % 8.50/4.54 & ( ord_less_eq @ ( set @ B ) @ B6 @ A3 ) ) % 8.50/4.54 => ( P @ ( image @ B @ A @ F2 @ B6 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5047_translation__Int,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ab_group_add @ A ) % 8.50/4.54 => ! [A2: A,S: set @ A,T2: set @ A] : % 8.50/4.54 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( inf_inf @ ( set @ A ) @ S @ T2 ) ) % 8.50/4.54 = ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5048_SUP__eq__const,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [I6: set @ B,F2: B > A,X3: A] : % 8.50/4.54 ( ( I6 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ! [I4: B] : % 8.50/4.54 ( ( member @ B @ I4 @ I6 ) % 8.50/4.54 => ( ( F2 @ I4 ) % 8.50/4.54 = X3 ) ) % 8.50/4.54 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ I6 ) ) % 8.50/4.54 = X3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5049_translation__diff,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ab_group_add @ A ) % 8.50/4.54 => ! [A2: A,S: set @ A,T2: set @ A] : % 8.50/4.54 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( minus_minus @ ( set @ A ) @ S @ T2 ) ) % 8.50/4.54 = ( minus_minus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5050_INF__eq__const,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [I6: set @ B,F2: B > A,X3: A] : % 8.50/4.54 ( ( I6 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ! [I4: B] : % 8.50/4.54 ( ( member @ B @ I4 @ I6 ) % 8.50/4.54 => ( ( F2 @ I4 ) % 8.50/4.54 = X3 ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) ) % 8.50/4.54 = X3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5051_finite__image__absD,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linordered_ring @ A ) % 8.50/4.54 => ! [S3: set @ A] : % 8.50/4.54 ( ( finite_finite2 @ A @ ( image @ A @ A @ ( abs_abs @ A ) @ S3 ) ) % 8.50/4.54 => ( finite_finite2 @ A @ S3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5052_image__Int__subset,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B3: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A3 @ B3 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ ( image @ B @ A @ F2 @ B3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5053_image__diff__subset,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B,B3: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) @ ( image @ B @ A @ F2 @ B3 ) ) @ ( image @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A3 @ B3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5054_Rat__induct__pos,axiom, % 8.50/4.54 ! [P: rat > $o,Q4: rat] : % 8.50/4.54 ( ! [A5: int,B5: int] : % 8.50/4.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 ) % 8.50/4.54 => ( P @ ( fract @ A5 @ B5 ) ) ) % 8.50/4.54 => ( P @ Q4 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5055_translation__Compl,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ab_group_add @ A ) % 8.50/4.54 => ! [A2: A,T2: set @ A] : % 8.50/4.54 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) ) % 8.50/4.54 = ( uminus_uminus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5056_quotient__of__eq,axiom, % 8.50/4.54 ! [A2: int,B2: int,P4: int,Q4: int] : % 8.50/4.54 ( ( ( quotient_of @ ( fract @ A2 @ B2 ) ) % 8.50/4.54 = ( product_Pair @ int @ int @ P4 @ Q4 ) ) % 8.50/4.54 => ( ( fract @ P4 @ Q4 ) % 8.50/4.54 = ( fract @ A2 @ B2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5057_SUP__eqI,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [A3: set @ B,F2: B > A,X3: A] : % 8.50/4.54 ( ! [I4: B] : % 8.50/4.54 ( ( member @ B @ I4 @ A3 ) % 8.50/4.54 => ( ord_less_eq @ A @ ( F2 @ I4 ) @ X3 ) ) % 8.50/4.54 => ( ! [Y4: A] : % 8.50/4.54 ( ! [I5: B] : % 8.50/4.54 ( ( member @ B @ I5 @ A3 ) % 8.50/4.54 => ( ord_less_eq @ A @ ( F2 @ I5 ) @ Y4 ) ) % 8.50/4.54 => ( ord_less_eq @ A @ X3 @ Y4 ) ) % 8.50/4.54 => ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 = X3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5058_SUP__mono,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,C: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [A3: set @ B,B3: set @ C,F2: B > A,G: C > A] : % 8.50/4.54 ( ! [N3: B] : % 8.50/4.54 ( ( member @ B @ N3 @ A3 ) % 8.50/4.54 => ? [X2: C] : % 8.50/4.54 ( ( member @ C @ X2 @ B3 ) % 8.50/4.54 & ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( G @ X2 ) ) ) ) % 8.50/4.54 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5059_SUP__least,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [A3: set @ B,F2: B > A,U: A] : % 8.50/4.54 ( ! [I4: B] : % 8.50/4.54 ( ( member @ B @ I4 @ A3 ) % 8.50/4.54 => ( ord_less_eq @ A @ ( F2 @ I4 ) @ U ) ) % 8.50/4.54 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5060_SUP__mono_H,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [F2: B > A,G: B > A,A3: set @ B] : % 8.50/4.54 ( ! [X5: B] : ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) % 8.50/4.54 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5061_SUP__upper,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [I: B,A3: set @ B,F2: B > A] : % 8.50/4.54 ( ( member @ B @ I @ A3 ) % 8.50/4.54 => ( ord_less_eq @ A @ ( F2 @ I ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5062_SUP__le__iff,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [F2: B > A,A3: set @ B,U: A] : % 8.50/4.54 ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U ) % 8.50/4.54 = ( ! [X: B] : % 8.50/4.54 ( ( member @ B @ X @ A3 ) % 8.50/4.54 => ( ord_less_eq @ A @ ( F2 @ X ) @ U ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5063_SUP__upper2,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [I: B,A3: set @ B,U: A,F2: B > A] : % 8.50/4.54 ( ( member @ B @ I @ A3 ) % 8.50/4.54 => ( ( ord_less_eq @ A @ U @ ( F2 @ I ) ) % 8.50/4.54 => ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5064_less__SUP__iff,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple5582772986160207858norder @ A ) % 8.50/4.54 => ! [A2: A,F2: B > A,A3: set @ B] : % 8.50/4.54 ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) % 8.50/4.54 = ( ? [X: B] : % 8.50/4.54 ( ( member @ B @ X @ A3 ) % 8.50/4.54 & ( ord_less @ A @ A2 @ ( F2 @ X ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5065_SUP__lessD,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [F2: B > A,A3: set @ B,Y: A,I: B] : % 8.50/4.54 ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ Y ) % 8.50/4.54 => ( ( member @ B @ I @ A3 ) % 8.50/4.54 => ( ord_less @ A @ ( F2 @ I ) @ Y ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5066_INF__eqI,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [A3: set @ B,X3: A,F2: B > A] : % 8.50/4.54 ( ! [I4: B] : % 8.50/4.54 ( ( member @ B @ I4 @ A3 ) % 8.50/4.54 => ( ord_less_eq @ A @ X3 @ ( F2 @ I4 ) ) ) % 8.50/4.54 => ( ! [Y4: A] : % 8.50/4.54 ( ! [I5: B] : % 8.50/4.54 ( ( member @ B @ I5 @ A3 ) % 8.50/4.54 => ( ord_less_eq @ A @ Y4 @ ( F2 @ I5 ) ) ) % 8.50/4.54 => ( ord_less_eq @ A @ Y4 @ X3 ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 = X3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5067_INF__mono,axiom, % 8.50/4.54 ! [C: $tType,A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [B3: set @ B,A3: set @ C,F2: C > A,G: B > A] : % 8.50/4.54 ( ! [M3: B] : % 8.50/4.54 ( ( member @ B @ M3 @ B3 ) % 8.50/4.54 => ? [X2: C] : % 8.50/4.54 ( ( member @ C @ X2 @ A3 ) % 8.50/4.54 & ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( G @ M3 ) ) ) ) % 8.50/4.54 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5068_INF__lower,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [I: B,A3: set @ B,F2: B > A] : % 8.50/4.54 ( ( member @ B @ I @ A3 ) % 8.50/4.54 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( F2 @ I ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5069_INF__mono_H,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [F2: B > A,G: B > A,A3: set @ B] : % 8.50/4.54 ( ! [X5: B] : ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) % 8.50/4.54 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5070_INF__lower2,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [I: B,A3: set @ B,F2: B > A,U: A] : % 8.50/4.54 ( ( member @ B @ I @ A3 ) % 8.50/4.54 => ( ( ord_less_eq @ A @ ( F2 @ I ) @ U ) % 8.50/4.54 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ U ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5071_le__INF__iff,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [U: A,F2: B > A,A3: set @ B] : % 8.50/4.54 ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) % 8.50/4.54 = ( ! [X: B] : % 8.50/4.54 ( ( member @ B @ X @ A3 ) % 8.50/4.54 => ( ord_less_eq @ A @ U @ ( F2 @ X ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5072_INF__greatest,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [A3: set @ B,U: A,F2: B > A] : % 8.50/4.54 ( ! [I4: B] : % 8.50/4.54 ( ( member @ B @ I4 @ A3 ) % 8.50/4.54 => ( ord_less_eq @ A @ U @ ( F2 @ I4 ) ) ) % 8.50/4.54 => ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5073_INF__less__iff,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple5582772986160207858norder @ A ) % 8.50/4.54 => ! [F2: B > A,A3: set @ B,A2: A] : % 8.50/4.54 ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ A2 ) % 8.50/4.54 = ( ? [X: B] : % 8.50/4.54 ( ( member @ B @ X @ A3 ) % 8.50/4.54 & ( ord_less @ A @ ( F2 @ X ) @ A2 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5074_less__INF__D,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [Y: A,F2: B > A,A3: set @ B,I: B] : % 8.50/4.54 ( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) % 8.50/4.54 => ( ( member @ B @ I @ A3 ) % 8.50/4.54 => ( ord_less @ A @ Y @ ( F2 @ I ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5075_finite__conv__nat__seg__image,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( finite_finite2 @ A ) % 8.50/4.54 = ( ^ [A6: set @ A] : % 8.50/4.54 ? [N: nat,F3: nat > A] : % 8.50/4.54 ( A6 % 8.50/4.54 = ( image @ nat @ A @ F3 % 8.50/4.54 @ ( collect @ nat % 8.50/4.54 @ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5076_nat__seg__image__imp__finite,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ A,F2: nat > A,N2: nat] : % 8.50/4.54 ( ( A3 % 8.50/4.54 = ( image @ nat @ A @ F2 % 8.50/4.54 @ ( collect @ nat % 8.50/4.54 @ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) ) ) % 8.50/4.54 => ( finite_finite2 @ A @ A3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5077_image__constant,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,X3: A,A3: set @ A,C2: B] : % 8.50/4.54 ( ( member @ A @ X3 @ A3 ) % 8.50/4.54 => ( ( image @ A @ B % 8.50/4.54 @ ^ [X: A] : C2 % 8.50/4.54 @ A3 ) % 8.50/4.54 = ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5078_image__constant__conv,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A3: set @ B,C2: A] : % 8.50/4.54 ( ( ( A3 % 8.50/4.54 = ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( image @ B @ A % 8.50/4.54 @ ^ [X: B] : C2 % 8.50/4.54 @ A3 ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.54 & ( ( A3 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( image @ B @ A % 8.50/4.54 @ ^ [X: B] : C2 % 8.50/4.54 @ A3 ) % 8.50/4.54 = ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5079_sum_Oimage__gen,axiom, % 8.50/4.54 ! [A: $tType,C: $tType,B: $tType] : % 8.50/4.54 ( ( comm_monoid_add @ A ) % 8.50/4.54 => ! [S3: set @ B,H2: B > A,G: B > C] : % 8.50/4.54 ( ( finite_finite2 @ B @ S3 ) % 8.50/4.54 => ( ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) % 8.50/4.54 = ( groups7311177749621191930dd_sum @ C @ A % 8.50/4.54 @ ^ [Y2: C] : % 8.50/4.54 ( groups7311177749621191930dd_sum @ B @ A @ H2 % 8.50/4.54 @ ( collect @ B % 8.50/4.54 @ ^ [X: B] : % 8.50/4.54 ( ( member @ B @ X @ S3 ) % 8.50/4.54 & ( ( G @ X ) % 8.50/4.54 = Y2 ) ) ) ) % 8.50/4.54 @ ( image @ B @ C @ G @ S3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5080_normalize__eq,axiom, % 8.50/4.54 ! [A2: int,B2: int,P4: int,Q4: int] : % 8.50/4.54 ( ( ( normalize @ ( product_Pair @ int @ int @ A2 @ B2 ) ) % 8.50/4.54 = ( product_Pair @ int @ int @ P4 @ Q4 ) ) % 8.50/4.54 => ( ( fract @ P4 @ Q4 ) % 8.50/4.54 = ( fract @ A2 @ B2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5081_prod_Oimage__gen,axiom, % 8.50/4.54 ! [A: $tType,C: $tType,B: $tType] : % 8.50/4.54 ( ( comm_monoid_mult @ A ) % 8.50/4.54 => ! [S3: set @ B,H2: B > A,G: B > C] : % 8.50/4.54 ( ( finite_finite2 @ B @ S3 ) % 8.50/4.54 => ( ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) % 8.50/4.54 = ( groups7121269368397514597t_prod @ C @ A % 8.50/4.54 @ ^ [Y2: C] : % 8.50/4.54 ( groups7121269368397514597t_prod @ B @ A @ H2 % 8.50/4.54 @ ( collect @ B % 8.50/4.54 @ ^ [X: B] : % 8.50/4.54 ( ( member @ B @ X @ S3 ) % 8.50/4.54 & ( ( G @ X ) % 8.50/4.54 = Y2 ) ) ) ) % 8.50/4.54 @ ( image @ B @ C @ G @ S3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5082_Gcd__mono,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( semiring_Gcd @ A ) % 8.50/4.54 => ! [A3: set @ B,F2: B > A,G: B > A] : % 8.50/4.54 ( ! [X5: B] : % 8.50/4.54 ( ( member @ B @ X5 @ A3 ) % 8.50/4.54 => ( dvd_dvd @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 8.50/4.54 => ( dvd_dvd @ A @ ( gcd_Gcd @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( gcd_Gcd @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5083_the__elem__image__unique,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A3: set @ A,F2: A > B,X3: A] : % 8.50/4.54 ( ( A3 % 8.50/4.54 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 => ( ! [Y4: A] : % 8.50/4.54 ( ( member @ A @ Y4 @ A3 ) % 8.50/4.54 => ( ( F2 @ Y4 ) % 8.50/4.54 = ( F2 @ X3 ) ) ) % 8.50/4.54 => ( ( the_elem @ B @ ( image @ A @ B @ F2 @ A3 ) ) % 8.50/4.54 = ( F2 @ X3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5084_le__SUP__iff,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple5582772986160207858norder @ A ) % 8.50/4.54 => ! [X3: A,F2: B > A,A3: set @ B] : % 8.50/4.54 ( ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) % 8.50/4.54 = ( ! [Y2: A] : % 8.50/4.54 ( ( ord_less @ A @ Y2 @ X3 ) % 8.50/4.54 => ? [X: B] : % 8.50/4.54 ( ( member @ B @ X @ A3 ) % 8.50/4.54 & ( ord_less @ A @ Y2 @ ( F2 @ X ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5085_INF__le__iff,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple5582772986160207858norder @ A ) % 8.50/4.54 => ! [F2: B > A,A3: set @ B,X3: A] : % 8.50/4.54 ( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ X3 ) % 8.50/4.54 = ( ! [Y2: A] : % 8.50/4.54 ( ( ord_less @ A @ X3 @ Y2 ) % 8.50/4.54 => ? [X: B] : % 8.50/4.54 ( ( member @ B @ X @ A3 ) % 8.50/4.54 & ( ord_less @ A @ ( F2 @ X ) @ Y2 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5086_SUP__eq__iff,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [I6: set @ B,C2: A,F2: B > A] : % 8.50/4.54 ( ( I6 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ! [I4: B] : % 8.50/4.54 ( ( member @ B @ I4 @ I6 ) % 8.50/4.54 => ( ord_less_eq @ A @ C2 @ ( F2 @ I4 ) ) ) % 8.50/4.54 => ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ I6 ) ) % 8.50/4.54 = C2 ) % 8.50/4.54 = ( ! [X: B] : % 8.50/4.54 ( ( member @ B @ X @ I6 ) % 8.50/4.54 => ( ( F2 @ X ) % 8.50/4.54 = C2 ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5087_cSUP__least,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( condit1219197933456340205attice @ A ) % 8.50/4.54 => ! [A3: set @ B,F2: B > A,M7: A] : % 8.50/4.54 ( ( A3 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ! [X5: B] : % 8.50/4.54 ( ( member @ B @ X5 @ A3 ) % 8.50/4.54 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ M7 ) ) % 8.50/4.54 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ M7 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5088_INF__eq__iff,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [I6: set @ B,F2: B > A,C2: A] : % 8.50/4.54 ( ( I6 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ! [I4: B] : % 8.50/4.54 ( ( member @ B @ I4 @ I6 ) % 8.50/4.54 => ( ord_less_eq @ A @ ( F2 @ I4 ) @ C2 ) ) % 8.50/4.54 => ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) ) % 8.50/4.54 = C2 ) % 8.50/4.54 = ( ! [X: B] : % 8.50/4.54 ( ( member @ B @ X @ I6 ) % 8.50/4.54 => ( ( F2 @ X ) % 8.50/4.54 = C2 ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5089_cINF__greatest,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( condit1219197933456340205attice @ A ) % 8.50/4.54 => ! [A3: set @ B,M: A,F2: B > A] : % 8.50/4.54 ( ( A3 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ! [X5: B] : % 8.50/4.54 ( ( member @ B @ X5 @ A3 ) % 8.50/4.54 => ( ord_less_eq @ A @ M @ ( F2 @ X5 ) ) ) % 8.50/4.54 => ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5090_card__image__le,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A3: set @ A,F2: A > B] : % 8.50/4.54 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.54 => ( ord_less_eq @ nat @ ( finite_card @ B @ ( image @ A @ B @ F2 @ A3 ) ) @ ( finite_card @ A @ A3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5091_Nats__diff,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linordered_idom @ A ) % 8.50/4.54 => ! [A2: A,B2: A] : % 8.50/4.54 ( ( member @ A @ A2 @ ( semiring_1_Nats @ A ) ) % 8.50/4.54 => ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) ) % 8.50/4.54 => ( ( ord_less_eq @ A @ B2 @ A2 ) % 8.50/4.54 => ( member @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5092_SUP__subset__mono,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [A3: set @ B,B3: set @ B,F2: B > A,G: B > A] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ B ) @ A3 @ B3 ) % 8.50/4.54 => ( ! [X5: B] : % 8.50/4.54 ( ( member @ B @ X5 @ A3 ) % 8.50/4.54 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 8.50/4.54 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5093_INF__superset__mono,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [B3: set @ B,A3: set @ B,F2: B > A,G: B > A] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ B ) @ B3 @ A3 ) % 8.50/4.54 => ( ! [X5: B] : % 8.50/4.54 ( ( member @ B @ X5 @ B3 ) % 8.50/4.54 => ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 8.50/4.54 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B3 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5094_SUP__empty,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [F2: B > A] : % 8.50/4.54 ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) ) % 8.50/4.54 = ( bot_bot @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5095_SUP__constant,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [A3: set @ B,C2: A] : % 8.50/4.54 ( ( ( A3 % 8.50/4.54 = ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Sup_Sup @ A % 8.50/4.54 @ ( image @ B @ A % 8.50/4.54 @ ^ [Y2: B] : C2 % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = ( bot_bot @ A ) ) ) % 8.50/4.54 & ( ( A3 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Sup_Sup @ A % 8.50/4.54 @ ( image @ B @ A % 8.50/4.54 @ ^ [Y2: B] : C2 % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = C2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5096_sum_Ogroup,axiom, % 8.50/4.54 ! [C: $tType,A: $tType,B: $tType] : % 8.50/4.54 ( ( comm_monoid_add @ A ) % 8.50/4.54 => ! [S3: set @ B,T4: set @ C,G: B > C,H2: B > A] : % 8.50/4.54 ( ( finite_finite2 @ B @ S3 ) % 8.50/4.54 => ( ( finite_finite2 @ C @ T4 ) % 8.50/4.54 => ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S3 ) @ T4 ) % 8.50/4.54 => ( ( groups7311177749621191930dd_sum @ C @ A % 8.50/4.54 @ ^ [Y2: C] : % 8.50/4.54 ( groups7311177749621191930dd_sum @ B @ A @ H2 % 8.50/4.54 @ ( collect @ B % 8.50/4.54 @ ^ [X: B] : % 8.50/4.54 ( ( member @ B @ X @ S3 ) % 8.50/4.54 & ( ( G @ X ) % 8.50/4.54 = Y2 ) ) ) ) % 8.50/4.54 @ T4 ) % 8.50/4.54 = ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5097_INF__inf__const2,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [I6: set @ B,F2: B > A,X3: A] : % 8.50/4.54 ( ( I6 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A % 8.50/4.54 @ ( image @ B @ A % 8.50/4.54 @ ^ [I3: B] : ( inf_inf @ A @ ( F2 @ I3 ) @ X3 ) % 8.50/4.54 @ I6 ) ) % 8.50/4.54 = ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) ) @ X3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5098_INF__inf__const1,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [I6: set @ B,X3: A,F2: B > A] : % 8.50/4.54 ( ( I6 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A % 8.50/4.54 @ ( image @ B @ A % 8.50/4.54 @ ^ [I3: B] : ( inf_inf @ A @ X3 @ ( F2 @ I3 ) ) % 8.50/4.54 @ I6 ) ) % 8.50/4.54 = ( inf_inf @ A @ X3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ I6 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5099_Nats__subset__Rats,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( field_char_0 @ A ) % 8.50/4.54 => ( ord_less_eq @ ( set @ A ) @ ( semiring_1_Nats @ A ) @ ( field_char_0_Rats @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5100_prod_Ogroup,axiom, % 8.50/4.54 ! [C: $tType,A: $tType,B: $tType] : % 8.50/4.54 ( ( comm_monoid_mult @ A ) % 8.50/4.54 => ! [S3: set @ B,T4: set @ C,G: B > C,H2: B > A] : % 8.50/4.54 ( ( finite_finite2 @ B @ S3 ) % 8.50/4.54 => ( ( finite_finite2 @ C @ T4 ) % 8.50/4.54 => ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S3 ) @ T4 ) % 8.50/4.54 => ( ( groups7121269368397514597t_prod @ C @ A % 8.50/4.54 @ ^ [Y2: C] : % 8.50/4.54 ( groups7121269368397514597t_prod @ B @ A @ H2 % 8.50/4.54 @ ( collect @ B % 8.50/4.54 @ ^ [X: B] : % 8.50/4.54 ( ( member @ B @ X @ S3 ) % 8.50/4.54 & ( ( G @ X ) % 8.50/4.54 = Y2 ) ) ) ) % 8.50/4.54 @ T4 ) % 8.50/4.54 = ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5101_quotient__of__Fract,axiom, % 8.50/4.54 ! [A2: int,B2: int] : % 8.50/4.54 ( ( quotient_of @ ( fract @ A2 @ B2 ) ) % 8.50/4.54 = ( normalize @ ( product_Pair @ int @ int @ A2 @ B2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5102_sum_Oreindex__nontrivial,axiom, % 8.50/4.54 ! [C: $tType,A: $tType,B: $tType] : % 8.50/4.54 ( ( comm_monoid_add @ A ) % 8.50/4.54 => ! [A3: set @ B,H2: B > C,G: C > A] : % 8.50/4.54 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.54 => ( ! [X5: B,Y4: B] : % 8.50/4.54 ( ( member @ B @ X5 @ A3 ) % 8.50/4.54 => ( ( member @ B @ Y4 @ A3 ) % 8.50/4.54 => ( ( X5 != Y4 ) % 8.50/4.54 => ( ( ( H2 @ X5 ) % 8.50/4.54 = ( H2 @ Y4 ) ) % 8.50/4.54 => ( ( G @ ( H2 @ X5 ) ) % 8.50/4.54 = ( zero_zero @ A ) ) ) ) ) ) % 8.50/4.54 => ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image @ B @ C @ H2 @ A3 ) ) % 8.50/4.54 = ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5103_INF__le__SUP,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [A3: set @ B,F2: B > A] : % 8.50/4.54 ( ( A3 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5104_prod_Oreindex__nontrivial,axiom, % 8.50/4.54 ! [C: $tType,A: $tType,B: $tType] : % 8.50/4.54 ( ( comm_monoid_mult @ A ) % 8.50/4.54 => ! [A3: set @ B,H2: B > C,G: C > A] : % 8.50/4.54 ( ( finite_finite2 @ B @ A3 ) % 8.50/4.54 => ( ! [X5: B,Y4: B] : % 8.50/4.54 ( ( member @ B @ X5 @ A3 ) % 8.50/4.54 => ( ( member @ B @ Y4 @ A3 ) % 8.50/4.54 => ( ( X5 != Y4 ) % 8.50/4.54 => ( ( ( H2 @ X5 ) % 8.50/4.54 = ( H2 @ Y4 ) ) % 8.50/4.54 => ( ( G @ ( H2 @ X5 ) ) % 8.50/4.54 = ( one_one @ A ) ) ) ) ) ) % 8.50/4.54 => ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image @ B @ C @ H2 @ A3 ) ) % 8.50/4.54 = ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5105_surj__card__le,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,F2: A > B] : % 8.50/4.54 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.54 => ( ( ord_less_eq @ ( set @ B ) @ B3 @ ( image @ A @ B @ F2 @ A3 ) ) % 8.50/4.54 => ( ord_less_eq @ nat @ ( finite_card @ B @ B3 ) @ ( finite_card @ A @ A3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5106_scaleR__image__atLeastAtMost,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V5355595471888546746vector @ A ) % 8.50/4.54 => ! [C2: real,X3: A,Y: A] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 ) % 8.50/4.54 => ( ( image @ A @ A @ ( real_V8093663219630862766scaleR @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ X3 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ Y ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5107_image__Suc__lessThan,axiom, % 8.50/4.54 ! [N2: nat] : % 8.50/4.54 ( ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5108_image__Suc__atMost,axiom, % 8.50/4.54 ! [N2: nat] : % 8.50/4.54 ( ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N2 ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5109_atLeast0__atMost__Suc__eq__insert__0,axiom, % 8.50/4.54 ! [N2: nat] : % 8.50/4.54 ( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) % 8.50/4.54 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5110_atLeast0__lessThan__Suc__eq__insert__0,axiom, % 8.50/4.54 ! [N2: nat] : % 8.50/4.54 ( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) % 8.50/4.54 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5111_lessThan__Suc__eq__insert__0,axiom, % 8.50/4.54 ! [N2: nat] : % 8.50/4.54 ( ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) % 8.50/4.54 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5112_atMost__Suc__eq__insert__0,axiom, % 8.50/4.54 ! [N2: nat] : % 8.50/4.54 ( ( set_ord_atMost @ nat @ ( suc @ N2 ) ) % 8.50/4.54 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5113_Fract__less__zero__iff,axiom, % 8.50/4.54 ! [B2: int,A2: int] : % 8.50/4.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.50/4.54 => ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( zero_zero @ rat ) ) % 8.50/4.54 = ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5114_zero__less__Fract__iff,axiom, % 8.50/4.54 ! [B2: int,A2: int] : % 8.50/4.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.50/4.54 => ( ( ord_less @ rat @ ( zero_zero @ rat ) @ ( fract @ A2 @ B2 ) ) % 8.50/4.54 = ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5115_one__less__Fract__iff,axiom, % 8.50/4.54 ! [B2: int,A2: int] : % 8.50/4.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.50/4.54 => ( ( ord_less @ rat @ ( one_one @ rat ) @ ( fract @ A2 @ B2 ) ) % 8.50/4.54 = ( ord_less @ int @ B2 @ A2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5116_Fract__less__one__iff,axiom, % 8.50/4.54 ! [B2: int,A2: int] : % 8.50/4.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.50/4.54 => ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( one_one @ rat ) ) % 8.50/4.54 = ( ord_less @ int @ A2 @ B2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5117_sum__image__le,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,C: $tType] : % 8.50/4.54 ( ( ordere6911136660526730532id_add @ B ) % 8.50/4.54 => ! [I6: set @ C,G: A > B,F2: C > A] : % 8.50/4.54 ( ( finite_finite2 @ C @ I6 ) % 8.50/4.54 => ( ! [I4: C] : % 8.50/4.54 ( ( member @ C @ I4 @ I6 ) % 8.50/4.54 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F2 @ I4 ) ) ) ) % 8.50/4.54 => ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image @ C @ A @ F2 @ I6 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F2 ) @ I6 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5118_zero__le__Fract__iff,axiom, % 8.50/4.54 ! [B2: int,A2: int] : % 8.50/4.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.50/4.54 => ( ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ ( fract @ A2 @ B2 ) ) % 8.50/4.54 = ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5119_Fract__le__zero__iff,axiom, % 8.50/4.54 ! [B2: int,A2: int] : % 8.50/4.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.50/4.54 => ( ( ord_less_eq @ rat @ ( fract @ A2 @ B2 ) @ ( zero_zero @ rat ) ) % 8.50/4.54 = ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5120_Fract__le__one__iff,axiom, % 8.50/4.54 ! [B2: int,A2: int] : % 8.50/4.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.50/4.54 => ( ( ord_less_eq @ rat @ ( fract @ A2 @ B2 ) @ ( one_one @ rat ) ) % 8.50/4.54 = ( ord_less_eq @ int @ A2 @ B2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5121_one__le__Fract__iff,axiom, % 8.50/4.54 ! [B2: int,A2: int] : % 8.50/4.54 ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) % 8.50/4.54 => ( ( ord_less_eq @ rat @ ( one_one @ rat ) @ ( fract @ A2 @ B2 ) ) % 8.50/4.54 = ( ord_less_eq @ int @ B2 @ A2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5122_image__mult__atLeastAtMost__if,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linordered_field @ A ) % 8.50/4.54 => ! [C2: A,X3: A,Y: A] : % 8.50/4.54 ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.54 => ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X3 ) @ ( times_times @ A @ C2 @ Y ) ) ) ) % 8.50/4.54 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.54 => ( ( ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.54 => ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y ) @ ( times_times @ A @ C2 @ X3 ) ) ) ) % 8.50/4.54 & ( ~ ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.54 => ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5123_image__mult__atLeastAtMost__if_H,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linordered_field @ A ) % 8.50/4.54 => ! [X3: A,Y: A,C2: A] : % 8.50/4.54 ( ( ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.54 => ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : ( times_times @ A @ X @ C2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X3 @ C2 ) @ ( times_times @ A @ Y @ C2 ) ) ) ) % 8.50/4.54 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : ( times_times @ A @ X @ C2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C2 ) @ ( times_times @ A @ X3 @ C2 ) ) ) ) ) ) % 8.50/4.54 & ( ~ ( ord_less_eq @ A @ X3 @ Y ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : ( times_times @ A @ X @ C2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5124_image__affinity__atLeastAtMost,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linordered_field @ A ) % 8.50/4.54 => ! [A2: A,B2: A,M: A,C2: A] : % 8.50/4.54 ( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.54 & ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 8.50/4.54 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) ) % 8.50/4.54 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5125_image__affinity__atLeastAtMost__diff,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linordered_field @ A ) % 8.50/4.54 => ! [A2: A,B2: A,M: A,C2: A] : % 8.50/4.54 ( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.54 & ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 8.50/4.54 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) ) % 8.50/4.54 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5126_image__affinity__atLeastAtMost__div,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linordered_field @ A ) % 8.50/4.54 => ! [A2: A,B2: A,M: A,C2: A] : % 8.50/4.54 ( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.54 & ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 8.50/4.54 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) ) % 8.50/4.54 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5127_image__affinity__atLeastAtMost__div__diff,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linordered_field @ A ) % 8.50/4.54 => ! [A2: A,B2: A,M: A,C2: A] : % 8.50/4.54 ( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.54 & ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) % 8.50/4.54 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) ) % 8.50/4.54 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M ) % 8.50/4.54 => ( ( image @ A @ A % 8.50/4.54 @ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 ) % 8.50/4.54 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5128_sum__fun__comp,axiom, % 8.50/4.54 ! [A: $tType,C: $tType,B: $tType] : % 8.50/4.54 ( ( semiring_1 @ C ) % 8.50/4.54 => ! [S3: set @ A,R: set @ B,G: A > B,F2: B > C] : % 8.50/4.54 ( ( finite_finite2 @ A @ S3 ) % 8.50/4.54 => ( ( finite_finite2 @ B @ R ) % 8.50/4.54 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ G @ S3 ) @ R ) % 8.50/4.54 => ( ( groups7311177749621191930dd_sum @ A @ C % 8.50/4.54 @ ^ [X: A] : ( F2 @ ( G @ X ) ) % 8.50/4.54 @ S3 ) % 8.50/4.54 = ( groups7311177749621191930dd_sum @ B @ C % 8.50/4.54 @ ^ [Y2: B] : % 8.50/4.54 ( times_times @ C % 8.50/4.54 @ ( semiring_1_of_nat @ C % 8.50/4.54 @ ( finite_card @ A % 8.50/4.54 @ ( collect @ A % 8.50/4.54 @ ^ [X: A] : % 8.50/4.54 ( ( member @ A @ X @ S3 ) % 8.50/4.54 & ( ( G @ X ) % 8.50/4.54 = Y2 ) ) ) ) ) % 8.50/4.54 @ ( F2 @ Y2 ) ) % 8.50/4.54 @ R ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5129_INF__nat__binary,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( counta3822494911875563373attice @ A ) % 8.50/4.54 => ! [A3: A,B3: A] : % 8.50/4.54 ( ( inf_inf @ A @ A3 % 8.50/4.54 @ ( complete_Inf_Inf @ A % 8.50/4.54 @ ( image @ nat @ A % 8.50/4.54 @ ^ [X: nat] : B3 % 8.50/4.54 @ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) ) % 8.50/4.54 = ( inf_inf @ A @ A3 @ B3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5130_empty__natural,axiom, % 8.50/4.54 ! [C: $tType,B: $tType,D: $tType,A: $tType,F2: A > C,G: D > B] : % 8.50/4.54 ( ( comp @ C @ ( set @ B ) @ A % 8.50/4.54 @ ^ [Uu3: C] : ( bot_bot @ ( set @ B ) ) % 8.50/4.54 @ F2 ) % 8.50/4.54 = ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image @ D @ B @ G ) % 8.50/4.54 @ ^ [Uu3: A] : ( bot_bot @ ( set @ D ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5131_listrel__def,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( listrel @ A @ B ) % 8.50/4.54 = ( ^ [R5: set @ ( product_prod @ A @ B )] : % 8.50/4.54 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) % 8.50/4.54 @ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o % 8.50/4.54 @ ( listrelp @ A @ B % 8.50/4.54 @ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R5 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5132_pair__imageI,axiom, % 8.50/4.54 ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set @ ( product_prod @ A @ B ),F2: A > B > C] : % 8.50/4.54 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A3 ) % 8.50/4.54 => ( member @ C @ ( F2 @ A2 @ B2 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F2 ) @ A3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5133_Gcd__abs__eq,axiom, % 8.50/4.54 ! [K5: set @ int] : % 8.50/4.54 ( ( gcd_Gcd @ int @ ( image @ int @ int @ ( abs_abs @ int ) @ K5 ) ) % 8.50/4.54 = ( gcd_Gcd @ int @ K5 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5134_finite__UN,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] : % 8.50/4.54 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.54 => ( ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) % 8.50/4.54 = ( ! [X: A] : % 8.50/4.54 ( ( member @ A @ X @ A3 ) % 8.50/4.54 => ( finite_finite2 @ B @ ( B3 @ X ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5135_UN__constant,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A3: set @ B,C2: set @ A] : % 8.50/4.54 ( ( ( A3 % 8.50/4.54 = ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Sup_Sup @ ( set @ A ) % 8.50/4.54 @ ( image @ B @ ( set @ A ) % 8.50/4.54 @ ^ [Y2: B] : C2 % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.54 & ( ( A3 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Sup_Sup @ ( set @ A ) % 8.50/4.54 @ ( image @ B @ ( set @ A ) % 8.50/4.54 @ ^ [Y2: B] : C2 % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = C2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5136_finite__UN__I,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] : % 8.50/4.54 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.54 => ( ! [A5: A] : % 8.50/4.54 ( ( member @ A @ A5 @ A3 ) % 8.50/4.54 => ( finite_finite2 @ B @ ( B3 @ A5 ) ) ) % 8.50/4.54 => ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5137_finite__INT,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B )] : % 8.50/4.54 ( ? [X2: A] : % 8.50/4.54 ( ( member @ A @ X2 @ I6 ) % 8.50/4.54 & ( finite_finite2 @ B @ ( A3 @ X2 ) ) ) % 8.50/4.54 => ( finite_finite2 @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5138_Gcd__int__eq,axiom, % 8.50/4.54 ! [N7: set @ nat] : % 8.50/4.54 ( ( gcd_Gcd @ int @ ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ N7 ) ) % 8.50/4.54 = ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ N7 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5139_UN__simps_I1_J,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,C5: set @ B,A2: A,B3: B > ( set @ A )] : % 8.50/4.54 ( ( ( C5 % 8.50/4.54 = ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Sup_Sup @ ( set @ A ) % 8.50/4.54 @ ( image @ B @ ( set @ A ) % 8.50/4.54 @ ^ [X: B] : ( insert @ A @ A2 @ ( B3 @ X ) ) % 8.50/4.54 @ C5 ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.54 & ( ( C5 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Sup_Sup @ ( set @ A ) % 8.50/4.54 @ ( image @ B @ ( set @ A ) % 8.50/4.54 @ ^ [X: B] : ( insert @ A @ A2 @ ( B3 @ X ) ) % 8.50/4.54 @ C5 ) ) % 8.50/4.54 = ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ C5 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5140_UN__singleton,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ A] : % 8.50/4.54 ( ( complete_Sup_Sup @ ( set @ A ) % 8.50/4.54 @ ( image @ A @ ( set @ A ) % 8.50/4.54 @ ^ [X: A] : ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = A3 ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5141_set__concat,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ ( list @ A )] : % 8.50/4.54 ( ( set2 @ A @ ( concat @ A @ Xs ) ) % 8.50/4.54 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xs ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5142_Gcd__nat__abs__eq,axiom, % 8.50/4.54 ! [K5: set @ int] : % 8.50/4.54 ( ( gcd_Gcd @ nat % 8.50/4.54 @ ( image @ int @ nat % 8.50/4.54 @ ^ [K3: int] : ( nat2 @ ( abs_abs @ int @ K3 ) ) % 8.50/4.54 @ K5 ) ) % 8.50/4.54 = ( nat2 @ ( gcd_Gcd @ int @ K5 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5143_SUP__UN__eq2,axiom, % 8.50/4.54 ! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] : % 8.50/4.54 ( ( complete_Sup_Sup @ ( A > B > $o ) % 8.50/4.54 @ ( image @ C @ ( A > B > $o ) % 8.50/4.54 @ ^ [I3: C,X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( R2 @ I3 ) ) % 8.50/4.54 @ S3 ) ) % 8.50/4.54 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5144_INF__INT__eq2,axiom, % 8.50/4.54 ! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] : % 8.50/4.54 ( ( complete_Inf_Inf @ ( A > B > $o ) % 8.50/4.54 @ ( image @ C @ ( A > B > $o ) % 8.50/4.54 @ ^ [I3: C,X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( R2 @ I3 ) ) % 8.50/4.54 @ S3 ) ) % 8.50/4.54 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5145_INF__Int__eq2,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,S3: set @ ( set @ ( product_prod @ A @ B ) )] : % 8.50/4.54 ( ( complete_Inf_Inf @ ( A > B > $o ) % 8.50/4.54 @ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o ) % 8.50/4.54 @ ^ [I3: set @ ( product_prod @ A @ B ),X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ I3 ) % 8.50/4.54 @ S3 ) ) % 8.50/4.54 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5146_SUP__Sup__eq2,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,S3: set @ ( set @ ( product_prod @ A @ B ) )] : % 8.50/4.54 ( ( complete_Sup_Sup @ ( A > B > $o ) % 8.50/4.54 @ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o ) % 8.50/4.54 @ ^ [I3: set @ ( product_prod @ A @ B ),X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ I3 ) % 8.50/4.54 @ S3 ) ) % 8.50/4.54 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5147_Sup__SUP__eq2,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( complete_Sup_Sup @ ( A > B > $o ) ) % 8.50/4.54 = ( ^ [S7: set @ ( A > B > $o ),X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S7 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5148_Inf__INT__eq2,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( complete_Inf_Inf @ ( A > B > $o ) ) % 8.50/4.54 = ( ^ [S7: set @ ( A > B > $o ),X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S7 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5149_UN__image__subset,axiom, % 8.50/4.54 ! [C: $tType,A: $tType,B: $tType,F2: B > ( set @ A ),G: C > ( set @ B ),X3: C,X8: set @ A] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F2 @ ( G @ X3 ) ) ) @ X8 ) % 8.50/4.54 = ( ord_less_eq @ ( set @ B ) @ ( G @ X3 ) % 8.50/4.54 @ ( collect @ B % 8.50/4.54 @ ^ [X: B] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ X ) @ X8 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5150_listrelp_ONil,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,R2: A > B > $o] : ( listrelp @ A @ B @ R2 @ ( nil @ A ) @ ( nil @ B ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5151_None__notin__image__Some,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ A] : % 8.50/4.54 ~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5152_INF__filter__not__bot,axiom, % 8.50/4.54 ! [I7: $tType,A: $tType,B3: set @ I7,F5: I7 > ( filter @ A )] : % 8.50/4.54 ( ! [X10: set @ I7] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ I7 ) @ X10 @ B3 ) % 8.50/4.54 => ( ( finite_finite2 @ I7 @ X10 ) % 8.50/4.54 => ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ I7 @ ( filter @ A ) @ F5 @ X10 ) ) % 8.50/4.54 != ( bot_bot @ ( filter @ A ) ) ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ I7 @ ( filter @ A ) @ F5 @ B3 ) ) % 8.50/4.54 != ( bot_bot @ ( filter @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5153_finite__int__iff__bounded,axiom, % 8.50/4.54 ( ( finite_finite2 @ int ) % 8.50/4.54 = ( ^ [S7: set @ int] : % 8.50/4.54 ? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S7 ) @ ( set_ord_lessThan @ int @ K3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5154_finite__int__iff__bounded__le,axiom, % 8.50/4.54 ( ( finite_finite2 @ int ) % 8.50/4.54 = ( ^ [S7: set @ int] : % 8.50/4.54 ? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S7 ) @ ( set_ord_atMost @ int @ K3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5155_UN__empty2,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A3: set @ B] : % 8.50/4.54 ( ( complete_Sup_Sup @ ( set @ A ) % 8.50/4.54 @ ( image @ B @ ( set @ A ) % 8.50/4.54 @ ^ [X: B] : ( bot_bot @ ( set @ A ) ) % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5156_UN__empty,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,B3: B > ( set @ A )] : % 8.50/4.54 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5157_UNION__empty__conv_I1_J,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B] : % 8.50/4.54 ( ( ( bot_bot @ ( set @ A ) ) % 8.50/4.54 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) ) % 8.50/4.54 = ( ! [X: B] : % 8.50/4.54 ( ( member @ B @ X @ A3 ) % 8.50/4.54 => ( ( B3 @ X ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5158_UNION__empty__conv_I2_J,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,B3: B > ( set @ A ),A3: set @ B] : % 8.50/4.54 ( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ A3 ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 = ( ! [X: B] : % 8.50/4.54 ( ( member @ B @ X @ A3 ) % 8.50/4.54 => ( ( B3 @ X ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5159_UN__subset__iff,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,A3: B > ( set @ A ),I6: set @ B,B3: set @ A] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) @ B3 ) % 8.50/4.54 = ( ! [X: B] : % 8.50/4.54 ( ( member @ B @ X @ I6 ) % 8.50/4.54 => ( ord_less_eq @ ( set @ A ) @ ( A3 @ X ) @ B3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5160_UN__upper,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A2: A,A3: set @ A,B3: A > ( set @ B )] : % 8.50/4.54 ( ( member @ A @ A2 @ A3 ) % 8.50/4.54 => ( ord_less_eq @ ( set @ B ) @ ( B3 @ A2 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5161_UN__least,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,A3: set @ A,B3: A > ( set @ B ),C5: set @ B] : % 8.50/4.54 ( ! [X5: A] : % 8.50/4.54 ( ( member @ A @ X5 @ A3 ) % 8.50/4.54 => ( ord_less_eq @ ( set @ B ) @ ( B3 @ X5 ) @ C5 ) ) % 8.50/4.54 => ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) @ C5 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5162_UN__mono,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,F2: A > ( set @ B ),G: A > ( set @ B )] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.54 => ( ! [X5: A] : % 8.50/4.54 ( ( member @ A @ X5 @ A3 ) % 8.50/4.54 => ( ord_less_eq @ ( set @ B ) @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 8.50/4.54 => ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ A3 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ B3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5163_INT__subset__iff,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,B3: set @ A,A3: B > ( set @ A ),I6: set @ B] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ B3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) ) % 8.50/4.54 = ( ! [X: B] : % 8.50/4.54 ( ( member @ B @ X @ I6 ) % 8.50/4.54 => ( ord_less_eq @ ( set @ A ) @ B3 @ ( A3 @ X ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5164_INT__anti__mono,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,F2: A > ( set @ B ),G: A > ( set @ B )] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.54 => ( ! [X5: A] : % 8.50/4.54 ( ( member @ A @ X5 @ A3 ) % 8.50/4.54 => ( ord_less_eq @ ( set @ B ) @ ( F2 @ X5 ) @ ( G @ X5 ) ) ) % 8.50/4.54 => ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F2 @ B3 ) ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ A3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5165_INT__greatest,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A3: set @ A,C5: set @ B,B3: A > ( set @ B )] : % 8.50/4.54 ( ! [X5: A] : % 8.50/4.54 ( ( member @ A @ X5 @ A3 ) % 8.50/4.54 => ( ord_less_eq @ ( set @ B ) @ C5 @ ( B3 @ X5 ) ) ) % 8.50/4.54 => ( ord_less_eq @ ( set @ B ) @ C5 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5166_INT__lower,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A2: A,A3: set @ A,B3: A > ( set @ B )] : % 8.50/4.54 ( ( member @ A @ A2 @ A3 ) % 8.50/4.54 => ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B3 @ A3 ) ) @ ( B3 @ A2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5167_in__image__insert__iff,axiom, % 8.50/4.54 ! [A: $tType,B3: set @ ( set @ A ),X3: A,A3: set @ A] : % 8.50/4.54 ( ! [C8: set @ A] : % 8.50/4.54 ( ( member @ ( set @ A ) @ C8 @ B3 ) % 8.50/4.54 => ~ ( member @ A @ X3 @ C8 ) ) % 8.50/4.54 => ( ( member @ ( set @ A ) @ A3 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X3 ) @ B3 ) ) % 8.50/4.54 = ( ( member @ A @ X3 @ A3 ) % 8.50/4.54 & ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5168_UN__extend__simps_I1_J,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,C5: set @ B,A2: A,B3: B > ( set @ A )] : % 8.50/4.54 ( ( ( C5 % 8.50/4.54 = ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ C5 ) ) ) % 8.50/4.54 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.54 & ( ( C5 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ C5 ) ) ) % 8.50/4.54 = ( complete_Sup_Sup @ ( set @ A ) % 8.50/4.54 @ ( image @ B @ ( set @ A ) % 8.50/4.54 @ ^ [X: B] : ( insert @ A @ A2 @ ( B3 @ X ) ) % 8.50/4.54 @ C5 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5169_INT__extend__simps_I2_J,axiom, % 8.50/4.54 ! [C: $tType,D: $tType,C5: set @ D,A3: set @ C,B3: D > ( set @ C )] : % 8.50/4.54 ( ( ( C5 % 8.50/4.54 = ( bot_bot @ ( set @ D ) ) ) % 8.50/4.54 => ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D @ ( set @ C ) @ B3 @ C5 ) ) ) % 8.50/4.54 = A3 ) ) % 8.50/4.54 & ( ( C5 % 8.50/4.54 != ( bot_bot @ ( set @ D ) ) ) % 8.50/4.54 => ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D @ ( set @ C ) @ B3 @ C5 ) ) ) % 8.50/4.54 = ( complete_Inf_Inf @ ( set @ C ) % 8.50/4.54 @ ( image @ D @ ( set @ C ) % 8.50/4.54 @ ^ [X: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B3 @ X ) ) % 8.50/4.54 @ C5 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5170_INT__extend__simps_I1_J,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,C5: set @ A,A3: A > ( set @ B ),B3: set @ B] : % 8.50/4.54 ( ( ( C5 % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 => ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ C5 ) ) @ B3 ) % 8.50/4.54 = B3 ) ) % 8.50/4.54 & ( ( C5 % 8.50/4.54 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 => ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ C5 ) ) @ B3 ) % 8.50/4.54 = ( complete_Inf_Inf @ ( set @ B ) % 8.50/4.54 @ ( image @ A @ ( set @ B ) % 8.50/4.54 @ ^ [X: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X ) @ B3 ) % 8.50/4.54 @ C5 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5171_Int__Inter__eq_I2_J,axiom, % 8.50/4.54 ! [A: $tType,B12: set @ ( set @ A ),A3: set @ A] : % 8.50/4.54 ( ( ( B12 % 8.50/4.54 = ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 8.50/4.54 => ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) @ A3 ) % 8.50/4.54 = A3 ) ) % 8.50/4.54 & ( ( B12 % 8.50/4.54 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 8.50/4.54 => ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) @ A3 ) % 8.50/4.54 = ( complete_Inf_Inf @ ( set @ A ) % 8.50/4.54 @ ( image @ ( set @ A ) @ ( set @ A ) % 8.50/4.54 @ ^ [B6: set @ A] : ( inf_inf @ ( set @ A ) @ B6 @ A3 ) % 8.50/4.54 @ B12 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5172_Int__Inter__eq_I1_J,axiom, % 8.50/4.54 ! [A: $tType,B12: set @ ( set @ A ),A3: set @ A] : % 8.50/4.54 ( ( ( B12 % 8.50/4.54 = ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 8.50/4.54 => ( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) ) % 8.50/4.54 = A3 ) ) % 8.50/4.54 & ( ( B12 % 8.50/4.54 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 8.50/4.54 => ( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) ) % 8.50/4.54 = ( complete_Inf_Inf @ ( set @ A ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 ) @ B12 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5173_Collect__split__mono__strong,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,X8: set @ A,A3: set @ ( product_prod @ A @ B ),Y7: set @ B,P: A > B > $o,Q: A > B > $o] : % 8.50/4.54 ( ( X8 % 8.50/4.54 = ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) ) % 8.50/4.54 => ( ( Y7 % 8.50/4.54 = ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) ) % 8.50/4.54 => ( ! [X5: A] : % 8.50/4.54 ( ( member @ A @ X5 @ X8 ) % 8.50/4.54 => ! [Xa3: B] : % 8.50/4.54 ( ( member @ B @ Xa3 @ Y7 ) % 8.50/4.54 => ( ( P @ X5 @ Xa3 ) % 8.50/4.54 => ( Q @ X5 @ Xa3 ) ) ) ) % 8.50/4.54 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) ) % 8.50/4.54 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5174_INT__extend__simps_I4_J,axiom, % 8.50/4.54 ! [G3: $tType,H4: $tType,C5: set @ H4,A3: set @ G3,B3: H4 > ( set @ G3 )] : % 8.50/4.54 ( ( ( C5 % 8.50/4.54 = ( bot_bot @ ( set @ H4 ) ) ) % 8.50/4.54 => ( ( minus_minus @ ( set @ G3 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image @ H4 @ ( set @ G3 ) @ B3 @ C5 ) ) ) % 8.50/4.54 = A3 ) ) % 8.50/4.54 & ( ( C5 % 8.50/4.54 != ( bot_bot @ ( set @ H4 ) ) ) % 8.50/4.54 => ( ( minus_minus @ ( set @ G3 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image @ H4 @ ( set @ G3 ) @ B3 @ C5 ) ) ) % 8.50/4.54 = ( complete_Inf_Inf @ ( set @ G3 ) % 8.50/4.54 @ ( image @ H4 @ ( set @ G3 ) % 8.50/4.54 @ ^ [X: H4] : ( minus_minus @ ( set @ G3 ) @ A3 @ ( B3 @ X ) ) % 8.50/4.54 @ C5 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5175_infinite__int__iff__infinite__nat__abs,axiom, % 8.50/4.54 ! [S3: set @ int] : % 8.50/4.54 ( ( ~ ( finite_finite2 @ int @ S3 ) ) % 8.50/4.54 = ( ~ ( finite_finite2 @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ S3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5176_UN__le__add__shift__strict,axiom, % 8.50/4.54 ! [A: $tType,M7: nat > ( set @ A ),K: nat,N2: nat] : % 8.50/4.54 ( ( complete_Sup_Sup @ ( set @ A ) % 8.50/4.54 @ ( image @ nat @ ( set @ A ) % 8.50/4.54 @ ^ [I3: nat] : ( M7 @ ( plus_plus @ nat @ I3 @ K ) ) % 8.50/4.54 @ ( set_ord_lessThan @ nat @ N2 ) ) ) % 8.50/4.54 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ N2 @ K ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5177_UN__le__add__shift,axiom, % 8.50/4.54 ! [A: $tType,M7: nat > ( set @ A ),K: nat,N2: nat] : % 8.50/4.54 ( ( complete_Sup_Sup @ ( set @ A ) % 8.50/4.54 @ ( image @ nat @ ( set @ A ) % 8.50/4.54 @ ^ [I3: nat] : ( M7 @ ( plus_plus @ nat @ I3 @ K ) ) % 8.50/4.54 @ ( set_ord_atMost @ nat @ N2 ) ) ) % 8.50/4.54 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or1337092689740270186AtMost @ nat @ K @ ( plus_plus @ nat @ N2 @ K ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5178_subset__subseqs,axiom, % 8.50/4.54 ! [A: $tType,X8: set @ A,Xs: list @ A] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ X8 @ ( set2 @ A @ Xs ) ) % 8.50/4.54 => ( member @ ( set @ A ) @ X8 @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5179_subseqs__powset,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A] : % 8.50/4.54 ( ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) % 8.50/4.54 = ( pow2 @ A @ ( set2 @ A @ Xs ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5180_conj__subset__def,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ A,P: A > $o,Q: A > $o] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ A3 % 8.50/4.54 @ ( collect @ A % 8.50/4.54 @ ^ [X: A] : % 8.50/4.54 ( ( P @ X ) % 8.50/4.54 & ( Q @ X ) ) ) ) % 8.50/4.54 = ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) % 8.50/4.54 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ Q ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5181_finite__Inf__Sup,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( finite8700451911770168679attice @ A ) % 8.50/4.54 => ! [A3: set @ ( set @ A )] : % 8.50/4.54 ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A3 ) ) % 8.50/4.54 @ ( complete_Sup_Sup @ A % 8.50/4.54 @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A ) % 8.50/4.54 @ ( collect @ ( set @ A ) % 8.50/4.54 @ ^ [Uu3: set @ A] : % 8.50/4.54 ? [F3: ( set @ A ) > A] : % 8.50/4.54 ( ( Uu3 % 8.50/4.54 = ( image @ ( set @ A ) @ A @ F3 @ A3 ) ) % 8.50/4.54 & ! [X: set @ A] : % 8.50/4.54 ( ( member @ ( set @ A ) @ X @ A3 ) % 8.50/4.54 => ( member @ A @ ( F3 @ X ) @ X ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5182_Inf__Sup__le,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( comple592849572758109894attice @ A ) % 8.50/4.54 => ! [A3: set @ ( set @ A )] : % 8.50/4.54 ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A3 ) ) % 8.50/4.54 @ ( complete_Sup_Sup @ A % 8.50/4.54 @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A ) % 8.50/4.54 @ ( collect @ ( set @ A ) % 8.50/4.54 @ ^ [Uu3: set @ A] : % 8.50/4.54 ? [F3: ( set @ A ) > A] : % 8.50/4.54 ( ( Uu3 % 8.50/4.54 = ( image @ ( set @ A ) @ A @ F3 @ A3 ) ) % 8.50/4.54 & ! [X: set @ A] : % 8.50/4.54 ( ( member @ ( set @ A ) @ X @ A3 ) % 8.50/4.54 => ( member @ A @ ( F3 @ X ) @ X ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5183_Sup__Inf__le,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [A3: set @ ( set @ A )] : % 8.50/4.54 ( ord_less_eq @ A % 8.50/4.54 @ ( complete_Sup_Sup @ A % 8.50/4.54 @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A ) % 8.50/4.54 @ ( collect @ ( set @ A ) % 8.50/4.54 @ ^ [Uu3: set @ A] : % 8.50/4.54 ? [F3: ( set @ A ) > A] : % 8.50/4.54 ( ( Uu3 % 8.50/4.54 = ( image @ ( set @ A ) @ A @ F3 @ A3 ) ) % 8.50/4.54 & ! [X: set @ A] : % 8.50/4.54 ( ( member @ ( set @ A ) @ X @ A3 ) % 8.50/4.54 => ( member @ A @ ( F3 @ X ) @ X ) ) ) ) ) ) % 8.50/4.54 @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5184_sum_OUNION__disjoint,axiom, % 8.50/4.54 ! [A: $tType,C: $tType,B: $tType] : % 8.50/4.54 ( ( comm_monoid_add @ A ) % 8.50/4.54 => ! [I6: set @ B,A3: B > ( set @ C ),G: C > A] : % 8.50/4.54 ( ( finite_finite2 @ B @ I6 ) % 8.50/4.54 => ( ! [X5: B] : % 8.50/4.54 ( ( member @ B @ X5 @ I6 ) % 8.50/4.54 => ( finite_finite2 @ C @ ( A3 @ X5 ) ) ) % 8.50/4.54 => ( ! [X5: B] : % 8.50/4.54 ( ( member @ B @ X5 @ I6 ) % 8.50/4.54 => ! [Xa3: B] : % 8.50/4.54 ( ( member @ B @ Xa3 @ I6 ) % 8.50/4.54 => ( ( X5 != Xa3 ) % 8.50/4.54 => ( ( inf_inf @ ( set @ C ) @ ( A3 @ X5 ) @ ( A3 @ Xa3 ) ) % 8.50/4.54 = ( bot_bot @ ( set @ C ) ) ) ) ) ) % 8.50/4.54 => ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ A3 @ I6 ) ) ) % 8.50/4.54 = ( groups7311177749621191930dd_sum @ B @ A % 8.50/4.54 @ ^ [X: B] : ( groups7311177749621191930dd_sum @ C @ A @ G @ ( A3 @ X ) ) % 8.50/4.54 @ I6 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5185_prod_OUNION__disjoint,axiom, % 8.50/4.54 ! [A: $tType,C: $tType,B: $tType] : % 8.50/4.54 ( ( comm_monoid_mult @ A ) % 8.50/4.54 => ! [I6: set @ B,A3: B > ( set @ C ),G: C > A] : % 8.50/4.54 ( ( finite_finite2 @ B @ I6 ) % 8.50/4.54 => ( ! [X5: B] : % 8.50/4.54 ( ( member @ B @ X5 @ I6 ) % 8.50/4.54 => ( finite_finite2 @ C @ ( A3 @ X5 ) ) ) % 8.50/4.54 => ( ! [X5: B] : % 8.50/4.54 ( ( member @ B @ X5 @ I6 ) % 8.50/4.54 => ! [Xa3: B] : % 8.50/4.54 ( ( member @ B @ Xa3 @ I6 ) % 8.50/4.54 => ( ( X5 != Xa3 ) % 8.50/4.54 => ( ( inf_inf @ ( set @ C ) @ ( A3 @ X5 ) @ ( A3 @ Xa3 ) ) % 8.50/4.54 = ( bot_bot @ ( set @ C ) ) ) ) ) ) % 8.50/4.54 => ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ A3 @ I6 ) ) ) % 8.50/4.54 = ( groups7121269368397514597t_prod @ B @ A % 8.50/4.54 @ ^ [X: B] : ( groups7121269368397514597t_prod @ C @ A @ G @ ( A3 @ X ) ) % 8.50/4.54 @ I6 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5186_card__UN__le,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B )] : % 8.50/4.54 ( ( finite_finite2 @ A @ I6 ) % 8.50/4.54 => ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) % 8.50/4.54 @ ( groups7311177749621191930dd_sum @ A @ nat % 8.50/4.54 @ ^ [I3: A] : ( finite_card @ B @ ( A3 @ I3 ) ) % 8.50/4.54 @ I6 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5187_listrelp__listrel__eq,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] : % 8.50/4.54 ( ( listrelp @ A @ B % 8.50/4.54 @ ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R2 ) ) % 8.50/4.54 = ( ^ [X: list @ A,Y2: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X @ Y2 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5188_Gcd__int__def,axiom, % 8.50/4.54 ( ( gcd_Gcd @ int ) % 8.50/4.54 = ( ^ [K6: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K6 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5189_card__UN__disjoint,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B )] : % 8.50/4.54 ( ( finite_finite2 @ A @ I6 ) % 8.50/4.54 => ( ! [X5: A] : % 8.50/4.54 ( ( member @ A @ X5 @ I6 ) % 8.50/4.54 => ( finite_finite2 @ B @ ( A3 @ X5 ) ) ) % 8.50/4.54 => ( ! [X5: A] : % 8.50/4.54 ( ( member @ A @ X5 @ I6 ) % 8.50/4.54 => ! [Xa3: A] : % 8.50/4.54 ( ( member @ A @ Xa3 @ I6 ) % 8.50/4.54 => ( ( X5 != Xa3 ) % 8.50/4.54 => ( ( inf_inf @ ( set @ B ) @ ( A3 @ X5 ) @ ( A3 @ Xa3 ) ) % 8.50/4.54 = ( bot_bot @ ( set @ B ) ) ) ) ) ) % 8.50/4.54 => ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) % 8.50/4.54 = ( groups7311177749621191930dd_sum @ A @ nat % 8.50/4.54 @ ^ [I3: A] : ( finite_card @ B @ ( A3 @ I3 ) ) % 8.50/4.54 @ I6 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5190_finite__mono__strict__prefix__implies__finite__fixpoint,axiom, % 8.50/4.54 ! [A: $tType,F2: nat > ( set @ A ),S3: set @ A] : % 8.50/4.54 ( ! [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ I4 ) @ S3 ) % 8.50/4.54 => ( ( finite_finite2 @ A @ S3 ) % 8.50/4.54 => ( ? [N8: nat] : % 8.50/4.54 ( ! [N3: nat] : % 8.50/4.54 ( ( ord_less_eq @ nat @ N3 @ N8 ) % 8.50/4.54 => ! [M3: nat] : % 8.50/4.54 ( ( ord_less_eq @ nat @ M3 @ N8 ) % 8.50/4.54 => ( ( ord_less @ nat @ M3 @ N3 ) % 8.50/4.54 => ( ord_less @ ( set @ A ) @ ( F2 @ M3 ) @ ( F2 @ N3 ) ) ) ) ) % 8.50/4.54 & ! [N3: nat] : % 8.50/4.54 ( ( ord_less_eq @ nat @ N8 @ N3 ) % 8.50/4.54 => ( ( F2 @ N8 ) % 8.50/4.54 = ( F2 @ N3 ) ) ) ) % 8.50/4.54 => ( ( F2 @ ( finite_card @ A @ S3 ) ) % 8.50/4.54 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ F2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5191_length__remdups__concat,axiom, % 8.50/4.54 ! [A: $tType,Xss: list @ ( list @ A )] : % 8.50/4.54 ( ( size_size @ ( list @ A ) @ ( remdups @ A @ ( concat @ A @ Xss ) ) ) % 8.50/4.54 = ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xss ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5192_INT__simps_I4_J,axiom, % 8.50/4.54 ! [G3: $tType,H4: $tType,C5: set @ H4,A3: set @ G3,B3: H4 > ( set @ G3 )] : % 8.50/4.54 ( ( ( C5 % 8.50/4.54 = ( bot_bot @ ( set @ H4 ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ ( set @ G3 ) % 8.50/4.54 @ ( image @ H4 @ ( set @ G3 ) % 8.50/4.54 @ ^ [X: H4] : ( minus_minus @ ( set @ G3 ) @ A3 @ ( B3 @ X ) ) % 8.50/4.54 @ C5 ) ) % 8.50/4.54 = ( top_top @ ( set @ G3 ) ) ) ) % 8.50/4.54 & ( ( C5 % 8.50/4.54 != ( bot_bot @ ( set @ H4 ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ ( set @ G3 ) % 8.50/4.54 @ ( image @ H4 @ ( set @ G3 ) % 8.50/4.54 @ ^ [X: H4] : ( minus_minus @ ( set @ G3 ) @ A3 @ ( B3 @ X ) ) % 8.50/4.54 @ C5 ) ) % 8.50/4.54 = ( minus_minus @ ( set @ G3 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image @ H4 @ ( set @ G3 ) @ B3 @ C5 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5193_top__apply,axiom, % 8.50/4.54 ! [C: $tType,D: $tType] : % 8.50/4.54 ( ( top @ C ) % 8.50/4.54 => ( ( top_top @ ( D > C ) ) % 8.50/4.54 = ( ^ [X: D] : ( top_top @ C ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5194_UNIV__I,axiom, % 8.50/4.54 ! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5195_finite__option__UNIV,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) ) % 8.50/4.54 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5196_finite__Plus__UNIV__iff,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) % 8.50/4.54 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5197_Int__UNIV,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ A,B3: set @ A] : % 8.50/4.54 ( ( ( inf_inf @ ( set @ A ) @ A3 @ B3 ) % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) % 8.50/4.54 = ( ( A3 % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) % 8.50/4.54 & ( B3 % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5198_range__mult,axiom, % 8.50/4.54 ! [A2: real] : % 8.50/4.54 ( ( ( A2 % 8.50/4.54 = ( zero_zero @ real ) ) % 8.50/4.54 => ( ( image @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) ) % 8.50/4.54 = ( insert @ real @ ( zero_zero @ real ) @ ( bot_bot @ ( set @ real ) ) ) ) ) % 8.50/4.54 & ( ( A2 % 8.50/4.54 != ( zero_zero @ real ) ) % 8.50/4.54 => ( ( image @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) ) % 8.50/4.54 = ( top_top @ ( set @ real ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5199_max__top,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order_top @ A ) % 8.50/4.54 => ! [X3: A] : % 8.50/4.54 ( ( ord_max @ A @ ( top_top @ A ) @ X3 ) % 8.50/4.54 = ( top_top @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5200_max__top2,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order_top @ A ) % 8.50/4.54 => ! [X3: A] : % 8.50/4.54 ( ( ord_max @ A @ X3 @ ( top_top @ A ) ) % 8.50/4.54 = ( top_top @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5201_min__top,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order_top @ A ) % 8.50/4.54 => ! [X3: A] : % 8.50/4.54 ( ( ord_min @ A @ ( top_top @ A ) @ X3 ) % 8.50/4.54 = X3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5202_min__top2,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order_top @ A ) % 8.50/4.54 => ! [X3: A] : % 8.50/4.54 ( ( ord_min @ A @ X3 @ ( top_top @ A ) ) % 8.50/4.54 = X3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5203_Pow__UNIV,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( pow2 @ A @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 = ( top_top @ ( set @ ( set @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5204_remdups__eq__nil__right__iff,axiom, % 8.50/4.54 ! [A: $tType,X3: list @ A] : % 8.50/4.54 ( ( ( nil @ A ) % 8.50/4.54 = ( remdups @ A @ X3 ) ) % 8.50/4.54 = ( X3 % 8.50/4.54 = ( nil @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5205_remdups__eq__nil__iff,axiom, % 8.50/4.54 ! [A: $tType,X3: list @ A] : % 8.50/4.54 ( ( ( remdups @ A @ X3 ) % 8.50/4.54 = ( nil @ A ) ) % 8.50/4.54 = ( X3 % 8.50/4.54 = ( nil @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5206_set__remdups,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A] : % 8.50/4.54 ( ( set2 @ A @ ( remdups @ A @ Xs ) ) % 8.50/4.54 = ( set2 @ A @ Xs ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5207_length__remdups__eq,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A] : % 8.50/4.54 ( ( ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) ) % 8.50/4.54 = ( size_size @ ( list @ A ) @ Xs ) ) % 8.50/4.54 = ( ( remdups @ A @ Xs ) % 8.50/4.54 = Xs ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5208_remdups__id__iff__distinct,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A] : % 8.50/4.54 ( ( ( remdups @ A @ Xs ) % 8.50/4.54 = Xs ) % 8.50/4.54 = ( distinct @ A @ Xs ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5209_distinct__remdups,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A] : ( distinct @ A @ ( remdups @ A @ Xs ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5210_Collect__const,axiom, % 8.50/4.54 ! [A: $tType,P: $o] : % 8.50/4.54 ( ( P % 8.50/4.54 => ( ( collect @ A % 8.50/4.54 @ ^ [S8: A] : P ) % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) ) % 8.50/4.54 & ( ~ P % 8.50/4.54 => ( ( collect @ A % 8.50/4.54 @ ^ [S8: A] : P ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5211_finite__Collect__not,axiom, % 8.50/4.54 ! [A: $tType,P: A > $o] : % 8.50/4.54 ( ( finite_finite2 @ A @ ( collect @ A @ P ) ) % 8.50/4.54 => ( ( finite_finite2 @ A % 8.50/4.54 @ ( collect @ A % 8.50/4.54 @ ^ [X: A] : % 8.50/4.54 ~ ( P @ X ) ) ) % 8.50/4.54 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5212_range__abs__Nats,axiom, % 8.50/4.54 ( ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) % 8.50/4.54 = ( semiring_1_Nats @ int ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5213_Collect__const__case__prod,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,P: $o] : % 8.50/4.54 ( ( P % 8.50/4.54 => ( ( collect @ ( product_prod @ A @ B ) % 8.50/4.54 @ ( product_case_prod @ A @ B @ $o % 8.50/4.54 @ ^ [A4: A,B4: B] : P ) ) % 8.50/4.54 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) % 8.50/4.54 & ( ~ P % 8.50/4.54 => ( ( collect @ ( product_prod @ A @ B ) % 8.50/4.54 @ ( product_case_prod @ A @ B @ $o % 8.50/4.54 @ ^ [A4: A,B4: B] : P ) ) % 8.50/4.54 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5214_range__add,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( group_add @ A ) % 8.50/4.54 => ! [A2: A] : % 8.50/4.54 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5215_surj__plus,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ab_group_add @ A ) % 8.50/4.54 => ! [A2: A] : % 8.50/4.54 ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5216_Sup__eq__top__iff,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( comple5582772986160207858norder @ A ) % 8.50/4.54 => ! [A3: set @ A] : % 8.50/4.54 ( ( ( complete_Sup_Sup @ A @ A3 ) % 8.50/4.54 = ( top_top @ A ) ) % 8.50/4.54 = ( ! [X: A] : % 8.50/4.54 ( ( ord_less @ A @ X @ ( top_top @ A ) ) % 8.50/4.54 => ? [Y2: A] : % 8.50/4.54 ( ( member @ A @ Y2 @ A3 ) % 8.50/4.54 & ( ord_less @ A @ X @ Y2 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5217_boolean__algebra_Ocompl__one,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.54 => ( ( uminus_uminus @ A @ ( top_top @ A ) ) % 8.50/4.54 = ( bot_bot @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5218_boolean__algebra_Ocompl__zero,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( boolea8198339166811842893lgebra @ A ) % 8.50/4.54 => ( ( uminus_uminus @ A @ ( bot_bot @ A ) ) % 8.50/4.54 = ( top_top @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5219_ball__UNIV,axiom, % 8.50/4.54 ! [A: $tType,P: A > $o] : % 8.50/4.54 ( ( ! [X: A] : % 8.50/4.54 ( ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 => ( P @ X ) ) ) % 8.50/4.54 = ( ! [X6: A] : ( P @ X6 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5220_Inf__UNIV,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 = ( bot_bot @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5221_ccInf__empty,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( counta3822494911875563373attice @ A ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 = ( top_top @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5222_Inf__empty,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 = ( top_top @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5223_Diff__UNIV,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ A] : % 8.50/4.54 ( ( minus_minus @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5224_surj__fn,axiom, % 8.50/4.54 ! [A: $tType,F2: A > A,N2: nat] : % 8.50/4.54 ( ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) % 8.50/4.54 => ( ( image @ A @ A @ ( compow @ ( A > A ) @ N2 @ F2 ) @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5225_finite__compl,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ A] : % 8.50/4.54 ( ( finite_finite2 @ A @ A3 ) % 8.50/4.54 => ( ( finite_finite2 @ A @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) % 8.50/4.54 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5226_Gcd__UNIV,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( semiring_Gcd @ A ) % 8.50/4.54 => ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 = ( one_one @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5227_length__remdups__leq,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5228_SUP__eq__top__iff,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple5582772986160207858norder @ A ) % 8.50/4.54 => ! [F2: B > A,A3: set @ B] : % 8.50/4.54 ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A3 ) ) % 8.50/4.54 = ( top_top @ A ) ) % 8.50/4.54 = ( ! [X: A] : % 8.50/4.54 ( ( ord_less @ A @ X @ ( top_top @ A ) ) % 8.50/4.54 => ? [Y2: B] : % 8.50/4.54 ( ( member @ B @ Y2 @ A3 ) % 8.50/4.54 & ( ord_less @ A @ X @ ( F2 @ Y2 ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5229_range__constant,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,X3: A] : % 8.50/4.54 ( ( image @ B @ A % 8.50/4.54 @ ^ [Uu3: B] : X3 % 8.50/4.54 @ ( top_top @ ( set @ B ) ) ) % 8.50/4.54 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5230_ccINF__empty,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( counta3822494911875563373attice @ A ) % 8.50/4.54 => ! [F2: B > A] : % 8.50/4.54 ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) ) % 8.50/4.54 = ( top_top @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5231_INT__constant,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,A3: set @ B,C2: set @ A] : % 8.50/4.54 ( ( ( A3 % 8.50/4.54 = ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ ( set @ A ) % 8.50/4.54 @ ( image @ B @ ( set @ A ) % 8.50/4.54 @ ^ [Y2: B] : C2 % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) ) % 8.50/4.54 & ( ( A3 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ ( set @ A ) % 8.50/4.54 @ ( image @ B @ ( set @ A ) % 8.50/4.54 @ ^ [Y2: B] : C2 % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = C2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5232_Inf__atMostLessThan,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [X3: A] : % 8.50/4.54 ( ( ord_less @ A @ ( top_top @ A ) @ X3 ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X3 ) ) % 8.50/4.54 = ( bot_bot @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5233_INT__simps_I1_J,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,C5: set @ A,A3: A > ( set @ B ),B3: set @ B] : % 8.50/4.54 ( ( ( C5 % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ ( set @ B ) % 8.50/4.54 @ ( image @ A @ ( set @ B ) % 8.50/4.54 @ ^ [X: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X ) @ B3 ) % 8.50/4.54 @ C5 ) ) % 8.50/4.54 = ( top_top @ ( set @ B ) ) ) ) % 8.50/4.54 & ( ( C5 % 8.50/4.54 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ ( set @ B ) % 8.50/4.54 @ ( image @ A @ ( set @ B ) % 8.50/4.54 @ ^ [X: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X ) @ B3 ) % 8.50/4.54 @ C5 ) ) % 8.50/4.54 = ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ C5 ) ) @ B3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5234_INT__simps_I2_J,axiom, % 8.50/4.54 ! [C: $tType,D: $tType,C5: set @ D,A3: set @ C,B3: D > ( set @ C )] : % 8.50/4.54 ( ( ( C5 % 8.50/4.54 = ( bot_bot @ ( set @ D ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ ( set @ C ) % 8.50/4.54 @ ( image @ D @ ( set @ C ) % 8.50/4.54 @ ^ [X: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B3 @ X ) ) % 8.50/4.54 @ C5 ) ) % 8.50/4.54 = ( top_top @ ( set @ C ) ) ) ) % 8.50/4.54 & ( ( C5 % 8.50/4.54 != ( bot_bot @ ( set @ D ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ ( set @ C ) % 8.50/4.54 @ ( image @ D @ ( set @ C ) % 8.50/4.54 @ ^ [X: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B3 @ X ) ) % 8.50/4.54 @ C5 ) ) % 8.50/4.54 = ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D @ ( set @ C ) @ B3 @ C5 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5235_INT__simps_I3_J,axiom, % 8.50/4.54 ! [E2: $tType,F: $tType,C5: set @ E2,A3: E2 > ( set @ F ),B3: set @ F] : % 8.50/4.54 ( ( ( C5 % 8.50/4.54 = ( bot_bot @ ( set @ E2 ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ ( set @ F ) % 8.50/4.54 @ ( image @ E2 @ ( set @ F ) % 8.50/4.54 @ ^ [X: E2] : ( minus_minus @ ( set @ F ) @ ( A3 @ X ) @ B3 ) % 8.50/4.54 @ C5 ) ) % 8.50/4.54 = ( top_top @ ( set @ F ) ) ) ) % 8.50/4.54 & ( ( C5 % 8.50/4.54 != ( bot_bot @ ( set @ E2 ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ ( set @ F ) % 8.50/4.54 @ ( image @ E2 @ ( set @ F ) % 8.50/4.54 @ ^ [X: E2] : ( minus_minus @ ( set @ F ) @ ( A3 @ X ) @ B3 ) % 8.50/4.54 @ C5 ) ) % 8.50/4.54 = ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E2 @ ( set @ F ) @ A3 @ C5 ) ) @ B3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5236_UNIV__option__conv,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( top_top @ ( set @ ( option @ A ) ) ) % 8.50/4.54 = ( insert @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5237_rangeI,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,X3: B] : ( member @ A @ ( F2 @ X3 ) @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5238_range__eqI,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,B2: A,F2: B > A,X3: B] : % 8.50/4.54 ( ( B2 % 8.50/4.54 = ( F2 @ X3 ) ) % 8.50/4.54 => ( member @ A @ B2 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5239_rangeE,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,B2: A,F2: B > A] : % 8.50/4.54 ( ( member @ A @ B2 @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) % 8.50/4.54 => ~ ! [X5: B] : % 8.50/4.54 ( B2 % 8.50/4.54 != ( F2 @ X5 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5240_range__composition,axiom, % 8.50/4.54 ! [A: $tType,C: $tType,B: $tType,F2: C > A,G: B > C] : % 8.50/4.54 ( ( image @ B @ A % 8.50/4.54 @ ^ [X: B] : ( F2 @ ( G @ X ) ) % 8.50/4.54 @ ( top_top @ ( set @ B ) ) ) % 8.50/4.54 = ( image @ C @ A @ F2 @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5241_INF__filter__bot__base,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,I6: set @ A,F5: A > ( filter @ B )] : % 8.50/4.54 ( ! [I4: A] : % 8.50/4.54 ( ( member @ A @ I4 @ I6 ) % 8.50/4.54 => ! [J2: A] : % 8.50/4.54 ( ( member @ A @ J2 @ I6 ) % 8.50/4.54 => ? [X2: A] : % 8.50/4.54 ( ( member @ A @ X2 @ I6 ) % 8.50/4.54 & ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X2 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ I4 ) @ ( F5 @ J2 ) ) ) ) ) ) % 8.50/4.54 => ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F5 @ I6 ) ) % 8.50/4.54 = ( bot_bot @ ( filter @ B ) ) ) % 8.50/4.54 = ( ? [X: A] : % 8.50/4.54 ( ( member @ A @ X @ I6 ) % 8.50/4.54 & ( ( F5 @ X ) % 8.50/4.54 = ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5242_remdups_Osimps_I1_J,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( remdups @ A @ ( nil @ A ) ) % 8.50/4.54 = ( nil @ A ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5243_Int__UNIV__right,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ A] : % 8.50/4.54 ( ( inf_inf @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 = A3 ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5244_Int__UNIV__left,axiom, % 8.50/4.54 ! [A: $tType,B3: set @ A] : % 8.50/4.54 ( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 ) % 8.50/4.54 = B3 ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5245_remdups__remdups,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A] : % 8.50/4.54 ( ( remdups @ A @ ( remdups @ A @ Xs ) ) % 8.50/4.54 = ( remdups @ A @ Xs ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5246_UNIV__eq__I,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ A] : % 8.50/4.54 ( ! [X5: A] : ( member @ A @ X5 @ A3 ) % 8.50/4.54 => ( ( top_top @ ( set @ A ) ) % 8.50/4.54 = A3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5247_UNIV__witness,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ? [X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5248_top_Oextremum__strict,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order_top @ A ) % 8.50/4.54 => ! [A2: A] : % 8.50/4.54 ~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5249_top_Onot__eq__extremum,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order_top @ A ) % 8.50/4.54 => ! [A2: A] : % 8.50/4.54 ( ( A2 % 8.50/4.54 != ( top_top @ A ) ) % 8.50/4.54 = ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5250_UNIV__def,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( top_top @ ( set @ A ) ) % 8.50/4.54 = ( collect @ A % 8.50/4.54 @ ^ [X: A] : $true ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5251_subset__UNIV,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5252_top_Oextremum__uniqueI,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order_top @ A ) % 8.50/4.54 => ! [A2: A] : % 8.50/4.54 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 ) % 8.50/4.54 => ( A2 % 8.50/4.54 = ( top_top @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5253_top_Oextremum__unique,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order_top @ A ) % 8.50/4.54 => ! [A2: A] : % 8.50/4.54 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 ) % 8.50/4.54 = ( A2 % 8.50/4.54 = ( top_top @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5254_top__greatest,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order_top @ A ) % 8.50/4.54 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5255_distinct__remdups__id,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A] : % 8.50/4.54 ( ( distinct @ A @ Xs ) % 8.50/4.54 => ( ( remdups @ A @ Xs ) % 8.50/4.54 = Xs ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5256_empty__not__UNIV,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( bot_bot @ ( set @ A ) ) % 8.50/4.54 != ( top_top @ ( set @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5257_atLeastAtMost__eq__UNIV__iff,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( bounded_lattice @ A ) % 8.50/4.54 => ! [X3: A,Y: A] : % 8.50/4.54 ( ( ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) % 8.50/4.54 = ( ( X3 % 8.50/4.54 = ( bot_bot @ A ) ) % 8.50/4.54 & ( Y % 8.50/4.54 = ( top_top @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5258_insert__UNIV,axiom, % 8.50/4.54 ! [A: $tType,X3: A] : % 8.50/4.54 ( ( insert @ A @ X3 @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5259_Inf__filter__not__bot,axiom, % 8.50/4.54 ! [A: $tType,B3: set @ ( filter @ A )] : % 8.50/4.54 ( ! [X10: set @ ( filter @ A )] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X10 @ B3 ) % 8.50/4.54 => ( ( finite_finite2 @ ( filter @ A ) @ X10 ) % 8.50/4.54 => ( ( complete_Inf_Inf @ ( filter @ A ) @ X10 ) % 8.50/4.54 != ( bot_bot @ ( filter @ A ) ) ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ ( filter @ A ) @ B3 ) % 8.50/4.54 != ( bot_bot @ ( filter @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5260_infinite__UNIV__int,axiom, % 8.50/4.54 ~ ( finite_finite2 @ int @ ( top_top @ ( set @ int ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5261_nat__not__finite,axiom, % 8.50/4.54 ~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5262_infinite__UNIV__nat,axiom, % 8.50/4.54 ~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5263_Finite__Set_Ofinite__set,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( finite_finite2 @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) ) % 8.50/4.54 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5264_finite__Prod__UNIV,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 => ( ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) % 8.50/4.54 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5265_finite__prod,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) % 8.50/4.54 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5266_finite__fun__UNIVD2,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) ) % 8.50/4.54 => ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5267_finite__UNIV,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( finite_finite @ A ) % 8.50/4.54 => ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5268_ex__new__if__finite,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ A] : % 8.50/4.54 ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 => ( ( finite_finite2 @ A @ A3 ) % 8.50/4.54 => ? [A5: A] : % 8.50/4.54 ~ ( member @ A @ A5 @ A3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5269_infinite__UNIV__char__0,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( semiring_char_0 @ A ) % 8.50/4.54 => ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5270_infinite__UNIV__listI,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ~ ( finite_finite2 @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5271_range__subsetD,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,F2: B > A,B3: set @ A,I: B] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) @ B3 ) % 8.50/4.54 => ( member @ A @ ( F2 @ I ) @ B3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5272_perfect__space__class_OUNIV__not__singleton,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( topolo8386298272705272623_space @ A ) % 8.50/4.54 => ! [X3: A] : % 8.50/4.54 ( ( top_top @ ( set @ A ) ) % 8.50/4.54 != ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5273_bot__finite__def,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( finite_lattice @ A ) % 8.50/4.54 => ( ( bot_bot @ A ) % 8.50/4.54 = ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5274_finite__range__Some,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( finite_finite2 @ ( option @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) % 8.50/4.54 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5275_not__UNIV__le__Icc,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( no_top @ A ) % 8.50/4.54 => ! [L: A,H2: A] : % 8.50/4.54 ~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5276_card__eq__UNIV__imp__eq__UNIV,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ A] : % 8.50/4.54 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 => ( ( ( finite_card @ A @ A3 ) % 8.50/4.54 = ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) % 8.50/4.54 => ( A3 % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5277_not__UNIV__le__Iic,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( no_top @ A ) % 8.50/4.54 => ! [H2: A] : % 8.50/4.54 ~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atMost @ A @ H2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5278_notin__range__Some,axiom, % 8.50/4.54 ! [A: $tType,X3: option @ A] : % 8.50/4.54 ( ( ~ ( member @ ( option @ A ) @ X3 @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) % 8.50/4.54 = ( X3 % 8.50/4.54 = ( none @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5279_Inter__UNIV,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( complete_Inf_Inf @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5280_Compl__UNIV__eq,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5281_Compl__empty__eq,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5282_int__in__range__abs,axiom, % 8.50/4.54 ! [N2: nat] : ( member @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5283_Compl__eq__Diff__UNIV,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( uminus_uminus @ ( set @ A ) ) % 8.50/4.54 = ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5284_Inter__empty,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( complete_Inf_Inf @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5285_full__SetCompr__eq,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A] : % 8.50/4.54 ( ( collect @ A % 8.50/4.54 @ ^ [U2: A] : % 8.50/4.54 ? [X: B] : % 8.50/4.54 ( U2 % 8.50/4.54 = ( F2 @ X ) ) ) % 8.50/4.54 = ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5286_finite__range__imageI,axiom, % 8.50/4.54 ! [C: $tType,A: $tType,B: $tType,G: B > A,F2: A > C] : % 8.50/4.54 ( ( finite_finite2 @ A @ ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) ) ) % 8.50/4.54 => ( finite_finite2 @ C % 8.50/4.54 @ ( image @ B @ C % 8.50/4.54 @ ^ [X: B] : ( F2 @ ( G @ X ) ) % 8.50/4.54 @ ( top_top @ ( set @ B ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5287_remove1__remdups,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A,X3: A] : % 8.50/4.54 ( ( distinct @ A @ Xs ) % 8.50/4.54 => ( ( remove1 @ A @ X3 @ ( remdups @ A @ Xs ) ) % 8.50/4.54 = ( remdups @ A @ ( remove1 @ A @ X3 @ Xs ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5288_finite__fun__UNIVD1,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) ) % 8.50/4.54 => ( ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) ) % 8.50/4.54 != ( suc @ ( zero_zero @ nat ) ) ) % 8.50/4.54 => ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5289_range__eq__singletonD,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,F2: B > A,A2: A,X3: B] : % 8.50/4.54 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) % 8.50/4.54 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) % 8.50/4.54 => ( ( F2 @ X3 ) % 8.50/4.54 = A2 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5290_INF__empty,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [F2: B > A] : % 8.50/4.54 ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ ( bot_bot @ ( set @ B ) ) ) ) % 8.50/4.54 = ( top_top @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5291_INF__constant,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 => ! [A3: set @ B,C2: A] : % 8.50/4.54 ( ( ( A3 % 8.50/4.54 = ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A % 8.50/4.54 @ ( image @ B @ A % 8.50/4.54 @ ^ [Y2: B] : C2 % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = ( top_top @ A ) ) ) % 8.50/4.54 & ( ( A3 % 8.50/4.54 != ( bot_bot @ ( set @ B ) ) ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A % 8.50/4.54 @ ( image @ B @ A % 8.50/4.54 @ ^ [Y2: B] : C2 % 8.50/4.54 @ A3 ) ) % 8.50/4.54 = C2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5292_Sup__finite__empty,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( finite_lattice @ A ) % 8.50/4.54 => ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 = ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5293_Inf__finite__empty,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( finite_lattice @ A ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 = ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5294_surj__Compl__image__subset,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A,A3: set @ B] : % 8.50/4.54 ( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) % 8.50/4.54 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A3 ) ) @ ( image @ B @ A @ F2 @ ( uminus_uminus @ ( set @ B ) @ A3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5295_length__remdups__card__conv,axiom, % 8.50/4.54 ! [A: $tType,Xs: list @ A] : % 8.50/4.54 ( ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) ) % 8.50/4.54 = ( finite_card @ A @ ( set2 @ A @ Xs ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5296_Nats__def,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( semiring_1 @ A ) % 8.50/4.54 => ( ( semiring_1_Nats @ A ) % 8.50/4.54 = ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5297_Collect__ex__eq,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,P: A > B > $o] : % 8.50/4.54 ( ( collect @ A % 8.50/4.54 @ ^ [X: A] : % 8.50/4.54 ? [X6: B] : ( P @ X @ X6 ) ) % 8.50/4.54 = ( complete_Sup_Sup @ ( set @ A ) % 8.50/4.54 @ ( image @ B @ ( set @ A ) % 8.50/4.54 @ ^ [Y2: B] : % 8.50/4.54 ( collect @ A % 8.50/4.54 @ ^ [X: A] : ( P @ X @ Y2 ) ) % 8.50/4.54 @ ( top_top @ ( set @ B ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5298_INT__empty,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,B3: B > ( set @ A )] : % 8.50/4.54 ( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) % 8.50/4.54 = ( top_top @ ( set @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5299_UN__UN__finite__eq,axiom, % 8.50/4.54 ! [A: $tType,A3: nat > ( set @ A )] : % 8.50/4.54 ( ( complete_Sup_Sup @ ( set @ A ) % 8.50/4.54 @ ( image @ nat @ ( set @ A ) % 8.50/4.54 @ ^ [N: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) % 8.50/4.54 @ ( top_top @ ( set @ nat ) ) ) ) % 8.50/4.54 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5300_inf__top_Osemilattice__neutr__order__axioms,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( bounde4346867609351753570nf_top @ A ) % 8.50/4.54 => ( semila1105856199041335345_order @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5301_finite__UNIV__card__ge__0,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5302_UN__finite__subset,axiom, % 8.50/4.54 ! [A: $tType,A3: nat > ( set @ A ),C5: set @ A] : % 8.50/4.54 ( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ C5 ) % 8.50/4.54 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) @ C5 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5303_UNIV__nat__eq,axiom, % 8.50/4.54 ( ( top_top @ ( set @ nat ) ) % 8.50/4.54 = ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5304_UN__finite2__eq,axiom, % 8.50/4.54 ! [A: $tType,A3: nat > ( set @ A ),B3: nat > ( set @ A ),K: nat] : % 8.50/4.54 ( ! [N3: nat] : % 8.50/4.54 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) % 8.50/4.54 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) ) % 8.50/4.54 => ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) % 8.50/4.54 = ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5305_INT__extend__simps_I3_J,axiom, % 8.50/4.54 ! [F: $tType,E2: $tType,C5: set @ E2,A3: E2 > ( set @ F ),B3: set @ F] : % 8.50/4.54 ( ( ( C5 % 8.50/4.54 = ( bot_bot @ ( set @ E2 ) ) ) % 8.50/4.54 => ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E2 @ ( set @ F ) @ A3 @ C5 ) ) @ B3 ) % 8.50/4.54 = ( minus_minus @ ( set @ F ) @ ( top_top @ ( set @ F ) ) @ B3 ) ) ) % 8.50/4.54 & ( ( C5 % 8.50/4.54 != ( bot_bot @ ( set @ E2 ) ) ) % 8.50/4.54 => ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E2 @ ( set @ F ) @ A3 @ C5 ) ) @ B3 ) % 8.50/4.54 = ( complete_Inf_Inf @ ( set @ F ) % 8.50/4.54 @ ( image @ E2 @ ( set @ F ) % 8.50/4.54 @ ^ [X: E2] : ( minus_minus @ ( set @ F ) @ ( A3 @ X ) @ B3 ) % 8.50/4.54 @ C5 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5306_card__range__greater__zero,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,F2: B > A] : % 8.50/4.54 ( ( finite_finite2 @ A @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) % 8.50/4.54 => ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5307_UN__finite2__subset,axiom, % 8.50/4.54 ! [A: $tType,A3: nat > ( set @ A ),B3: nat > ( set @ A ),K: nat] : % 8.50/4.54 ( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) ) % 8.50/4.54 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5308_range__mod,axiom, % 8.50/4.54 ! [N2: nat] : % 8.50/4.54 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.54 => ( ( image @ nat @ nat % 8.50/4.54 @ ^ [M2: nat] : ( modulo_modulo @ nat @ M2 @ N2 ) % 8.50/4.54 @ ( top_top @ ( set @ nat ) ) ) % 8.50/4.54 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5309_cclfp__def,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( counta3822494911875563373attice @ A ) % 8.50/4.54 => ( ( order_532582986084564980_cclfp @ A ) % 8.50/4.54 = ( ^ [F3: A > A] : % 8.50/4.54 ( complete_Sup_Sup @ A % 8.50/4.54 @ ( image @ nat @ A % 8.50/4.54 @ ^ [I3: nat] : ( compow @ ( A > A ) @ I3 @ F3 @ ( bot_bot @ A ) ) % 8.50/4.54 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5310_root__def,axiom, % 8.50/4.54 ( root % 8.50/4.54 = ( ^ [N: nat,X: real] : % 8.50/4.54 ( if @ real % 8.50/4.54 @ ( N % 8.50/4.54 = ( zero_zero @ nat ) ) % 8.50/4.54 @ ( zero_zero @ real ) % 8.50/4.54 @ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) ) % 8.50/4.54 @ ^ [Y2: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N ) ) % 8.50/4.54 @ X ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5311_Id__on__def,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( id_on @ A ) % 8.50/4.54 = ( ^ [A6: set @ A] : % 8.50/4.54 ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) % 8.50/4.54 @ ( image @ A @ ( set @ ( product_prod @ A @ A ) ) % 8.50/4.54 @ ^ [X: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) % 8.50/4.54 @ A6 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5312_Id__onI,axiom, % 8.50/4.54 ! [A: $tType,A2: A,A3: set @ A] : % 8.50/4.54 ( ( member @ A @ A2 @ A3 ) % 8.50/4.54 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5313_Id__on__empty,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5314_top__empty__eq2,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( top_top @ ( A > B > $o ) ) % 8.50/4.54 = ( ^ [X: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5315_top__set__def,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( top_top @ ( set @ A ) ) % 8.50/4.54 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5316_less__filter__def,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ord_less @ ( filter @ A ) ) % 8.50/4.54 = ( ^ [F9: filter @ A,F10: filter @ A] : % 8.50/4.54 ( ( ord_less_eq @ ( filter @ A ) @ F9 @ F10 ) % 8.50/4.54 & ~ ( ord_less_eq @ ( filter @ A ) @ F10 @ F9 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5317_Id__on__iff,axiom, % 8.50/4.54 ! [A: $tType,X3: A,Y: A,A3: set @ A] : % 8.50/4.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( id_on @ A @ A3 ) ) % 8.50/4.54 = ( ( X3 = Y ) % 8.50/4.54 & ( member @ A @ X3 @ A3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5318_Id__on__eqI,axiom, % 8.50/4.54 ! [A: $tType,A2: A,B2: A,A3: set @ A] : % 8.50/4.54 ( ( A2 = B2 ) % 8.50/4.54 => ( ( member @ A @ A2 @ A3 ) % 8.50/4.54 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id_on @ A @ A3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5319_Id__onE,axiom, % 8.50/4.54 ! [A: $tType,C2: product_prod @ A @ A,A3: set @ A] : % 8.50/4.54 ( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A3 ) ) % 8.50/4.54 => ~ ! [X5: A] : % 8.50/4.54 ( ( member @ A @ X5 @ A3 ) % 8.50/4.54 => ( C2 % 8.50/4.54 != ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5320_these__insert__Some,axiom, % 8.50/4.54 ! [A: $tType,X3: A,A3: set @ ( option @ A )] : % 8.50/4.54 ( ( these @ A @ ( insert @ ( option @ A ) @ ( some @ A @ X3 ) @ A3 ) ) % 8.50/4.54 = ( insert @ A @ X3 @ ( these @ A @ A3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5321_DERIV__even__real__root,axiom, % 8.50/4.54 ! [N2: nat,X3: real] : % 8.50/4.54 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.54 => ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.50/4.54 => ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) % 8.50/4.54 => ( has_field_derivative @ real @ ( root @ N2 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N2 ) ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5322_DERIV__real__root__generic,axiom, % 8.50/4.54 ! [N2: nat,X3: real,D6: real] : % 8.50/4.54 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.54 => ( ( X3 % 8.50/4.54 != ( zero_zero @ real ) ) % 8.50/4.54 => ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.50/4.54 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.54 => ( D6 % 8.50/4.54 = ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) % 8.50/4.54 => ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.50/4.54 => ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) % 8.50/4.54 => ( D6 % 8.50/4.54 = ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ) % 8.50/4.54 => ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.50/4.54 => ( D6 % 8.50/4.54 = ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) % 8.50/4.54 => ( has_field_derivative @ real @ ( root @ N2 ) @ D6 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5323_top1I,axiom, % 8.50/4.54 ! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5324_top2I,axiom, % 8.50/4.54 ! [A: $tType,B: $tType,X3: A,Y: B] : ( top_top @ ( A > B > $o ) @ X3 @ Y ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5325_these__empty,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5326_at__within__empty,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( topolo4958980785337419405_space @ A ) % 8.50/4.54 => ! [A2: A] : % 8.50/4.54 ( ( topolo174197925503356063within @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 = ( bot_bot @ ( filter @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5327_these__image__Some__eq,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ A] : % 8.50/4.54 ( ( these @ A @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ) % 8.50/4.54 = A3 ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5328_these__insert__None,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ ( option @ A )] : % 8.50/4.54 ( ( these @ A @ ( insert @ ( option @ A ) @ ( none @ A ) @ A3 ) ) % 8.50/4.54 = ( these @ A @ A3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5329_DERIV__at__within__shift__lemma,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: A > A,Y: A,Z2: A,X3: A,S3: set @ A] : % 8.50/4.54 ( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z2 @ X3 ) @ ( image @ A @ A @ ( plus_plus @ A @ Z2 ) @ S3 ) ) ) % 8.50/4.54 => ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F2 @ ( plus_plus @ A @ Z2 ) ) @ Y @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5330_DERIV__at__within__shift,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: A > A,Y: A,Z2: A,X3: A,S3: set @ A] : % 8.50/4.54 ( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z2 @ X3 ) @ ( image @ A @ A @ ( plus_plus @ A @ Z2 ) @ S3 ) ) ) % 8.50/4.54 = ( has_field_derivative @ A % 8.50/4.54 @ ^ [X: A] : ( F2 @ ( plus_plus @ A @ Z2 @ X ) ) % 8.50/4.54 @ Y % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5331_DERIV__neg__dec__right,axiom, % 8.50/4.54 ! [F2: real > real,L: real,X3: real] : % 8.50/4.54 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ L @ ( zero_zero @ real ) ) % 8.50/4.54 => ? [D3: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 8.50/4.54 & ! [H5: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 8.50/4.54 => ( ( ord_less @ real @ H5 @ D3 ) % 8.50/4.54 => ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5332_DERIV__pos__inc__right,axiom, % 8.50/4.54 ! [F2: real > real,L: real,X3: real] : % 8.50/4.54 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( zero_zero @ real ) @ L ) % 8.50/4.54 => ? [D3: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 8.50/4.54 & ! [H5: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 8.50/4.54 => ( ( ord_less @ real @ H5 @ D3 ) % 8.50/4.54 => ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5333_DERIV__pos__inc__left,axiom, % 8.50/4.54 ! [F2: real > real,L: real,X3: real] : % 8.50/4.54 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( zero_zero @ real ) @ L ) % 8.50/4.54 => ? [D3: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 8.50/4.54 & ! [H5: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 8.50/4.54 => ( ( ord_less @ real @ H5 @ D3 ) % 8.50/4.54 => ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5334_DERIV__neg__dec__left,axiom, % 8.50/4.54 ! [F2: real > real,L: real,X3: real] : % 8.50/4.54 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ L @ ( zero_zero @ real ) ) % 8.50/4.54 => ? [D3: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 8.50/4.54 & ! [H5: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 8.50/4.54 => ( ( ord_less @ real @ H5 @ D3 ) % 8.50/4.54 => ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5335_DERIV__neg__imp__decreasing,axiom, % 8.50/4.54 ! [A2: real,B2: real,F2: real > real] : % 8.50/4.54 ( ( ord_less @ real @ A2 @ B2 ) % 8.50/4.54 => ( ! [X5: real] : % 8.50/4.54 ( ( ord_less_eq @ real @ A2 @ X5 ) % 8.50/4.54 => ( ( ord_less_eq @ real @ X5 @ B2 ) % 8.50/4.54 => ? [Y5: real] : % 8.50/4.54 ( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) % 8.50/4.54 & ( ord_less @ real @ Y5 @ ( zero_zero @ real ) ) ) ) ) % 8.50/4.54 => ( ord_less @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5336_DERIV__pos__imp__increasing,axiom, % 8.50/4.54 ! [A2: real,B2: real,F2: real > real] : % 8.50/4.54 ( ( ord_less @ real @ A2 @ B2 ) % 8.50/4.54 => ( ! [X5: real] : % 8.50/4.54 ( ( ord_less_eq @ real @ A2 @ X5 ) % 8.50/4.54 => ( ( ord_less_eq @ real @ X5 @ B2 ) % 8.50/4.54 => ? [Y5: real] : % 8.50/4.54 ( ( has_field_derivative @ real @ F2 @ Y5 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) % 8.50/4.54 & ( ord_less @ real @ ( zero_zero @ real ) @ Y5 ) ) ) ) % 8.50/4.54 => ( ord_less @ real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5337_DERIV__subset,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: A > A,F11: A,X3: A,S: set @ A,T2: set @ A] : % 8.50/4.54 ( ( has_field_derivative @ A @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( ( ord_less_eq @ ( set @ A ) @ T2 @ S ) % 8.50/4.54 => ( has_field_derivative @ A @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5338_has__field__derivative__subset,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: A > A,Y: A,X3: A,S: set @ A,T2: set @ A] : % 8.50/4.54 ( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( ( ord_less_eq @ ( set @ A ) @ T2 @ S ) % 8.50/4.54 => ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5339_DERIV__shift,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: A > A,Y: A,X3: A,Z2: A] : % 8.50/4.54 ( ( has_field_derivative @ A @ F2 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ X3 @ Z2 ) @ ( top_top @ ( set @ A ) ) ) ) % 8.50/4.54 = ( has_field_derivative @ A % 8.50/4.54 @ ^ [X: A] : ( F2 @ ( plus_plus @ A @ X @ Z2 ) ) % 8.50/4.54 @ Y % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5340_at__discrete,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( topolo8865339358273720382pology @ A ) % 8.50/4.54 => ( ( topolo174197925503356063within @ A ) % 8.50/4.54 = ( ^ [X: A,S7: set @ A] : ( bot_bot @ ( filter @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5341_DERIV__inverse_H,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: A > A,D6: A,X3: A,S: set @ A] : % 8.50/4.54 ( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( ( ( F2 @ X3 ) % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( has_field_derivative @ A % 8.50/4.54 @ ^ [X: A] : ( inverse_inverse @ A @ ( F2 @ X ) ) % 8.50/4.54 @ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) @ D6 ) @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5342_has__real__derivative__pos__inc__right,axiom, % 8.50/4.54 ! [F2: real > real,L: real,X3: real,S3: set @ real] : % 8.50/4.54 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S3 ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( zero_zero @ real ) @ L ) % 8.50/4.54 => ? [D3: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 8.50/4.54 & ! [H5: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 8.50/4.54 => ( ( member @ real @ ( plus_plus @ real @ X3 @ H5 ) @ S3 ) % 8.50/4.54 => ( ( ord_less @ real @ H5 @ D3 ) % 8.50/4.54 => ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5343_has__real__derivative__neg__dec__right,axiom, % 8.50/4.54 ! [F2: real > real,L: real,X3: real,S3: set @ real] : % 8.50/4.54 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S3 ) ) % 8.50/4.54 => ( ( ord_less @ real @ L @ ( zero_zero @ real ) ) % 8.50/4.54 => ? [D3: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 8.50/4.54 & ! [H5: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 8.50/4.54 => ( ( member @ real @ ( plus_plus @ real @ X3 @ H5 ) @ S3 ) % 8.50/4.54 => ( ( ord_less @ real @ H5 @ D3 ) % 8.50/4.54 => ( ord_less @ real @ ( F2 @ ( plus_plus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5344_has__real__derivative__neg__dec__left,axiom, % 8.50/4.54 ! [F2: real > real,L: real,X3: real,S3: set @ real] : % 8.50/4.54 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S3 ) ) % 8.50/4.54 => ( ( ord_less @ real @ L @ ( zero_zero @ real ) ) % 8.50/4.54 => ? [D3: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 8.50/4.54 & ! [H5: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 8.50/4.54 => ( ( member @ real @ ( minus_minus @ real @ X3 @ H5 ) @ S3 ) % 8.50/4.54 => ( ( ord_less @ real @ H5 @ D3 ) % 8.50/4.54 => ( ord_less @ real @ ( F2 @ X3 ) @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5345_has__real__derivative__pos__inc__left,axiom, % 8.50/4.54 ! [F2: real > real,L: real,X3: real,S3: set @ real] : % 8.50/4.54 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ S3 ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( zero_zero @ real ) @ L ) % 8.50/4.54 => ? [D3: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 ) % 8.50/4.54 & ! [H5: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ H5 ) % 8.50/4.54 => ( ( member @ real @ ( minus_minus @ real @ X3 @ H5 ) @ S3 ) % 8.50/4.54 => ( ( ord_less @ real @ H5 @ D3 ) % 8.50/4.54 => ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X3 @ H5 ) ) @ ( F2 @ X3 ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5346_DERIV__const,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [K: A,F5: filter @ A] : % 8.50/4.54 ( has_field_derivative @ A % 8.50/4.54 @ ^ [X: A] : K % 8.50/4.54 @ ( zero_zero @ A ) % 8.50/4.54 @ F5 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5347_field__differentiable__add,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: A > A,F11: A,F5: filter @ A,G: A > A,G7: A] : % 8.50/4.54 ( ( has_field_derivative @ A @ F2 @ F11 @ F5 ) % 8.50/4.54 => ( ( has_field_derivative @ A @ G @ G7 @ F5 ) % 8.50/4.54 => ( has_field_derivative @ A % 8.50/4.54 @ ^ [Z6: A] : ( plus_plus @ A @ ( F2 @ Z6 ) @ ( G @ Z6 ) ) % 8.50/4.54 @ ( plus_plus @ A @ F11 @ G7 ) % 8.50/4.54 @ F5 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5348_DERIV__add,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: A > A,D6: A,X3: A,S: set @ A,G: A > A,E5: A] : % 8.50/4.54 ( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( has_field_derivative @ A % 8.50/4.54 @ ^ [X: A] : ( plus_plus @ A @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ( plus_plus @ A @ D6 @ E5 ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5349_DERIV__mult_H,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: A > A,D6: A,X3: A,S: set @ A,G: A > A,E5: A] : % 8.50/4.54 ( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( has_field_derivative @ A % 8.50/4.54 @ ^ [X: A] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X3 ) @ E5 ) @ ( times_times @ A @ D6 @ ( G @ X3 ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5350_DERIV__mult,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: A > A,Da: A,X3: A,S: set @ A,G: A > A,Db: A] : % 8.50/4.54 ( ( has_field_derivative @ A @ F2 @ Da @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( has_field_derivative @ A % 8.50/4.54 @ ^ [X: A] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ( plus_plus @ A @ ( times_times @ A @ Da @ ( G @ X3 ) ) @ ( times_times @ A @ Db @ ( F2 @ X3 ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5351_DERIV__divide,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: A > A,D6: A,X3: A,S: set @ A,G: A > A,E5: A] : % 8.50/4.54 ( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( ( ( G @ X3 ) % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( has_field_derivative @ A % 8.50/4.54 @ ^ [X: A] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D6 @ ( G @ X3 ) ) @ ( times_times @ A @ ( F2 @ X3 ) @ E5 ) ) @ ( times_times @ A @ ( G @ X3 ) @ ( G @ X3 ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5352_in__these__eq,axiom, % 8.50/4.54 ! [A: $tType,X3: A,A3: set @ ( option @ A )] : % 8.50/4.54 ( ( member @ A @ X3 @ ( these @ A @ A3 ) ) % 8.50/4.54 = ( member @ ( option @ A ) @ ( some @ A @ X3 ) @ A3 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5353_at__neq__bot,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( topolo8386298272705272623_space @ A ) % 8.50/4.54 => ! [A2: A] : % 8.50/4.54 ( ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 != ( bot_bot @ ( filter @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5354_at__le,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( topolo4958980785337419405_space @ A ) % 8.50/4.54 => ! [S: set @ A,T2: set @ A,X3: A] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ S @ T2 ) % 8.50/4.54 => ( ord_less_eq @ ( filter @ A ) @ ( topolo174197925503356063within @ A @ X3 @ S ) @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5355_trivial__limit__at__left__real,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ( dense_order @ A ) % 8.50/4.54 & ( no_bot @ A ) % 8.50/4.54 & ( topolo1944317154257567458pology @ A ) ) % 8.50/4.54 => ! [X3: A] : % 8.50/4.54 ( ( topolo174197925503356063within @ A @ X3 @ ( set_ord_lessThan @ A @ X3 ) ) % 8.50/4.54 != ( bot_bot @ ( filter @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5356_MVT2,axiom, % 8.50/4.54 ! [A2: real,B2: real,F2: real > real,F11: real > real] : % 8.50/4.54 ( ( ord_less @ real @ A2 @ B2 ) % 8.50/4.54 => ( ! [X5: real] : % 8.50/4.54 ( ( ord_less_eq @ real @ A2 @ X5 ) % 8.50/4.54 => ( ( ord_less_eq @ real @ X5 @ B2 ) % 8.50/4.54 => ( has_field_derivative @ real @ F2 @ ( F11 @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) ) % 8.50/4.54 => ? [Z3: real] : % 8.50/4.54 ( ( ord_less @ real @ A2 @ Z3 ) % 8.50/4.54 & ( ord_less @ real @ Z3 @ B2 ) % 8.50/4.54 & ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) % 8.50/4.54 = ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( F11 @ Z3 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5357_DERIV__local__const,axiom, % 8.50/4.54 ! [F2: real > real,L: real,X3: real,D2: real] : % 8.50/4.54 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 ) % 8.50/4.54 => ( ! [Y4: real] : % 8.50/4.54 ( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y4 ) ) @ D2 ) % 8.50/4.54 => ( ( F2 @ X3 ) % 8.50/4.54 = ( F2 @ Y4 ) ) ) % 8.50/4.54 => ( L % 8.50/4.54 = ( zero_zero @ real ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5358_DERIV__ln,axiom, % 8.50/4.54 ! [X3: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.54 => ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( inverse_inverse @ real @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5359_DERIV__cos__add,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ( real_Vector_banach @ A ) % 8.50/4.54 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.54 => ! [K: A,Xa2: A] : % 8.50/4.54 ( has_field_derivative @ A % 8.50/4.54 @ ^ [X: A] : ( cos @ A @ ( plus_plus @ A @ X @ K ) ) % 8.50/4.54 @ ( uminus_uminus @ A @ ( sin @ A @ ( plus_plus @ A @ Xa2 @ K ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ Xa2 @ ( top_top @ ( set @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5360_DERIV__power__Suc,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: A > A,D6: A,X3: A,S: set @ A,N2: nat] : % 8.50/4.54 ( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( has_field_derivative @ A % 8.50/4.54 @ ^ [X: A] : ( power_power @ A @ ( F2 @ X ) @ ( suc @ N2 ) ) % 8.50/4.54 @ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( times_times @ A @ D6 @ ( power_power @ A @ ( F2 @ X3 ) @ N2 ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5361_DERIV__inverse,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [X3: A,S: set @ A] : % 8.50/4.54 ( ( X3 % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5362_DERIV__power,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: A > A,D6: A,X3: A,S: set @ A,N2: nat] : % 8.50/4.54 ( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( has_field_derivative @ A % 8.50/4.54 @ ^ [X: A] : ( power_power @ A @ ( F2 @ X ) @ N2 ) % 8.50/4.54 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( times_times @ A @ D6 @ ( power_power @ A @ ( F2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5363_DERIV__local__max,axiom, % 8.50/4.54 ! [F2: real > real,L: real,X3: real,D2: real] : % 8.50/4.54 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 ) % 8.50/4.54 => ( ! [Y4: real] : % 8.50/4.54 ( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y4 ) ) @ D2 ) % 8.50/4.54 => ( ord_less_eq @ real @ ( F2 @ Y4 ) @ ( F2 @ X3 ) ) ) % 8.50/4.54 => ( L % 8.50/4.54 = ( zero_zero @ real ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5364_DERIV__local__min,axiom, % 8.50/4.54 ! [F2: real > real,L: real,X3: real,D2: real] : % 8.50/4.54 ( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 ) % 8.50/4.54 => ( ! [Y4: real] : % 8.50/4.54 ( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y4 ) ) @ D2 ) % 8.50/4.54 => ( ord_less_eq @ real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) ) % 8.50/4.54 => ( L % 8.50/4.54 = ( zero_zero @ real ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5365_DERIV__ln__divide,axiom, % 8.50/4.54 ! [X3: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.54 => ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5366_DERIV__pow,axiom, % 8.50/4.54 ! [N2: nat,X3: real,S: set @ real] : % 8.50/4.54 ( has_field_derivative @ real % 8.50/4.54 @ ^ [X: real] : ( power_power @ real @ X @ N2 ) % 8.50/4.54 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ X3 @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ real @ X3 @ S ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5367_at__within__Icc__at,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( topolo2564578578187576103pology @ A ) % 8.50/4.54 => ! [A2: A,X3: A,B2: A] : % 8.50/4.54 ( ( ord_less @ A @ A2 @ X3 ) % 8.50/4.54 => ( ( ord_less @ A @ X3 @ B2 ) % 8.50/4.54 => ( ( topolo174197925503356063within @ A @ X3 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5368_at__within__Icc__at__left,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( topolo2564578578187576103pology @ A ) % 8.50/4.54 => ! [A2: A,B2: A] : % 8.50/4.54 ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.54 => ( ( topolo174197925503356063within @ A @ B2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5369_trivial__limit__at__left__bot,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ( order_bot @ A ) % 8.50/4.54 & ( topolo1944317154257567458pology @ A ) ) % 8.50/4.54 => ( ( topolo174197925503356063within @ A @ ( bot_bot @ A ) @ ( set_ord_lessThan @ A @ ( bot_bot @ A ) ) ) % 8.50/4.54 = ( bot_bot @ ( filter @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5370_DERIV__quotient,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: A > A,D2: A,X3: A,S: set @ A,G: A > A,E3: A] : % 8.50/4.54 ( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( ( has_field_derivative @ A @ G @ E3 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( ( ( G @ X3 ) % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( has_field_derivative @ A % 8.50/4.54 @ ^ [Y2: A] : ( divide_divide @ A @ ( F2 @ Y2 ) @ ( G @ Y2 ) ) % 8.50/4.54 @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D2 @ ( G @ X3 ) ) @ ( times_times @ A @ E3 @ ( F2 @ X3 ) ) ) @ ( power_power @ A @ ( G @ X3 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5371_DERIV__inverse__fun,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: A > A,D2: A,X3: A,S: set @ A] : % 8.50/4.54 ( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( ( ( F2 @ X3 ) % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( has_field_derivative @ A % 8.50/4.54 @ ^ [X: A] : ( inverse_inverse @ A @ ( F2 @ X ) ) % 8.50/4.54 @ ( uminus_uminus @ A @ ( times_times @ A @ D2 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F2 @ X3 ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5372_termdiffs__sums__strong,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ( real_Vector_banach @ A ) % 8.50/4.54 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.54 => ! [K5: real,C2: nat > A,F2: A > A,F11: A,Z2: A] : % 8.50/4.54 ( ! [Z3: A] : % 8.50/4.54 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 ) % 8.50/4.54 => ( sums @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ Z3 @ N ) ) % 8.50/4.54 @ ( F2 @ Z3 ) ) ) % 8.50/4.54 => ( ( has_field_derivative @ A @ F2 @ F11 @ ( topolo174197925503356063within @ A @ Z2 @ ( top_top @ ( set @ A ) ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K5 ) % 8.50/4.54 => ( sums @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) % 8.50/4.54 @ F11 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5373_has__real__derivative__powr,axiom, % 8.50/4.54 ! [Z2: real,R2: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ Z2 ) % 8.50/4.54 => ( has_field_derivative @ real % 8.50/4.54 @ ^ [Z6: real] : ( powr @ real @ Z6 @ R2 ) % 8.50/4.54 @ ( times_times @ real @ R2 @ ( powr @ real @ Z2 @ ( minus_minus @ real @ R2 @ ( one_one @ real ) ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5374_termdiffs__strong_H,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ( real_Vector_banach @ A ) % 8.50/4.54 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.54 => ! [K5: real,C2: nat > A,Z2: A] : % 8.50/4.54 ( ! [Z3: A] : % 8.50/4.54 ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 ) % 8.50/4.54 => ( summable @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ Z3 @ N ) ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K5 ) % 8.50/4.54 => ( has_field_derivative @ A % 8.50/4.54 @ ^ [Z6: A] : % 8.50/4.54 ( suminf @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ Z6 @ N ) ) ) % 8.50/4.54 @ ( suminf @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ Z2 @ N ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ Z2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5375_termdiffs__strong,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ( real_Vector_banach @ A ) % 8.50/4.54 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.54 => ! [C2: nat > A,K5: A,X3: A] : % 8.50/4.54 ( ( summable @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ K5 @ N ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) ) % 8.50/4.54 => ( has_field_derivative @ A % 8.50/4.54 @ ^ [X: A] : % 8.50/4.54 ( suminf @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ X @ N ) ) ) % 8.50/4.54 @ ( suminf @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5376_termdiffs,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ( real_Vector_banach @ A ) % 8.50/4.54 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.54 => ! [C2: nat > A,K5: A,X3: A] : % 8.50/4.54 ( ( summable @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ K5 @ N ) ) ) % 8.50/4.54 => ( ( summable @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ K5 @ N ) ) ) % 8.50/4.54 => ( ( summable @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N ) @ ( power_power @ A @ K5 @ N ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) ) % 8.50/4.54 => ( has_field_derivative @ A % 8.50/4.54 @ ^ [X: A] : % 8.50/4.54 ( suminf @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ X @ N ) ) ) % 8.50/4.54 @ ( suminf @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N ) @ ( power_power @ A @ X3 @ N ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5377_DERIV__log,axiom, % 8.50/4.54 ! [X3: real,B2: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.54 => ( has_field_derivative @ real @ ( log @ B2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( times_times @ real @ ( ln_ln @ real @ B2 ) @ X3 ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5378_DERIV__fun__powr,axiom, % 8.50/4.54 ! [G: real > real,M: real,X3: real,R2: real] : % 8.50/4.54 ( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) % 8.50/4.54 => ( has_field_derivative @ real % 8.50/4.54 @ ^ [X: real] : ( powr @ real @ ( G @ X ) @ R2 ) % 8.50/4.54 @ ( times_times @ real @ ( times_times @ real @ R2 @ ( powr @ real @ ( G @ X3 ) @ ( minus_minus @ real @ R2 @ ( semiring_1_of_nat @ real @ ( one_one @ nat ) ) ) ) ) @ M ) % 8.50/4.54 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5379_DERIV__powr,axiom, % 8.50/4.54 ! [G: real > real,M: real,X3: real,F2: real > real,R2: real] : % 8.50/4.54 ( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) % 8.50/4.54 => ( ( has_field_derivative @ real @ F2 @ R2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) % 8.50/4.54 => ( has_field_derivative @ real % 8.50/4.54 @ ^ [X: real] : ( powr @ real @ ( G @ X ) @ ( F2 @ X ) ) % 8.50/4.54 @ ( times_times @ real @ ( powr @ real @ ( G @ X3 ) @ ( F2 @ X3 ) ) @ ( plus_plus @ real @ ( times_times @ real @ R2 @ ( ln_ln @ real @ ( G @ X3 ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ M @ ( F2 @ X3 ) ) @ ( G @ X3 ) ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5380_DERIV__tan,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ( real_Vector_banach @ A ) % 8.50/4.54 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.54 => ! [X3: A] : % 8.50/4.54 ( ( ( cos @ A @ X3 ) % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( has_field_derivative @ A @ ( tan @ A ) @ ( inverse_inverse @ A @ ( power_power @ A @ ( cos @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5381_arcosh__real__has__field__derivative,axiom, % 8.50/4.54 ! [X3: real,A3: set @ real] : % 8.50/4.54 ( ( ord_less @ real @ ( one_one @ real ) @ X3 ) % 8.50/4.54 => ( has_field_derivative @ real @ ( arcosh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ A3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5382_DERIV__real__sqrt,axiom, % 8.50/4.54 ! [X3: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.54 => ( has_field_derivative @ real @ sqrt @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5383_DERIV__cot,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ( real_Vector_banach @ A ) % 8.50/4.54 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.54 => ! [X3: A] : % 8.50/4.54 ( ( ( sin @ A @ X3 ) % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( has_field_derivative @ A @ ( cot @ A ) @ ( uminus_uminus @ A @ ( inverse_inverse @ A @ ( power_power @ A @ ( sin @ A @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5384_Option_Othese__def,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( these @ A ) % 8.50/4.54 = ( ^ [A6: set @ ( option @ A )] : % 8.50/4.54 ( image @ ( option @ A ) @ A @ ( the2 @ A ) % 8.50/4.54 @ ( collect @ ( option @ A ) % 8.50/4.54 @ ^ [X: option @ A] : % 8.50/4.54 ( ( member @ ( option @ A ) @ X @ A6 ) % 8.50/4.54 & ( X % 8.50/4.54 != ( none @ A ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5385_has__field__derivative__tanh,axiom, % 8.50/4.54 ! [A10: $tType] : % 8.50/4.54 ( ( ( real_Vector_banach @ A10 ) % 8.50/4.54 & ( real_V3459762299906320749_field @ A10 ) ) % 8.50/4.54 => ! [G: A10 > A10,X3: A10,Db: A10,S: set @ A10] : % 8.50/4.54 ( ( ( cosh @ A10 @ ( G @ X3 ) ) % 8.50/4.54 != ( zero_zero @ A10 ) ) % 8.50/4.54 => ( ( has_field_derivative @ A10 @ G @ Db @ ( topolo174197925503356063within @ A10 @ X3 @ S ) ) % 8.50/4.54 => ( has_field_derivative @ A10 % 8.50/4.54 @ ^ [X: A10] : ( tanh @ A10 @ ( G @ X ) ) % 8.50/4.54 @ ( times_times @ A10 @ ( minus_minus @ A10 @ ( one_one @ A10 ) @ ( power_power @ A10 @ ( tanh @ A10 @ ( G @ X3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db ) % 8.50/4.54 @ ( topolo174197925503356063within @ A10 @ X3 @ S ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5386_DERIV__real__sqrt__generic,axiom, % 8.50/4.54 ! [X3: real,D6: real] : % 8.50/4.54 ( ( X3 % 8.50/4.54 != ( zero_zero @ real ) ) % 8.50/4.54 => ( ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.54 => ( D6 % 8.50/4.54 = ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.54 => ( ( ( ord_less @ real @ X3 @ ( zero_zero @ real ) ) % 8.50/4.54 => ( D6 % 8.50/4.54 = ( divide_divide @ real @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( sqrt @ X3 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) % 8.50/4.54 => ( has_field_derivative @ real @ sqrt @ D6 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5387_artanh__real__has__field__derivative,axiom, % 8.50/4.54 ! [X3: real,A3: set @ real] : % 8.50/4.54 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 8.50/4.54 => ( has_field_derivative @ real @ ( artanh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ A3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5388_DERIV__real__root,axiom, % 8.50/4.54 ! [N2: nat,X3: real] : % 8.50/4.54 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.54 => ( ( ord_less @ real @ ( zero_zero @ real ) @ X3 ) % 8.50/4.54 => ( has_field_derivative @ real @ ( root @ N2 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5389_DERIV__arcsin,axiom, % 8.50/4.54 ! [X3: real] : % 8.50/4.54 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 8.50/4.54 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 8.50/4.54 => ( has_field_derivative @ real @ arcsin @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5390_these__not__empty__eq,axiom, % 8.50/4.54 ! [A: $tType,B3: set @ ( option @ A )] : % 8.50/4.54 ( ( ( these @ A @ B3 ) % 8.50/4.54 != ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 = ( ( B3 % 8.50/4.54 != ( bot_bot @ ( set @ ( option @ A ) ) ) ) % 8.50/4.54 & ( B3 % 8.50/4.54 != ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5391_these__empty__eq,axiom, % 8.50/4.54 ! [A: $tType,B3: set @ ( option @ A )] : % 8.50/4.54 ( ( ( these @ A @ B3 ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 = ( ( B3 % 8.50/4.54 = ( bot_bot @ ( set @ ( option @ A ) ) ) ) % 8.50/4.54 | ( B3 % 8.50/4.54 = ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5392_Maclaurin__all__le__objl,axiom, % 8.50/4.54 ! [Diff: nat > real > real,F2: real > real,X3: real,N2: nat] : % 8.50/4.54 ( ( ( ( Diff @ ( zero_zero @ nat ) ) % 8.50/4.54 = F2 ) % 8.50/4.54 & ! [M3: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) % 8.50/4.54 => ? [T7: real] : % 8.50/4.54 ( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) ) % 8.50/4.54 & ( ( F2 @ X3 ) % 8.50/4.54 = ( plus_plus @ real % 8.50/4.54 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.54 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X3 @ M2 ) ) % 8.50/4.54 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.54 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5393_Maclaurin__all__le,axiom, % 8.50/4.54 ! [Diff: nat > real > real,F2: real > real,X3: real,N2: nat] : % 8.50/4.54 ( ( ( Diff @ ( zero_zero @ nat ) ) % 8.50/4.54 = F2 ) % 8.50/4.54 => ( ! [M3: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) % 8.50/4.54 => ? [T7: real] : % 8.50/4.54 ( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) ) % 8.50/4.54 & ( ( F2 @ X3 ) % 8.50/4.54 = ( plus_plus @ real % 8.50/4.54 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.54 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X3 @ M2 ) ) % 8.50/4.54 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.54 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5394_Some__image__these__eq,axiom, % 8.50/4.54 ! [A: $tType,A3: set @ ( option @ A )] : % 8.50/4.54 ( ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A3 ) ) % 8.50/4.54 = ( collect @ ( option @ A ) % 8.50/4.54 @ ^ [X: option @ A] : % 8.50/4.54 ( ( member @ ( option @ A ) @ X @ A3 ) % 8.50/4.54 & ( X % 8.50/4.54 != ( none @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5395_DERIV__odd__real__root,axiom, % 8.50/4.54 ! [N2: nat,X3: real] : % 8.50/4.54 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) % 8.50/4.54 => ( ( X3 % 8.50/4.54 != ( zero_zero @ real ) ) % 8.50/4.54 => ( has_field_derivative @ real @ ( root @ N2 ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( power_power @ real @ ( root @ N2 @ X3 ) @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5396_Maclaurin__minus,axiom, % 8.50/4.54 ! [H2: real,N2: nat,Diff: nat > real > real,F2: real > real] : % 8.50/4.54 ( ( ord_less @ real @ H2 @ ( zero_zero @ real ) ) % 8.50/4.54 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.54 => ( ( ( Diff @ ( zero_zero @ nat ) ) % 8.50/4.54 = F2 ) % 8.50/4.54 => ( ! [M3: nat,T7: real] : % 8.50/4.54 ( ( ( ord_less @ nat @ M3 @ N2 ) % 8.50/4.54 & ( ord_less_eq @ real @ H2 @ T7 ) % 8.50/4.54 & ( ord_less_eq @ real @ T7 @ ( zero_zero @ real ) ) ) % 8.50/4.54 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 8.50/4.54 => ? [T7: real] : % 8.50/4.54 ( ( ord_less @ real @ H2 @ T7 ) % 8.50/4.54 & ( ord_less @ real @ T7 @ ( zero_zero @ real ) ) % 8.50/4.54 & ( ( F2 @ H2 ) % 8.50/4.54 = ( plus_plus @ real % 8.50/4.54 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.54 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) ) % 8.50/4.54 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.54 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5397_Maclaurin2,axiom, % 8.50/4.54 ! [H2: real,Diff: nat > real > real,F2: real > real,N2: nat] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ H2 ) % 8.50/4.54 => ( ( ( Diff @ ( zero_zero @ nat ) ) % 8.50/4.54 = F2 ) % 8.50/4.54 => ( ! [M3: nat,T7: real] : % 8.50/4.54 ( ( ( ord_less @ nat @ M3 @ N2 ) % 8.50/4.54 & ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 ) % 8.50/4.54 & ( ord_less_eq @ real @ T7 @ H2 ) ) % 8.50/4.54 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 8.50/4.54 => ? [T7: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ T7 ) % 8.50/4.54 & ( ord_less_eq @ real @ T7 @ H2 ) % 8.50/4.54 & ( ( F2 @ H2 ) % 8.50/4.54 = ( plus_plus @ real % 8.50/4.54 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.54 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) ) % 8.50/4.54 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.54 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ H2 @ N2 ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5398_Maclaurin,axiom, % 8.50/4.54 ! [H2: real,N2: nat,Diff: nat > real > real,F2: real > real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ H2 ) % 8.50/4.54 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.54 => ( ( ( Diff @ ( zero_zero @ nat ) ) % 8.50/4.54 = F2 ) % 8.50/4.54 => ( ! [M3: nat,T7: real] : % 8.50/4.54 ( ( ( ord_less @ nat @ M3 @ N2 ) % 8.50/4.54 & ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 ) % 8.50/4.54 & ( ord_less_eq @ real @ T7 @ H2 ) ) % 8.50/4.54 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 8.50/4.54 => ? [T7: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ T7 ) % 8.50/4.54 & ( ord_less @ real @ T7 @ H2 ) % 8.50/4.54 & ( ( F2 @ H2 ) % 8.50/4.54 = ( plus_plus @ real % 8.50/4.54 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.54 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) ) % 8.50/4.54 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.54 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5399_Maclaurin__all__lt,axiom, % 8.50/4.54 ! [Diff: nat > real > real,F2: real > real,N2: nat,X3: real] : % 8.50/4.54 ( ( ( Diff @ ( zero_zero @ nat ) ) % 8.50/4.54 = F2 ) % 8.50/4.54 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.54 => ( ( X3 % 8.50/4.54 != ( zero_zero @ real ) ) % 8.50/4.54 => ( ! [M3: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) % 8.50/4.54 => ? [T7: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T7 ) ) % 8.50/4.54 & ( ord_less @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) ) % 8.50/4.54 & ( ( F2 @ X3 ) % 8.50/4.54 = ( plus_plus @ real % 8.50/4.54 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.54 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X3 @ M2 ) ) % 8.50/4.54 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.54 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5400_Maclaurin__bi__le,axiom, % 8.50/4.54 ! [Diff: nat > real > real,F2: real > real,N2: nat,X3: real] : % 8.50/4.54 ( ( ( Diff @ ( zero_zero @ nat ) ) % 8.50/4.54 = F2 ) % 8.50/4.54 => ( ! [M3: nat,T7: real] : % 8.50/4.54 ( ( ( ord_less @ nat @ M3 @ N2 ) % 8.50/4.54 & ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) ) ) % 8.50/4.54 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 8.50/4.54 => ? [T7: real] : % 8.50/4.54 ( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X3 ) ) % 8.50/4.54 & ( ( F2 @ X3 ) % 8.50/4.54 = ( plus_plus @ real % 8.50/4.54 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.54 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X3 @ M2 ) ) % 8.50/4.54 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.54 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ X3 @ N2 ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5401_Taylor,axiom, % 8.50/4.54 ! [N2: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real,X3: real] : % 8.50/4.54 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.54 => ( ( ( Diff @ ( zero_zero @ nat ) ) % 8.50/4.54 = F2 ) % 8.50/4.54 => ( ! [M3: nat,T7: real] : % 8.50/4.54 ( ( ( ord_less @ nat @ M3 @ N2 ) % 8.50/4.54 & ( ord_less_eq @ real @ A2 @ T7 ) % 8.50/4.54 & ( ord_less_eq @ real @ T7 @ B2 ) ) % 8.50/4.54 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 8.50/4.54 => ( ( ord_less_eq @ real @ A2 @ C2 ) % 8.50/4.54 => ( ( ord_less_eq @ real @ C2 @ B2 ) % 8.50/4.54 => ( ( ord_less_eq @ real @ A2 @ X3 ) % 8.50/4.54 => ( ( ord_less_eq @ real @ X3 @ B2 ) % 8.50/4.54 => ( ( X3 != C2 ) % 8.50/4.54 => ? [T7: real] : % 8.50/4.54 ( ( ( ord_less @ real @ X3 @ C2 ) % 8.50/4.54 => ( ( ord_less @ real @ X3 @ T7 ) % 8.50/4.54 & ( ord_less @ real @ T7 @ C2 ) ) ) % 8.50/4.54 & ( ~ ( ord_less @ real @ X3 @ C2 ) % 8.50/4.54 => ( ( ord_less @ real @ C2 @ T7 ) % 8.50/4.54 & ( ord_less @ real @ T7 @ X3 ) ) ) % 8.50/4.54 & ( ( F2 @ X3 ) % 8.50/4.54 = ( plus_plus @ real % 8.50/4.54 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.54 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ C2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X3 @ C2 ) @ M2 ) ) % 8.50/4.54 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.54 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X3 @ C2 ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5402_Taylor__up,axiom, % 8.50/4.54 ! [N2: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real] : % 8.50/4.54 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.54 => ( ( ( Diff @ ( zero_zero @ nat ) ) % 8.50/4.54 = F2 ) % 8.50/4.54 => ( ! [M3: nat,T7: real] : % 8.50/4.54 ( ( ( ord_less @ nat @ M3 @ N2 ) % 8.50/4.54 & ( ord_less_eq @ real @ A2 @ T7 ) % 8.50/4.54 & ( ord_less_eq @ real @ T7 @ B2 ) ) % 8.50/4.54 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 8.50/4.54 => ( ( ord_less_eq @ real @ A2 @ C2 ) % 8.50/4.54 => ( ( ord_less @ real @ C2 @ B2 ) % 8.50/4.54 => ? [T7: real] : % 8.50/4.54 ( ( ord_less @ real @ C2 @ T7 ) % 8.50/4.54 & ( ord_less @ real @ T7 @ B2 ) % 8.50/4.54 & ( ( F2 @ B2 ) % 8.50/4.54 = ( plus_plus @ real % 8.50/4.54 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.54 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ C2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ M2 ) ) % 8.50/4.54 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.54 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ N2 ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5403_Taylor__down,axiom, % 8.50/4.54 ! [N2: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real] : % 8.50/4.54 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.54 => ( ( ( Diff @ ( zero_zero @ nat ) ) % 8.50/4.54 = F2 ) % 8.50/4.54 => ( ! [M3: nat,T7: real] : % 8.50/4.54 ( ( ( ord_less @ nat @ M3 @ N2 ) % 8.50/4.54 & ( ord_less_eq @ real @ A2 @ T7 ) % 8.50/4.54 & ( ord_less_eq @ real @ T7 @ B2 ) ) % 8.50/4.54 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ A2 @ C2 ) % 8.50/4.54 => ( ( ord_less_eq @ real @ C2 @ B2 ) % 8.50/4.54 => ? [T7: real] : % 8.50/4.54 ( ( ord_less @ real @ A2 @ T7 ) % 8.50/4.54 & ( ord_less @ real @ T7 @ C2 ) % 8.50/4.54 & ( ( F2 @ A2 ) % 8.50/4.54 = ( plus_plus @ real % 8.50/4.54 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.54 @ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ C2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ M2 ) ) % 8.50/4.54 @ ( set_ord_lessThan @ nat @ N2 ) ) % 8.50/4.54 @ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N2 @ T7 ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ N2 ) ) ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5404_Maclaurin__lemma2,axiom, % 8.50/4.54 ! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B3: real] : % 8.50/4.54 ( ! [M3: nat,T7: real] : % 8.50/4.54 ( ( ( ord_less @ nat @ M3 @ N2 ) % 8.50/4.54 & ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 ) % 8.50/4.54 & ( ord_less_eq @ real @ T7 @ H2 ) ) % 8.50/4.54 => ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) ) % 8.50/4.54 => ( ( N2 % 8.50/4.54 = ( suc @ K ) ) % 8.50/4.54 => ! [M6: nat,T9: real] : % 8.50/4.54 ( ( ( ord_less @ nat @ M6 @ N2 ) % 8.50/4.54 & ( ord_less_eq @ real @ ( zero_zero @ real ) @ T9 ) % 8.50/4.54 & ( ord_less_eq @ real @ T9 @ H2 ) ) % 8.50/4.54 => ( has_field_derivative @ real % 8.50/4.54 @ ^ [U2: real] : % 8.50/4.54 ( minus_minus @ real @ ( Diff @ M6 @ U2 ) % 8.50/4.54 @ ( plus_plus @ real % 8.50/4.54 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.54 @ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M6 @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ U2 @ P6 ) ) % 8.50/4.54 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ M6 ) ) ) % 8.50/4.54 @ ( times_times @ real @ B3 @ ( divide_divide @ real @ ( power_power @ real @ U2 @ ( minus_minus @ nat @ N2 @ M6 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N2 @ M6 ) ) ) ) ) ) % 8.50/4.54 @ ( minus_minus @ real @ ( Diff @ ( suc @ M6 ) @ T9 ) % 8.50/4.54 @ ( plus_plus @ real % 8.50/4.54 @ ( groups7311177749621191930dd_sum @ nat @ real % 8.50/4.54 @ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M6 ) @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ T9 @ P6 ) ) % 8.50/4.54 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ M6 ) ) ) ) % 8.50/4.54 @ ( times_times @ real @ B3 @ ( divide_divide @ real @ ( power_power @ real @ T9 @ ( minus_minus @ nat @ N2 @ ( suc @ M6 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N2 @ ( suc @ M6 ) ) ) ) ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ real @ T9 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5405_DERIV__arctan__series,axiom, % 8.50/4.54 ! [X3: real] : % 8.50/4.54 ( ( ord_less @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 8.50/4.54 => ( has_field_derivative @ real % 8.50/4.54 @ ^ [X9: real] : % 8.50/4.54 ( suminf @ real % 8.50/4.54 @ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X9 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) % 8.50/4.54 @ ( suminf @ real % 8.50/4.54 @ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( power_power @ real @ X3 @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5406_DERIV__arccos,axiom, % 8.50/4.54 ! [X3: real] : % 8.50/4.54 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 8.50/4.54 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 8.50/4.54 => ( has_field_derivative @ real @ arccos @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5407_DERIV__power__series_H,axiom, % 8.50/4.54 ! [R: real,F2: nat > real,X0: real] : % 8.50/4.54 ( ! [X5: real] : % 8.50/4.54 ( ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) ) % 8.50/4.54 => ( summable @ real % 8.50/4.54 @ ^ [N: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N ) @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) @ ( power_power @ real @ X5 @ N ) ) ) ) % 8.50/4.54 => ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( zero_zero @ real ) @ R ) % 8.50/4.54 => ( has_field_derivative @ real % 8.50/4.54 @ ^ [X: real] : % 8.50/4.54 ( suminf @ real % 8.50/4.54 @ ^ [N: nat] : ( times_times @ real @ ( F2 @ N ) @ ( power_power @ real @ X @ ( suc @ N ) ) ) ) % 8.50/4.54 @ ( suminf @ real % 8.50/4.54 @ ^ [N: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N ) @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) @ ( power_power @ real @ X0 @ N ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5408_has__derivative__arcsin,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 => ! [G: A > real,X3: A,G7: A > real,S: set @ A] : % 8.50/4.54 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X3 ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( G @ X3 ) @ ( one_one @ real ) ) % 8.50/4.54 => ( ( has_derivative @ A @ real @ G @ G7 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( has_derivative @ A @ real % 8.50/4.54 @ ^ [X: A] : ( arcsin @ ( G @ X ) ) % 8.50/4.54 @ ^ [X: A] : ( times_times @ real @ ( G7 @ X ) @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5409_greaterThanLessThan__iff,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ord @ A ) % 8.50/4.54 => ! [I: A,L: A,U: A] : % 8.50/4.54 ( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U ) ) % 8.50/4.54 = ( ( ord_less @ A @ L @ I ) % 8.50/4.54 & ( ord_less @ A @ I @ U ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5410_greaterThanLessThan__empty__iff2,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( dense_linorder @ A ) % 8.50/4.54 => ! [A2: A,B2: A] : % 8.50/4.54 ( ( ( bot_bot @ ( set @ A ) ) % 8.50/4.54 = ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) % 8.50/4.54 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5411_greaterThanLessThan__empty__iff,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( dense_linorder @ A ) % 8.50/4.54 => ! [A2: A,B2: A] : % 8.50/4.54 ( ( ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) % 8.50/4.54 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5412_greaterThanLessThan__empty,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order @ A ) % 8.50/4.54 => ! [L: A,K: A] : % 8.50/4.54 ( ( ord_less_eq @ A @ L @ K ) % 8.50/4.54 => ( ( set_or5935395276787703475ssThan @ A @ K @ L ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5413_infinite__Ioo__iff,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( dense_linorder @ A ) % 8.50/4.54 => ! [A2: A,B2: A] : % 8.50/4.54 ( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) % 8.50/4.54 = ( ord_less @ A @ A2 @ B2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5414_cSup__greaterThanLessThan,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ( condit6923001295902523014norder @ A ) % 8.50/4.54 & ( dense_linorder @ A ) ) % 8.50/4.54 => ! [Y: A,X3: A] : % 8.50/4.54 ( ( ord_less @ A @ Y @ X3 ) % 8.50/4.54 => ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X3 ) ) % 8.50/4.54 = X3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5415_Sup__greaterThanLessThan,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 & ( dense_linorder @ A ) ) % 8.50/4.54 => ! [X3: A,Y: A] : % 8.50/4.54 ( ( ord_less @ A @ X3 @ Y ) % 8.50/4.54 => ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X3 @ Y ) ) % 8.50/4.54 = Y ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5416_cInf__greaterThanLessThan,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ( condit6923001295902523014norder @ A ) % 8.50/4.54 & ( dense_linorder @ A ) ) % 8.50/4.54 => ! [Y: A,X3: A] : % 8.50/4.54 ( ( ord_less @ A @ Y @ X3 ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X3 ) ) % 8.50/4.54 = Y ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5417_Inf__greaterThanLessThan,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ( comple6319245703460814977attice @ A ) % 8.50/4.54 & ( dense_linorder @ A ) ) % 8.50/4.54 => ! [X3: A,Y: A] : % 8.50/4.54 ( ( ord_less @ A @ X3 @ Y ) % 8.50/4.54 => ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X3 @ Y ) ) % 8.50/4.54 = X3 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5418_has__derivative__subset,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 & ( real_V822414075346904944vector @ B ) ) % 8.50/4.54 => ! [F2: A > B,F11: A > B,X3: A,S: set @ A,T2: set @ A] : % 8.50/4.54 ( ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( ( ord_less_eq @ ( set @ A ) @ T2 @ S ) % 8.50/4.54 => ( has_derivative @ A @ B @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ T2 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5419_has__derivative__scaleR,axiom, % 8.50/4.54 ! [C: $tType,D: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ D ) % 8.50/4.54 & ( real_V822414075346904944vector @ C ) ) % 8.50/4.54 => ! [F2: D > real,F11: D > real,X3: D,S: set @ D,G: D > C,G7: D > C] : % 8.50/4.54 ( ( has_derivative @ D @ real @ F2 @ F11 @ ( topolo174197925503356063within @ D @ X3 @ S ) ) % 8.50/4.54 => ( ( has_derivative @ D @ C @ G @ G7 @ ( topolo174197925503356063within @ D @ X3 @ S ) ) % 8.50/4.54 => ( has_derivative @ D @ C % 8.50/4.54 @ ^ [X: D] : ( real_V8093663219630862766scaleR @ C @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ^ [H: D] : ( plus_plus @ C @ ( real_V8093663219630862766scaleR @ C @ ( F2 @ X3 ) @ ( G7 @ H ) ) @ ( real_V8093663219630862766scaleR @ C @ ( F11 @ H ) @ ( G @ X3 ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ D @ X3 @ S ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5420_has__derivative__add,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 & ( real_V822414075346904944vector @ B ) ) % 8.50/4.54 => ! [F2: A > B,F11: A > B,F5: filter @ A,G: A > B,G7: A > B] : % 8.50/4.54 ( ( has_derivative @ A @ B @ F2 @ F11 @ F5 ) % 8.50/4.54 => ( ( has_derivative @ A @ B @ G @ G7 @ F5 ) % 8.50/4.54 => ( has_derivative @ A @ B % 8.50/4.54 @ ^ [X: A] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ^ [X: A] : ( plus_plus @ B @ ( F11 @ X ) @ ( G7 @ X ) ) % 8.50/4.54 @ F5 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5421_has__derivative__const,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 & ( real_V822414075346904944vector @ B ) ) % 8.50/4.54 => ! [C2: B,F5: filter @ A] : % 8.50/4.54 ( has_derivative @ A @ B % 8.50/4.54 @ ^ [X: A] : C2 % 8.50/4.54 @ ^ [X: A] : ( zero_zero @ B ) % 8.50/4.54 @ F5 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5422_infinite__Ioo,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( dense_linorder @ A ) % 8.50/4.54 => ! [A2: A,B2: A] : % 8.50/4.54 ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.54 => ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5423_has__derivative__zero__unique,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 & ( real_V822414075346904944vector @ B ) ) % 8.50/4.54 => ! [F5: A > B,X3: A] : % 8.50/4.54 ( ( has_derivative @ A @ B % 8.50/4.54 @ ^ [X: A] : ( zero_zero @ B ) % 8.50/4.54 @ F5 % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) % 8.50/4.54 => ( F5 % 8.50/4.54 = ( ^ [H: A] : ( zero_zero @ B ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5424_has__derivative__mult,axiom, % 8.50/4.54 ! [A: $tType,D: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ D ) % 8.50/4.54 & ( real_V4412858255891104859lgebra @ A ) ) % 8.50/4.54 => ! [F2: D > A,F11: D > A,X3: D,S: set @ D,G: D > A,G7: D > A] : % 8.50/4.54 ( ( has_derivative @ D @ A @ F2 @ F11 @ ( topolo174197925503356063within @ D @ X3 @ S ) ) % 8.50/4.54 => ( ( has_derivative @ D @ A @ G @ G7 @ ( topolo174197925503356063within @ D @ X3 @ S ) ) % 8.50/4.54 => ( has_derivative @ D @ A % 8.50/4.54 @ ^ [X: D] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ^ [H: D] : ( plus_plus @ A @ ( times_times @ A @ ( F2 @ X3 ) @ ( G7 @ H ) ) @ ( times_times @ A @ ( F11 @ H ) @ ( G @ X3 ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ D @ X3 @ S ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5425_has__derivative__in__compose2,axiom, % 8.50/4.54 ! [B: $tType,A: $tType,C: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ C ) % 8.50/4.54 & ( real_V822414075346904944vector @ A ) % 8.50/4.54 & ( real_V822414075346904944vector @ B ) ) % 8.50/4.54 => ! [T2: set @ A,G: A > B,G7: A > A > B,F2: C > A,S: set @ C,X3: C,F11: C > A] : % 8.50/4.54 ( ! [X5: A] : % 8.50/4.54 ( ( member @ A @ X5 @ T2 ) % 8.50/4.54 => ( has_derivative @ A @ B @ G @ ( G7 @ X5 ) @ ( topolo174197925503356063within @ A @ X5 @ T2 ) ) ) % 8.50/4.54 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F2 @ S ) @ T2 ) % 8.50/4.54 => ( ( member @ C @ X3 @ S ) % 8.50/4.54 => ( ( has_derivative @ C @ A @ F2 @ F11 @ ( topolo174197925503356063within @ C @ X3 @ S ) ) % 8.50/4.54 => ( has_derivative @ C @ B % 8.50/4.54 @ ^ [X: C] : ( G @ ( F2 @ X ) ) % 8.50/4.54 @ ^ [Y2: C] : ( G7 @ ( F2 @ X3 ) @ ( F11 @ Y2 ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ C @ X3 @ S ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5426_greaterThanLessThan__subseteq__greaterThanLessThan,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( dense_linorder @ A ) % 8.50/4.54 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D2 ) ) % 8.50/4.54 = ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.54 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 8.50/4.54 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5427_ivl__disj__int__two_I4_J,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order @ A ) % 8.50/4.54 => ! [L: A,M: A,U: A] : % 8.50/4.54 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5428_ivl__disj__int__two_I5_J,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order @ A ) % 8.50/4.54 => ! [L: A,M: A,U: A] : % 8.50/4.54 ( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5429_ivl__disj__int__two_I1_J,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order @ A ) % 8.50/4.54 => ! [L: A,M: A,U: A] : % 8.50/4.54 ( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5430_ivl__disj__int__one_I1_J,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( order @ A ) % 8.50/4.54 => ! [L: A,U: A] : % 8.50/4.54 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) ) % 8.50/4.54 = ( bot_bot @ ( set @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5431_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( dense_linorder @ A ) % 8.50/4.54 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) ) % 8.50/4.54 = ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.54 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 8.50/4.54 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5432_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( dense_linorder @ A ) % 8.50/4.54 => ! [A2: A,B2: A,C2: A,D2: A] : % 8.50/4.54 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) ) % 8.50/4.54 = ( ( ord_less @ A @ A2 @ B2 ) % 8.50/4.54 => ( ( ord_less_eq @ A @ C2 @ A2 ) % 8.50/4.54 & ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5433_atLeastAtMost__diff__ends,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( linorder @ A ) % 8.50/4.54 => ! [A2: A,B2: A] : % 8.50/4.54 ( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) % 8.50/4.54 = ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5434_has__derivative__divide_H,axiom, % 8.50/4.54 ! [A: $tType,C: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ C ) % 8.50/4.54 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.54 => ! [F2: C > A,F11: C > A,X3: C,S3: set @ C,G: C > A,G7: C > A] : % 8.50/4.54 ( ( has_derivative @ C @ A @ F2 @ F11 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) % 8.50/4.54 => ( ( has_derivative @ C @ A @ G @ G7 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) % 8.50/4.54 => ( ( ( G @ X3 ) % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( has_derivative @ C @ A % 8.50/4.54 @ ^ [X: C] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ^ [H: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F11 @ H ) @ ( G @ X3 ) ) @ ( times_times @ A @ ( F2 @ X3 ) @ ( G7 @ H ) ) ) @ ( times_times @ A @ ( G @ X3 ) @ ( G @ X3 ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5435_arccos__less__arccos,axiom, % 8.50/4.54 ! [X3: real,Y: real] : % 8.50/4.54 ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 8.50/4.54 => ( ( ord_less @ real @ X3 @ Y ) % 8.50/4.54 => ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) ) % 8.50/4.54 => ( ord_less @ real @ ( arccos @ Y ) @ ( arccos @ X3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5436_arccos__less__mono,axiom, % 8.50/4.54 ! [X3: real,Y: real] : % 8.50/4.54 ( ( ord_less_eq @ real @ ( abs_abs @ real @ X3 ) @ ( one_one @ real ) ) % 8.50/4.54 => ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( arccos @ X3 ) @ ( arccos @ Y ) ) % 8.50/4.54 = ( ord_less @ real @ Y @ X3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5437_has__derivative__inverse,axiom, % 8.50/4.54 ! [A: $tType,C: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ C ) % 8.50/4.54 & ( real_V8999393235501362500lgebra @ A ) ) % 8.50/4.54 => ! [F2: C > A,X3: C,F11: C > A,S3: set @ C] : % 8.50/4.54 ( ( ( F2 @ X3 ) % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( ( has_derivative @ C @ A @ F2 @ F11 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) % 8.50/4.54 => ( has_derivative @ C @ A % 8.50/4.54 @ ^ [X: C] : ( inverse_inverse @ A @ ( F2 @ X ) ) % 8.50/4.54 @ ^ [H: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) @ ( F11 @ H ) ) @ ( inverse_inverse @ A @ ( F2 @ X3 ) ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5438_has__derivative__inverse_H,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.54 => ! [X3: A,S3: set @ A] : % 8.50/4.54 ( ( X3 % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( has_derivative @ A @ A @ ( inverse_inverse @ A ) % 8.50/4.54 @ ^ [H: A] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ X3 ) @ H ) @ ( inverse_inverse @ A @ X3 ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5439_DERIV__isconst3,axiom, % 8.50/4.54 ! [A2: real,B2: real,X3: real,Y: real,F2: real > real] : % 8.50/4.54 ( ( ord_less @ real @ A2 @ B2 ) % 8.50/4.54 => ( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) ) % 8.50/4.54 => ( ( member @ real @ Y @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) ) % 8.50/4.54 => ( ! [X5: real] : % 8.50/4.54 ( ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) ) % 8.50/4.54 => ( has_field_derivative @ real @ F2 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) % 8.50/4.54 => ( ( F2 @ X3 ) % 8.50/4.54 = ( F2 @ Y ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5440_arccos__lt__bounded,axiom, % 8.50/4.54 ! [Y: real] : % 8.50/4.54 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y ) % 8.50/4.54 => ( ( ord_less @ real @ Y @ ( one_one @ real ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) ) % 8.50/4.54 & ( ord_less @ real @ ( arccos @ Y ) @ pi ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5441_has__derivative__ln,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 => ! [G: A > real,X3: A,G7: A > real,S: set @ A] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) % 8.50/4.54 => ( ( has_derivative @ A @ real @ G @ G7 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( has_derivative @ A @ real % 8.50/4.54 @ ^ [X: A] : ( ln_ln @ real @ ( G @ X ) ) % 8.50/4.54 @ ^ [X: A] : ( times_times @ real @ ( G7 @ X ) @ ( inverse_inverse @ real @ ( G @ X3 ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5442_sin__arccos__nonzero,axiom, % 8.50/4.54 ! [X3: real] : % 8.50/4.54 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X3 ) % 8.50/4.54 => ( ( ord_less @ real @ X3 @ ( one_one @ real ) ) % 8.50/4.54 => ( ( sin @ real @ ( arccos @ X3 ) ) % 8.50/4.54 != ( zero_zero @ real ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5443_has__derivative__arccos,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 => ! [G: A > real,X3: A,G7: A > real,S: set @ A] : % 8.50/4.54 ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X3 ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( G @ X3 ) @ ( one_one @ real ) ) % 8.50/4.54 => ( ( has_derivative @ A @ real @ G @ G7 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( has_derivative @ A @ real % 8.50/4.54 @ ^ [X: A] : ( arccos @ ( G @ X ) ) % 8.50/4.54 @ ^ [X: A] : ( times_times @ real @ ( G7 @ X ) @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5444_has__derivative__divide,axiom, % 8.50/4.54 ! [A: $tType,C: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ C ) % 8.50/4.54 & ( real_V8999393235501362500lgebra @ A ) ) % 8.50/4.54 => ! [F2: C > A,F11: C > A,X3: C,S3: set @ C,G: C > A,G7: C > A] : % 8.50/4.54 ( ( has_derivative @ C @ A @ F2 @ F11 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) % 8.50/4.54 => ( ( has_derivative @ C @ A @ G @ G7 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) % 8.50/4.54 => ( ( ( G @ X3 ) % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( has_derivative @ C @ A % 8.50/4.54 @ ^ [X: C] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ^ [H: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F2 @ X3 ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G @ X3 ) ) @ ( G7 @ H ) ) @ ( inverse_inverse @ A @ ( G @ X3 ) ) ) ) @ ( divide_divide @ A @ ( F11 @ H ) @ ( G @ X3 ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ C @ X3 @ S3 ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5445_has__derivative__prod,axiom, % 8.50/4.54 ! [B: $tType,I7: $tType,A: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 & ( real_V3459762299906320749_field @ B ) ) % 8.50/4.54 => ! [I6: set @ I7,F2: I7 > A > B,F11: I7 > A > B,X3: A,S3: set @ A] : % 8.50/4.54 ( ! [I4: I7] : % 8.50/4.54 ( ( member @ I7 @ I4 @ I6 ) % 8.50/4.54 => ( has_derivative @ A @ B @ ( F2 @ I4 ) @ ( F11 @ I4 ) @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) % 8.50/4.54 => ( has_derivative @ A @ B % 8.50/4.54 @ ^ [X: A] : % 8.50/4.54 ( groups7121269368397514597t_prod @ I7 @ B % 8.50/4.54 @ ^ [I3: I7] : ( F2 @ I3 @ X ) % 8.50/4.54 @ I6 ) % 8.50/4.54 @ ^ [Y2: A] : % 8.50/4.54 ( groups7311177749621191930dd_sum @ I7 @ B % 8.50/4.54 @ ^ [I3: I7] : % 8.50/4.54 ( times_times @ B @ ( F11 @ I3 @ Y2 ) % 8.50/4.54 @ ( groups7121269368397514597t_prod @ I7 @ B % 8.50/4.54 @ ^ [J3: I7] : ( F2 @ J3 @ X3 ) % 8.50/4.54 @ ( minus_minus @ ( set @ I7 ) @ I6 @ ( insert @ I7 @ I3 @ ( bot_bot @ ( set @ I7 ) ) ) ) ) ) % 8.50/4.54 @ I6 ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5446_has__derivative__powr,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 => ! [G: A > real,G7: A > real,X3: A,X8: set @ A,F2: A > real,F11: A > real] : % 8.50/4.54 ( ( has_derivative @ A @ real @ G @ G7 @ ( topolo174197925503356063within @ A @ X3 @ X8 ) ) % 8.50/4.54 => ( ( has_derivative @ A @ real @ F2 @ F11 @ ( topolo174197925503356063within @ A @ X3 @ X8 ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) % 8.50/4.54 => ( ( member @ A @ X3 @ X8 ) % 8.50/4.54 => ( has_derivative @ A @ real % 8.50/4.54 @ ^ [X: A] : ( powr @ real @ ( G @ X ) @ ( F2 @ X ) ) % 8.50/4.54 @ ^ [H: A] : ( times_times @ real @ ( powr @ real @ ( G @ X3 ) @ ( F2 @ X3 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( F11 @ H ) @ ( ln_ln @ real @ ( G @ X3 ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G7 @ H ) @ ( F2 @ X3 ) ) @ ( G @ X3 ) ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ X8 ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5447_has__derivative__real__sqrt,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 => ! [G: A > real,X3: A,G7: A > real,S: set @ A] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) % 8.50/4.54 => ( ( has_derivative @ A @ real @ G @ G7 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) % 8.50/4.54 => ( has_derivative @ A @ real % 8.50/4.54 @ ^ [X: A] : ( sqrt @ ( G @ X ) ) % 8.50/4.54 @ ^ [X: A] : ( times_times @ real @ ( G7 @ X ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G @ X3 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ X3 @ S ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5448_termdiffs__aux,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ( real_Vector_banach @ A ) % 8.50/4.54 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.54 => ! [C2: nat > A,K5: A,X3: A] : % 8.50/4.54 ( ( summable @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N ) @ ( power_power @ A @ K5 @ N ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) ) % 8.50/4.54 => ( filterlim @ A @ A % 8.50/4.54 @ ^ [H: A] : % 8.50/4.54 ( suminf @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X3 @ H ) @ N ) @ ( power_power @ A @ X3 @ N ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ X3 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5449_isCont__powser,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( ( real_Vector_banach @ A ) % 8.50/4.54 & ( real_V3459762299906320749_field @ A ) ) % 8.50/4.54 => ! [C2: nat > A,K5: A,X3: A] : % 8.50/4.54 ( ( summable @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ K5 @ N ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) ) % 8.50/4.54 => ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 @ ^ [X: A] : % 8.50/4.54 ( suminf @ A % 8.50/4.54 @ ^ [N: nat] : ( times_times @ A @ ( C2 @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5450_isCont__powser_H,axiom, % 8.50/4.54 ! [Aa: $tType,A: $tType] : % 8.50/4.54 ( ( ( topological_t2_space @ A ) % 8.50/4.54 & ( real_Vector_banach @ Aa ) % 8.50/4.54 & ( real_V3459762299906320749_field @ Aa ) ) % 8.50/4.54 => ! [A2: A,F2: A > Aa,C2: nat > Aa,K5: Aa] : % 8.50/4.54 ( ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 ) % 8.50/4.54 => ( ( summable @ Aa % 8.50/4.54 @ ^ [N: nat] : ( times_times @ Aa @ ( C2 @ N ) @ ( power_power @ Aa @ K5 @ N ) ) ) % 8.50/4.54 => ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ Aa @ ( F2 @ A2 ) ) @ ( real_V7770717601297561774m_norm @ Aa @ K5 ) ) % 8.50/4.54 => ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 @ ^ [X: A] : % 8.50/4.54 ( suminf @ Aa % 8.50/4.54 @ ^ [N: nat] : ( times_times @ Aa @ ( C2 @ N ) @ ( power_power @ Aa @ ( F2 @ X ) @ N ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5451_finite__greaterThanLessThan,axiom, % 8.50/4.54 ! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5452_finite__greaterThanLessThan__int,axiom, % 8.50/4.54 ! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5453_card__greaterThanLessThan,axiom, % 8.50/4.54 ! [L: nat,U: nat] : % 8.50/4.54 ( ( finite_card @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) % 8.50/4.54 = ( minus_minus @ nat @ U @ ( suc @ L ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5454_tendsto__mult__left__iff,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( ( field @ A ) % 8.50/4.54 & ( topolo4211221413907600880p_mult @ A ) ) % 8.50/4.54 => ! [C2: A,F2: B > A,L: A,F5: filter @ B] : % 8.50/4.54 ( ( C2 % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( ( filterlim @ B @ A % 8.50/4.54 @ ^ [X: B] : ( times_times @ A @ C2 @ ( F2 @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L ) ) % 8.50/4.54 @ F5 ) % 8.50/4.54 = ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5455_tendsto__mult__right__iff,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( ( field @ A ) % 8.50/4.54 & ( topolo4211221413907600880p_mult @ A ) ) % 8.50/4.54 => ! [C2: A,F2: B > A,L: A,F5: filter @ B] : % 8.50/4.54 ( ( C2 % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( ( filterlim @ B @ A % 8.50/4.54 @ ^ [X: B] : ( times_times @ A @ ( F2 @ X ) @ C2 ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C2 ) ) % 8.50/4.54 @ F5 ) % 8.50/4.54 = ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5456_power__tendsto__0__iff,axiom, % 8.50/4.54 ! [A: $tType,N2: nat,F2: A > real,F5: filter @ A] : % 8.50/4.54 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.54 => ( ( filterlim @ A @ real % 8.50/4.54 @ ^ [X: A] : ( power_power @ real @ ( F2 @ X ) @ N2 ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 8.50/4.54 @ F5 ) % 8.50/4.54 = ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5457_isCont__Pair,axiom, % 8.50/4.54 ! [C: $tType,B: $tType,A: $tType] : % 8.50/4.54 ( ( ( topological_t2_space @ A ) % 8.50/4.54 & ( topolo4958980785337419405_space @ B ) % 8.50/4.54 & ( topolo4958980785337419405_space @ C ) ) % 8.50/4.54 => ! [A2: A,F2: A > B,G: A > C] : % 8.50/4.54 ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 ) % 8.50/4.54 => ( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G ) % 8.50/4.54 => ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) % 8.50/4.54 @ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5458_isCont__LIM__compose2,axiom, % 8.50/4.54 ! [C: $tType,B: $tType,A: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 & ( topolo4958980785337419405_space @ B ) % 8.50/4.54 & ( topolo4958980785337419405_space @ C ) ) % 8.50/4.54 => ! [A2: A,F2: A > B,G: B > C,L: C] : % 8.50/4.54 ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F2 ) % 8.50/4.54 => ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ L ) @ ( topolo174197925503356063within @ B @ ( F2 @ A2 ) @ ( top_top @ ( set @ B ) ) ) ) % 8.50/4.54 => ( ? [D4: real] : % 8.50/4.54 ( ( ord_less @ real @ ( zero_zero @ real ) @ D4 ) % 8.50/4.54 & ! [X5: A] : % 8.50/4.54 ( ( ( X5 != A2 ) % 8.50/4.54 & ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X5 @ A2 ) ) @ D4 ) ) % 8.50/4.54 => ( ( F2 @ X5 ) % 8.50/4.54 != ( F2 @ A2 ) ) ) ) % 8.50/4.54 => ( filterlim @ A @ C % 8.50/4.54 @ ^ [X: A] : ( G @ ( F2 @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ C @ L ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5459_isCont__iff,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 & ( topolo4958980785337419405_space @ B ) ) % 8.50/4.54 => ! [X3: A,F2: A > B] : % 8.50/4.54 ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) @ F2 ) % 8.50/4.54 = ( filterlim @ A @ B % 8.50/4.54 @ ^ [H: A] : ( F2 @ ( plus_plus @ A @ X3 @ H ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X3 ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5460_LIM__not__zero,axiom, % 8.50/4.54 ! [Aa: $tType,A: $tType] : % 8.50/4.54 ( ( ( topolo8386298272705272623_space @ A ) % 8.50/4.54 & ( zero @ Aa ) % 8.50/4.54 & ( topological_t2_space @ Aa ) ) % 8.50/4.54 => ! [K: Aa,A2: A] : % 8.50/4.54 ( ( K % 8.50/4.54 != ( zero_zero @ Aa ) ) % 8.50/4.54 => ~ ( filterlim @ A @ Aa % 8.50/4.54 @ ^ [X: A] : K % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5461_tendsto__within__subset,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( topolo4958980785337419405_space @ A ) % 8.50/4.54 => ! [F2: A > B,L: filter @ B,X3: A,S3: set @ A,T4: set @ A] : % 8.50/4.54 ( ( filterlim @ A @ B @ F2 @ L @ ( topolo174197925503356063within @ A @ X3 @ S3 ) ) % 8.50/4.54 => ( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 ) % 8.50/4.54 => ( filterlim @ A @ B @ F2 @ L @ ( topolo174197925503356063within @ A @ X3 @ T4 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5462_tendsto__arcosh,axiom, % 8.50/4.54 ! [B: $tType,F2: B > real,A2: real,F5: filter @ B] : % 8.50/4.54 ( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 ) % 8.50/4.54 => ( ( ord_less @ real @ ( one_one @ real ) @ A2 ) % 8.50/4.54 => ( filterlim @ B @ real % 8.50/4.54 @ ^ [X: B] : ( arcosh @ real @ ( F2 @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) ) % 8.50/4.54 @ F5 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5463_tendsto__const__iff,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( topological_t2_space @ A ) % 8.50/4.54 => ! [F5: filter @ B,A2: A,B2: A] : % 8.50/4.54 ( ( F5 % 8.50/4.54 != ( bot_bot @ ( filter @ B ) ) ) % 8.50/4.54 => ( ( filterlim @ B @ A % 8.50/4.54 @ ^ [X: B] : A2 % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ A @ B2 ) % 8.50/4.54 @ F5 ) % 8.50/4.54 = ( A2 = B2 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5464_tendsto__unique,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( topological_t2_space @ A ) % 8.50/4.54 => ! [F5: filter @ B,F2: B > A,A2: A,B2: A] : % 8.50/4.54 ( ( F5 % 8.50/4.54 != ( bot_bot @ ( filter @ B ) ) ) % 8.50/4.54 => ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 8.50/4.54 => ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 ) % 8.50/4.54 => ( A2 = B2 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5465_tendsto__bot,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( topolo4958980785337419405_space @ A ) % 8.50/4.54 => ! [F2: B > A,A2: A] : ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( bot_bot @ ( filter @ B ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5466_nhds__neq__bot,axiom, % 8.50/4.54 ! [A: $tType] : % 8.50/4.54 ( ( topolo4958980785337419405_space @ A ) % 8.50/4.54 => ! [A2: A] : % 8.50/4.54 ( ( topolo7230453075368039082e_nhds @ A @ A2 ) % 8.50/4.54 != ( bot_bot @ ( filter @ A ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5467_continuous__bot,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( ( topological_t2_space @ A ) % 8.50/4.54 & ( topolo4958980785337419405_space @ B ) ) % 8.50/4.54 => ! [F2: A > B] : ( topolo3448309680560233919inuous @ A @ B @ ( bot_bot @ ( filter @ A ) ) @ F2 ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5468_continuous__trivial__limit,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( ( topological_t2_space @ A ) % 8.50/4.54 & ( topolo4958980785337419405_space @ B ) ) % 8.50/4.54 => ! [Net: filter @ A,F2: A > B] : % 8.50/4.54 ( ( Net % 8.50/4.54 = ( bot_bot @ ( filter @ A ) ) ) % 8.50/4.54 => ( topolo3448309680560233919inuous @ A @ B @ Net @ F2 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5469_LIM__offset__zero__cancel,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 & ( topolo4958980785337419405_space @ B ) ) % 8.50/4.54 => ! [F2: A > B,A2: A,L5: B] : % 8.50/4.54 ( ( filterlim @ A @ B % 8.50/4.54 @ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) % 8.50/4.54 => ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5470_LIM__offset__zero,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 & ( topolo4958980785337419405_space @ B ) ) % 8.50/4.54 => ! [F2: A > B,L5: B,A2: A] : % 8.50/4.54 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 8.50/4.54 => ( filterlim @ A @ B % 8.50/4.54 @ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5471_LIM__isCont__iff,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 & ( topolo4958980785337419405_space @ B ) ) % 8.50/4.54 => ! [F2: A > B,A2: A] : % 8.50/4.54 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 8.50/4.54 = ( filterlim @ A @ B % 8.50/4.54 @ ^ [H: A] : ( F2 @ ( plus_plus @ A @ A2 @ H ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A2 ) ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5472_tendsto__null__power,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( real_V2822296259951069270ebra_1 @ B ) % 8.50/4.54 => ! [F2: A > B,F5: filter @ A,N2: nat] : % 8.50/4.54 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) % 8.50/4.54 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) % 8.50/4.54 => ( filterlim @ A @ B % 8.50/4.54 @ ^ [X: A] : ( power_power @ B @ ( F2 @ X ) @ N2 ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 8.50/4.54 @ F5 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5473_LIM__offset,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 & ( topolo4958980785337419405_space @ B ) ) % 8.50/4.54 => ! [F2: A > B,L5: B,A2: A,K: A] : % 8.50/4.54 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) % 8.50/4.54 => ( filterlim @ A @ B % 8.50/4.54 @ ^ [X: A] : ( F2 @ ( plus_plus @ A @ X @ K ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) % 8.50/4.54 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ K ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5474_tendsto__Pair,axiom, % 8.50/4.54 ! [C: $tType,B: $tType,A: $tType] : % 8.50/4.54 ( ( ( topolo4958980785337419405_space @ B ) % 8.50/4.54 & ( topolo4958980785337419405_space @ C ) ) % 8.50/4.54 => ! [F2: A > B,A2: B,F5: filter @ A,G: A > C,B2: C] : % 8.50/4.54 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F5 ) % 8.50/4.54 => ( ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ B2 ) @ F5 ) % 8.50/4.54 => ( filterlim @ A @ ( product_prod @ B @ C ) % 8.50/4.54 @ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ B2 ) ) % 8.50/4.54 @ F5 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5475_continuous__Pair,axiom, % 8.50/4.54 ! [C: $tType,B: $tType,A: $tType] : % 8.50/4.54 ( ( ( topological_t2_space @ A ) % 8.50/4.54 & ( topolo4958980785337419405_space @ B ) % 8.50/4.54 & ( topolo4958980785337419405_space @ C ) ) % 8.50/4.54 => ! [F5: filter @ A,F2: A > B,G: A > C] : % 8.50/4.54 ( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 ) % 8.50/4.54 => ( ( topolo3448309680560233919inuous @ A @ C @ F5 @ G ) % 8.50/4.54 => ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F5 % 8.50/4.54 @ ^ [X: A] : ( product_Pair @ B @ C @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5476_Lim__transform__eq,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 => ! [F2: B > A,G: B > A,F5: filter @ B,A2: A] : % 8.50/4.54 ( ( filterlim @ B @ A % 8.50/4.54 @ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 8.50/4.54 @ F5 ) % 8.50/4.54 => ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 8.50/4.54 = ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5477_LIM__zero__cancel,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( real_V822414075346904944vector @ B ) % 8.50/4.54 => ! [F2: A > B,L: B,F5: filter @ A] : % 8.50/4.54 ( ( filterlim @ A @ B % 8.50/4.54 @ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ L ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 8.50/4.54 @ F5 ) % 8.50/4.54 => ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5478_Lim__transform2,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 => ! [F2: B > A,A2: A,F5: filter @ B,G: B > A] : % 8.50/4.54 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 8.50/4.54 => ( ( filterlim @ B @ A % 8.50/4.54 @ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 8.50/4.54 @ F5 ) % 8.50/4.54 => ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5479_Lim__transform,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 => ! [G: B > A,A2: A,F5: filter @ B,F2: B > A] : % 8.50/4.54 ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 8.50/4.54 => ( ( filterlim @ B @ A % 8.50/4.54 @ ^ [X: B] : ( minus_minus @ A @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 8.50/4.54 @ F5 ) % 8.50/4.54 => ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5480_LIM__zero__iff,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( real_V822414075346904944vector @ B ) % 8.50/4.54 => ! [F2: A > B,L: B,F5: filter @ A] : % 8.50/4.54 ( ( filterlim @ A @ B % 8.50/4.54 @ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ L ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 8.50/4.54 @ F5 ) % 8.50/4.54 = ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5481_LIM__zero,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( real_V822414075346904944vector @ B ) % 8.50/4.54 => ! [F2: A > B,L: B,F5: filter @ A] : % 8.50/4.54 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) % 8.50/4.54 => ( filterlim @ A @ B % 8.50/4.54 @ ^ [X: A] : ( minus_minus @ B @ ( F2 @ X ) @ L ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 8.50/4.54 @ F5 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5482_tendsto__mult__zero,axiom, % 8.50/4.54 ! [A: $tType,D: $tType] : % 8.50/4.54 ( ( real_V4412858255891104859lgebra @ A ) % 8.50/4.54 => ! [F2: D > A,F5: filter @ D,G: D > A] : % 8.50/4.54 ( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 ) % 8.50/4.54 => ( ( filterlim @ D @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 ) % 8.50/4.54 => ( filterlim @ D @ A % 8.50/4.54 @ ^ [X: D] : ( times_times @ A @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 8.50/4.54 @ F5 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5483_tendsto__mult__left__zero,axiom, % 8.50/4.54 ! [A: $tType,D: $tType] : % 8.50/4.54 ( ( real_V4412858255891104859lgebra @ A ) % 8.50/4.54 => ! [F2: D > A,F5: filter @ D,C2: A] : % 8.50/4.54 ( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 ) % 8.50/4.54 => ( filterlim @ D @ A % 8.50/4.54 @ ^ [X: D] : ( times_times @ A @ ( F2 @ X ) @ C2 ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 8.50/4.54 @ F5 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5484_tendsto__mult__right__zero,axiom, % 8.50/4.54 ! [A: $tType,D: $tType] : % 8.50/4.54 ( ( real_V4412858255891104859lgebra @ A ) % 8.50/4.54 => ! [F2: D > A,F5: filter @ D,C2: A] : % 8.50/4.54 ( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 ) % 8.50/4.54 => ( filterlim @ D @ A % 8.50/4.54 @ ^ [X: D] : ( times_times @ A @ C2 @ ( F2 @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 8.50/4.54 @ F5 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5485_tendsto__add__zero,axiom, % 8.50/4.54 ! [B: $tType,D: $tType] : % 8.50/4.54 ( ( topolo6943815403480290642id_add @ B ) % 8.50/4.54 => ! [F2: D > B,F5: filter @ D,G: D > B] : % 8.50/4.54 ( ( filterlim @ D @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) % 8.50/4.54 => ( ( filterlim @ D @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) % 8.50/4.54 => ( filterlim @ D @ B % 8.50/4.54 @ ^ [X: D] : ( plus_plus @ B @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) % 8.50/4.54 @ F5 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5486_tendsto__inverse,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( real_V8999393235501362500lgebra @ A ) % 8.50/4.54 => ! [F2: B > A,A2: A,F5: filter @ B] : % 8.50/4.54 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 8.50/4.54 => ( ( A2 % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( filterlim @ B @ A % 8.50/4.54 @ ^ [X: B] : ( inverse_inverse @ A @ ( F2 @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A2 ) ) % 8.50/4.54 @ F5 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5487_tendsto__norm__zero,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( real_V822414075346904944vector @ B ) % 8.50/4.54 => ! [F2: A > B,F5: filter @ A] : % 8.50/4.54 ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) % 8.50/4.54 => ( filterlim @ A @ real % 8.50/4.54 @ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 8.50/4.54 @ F5 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5488_tendsto__norm__zero__iff,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( real_V822414075346904944vector @ B ) % 8.50/4.54 => ! [F2: A > B,F5: filter @ A] : % 8.50/4.54 ( ( filterlim @ A @ real % 8.50/4.54 @ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 8.50/4.54 @ F5 ) % 8.50/4.54 = ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5489_tendsto__norm__zero__cancel,axiom, % 8.50/4.54 ! [B: $tType,A: $tType] : % 8.50/4.54 ( ( real_V822414075346904944vector @ B ) % 8.50/4.54 => ! [F2: A > B,F5: filter @ A] : % 8.50/4.54 ( ( filterlim @ A @ real % 8.50/4.54 @ ^ [X: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) % 8.50/4.54 @ F5 ) % 8.50/4.54 => ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5490_tendsto__sgn,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( real_V822414075346904944vector @ A ) % 8.50/4.54 => ! [F2: B > A,L: A,F5: filter @ B] : % 8.50/4.54 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) % 8.50/4.54 => ( ( L % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( filterlim @ B @ A % 8.50/4.54 @ ^ [X: B] : ( sgn_sgn @ A @ ( F2 @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L ) ) % 8.50/4.54 @ F5 ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5491_tendsto__divide__zero,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: B > A,F5: filter @ B,C2: A] : % 8.50/4.54 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 ) % 8.50/4.54 => ( filterlim @ B @ A % 8.50/4.54 @ ^ [X: B] : ( divide_divide @ A @ ( F2 @ X ) @ C2 ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) % 8.50/4.54 @ F5 ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5492_tendsto__divide,axiom, % 8.50/4.54 ! [A: $tType,B: $tType] : % 8.50/4.54 ( ( real_V3459762299906320749_field @ A ) % 8.50/4.54 => ! [F2: B > A,A2: A,F5: filter @ B,G: B > A,B2: A] : % 8.50/4.54 ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) % 8.50/4.54 => ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 ) % 8.50/4.54 => ( ( B2 % 8.50/4.54 != ( zero_zero @ A ) ) % 8.50/4.54 => ( filterlim @ B @ A % 8.50/4.54 @ ^ [X: B] : ( divide_divide @ A @ ( F2 @ X ) @ ( G @ X ) ) % 8.50/4.54 @ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A2 @ B2 ) ) % 8.50/4.54 @ F5 ) ) ) ) ) ). % 8.50/4.54 % 8.50/4.54 thf(fact_5493_continuous__add,axiom, % 8.50/4.54 ! [B: $tType,D: $tType] : % 8.50/4.54 ( ( ( topological_t2_space @ D ) % 8.50/4.54 & ( topolo6943815403480290642id_add @ B ) ) % 8.50/4.54 => ! [F5: filter @ D,F2: D > B,G: D > B] : % 8.50/4.54 ( ( topolo3448309680560233919inuous @ D @ B @ F5 @ F2 ) % 8.50/4.54 => ( ( topolo3448309680560233919inuous @ D @ B @ F5 @ G ) % 8.50/4.54 =>